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1510.04030 | {'1510.04030-1-0-0': 'We consider cosmological inflation generated by a scalar field slowly rolling off from a de Sitter maximum of its potential.', '1510.04030-1-0-1': 'We construct the most general model of this kind in which the scalar potential can be written as the sum of two exponentials.', '1510.04030-1-0-2': 'The minimally coupled Einstein-scalar gravity theory obtained in this way is the cosmological version of a two-scale generalisation of known holographic models, allowing for solitonic solutions interpolating between an AdS spacetime in the infrared and scaling solutions in the ultraviolet.', '1510.04030-1-0-3': 'We then investigate cosmological inflation in the slow-roll approximation.', '1510.04030-1-0-4': 'Our model reproduces correctly, for a wide range of its parameters, the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-1-0-5': 'Moreover, it predicts inflation at energy scales of four to five orders of magnitude below the Planck scale.', '1510.04030-1-0-6': 'At the onset of inflation, the mass of the tachyonic excitation, i.e. of the inflaton, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-1-1-0': '# Introduction', '1510.04030-1-2-0': 'Nowadays, inflationary cosmology [CITATION] represents the easiest way to solve the problems of the standard Friedmann-Robertson-Walker (FRW) cosmology, such as the horizon and flatness problems.', '1510.04030-1-3-0': 'The simplest way to generate inflation is to minimally couple Einstein gravity to a scalar field (the inflaton) with a self-interaction potential.', '1510.04030-1-3-1': 'There exist a plethora of models that can be classified in three sets according to the features of the potential: the large-field, small-field and hybrid potentials [CITATION].', '1510.04030-1-3-2': 'Other alternatives include more scalar fields, as in the curvaton mechanism [CITATION].', '1510.04030-1-3-3': 'Nevertheless, the most recent data of the Planck satellite exclude non-Gaussian perturbations and give a striking experimental confirmation of the simplest single-field inflationary scenario [CITATION].', '1510.04030-1-3-4': 'The Planck data favour the small-field models and in particular the Starobinsky model [CITATION], or more in general the so-called cosmological attractors [CITATION], characterised by a "red" power spectrum for primordial perturbations and a small tensor/scalar amplitude ratio.', '1510.04030-1-4-0': 'Small-field models can be realised in two different ways: (1) inflation is generated by the rolling down of the scalar field from an asymptotically constant value to a minimum, e.g. the Starobinsky model; (2) the scalar field rolls off from a local maximum to a local minimum of a potential that is typical of spontaneous symmetry breaking and phase transitions, e.g. quartic potentials, natural inflation models [CITATION] and Coleman-Weinberg potentials [CITATION].', '1510.04030-1-5-0': 'The accuracy of the observational data concerning the power spectrum of primordial quantum fluctuations represents an efficient guide to select inflation models.', '1510.04030-1-5-1': 'But, despite the recent remarkable improvements, the important questions about the microscopic origin of the inflaton and about the physics before inflation are still unanswered.', '1510.04030-1-5-2': 'This lack of knowledge does not allow to single out a unique inflationary model, i.e. a specific form of the potential.', '1510.04030-1-5-3': 'In fact, although the Planck data can be used to strongly constrain the inflationary model, mainly through the values of the spectral index [MATH] and the tensor/scalar amplitude ratio [MATH], they are not sufficient to select a unique model.', '1510.04030-1-6-0': 'In view of this situation, it is natural to look for hints coming from somewhere else in gravitational physics, for instance supergravity and string theory [CITATION].', '1510.04030-1-6-1': 'In recent times, minimally coupled Einstein-scalar gravity have been intensively investigated for holographic applications [CITATION].', '1510.04030-1-6-2': 'A class of Einstein-scalar gravity models of particular interest are those allowing for solitonic solutions interpolating between anti-de Sitter (AdS) vacua and domain wall (DW) solutions with specific scaling symmetries (scale-covariant symmetry).', '1510.04030-1-6-3': 'The holographically dual QFT has scaling symmetries, which have a nice interpretation in terms of features of phase transitions in condensed matter systems (hyperscaling violation).', '1510.04030-1-6-4': 'These solitonic solutions are naturally related to cosmological solutions by the so called DW/cosmology duality, a sort of analytic continuation, which maps the soliton in a FRW solution [CITATION].', '1510.04030-1-7-0': 'The cosmological duals of solitons which interpolate between an AdS spacetime at large distances of the bulk theory (the ultraviolet of the dual QFT) and a scale covariant geometry at small distances in the bulk theory (the infrared of the dual QFT) are natural candidates for describing dark energy [CITATION].', '1510.04030-1-7-1': 'On the other hand, the cosmological duals of solitons interpolating between AdS in the infrared and scale covariant geometries in the ultraviolet [CITATION] may be relevant for describing inflation.', '1510.04030-1-7-2': 'It has been shown that the cosmological solutions of this class of models generate inflation as the scalar field rolls down from a de Sitter (dS) spacetime [CITATION].', '1510.04030-1-7-3': 'As such, these inflationary models belong to the class of small-field potentials and inflation can be described as an instability of the de Sitter spacetime rolling down to a scaling solution.', '1510.04030-1-8-0': 'The structure of the paper is as follows.', '1510.04030-1-8-1': 'In Sect. [REF] we construct the most general model in which inflation is generated by a scalar field slowly rolling off from a de Sitter maximum of the potential, requiring the potential to be the sum of two exponentials.', '1510.04030-1-8-2': 'We show that the minimally coupled Einstein-scalar gravity theory constructed in this way is the cosmological version of a two-scale generalisation of the holographic models of Refs. [CITATION].', '1510.04030-1-8-3': 'In Sect. [REF] we discuss the cosmological solution of our model.', '1510.04030-1-8-4': 'Inflation and the spectral parameters of the power spectrum of primordial perturbations are discussed in Sect. [REF] using the slow-roll approximation.', '1510.04030-1-8-5': 'In Sect. [REF] we compare the theoretical predictions of our model with observations.', '1510.04030-1-8-6': 'Finally, in Sect. [REF] we state our conclusions and in [REF] we briefly repeat our calculations for a model in which the potential has a constant additive term.', '1510.04030-1-9-0': '# The model', '1510.04030-1-10-0': 'The simplest way to fuel inflation into a cosmological scenario is to couple, minimally, Einstein gravity to a scalar field [MATH] with an appropriate self-interaction potential [MATH]: A=d^4x-g(m_P^216R -12()^2-V ()).', '1510.04030-1-11-0': 'In this paper we focus on inflation generated by a scalar field rolling off from a maximum of [MATH].', '1510.04030-1-11-1': 'This class of models is very natural from a physical point of view because inflation can be thought of just as an instability of the de Sitter spacetime, generated by a scalar perturbation.', '1510.04030-1-12-0': 'Our first goal is to construct the general form of the potential belonging to this class.', '1510.04030-1-12-1': 'Without loss of generality we can assume that the maximum of the potential occurs at [MATH], so that the basic necessary conditions to be imposed on the potential read V (0)>0,V\' (0)=0,V" (0)<0.', '1510.04030-1-12-2': 'Obviously, the previous conditions are very loose and do not select any specific form of [MATH].', '1510.04030-1-12-3': 'We further constrain the form of the potential by requiring it to be a linear combination of two exponentials.', '1510.04030-1-12-4': 'This is a rather strong assumption, but is supported by several arguments.', '1510.04030-1-12-5': 'Exponential potentials for scalar field appear quite generically in a variety of situations: compactifications of extra dimensions, [MATH] gravity theories (which on-shell are equivalent to Einstein-scalar gravity) and low-energy effective string theory.', '1510.04030-1-12-6': 'Moreover, exponential potentials have been shown to be the source of brane solutions of Einstein-scalar gravity called domain walls (DW) [CITATION], which can be analytically continued into FRW cosmological solutions [CITATION].', '1510.04030-1-13-0': 'We are therefore led to consider the following general form of the inflation potential V () =^2(a_1e^b_1+a_2 e^b_2), where [MATH] and [MATH] are some length scales, whose physical meaning will be clarified in short, and [MATH] are some dimensionless constants characterising the model.', '1510.04030-1-13-1': 'They are constrained by [REF], giving a_1+a_2>0,a_1b_1=- a_2b_2,a_1b_1^2+a_2b_2^2 <0.', '1510.04030-1-13-2': 'Modulo trivial symmetries interchanging the two exponentials in the potential, the most general solution of the previous equations is [MATH], where we have defined a new dimensionless parameter [MATH].', '1510.04030-1-13-3': 'The parameter rescaling [MATH] brings the potential in the form V () =2^23(e^3-^2 e^3/), where [MATH].', '1510.04030-1-13-4': 'The potential [REF] is a two-scales generalisation of the model proposed in Ref. [CITATION] to which it reduces for the particular value of the parameter [MATH].', '1510.04030-1-13-5': 'The cosmology of this latter model has been investigated in Ref. [CITATION].', '1510.04030-1-13-6': 'We will see in the next section that for generic values of the parameter [MATH] the cosmological equations resulting from the model [REF] do not give rise to an exactly integrable system.', '1510.04030-1-14-0': 'The potential [REF] is invariant both under the transformation [MATH], which corresponds to interchanging the two exponentials in the potential [REF] and under the transformation [MATH].', '1510.04030-1-14-1': 'This symmetries allow us to limit our consideration to [MATH].', '1510.04030-1-14-2': 'The two limiting cases [MATH] correspond respectively to a pure exponential and to a potential behaving at leading order as [MATH].', '1510.04030-1-14-3': 'The potential [MATH] has a maximum at [MATH] corresponding to an unstable de Sitter solution with [MATH] and a corresponding tachyonic excitation, the inflaton.', '1510.04030-1-15-0': 'The potential [MATH] is depicted in [REF] for selected values of the parameters [MATH], [MATH] and [MATH].', '1510.04030-1-16-0': 'One can therefore use this model to describe inflation as generated by an unstable de Sitter solution.', '1510.04030-1-16-1': 'Inflation starts as a quantum fluctuation of the de Sitter solution and is initially driven by a tachyonic excitation of the de Sitter spacetime and proceeds as the scalar field rolls off from the maximum of the potential.', '1510.04030-1-17-0': '## Physical scales', '1510.04030-1-18-0': 'Besides the Planck length [MATH], the model is parametrised by the two length scales [MATH] and [MATH] and by the dimensionless parameter [MATH].', '1510.04030-1-18-1': 'The presence of two length scales is a characteristic feature of small-field models of inflation.', '1510.04030-1-18-2': 'In the present context the two scales have a simple interpretation in terms of geometric properties of the function [MATH].', '1510.04030-1-18-3': 'They give, respectively, the height and the curvature of the [MATH] maximum of the function [MATH].', '1510.04030-1-18-4': 'Correspondingly, [MATH] and [MATH] determine the two physical scales relevant for inflation: the vacuum energy [MATH] at the beginning of inflation and the inflaton mass squared [MATH].', '1510.04030-1-18-5': 'We have M^2_I = V"(0)=-2^2^2= -323^4h^2m_P^2,E_V= [V (0)]^1/4= (2/3)^1/4 m_P,h= 43(l_P),^1/2m_P, where we have introduced the two dimensionless parameters [MATH] and [MATH] representing the measures of [MATH] and [MATH] in Planck units.', '1510.04030-1-19-0': 'Conversely, [MATH] is a purely dimensionless parameter and plays a role which is drastically different from [MATH] and [MATH].', '1510.04030-1-19-1': 'It is not linked to any physical scale of the model but quantifies the deviation of the potential from a pure exponential behaviour attained for [MATH] near to 0.', '1510.04030-1-20-0': 'In the following we use instead of the negative quantity [MATH], the inflaton mass defined as [MATH].', '1510.04030-1-21-0': '# Cosmological solutions', '1510.04030-1-22-0': 'The cosmology of our model can be investigated using the parametrisation for the metric and the scalar field used in Ref. [CITATION]: ds^2=- e^2 b()d^2+ e^23 b()dS_(3)^2,where we are considering a 3D flat universe and [MATH] is the time coordinate.', '1510.04030-1-22-1': 'In fact, according to observation the universe must be flat and in most models we have [MATH].', '1510.04030-1-23-0': 'Writing the metric in the usual FRW form ds^2=- dt^2+ a(t)^2 dS_(3)^2, one can easily find the cosmic time [MATH] and the the scale factor [MATH]: [MATH].', '1510.04030-1-23-1': 'Using the parametrisation [REF] the field equations stemming from the action [REF] take the form [EQUATION]', '1510.04030-1-23-2': 'The dS spacetime with constant inflaton is an exact solution of the previous system.', '1510.04030-1-23-3': 'In the parametrisation [REF] the dS solution is given by [MATH], whereas in the FRW form [REF] we have the usual exponential form for the scale factor: a=e^8 l_P t/3.', '1510.04030-1-23-4': 'This solution describes a scalar field sitting forever at the maximum of the potential, generating an exact exponential expansion of the universe, i.e. never ending inflation.', '1510.04030-1-24-0': 'The most interesting cosmological solutions are those describing inflation lasting for a finite amount of time.', '1510.04030-1-24-1': 'In this case the scalar rolls off from the maximum of [MATH], generating a quasi-exponential expansion of the universe as long as the potential energy of the scalar dominates the kinetic one.', '1510.04030-1-24-2': 'This kind of solutions would be the cosmological counterpart of the solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-1-25-0': 'Searching for these solutions, one can try, following Ref. [CITATION], to decouple the system [REF] by defining linear combinations of [MATH] and [MATH]: [MATH].', '1510.04030-1-25-1': 'However, one can easily realise that the decoupling works only for the particular value of the parameter [MATH] (corresponding to [MATH]).', '1510.04030-1-25-2': 'For this value of [MATH] the Einstein-scalar gravity models give rise to exactly integrable models both in the case of static (brane) [CITATION] and cosmological solutions [CITATION].', '1510.04030-1-25-3': 'In the static case we have solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-1-25-4': 'Analogously, in the cosmological case we have exact solutions which can be used to model inflation [CITATION].', '1510.04030-1-26-0': 'For generic values of the parameter [MATH] the system [REF] does not decouple, is not exactly integrable and cosmological solution cannot be found in analytic form.', '1510.04030-1-27-0': 'Approximate solutions of the field equations [REF] can be found for some limiting cases.', '1510.04030-1-27-1': 'Of particular interest is the case of small [MATH].', '1510.04030-1-27-2': 'For [MATH] the potential [REF] behaves exponentially, V () 2^2^23e^3/, the system can be solved analytically and we have scaling (power-law) solutions, which are obtained from scale-covariant (DW) solutions [CITATION] using the transformation [MATH].', '1510.04030-1-27-3': 'In the gauge [REF] this scaling solution has the form at^h^2^2,e^2t^-hl_P .', '1510.04030-1-28-0': '# Inflation and slow-roll approximation', '1510.04030-1-29-0': 'Lacking exact solutions to investigate the cosmology of our model [REF], we work in the slow-roll approximation [CITATION].', '1510.04030-1-29-1': 'In this regime the potential energy of the scalar field dominates over the kinetic energy and the universe has a quasi-exponential accelerated expansion as the scalar field slowly rolls off from the maximum of the potential.', '1510.04030-1-29-2': 'Following the usual approach, we introduce the slow-roll parameters [MATH] and [MATH], m_P^216(V\'V)^2,m_P^28V"V-.', '1510.04030-1-29-3': 'We will have inflation as long as [MATH].', '1510.04030-1-29-4': 'The slow-roll approximation is valid as long as [MATH].', '1510.04030-1-29-5': 'For [MATH] the solution is exactly de Sitter, whereas inflation ends when [MATH].', '1510.04030-1-30-0': 'The potential [REF] is not a monotonic function of the scalar field [MATH] but has a maximum at [MATH] and [MATH] for [MATH], whereas [MATH] for [MATH] (see [REF]).', '1510.04030-1-30-1': 'We have therefore two alternative branches that we can use to generate inflation, i.e. I: [MATH] and II: [MATH].', '1510.04030-1-30-2': 'In the following, we mainly consider the first branch.', '1510.04030-1-30-3': 'In [REF] we discuss briefly branch II and show that it cannot be compatible with observations.', '1510.04030-1-31-0': 'Let us now introduce the variable Y= e^3/.', '1510.04030-1-31-1': 'In this parametrisation the branch under consideration corresponds to Y1.', '1510.04030-1-31-2': 'The slow-roll parameter [MATH] is zero on the maximum of the potential at [MATH]), whereas [MATH] for [MATH], where Y_0=+h+^2 h. For [MATH] we have inflation, while for [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-1-31-3': 'One can easily check that [MATH], so that during inflation we always have [MATH] and we can easily satisfy the first slow-roll condition [MATH].', '1510.04030-1-31-4': 'On the other hand, the parameter [MATH], which gives a measure of the curvature of the potential, is not small, but we have [MATH].', '1510.04030-1-31-5': 'It follows that the simplest way to satisfy the second slow-roll condition, [MATH], is to choose h10, in this way we can have [MATH] as well as [MATH].', '1510.04030-1-31-6': 'As already noted, the model discussed in Ref. [CITATION] does not satisfy [REF] because is characterised by [MATH].', '1510.04030-1-32-0': "In the slow-roll regime, the universe expands quasi-exponentially and the number of [MATH]-folds [MATH], which determines the duration of inflation, is determined by N=-dt H=8m_P^2__0^_1dVV', where [MATH] are, respectively, the inflaton-field values at the end and beginning of inflation and [MATH] is the Hubble parameter.", '1510.04030-1-33-0': 'Using the definition [REF] and the expression [MATH] for [MATH] at the end of inflation, [REF] gives the function [MATH] in implicit form, Y^1/Y-1= e^2N/h^2 A,A := (+1h)(+h+^2 h)^1/.', '1510.04030-1-33-1': 'In the case of the dS solution [REF] the scalar field remains constant (the inflaton sits on the top of the potential), and we have [MATH] (eternal inflation).', '1510.04030-1-33-2': 'Obviously this configuration is highly unstable.', '1510.04030-1-33-3': 'A small perturbation of the scalar field starts the slow-roll of the inflaton along the slope and a finite value of [MATH] is generated.', '1510.04030-1-33-4': 'If this fluctuation is small enough we can solve approximately [REF] for [MATH] near [MATH].', '1510.04030-1-33-5': 'We get at leading order, Y= 1+ A^-1 e^-2N/h^2.', '1510.04030-1-34-0': 'One can easily check that [MATH] with [MATH] for [MATH] and [MATH] for [MATH].', '1510.04030-1-34-1': 'Moreover, in the range [MATH], [MATH] is a monotonically decreasing function of [MATH] which depends very weakly on [MATH].', '1510.04030-1-34-2': 'It follows immediately that [REF] is a good approximation for [MATH] not too close to 0, whenever [MATH].', '1510.04030-1-34-3': 'When [MATH] the approximation [REF] holds irrespectively of the value of [MATH].', '1510.04030-1-35-0': '## Perturbations and spectral parameters', '1510.04030-1-36-0': 'One of the most striking predictions of inflation concerns the spectrum of perturbations in the early universe [CITATION].', '1510.04030-1-36-1': 'During inflation the horizon shrinks and the primordial perturbations, which were causally connected are redshifted to superhorizon scales.', '1510.04030-1-36-2': 'Conversely, in the matter-radiation dominated era the horizon grows, the perturbations fall back in the horizon so that they can act as seeds for structure formation and anisotropy in the universe.', '1510.04030-1-36-3': 'The information about these primordial fluctuations is therefore encoded in the anisotropies of the CMB.', '1510.04030-1-37-0': 'Primordial quantum fluctuations are described in terms of two-point correlation functions for scalar and tensor modes in Fourier space and the associated power spectrum.', '1510.04030-1-37-1': 'In the slow-roll approximation, the power spectrum has a power-law behaviour and is usually characterised by four parameters: the amplitudes of scalar perturbations [MATH], the ratio [MATH] of the amplitudes of tensor and scalar perturbations and their spectral indices [MATH] and [MATH].', '1510.04030-1-37-2': 'These parameters are function of the number of [MATH]-folds [MATH] and can be expressed in terms of the potential [MATH] and the slow-roll parameters [REF] as follows [EQUATION] where [MATH] is defined by ll.', '1510.04030-1-37-3': 'Using nv and sr we can express the spectral parameters as a function of [MATH]: [EQUATION] where [MATH] is defined, implicitly, by klh.', '1510.04030-1-38-0': 'For [MATH] we can use the approximate expansion for [MATH] given by qrt and we get at leading order in the [MATH] expansion, [EQUATION]', '1510.04030-1-38-1': 'One important feature of param1 is the exponential dependence on [MATH].', '1510.04030-1-38-2': 'This must be compared with the typical behaviour of the Starobinsky model and more in general of cosmological attractor models, where one typically obtains [MATH] and [MATH] (see [CITATION] and references therein).', '1510.04030-1-39-0': '# Comparison with observation', '1510.04030-1-40-0': 'In this section we compare the theoretical results of our model for the spectral parameters [MATH], [MATH] and [MATH] with the most recent results of observations, in particular the joint analysis of BICEP2/Keck Array and Planck data [CITATION].', '1510.04030-1-41-0': 'The spectral parameters are functions of the number of the [MATH]-folds [MATH] and depend on the three dimensionless parameters [MATH], [MATH] and [MATH].', '1510.04030-1-41-1': 'Because [MATH] enters only in the normalisation of the power spectrum [MATH], whereas [MATH] and [MATH] depend on [MATH] and [MATH] only we will use the following strategy: we will first determine using r,ns and the experimental results for [MATH] and [MATH], the allowed range of the parameters [MATH] and [MATH].', '1510.04030-1-41-2': 'We will then use pr and the experimental results for [MATH] to determine the corresponding values of the parameter [MATH].', '1510.04030-1-41-3': 'Finally we use pm to determine the vacuum energy [MATH] and the inflaton mass [MATH].', '1510.04030-1-42-0': 'For [MATH], [MATH] and [MATH] we use the most recent results [CITATION], i.e. [MATH], [MATH] and [MATH].', '1510.04030-1-42-1': 'Since there is only a lower bound for the number of [MATH]-folds [MATH], [MATH], we use for [MATH] a quite broad range of values, [MATH].', '1510.04030-1-43-0': 'The calculations have to be performed numerically because the function [MATH] appearing in param is not known, but is defined implicitly by klh.', '1510.04030-1-44-0': 'A possible way to avoid numerical computations is to work in a regime where [MATH], so that we can trust the approximate solution for [MATH] given by qrt and the resulting expressions for [MATH], [MATH] and [MATH] given by param1.', '1510.04030-1-44-1': 'Unfortunately, since we need at least [MATH], in order to have [MATH], we must take values of [MATH].', '1510.04030-1-44-2': 'For instance for [MATH], [MATH] we have [MATH].', '1510.04030-1-44-3': 'It follows that the approximate expressions [REF] can only be used in a regime of very large [MATH], for which we do not have a direct access to observations.', '1510.04030-1-45-0': 'The results of our numerical computations are shown in the two sets of region plots shown in [REF].', '1510.04030-1-46-0': '## Spectral parameters', '1510.04030-1-47-0': 'In [REF] we show the numerical results obtained from [REF].', '1510.04030-1-47-1': 'We plot the spectral parameters [MATH] (plot on the left) and [MATH] (plot on the right) as functions of [MATH] and [MATH] for four selected values of the parameter [MATH].', '1510.04030-1-47-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the scale of colour shown on the right of every plot.', '1510.04030-1-47-3': 'The coloured regions in the plots in [REF] give the range of values of [MATH] and [MATH] for which we have values of [MATH] and [MATH] compatible with the experimental measurements.', '1510.04030-1-48-0': 'In general, higher values of [MATH] correspond to higher values of [MATH].', '1510.04030-1-48-1': 'Moreover, [MATH] depends very weakly on [MATH], and for values of [MATH] near to [MATH] is almost independent of [MATH].', '1510.04030-1-48-2': 'However, as [MATH] grows, differences appear.', '1510.04030-1-48-3': 'Whereas for [MATH] almost all values of [MATH] are admissible with [MATH], for [MATH], the [MATH]-strip gets thinner for values of [MATH] near to [MATH] and narrows towards [MATH].', '1510.04030-1-48-4': 'If [MATH], values of [MATH] near to [MATH] are no longer suitable for reproducing the data.', '1510.04030-1-48-5': 'Such [MATH]-region corresponds to [MATH] for [MATH] and reduces for bigger [MATH].', '1510.04030-1-49-0': 'The tensor/scalar ratio [MATH] shows a different behaviour.', '1510.04030-1-49-1': 'It depends weakly on [MATH] but strongly on [MATH], with higher values of [MATH] corresponding to higher values of [MATH].', '1510.04030-1-49-2': 'Moreover, increasing the value of [MATH] pushes the value of [MATH] towards the upper bound [MATH].', '1510.04030-1-49-3': 'For [MATH] we predict [MATH] independently of the value of [MATH], and then, in order to have values of order [MATH] we need both [MATH] and [MATH] near its upper value.', '1510.04030-1-50-0': '## Vacuum energy and inflaton mass', '1510.04030-1-51-0': 'In [REF] we show the numerical results obtained from [REF].', '1510.04030-1-51-1': 'We plot the vacuum energy [MATH] (left) and the inflaton mass [MATH] (right) as functions of [MATH] and [MATH], again for [MATH].', '1510.04030-1-51-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the scale of colour shown on the right of every plot.', '1510.04030-1-52-0': 'The regions plotted in [REF] are the same as those plotted in [REF], i.e. they represent the range of values of [MATH] and [MATH] allowed by the experimental data.', '1510.04030-1-52-1': 'Because we do not have stringent experimental bounds on [MATH] and [MATH], we are interested just in the order of magnitude of these quantities.', '1510.04030-1-52-2': 'We observe that the order of magnitude of [MATH] and [MATH] depends very weakly on both [MATH] and [MATH].', '1510.04030-1-52-3': 'Also the dependence on [MATH] is quite weak, as long as we take values of [MATH] not too close to [MATH].', '1510.04030-1-52-4': 'Thus for [MATH] not too close to [MATH], the vacuum energy remains about [MATH] to [MATH] Planck masses, whereas the inflaton mass is between [MATH] and [MATH] Planck masses.', '1510.04030-1-52-5': 'On the other hand both [MATH] and [MATH] shrink drastically when we move close to [MATH].', '1510.04030-1-53-0': '## Other branch of the potential', '1510.04030-1-54-0': 'Until now we have considered the slow-roll regime for the branch I of the potential, i.e. [MATH].', '1510.04030-1-54-1': 'Let us briefly consider branch II, i.e. [MATH].', '1510.04030-1-54-2': 'Investigation of this branch is of particular interest because the most interesting cosmological solutions one can obtain for the exact solvable model with [MATH] are defined in the branch II of the potential [CITATION].', '1510.04030-1-55-0': 'In terms of the parametrisation [REF], region II corresponds to [MATH].', '1510.04030-1-55-1': 'The slow-roll parameters [MATH] and [MATH] are still given by [REF] but now the condition for inflation [MATH] requires (1-h)Y1, which can be satisfied only if [MATH].', '1510.04030-1-55-2': 'It follows that [MATH].', '1510.04030-1-55-3': 'One can easily see from [REF] that these values of [MATH] are not only incompatible with the slow-roll condition [MATH], but are also completely ruled out by the experimental constraints on [MATH].', '1510.04030-1-56-0': '# Conclusion', '1510.04030-1-57-0': 'In this paper we have constructed the most general Einstein-scalar gravity model in which the potential is given by the sum of two exponentials and inflation is generated by a scalar field [MATH] rolling off from the de Sitter maximum of the potential [MATH].', '1510.04030-1-57-1': 'These models are the cosmological counterpart of holographic models used to describe hyperscaling violation in the ultraviolet [CITATION].', '1510.04030-1-57-2': 'We have investigated inflation in the slow-roll approximation.', '1510.04030-1-57-3': 'Our model predicts inflation at energy scales of four to five orders of magnitude below the Planck scale, whereas the inflaton mass, at the onset of inflation, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-1-57-4': 'We have shown that our model reproduces correctly, for a wide range of its parameters the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-1-58-0': 'The proposed inflationary model belongs to the wide class of small-field models, which also include the Starobinsky model and, more generally, the cosmological attractor models.', '1510.04030-1-58-1': 'Our model shares with those several features: (1) the potential is built as a combination of exponentials, it predicts (2) an energy scale of inflation four order of magnitude below the Planck mass, (3) a "red" power spectrum and (4) a small tensor/scalar amplitude ratio.', '1510.04030-1-58-2': 'On the other hand, our model differs from the Starobinsky one in a crucial aspect: inflation is not generated, as in Starobinsky model, by a scalar field rolling off from an asymptotically constant potential, but rather from a local maximum of the potential.', '1510.04030-1-58-3': 'This property allows us to interpret the inflaton as a tachyonic excitation of the dS vacuum and to introduce a second scale of energy in the theory, the mass scale [MATH], which is 7-8 order of magnitude below the Planck mass.', '1510.04030-1-58-4': 'This hierarchy of scales opens the intriguing possibility that the origin of the inflaton could be explained by the physics at energy scales 7-8 order of magnitude below the Planck mass.', '1510.04030-1-59-0': 'We close with a brief comment about the reheating phase and the transition from inflation to the radiation/matter dominated era.', '1510.04030-1-59-1': 'During reheating the energy is transferred from the inflaton to matter fields.', '1510.04030-1-59-2': 'This means that there must exist a region in which the kinetic energy of the inflaton dominates over its potential energy, e.g. a local minimum of the potential.', '1510.04030-1-59-3': 'It is evident from [REF] that the potential [REF] does not have such a region and hence it cannot be used to describe reheating.', '1510.04030-1-59-4': 'Thus, in order to describe reheating our potential must be matched with continuity at the end of inflation with some other branch of a potential exhibiting a local minimum.', '1510.04030-1-59-5': 'This can be done very easily.', '1510.04030-1-59-6': 'In the [MATH]-parametrisation the point [MATH] given by [REF], at which the universe exits inflation, is always on the left of the point [MATH] at which [MATH] cuts the horizontal axis, i.e. we have [MATH] and [MATH].', '1510.04030-1-59-7': 'Since the slow-roll approximation is badly broken at [MATH], the matching with the branch of the potential with the local minimum must be performed at a point [MATH].', '1510.04030-1-60-0': '# [MATH] model Cosh phi model', '1510.04030-1-61-0': 'The model [REF] is the most general form of the potential one can obtain imposing conditions [REF] and assuming that [MATH] is built as a combination of two exponential without an additive constant term.', '1510.04030-1-61-1': 'When such a constant term (which we call [MATH]) is present, only the first equation in [REF] has to be modified and becomes [MATH], whereas the second and third equations remain unchanged.', '1510.04030-1-61-2': 'A general solution of the ensuing system is given by [MATH].', '1510.04030-1-62-0': 'A simple example of this class of potentials is given by V () =^2(2- ) This potential gives a further example of inflation generated by an unstable de-Sitter vacuum.', '1510.04030-1-63-0': 'The potential [REF] has a maximum at [MATH], corresponding to an unstable de Sitter solution with [MATH], and a corresponding tachyonic excitation.', '1510.04030-1-63-1': 'For [MATH], the potential behaves as purely exponential.', '1510.04030-1-64-0': 'The vacuum energy and inflaton mass, expressed in terms of [MATH] and [MATH], defined as in [REF], are M^2_I = -163^4h^2m_P^2,E_V= ,m_P.', '1510.04030-1-65-0': 'Introducing the variable [MATH], the slow-roll parameters [MATH] and [MATH] take the form 13 h^2(Y^2-1Y^2-4Y+1)^2,23 h^2Y^2+1Y^2-4Y+1-.', '1510.04030-1-65-1': 'The slow-roll parameter [MATH] is zero on the maximum of the potential ([MATH]).', '1510.04030-1-65-2': 'Moreover, we have [MATH] for [MATH], where Y_0=2 3h+ 1+9 h^23h+1.', '1510.04030-1-65-3': 'For [MATH] we have inflation, whereas for [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-1-65-4': 'One can easily check that during inflation we always have [MATH].', '1510.04030-1-65-5': 'Conversely, the parameter [MATH], which gives a measure of the curvature of the potential, is not small in general, but is of order [MATH].', '1510.04030-1-66-0': 'Also for these models the simplest way to satisfy the usual slow-roll conditions for inflation, [MATH], is to choose [MATH], so that [MATH] as well as [MATH].', '1510.04030-1-67-0': 'In the slow-roll approximation the spectral parameters [MATH], [MATH] and [MATH] expressed in terms of [MATH] are, [EQUATION] where [MATH] is defined implicitly as a function of [MATH] by klh1.', '1510.04030-1-68-0': 'fig:cosh shows that there exists a region in the parameter space [MATH] where the model correctly reproduces the results of observation [CITATION].', '1510.04030-1-68-1': 'Moreover, it predicts the vacuum energy to be three to four orders of magnitude below the Planck scale and the mass of the inflaton six to seven orders of magnitude smaller than the Planck mass.', '1510.04030-1-69-0': 'EF acknowledges financial support provided under the European Union\'s H2020 ERC Consolidator Grant "Matter and strong-field gravity: New frontiers in Einstein\'s theory" grant agreement no.', '1510.04030-1-69-1': 'MaGRaTh-646597.'} | {'1510.04030-2-0-0': 'We consider cosmological inflation generated by a scalar field slowly rolling off from a de Sitter maximum of its potential.', '1510.04030-2-0-1': 'We construct the most general model of this kind in which the scalar potential can be written as the sum of two exponentials.', '1510.04030-2-0-2': 'The minimally coupled Einstein-scalar gravity theory obtained in this way is the cosmological version of a two-scale generalisation of known holographic models, allowing for solitonic solutions interpolating between an AdS spacetime in the infrared and scaling solutions in the ultraviolet.', '1510.04030-2-0-3': 'We then investigate cosmological inflation in the slow-roll approximation.', '1510.04030-2-0-4': 'Our model reproduces correctly, for a wide range of its parameters, the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-2-0-5': 'Moreover, it predicts inflation at energy scales of four to five orders of magnitude below the Planck scale.', '1510.04030-2-0-6': 'At the onset of inflation, the mass of the tachyonic excitation, i.e. of the inflaton, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-2-1-0': '# Introduction', '1510.04030-2-2-0': 'Nowadays, inflationary cosmology [CITATION] represents the easiest way to solve the problems of the standard Friedmann-Robertson-Walker (FRW) cosmology, such as the horizon and flatness problems.', '1510.04030-2-3-0': 'The simplest way to generate inflation is to minimally couple Einstein gravity to a scalar field (the inflaton) with a self-interaction potential.', '1510.04030-2-3-1': 'There exist a plethora of models that can be classified in three sets according to the features of the potential: the large-field, small-field and hybrid potentials [CITATION].', '1510.04030-2-3-2': 'Other alternatives include more scalar fields, as in the curvaton mechanism [CITATION].', '1510.04030-2-3-3': 'Nevertheless, the most recent data of the Planck satellite exclude non-Gaussian perturbations and give a striking experimental confirmation of the simplest single-field inflationary scenario [CITATION].', '1510.04030-2-3-4': 'The Planck data favour the small-field models and in particular the Starobinsky model [CITATION], or more in general the so-called cosmological attractors [CITATION], characterised by a "red" power spectrum for primordial perturbations and a small tensor/scalar amplitude ratio.', '1510.04030-2-4-0': 'Small-field models can be realised in two different ways: (1) inflation is generated by the rolling down of the scalar field from an asymptotically constant value to a minimum, e.g. the Starobinsky model; (2) the scalar field rolls off from a local maximum to a local minimum of a potential that is typical of spontaneous symmetry breaking and phase transitions, e.g. quartic potentials, natural inflation models [CITATION] and Coleman-Weinberg potentials [CITATION].', '1510.04030-2-5-0': 'The accuracy of the observational data concerning the power spectrum of primordial quantum fluctuations represents an efficient guide to select inflation models.', '1510.04030-2-5-1': 'But, despite the recent remarkable improvements, the important questions about the microscopic origin of the inflaton and about the physics before inflation are still unanswered.', '1510.04030-2-5-2': 'This lack of knowledge does not allow to single out a unique inflationary model, i.e. a specific form of the potential.', '1510.04030-2-5-3': 'In fact, although the Planck data can be used to strongly constrain the inflationary model, mainly through the values of the spectral index [MATH] and the tensor/scalar amplitude ratio [MATH], they are not sufficient to select a unique model.', '1510.04030-2-6-0': 'In view of this situation, it is natural to look for hints coming from somewhere else in gravitational physics, for instance supergravity and string theory [CITATION].', '1510.04030-2-6-1': 'In recent times, minimally coupled Einstein-scalar gravity have been intensively investigated for holographic applications [CITATION].', '1510.04030-2-6-2': 'A class of Einstein-scalar gravity models of particular interest are those allowing for solitonic solutions interpolating between anti-de Sitter (AdS) vacua and domain wall (DW) solutions with specific scaling symmetries (scale-covariant symmetry).', '1510.04030-2-6-3': 'The holographically dual QFT has scaling symmetries, which have a nice interpretation in terms of features of phase transitions in condensed matter systems (hyperscaling violation).', '1510.04030-2-6-4': 'These solitonic solutions are naturally related to cosmological solutions by the so called DW/cosmology duality, a sort of analytic continuation, which maps the soliton in a FRW solution [CITATION].', '1510.04030-2-7-0': 'The cosmological duals of solitons which interpolate between an AdS spacetime at large distances of the bulk theory (the ultraviolet of the dual QFT) and a scale covariant geometry at small distances in the bulk theory (the infrared of the dual QFT) are natural candidates for describing dark energy [CITATION].', '1510.04030-2-7-1': 'On the other hand, the cosmological duals of solitons interpolating between AdS in the infrared and scale covariant geometries in the ultraviolet [CITATION] may be relevant for describing inflation.', '1510.04030-2-7-2': 'It has been shown that the cosmological solutions of this class of models generate inflation as the scalar field rolls down from a de Sitter (dS) spacetime [CITATION].', '1510.04030-2-7-3': 'As such, these inflationary models belong to the class of small-field potentials and inflation can be described as an instability of the de Sitter spacetime rolling down to a scaling solution.', '1510.04030-2-8-0': 'The structure of the paper is as follows.', '1510.04030-2-8-1': 'In Sect. [REF] we construct the most general model in which inflation is generated by a scalar field slowly rolling off from a de Sitter maximum of the potential, requiring the potential to be the sum of two exponentials.', '1510.04030-2-8-2': 'We show that the minimally coupled Einstein-scalar gravity theory constructed in this way is the cosmological version of a two-scale generalisation of the holographic models of Refs. [CITATION].', '1510.04030-2-8-3': 'In Sect. [REF] we discuss the cosmological solution of our model.', '1510.04030-2-8-4': 'Inflation and the spectral parameters of the power spectrum of primordial perturbations are discussed in Sect. [REF] using the slow-roll approximation.', '1510.04030-2-8-5': 'In Sect. [REF] we compare the theoretical predictions of our model with observations.', '1510.04030-2-8-6': 'Finally, in Sect. [REF] we state our conclusions and in [REF] we briefly repeat our calculations for a model in which the potential has a constant additive term.', '1510.04030-2-9-0': '# The model', '1510.04030-2-10-0': 'The simplest way to fuel inflation into a cosmological scenario is to couple, minimally, Einstein gravity to a scalar field [MATH] with an appropriate self-interaction potential [MATH]: A=d^4x-g(m_P^216R -12()^2-V ()).', '1510.04030-2-11-0': 'In this paper we focus on inflation generated by a scalar field rolling off from a maximum of [MATH].', '1510.04030-2-11-1': 'This class of models is very natural from a physical point of view because inflation can be thought of just as an instability of the de Sitter spacetime, generated by a scalar perturbation.', '1510.04030-2-12-0': 'Our first goal is to construct the general form of the potential belonging to this class.', '1510.04030-2-12-1': 'Without loss of generality we can assume that the maximum of the potential occurs at [MATH], so that the basic necessary conditions to be imposed on the potential read V (0)>0,V\' (0)=0,V" (0)<0.', '1510.04030-2-12-2': 'Obviously, the previous conditions are very loose and do not select any specific form of [MATH].', '1510.04030-2-12-3': 'We further constrain the form of the potential by requiring it to be a linear combination of two exponentials.', '1510.04030-2-12-4': 'This is a rather strong assumption, but is supported by several arguments.', '1510.04030-2-12-5': 'Exponential potentials for scalar field appear quite generically in a variety of situations: compactifications of extra dimensions, [MATH] gravity theories (which on-shell are equivalent to Einstein-scalar gravity) and low-energy effective string theory.', '1510.04030-2-12-6': 'Double exponential potential appear in the context of dimensional reduction of gravity with non-trivial four-form flux on a maximally symmetric internal space (see e.g. Ref. [CITATION] and references therein).', '1510.04030-2-12-7': 'Moreover, double exponential potentials have been shown to be the source of brane solutions of Einstein-scalar gravity called domain walls (DW) [CITATION], which can be analytically continued into FRW cosmological solutions [CITATION].', '1510.04030-2-13-0': 'We are therefore led to consider the following general form of the inflation potential V () =^2(a_1e^b_1+a_2 e^b_2), where [MATH] and [MATH] are some length scales, whose physical meaning will be clarified in short, and [MATH] are some dimensionless constants characterising the model.', '1510.04030-2-13-1': 'They are constrained by [REF], giving a_1+a_2>0,a_1b_1=- a_2b_2,a_1b_1^2+a_2b_2^2 <0.', '1510.04030-2-13-2': 'Modulo trivial symmetries interchanging the two exponentials in the potential, the most general solution of the previous equations is [MATH], where we have defined a new dimensionless parameter [MATH].', '1510.04030-2-13-3': 'The parameter rescaling [MATH] brings the potential in the form V () =2^23(e^3-^2 e^3/), where [MATH].', '1510.04030-2-13-4': 'The potential [REF] is a two-scales generalisation of the model proposed in Ref. [CITATION] to which it reduces for the particular value of the parameter [MATH].', '1510.04030-2-13-5': 'The cosmology of this latter model has been investigated in Ref. [CITATION].', '1510.04030-2-13-6': 'We will see in the next section that for generic values of the parameter [MATH] the cosmological equations resulting from the model [REF] do not give rise to an exactly integrable system.', '1510.04030-2-14-0': 'The potential [REF] is invariant both under the transformation [MATH], which corresponds to interchanging the two exponentials in the potential [REF] and under the transformation [MATH].', '1510.04030-2-14-1': 'This symmetries allow us to limit our consideration to [MATH].', '1510.04030-2-14-2': 'The two limiting cases [MATH] correspond respectively to a pure exponential and to a potential behaving at leading order as [MATH].', '1510.04030-2-14-3': 'The potential [MATH] has a maximum at [MATH] corresponding to an unstable de Sitter solution with [MATH] and a corresponding tachyonic excitation, the inflaton.', '1510.04030-2-15-0': 'The potential [MATH] is depicted in [REF] for selected values of the parameters [MATH], [MATH] and [MATH].', '1510.04030-2-16-0': 'One can therefore use this model to describe inflation as generated by an unstable de Sitter solution.', '1510.04030-2-16-1': 'Inflation starts as a quantum fluctuation of the de Sitter solution and is initially driven by a tachyonic excitation of the de Sitter spacetime and proceeds as the scalar field rolls off from the maximum of the potential.', '1510.04030-2-17-0': '## Physical scales', '1510.04030-2-18-0': 'Besides the Planck length [MATH], the model is parametrised by the two length scales [MATH] and [MATH] and by the dimensionless parameter [MATH].', '1510.04030-2-18-1': 'The presence of two length scales is a characteristic feature of small-field models of inflation.', '1510.04030-2-18-2': 'In the present context the two scales have a simple interpretation in terms of geometric properties of the function [MATH].', '1510.04030-2-18-3': 'They give, respectively, the height and the curvature of the [MATH] maximum of the function [MATH].', '1510.04030-2-18-4': 'Correspondingly, [MATH] and [MATH] determine the two physical scales relevant for inflation: the vacuum energy [MATH] at the beginning of inflation and the inflaton mass squared [MATH].', '1510.04030-2-18-5': 'We have M^2_I = V"(0)=-2^2^2= -323^4h^2m_P^2,E_V= [V (0)]^1/4= (2/3)^1/4 m_P,h= 43(l_P),^1/2m_P, where we have introduced the two dimensionless parameters [MATH] and [MATH] representing the measures of [MATH] and [MATH] in Planck units.', '1510.04030-2-19-0': 'Conversely, [MATH] is a purely dimensionless parameter and plays a role which is drastically different from [MATH] and [MATH].', '1510.04030-2-19-1': 'It is not linked to any physical scale of the model but quantifies the deviation of the potential from a pure exponential behaviour attained for [MATH] near to 0.', '1510.04030-2-20-0': 'In the following we use instead of the negative quantity [MATH], the inflaton mass defined as [MATH].', '1510.04030-2-21-0': '# Cosmological solutions', '1510.04030-2-22-0': 'The cosmology of our model can be investigated using the parametrisation for the metric and the scalar field used in Ref. [CITATION]: ds^2=- e^2 b()d^2+ e^23 b()dS_(3)^2,where we are considering a 3D flat universe and [MATH] is the time coordinate.', '1510.04030-2-22-1': 'In fact, according to observation the universe must be flat and in most models we have [MATH].', '1510.04030-2-23-0': 'Writing the metric in the usual FRW form ds^2=- dt^2+ a(t)^2 dS_(3)^2, one can easily find the cosmic time [MATH] and the the scale factor [MATH]: [MATH].', '1510.04030-2-23-1': 'Using the parametrisation [REF] the field equations stemming from the action [REF] take the form [EQUATION]', '1510.04030-2-23-2': 'The dS spacetime with constant inflaton is an exact solution of the previous system.', '1510.04030-2-23-3': 'In the parametrisation [REF] the dS solution is given by [MATH], whereas in the FRW form [REF] we have the usual exponential form for the scale factor: a=e^8 l_P t/3.', '1510.04030-2-23-4': 'This solution describes a scalar field sitting forever at the maximum of the potential, generating an exact exponential expansion of the universe, i.e. never ending inflation.', '1510.04030-2-24-0': 'The most interesting cosmological solutions are those describing inflation lasting for a finite amount of time.', '1510.04030-2-24-1': 'In this case the scalar rolls off from the maximum of [MATH], generating a quasi-exponential expansion of the universe as long as the potential energy of the scalar dominates the kinetic one.', '1510.04030-2-24-2': 'This kind of solutions would be the cosmological counterpart of the solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-2-25-0': 'Searching for these solutions, one can try, following Ref. [CITATION], to decouple the system [REF] by defining linear combinations of [MATH] and [MATH]: [MATH].', '1510.04030-2-25-1': 'However, one can easily realise that the decoupling works only for the particular value of the parameter [MATH] (corresponding to [MATH]).', '1510.04030-2-25-2': 'For this value of [MATH] the Einstein-scalar gravity models give rise to exactly integrable models both in the case of static (brane) [CITATION] and cosmological solutions [CITATION].', '1510.04030-2-25-3': 'In the static case we have solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-2-25-4': 'Analogously, in the cosmological case we have exact solutions which can be used to model inflation [CITATION].', '1510.04030-2-26-0': 'For generic values of the parameter [MATH] the system [REF] does not decouple, is not exactly integrable and cosmological solution cannot be found in analytic form.', '1510.04030-2-27-0': 'Approximate solutions of the field equations [REF] can be found for some limiting cases.', '1510.04030-2-27-1': 'Of particular interest is the case of small [MATH].', '1510.04030-2-27-2': 'For [MATH] the potential [REF] behaves exponentially, V () 2^2^23e^3/, the system can be solved analytically and we have scaling (power-law) solutions, which are obtained from scale-covariant (DW) solutions [CITATION] using the transformation [MATH].', '1510.04030-2-27-3': 'In the gauge [REF] this scaling solution has the form at^h^2^2,e^2t^-hl_P .', '1510.04030-2-28-0': '# Inflation and slow-roll approximation', '1510.04030-2-29-0': 'Lacking exact solutions to investigate the cosmology of our model [REF], we work in the slow-roll approximation [CITATION].', '1510.04030-2-29-1': 'In this regime the potential energy of the scalar field dominates over the kinetic energy and the universe has a quasi-exponential accelerated expansion as the scalar field slowly rolls off from the maximum of the potential.', '1510.04030-2-29-2': 'Following the usual approach, we introduce the slow-roll parameters [MATH] and [MATH], m_P^216(V\'V)^2,m_P^28V"V-.', '1510.04030-2-29-3': 'We will have inflation as long as [MATH].', '1510.04030-2-29-4': 'The slow-roll approximation is valid as long as [MATH].', '1510.04030-2-29-5': 'For [MATH] the solution is exactly de Sitter, whereas inflation ends when [MATH].', '1510.04030-2-30-0': 'The potential [REF] is not a monotonic function of the scalar field [MATH] but has a maximum at [MATH] and [MATH] for [MATH], whereas [MATH] for [MATH] (see [REF]).', '1510.04030-2-30-1': 'We have therefore two alternative branches that we can use to generate inflation, i.e. I: [MATH] and II: [MATH].', '1510.04030-2-30-2': 'In the following, we mainly consider the first branch.', '1510.04030-2-30-3': 'In [REF] we discuss briefly branch II and show that it cannot be compatible with observations.', '1510.04030-2-31-0': 'Let us now introduce the variable Y= e^3/.', '1510.04030-2-31-1': 'In this parametrisation the branch under consideration corresponds to Y1.', '1510.04030-2-31-2': 'The slow-roll parameter [MATH] is zero on the maximum of the potential at [MATH]), whereas [MATH] for [MATH], where Y_0=+h+^2 h. For [MATH] we have inflation, while for [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-2-31-3': 'One can easily check that [MATH], so that during inflation we always have [MATH] and we can easily satisfy the first slow-roll condition [MATH].', '1510.04030-2-31-4': 'On the other hand, the parameter [MATH], which gives a measure of the curvature of the potential, is not small, but we have [MATH].', '1510.04030-2-31-5': 'It follows that the simplest way to satisfy the second slow-roll condition, [MATH], is to choose h10, in this way we can have [MATH] as well as [MATH].', '1510.04030-2-31-6': 'As already noted, the model discussed in Ref. [CITATION] does not satisfy [REF] because is characterised by [MATH].', '1510.04030-2-32-0': "In the slow-roll regime, the universe expands quasi-exponentially and the number of [MATH]-folds [MATH], which determines the duration of inflation, is determined by N=-dt H=8m_P^2__0^_1dVV', where [MATH] are, respectively, the inflaton-field values at the end and beginning of inflation and [MATH] is the Hubble parameter.", '1510.04030-2-33-0': 'Using the definition [REF] and the expression [MATH] for [MATH] at the end of inflation, [REF] gives the function [MATH] in implicit form, Y^1/Y-1= e^2N/h^2 A,A := (+1h)(+h+^2 h)^1/.', '1510.04030-2-33-1': 'In the case of the dS solution [REF] the scalar field remains constant (the inflaton sits on the top of the potential), and we have [MATH] (eternal inflation).', '1510.04030-2-33-2': 'Obviously this configuration is highly unstable.', '1510.04030-2-33-3': 'A small perturbation of the scalar field starts the slow-roll of the inflaton along the slope and a finite value of [MATH] is generated.', '1510.04030-2-33-4': 'If this fluctuation is small enough we can solve approximately [REF] for [MATH] near [MATH].', '1510.04030-2-33-5': 'We get at leading order, Y= 1+ A^-1 e^-2N/h^2.', '1510.04030-2-34-0': 'One can easily check that [MATH] with [MATH] for [MATH] and [MATH] for [MATH].', '1510.04030-2-34-1': 'Moreover, in the range [MATH], [MATH] is a monotonically decreasing function of [MATH] which depends very weakly on [MATH].', '1510.04030-2-34-2': 'It follows immediately that [REF] is a good approximation for [MATH] not too close to 0, whenever [MATH].', '1510.04030-2-34-3': 'When [MATH] the approximation [REF] holds irrespectively of the value of [MATH].', '1510.04030-2-35-0': '## Perturbations and spectral parameters', '1510.04030-2-36-0': 'One of the most striking predictions of inflation concerns the spectrum of perturbations in the early universe [CITATION].', '1510.04030-2-36-1': 'During inflation the horizon shrinks and the primordial perturbations, which were causally connected are redshifted to superhorizon scales.', '1510.04030-2-36-2': 'Conversely, in the matter-radiation dominated era the horizon grows, the perturbations fall back in the horizon so that they can act as seeds for structure formation and anisotropy in the universe.', '1510.04030-2-36-3': 'The information about these primordial fluctuations is therefore encoded in the anisotropies of the CMB.', '1510.04030-2-37-0': 'Primordial quantum fluctuations are described in terms of two-point correlation functions for scalar and tensor modes in Fourier space and the associated power spectrum.', '1510.04030-2-37-1': 'In the slow-roll approximation, the power spectrum has a power-law behaviour and is usually characterised by four parameters: the amplitudes of scalar perturbations [MATH], the ratio [MATH] of the amplitudes of tensor and scalar perturbations and their spectral indices [MATH] and [MATH].', '1510.04030-2-37-2': 'These parameters are function of the number of [MATH]-folds [MATH] and can be expressed in terms of the potential [MATH] and the slow-roll parameters [REF] as follows [EQUATION] where [MATH] is defined by ll.', '1510.04030-2-37-3': 'Using nv and sr we can express the spectral parameters as a function of [MATH]: [EQUATION] where [MATH] is defined, implicitly, by klh.', '1510.04030-2-38-0': 'For [MATH] we can use the approximate expansion for [MATH] given by qrt and we get at leading order in the [MATH] expansion, [EQUATION]', '1510.04030-2-38-1': 'One important feature of param1 is the exponential dependence on [MATH].', '1510.04030-2-38-2': 'This must be compared with the typical behaviour of the Starobinsky model and more in general of cosmological attractor models, where one typically obtains [MATH] and [MATH] (see [CITATION] and references therein).', '1510.04030-2-39-0': '# Comparison with observation', '1510.04030-2-40-0': 'In this section we compare the theoretical results of our model for the spectral parameters [MATH], [MATH] and [MATH] with the most recent results of observations, in particular the joint analysis of BICEP2/Keck Array and Planck data [CITATION].', '1510.04030-2-41-0': 'The spectral parameters are functions of the number of the [MATH]-folds [MATH] and depend on the three dimensionless parameters [MATH], [MATH] and [MATH].', '1510.04030-2-41-1': 'Because [MATH] enters only in the normalisation of the power spectrum [MATH], whereas [MATH] and [MATH] depend on [MATH] and [MATH] only we will use the following strategy: we will first determine using r,ns and the experimental results for [MATH] and [MATH], the allowed range of the parameters [MATH] and [MATH].', '1510.04030-2-41-2': 'We will then use pr and the experimental results for [MATH] to determine the corresponding values of the parameter [MATH].', '1510.04030-2-41-3': 'Finally we use pm to determine the vacuum energy [MATH] and the inflaton mass [MATH].', '1510.04030-2-42-0': 'For [MATH], [MATH] and [MATH] we use the most recent results [CITATION], i.e. [MATH], [MATH] and [MATH].', '1510.04030-2-42-1': 'Since there is only a lower bound for the number of [MATH]-folds [MATH], [MATH], we use for [MATH] a quite broad range of values, [MATH].', '1510.04030-2-43-0': 'The calculations have to be performed numerically because the function [MATH] appearing in param is not known, but is defined implicitly by klh.', '1510.04030-2-44-0': 'A possible way to avoid numerical computations is to work in a regime where [MATH], so that we can trust the approximate solution for [MATH] given by qrt and the resulting expressions for [MATH], [MATH] and [MATH] given by param1.', '1510.04030-2-44-1': 'Unfortunately, since we need at least [MATH], in order to have [MATH], we must take values of [MATH].', '1510.04030-2-44-2': 'For instance for [MATH], [MATH] we have [MATH].', '1510.04030-2-44-3': 'It follows that the approximate expressions [REF] can only be used in a regime of very large [MATH], for which we do not have a direct access to observations.', '1510.04030-2-45-0': 'The results of our numerical computations are shown in the two sets of region plots shown in [REF].', '1510.04030-2-46-0': '## Spectral parameters', '1510.04030-2-47-0': 'In [REF] we show the numerical results obtained from [REF].', '1510.04030-2-47-1': 'We plot the spectral parameters [MATH] (plot on the left) and [MATH] (plot on the right) as functions of [MATH] and [MATH] for four selected values of the parameter [MATH].', '1510.04030-2-47-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the scale of colour shown on the right of every plot.', '1510.04030-2-47-3': 'The coloured regions in the plots in [REF] give the range of values of [MATH] and [MATH] for which we have values of [MATH] and [MATH] compatible with the experimental measurements.', '1510.04030-2-48-0': 'In general, higher values of [MATH] correspond to higher values of [MATH].', '1510.04030-2-48-1': 'Moreover, [MATH] depends very weakly on [MATH], and for values of [MATH] near to [MATH] is almost independent of [MATH].', '1510.04030-2-48-2': 'However, as [MATH] grows, differences appear.', '1510.04030-2-48-3': 'Whereas for [MATH] almost all values of [MATH] are admissible with [MATH], for [MATH], the [MATH]-strip gets thinner for values of [MATH] near to [MATH] and narrows towards [MATH].', '1510.04030-2-48-4': 'If [MATH], values of [MATH] near to [MATH] are no longer suitable for reproducing the data.', '1510.04030-2-48-5': 'Such [MATH]-region corresponds to [MATH] for [MATH] and reduces for bigger [MATH].', '1510.04030-2-49-0': 'The tensor/scalar ratio [MATH] shows a different behaviour.', '1510.04030-2-49-1': 'It depends weakly on [MATH] but strongly on [MATH], with higher values of [MATH] corresponding to higher values of [MATH].', '1510.04030-2-49-2': 'Moreover, increasing the value of [MATH] pushes the value of [MATH] towards the upper bound [MATH].', '1510.04030-2-49-3': 'For [MATH] we predict [MATH] independently of the value of [MATH], and then, in order to have values of order [MATH] we need both [MATH] and [MATH] near its upper value.', '1510.04030-2-50-0': '## Vacuum energy and inflaton mass', '1510.04030-2-51-0': 'In [REF] we show the numerical results obtained from [REF].', '1510.04030-2-51-1': 'We plot the vacuum energy [MATH] (left) and the inflaton mass [MATH] (right) as functions of [MATH] and [MATH], again for [MATH].', '1510.04030-2-51-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the scale of colour shown on the right of every plot.', '1510.04030-2-52-0': 'The regions plotted in [REF] are the same as those plotted in [REF], i.e. they represent the range of values of [MATH] and [MATH] allowed by the experimental data.', '1510.04030-2-52-1': 'Because we do not have stringent experimental bounds on [MATH] and [MATH], we are interested just in the order of magnitude of these quantities.', '1510.04030-2-52-2': 'We observe that the order of magnitude of [MATH] and [MATH] depends very weakly on both [MATH] and [MATH].', '1510.04030-2-52-3': 'Also the dependence on [MATH] is quite weak, as long as we take values of [MATH] not too close to [MATH].', '1510.04030-2-52-4': 'Thus for [MATH] not too close to [MATH], the vacuum energy remains about [MATH] to [MATH] Planck masses, whereas the inflaton mass is between [MATH] and [MATH] Planck masses.', '1510.04030-2-52-5': 'On the other hand both [MATH] and [MATH] shrink drastically when we move close to [MATH].', '1510.04030-2-53-0': '## Other branch of the potential', '1510.04030-2-54-0': 'Until now we have considered the slow-roll regime for the branch I of the potential, i.e. [MATH].', '1510.04030-2-54-1': 'Let us briefly consider branch II, i.e. [MATH].', '1510.04030-2-54-2': 'Investigation of this branch is of particular interest because the most interesting cosmological solutions one can obtain for the exact solvable model with [MATH] are defined in the branch II of the potential [CITATION].', '1510.04030-2-55-0': 'In terms of the parametrisation [REF], region II corresponds to [MATH].', '1510.04030-2-55-1': 'The slow-roll parameters [MATH] and [MATH] are still given by [REF] but now the condition for inflation [MATH] requires (1-h)Y1, which can be satisfied only if [MATH].', '1510.04030-2-55-2': 'It follows that [MATH].', '1510.04030-2-55-3': 'One can easily see from [REF] that these values of [MATH] are not only incompatible with the slow-roll condition [MATH], but are also completely ruled out by the experimental constraints on [MATH].', '1510.04030-2-56-0': '# Conclusion', '1510.04030-2-57-0': 'In this paper we have constructed the most general Einstein-scalar gravity model in which the potential is given by the sum of two exponentials and inflation is generated by a scalar field [MATH] rolling off from the de Sitter maximum of the potential [MATH].', '1510.04030-2-57-1': 'These models are the cosmological counterpart of holographic models used to describe hyperscaling violation in the ultraviolet [CITATION].', '1510.04030-2-57-2': 'We have investigated inflation in the slow-roll approximation.', '1510.04030-2-57-3': 'Our model predicts inflation at energy scales of four to five orders of magnitude below the Planck scale, whereas the inflaton mass, at the onset of inflation, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-2-57-4': 'We have shown that our model reproduces correctly, for a wide range of its parameters the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-2-58-0': 'The proposed inflationary model belongs to the wide class of small-field models, which also include the Starobinsky model and, more generally, the cosmological attractor models.', '1510.04030-2-58-1': 'Our model shares with those several features: (1) the potential is built as a combination of exponentials, it predicts (2) an energy scale of inflation four order of magnitude below the Planck mass, (3) a "red" power spectrum and (4) a small tensor/scalar amplitude ratio.', '1510.04030-2-58-2': 'On the other hand, our model differs from the Starobinsky one in a crucial aspect: inflation is not generated, as in Starobinsky model, by a scalar field rolling off from an asymptotically constant potential, but rather from a local maximum of the potential.', '1510.04030-2-58-3': 'This property allows us to interpret the inflaton as a tachyonic excitation of the dS vacuum and to introduce a second scale of energy in the theory, the mass scale [MATH], which is 7-8 order of magnitude below the Planck mass.', '1510.04030-2-58-4': 'This hierarchy of scales opens the intriguing possibility that the origin of the inflaton could be explained by the physics at energy scales 7-8 order of magnitude below the Planck mass.', '1510.04030-2-59-0': 'We close with a brief comment about the reheating phase and the transition from inflation to the radiation/matter dominated era.', '1510.04030-2-59-1': 'During reheating the energy is transferred from the inflaton to matter fields.', '1510.04030-2-59-2': 'This means that there must exist a region in which the kinetic energy of the inflaton dominates over its potential energy, e.g. a local minimum of the potential.', '1510.04030-2-59-3': 'It is evident from [REF] that the potential [REF] does not have such a region and hence it cannot be used to describe reheating.', '1510.04030-2-59-4': 'Thus, in order to describe reheating our potential must be matched with continuity at the end of inflation with some other branch of a potential exhibiting a local minimum.', '1510.04030-2-59-5': 'This can be done very easily.', '1510.04030-2-59-6': 'In the [MATH]-parametrisation the point [MATH] given by [REF], at which the universe exits inflation, is always on the left of the point [MATH] at which [MATH] cuts the horizontal axis, i.e. we have [MATH] and [MATH].', '1510.04030-2-59-7': 'Since the slow-roll approximation is badly broken at [MATH], the matching with the branch of the potential with the local minimum must be performed at a point [MATH].', '1510.04030-2-60-0': '# [MATH] model Cosh phi model', '1510.04030-2-61-0': 'The model [REF] is the most general form of the potential one can obtain imposing conditions [REF] and assuming that [MATH] is built as a combination of two exponential without an additive constant term.', '1510.04030-2-61-1': 'When such a constant term (which we call [MATH]) is present, only the first equation in [REF] has to be modified and becomes [MATH], whereas the second and third equations remain unchanged.', '1510.04030-2-61-2': 'A general solution of the ensuing system is given by [MATH].', '1510.04030-2-62-0': 'A simple example of this class of potentials is given by V () =^2(2- ) This potential gives a further example of inflation generated by an unstable de-Sitter vacuum.', '1510.04030-2-63-0': 'The potential [REF] has a maximum at [MATH], corresponding to an unstable de Sitter solution with [MATH], and a corresponding tachyonic excitation.', '1510.04030-2-63-1': 'For [MATH], the potential behaves as purely exponential.', '1510.04030-2-64-0': 'The vacuum energy and inflaton mass, expressed in terms of [MATH] and [MATH], defined as in [REF], are M^2_I = -163^4h^2m_P^2,E_V= ,m_P.', '1510.04030-2-65-0': 'Introducing the variable [MATH], the slow-roll parameters [MATH] and [MATH] take the form 13 h^2(Y^2-1Y^2-4Y+1)^2,23 h^2Y^2+1Y^2-4Y+1-.', '1510.04030-2-65-1': 'The slow-roll parameter [MATH] is zero on the maximum of the potential ([MATH]).', '1510.04030-2-65-2': 'Moreover, we have [MATH] for [MATH], where Y_0=2 3h+ 1+9 h^23h+1.', '1510.04030-2-65-3': 'For [MATH] we have inflation, whereas for [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-2-65-4': 'One can easily check that during inflation we always have [MATH].', '1510.04030-2-65-5': 'Conversely, the parameter [MATH], which gives a measure of the curvature of the potential, is not small in general, but is of order [MATH].', '1510.04030-2-66-0': 'Also for these models the simplest way to satisfy the usual slow-roll conditions for inflation, [MATH], is to choose [MATH], so that [MATH] as well as [MATH].', '1510.04030-2-67-0': 'In the slow-roll approximation the spectral parameters [MATH], [MATH] and [MATH] expressed in terms of [MATH] are, [EQUATION] where [MATH] is defined implicitly as a function of [MATH] by klh1.', '1510.04030-2-68-0': 'fig:cosh shows that there exists a region in the parameter space [MATH] where the model correctly reproduces the results of observation [CITATION].', '1510.04030-2-68-1': 'Moreover, it predicts the vacuum energy to be three to four orders of magnitude below the Planck scale and the mass of the inflaton six to seven orders of magnitude smaller than the Planck mass.', '1510.04030-2-69-0': 'EF acknowledges financial support provided under the European Union\'s H2020 ERC Consolidator Grant "Matter and strong-field gravity: New frontiers in Einstein\'s theory" grant agreement no.', '1510.04030-2-69-1': 'MaGRaTh-646597.'} | [['1510.04030-1-57-0', '1510.04030-2-57-0'], ['1510.04030-1-57-1', '1510.04030-2-57-1'], ['1510.04030-1-57-2', '1510.04030-2-57-2'], ['1510.04030-1-57-3', '1510.04030-2-57-3'], ['1510.04030-1-57-4', '1510.04030-2-57-4'], ['1510.04030-1-69-0', '1510.04030-2-69-0'], ['1510.04030-1-59-0', '1510.04030-2-59-0'], ['1510.04030-1-59-1', '1510.04030-2-59-1'], ['1510.04030-1-59-2', '1510.04030-2-59-2'], ['1510.04030-1-59-3', '1510.04030-2-59-3'], ['1510.04030-1-59-4', '1510.04030-2-59-4'], ['1510.04030-1-59-5', '1510.04030-2-59-5'], ['1510.04030-1-59-6', '1510.04030-2-59-6'], ['1510.04030-1-59-7', '1510.04030-2-59-7'], ['1510.04030-1-67-0', '1510.04030-2-67-0'], ['1510.04030-1-0-0', '1510.04030-2-0-0'], ['1510.04030-1-0-1', '1510.04030-2-0-1'], ['1510.04030-1-0-2', '1510.04030-2-0-2'], ['1510.04030-1-0-3', '1510.04030-2-0-3'], ['1510.04030-1-0-4', '1510.04030-2-0-4'], 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'1510.04030-2-15-0'], ['1510.04030-1-38-0', '1510.04030-2-38-0'], ['1510.04030-1-38-1', '1510.04030-2-38-1'], ['1510.04030-1-38-2', '1510.04030-2-38-2'], ['1510.04030-1-8-0', '1510.04030-2-8-0'], ['1510.04030-1-8-1', '1510.04030-2-8-1'], ['1510.04030-1-8-2', '1510.04030-2-8-2'], ['1510.04030-1-8-3', '1510.04030-2-8-3'], ['1510.04030-1-8-4', '1510.04030-2-8-4'], ['1510.04030-1-8-5', '1510.04030-2-8-5'], ['1510.04030-1-8-6', '1510.04030-2-8-6'], ['1510.04030-1-20-0', '1510.04030-2-20-0'], ['1510.04030-1-29-0', '1510.04030-2-29-0'], ['1510.04030-1-29-1', '1510.04030-2-29-1'], ['1510.04030-1-29-2', '1510.04030-2-29-2'], ['1510.04030-1-29-3', '1510.04030-2-29-3'], ['1510.04030-1-29-4', '1510.04030-2-29-4'], ['1510.04030-1-29-5', '1510.04030-2-29-5'], ['1510.04030-1-22-0', '1510.04030-2-22-0'], ['1510.04030-1-22-1', '1510.04030-2-22-1'], ['1510.04030-1-7-0', '1510.04030-2-7-0'], ['1510.04030-1-7-1', '1510.04030-2-7-1'], ['1510.04030-1-7-2', '1510.04030-2-7-2'], ['1510.04030-1-7-3', '1510.04030-2-7-3'], ['1510.04030-1-16-0', '1510.04030-2-16-0'], ['1510.04030-1-16-1', '1510.04030-2-16-1'], ['1510.04030-1-31-0', '1510.04030-2-31-0'], ['1510.04030-1-31-1', '1510.04030-2-31-1'], ['1510.04030-1-31-2', '1510.04030-2-31-2'], ['1510.04030-1-31-3', '1510.04030-2-31-3'], ['1510.04030-1-31-4', '1510.04030-2-31-4'], ['1510.04030-1-31-5', '1510.04030-2-31-5'], ['1510.04030-1-31-6', '1510.04030-2-31-6'], ['1510.04030-1-63-0', '1510.04030-2-63-0'], ['1510.04030-1-63-1', '1510.04030-2-63-1'], ['1510.04030-1-61-0', '1510.04030-2-61-0'], ['1510.04030-1-61-1', '1510.04030-2-61-1'], ['1510.04030-1-61-2', '1510.04030-2-61-2'], ['1510.04030-1-10-0', '1510.04030-2-10-0'], ['1510.04030-1-34-0', '1510.04030-2-34-0'], ['1510.04030-1-34-1', '1510.04030-2-34-1'], ['1510.04030-1-34-2', '1510.04030-2-34-2'], ['1510.04030-1-34-3', '1510.04030-2-34-3'], ['1510.04030-1-37-0', '1510.04030-2-37-0'], ['1510.04030-1-37-1', '1510.04030-2-37-1'], ['1510.04030-1-37-2', '1510.04030-2-37-2'], 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'1510.04030-3-13-1'], ['1510.04030-2-12-2', '1510.04030-3-13-2'], ['1510.04030-2-12-3', '1510.04030-3-13-3'], ['1510.04030-2-12-4', '1510.04030-3-13-4'], ['1510.04030-2-27-3', '1510.04030-3-28-2'], ['1510.04030-2-36-1', '1510.04030-3-38-1'], ['1510.04030-2-36-2', '1510.04030-3-38-2'], ['1510.04030-2-41-0', '1510.04030-3-52-0'], ['1510.04030-2-41-2', '1510.04030-3-52-2'], ['1510.04030-2-58-1', '1510.04030-3-70-2'], ['1510.04030-2-58-2', '1510.04030-3-70-3'], ['1510.04030-2-58-3', '1510.04030-3-70-4'], ['1510.04030-2-58-4', '1510.04030-3-70-5'], ['1510.04030-2-51-1', '1510.04030-3-62-1'], ['1510.04030-2-51-2', '1510.04030-3-62-2'], ['1510.04030-2-13-6', '1510.04030-3-14-6'], ['1510.04030-2-19-0', '1510.04030-3-21-0'], ['1510.04030-2-25-2', '1510.04030-3-26-2'], ['1510.04030-2-25-3', '1510.04030-3-26-3'], ['1510.04030-2-25-4', '1510.04030-3-26-4'], ['1510.04030-2-5-0', '1510.04030-3-5-0'], ['1510.04030-2-5-1', '1510.04030-3-5-1'], ['1510.04030-2-5-3', '1510.04030-3-5-3'], 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'1510.04030-3-72-5'], ['1510.04030-2-59-7', '1510.04030-3-72-7'], ['1510.04030-2-8-0', '1510.04030-3-9-0'], ['1510.04030-2-10-0', '1510.04030-3-11-0'], ['1510.04030-2-15-0', '1510.04030-3-16-0'], ['1510.04030-2-33-0', '1510.04030-3-35-0'], ['1510.04030-2-33-1', '1510.04030-3-35-1'], ['1510.04030-2-33-2', '1510.04030-3-35-2'], ['1510.04030-2-33-4', '1510.04030-3-35-4'], ['1510.04030-2-34-0', '1510.04030-3-36-0'], ['1510.04030-2-34-1', '1510.04030-3-36-1'], ['1510.04030-2-34-2', '1510.04030-3-36-2'], ['1510.04030-2-34-3', '1510.04030-3-36-3'], ['1510.04030-2-42-0', '1510.04030-3-53-0'], ['1510.04030-2-18-1', '1510.04030-3-19-1'], ['1510.04030-2-18-2', '1510.04030-3-19-2'], ['1510.04030-2-18-3', '1510.04030-3-19-3'], ['1510.04030-2-18-4', '1510.04030-3-19-4'], ['1510.04030-2-61-1', '1510.04030-3-74-1'], ['1510.04030-2-54-1', '1510.04030-3-66-1'], ['1510.04030-2-66-0', '1510.04030-3-78-0'], ['1510.04030-2-40-0', '1510.04030-3-51-0'], ['1510.04030-2-65-1', '1510.04030-3-77-1'], ['1510.04030-2-65-3', '1510.04030-3-77-3'], ['1510.04030-2-65-4', '1510.04030-3-77-4'], ['1510.04030-2-65-5', '1510.04030-3-77-5'], ['1510.04030-2-3-0', '1510.04030-3-3-0'], ['1510.04030-2-3-2', '1510.04030-3-3-2'], ['1510.04030-2-31-0', '1510.04030-3-32-0'], ['1510.04030-2-31-3', '1510.04030-3-33-1'], ['1510.04030-2-31-4', '1510.04030-3-33-2'], ['1510.04030-2-31-5', '1510.04030-3-33-3'], ['1510.04030-2-6-0', '1510.04030-3-6-0'], ['1510.04030-2-6-3', '1510.04030-3-7-2'], ['1510.04030-2-63-0', '1510.04030-3-76-0'], ['1510.04030-2-7-2', '1510.04030-3-8-2'], ['1510.04030-2-51-0', '1510.04030-3-62-0']] | [['1510.04030-2-38-1', '1510.04030-3-47-1'], ['1510.04030-1-12-6', '1510.04030-2-12-7'], ['1510.04030-2-29-3', '1510.04030-3-30-4'], ['1510.04030-2-38-2', '1510.04030-3-47-2'], ['1510.04030-2-45-0', '1510.04030-3-55-0'], ['1510.04030-2-52-2', '1510.04030-3-63-1'], ['1510.04030-2-57-1', '1510.04030-3-69-1'], ['1510.04030-2-12-5', '1510.04030-3-13-5'], ['1510.04030-2-12-6', '1510.04030-3-13-6'], ['1510.04030-2-12-7', '1510.04030-3-13-7'], ['1510.04030-2-27-0', '1510.04030-3-28-0'], ['1510.04030-2-27-2', '1510.04030-3-28-1'], ['1510.04030-2-36-0', '1510.04030-3-38-0'], ['1510.04030-2-36-3', '1510.04030-3-38-3'], ['1510.04030-2-41-1', '1510.04030-3-52-1'], ['1510.04030-2-41-3', '1510.04030-3-52-3'], ['1510.04030-2-26-0', '1510.04030-3-27-0'], ['1510.04030-2-13-0', '1510.04030-3-14-0'], ['1510.04030-2-13-2', '1510.04030-3-14-2'], ['1510.04030-2-13-3', '1510.04030-3-14-3'], ['1510.04030-2-13-4', '1510.04030-3-14-4'], ['1510.04030-2-13-5', '1510.04030-3-14-5'], ['1510.04030-2-19-1', '1510.04030-3-21-1'], ['1510.04030-2-25-1', '1510.04030-3-26-1'], ['1510.04030-2-5-2', '1510.04030-3-5-2'], ['1510.04030-2-14-1', '1510.04030-3-15-1'], ['1510.04030-2-14-3', '1510.04030-3-15-3'], ['1510.04030-2-2-0', '1510.04030-3-2-0'], ['1510.04030-2-68-1', '1510.04030-3-83-1'], ['1510.04030-2-37-1', '1510.04030-3-39-1'], ['1510.04030-2-0-2', '1510.04030-3-0-2'], ['1510.04030-2-11-1', '1510.04030-3-12-1'], ['1510.04030-2-49-0', '1510.04030-3-60-0'], ['1510.04030-2-55-0', '1510.04030-3-67-0'], ['1510.04030-2-16-0', '1510.04030-3-17-0'], ['1510.04030-2-16-1', '1510.04030-3-17-1'], ['1510.04030-2-59-6', '1510.04030-3-72-6'], ['1510.04030-2-8-2', '1510.04030-3-9-2'], ['1510.04030-2-8-3', '1510.04030-3-9-3'], ['1510.04030-2-8-4', '1510.04030-3-9-4'], ['1510.04030-2-8-5', '1510.04030-3-9-5'], ['1510.04030-2-8-6', '1510.04030-3-9-6'], ['1510.04030-2-43-0', '1510.04030-3-54-0'], ['1510.04030-2-44-2', '1510.04030-3-49-2'], ['1510.04030-2-44-3', '1510.04030-3-49-3'], ['1510.04030-2-32-0', '1510.04030-3-34-0'], ['1510.04030-2-30-1', '1510.04030-3-31-2'], ['1510.04030-2-30-3', '1510.04030-3-31-3'], ['1510.04030-2-33-3', '1510.04030-3-35-3'], ['1510.04030-2-18-0', '1510.04030-3-19-0'], ['1510.04030-2-61-0', '1510.04030-3-74-0'], ['1510.04030-2-61-2', '1510.04030-3-74-2'], ['1510.04030-2-54-0', '1510.04030-3-66-0'], ['1510.04030-2-54-2', '1510.04030-3-66-2'], ['1510.04030-2-47-2', '1510.04030-3-58-2'], ['1510.04030-2-31-2', '1510.04030-3-33-0'], ['1510.04030-2-31-6', '1510.04030-3-33-4'], ['1510.04030-2-6-1', '1510.04030-3-7-0'], ['1510.04030-2-6-2', '1510.04030-3-7-1'], ['1510.04030-2-6-4', '1510.04030-3-7-3'], ['1510.04030-2-63-1', '1510.04030-3-76-1'], ['1510.04030-2-64-0', '1510.04030-3-76-2'], ['1510.04030-2-7-0', '1510.04030-3-8-0'], ['1510.04030-2-7-1', '1510.04030-3-8-1'], ['1510.04030-2-47-0', '1510.04030-3-58-0']] | [] | [['1510.04030-2-52-4', '1510.04030-3-63-3'], ['1510.04030-2-58-0', '1510.04030-3-70-1'], ['1510.04030-2-25-0', '1510.04030-3-26-0'], ['1510.04030-2-14-2', '1510.04030-3-15-2'], ['1510.04030-2-68-0', '1510.04030-3-83-0'], ['1510.04030-2-0-1', '1510.04030-3-0-1'], ['1510.04030-2-23-0', '1510.04030-3-24-0'], ['1510.04030-2-23-3', '1510.04030-3-24-1'], ['1510.04030-2-49-3', '1510.04030-3-60-2'], ['1510.04030-2-30-0', '1510.04030-3-31-0'], ['1510.04030-2-42-1', '1510.04030-3-53-1'], ['1510.04030-2-62-0', '1510.04030-3-75-0'], ['1510.04030-2-62-0', '1510.04030-3-75-1'], ['1510.04030-2-47-1', '1510.04030-3-58-1'], ['1510.04030-2-3-1', '1510.04030-3-3-1'], ['1510.04030-2-3-3', '1510.04030-3-4-0'], ['1510.04030-2-3-4', '1510.04030-3-4-0'], ['1510.04030-2-31-1', '1510.04030-3-32-1'], ['1510.04030-2-4-0', '1510.04030-3-8-4'], ['1510.04030-2-4-0', '1510.04030-3-8-7'], ['1510.04030-2-7-3', '1510.04030-3-8-3']] | [] | ['1510.04030-1-13-1', '1510.04030-1-18-5', '1510.04030-1-33-5', '1510.04030-1-65-0', '1510.04030-1-65-2', '1510.04030-1-69-1', '1510.04030-2-13-1', '1510.04030-2-18-5', '1510.04030-2-33-5', '1510.04030-2-65-0', '1510.04030-2-65-2', '1510.04030-2-69-1', '1510.04030-3-14-1', '1510.04030-3-19-5', '1510.04030-3-35-5', '1510.04030-3-39-2', '1510.04030-3-40-0', '1510.04030-3-41-0', '1510.04030-3-42-0', '1510.04030-3-43-0', '1510.04030-3-44-0', '1510.04030-3-45-0', '1510.04030-3-46-0', '1510.04030-3-47-0', '1510.04030-3-77-0', '1510.04030-3-77-2', '1510.04030-3-79-0', '1510.04030-3-80-0', '1510.04030-3-81-0', '1510.04030-3-82-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1510.04030 | {'1510.04030-3-0-0': 'We consider cosmological inflation generated by a scalar field slowly rolling off from a de Sitter maximum of its potential.', '1510.04030-3-0-1': 'The models belong to the class of hilltop models and represent the most general model of this kind in which the scalar potential can be written as the sum of two exponentials.', '1510.04030-3-0-2': 'The minimally coupled Einstein-scalar gravity theory obtained in this way is the cosmological version of a two-scale generalization of known holographic models, allowing for solitonic solutions interpolating between an AdS spacetime in the infrared and scaling solutions in the ultraviolet.', '1510.04030-3-0-3': 'We then investigate cosmological inflation in the slow-roll approximation.', '1510.04030-3-0-4': 'Our model reproduces correctly, for a wide range of its parameters, the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-3-0-5': 'Moreover, it predicts inflation at energy scales of four to five orders of magnitude below the Planck scale.', '1510.04030-3-0-6': 'At the onset of inflation, the mass of the tachyonic excitation, i.e. of the inflaton, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-3-1-0': '# Introduction', '1510.04030-3-2-0': 'Nowadays, inflationary cosmology [CITATION] represents the easiest way to solve the problems of the standard Friedmann-Robertson-Walker (FRW) cosmology, such as the horizon and flatness problems - for a review, see e.g. Ref. [CITATION].', '1510.04030-3-3-0': 'The simplest way to generate inflation is to minimally couple Einstein gravity to a scalar field (the inflaton) with a self-interaction potential.', '1510.04030-3-3-1': 'There exist a plethora of single field inflationary models that can be classified according to the features of the potential [CITATION].', '1510.04030-3-3-2': 'Other alternatives include more scalar fields, as in the curvaton mechanism [CITATION].', '1510.04030-3-4-0': 'Nevertheless, the most recent data of the Planck satellite exclude non-Gaussian perturbations and give a striking experimental confirmation of the simplest single-field inflationary scenario [CITATION], and in particular the Starobinsky model [CITATION], or more in general, the so-called cosmological attractors [CITATION], characterized by a "red" power spectrum for primordial perturbations and a small tensor/ scalar amplitude ratio.', '1510.04030-3-5-0': 'The accuracy of the observational data concerning the power spectrum of primordial quantum fluctuations represents an efficient guide to select inflation models.', '1510.04030-3-5-1': 'But, despite the recent remarkable improvements, the important questions about the microscopic origin of the inflaton and about the physics before inflation are still unanswered.', '1510.04030-3-5-2': 'This lack of knowledge does not allow one to single out a unique inflationary model, i.e. a specific form of the potential.', '1510.04030-3-5-3': 'In fact, although the Planck data can be used to strongly constrain the inflationary model, mainly through the values of the spectral index [MATH] and the tensor/scalar amplitude ratio [MATH], they are not sufficient to select a unique model.', '1510.04030-3-6-0': 'In view of this situation, it is natural to look for hints coming from somewhere else in gravitational physics, for instance supergravity and string theory [CITATION].', '1510.04030-3-7-0': 'In recent times, minimally coupled Einstein-scalar gravity has been intensively investigated for holographic applications [CITATION].', '1510.04030-3-7-1': 'A class of Einstein-scalar gravity models of particular interest are those allowing for solitonic solutions interpolating between anti de Sitter (AdS) vacua and domain wall (DW) solutions with scale-covariant symmetries.', '1510.04030-3-7-2': 'The holographically dual QFT has scaling symmetries, which have a nice interpretation in terms of features of phase transitions in condensed matter systems (hyperscaling violation).', '1510.04030-3-7-3': 'These solitonic solutions are naturally related to cosmological solutions by the so-called DW/cosmology duality, a sort of analytic continuation, which maps the soliton in a FRW solution [CITATION].', '1510.04030-3-8-0': 'The cosmological duals of solitons which interpolate between an AdS spacetime at large distances of the bulk theory (the ultraviolet of the dual QFT) and a scale-covariant geometry at small distances in the bulk theory (the infrared of the dual QFT) are natural candidates for describing dark energy [CITATION].', '1510.04030-3-8-1': 'On the other hand, the cosmological duals of solitons interpolating between AdS in the infrared and scale-covariant geometries in the ultraviolet [CITATION] may be relevant for describing inflation.', '1510.04030-3-8-2': 'It has been shown that the cosmological solutions of this class of models generate inflation as the scalar field rolls down from a de Sitter (dS) spacetime [CITATION].', '1510.04030-3-8-3': 'In this context, inflation can be described as an instability of the dS spacetime rolling down to a scaling solution.', '1510.04030-3-8-4': 'Such models are known as hilltop models [CITATION] and inflation is generated by a scalar field rolling off from a local maximum to the potential.', '1510.04030-3-8-5': 'In such a scenario, since inflation starts from a local maximum, the slow-roll conditions can be satisfied more easily.', '1510.04030-3-8-6': 'On the experimental side, hilltop models are a subset of the small-field models, which are characterized by a potential with negative curvature.', '1510.04030-3-8-7': 'This behaviour of the potential is typical of spontaneous symmetry breaking and phase transitions, e.g. quartic potentials, natural inflation models [CITATION] and Coleman-Weinberg potentials [CITATION].', '1510.04030-3-8-8': 'Although hilltop models have been widely used to generate cosmological inflation, in most of them the potential is constructed using powers of the the inflaton field.', '1510.04030-3-8-9': 'To our knowledge, little attention has been given to hilltop models in which the potential is built as a combination of two exponentials.', '1510.04030-3-8-10': 'In this paper we discus the most general, holographically motivated, hilltop model, for which the potential can be written as the sum of two exponentials.', '1510.04030-3-8-11': 'We will show that although near the maximum our model has the well-known behaviour of hilltop models with a parabolic potential, at late times it gives predictions of the spectral parameters of the CMB radiation, which are specific for a two-exponential potential.', '1510.04030-3-9-0': 'The structure of the paper is as follows.', '1510.04030-3-9-1': 'In [REF] we generalize the model proposed in Ref. [CITATION] and we construct the most general potential given by the sum of two exponentials.', '1510.04030-3-9-2': 'We show that the minimally coupled Einstein-scalar gravity theory constructed in this way is the cosmological version of a two-scale generalization of the holographic models of Refs. [CITATION].', '1510.04030-3-9-3': 'In sect:s3 we discuss the cosmological solution of our model.', '1510.04030-3-9-4': 'Inflation and the spectral parameters of the power spectrum of primordial perturbations are discussed in sect:slowroll using the slow-roll approximation.', '1510.04030-3-9-5': 'In sect:s5 we compare the theoretical predictions of our model with observations.', '1510.04030-3-9-6': 'Finally, in sect:s6 we state our conclusions and in [REF] we briefly repeat our calculations for a model in which the potential has a constant additive term.', '1510.04030-3-10-0': '# The model', '1510.04030-3-11-0': 'The simplest way to fuel inflation into a cosmological scenario is to couple, minimally, Einstein gravity to a scalar field [MATH] with an appropriate self-interaction potential [MATH]: A=d^4x-g(m_P^216R -12()^2-V ()).', '1510.04030-3-12-0': 'In this paper we focus on inflation generated by a scalar field rolling off from a maximum of [MATH].', '1510.04030-3-12-1': 'This class of models is very natural from a physical point of view because inflation can be thought of just as an instability of the dS spacetime, generated by a scalar perturbation.', '1510.04030-3-13-0': 'Our first goal is to construct the general form of the potential belonging to this class.', '1510.04030-3-13-1': 'Without loss of generality we can assume that the maximum of the potential occurs at [MATH], so that the basic necessary conditions to be imposed on the potential read V (0)>0,V\' (0)=0,V" (0)<0.', '1510.04030-3-13-2': 'Obviously, the previous conditions are very loose and do not select any specific form of [MATH].', '1510.04030-3-13-3': 'We further constrain the form of the potential by requiring it to be a linear combination of two exponentials.', '1510.04030-3-13-4': 'This is a rather strong assumption, but is supported by several arguments.', '1510.04030-3-13-5': 'Exponential potentials for scalar field appear quite generically in a variety of situations: compactification of extra dimensions, [MATH] gravity theories (which on-shell are equivalent to Einstein-scalar gravity) and low-energy effective string theory.', '1510.04030-3-13-6': 'The double exponential potential appears in the context of dimensional reduction of gravity with non-trivial four-form flux on a maximally symmetric internal space - see e.g. Ref. [CITATION] and references therein.', '1510.04030-3-13-7': 'Moreover, exponential potentials have been shown to be the source of brane solutions of Einstein-scalar gravity called DWs [CITATION], which can be analytically continued into FRW cosmological solutions [CITATION].', '1510.04030-3-14-0': 'We are therefore led to consider the following general form of the inflation potential V () =^2(a_1e^b_1+a_2 e^b_2), where [MATH] and [MATH] are some length scales, whose physical meaning will be clarified in short, and [MATH] are some dimensionless constants characterizing the model.', '1510.04030-3-14-1': 'They are constrained by [REF], giving a_1+a_2>0,a_1b_1=- a_2b_2,a_1b_1^2+a_2b_2^2 <0.', '1510.04030-3-14-2': 'Modulo trivial symmetries interchanging the two exponentials in the potential, the most general solution of the previous equations is [MATH], [MATH], [MATH], [MATH] and [MATH], where we have defined a new dimensionless parameter [MATH].', '1510.04030-3-14-3': 'The parameter rescaling [MATH], [MATH] brings the potential in the form V () =2^23(e^3-^2 e^3/), where [MATH].', '1510.04030-3-14-4': 'The potential [REF] is a two-scale generalization of the model proposed in Refs. [CITATION], to which it reduces for the particular value of the parameter [MATH].', '1510.04030-3-14-5': 'The cosmology of the latter model has been investigated in Ref. [CITATION].', '1510.04030-3-14-6': 'We will see in the next section that for generic values of the parameter [MATH] the cosmological equations resulting from the model [REF] do not give rise to an exactly integrable system.', '1510.04030-3-15-0': 'The potential [REF] is invariant both under the transformation [MATH], which corresponds to interchanging the two exponentials in the potential [REF] and under the transformation [MATH].', '1510.04030-3-15-1': 'These symmetries allow us to limit our consideration to [MATH].', '1510.04030-3-15-2': 'The two limiting cases [MATH] correspond, respectively, to a pure exponential and to a potential behaving at leading order as V=(2^2/3)(1-3) e^3.', '1510.04030-3-15-3': 'The potential [MATH] has a maximum at [MATH] corresponding to an unstable dS solution with [MATH] and a corresponding tachyonic excitation, the inflaton.', '1510.04030-3-16-0': 'The potential [MATH] is depicted in [REF] for selected values of the parameters [MATH], [MATH] and [MATH].', '1510.04030-3-17-0': 'One can therefore use this model to describe inflation as generated by an unstable dS solution.', '1510.04030-3-17-1': 'Inflation starts as a quantum fluctuation of the dS solution and is initially driven by a tachyonic excitation of the dS spacetime and proceeds as the scalar field rolls off from the maximum of the potential.', '1510.04030-3-18-0': '## Physical scales', '1510.04030-3-19-0': 'Besides the Planck length [MATH], the model is parametrized by the two length scales [MATH] and [MATH] and by the dimensionless parameter [MATH].', '1510.04030-3-19-1': 'The presence of two length scales is a characteristic feature of small-field models of inflation.', '1510.04030-3-19-2': 'In the present context the two scales have a simple interpretation in terms of geometric properties of the function [MATH].', '1510.04030-3-19-3': 'They give, respectively, the height and the curvature of the [MATH] maximum of the function [MATH].', '1510.04030-3-19-4': 'Correspondingly, [MATH] and [MATH] determine the two physical scales relevant for inflation: the vacuum energy [MATH] at the beginning of inflation and the inflaton mass squared [MATH].', '1510.04030-3-19-5': 'We have M^2_I &= V" (0)=-2^2^2= -323^4h^2m_P^2,', '1510.04030-3-20-0': 'E_V &= [V (0)]^1/4= (2/3)^1/4m_P, where we have introduced the two dimensionless parameters [MATH] and [MATH], h= 43(l_P),^1/2m_P, representing the measures of [MATH] and [MATH] in Planck units.', '1510.04030-3-21-0': 'Conversely, [MATH] is a purely dimensionless parameter and plays a role which is drastically different from [MATH] and [MATH].', '1510.04030-3-21-1': 'It is not linked to any physical scale of the model but quantifies the deviation of the potential from a pure exponential behaviour attained for [MATH] near 0.', '1510.04030-3-22-0': 'In the following we use instead of the negative quantity [MATH], the inflaton mass defined as [MATH].', '1510.04030-3-23-0': '# Cosmological solutions', '1510.04030-3-24-0': 'The cosmology of our model can be investigated using the usual FRW parametrization of the metric ds^2=- dt^2+ a(t)^2 dS_(3)^2, The dS spacetime with constant inflaton is an exact solution of cosmological equations.', '1510.04030-3-24-1': 'The dS solution has the usual exponential form, with the scale factor given by a=e^8 l_P Lt/3.', '1510.04030-3-24-2': 'This solution describes a scalar field sitting forever at the maximum of the potential, generating an exact exponential expansion of the universe, i.e. never ending inflation.', '1510.04030-3-25-0': 'The most interesting cosmological solutions are those describing inflation lasting for a finite amount of time.', '1510.04030-3-25-1': 'In this case the scalar rolls off from the maximum of [MATH], generating a quasi-exponential expansion of the universe as long as the potential energy of the scalar dominates the kinetic one.', '1510.04030-3-25-2': 'This kind of solutions would be the cosmological counterpart of the solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-3-26-0': 'Searching for these solutions, following Ref. [CITATION], one can try to find exact cosmological solutions by using a different parametrization for the time variable and linear combinations of the fields in such way that the equations for the scalar field and for the scale factor decouple.', '1510.04030-3-26-1': 'However, one can easily realize that the decoupling works only for the particular value of the parameter [MATH] (corresponding to [MATH]).', '1510.04030-3-26-2': 'For this value of [MATH] the Einstein-scalar gravity models give rise to exactly integrable models both in the case of static (brane) [CITATION] and cosmological solutions [CITATION].', '1510.04030-3-26-3': 'In the static case we have solitonic solutions interpolating between an AdS spacetime in the infrared and a DW in the ultraviolet [CITATION].', '1510.04030-3-26-4': 'Analogously, in the cosmological case we have exact solutions which can be used to model inflation [CITATION].', '1510.04030-3-27-0': 'For generic values of the parameter [MATH] the Einstein-scalar system does not decouple, is not exactly integrable and a cosmological solution cannot be found in analytic form.', '1510.04030-3-28-0': 'Approximate solutions of the field equations can be found for some limiting cases.', '1510.04030-3-28-1': 'Of particular interest is the case of small [MATH], for which the potential [REF] behaves exponentially, V () 2^2^23e^3/, the system can be solved analytically and we have scaling (power-law) solutions, which are obtained from scale-covariant (DW) solutions [CITATION] using the transformation [MATH], [MATH].', '1510.04030-3-28-2': 'In the gauge [REF] this scaling solution has the form at^h^2^2,e^2t^-hl_P .', '1510.04030-3-29-0': '# Inflation and slow-roll approximation', '1510.04030-3-30-0': 'Lacking exact solutions to investigate the cosmology of our model [REF], we work in the slow-roll approximation [CITATION].', '1510.04030-3-30-1': 'In this regime the potential energy of the scalar field dominates over the kinetic energy and the universe has a quasi-exponential accelerated expansion as the scalar field slowly rolls off from the maximum of the potential.', '1510.04030-3-30-2': 'Following the usual approach, we introduce the slow-roll parameters [MATH] and [MATH], m_P^216(V\'V)^2,m_P^28V"V-.', '1510.04030-3-30-3': 'Necessary conditions for the slow-roll approximation to be valid are [MATH].', '1510.04030-3-30-4': 'We have inflation as long as [MATH].', '1510.04030-3-30-5': 'The parameter [MATH] is zero on the maximum of the potential ([MATH]) and the solution is exactly dS, whereas inflation ends when [MATH].', '1510.04030-3-31-0': 'The potential [REF] is not a monotonic function of the scalar field [MATH]: it goes to zero as [MATH], has a maximum at [MATH], and crosses the axis for [MATH]; [MATH] as [MATH] (see [REF]).', '1510.04030-3-31-1': 'Since slow-roll inflation cannot occur for a negative inflaton potential, our model is valid up to [MATH] and the potential must be modified for values of [MATH] greater than [MATH].', '1510.04030-3-31-2': 'We have then two alternative branches that we can use to generate inflation, i.e. I: [MATH] and II: [MATH].', '1510.04030-3-31-3': 'In the following, we investigate the first branch and in [REF] we briefly discuss branch II and show that it cannot be compatible with observations.', '1510.04030-3-32-0': 'Let us now introduce the variable Y= e^3/.', '1510.04030-3-32-1': 'In this parametrization the branch under consideration corresponds to [MATH].', '1510.04030-3-33-0': 'The slow-roll parameter [MATH] is zero for [MATH], whereas [MATH] for [MATH], where Y_0=+h+^2 h. For [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-3-33-1': 'One can easily check that [MATH], so that during inflation we always have [MATH] and we can easily satisfy the first slow-roll condition [MATH].', '1510.04030-3-33-2': 'On the other hand, the parameter [MATH], which gives a measure of the curvature of the potential, is not small, but we have [MATH].', '1510.04030-3-33-3': 'It follows that the simplest way to satisfy the second slow-roll condition, [MATH], is to choose h10, in this way we can have [MATH] as well as [MATH].', '1510.04030-3-33-4': 'As already noted, the model discussed in Ref. [CITATION] does not satisfy [REF] because it is characterized by [MATH].', '1510.04030-3-34-0': "In the slow-roll regime, the universe expands quasi-exponentially and the number of [MATH]-folds [MATH], which determines the duration of inflation, is determined by N=-dt H=8m_P^2__e^_bdVV', where [MATH] are, respectively, the inflaton-field values at the end and beginning of inflation and [MATH] is the Hubble parameter.", '1510.04030-3-35-0': 'Using the definition [REF] and the expression [MATH] for [MATH] at the end of inflation, [REF] gives the function [MATH] in implicit form, Y^1/Y-1= e^2N/h^2 A,A := (+1h)(+h+^2 h)^1/.', '1510.04030-3-35-1': 'In the case of the dS solution [REF] the scalar field remains constant (the inflaton sits on the top of the potential), and we have [MATH] (eternal inflation).', '1510.04030-3-35-2': 'Obviously this configuration is highly unstable.', '1510.04030-3-35-3': 'A small perturbation of the scalar field starts the slow roll of the inflaton along the slope and a finite value of [MATH] is generated.', '1510.04030-3-35-4': 'If this fluctuation is small enough we can solve approximately [REF] for [MATH] near [MATH].', '1510.04030-3-35-5': 'We get at leading order, Y= 1+ A^-1 e^-2N/h^2.', '1510.04030-3-36-0': 'One can easily check that [MATH] with [MATH] for [MATH] and [MATH] for [MATH].', '1510.04030-3-36-1': 'Moreover, in the range [MATH], [MATH] is a monotonically decreasing function of [MATH] which depends very weakly on [MATH].', '1510.04030-3-36-2': 'It follows immediately that [REF] is a good approximation for [MATH] not too close to 0, whenever [MATH].', '1510.04030-3-36-3': 'When [MATH] the approximation [REF] holds irrespectively of the value of [MATH].', '1510.04030-3-37-0': '## Perturbations and spectral parameters', '1510.04030-3-38-0': 'One of the most striking predictions of inflation concerns the spectrum of tensor and scalar perturbations in the early universe [CITATION].', '1510.04030-3-38-1': 'During inflation the horizon shrinks and the primordial perturbations, which were causally connected are redshifted to superhorizon scales.', '1510.04030-3-38-2': 'Conversely, in the matter-radiation dominated era the horizon grows, the perturbations fall back in the horizon so that they can act as seeds for structure formation and anisotropy in the universe.', '1510.04030-3-38-3': 'The information as regards these primordial fluctuations is therefore encoded in the anisotropies of the CMB.', '1510.04030-3-39-0': 'Primordial quantum fluctuations are described in terms of two-point correlation functions for scalar and tensor modes in Fourier space and the associated power spectrum.', '1510.04030-3-39-1': 'In the slow-roll approximation, the power spectrum has a power-law behaviour and is usually characterized by four parameters: the amplitudes of scalar perturbations [MATH], the ratio [MATH] of the amplitudes of tensor and scalar perturbations and their spectral indices [MATH] and [MATH].', '1510.04030-3-39-2': "These parameters are functions of the number of [MATH]-folds [MATH] and can be expressed in terms of the potential [MATH] and the slow-roll parameters [REF] as follows: [param0] &P^1/2_R (N) = 4243 m_P^3V((N))^3/2V'((N)),", '1510.04030-3-40-0': '&r (N) =-8n_T((N)) = 16 ((N)),', '1510.04030-3-41-0': '&n_s (N) = 1 -4((N)) +2((N)), where [MATH] is defined by [REF].', '1510.04030-3-42-0': 'Using Eqs. [REF] and [REF] we can express the spectral parameters as a function of [MATH]: [param] &P^1/2_R (N) =4 h ^23[1-^2 Y (N)]^3/21-Y (N)Y(N)^^2/2,', '1510.04030-3-43-0': '&r (N) = 16^2h^2(1-Y (N)1-^2 Y (N))^2,', '1510.04030-3-44-0': '&n_s (N) = 1- 6^2h^2[1-Y(N)1-^2 Y(N)]^2+4h^2^2-Y (N)1-^2 Y (N), where [MATH] is defined, implicitly, by [REF].', '1510.04030-3-45-0': 'For [MATH] we can use the approximate expansion for [MATH] given by [REF] and we get, at leading order in the [MATH] expansion, [param1] &P^1/2_R (N) =4A3 h ^2 e^2N/h^2,', '1510.04030-3-46-0': '&r (N) =(4Ah)^2 e^-4N/h^2,', '1510.04030-3-47-0': '&n_s (N) =1-4h^2(1+1+^2Ae^-2N/h^2).', '1510.04030-3-47-1': 'One important feature of [REF] is the exponential dependence on [MATH].', '1510.04030-3-47-2': 'This must be compared with the typical behaviour of the Starobinsky model and more in general of cosmological attractor models, where one typically obtains [MATH] and [MATH] - see e.g. Ref. [CITATION] and references therein.', '1510.04030-3-48-0': 'It is easy to check that the exponential behaviour of the spectral parameters [REF] is an universal feature of hilltop models characterized by a quadratic maximum.', '1510.04030-3-48-1': 'It is a consequence of the local behaviour of the potential near [MATH].', '1510.04030-3-48-2': 'In fact [REF] can also be obtained by considering a potential [MATH], with [MATH] given by [REF].', '1510.04030-3-48-3': 'This is consistent with the fact that for [MATH] very large, inflation occurs near to the maximum of the potential, where [MATH] can be approximated by the previous form.', '1510.04030-3-49-0': 'Notice that the condition [MATH] alone does not does not guarantee the potential to be well approximated by the parabolic one.', '1510.04030-3-49-1': 'Since we need at least [MATH], such limit is obtained for [MATH].', '1510.04030-3-49-2': 'For instance for [MATH] and [MATH] we have [MATH].', '1510.04030-3-49-3': 'It follows that the approximate expressions [REF] can only be used in a regime of very large [MATH], for which we do not have a direct access to observations, and, therefore, in the following we will be using expressions [REF].', '1510.04030-3-50-0': '# Comparison with observation', '1510.04030-3-51-0': 'In this section we compare the theoretical results of our model for the spectral parameters [MATH], [MATH] and [MATH] with the most recent results of observations, in particular the joint analysis of BICEP2/Keck Array and Planck data [CITATION].', '1510.04030-3-52-0': 'The spectral parameters are functions of the number of the [MATH]-folds [MATH] and depend on the three dimensionless parameters [MATH], [MATH] and [MATH].', '1510.04030-3-52-1': 'Because [MATH] enters only in the normalization of the power spectrum [MATH], whereas [MATH] and [MATH] depend on [MATH] and [MATH] only we will use the following strategy: we will first determine using r,ns and the experimental results for [MATH] and [MATH], the allowed range of the parameters [MATH] and [MATH] for a given value of [MATH]-folds [MATH].', '1510.04030-3-52-2': 'We will then use pr and the experimental results for [MATH] to determine the corresponding values of the parameter [MATH].', '1510.04030-3-52-3': 'Finally we use pm-mass,pm-energy to determine the vacuum energy [MATH] and the inflaton mass [MATH].', '1510.04030-3-53-0': 'For [MATH], [MATH] and [MATH] we use the most recent results [CITATION], i.e. [MATH], [MATH] and [MATH].', '1510.04030-3-53-1': 'Since the perturbations we are observing today with momentum of the order of the horizon radius exited the horizon during inflation at [MATH], we will consider only values of [MATH] in this range.', '1510.04030-3-54-0': 'The calculations have to be performed numerically because the function [MATH] appearing in param is not known, but it is defined implicitly by klh.', '1510.04030-3-54-1': 'As we said in the previous section, a possible way to avoid numerical computations is to work in a regime where [MATH] and then param1 hold.', '1510.04030-3-54-2': 'But unfortunately, these expressions are valid in the large [MATH] regime, not accessible to observations.', '1510.04030-3-55-0': 'The results of our numerical computations are shown in the two sets of density plots in [REF].', '1510.04030-3-55-1': 'Once we have chosen the value of [MATH], the coloured region in such plots represent the range of values of [MATH] and [MATH] for which we have values of [MATH] and [MATH] compatible with the experimental measurements.', '1510.04030-3-56-0': 'Note that the allowed region of parameters [MATH] is quite independent from [MATH], at least for [MATH] in the range [MATH].', '1510.04030-3-57-0': '## Spectral parameters', '1510.04030-3-58-0': 'In [REF] we show the numerical results obtained from Eqs. [REF] and [REF].', '1510.04030-3-58-1': 'We plot the tensor/scalar ratio [MATH] (left) and the spectral index [MATH] (right) as functions of [MATH] and [MATH] for four selected values of [MATH].', '1510.04030-3-58-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the colour scale shown on the right of every plot.', '1510.04030-3-59-0': 'In general, higher values of [MATH] correspond to higher values of [MATH].', '1510.04030-3-59-1': 'Moreover, when [MATH] is not too close to zero [MATH] depends very weakly on [MATH].', '1510.04030-3-59-2': 'For [MATH] close to zero [MATH] is allowed to vary from [MATH] up to [MATH] and farther.', '1510.04030-3-59-3': 'As [MATH] increases the allowed range of [MATH] shrinks monotonically and is restricted to [MATH] for [MATH] close to [MATH].', '1510.04030-3-60-0': 'The tensor/scalar ratio [MATH] shows a different pattern.', '1510.04030-3-60-1': 'For [MATH] close to zero it depends strongly on [MATH].', '1510.04030-3-60-2': 'Whereas for values of [MATH] not too close to zero, it depends weakly on both parameters [MATH] and [MATH].', '1510.04030-3-60-3': 'Also in this case we observe the monotonic shrinking of the allowed values of [MATH] for growing values of [MATH].', '1510.04030-3-61-0': '## Vacuum energy and inflaton mass', '1510.04030-3-62-0': 'In [REF] we show the numerical results obtained from [REF].', '1510.04030-3-62-1': 'We plot the vacuum energy [MATH] (left) and the inflaton mass [MATH] (right) as functions of [MATH] and [MATH], again for [MATH].', '1510.04030-3-62-2': 'The corresponding values of [MATH] and [MATH] are given in terms of the scale of colour shown on the right of every plot.', '1510.04030-3-63-0': 'Because we do not have stringent experimental bounds on [MATH] and [MATH], we are interested just in the order of magnitude of these quantities.', '1510.04030-3-63-1': 'We observe that the order of magnitude of [MATH] depends very weakly on [MATH] and [MATH].', '1510.04030-3-63-2': 'Also the dependence on [MATH] is quite weak, as long as we take values of [MATH] not too close to [MATH].', '1510.04030-3-63-3': 'Thus, for [MATH] not too close to [MATH], the vacuum energy remains about [MATH] to [MATH] Planck masses.', '1510.04030-3-64-0': 'On the other hand, the inflaton mass is more sensitive to [MATH].', '1510.04030-3-64-1': 'Its order of magnitude is between [MATH] and [MATH] Planck masses but for values of [MATH] near to [MATH] we have smaller values of [MATH].', '1510.04030-3-65-0': '## Other branch of the potential', '1510.04030-3-66-0': 'Until now we have considered the slow-roll regime for branch I of the potential, i.e. [MATH].', '1510.04030-3-66-1': 'Let us briefly consider branch II, i.e. [MATH].', '1510.04030-3-66-2': 'Investigation of this branch is of particular interest because the most interesting cosmological solutions one can obtain for the exact solvable model with [MATH] are defined in branch II of the potential [CITATION].', '1510.04030-3-67-0': 'In terms of the parametrization [REF], region II corresponds to [MATH].', '1510.04030-3-67-1': 'The slow-roll parameters [MATH] and [MATH] are still given by [REF] but now the condition for inflation [MATH] requires (1-h)Y1, which can be satisfied only if [MATH].', '1510.04030-3-67-2': 'It follows that [MATH].', '1510.04030-3-67-3': 'One can easily see from [REF] that these values of [MATH] are not only incompatible with the slow-roll condition [MATH], but are also completely ruled out by the experimental constraints on [MATH].', '1510.04030-3-68-0': '# Conclusion', '1510.04030-3-69-0': 'In this paper we have constructed the most general Einstein-scalar gravity model in which the potential is given by the sum of two exponentials and inflation is generated by a scalar field [MATH] rolling off from the de Sitter maximum of the potential [MATH].', '1510.04030-3-69-1': 'These models are the cosmological counterparts of holographic models used to describe hyperscaling violation in the ultraviolet [CITATION].', '1510.04030-3-69-2': 'We have investigated inflation in the slow-roll approximation.', '1510.04030-3-69-3': 'Our model predicts inflation at energy scales of four to five orders of magnitude below the Planck scale, whereas the inflaton mass, at the onset of inflation, turns out to be seven to eight orders of magnitude smaller than the Planck mass.', '1510.04030-3-69-4': 'We have shown that our model reproduces correctly, for a wide range of its parameters the most recent experimental data for the power spectrum of primordial perturbations.', '1510.04030-3-70-0': 'The proposed inflationary model belongs to the class of models in which the potential has a dS regime.', '1510.04030-3-70-1': 'This class of models includes the Starobinsky model and, more generally, the cosmological attractor models.', '1510.04030-3-70-2': 'Our model shares with those several features: (1) the potential is built as a combination of exponentials, it predicts (2) an energy scale of inflation four order of magnitude below the Planck mass, (3) a "red" power spectrum and (4) a small tensor/scalar amplitude ratio.', '1510.04030-3-70-3': 'On the other hand, our model differs from the Starobinsky one in a crucial aspect: inflation is not generated, as in Starobinsky model, by a scalar field rolling off from an asymptotically constant potential, but rather from a local maximum of the potential.', '1510.04030-3-70-4': 'This property allows us to interpret the inflaton as a tachyonic excitation of the dS vacuum and to introduce a second scale of energy in the theory, the mass scale [MATH], which is 7-8 order of magnitude below the Planck mass.', '1510.04030-3-70-5': 'This hierarchy of scales opens the intriguing possibility that the origin of the inflaton could be explained by the physics at energy scales 7-8 order of magnitude below the Planck mass.', '1510.04030-3-71-0': 'Our model belongs to the general class of hilltop models and shares with the latter the local behaviour near the maximum of the potential.', '1510.04030-3-71-1': 'However, in our model the potential that is constructed has the sum of two exponentials, therefore the global behaviour of our inflationary model is sensibly different from usual hilltop models constructed using powers of the inflaton.', '1510.04030-3-71-2': 'In particular, this results in different predictions for the spectral parameters [MATH] and [MATH] in the region of the [MATH]-folds [MATH] accessible to observations.', '1510.04030-3-72-0': 'We close with a brief comment about the reheating phase and the transition from inflation to the radiation/matter dominated era.', '1510.04030-3-72-1': 'During reheating the energy is transferred from the inflaton to matter fields.', '1510.04030-3-72-2': 'This means that there must exist a region in which the kinetic energy of the inflaton dominates over its potential energy, e.g. a local minimum of the potential.', '1510.04030-3-72-3': 'It is evident from [REF] that the potential [REF] does not have such a region and hence it cannot be used to describe reheating.', '1510.04030-3-72-4': 'Thus, in order to describe reheating our potential must be matched with continuity at the end of inflation with some other branch of a potential exhibiting a local minimum.', '1510.04030-3-72-5': 'This can be done very easily.', '1510.04030-3-72-6': 'In the [MATH]-parametrization the point [MATH] given by [REF], at which the universe exits inflation, is always on the left of the point [MATH] at which [MATH] cuts the horizontal axis, i.e. we have [MATH] and [MATH].', '1510.04030-3-72-7': 'Since the slow-roll approximation is badly broken at [MATH], the matching with the branch of the potential with the local minimum must be performed at a point [MATH].', '1510.04030-3-73-0': '# [MATH] model Appendix A: Cosh(phi) model', '1510.04030-3-74-0': 'The model [REF] is the most general form of the potential one can obtain imposing conditions [REF] and assuming that [MATH] is built as a combination of two exponentials without an additive constant term.', '1510.04030-3-74-1': 'When such a constant term (which we call [MATH]) is present, only the first equation in [REF] has to be modified and becomes [MATH], whereas the second and third equations remain unchanged.', '1510.04030-3-74-2': 'A general solution of the ensuing system is given by [MATH], [MATH], [MATH], [MATH] and [MATH].', '1510.04030-3-75-0': 'A simple example of this class of potentials is given by V () =^2(2- ).', '1510.04030-3-75-1': 'This potential gives a further example of inflation generated by an unstable de Sitter vacuum.', '1510.04030-3-76-0': 'The potential [REF] has a maximum at [MATH], corresponding to an unstable de Sitter solution with [MATH], and a corresponding tachyonic excitation.', '1510.04030-3-76-1': 'For [MATH], the potential behaves as a pure exponential.', '1510.04030-3-76-2': 'The vacuum energy and inflaton mass, expressed in terms of [MATH] and [MATH], defined as in [REF], are M^2_I = -163^4h^2m_P^2,E_V= m_P.', '1510.04030-3-77-0': 'Introducing the variable [MATH], the slow-roll parameters [MATH] and [MATH] take the form 13 h^2(Y^2-1Y^2-4Y+1)^2, 23 h^2Y^2+1Y^2-4Y+1-.', '1510.04030-3-77-1': 'The slow-roll parameter [MATH] is zero on the maximum of the potential ([MATH]).', '1510.04030-3-77-2': 'Moreover, we have [MATH] for [MATH], where Y_0=2 3h+ 1+9 h^23h+1.', '1510.04030-3-77-3': 'For [MATH] we have inflation, whereas for [MATH] we have [MATH] and the universe exits inflation.', '1510.04030-3-77-4': 'One can easily check that during inflation we always have [MATH].', '1510.04030-3-77-5': 'Conversely, the parameter [MATH], which gives a measure of the curvature of the potential, is not small in general, but is of order [MATH].', '1510.04030-3-78-0': 'Also for these models the simplest way to satisfy the usual slow-roll conditions for inflation, [MATH], is to choose [MATH], so that [MATH] as well as [MATH].', '1510.04030-3-79-0': 'The number of [MATH]-folds [MATH] is given by (1+Y)(Y(Y-1))^1/3= Ae^2N/9h^2,A := 1+Y_0(Y_0(Y_0-1))^1/3.', '1510.04030-3-80-0': 'In the slow-roll approximation the spectral parameters [MATH], [MATH] and [MATH] expressed in terms of [MATH] are, [paramcosh] &P^1/2_R (N) =2 h ^2(4Y -Y^2-1)^3/2(Y^2-1) Y^1/2', '1510.04030-3-81-0': '&r (N) = 16 (N)= 163 h^2(Y^2-14Y -Y^2-1)^2,', '1510.04030-3-82-0': '&n_s (N) = 1 -4(N) +2(N) &n_s (N) = 1- 38r (N)+43 h^2Y^2+1Y^2-4Y+1, where [MATH] is defined implicitly as a function of [MATH] by [REF].', '1510.04030-3-83-0': '[REF] shows that there exist values of [MATH] for which the model correctly reproduces the results of observation [CITATION] with [MATH].', '1510.04030-3-83-1': 'Moreover, it predicts the vacuum energy to be four orders of magnitude below the Planck scale and the mass of the inflaton seven orders of magnitude smaller than the Planck mass.'} | null | null | null | null |
physics-0607100 | {'physics-0607100-1-0-0': 'Detecting community structure is fundamental to clarify the link between structure and function in complex networks and is used for practical applications in many disciplines.', 'physics-0607100-1-0-1': 'A successful method relies on the optimization of a quantity called modularity [Newman and Girvan, Phys.', 'physics-0607100-1-0-2': 'Rev. E 69, 026113 (2004)], which is a quality index of a partition of a network into communities.', 'physics-0607100-1-0-3': 'We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total number [MATH] of links of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined.', 'physics-0607100-1-0-4': 'The probability that a module conceals well-defined substructures is the highest if the number of links internal to the module is of the order of [MATH] or smaller.', 'physics-0607100-1-0-5': 'We discuss the practical consequences of this result by analyzing partitions obtained through modularity optimization in artificial and real networks.', 'physics-0607100-1-1-0': '# Introduction', 'physics-0607100-1-2-0': 'Community detection in complex networks has attracted a lot of attention in the last years (for a review see [CITATION]).', 'physics-0607100-1-2-1': 'The main reason is that complex networks [CITATION] are made of a large number of nodes and that so far most of the quantitative investigations were focusing on statistical properties disregarding the roles played by specific subgraphs.', 'physics-0607100-1-2-2': 'Detecting communities (or modules) can then be a way to identify relevant substructures that may also correspond to important functions.', 'physics-0607100-1-2-3': 'In the case of the World Wide Web, for instance, communities are sets of Web pages dealing with the same topic [CITATION].', 'physics-0607100-1-2-4': 'Relevant community structures were also found in social networks [CITATION], biochemical networks [CITATION], the Internet [CITATION], food webs [CITATION], and in networks of sexual contacts [CITATION].', 'physics-0607100-1-3-0': 'Loosely speaking a community is a subgraph of a network whose nodes are more tightly connected with each other than with nodes outside the subgraph.', 'physics-0607100-1-3-1': 'A decisive advance in community detection was made by Newman and Girvan [CITATION], who introduced a quantitative measure for the quality of a partition of a network into communities, the so-called modularity.', 'physics-0607100-1-3-2': 'This measure essentially compares the number of links inside a given module with the expected value for a randomized graph of the same size and degree sequence.', 'physics-0607100-1-3-3': 'If one takes modularity as the relevant quality function, the problem of community detection becomes equivalent to modularity optimization.', 'physics-0607100-1-3-4': 'The latter is not trivial, as the number of possible partitions of a network in clusters increases exponentially with the size of the network, making exhaustive optimization computationally unreachable even for relatively small graphs.', 'physics-0607100-1-3-5': 'Therefore, a number of algorithms have been devised in order to find a good optimization with the least computational cost.', 'physics-0607100-1-3-6': 'The fastest available procedures uses greedy techniques [CITATION] and extremal optimization [CITATION], and are at present time the only algorithms capable to detect communities on large networks.', 'physics-0607100-1-3-7': 'More accurate results are obtained through simulated annealing [CITATION], although this method is computationally very expensive.', 'physics-0607100-1-4-0': 'Modularity optimization seems thus to be a very effective method to detect communities, both in real and in artificially generated networks.', 'physics-0607100-1-4-1': 'The modularity itself has however not yet been thoroughly investigated and only a few general properties are known.', 'physics-0607100-1-4-2': 'For example, it is known that the modularity value of a partition does not have a meaning by itself, but only if compared with the corresponding modularity expected for a random graph of the same size [CITATION], as the latter may attain very high values, due to fluctuations [CITATION].', 'physics-0607100-1-5-0': 'In this paper we focus on communities defined by modularity.', 'physics-0607100-1-5-1': 'We will show that modularity contains an intrinsic scale which depends on the number of links of the network, and that modules smaller than that scale may not be resolved, even if they were complete graphs connected by single bridges.', 'physics-0607100-1-5-2': 'The resolution limit of modularity actually depends on the degree of interconnectedness between pairs of communities and can reach values of the order of the size of the whole network.', 'physics-0607100-1-5-3': 'It is thus a priori impossible to tell whether a module (large or small), obtained through modularity optimization, is indeed a single module or a cluster of smaller modules.', 'physics-0607100-1-5-4': 'This result thus introduces some caveats in the use of modularity to detect community structure.', 'physics-0607100-1-6-0': 'In Section [REF] we recall the notion of modularity and discuss some of its properties.', 'physics-0607100-1-6-1': 'Section [REF] deals with the problem of finding the most modular network with a given number of nodes and links.', 'physics-0607100-1-6-2': 'In Section [REF] we show how the resolution limit of modularity arises.', 'physics-0607100-1-6-3': 'In Section [REF] we illustrate the problem with some artificially generated networks, and extend the discussion to real networks.', 'physics-0607100-1-6-4': 'Our conclusions are presented in Section [REF].', 'physics-0607100-1-7-0': '# Modularity', 'physics-0607100-1-8-0': 'The modularity of a partition of a network in modules can be written as [CITATION] [EQUATION] where the sum is over the [MATH] modules of the partition, [MATH] is the number of links inside module [MATH], [MATH] is the total number of links in the network, and [MATH] is the total degree of the nodes in module [MATH].', 'physics-0607100-1-8-1': 'The first term of the summands in Eq. ([REF]) is the fraction of links inside module [MATH]; the second term instead represents the expected fraction of links in that module if links were located at random in the network (under the only constraint that the degree sequence coincides with that in the original graph).', 'physics-0607100-1-8-2': 'If for a subgraph [MATH] of a network the first term is much larger than the second, it means that there are many more links inside [MATH] than one would expect by random chance, so [MATH] is indeed a module.', 'physics-0607100-1-8-3': 'The comparison with the null model represented by the randomized network leads to the quantitative definition of community embedded in the ansatz of Eq. ([REF]).', 'physics-0607100-1-8-4': 'We conclude that, in a modularity-based framework, a subgraph [MATH] with [MATH] internal links and total degree [MATH] is a module if [EQUATION]', 'physics-0607100-1-8-5': 'Let us express the number of links [MATH] joining nodes of the module [MATH] to the rest of the network in terms of [MATH], i.e. [MATH] with [MATH].', 'physics-0607100-1-8-6': 'So, [MATH] and the condition ([REF]) becomes [EQUATION] from which, rearranging terms, one obtains [EQUATION]', 'physics-0607100-1-8-7': 'If [MATH], the subgraph [MATH] is a disconnected part of the network and is a module if [MATH] which is always true.', 'physics-0607100-1-8-8': 'If [MATH] is strictly positive, Eq. ([REF]) sets an upper limit to the number of internal links that [MATH] must have in order to be a module.', 'physics-0607100-1-8-9': 'This is a little odd, because it means that the definition of community implied by modularity depends on the size of the whole network, instead of involving a "local" comparison between the number of internal and external links of the module.', 'physics-0607100-1-8-10': 'For [MATH] one has [MATH], which means that the total degree internal to the subgraph is larger than its external degree, i.e. [MATH].', 'physics-0607100-1-8-11': 'The attributes "internal" and "external" here mean that the degree is calculated considering only the internal or the external links, respectively.', 'physics-0607100-1-8-12': 'In this case, the subgraph [MATH] would be a community according to the "weak" definition given by Radicchi et al. [CITATION].', 'physics-0607100-1-9-0': 'From Eq. ([REF]), we then get that such a subgraph would be a community both within the modularity framework and according to the weak definition of Radicchi et al. if [EQUATION]', 'physics-0607100-1-9-1': 'In Section [REF] we shall consider modules of this kind.', 'physics-0607100-1-10-0': 'According to Eq. ([REF]), a partition of a network into actual modules would have a positive modularity, as all summands in Eq. ([REF]) are positive.', 'physics-0607100-1-10-1': 'On the other hand, for particular partitions, one could bump into values of [MATH] which are negative.', 'physics-0607100-1-10-2': 'The network itself, meant as a partition with a single module, has modularity zero: in this case, in fact, [MATH], [MATH], and the only two terms of the unique module in [MATH] cancel each other.', 'physics-0607100-1-10-3': 'Usually, a value of [MATH] larger than [MATH] is a clear indication that the subgraphs of the corresponding partition are modules.', 'physics-0607100-1-10-4': 'However, the maximal modularity differs from a network to another and depends on the number of links of the network.', 'physics-0607100-1-10-5': 'In the next section we shall derive the expression of the maximal possible value [MATH] that [MATH] can attain on a network with [MATH] links.', 'physics-0607100-1-10-6': 'We will prove that the upper limit for the value of modularity for any network is [MATH] and we will see why the modularity is not scale independent.', 'physics-0607100-1-11-0': '# The most modular network', 'physics-0607100-1-12-0': 'In this section we discuss of the most modular network which will introduce naturally the problem of scales in modularity optimization.', 'physics-0607100-1-12-1': "In Ref. [CITATION], the authors consider the interesting example of a network made of [MATH] identical complete graphs (or 'cliques'), disjoint from each other.", 'physics-0607100-1-12-2': 'In this case, the modularity is maximal for the partition of the network in the cliques and is given by the sum of [MATH] equal terms.', 'physics-0607100-1-12-3': 'In each clique there are [MATH] links, and the total degree is [MATH], as there are no links connecting nodes of the clique to the other cliques.', 'physics-0607100-1-12-4': 'We thus obtain [EQUATION] which converges to [MATH] when the number of cliques goes to infinity.', 'physics-0607100-1-12-5': 'We remark that for this result to hold it is not necessary that the [MATH] connected components be cliques.', 'physics-0607100-1-12-6': 'The number of nodes of the network and within the modules does not affect modularity.', 'physics-0607100-1-12-7': 'If we have [MATH] modules, we just need to have [MATH] links inside the modules, as long as this is compatible with topological constraints, like connectedness.', 'physics-0607100-1-12-8': 'In this way, a network composed by [MATH] identical trees (in graph theory, a forest) has the same maximal modularity reported in Eq. ([REF]), although it has a far smaller number of links as compared with the case of the densely connected cliques (for a given number of nodes).', 'physics-0607100-1-13-0': 'A further interesting question is how to design a connected network with [MATH] nodes and [MATH] links which maximizes modularity.', 'physics-0607100-1-13-1': 'To address this issue, we proceed in two steps: first, we consider the maximal value [MATH] for a partition into a fixed number [MATH] of modules; after that, we look for the number [MATH] that maximizes [MATH].', 'physics-0607100-1-14-0': 'Let us first consider a partition into [MATH] modules.', 'physics-0607100-1-14-1': 'Ideally, to maximize the contribution to modularity of each module, we should reduce as much as possible the number of links connecting modules.', 'physics-0607100-1-14-2': 'If we want to keep the network connected, the smallest number of inter-community links must be [MATH].', 'physics-0607100-1-14-3': 'For the sake of clarity, and to simplify the mathematical expressions (without affecting the final result), we assume instead that there are [MATH] links between the modules, so that we can arrange the latter in the simple ring-like configuration illustrated in Fig. [REF].', 'physics-0607100-1-15-0': 'The modularity of such a network is [EQUATION] where [EQUATION]', 'physics-0607100-1-15-1': 'It is easy to see that the expression of Eq. ([REF]) reaches its maximum when all modules contain the same number of links, i.e. [MATH].', 'physics-0607100-1-15-2': 'The maximum is then given by [EQUATION]', 'physics-0607100-1-15-3': 'We have now to find the maximum of [MATH] when the number of modules [MATH] is variable.', 'physics-0607100-1-15-4': 'For this purpose we treat [MATH] as a real variable and take the derivative of [MATH] with respect to [MATH] [EQUATION] which vanishes when [MATH].', 'physics-0607100-1-15-5': 'This point indeed corresponds to the absolute maximum [MATH] of the function [MATH].', 'physics-0607100-1-15-6': 'This result coincides with the one found by the authors of [CITATION] for a one-dimensional lattice, but our proof is completely general and does not require preliminary assumptions on the type of network and modules.', 'physics-0607100-1-16-0': 'Since [MATH] is not a real number, the actual maximum is reached when [MATH] equals one of the two integers closest to [MATH], but that is not important for our purpose, so from now on we shall stick to the real-valued expressions, their meaning being clear.', 'physics-0607100-1-16-1': 'The maximal modularity is then [EQUATION] and approaches [MATH] if the total number of links [MATH] goes to infinity.', 'physics-0607100-1-16-2': 'The corresponding number of links in each module is [MATH].', 'physics-0607100-1-16-3': 'The fact that all modules have the same number of links does not imply that they have as well the same number of nodes.', 'physics-0607100-1-16-4': 'Again, modularity does not depend on the distribution of the nodes among the modules as long as the topological constraints are satisfied.', 'physics-0607100-1-16-5': 'For instance, if we assume that the modules are connected graphs, there must be at most [MATH] nodes in each module.', 'physics-0607100-1-16-6': 'The crucial point here is that modularity seems to have some intrinsic scale of order [MATH], which constrains the number and the size of the modules.', 'physics-0607100-1-16-7': 'For a given total number of nodes and links we could build many more than [MATH] modules, but the corresponding network would be less "modular", namely with a value of the modularity lower than the maximum of Eq. ([REF]).', 'physics-0607100-1-16-8': 'This fact is the basic reason why small modules may not be resolved through modularity optimization, as it will be clear in the next section.', 'physics-0607100-1-17-0': '# The resolution limit', 'physics-0607100-1-18-0': 'We analyze a network with [MATH] links and with at least three modules (Fig. [REF]).', 'physics-0607100-1-18-1': 'We focus on a pair of modules, [MATH] and [MATH], and distinguish three types of links: those internal to each of the two communities ([MATH] and [MATH], respectively), between [MATH] and [MATH]) and between the two communities and the rest of the network [MATH] and [MATH]).', 'physics-0607100-1-18-2': 'In order to simplify the calculations we express the numbers of external links in terms of [MATH] and [MATH], so [MATH], [MATH] and [MATH], with [MATH].', 'physics-0607100-1-18-3': 'Since [MATH] and [MATH] are modules by construction, we also have [MATH], [MATH] and [MATH] (see Section [REF]).', 'physics-0607100-1-19-0': 'Now we consider two partitions [MATH] and [MATH] of the network.', 'physics-0607100-1-19-1': 'In partition [MATH], [MATH] and [MATH] are taken as separate modules, and in partition [MATH] they are considered as a single community.', 'physics-0607100-1-19-2': 'The split of the rest of the network is arbitrary but identical in both partitions.', 'physics-0607100-1-19-3': 'We want to compare the modularity values [MATH] and [MATH] of the two partitions.', 'physics-0607100-1-19-4': 'Since the modularity is a sum over the modules, the contribution of [MATH] is the same in both partitions and is denoted by [MATH].', 'physics-0607100-1-19-5': 'From Eq. ([REF]) we obtain [EQUATION]', 'physics-0607100-1-19-6': 'The difference [MATH] is [EQUATION]', 'physics-0607100-1-19-7': 'As [MATH] and [MATH] are both modules by construction, we would expect that the modularity should be larger for the partition where the two modules are separated, i.e. [MATH], which in turn implies [MATH].', 'physics-0607100-1-19-8': 'From Eq. ([REF]) we see that [MATH] is negative if [EQUATION]', 'physics-0607100-1-19-9': 'We see that if [MATH], i.e. if there are no links between [MATH] and [MATH], the above condition is trivially satisfied.', 'physics-0607100-1-19-10': 'Instead, if the two modules are connected to each other, something interesting happens.', 'physics-0607100-1-19-11': 'Each of the coefficients [MATH], [MATH], [MATH], [MATH] cannot exceed [MATH] and [MATH] and [MATH] are both smaller than [MATH] by construction but can be taken as small as we wish with respect to [MATH].', 'physics-0607100-1-19-12': 'In this way, it is possible to choose [MATH] and [MATH] such that the inequality of Eq. ([REF]) is not satisfied.', 'physics-0607100-1-19-13': 'In such a situation we can have [MATH] and the modularity of the configuration where the two modules are considered as a single community is larger than the partition where the two modules are clearly identified.', 'physics-0607100-1-19-14': 'This implies that by looking for the maximal modularity, there is the risk to miss important structures at smaller scales.', 'physics-0607100-1-19-15': 'To give an idea of the size of [MATH] and [MATH] at which modularity optimization could fail, we consider for simplicity the case in which [MATH] and [MATH] have the same number of links, i.e. [MATH].', 'physics-0607100-1-19-16': 'The condition on [MATH] for the modularity to miss the two modules also depends on the fuzziness of the modules, as expressed by the values of the parameters [MATH], [MATH], [MATH], [MATH].', 'physics-0607100-1-19-17': 'In order to find the range of potentially "dangerous" values of [MATH], we consider the two extreme cases in which', 'physics-0607100-1-20-0': 'In the first case, the maximum value that the coefficient of [MATH] can take in Eq. ([REF]) is [MATH], when [MATH] and [MATH], [MATH], so we obtain that Eq. ([REF]) may not be satisfied for [EQUATION] which is a scale of the order of the size of the whole network.', 'physics-0607100-1-20-1': 'In this way, even a pair of large communities may not be resolved if they share enough links with the nodes outside them (in this case we speak of "fuzzy" communities).', 'physics-0607100-1-20-2': 'A more striking result emerges when we consider the other limit, i.e. when [MATH].', 'physics-0607100-1-20-3': 'In this case it is easy to check that Eq. ([REF]) is not satisfied for values of the number of links inside the modules satisfying [EQUATION]', 'physics-0607100-1-20-4': 'If we now assume that we have two (interconnected) modules with the same number of internal links [MATH], the discussion above implies that the modules cannot be resolved through modularity optimization, not even if they were complete graphs connected by a single link.', 'physics-0607100-1-20-5': 'As we have seen from Eq. ([REF]), it is possible to miss modules of larger size, if they share more links with the rest of the network (and with each other).', 'physics-0607100-1-20-6': 'For [MATH] the conclusion is similar but the scales [MATH] are modified by simple factors.', 'physics-0607100-1-21-0': '# Consequences', 'physics-0607100-1-22-0': 'We begin with a very schematic example, for illustrative purposes.', 'physics-0607100-1-22-1': 'In Fig. [REF](A) we show a network consisting of a ring of cliques, connected through single links.', 'physics-0607100-1-22-2': 'Each clique is a complete graph [MATH] with [MATH] nodes and has [MATH] links.', 'physics-0607100-1-22-3': 'If we assume that there are [MATH] cliques, with [MATH] even, the network has a total of [MATH] nodes and [MATH] links.', 'physics-0607100-1-23-0': 'The network has a clear modular structure where the communities correspond to single cliques and we expect that any detection algorithm should be able to detect these communities.', 'physics-0607100-1-23-1': 'The modularity [MATH] of this natural partition can be easily calculated and equals [EQUATION]', 'physics-0607100-1-23-2': 'On the other hand, the modularity [MATH] of the partition in which pairs of consecutive cliques are considered as single communities (as shown by the dotted lines in Fig. [REF](A)) is [EQUATION]', 'physics-0607100-1-23-3': 'The condition [MATH] is satisfied if and only if [EQUATION]', 'physics-0607100-1-23-4': 'In this example, [MATH] and [MATH] are independent variables and we can choose them such that the inequality of formula ([REF]) is not satistied.', 'physics-0607100-1-23-5': 'For instance, for [MATH] and [MATH], [MATH] and [MATH].', 'physics-0607100-1-23-6': 'An efficient algorithm looking for the maximum of the modularity would find the configuration with pairs of cliques and not the actual modules.', 'physics-0607100-1-23-7': 'The difference [MATH] would be even larger if [MATH] increases, for [MATH] fixed.', 'physics-0607100-1-24-0': 'The example we considered was particularly simple and hardly represents situations found in real networks.', 'physics-0607100-1-24-1': 'However, the initial configuration that we considered in the previous section (Fig. [REF]) is absolutely general, and the results make us free to design arbitrarily many networks with obvious community structures in which modularity optimization does not recognize (some of) the real modules.', 'physics-0607100-1-24-2': 'Another example is shown in Fig. [REF](B).', 'physics-0607100-1-24-3': 'The circles represent again cliques, i.e. complete graphs: the two on the left have [MATH] nodes each, the other two [MATH] nodes.', 'physics-0607100-1-24-4': 'If we take [MATH] and [MATH], the maximal modularity of the network corresponds to the partition in which the two smaller cliques are merged [as shown by the dotted line in Fig. [REF](B)].', 'physics-0607100-1-24-5': 'This trend of the optimal modularity to group small modules has already been remarked in [CITATION], but as a result of empirical studies on special networks, without any complete explanation.', 'physics-0607100-1-25-0': 'In general, we cannot make any definite statement about modules found through modularity optimization without a method which verifies whether the modules are indeed single communities or a combination of communities.', 'physics-0607100-1-25-1': 'It is then necessary to inspect the structure of each of the modules found.', 'physics-0607100-1-25-2': 'As an example, we take the network of Fig. [REF](A), with [MATH] identical cliques, where each clique is a [MATH] with [MATH].', 'physics-0607100-1-25-3': 'As already said above, modularity optimization would find modules which are pairs of connected cliques.', 'physics-0607100-1-25-4': "By inspecting each of the modules of the 'first generation' (by optimizing modularity, for example), we would ultimately find that each module is actually a set of two cliques.", 'physics-0607100-1-26-0': 'We thus have seen that modules identified through modularity optimization may actually be combinations of smaller modules.', 'physics-0607100-1-26-1': 'During the process of modularity optimization, it is favorable to merge connected modules if they are sufficiently small.', 'physics-0607100-1-27-0': 'We have seen in the previous Section that any two interconnected modules, fuzzy or not, are merged if the number of links inside each of them does not exceed [MATH].', 'physics-0607100-1-27-1': 'This means that the largest structure one can form by merging a pair of modules of any type (including cliques) has at least [MATH] internal links.', 'physics-0607100-1-27-2': 'By reversing the argument, we conclude that if modularity optimization finds a module [MATH] with [MATH] internal links, it may be that the latter is a combination of two or more smaller communities if [EQUATION]', 'physics-0607100-1-27-3': 'This example is an extreme case, in which the internal partition of [MATH] can be arbitrary, as long as the pieces are modules in the sense discussed in Section [REF].', 'physics-0607100-1-27-4': 'Under the condition ([REF]), the module could in principle be a cluster of loosely interconnected complete graphs.', 'physics-0607100-1-28-0': 'On the other hand, the upper limit of [MATH] can be much larger than [MATH], if the substructures are on average more interconnected with each other, as we have seen in Section [REF].', 'physics-0607100-1-28-1': 'In fact, fuzzy modules can be combined with each other even if they contain many more than [MATH] links.', 'physics-0607100-1-28-2': 'The more interconnected the modules, the larger will be the resulting supermodule.', 'physics-0607100-1-28-3': 'In the extreme case in which all submodules are very fuzzy, the size [MATH] of the supermodule could be in principle as large as that of the whole network, i.e. [MATH].', 'physics-0607100-1-28-4': 'This result comes from the extreme case where the network is split in two very fuzzy communities, with [MATH] internal links each and [MATH] between them.', 'physics-0607100-1-28-5': 'By virtue of Eq. ([REF]), it is favorable (or just as good) to merge the two modules and the resulting structure is the whole network.', 'physics-0607100-1-28-6': 'This limit [MATH] is of course always satisfied but suggests here that it is important to carefully analyze all modules found through modularity optimization, regardless of their size.', 'physics-0607100-1-29-0': 'The probability that a very large module conceals substructures is however small, because that could only happen if all hidden submodules are very fuzzy communities, which is unlikely.', 'physics-0607100-1-29-1': 'Instead, modules with a size [MATH] or smaller can result from an arbitrary merge of smaller structures, which may go from loosely interconnected cliques to very fuzzy communities.', 'physics-0607100-1-29-2': 'Modularity optimization is most likely to fail in these cases.', 'physics-0607100-1-30-0': 'In order to illustrate this theoretical discussion, we analyze five examples of real networks:', 'physics-0607100-1-31-0': 'the transcriptional regulation network of Saccharomyces cerevisiae (yeast); the transcriptional regulation network of Escherichia coli; a network of electronic circuits; a social network; the neural network of Caenorhabditis Elegans.', 'physics-0607100-1-32-0': "We downloaded the lists of edges of the first four networks from Uri Alon's Website [CITATION], whereas the last one was downloaded from the WebSite of the Collective Dynamics Group at Columbia University [CITATION].", 'physics-0607100-1-33-0': 'In the transcriptional regulation networks, nodes represent operons, i.e. groups of genes that are transcribed on to the same mRNA and an edge is set between two nodes A and B if A activates B.', 'physics-0607100-1-33-1': 'These systems have been previously studied to identify motifs in complex networks [CITATION].', 'physics-0607100-1-33-2': 'There are [MATH] nodes, [MATH] links for yeast, [MATH] nodes and [MATH] links for E. coli.', 'physics-0607100-1-33-3': 'Electronic circuits can be viewed as networks in which vertices are electronic components (like capacitors, diodes, etc.) and connections are wires.', 'physics-0607100-1-33-4': "Our network maps one of the benchmark circuits of the so-called ISCAS'89 set; it has [MATH] nodes, [MATH] links.", 'physics-0607100-1-33-5': 'In the social network we considered, nodes are people of a group and links represent positive sentiments directed from one person to another, based on questionnaires: it has [MATH] nodes and [MATH] links.', 'physics-0607100-1-33-6': 'Finally, the neural network of C. elegans is made of [MATH] nodes (neurons), connected through [MATH] links (synapsis, gap junctions).', 'physics-0607100-1-33-7': 'We remark that most of these networks are directed, here we considered them as undirected.', 'physics-0607100-1-34-0': 'First, we look for the modularity maximum by using simulated annealing.', 'physics-0607100-1-34-1': 'We adopt the same recipe introduced in Ref. [CITATION], which makes the optimization procedure very effective.', 'physics-0607100-1-34-2': 'There are two types of moves to pass from a network partition to the next: individual moves, where a single node is passed from a community to another, and collective moves, where a pair of communities is merged into a single one or, vice versa, a community is split into two parts.', 'physics-0607100-1-34-3': 'Each iteration at the same temperature consists of a succession of [MATH] individual and [MATH] collective moves, where [MATH] is the total number of nodes of the network.', 'physics-0607100-1-34-4': 'The initial temperature [MATH] and the temperature reduction factor [MATH] are arbitrarily tuned to find the highest possible modularity: in most cases we took [MATH] and [MATH] between [MATH] and [MATH].', 'physics-0607100-1-35-0': 'We found that all networks are characterized by high modularity peaks, with [MATH] ranging from [MATH] (C. elegans) to [MATH] (E. coli).', 'physics-0607100-1-35-1': 'The corresponding optimal partitions consist of [MATH] (yeast), [MATH] (E. coli), [MATH] (electronic), [MATH] (social) and [MATH] (C. elegans) modules (for E. coli our results differ but are not inconsistent with those obtained in [CITATION] for a slighly different database; these differences however do not affect our conclusions).', 'physics-0607100-1-35-2': 'In order to check if the communities have a substructure, we used again modularity optimization, by constraining it to each of the modules found.', 'physics-0607100-1-35-3': 'In all cases, we found that most modules displayed themselves a clear community structure, with very high values of [MATH].', 'physics-0607100-1-35-4': 'The total number of submodules is [MATH] (yeast), [MATH] (E. coli), [MATH] (electronic), [MATH] (social) and [MATH] (C. elegans), and is far larger than the corresponding number at the modularity peaks.', 'physics-0607100-1-35-5': 'The analysis of course is necessarily biased by the fact that we neglect all links between the original communities, and it may be that the submodules we found are not real modules for the original network.', 'physics-0607100-1-35-6': 'In order to verify that, we need to check whether the condition of Eq. ([REF]) is satisfied or not for each submodule and we found that it is the case.', 'physics-0607100-1-35-7': 'A further inspection of the communities found through modularity optimization thus reveals that they are, in fact, clusters of smaller modules.', 'physics-0607100-1-35-8': 'The modularity values corresponding to the partitions of the networks in the submodules are clearly smaller than the peak modularities that we originally found through simulated annealing (see Table [REF]).', 'physics-0607100-1-35-9': 'By restricting modularity optimization to a module we have no guarantee that we accurately detect its substructure and that this is a safe way to proceed.', 'physics-0607100-1-35-10': 'Nevertheless, we have verified that all substructures we detected are indeed modules, so our results show that the search for the modularity optimum is not equivalent to the detection of communities defined through Eq. ([REF]).', 'physics-0607100-1-36-0': 'The networks we have examined are fairly small but the problem we pointed out can only get worse if we increase the network size, especially when small communities coexist with large ones and the module size distribution is broad, which happens in many cases [CITATION].', 'physics-0607100-1-36-1': 'As an example, we take the recommendation network of the online seller Amazon.com.', 'physics-0607100-1-36-2': 'While buying a product, Amazon recommends items which have been purchased by people who bought the same product.', 'physics-0607100-1-36-3': 'In this way it is possible to build a network in which the nodes are the items (books, music), and there is an edge between two items [MATH] and [MATH] if [MATH] was frequently purchased by buyers of [MATH].', 'physics-0607100-1-36-4': 'Such a network was examined in Ref. [CITATION] and is very large, with [MATH] nodes and [MATH] edges.', 'physics-0607100-1-36-5': 'The authors analyzed the community structure by greedy modularity optimization which is not necessarily accurate but represents the only strategy currently available for large networks.', 'physics-0607100-1-36-6': 'They identified [MATH] communities whose size distribution is well approximated by a power law with exponent [MATH].', 'physics-0607100-1-36-7': 'From the size distribution, we estimated that over [MATH] of the modules have sizes below the limit of Eq. ([REF]), which implies that basically all modules need to be further investigated.', 'physics-0607100-1-37-0': '# Conclusions', 'physics-0607100-1-38-0': 'In this paper we analyzed in detail modularity and its applicability to community detection.', 'physics-0607100-1-38-1': 'We found that the definition of community implied by modularity is actually not consistent with its optimization, as the latter may favour network partitions in which groups of modules are combined into larger communities.', 'physics-0607100-1-38-2': 'This is due to the fact that, by enforcing modularity optimization, the possible partitions of the system are explored at a coarse level, so that modules smaller than some scale may not be resolved.', 'physics-0607100-1-38-3': 'The resolution limit of modularity does not rely on particular network structures, but only on the comparison between the sizes of interconnected communities and that of the whole network, where the sizes are measured by the number of links.', 'physics-0607100-1-39-0': 'The origin of the resolution scale lies in the fact that modularity is a sum of terms, where each term corresponds to a module.', 'physics-0607100-1-39-1': 'Finding the maximal modularity is then equivalent to look for the ideal tradeoff between the number of terms in the sum, i.e. the number of modules, which is an unknown, and the value of each term.', 'physics-0607100-1-39-2': 'An increase of the number of modules does not necessarily correspond to an increase in modularity because the modules would be smaller and so would be each term of the sum.', 'physics-0607100-1-39-3': 'This is why for some characteristic number of terms the modularity has a peak (see Section [REF]).', 'physics-0607100-1-39-4': 'The problem is that this "optimal" partition, imposed by mathematics, is not necessarily correlated with the actual community structure of the network, where communities may be very heterogeneous in size, especially if the network is large.', 'physics-0607100-1-40-0': 'Our result implies that modularity optimization might miss some important substructures of a network.', 'physics-0607100-1-40-1': 'From our discussion we deduce that it is not possible to exclude that modules of virtually any size may be clusters of modules, although the problem is most likely to occur for modules with a number of internal links of the order of [MATH] or smaller.', 'physics-0607100-1-40-2': 'For this reason, it is crucial to check the structure of all detected modules, for instance by constraining modularity optimization on each single module.', 'physics-0607100-1-41-0': 'The fact that quality functions such as the modularity have an intrinsic resolution limit calls for a new theoretical framework which focuses on a local definition of community, regardless of its size.', 'physics-0607100-1-41-1': 'Quality functions are still helpful, but their role should be probably limited to the comparison of partitions with the same number of modules.'} | {'physics-0607100-2-0-0': 'Detecting community structure is fundamental to clarify the link between structure and function in complex networks and is used for practical applications in many disciplines.', 'physics-0607100-2-0-1': 'A successful method relies on the optimization of a quantity called modularity [Newman and Girvan, Phys.', 'physics-0607100-2-0-2': 'Rev. E 69, 026113 (2004)], which is a quality index of a partition of a network into communities.', 'physics-0607100-2-0-3': 'We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total number [MATH] of links of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined.', 'physics-0607100-2-0-4': 'The probability that a module conceals well-defined substructures is the highest if the number of links internal to the module is of the order of [MATH] or smaller.', 'physics-0607100-2-0-5': 'We discuss the practical consequences of this result by analyzing partitions obtained through modularity optimization in artificial and real networks.', 'physics-0607100-2-1-0': '# Introduction', 'physics-0607100-2-2-0': 'Community detection in complex networks has attracted a lot of attention in the last years (for a review see [CITATION]).', 'physics-0607100-2-2-1': 'The main reason is that complex networks [CITATION] are made of a large number of nodes and that so far most of the quantitative investigations were focusing on statistical properties disregarding the roles played by specific subgraphs.', 'physics-0607100-2-2-2': 'Detecting communities (or modules) can then be a way to identify relevant substructures that may also correspond to important functions.', 'physics-0607100-2-2-3': 'In the case of the World Wide Web, for instance, communities are sets of Web pages dealing with the same topic [CITATION].', 'physics-0607100-2-2-4': 'Relevant community structures were also found in social networks [CITATION], biochemical networks [CITATION], the Internet [CITATION], food webs [CITATION], and in networks of sexual contacts [CITATION].', 'physics-0607100-2-3-0': 'Loosely speaking a community is a subgraph of a network whose nodes are more tightly connected with each other than with nodes outside the subgraph.', 'physics-0607100-2-3-1': 'A decisive advance in community detection was made by Newman and Girvan [CITATION], who introduced a quantitative measure for the quality of a partition of a network into communities, the so-called modularity.', 'physics-0607100-2-3-2': 'This measure essentially compares the number of links inside a given module with the expected value for a randomized graph of the same size and degree sequence.', 'physics-0607100-2-3-3': 'If one takes modularity as the relevant quality function, the problem of community detection becomes equivalent to modularity optimization.', 'physics-0607100-2-3-4': 'The latter is not trivial, as the number of possible partitions of a network in clusters increases exponentially with the size of the network, making exhaustive optimization computationally unreachable even for relatively small graphs.', 'physics-0607100-2-3-5': 'Therefore, a number of algorithms have been devised in order to find a good optimization with the least computational cost.', 'physics-0607100-2-3-6': 'The fastest available procedures uses greedy techniques [CITATION] and extremal optimization [CITATION], and are at present time the only algorithms capable to detect communities on large networks.', 'physics-0607100-2-3-7': 'More accurate results are obtained through simulated annealing [CITATION], although this method is computationally very expensive.', 'physics-0607100-2-4-0': 'Modularity optimization seems thus to be a very effective method to detect communities, both in real and in artificially generated networks.', 'physics-0607100-2-4-1': 'The modularity itself has however not yet been thoroughly investigated and only a few general properties are known.', 'physics-0607100-2-4-2': 'For example, it is known that the modularity value of a partition does not have a meaning by itself, but only if compared with the corresponding modularity expected for a random graph of the same size [CITATION], as the latter may attain very high values, due to fluctuations [CITATION].', 'physics-0607100-2-5-0': 'In this paper we focus on communities defined by modularity.', 'physics-0607100-2-5-1': 'We will show that modularity contains an intrinsic scale which depends on the number of links of the network, and that modules smaller than that scale may not be resolved, even if they were complete graphs connected by single bridges.', 'physics-0607100-2-5-2': 'The resolution limit of modularity actually depends on the degree of interconnectedness between pairs of communities and can reach values of the order of the size of the whole network.', 'physics-0607100-2-5-3': 'It is thus a priori impossible to tell whether a module (large or small), obtained through modularity optimization, is indeed a single module or a cluster of smaller modules.', 'physics-0607100-2-5-4': 'This result thus introduces some caveats in the use of modularity to detect community structure.', 'physics-0607100-2-6-0': 'In Section [REF] we recall the notion of modularity and discuss some of its properties.', 'physics-0607100-2-6-1': 'Section [REF] deals with the problem of finding the most modular network with a given number of nodes and links.', 'physics-0607100-2-6-2': 'In Section [REF] we show how the resolution limit of modularity arises.', 'physics-0607100-2-6-3': 'In Section [REF] we illustrate the problem with some artificially generated networks, and extend the discussion to real networks.', 'physics-0607100-2-6-4': 'Our conclusions are presented in Section [REF].', 'physics-0607100-2-7-0': '# Modularity', 'physics-0607100-2-8-0': 'The modularity of a partition of a network in modules can be written as [CITATION] [EQUATION] where the sum is over the [MATH] modules of the partition, [MATH] is the number of links inside module [MATH], [MATH] is the total number of links in the network, and [MATH] is the total degree of the nodes in module [MATH].', 'physics-0607100-2-8-1': 'The first term of the summands in Eq. ([REF]) is the fraction of links inside module [MATH]; the second term instead represents the expected fraction of links in that module if links were located at random in the network (under the only constraint that the degree sequence coincides with that in the original graph).', 'physics-0607100-2-8-2': 'If for a subgraph [MATH] of a network the first term is much larger than the second, it means that there are many more links inside [MATH] than one would expect by random chance, so [MATH] is indeed a module.', 'physics-0607100-2-8-3': 'The comparison with the null model represented by the randomized network leads to the quantitative definition of community embedded in the ansatz of Eq. ([REF]).', 'physics-0607100-2-8-4': 'We conclude that, in a modularity-based framework, a subgraph [MATH] with [MATH] internal links and total degree [MATH] is a module if [EQUATION]', 'physics-0607100-2-8-5': 'Let us express the number of links [MATH] joining nodes of the module [MATH] to the rest of the network in terms of [MATH], i.e. [MATH] with [MATH].', 'physics-0607100-2-8-6': 'So, [MATH] and the condition ([REF]) becomes [EQUATION] from which, rearranging terms, one obtains [EQUATION]', 'physics-0607100-2-8-7': 'If [MATH], the subgraph [MATH] is a disconnected part of the network and is a module if [MATH] which is always true.', 'physics-0607100-2-8-8': 'If [MATH] is strictly positive, Eq. ([REF]) sets an upper limit to the number of internal links that [MATH] must have in order to be a module.', 'physics-0607100-2-8-9': 'This is a little odd, because it means that the definition of community implied by modularity depends on the size of the whole network, instead of involving a "local" comparison between the number of internal and external links of the module.', 'physics-0607100-2-8-10': 'For [MATH] one has [MATH], which means that the total degree internal to the subgraph is larger than its external degree, i.e. [MATH].', 'physics-0607100-2-8-11': 'The attributes "internal" and "external" here mean that the degree is calculated considering only the internal or the external links, respectively.', 'physics-0607100-2-8-12': 'In this case, the subgraph [MATH] would be a community according to the "weak" definition given by Radicchi et al. [CITATION].', 'physics-0607100-2-9-0': 'For [MATH] the right-hand-side of inequality ([REF]) is in the interval [MATH].', 'physics-0607100-2-9-1': 'A subgraph of size [MATH] would then be a community both within the modularity framework and according to the weak definition of Radicchi et al. if [MATH] and [MATH] is less than a quantity in the interval [MATH].', 'physics-0607100-2-9-2': 'Sufficient conditions for which these constraints are always met are then [EQUATION]', 'physics-0607100-2-9-3': 'In Section [REF] we shall only consider modules of this kind.', 'physics-0607100-2-10-0': 'According to Eq. ([REF]), a partition of a network into actual modules would have a positive modularity, as all summands in Eq. ([REF]) are positive.', 'physics-0607100-2-10-1': 'On the other hand, for particular partitions, one could bump into values of [MATH] which are negative.', 'physics-0607100-2-10-2': 'The network itself, meant as a partition with a single module, has modularity zero: in this case, in fact, [MATH], [MATH], and the only two terms of the unique module in [MATH] cancel each other.', 'physics-0607100-2-10-3': 'Usually, a value of [MATH] larger than [MATH] is a clear indication that the subgraphs of the corresponding partition are modules.', 'physics-0607100-2-10-4': 'However, the maximal modularity differs from a network to another and depends on the number of links of the network.', 'physics-0607100-2-10-5': 'In the next section we shall derive the expression of the maximal possible value [MATH] that [MATH] can attain on a network with [MATH] links.', 'physics-0607100-2-10-6': 'We will prove that the upper limit for the value of modularity for any network is [MATH] and we will see why the modularity is not scale independent.', 'physics-0607100-2-11-0': '# The most modular network', 'physics-0607100-2-12-0': 'In this section we discuss of the most modular network which will introduce naturally the problem of scales in modularity optimization.', 'physics-0607100-2-12-1': "In Ref. [CITATION], the authors consider the interesting example of a network made of [MATH] identical complete graphs (or 'cliques'), disjoint from each other.", 'physics-0607100-2-12-2': 'In this case, the modularity is maximal for the partition of the network in the cliques and is given by the sum of [MATH] equal terms.', 'physics-0607100-2-12-3': 'In each clique there are [MATH] links, and the total degree is [MATH], as there are no links connecting nodes of the clique to the other cliques.', 'physics-0607100-2-12-4': 'We thus obtain [EQUATION] which converges to [MATH] when the number of cliques goes to infinity.', 'physics-0607100-2-12-5': 'We remark that for this result to hold it is not necessary that the [MATH] connected components be cliques.', 'physics-0607100-2-12-6': 'The number of nodes of the network and within the modules does not affect modularity.', 'physics-0607100-2-12-7': 'If we have [MATH] modules, we just need to have [MATH] links inside the modules, as long as this is compatible with topological constraints, like connectedness.', 'physics-0607100-2-12-8': 'In this way, a network composed by [MATH] identical trees (in graph theory, a forest) has the same maximal modularity reported in Eq. ([REF]), although it has a far smaller number of links as compared with the case of the densely connected cliques (for a given number of nodes).', 'physics-0607100-2-13-0': 'A further interesting question is how to design a connected network with [MATH] nodes and [MATH] links which maximizes modularity.', 'physics-0607100-2-13-1': 'To address this issue, we proceed in two steps: first, we consider the maximal value [MATH] for a partition into a fixed number [MATH] of modules; after that, we look for the number [MATH] that maximizes [MATH].', 'physics-0607100-2-14-0': 'Let us first consider a partition into [MATH] modules.', 'physics-0607100-2-14-1': 'Ideally, to maximize the contribution to modularity of each module, we should reduce as much as possible the number of links connecting modules.', 'physics-0607100-2-14-2': 'If we want to keep the network connected, the smallest number of inter-community links must be [MATH].', 'physics-0607100-2-14-3': 'For the sake of clarity, and to simplify the mathematical expressions (without affecting the final result), we assume instead that there are [MATH] links between the modules, so that we can arrange the latter in the simple ring-like configuration illustrated in Fig. [REF].', 'physics-0607100-2-15-0': 'The modularity of such a network is [EQUATION] where [EQUATION]', 'physics-0607100-2-15-1': 'It is easy to see that the expression of Eq. ([REF]) reaches its maximum when all modules contain the same number of links, i.e. [MATH].', 'physics-0607100-2-15-2': 'The maximum is then given by [EQUATION]', 'physics-0607100-2-15-3': 'We have now to find the maximum of [MATH] when the number of modules [MATH] is variable.', 'physics-0607100-2-15-4': 'For this purpose we treat [MATH] as a real variable and take the derivative of [MATH] with respect to [MATH] [EQUATION] which vanishes when [MATH].', 'physics-0607100-2-15-5': 'This point indeed corresponds to the absolute maximum [MATH] of the function [MATH].', 'physics-0607100-2-15-6': 'This result coincides with the one found by the authors of [CITATION] for a one-dimensional lattice, but our proof is completely general and does not require preliminary assumptions on the type of network and modules.', 'physics-0607100-2-16-0': 'Since [MATH] is not a real number, the actual maximum is reached when [MATH] equals one of the two integers closest to [MATH], but that is not important for our purpose, so from now on we shall stick to the real-valued expressions, their meaning being clear.', 'physics-0607100-2-16-1': 'The maximal modularity is then [EQUATION] and approaches [MATH] if the total number of links [MATH] goes to infinity.', 'physics-0607100-2-16-2': 'The corresponding number of links in each module is [MATH].', 'physics-0607100-2-16-3': 'The fact that all modules have the same number of links does not imply that they have as well the same number of nodes.', 'physics-0607100-2-16-4': 'Again, modularity does not depend on the distribution of the nodes among the modules as long as the topological constraints are satisfied.', 'physics-0607100-2-16-5': 'For instance, if we assume that the modules are connected graphs, there must be at most [MATH] nodes in each module.', 'physics-0607100-2-16-6': 'The crucial point here is that modularity seems to have some intrinsic scale of order [MATH], which constrains the number and the size of the modules.', 'physics-0607100-2-16-7': 'For a given total number of nodes and links we could build many more than [MATH] modules, but the corresponding network would be less "modular", namely with a value of the modularity lower than the maximum of Eq. ([REF]).', 'physics-0607100-2-16-8': 'This fact is the basic reason why small modules may not be resolved through modularity optimization, as it will be clear in the next section.', 'physics-0607100-2-17-0': '# The resolution limit', 'physics-0607100-2-18-0': 'We analyze a network with [MATH] links and with at least three modules, in the sense of the definition of formula ([REF]) (Fig. [REF]).', 'physics-0607100-2-18-1': 'We focus on a pair of modules, [MATH] and [MATH], and distinguish three types of links: those internal to each of the two communities ([MATH] and [MATH], respectively), between [MATH] and [MATH]) and between the two communities and the rest of the network [MATH] and [MATH]).', 'physics-0607100-2-18-2': 'In order to simplify the calculations we express the numbers of external links in terms of [MATH] and [MATH], so [MATH], [MATH] and [MATH], with [MATH].', 'physics-0607100-2-18-3': 'Since [MATH] and [MATH] are modules by construction, we also have [MATH], [MATH] and [MATH] (see Section [REF]).', 'physics-0607100-2-19-0': 'Now we consider two partitions [MATH] and [MATH] of the network.', 'physics-0607100-2-19-1': 'In partition [MATH], [MATH] and [MATH] are taken as separate modules, and in partition [MATH] they are considered as a single community.', 'physics-0607100-2-19-2': 'The split of the rest of the network is arbitrary but identical in both partitions.', 'physics-0607100-2-19-3': 'We want to compare the modularity values [MATH] and [MATH] of the two partitions.', 'physics-0607100-2-19-4': 'Since the modularity is a sum over the modules, the contribution of [MATH] is the same in both partitions and is denoted by [MATH].', 'physics-0607100-2-19-5': 'From Eq. ([REF]) we obtain [EQUATION]', 'physics-0607100-2-19-6': 'The difference [MATH] is [EQUATION]', 'physics-0607100-2-19-7': 'As [MATH] and [MATH] are both modules by construction, we would expect that the modularity should be larger for the partition where the two modules are separated, i.e. [MATH], which in turn implies [MATH].', 'physics-0607100-2-19-8': 'From Eq. ([REF]) we see that [MATH] is negative if [EQUATION]', 'physics-0607100-2-19-9': 'We see that if [MATH], i.e. if there are no links between [MATH] and [MATH], the above condition is trivially satisfied.', 'physics-0607100-2-19-10': 'Instead, if the two modules are connected to each other, something interesting happens.', 'physics-0607100-2-19-11': 'Each of the coefficients [MATH], [MATH], [MATH], [MATH] cannot exceed [MATH] and [MATH] and [MATH] are both smaller than [MATH] by construction but can be taken as small as we wish with respect to [MATH].', 'physics-0607100-2-19-12': 'In this way, it is possible to choose [MATH] and [MATH] such that the inequality of Eq. ([REF]) is not satisfied.', 'physics-0607100-2-19-13': 'In such a situation we can have [MATH] and the modularity of the configuration where the two modules are considered as a single community is larger than the partition where the two modules are clearly identified.', 'physics-0607100-2-19-14': 'This implies that by looking for the maximal modularity, there is the risk to miss important structures at smaller scales.', 'physics-0607100-2-19-15': 'To give an idea of the size of [MATH] and [MATH] at which modularity optimization could fail, we consider for simplicity the case in which [MATH] and [MATH] have the same number of links, i.e. [MATH].', 'physics-0607100-2-19-16': 'The condition on [MATH] for the modularity to miss the two modules also depends on the fuzziness of the modules, as expressed by the values of the parameters [MATH], [MATH], [MATH], [MATH].', 'physics-0607100-2-19-17': 'In order to find the range of potentially "dangerous" values of [MATH], we consider the two extreme cases in which', 'physics-0607100-2-20-0': 'In the first case, the maximum value that the coefficient of [MATH] can take in Eq. ([REF]) is [MATH], when [MATH] and [MATH], [MATH], so we obtain that Eq. ([REF]) may not be satisfied for [EQUATION] which is a scale of the order of the size of the whole network.', 'physics-0607100-2-20-1': 'In this way, even a pair of large communities may not be resolved if they share enough links with the nodes outside them (in this case we speak of "fuzzy" communities).', 'physics-0607100-2-20-2': 'A more striking result emerges when we consider the other limit, i.e. when [MATH].', 'physics-0607100-2-20-3': 'In this case it is easy to check that Eq. ([REF]) is not satisfied for values of the number of links inside the modules satisfying [EQUATION]', 'physics-0607100-2-20-4': 'If we now assume that we have two (interconnected) modules with the same number of internal links [MATH], the discussion above implies that the modules cannot be resolved through modularity optimization, not even if they were complete graphs connected by a single link.', 'physics-0607100-2-20-5': 'As we have seen from Eq. ([REF]), it is possible to miss modules of larger size, if they share more links with the rest of the network (and with each other).', 'physics-0607100-2-20-6': 'For [MATH] the conclusion is similar but the scales [MATH] are modified by simple factors.', 'physics-0607100-2-21-0': '# Consequences', 'physics-0607100-2-22-0': 'We begin with a very schematic example, for illustrative purposes.', 'physics-0607100-2-22-1': 'In Fig. [REF](A) we show a network consisting of a ring of cliques, connected through single links.', 'physics-0607100-2-22-2': 'Each clique is a complete graph [MATH] with [MATH] nodes and has [MATH] links.', 'physics-0607100-2-22-3': 'If we assume that there are [MATH] cliques, with [MATH] even, the network has a total of [MATH] nodes and [MATH] links.', 'physics-0607100-2-23-0': 'The network has a clear modular structure where the communities correspond to single cliques and we expect that any detection algorithm should be able to detect these communities.', 'physics-0607100-2-23-1': 'The modularity [MATH] of this natural partition can be easily calculated and equals [EQUATION]', 'physics-0607100-2-23-2': 'On the other hand, the modularity [MATH] of the partition in which pairs of consecutive cliques are considered as single communities (as shown by the dotted lines in Fig. [REF](A)) is [EQUATION]', 'physics-0607100-2-23-3': 'The condition [MATH] is satisfied if and only if [EQUATION]', 'physics-0607100-2-23-4': 'In this example, [MATH] and [MATH] are independent variables and we can choose them such that the inequality of formula ([REF]) is not satistied.', 'physics-0607100-2-23-5': 'For instance, for [MATH] and [MATH], [MATH] and [MATH].', 'physics-0607100-2-23-6': 'An efficient algorithm looking for the maximum of the modularity would find the configuration with pairs of cliques and not the actual modules.', 'physics-0607100-2-23-7': 'The difference [MATH] would be even larger if [MATH] increases, for [MATH] fixed.', 'physics-0607100-2-24-0': 'The example we considered was particularly simple and hardly represents situations found in real networks.', 'physics-0607100-2-24-1': 'However, the initial configuration that we considered in the previous section (Fig. [REF]) is absolutely general, and the results make us free to design arbitrarily many networks with obvious community structures in which modularity optimization does not recognize (some of) the real modules.', 'physics-0607100-2-24-2': 'Another example is shown in Fig. [REF](B).', 'physics-0607100-2-24-3': 'The circles represent again cliques, i.e. complete graphs: the two on the left have [MATH] nodes each, the other two [MATH] nodes.', 'physics-0607100-2-24-4': 'If we take [MATH] and [MATH], the maximal modularity of the network corresponds to the partition in which the two smaller cliques are merged [as shown by the dotted line in Fig. [REF](B)].', 'physics-0607100-2-24-5': 'This trend of the optimal modularity to group small modules has already been remarked in [CITATION], but as a result of empirical studies on special networks, without any complete explanation.', 'physics-0607100-2-25-0': 'In general, we cannot make any definite statement about modules found through modularity optimization without a method which verifies whether the modules are indeed single communities or a combination of communities.', 'physics-0607100-2-25-1': 'It is then necessary to inspect the structure of each of the modules found.', 'physics-0607100-2-25-2': 'As an example, we take the network of Fig. [REF](A), with [MATH] identical cliques, where each clique is a [MATH] with [MATH].', 'physics-0607100-2-25-3': 'As already said above, modularity optimization would find modules which are pairs of connected cliques.', 'physics-0607100-2-25-4': "By inspecting each of the modules of the 'first generation' (by optimizing modularity, for example), we would ultimately find that each module is actually a set of two cliques.", 'physics-0607100-2-26-0': 'We thus have seen that modules identified through modularity optimization may actually be combinations of smaller modules.', 'physics-0607100-2-26-1': 'During the process of modularity optimization, it is favorable to merge connected modules if they are sufficiently small.', 'physics-0607100-2-27-0': 'We have seen in the previous Section that any two interconnected modules, fuzzy or not, are merged if the number of links inside each of them does not exceed [MATH].', 'physics-0607100-2-27-1': 'This means that the largest structure one can form by merging a pair of modules of any type (including cliques) has at least [MATH] internal links.', 'physics-0607100-2-27-2': 'By reversing the argument, we conclude that if modularity optimization finds a module [MATH] with [MATH] internal links, it may be that the latter is a combination of two or more smaller communities if [EQUATION]', 'physics-0607100-2-27-3': 'This example is an extreme case, in which the internal partition of [MATH] can be arbitrary, as long as the pieces are modules in the sense discussed in Section [REF].', 'physics-0607100-2-27-4': 'Under the condition ([REF]), the module could in principle be a cluster of loosely interconnected complete graphs.', 'physics-0607100-2-28-0': 'On the other hand, the upper limit of [MATH] can be much larger than [MATH], if the substructures are on average more interconnected with each other, as we have seen in Section [REF].', 'physics-0607100-2-28-1': 'In fact, fuzzy modules can be combined with each other even if they contain many more than [MATH] links.', 'physics-0607100-2-28-2': 'The more interconnected the modules, the larger will be the resulting supermodule.', 'physics-0607100-2-28-3': 'In the extreme case in which all submodules are very fuzzy, the size [MATH] of the supermodule could be in principle as large as that of the whole network, i.e. [MATH].', 'physics-0607100-2-28-4': 'This result comes from the extreme case where the network is split in two very fuzzy communities, with [MATH] internal links each and [MATH] between them.', 'physics-0607100-2-28-5': 'By virtue of Eq. ([REF]), it is favorable (or just as good) to merge the two modules and the resulting structure is the whole network.', 'physics-0607100-2-28-6': 'This limit [MATH] is of course always satisfied but suggests here that it is important to carefully analyze all modules found through modularity optimization, regardless of their size.', 'physics-0607100-2-29-0': 'The probability that a very large module conceals substructures is however small, because that could only happen if all hidden submodules are very fuzzy communities, which is unlikely.', 'physics-0607100-2-29-1': 'Instead, modules with a size [MATH] or smaller can result from an arbitrary merge of smaller structures, which may go from loosely interconnected cliques to very fuzzy communities.', 'physics-0607100-2-29-2': 'Modularity optimization is most likely to fail in these cases.', 'physics-0607100-2-30-0': 'In order to illustrate this theoretical discussion, we analyze five examples of real networks:', 'physics-0607100-2-31-0': 'the transcriptional regulation network of Saccharomyces cerevisiae (yeast); the transcriptional regulation network of Escherichia coli; a network of electronic circuits; a social network; the neural network of Caenorhabditis Elegans.', 'physics-0607100-2-32-0': "We downloaded the lists of edges of the first four networks from Uri Alon's Website [CITATION], whereas the last one was downloaded from the WebSite of the Collective Dynamics Group at Columbia University [CITATION].", 'physics-0607100-2-33-0': 'In the transcriptional regulation networks, nodes represent operons, i.e. groups of genes that are transcribed on to the same mRNA and an edge is set between two nodes A and B if A activates B.', 'physics-0607100-2-33-1': 'These systems have been previously studied to identify motifs in complex networks [CITATION].', 'physics-0607100-2-33-2': 'There are [MATH] nodes, [MATH] links for yeast, [MATH] nodes and [MATH] links for E. coli.', 'physics-0607100-2-33-3': 'Electronic circuits can be viewed as networks in which vertices are electronic components (like capacitors, diodes, etc.) and connections are wires.', 'physics-0607100-2-33-4': "Our network maps one of the benchmark circuits of the so-called ISCAS'89 set; it has [MATH] nodes, [MATH] links.", 'physics-0607100-2-33-5': 'In the social network we considered, nodes are people of a group and links represent positive sentiments directed from one person to another, based on questionnaires: it has [MATH] nodes and [MATH] links.', 'physics-0607100-2-33-6': 'Finally, the neural network of C. elegans is made of [MATH] nodes (neurons), connected through [MATH] links (synapsis, gap junctions).', 'physics-0607100-2-33-7': 'We remark that most of these networks are directed, here we considered them as undirected.', 'physics-0607100-2-34-0': 'First, we look for the modularity maximum by using simulated annealing.', 'physics-0607100-2-34-1': 'We adopt the same recipe introduced in Ref. [CITATION], which makes the optimization procedure very effective.', 'physics-0607100-2-34-2': 'There are two types of moves to pass from a network partition to the next: individual moves, where a single node is passed from a community to another, and collective moves, where a pair of communities is merged into a single one or, vice versa, a community is split into two parts.', 'physics-0607100-2-34-3': 'Each iteration at the same temperature consists of a succession of [MATH] individual and [MATH] collective moves, where [MATH] is the total number of nodes of the network.', 'physics-0607100-2-34-4': 'The initial temperature [MATH] and the temperature reduction factor [MATH] are arbitrarily tuned to find the highest possible modularity: in most cases we took [MATH] and [MATH] between [MATH] and [MATH].', 'physics-0607100-2-35-0': 'We found that all networks are characterized by high modularity peaks, with [MATH] ranging from [MATH] (C. elegans) to [MATH] (E. coli).', 'physics-0607100-2-35-1': 'The corresponding optimal partitions consist of [MATH] (yeast), [MATH] (E. coli), [MATH] (electronic), [MATH] (social) and [MATH] (C. elegans) modules (for E. coli our results differ but are not inconsistent with those obtained in [CITATION] for a slighly different database; these differences however do not affect our conclusions).', 'physics-0607100-2-35-2': 'In order to check if the communities have a substructure, we used again modularity optimization, by constraining it to each of the modules found.', 'physics-0607100-2-35-3': 'In all cases, we found that most modules displayed themselves a clear community structure, with very high values of [MATH].', 'physics-0607100-2-35-4': 'The total number of submodules is [MATH] (yeast), [MATH] (E. coli), [MATH] (electronic), [MATH] (social) and [MATH] (C. elegans), and is far larger than the corresponding number at the modularity peaks.', 'physics-0607100-2-35-5': 'The analysis of course is necessarily biased by the fact that we neglect all links between the original communities, and it may be that the submodules we found are not real modules for the original network.', 'physics-0607100-2-35-6': 'In order to verify that, we need to check whether the condition of Eq. ([REF]) is satisfied or not for each submodule and we found that it is the case.', 'physics-0607100-2-35-7': 'A further inspection of the communities found through modularity optimization thus reveals that they are, in fact, clusters of smaller modules.', 'physics-0607100-2-35-8': 'The modularity values corresponding to the partitions of the networks in the submodules are clearly smaller than the peak modularities that we originally found through simulated annealing (see Table [REF]).', 'physics-0607100-2-35-9': 'By restricting modularity optimization to a module we have no guarantee that we accurately detect its substructure and that this is a safe way to proceed.', 'physics-0607100-2-35-10': 'Nevertheless, we have verified that all substructures we detected are indeed modules, so our results show that the search for the modularity optimum is not equivalent to the detection of communities defined through Eq. ([REF]).', 'physics-0607100-2-36-0': 'The networks we have examined are fairly small but the problem we pointed out can only get worse if we increase the network size, especially when small communities coexist with large ones and the module size distribution is broad, which happens in many cases [CITATION].', 'physics-0607100-2-36-1': 'As an example, we take the recommendation network of the online seller Amazon.com.', 'physics-0607100-2-36-2': 'While buying a product, Amazon recommends items which have been purchased by people who bought the same product.', 'physics-0607100-2-36-3': 'In this way it is possible to build a network in which the nodes are the items (books, music), and there is an edge between two items [MATH] and [MATH] if [MATH] was frequently purchased by buyers of [MATH].', 'physics-0607100-2-36-4': 'Such a network was examined in Ref. [CITATION] and is very large, with [MATH] nodes and [MATH] edges.', 'physics-0607100-2-36-5': 'The authors analyzed the community structure by greedy modularity optimization which is not necessarily accurate but represents the only strategy currently available for large networks.', 'physics-0607100-2-36-6': 'They identified [MATH] communities whose size distribution is well approximated by a power law with exponent [MATH].', 'physics-0607100-2-36-7': 'From the size distribution, we estimated that over [MATH] of the modules have sizes below the limit of Eq. ([REF]), which implies that basically all modules need to be further investigated.', 'physics-0607100-2-37-0': '# Conclusions', 'physics-0607100-2-38-0': 'In this article we have analyzed in detail modularity and its applicability to community detection.', 'physics-0607100-2-38-1': 'We have found that the definition of community implied by modularity is actually not consistent with its optimization which may favour network partitions with groups of modules combined into larger communities.', 'physics-0607100-2-38-2': 'We could say that, by enforcing modularity optimization, the possible partitions of the system are explored at a coarse level, so that modules smaller than some scale may not be resolved.', 'physics-0607100-2-38-3': 'The resolution limit of modularity does not rely on particular network structures, but only on the comparison between the sizes of interconnected communities and that of the whole network, where the sizes are measured by the number of links.', 'physics-0607100-2-39-0': 'The origin of the resolution scale lies in the fact that modularity is a sum of terms, where each term corresponds to a module.', 'physics-0607100-2-39-1': 'Finding the maximal modularity is then equivalent to look for the ideal tradeoff between the number of terms in the sum, i.e. the number of modules, and the value of each term.', 'physics-0607100-2-39-2': 'An increase of the number of modules does not necessarily correspond to an increase in modularity because the modules would be smaller and so would be each term of the sum.', 'physics-0607100-2-39-3': 'This is why for some characteristic number of terms the modularity has a peak.', 'physics-0607100-2-39-4': 'The problem is that this "optimal" partition, imposed by mathematics, is not necessarily correlated with the actual community structure of the network, where communities may be very heterogeneous in size, especially if the network is large.', 'physics-0607100-2-40-0': 'Our result implies that modularity optimization might miss important substructures of a network, as we have confirmed in real world examples.', 'physics-0607100-2-40-1': 'From our discussion we deduce that it is not possible to exclude that modules of virtually any size may be clusters of modules, although the problem is most likely to occur for modules with a number of internal links of the order of [MATH] or smaller.', 'physics-0607100-2-40-2': 'For this reason, it is crucial to check the structure of all detected modules, for instance by constraining modularity optimization on each single module, a procedure which is not safe but may give useful indications.', 'physics-0607100-2-41-0': 'The fact that quality functions such as the modularity have an intrinsic resolution limit calls for a new theoretical framework which focuses on a local definition of community, regardless of its size.', 'physics-0607100-2-41-1': 'Quality functions are still helpful, but their role should be probably limited to the 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'physics-0607100-2-30-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/physics/0607100 | null | null | null | null | null |
hep-ph-0201037 | {'hep-ph-0201037-1-0-0': 'The general formula for bound-continuous transition form factors are derived.', 'hep-ph-0201037-1-0-1': 'It is shown that these form factors are represented in the form of superposition of finite number of terms with simple analytical structure.', 'hep-ph-0201037-1-1-0': 'In the previous paper [CITATION] it have been shown that the form factors of transitions from [MATH] - states of hydrogenlike atoms [CITATION] to the state of continuous spectra with definite value of relative momenta [MATH] may be expressed in the terms of the classical polynomials in a rather simple way.', 'hep-ph-0201037-1-1-1': 'Below this result is generalized for the case of transition from arbitrary initial bound states.', 'hep-ph-0201037-1-2-0': 'The transition form factors are defined as follows [EQUATION] where [MATH] are the wave functions of initial (final) states.', 'hep-ph-0201037-1-3-0': 'According to [CITATION] (see also [CITATION]) the final state wave function must be choose in the form [EQUATION]', 'hep-ph-0201037-1-3-1': 'For arbitrary initial bound state [EQUATION] where [MATH] are the associated Laguerre polynomials.', 'hep-ph-0201037-1-4-0': 'Making use of the recurrence relations [CITATION] [EQUATION] and the representation of Laguerre polynomials in terms of the generating function [EQUATION] where operator [MATH] is defined as follows [EQUATION] let us rewrite the radial part of initial state wave function in the form [EQUATION] more convenient for further calculations.', 'hep-ph-0201037-1-5-0': 'Then it is not difficult to see that transition form factors ([REF]) may be represent as linear combination of the quantities [EQUATION].', 'hep-ph-0201037-1-6-0': 'In order to calculate ([REF]) it is useful to represent the hypergeometrical function in ([REF]) in the form [EQUATION]', 'hep-ph-0201037-1-6-1': 'Using the following relations [EQUATION] where [EQUATION] after simple calculations we find [EQUATION]', 'hep-ph-0201037-1-6-2': 'It is easily to check that [EQUATION]', 'hep-ph-0201037-1-6-3': 'Further according [CITATION] [EQUATION]', 'hep-ph-0201037-1-6-4': 'Taking into account ([REF]), ([REF]) it is easily seen that ([REF]) is the superposition of the quantities [EQUATION] where [MATH] are defined as follows [EQUATION].', 'hep-ph-0201037-1-7-0': 'Further calculations are the same as in the paper [CITATION].', 'hep-ph-0201037-1-8-0': 'Omitting the simple but cumbersome algebra let us present the final expression for transition form factors [EQUATION]', 'hep-ph-0201037-1-9-0': 'Thus, the form factors for transition form arbitrary bound states of hydrogenlike atoms to the "[MATH]" of continuous spectra are represented in the form of superposition of finite number of terms with simple analytical structure and can be evaluated numerically with arbitrary degree of accuracy.', 'hep-ph-0201037-1-10-0': 'The equations ([REF])-([REF]) are the generalization of the results of the papers [CITATION].'} | {'hep-ph-0201037-2-0-0': 'A general formula for bound-continuous transition form factors is derived.', 'hep-ph-0201037-2-0-1': 'It is shown that these form factors can be represented in the form of superposition of finite number of terms with simple analytic structure.', 'hep-ph-0201037-2-1-0': 'In the previous paper [CITATION] it have been shown that the form factors of transitions from [MATH] - states of hydrogenlike atoms [CITATION] to the state of continuous spectra with definite value of relative momenta [MATH] may be expressed in the terms of the classical polynomials in a rather simple way.', 'hep-ph-0201037-2-1-1': 'Below this result is generalized for the case of transition from arbitrary initial bound states.', 'hep-ph-0201037-2-2-0': 'The transition form factors are defined as follows [EQUATION] where [MATH] are the wave functions of initial (final) states.', 'hep-ph-0201037-2-3-0': 'According to [CITATION] (see also [CITATION]) the final state wave function must be choose in the form [EQUATION]', 'hep-ph-0201037-2-3-1': 'For arbitrary initial bound state [EQUATION] where [MATH] are the associated Laguerre polynomials.', 'hep-ph-0201037-2-4-0': 'Making use of the recurrence relations [CITATION] [EQUATION] and the representation of Laguerre polynomials in terms of the generating function [EQUATION] where operator [MATH] is defined as follows [EQUATION] let us rewrite the radial part of initial state wave function in the form [EQUATION] more convenient for further calculations.', 'hep-ph-0201037-2-5-0': 'Then it is not difficult to see that transition form factors ([REF]) may be represent as linear combination of the quantities [EQUATION].', 'hep-ph-0201037-2-6-0': 'In order to calculate ([REF]) it is useful to represent the hypergeometrical function in ([REF]) in the form [EQUATION]', 'hep-ph-0201037-2-6-1': 'Using the following relations [EQUATION] where [EQUATION] after simple calculations we find [EQUATION]', 'hep-ph-0201037-2-6-2': 'It is easily to check that [EQUATION]', 'hep-ph-0201037-2-6-3': 'Further according [CITATION] [EQUATION]', 'hep-ph-0201037-2-6-4': 'Taking into account ([REF]), ([REF]) it is easily seen that ([REF]) is the superposition of the quantities [EQUATION] where [MATH] are defined as follows [EQUATION].', 'hep-ph-0201037-2-7-0': 'Further calculations are the same as in the paper [CITATION].', 'hep-ph-0201037-2-8-0': 'Omitting the simple but cumbersome algebra let us present the final expression for transition form factors [EQUATION]', 'hep-ph-0201037-2-9-0': 'Thus, the form factors for transition form arbitrary bound states of hydrogenlike atoms to the "[MATH]" of continuous spectra are represented in the form of superposition of finite number of terms with simple analytical structure and can be evaluated numerically with arbitrary degree of accuracy.', 'hep-ph-0201037-2-10-0': 'The equations ([REF])-([REF]) are the generalization of the results of the papers [CITATION].'} | [['hep-ph-0201037-1-3-0', 'hep-ph-0201037-2-3-0'], ['hep-ph-0201037-1-3-1', 'hep-ph-0201037-2-3-1'], ['hep-ph-0201037-1-5-0', 'hep-ph-0201037-2-5-0'], ['hep-ph-0201037-1-2-0', 'hep-ph-0201037-2-2-0'], ['hep-ph-0201037-1-10-0', 'hep-ph-0201037-2-10-0'], ['hep-ph-0201037-1-8-0', 'hep-ph-0201037-2-8-0'], ['hep-ph-0201037-1-9-0', 'hep-ph-0201037-2-9-0'], ['hep-ph-0201037-1-4-0', 'hep-ph-0201037-2-4-0'], ['hep-ph-0201037-1-6-0', 'hep-ph-0201037-2-6-0'], ['hep-ph-0201037-1-6-1', 'hep-ph-0201037-2-6-1'], ['hep-ph-0201037-1-6-2', 'hep-ph-0201037-2-6-2'], ['hep-ph-0201037-1-6-3', 'hep-ph-0201037-2-6-3'], ['hep-ph-0201037-1-6-4', 'hep-ph-0201037-2-6-4'], ['hep-ph-0201037-1-1-0', 'hep-ph-0201037-2-1-0'], ['hep-ph-0201037-1-1-1', 'hep-ph-0201037-2-1-1'], ['hep-ph-0201037-2-6-1', 'hep-ph-0201037-3-6-1'], ['hep-ph-0201037-2-0-0', 'hep-ph-0201037-3-0-0'], ['hep-ph-0201037-2-3-1', 'hep-ph-0201037-3-3-1'], ['hep-ph-0201037-2-1-0', 'hep-ph-0201037-3-1-0'], ['hep-ph-0201037-2-1-1', 'hep-ph-0201037-3-1-1'], ['hep-ph-0201037-1-0-0', 'hep-ph-0201037-2-0-0'], ['hep-ph-0201037-1-0-1', 'hep-ph-0201037-2-0-1'], ['hep-ph-0201037-2-6-0', 'hep-ph-0201037-3-6-0'], ['hep-ph-0201037-2-6-2', 'hep-ph-0201037-3-6-2'], ['hep-ph-0201037-2-6-4', 'hep-ph-0201037-3-6-4'], ['hep-ph-0201037-2-9-0', 'hep-ph-0201037-3-9-0'], ['hep-ph-0201037-2-0-1', 'hep-ph-0201037-3-0-1'], ['hep-ph-0201037-2-3-0', 'hep-ph-0201037-3-3-0'], ['hep-ph-0201037-2-8-0', 'hep-ph-0201037-3-8-0'], ['hep-ph-0201037-2-4-0', 'hep-ph-0201037-3-4-0'], ['hep-ph-0201037-2-5-0', 'hep-ph-0201037-3-5-0'], ['hep-ph-0201037-2-6-3', 'hep-ph-0201037-3-6-3'], ['hep-ph-0201037-2-2-0', 'hep-ph-0201037-3-2-0'], ['hep-ph-0201037-2-2-0', 'hep-ph-0201037-3-2-1']] | [['hep-ph-0201037-1-3-0', 'hep-ph-0201037-2-3-0'], ['hep-ph-0201037-1-3-1', 'hep-ph-0201037-2-3-1'], ['hep-ph-0201037-1-5-0', 'hep-ph-0201037-2-5-0'], ['hep-ph-0201037-1-2-0', 'hep-ph-0201037-2-2-0'], ['hep-ph-0201037-1-10-0', 'hep-ph-0201037-2-10-0'], ['hep-ph-0201037-1-8-0', 'hep-ph-0201037-2-8-0'], ['hep-ph-0201037-1-9-0', 'hep-ph-0201037-2-9-0'], ['hep-ph-0201037-1-4-0', 'hep-ph-0201037-2-4-0'], ['hep-ph-0201037-1-6-0', 'hep-ph-0201037-2-6-0'], ['hep-ph-0201037-1-6-1', 'hep-ph-0201037-2-6-1'], ['hep-ph-0201037-1-6-2', 'hep-ph-0201037-2-6-2'], ['hep-ph-0201037-1-6-3', 'hep-ph-0201037-2-6-3'], ['hep-ph-0201037-1-6-4', 'hep-ph-0201037-2-6-4'], ['hep-ph-0201037-1-1-0', 'hep-ph-0201037-2-1-0'], ['hep-ph-0201037-1-1-1', 'hep-ph-0201037-2-1-1'], ['hep-ph-0201037-2-6-1', 'hep-ph-0201037-3-6-1'], ['hep-ph-0201037-2-0-0', 'hep-ph-0201037-3-0-0'], ['hep-ph-0201037-2-3-1', 'hep-ph-0201037-3-3-1'], ['hep-ph-0201037-2-1-0', 'hep-ph-0201037-3-1-0'], ['hep-ph-0201037-2-1-1', 'hep-ph-0201037-3-1-1']] | [['hep-ph-0201037-1-0-0', 'hep-ph-0201037-2-0-0'], ['hep-ph-0201037-1-0-1', 'hep-ph-0201037-2-0-1'], ['hep-ph-0201037-2-6-0', 'hep-ph-0201037-3-6-0'], ['hep-ph-0201037-2-6-2', 'hep-ph-0201037-3-6-2'], ['hep-ph-0201037-2-6-4', 'hep-ph-0201037-3-6-4'], ['hep-ph-0201037-2-9-0', 'hep-ph-0201037-3-9-0'], ['hep-ph-0201037-2-0-1', 'hep-ph-0201037-3-0-1'], ['hep-ph-0201037-2-3-0', 'hep-ph-0201037-3-3-0'], ['hep-ph-0201037-2-8-0', 'hep-ph-0201037-3-8-0'], ['hep-ph-0201037-2-4-0', 'hep-ph-0201037-3-4-0'], ['hep-ph-0201037-2-5-0', 'hep-ph-0201037-3-5-0']] | [] | [['hep-ph-0201037-2-6-3', 'hep-ph-0201037-3-6-3'], ['hep-ph-0201037-2-2-0', 'hep-ph-0201037-3-2-0'], ['hep-ph-0201037-2-2-0', 'hep-ph-0201037-3-2-1']] | [] | ['hep-ph-0201037-1-7-0', 'hep-ph-0201037-2-7-0', 'hep-ph-0201037-3-7-0', 'hep-ph-0201037-3-10-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/0201037 | {'hep-ph-0201037-3-0-0': 'A general formula for bound-continuous transition form factors is derived.', 'hep-ph-0201037-3-0-1': 'It is shown that these form factors can be represented in the form of finite sum of terms with simple analytical structure.', 'hep-ph-0201037-3-1-0': 'In the previous paper [CITATION] it have been shown that the form factors of transitions from [MATH] - states of hydrogenlike atoms [CITATION] to the state of continuous spectra with definite value of relative momenta [MATH] may be expressed in the terms of the classical polynomials in a rather simple way.', 'hep-ph-0201037-3-1-1': 'Below this result is generalized for the case of transition from arbitrary initial bound states.', 'hep-ph-0201037-3-2-0': 'The transition form factors are defined as follows: [EQUATION]', 'hep-ph-0201037-3-2-1': 'Here, [MATH] are the wave functions of initial (final) states.', 'hep-ph-0201037-3-3-0': 'According to [CITATION] (see also [CITATION]), the final state wave function must be choose in the form [EQUATION]', 'hep-ph-0201037-3-3-1': 'For arbitrary initial bound state [EQUATION] where [MATH] are the associated Laguerre polynomials.', 'hep-ph-0201037-3-4-0': 'Making use of the recurrence relations [CITATION] [EQUATION] and the representation of the Laguerre polynomials in terms of the generating function [EQUATION] where operator [MATH] is defined as follows [EQUATION] let us rewrite the radial part of initial state wave function in the form [EQUATION] more convenient for the further calculations.', 'hep-ph-0201037-3-5-0': 'Then it is not difficult to see that transition form factors ([REF]) may be represent as a linear combination of the quantities [EQUATION].', 'hep-ph-0201037-3-6-0': 'In order to calculate ([REF]), it is useful to represent the hypergeometrical function in ([REF]) in the form [EQUATION]', 'hep-ph-0201037-3-6-1': 'Using the following relations [EQUATION] where [EQUATION] after simple calculations we find [EQUATION]', 'hep-ph-0201037-3-6-2': 'It is easy to check that [EQUATION]', 'hep-ph-0201037-3-6-3': 'Further, according to [CITATION], we get [EQUATION]', 'hep-ph-0201037-3-6-4': 'Taking into account ([REF]) and ([REF]), it is easy to see that ([REF]) is the superposition of the quantities [EQUATION] where [MATH] are defined as follows: [EQUATION].', 'hep-ph-0201037-3-7-0': 'The further calculations are the same as in [CITATION].', 'hep-ph-0201037-3-8-0': 'Omitting the simple but cumbersome algebra, let us present the final expression for transition form factors: [EQUATION]', 'hep-ph-0201037-3-9-0': 'Thus, the form factors for transition from arbitrary bound states of hydrogenlike atoms to the "[MATH]" of continuous spectra are represented as the superposition of finite number of terms with simple analytical structure and can be evaluated numerically with arbitrary degree of accuracy.', 'hep-ph-0201037-3-10-0': 'Eqs. ([REF])-([REF]) are the generalization of the results of [CITATION].'} | null | null | null | null |
1008.5155 | {'1008.5155-1-0-0': 'We report Coulomb drag measurements between tunable vertically-coupled quantum wires.', '1008.5155-1-0-1': 'The Coulomb drag signal is mapped out versus the number of subbands occupied in each wire, and a phase diagram for a positive versus negative drag is established.', '1008.5155-1-0-2': 'When the Coulomb drag is negative, one-dimensional subbands are observed as plateau-like features in the drag signal.', '1008.5155-1-0-3': 'Negative Coulomb drag is also observed in two regimes: one at low electronic density when the drag wire is depleted, and one at higher electronic density when the drag wire has more than a single one-dimensional subband occupied.', '1008.5155-1-0-4': 'A discussion of the negative drag signal in terms of electron-hole asymmetry and localization is presented.', '1008.5155-1-1-0': 'Understanding the role played by electron-electron interactions has been one of the driving forces leading to the development of coupled nanostructures designed for Coulomb drag measurements.', '1008.5155-1-1-1': 'The Coulomb drag experiment consists of coupling two independent circuits by proximity, and sending a small current [MATH] through one of the (drive) circuits.', '1008.5155-1-1-2': 'Under the condition of no current flow, a voltage [MATH] develops across the second (drag) circuit.', '1008.5155-1-1-3': 'This drag voltage, emanating from electron scattering between the two circuits, defines a drag resistance [MATH] that is a direct probe of electron-electron interactions.', '1008.5155-1-1-4': 'The sign convention on [MATH] here implies that a positive (negative) drag will result from electrons in the drag circuit scattered in the same (opposite) direction than the electron flow in the drive circuit.', '1008.5155-1-1-5': 'The onset of drag voltage in a passive circuit relies on electron-hole asymmetry as charges are effectively carried both by electron-like and hole-like excitations.', '1008.5155-1-1-6': 'Without such electron-hole asymmetry, the contribution to Coulomb drag from electron-like and hole-like excitations would cancel each other, yielding no net drag signal.', '1008.5155-1-1-7': 'Recent experiments performed in quantum confined geometries such as quantum dots [CITATION] or quantum point contacts [CITATION] have highlighted additional contributions to transport measurements that have been attributed to a gate voltage dependent enhancement of such electron-hole asymmetry as well as to shot noise rectification, depending on the operating voltage regime [CITATION].', '1008.5155-1-2-0': 'Coulomb drag measurements were first performed in two-dimensional systems [CITATION].', '1008.5155-1-2-1': 'Subsequent studies, in agreement with the predictions from Fermi liquid theory (FL), established the drag resistivity to follow a [MATH] dependence, [MATH], where [MATH] is the electron density of each layer and [MATH] the interlayer separation.', '1008.5155-1-2-2': 'The [MATH] dependence can be understood in terms of simple scattering and thermal broadening arguments as each layer contributes [MATH] to the drag.', '1008.5155-1-2-3': 'Accordingly, a linear temperature dependence for Coulomb drag is expected within FL theory in one dimension for identical wires [CITATION].', '1008.5155-1-2-4': 'Within Tomonaga-Luttinger liquid theory (TLL), the situation is more complex as two competing mechanisms, backscattering [CITATION] and forward scattering [CITATION], contribute to Coulomb drag; their respective strengths depend on [MATH] is the Fermi wavevector), the temperature, and the mismatch in the electron densities of each wire.', '1008.5155-1-2-5': 'A key point is that these theories predict a positive drag signal between two wires with negative charge carriers, i.e. [MATH].', '1008.5155-1-2-6': 'However, a negative drag signal ([MATH]) was recently reported between lateral 1D-1D systems at low density [CITATION] and heuristically attributed to Wigner crystallization in one-dimension.', '1008.5155-1-2-7': 'This experimental observation certainly demonstrates that our theoretical knowledge of Coulomb drag in one-dimensional systems is not complete.', '1008.5155-1-3-0': 'Despite the large amount of theoretical work on the subject, very few experiments have actually measured 1D-1D Coulomb drag [CITATION].', '1008.5155-1-3-1': 'These experiments have so far all been realized in a lateral geometry in a regime where both wires had a very similar 1D subband occupancy, where the interwire separation was large (d [MATH] 200 nm), and where the barrier between the wires was soft (electrostatic).', '1008.5155-1-3-2': 'In order to address those limitations, we fabricated coupled quantum wires in a vertical geometry with the barrier grown by molecular-beam epitaxy.', '1008.5155-1-3-3': 'This design allows the mapping of the drag signal over a large range of density and subband occupancy in both wires, allowing for the precise determination of positive and negative drag regimes.', '1008.5155-1-3-4': 'Surprisingly, negative Coulomb drag is observed in two distinct regimes: one with the drag wire depleted and one with the drag wire conducting.', '1008.5155-1-3-5': 'Both of these negative drag regimes occur with the drive wire having several 1D subbands occupied.', '1008.5155-1-3-6': 'This contrasts with previous observations [CITATION] where negative Coulomb drag was only observed when the conductance in both wires was lower than the conductance of the first plateau.', '1008.5155-1-4-0': 'The vertically-coupled double quantum wires used in this Letter (see Fig. 1) are patterned on a n-doped GaAs/AlGaAs electron bilayer heterostructure (wafer EA0975).', '1008.5155-1-4-1': 'The two 18 nm wide quantum wells are separated by a 15 nm wide Al[MATH]Ga[MATH]As barrier.', '1008.5155-1-4-2': 'The electron density is 1.1 (1.4) [MATH] cm[MATH] for the upper (lower) 2DEG, yielding a combined mobility of [MATH] cm[MATH] / V[MATH] s.', '1008.5155-1-4-3': 'A set of two split gates (a pinch-off gate and a plunger gate) is defined using electron beam lithography on the upper side of the sample.', '1008.5155-1-4-4': 'Using an epoxy-bound-and-stop-etch (EBASE) process [CITATION], the sample is flipped and thinned and another set of gates is defined on the lower side of the sample.', '1008.5155-1-4-5': 'Using atomic layer deposition, a thin dielectric layer (Al[MATH]O[MATH]) is deposited between the lower quantum well and the lower gates to prevent electrical leakage.', '1008.5155-1-4-6': 'Due to the presence of this layer, both wires are not expected to be identically confined as the top gates are 154 nm away from the wire whereas the bottom gates are 214 nm away from the wire.', '1008.5155-1-4-7': 'The wires used are 5 [MATH] long and 0.5 [MATH] wide.', '1008.5155-1-4-8': 'From the alignment uncertainty, we estimate an interwire separation bounded between 33 nm and 100 nm.', '1008.5155-1-4-9': 'All transport measurements are performed at a temperature of 330 mK in a [MATH]He refrigerator.', '1008.5155-1-4-10': 'The conductance in each quantum wire is measured independently and simultaneously using two-wire measurements with an excitation voltage of 50 [MATH] at a frequency of 9 Hz for the lower wire and 13 Hz for the upper wire.', '1008.5155-1-4-11': 'The Coulomb drag measurements are performed with a 4.5 nA current at 9 Hz, and it was verified that [MATH] remains the same over a large range of excitation current and frequency (0.3 - 11 nA in current and 1 Hz - 50 Hz in frequency) as well as upon inverting the drag and drive layers, provided that neither wire was depleted (i.e. for [MATH]).', '1008.5155-1-4-12': 'For the Coulomb drag results presented here, current is driven in the lower (drive) wire and voltage is measured in the upper (drag) wire.', '1008.5155-1-5-0': 'In the Laudauer formalism, the conductance for quantum transport assumes the value [MATH] where [MATH] is the number of conduction channels and [MATH] is the electron transmission probability for each channel.', '1008.5155-1-5-1': 'In the ballistic regime, electron transmission is unhindered and [MATH] while an increase in scattering along the wire causes [MATH] in the non-ballistic regime.', '1008.5155-1-5-2': 'Due to their length, the wires presented in this Letter are in this non-ballistic regime and have conductance plateaus at values less than [MATH].', '1008.5155-1-5-3': 'These plateaus can be mapped out using the plunger gates capacitively coupled to both wires.', '1008.5155-1-5-4': 'It is therefore possible to observe 1D subbands depopulation in each wire by sweeping a single plunger gate (i.e. the lower plunger gate).', '1008.5155-1-5-5': 'In addition, by varying both the upper (UPL) and the lower plunger gates (LPL), a large combination of 1D subbands is accessible and a mapping of the conductance as a function of UPL and LPL becomes possible.', '1008.5155-1-5-6': 'To highlight the position of the subbands in each wire, the numerical derivatives of the conductance with respect to the LPL voltage are presented in Fig. 2(a) and (b).', '1008.5155-1-5-7': 'In these figures, 1D subbands can easily be identified as the derivative approaches zero for every subband crossing, appearing as blue and black stripes in the differential conductance maps, e.g. the dark features denoted by I, II and III in Fig. 2(b).', '1008.5155-1-5-8': 'The wide black and blue region labeled A in Fig. 2(a) denotes the low-density regime where the upper (drag) quantum wire is depleted.', '1008.5155-1-5-9': 'The white region in the upper left corner of both figures is a non-conducting regime for both wires where data was not taken.', '1008.5155-1-6-0': 'The drag resistance (left-axis) along with the conductance in the lower and upper wires (right-axis) is presented in Fig. 3.', '1008.5155-1-6-1': 'This data was taken during a cooldown different than the data shown in Fig. 2 and Fig. 4.', '1008.5155-1-6-2': 'The drag resistance [MATH] shows a strong peak at LPL = [MATH] V as the first 1D subband of the drag (upper) wire depopulates.', '1008.5155-1-6-3': 'As the upper wire density is lowered further, [MATH] transits towards a strongly negative regime.', '1008.5155-1-6-4': 'In this regime, subband crossings of the drive wire are clearly observed and shown in the plot by the grey lines denoted by i, ii and iii.', '1008.5155-1-6-5': 'We note that in the drag configuration, the tunneling resistance between the wires is [MATH]25 M[MATH], and these features are not due to current leakage.', '1008.5155-1-6-6': 'These subband characteristics of the drive wire are however not observed in the positive drag regime.', '1008.5155-1-6-7': 'Perhaps the most striking feature of the data is the presence of a re-entrant negative drag signal occurring between LPL = [MATH] V and LPL = [MATH] V, delimited by the grey lines noted iv and v in Fig. 3, and magnified in the inset.', '1008.5155-1-6-8': 'This high-density negative drag occurs when both wires have [MATH] and [MATH]).', '1008.5155-1-6-9': 'To the best of our knowledge, this is the first report of a 1D-1D negative Coulomb drag when both wires have conductance larger than that corresponding to one subband occupation.', '1008.5155-1-7-0': 'The complete mapping (versus LPL and UPL) of the drag resistance [MATH], as well as its numerical derivative [MATH], are shown in Fig. 4(a) and (b) respectively.', '1008.5155-1-7-1': 'These maps allow the tracking of the Coulomb drag features observed in Fig. 3 over a large range of 1D subband occupancy in each wire.', '1008.5155-1-7-2': 'The strong peak observed in [MATH] as the first 1D subband of the drag wire depopulates occurs over the entire wires density range.', '1008.5155-1-7-3': 'It appear as stripes of negative slopes (blue) near the yellow line in Fig. 4(b), but can also be observed from the darker red coloring before the drag signal becomes negative (blue) in Fig. 4(a).', '1008.5155-1-7-4': 'Such peaks at the opening of a conduction channel were predicted to occur due to an enhancement of electron-hole asymmetry [CITATION].', '1008.5155-1-7-5': 'Plateaus in the low-density negative drag regime resulting from the drive wire 1D subband crossings can also be observed in Fig. 4(b) (features labeled I, II and III in Fig. 2(b) and in Fig. 4(b)).', '1008.5155-1-7-6': 'The transition from positive (red) to negative (blue) drag is best tracked in Fig. 4(a) and always occurs near the depletion point of the drag wire (yellow lines in Fig. 4).', '1008.5155-1-7-7': 'Unlike the other features mentioned previously, the high-density negative drag regime only occurs in a narrow window when UPL is between [MATH] V and [MATH] V and for LPL values corresponding to the first 1D subband crossing in the upper (drag) quantum wire.', '1008.5155-1-7-8': 'This corresponds to region B in Fig. 4(a) and (b).', '1008.5155-1-7-9': 'A typical curve of this negative drag regime is shown in the inset of Fig. 4(a) for UPL = [MATH] V.', '1008.5155-1-8-0': 'Previous theoretical studies of Coulomb drag in the diffusive transport regime [CITATION] and in the non-linear regime [CITATION] have predicted the existence of negative Coulomb drag when electrons are the sole charge carrier type.', '1008.5155-1-8-1': 'In the diffusive regime, negative Coulomb drag is expected for wires longer than the characteristic phonon assisted 1D transport length, [MATH], where [MATH] is the phonon assisted 1D transport time.', '1008.5155-1-8-2': 'This length is estimated to be [MATH]m [CITATION] which is much longer than the actual wire size, 5 [MATH]m, and therefore unlikely to play a role.', '1008.5155-1-8-3': 'Likewise, it was verified that the drag voltage scales linearly with current excitation, both in the positive and negative drag regimes.', '1008.5155-1-8-4': 'Therefore, diffusive transport and non-linear drag are unlikely to explain the negative drag reported in this Letter.', '1008.5155-1-8-5': 'Negative Coulomb drag between parallel quantum wires was previously attributed to Wigner crystallization [CITATION].', '1008.5155-1-8-6': 'While this mechanism could explain the onset of negative Coulomb drag in the low-density regime, it cannot explain the high-density negative drag observed with [MATH] in both wires.', '1008.5155-1-8-7': 'Indeed, Wigner crystallization should occur at a value of [MATH] [CITATION], where [MATH] is the effective Bohr atomic radius.', '1008.5155-1-8-8': 'In the high-density negative drag regime, we estimate [MATH] and therefore the electrons are unlikely to form a Wigner crystal.', '1008.5155-1-9-0': 'An enhancement of electron-hole asymmetry near the depletion of a 1D subband was predicted to cause peaks in the drag signal [CITATION], as can be observed in our data when [MATH] = 1.', '1008.5155-1-9-1': 'Compared to bulk systems, this particle-hole asymmetry is stronger in mesoscopic and quantum circuits with spatial dimension less than the temperature length [MATH] and the voltage length [MATH].', '1008.5155-1-9-2': 'In our Coulomb drag circuit in the high density negative drag regime, we estimate [MATH] at 0.33 K and [MATH], which are of the same order of magnitude as our quantum wire dimensions.', '1008.5155-1-9-3': 'In the linear drag regime, as is the case in this work, we expect a positive drag signal for monotonically increasing transmission probabilities, i.e. for monotonically increasing wire conductances.', '1008.5155-1-9-4': 'On the other hand, negative Coulomb drag might occur in the case where the conductance in the drag wire is non-monotonic.', '1008.5155-1-9-5': 'Indeed, such a non-monotonic behavior in the drag wire conductance is observed concomitant with the re-entrant negative drag (see inset of Fig. 3).', '1008.5155-1-9-6': 'We speculate that this observation might be responsible (at least in part) for the negative drag observed in the high-density regime.', '1008.5155-1-10-0': 'To conclude, we have presented the first drag measurements between vertically-coupled quantum wires with in situ control over the subbands occupation of each wire.', '1008.5155-1-10-1': 'These stacked wires form quantum circuits with their own current path and gating.', '1008.5155-1-10-2': 'Our data shows a clear correlation observed between the low density negative drag and the depletion of the drag wire.', '1008.5155-1-10-3': 'While in this regime localization might explain the negative drag, the same heuristic explanation cannot explain the re-entrance of the negative drag signal observed at higher electronic density where the drag wire is conducting.', '1008.5155-1-10-4': 'We hope that this observation will motivate further theoretical work aimed at understanding the exact role played by electron interactions and electron-hole asymmetry in 1D-1D coupled electronic systems.', '1008.5155-1-11-0': 'We acknowledge the outstanding technical assistance of Denise Tibbetts and James Hedberg.', '1008.5155-1-11-1': 'We thank Aashish Clerk for inspiring discussions.', '1008.5155-1-11-2': 'This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. DOE, Office of Basic Energy Sciences user facility.', '1008.5155-1-11-3': "Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000."} | {'1008.5155-2-0-0': 'We report Coulomb drag measurements between tunable vertically-coupled quantum wires.', '1008.5155-2-0-1': 'The Coulomb drag signal is mapped out versus the number of subbands occupied in each wire, and a phase diagram for a positive versus negative drag is established.', '1008.5155-2-0-2': 'When the Coulomb drag is negative, one-dimensional subbands are observed as plateau-like features in the drag signal.', '1008.5155-2-0-3': 'Negative Coulomb drag is also observed in two regimes: one at low electronic density when the drag wire is depleted, and one at higher electronic density when the drag wire has more than a single one-dimensional subband occupied.', '1008.5155-2-0-4': 'A discussion of the negative drag signal in terms of electron-hole asymmetry and localization is presented.', '1008.5155-2-1-0': 'Understanding the role played by electron-electron interactions has been one of the driving forces leading to the development of coupled nanostructures designed for Coulomb drag measurements.', '1008.5155-2-1-1': 'The Coulomb drag experiment consists of coupling two independent circuits by proximity, and sending a small current [MATH] through one of the (drive) circuits.', '1008.5155-2-1-2': 'Under the condition of no current flow, a voltage [MATH] develops across the second (drag) circuit.', '1008.5155-2-1-3': 'This drag voltage, emanating from electron scattering between the two circuits, defines a drag resistance [MATH] that is a direct probe of electron-electron interactions.', '1008.5155-2-1-4': 'The sign convention on [MATH] here implies that a positive (negative) drag will result from electrons in the drag circuit scattered in the same (opposite) direction than the electron flow in the drive circuit.', '1008.5155-2-1-5': 'The onset of drag voltage in a passive circuit relies on electron-hole asymmetry as charges are effectively carried both by electron-like and hole-like excitations.', '1008.5155-2-1-6': 'Without such electron-hole asymmetry, the contribution to Coulomb drag from electron-like and hole-like excitations would cancel each other, yielding no net drag signal.', '1008.5155-2-1-7': 'Recent experiments performed in quantum confined geometries such as quantum dots [CITATION] or quantum point contacts [CITATION] have highlighted additional contributions to transport measurements that have been attributed to a gate voltage dependent enhancement of such electron-hole asymmetry as well as to shot noise rectification, depending on the operating voltage regime [CITATION].', '1008.5155-2-2-0': 'Coulomb drag measurements were first performed in two-dimensional systems [CITATION].', '1008.5155-2-2-1': 'Subsequent studies, in agreement with the predictions from Fermi liquid theory (FL), established the drag resistivity to follow a [MATH] dependence, [MATH], where [MATH] is the electron density of each layer and [MATH] the interlayer separation.', '1008.5155-2-2-2': 'The [MATH] dependence can be understood in terms of simple scattering and thermal broadening arguments as each layer contributes [MATH] to the drag.', '1008.5155-2-2-3': 'Accordingly, a linear temperature dependence for Coulomb drag is expected within FL theory in one dimension for identical wires [CITATION].', '1008.5155-2-2-4': 'Within Tomonaga-Luttinger liquid theory (TLL), the situation is more complex as two competing mechanisms, backscattering [CITATION] and forward scattering [CITATION], contribute to Coulomb drag; their respective strengths depend on [MATH] is the Fermi wavevector), the temperature, and the mismatch in the electron densities of each wire.', '1008.5155-2-2-5': 'A key point is that these theories predict a positive drag signal between two wires with negative charge carriers, i.e. [MATH].', '1008.5155-2-2-6': 'However, a negative drag signal ([MATH]) was recently reported between lateral 1D-1D systems at low density [CITATION] and heuristically attributed to Wigner crystallization in one-dimension.', '1008.5155-2-2-7': 'This experimental observation certainly demonstrates that our theoretical knowledge of Coulomb drag in one-dimensional systems is not complete.', '1008.5155-2-3-0': 'Despite the large amount of theoretical work on the subject, very few experiments have actually measured 1D-1D Coulomb drag [CITATION].', '1008.5155-2-3-1': 'These experiments have so far all been realized in a lateral geometry in a regime where both wires had a very similar 1D subband occupancy, where the interwire separation was large (d [MATH] 200 nm), and where the barrier between the wires was soft (electrostatic).', '1008.5155-2-3-2': 'In order to address those limitations, we fabricated coupled quantum wires in a vertical geometry with the barrier grown by molecular-beam epitaxy.', '1008.5155-2-3-3': 'This design allows the mapping of the drag signal over a large range of density and subband occupancy in both wires, allowing for the precise determination of positive and negative drag regimes.', '1008.5155-2-3-4': 'Surprisingly, negative Coulomb drag is observed in two distinct regimes: one with the drag wire depleted and one with the drag wire conducting.', '1008.5155-2-3-5': 'Both of these negative drag regimes occur with the drive wire having several 1D subbands occupied.', '1008.5155-2-3-6': 'This contrasts with previous observations [CITATION] where negative Coulomb drag was only observed when the conductance in both wires was lower than the conductance of the first plateau.', '1008.5155-2-4-0': 'The vertically-coupled double quantum wires used in this Letter (see Fig. 1) are patterned on a n-doped GaAs/AlGaAs electron bilayer heterostructure (wafer EA0975).', '1008.5155-2-4-1': 'The two 18 nm wide quantum wells are separated by a 15 nm wide Al[MATH]Ga[MATH]As barrier.', '1008.5155-2-4-2': 'The electron density is 1.1 (1.4) [MATH] cm[MATH] for the upper (lower) 2DEG, yielding a combined mobility of [MATH] cm[MATH] / V[MATH] s.', '1008.5155-2-4-3': 'A set of two split gates (a pinch-off gate and a plunger gate) is defined using electron beam lithography on the upper side of the sample.', '1008.5155-2-4-4': 'Using an epoxy-bound-and-stop-etch (EBASE) process [CITATION], the sample is flipped and thinned and another set of gates is defined on the lower side of the sample.', '1008.5155-2-4-5': 'Using atomic layer deposition, a thin dielectric layer (Al[MATH]O[MATH]) is deposited between the lower quantum well and the lower gates to prevent electrical leakage.', '1008.5155-2-4-6': 'Due to the presence of this layer, both wires are not expected to be identically confined as the top gates are 154 nm away from the wire whereas the bottom gates are 214 nm away from the wire.', '1008.5155-2-4-7': 'The wires used are 5 [MATH] long and 0.5 [MATH] wide.', '1008.5155-2-4-8': 'From the alignment uncertainty, we estimate an interwire separation bounded between 33 nm and 100 nm.', '1008.5155-2-4-9': 'All transport measurements are performed at a temperature of 330 mK in a [MATH]He refrigerator.', '1008.5155-2-4-10': 'The conductance in each quantum wire is measured independently and simultaneously using two-wire measurements with an excitation voltage of 50 [MATH] at a frequency of 9 Hz for the lower wire and 13 Hz for the upper wire.', '1008.5155-2-4-11': 'The Coulomb drag measurements are performed with a 4.5 nA current at 9 Hz, and it was verified that [MATH] remains the same over a large range of excitation current and frequency (0.3 - 11 nA in current and 1 Hz - 50 Hz in frequency) as well as upon inverting the drag and drive layers, provided that neither wire was depleted (i.e. for [MATH]).', '1008.5155-2-4-12': 'For the Coulomb drag results presented here, current is driven in the lower (drive) wire and voltage is measured in the upper (drag) wire.', '1008.5155-2-5-0': 'In the Laudauer formalism, the conductance for quantum transport assumes the value [MATH] where [MATH] is the number of conduction channels and [MATH] is the electron transmission probability for each channel.', '1008.5155-2-5-1': 'In the ballistic regime, electron transmission is unhindered and [MATH] while an increase in scattering along the wire causes [MATH] in the non-ballistic regime.', '1008.5155-2-5-2': 'Due to their length, the wires presented in this Letter are in this non-ballistic regime and have conductance plateaus at values less than [MATH].', '1008.5155-2-5-3': 'These plateaus can be mapped out using the plunger gates capacitively coupled to both wires.', '1008.5155-2-5-4': 'It is therefore possible to observe 1D subbands depopulation in each wire by sweeping a single plunger gate (i.e. the lower plunger gate).', '1008.5155-2-5-5': 'In addition, by varying both the upper (UPL) and the lower plunger gates (LPL), a large combination of 1D subbands is accessible and a mapping of the conductance as a function of UPL and LPL becomes possible.', '1008.5155-2-5-6': 'To highlight the position of the subbands in each wire, the numerical derivatives of the conductance with respect to the LPL voltage are presented in Fig. 2(a) and (b).', '1008.5155-2-5-7': 'In these figures, 1D subbands can easily be identified as the derivative approaches zero for every subband crossing, appearing as blue and black stripes in the differential conductance maps, e.g. the dark features denoted by I, II and III in Fig. 2(b).', '1008.5155-2-5-8': 'The wide black and blue region labeled A in Fig. 2(a) denotes the low-density regime where the upper (drag) quantum wire is depleted.', '1008.5155-2-5-9': 'The white region in the upper left corner of both figures is a non-conducting regime for both wires where data was not taken.', '1008.5155-2-6-0': 'The drag resistance (left-axis) along with the conductance in the lower and upper wires (right-axis) is presented in Fig. 3.', '1008.5155-2-6-1': 'This data was taken during a cooldown different than the data shown in Fig. 2 and Fig. 4.', '1008.5155-2-6-2': 'The drag resistance [MATH] shows a strong peak at LPL = [MATH] V as the first 1D subband of the drag (upper) wire depopulates.', '1008.5155-2-6-3': 'As the upper wire density is lowered further, [MATH] transits towards a strongly negative regime.', '1008.5155-2-6-4': 'In this regime, subband crossings of the drive wire are clearly observed and shown in the plot by the grey lines denoted by i, ii and iii.', '1008.5155-2-6-5': 'We note that in the drag configuration, the tunneling resistance between the wires is [MATH]25 M[MATH], and these features are not due to current leakage.', '1008.5155-2-6-6': 'These subband characteristics of the drive wire are however not observed in the positive drag regime.', '1008.5155-2-6-7': 'Perhaps the most striking feature of the data is the presence of a re-entrant negative drag signal occurring between LPL = [MATH] V and LPL = [MATH] V, delimited by the grey lines noted iv and v in Fig. 3, and magnified in the inset.', '1008.5155-2-6-8': 'This high-density negative drag occurs when both wires have [MATH] and [MATH]).', '1008.5155-2-6-9': 'To the best of our knowledge, this is the first report of a 1D-1D negative Coulomb drag when both wires have conductance larger than that corresponding to one subband occupation.', '1008.5155-2-7-0': 'The complete mapping (versus LPL and UPL) of the drag resistance [MATH], as well as its numerical derivative [MATH], are shown in Fig. 4(a) and (b) respectively.', '1008.5155-2-7-1': 'These maps allow the tracking of the Coulomb drag features observed in Fig. 3 over a large range of 1D subband occupancy in each wire.', '1008.5155-2-7-2': 'The strong peak observed in [MATH] as the first 1D subband of the drag wire depopulates occurs over the entire wires density range.', '1008.5155-2-7-3': 'It appear as stripes of negative slopes (blue) near the yellow line in Fig. 4(b), but can also be observed from the darker red coloring before the drag signal becomes negative (blue) in Fig. 4(a).', '1008.5155-2-7-4': 'Such peaks at the opening of a conduction channel were predicted to occur due to an enhancement of electron-hole asymmetry [CITATION].', '1008.5155-2-7-5': 'Plateaus in the low-density negative drag regime resulting from the drive wire 1D subband crossings can also be observed in Fig. 4(b) (features labeled I, II and III in Fig. 2(b) and in Fig. 4(b)).', '1008.5155-2-7-6': 'The transition from positive (red) to negative (blue) drag is best tracked in Fig. 4(a) and always occurs near the depletion point of the drag wire (yellow lines in Fig. 4).', '1008.5155-2-7-7': 'Unlike the other features mentioned previously, the high-density negative drag regime only occurs in a narrow window when UPL is between [MATH] V and [MATH] V and for LPL values corresponding to the first 1D subband crossing in the upper (drag) quantum wire.', '1008.5155-2-7-8': 'This corresponds to region B in Fig. 4(a) and (b).', '1008.5155-2-7-9': 'A typical curve of this negative drag regime is shown in the inset of Fig. 4(a) for UPL = [MATH] V.', '1008.5155-2-8-0': 'Previous theoretical studies of Coulomb drag in the diffusive transport regime [CITATION] and in the non-linear regime [CITATION] have predicted the existence of negative Coulomb drag when electrons are the sole charge carrier type.', '1008.5155-2-8-1': 'In the diffusive regime, negative Coulomb drag is expected for wires longer than the characteristic phonon assisted 1D transport length, [MATH], where [MATH] is the phonon assisted 1D transport time.', '1008.5155-2-8-2': 'This length is estimated to be [MATH]m [CITATION] which is much longer than the actual wire size, 5 [MATH]m, and therefore unlikely to play a role.', '1008.5155-2-8-3': 'Likewise, it was verified that the drag voltage scales linearly with current excitation, both in the positive and negative drag regimes.', '1008.5155-2-8-4': 'Therefore, diffusive transport and non-linear drag are unlikely to explain the negative drag reported in this Letter.', '1008.5155-2-8-5': 'Negative Coulomb drag between parallel quantum wires was previously attributed to Wigner crystallization [CITATION].', '1008.5155-2-8-6': 'While this mechanism could explain the onset of negative Coulomb drag in the low-density regime, it cannot explain the high-density negative drag observed with [MATH] in both wires.', '1008.5155-2-8-7': 'Indeed, Wigner crystallization should occur at a value of [MATH] [CITATION], where [MATH] is the effective Bohr atomic radius.', '1008.5155-2-8-8': 'In the high-density negative drag regime, we estimate [MATH] and therefore the electrons are unlikely to form a Wigner crystal.', '1008.5155-2-9-0': 'An enhancement of electron-hole asymmetry near the depletion of a 1D subband was predicted to cause peaks in the drag signal [CITATION], as can be observed in our data when [MATH] = 1.', '1008.5155-2-9-1': 'Compared to bulk systems, this particle-hole asymmetry is stronger in mesoscopic and quantum circuits with spatial dimension less than the temperature length [MATH] and the voltage length [MATH].', '1008.5155-2-9-2': 'In our Coulomb drag circuit in the high density negative drag regime, we estimate [MATH] at 0.33 K and [MATH], which are of the same order of magnitude as our quantum wire dimensions.', '1008.5155-2-9-3': 'In the linear drag regime, as is the case in this work, we expect a positive drag signal for monotonically increasing transmission probabilities, i.e. for monotonically increasing wire conductances.', '1008.5155-2-9-4': 'On the other hand, negative Coulomb drag might occur in the case where the conductance in the drag wire is non-monotonic.', '1008.5155-2-9-5': 'Indeed, such a non-monotonic behavior in the drag wire conductance is observed concomitant with the re-entrant negative drag (see inset of Fig. 3).', '1008.5155-2-9-6': 'We speculate that this observation might be responsible (at least in part) for the negative drag observed in the high-density regime.', '1008.5155-2-10-0': 'To conclude, we have presented the first drag measurements between vertically-coupled quantum wires with in situ control over the subbands occupation of each wire.', '1008.5155-2-10-1': 'These stacked wires form quantum circuits with their own current path and gating.', '1008.5155-2-10-2': 'Our data shows a clear correlation observed between the low density negative drag and the depletion of the drag wire.', '1008.5155-2-10-3': 'While in this regime localization might explain the negative drag, the same heuristic explanation cannot explain the re-entrance of the negative drag signal observed at higher electronic density where the drag wire is conducting.', '1008.5155-2-10-4': 'We hope that this observation will motivate further theoretical work aimed at understanding the exact role played by electron interactions and electron-hole asymmetry in 1D-1D coupled electronic systems.', '1008.5155-2-11-0': 'We acknowledge the outstanding technical assistance of Denise Tibbetts and James Hedberg.', '1008.5155-2-11-1': 'We thank Aashish Clerk for inspiring discussions.', '1008.5155-2-11-2': 'This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. DOE, Office of Basic Energy Sciences user facility.', '1008.5155-2-11-3': "Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000."} | [['1008.5155-1-4-0', '1008.5155-2-4-0'], 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['1008.5155-1-6-8', '1008.5155-2-6-8'], ['1008.5155-1-6-9', '1008.5155-2-6-9']] | [['1008.5155-2-5-1', '1008.5155-3-6-0']] | [] | [['1008.5155-2-5-2', '1008.5155-3-6-1'], ['1008.5155-2-5-7', '1008.5155-3-6-7']] | [['1008.5155-2-0-3', '1008.5155-3-8-3'], ['1008.5155-2-1-2', '1008.5155-3-7-2'], ['1008.5155-2-1-3', '1008.5155-3-7-2'], ['1008.5155-2-4-1', '1008.5155-3-13-0'], ['1008.5155-2-8-6', '1008.5155-3-8-5']] | ['1008.5155-3-11-0', '1008.5155-3-12-0', '1008.5155-3-14-0', '1008.5155-3-16-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1008.5155 | {'1008.5155-3-0-0': 'Electron interactions in and between wires become increasingly complex and important as circuits are scaled to nanometre sizes, or employ reduced-dimensional conductors [CITATION] like carbon nanotubes [CITATION], nanowires [CITATION] and gated high mobility 2D electron systems [CITATION].', '1008.5155-3-0-1': 'This is because the screening of the long-range Coulomb potential of individual carriers is weakened in these systems, which can lead to phenomenon such as Coulomb drag: a current in one wire induces a voltage in a second wire through Coulomb interactions alone.', '1008.5155-3-0-2': 'Previous experiments have observed electron drag in wires separated by a soft electrostatic barrier [MATH] 80 nm [CITATION].', '1008.5155-3-0-3': 'Here, we measure both positive and negative drag between adjacent vertical quantum wires that are separated by [MATH] 15 nm and have independent contacts, which allows their electron densities to be tuned independently.', '1008.5155-3-0-4': 'We map out the drag signal versus the number of electron subbands occupied in each wire, and interpret the results in terms of momentum-transfer and charge-fluctuation induced transport models.', '1008.5155-3-0-5': 'For wires of significantly different subband occupancies, the positive drag effect can be as large as [MATH].', '1008.5155-3-1-0': 'Our report addresses the fundamental issues as to what one might expect when coupling quantum circuits in close proximity at the nanoscale.', '1008.5155-3-1-1': 'As the transport channel size is reduced towards the one-dimensional limit, charge flow across the channel becomes increasingly dominated by quantum processes.', '1008.5155-3-1-2': 'Due to the long-range nature of the Coulomb potential, coupling two quantum circuits in close proximity (separated by a hard barrier of width [MATH]) may have profound effects on the current flow and on the equilibrium charge distribution in one wire when current is driven in another wire.', '1008.5155-3-1-3': 'First, when [MATH] is only a few nanometres, tunneling may occur between the two circuits and induce a current.', '1008.5155-3-1-4': 'This tunneling current is strongly suppressed with increasing [MATH].', '1008.5155-3-1-5': 'Even after tunneling becomes negligible, a non-zero potential across one circuit may appear when current flows in the second circuit as a result of Coulomb interactions.', '1008.5155-3-1-6': "This resulting 'drag signal' depends critically on the inter-circuit separation, the electronic wire density, and electron-electron interactions in the wire.", '1008.5155-3-1-7': 'We have fabricated a device that allows for a large degree of electrical control and tunability between two one-dimensional (1D) quantum circuits, thereby providing us with the platform to study in detail the Coulomb drag signal emanating between two quantum wires separated by only a few tens of nanometre.', '1008.5155-3-2-0': 'The ability to independently control the density of each component in a quantum circuit is an important asset needed to thoroughly characterize the interactions between closely spaced low-dimensional structures.', '1008.5155-3-2-1': 'Designs allowing such independent characterization have successfully been implemented in electron-electron [CITATION], hole-hole [CITATION] and electron-hole [CITATION] bilayer systems and have led to a great understanding of the interaction mechanisms in two dimensions.', '1008.5155-3-2-2': 'Several experiments have been realized in strongly coupled one-dimensional systems consisting of carbon nanotubes [CITATION], nanowires [CITATION] or quantum wires networks coupled vertically [CITATION] or laterally [CITATION].', '1008.5155-3-2-3': 'However, with the exception of laterally coupled quantum wires, none of these one-dimensional networks have been realized with independent electrical contacts and independently tunable density allowing for the measurement of each system component.', '1008.5155-3-2-4': 'This is in contrast to our vertically-coupled electrical design where each wire has its own ohmic contacts and capacitively coupled gates.', '1008.5155-3-3-0': 'There are two distinct approaches that can be taken when designing quantum wires coupled by proximity.', '1008.5155-3-3-1': 'The first approach is to couple the wires laterally using an electrostatic gate to separate both circuits.', '1008.5155-3-3-2': 'Such design allows fabrication of a quantum structures with independent contacts and tunable density, as first demonstrated by Debray et al. [CITATION], and subsequently by Yamamoto et al.[CITATION] The depth of 2D systems in GaAs heterostructures (typically 80 nm or greater) and the fringing fields of surface defined gates impose that the effective barrier between lateral one-dimensional circuits are soft and are no less than 80 nm.', '1008.5155-3-3-3': 'Thus, in order to construct coupled circuits in the 10 nm range, one must couple the circuits vertically.', '1008.5155-3-3-4': 'In this design, a hard barrier is introduced during the material growth process, allowing coupling electrical circuits over distances of only a few nanometres.', '1008.5155-3-3-5': 'The price to pay in this approach is the complex fabrication process [CITATION] required for defining quantum wires with independent electrical contacts, as sketched in Fig. 1a-d.', '1008.5155-3-4-0': 'Fig. 2a shows a schematic of the interacting region of such a device.', '1008.5155-3-4-1': 'This design was modified from previous work [CITATION] so as to enable the measurement of both wires individually, simultaneously and independently.', '1008.5155-3-4-2': 'Applying a suitable voltage on the gates selectively depletes the two-dimensional electron gases (2DEGs) such that two independently contacted quantum wires are created, as shown in Fig. 2b and Fig. 2d.', '1008.5155-3-4-3': 'In this regime, only 1D regions are vertically coupled and parasitic 2D coupling is minimized.', '1008.5155-3-4-4': 'A scanning electron microscope picture of the device is shown in Fig. 2c where [MATH] long and [MATH] wide wires are observed.', '1008.5155-3-4-5': 'It is important for the alignment between the upper and the lower gates to be lesser than, or equal to 30 nm in the direction perpendicular to the wires to ensure a sub 50 nm effective center-to-center distance between the wires.', '1008.5155-3-4-6': 'For the device presented here, this alignment error was less than 25 nm, leading to an effective center-to-center distance between the wires bounded between 33 nm and 41 nm.', '1008.5155-3-4-7': 'Finally, the accuracy of the alignment in the direction parallel to the wires is not as crucial as it only affects the 1D-1D interacting length, which is determined to be 2.8 [MATH] long in this device.', '1008.5155-3-5-0': 'Fig. 3 shows the evolution of the conductance at a temperature of [MATH] K in each quantum wire as a function of gate voltage.', '1008.5155-3-5-1': 'Fig. 3a was taken at a fixed UPL = -0.23 V where both wires have almost identical 1D subband occupancies for a given LPL voltage.', '1008.5155-3-5-2': 'For a fixed UPL = -0.34 V, the wires subband occupancies differ greatly, as depicted in Fig. 3b.', '1008.5155-3-5-3': 'These measurements demonstrate that our design allows control of the number of occupied subbands in each circuit.', '1008.5155-3-6-0': 'In the ballistic regime, electron transmission is unhindered and [MATH] in the quantum transport conductance [MATH], while an increase in scattering along the wire causes [MATH] in the non-ballistic regime.', '1008.5155-3-6-1': 'The wires presented here are in the non-ballistic regime and the spacing between the conductance plateau features is less than [MATH].', '1008.5155-3-6-2': 'Correcting for the contact resistance by subtracting a series resistance to the quantum wires yields an even conductance spacing between the plateau-like features, albeit one smaller than [MATH], as shown in Fig. 3a and 3b.', '1008.5155-3-6-3': 'A 1.25 k[MATH] (5.00 k[MATH]) contact resistance was subtracted from the lower (upper) wire conductance.', '1008.5155-3-6-4': 'This higher value of the contact resistance is due to partial depletion of the upper 2DEG when the gates are biased for the drag measurement (see Methods).', '1008.5155-3-6-5': 'An even conductance spacing of plateau-like features at values lower than [MATH] in quasi one-dimensional structures was previously observed [CITATION] and has been found not to affect the 1D nature of the quantum wires.', '1008.5155-3-6-6': 'We also show in Fig 3c the derivative of the conductance of the lower wire as a function of LPL voltage.', '1008.5155-3-6-7': 'Plateau-like features are observed when the derivative approaches zero, appearing as black and blue stripes in this mapping.', '1008.5155-3-6-8': 'This tracking of the plateau-like features, combined with their even conductance spacing, strongly supports the existence of well-defined one-dimensional subbands.', '1008.5155-3-7-0': 'Coupling two independent electrical circuits by proximity may lead to signals in one circuit whose origin emanates entirely from the neighboring circuit such as Coulomb drag.', '1008.5155-3-7-1': 'To measure this drag effect, a current [MATH] is set in one of the (drive) circuits.', '1008.5155-3-7-2': 'Under the condition of no current flow, a voltage [MATH] develops across the second (drag) circuit, defining a drag resistance [MATH] that is a direct probe of electron-electron interactions.', '1008.5155-3-7-3': 'Coulomb drag is distinct from rectification and ratchets mechanisms where a voltage develops due to a neighboring current flow whose I-V characteristics are highly non-linear (with respect to [MATH]) and non-symmetric with respect to probe inversion.', '1008.5155-3-7-4': 'In contrast, Coulomb drag is an equilibrium phenomena that is linear, invertible with respect to probe symmetry, mutual, and present in ballistic and non-ballistic circuits.', '1008.5155-3-8-0': 'The drag resistance measured in our quantum circuit is shown in Fig. 4a, along with the conductance of each wire.', '1008.5155-3-8-1': 'Coulomb drag peaks are observed concomitant with the opening of 1D subbands in either wire (see dotted lines in Fig. 4a).', '1008.5155-3-8-2': 'Momentum matching between both wires can explain the presence of the positive drag peaks when the wires have similar subband occupancies [CITATION], but an enhancement of the electron-hole asymmetry as 1D channels open in the quantum wires [CITATION], appears more probable to explain the presence of positive peaks when the wires have different subband occupancies.', '1008.5155-3-8-3': 'In addition, negative Coulomb drag is observed in two clearly distinct regimes : one at low electronic density when the drag wire is close to or beyond depletion, and one at higher electronic density when [MATH].', '1008.5155-3-8-4': 'Negative Coulomb drag has been previously observed at low density (i.e. for [MATH] in both wires)[CITATION] and attributed to one-dimensional Wigner crystallization.', '1008.5155-3-8-5': 'While Wigner crystallization could explain the low-density negative drag reported in this Letter, it cannot explain the high-density negative drag.', '1008.5155-3-8-6': 'Negative Coulomb drag has been predicted to occur following a charge-fluctuation induced Coulomb drag model in asymmetric mesoscopic circuits [CITATION], but more work is required to assess its consistency over the whole space-phase of 1D Coulomb drag.', '1008.5155-3-9-0': 'We show in Fig. 4b the temperature dependence between [MATH] and [MATH] in both the high-density negative drag and the positive drag regimes.', '1008.5155-3-9-1': 'In either case, the drag resistance shows no saturation down to the lowest temperature probed in this experiment, confirming the thermal equilibrium of the electrons in the quantum wires with the apparatus.', '1008.5155-3-9-2': 'The re-entrant negative drag signal disappears at [MATH], which is consistent with the system leaving the mesoscopic regime as the temperature length [MATH] is lowered from [MATH] at 0.33 K to [MATH] at 1.2 K, and becomes shorter than the system size.', '1008.5155-3-9-3': 'Fig. 4c and 4d show the linearity of the drag voltage with drive current (for small enough drive voltages, i.e. empirically for [MATH]) and the probe symmetry of the drag signal, confirming that the signals observed are consistent with Coulomb drag.', '1008.5155-3-9-4': 'For wires with a similar subband occupancy presented in Fig. 4a, the drag effect is [MATH] of the drive voltage value.', '1008.5155-3-9-5': 'However, in wires with significantly different subband occupancy, this effect can be as large as [MATH].', '1008.5155-3-10-0': 'Coulomb drag between nanoelectronic circuits will become increasingly important as nano-circuitry becomes coupled by proximity.', '1008.5155-3-10-1': 'As nanostructure cross-sections become comparable to the 3D screening length, the effective 1D screening length is expected to become large[CITATION].', '1008.5155-3-10-2': 'Using typical doping values for silicon nanowires [CITATION], the bulk Thomas-Fermi screening length [MATH], where [MATH] is the electron change, [MATH] is the electron density, [MATH] is the silicon dielectric constant and [MATH] is the Fermi energy, is estimated to be [MATH] 4 nm.', '1008.5155-3-10-3': 'Therefore, as nanowire diameter approaches this length scale, the previously screened Coulomb interactions will induce Coulomb drag signals in circuit elements located in close proximity.', '1008.5155-3-10-4': 'This drag effect is found to be as large as [MATH] of the drive voltage value, or up to [MATH]V, for the structures presented in this Letter, which is far from negligible.', '1008.5155-3-10-5': 'An understanding of one-dimensional Coulomb drag phenomenon in model systems such as quantum wires will ultimately prove to be an essential asset to understand the coupling between independently addressed conductors at the nanoscale, for example coupled nanowires for nanoprocessing [CITATION].', '1008.5155-3-11-0': 'Methods', '1008.5155-3-12-0': 'Device fabrication', '1008.5155-3-13-0': 'The wires are patterned on a n-doped GaAs/AlGaAs electron bilayer heterostructure where two 18 nm wide quantum wells are separated by a 15 nm wide Al[MATH]Ga[MATH]As barrier.', '1008.5155-3-13-1': 'After a mesa-structure is wet-etched using phosphoric acid into the double quantum well heterostrocture, Ge-Au-Ni-Au ohmic contacts are deposited on the structure (Fig. 1a).', '1008.5155-3-13-2': 'Following an annealing at [MATH] for 60 seconds, a set of two Ti-Au split-gates, consisting of a T-shaped pinch-off gate and of a plunger gate, is defined on the surface of the heterostructure.', '1008.5155-3-13-3': 'The off-mesa patterning is defined using photo-lithography while electron-beam lithography is used to pattern the gates on-mesa (Fig 1b).', '1008.5155-3-13-4': 'The thickness of the gates is 160 nm off-mesa and 60 nm on-mesa.', '1008.5155-3-13-5': 'A set of four alignment marks is also patterned simultaneously to the patterning of the e-beam lithography defined top gates.', '1008.5155-3-13-6': 'These marks are used to align the lower gates to the upper ones.', '1008.5155-3-13-7': 'Once the upper side processing is completed, bare GaAs is glued on top of the substrate and the sample is flipped, mechanically lapped and chemically etched until the lower 2DEG is only [MATH] 150 nm away from the lower surface (which is now on top of the device, as show in Fig. 1c), following an EBASE technique [CITATION].', '1008.5155-3-13-8': 'Two stop-etch layers are incorporated in the original heterostructure : a larger AlGaAs stop-etch layer and a thinner GaAs stop-etch layer.', '1008.5155-3-13-9': 'The AlGaAs stop-etch layer purpose is to flatten out the unevenness arising from the lapping process during the subsequent citric wet-etching.', '1008.5155-3-13-10': 'Indeed, the citric acid etch rate is greatly reduced in AlGaAs compared to GaAs, allowing to smooth the surface of the device after the mechanical lapping.', '1008.5155-3-13-11': 'After the citric etch, the remainder of the AlGaAs stop-etch layer is etched using hydrofluoric acid, leaving only the thin GaAs stop-etc layer, which is grown to prevent over-etching during the hydrofluoric etch.', '1008.5155-3-13-12': 'To ensure that no off-mesa leakage occurs between the upper and the lower gates, a thin 60 nm layer of Al[MATH]O[MATH] is deposited on the top of the device using atomic layer deposition.', '1008.5155-3-13-13': 'Using phosphoric acid, vias are then etched through the surface to enable electrical connection to the ohmic contacts and to the upper split gates on the buried surface of the device.', '1008.5155-3-13-14': 'Finally, using a combination of photolithography and electron-beam lithography, another set of two Ti-Au spit gates is defined on the lower-side of the sample and aligned with the upper gates using the previously deposited alignment marks buried underneath the surface.', '1008.5155-3-13-15': 'It is possible to observe these marks using a SEM or an e-beam lithography tool with an accelerated voltage greater than or equal to 30 keV, and therefore precisely align the lower and the upper gates.', '1008.5155-3-13-16': 'The end result is presented in Fig. 1d.', '1008.5155-3-14-0': 'Device operation', '1008.5155-3-15-0': 'The pinch-off gates are first adjusted such that they principally deplete the 2DEG closest to them.', '1008.5155-3-15-1': 'While each pinch-off gate can deplete both 2DEGs for sufficiently large applied negative voltage, a 0.3 V (0.15 V) wide plateau (where the conductance across the device is roughly constant) is observed when sweeping the upper (lower) pinch-off gate.', '1008.5155-3-15-2': 'On this plateau, the 2DEG closest to the gate is fully depleted whereas the other one is only partially depleted.', '1008.5155-3-15-3': 'For the device presented in this Letter, the lower gates create a larger partial depletion than the upper gates, causing the contact resistance to the upper wire to be larger than the contact resistance of the lower wire.', '1008.5155-3-15-4': 'The positioning within the plateaus is adjusted such that the tunneling resistance between both layers is larger than 25 [MATH].', '1008.5155-3-15-5': 'In such experimental configuration, there is minimal tunneling between the upper and the lower layer contacts.', '1008.5155-3-15-6': 'Indeed, the depletion mechanism of the pinch-off gates results in a coupling of each side of the device to a single layer, allowing simultaneous and independent measurement of both layers, unlike in the device presented by Bielejec et al.[CITATION] Subsequently, adjusting both the lower and the upper plunger gate voltages allows for the independent tuning of the subband occupancy in each independent wire.', '1008.5155-3-16-0': 'Device characterization', '1008.5155-3-17-0': 'Measurements performed on the sample post processing with the split-gates grounded yielded an electronic density of 1.1 (1.4) [MATH] cm[MATH] for the upper (lower) 2DEG, and a combined mobility of [MATH] cm[MATH] / V[MATH] s. Transport measurements on individual quantum wires were performed in a [MATH] refrigerator at a temperature of 330 mK using a constant 50[MATH]V excitation at 9 Hz in the lower wire and at 13 Hz in the upper wire in a two-contact configuration.', '1008.5155-3-17-1': 'The Coulomb drag measurements were performed in a constant current mode where 4.5 nA at 9 Hz was sent through the drive wire.', '1008.5155-3-17-2': 'In this configuration, the out-of-phase current was always much smaller than the in-phase current.'} | null | null | null | null |
1701.02957 | {'1701.02957-1-0-0': 'We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel.', '1701.02957-1-1-0': 'Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial.', '1701.02957-1-1-1': 'The established pre-factor is essentially optimal because it matches the best known random coding upper bound in the classical case.', '1701.02957-1-1-2': 'Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.', '1701.02957-1-2-0': '# Introduction', '1701.02957-1-3-0': 'The probability of decoding error is one of the fundamental criteria for evaluating the performance of a communication system.', '1701.02957-1-3-1': "In Shannon's seminal work [CITATION], he pioneered the study of the noisy coding theorem, which states that the error probability can be made arbitrarily small as the coding blocklength grows when the coding rate [MATH] is below the channel capacity [MATH].", '1701.02957-1-3-2': 'Later, Shannon [CITATION] made a further step in exploring the exponential dependency of the optimal error probability [MATH] on the blocklength [MATH] and rate [MATH], and defined the reliability function as follows: given a fixed coding rate [MATH],', '1701.02957-1-4-0': '[MATH].', '1701.02957-1-5-0': 'The quantity [MATH] then provides a measure of how rapidly the error probability approaches zero with an increase in blocklength.', '1701.02957-1-5-1': 'This asymptotic characterization of the optimal error probability under a fixed rate is hence called the error exponent analysis.', '1701.02957-1-5-2': 'For a classical channel, the upper bounds of the optimal error can be established using a random coding argument [CITATION].', '1701.02957-1-5-3': 'On the other hand, the lower bound was first developed by Shannon, Gallager, and Berlekamp [CITATION] and was called the sphere-packing bound.', '1701.02957-1-5-4': 'Alternative approaches by Haroutunian [CITATION] and Blahut [CITATION] were subsequently proposed.', '1701.02957-1-6-0': 'In recent years, much attention has been paid to the finite blocklength regime [CITATION].', '1701.02957-1-6-1': 'Altug and Wagner employed strong large deviation techniques [CITATION] to prove a sphere-packing bound with a finite blocklength [MATH].', '1701.02957-1-6-2': 'Moreover, the pre-factor of the bound was significantly refined from the order of the subexponential [MATH] [CITATION] to the polynomial [CITATION].', '1701.02957-1-6-3': 'This refinement is substantial especially at rates near capacity, where the error-exponent function is close to zero; hence, the pre-factor dominants the bound [CITATION].', '1701.02957-1-7-0': 'Error exponent analysis in classical-quantum (c-q) channels is much more difficult because of the noncommutative nature of quantum mechanics.', '1701.02957-1-7-1': 'Burnashev and Holevo [CITATION] investigated reliability functions in c-q channels and proved the random coding upper bound for pure-state channels.', '1701.02957-1-7-2': "Winter [CITATION] adopted Haroutunian's method to derive a sphere-packing bound for c-q channels in the form of relative entropy functions [CITATION].", '1701.02957-1-7-3': "Dalai [CITATION] employed Shannon-Gallager-Berlekamp's approach to establish a sphere-packing bound with Gallager's expression [CITATION].", '1701.02957-1-7-4': 'It was later pointed out that these two sphere-packing exponents are not equal for general c-q channels [CITATION].', '1701.02957-1-7-5': 'In this work, we initiate the study of the refined sphere-packing bound in the quantum scenario.', '1701.02957-1-8-0': 'In particular, we consider a "symmetric c-q channel" (see Section [REF] for a detailed definition), which is an important class of covariant channels (e.g. [CITATION]), and establish a sphere-packing bound with the pre-factor improved from the order of the subexponential in Dalai\'s result [CITATION] to the polynomial.', '1701.02957-1-8-1': "Our result recovers Altug and Wagner's work [CITATION] for classical symmetric channels including the binary symmetric channel and binary erasure channel.", '1701.02957-1-8-2': 'Furthermore, the proved pre-factor matches that of the best known random coding upper bound [CITATION] in the classical case.', '1701.02957-1-8-3': 'Hence, our result yields the exact asymptotics for the sphere-packing bound in symmetric c-q channels.', '1701.02957-1-8-4': 'The main ingredients in our proof are a tight concentration inequality from Bahadur and Ranga Rao [CITATION], [CITATION] (see Appendix [REF]) and the major properties of the sphere-packing exponent [CITATION].', '1701.02957-1-8-5': 'We remark that the result obtained in this paper might enable analysis in the medium error probability regime of a classical-quantum channel [CITATION].', '1701.02957-1-8-6': 'We leave the case for general c-q channels as future work [CITATION].', '1701.02957-1-9-0': 'This paper is organized as follows.', '1701.02957-1-9-1': 'We introduce the necessary notation and state our main result in Section [REF].', '1701.02957-1-9-2': 'Section [REF] includes the crucial properties of the error-exponent function.', '1701.02957-1-9-3': 'We provide the proof of the main result in Section [REF].', '1701.02957-1-9-4': 'Section [REF] concludes this paper.', '1701.02957-1-10-0': '# Notation and Main Result', '1701.02957-1-11-0': '## Notation', '1701.02957-1-12-0': 'Throughout this paper, we consider a finite-dimensional Hilbert space [MATH].', '1701.02957-1-13-0': 'The set of density operators (i.e. positive semi-definite operators with unit trace) [MATH] are defined as [MATH].', '1701.02957-1-14-0': 'For [MATH], we write [MATH] if [MATH], where [MATH] denotes the support of [MATH].', '1701.02957-1-14-1': 'The identity operator on [MATH] is denoted by [MATH].', '1701.02957-1-14-2': 'When there is no possibility of confusion, we skip the subscript [MATH].', '1701.02957-1-14-3': 'We use [MATH] as the trace function.', '1701.02957-1-15-0': 'Let [MATH], [MATH], and [MATH] denote the set of integers, real numbers, and positive real numbers" respectively.', '1701.02957-1-15-1': 'Define [MATH] for [MATH].', '1701.02957-1-16-0': 'Given a pair of positive semi-definite operators [MATH], we define the (quantum) relative entropy as', '1701.02957-1-17-0': '[MATH],', '1701.02957-1-18-0': 'when [MATH], and [MATH] otherwise.', '1701.02957-1-19-0': 'For every [MATH], we define the (Petz) quantum Renyi divergences', '1701.02957-1-20-0': '[MATH].', '1701.02957-1-21-0': 'For [MATH], [MATH].', '1701.02957-1-22-0': 'Let [MATH] be a finite alphabet, and let [MATH] be the set of probability distributions on [MATH].', '1701.02957-1-22-1': 'In particular, we denote by [MATH] the uniform distribution on [MATH].', '1701.02957-1-22-2': 'A classical-quantum (c-q) channel [MATH] maps elements of the finite set [MATH] to the density operators in [MATH], i.e., [MATH].', '1701.02957-1-23-0': 'Let [MATH] be a finite alphabetical set with size [MATH].', '1701.02957-1-23-1': 'An ([MATH]-block) encoder is a map [MATH] that encodes each message [MATH] to a codeword [MATH].', '1701.02957-1-23-2': 'The codeword [MATH] is then mapped to a state', '1701.02957-1-24-0': '[MATH].', '1701.02957-1-25-0': 'The decoder is described by a positive operator-valued measurement (POVM) [MATH] on [MATH], where [MATH] and [MATH].', '1701.02957-1-25-1': 'The pair [MATH] is called a code with rate [MATH].', '1701.02957-1-25-2': 'The error probability of sending a message [MATH] with the code [MATH] is [MATH].', '1701.02957-1-25-3': 'We use [MATH] and [MATH] to denote the maximal error probability and the average error probability, respectively.', '1701.02957-1-26-0': 'Given a sequence [MATH], we denote by', '1701.02957-1-27-0': '[MATH]', '1701.02957-1-28-0': 'the empirical distribution of [MATH].', '1701.02957-1-29-0': 'Throughout this paper, we consider a symmetric c-q channel defined as [EQUATION] where [MATH] is an arbitrary density operator, and [MATH]', '1701.02957-1-30-0': 'satisfies [MATH].', '1701.02957-1-30-1': 'We define the following conditional entropic quantities for the channel [MATH] with [MATH]:', '1701.02957-1-31-0': '[MATH].', '1701.02957-1-32-0': 'The mutual information of the c-q channel [MATH] with prior distribution [MATH] is defined as', '1701.02957-1-33-0': '[MATH],', '1701.02957-1-34-0': 'where [MATH], [MATH].', '1701.02957-1-35-0': 'The (classical) capacity of the channel [MATH] is denoted by', '1701.02957-1-36-0': '[MATH].', '1701.02957-1-37-0': 'Let [EQUATION] where we denote by', '1701.02957-1-38-0': '[MATH]', '1701.02957-1-39-0': 'an auxiliary function [CITATION].', '1701.02957-1-39-1': 'The sphere-packing exponent is defined by [EQUATION] where the last equality follows from [CITATION].', '1701.02957-1-39-2': 'Further, we define a rate [CITATION], [CITATION]: [EQUATION]', '1701.02957-1-39-3': 'It follows that [MATH] for any [MATH] (see also [CITATION] and [CITATION]).', '1701.02957-1-40-0': 'Consider a binary hypothesis whose null and alternative hypotheses are [MATH] and [MATH], respectively.', '1701.02957-1-40-1': 'The type-I error and type-II error of the hypothesis testing, for an operator [MATH], are defined as', '1701.02957-1-41-0': '[MATH], and [MATH].', '1701.02957-1-42-0': 'There is a trade-off between these two errors.', '1701.02957-1-42-1': 'Thus, we can define the minimum type-I error, when the type-II error is below [MATH], as [EQUATION]', '1701.02957-1-43-0': '## Main Result', '1701.02957-1-44-0': 'Let us now consider any symmetric c-q channel with capacity [MATH].', '1701.02957-1-45-0': '[Exact Sphere-Packing Bound] For any rate [MATH], there exist an [MATH] such that for all codes [MATH] of length [MATH], we have [EQUATION] where [MATH].', '1701.02957-1-46-0': '# Properties of the Sphere-Packing Exponent', '1701.02957-1-47-0': '[Optimal Input Distribution] For any [MATH], the distribution [MATH] is a maximizer of [MATH] and [MATH].', '1701.02957-1-48-0': 'We first prove that [MATH] attains [MATH].', '1701.02957-1-48-1': 'From Eq. [REF], it is not hard to verify that [MATH] for all [MATH].', '1701.02957-1-48-2': 'Hence, it follows that [EQUATION] for all [MATH].', '1701.02957-1-48-3': 'The above equation shows that', '1701.02957-1-49-0': 'the distribution [MATH] that maximizes [MATH], [MATH] [CITATION].', '1701.02957-1-50-0': 'Then we have [EQUATION]', '1701.02957-1-50-1': "Further, Jensen's inequality implies that [MATH], which completes the proof.", '1701.02957-1-51-0': '[Saddle-Point Property] Consider any [MATH] and [MATH].', '1701.02957-1-52-0': 'Let', '1701.02957-1-53-0': '[MATH].', '1701.02957-1-53-1': 'We define [EQUATION] on [MATH], and let', '1701.02957-1-54-0': '[MATH].', '1701.02957-1-55-0': 'The following holds', '1701.02957-1-56-0': 'The proof is provided in Appendix [REF].', '1701.02957-1-57-0': '[Representation] For any [MATH], let [MATH] be the saddle-point of [MATH].', '1701.02957-1-57-1': 'It follows that [EQUATION]', '1701.02957-1-58-0': 'Since Lemma [REF] implies that [MATH] attains [MATH], one observes from the definition of [MATH] that all the quantities [MATH], [MATH] are equal.', '1701.02957-1-58-1': 'By item (iii) of Lemma [REF], we obtain a representation of [MATH] in Eq. [REF].', '1701.02957-1-58-2': 'The optimal [MATH] follows from [CITATION].', '1701.02957-1-59-0': '[Invariance] For any [MATH], we have [EQUATION] where [MATH] and [MATH] are defined in Eq. [REF].', '1701.02957-1-60-0': 'Following the argument in Lemma [REF] and recalling Eq. [REF] in Lemma [REF], one can verify that [MATH] for all [MATH].', '1701.02957-1-60-1': 'Further, we obtain [MATH] for [MATH] from the result in [CITATION].', '1701.02957-1-61-0': '# Proof of the Main Result', '1701.02957-1-62-0': 'For rates in the range [MATH], we have [MATH].', '1701.02957-1-62-1': 'The bound in Eq. [REF] obviously holds.', '1701.02957-1-62-2': 'Hence, we consider the case of [MATH] and fix the rate throughout the proof.', '1701.02957-1-63-0': 'We first pose the channel coding problem into a binary hypothesis testing through Lemma [REF], which originates from Blahut [CITATION] for the classical case.', '1701.02957-1-63-1': '[Hypothesis Testing Reduction] For any code [MATH] with message size [MATH], there exists an [MATH] such that [EQUATION]', '1701.02957-1-64-0': 'The proof is provided in Appendix [REF].', '1701.02957-1-65-0': 'Let us now commence with the proof of Theorem [REF].', '1701.02957-1-65-1': 'Fix arbitrary [MATH].', '1701.02957-1-65-2': 'Let [MATH] and [MATH].', '1701.02957-1-65-3': 'The choice of the rate back-off term [MATH] will become evident later.', '1701.02957-1-65-4': 'Choose [MATH] such that [MATH].', '1701.02957-1-65-5': 'Let [MATH] be defined in Eq. [REF], and from Lemma [REF], we have [EQUATION]', '1701.02957-1-65-6': 'In the following, we provide a lower bound for the type-I error [MATH].', '1701.02957-1-65-7': 'Let [MATH] and [MATH], where [MATH] are Nussbaum-Szkola distributions [CITATION] of [MATH] for every [MATH].', '1701.02957-1-65-8': 'Since [MATH], for all [MATH], we shorthand', '1701.02957-1-66-0': '[MATH], where [MATH] is the empirical distribution of [MATH].', '1701.02957-1-67-0': 'Moreover, item (iii) in Lemma [REF] implies that the state [MATH] dominates all the channel outputs: [MATH], for all [MATH], Hence, we have [MATH].', '1701.02957-1-67-1': 'Subsequently, for every [MATH], we let [MATH], for all [MATH].', '1701.02957-1-68-0': "We apply Nagaoka's argument [CITATION] by choosing [MATH] to yield, for any [MATH], [EQUATION] where", '1701.02957-1-69-0': '[MATH], [MATH],', '1701.02957-1-70-0': 'and', '1701.02957-1-71-0': '[MATH].', '1701.02957-1-72-0': "Next, we employ Bahadur-Ranga Rao's concentration inequality, Theorem [REF] in Appendix [REF], to further lower bound [MATH] and [MATH].", '1701.02957-1-72-1': 'Before proceeding, we need to introduce some notation.', '1701.02957-1-72-2': 'We define the tilted distributions, for every [MATH], [MATH], and [MATH] by [EQUATION]', '1701.02957-1-72-3': 'Let [EQUATION]', '1701.02957-1-72-4': 'Since [MATH] and [MATH] are mutually absolutely continuous, the maps [MATH], [MATH] are differentiable for all [MATH].', '1701.02957-1-72-5': 'One can immediately verify the following partial derivatives with respect to [MATH]: [EQUATION]', '1701.02957-1-72-6': 'With [MATH] in Eq. [REF], we can define [EQUATION] where [MATH] in Eq. ([REF]) is the Fenchel-Legendre transform of [MATH].', '1701.02957-1-72-7': "The quantities [MATH] would appear in the lower bounds of [MATH] and [MATH] obtained by Bahadur-Randga Rao's inequality as shown later.", '1701.02957-1-73-0': 'In the following, we relate the Fenchel-Legendre transform [MATH] to the desired error-exponent function [MATH].', '1701.02957-1-74-0': 'Such a relationship is stated in Lemma [REF]; the proof is provided in Appendix [REF].', '1701.02957-1-75-0': 'Under the prevailing assumptions and for all [MATH], the following holds:', '1701.02957-1-76-0': 'Item (iii) in Lemma [REF] shows that the optimizer [MATH] in Eq. [REF] always lies in the compact set [MATH].', '1701.02957-1-76-1': 'Further, Eqs. [REF] and [REF] ensure that', '1701.02957-1-77-0': '[MATH], [MATH], [MATH].', '1701.02957-1-78-0': 'We define the following quantities: [EQUATION] [MATH]; and [MATH].', '1701.02957-1-78-1': 'Note that for every [MATH], [MATH] and [MATH] are continuous functions on [MATH] from the definitions in Eqs. [REF], [REF] (see also [CITATION]).', '1701.02957-1-78-2': 'The maximization and minimization in the above definitions are well-defined and finite.', '1701.02957-1-78-3': 'Moreover, Lemma [REF] guarantees that [MATH] is bounded away from zero.', '1701.02957-1-79-0': '[Positivity] For any [MATH] and [MATH], [MATH], for all [MATH].', '1701.02957-1-80-0': 'Assume [MATH] is zero for some [MATH].', '1701.02957-1-80-1': 'This is equivalent to [EQUATION]', '1701.02957-1-80-2': 'Summing the right-hand side of Eq. [REF] over [MATH] gives', '1701.02957-1-81-0': '[MATH].', '1701.02957-1-82-0': 'Then, Eqs. [REF] and the above equation imply that [EQUATION] where we use the fact that [MATH]; see Eq. [REF]).', '1701.02957-1-83-0': 'However, Lemma [REF] implies that [MATH], which leads to a contradiction.', '1701.02957-1-84-0': 'Now, we are ready to derive the lower bounds to [MATH] and [MATH].', '1701.02957-1-84-1': 'Let [MATH] be sufficiently large such that for all [MATH], [EQUATION]', '1701.02957-1-84-2': "Applying Bahadur-Randga Rao's inequality (Theorem [REF]) to [MATH] with the probability measure [MATH], and [MATH] gives [EQUATION] where", '1701.02957-1-85-0': '[MATH].', '1701.02957-1-86-0': 'Similarly, applying Theorem [REF] to [MATH] with the probability measure [MATH], and [MATH] yields [EQUATION]', '1701.02957-1-86-1': 'Continuing from Eq. ([REF]) and item (i) in Lemma [REF] gives [EQUATION]', '1701.02957-1-86-2': 'Eq. ([REF]) together with item (iii) in Lemma [REF] yields [EQUATION]', '1701.02957-1-86-3': 'Let [MATH] such that [MATH], for all [MATH].', '1701.02957-1-86-4': 'Then Eq. [REF] implies that [MATH].', '1701.02957-1-86-5': 'Thus, we can bound the left-hand side of Eq. ([REF]) from below by [MATH].', '1701.02957-1-86-6': 'For any test [MATH] such that', '1701.02957-1-87-0': '[MATH],', '1701.02957-1-88-0': 'we have [EQUATION] where the last equality follows from Lemma [REF].', '1701.02957-1-89-0': 'Finally, it remains to remove the back-off term [MATH] in Eq. [REF].', '1701.02957-1-90-0': "By Taylor's theorem, we have [EQUATION] for some [MATH] and [MATH].", '1701.02957-1-90-1': 'Further, one can calculate that [EQUATION] where [MATH].', '1701.02957-1-90-2': 'From item (iii) in Lemma [REF], it follows that both [MATH] and [MATH] are both positive and finite for [MATH] and [MATH].', '1701.02957-1-90-3': 'Together with the fact that [MATH], we have [MATH].', '1701.02957-1-91-0': "We apply Taylor's expansion on the function [MATH] again to yield [EQUATION] where the first equality holds for some [MATH], and the last line follows from the definition [MATH].", '1701.02957-1-91-1': 'Finally, by combining Eqs. [REF], [REF], and [REF], we obtain the desired Eq. [REF] for sufficiently large [MATH].', '1701.02957-1-92-0': '# Discussion', '1701.02957-1-93-0': 'In this work, we establish a sphere-packing bound with a refined polynomial pre-factor that coincides with the best classical results [CITATION] to date.', '1701.02957-1-93-1': 'As discussed by Altug and Wagner [CITATION], the pre-factor is correct for binary symmetric channels but slightly worse for binary erasure channels (in the order of [MATH]).', '1701.02957-1-93-2': 'On the other hand, our pre-factor matches the recent result of the random coding upper bound [CITATION], where the pre-factor has been shown to be exact.', '1701.02957-1-93-3': 'Hence, we conjecture that the established result is optimal for general symmetric c-q channels.', '1701.02957-1-94-0': 'This work admits variety of potential extensions.', '1701.02957-1-94-1': 'First, the symmetric c-q channel studied in this paper is a covariant channel with a cyclic group: [EQUATION] where [MATH] and [MATH] are the unitary representations on [MATH] and [MATH] such that [MATH] and [MATH].', '1701.02957-1-94-2': 'It would be interesting to investigate whether the refined sphere-packing bound can be extended to covariant quantum channels [MATH] with arbitrary compact groups.', '1701.02957-1-94-3': 'Second, the random coding bound in the quantum case has been proved only for pure-state channels [CITATION].', '1701.02957-1-94-4': 'It is promising to prove the bound for this class of c-q channels by employing the symmetry property.', '1701.02957-1-94-5': 'Finally, the refinement provides a new possibility for moderate deviation analysis in c-q channels [CITATION], which is left as future work.'} | {'1701.02957-2-0-0': 'We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel.', '1701.02957-2-1-0': 'Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial.', '1701.02957-2-1-1': 'This established pre-factor is essentially optimal because it matches the best known random coding upper bound in the classical case.', '1701.02957-2-1-2': 'Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.', '1701.02957-2-2-0': '# Introduction', '1701.02957-2-3-0': 'The probability of decoding error is one of the fundamental criteria for evaluating the performance of a communication system.', '1701.02957-2-3-1': "In Shannon's seminal work [CITATION], he pioneered the study of the noisy coding theorem, which states that the error probability can be made arbitrarily small as the coding blocklength grows when the coding rate [MATH] is below the channel capacity [MATH].", '1701.02957-2-3-2': 'Later, Shannon [CITATION] made a further step in exploring the exponential dependency of the optimal error probability [MATH] on the blocklength [MATH] and rate [MATH], and defined the reliability function as follows: given a fixed coding rate [MATH],', '1701.02957-2-4-0': '[MATH].', '1701.02957-2-5-0': 'The quantity [MATH] then provides a measure of how rapidly the error probability approaches zero with an increase in blocklength.', '1701.02957-2-5-1': 'This asymptotic characterization of the optimal error probability under a fixed rate is hence called the error exponent analysis.', '1701.02957-2-5-2': 'For a classical channel, the upper bounds of the optimal error can be established using a random coding argument [CITATION].', '1701.02957-2-5-3': 'On the other hand, the lower bound was first developed by Shannon, Gallager, and Berlekamp [CITATION] and was called the sphere-packing bound.', '1701.02957-2-5-4': 'Alternative approaches by Haroutunian [CITATION] and Blahut [CITATION] were subsequently proposed.', '1701.02957-2-6-0': 'In recent years, much attention has been paid to the finite blocklength regime [CITATION].', '1701.02957-2-6-1': 'Altug and Wagner employed strong large deviation techniques [CITATION] to prove a sphere-packing bound with a finite blocklength [MATH].', '1701.02957-2-6-2': 'Moreover, the pre-factor of the bound was significantly refined from the order of the subexponential [MATH] [CITATION] to the polynomial [CITATION].', '1701.02957-2-6-3': 'This refinement is substantial especially at rates near capacity, where the error-exponent function is close to zero; hence, the pre-factor dominants the bound [CITATION].', '1701.02957-2-7-0': 'Error exponent analysis in classical-quantum (c-q) channels is much more difficult because of the noncommutative nature of quantum mechanics.', '1701.02957-2-7-1': 'Burnashev and Holevo [CITATION] investigated reliability functions in c-q channels and proved the random coding upper bound for pure-state channels.', '1701.02957-2-7-2': "Winter [CITATION] adopted Haroutunian's method to derive a sphere-packing bound for c-q channels in the form of relative entropy functions [CITATION].", '1701.02957-2-7-3': "Dalai [CITATION] employed Shannon-Gallager-Berlekamp's approach to establish a sphere-packing bound with Gallager's expression [CITATION].", '1701.02957-2-7-4': 'It was later pointed out that these two sphere-packing exponents are not equal for general c-q channels [CITATION].', '1701.02957-2-7-5': 'In this work, we initiate the study of the refined sphere-packing bound in the quantum scenario.', '1701.02957-2-8-0': 'In particular, we consider a "symmetric c-q channel" (see Section [REF] for a detailed definition), which is an important class of covariant channels (e.g. [CITATION]), and establish a sphere-packing bound with the pre-factor improved from the order of the subexponential in Dalai\'s result [CITATION] to the polynomial.', '1701.02957-2-8-1': "Our result recovers Altug and Wagner's work [CITATION] for classical symmetric channels including the binary symmetric channel and binary erasure channel.", '1701.02957-2-8-2': 'Furthermore, the proved pre-factor matches that of the best known random coding upper bound [CITATION] in the classical case.', '1701.02957-2-8-3': 'Hence, our result yields the exact asymptotics for the sphere-packing bound in symmetric c-q channels.', '1701.02957-2-8-4': 'The main ingredients in our proof are a tight concentration inequality from Bahadur and Ranga Rao [CITATION], [CITATION] (see Appendix [REF]) and the major properties of the sphere-packing exponent [CITATION].', '1701.02957-2-8-5': 'We remark that the result obtained in this paper might enable analysis in the medium error probability regime of a classical-quantum channel [CITATION].', '1701.02957-2-8-6': 'We leave the case for general c-q channels as future work [CITATION].', '1701.02957-2-9-0': 'This paper is organized as follows.', '1701.02957-2-9-1': 'We introduce the necessary notation and state our main result in Section [REF].', '1701.02957-2-9-2': 'Section [REF] includes the crucial properties of the error-exponent function.', '1701.02957-2-9-3': 'We provide the proof of the main result in Section [REF].', '1701.02957-2-9-4': 'Section [REF] concludes this paper.', '1701.02957-2-10-0': '# Notation and Main Result', '1701.02957-2-11-0': '## Notation', '1701.02957-2-12-0': 'Throughout this paper, we consider a finite-dimensional Hilbert space [MATH].', '1701.02957-2-13-0': 'The set of density operators (i.e. positive semi-definite operators with unit trace) on [MATH] are defined as [MATH].', '1701.02957-2-14-0': 'For [MATH], we write [MATH] if [MATH], where [MATH] denotes the support of [MATH].', '1701.02957-2-14-1': 'The identity operator on [MATH] is denoted by [MATH].', '1701.02957-2-14-2': 'When there is no possibility of confusion, we skip the subscript [MATH].', '1701.02957-2-14-3': 'We use [MATH] as the trace function.', '1701.02957-2-15-0': 'Let [MATH], [MATH], and [MATH] denote the set of integers, real numbers, and positive real numbers" respectively.', '1701.02957-2-15-1': 'Define [MATH] for [MATH].', '1701.02957-2-16-0': 'Given a pair of positive semi-definite operators [MATH], we define the (quantum) relative entropy as', '1701.02957-2-17-0': '[MATH],', '1701.02957-2-18-0': 'when [MATH], and [MATH] otherwise.', '1701.02957-2-19-0': 'For every [MATH], we define the (Petz) quantum Renyi divergences', '1701.02957-2-20-0': '[MATH].', '1701.02957-2-21-0': 'For [MATH], [MATH].', '1701.02957-2-22-0': 'Let [MATH] be a finite alphabet, and let [MATH] be the set of probability distributions on [MATH].', '1701.02957-2-22-1': 'In particular, we denote by [MATH] the uniform distribution on [MATH].', '1701.02957-2-22-2': 'A classical-quantum (c-q) channel [MATH] maps elements of the finite set [MATH] to the density operators in [MATH], i.e., [MATH].', '1701.02957-2-23-0': 'Let [MATH] be a finite alphabetical set with size [MATH].', '1701.02957-2-23-1': 'An ([MATH]-block) encoder is a map [MATH] that encodes each message [MATH] to a codeword [MATH].', '1701.02957-2-23-2': 'The codeword [MATH] is then mapped to a state', '1701.02957-2-24-0': '[MATH].', '1701.02957-2-25-0': 'The decoder is described by a positive operator-valued measurement (POVM) [MATH] on [MATH], where [MATH] and [MATH].', '1701.02957-2-25-1': 'The pair [MATH] is called a code with rate [MATH].', '1701.02957-2-25-2': 'The error probability of sending a message [MATH] with the code [MATH] is [MATH].', '1701.02957-2-25-3': 'We use [MATH] and [MATH] to denote the maximal error probability and the average error probability, respectively.', '1701.02957-2-26-0': 'Given a sequence [MATH], we denote by', '1701.02957-2-27-0': '[MATH]', '1701.02957-2-28-0': 'the empirical distribution of [MATH].', '1701.02957-2-29-0': 'Throughout this paper, we consider a symmetric c-q channel defined as [EQUATION] where [MATH] is an arbitrary density operator, and [MATH]', '1701.02957-2-30-0': 'satisfies [MATH].', '1701.02957-2-30-1': 'We define the following conditional entropic quantities for the channel [MATH] with [MATH]:', '1701.02957-2-31-0': '[MATH].', '1701.02957-2-32-0': 'The mutual information of the c-q channel [MATH] with prior distribution [MATH] is defined as', '1701.02957-2-33-0': '[MATH],', '1701.02957-2-34-0': 'where [MATH], [MATH].', '1701.02957-2-35-0': 'The (classical) capacity of the channel [MATH] is denoted by', '1701.02957-2-36-0': '[MATH].', '1701.02957-2-37-0': 'Let [EQUATION] where we denote by', '1701.02957-2-38-0': '[MATH]', '1701.02957-2-39-0': 'an auxiliary function [CITATION].', '1701.02957-2-39-1': 'The sphere-packing exponent is defined by [EQUATION] where the last equality follows from [CITATION].', '1701.02957-2-39-2': 'Further, we define a rate [CITATION], [CITATION]: [EQUATION]', '1701.02957-2-39-3': 'It follows that [MATH] for any [MATH] (see also [CITATION] and [CITATION]).', '1701.02957-2-40-0': 'Consider a binary hypothesis whose null and alternative hypotheses are [MATH] and [MATH], respectively.', '1701.02957-2-40-1': 'The type-I error and type-II error of the hypothesis testing, for an operator [MATH], are defined as', '1701.02957-2-41-0': '[MATH], and [MATH].', '1701.02957-2-42-0': 'There is a trade-off between these two errors.', '1701.02957-2-42-1': 'Thus, we can define the minimum type-I error, when the type-II error is below [MATH], as [EQUATION]', '1701.02957-2-43-0': '## Main Result', '1701.02957-2-44-0': 'Let us now consider any symmetric c-q channel with capacity [MATH].', '1701.02957-2-45-0': '[Exact Sphere-Packing Bound] For any rate [MATH], there exist an [MATH] such that for all codes [MATH] of length [MATH], we have [EQUATION] where [MATH].', '1701.02957-2-46-0': '# Properties of the Sphere-Packing Exponent', '1701.02957-2-47-0': '[Optimal Input Distribution] For any [MATH], the distribution [MATH] is a maximizer of [MATH] and [MATH].', '1701.02957-2-48-0': 'We first prove that [MATH] attains [MATH].', '1701.02957-2-48-1': 'From Eq. [REF], it is not hard to verify that [MATH] for all [MATH].', '1701.02957-2-48-2': 'Hence, it follows that [EQUATION] for all [MATH].', '1701.02957-2-48-3': 'The above equation shows that', '1701.02957-2-49-0': 'the distribution [MATH] that maximizes [MATH], [MATH] [CITATION].', '1701.02957-2-50-0': 'Then we have [EQUATION]', '1701.02957-2-50-1': "Further, Jensen's inequality implies that [MATH], which completes the proof.", '1701.02957-2-51-0': '[Saddle-Point Property] Consider any [MATH] and [MATH].', '1701.02957-2-52-0': 'Let', '1701.02957-2-53-0': '[MATH].', '1701.02957-2-53-1': 'We define [EQUATION] on [MATH], and let', '1701.02957-2-54-0': '[MATH].', '1701.02957-2-55-0': 'The following holds', '1701.02957-2-56-0': 'The proof is provided in Appendix [REF].', '1701.02957-2-57-0': '[Representation] For any [MATH], let [MATH] be the saddle-point of [MATH].', '1701.02957-2-57-1': 'It follows that [EQUATION]', '1701.02957-2-58-0': 'Since Lemma [REF] implies that [MATH] attains [MATH], one observes from the definition of [MATH] that all the quantities [MATH], [MATH] are equal.', '1701.02957-2-58-1': 'By item (iii) of Lemma [REF], we obtain a representation of [MATH] in Eq. [REF].', '1701.02957-2-58-2': 'The optimal [MATH] follows from [CITATION].', '1701.02957-2-59-0': '[Invariance] For any [MATH], we have [EQUATION] where [MATH] and [MATH] are defined in Eq. [REF].', '1701.02957-2-60-0': 'Following the argument in Lemma [REF] and recalling Eq. [REF] in Lemma [REF], one can verify that [MATH] for all [MATH].', '1701.02957-2-60-1': 'Further, we obtain [MATH] for [MATH] from the result in [CITATION].', '1701.02957-2-61-0': '# Proof of the Main Result', '1701.02957-2-62-0': 'For rates in the range [MATH], we have [MATH].', '1701.02957-2-62-1': 'The bound in Eq. [REF] obviously holds.', '1701.02957-2-62-2': 'Hence, we consider the case of [MATH] and fix the rate throughout the proof.', '1701.02957-2-63-0': 'We first pose the channel coding problem into a binary hypothesis testing through Lemma [REF], which originates from Blahut [CITATION] for the classical case.', '1701.02957-2-63-1': '[Hypothesis Testing Reduction] For any code [MATH] with message size [MATH], there exists an [MATH] such that [EQUATION]', '1701.02957-2-64-0': 'The proof is provided in Appendix [REF].', '1701.02957-2-65-0': 'Let us now commence with the proof of Theorem [REF].', '1701.02957-2-65-1': 'Fix arbitrary [MATH].', '1701.02957-2-65-2': 'Let [MATH] and [MATH].', '1701.02957-2-65-3': 'The choice of the rate back-off term [MATH] will become evident later.', '1701.02957-2-65-4': 'Choose [MATH] such that [MATH].', '1701.02957-2-65-5': 'Let [MATH] be defined in Eq. [REF], and from Lemma [REF], we have [EQUATION]', '1701.02957-2-65-6': 'In the following, we provide a lower bound for the type-I error [MATH].', '1701.02957-2-65-7': 'Let [MATH] and [MATH], where [MATH] are Nussbaum-Szkola distributions [CITATION] of [MATH] for every [MATH].', '1701.02957-2-65-8': 'Since [MATH], for all [MATH], we shorthand', '1701.02957-2-66-0': '[MATH], where [MATH] is the empirical distribution of [MATH].', '1701.02957-2-67-0': 'Moreover, item (iii) in Lemma [REF] implies that the state [MATH] dominates all the channel outputs: [MATH], for all [MATH], Hence, we have [MATH].', '1701.02957-2-67-1': 'Subsequently, for every [MATH], we let [MATH], for all [MATH].', '1701.02957-2-68-0': "We apply Nagaoka's argument [CITATION] by choosing [MATH] to yield, for any [MATH], [EQUATION] where", '1701.02957-2-69-0': '[MATH], [MATH],', '1701.02957-2-70-0': 'and', '1701.02957-2-71-0': '[MATH].', '1701.02957-2-72-0': "Next, we employ Bahadur-Ranga Rao's concentration inequality, Theorem [REF] in Appendix [REF], to further lower bound [MATH] and [MATH].", '1701.02957-2-72-1': 'Before proceeding, we need to introduce some notation.', '1701.02957-2-72-2': 'We define the tilted distributions, for every [MATH], [MATH], and [MATH] by [EQUATION]', '1701.02957-2-72-3': 'Let [EQUATION]', '1701.02957-2-72-4': 'Since [MATH] and [MATH] are mutually absolutely continuous, the maps [MATH], [MATH] are differentiable for all [MATH].', '1701.02957-2-72-5': 'One can immediately verify the following partial derivatives with respect to [MATH]: [EQUATION]', '1701.02957-2-72-6': 'With [MATH] in Eq. [REF], we can define [EQUATION] where [MATH] in Eq. ([REF]) is the Fenchel-Legendre transform of [MATH].', '1701.02957-2-72-7': "The quantities [MATH] would appear in the lower bounds of [MATH] and [MATH] obtained by Bahadur-Randga Rao's inequality as shown later.", '1701.02957-2-73-0': 'In the following, we relate the Fenchel-Legendre transform [MATH] to the desired error-exponent function [MATH].', '1701.02957-2-74-0': 'Such a relationship is stated in Lemma [REF]; the proof is provided in Appendix [REF].', '1701.02957-2-75-0': 'Under the prevailing assumptions and for all [MATH], the following holds:', '1701.02957-2-76-0': 'Item (iii) in Lemma [REF] shows that the optimizer [MATH] in Eq. [REF] always lies in the compact set [MATH].', '1701.02957-2-76-1': 'Further, Eqs. [REF] and [REF] ensure that', '1701.02957-2-77-0': '[MATH], [MATH], [MATH].', '1701.02957-2-78-0': 'We define the following quantities: [EQUATION] [MATH]; and [MATH].', '1701.02957-2-78-1': 'Note that for every [MATH], [MATH] and [MATH] are continuous functions on [MATH] from the definitions in Eqs. [REF], [REF] (see also [CITATION]).', '1701.02957-2-78-2': 'The maximization and minimization in the above definitions are well-defined and finite.', '1701.02957-2-78-3': 'Moreover, Lemma [REF] guarantees that [MATH] is bounded away from zero.', '1701.02957-2-79-0': '[Positivity] For any [MATH] and [MATH], [MATH], for all [MATH].', '1701.02957-2-80-0': 'Assume [MATH] is zero for some [MATH].', '1701.02957-2-80-1': 'This is equivalent to [EQUATION]', '1701.02957-2-80-2': 'Summing the right-hand side of Eq. [REF] over [MATH] gives', '1701.02957-2-81-0': '[MATH].', '1701.02957-2-82-0': 'Then, Eqs. [REF] and the above equation imply that [EQUATION] where we use the fact that [MATH]; see Eq. [REF]).', '1701.02957-2-83-0': 'However, Lemma [REF] implies that [MATH], which leads to a contradiction.', '1701.02957-2-84-0': 'Now, we are ready to derive the lower bounds to [MATH] and [MATH].', '1701.02957-2-84-1': 'Let [MATH] be sufficiently large such that for all [MATH], [EQUATION]', '1701.02957-2-84-2': "Applying Bahadur-Randga Rao's inequality (Theorem [REF]) to [MATH] with the probability measure [MATH], and [MATH] gives [EQUATION] where", '1701.02957-2-85-0': '[MATH].', '1701.02957-2-86-0': 'Similarly, applying Theorem [REF] to [MATH] with the probability measure [MATH], and [MATH] yields [EQUATION]', '1701.02957-2-86-1': 'Continuing from Eq. ([REF]) and item (i) in Lemma [REF] gives [EQUATION]', '1701.02957-2-86-2': 'Eq. ([REF]) together with item (iii) in Lemma [REF] yields [EQUATION]', '1701.02957-2-86-3': 'Let [MATH] such that [MATH], for all [MATH].', '1701.02957-2-86-4': 'Then Eq. [REF] implies that [MATH].', '1701.02957-2-86-5': 'Thus, we can bound the left-hand side of Eq. ([REF]) from below by [MATH].', '1701.02957-2-86-6': 'For any test [MATH] such that', '1701.02957-2-87-0': '[MATH],', '1701.02957-2-88-0': 'we have [EQUATION] where the last equality follows from Lemma [REF].', '1701.02957-2-89-0': 'Finally, it remains to remove the back-off term [MATH] in Eq. [REF].', '1701.02957-2-90-0': "By Taylor's theorem, we have [EQUATION] for some [MATH] and [MATH].", '1701.02957-2-90-1': 'Further, one can calculate that [EQUATION] where [MATH].', '1701.02957-2-90-2': 'From item (iii) in Lemma [REF], it follows that both [MATH] and [MATH] are both positive and finite for [MATH] and [MATH].', '1701.02957-2-90-3': 'Together with the fact that [MATH], we have [MATH].', '1701.02957-2-91-0': "We apply Taylor's expansion on the function [MATH] again to yield [EQUATION] where the first equality holds for some [MATH], and the last line follows from the definition [MATH].", '1701.02957-2-91-1': 'Finally, by combining Eqs. [REF], [REF], and [REF], we obtain the desired Eq. [REF] for sufficiently large [MATH].', '1701.02957-2-92-0': '# Discussion', '1701.02957-2-93-0': 'In this work, we establish a sphere-packing bound with a refined polynomial pre-factor that coincides with the best classical results [CITATION] to date.', '1701.02957-2-93-1': 'As discussed by Altug and Wagner [CITATION], the pre-factor is correct for binary symmetric channels but slightly worse for binary erasure channels (in the order of [MATH]).', '1701.02957-2-93-2': 'On the other hand, our pre-factor matches the recent result of the random coding upper bound [CITATION], where the pre-factor has been shown to be exact.', '1701.02957-2-93-3': 'Hence, we conjecture that the established result is optimal for general symmetric c-q channels.', '1701.02957-2-94-0': 'This work admits variety of potential extensions.', '1701.02957-2-94-1': 'First, the symmetric c-q channel studied in this paper is a covariant channel with a cyclic group: [EQUATION] where [MATH] and [MATH] are the unitary representations on [MATH] and [MATH] such that [MATH] and [MATH].', '1701.02957-2-94-2': 'It would be interesting to investigate whether the refined sphere-packing bound can be extended to covariant quantum channels [MATH] with arbitrary compact groups.', '1701.02957-2-94-3': 'Second, the random coding bound in the quantum case has been proved only for pure-state channels [CITATION].', '1701.02957-2-94-4': 'It is promising to prove the bound for this class of c-q channels by employing the symmetry property.', '1701.02957-2-94-5': 'Finally, the refinement provides a new possibility for moderate deviation analysis in c-q channels [CITATION], which is left as future work.'} | [['1701.02957-1-5-0', '1701.02957-2-5-0'], ['1701.02957-1-5-1', '1701.02957-2-5-1'], ['1701.02957-1-5-2', '1701.02957-2-5-2'], ['1701.02957-1-5-3', '1701.02957-2-5-3'], ['1701.02957-1-5-4', '1701.02957-2-5-4'], ['1701.02957-1-91-0', '1701.02957-2-91-0'], ['1701.02957-1-91-1', '1701.02957-2-91-1'], 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'1701.02957-2-12-0', '1701.02957-2-17-0', '1701.02957-2-18-0', '1701.02957-2-19-0', '1701.02957-2-20-0', '1701.02957-2-21-0', '1701.02957-2-24-0', '1701.02957-2-26-0', '1701.02957-2-27-0', '1701.02957-2-28-0', '1701.02957-2-30-0', '1701.02957-2-30-1', '1701.02957-2-31-0', '1701.02957-2-33-0', '1701.02957-2-34-0', '1701.02957-2-35-0', '1701.02957-2-36-0', '1701.02957-2-37-0', '1701.02957-2-38-0', '1701.02957-2-41-0', '1701.02957-2-49-0', '1701.02957-2-51-0', '1701.02957-2-52-0', '1701.02957-2-53-0', '1701.02957-2-53-1', '1701.02957-2-54-0', '1701.02957-2-55-0', '1701.02957-2-56-0', '1701.02957-2-59-0', '1701.02957-2-64-0', '1701.02957-2-65-1', '1701.02957-2-65-2', '1701.02957-2-66-0', '1701.02957-2-69-0', '1701.02957-2-70-0', '1701.02957-2-71-0', '1701.02957-2-72-3', '1701.02957-2-75-0', '1701.02957-2-77-0', '1701.02957-2-79-0', '1701.02957-2-81-0', '1701.02957-2-85-0', '1701.02957-2-87-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1701.02957 | null | null | null | null | null |
1910.07486 | {'1910.07486-1-0-0': 'State-of-the-art computer vision approaches rely on huge amounts of annotated data.', '1910.07486-1-0-1': 'The collection of such data is a time consuming process since it is mainly performed by humans.', '1910.07486-1-0-2': 'The literature shows that semi-automatic annotation approaches can significantly speed up the annotation process by the automatic generation of annotation proposals to support the annotator.', '1910.07486-1-0-3': 'In this paper we present a framework that allows for a quick and flexible design of semi-automatic annotation pipelines.', '1910.07486-1-0-4': 'We show that a good design of the process will speed up the collection of annotations.', '1910.07486-1-0-5': 'Our contribution is a new approach to image annotation that allows for the combination of different annotation tools and machine learning algorithms in one process.', '1910.07486-1-0-6': 'We further present potential applications of our approach.', '1910.07486-1-0-7': 'The source code of our framework called LOST (Label Objects and Save Time) is available at: https://github.com/l3p-cv/lost.', '1910.07486-1-1-0': '# Introduction', '1910.07486-1-2-0': 'A huge amount of annotated data is the key to success in machine learning and computer vision.', '1910.07486-1-2-1': 'However the annotation process is still extremely elaborate, since humans or even experts in a specific field are required.', '1910.07486-1-2-2': 'Therefore a good annotation tool and smart annotation strategies are essential to build large datasets of sufficient quality.', '1910.07486-1-3-0': 'In recent years the community focused on three main points to save time and improve annotation quality while collecting datasets for computer vision research.', '1910.07486-1-4-0': '1) Crowdsourcing approaches as presented in [CITATION] have been utilized to collect huge amounts of annotations via crowdsourcing platforms such as Amazon Mechanical Turk.', '1910.07486-1-4-1': 'With this strategy the overall time for dataset collection is reduced significantly by employing a large number of annotators.', '1910.07486-1-5-0': '2) A second focus of the community was to optimize the annotation process itself by supporting the human annotator.', '1910.07486-1-5-1': 'The main idea here is to reduce the human interaction with the annotation tool to save time, while maintaining the quality of the annotations [CITATION].', '1910.07486-1-6-0': '3) The third main focus was on the development of annotation tools and their user interfaces, since the user experience with such a tool is important for the annotators motivation and the quality of the annotations.', '1910.07486-1-6-1': 'A wide variety of annotation tools is described in the literature [CITATION].', '1910.07486-1-7-0': 'Our contribution is a flexible pipeline concept to model semi-automatic image and video annotation.', '1910.07486-1-7-1': 'This approach allows to combine multiple annotation tools and machine learning algorithms as one process in a building block style.', '1910.07486-1-7-2': 'We visualize the whole annotation process in a web-based user interface.', '1910.07486-1-7-3': 'Furthermore, we provide an annotation interface to assign labels to clusters of images or annotations e.g. boxes (see Section [REF]).', '1910.07486-1-8-0': 'The open source implementation called LOST (Label Objects and Save Time) is available on GitHub (https://github.com/l3p-cv/lost).', '1910.07486-1-8-1': 'This implementation contains an annotation process visualization, two annotation tools, a tree-based label management and an annotator management.', '1910.07486-1-8-2': 'Our tool allows researchers to design and run their own annotation pipelines in a quick and consistent way.', '1910.07486-1-8-3': 'Furthermore each developer can inject his own Python scripts to gain full control over the process.', '1910.07486-1-8-4': 'A single instance of LOST can be easily set up with docker, to be used as stand alone application on a single machine.', '1910.07486-1-8-5': 'LOST can also be set up as a cloud application to allow collaborative annotation via the web.', '1910.07486-1-8-6': 'If required, LOST is able to distribute computational workload across multiple machines.', '1910.07486-1-9-0': '# Related Work', '1910.07486-1-10-0': '## Approaches to Support the Annotator', '1910.07486-1-11-0': 'The authors of [CITATION] use point annotations to train object detection models.', '1910.07486-1-11-1': 'Papadopoulus et al. [CITATION] report a speed up of the total annotation time by a factor of 9 to 18 compared to traditional bounding box annotations.', '1910.07486-1-11-2': 'The performance of their detectors is close to a detector trained with hand drawn bounding box annotations.', '1910.07486-1-11-3': 'Russakovsky et al. [CITATION] use point supervision to create a segmentation model that is more accurate than models trained with full supervision given a fixed time budget.', '1910.07486-1-12-0': 'The authors of [CITATION] generate bounding box proposals to ask the annotators if a box is correct or not.', '1910.07486-1-12-1': 'A bounding box is considered as correct if the intersection over union [CITATION] with a tight box around the object is greater than [MATH].', '1910.07486-1-12-2': 'After the annotators verification step, the object detector is retrained with the new boxes.', '1910.07486-1-12-3': 'This approach reduces the human annotation time by a factor of [MATH] to [MATH], while achieving a mAP of [MATH] on Pascal VOC 2007.', '1910.07486-1-12-4': 'When training with full supervision the authors achieve a mAP of [MATH].', '1910.07486-1-12-5': 'In [CITATION] an agent is trained to select the best strategy for bounding box annotation.', '1910.07486-1-12-6': 'Two strategies are considered to be selected by the agent: either box verification [CITATION] as described above or manual box drawing.', '1910.07486-1-13-0': '## Annotation Tools/ Interfaces', '1910.07486-1-14-0': 'Russel et al. [CITATION] present a general purpose web-based image annotation tool called LabelMe to create polygon annotations.', '1910.07486-1-14-1': 'Vondrick et al. [CITATION] propose a web-based tool called VATIC for semi-automatic video annotation in a crowdsourcing setup.', '1910.07486-1-14-2': 'They use bounding box annotations and link these boxes to create ground-truth tracks for video sequences within an optimized user interface.', '1910.07486-1-14-3': 'Similar to VATIC, iVAT [CITATION] and ViTBAT [CITATION] are tools for semi-automatic video annotation.', '1910.07486-1-14-4': 'In contrast to VATIC these tools are not web-based.', '1910.07486-1-14-5': 'Polygon-RNN [CITATION] follows an interactive approach for faster polygon annotation.', '1910.07486-1-14-6': 'A recurrent neural network is utilized to support the human annotator by iteratively sending annotation proposals to the user interface.', '1910.07486-1-14-7': 'Qin et al. [CITATION] present the semi-automatic tool ByLabel that supports the annotator in segmentation tasks.', '1910.07486-1-14-8': 'The tool FreeLabel [CITATION] is also designed to collect segmentation masks.', '1910.07486-1-14-9': 'It uses scribble annotations as seeds for the region growing algorithm to create a semi-automatic segmentation result.', '1910.07486-1-15-0': 'BIIGLE [CITATION] is a web-based tool that is especially designed for the annotation and exploration of marine image collections.', '1910.07486-1-15-1': 'It provides different annotation and review interfaces, a project management and a user management.', '1910.07486-1-15-2': 'In difference to the other tools it also implements a label tree management.', '1910.07486-1-15-3': 'In contrast to LOST, Biigle does not implement a flexible pipeline system.', '1910.07486-1-16-0': 'Each of the above mentioned annotation tools was designed with a specific application and annotation process in mind.', '1910.07486-1-16-1': 'Due to that, these annotation tools have hard-coded a specific process/ algorithm and model only a single use case.', '1910.07486-1-16-2': 'This leads to the fact that every time a new annotation approach was tested a new tool was implemented.', '1910.07486-1-16-3': 'In contrast to that, our framework is able to model multiple semi-automatic annotation approaches, e.g. [CITATION], in a consistent and fast way.', '1910.07486-1-16-4': 'In this sense our proposed framework is a generalization of single purpose annotation tools.', '1910.07486-1-17-0': '# Approach', '1910.07486-1-18-0': 'We propose a framework for semi-automatic image annotation.', '1910.07486-1-18-1': 'This framework allows any combination of machine learning algorithms and annotation interfaces in a building block style.', '1910.07486-1-19-0': 'Figure [REF] represents a simple example, while much more complex processes can be modeled.', '1910.07486-1-19-1': 'In general an annotation process is defined as a directed graph.', '1910.07486-1-19-2': 'Each node in a graph represents one building block and the connections between the nodes define the order in which the building blocks are processed.', '1910.07486-1-19-3': 'Also information, such as annotations, can be exchanged between connected elements and accessed via the framework API.', '1910.07486-1-20-0': 'The basic building blocks of an annotation pipeline are datasources, annotation tasks and scripts that implement different algorithms.', '1910.07486-1-20-1': 'By means of these building blocks, annotation tasks for humans can be combined with machine learning and other algorithms in a flexible way.', '1910.07486-1-21-0': '## Flexibility', '1910.07486-1-22-0': 'The main feature of LOST is a flexible pipeline concept.', '1910.07486-1-22-1': 'An example for its flexibility is the combination of different annotation interfaces in one pipeline.', '1910.07486-1-22-2': 'When combining a single image annotation interface (SIA, Section [REF]) with a multi image annotation interface (MIA, Section [REF]), annotation tasks can be split into object localization and class label assignment.', '1910.07486-1-22-3': 'See Section [REF] for a detailed description of a two-stage annotation approach where SIA is used to draw bounding boxes and MIA to assign a class label to each box.', '1910.07486-1-23-0': 'Flexible pipelines allow also to combine any kind of machine learning algorithm with an annotation interface to realize semi-automatic annotation approaches.', '1910.07486-1-23-1': 'For example, SIA can be combined with a script that implements an object detector that generates bounding box proposals.', '1910.07486-1-23-2': 'MIA can be connected to a script element that implements an algorithm that clusters images based on their visual similarity in order to speed up the annotation.', '1910.07486-1-23-3': 'See Section [REF] for a semi-automatic annotation pipeline that combines SIA and MIA with machine learning algorithms.', '1910.07486-1-24-0': 'LOST allows also to model iterative annotation processes when adding loop elements to a pipeline (see Section [REF]).', '1910.07486-1-24-1': 'In this way lifelong learning [CITATION] and active learning [CITATION] approaches can be realized with LOST.', '1910.07486-1-25-0': '## Building Blocks', '1910.07486-1-26-0': 'An annotation pipeline (annotation process) can be composed of six different building block types.', '1910.07486-1-26-1': 'These are datasources, scripts, annotation tasks, loops, data exports and visualizations.', '1910.07486-1-26-2': 'After a pipeline was designed as a composition of the different building blocks, it can be loaded into the LOST framework.', '1910.07486-1-26-3': 'When starting (instantiating) a pipeline , each element can be parameterized.', '1910.07486-1-26-4': 'For example, for a datasource a set of images will be selected.', '1910.07486-1-26-5': 'Another common example is the selection of a user or group that will perform an annotation task to parameterize an annotation task-element in a pipeline.', '1910.07486-1-27-0': 'A datasource represents a set of images or videos that can be used by connected elements in the pipeline, for example by one or many scripts.', '1910.07486-1-27-1': 'Such a script-element is an arbitrary algorithm implemented in Python that communicates with connected elements via the framework API.', '1910.07486-1-27-2': 'The main purpose of a script is to generate object proposals or to cluster images for semi-automatic annotation.', '1910.07486-1-27-3': 'An example of such an object proposal could be a bounding box generated by a RetinaNet [CITATION] object detector.', '1910.07486-1-27-4': 'When a script has generated proposals for all images that should be annotated it will send them to an annotation task, where the proposals will be displayed to a human annotator.', '1910.07486-1-28-0': 'An annotation task-element links users, annotation tools and labels.', '1910.07486-1-28-1': 'There are two types of annotation tools that can be used in our current implementation.', '1910.07486-1-28-2': 'The first tool was designed to annotate single images (Section [REF]) and with the second tool clusters of images can be annotated (Section [REF]).', '1910.07486-1-28-3': 'Labels are represented as trees (Section [REF]).', '1910.07486-1-28-4': 'In this way we are able to model label hierarchies.', '1910.07486-1-29-0': 'Loop-blocks can be used for iterative annotation processes where parts of a pipeline need to be executed multiple times until a certain criterion is fulfilled.', '1910.07486-1-29-1': 'Loops are often useful to model active learning or continuous learning approaches.', '1910.07486-1-29-2': 'See Section [REF] for an example how a loop can be used in a pipeline.', '1910.07486-1-30-0': 'The last two element types that can be used are data exports and visualization-elements.', '1910.07486-1-30-1': 'Data exports are used to provide any files created by scripts via the GUI for download.', '1910.07486-1-30-2': 'Similar to data exports, visualizations display images created by scripts within the web interface.', '1910.07486-1-31-0': '## Multi Image Annotation (MIA) Interface', '1910.07486-1-32-0': 'MIA serves to assign labels to clusters of images or annotations.', '1910.07486-1-32-1': 'The main idea is that visual similar objects are likely to get the same label.', '1910.07486-1-32-2': 'This idea is related to the cluster-based approach to fish annotation proposed by [CITATION].', '1910.07486-1-33-0': 'In an ideal world a cluster contains only images that belong to the same class.', '1910.07486-1-33-1': 'A human annotator has the task of sorting out images that do not belong to the cluster.', '1910.07486-1-33-2': 'Therefore, the same label can be assigned to large number of images at the same time.', '1910.07486-1-33-3': 'In the same way labels can be assigned to point, box, line and polygon annotations.', '1910.07486-1-34-0': 'LOST provides the first open source implementation of a MIA interface.', '1910.07486-1-34-1': 'Due to the pipeline concept, MIA can be combined with a SIA interface in one annotation process (see Section [REF]).', '1910.07486-1-34-2': 'In this way, class labels and object localizations can be annotated in specialized interfaces for each task to speed up annotation.', '1910.07486-1-34-3': 'Furthermore, MIA can be combined with different cluster algorithms by connecting it to a script element that will sort annotations or images into clusters.', '1910.07486-1-35-0': '## Single Image Annotation (SIA) Interface', '1910.07486-1-36-0': 'SIA is designed to create polygons, points, lines and bounding box annotations.', '1910.07486-1-36-1': 'To each annotation and the whole image, a class label can be assigned.', '1910.07486-1-36-2': 'Also the assignment of multiple class labels is possible.', '1910.07486-1-36-3': 'Furthermore, the tool is configurable to allow or deny different types of user actions and annotations depending on the use case.', '1910.07486-1-36-4': 'For example, you can specify that only class label assignment is possible, but no other modifications are allowed.', '1910.07486-1-36-5': 'When combining SIA with an algorithm that generates annotation proposals semi-automatic annotation can be realized.', '1910.07486-1-36-6': 'See Section [REF] for an example.', '1910.07486-1-37-0': '## Label Management', '1910.07486-1-38-0': 'Labels are managed in label trees to model label hierarchies.', '1910.07486-1-38-1': 'Multiple label trees can be created and edited.', '1910.07486-1-38-2': 'When starting an annotation task a whole label tree or a composition of subtrees can be selected as possible labels.', '1910.07486-1-38-3': 'During the annotation task the annotator can assign one of the possible labels to an annotation.', '1910.07486-1-39-0': '## Comparison to State-of-the-Art Annotation Tools', '1910.07486-1-40-0': 'Table [REF] presents an overview of the key ideas of LOST in comparison to other open source tools.', '1910.07486-1-40-1': 'LOST is the only tool with a flexible pipeline system, where multiple annotation interfaces and algorithms can be combined in one process.', '1910.07486-1-40-2': 'There are many tools that where build on web technologies to enable collaborative annotation.', '1910.07486-1-41-0': 'Table [REF] shows also three different annotation interface designs and if they are implemented for a specific tool:', '1910.07486-1-42-0': '1) SIA interfaces are used to annotate single images with different annotations like points, boxes, lines, polygons, etc.', '1910.07486-1-42-1': 'Most tools do focus on a SIA interface.', '1910.07486-1-43-0': '2) MIA interfaces are used to present clusters of similar images and assign a label to a whole cluster of images.', '1910.07486-1-43-1': 'This idea of MIA was described by Boom et al. [CITATION].', '1910.07486-1-43-2': 'In the presented comparison LOST is the only tool that has implemented this type of interface.', '1910.07486-1-44-0': '3)Image sequence annotation (ISA) interfaces are especially designed to annotate video sequences with tracking information.', '1910.07486-1-44-1': 'In most cases these interfaces are extensions of a SIA interface.', '1910.07486-1-44-2': 'In the current version, LOST has no ISA interface to annotate tracks.', '1910.07486-1-44-3': 'The implementation of ISA is planned for the near future.', '1910.07486-1-45-0': 'Another difference between LOST and the other tools is that user-defined python scripts can be executed as part of an annotation pipeline.', '1910.07486-1-45-1': 'In such scripts any python code can be implemented and information with other pipeline elements can be exchanged to get full control over the process.', '1910.07486-1-46-0': '# Case Studies', '1910.07486-1-47-0': 'In this Section we show how annotation pipelines can be modeled and executed within our web-based framework.', '1910.07486-1-48-0': 'All case studies are performed and analyzed on the Pascal VOC2012 [CITATION] dataset.', '1910.07486-1-48-1': 'We use [MATH] of the VOC validation split to select the images for annotation and [MATH] for evaluation of trained models.', '1910.07486-1-48-2': 'The VOC training split is used for initial training.', '1910.07486-1-48-3': 'For our experiments we utilize the keras implementations of the respective models.', '1910.07486-1-48-4': 'The experiments are executed inside a NVIDIA Docker container integrated in LOST.', '1910.07486-1-48-5': 'The server is equipped with a NVIDIA Geforce GTX1080Ti, an Intel i7-8700K and 16GB DDR4 RAM.', '1910.07486-1-49-0': '## Single-Stage Annotation', '1910.07486-1-50-0': 'In our first experiment we create a baseline for simple bounding box annotation in the proposed framework as it is possible in most annotation tools.', '1910.07486-1-50-1': 'In order to do that, we use 200 images randomly selected from the Pascal VOC validation set.', '1910.07486-1-50-2': 'In this selection we guarantee that each class is present in at least 10 images.', '1910.07486-1-50-3': 'Two annotators perform bounding boxes annotation according to the VOC annotation guidelines on each image.', '1910.07486-1-50-4': 'Box drawing and class assignment is performed with the SIA tool (Section [REF]).', '1910.07486-1-51-0': 'Results', '1910.07486-1-52-0': 'Annotator1 achieved a mAP of [MATH] and Annotator2 a mAP of [MATH] compared to the Pascal VOC ground truth annotations, when using an intersection over union threshold of [MATH].', '1910.07486-1-52-1': 'Among each other the annotators agreed on [MATH] of the annotated boxes.', '1910.07486-1-52-2': 'This show that even among human annotators the level of agreement was below [MATH] mAP in our experiment.', '1910.07486-1-52-3': 'Which is an interesting fact when considering that object detectors are trained with human data.', '1910.07486-1-53-0': 'On average the annotators needed [MATH] seconds to draw a bounding box and assign a class label to this box.', '1910.07486-1-53-1': 'The average time to annotate an image was [MATH] seconds.', '1910.07486-1-53-2': 'For the annotation of [MATH] images [MATH] minutes of annotation time was required.', '1910.07486-1-54-0': '## Two-Stage Annotation', '1910.07486-1-55-0': 'In this experiment we combine the single image annotation (SIA) with the multi image annotation tool (MIA) in a two-stage annotation process.', '1910.07486-1-55-1': 'In the first stage the annotator has the task to draw bounding boxes according to the VOC annotation guidelines.', '1910.07486-1-55-2': 'In this stage no class labels will be assigned.', '1910.07486-1-55-3': 'In the second stage all boxes from the first stage will be clustered according to their visual similarity and presented in the MIA interface.', '1910.07486-1-55-4': 'The annotators task here is to assign a label to the whole cluster.', '1910.07486-1-55-5': 'If an image does not belong to that cluster, it should be removed by the annotator.', '1910.07486-1-55-6': 'For example, if there are 19 persons and one cat in the view, the annotator should remove the cat and select the label person for the remaining images.', '1910.07486-1-56-0': 'For this experiment, a similar setup as in Section [REF] is used.', '1910.07486-1-56-1': 'The same 200 images from the VOC validation set are annotated by a human annotator, while this time two annotation stages are performed.', '1910.07486-1-57-0': 'We test two clustering strategies.', '1910.07486-1-57-1': 'First we extract CNN features for all annotated boxes from stage one and use the K-Means algorithm to cluster all images.', '1910.07486-1-57-2': 'For feature extraction we use the last pooling layer of ResNet50 [CITATION] that was pretrained on ImageNet [CITATION].', '1910.07486-1-58-0': 'As second clustering approach we fine-tune ResNet50 on a small subset of the Pascal VOC training set and utilize the networks predictions directly for clustering.', '1910.07486-1-59-0': 'Results In the first stage the annotator used the SIA annotation tool and needed [MATH] seconds (see Figure [REF], SIA exp2) to draw a bounding box and [MATH] seconds to draw boxes for all objects in an image.', '1910.07486-1-59-1': 'On average he annotated [MATH] boxes per image, while the total annotation time for 200 images was [MATH] minutes in the first annotation stage.', '1910.07486-1-60-0': 'Figure [REF] shows also the average annotation time per box for class label assignment in the second annotation stage (MIA exp2).', '1910.07486-1-60-1': 'The x-axis indicates the different cluster methods, where K-Means and ResNet50 were used.', '1910.07486-1-60-2': 'ResNet50 was fine-tuned with [MATH] images, [MATH] images, [MATH] images and [MATH] images of the Pascal VOC training split.', '1910.07486-1-60-3': 'We see that the two-stage annotation process (Figure [REF], SIA+MIA exp2) is faster than single stage annotation if the clustering algorithm works well.', '1910.07486-1-60-4': 'Only when ResNet50 was fine-tuned with [MATH] of the training data, the two-stage process consumed the same time as one-stage.', '1910.07486-1-60-5': 'The mAP of the created boxes compared to the VOC ground-truth data was almost equal and around [MATH] for all approaches.', '1910.07486-1-60-6': 'With the fastest two-stage approach it took [MATH] minutes to annotate [MATH] images, where single-stage annotation took [MATH] minutes.', '1910.07486-1-61-0': 'When considering that class label assignment plus box drawing with SIA needs [MATH] seconds and box drawing only needs [MATH]s, we know that pure class label assignment with SIA takes [MATH] seconds.', '1910.07486-1-61-1': 'The fastest annotation approach with MIA takes [MATH] seconds per box, which is a speed up in class label assignment by a factor of two.', '1910.07486-1-62-0': 'Use cases The main idea here was to break down the complex task of simultaneous bounding box drawing and class label assignment into two separate tasks that are simpler, while supporting the annotator with a preclustering during class label assignment.', '1910.07486-1-62-1': 'In this way it was also possible to split the annotation work into a simple and an expert task, where in most cases expert knowledge is required for class label assignment as in many biological [CITATION] or medical applications [CITATION].', '1910.07486-1-62-2': 'We saw that when using the MIA annotation tool and a good clustering algorithm the time for class label assignment was reduced by a factor of two compared to label assignment with SIA.', '1910.07486-1-62-3': 'This allows for saving expensive annotation time for expert tasks.', '1910.07486-1-62-4': 'For the experts, who will not need to draw a box, annotation time is reduced by a factor of [MATH] to [MATH].', '1910.07486-1-63-0': '## Two-Stage Annotation in the loop', '1910.07486-1-64-0': 'In this experiment we show how to implement an iterative annotation process within our framework.', '1910.07486-1-64-1': 'As in Section [REF] we model a two-stage annotation pipeline composed of a single image annotation task and a clustered image annotation view.', '1910.07486-1-64-2': 'In difference to the previous experiment we use semi-automatic support in both annotation stages, do not use any VOC data for pretraining and put everything into a loop.', '1910.07486-1-64-3': 'In other words we assume that we have no annotated data in the beginning and try to get a better automatic support for the annotator over time.', '1910.07486-1-65-0': 'As in the previous experiments we use images from the Pascal VOC2012 validation dataset for annotation.', '1910.07486-1-65-1': 'Due to the iterative setup we use [MATH] images per iteration that will be processed by two human annotators.', '1910.07486-1-65-2': 'In contrast to the experiment in Section [REF], both annotators work on the same annotation task to split the workload.', '1910.07486-1-66-0': 'Figure [REF] shows a high level view of the annotation pipeline.', '1910.07486-1-66-1': 'In the first annotation stage RetinaNet [CITATION] is used for bounding box proposal generation in order to support the human annotator.', '1910.07486-1-66-2': 'As proposals we use all boxes with a confidence value above [MATH].', '1910.07486-1-66-3': 'After each iteration, RetinaNet will be retrained with all annotations from previous iterations.', '1910.07486-1-66-4': 'When all images have been processed by RetinaNet a SIA task will be performed by the human annotators.', '1910.07486-1-66-5': 'Since there are no annotations in the first iteration, no box proposals are generated in the first iteration.', '1910.07486-1-66-6': 'The annotators are instructed to draw bounding boxes around all VOC2012 objects in the images.', '1910.07486-1-67-0': 'In the second annotation stage ResNet50 [CITATION] is used to cluster all bounding boxes by class.', '1910.07486-1-67-1': 'We use pretrained ImageNet weights for initialization and fine-tune ResNet50 after each iteration with all annotations from previous iterations.', '1910.07486-1-67-2': 'In the first iteration where no class label annotations are available, we use ImageNet class predictions for clustering.', '1910.07486-1-67-3': 'The idea here is that visual similar images will get the same class label, even if the predicted class is not part of Pascal VOC.', '1910.07486-1-67-4': 'After that, the MIA tool is used to correct the proposed clusters and to assign class labels to the clustered box annotations.', '1910.07486-1-67-5': 'When the second stage was processed, the next loop iteration will start.', '1910.07486-1-68-0': 'Results', '1910.07486-1-69-0': 'Figure [REF] presents the average annotation time per box per iteration.', '1910.07486-1-69-1': 'We see times for box drawing in the first annotation stage (SIA exp3), the times for class label assignment in the second stage(MIA exp3), the total time per box for the looped two-stage annotation approach (Total exp3) and the single-stage annotation approach for comparison (SIA only exp1).', '1910.07486-1-69-2': 'In the first iteration, when no box proposals are generated and the cluster algorithm is not fine-tuned to the VOC dataset, the looped two-stage approach is slower than single-stage annotation.', '1910.07486-1-69-3': 'But in the following iterations when RetinaNet and ResNet50 are fine-tuned with the annotations of the previous iterations, the looped two-stage annotation process gets faster than single-stage annotation.', '1910.07486-1-70-0': 'Figure [REF] shows the corresponding mAPs of the human-annotated boxes (Annotator) and the detector performance of RetinaNet per iteration.', '1910.07486-1-70-1': 'While the detector performance increases, the performance of the created annotations seems to be stable around a mean of [MATH] mAP with a standard deviation of [MATH].', '1910.07486-1-70-2': 'This deviation most likely reflects the annotators attention level and the difficulty of the images that have been annotated.', '1910.07486-1-71-0': 'Use cases We found that a looped two-stage annotation approach would be beneficial if there are no ground-truth data available in the beginning of the annotation process.', '1910.07486-1-71-1': 'This approach creates annotations that have an equal quality compared to the single-stage approach and trains a detector on-the-fly, while taking less annotator time than single-stage annotation.', '1910.07486-1-71-2': 'Since it is modeled as two-stage process, the annotation work can be split in a simple and an expert task as in Section [REF].', '1910.07486-1-71-3': 'It is notable that the training process of the machine learning models takes additional time compared to annotation approaches without machine learning elements, but we found that this is no problem when performing the training in time slots when the annotator needs to rest anyway e. g. over night.', '1910.07486-1-72-0': '# Conclusion', '1910.07486-1-73-0': 'To the best of our knowledge, we present the first framework for a flexible design and instantiation of image annotation pipelines.', '1910.07486-1-73-1': 'Our approach enables the combination of different annotation tools and machine learning algorithms in one process.', '1910.07486-1-73-2': 'We also provide an annotation interface called MIA (multi image annotation) to annotate whole clusters of images at the same time.', '1910.07486-1-74-0': 'Our case studies show how our framework can be used to model machine learning based semi-automatic annotation pipelines and iterative annotation approaches.', '1910.07486-1-74-1': 'In Section [REF] we found that simple clustering in combination with the MIA annotation interface can speed up class label assignment by a factor of two compared to single stage annotation.', '1910.07486-1-74-2': 'We also show that an annotation task can be split in an expert and a simple task, which can significantly reduce expensive expert annotation time.', '1910.07486-1-74-3': 'We further show in Section [REF] that a looped two-stage approach is beneficial when no annotation data is available in the beginning.', '1910.07486-1-74-4': 'The quality of the created annotations is kept high while the time spent for box annotation gets smaller over time.', '1910.07486-1-75-0': 'In future, we plan to release the missing image sequence annotation interface (ISA) that is specialized to annotate tracks.', '1910.07486-1-75-1': 'We also want to implement an interface to Mechanical Turk for crowdsourcing applications.'} | {'1910.07486-2-0-0': 'State-of-the-art computer vision approaches rely on huge amounts of annotated data.', '1910.07486-2-0-1': 'The collection of such data is a time consuming process since it is mainly performed by humans.', '1910.07486-2-0-2': 'The literature shows that semi-automatic annotation approaches can significantly speed up the annotation process by the automatic generation of annotation proposals to support the annotator.', '1910.07486-2-0-3': 'In this paper we present a framework that allows for a quick and flexible design of semi-automatic annotation pipelines.', '1910.07486-2-0-4': 'We show that a good design of the process will speed up the collection of annotations.', '1910.07486-2-0-5': 'Our contribution is a new approach to image annotation that allows for the combination of different annotation tools and machine learning algorithms in one process.', '1910.07486-2-0-6': 'We further present potential applications of our approach.', '1910.07486-2-0-7': 'The source code of our framework called LOST (Label Objects and Save Time) is available at: https://github.com/l3p-cv/lost.', '1910.07486-2-1-0': '# Introduction', '1910.07486-2-2-0': 'A huge amount of annotated data is the key to success in machine learning and computer vision.', '1910.07486-2-2-1': 'However the annotation process is still extremely elaborate, since humans or even experts in a specific field are required.', '1910.07486-2-2-2': 'Therefore a good annotation tool and smart annotation strategies are essential to build large datasets of sufficient quality.', '1910.07486-2-3-0': 'In recent years the community focused on three main points to save time and improve annotation quality while collecting datasets for computer vision research.', '1910.07486-2-4-0': '1) Crowdsourcing approaches as presented in [CITATION] have been utilized to collect huge amounts of annotations via crowdsourcing platforms such as Amazon Mechanical Turk.', '1910.07486-2-4-1': 'With this strategy the overall time for dataset collection is reduced significantly by employing a large number of annotators.', '1910.07486-2-5-0': '2) A second focus of the community was to optimize the annotation process itself by supporting the human annotator.', '1910.07486-2-5-1': 'The main idea here is to reduce the human interaction with the annotation tool to save time, while maintaining the quality of the annotations [CITATION].', '1910.07486-2-6-0': '3) The third main focus was on the development of annotation tools and their user interfaces, since the user experience with such a tool is important for the annotators motivation and the quality of the annotations.', '1910.07486-2-6-1': 'A wide variety of annotation tools is described in the literature [CITATION].', '1910.07486-2-7-0': 'Our contribution is a flexible pipeline concept to model semi-automatic image and video annotation.', '1910.07486-2-7-1': 'This approach allows to combine multiple annotation tools and machine learning algorithms as one process in a building block style.', '1910.07486-2-7-2': 'We visualize the whole annotation process in a web-based user interface.', '1910.07486-2-7-3': 'Furthermore, we provide an annotation interface to assign labels to clusters of images or annotations e.g. boxes (see Section [REF]).', '1910.07486-2-8-0': 'The open source implementation called LOST (Label Objects and Save Time) is available on GitHub (https://github.com/l3p-cv/lost).', '1910.07486-2-8-1': 'This implementation contains an annotation process visualization, two annotation tools, a tree-based label management and an annotator management.', '1910.07486-2-8-2': 'Our tool allows researchers to design and run their own annotation pipelines in a quick and consistent way.', '1910.07486-2-8-3': 'Furthermore each developer can inject his own Python scripts to gain full control over the process.', '1910.07486-2-8-4': 'A single instance of LOST can be easily set up with docker, to be used as stand alone application on a single machine.', '1910.07486-2-8-5': 'LOST can also be set up as a cloud application to allow collaborative annotation via the web.', '1910.07486-2-8-6': 'If required, LOST is able to distribute computational workload across multiple machines.', '1910.07486-2-9-0': '# Related Work', '1910.07486-2-10-0': '## Approaches to Support the Annotator', '1910.07486-2-11-0': 'The authors of [CITATION] use point annotations to train object detection models.', '1910.07486-2-11-1': 'Papadopoulus et al. [CITATION] report a speed up of the total annotation time by a factor of 9 to 18 compared to traditional bounding box annotations.', '1910.07486-2-11-2': 'The performance of their detectors is close to a detector trained with hand drawn bounding box annotations.', '1910.07486-2-11-3': 'Russakovsky et al. [CITATION] use point supervision to create a segmentation model that is more accurate than models trained with full supervision given a fixed time budget.', '1910.07486-2-12-0': 'The authors of [CITATION] generate bounding box proposals to ask the annotators if a box is correct or not.', '1910.07486-2-12-1': 'A bounding box is considered as correct if the intersection over union [CITATION] with a tight box around the object is greater than [MATH].', '1910.07486-2-12-2': 'After the annotators verification step, the object detector is retrained with the new boxes.', '1910.07486-2-12-3': 'This approach reduces the human annotation time by a factor of [MATH] to [MATH], while achieving a mAP of [MATH] on Pascal VOC 2007.', '1910.07486-2-12-4': 'When training with full supervision the authors achieve a mAP of [MATH].', '1910.07486-2-12-5': 'In [CITATION] an agent is trained to select the best strategy for bounding box annotation.', '1910.07486-2-12-6': 'Two strategies are considered to be selected by the agent: either box verification [CITATION] as described above or manual box drawing.', '1910.07486-2-13-0': '## Annotation Tools/ Interfaces', '1910.07486-2-14-0': 'Russel et al. [CITATION] present a general purpose web-based image annotation tool called LabelMe to create polygon annotations.', '1910.07486-2-14-1': 'Vondrick et al. [CITATION] propose a web-based tool called VATIC for semi-automatic video annotation in a crowdsourcing setup.', '1910.07486-2-14-2': 'They use bounding box annotations and link these boxes to create ground-truth tracks for video sequences within an optimized user interface.', '1910.07486-2-14-3': 'Similar to VATIC, iVAT [CITATION] and ViTBAT [CITATION] are tools for semi-automatic video annotation.', '1910.07486-2-14-4': 'In contrast to VATIC these tools are not web-based.', '1910.07486-2-14-5': 'Polygon-RNN [CITATION] follows an interactive approach for faster polygon annotation.', '1910.07486-2-14-6': 'A recurrent neural network is utilized to support the human annotator by iteratively sending annotation proposals to the user interface.', '1910.07486-2-14-7': 'Qin et al. [CITATION] present the semi-automatic tool ByLabel that supports the annotator in segmentation tasks.', '1910.07486-2-14-8': 'The tool FreeLabel [CITATION] is also designed to collect segmentation masks.', '1910.07486-2-14-9': 'It uses scribble annotations as seeds for the region growing algorithm to create a semi-automatic segmentation result.', '1910.07486-2-15-0': 'BIIGLE [CITATION] is a web-based tool that is especially designed for the annotation and exploration of marine image collections.', '1910.07486-2-15-1': 'It provides different annotation and review interfaces, a project management and a user management.', '1910.07486-2-15-2': 'In difference to the other tools it also implements a label tree management.', '1910.07486-2-15-3': 'In contrast to LOST, Biigle does not implement a flexible pipeline system.', '1910.07486-2-16-0': 'Each of the above mentioned annotation tools was designed with a specific application and annotation process in mind.', '1910.07486-2-16-1': 'Due to that, these annotation tools have hard-coded a specific process/ algorithm and model only a single use case.', '1910.07486-2-16-2': 'This leads to the fact that every time a new annotation approach was tested a new tool was implemented.', '1910.07486-2-16-3': 'In contrast to that, our framework is able to model multiple semi-automatic annotation approaches, e.g. [CITATION], in a consistent and fast way.', '1910.07486-2-16-4': 'In this sense our proposed framework is a generalization of single purpose annotation tools.', '1910.07486-2-17-0': '# Approach', '1910.07486-2-18-0': 'We propose a framework for semi-automatic image annotation.', '1910.07486-2-18-1': 'This framework allows any combination of machine learning algorithms and annotation interfaces in a building block style.', '1910.07486-2-19-0': 'Figure [REF] represents a simple example, while much more complex processes can be modeled.', '1910.07486-2-19-1': 'In general an annotation process is defined as a directed graph.', '1910.07486-2-19-2': 'Each node in a graph represents one building block and the connections between the nodes define the order in which the building blocks are processed.', '1910.07486-2-19-3': 'Also information, such as annotations, can be exchanged between connected elements and accessed via the framework API.', '1910.07486-2-20-0': 'The basic building blocks of an annotation pipeline are datasources, annotation tasks and scripts that implement different algorithms.', '1910.07486-2-20-1': 'By means of these building blocks, annotation tasks for humans can be combined with machine learning and other algorithms in a flexible way.', '1910.07486-2-21-0': '## Flexibility', '1910.07486-2-22-0': 'The main feature of LOST is a flexible pipeline concept.', '1910.07486-2-22-1': 'An example for its flexibility is the combination of different annotation interfaces in one pipeline.', '1910.07486-2-22-2': 'When combining a single image annotation interface (SIA, Section [REF]) with a multi image annotation interface (MIA, Section [REF]), annotation tasks can be split into object localization and class label assignment.', '1910.07486-2-22-3': 'See Section [REF] for a detailed description of a two-stage annotation approach where SIA is used to draw bounding boxes and MIA to assign a class label to each box.', '1910.07486-2-23-0': 'Flexible pipelines allow also to combine any kind of machine learning algorithm with an annotation interface to realize semi-automatic annotation approaches.', '1910.07486-2-23-1': 'For example, SIA can be combined with a script that implements an object detector that generates bounding box proposals.', '1910.07486-2-23-2': 'MIA can be connected to a script element that implements an algorithm that clusters images based on their visual similarity in order to speed up the annotation.', '1910.07486-2-23-3': 'See Section [REF] for a semi-automatic annotation pipeline that combines SIA and MIA with machine learning algorithms.', '1910.07486-2-24-0': 'LOST allows also to model iterative annotation processes when adding loop elements to a pipeline (see Section [REF]).', '1910.07486-2-24-1': 'In this way lifelong learning [CITATION] and active learning [CITATION] approaches can be realized with LOST.', '1910.07486-2-25-0': '## Building Blocks', '1910.07486-2-26-0': 'An annotation pipeline (annotation process) can be composed of six different building block types.', '1910.07486-2-26-1': 'These are datasources, scripts, annotation tasks, loops, data exports and visualizations.', '1910.07486-2-26-2': 'After a pipeline was designed as a composition of the different building blocks, it can be loaded into the LOST framework.', '1910.07486-2-26-3': 'When starting (instantiating) a pipeline , each element can be parameterized.', '1910.07486-2-26-4': 'For example, for a datasource a set of images will be selected.', '1910.07486-2-26-5': 'Another common example is the selection of a user or group that will perform an annotation task to parameterize an annotation task-element in a pipeline.', '1910.07486-2-27-0': 'A datasource represents a set of images or videos that can be used by connected elements in the pipeline, for example by one or many scripts.', '1910.07486-2-27-1': 'Such a script-element is an arbitrary algorithm implemented in Python that communicates with connected elements via the framework API.', '1910.07486-2-27-2': 'The main purpose of a script is to generate object proposals or to cluster images for semi-automatic annotation.', '1910.07486-2-27-3': 'An example of such an object proposal could be a bounding box generated by a RetinaNet [CITATION] object detector.', '1910.07486-2-27-4': 'When a script has generated proposals for all images that should be annotated it will send them to an annotation task, where the proposals will be displayed to a human annotator.', '1910.07486-2-28-0': 'An annotation task-element links users, annotation tools and labels.', '1910.07486-2-28-1': 'There are two types of annotation tools that can be used in our current implementation.', '1910.07486-2-28-2': 'The first tool was designed to annotate single images (Section [REF]) and with the second tool clusters of images can be annotated (Section [REF]).', '1910.07486-2-28-3': 'Labels are represented as trees (Section [REF]).', '1910.07486-2-28-4': 'In this way we are able to model label hierarchies.', '1910.07486-2-29-0': 'Loop-blocks can be used for iterative annotation processes where parts of a pipeline need to be executed multiple times until a certain criterion is fulfilled.', '1910.07486-2-29-1': 'Loops are often useful to model active learning or continuous learning approaches.', '1910.07486-2-29-2': 'See Section [REF] for an example how a loop can be used in a pipeline.', '1910.07486-2-30-0': 'The last two element types that can be used are data exports and visualization-elements.', '1910.07486-2-30-1': 'Data exports are used to provide any files created by scripts via the GUI for download.', '1910.07486-2-30-2': 'Similar to data exports, visualizations display images created by scripts within the web interface.', '1910.07486-2-31-0': '## Multi Image Annotation (MIA) Interface', '1910.07486-2-32-0': 'MIA serves to assign labels to clusters of images or annotations.', '1910.07486-2-32-1': 'The main idea is that visual similar objects are likely to get the same label.', '1910.07486-2-32-2': 'This idea is related to the cluster-based approach to fish annotation proposed by [CITATION].', '1910.07486-2-33-0': 'In an ideal world a cluster contains only images that belong to the same class.', '1910.07486-2-33-1': 'A human annotator has the task of sorting out images that do not belong to the cluster.', '1910.07486-2-33-2': 'Therefore, the same label can be assigned to large number of images at the same time.', '1910.07486-2-33-3': 'In the same way labels can be assigned to point, box, line and polygon annotations.', '1910.07486-2-34-0': 'LOST provides the first open source implementation of a MIA interface.', '1910.07486-2-34-1': 'Due to the pipeline concept, MIA can be combined with a SIA interface in one annotation process (see Section [REF]).', '1910.07486-2-34-2': 'In this way, class labels and object localizations can be annotated in specialized interfaces for each task to speed up annotation.', '1910.07486-2-34-3': 'Furthermore, MIA can be combined with different cluster algorithms by connecting it to a script element that will sort annotations or images into clusters.', '1910.07486-2-35-0': '## Single Image Annotation (SIA) Interface', '1910.07486-2-36-0': 'SIA is designed to create polygons, points, lines and bounding box annotations.', '1910.07486-2-36-1': 'To each annotation and the whole image, a class label can be assigned.', '1910.07486-2-36-2': 'Also the assignment of multiple class labels is possible.', '1910.07486-2-36-3': 'Furthermore, the tool is configurable to allow or deny different types of user actions and annotations depending on the use case.', '1910.07486-2-36-4': 'For example, you can specify that only class label assignment is possible, but no other modifications are allowed.', '1910.07486-2-36-5': 'When combining SIA with an algorithm that generates annotation proposals semi-automatic annotation can be realized.', '1910.07486-2-36-6': 'See Section [REF] for an example.', '1910.07486-2-37-0': '## Label Management', '1910.07486-2-38-0': 'Labels are managed in label trees to model label hierarchies.', '1910.07486-2-38-1': 'Multiple label trees can be created and edited.', '1910.07486-2-38-2': 'When starting an annotation task a whole label tree or a composition of subtrees can be selected as possible labels.', '1910.07486-2-38-3': 'During the annotation task the annotator can assign one of the possible labels to an annotation.', '1910.07486-2-39-0': '## Comparison to State-of-the-Art Annotation Tools', '1910.07486-2-40-0': 'Table [REF] presents an overview of the key ideas of LOST in comparison to other open source tools.', '1910.07486-2-40-1': 'LOST is the only tool with a flexible pipeline system, where multiple annotation interfaces and algorithms can be combined in one process.', '1910.07486-2-40-2': 'There are many tools that where build on web technologies to enable collaborative annotation.', '1910.07486-2-41-0': 'Table [REF] shows also three different annotation interface designs and if they are implemented for a specific tool:', '1910.07486-2-42-0': '1) SIA interfaces are used to annotate single images with different annotations like points, boxes, lines, polygons, etc.', '1910.07486-2-42-1': 'Most tools do focus on a SIA interface.', '1910.07486-2-43-0': '2) MIA interfaces are used to present clusters of similar images and assign a label to a whole cluster of images.', '1910.07486-2-43-1': 'This idea of MIA was described by Boom et al. [CITATION].', '1910.07486-2-43-2': 'In the presented comparison LOST is the only tool that has implemented this type of interface.', '1910.07486-2-44-0': '3)Image sequence annotation (ISA) interfaces are especially designed to annotate video sequences with tracking information.', '1910.07486-2-44-1': 'In most cases these interfaces are extensions of a SIA interface.', '1910.07486-2-44-2': 'In the current version, LOST has no ISA interface to annotate tracks.', '1910.07486-2-44-3': 'The implementation of ISA is planned for the near future.', '1910.07486-2-45-0': 'Another difference between LOST and the other tools is that user-defined python scripts can be executed as part of an annotation pipeline.', '1910.07486-2-45-1': 'In such scripts any python code can be implemented and information with other pipeline elements can be exchanged to get full control over the process.', '1910.07486-2-46-0': '# Case Studies', '1910.07486-2-47-0': 'In this Section we show how annotation pipelines can be modeled and executed within our web-based framework.', '1910.07486-2-48-0': 'All case studies are performed and analyzed on the Pascal VOC2012 [CITATION] dataset.', '1910.07486-2-48-1': 'We use [MATH] of the VOC validation split to select the images for annotation and [MATH] for evaluation of trained models.', '1910.07486-2-48-2': 'The VOC training split is used for initial training.', '1910.07486-2-48-3': 'For our experiments we utilize the keras implementations of the respective models.', '1910.07486-2-48-4': 'The experiments are executed inside a NVIDIA Docker container integrated in LOST.', '1910.07486-2-48-5': 'The server is equipped with a NVIDIA Geforce GTX1080Ti, an Intel i7-8700K and 16GB DDR4 RAM.', '1910.07486-2-49-0': '## Single-Stage Annotation', '1910.07486-2-50-0': 'In our first experiment we create a baseline for simple bounding box annotation in the proposed framework as it is possible in most annotation tools.', '1910.07486-2-50-1': 'In order to do that, we use 200 images randomly selected from the Pascal VOC validation set.', '1910.07486-2-50-2': 'In this selection we guarantee that each class is present in at least 10 images.', '1910.07486-2-50-3': 'Two annotators perform bounding boxes annotation according to the VOC annotation guidelines on each image.', '1910.07486-2-50-4': 'Box drawing and class assignment is performed with the SIA tool (Section [REF]).', '1910.07486-2-51-0': 'Results', '1910.07486-2-52-0': 'Annotator1 achieved a mAP of [MATH] and Annotator2 a mAP of [MATH] compared to the Pascal VOC ground truth annotations, when using an intersection over union threshold of [MATH].', '1910.07486-2-52-1': 'Among each other the annotators agreed on [MATH] of the annotated boxes.', '1910.07486-2-52-2': 'This show that even among human annotators the level of agreement was below [MATH] mAP in our experiment.', '1910.07486-2-52-3': 'Which is an interesting fact when considering that object detectors are trained with human data.', '1910.07486-2-53-0': 'On average the annotators needed [MATH] seconds to draw a bounding box and assign a class label to this box.', '1910.07486-2-53-1': 'The average time to annotate an image was [MATH] seconds.', '1910.07486-2-53-2': 'For the annotation of [MATH] images [MATH] minutes of annotation time was required.', '1910.07486-2-54-0': '## Two-Stage Annotation', '1910.07486-2-55-0': 'In this experiment we combine the single image annotation (SIA) with the multi image annotation tool (MIA) in a two-stage annotation process.', '1910.07486-2-55-1': 'In the first stage the annotator has the task to draw bounding boxes according to the VOC annotation guidelines.', '1910.07486-2-55-2': 'In this stage no class labels will be assigned.', '1910.07486-2-55-3': 'In the second stage all boxes from the first stage will be clustered according to their visual similarity and presented in the MIA interface.', '1910.07486-2-55-4': 'The annotators task here is to assign a label to the whole cluster.', '1910.07486-2-55-5': 'If an image does not belong to that cluster, it should be removed by the annotator.', '1910.07486-2-55-6': 'For example, if there are 19 persons and one cat in the view, the annotator should remove the cat and select the label person for the remaining images.', '1910.07486-2-56-0': 'For this experiment, a similar setup as in Section [REF] is used.', '1910.07486-2-56-1': 'The same 200 images from the VOC validation set are annotated by a human annotator, while this time two annotation stages are performed.', '1910.07486-2-57-0': 'We test two clustering strategies.', '1910.07486-2-57-1': 'First we extract CNN features for all annotated boxes from stage one and use the K-Means algorithm to cluster all images.', '1910.07486-2-57-2': 'For feature extraction we use the last pooling layer of ResNet50 [CITATION] that was pretrained on ImageNet [CITATION].', '1910.07486-2-58-0': 'As second clustering approach we fine-tune ResNet50 on a small subset of the Pascal VOC training set and utilize the networks predictions directly for clustering.', '1910.07486-2-59-0': 'Results In the first stage the annotator used the SIA annotation tool and needed [MATH] seconds (see Figure [REF], SIA exp2) to draw a bounding box and [MATH] seconds to draw boxes for all objects in an image.', '1910.07486-2-59-1': 'On average he annotated [MATH] boxes per image, while the total annotation time for 200 images was [MATH] minutes in the first annotation stage.', '1910.07486-2-60-0': 'Figure [REF] shows also the average annotation time per box for class label assignment in the second annotation stage (MIA exp2).', '1910.07486-2-60-1': 'The x-axis indicates the different cluster methods, where K-Means and ResNet50 were used.', '1910.07486-2-60-2': 'ResNet50 was fine-tuned with [MATH] images, [MATH] images, [MATH] images and [MATH] images of the Pascal VOC training split.', '1910.07486-2-60-3': 'We see that the two-stage annotation process (Figure [REF], SIA+MIA exp2) is faster than single stage annotation if the clustering algorithm works well.', '1910.07486-2-60-4': 'Only when ResNet50 was fine-tuned with [MATH] of the training data, the two-stage process consumed the same time as one-stage.', '1910.07486-2-60-5': 'The mAP of the created boxes compared to the VOC ground-truth data was almost equal and around [MATH] for all approaches.', '1910.07486-2-60-6': 'With the fastest two-stage approach it took [MATH] minutes to annotate [MATH] images, where single-stage annotation took [MATH] minutes.', '1910.07486-2-61-0': 'When considering that class label assignment plus box drawing with SIA needs [MATH] seconds and box drawing only needs [MATH]s, we know that pure class label assignment with SIA takes [MATH] seconds.', '1910.07486-2-61-1': 'The fastest annotation approach with MIA takes [MATH] seconds per box, which is a speed up in class label assignment by a factor of two.', '1910.07486-2-62-0': 'Use cases The main idea here was to break down the complex task of simultaneous bounding box drawing and class label assignment into two separate tasks that are simpler, while supporting the annotator with a preclustering during class label assignment.', '1910.07486-2-62-1': 'In this way it was also possible to split the annotation work into a simple and an expert task, where in most cases expert knowledge is required for class label assignment as in many biological [CITATION] or medical applications [CITATION].', '1910.07486-2-62-2': 'We saw that when using the MIA annotation tool and a good clustering algorithm the time for class label assignment was reduced by a factor of two compared to label assignment with SIA.', '1910.07486-2-62-3': 'This allows for saving expensive annotation time for expert tasks.', '1910.07486-2-62-4': 'For the experts, who will not need to draw a box, annotation time is reduced by a factor of [MATH] to [MATH].', '1910.07486-2-63-0': '## Two-Stage Annotation in the loop', '1910.07486-2-64-0': 'In this experiment we show how to implement an iterative annotation process within our framework.', '1910.07486-2-64-1': 'As in Section [REF] we model a two-stage annotation pipeline composed of a single image annotation task and a clustered image annotation view.', '1910.07486-2-64-2': 'In difference to the previous experiment we use semi-automatic support in both annotation stages, do not use any VOC data for pretraining and put everything into a loop.', '1910.07486-2-64-3': 'In other words we assume that we have no annotated data in the beginning and try to get a better automatic support for the annotator over time.', '1910.07486-2-65-0': 'As in the previous experiments we use images from the Pascal VOC2012 validation dataset for annotation.', '1910.07486-2-65-1': 'Due to the iterative setup we use [MATH] images per iteration that will be processed by two human annotators.', '1910.07486-2-65-2': 'In contrast to the experiment in Section [REF], both annotators work on the same annotation task to split the workload.', '1910.07486-2-66-0': 'Figure [REF] shows a high level view of the annotation pipeline.', '1910.07486-2-66-1': 'In the first annotation stage RetinaNet [CITATION] is used for bounding box proposal generation in order to support the human annotator.', '1910.07486-2-66-2': 'As proposals we use all boxes with a confidence value above [MATH].', '1910.07486-2-66-3': 'After each iteration, RetinaNet will be retrained with all annotations from previous iterations.', '1910.07486-2-66-4': 'When all images have been processed by RetinaNet a SIA task will be performed by the human annotators.', '1910.07486-2-66-5': 'Since there are no annotations in the first iteration, no box proposals are generated in the first iteration.', '1910.07486-2-66-6': 'The annotators are instructed to draw bounding boxes around all VOC2012 objects in the images.', '1910.07486-2-67-0': 'In the second annotation stage ResNet50 [CITATION] is used to cluster all bounding boxes by class.', '1910.07486-2-67-1': 'We use pretrained ImageNet weights for initialization and fine-tune ResNet50 after each iteration with all annotations from previous iterations.', '1910.07486-2-67-2': 'In the first iteration where no class label annotations are available, we use ImageNet class predictions for clustering.', '1910.07486-2-67-3': 'The idea here is that visual similar images will get the same class label, even if the predicted class is not part of Pascal VOC.', '1910.07486-2-67-4': 'After that, the MIA tool is used to correct the proposed clusters and to assign class labels to the clustered box annotations.', '1910.07486-2-67-5': 'When the second stage was processed, the next loop iteration will start.', '1910.07486-2-68-0': 'Results', '1910.07486-2-69-0': 'Figure [REF] presents the average annotation time per box per iteration.', '1910.07486-2-69-1': 'We see times for box drawing in the first annotation stage (SIA exp3), the times for class label assignment in the second stage(MIA exp3), the total time per box for the looped two-stage annotation approach (Total exp3) and the single-stage annotation approach for comparison (SIA only exp1).', '1910.07486-2-69-2': 'In the first iteration, when no box proposals are generated and the cluster algorithm is not fine-tuned to the VOC dataset, the looped two-stage approach is slower than single-stage annotation.', '1910.07486-2-69-3': 'But in the following iterations when RetinaNet and ResNet50 are fine-tuned with the annotations of the previous iterations, the looped two-stage annotation process gets faster than single-stage annotation.', '1910.07486-2-70-0': 'Figure [REF] shows the corresponding mAPs of the human-annotated boxes (Annotator) and the detector performance of RetinaNet per iteration.', '1910.07486-2-70-1': 'While the detector performance increases, the performance of the created annotations seems to be stable around a mean of [MATH] mAP with a standard deviation of [MATH].', '1910.07486-2-70-2': 'This deviation most likely reflects the annotators attention level and the difficulty of the images that have been annotated.', '1910.07486-2-71-0': 'Use cases We found that a looped two-stage annotation approach would be beneficial if there are no ground-truth data available in the beginning of the annotation process.', '1910.07486-2-71-1': 'This approach creates annotations that have an equal quality compared to the single-stage approach and trains a detector on-the-fly, while taking less annotator time than single-stage annotation.', '1910.07486-2-71-2': 'Since it is modeled as two-stage process, the annotation work can be split in a simple and an expert task as in Section [REF].', '1910.07486-2-71-3': 'It is notable that the training process of the machine learning models takes additional time compared to annotation approaches without machine learning elements, but we found that this is no problem when performing the training in time slots when the annotator needs to rest anyway e. g. over night.', '1910.07486-2-72-0': '# Conclusion', '1910.07486-2-73-0': 'To the best of our knowledge, we present the first framework for a flexible design and instantiation of image annotation pipelines.', '1910.07486-2-73-1': 'Our approach enables the combination of different annotation tools and machine learning algorithms in one process.', '1910.07486-2-73-2': 'We also provide an annotation interface called MIA (multi image annotation) to annotate whole clusters of images at the same time.', '1910.07486-2-74-0': 'Our case studies show how our framework can be used to model machine learning based semi-automatic annotation pipelines and iterative annotation approaches.', '1910.07486-2-74-1': 'In Section [REF] we found that simple clustering in combination with the MIA annotation interface can speed up class label assignment by a factor of two compared to single stage annotation.', '1910.07486-2-74-2': 'We also show that an annotation task can be split in an expert and a simple task, which can significantly reduce expensive expert annotation time.', '1910.07486-2-74-3': 'We further show in Section [REF] that a looped two-stage approach is beneficial when no annotation data is available in the beginning.', '1910.07486-2-74-4': 'The quality of the created annotations is kept high while the time spent for box annotation gets smaller over time.', '1910.07486-2-75-0': 'In future, we plan to release the missing image sequence annotation interface (ISA) that is specialized to annotate tracks.', '1910.07486-2-75-1': 'We also want to implement an interface to Mechanical Turk for crowdsourcing applications.'} | [['1910.07486-1-0-0', '1910.07486-2-0-0'], ['1910.07486-1-0-1', '1910.07486-2-0-1'], ['1910.07486-1-0-2', '1910.07486-2-0-2'], ['1910.07486-1-0-3', '1910.07486-2-0-3'], ['1910.07486-1-0-4', '1910.07486-2-0-4'], ['1910.07486-1-0-5', '1910.07486-2-0-5'], ['1910.07486-1-0-6', '1910.07486-2-0-6'], ['1910.07486-1-0-7', '1910.07486-2-0-7'], ['1910.07486-1-53-0', 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['1910.07486-1-3-0', '1910.07486-2-3-0'], ['1910.07486-1-32-0', '1910.07486-2-32-0'], ['1910.07486-1-32-1', '1910.07486-2-32-1'], ['1910.07486-1-32-2', '1910.07486-2-32-2'], ['1910.07486-1-34-0', '1910.07486-2-34-0'], ['1910.07486-1-34-1', '1910.07486-2-34-1'], ['1910.07486-1-34-2', '1910.07486-2-34-2'], ['1910.07486-1-34-3', '1910.07486-2-34-3'], ['1910.07486-1-44-0', '1910.07486-2-44-0'], ['1910.07486-1-44-1', '1910.07486-2-44-1'], ['1910.07486-1-44-2', '1910.07486-2-44-2'], ['1910.07486-1-44-3', '1910.07486-2-44-3'], ['1910.07486-1-7-0', '1910.07486-2-7-0'], ['1910.07486-1-7-1', '1910.07486-2-7-1'], ['1910.07486-1-7-2', '1910.07486-2-7-2'], ['1910.07486-1-7-3', '1910.07486-2-7-3'], ['1910.07486-1-11-0', '1910.07486-2-11-0'], ['1910.07486-1-11-1', '1910.07486-2-11-1'], ['1910.07486-1-11-2', '1910.07486-2-11-2'], ['1910.07486-1-11-3', '1910.07486-2-11-3'], ['1910.07486-1-33-0', '1910.07486-2-33-0'], ['1910.07486-1-33-1', '1910.07486-2-33-1'], ['1910.07486-1-33-2', '1910.07486-2-33-2'], ['1910.07486-1-33-3', '1910.07486-2-33-3'], ['1910.07486-1-23-0', '1910.07486-2-23-0'], ['1910.07486-1-23-1', '1910.07486-2-23-1'], ['1910.07486-1-23-2', '1910.07486-2-23-2'], ['1910.07486-1-23-3', '1910.07486-2-23-3'], ['1910.07486-1-38-0', '1910.07486-2-38-0'], ['1910.07486-1-38-1', '1910.07486-2-38-1'], ['1910.07486-1-38-2', '1910.07486-2-38-2'], ['1910.07486-1-38-3', '1910.07486-2-38-3'], ['1910.07486-1-69-0', '1910.07486-2-69-0'], ['1910.07486-1-69-1', '1910.07486-2-69-1'], ['1910.07486-1-69-2', '1910.07486-2-69-2'], ['1910.07486-1-69-3', '1910.07486-2-69-3'], ['1910.07486-1-24-0', '1910.07486-2-24-0'], ['1910.07486-1-24-1', '1910.07486-2-24-1'], ['1910.07486-1-43-0', '1910.07486-2-43-0'], ['1910.07486-1-43-1', '1910.07486-2-43-1'], ['1910.07486-1-43-2', '1910.07486-2-43-2'], ['1910.07486-1-2-0', '1910.07486-2-2-0'], ['1910.07486-1-2-1', '1910.07486-2-2-1'], ['1910.07486-1-2-2', '1910.07486-2-2-2'], ['1910.07486-1-4-0', '1910.07486-2-4-0'], ['1910.07486-1-4-1', '1910.07486-2-4-1'], ['1910.07486-1-8-0', '1910.07486-2-8-0'], ['1910.07486-1-8-1', '1910.07486-2-8-1'], ['1910.07486-1-8-2', '1910.07486-2-8-2'], ['1910.07486-1-8-3', '1910.07486-2-8-3'], ['1910.07486-1-8-4', '1910.07486-2-8-4'], ['1910.07486-1-8-5', '1910.07486-2-8-5'], ['1910.07486-1-8-6', '1910.07486-2-8-6'], ['1910.07486-1-75-0', '1910.07486-2-75-0'], ['1910.07486-1-75-1', '1910.07486-2-75-1'], ['1910.07486-1-18-0', '1910.07486-2-18-0'], ['1910.07486-1-18-1', '1910.07486-2-18-1'], ['1910.07486-1-12-0', '1910.07486-2-12-0'], ['1910.07486-1-12-1', '1910.07486-2-12-1'], ['1910.07486-1-12-2', '1910.07486-2-12-2'], ['1910.07486-1-12-3', '1910.07486-2-12-3'], ['1910.07486-1-12-4', '1910.07486-2-12-4'], ['1910.07486-1-12-5', '1910.07486-2-12-5'], ['1910.07486-1-12-6', '1910.07486-2-12-6'], ['1910.07486-1-55-0', '1910.07486-2-55-0'], ['1910.07486-1-55-1', '1910.07486-2-55-1'], ['1910.07486-1-55-2', '1910.07486-2-55-2'], ['1910.07486-1-55-3', '1910.07486-2-55-3'], ['1910.07486-1-55-4', '1910.07486-2-55-4'], ['1910.07486-1-55-5', '1910.07486-2-55-5'], ['1910.07486-1-55-6', '1910.07486-2-55-6'], ['1910.07486-1-16-0', '1910.07486-2-16-0'], ['1910.07486-1-16-1', '1910.07486-2-16-1'], ['1910.07486-1-16-2', '1910.07486-2-16-2'], ['1910.07486-1-16-3', '1910.07486-2-16-3'], ['1910.07486-1-16-4', '1910.07486-2-16-4'], ['1910.07486-1-45-0', '1910.07486-2-45-0'], ['1910.07486-1-45-1', '1910.07486-2-45-1'], ['1910.07486-1-47-0', '1910.07486-2-47-0'], ['1910.07486-1-57-0', '1910.07486-2-57-0'], ['1910.07486-1-57-1', '1910.07486-2-57-1'], ['1910.07486-1-57-2', '1910.07486-2-57-2'], ['1910.07486-1-64-0', '1910.07486-2-64-0'], ['1910.07486-1-64-1', '1910.07486-2-64-1'], ['1910.07486-1-64-2', '1910.07486-2-64-2'], ['1910.07486-1-64-3', '1910.07486-2-64-3'], ['1910.07486-1-73-0', '1910.07486-2-73-0'], ['1910.07486-1-73-1', '1910.07486-2-73-1'], ['1910.07486-1-73-2', '1910.07486-2-73-2'], ['1910.07486-1-42-0', '1910.07486-2-42-0'], ['1910.07486-1-42-1', '1910.07486-2-42-1'], ['1910.07486-1-61-0', '1910.07486-2-61-0'], ['1910.07486-1-61-1', '1910.07486-2-61-1'], ['1910.07486-1-74-0', '1910.07486-2-74-0'], ['1910.07486-1-74-1', '1910.07486-2-74-1'], ['1910.07486-1-74-2', '1910.07486-2-74-2'], ['1910.07486-1-74-3', '1910.07486-2-74-3'], ['1910.07486-1-74-4', '1910.07486-2-74-4'], ['1910.07486-1-30-0', '1910.07486-2-30-0'], ['1910.07486-1-30-1', '1910.07486-2-30-1'], ['1910.07486-1-30-2', '1910.07486-2-30-2']] | [] | [] | [] | [] | ['1910.07486-1-41-0', '1910.07486-1-51-0', '1910.07486-1-68-0', '1910.07486-2-41-0', '1910.07486-2-51-0', '1910.07486-2-68-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1910.07486 | null | null | null | null | null |
astro-ph-0605019 | {'astro-ph-0605019-1-0-0': 'The Wide Field Imager Lyman-Alpha Search (WFILAS) for Galaxies at Redshift [MATH]5.7', 'astro-ph-0605019-1-1-0': 'II.', 'astro-ph-0605019-1-1-1': 'Survey Design and Sample Analysis', 'astro-ph-0605019-1-2-0': 'Wide-field narrowband surveys are an efficient way of searching large volumes of high-redshift space for distant galaxies.', 'astro-ph-0605019-1-2-1': 'We describe the Wide Field Imager Lyman-Alpha Search (WFILAS) over 0.74sq. degree for bright emission-line galaxies at [MATH].', 'astro-ph-0605019-1-2-2': 'WFILAS uses deep images taken with the Wide Field Imager (WFI) on the ESO/MPI 2.2m telescope in three narrowband (70), one encompassing intermediate band (220) and two broadband filters, [MATH] and [MATH].', 'astro-ph-0605019-1-2-3': 'We use the novel technique of an encompassing intermediate band filter to exclude false detections.', 'astro-ph-0605019-1-2-4': 'Images taken with broadband [MATH] and [MATH] filters are used to remove low redshift galaxies from our sample.', 'astro-ph-0605019-1-2-5': 'We present a sample of seven Ly[MATH] emitting galaxy candidates, two of which are spectroscopically confirmed.', 'astro-ph-0605019-1-2-6': 'Compared to other surveys all our candidates are bright, the results of this survey complements other narrowband surveys at this redshift.', 'astro-ph-0605019-1-2-7': 'Most of our candidates are in the regime of bright luminosities, beyond the reach of less voluminous surveys.', 'astro-ph-0605019-1-2-8': 'Adding our candidates to those of another survey increases the derived luminosity density by [MATH]30%.', 'astro-ph-0605019-1-2-9': 'We also find potential clustering in the Chandra Deep Field South, supporting overdensities discovered by other surveys.', 'astro-ph-0605019-1-2-10': 'Based on a FORS2/VLT spectrum we additionally present the analysis of the second confirmed Ly[MATH] emitting galaxy in our sample.', 'astro-ph-0605019-1-2-11': 'We find that it is the brightest Ly[MATH] emitting galaxy (1[MATH]ergs[MATH]cm[MATH]) at this redshift to date and the second confirmed candidate of our survey.', 'astro-ph-0605019-1-2-12': 'Both objects exhibit the presence of a possible second Ly[MATH] component redward of the line.', 'astro-ph-0605019-1-3-0': 'WFILAS for Galaxies at [MATH]5.7', 'astro-ph-0605019-1-4-0': '# Introduction', 'astro-ph-0605019-1-5-0': 'Detections of both galaxies and QSOs at [MATH] indicate that the Universe was largely reionised at that epoch.', 'astro-ph-0605019-1-5-1': 'The recent three-year WMAP results combined with other cosmological surveys suggest an epoch of reionisation aroun [MATH] , consistent with both QSO results and the epoch predicted by structure formation models .', 'astro-ph-0605019-1-5-2': 'While the UV contributions of QSOs and AGN are almost certainly not responsible for reionisation , faint star forming galaxies need to exist in extraordinary numbers if they are to be the cause .', 'astro-ph-0605019-1-5-3': 'However, analyses of the Hubble Ultra Deep Field failed to find sufficient numbers of faint galaxies to support this idea .', 'astro-ph-0605019-1-5-4': 'Therefore, it is crucial to investigate what the contribution to the ionising UV flux is from young stellar populations of star forming galaxies.', 'astro-ph-0605019-1-6-0': 'Broadly speaking, two classes of star-forming galaxy dominate high redshift surveys: Lyman Break Galaxies (LBGs) and Lyman-[MATH] Emitters (LAEs).', 'astro-ph-0605019-1-6-1': 'LBG surveys, which now number in the thousands of objects at [MATH]=3 to 5, find clumpy source distributions and a two-point angular correlation function indicative of strong clustering .', 'astro-ph-0605019-1-6-2': 'LAEs also show evidence for clustering although many of the LAE surveys target fields surrounding known sources such as proto-clusters, radio galaxies and QSOs .', 'astro-ph-0605019-1-6-3': 'On average, LAEs number 1.5[MATH][MATH]deg[MATH] per unit redshift down to 1.5[MATH][MATH]ergs[MATH]cm[MATH] at [MATH] and 4.5 .', 'astro-ph-0605019-1-6-4': 'Also, their consistently small size ([MATH]0.6[MATH]kpc) suggests they are subgalactic clumps residing in the wind-driven outflows of larger unseen hosts .', 'astro-ph-0605019-1-6-5': 'Such mechanisms provide a straightforward means of UV photon escape from the host galaxy, efficiently reionising the surrounding IGM in a way than ordinary LBGs can not.', 'astro-ph-0605019-1-7-0': 'The most efficient way to find LAEs is through imaging surveys using a combination of broad- and narrowband filters.', 'astro-ph-0605019-1-7-1': 'The advent of wide field cameras has allowed systematic imaging searches that have been carried out to build up samples of candidate LAEs at high redshifts .', 'astro-ph-0605019-1-7-2': 'The availability of high throughput spectrographs on 8 to 10m-class telescopes has enabled the spectroscopic confirmation of these galaxies.', 'astro-ph-0605019-1-7-3': 'Such direct imaging searches typically cover 10[MATH]-10[MATH] times the volume of blind long-slit spectroscopic searches .', 'astro-ph-0605019-1-7-4': 'Furthermore, candidates from narrowband surveys always have an identifiable emission feature that is well separated from sky lines courtesy of the filter design.', 'astro-ph-0605019-1-7-5': "This is in contrast to other methods, including the widely-used 'dropout' technique .", 'astro-ph-0605019-1-8-0': 'The narrowband filter design leads to a higher candidate LAE selection efficiency than other techniques.', 'astro-ph-0605019-1-8-1': 'The only way to secure the identification of the emission line is spectroscopic follow-up.', 'astro-ph-0605019-1-8-2': 'The most common low redshift interlopers are the emission line doublets of [Oii] [MATH]3726,3728 and [Oiii] [MATH]4959,5007.', 'astro-ph-0605019-1-8-3': 'These can be identified by obtaining spectra with a resolution [MATH] to separate the line pair.', 'astro-ph-0605019-1-8-4': 'Other emission lines, such as H[MATH] and H[MATH], can be identified by neighbouring lines.', 'astro-ph-0605019-1-8-5': 'The narrowband technique has been successfully applied by many authors in order to discover galaxies at redshift 5[MATH]6 and to locate galaxies at redshift 6[MATH]7 .', 'astro-ph-0605019-1-8-6': 'Likewise, we employ the narrowband technique in the Wide Field Imager Lyman-Alpha Search (WFILAS) to find galaxies at [MATH].', 'astro-ph-0605019-1-8-7': 'In Paper I in this series , we described a compact LAE at [MATH] discovered by our survey.', 'astro-ph-0605019-1-9-0': 'In this Paper, we describe the survey design and sample analysis of WFILAS.', 'astro-ph-0605019-1-9-1': 'In Sect. [REF] we describe the scope of the survey and the observing strategy.', 'astro-ph-0605019-1-9-2': 'The data reduction is described in Sect. [REF].', 'astro-ph-0605019-1-9-3': 'Section [REF] outlines the candidate selection and Sect. [REF] outlines sample properties and comparison to other surveys.', 'astro-ph-0605019-1-9-4': 'We discuss the spectroscopic follow-up of two candidates in Sect. [REF].', 'astro-ph-0605019-1-9-5': 'Throughout this paper we assume a flat Universe with [MATH] and a Hubble constant [MATH]kms[MATH]Mpc[MATH].', 'astro-ph-0605019-1-9-6': 'All quoted magnitudes are in the AB system .', 'astro-ph-0605019-1-10-0': '# WFILAS Survey Design and Observations', 'astro-ph-0605019-1-11-0': 'The sky area surveyed by the WFILAS is [MATH]0.74sq. degree.', 'astro-ph-0605019-1-11-1': 'We observed three fields in broadbands [MATH], [MATH] and in an intermediate width filter centred at 815nm encompassing three narrowband filters (Fig. [REF]).', 'astro-ph-0605019-1-11-2': 'The adoption of an additional intermediate width filter encompassing the multiple narrowband width filters is a novel approach compared to previous narrowband surveys.', 'astro-ph-0605019-1-11-3': 'The application of the intermediate band filter enables us to drastically reduce the number of spurious detections in the narrowband filters.', 'astro-ph-0605019-1-11-4': 'The narrow width of the narrowband filters (FWHM=7nm) gives a prominent appearance to emission line objects.', 'astro-ph-0605019-1-11-5': 'Furthermore, the three chosen fields are spread across the sky to enable us to average out variations in cosmic variance.', 'astro-ph-0605019-1-11-6': 'Our search has covered one of the largest co-moving volumes compared to other surveys.', 'astro-ph-0605019-1-11-7': 'Table [REF] compares WFILAS with other published surveys.', 'astro-ph-0605019-1-12-0': 'The observations were taken with the Wide Field Imager on the ESO/MPI 2.2m telescope at the Cerro La Silla Observatory, Chile.', 'astro-ph-0605019-1-12-1': 'The data were taken over 65 separate nights from 2001 January 19 to 2003 December 1.', 'astro-ph-0605019-1-12-2': 'The WFI is a mosaic of eight ([MATH]) 2k [MATH] 4k CCDs arranged to give a field of view of 34[MATH].', 'astro-ph-0605019-1-12-3': 'The pixels are 0238 on a side.', 'astro-ph-0605019-1-13-0': 'As WFILAS was planned as joint project of ESO Santiago and the COMBO-17 team at MPIA Heidelberg, three fields were selected to overlap with the COMBO-17 survey, i.e. their extended Chandra Deep Field South (CDFS), SGP (South Galactic Pole) and S11 fields.', 'astro-ph-0605019-1-13-1': 'The coordinates of the field centres and the exposure times in each of the filters for each field are given in Table [REF].', 'astro-ph-0605019-1-13-2': 'All three fields are at high Galactic latitude ([MATH]) and have extinctions less than [MATH]=0.022 mag .', 'astro-ph-0605019-1-14-0': 'We employ standard broadband [MATH] and [MATH] filters.', 'astro-ph-0605019-1-14-1': 'The intermediate band (FWHM = 22nm) observatory filter is centred at 815nm.', 'astro-ph-0605019-1-14-2': 'The three custom made narrowband (FWHM = 7nm) filters are centered at 810nm, 817nm and 824nm.', 'astro-ph-0605019-1-14-3': 'The transmission profiles of the filters are shown in Fig. [REF].', 'astro-ph-0605019-1-14-4': 'The intermediate and narrowband filters are designed to fit in the atmospheric 815nm OH-airglow window, where the brightness of the sky background is low and hence favourable to detect Ly[MATH] emission at redshift [MATH]5.7.', 'astro-ph-0605019-1-14-5': 'The data taken with the intermediate band filter confirm detections of the Ly[MATH] line in one of the narrowband filters.', 'astro-ph-0605019-1-14-6': 'The broadband [MATH] and [MATH] data, which were taken from the COMBO-17 survey , are used to confirm the absence of continuum blueward of the Ly[MATH] line and to avoid sample contamination by lower redshift emission line galaxies (e.g. H[MATH] at [MATH], or [Oii] at [MATH]).', 'astro-ph-0605019-1-15-0': 'To establish the photometric zero-point of the intermediate and narrowband filters two spectrophotometric standard stars were observed.', 'astro-ph-0605019-1-16-0': 'Between 10-50 exposures were taken for each intermediate and narrowband filter for each field.', 'astro-ph-0605019-1-16-1': 'The exposure times varied between 1000 and 1800sec per frame, with a typical exposure time of around 1600sec. All frames are background-limited despite the low night sky emission in this spectral region.', 'astro-ph-0605019-1-16-2': 'The median, first and last decile of both seeing and background are given in Table [REF].', 'astro-ph-0605019-1-17-0': '# Data Reduction', 'astro-ph-0605019-1-18-0': 'The data were processed with standard IRAF routines (MSCRED TASK) and our own specially designed scripts.', 'astro-ph-0605019-1-18-1': 'The initial steps in the reduction process consist of removing the zero level offset with bias frames, normalising pixel-to-pixel sensitivity differences with twilight flatfield frames and removal of fringes with fringe frames.', 'astro-ph-0605019-1-18-2': 'During these steps, the 8 CCDs that make up a single WFI image are treated independently.', 'astro-ph-0605019-1-18-3': 'These processes are described in detail below.', 'astro-ph-0605019-1-19-0': 'Normally, the overscan region of the science frames can be used to remove the zero level offset.', 'astro-ph-0605019-1-19-1': 'However, it was noticed that the bias frames contained significant intermediate scale structure (10-30 pixels).', 'astro-ph-0605019-1-19-2': 'To remove this, bias frames were taken on every day of our observations and averaged into a bias frame for that day.', 'astro-ph-0605019-1-19-3': 'In order to minimise the noise added to the data by subtracting the bias, the bias frames were smoothed by 5 pixels and 30 pixels in horizontal and vertical direction of the CCDs, respectively, and subsequently medianed.', 'astro-ph-0605019-1-19-4': 'The structures are stable over periods of several months.', 'astro-ph-0605019-1-19-5': 'Therefore, it was possible to use bias frames from different nights without degrading the quality of the data.', 'astro-ph-0605019-1-20-0': 'Typically, five twilight flatfield frames were taken in one night for one or more filters.', 'astro-ph-0605019-1-20-1': 'The frames were medianed and the science data was divided by the median.', 'astro-ph-0605019-1-20-2': 'Hence pixel-to-pixel sensitivity differences were removed.', 'astro-ph-0605019-1-20-3': 'The structure in the individual flatfield frames was stable over a period of several weeks.', 'astro-ph-0605019-1-20-4': 'Frames taken on different nights could thus be reused.', 'astro-ph-0605019-1-20-5': 'Any differences between flatfield frames were due to the appearance or disappearance of dust features, or large scale illumination differences.', 'astro-ph-0605019-1-20-6': 'The differences rarely amounted to more than a few percent.', 'astro-ph-0605019-1-21-0': 'The raw data in the intermediate and narrowband filters show fringe patterns with amplitudes of up to 10% which was only partially removed after the data had been flatfielded.', 'astro-ph-0605019-1-21-1': 'To entirely remove the fringe pattern, we subtracted a fringe frame created from 10-30 science frames.', 'astro-ph-0605019-1-21-2': 'The fringing is very stable over time, so we were able to use data spanning several months.', 'astro-ph-0605019-1-21-3': 'Certain science frames still show fringe patterns because they are contaminated by either moonlight or twilight.', 'astro-ph-0605019-1-21-4': 'Residual differences in the level of the background between the different CCDs were removed by subtracting the median background level from each CCD.', 'astro-ph-0605019-1-22-0': 'To produce the final deep images we only used images with a seeing of less than 5 pixels (=12) and without significant residual fringing.', 'astro-ph-0605019-1-22-1': 'To make the combining of the images possible, we had to apply an astrometric correction based on stars from the USNO CCD Astrograph Catalogue 2 in the three observed fields.', 'astro-ph-0605019-1-22-2': 'The frames have a set pixel scale of 0238 pixel[MATH] with North up and East left.', 'astro-ph-0605019-1-22-3': "The images were weighted according to their exposure time and combined using the IRAF 'mscstack' routine rejecting deviant pixels.", 'astro-ph-0605019-1-22-4': 'Table [REF] summarises the depth, image quality and total exposure time, for each coadded frame.', 'astro-ph-0605019-1-23-0': '# Sample Selection and Completeness', 'astro-ph-0605019-1-24-0': '## Photometry and Noise Characteristics', 'astro-ph-0605019-1-25-0': 'Initial source catalogues were created for each of the 8 narrowband images.', 'astro-ph-0605019-1-25-1': 'Each catalogue contains the photometry for the sources in all 6 filters.', 'astro-ph-0605019-1-25-2': 'We used the SExtractor source detection software .', 'astro-ph-0605019-1-25-3': 'Sources were selected when at least 5 pixels were 0.8[MATH] above the noise level in the narrowband image used for detection.', 'astro-ph-0605019-1-25-4': 'The photometry was measured in two apertures, 6 and 10 pixels in diameter (=14 and 24, respectively).', 'astro-ph-0605019-1-25-5': 'The 6 pixel aperture was used to maximise the signal-to-noise of the flux of the objects, while the larger 10 pixel aperture was used for the more accurate determination of the total flux and hence the star formation rate.', 'astro-ph-0605019-1-26-0': 'Some authors have found that SExtractor underestimates flux uncertainties .', 'astro-ph-0605019-1-26-1': 'SExtractor estimates the uncertainties using various assumptions that are often not valid (e.g. perfect flatfielding, perfect sky subtraction).', 'astro-ph-0605019-1-26-2': 'The pixel-to-pixel noise in our data is slightly correlated because the scatter in the counts summed in 6 pixel apertures is about 10% higher than what one would derive from the measured pixel-to-pixel RMS.', 'astro-ph-0605019-1-27-0': 'We devised a method to correct the uncertainties given by SExtractor to their true values as follows.', 'astro-ph-0605019-1-27-1': 'First, sources with flux in all filters and their [MATH] magnitude between 16 and 23 were selected.', 'astro-ph-0605019-1-27-2': 'Sources brighter than [MATH],=16 are typically saturated, while those fainter than [MATH],=23 are incomplete (see Sect. [REF] for a further discussion of incompleteness).', 'astro-ph-0605019-1-27-3': 'The colour (where [MATH] is any of narrowband filters [MATH], [MATH], or [MATH]) is the same for any flat continuum source.', 'astro-ph-0605019-1-27-4': 'Therefore, the spread in the colour will be the same as the true flux uncertainty from the two contributing filters.', 'astro-ph-0605019-1-27-5': 'Next, the sources were binned into 200-source bins based on their [MATH] magnitude.', 'astro-ph-0605019-1-27-6': 'In Fig. [REF] we plot the colour versus the [MATH] magnitude of one of our S11 catalogues.', 'astro-ph-0605019-1-27-7': 'Mean values for the colour, [MATH], [MATH] magnitude and the mean of the SExtractor uncertainty were calculated for each bin.', 'astro-ph-0605019-1-27-8': 'The uncertainty in the colour for each object was determined by adding the uncertainty of [MATH] and [MATH] in quadrature ([MATH]).', 'astro-ph-0605019-1-27-9': 'The interval in which 68.3% of the objects were closest to this mean colour was used to infer the actual 1[MATH] colour uncertainty.', 'astro-ph-0605019-1-27-10': 'We assumed that the ratio between the old uncertainties [MATH] and [MATH] was the same for the new uncertainties [MATH] and [MATH].', 'astro-ph-0605019-1-27-11': 'We related between the new and old uncertainty in the intermediate and narrowband flux using the function [MATH], where [MATH] is the zero-offset for the uncertainty in the flux of bright sources and [MATH] is the ratio between the new and old uncertainty for the flux of the faintest sources.', 'astro-ph-0605019-1-27-12': 'The parameters [MATH] and [MATH] correspond to imperfections in the photometry and wrongly assumed background by SExtractor, respectively.', 'astro-ph-0605019-1-28-0': 'Typically, the correction factors are moderate (between [MATH]50%) for the faint sources in the catalogues.', 'astro-ph-0605019-1-28-1': 'Even though the correction factors are moderate, we assume that the corrections for the uncertainties in the broadband [MATH] and [MATH] are irrelevant, since they are used in a different way than the intermediate and narrowband images (see Sect. [REF]).', 'astro-ph-0605019-1-29-0': '## Selection criteria', 'astro-ph-0605019-1-30-0': 'The following four criteria were applied to select our candidate LAEs from the eight initial source catalogues:', 'astro-ph-0605019-1-31-0': 'The narrowband image used as the detection image must have the most flux of all the narrowband images and the source must have a 4[MATH] detection or better, The narrowband image with the least flux needs to be a non-detection, i.e. less than 2[MATH], There must be at least a 2[MATH] detection in the intermediate band image, None of the broadband images, i.e. neither [MATH] nor [MATH], must have a detection above 2[MATH].', 'astro-ph-0605019-1-32-0': 'Table [REF] contains the values of the 2[MATH] detection thresholds of the images used for the 6 pixel aperture.', 'astro-ph-0605019-1-32-1': 'In total 33 candidates were selected using the above criteria.', 'astro-ph-0605019-1-32-2': 'Visual inspection showed that 26 sources arose from artefacts of which the vast majority were out-of-focus ghost rings from bright stars.', 'astro-ph-0605019-1-32-3': 'The final sample contains seven candidate LAEs.', 'astro-ph-0605019-1-33-0': 'We note here the importance of the usage of the intermediate band filter.', 'astro-ph-0605019-1-33-1': 'If we were to reapply all the criteria except for criterion 3, i.e. we do not use the intermediate band images, we would obtain 284 candidates instead of the 33 for visual inspection.', 'astro-ph-0605019-1-34-0': 'The AB-magnitudes, derived line fluxes and luminosities for the candidates are shown in Table [REF].', 'astro-ph-0605019-1-34-1': 'To convert between AB-mangitudes and line flux in ergs[MATH]cm[MATH] we use the following relation: [EQUATION] where [MATH] and [MATH] are the FWHM and the central wavelength of the narrowband filter in , respectively, and [MATH] the AB-magnitude of the object.', 'astro-ph-0605019-1-34-2': 'In Fig. [REF] the thumbnails of the seven candidate LAEs at [MATH] are shown.', 'astro-ph-0605019-1-34-3': 'We defer a more detailed discussion about the sample properties to Sect. [REF].', 'astro-ph-0605019-1-35-0': '## Completeness corrections', 'astro-ph-0605019-1-36-0': 'From the Hubble Deep Field (HDF) galaxy number-count data for the F814W filter we computed completeness corrections for our eight source catalogues.', 'astro-ph-0605019-1-36-1': 'The HDF counts are determined over the magnitude range [MATH]29, and agree well with our galaxy counts over all narrowband filters in the range [MATH]24.', 'astro-ph-0605019-1-36-2': 'Figure [REF] shows the counts for the F814W filter in the HDF and for the [MATH] filter in the S11 field.', 'astro-ph-0605019-1-36-3': 'Figure [REF] also shows the linear fit used as the basis for the calculation of the detection completeness.', 'astro-ph-0605019-1-36-4': 'The fit is done to the combined number count data over two intervals: [MATH],=[20,22.5], where the WFILAS counts are complete, and [MATH]=[22.5,25], where the HDF counts are linear.', 'astro-ph-0605019-1-37-0': 'Detection completeness is defined as the ratio of WFILAS sources to the number expected from the number-count relation.', 'astro-ph-0605019-1-37-1': 'Figure [REF] shows the derived detection completeness for each filter-field combination used for WFILAS.', 'astro-ph-0605019-1-37-2': 'The differences are mainly due to unequal exposure times, although filter throughput and image quality also play a role.', 'astro-ph-0605019-1-37-3': 'These could explain the overall lower sensitivity of the [MATH] filter, as can be inferred from Fig. [REF].', 'astro-ph-0605019-1-37-4': 'Additionally, we correct for detection completenesss arising due to the intermediate band selection criterion.', 'astro-ph-0605019-1-37-5': 'We constructed a noise image by stacking the intermediate band images without registering.', 'astro-ph-0605019-1-37-6': 'The completeness is defined as the rate of recovery of artificially inserted objects.', 'astro-ph-0605019-1-38-0': 'Given the different sensitivities of each filter-field combination, we define a homogeneous subsample of our initial candidate sample, using the candidates from our four most sensitive field-filter combinations.', 'astro-ph-0605019-1-38-1': "We call this our 'complete' sample (4 of the 7 LAEs; marked in Table [REF]), because once defined, we use the curves in Fig. [REF] to correct the detected candidate numbers for incompleteness, in contrast to our initial 'incomplete' sample (all 7 LAEs).", 'astro-ph-0605019-1-38-2': 'The purpose of the subsample is that it lies within a uniform flux limit.', 'astro-ph-0605019-1-38-3': 'Figure [REF] shows that our four best filter-field combinations consist of the [MATH] and [MATH] filters in both the CDFS and S11 fields.', 'astro-ph-0605019-1-38-4': 'These four field-filter combinations reach at least 50% completeness at [MATH]=23.38, or 5.1[MATH]ergs[MATH]cm[MATH].', 'astro-ph-0605019-1-38-5': 'We take this as the flux limit of our complete sample.', 'astro-ph-0605019-1-38-6': 'As such, the number density derived from the complete sample is a more accurate measure of the density of sources down to the nominated flux limit than the number density of the incomplete sample.', 'astro-ph-0605019-1-38-7': "Figure [REF] shows the luminosity distribution of the complete sample alongside our initial candidate list, which we call the 'incomplete' sample.", 'astro-ph-0605019-1-38-8': 'It shows that in using completeness corrections our detected source density is up by 50%.', 'astro-ph-0605019-1-39-0': '# [MATH] Candidate LAE Catalogue', 'astro-ph-0605019-1-40-0': 'In the previous Sect. we introduced two sets of candidate LAEs: the full (but incomplete) sample of seven candidate LAEs and a subsample thereof, complete to [MATH]=5.1[MATH]ergs[MATH]cm[MATH] (the complete sample).', 'astro-ph-0605019-1-40-1': 'The flux limit of the incomplete sample is almost twice the limit of the complete sample (3.4[MATH]ergs[MATH]cm[MATH]).', 'astro-ph-0605019-1-41-0': 'To examine the luminosity distribution of our sample we use the Schechter function , as it is a good representation of the data at bright luminosities.', 'astro-ph-0605019-1-41-1': 'From this, the luminosity density [MATH] of a distribution with a limiting luminosity [MATH] is given by [EQUATION] where [MATH] and [MATH] represent the slope of the faint end of the Schechter function and the normalisation constant of the galaxy density, respectively.', 'astro-ph-0605019-1-41-2': '[MATH] is the incomplete gamma-function.', 'astro-ph-0605019-1-41-3': 'Currently, the luminosity function for LAEs at [MATH] is poorly defined and authors commonly adopt either one or two of the three parameters from low redshift surveys to calculate the third.', 'astro-ph-0605019-1-42-0': 'We examine the influence of non-detections of bright ([MATH]) LAEs for the total Ly[MATH] luminosity density by employing the same method as [CITATION], another narrowband imaging survey aimed at finding LAEs at [MATH].', 'astro-ph-0605019-1-42-1': 'In the interest of comparison, we follow [CITATION] exactly and adopt the [CITATION] values for [MATH] (-1.53) and [MATH]3).', 'astro-ph-0605019-1-42-2': 'Their approach was to solve eq. ([REF]) for [MATH], instead of fitting a Schechter function.', 'astro-ph-0605019-1-42-3': 'Fixing [MATH] and allowing [MATH] and [MATH] to vary imposes a strong prior on the final fit, it allows us to compare directly to the results of [CITATION] by preserving their method.', 'astro-ph-0605019-1-42-4': 'The luminosity density [MATH] was calculated by summing the luminosity of all candidates (corrected for completeness) and divided by the corresponding survey volume.', 'astro-ph-0605019-1-42-5': 'With the given survey limits the equation can be solved for [MATH].', 'astro-ph-0605019-1-42-6': 'Eq. ([REF]) yields the total luminosity density when [MATH].', 'astro-ph-0605019-1-42-7': 'We have done this for three cases: for the candidates ofauthorAjiki03 (case A), the complete sample of our candidates (case B) and a combined sample of these two surveys (case C).', 'astro-ph-0605019-1-42-8': 'For our complete sample we derive a higher [MATH] (+0.12dex; case B) than [CITATION] which implies an increase of the luminosity density [MATH] of [MATH]30.', 'astro-ph-0605019-1-42-9': 'If we scale the luminosity contribution of the candidates fromauthorAjiki03 to our volume and combine the two samples, [MATH] is higher ([MATH]; case C).', 'astro-ph-0605019-1-42-10': 'Table [REF] summarises the results.', 'astro-ph-0605019-1-42-11': 'Detecting LAEs of such bright luminosity at this redshift demonstrates the necessity of wide field surveys, such as WFILAS, to provide a sample of LAEs at the bright end.', 'astro-ph-0605019-1-43-0': 'As a second approach, we tried fitting a Schechter function to the combined WFILAS and [CITATION] dataset, using a minimised [MATH] fit (Fig. [REF]).', 'astro-ph-0605019-1-43-1': 'We did not use the two lowest luminosity bins of [CITATION] to constrain the fit because these force the function to decline at the faint end.', 'astro-ph-0605019-1-43-2': 'Instead, we set the faint end slope to [MATH], similar to the H[MATH] luminosity function at [MATH] from [CITATION], on which [CITATION] based their work.', 'astro-ph-0605019-1-43-3': 'Figure [REF]b shows a strong correlation between [MATH] and [MATH] due to the slow turn-over at the bright end.', 'astro-ph-0605019-1-44-0': 'From the fitting there are three results to conclude.', 'astro-ph-0605019-1-44-1': 'Firstly, incorporating the four completeness-corrected WFILAS galaxies into the [CITATION] galaxies better constrains the bright end of the luminosity function.', 'astro-ph-0605019-1-44-2': 'Furthermore, it seems that the current generation of surveys is only just reaching the volume coverage necessary to discover LAEs with [MATH].', 'astro-ph-0605019-1-44-3': 'The histogram in Fig. [REF] shows a decreasing number of sources at the faint end.', 'astro-ph-0605019-1-44-4': 'At face value, this could suggest that the ionising flux of the less luminous sources may be insufficient to escape the slowly expanding envelope of neutral hydrogen that surrounds the Hii region in the LAE.', 'astro-ph-0605019-1-44-5': 'Consequently, the sources are undetected and the faint end of the luminosity distribution decreases.', 'astro-ph-0605019-1-44-6': 'However, it is difficult to detect faint LAEs and so the possibility of detection incompleteness cannot be ruled out.', 'astro-ph-0605019-1-45-0': 'Figure [REF] shows the sky distribution of our candidates in each field.', 'astro-ph-0605019-1-45-1': 'All candidates but one are in the CDFS and S11 fields.', 'astro-ph-0605019-1-45-2': 'The only candidate in the SGP field is brighter than the candidates in the other fields (line flux [MATH]ergs[MATH]cm[MATH]).', 'astro-ph-0605019-1-45-3': 'The reason for this is that the [MATH] filter for the SGP field has a shorter exposure time and lower signal-to-noise than the other fields.', 'astro-ph-0605019-1-46-0': 'In the CDFS field we note that our three candidates appear to be spatially clustered.', 'astro-ph-0605019-1-46-1': 'Additionally, we note that the confirmed [MATH]-drop galaxy of [CITATION] is at the same redshift as the WFILAS candidates in this field, just like four candidate LAEs from a narrowband survey by [CITATION].', 'astro-ph-0605019-1-46-2': 'We did not detect these four candidates since they are fainter than the detection limits of WFILAS in this field.', 'astro-ph-0605019-1-46-3': '[CITATION] have also done a narrowband survey of the CDFS field.', 'astro-ph-0605019-1-46-4': 'They also find evidence for an overdensity of [MATH] sources in this field.', 'astro-ph-0605019-1-46-5': 'Similarly, [CITATION] find an overdensity at redshift 5.9[MATH]0.2 in the HUDF.', 'astro-ph-0605019-1-47-0': '# Confirmed LAEs', 'astro-ph-0605019-1-48-0': 'In [CITATION] we reported the spectroscopic follow-up of one of the candidates, J114334.98[MATH]014433.7 (S1113368 in that paper, hereafter S115236).', 'astro-ph-0605019-1-48-1': 'It was confirmed to be a LAE at [MATH].', 'astro-ph-0605019-1-48-2': 'Here we present the spectral confirmation of a new candidate, J004525.38[MATH]292402.8 (hereafter SGP8884), at [MATH].', 'astro-ph-0605019-1-48-3': 'We also show its pre-imaging and compare its Ly[MATH] profile to S115236.', 'astro-ph-0605019-1-48-4': 'SGP8884 and S115236 are the only two out of the seven candidates presented in this paper for which we have obtained spectra.', 'astro-ph-0605019-1-49-0': '## Spectral data reduction', 'astro-ph-0605019-1-50-0': 'A pre-image with an intermediate band filter (FWHM = 13nm) centered at 815nm was taken with VLT/FORS2 on 2005 August 9.', 'astro-ph-0605019-1-50-1': 'The 0252pix[MATH] plate scale undersamples the [MATH]0.5[MATH] stellar point spread function of the frames which were taken during excellent seeing.', 'astro-ph-0605019-1-50-2': 'SGP8884 is unresolved, implying that the FWHM of the emitting region is [MATH]2.2kpc.', 'astro-ph-0605019-1-50-3': 'A 38[MATH],[MATH] region around the object is shown in Fig. [REF].', 'astro-ph-0605019-1-51-0': 'The spectroscopy consists of four exposures of 900 seconds, taken on 2005 October 3 with FORS2 using the 1028z grism and a 1[MATH] slit.', 'astro-ph-0605019-1-51-1': 'The frames were overscan subtracted and flatfielded.', 'astro-ph-0605019-1-51-2': 'They were combined by summing individual frames, thereby removing cosmic rays in the process.', 'astro-ph-0605019-1-52-0': 'The spectrum was flux calibrated using a standard star (HD49798) taken with a 5[MATH] slit and corrected for slit-loss.', 'astro-ph-0605019-1-52-1': 'This was calculated assuming a gaussian source profile with a FWHM of 072 as measured from the spatial direction of the spectrum.', 'astro-ph-0605019-1-52-2': 'The flux lost due to the 1[MATH] slit was calculated and added to the spectrum of the object.', 'astro-ph-0605019-1-53-0': '## Line fitting', 'astro-ph-0605019-1-54-0': 'Figure [REF] shows the reduced spectrum of SGP8884 alongside its best model fit.', 'astro-ph-0605019-1-54-1': 'The spectrum has an asymmetric line profile, similar to our previously confirmed candidate LAE .', 'astro-ph-0605019-1-54-2': 'It unlikely originates from a redshifted [Oii] line at [MATH] because the resolution of our spectrum is high enough to resolve the [Oii] [MATH]3726,3728.', 'astro-ph-0605019-1-54-3': 'Figure [REF] shows the spectrum of one such [Oii] emitter at [MATH] which was included in the same observations as SGP8884.', 'astro-ph-0605019-1-54-4': 'Furthermore, we do not find any other spectral features in our spectrum, such as H[MATH] or [Nii], which could classify the emission coming from a lower redshift galaxy.', 'astro-ph-0605019-1-54-5': 'Hence, we identify the line as Ly[MATH] at [MATH].', 'astro-ph-0605019-1-54-6': 'With a total spectral line flux of (1.0[MATH]0.1)[MATH]ergs[MATH]cm[MATH] (slit-loss corrected), SGP8884 is the brightest LAE at redshift [MATH]5.7 to date.', 'astro-ph-0605019-1-54-7': 'The line flux derived from the spectrum is consistent with the flux derived from narrowband photometry (9.5[MATH]1.4)[MATH][MATH]ergs[MATH]cm[MATH], which is given in Table [REF].', 'astro-ph-0605019-1-54-8': 'The spectral line flux corresponds to a line luminosity of [MATH]=3.5[MATH][MATH]ergs[MATH] and a star formation rate of 32[MATH]yr[MATH], using the star formation conversion rate of [CITATION].', 'astro-ph-0605019-1-54-9': 'If we adopt [MATH]16pixels (=32kpc[MATH]) as an upper limit to the size of the emitting region, we derive a star formation rate surface density of [MATH][MATH]yr[MATH]kpc[MATH].', 'astro-ph-0605019-1-55-0': 'Following earlier works we fitted a single component model to the Ly[MATH] line SGP8884.', 'astro-ph-0605019-1-55-1': 'The model consists of a truncated gaussian with complete absorption blueward of the Ly[MATH] line centre.', 'astro-ph-0605019-1-55-2': 'We find an excess of flux in the observed data compared to the model around 8110.', 'astro-ph-0605019-1-55-3': 'This suggests the presence of a second line component redward of the main peak.', 'astro-ph-0605019-1-55-4': 'To test this, we measured the mean continuum levels, both red- and blueward of the line, as well as across the red-flanking region of the line.', 'astro-ph-0605019-1-55-5': 'The continuum is calculated as the weighted mean of the flux density over this region.', 'astro-ph-0605019-1-55-6': 'This yields for continuum in the red-flanking region a flux density of (3.2[MATH]0.8)[MATH]ergs[MATH]cm[MATH][MATH].', 'astro-ph-0605019-1-55-7': 'Red- and blueward of the Ly[MATH] line the continuum is (-1.0[MATH]0.8)[MATH]ergs[MATH]cm[MATH][MATH] and (0.9[MATH]0.6)[MATH]ergs[MATH]cm[MATH][MATH], respectively.', 'astro-ph-0605019-1-55-8': 'These continuum levels are indicated by the heavy bold lines in Fig. [REF].', 'astro-ph-0605019-1-55-9': 'The lower limit for the rest frame equivalent width derived from the continuum of the red flank is 46.', 'astro-ph-0605019-1-55-10': 'The rest frame equivalent width derived from the 2[MATH] upper limit of the continuum redward of the line is 125.', 'astro-ph-0605019-1-56-0': 'To see if the excess of flux in the red flank of the Ly[MATH] line can be explained by an outflow, we fit a second gaussian component to the spectrum of SGP8884, as we did to the spectrum of S115236 in [CITATION].', 'astro-ph-0605019-1-56-1': 'This yields an extremely faint and broad second component ([MATH][MATH]ergs[MATH]cm[MATH][MATH] and FWHM[MATH]1700kms[MATH]).', 'astro-ph-0605019-1-56-2': 'The precise parameters for the red component are difficult to constrain given its faint and broad profile.', 'astro-ph-0605019-1-56-3': 'The parameters from the single component model for SGP8884 and the single and double component models for S115236 are given in Table [REF].', 'astro-ph-0605019-1-57-0': '## Discussion/Comparison', 'astro-ph-0605019-1-58-0': 'The Ly[MATH] emission we see is due to intense star formation rates synonymous with local starburst galaxies.', 'astro-ph-0605019-1-58-1': 'Star formation rates per unit area in excess of 0.1[MATH]yr[MATH]kpc[MATH] are prone to produce large scale outflows of neutral hydrogen from a galaxy, powered by the supernovae and stellar winds of massive stars .', 'astro-ph-0605019-1-58-2': 'The most efficient way for Ly[MATH] to escape from the compact star forming regions is due to scattering of the photons by the entrained neutral hydrogren .', 'astro-ph-0605019-1-58-3': 'The kinematics and orientation of the outflowing neutral hydrogen can alter the Ly[MATH] profile by absorbing photons bluer if along the line of sight, or backscattering redder than Ly[MATH] if behind and receding .', 'astro-ph-0605019-1-58-4': 'Ly[MATH] emission can also arise when large scale shocks from starburst winds impinge on clumps ([MATH]100pc) of condensed gas accreting onto the halo .', 'astro-ph-0605019-1-59-0': 'Most examples of asymmetric Ly[MATH] emission at [MATH] show an extended tail implying backscattering over a fairly wide range of velocities beyond the central Ly[MATH] emission .', 'astro-ph-0605019-1-59-1': 'The limiting physical size of SGP8884 (FWHM[MATH]2.2kpc) is consistent with the scale of emitting regions in the local starburst galaxy M82 which span 0.5 to 1kpc .', 'astro-ph-0605019-1-59-2': 'This, and the scale of its outflow, make it fairly typical of both the starbursting sources seen at [MATH] and their local counterparts.', 'astro-ph-0605019-1-60-0': 'The tentative discovery of a second component in S115236 could be explained by either an expanding shell of neutral hydrogen , or by infall of the IGM onto the LAE .', 'astro-ph-0605019-1-60-1': 'The flux of the intrinsic Ly[MATH] line depends heavily on the model.', 'astro-ph-0605019-1-60-2': 'It is suggested that the total intrinsic Ly[MATH] flux emerging from these sources is underestimated by an order of magnitude (e.g. [CITATION]).', 'astro-ph-0605019-1-60-3': 'Therefore, the star formation rates derived from the observed Ly[MATH] lines could be heavily underestimated.', 'astro-ph-0605019-1-61-0': 'Figure [REF] shows a comparison between the line profiles of the two LAEs discovered with WFILAS.', 'astro-ph-0605019-1-61-1': 'S115236 differs from SGP8884 in that a clear peak, [MATH]90kms[MATH] wide, is seen [MATH]400kms[MATH] redward of Ly[MATH] .', 'astro-ph-0605019-1-61-2': 'The red component is narrower ([MATH]15) and relatively stronger than SGP8884.', 'astro-ph-0605019-1-61-3': 'The difference in the width of the red component is even more pronounced ([MATH]30) when we compare the main peak of the two-component fits to the spectrum of S115236 to the single peak of the one-component fits to the spectrum of SGP8884.', 'astro-ph-0605019-1-61-4': 'This can clearly be seen in panels a and e of Fig. [REF].', 'astro-ph-0605019-1-62-0': 'Ultimately, such outflows are thought to be responsible for the chemical enrichment of the IGM by [MATH] .', 'astro-ph-0605019-1-62-1': 'Outflows are a process facilitating the escape of UV photons, which are the origin for the UV background .', 'astro-ph-0605019-1-63-0': '# Summary', 'astro-ph-0605019-1-64-0': 'In this paper we have presented the Wide Field Imager Lyman-Alpha Search (WFILAS), which uses a combination of narrow-, intermediate and broadband filters on the ESO/MPI 2.2m telescope to search for LAEs at redshift [MATH].', 'astro-ph-0605019-1-64-1': 'This search has resulted in seven bright ([MATH][MATH]1.1[MATH]ergs[MATH]) candidate galaxies across three fields spanning almost 0.8 sq. degree.', 'astro-ph-0605019-1-65-0': 'Most of our candidates are in the regimes of bright luminosities, beyond the reach of less voluminous surveys.', 'astro-ph-0605019-1-65-1': 'Adding our candidates to those of earlier such surveys results in an integrated luminosity density [MATH]30% higher than found by such surveys alone.', 'astro-ph-0605019-1-65-2': 'We also find potential clustering in our CDFS field, supporting overdensities discovered by other surveys.', 'astro-ph-0605019-1-65-3': 'Spectroscopic follow-up for confirmation in this area will be crucial.', 'astro-ph-0605019-1-66-0': 'Two candidates have been confirmed to be LAEs at [MATH] by means of spectroscopy.', 'astro-ph-0605019-1-66-1': 'One of these galaxies is the brightest LAEs at this redshift.', 'astro-ph-0605019-1-66-2': 'The broad, asymmetric profiles of the Ly[MATH] line of both objects are consistent with neutral hydrogen backscattering of a central starbursting source.', 'astro-ph-0605019-1-67-0': 'The authors wish to thank the Max-Planck-Institut fur Astronomie and the DDT grant of the European Southern Observatory for providing the narrow band filters which are crucial to the WFILAS survey.', 'astro-ph-0605019-1-67-1': 'The broadband and part of the intermediate band data were kindly provided by the COMBO-17 team .', 'astro-ph-0605019-1-67-2': 'We also like to thank the anonymous referee for his/her useful suggestions and comments.', 'astro-ph-0605019-1-67-3': 'EW wishes to thank A. Frebel for her useful comments and discussions regarding this paper and the Astronomical Society of Australia Travel Grant.', 'astro-ph-0605019-1-67-4': 'DHJ is supported as a Research Associate by the Australian Research Council Discovery-Projects Grant (DP-0208876), administered by the Australian National University.', 'astro-ph-0605019-1-67-5': 'CW is supported by a PPARC Advanced Fellowship.'} | {'astro-ph-0605019-2-0-0': 'The Wide Field Imager Lyman-Alpha Search (WFILAS) for Galaxies at Redshift [MATH]5.7', 'astro-ph-0605019-2-1-0': 'II.', 'astro-ph-0605019-2-1-1': 'Survey Design and Sample Analysis', 'astro-ph-0605019-2-2-0': 'Wide-field narrowband surveys are an efficient way of searching large volumes of high-redshift space for distant galaxies.', 'astro-ph-0605019-2-2-1': 'We describe the Wide Field Imager Lyman-Alpha Search (WFILAS) over 0.74sq. degree for bright emission-line galaxies at [MATH].', 'astro-ph-0605019-2-2-2': 'WFILAS uses deep images taken with the Wide Field Imager (WFI) on the ESO/MPI 2.2m telescope in three narrowband (70), one encompassing intermediate band (220) and two broadband filters, [MATH] and [MATH].', 'astro-ph-0605019-2-2-3': 'We use the novel technique of an encompassing intermediate band filter to exclude false detections.', 'astro-ph-0605019-2-2-4': 'Images taken with broadband [MATH] and [MATH] filters are used to remove low redshift galaxies from our sample.', 'astro-ph-0605019-2-2-5': 'We present a sample of seven Ly[MATH] emitting galaxy candidates, two of which are spectroscopically confirmed.', 'astro-ph-0605019-2-2-6': 'Compared to other surveys all our candidates are bright, the results of this survey complements other narrowband surveys at this redshift.', 'astro-ph-0605019-2-2-7': 'Most of our candidates are in the regime of bright luminosities, beyond the reach of less voluminous surveys.', 'astro-ph-0605019-2-2-8': 'Adding our candidates to those of another survey increases the derived luminosity density by [MATH]30%.', 'astro-ph-0605019-2-2-9': 'We also find potential clustering in the Chandra Deep Field South, supporting overdensities discovered by other surveys.', 'astro-ph-0605019-2-2-10': 'Based on a FORS2/VLT spectrum we additionally present the analysis of the second confirmed Ly[MATH] emitting galaxy in our sample.', 'astro-ph-0605019-2-2-11': 'We find that it is the brightest Ly[MATH] emitting galaxy (1[MATH]ergs[MATH]cm[MATH]) at this redshift to date and the second confirmed candidate of our survey.', 'astro-ph-0605019-2-2-12': 'Both objects exhibit the presence of a possible second Ly[MATH] component redward of the line.', 'astro-ph-0605019-2-3-0': 'WFILAS for Galaxies at [MATH]5.7.', 'astro-ph-0605019-2-3-1': 'II.', 'astro-ph-0605019-2-4-0': '# Introduction', 'astro-ph-0605019-2-5-0': 'Detections of both galaxies and QSOs at [MATH] indicate that the Universe was largely reionised at that epoch.', 'astro-ph-0605019-2-5-1': 'The recent three-year WMAP results combined with other cosmological surveys suggest an epoch of reionisation around [MATH] , consistent with both QSO results and the epoch predicted by structure formation models .', 'astro-ph-0605019-2-5-2': 'While the UV contributions of QSOs and AGN are almost certainly not responsible for reionisation , faint star forming galaxies need to exist in extraordinary numbers if they are to be the cause .', 'astro-ph-0605019-2-5-3': 'However, analyses of the Hubble Ultra Deep Field failed to find sufficient numbers of faint galaxies to support this idea .', 'astro-ph-0605019-2-5-4': 'Therefore, it is crucial to investigate what the contribution to the ionising UV flux is from young stellar populations of star forming galaxies.', 'astro-ph-0605019-2-6-0': 'Broadly speaking, two classes of star-forming galaxy dominate high redshift surveys: Lyman Break Galaxies (LBGs) and Lyman-[MATH] Emitters (LAEs).', 'astro-ph-0605019-2-6-1': 'LBG surveys, which now number in the thousands of objects at [MATH]=3 to 5, find clumpy source distributions and a two-point angular correlation function indicative of strong clustering .', 'astro-ph-0605019-2-6-2': 'LAEs also show evidence for clustering although many of the LAE surveys target fields surrounding known sources such as proto-clusters, radio galaxies and QSOs .', 'astro-ph-0605019-2-6-3': 'On average, LAEs number 1.5[MATH][MATH]deg[MATH] per unit redshift down to 1.5[MATH][MATH]ergs[MATH]cm[MATH] at [MATH] and 4.5 .', 'astro-ph-0605019-2-6-4': 'Also, their consistently small size ([MATH]0.6[MATH]kpc) suggests they are subgalactic clumps residing in the wind-driven outflows of larger unseen hosts .', 'astro-ph-0605019-2-6-5': 'Such mechanisms provide a straightforward means of UV photon escape from the host galaxy, efficiently reionising the surrounding IGM in a way than ordinary LBGs can not.', 'astro-ph-0605019-2-7-0': 'The most efficient way to find LAEs is through imaging surveys using a combination of broad- and narrowband filters.', 'astro-ph-0605019-2-7-1': 'The advent of wide field cameras has allowed systematic imaging searches that have been carried out to build up samples of candidate LAEs at high redshifts .', 'astro-ph-0605019-2-7-2': 'The availability of high throughput spectrographs on 8 to 10m-class telescopes has enabled the spectroscopic confirmation of these galaxies.', 'astro-ph-0605019-2-7-3': 'Such direct imaging searches typically cover 10[MATH]-10[MATH] times the volume of blind long-slit spectroscopic searches .', 'astro-ph-0605019-2-7-4': 'Furthermore, candidates from narrowband surveys always have an identifiable emission feature that is well separated from sky lines courtesy of the filter design.', 'astro-ph-0605019-2-7-5': 'This is in contrast to other methods, including the widely-used "dropout" technique .', 'astro-ph-0605019-2-8-0': 'The narrowband filter design leads to a higher candidate LAE selection efficiency than other techniques.', 'astro-ph-0605019-2-8-1': 'The only way to secure the identification of the emission line is spectroscopic follow-up.', 'astro-ph-0605019-2-8-2': 'The most common low redshift interlopers are the emission line doublets of [Oii] [MATH]3726,3728 and [Oiii] [MATH]4959,5007.', 'astro-ph-0605019-2-8-3': 'These can be identified by obtaining spectra with a resolution [MATH] to separate the line pair.', 'astro-ph-0605019-2-8-4': 'Other emission lines, such as H[MATH] and H[MATH], can be identified by neighbouring lines.', 'astro-ph-0605019-2-8-5': 'The narrowband technique has been successfully applied by many authors in order to discover galaxies at redshift 5[MATH]6 and to locate galaxies at redshift 6[MATH]7 .', 'astro-ph-0605019-2-8-6': 'Likewise, we employ the narrowband technique in the Wide Field Imager Lyman-Alpha Search (WFILAS) to find galaxies at [MATH].', 'astro-ph-0605019-2-8-7': 'In Paper I in this series , we described a compact LAE at [MATH] discovered by our survey.', 'astro-ph-0605019-2-9-0': 'In this Paper, we describe the survey design and sample analysis of WFILAS.', 'astro-ph-0605019-2-9-1': 'In Sect. [REF] we describe the scope of the survey and the observing strategy.', 'astro-ph-0605019-2-9-2': 'The data reduction is described in Sect. [REF].', 'astro-ph-0605019-2-9-3': 'Section [REF] outlines the candidate selection and Sect. [REF] outlines sample properties and comparison to other surveys.', 'astro-ph-0605019-2-9-4': 'We discuss the spectroscopic follow-up of two candidates in Sect. [REF].', 'astro-ph-0605019-2-9-5': 'Throughout this paper we assume a flat Universe with [MATH] and a Hubble constant [MATH]kms[MATH]Mpc[MATH].', 'astro-ph-0605019-2-9-6': 'All quoted magnitudes are in the AB system .', 'astro-ph-0605019-2-10-0': '# WFILAS Survey Design and Observations', 'astro-ph-0605019-2-11-0': 'The sky area surveyed by the WFILAS is [MATH]0.74sq. degree.', 'astro-ph-0605019-2-11-1': 'We observed three fields in broadbands [MATH], [MATH] and in an intermediate width filter centred at 815nm encompassing three narrowband filters (Fig. [REF]).', 'astro-ph-0605019-2-11-2': 'The adoption of an additional intermediate width filter encompassing the multiple narrowband width filters is a novel approach compared to previous narrowband surveys.', 'astro-ph-0605019-2-11-3': 'The application of the intermediate band filter enables us to drastically reduce the number of spurious detections in the narrowband filters.', 'astro-ph-0605019-2-11-4': 'The narrow width of the narrowband filters (FWHM=7nm) gives a prominent appearance to emission line objects.', 'astro-ph-0605019-2-11-5': 'Furthermore, the three chosen fields are spread across the sky to enable us to average out variations in cosmic variance.', 'astro-ph-0605019-2-11-6': 'Our search has covered one of the largest co-moving volumes compared to other surveys.', 'astro-ph-0605019-2-11-7': 'Table [REF] compares WFILAS with other published surveys.', 'astro-ph-0605019-2-12-0': 'The observations were taken with the Wide Field Imager on the ESO/MPI 2.2m telescope at the Cerro La Silla Observatory, Chile.', 'astro-ph-0605019-2-12-1': 'The data were taken over 65 separate nights from 2001 January 19 to 2003 December 1.', 'astro-ph-0605019-2-12-2': 'The WFI is a mosaic of eight ([MATH]) 2k [MATH] 4k CCDs arranged to give a field of view of 34[MATH].', 'astro-ph-0605019-2-12-3': 'The pixels are 0238 on a side.', 'astro-ph-0605019-2-13-0': 'As WFILAS was planned as joint project of ESO Santiago and the COMBO-17 team at MPIA Heidelberg, three fields were selected to overlap with the COMBO-17 survey, i.e. their extended Chandra Deep Field South (CDFS), SGP (South Galactic Pole) and S11 fields.', 'astro-ph-0605019-2-13-1': 'The coordinates of the field centres and the exposure times in each of the filters for each field are given in Table [REF].', 'astro-ph-0605019-2-13-2': 'All three fields are at high Galactic latitude ([MATH]) and have extinctions less than [MATH]=0.022 mag .', 'astro-ph-0605019-2-14-0': 'We employ standard broadband [MATH] and [MATH] filters.', 'astro-ph-0605019-2-14-1': 'The intermediate band (FWHM = 22nm) observatory filter is centred at 815nm.', 'astro-ph-0605019-2-14-2': 'The three custom made narrowband (FWHM = 7nm) filters are centred at 810nm, 817nm and 824nm.', 'astro-ph-0605019-2-14-3': 'The transmission profiles of the filters are shown in Fig. [REF].', 'astro-ph-0605019-2-14-4': 'The intermediate and narrowband filters are designed to fit in the atmospheric 815nm OH-airglow window, where the brightness of the sky background is low and hence favourable to detect Ly[MATH] emission at redshift [MATH]5.7.', 'astro-ph-0605019-2-14-5': 'The data taken with the intermediate band filter confirm detections of the Ly[MATH] line in one of the narrowband filters.', 'astro-ph-0605019-2-14-6': 'The broadband [MATH] and [MATH] data, which were taken from the COMBO-17 survey , are used to confirm the absence of continuum blueward of the Ly[MATH] line and to avoid sample contamination by lower redshift emission line galaxies (e.g. H[MATH] at [MATH], or [Oii] at [MATH]).', 'astro-ph-0605019-2-15-0': 'To establish the photometric zero-point of the intermediate and narrowband filters two spectrophotometric standard stars were observed.', 'astro-ph-0605019-2-16-0': 'Between 10-50 exposures were taken for each intermediate and narrowband filter for each field.', 'astro-ph-0605019-2-16-1': 'The exposure times varied between 1000 and 1800sec per frame, with a typical exposure time of around 1600sec. All frames are background-limited despite the low night sky emission in this spectral region.', 'astro-ph-0605019-2-16-2': 'The median, first and last decile of both seeing and background are given in Table [REF].', 'astro-ph-0605019-2-17-0': '# Data Reduction', 'astro-ph-0605019-2-18-0': 'The data were processed with standard IRAF routines (MSCRED TASK) and our own specially designed scripts.', 'astro-ph-0605019-2-18-1': 'The initial steps in the reduction process consist of removing the zero level offset with bias frames, normalising pixel-to-pixel sensitivity differences with twilight flatfield frames and removal of fringes with fringe frames.', 'astro-ph-0605019-2-18-2': 'During these steps, the 8 CCDs that make up a single WFI image are treated independently.', 'astro-ph-0605019-2-18-3': 'These processes are described in detail below.', 'astro-ph-0605019-2-19-0': 'Normally, the overscan region of the science frames can be used to remove the zero level offset.', 'astro-ph-0605019-2-19-1': 'However, it was noticed that the bias frames contained significant intermediate scale structure (10-30 pixels).', 'astro-ph-0605019-2-19-2': 'To remove this, bias frames were taken on every day of our observations and averaged into a bias frame for that day.', 'astro-ph-0605019-2-19-3': 'In order to minimise the noise added to the data by subtracting the bias, the bias frames were smoothed by 5 pixels and 30 pixels in horizontal and vertical direction of the CCDs, respectively, and subsequently medianed.', 'astro-ph-0605019-2-19-4': 'The structures are stable over periods of several months.', 'astro-ph-0605019-2-19-5': 'Therefore, it was possible to use bias frames from different nights without degrading the quality of the data.', 'astro-ph-0605019-2-20-0': 'Typically, five twilight flatfield frames were taken in one night for one or more filters.', 'astro-ph-0605019-2-20-1': 'The frames were medianed and the science data was divided by the median.', 'astro-ph-0605019-2-20-2': 'Hence pixel-to-pixel sensitivity differences were removed.', 'astro-ph-0605019-2-20-3': 'The structure in the individual flatfield frames was stable over a period of several weeks.', 'astro-ph-0605019-2-20-4': 'Frames taken on different nights could thus be reused.', 'astro-ph-0605019-2-20-5': 'Any differences between flatfield frames were due to the appearance or disappearance of dust features, or large scale illumination differences.', 'astro-ph-0605019-2-20-6': 'The differences rarely amounted to more than a few percent.', 'astro-ph-0605019-2-21-0': 'The raw data in the intermediate and narrowband filters show fringe patterns with amplitudes of up to 10% which was only partially removed after the data had been flatfielded.', 'astro-ph-0605019-2-21-1': 'To entirely remove the fringe pattern, we subtracted a fringe frame created from 10-30 science frames.', 'astro-ph-0605019-2-21-2': 'The fringing is very stable over time, so we were able to use data spanning several months.', 'astro-ph-0605019-2-21-3': 'Certain science frames still show fringe patterns because they are contaminated by either moonlight or twilight.', 'astro-ph-0605019-2-21-4': 'Residual differences in the level of the background between the different CCDs were removed by subtracting the median background level from each CCD.', 'astro-ph-0605019-2-22-0': 'To produce the final deep images we only used images with a seeing of less than 5 pixels (=12) and without significant residual fringing.', 'astro-ph-0605019-2-22-1': 'To make the combining of the images possible, we had to apply an astrometric correction based on stars from the USNO CCD Astrograph Catalogue 2 in the three observed fields.', 'astro-ph-0605019-2-22-2': 'The frames have a set pixel scale of 0238 pixel[MATH] with North up and East left.', 'astro-ph-0605019-2-22-3': 'The images were weighted according to their exposure time and combined using the IRAF "mscstack" routine rejecting deviant pixels.', 'astro-ph-0605019-2-22-4': 'Table [REF] summarises the depth, image quality and total exposure time, for each coadded frame.', 'astro-ph-0605019-2-23-0': '# Sample Selection and Completeness', 'astro-ph-0605019-2-24-0': '## Photometry and Noise Characteristics', 'astro-ph-0605019-2-25-0': 'Initial source catalogues were created for each of the 8 narrowband images.', 'astro-ph-0605019-2-25-1': 'Each catalogue contains the photometry for the sources in all 6 filters.', 'astro-ph-0605019-2-25-2': 'We used the SExtractor source detection software .', 'astro-ph-0605019-2-25-3': 'Sources were selected when at least 5 pixels were 0.8[MATH] above the noise level in the narrowband image used for detection.', 'astro-ph-0605019-2-25-4': 'The photometry was measured in two apertures, 6 and 10 pixels in diameter (=14 and 24, respectively).', 'astro-ph-0605019-2-25-5': 'The 6 pixel aperture was used to maximise the signal-to-noise of the flux of the objects, while the larger 10 pixel aperture was used for the more accurate determination of the total flux and hence the star formation rate.', 'astro-ph-0605019-2-26-0': 'Some authors have found that SExtractor underestimates flux uncertainties .', 'astro-ph-0605019-2-26-1': 'SExtractor estimates the uncertainties using various assumptions that are often not valid (e.g. perfect flatfielding, perfect sky subtraction).', 'astro-ph-0605019-2-26-2': 'The pixel-to-pixel noise in our data is slightly correlated because the scatter in the counts summed in 6 pixel apertures is about 10% higher than what one would derive from the measured pixel-to-pixel RMS.', 'astro-ph-0605019-2-27-0': 'We devised a method to correct the uncertainties given by SExtractor to their true values as follows.', 'astro-ph-0605019-2-27-1': 'First, sources with flux in all filters and their [MATH] magnitude between 16 and 23 were selected.', 'astro-ph-0605019-2-27-2': 'Sources brighter than [MATH],=16 are typically saturated, while those fainter than [MATH],=23 are incomplete (see Sect. [REF] for a further discussion of incompleteness).', 'astro-ph-0605019-2-27-3': 'The colour (where [MATH] is any of narrowband filters [MATH], [MATH], or [MATH]) is the same for any flat continuum source.', 'astro-ph-0605019-2-27-4': 'Therefore, the spread in the colour will be the same as the true flux uncertainty from the two contributing filters.', 'astro-ph-0605019-2-27-5': 'Next, the sources were binned into 200-source bins based on their [MATH] magnitude.', 'astro-ph-0605019-2-27-6': 'In Fig. [REF] we plot the colour versus the [MATH] magnitude of one of our S11 catalogues.', 'astro-ph-0605019-2-27-7': 'Mean values for the colour, [MATH], [MATH] magnitude and the mean of the SExtractor uncertainty were calculated for each bin.', 'astro-ph-0605019-2-27-8': 'The uncertainty in the colour for each object was determined by adding the uncertainty of [MATH] and [MATH] in quadrature ([MATH]).', 'astro-ph-0605019-2-27-9': 'The interval in which 68.3% of the objects were closest to this mean colour was used to infer the actual 1[MATH] colour uncertainty.', 'astro-ph-0605019-2-27-10': 'We assumed that the ratio between the old uncertainties [MATH] and [MATH] was the same for the new uncertainties [MATH] and [MATH].', 'astro-ph-0605019-2-27-11': 'We related between the new and old uncertainty in the intermediate and narrowband flux using the function [MATH], where [MATH] is the zero-offset for the uncertainty in the flux of bright sources and [MATH] is the ratio between the new and old uncertainty for the flux of the faintest sources.', 'astro-ph-0605019-2-27-12': 'The parameters [MATH] and [MATH] correspond to imperfections in the photometry and wrongly assumed background by SExtractor, respectively.', 'astro-ph-0605019-2-28-0': 'Typically, the correction factors are moderate (between [MATH]50%) for the faint sources in the catalogues.', 'astro-ph-0605019-2-28-1': 'Even though the correction factors are moderate, we assume that the corrections for the uncertainties in the broadband [MATH] and [MATH] are irrelevant, since they are used in a different way than the intermediate and narrowband images (see Sect. [REF]).', 'astro-ph-0605019-2-29-0': '## Selection criteria', 'astro-ph-0605019-2-30-0': 'The following four criteria were applied to select our candidate LAEs from the eight initial source catalogues:', 'astro-ph-0605019-2-31-0': 'the narrowband image used as the detection image must have the most flux of all the narrowband images and the source must have a 4[MATH] detection or better; the narrowband image with the least flux needs to be a non-detection, i.e. less than 2[MATH]; there must be at least a 2[MATH] detection in the intermediate band image; none of the broadband images, i.e. neither [MATH] nor [MATH], must have a detection above 2[MATH].', 'astro-ph-0605019-2-32-0': 'Table [REF] contains the values of the 2[MATH] detection thresholds of the images used for the 6 pixel aperture.', 'astro-ph-0605019-2-32-1': 'In total 33 candidates were selected using the above criteria.', 'astro-ph-0605019-2-32-2': 'Visual inspection showed that 26 sources arose from artefacts of which the vast majority were out-of-focus ghost rings from bright stars.', 'astro-ph-0605019-2-32-3': 'The final sample contains seven candidate LAEs.', 'astro-ph-0605019-2-33-0': 'We note here the importance of the usage of the intermediate band filter.', 'astro-ph-0605019-2-33-1': 'If we were to reapply all the criteria except for criterion 3, i.e. we do not use the intermediate band images, we would obtain 284 candidates instead of the 33 for visual inspection.', 'astro-ph-0605019-2-34-0': 'The AB-magnitudes, derived line fluxes and luminosities for the candidates are shown in Table [REF].', 'astro-ph-0605019-2-34-1': 'To convert between AB-magnitudes and line flux in ergs[MATH]cm[MATH] we use the following relation: [EQUATION] where [MATH] and [MATH] are the FWHM and the central wavelength of the narrowband filter in , respectively, and [MATH] the AB-magnitude of the object.', 'astro-ph-0605019-2-34-2': 'In Fig. [REF] the thumbnails of the seven candidate LAEs at [MATH] are shown.', 'astro-ph-0605019-2-34-3': 'We defer a more detailed discussion about the sample properties to Sect. [REF].', 'astro-ph-0605019-2-35-0': '## Completeness corrections', 'astro-ph-0605019-2-36-0': 'From the Hubble Deep Field (HDF) galaxy number-count data for the F814W filter we computed completeness corrections for our eight source catalogues.', 'astro-ph-0605019-2-36-1': 'The HDF counts are determined over the magnitude range [MATH]29, and agree well with our galaxy counts over all narrowband filters in the range [MATH]24.', 'astro-ph-0605019-2-36-2': 'Figure [REF] shows the counts for the F814W filter in the HDF and for the [MATH] filter in the S11 field.', 'astro-ph-0605019-2-36-3': 'Figure [REF] also shows the linear fit used as the basis for the calculation of the detection completeness.', 'astro-ph-0605019-2-36-4': 'The fit is done to the combined number count data over two intervals: [MATH],=[20,22.5], where the WFILAS counts are complete, and [MATH]=[22.5,25], where the HDF counts are linear.', 'astro-ph-0605019-2-37-0': 'Detection completeness is defined as the ratio of WFILAS sources to the number expected from the number-count relation.', 'astro-ph-0605019-2-37-1': 'Figure [REF] shows the derived detection completeness for each filter-field combination used for WFILAS.', 'astro-ph-0605019-2-37-2': 'The differences are mainly due to unequal exposure times, although filter throughput and image quality also play a role.', 'astro-ph-0605019-2-37-3': 'These could explain the overall lower sensitivity of the [MATH] filter, as can be inferred from Fig. [REF].', 'astro-ph-0605019-2-37-4': 'Additionally, we correct for detection completeness arising due to the intermediate band selection criterion.', 'astro-ph-0605019-2-37-5': 'We constructed a noise image by stacking the intermediate band images without registering.', 'astro-ph-0605019-2-37-6': 'The completeness is defined as the rate of recovery of artificially inserted objects.', 'astro-ph-0605019-2-38-0': 'Given the different sensitivities of each filter-field combination, we define a homogeneous subsample of our initial candidate sample, using the candidates from our four most sensitive field-filter combinations.', 'astro-ph-0605019-2-38-1': 'We call this our "complete" sample (4 of the 7 LAEs; marked in Table [REF]), because once defined, we use the curves in Fig. [REF] to correct the detected candidate numbers for incompleteness, in contrast to our initial "incomplete" sample (all 7 LAEs).', 'astro-ph-0605019-2-38-2': 'The purpose of the subsample is that it lies within a uniform flux limit.', 'astro-ph-0605019-2-38-3': 'Figure [REF] shows that our four best filter-field combinations consist of the [MATH] and [MATH] filters in both the CDFS and S11 fields.', 'astro-ph-0605019-2-38-4': 'These four field-filter combinations reach at least 50% completeness at [MATH]=23.38, or 5.1[MATH]ergs[MATH]cm[MATH].', 'astro-ph-0605019-2-38-5': 'We take this as the flux limit of our complete sample.', 'astro-ph-0605019-2-38-6': 'As such, the number density derived from the complete sample is a more accurate measure of the density of sources down to the nominated flux limit than the number density of the incomplete sample.', 'astro-ph-0605019-2-38-7': 'Figure [REF] shows the luminosity distribution of the complete sample alongside our initial candidate list, which we call the "incomplete" sample.', 'astro-ph-0605019-2-38-8': 'It shows that in using completeness corrections our detected source density is up by 50%.', 'astro-ph-0605019-2-39-0': '# [MATH] Candidate LAE Catalogue', 'astro-ph-0605019-2-40-0': 'In the previous Sect. we introduced two sets of candidate LAEs: the full (but incomplete) sample of seven candidate LAEs and a subsample thereof, complete to [MATH]=5.1[MATH]ergs[MATH]cm[MATH] (the complete sample).', 'astro-ph-0605019-2-40-1': 'The flux limit of the incomplete sample is almost twice the limit of the complete sample (3.4[MATH]ergs[MATH]cm[MATH]).', 'astro-ph-0605019-2-41-0': 'To examine the luminosity distribution of our sample we use the Schechter function , as it is a good representation of the data at bright luminosities.', 'astro-ph-0605019-2-41-1': 'From this, the luminosity density [MATH] of a distribution with a limiting luminosity [MATH] is given by [EQUATION] where [MATH] and [MATH] represent the slope of the faint end of the Schechter function and the normalisation constant of the galaxy density, respectively.', 'astro-ph-0605019-2-41-2': '[MATH] is the incomplete gamma-function.', 'astro-ph-0605019-2-41-3': 'Currently, the luminosity function for LAEs at [MATH] is poorly defined and authors commonly adopt either one or two of the three parameters from low redshift surveys to calculate the third.', 'astro-ph-0605019-2-42-0': 'We examine the influence of non-detections of bright ([MATH]) LAEs for the total Ly[MATH] luminosity density by employing the same method as [CITATION], another narrowband imaging survey aimed at finding LAEs at [MATH].', 'astro-ph-0605019-2-42-1': 'In the interest of comparison, we follow [CITATION] exactly and adopt the [CITATION] values for [MATH] (-1.53) and [MATH]3).', 'astro-ph-0605019-2-42-2': 'Their approach was to solve Eq. ([REF]) for [MATH], instead of fitting a Schechter function.', 'astro-ph-0605019-2-42-3': 'Fixing [MATH] and allowing [MATH] and [MATH] to vary imposes a strong prior on the final fit, it allows us to compare directly to the results of [CITATION] by preserving their method.', 'astro-ph-0605019-2-42-4': 'The luminosity density [MATH] was calculated by summing the luminosity of all candidates (corrected for completeness) and divided by the corresponding survey volume.', 'astro-ph-0605019-2-42-5': 'With the given survey limits the equation can be solved for [MATH].', 'astro-ph-0605019-2-42-6': 'Equation ([REF]) yields the total luminosity density when [MATH].', 'astro-ph-0605019-2-42-7': 'We have done this for three cases: for the candidates ofauthorAjiki03 (case A), the complete sample of our candidates (case B) and a combined sample of these two surveys (case C).', 'astro-ph-0605019-2-42-8': 'For our complete sample we derive a higher [MATH] (+0.12dex; case B) than [CITATION] which implies an increase of the luminosity density [MATH] of [MATH]30.', 'astro-ph-0605019-2-42-9': 'If we scale the luminosity contribution of the candidates fromauthorAjiki03 to our volume and combine the two samples, [MATH] is higher ([MATH]; case C).', 'astro-ph-0605019-2-42-10': 'Table [REF] summarises the results.', 'astro-ph-0605019-2-42-11': 'Detecting LAEs of such bright luminosity at this redshift demonstrates the necessity of wide field surveys, such as WFILAS, to provide a sample of LAEs at the bright end.', 'astro-ph-0605019-2-43-0': 'As a second approach, we tried fitting a Schechter function to the combined WFILAS and [CITATION] dataset, using a minimised [MATH] fit (Fig. [REF]).', 'astro-ph-0605019-2-43-1': 'We did not use the two lowest luminosity bins of [CITATION] to constrain the fit because these force the function to decline at the faint end.', 'astro-ph-0605019-2-43-2': 'Instead, we set the faint end slope to [MATH], similar to the H[MATH] luminosity function at [MATH] from [CITATION], on which [CITATION] based their work.', 'astro-ph-0605019-2-43-3': 'Figure [REF]b shows a strong correlation between [MATH] and [MATH] due to the slow turn-over at the bright end.', 'astro-ph-0605019-2-44-0': 'From the fitting there are three results to conclude.', 'astro-ph-0605019-2-44-1': 'Firstly, incorporating the four completeness-corrected WFILAS galaxies into the [CITATION] galaxies better constrains the bright end of the luminosity function.', 'astro-ph-0605019-2-44-2': 'Furthermore, it seems that the current generation of surveys is only just reaching the volume coverage necessary to discover LAEs with [MATH].', 'astro-ph-0605019-2-44-3': 'The histogram in Fig. [REF] shows a decreasing number of sources at the faint end.', 'astro-ph-0605019-2-44-4': 'At face value, this could suggest that the ionising flux of the less luminous sources may be insufficient to escape the slowly expanding envelope of neutral hydrogen that surrounds the Hii region in the LAE.', 'astro-ph-0605019-2-44-5': 'Consequently, the sources are undetected and the faint end of the luminosity distribution decreases.', 'astro-ph-0605019-2-44-6': 'However, it is difficult to detect faint LAEs and so the possibility of detection incompleteness cannot be ruled out.', 'astro-ph-0605019-2-45-0': 'Figure [REF] shows the sky distribution of our candidates in each field.', 'astro-ph-0605019-2-45-1': 'All candidates but one are in the CDFS and S11 fields.', 'astro-ph-0605019-2-45-2': 'The only candidate in the SGP field is brighter than the candidates in the other fields (line flux [MATH]ergs[MATH]cm[MATH]).', 'astro-ph-0605019-2-45-3': 'The reason for this is that the [MATH] filter for the SGP field has a shorter exposure time and lower signal-to-noise than the other fields.', 'astro-ph-0605019-2-46-0': 'In the CDFS field we note that our three candidates appear to be spatially clustered.', 'astro-ph-0605019-2-46-1': 'Additionally, we note that the confirmed [MATH]-drop galaxy of [CITATION] is at the same redshift as the WFILAS candidates in this field, just like four candidate LAEs from a narrowband survey by [CITATION].', 'astro-ph-0605019-2-46-2': 'We did not detect these four candidates since they are fainter than the detection limits of WFILAS in this field.', 'astro-ph-0605019-2-46-3': '[CITATION] have also done a narrowband survey of the CDFS field.', 'astro-ph-0605019-2-46-4': 'They also find evidence for an overdensity of [MATH] sources in this field.', 'astro-ph-0605019-2-46-5': 'Similarly, [CITATION] find an overdensity at redshift 5.9[MATH]0.2 in the HUDF.', 'astro-ph-0605019-2-47-0': '# Confirmed LAEs', 'astro-ph-0605019-2-48-0': 'In [CITATION] we reported the spectroscopic follow-up of one of the candidates, J114334.98[MATH]014433.7 (S1113368 in that paper, hereafter S115236).', 'astro-ph-0605019-2-48-1': 'It was confirmed to be a LAE at [MATH].', 'astro-ph-0605019-2-48-2': 'Here we present the spectral confirmation of a new candidate, J004525.38[MATH]292402.8 (hereafter SGP8884), at [MATH].', 'astro-ph-0605019-2-48-3': 'We also show its pre-imaging and compare its Ly[MATH] profile to S115236.', 'astro-ph-0605019-2-48-4': 'SGP8884 and S115236 are the only two out of the seven candidates presented in this paper for which we have obtained spectra.', 'astro-ph-0605019-2-49-0': '## Spectral data reduction', 'astro-ph-0605019-2-50-0': 'A pre-image with an intermediate band filter (FWHM = 13nm) centred at 815nm was taken with VLT/FORS2 on 2005 August 9.', 'astro-ph-0605019-2-50-1': 'The 0252pix[MATH] plate scale undersamples the [MATH]0.5[MATH] stellar point spread function of the frames which were taken during excellent seeing.', 'astro-ph-0605019-2-50-2': 'SGP8884 is unresolved, implying that the FWHM of the emitting region is [MATH]2.2kpc.', 'astro-ph-0605019-2-50-3': 'A 38[MATH],[MATH] region around the object is shown in Fig. [REF].', 'astro-ph-0605019-2-51-0': 'The spectroscopy consists of four exposures of 900s, taken on 2005 October 3 with FORS2 using the 1028z grism and a 1[MATH] slit.', 'astro-ph-0605019-2-51-1': 'The frames were overscan subtracted and flatfielded.', 'astro-ph-0605019-2-51-2': 'They were combined by summing individual frames, thereby removing cosmic rays in the process.', 'astro-ph-0605019-2-52-0': 'The spectrum was flux calibrated using a standard star (HD49798) taken with a 5[MATH] slit and corrected for slit-loss.', 'astro-ph-0605019-2-52-1': 'This was calculated assuming a Gaussian source profile with a FWHM of 072 as measured from the spatial direction of the spectrum.', 'astro-ph-0605019-2-52-2': 'The flux lost due to the 1[MATH] slit was calculated and added to the spectrum of the object.', 'astro-ph-0605019-2-53-0': '## Line fitting', 'astro-ph-0605019-2-54-0': 'Figure [REF] shows the reduced spectrum of SGP8884 alongside its best model fit.', 'astro-ph-0605019-2-54-1': 'The spectrum has an asymmetric line profile, similar to our previously confirmed candidate LAE .', 'astro-ph-0605019-2-54-2': 'It unlikely originates from a redshifted [Oii] line at [MATH] because the resolution of our spectrum is high enough to resolve the [Oii] [MATH]3726,3728.', 'astro-ph-0605019-2-54-3': 'Figure [REF] shows the spectrum of one such [Oii] emitter at [MATH] which was included in the same observations as SGP8884.', 'astro-ph-0605019-2-54-4': 'Furthermore, we do not find any other spectral features in our spectrum, such as H[MATH] or [Nii], which could classify the emission coming from a lower redshift galaxy.', 'astro-ph-0605019-2-54-5': 'Hence, we identify the line as Ly[MATH] at [MATH].', 'astro-ph-0605019-2-54-6': 'With a total spectral line flux of (1.0[MATH]0.1)[MATH]ergs[MATH]cm[MATH] (slit-loss corrected), SGP8884 is the brightest LAE at redshift [MATH]5.7 to date.', 'astro-ph-0605019-2-54-7': 'The line flux derived from the spectrum is consistent with the flux derived from narrowband photometry (9.5[MATH]1.4)[MATH][MATH]ergs[MATH]cm[MATH], which is given in Table [REF].', 'astro-ph-0605019-2-54-8': 'The spectral line flux corresponds to a line luminosity of [MATH]=3.5[MATH][MATH]ergs[MATH] and a star formation rate of 32[MATH]yr[MATH], using the star formation conversion rate of [CITATION].', 'astro-ph-0605019-2-54-9': 'If we adopt [MATH]16pixels (=32kpc[MATH]) as an upper limit to the size of the emitting region, we derive a star formation rate surface density of [MATH][MATH]yr[MATH]kpc[MATH].', 'astro-ph-0605019-2-55-0': 'Following earlier works we fitted a single component model to the Ly[MATH] line SGP8884.', 'astro-ph-0605019-2-55-1': 'The model consists of a truncated Gaussian with complete absorption blueward of the Ly[MATH] line centre.', 'astro-ph-0605019-2-55-2': 'We find an excess of flux in the observed data compared to the model around 8110.', 'astro-ph-0605019-2-55-3': 'This suggests the presence of a second line component redward of the main peak.', 'astro-ph-0605019-2-55-4': 'To test this, we measured the mean continuum levels, both red- and blueward of the line, as well as across the red-flanking region of the line.', 'astro-ph-0605019-2-55-5': 'The continuum is calculated as the weighted mean of the flux density over this region.', 'astro-ph-0605019-2-55-6': 'This yields for continuum in the red-flanking region a flux density of (3.2[MATH]0.8)[MATH]ergs[MATH]cm[MATH][MATH].', 'astro-ph-0605019-2-55-7': 'Red- and blueward of the Ly[MATH] line the continuum is (-1.0[MATH]0.8)[MATH]ergs[MATH]cm[MATH][MATH] and (0.9[MATH]0.6)[MATH]ergs[MATH]cm[MATH][MATH], respectively.', 'astro-ph-0605019-2-55-8': 'These continuum levels are indicated by the heavy bold lines in Fig. [REF].', 'astro-ph-0605019-2-55-9': 'The lower limit for the rest frame equivalent width derived from the continuum of the red flank is 46.', 'astro-ph-0605019-2-55-10': 'The rest frame equivalent width derived from the 2[MATH] upper limit of the continuum redward of the line is 125.', 'astro-ph-0605019-2-56-0': 'To see if the excess of flux in the red flank of the Ly[MATH] line can be explained by an outflow, we fit a second Gaussian component to the spectrum of SGP8884, as we did to the spectrum of S115236 in [CITATION].', 'astro-ph-0605019-2-56-1': 'This yields an extremely faint and broad second component ([MATH][MATH]ergs[MATH]cm[MATH][MATH] and FWHM[MATH]1700kms[MATH]).', 'astro-ph-0605019-2-56-2': 'The precise parameters for the red component are difficult to constrain given its faint and broad profile.', 'astro-ph-0605019-2-56-3': 'The parameters from the single component model for SGP8884 and the single and double component models for S115236 are given in Table [REF].', 'astro-ph-0605019-2-57-0': '## Discussion/Comparison', 'astro-ph-0605019-2-58-0': 'The Ly[MATH] emission we see is due to intense star formation rates synonymous with local starburst galaxies.', 'astro-ph-0605019-2-58-1': 'Star formation rates per unit area in excess of 0.1[MATH]yr[MATH]kpc[MATH] are prone to produce large scale outflows of neutral hydrogen from a galaxy, powered by the supernovae and stellar winds of massive stars .', 'astro-ph-0605019-2-58-2': 'The most efficient way for Ly[MATH] to escape from the compact star forming regions is due to scattering of the photons by the entrained neutral hydrogen .', 'astro-ph-0605019-2-58-3': 'The kinematics and orientation of the outflowing neutral hydrogen can alter the Ly[MATH] profile by absorbing photons bluer if along the line of sight, or backscattering redder than Ly[MATH] if behind and receding .', 'astro-ph-0605019-2-58-4': 'Ly[MATH] emission can also arise when large scale shocks from starburst winds impinge on clumps ([MATH]100pc) of condensed gas accreting onto the halo .', 'astro-ph-0605019-2-59-0': 'Most examples of asymmetric Ly[MATH] emission at [MATH] show an extended tail implying backscattering over a fairly wide range of velocities beyond the central Ly[MATH] emission .', 'astro-ph-0605019-2-59-1': 'The limiting physical size of SGP8884 (FWHM[MATH]2.2kpc) is consistent with the scale of emitting regions in the local starburst galaxy M82 which span 0.5 to 1kpc .', 'astro-ph-0605019-2-59-2': 'This, and the scale of its outflow, make it fairly typical of both the starbursting sources seen at [MATH] and their local counterparts.', 'astro-ph-0605019-2-60-0': 'The tentative discovery of a second component in S115236 could be explained by either an expanding shell of neutral hydrogen , or by infall of the IGM onto the LAE .', 'astro-ph-0605019-2-60-1': 'The flux of the intrinsic Ly[MATH] line depends heavily on the model.', 'astro-ph-0605019-2-60-2': 'It is suggested that the total intrinsic Ly[MATH] flux emerging from these sources is underestimated by an order of magnitude (e.g. [CITATION]).', 'astro-ph-0605019-2-60-3': 'Therefore, the star formation rates derived from the observed Ly[MATH] lines could be heavily underestimated.', 'astro-ph-0605019-2-61-0': 'Figure [REF] shows a comparison between the line profiles of the two LAEs discovered with WFILAS.', 'astro-ph-0605019-2-61-1': 'S115236 differs from SGP8884 in that a clear peak, [MATH]90kms[MATH] wide, is seen [MATH]400kms[MATH] redward of Ly[MATH] .', 'astro-ph-0605019-2-61-2': 'The red component is narrower ([MATH]15) and relatively stronger than SGP8884.', 'astro-ph-0605019-2-61-3': 'The difference in the width of the red component is even more pronounced ([MATH]30) when we compare the main peak of the two-component fits to the spectrum of S115236 to the single peak of the one-component fits to the spectrum of SGP8884.', 'astro-ph-0605019-2-61-4': 'This can clearly be seen in panels a and e of Fig. [REF].', 'astro-ph-0605019-2-62-0': 'Ultimately, such outflows are thought to be responsible for the chemical enrichment of the IGM by [MATH] .', 'astro-ph-0605019-2-62-1': 'Outflows are a process facilitating the escape of UV photons, which are the origin for the UV background .', 'astro-ph-0605019-2-63-0': '# Summary', 'astro-ph-0605019-2-64-0': 'In this paper we have presented the Wide Field Imager Lyman-Alpha Search (WFILAS), which uses a combination of narrow-, intermediate and broadband filters on the ESO/MPI 2.2m telescope to search for LAEs at redshift [MATH].', 'astro-ph-0605019-2-64-1': 'This search has resulted in seven bright ([MATH][MATH]1.1[MATH]ergs[MATH]) candidate galaxies across three fields spanning almost 0.8 sq. degree.', 'astro-ph-0605019-2-65-0': 'Most of our candidates are in the regimes of bright luminosities, beyond the reach of less voluminous surveys.', 'astro-ph-0605019-2-65-1': 'Adding our candidates to those of earlier such surveys results in an integrated luminosity density [MATH]30% higher than found by such surveys alone.', 'astro-ph-0605019-2-65-2': 'We also find potential clustering in our CDFS field, supporting overdensities discovered by other surveys.', 'astro-ph-0605019-2-65-3': 'Spectroscopic follow-up for confirmation in this area will be crucial.', 'astro-ph-0605019-2-66-0': 'Two candidates have been confirmed to be LAEs at [MATH] by means of spectroscopy.', 'astro-ph-0605019-2-66-1': 'One of these galaxies is the brightest LAEs at this redshift.', 'astro-ph-0605019-2-66-2': 'The broad, asymmetric profiles of the Ly[MATH] line of both objects are consistent with neutral hydrogen backscattering of a central starbursting source.', 'astro-ph-0605019-2-67-0': 'The authors wish to thank the Max-Planck-Institut fur Astronomie and the DDT grant of the European Southern Observatory for providing the narrow band filters which are crucial to the WFILAS survey.', 'astro-ph-0605019-2-67-1': 'The broadband and part of the intermediate band data were kindly provided by the COMBO-17 team .', 'astro-ph-0605019-2-67-2': 'We also like to thank the anonymous referee for his/her useful suggestions and comments.', 'astro-ph-0605019-2-67-3': 'E.W. wishes to thank A. Frebel for her useful comments and discussions regarding this paper and the Astronomical Society of Australia Travel Grant.', 'astro-ph-0605019-2-67-4': 'D.H.J. is supported as a Research Associate by the Australian Research Council Discovery-Projects Grant (DP-0208876), administered by the Australian National University.', 'astro-ph-0605019-2-67-5': 'C.W. is supported by a PPARC Advanced Fellowship.'} | [['astro-ph-0605019-1-5-0', 'astro-ph-0605019-2-5-0'], ['astro-ph-0605019-1-5-2', 'astro-ph-0605019-2-5-2'], ['astro-ph-0605019-1-5-3', 'astro-ph-0605019-2-5-3'], ['astro-ph-0605019-1-5-4', 'astro-ph-0605019-2-5-4'], ['astro-ph-0605019-1-67-0', 'astro-ph-0605019-2-67-0'], ['astro-ph-0605019-1-67-1', 'astro-ph-0605019-2-67-1'], ['astro-ph-0605019-1-67-2', 'astro-ph-0605019-2-67-2'], ['astro-ph-0605019-1-51-1', 'astro-ph-0605019-2-51-1'], ['astro-ph-0605019-1-51-2', 'astro-ph-0605019-2-51-2'], ['astro-ph-0605019-1-66-0', 'astro-ph-0605019-2-66-0'], ['astro-ph-0605019-1-66-1', 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'astro-ph-0605019-2-14-0'], ['astro-ph-0605019-1-14-1', 'astro-ph-0605019-2-14-1'], ['astro-ph-0605019-1-14-3', 'astro-ph-0605019-2-14-3'], ['astro-ph-0605019-1-14-4', 'astro-ph-0605019-2-14-4'], ['astro-ph-0605019-1-14-5', 'astro-ph-0605019-2-14-5'], ['astro-ph-0605019-1-14-6', 'astro-ph-0605019-2-14-6'], ['astro-ph-0605019-1-21-0', 'astro-ph-0605019-2-21-0'], ['astro-ph-0605019-1-21-1', 'astro-ph-0605019-2-21-1'], ['astro-ph-0605019-1-21-2', 'astro-ph-0605019-2-21-2'], ['astro-ph-0605019-1-21-3', 'astro-ph-0605019-2-21-3'], ['astro-ph-0605019-1-21-4', 'astro-ph-0605019-2-21-4'], ['astro-ph-0605019-1-9-0', 'astro-ph-0605019-2-9-0'], ['astro-ph-0605019-1-9-1', 'astro-ph-0605019-2-9-1'], ['astro-ph-0605019-1-9-2', 'astro-ph-0605019-2-9-2'], ['astro-ph-0605019-1-9-3', 'astro-ph-0605019-2-9-3'], ['astro-ph-0605019-1-9-4', 'astro-ph-0605019-2-9-4'], ['astro-ph-0605019-1-9-5', 'astro-ph-0605019-2-9-5'], ['astro-ph-0605019-1-9-6', 'astro-ph-0605019-2-9-6'], ['astro-ph-0605019-1-2-0', 'astro-ph-0605019-2-2-0'], ['astro-ph-0605019-1-2-1', 'astro-ph-0605019-2-2-1'], ['astro-ph-0605019-1-2-2', 'astro-ph-0605019-2-2-2'], ['astro-ph-0605019-1-2-3', 'astro-ph-0605019-2-2-3'], ['astro-ph-0605019-1-2-4', 'astro-ph-0605019-2-2-4'], ['astro-ph-0605019-1-2-5', 'astro-ph-0605019-2-2-5'], ['astro-ph-0605019-1-2-6', 'astro-ph-0605019-2-2-6'], ['astro-ph-0605019-1-2-7', 'astro-ph-0605019-2-2-7'], ['astro-ph-0605019-1-2-8', 'astro-ph-0605019-2-2-8'], ['astro-ph-0605019-1-2-9', 'astro-ph-0605019-2-2-9'], ['astro-ph-0605019-1-2-10', 'astro-ph-0605019-2-2-10'], ['astro-ph-0605019-1-2-11', 'astro-ph-0605019-2-2-11'], ['astro-ph-0605019-1-2-12', 'astro-ph-0605019-2-2-12'], ['astro-ph-0605019-1-50-1', 'astro-ph-0605019-2-50-1'], ['astro-ph-0605019-1-50-2', 'astro-ph-0605019-2-50-2'], ['astro-ph-0605019-1-50-3', 'astro-ph-0605019-2-50-3'], ['astro-ph-0605019-1-43-0', 'astro-ph-0605019-2-43-0'], ['astro-ph-0605019-1-43-1', 'astro-ph-0605019-2-43-1'], ['astro-ph-0605019-1-43-2', 'astro-ph-0605019-2-43-2'], ['astro-ph-0605019-1-43-3', 'astro-ph-0605019-2-43-3'], ['astro-ph-0605019-1-26-0', 'astro-ph-0605019-2-26-0'], ['astro-ph-0605019-1-26-1', 'astro-ph-0605019-2-26-1'], ['astro-ph-0605019-1-26-2', 'astro-ph-0605019-2-26-2'], ['astro-ph-0605019-1-62-0', 'astro-ph-0605019-2-62-0'], ['astro-ph-0605019-1-62-1', 'astro-ph-0605019-2-62-1'], ['astro-ph-0605019-1-6-0', 'astro-ph-0605019-2-6-0'], ['astro-ph-0605019-1-6-1', 'astro-ph-0605019-2-6-1'], ['astro-ph-0605019-1-6-2', 'astro-ph-0605019-2-6-2'], ['astro-ph-0605019-1-6-3', 'astro-ph-0605019-2-6-3'], ['astro-ph-0605019-1-6-4', 'astro-ph-0605019-2-6-4'], ['astro-ph-0605019-1-6-5', 'astro-ph-0605019-2-6-5'], ['astro-ph-0605019-1-34-0', 'astro-ph-0605019-2-34-0'], ['astro-ph-0605019-1-34-2', 'astro-ph-0605019-2-34-2'], ['astro-ph-0605019-1-34-3', 'astro-ph-0605019-2-34-3'], ['astro-ph-0605019-1-15-0', 'astro-ph-0605019-2-15-0'], ['astro-ph-0605019-1-8-0', 'astro-ph-0605019-2-8-0'], ['astro-ph-0605019-1-8-1', 'astro-ph-0605019-2-8-1'], ['astro-ph-0605019-1-8-2', 'astro-ph-0605019-2-8-2'], ['astro-ph-0605019-1-8-3', 'astro-ph-0605019-2-8-3'], ['astro-ph-0605019-1-8-4', 'astro-ph-0605019-2-8-4'], ['astro-ph-0605019-1-8-5', 'astro-ph-0605019-2-8-5'], ['astro-ph-0605019-1-8-6', 'astro-ph-0605019-2-8-6'], ['astro-ph-0605019-1-8-7', 'astro-ph-0605019-2-8-7']] | [['astro-ph-0605019-1-5-1', 'astro-ph-0605019-2-5-1'], ['astro-ph-0605019-1-67-3', 'astro-ph-0605019-2-67-3'], ['astro-ph-0605019-1-67-4', 'astro-ph-0605019-2-67-4'], ['astro-ph-0605019-1-67-5', 'astro-ph-0605019-2-67-5'], ['astro-ph-0605019-1-51-0', 'astro-ph-0605019-2-51-0'], ['astro-ph-0605019-1-58-2', 'astro-ph-0605019-2-58-2'], ['astro-ph-0605019-1-52-1', 'astro-ph-0605019-2-52-1'], ['astro-ph-0605019-1-55-1', 'astro-ph-0605019-2-55-1'], ['astro-ph-0605019-1-31-0', 'astro-ph-0605019-2-31-0'], ['astro-ph-0605019-1-38-1', 'astro-ph-0605019-2-38-1'], ['astro-ph-0605019-1-38-7', 'astro-ph-0605019-2-38-7'], ['astro-ph-0605019-1-42-2', 'astro-ph-0605019-2-42-2'], ['astro-ph-0605019-1-42-6', 'astro-ph-0605019-2-42-6'], ['astro-ph-0605019-1-7-5', 'astro-ph-0605019-2-7-5'], ['astro-ph-0605019-1-56-0', 'astro-ph-0605019-2-56-0'], ['astro-ph-0605019-1-37-4', 'astro-ph-0605019-2-37-4'], ['astro-ph-0605019-1-22-3', 'astro-ph-0605019-2-22-3'], ['astro-ph-0605019-1-14-2', 'astro-ph-0605019-2-14-2'], ['astro-ph-0605019-1-50-0', 'astro-ph-0605019-2-50-0'], ['astro-ph-0605019-1-34-1', 'astro-ph-0605019-2-34-1']] | [] | [] | [] | ['astro-ph-0605019-1-1-0', 'astro-ph-0605019-1-1-1', 'astro-ph-0605019-1-3-0', 'astro-ph-0605019-1-30-0', 'astro-ph-0605019-1-48-0', 'astro-ph-0605019-1-55-7', 'astro-ph-0605019-2-1-0', 'astro-ph-0605019-2-1-1', 'astro-ph-0605019-2-3-0', 'astro-ph-0605019-2-3-1', 'astro-ph-0605019-2-30-0', 'astro-ph-0605019-2-48-0', 'astro-ph-0605019-2-55-7'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/0605019 | null | null | null | null | null |
1810.03626 | {'1810.03626-1-0-0': 'We consider the production of a new MeV-scale fermion in coherent elastic neutrino-nucleus scattering.', '1810.03626-1-0-1': 'The effect on the measurable nucleon recoil spectrum is calculated.', '1810.03626-1-0-2': 'Assuming that the new fermion couples to neutrinos and quarks via a singlet scalar, we set limits on its mass and coupling using COHERENT data and also determine the sensitivity of the CONUS experiment.', '1810.03626-1-0-3': 'We investigate the possible connection of the new fermion to neutrino mass generation.', '1810.03626-1-0-4': 'The possibility of the new fermion being the dark matter particle is also studied.', '1810.03626-1-1-0': '# Introduction', '1810.03626-1-2-0': 'Despite being the most elusive Standard model (SM) particles, neutrinos have been detected in a number of charged- and neutral-current processes.', '1810.03626-1-2-1': 'The recent measurement [CITATION] of coherent elastic neutrino-nucleus scattering (CE[MATH]S) [CITATION] yields a novel channel where, for the first time, the interaction of low energy neutrinos with nuclei as a whole is probed.', '1810.03626-1-2-2': 'This serves not only as a handle to probe SM and nuclear physics parameters, but also as a robust probe of new physics.', '1810.03626-1-2-3': 'In particular, light sterile neutrinos [CITATION], non-standard interactions of both quarks and leptons [CITATION] as well as neutrino magnetic moments [CITATION] can be searched for.', '1810.03626-1-3-0': 'The basic requirement for the coherent neutrino-nucleus scattering is the smallness of the momentum transfer.', '1810.03626-1-3-1': 'Namely, in case it exceeds the inverse size of the nucleus, one can in principle determine on which nucleon the scattering occurred and this is what breaks the coherence.', '1810.03626-1-3-2': 'It is also important that the quantum state of the nucleus does not alter in the scattering because, otherwise, the nuclear excitations in such processes would allow individual nucleons to be tagged which would again directly break the condition for the coherent scattering [CITATION].', '1810.03626-1-4-0': 'On the other hand, the production of new light particles does not a priori violate the coherence as long as the above conditions for nuclei are satisfied.', '1810.03626-1-4-1': 'Hence, in this work we explore an interesting new possibility for coherent elastic scattering process, namely [EQUATION].', '1810.03626-1-4-2': 'Here a light MeV-scale fermion (dubbed [MATH]) is produced from the interaction of the incoming neutrino [MATH] with a nucleus [MATH].', '1810.03626-1-4-3': 'We are interested, given the lack of evidence for new physics at high energy, in MeV-scale particles as this is the typical energy scale of CE[MATH]S, where naturally the most interesting phenomenology arises.', '1810.03626-1-5-0': 'Assuming in a minimal setup that the interaction of the new fermion [MATH] with neutrinos and quarks is mediated by a scalar singlet [MATH], we derive limits on the masses of [MATH] and [MATH] and their coupling to neutrinos and the nucleus.', '1810.03626-1-5-1': 'Existing and expected data from the running experiments COHERENT [CITATION] and CONUS [CITATION] is used, and the results are compared to existing terrestrial and astrophysical limits.', '1810.03626-1-6-0': 'Any new fermion that couples to light neutrinos needs to be considered regarding its role in the generation of neutrino mass, and we demonstrate that a straightforward extension of the type-I seesaw mechanism can indeed generate the observable magnitude of neutrino masses, as well as be testable in CE[MATH]S. Moreover, any new particle beyond the Standard Model is an attractive candidate for dark matter (DM), therefore we investigate in such a setup whether [MATH] can be such a popular MeV-scale DM candidate (see e.g. Refs. [CITATION] for recent studies).', '1810.03626-1-6-1': 'We find that for the size of the couplings to which CE[MATH]S experiments are sensitive, the DM abundance can match the observed value in case there was an entropy injection episode between the QCD phase transition and Big Bang Nucleosynthesis (BBN).', '1810.03626-1-7-0': 'The paper is organized as follows.', '1810.03626-1-7-1': 'In [REF] we derive bounds on the relevant couplings and masses within our framework of [MATH] coherent scattering without restricting the discussion to a specific model.', '1810.03626-1-7-2': 'We also obtain the corresponding recoil spectra of [MATH] in case a massive particle [MATH] is emitted in the final state.', '1810.03626-1-7-3': 'In sec:model we discuss a minimal UV-complete setup in which the MeV-scale [MATH] is related to neutrino mass generation.', '1810.03626-1-7-4': 'sec:DM is devoted to the assumption that [MATH] is the DM particle, in which we scrutinize its production in the early Universe.', '1810.03626-1-7-5': 'Finally, in sec:summary we conclude.', '1810.03626-1-8-0': '# Probing MeV-scale Particle in CE[MATH] S', '1810.03626-1-9-0': 'In this section we investigate the phenomenological aspects of [MATH] coherent scattering by assuming only the following interaction [EQUATION] where [MATH] and [MATH] parametrize the strength of the Yukawa interaction of a mediator particle [MATH] with [MATH]-[MATH] and the nucleus, respectively.', '1810.03626-1-9-1': 'In principle, the mediator for a [MATH] process with fermions on the external legs can be a scalar or vector boson; we will consider scalar mediators here, though the discussion in Section [REF] is independent on this.', '1810.03626-1-9-2': 'Furthermore, we do not require significant mixing between active neutrinos and [MATH] for coherent scattering, and hence the exchange of SM gauge bosons is suppressed.', '1810.03626-1-9-3': 'Model building options for generating interactions of a scalar singlet [MATH] with quarks, and hence eventually nuclei, are presented for instance in Ref. [CITATION].', '1810.03626-1-9-4': 'The process under our consideration is shown in a diagrammatic form in [REF].', '1810.03626-1-10-0': '## Prerequisites for obtaining the cross sections', '1810.03626-1-11-0': 'Due to the mass of [MATH], the process [MATH] has different kinematics than CE[MATH]S. Hence, as a starting point, we derive some relations for the kinematics of this process that will be used throughout the paper.', '1810.03626-1-11-1': 'The notation of various quantities is given as follows:', '1810.03626-1-12-0': 'Using the above notations, we can explicitly express the 4-momenta: [EQUATION]', '1810.03626-1-12-1': 'When computing the cross section, scalar products of the external momenta (e.g. [MATH], [MATH], [MATH], etc.) will be used.', '1810.03626-1-12-2': 'All scalar products of [MATH], [MATH] and [MATH] and [MATH] can be expressed in terms of these three 4-momenta) read: [EQUATION]', '1810.03626-1-12-3': 'We obtained [MATH] by squaring both sides of [MATH] and using Eq. ([REF]): [EQUATION].', '1810.03626-1-12-4': 'Applying the same to [MATH] and [MATH] and using [MATH], we obtained [MATH] and [MATH] given in Eq. ([REF]).', '1810.03626-1-13-0': 'One can also use the explicit forms of [MATH], [MATH] and [MATH] in Eqs. ([REF])-([REF]) to compute these scalar products directly, e.g., [EQUATION]', '1810.03626-1-13-1': 'We can compare this result with Eq. ([REF]) and obtain [EQUATION] which reveals the relation between [MATH] and [MATH].', '1810.03626-1-13-2': 'In Fig. [REF] we plot the relation for some specific values of [MATH] in order to illustrate how [MATH] varies with [MATH].', '1810.03626-1-13-3': 'Typically (for nonzero [MATH] as a function of [MATH] has a minimum corresponding to the maximal scattering angle [MATH].', '1810.03626-1-14-0': 'By solving [MATH] we obtain [EQUATION]', '1810.03626-1-14-1': 'For [MATH]), [MATH] increases (decreases) with [MATH].', '1810.03626-1-14-2': 'Therefore, [MATH] should be in the range [EQUATION] and due to the upper bound, [MATH] can reach values in the range [EQUATION] where [MATH] and [MATH] are determined by setting the left-hand side of Eq. ([REF]) to [MATH] and solving the equation with respect to [MATH].', '1810.03626-1-14-3': 'The solutions are [EQUATION]', '1810.03626-1-14-4': 'One can check that [REF] has the following massless limit [EQUATION] which is consistent with the standard results of coherent elastic neutrino scattering.', '1810.03626-1-15-0': 'Another important quantity is the minimal neutrino energy [MATH] necessary to create a massive particle [MATH]: [EQUATION] which is obtained by solving [MATH].', '1810.03626-1-15-1': 'If [MATH] is lower than [MATH], [MATH] cannot be produced in the scattering.', '1810.03626-1-15-2': 'In the limit when [MATH] can just be produced, we have [EQUATION]', '1810.03626-1-15-3': 'An interesting difference between the cases of massive and massless [MATH] occurs at [MATH].', '1810.03626-1-15-4': 'From Eq. ([REF]) one can obtain [EQUATION] which implies that for massive [MATH], the nucleus after scattering moves along the same direction as the incoming neutrino ([MATH]); for massless [MATH], it moves in the perpendicular direction ([MATH]).', '1810.03626-1-15-5': 'Although the difference is theoretically distinct, experimentally for very small [MATH] it is difficult to observe such a difference due to rather small recoil energies.', '1810.03626-1-16-0': '## Cross sections', '1810.03626-1-17-0': 'The exchanged scalar [MATH] is generally assumed to be massive with its mass denoted by [MATH].', '1810.03626-1-17-1': 'We evaluate the cross section without assuming [MATH] or [MATH].', '1810.03626-1-17-2': 'The heavy/light mass limits will be discussed below.', '1810.03626-1-18-0': 'From the Feynman diagram in [REF] and the relevant Lagrangian ([REF]), one can straightforwardly write down the scattering amplitudes for (anti)neutrino initial state [EQUATION] where spinor superscripts denote spin and we have inserted the left-/right-handed projectors [MATH] and [MATH] since the neutrino sources can only emit left-handed neutrinos or right-handed antineutrinos.', '1810.03626-1-18-1': 'Using FeynCalc [CITATION] we compute [MATH] for both cases.', '1810.03626-1-18-2': 'The result is identical for both neutrino and antineutrino scattering, namely [EQUATION] with the combined coupling constant [EQUATION]', '1810.03626-1-18-3': 'The dimensionless quantity [MATH] is typically [MATH] and reads [EQUATION]', '1810.03626-1-18-4': 'We will in what follows set limits using experiments with different nuclear targets.', '1810.03626-1-18-5': 'To reduce the dependence of the limits on the type of the nucleus we define [EQUATION] where [MATH] is the nucleon number (sum of neutron and proton numbers).', '1810.03626-1-18-6': 'Since [MATH] has been factored out, [MATH] has little dependence on the type of nuclei.', '1810.03626-1-18-7': 'For example, for Ge and CsI detectors we obtain [EQUATION] where the Yukawa couplings of the scalar [MATH] to neutrons and protons are denoted with [MATH] and [MATH] respectively.', '1810.03626-1-18-8': 'Clearly, [MATH] for Ge (employed at the CONUS experiment) is approximately the same as [MATH] for CsI (currently employed at the COHERENT experiment).', '1810.03626-1-19-0': 'The differential cross section, according to Eq. ([REF]), reads [EQUATION]', '1810.03626-1-19-1': 'One can straightforwardly check that in the limit [MATH] the result in Eq. ([REF]) is consistent with the standard cross section of elastic neutrino scattering [CITATION].', '1810.03626-1-20-0': '## Signals and constraints', '1810.03626-1-21-0': 'Now let us study the signal of our new fermion [MATH] in CE[MATH]S experiments.', '1810.03626-1-21-1': 'We will focus on two experiments, namely COHERENT [CITATION] and CONUS [CITATION].', '1810.03626-1-21-2': 'For the former, we will present the limits on the relevant parameters in [MATH] scattering based on the recent data release, whereas for the latter experiment we obtain sensitivities.', '1810.03626-1-22-0': 'The COHERENT experiment is based on neutrino emission from the Spallation Neutron Source at Oak Ridge National Laboratory.', '1810.03626-1-22-1': 'A crystal scintillator detector with 14.6 kg CsI was used in its recent measurement of CE[MATH]S and the SM signal has been observed with 6.7[MATH] confidence.', '1810.03626-1-22-2': 'The neutrinos are produced via [MATH] decay ([MATH]) and subsequently [MATH] decay ([MATH]).', '1810.03626-1-22-3': 'In this experiment, both [MATH] and [MATH] approximately decay at rest, which allows us to obtain the analytical expressions for neutrino spectra [CITATION] [EQUATION] which contains a monochromatic component [MATH] (i.e. all [MATH] have the same energy [MATH]).', '1810.03626-1-23-0': 'The CONUS experiment measures CE[MATH]S of reactor neutrinos ([MATH]) from a 3.9 GW nuclear power plant in Brokdorf, Germany.', '1810.03626-1-23-1': 'The detector is a Germanium semiconductor containing 4 kg of natural Ge ([MATH] in average), which is set at a distance of [MATH] meters from the reactor.', '1810.03626-1-23-2': 'To compute the event rates we adopt the reactor neutrino flux computed in [CITATION] and normalize the total flux to [MATH].', '1810.03626-1-23-3': 'CONUS data taking has started in April 2018 and recently a preliminary 2.4[MATH] statistical significance for observing the process was announced [CITATION].', '1810.03626-1-24-0': 'The event numbers in both experiments can be computed in the following way: in the [MATH]-th recoil energy bin ([MATH]), the total event number [MATH] consists of the SM contribution [MATH] and the new physics contribution [MATH], i.e. [EQUATION] which are computed by [EQUATION] where [EQUATION]', '1810.03626-1-24-1': 'Here [MATH] is the neutrino spectrum, [MATH] is the number of nuclei in the detector and [MATH] is the data taking period.', '1810.03626-1-24-2': 'The explicit expressions of [MATH], [MATH] and [MATH] are given in [REF].', '1810.03626-1-24-3': 'We note that we have included a form factor [MATH] in the cross section for the COHERENT experiment, where we take the parametrization given in Ref. [CITATION], see Fig. 1(a) therein.', '1810.03626-1-24-4': 'For the CsI detectors used in COHERENT, since the atomic number of Xe (54) is between Cs (55) and I (53), it is a good approximation to use the Xe form factor for both Cs and I. For reactor neutrinos, we can set [MATH] due to the low recoil energy.', '1810.03626-1-25-0': 'Using the above equations, we can compute the event numbers and study the signal of new physics in these two experiments.', '1810.03626-1-25-1': 'In Fig. [REF], we present the event distributions for several choices of [MATH] parameters together with the ratio of [MATH] for both CONUS (left) and COHERENT (right).', '1810.03626-1-25-2': 'We selectively choose several values for [MATH] (0 MeV and 3 MeV for CONUS; 3 MeV and 30 MeV for COHERENT) to illustrate the effect of [MATH] on CE[MATH]S. Light and heavy mediator cases have been illustrated by considering both [MATH] MeV and [MATH] MeV.', '1810.03626-1-25-3': 'The kinks of the red and blue curves appearing in the right panel at [MATH] keV are caused by the monochromatic [MATH] in COHERENT.', '1810.03626-1-25-4': 'The green and black curves correspond to [MATH] MeV.', '1810.03626-1-25-5': 'Since the monochromatic [MATH] neutrinos of [MATH] MeV energy do not have sufficient energy to produce [MATH] there are no similar kinks in these two curves.', '1810.03626-1-26-0': 'By comparing [MATH] with the observed event numbers we can obtain the constraints on the [MATH] coupling to neutrinos and nuclei.', '1810.03626-1-26-1': 'For COHERENT, the observed event numbers have been published in Ref. [CITATION] which can be used directly in our data fitting procedure.', '1810.03626-1-26-2': 'The recoil threshold in COHERENT is controlled by the signal acceptance fraction (see Fig. S9 of Ref. [CITATION]) which drops down quickly when the number of photoelectrons [MATH]) is less than [MATH], and approximately vanishes when [MATH].', '1810.03626-1-26-3': 'Therefore, in fitting the COHERENT data we import the signal acceptance fraction directly instead of setting a distinct threshold.', '1810.03626-1-26-4': 'The systematic and statistical uncertainties have been combined and provided in Fig. 3 of Ref. [CITATION], and are employed directly in our data fitting.', '1810.03626-1-27-0': 'The CONUS data has not been published, and hence we assume that their findings will be compatible with the SM prediction after [MATH] year of data taking with a [MATH] kg detector (thus [MATH] year [MATH] kg exposure).', '1810.03626-1-27-1': 'This allows us to compute sensitivity of CONUS on the production of [MATH].', '1810.03626-1-27-2': 'More explicitly, we adopt the following [MATH]-function comparing [MATH] with [MATH]: [EQUATION]', '1810.03626-1-27-3': 'Here [MATH] is a rescaling factor with an uncertainty [MATH] which mainly comes from the overall uncertainty of the neutrino flux.', '1810.03626-1-27-4': 'In addition, other systematic uncertainties may change the shape of the event spectrum, which is parametrized by [MATH] and assumed to be [MATH].', '1810.03626-1-27-5': 'The background [MATH] in each bin is 1 [MATH].', '1810.03626-1-27-6': 'For the nucleus recoil threshold we take 1.2 keV.', '1810.03626-1-28-0': 'The results are presented in Fig. [REF] where we show the constraints in the [MATH] plane (with [MATH] fixed) and the [MATH] plane (with [MATH] fixed).', '1810.03626-1-28-1': 'In the [MATH] panels, the bounds are almost flat when [MATH] MeV (CONUS) or [MATH] MeV (COHERENT), which can be understood from Eq. ([REF]) where, for small [MATH], [MATH] dominates over [MATH] in the denominator.', '1810.03626-1-28-2': 'Similarly, in the [MATH] plots, the bounds are also approximately flat for small [MATH] which can be understood from the [MATH] factor in Eq. ([REF]).', '1810.03626-1-28-3': 'However, the large mass behaviors are different for [MATH] and [MATH].', '1810.03626-1-28-4': 'As shown in the left panels of Fig. [REF], the curves are approximatively linear for large [MATH] because in this case the cross section is proportional to [MATH].', '1810.03626-1-28-5': 'On the other hand, large [MATH] can only be constrained by the events with high [MATH].', '1810.03626-1-28-6': 'If [MATH] is larger than the maximal value of [MATH] of the neutrino flux, then there will be no constraint at all because neutrinos do not have sufficient energy to produce [MATH].', '1810.03626-1-28-7': 'For reactor neutrinos, the event rate above [MATH] MeV is essentially too low to have a significant impact and hence the sensitivity to the new physics scenario diminishes.', '1810.03626-1-28-8': 'Therefore, the CONUS curves in the right panel rise up quickly around 6 MeV.', '1810.03626-1-28-9': 'For COHERENT, the maximal [MATH] is about [MATH] MeV (half of [MATH]) but, unlike in CONUS, the flux is not suppressed when [MATH] is approaching [MATH] MeV, so the curves do not rise so quickly when [MATH] is close to the maximal [MATH].', '1810.03626-1-29-0': 'In the future, the measurement of CE[MATH]NS will be significantly improved by lower thresholds, larger fiducial masses, and longer exposure times, etc.', '1810.03626-1-29-1': 'For reactor neutrinos, lower thresholds can increase the statistics drastically because the current threshold actually only allows CONUS to measure the high energy tail of the reactor neutrino flux.', '1810.03626-1-29-2': 'For COHERENT, using lower threshold detectors will not improve the measurement significantly.', '1810.03626-1-29-3': 'We will consider here the following two benchmark configurations to illustrate the future sensitivities of CE[MATH]S experiments.', '1810.03626-1-29-4': 'The first one is running CONUS for 5 years with 100 kg Ge, and a considerably improved threshold down to [MATH] keV.', '1810.03626-1-29-5': 'In addition, the theoretical uncertainties of reactor neutrino flux are assumed to be reduced by a factor of 2.', '1810.03626-1-29-6': 'The second is (instead of doing a very detailled study of various other detectors and target materials that are planed [CITATION]) increasing the statistics of COHERENT by a factor of 100, which could be achieved by, e.g., a 20 times larger fiducial mass with 5 times longer exposure.', '1810.03626-1-29-7': 'The systematic uncertainties are correspondingly reduced so that we assume the overall uncertainty is reduced by a factor of [MATH].', '1810.03626-1-29-8': 'In Fig. [REF], we show the sensitivities of these two future experiments together with their current constraints/sensitivities.', '1810.03626-1-29-9': 'Here, [MATH] is set at 5 MeV as a benchmark value.', '1810.03626-1-30-0': 'Let us now discuss other limits on the scenario under study.', '1810.03626-1-30-1': 'Regarding CE[MATH]S, aspects of light scalars coupling to neutrinos and nuclei were explored in Ref. [CITATION].', '1810.03626-1-30-2': 'Since in our framework a massive MeV-scale fermion [MATH] is involved, most limits are expected to be weaker than the ones collected in Ref. [CITATION], where only couplings to nuclei and light neutrinos were considered.', '1810.03626-1-30-3': 'It is in addition more complicated to obtain precise limits, so we focus here on giving reasonably robust estimates.', '1810.03626-1-30-4': 'It was found in Ref. [CITATION] that all limits from terrestrial experiments, e.g. [MATH]-Pb scattering and meson decay experiments, are weaker than the bound from COHERENT as well as the CONUS sensitivity.', '1810.03626-1-30-5': 'BBN constraints, however, are relevant for [MATH] MeV-scale [MATH], and thus the [MATH] curves in fig:constraints should be interpreted as an illustration to show the strength of the limit in the small mass regime.', '1810.03626-1-30-6': 'When considering the [MATH] density evolution in the early Universe (see sec:DM), we will actually take [MATH] MeV.', '1810.03626-1-31-0': 'We should also mention limits from Supernova 1987A.', '1810.03626-1-31-1': 'If efficiently produced, the light states can carry a significant amount of energy from the Supernova core.', '1810.03626-1-31-2': 'In such case, the amount of energy carried by active neutrinos would be too small to match the observation of Supernova 1987A and hence a limit can be set.', '1810.03626-1-31-3': 'The leading process for the production of [MATH] is [MATH] via [MATH]-channel [MATH] exchange and it is suppressed by the fourth power of the small coupling [MATH].', '1810.03626-1-31-4': 'As [MATH] is concerned, in Ref. [CITATION] the authors presented, within a specific model, that the cross section for scattering of a new light fermion on protons and electrons is constrained by Supernova 1987A cooling arguments to values comparable to the corresponding cross sections for neutrinos.', '1810.03626-1-31-5': 'This can be understood as follows: if a novel fermion acts as a fourth neutrino species inside of the star, it will carry away energy comparable to the one carried away by the individual active neutrino species.', '1810.03626-1-31-6': 'This suggests that [MATH] of the energy budget would be carried away by [MATH].', '1810.03626-1-31-7': 'Given the astrophysical uncertainties associated to Supernova 1987A, exotic particles can carry away up to 50 of the total energy of the collapse [CITATION].', '1810.03626-1-31-8': 'This corresponds again roughly to a new cross section of similar magnitude as the SM one.', '1810.03626-1-31-9': 'The reachable parameter values from [REF] fulfill this constraint.', '1810.03626-1-31-10': 'Hence, we infer that the cooling arguments are not excluding the relevant parameter space.', '1810.03626-1-31-11': 'Finally, note that [MATH], being an MeV-scale particle, can not be resonantly produced through an MSW effect [CITATION].', '1810.03626-1-31-12': 'Such effect is very relevant for keV-scale particles for which strong limits can be derived [CITATION].', '1810.03626-1-32-0': '# [MATH] and Neutrino Mass Generation', '1810.03626-1-33-0': 'In this section we will discuss the possible connection of the new fermion [MATH] to neutrino mass generation.', '1810.03626-1-33-1': 'Any fermion that couples to light active neutrinos must be investigated with regard to its contribution to neutrino mass.', '1810.03626-1-34-0': 'Let us first discuss the nature of the scalar [MATH] that appears in our framework.', '1810.03626-1-34-1': 'Given our preference for light [MATH], such construction is not achievable with representations higher than singlets.', '1810.03626-1-34-2': 'Namely, [MATH] can obviously not be the SM Higgs due to its tiny couplings with [MATH] and [MATH] quark as well as its heavy mass which would further suppress the strength of the CE[MATH]S process.', '1810.03626-1-34-3': 'If we replace the SM Higgs by a novel Higgs doublet [MATH] with possibly larger couplings to quarks (and hence nuclei), we face the problem of a neccessary huge mass splitting between the light neutral component and the charged ones, which have not been seen.', '1810.03626-1-34-4': 'An option would be to consider the following gauge invariant [MATH]-dimensional operator in the effective theory formalism [EQUATION] with singlets [MATH] and [MATH], where [MATH] represents the scale of new physics.', '1810.03626-1-34-5': 'After electroweak symmetry breaking this operator yields an interaction term [EQUATION]', '1810.03626-1-34-6': 'By assuming furthermore non-vanishing interactions between [MATH] and nuclei (or light quarks), the CE[MATH]S occurs through the process shown in [REF].', '1810.03626-1-34-7': 'We will now discuss a minimal model containing SM singlets only, which will generate the effective Lagrangian in Eq. ([REF]).', '1810.03626-1-35-0': '## The Model', '1810.03626-1-36-0': 'We supplement the SM particle content with [EQUATION] where [MATH] and [MATH] are Majorana fermions and [MATH] is a real scalar.', '1810.03626-1-36-1': 'The quantum numbers under the SM gauge group [MATH] are indicated in brackets, and clearly no charged degrees of freedom are introduced.', '1810.03626-1-36-2': 'One of the goals of this section is to demonstrate that the neutrino masses can be generated from this extended fermion sector via a modified type-I seesaw mechanism [CITATION].', '1810.03626-1-36-3': 'This means that we would require at least two generations of novel fermions which participate in this mechanism, such that at most one light neutrino is massless.', '1810.03626-1-36-4': 'Still, for simplicity, throughout this section we will focus on the [MATH]-generation case which can be straightforwardly extended.', '1810.03626-1-36-5': 'Similarly, we will also restrict our discussion to one active neutrino flavor state, namely for definitness the electron (anti)neutrino [MATH].', '1810.03626-1-37-0': 'The relevant part of the Lagrangian reads [EQUATION] where [MATH] are the Yukawa couplings, [MATH] and [MATH] are Majorana masses of [MATH] and [MATH] fields, respectively, and [MATH] is the Dirac mass which is allowed by gauge symmetries.', '1810.03626-1-38-0': 'This Lagrangian is a minimal UV complete realization of eq:5D,eq:eff_coupling with fermion singlet [MATH] interacting with the fields given in both brackets of eq:5D through Yukawa couplings [MATH] and [MATH] (see eq:lagr).', '1810.03626-1-38-1': 'We will show that the allowed values of [MATH] exceed the characteristic momentum exchange [MATH] in CE[MATH]S experiments, which justifies the analysis setup in sec:production.', '1810.03626-1-38-2': 'If [MATH], we can easily relate the parameters of the full theory with [MATH] and obtain [MATH].', '1810.03626-1-38-3': 'If that was not the case, the topology shown in [REF] including [MATH] as the dynamical degree of freedom should be considered.', '1810.03626-1-39-0': 'More importantly, within the presented model, we will demonstrate the existence of parameter space that can be probed by CE[MATH]S experiments, generates neutrino masses in the right ballpark, and is not excluded from the new physics searches at neutrino oscillation facilities, beam dump experiments, colliders, etc.', '1810.03626-1-39-1': 'This indicates the importance of the CE[MATH]S in future new physics searches as there are scenarios where it could yield the strongest limits or perhaps even lead to new discoveries.', '1810.03626-1-40-0': 'After electroweak symmetry breaking, the neutral fermion mass matrix reads [EQUATION] where [MATH] GeV and we assumed that [MATH] does not develop a non-vanishing vacuum expectation value.', '1810.03626-1-41-0': 'We furthermore assume for the mass matrix given in [REF] that [MATH].', '1810.03626-1-41-1': 'In this way, the mixing between [MATH] and [MATH] is suppressed and hence, the masses of heavy new fermions essentially match the parameters in the flavor basis, [MATH] and [MATH].', '1810.03626-1-41-2': 'Contrary, if [MATH], the two physical masses would be of similar size which is not wanted in our scenario.', '1810.03626-1-42-0': 'We start by performing a rotation in the [MATH]-[MATH] plane by an angle [MATH].', '1810.03626-1-42-1': 'As discussed above, [MATH] is the physical mass of a particle produced in CE[MATH]S experiments.', '1810.03626-1-42-2': 'We take [MATH] MeV as an illustrative number.', '1810.03626-1-42-3': 'The bounds on the mixing of active neutrinos with heavy fermions have been extensively studies in the literature.', '1810.03626-1-42-4': 'From Refs. [CITATION] we infer that the constraint on the mixing between [MATH] and [MATH] for [MATH] MeV reads [EQUATION] and is set by neutrinoless double beta decay experiments.', '1810.03626-1-42-5': 'Weaker limits apply for the other flavors, which therefore can be accommodated more easily.', '1810.03626-1-42-6': 'The mass matrix after the [MATH]-[MATH] rotation reads approximately [EQUATION] from where one can infer that [MATH] may serve as a potential source of neutrino mass.', '1810.03626-1-42-7': 'By taking the upper value of [MATH] in eq:limit we obtain [MATH] eV which matches the required order of magnitude for neutrino mass.', '1810.03626-1-43-0': 'It was demonstrated in sec:production that the numerical analysis of CE[MATH]S for [MATH] MeV yields the limit [EQUATION] where [MATH] was introduced in eq:eft-lagrangian and [MATH] roughly corresponds to the coupling strength to individual quarks.', '1810.03626-1-43-1': 'The values indicated in square brackets represent the range in which the bound, depending on specific values of [MATH] and [MATH], is set (see fig:constraints).', '1810.03626-1-43-2': 'We assume [MATH], i.e. similar size of the [MATH] coupling to quarks and fermions, such that [EQUATION] approximatively holds.', '1810.03626-1-43-3': 'Having now a feeling for the numbers in eq:neutral_mass2 , we continue the diagonalization.', '1810.03626-1-43-4': 'Performing a rotation in the [MATH]-[MATH] plane by an angle [MATH] and using this expression as well as eq:Y2 we can relate the mixing angle with the upper limit from CE[MATH]S experiments [EQUATION]', '1810.03626-1-43-5': 'Clearly, [MATH] must not be tiny as otherwise the large mixing would pose a problem for [MATH].', '1810.03626-1-43-6': 'We can safely assume [MATH] because it parametrizes the strength of the interaction between three hidden particles and moreover [MATH] does not mix with the SM Higgs.', '1810.03626-1-43-7': 'By inserting [MATH] GeV in [REF] we obtain the relation [EQUATION]', '1810.03626-1-43-8': 'The seesaw contribution to the neutrino mass from mixing between [MATH] and [MATH] is then [EQUATION] which gives the upper bound on [MATH] from neutrino mass considerations [EQUATION]', '1810.03626-1-43-9': 'From eq:y1-M1,eq:y_1 we infer that the [MATH] values which can contribute to this neutrino mass generation are in the [MATH] GeV mass range.', '1810.03626-1-43-10': 'Finally, we need to check if the mixing given in eq:mix_eN is compatible with such masses.', '1810.03626-1-43-11': 'To this end, we again employ the limits from Refs. [CITATION] and infer that [MATH] GeV is fully consistent, whereas the smaller values are marginally allowed, i.e. in tension with the constraints from neutrinoless double beta decay experiments, big bang nucleosynthesis as well as the PS191 [CITATION] beam dump experiment.', '1810.03626-1-43-12': 'Interestingly, GeV-scale [MATH] will be testable at DUNE [CITATION] and SHiP [CITATION].', '1810.03626-1-44-0': 'In summary, parameters that are compatible with all available laboratory constraints and give an observable signal in coherent scattering experiments, give a neutrino mass of order [EQUATION] which is compatible with observation.', '1810.03626-1-44-1': 'Here, [MATH] denotes relative contribution to the active neutrino mass from [MATH] and [MATH].', '1810.03626-1-45-0': '# [MATH] as Dark Matter Particle', '1810.03626-1-46-0': 'Limits on [MATH] from terrestrial experiments as well as astrophysics were discussed in [REF].', '1810.03626-1-46-1': 'This section is devoted to the evaluation of the cosmic abundance of [MATH].', '1810.03626-1-46-2': 'As a first observation, we note that the smallness of [MATH] relative to the electroweak scale implies that the production of DM from freeze-out might not yield desired results and signifies the preference for non-thermal production.', '1810.03626-1-46-3': 'Note further that within our framework there is no possibility for [MATH] to decay into a pair of electrons or neutrinos.', '1810.03626-1-46-4': 'However, there could be tree-level ([MATH]) and radiative decays ([MATH]).', '1810.03626-1-46-5': 'From the expressions for the decay rates of these processes [CITATION] we infer that in order to ensure [MATH] stability, [MATH] for [MATH] MeV.', '1810.03626-1-46-6': 'The implied tiny value of [MATH] does not jeopardize our new physics scenario at CE[MATH]S experiments because the rate for [MATH] depends on [MATH] and [MATH] couplings, but not on [MATH].', '1810.03626-1-46-7': 'Note however from Eq. ([REF]) that neutrino mass will be dominated by [MATH] in this case.', '1810.03626-1-47-0': 'If [MATH] is the DM particle, CE[MATH]S experiments would yield an entirely novel method for searching (MeV-scale) DM.', '1810.03626-1-47-1': 'Note that dark matter in CE[MATH]S experiments was discussed already in [CITATION], where kinetically mixed dark photons decay into DM pairs which subsequently scatters at CE[MATH]S experiments (note that the neutrino sources of those experiments also generate photons).', '1810.03626-1-47-2': 'Here, we will propose a novel framework by demonstrating that the DM particle can be produced directly via CE[MATH]S and the effects of such process are imprinted in the measurable recoils of the nuclei.', '1810.03626-1-47-3': 'Interestingly, this would resemble DM search in direct detection experiments because in both cases the observable signal is the nuclear recoil.', '1810.03626-1-47-4': 'While nuclear recoil in direct detection experiments stems from DM-nucleus interaction, in CE[MATH]S such effect is caused by neutrinos.', '1810.03626-1-47-5': 'Thus, the main goal would be to distinguish between the SM coherent neutrino scattering events and "new physics" events in which [MATH] is produced.', '1810.03626-1-47-6': 'We have shown in [REF] that the distributions of the recoil energy corresponding to these two cases can be vastly different and it is possible to discriminate between them.', '1810.03626-1-47-7': 'These arguments strongly motivate the search for DM at CE[MATH]S experiments if [MATH] can indeed play the role of DM.', '1810.03626-1-47-8': 'Hence, in what follows we will discuss if and how [MATH] can be produced in right amounts.', '1810.03626-1-47-9': 'We rely on the UV complete model introduced in sec:model which was already shown to be successful in generating non-vanishing neutrino masses.', '1810.03626-1-48-0': 'In order to realize the CE[MATH]S process shown in fig1 we require [MATH] to interact with quarks as well as with [MATH] and neutrinos.', '1810.03626-1-48-1': 'The Lagrangian for these interactions reads [EQUATION] where [MATH] and [MATH] are the couplings to quarks and leptons, respectively.', '1810.03626-1-48-2': 'We consider the case in which neither of these couplings is weaker than [MATH].', '1810.03626-1-48-3': 'For an UV complete model from which [REF] stems after electroweak symmetry breaking, we refer the reader to [REF].', '1810.03626-1-49-0': 'For [MATH] MeV and [MATH], the decay and inverse decay widths of the process [MATH] are much larger than the value of the Hubble parameter across the relevant temperatures.', '1810.03626-1-49-1': 'Thus, [MATH] is in thermal equilibrium with the SM bath.', '1810.03626-1-49-2': 'At the temperatures below the QCD phase transition ([MATH] MeV) quarks are no longer relevant degrees of freedom.', '1810.03626-1-49-3': 'The vast majority of the formed mesons and baryons are non-relativistic and hence effectively disappear from the SM thermal bath shortly below [MATH] (the only exception are pions which are somewhat lighter).', '1810.03626-1-49-4': 'The [MATH] interaction which was keeping [MATH] in thermal equilibrium is not relevant below [MATH].', '1810.03626-1-49-5': 'However, even below [MATH], [MATH] can be in thermal equilibrium through the process [MATH], provided there is a sufficiently large Yukawa coupling [MATH].', '1810.03626-1-49-6': 'Such process occurs via [MATH]-channel exchange of [MATH] and is governed exclusively by the second term in [REF].', '1810.03626-1-49-7': 'Note that [MATH] should decay shortly below [MATH] into [MATH] and neutrinos in order not to violate any of the BBN predictions.', '1810.03626-1-49-8': 'For the parameter values we will take below this is indeed the case.', '1810.03626-1-49-9': 'It is also worthwhile to mention that the [MATH], introduced in [REF] to UV-complete the model, decay well above [MATH] due to their large [MATH] coupling.', '1810.03626-1-49-10': 'On the other hand, the stability of [MATH] is ensured by assuming a small value of [MATH] which sufficiently suppresses [MATH] and [MATH] decays.', '1810.03626-1-50-0': 'Above [MATH], [MATH] is in thermal equilibrium with [MATH] and hence also with the SM bath, due to rapid [MATH] processes.', '1810.03626-1-50-1': 'Whether [MATH] remains in contact with the SM bath at lower temperatures (after [MATH] decays) depends on the strength of the [MATH] process which occurs via [MATH]-channel exchange of [MATH] (hence is proportional to [MATH]).', '1810.03626-1-51-0': 'In order to accurately determine the present-day abundance of [MATH] we solve the following Boltzmann equations [CITATION] [EQUATION]', '1810.03626-1-51-1': 'For a particle denoted with [MATH] we define the yield as [MATH] (superscript eq denotes equilibrium value), where [MATH] and [MATH] are number and entropy density, respectively.', '1810.03626-1-51-2': 'In addition, [MATH], [MATH] is the Hubble parameter, [MATH] is the thermally averaged cross section for a given process (evaluated following Ref. [CITATION]), [MATH], [MATH] and [MATH] are modified Bessel functions.', '1810.03626-1-51-3': 'The initial conditions are [MATH] and [MATH], since we argued above that the sizable coupling to free quarks leaves the particles in thermal equilibrium at temperatures above [MATH].', '1810.03626-1-52-0': 'The solution for [MATH] is shown in [REF].', '1810.03626-1-52-1': 'In both panels we set [MATH] MeV and show the results for [MATH] which are in the ballpark of testable values at CE[MATH]S experiments.', '1810.03626-1-52-2': 'In the (left) right panel, [MATH] is fixed to [MATH] MeV ([MATH] MeV).', '1810.03626-1-52-3': 'Much smaller values of [MATH] are not considered due to the requirement [MATH] which forbids [MATH] to decay and is thus essential to render our DM candidate stable.', '1810.03626-1-52-4': 'We discuss and interpret the results from the figure in what follows.', '1810.03626-1-53-0': 'In both panels we observe the standard thermal freeze-out of [MATH] for [MATH].', '1810.03626-1-53-1': 'The interaction which keeps [MATH] in thermal equilibrium with the SM is [MATH], and the strength of this process increases by reducing [MATH].', '1810.03626-1-53-2': 'Hence, for smaller [MATH], the stronger interaction indicates that [MATH] stays longer in the thermal equilibrium and undergoes freeze-out at later times (smaller temperatures) which implies smaller final abundance of [MATH].', '1810.03626-1-53-3': 'This is visible from fig:DM where the final value of [MATH] for [MATH] (blue line) in the left panel (lighter [MATH]) is much smaller that the one shown in the right panel.', '1810.03626-1-53-4': 'By using the relation between the yield and the relic abundance [EQUATION] it is clear from both panels of fig:DM that [MATH] is strongly overproduced with respect to the observed DM abundance [MATH] [CITATION].', '1810.03626-1-53-5': 'Moreover, the freeze-out temperatures are smaller than [MATH] MeV which means that any new physics contribution invoked to deplete [MATH] is strongly constrained by BBN considerations.', '1810.03626-1-54-0': 'The freeze-out also occurs for [MATH] MeV and [MATH] (red line in the left panel).', '1810.03626-1-54-1': 'In this case, initially, the interaction between [MATH] and the SM is not sufficiently strong to follow the sudden change of [MATH] which is induced by a rapid change of SM degrees of freedom present in the thermal bath at [MATH] MeV.', '1810.03626-1-54-2': 'However, [MATH] eventually reaches the equilibrium value and the freeze-out occurs leaving [MATH] with even larger abundance than in the previously discussed scenarios.', '1810.03626-1-54-3': 'For [MATH] MeV and [MATH] (red line in the right panel) as well as for both cases with [MATH], [MATH] does not reach equilibrium.', '1810.03626-1-54-4': 'This means that the production occurs via "freeze-in" [CITATION] where the weak interaction with the SM bath leads to a gradual accumulation of [MATH] abundance.', '1810.03626-1-54-5': 'From [REF] it is obvious that the final [MATH] in all "freeze-in" scenarios is chiefly set by the initial condition (non-vanishing [MATH] at QCD phase transition) and the freeze-in contribution yields only a subdominant effect.', '1810.03626-1-55-0': 'Even though the DM abundance is still too large, in the "freeze-in" scenarios where [MATH] is not in thermal equilibrium at [MATH] MeV temperatures, a late-time entropy injection episode below [MATH] can reduce the [MATH] abundance significantly.', '1810.03626-1-55-1': 'Such entropy injection can be achieved for instance via decays of heavy scalars (see Ref. [CITATION] and references therein).', '1810.03626-1-55-2': 'It is clear that [MATH] needs to be diluted by [MATH] in order to meet observation.', '1810.03626-1-55-3': 'Let us note that alternatives with respect to late-time entropy injection have also been considered.', '1810.03626-1-55-4': 'For instance, a heavier, overproduced particle could decay into lighter states which may be DM [CITATION].', '1810.03626-1-55-5': 'This helps because, as may be inferred from eq:rel, the abundance of DM is proportional to the mass of a DM particle.', '1810.03626-1-55-6': 'A detailed analysis of such a scenario is beyond the scope of this project as it would require a significant extension of the Boltzmann equations given in eq:Boltzmann.', '1810.03626-1-55-7': 'In such extension, the late-time entropy injection would be avoided.', '1810.03626-1-55-8': 'We also note that there may be more models in which MeV-scale particle is produced in right amounts, without a necessity for late-time entropy injection.', '1810.03626-1-56-0': 'In conclusion, for the couplings that can be probed in CONUS or COHERENT [MATH], we explored whether in our model [MATH] can be DM, i.e. produced in right amounts in the early Universe.', '1810.03626-1-56-1': 'It turns out (see again [REF]) that this is not achievable without extending the minimal model presented in [REF].', '1810.03626-1-56-2': 'Namely, we find that a late time entropy injection episode is necessary to sufficiently deplete the abundance of [MATH] and render it a viable DM candidate.', '1810.03626-1-57-0': 'Since [MATH] in our model does not scatter on nuclei and electrons at tree-level, the bounds from direct detection are not strong.', '1810.03626-1-57-1': 'Moreover, for the considered range of couplings and masses, the CE[MATH]S cross section is several orders of magnitude smaller than those that can be currently tested at direct DM detection facilities for MeV-scale DM [CITATION].', '1810.03626-1-57-2': 'Given the absence of the annihilation channel into [MATH] pairs, the constraints from CMB [CITATION] are also not probing the parameter space to which CE[MATH]S experiments are sensitive.', '1810.03626-1-58-0': '# Summary and Conclusions', '1810.03626-1-59-0': 'Coherent neutrino-nucleus scattering is a new window to probe physics within and beyond the Standard Model.', '1810.03626-1-59-1': 'We have noted here that the final state fermion does not necessarily have to be a light active neutrino.', '1810.03626-1-59-2': 'Instead, we have entertained the possibility that an MeV-scale fermion [MATH] is produced in the process, which will lead to a significant modification of the observable recoil spectrum.', '1810.03626-1-59-3': 'We have set limits on the parameters that are involved when the couplings of the neutrino-[MATH] pair to quarks is mediated by a light scalar.', '1810.03626-1-59-4': 'The measurable couplings are well compatible with neutrino mass generation.', '1810.03626-1-59-5': 'Furthermore, the new fermion [MATH] may be the DM particle, which we have shown to be typically requiring an injection of entropy in the early Universe after the QCD phase transition.', '1810.03626-1-60-0': 'Thus, exotic physics in coherent neutrino-nucleus scattering has a variety of interesting implications in neutrino physics and cosmology.', '1810.03626-1-60-1': 'The present analysis is only one example of the exciting prospects that this new window to physics has given us the opportunity to probe.'} | {'1810.03626-2-0-0': 'We consider the production of a new MeV-scale fermion in coherent elastic neutrino-nucleus scattering.', '1810.03626-2-0-1': 'The effect on the measurable nucleon recoil spectrum is calculated.', '1810.03626-2-0-2': 'Assuming that the new fermion couples to neutrinos and quarks via a singlet scalar, we set limits on its mass and coupling using COHERENT data and also determine the sensitivity of the CONUS experiment.', '1810.03626-2-0-3': 'We investigate the possible connection of the new fermion to neutrino mass generation.', '1810.03626-2-0-4': 'The possibility of the new fermion being the dark matter particle is also studied.', '1810.03626-2-1-0': '# Introduction', '1810.03626-2-2-0': 'Despite being the most elusive Standard Model (SM) particles, neutrinos have been detected in a number of charged- and neutral-current processes.', '1810.03626-2-2-1': 'The recent measurement [CITATION] of coherent elastic neutrino-nucleus scattering (CE[MATH]S) [CITATION] yields a novel channel where, for the first time, the interaction of low energy neutrinos with nuclei as a whole is probed.', '1810.03626-2-2-2': 'This serves not only as a handle to probe SM and nuclear physics parameters, but also as a robust probe of new physics.', '1810.03626-2-2-3': 'In particular, light sterile neutrinos [CITATION], non-standard interactions of both quarks and leptons [CITATION] as well as neutrino magnetic moments [CITATION] can be searched for.', '1810.03626-2-3-0': 'The basic requirement for the coherent neutrino-nucleus scattering is the smallness of the momentum transfer.', '1810.03626-2-3-1': 'Namely, in case it exceeds the inverse size of the nucleus, one can in principle determine on which nucleon the scattering occurred and this is what breaks the coherence.', '1810.03626-2-3-2': 'It is also important that the quantum state of the nucleus does not alter in the scattering because, otherwise, the nuclear excitations in such processes would allow individual nucleons to be tagged which would again directly break the condition for the coherent scattering [CITATION].', '1810.03626-2-4-0': 'On the other hand, the production of new light particles does not a priori violate the coherence as long as the above conditions for nuclei are satisfied.', '1810.03626-2-4-1': 'Hence, in this work we explore an interesting new possibility for coherent elastic scattering process, namely [EQUATION].', '1810.03626-2-4-2': 'Here a light MeV-scale fermion (dubbed [MATH]) is produced from the interaction of the incoming neutrino [MATH] with a nucleus [MATH].', '1810.03626-2-4-3': 'We are interested, given the lack of evidence for new physics at high energy, in MeV-scale particles as this is the typical energy scale of CE[MATH]S, where naturally the most interesting phenomenology arises.', '1810.03626-2-5-0': 'Assuming in a minimal setup that the interaction of the new fermion [MATH] with neutrinos and quarks is mediated by a scalar singlet [MATH], we derive limits on the masses of [MATH] and [MATH] and their coupling to neutrinos and the nucleus.', '1810.03626-2-5-1': 'Existing and expected data from the running experiments COHERENT [CITATION] and CONUS [CITATION] is used, and the results are compared to existing terrestrial and astrophysical limits.', '1810.03626-2-5-2': 'In the near future, other upcoming experiments including [MATH]-cleus [CITATION], CONNIE [CITATION], MINER [CITATION], TEXONO [CITATION], [MATH]GEN [CITATION] and Ricochet [CITATION] will also be able to measure the CE[MATH]NS process.', '1810.03626-2-6-0': 'Any new fermion that couples to light neutrinos needs to be considered regarding its role in the generation of neutrino mass, and we demonstrate that a straightforward extension of the type-I seesaw mechanism can indeed generate the observable magnitude of neutrino masses, as well as be testable in CE[MATH]S. Moreover, any new particle beyond the Standard Model is an attractive candidate for dark matter (DM), therefore we investigate in such a setup whether [MATH] can be such a popular MeV-scale DM candidate (see e.g. Refs. [CITATION] for recent studies).', '1810.03626-2-6-1': 'We find that for the size of the couplings to which CE[MATH]S experiments are sensitive, the DM abundance can match the observed value in case there was an entropy injection episode between the QCD phase transition and Big Bang Nucleosynthesis (BBN).', '1810.03626-2-7-0': 'The paper is organized as follows.', '1810.03626-2-7-1': 'In [REF] we derive bounds on the relevant couplings and masses within our framework of [MATH] coherent scattering without restricting the discussion to a specific model.', '1810.03626-2-7-2': 'We also obtain the corresponding recoil spectra of [MATH] in case a massive particle [MATH] is emitted in the final state.', '1810.03626-2-7-3': 'In sec:model we discuss a minimal UV-complete setup in which the MeV-scale [MATH] is related to neutrino mass generation.', '1810.03626-2-7-4': 'sec:DM is devoted to the assumption that [MATH] is the DM particle, in which we scrutinize its production in the early Universe.', '1810.03626-2-7-5': 'Finally, in sec:summary we conclude.', '1810.03626-2-8-0': '# Probing MeV-scale Particle in CE[MATH] S', '1810.03626-2-9-0': 'In this section we investigate the phenomenological aspects of [MATH] coherent scattering by assuming only the following interaction [EQUATION] where [MATH] and [MATH] parametrize the strength of the Yukawa interaction of a mediator particle [MATH] with [MATH]-[MATH] and the nucleus, respectively.', '1810.03626-2-9-1': 'In principle, the mediator for a [MATH] process with fermions on the external legs can be a scalar or vector boson; we will consider scalar mediators here, though the discussion in Section [REF] is independent on this.', '1810.03626-2-9-2': 'Furthermore, we do not require significant mixing between active neutrinos and [MATH] for coherent scattering, and hence the exchange of SM gauge bosons is suppressed.', '1810.03626-2-9-3': 'Model building options for generating interactions of a scalar singlet [MATH] with quarks, and hence eventually nuclei, are presented for instance in Ref. [CITATION].', '1810.03626-2-9-4': 'The process under our consideration is shown in a diagrammatic form in [REF].', '1810.03626-2-10-0': '## Prerequisites for obtaining the cross sections', '1810.03626-2-11-0': 'Due to the mass of [MATH], the process [MATH] has different kinematics than CE[MATH]S. Hence, as a starting point, we derive some relations for the kinematics of this process that will be used throughout the paper.', '1810.03626-2-11-1': 'The notation of various quantities is given as follows:', '1810.03626-2-12-0': 'Using the above notations, we can explicitly express the 4-momenta: [EQUATION]', '1810.03626-2-12-1': 'When computing the cross section, scalar products of the external momenta (e.g. [MATH], [MATH], [MATH], etc.) will be used.', '1810.03626-2-12-2': 'All scalar products of [MATH], [MATH] and [MATH] and [MATH] can be expressed in terms of these three 4-momenta) read: [EQUATION]', '1810.03626-2-12-3': 'We obtained [MATH] by squaring both sides of [MATH] and using Eq. ([REF]): [EQUATION].', '1810.03626-2-12-4': 'Applying the same to [MATH] and [MATH] and using [MATH], we obtained [MATH] and [MATH] given in Eq. ([REF]).', '1810.03626-2-13-0': 'One can also use the explicit forms of [MATH], [MATH] and [MATH] in Eqs. ([REF])-([REF]) to compute these scalar products directly, e.g., [EQUATION]', '1810.03626-2-13-1': 'We can compare this result with Eq. ([REF]) and obtain [EQUATION] which reveals the relation between [MATH] and [MATH].', '1810.03626-2-13-2': 'In Fig. [REF] we plot the relation for some specific values of [MATH] in order to illustrate how [MATH] varies with [MATH].', '1810.03626-2-13-3': 'Typically (for nonzero [MATH] as a function of [MATH] has a minimum corresponding to the maximal scattering angle [MATH].', '1810.03626-2-14-0': 'By solving [MATH] we obtain [EQUATION]', '1810.03626-2-14-1': 'For [MATH]), [MATH] increases (decreases) with [MATH].', '1810.03626-2-14-2': 'Therefore, [MATH] should be in the range [EQUATION] and due to the upper bound, [MATH] can reach values in the range [EQUATION] where [MATH] and [MATH] are determined by setting the left-hand side of Eq. ([REF]) to [MATH] and solving the equation with respect to [MATH].', '1810.03626-2-14-3': 'The solutions are [EQUATION]', '1810.03626-2-14-4': 'One can check that [REF] has the following massless limit [EQUATION] which is consistent with the standard results of coherent elastic neutrino scattering.', '1810.03626-2-15-0': 'Another important quantity is the minimal neutrino energy [MATH] necessary to create a massive particle [MATH]: [EQUATION] which is obtained by solving [MATH].', '1810.03626-2-15-1': 'If [MATH] is lower than [MATH], [MATH] cannot be produced in the scattering.', '1810.03626-2-15-2': 'In the limit when [MATH] can just be produced, we have [EQUATION]', '1810.03626-2-15-3': 'An interesting difference between the cases of massive and massless [MATH] occurs at [MATH].', '1810.03626-2-15-4': 'From Eq. ([REF]) one can obtain [EQUATION] which implies that in the minimal recoil limit for massive [MATH] the nucleus after scattering moves along the same direction as the incoming neutrino ([MATH]), while for massless [MATH] it moves in the perpendicular direction ([MATH]).', '1810.03626-2-16-0': 'We would like to clarify here that we are discussing [MATH] approaching [MATH] instead of being exactly equal to [MATH], because for [MATH], according to Eq. ([REF]), [MATH] is exactly zero.', '1810.03626-2-16-1': 'If [MATH], strictly speaking, [MATH] is not well defined because it implies that the nucleus after scattering stays at rest.', '1810.03626-2-16-2': 'If [MATH] is approaching [MATH] but remains nonzero, then [MATH] indeed is very close to zero for [MATH].', '1810.03626-2-16-3': 'For [MATH] fixed at a very small but nonzero value, when [MATH] increases from zero to nonzero values, [MATH] will rise steeply (depending on the smallness of [MATH]) to 1-as shown in Fig. [REF].', '1810.03626-2-16-4': 'Therefore there is no inconsistency in the minimal recoil limit of [MATH] and [MATH].', '1810.03626-2-16-5': 'Although the [MATH]-[MATH] relation in the minimal recoil limit is very sensitive to small [MATH], experimentally it is difficult to observe this behavior due to rather small recoil energies.', '1810.03626-2-17-0': '## Cross sections', '1810.03626-2-18-0': 'The exchanged scalar [MATH] is generally assumed to be massive with its mass denoted by [MATH].', '1810.03626-2-18-1': 'We evaluate the cross section without assuming [MATH] or [MATH].', '1810.03626-2-18-2': 'The heavy/light mass limits will be discussed below.', '1810.03626-2-19-0': 'From the Feynman diagram in [REF] and the relevant Lagrangian ([REF]), one can straightforwardly write down the scattering amplitudes for (anti)neutrino initial state [EQUATION] where spinor superscripts denote spin and we have inserted the left-/right-handed projectors [MATH] and [MATH] since the neutrino sources can only emit left-handed neutrinos or right-handed antineutrinos.', '1810.03626-2-19-1': 'Using FeynCalc [CITATION] we compute [MATH] for both cases.', '1810.03626-2-19-2': 'The result is identical for both neutrino and antineutrino scattering, namely [EQUATION] with the combined coupling constant [EQUATION]', '1810.03626-2-19-3': 'The dimensionless quantity [MATH] is typically [MATH] and reads [EQUATION]', '1810.03626-2-19-4': 'We will in what follows set limits using experiments with different nuclear targets.', '1810.03626-2-19-5': 'To reduce the dependence of the limits on the type of the nucleus we define [EQUATION] where [MATH] is the nucleon number (sum of neutron and proton numbers).', '1810.03626-2-19-6': 'Since [MATH] has been factored out, [MATH] has little dependence on the type of nuclei.', '1810.03626-2-19-7': 'For example, for Ge and CsI detectors we obtain [EQUATION] where the Yukawa couplings of the scalar [MATH] to neutrons and protons are denoted with [MATH] and [MATH] respectively.', '1810.03626-2-19-8': 'Clearly, [MATH] for Ge (employed at the CONUS experiment) is approximately the same as [MATH] for CsI (currently employed at the COHERENT experiment).', '1810.03626-2-20-0': 'The differential cross section, according to Eq. ([REF]), reads [EQUATION]', '1810.03626-2-20-1': 'One can straightforwardly check that in the limit [MATH] the result in Eq. ([REF]) is consistent with the standard cross section of elastic neutrino scattering [CITATION].', '1810.03626-2-21-0': '## Signals and constraints', '1810.03626-2-22-0': 'Now let us study the signal of our new fermion [MATH] in CE[MATH]S experiments.', '1810.03626-2-22-1': 'We will focus on two experiments, namely COHERENT [CITATION] and CONUS [CITATION].', '1810.03626-2-22-2': 'For the former, we will present the limits on the relevant parameters in [MATH] scattering based on the recent data release, whereas for the latter experiment we obtain sensitivities.', '1810.03626-2-23-0': 'The COHERENT experiment is based on neutrino emission from the Spallation Neutron Source at Oak Ridge National Laboratory.', '1810.03626-2-23-1': 'A crystal scintillator detector with 14.6 kg CsI was used in its recent measurement of CE[MATH]S and the SM signal has been observed with 6.7[MATH] confidence.', '1810.03626-2-23-2': 'The neutrinos are produced via [MATH] decay ([MATH]) and subsequently [MATH] decay ([MATH]).', '1810.03626-2-23-3': 'In this experiment, both [MATH] and [MATH] approximately decay at rest, which allows us to obtain the analytical expressions for neutrino spectra [CITATION] [EQUATION] which contains a monochromatic component [MATH] (i.e. all [MATH] have the same energy [MATH]).', '1810.03626-2-24-0': 'The CONUS experiment measures CE[MATH]S of reactor neutrinos ([MATH]) from a 3.9 GW nuclear power plant in Brokdorf, Germany.', '1810.03626-2-24-1': 'The detector is a Germanium semiconductor containing 4 kg of natural Ge ([MATH] in average), which is set at a distance of [MATH] meters from the reactor.', '1810.03626-2-24-2': 'To compute the event rates we adopt the reactor neutrino flux computed in [CITATION] and normalize the total flux to [MATH].', '1810.03626-2-24-3': 'CONUS data taking has started in April 2018 and recently a preliminary 2.4[MATH] statistical significance for observing the process was announced [CITATION].', '1810.03626-2-25-0': 'The event numbers in both experiments can be computed in the following way: in the [MATH]-th recoil energy bin ([MATH]), the total event number [MATH] consists of the SM contribution [MATH] and the new physics contribution [MATH], i.e. [EQUATION] which are computed by [EQUATION] where [EQUATION]', '1810.03626-2-25-1': 'Here [MATH] is the neutrino spectrum, [MATH] is the number of nuclei in the detector and [MATH] is the data taking period.', '1810.03626-2-25-2': 'The explicit expressions of [MATH], [MATH] and [MATH] are given in [REF].', '1810.03626-2-25-3': 'We note that we have included a form factor [MATH] in the cross section for the COHERENT experiment, where we take the parametrization given in Ref. [CITATION], see Fig. 1(a) therein.', '1810.03626-2-25-4': 'For the CsI detectors used in COHERENT, since the atomic number of Xe (54) is between Cs (55) and I (53), it is a good approximation to use the Xe form factor for both Cs and I. For reactor neutrinos, we can set [MATH] due to the low recoil energy.', '1810.03626-2-26-0': 'Using the above equations, we can compute the event numbers and study the signal of new physics in these two experiments.', '1810.03626-2-26-1': 'In Fig. [REF], we present the event distributions for several choices of [MATH] parameters together with the ratio of [MATH] for both CONUS (left) and COHERENT (right).', '1810.03626-2-26-2': 'We selectively choose several values for [MATH] (0 MeV and 3 MeV for CONUS; 3 MeV and 30 MeV for COHERENT) to illustrate the effect of [MATH] on CE[MATH]S. Light and heavy mediator cases have been illustrated by considering both [MATH] MeV and [MATH] MeV.', '1810.03626-2-26-3': 'The kinks of the red and blue curves appearing in the right panel at [MATH] keV are caused by the monochromatic [MATH] in COHERENT.', '1810.03626-2-26-4': 'The green and black curves correspond to [MATH] MeV.', '1810.03626-2-26-5': 'Since the monochromatic [MATH] neutrinos of [MATH] MeV energy do not have sufficient energy to produce [MATH] there are no similar kinks in these two curves.', '1810.03626-2-27-0': 'By comparing [MATH] with the observed event numbers we can obtain the constraints on the [MATH] coupling to neutrinos and nuclei.', '1810.03626-2-27-1': 'For COHERENT, the observed event numbers have been published in Ref. [CITATION] which can be used directly in our data fitting procedure.', '1810.03626-2-27-2': 'The recoil threshold in COHERENT is controlled by the signal acceptance fraction (see Fig. S9 of Ref. [CITATION]) which drops down quickly when the number of photoelectrons [MATH]) is less than [MATH], and approximately vanishes when [MATH].', '1810.03626-2-27-3': 'Therefore, in fitting the COHERENT data we import the signal acceptance fraction directly instead of setting a distinct threshold.', '1810.03626-2-27-4': 'The systematic and statistical uncertainties have been combined and provided in Fig. 3 of Ref. [CITATION], and are employed directly in our data fitting.', '1810.03626-2-28-0': 'The CONUS data has not been published, and hence we assume that their findings will be compatible with the SM prediction after [MATH] year of data taking with a [MATH] kg detector (thus [MATH] year [MATH] kg exposure).', '1810.03626-2-28-1': 'This allows us to compute sensitivity of CONUS on the production of [MATH].', '1810.03626-2-28-2': 'More explicitly, we adopt the following [MATH]-function comparing [MATH] with [MATH]: [EQUATION]', '1810.03626-2-28-3': 'Here [MATH] is a rescaling factor with an uncertainty [MATH] which mainly comes from the overall uncertainty of the neutrino flux.', '1810.03626-2-28-4': 'In addition, other systematic uncertainties may change the shape of the event spectrum, which is parametrized by [MATH] and assumed to be [MATH].', '1810.03626-2-28-5': 'The flux uncertainties used here are somewhat optimistic.', '1810.03626-2-28-6': 'According to the previous theoretical calculations [CITATION], the flux uncertainty at 5 MeV is about 3%.', '1810.03626-2-28-7': 'In the next few years, both the theoretical understanding and experimental measurements will be considerably improved [CITATION] so that the flux will be determined more precisely.', '1810.03626-2-28-8': 'The background [MATH] in each bin is 1 [MATH].', '1810.03626-2-28-9': 'For the nucleus recoil threshold we take 1.2 keV.', '1810.03626-2-29-0': 'The results are presented in Fig. [REF] where we show the constraints in the [MATH] plane (with [MATH] fixed) and the [MATH] plane (with [MATH] fixed).', '1810.03626-2-29-1': 'In the [MATH] panels, the bounds are almost flat when [MATH] MeV (CONUS) or [MATH] MeV (COHERENT), which can be understood from Eq. ([REF]) where, for small [MATH], [MATH] dominates over [MATH] in the denominator.', '1810.03626-2-29-2': 'Similarly, in the [MATH] plots, the bounds are also approximately flat for small [MATH] which can be understood from the [MATH] factor in Eq. ([REF]).', '1810.03626-2-29-3': 'However, the large mass behaviors are different for [MATH] and [MATH].', '1810.03626-2-29-4': 'As shown in the left panels of Fig. [REF], the curves are approximatively linear for large [MATH] because in this case the cross section is proportional to [MATH].', '1810.03626-2-29-5': 'On the other hand, large [MATH] can only be constrained by the events with high [MATH].', '1810.03626-2-29-6': 'If [MATH] is larger than the maximal value of [MATH] of the neutrino flux, then there will be no constraint at all because neutrinos do not have sufficient energy to produce [MATH].', '1810.03626-2-29-7': 'For reactor neutrinos, the event rate above [MATH] MeV is essentially too low to have a significant impact and hence the sensitivity to the new physics scenario diminishes.', '1810.03626-2-29-8': 'Therefore, the CONUS curves in the right panel rise up quickly around 6 MeV.', '1810.03626-2-29-9': 'For COHERENT, the maximal [MATH] is about [MATH] MeV (half of [MATH]) but, unlike in CONUS, the flux is not suppressed when [MATH] is approaching [MATH] MeV, so the curves do not rise so quickly when [MATH] is close to the maximal [MATH].', '1810.03626-2-30-0': 'In the future, the measurement of CE[MATH]NS will be significantly improved by lower thresholds, larger fiducial masses, and longer exposure times, etc.', '1810.03626-2-30-1': 'For reactor neutrinos, lower thresholds can increase the statistics drastically because the current threshold actually only allows CONUS to measure the high energy tail of the reactor neutrino flux.', '1810.03626-2-30-2': 'For COHERENT, using lower threshold detectors will not improve the measurement significantly.', '1810.03626-2-30-3': 'This is because, unlike reactor neutrinos, the majority of the neutrinos produced by [MATH] or [MATH] decays are not in the low-energy region-see Fig. [REF] for comparison.', '1810.03626-2-30-4': 'Consequently, lower thresholds for COHERENT cannot enhance the event numbers considerably and thus cannot improve the sensitivity significantly.', '1810.03626-2-30-5': 'We will consider here the following two benchmark configurations to illustrate the future sensitivities of CE[MATH]S experiments.', '1810.03626-2-30-6': 'The first one is running CONUS for 5 years with 100 kg Ge, and a considerably improved threshold down to [MATH] keV.', '1810.03626-2-30-7': 'In addition, the theoretical uncertainties of reactor neutrino flux are assumed to be reduced by a factor of 2.', '1810.03626-2-30-8': 'The second is (instead of doing a very detailled study of various other detectors and target materials that are planed [CITATION]) increasing the statistics of COHERENT by a factor of 100, which could be achieved by, e.g., a 20 times larger fiducial mass with 5 times longer exposure.', '1810.03626-2-30-9': 'The systematic uncertainties are correspondingly reduced so that we assume the overall uncertainty is reduced by a factor of [MATH].', '1810.03626-2-30-10': 'In Fig. [REF], we show the sensitivities of these two future experiments together with their current constraints/sensitivities.', '1810.03626-2-30-11': 'Here, [MATH] is set at 5 MeV as a benchmark value.', '1810.03626-2-31-0': 'Let us now discuss other limits on the scenario under study.', '1810.03626-2-31-1': 'Regarding CE[MATH]S, aspects of light scalars coupling to neutrinos and nuclei were explored in Ref. [CITATION].', '1810.03626-2-31-2': 'Since in our framework a massive MeV-scale fermion [MATH] is involved, most limits are expected to be weaker than the ones collected in Ref. [CITATION], where only couplings to nuclei and light neutrinos were considered.', '1810.03626-2-31-3': 'It is in addition more complicated to obtain precise limits, so we focus here on giving reasonably robust estimates.', '1810.03626-2-31-4': 'It was found in Ref. [CITATION] that all limits from terrestrial experiments, e.g. [MATH]-Pb scattering and meson decay experiments, are weaker than the bound from COHERENT as well as the CONUS sensitivity.', '1810.03626-2-31-5': 'BBN constraints, however, are relevant for [MATH] MeV-scale [MATH], and thus the [MATH] curves in fig:constraints should be interpreted as an illustration to show the strength of the limit in the small mass regime.', '1810.03626-2-31-6': 'When considering the [MATH] density evolution in the early Universe (see sec:DM), we will actually take [MATH] MeV.', '1810.03626-2-32-0': 'We should also mention limits from Supernova 1987A.', '1810.03626-2-32-1': 'If efficiently produced, the light states can carry a significant amount of energy from the Supernova core.', '1810.03626-2-32-2': 'In such case, the amount of energy carried by active neutrinos would be too small to match the observation of Supernova 1987A and hence a limit can be set.', '1810.03626-2-32-3': 'The leading process for the production of [MATH] is [MATH] via [MATH]-channel [MATH] exchange and it is suppressed by the fourth power of the small coupling [MATH].', '1810.03626-2-32-4': 'As [MATH] is concerned, in Ref. [CITATION] the authors presented, within a specific model, that the cross section for scattering of a new light fermion on protons and electrons is constrained by Supernova 1987A cooling arguments to values comparable to the corresponding cross sections for neutrinos.', '1810.03626-2-32-5': 'This can be understood as follows: if a novel fermion acts as a fourth neutrino species inside of the star, it will carry away energy comparable to the one carried away by the individual active neutrino species.', '1810.03626-2-32-6': 'This suggests that [MATH] of the energy budget would be carried away by [MATH].', '1810.03626-2-32-7': 'Given the astrophysical uncertainties associated to Supernova 1987A, exotic particles can carry away up to 50 of the total energy of the collapse [CITATION].', '1810.03626-2-32-8': 'This corresponds again roughly to a new cross section of similar magnitude as the SM one.', '1810.03626-2-32-9': 'The reachable parameter values from [REF] fulfill this constraint.', '1810.03626-2-32-10': 'Hence, we infer that the cooling arguments are not excluding the relevant parameter space.', '1810.03626-2-32-11': 'Finally, note that [MATH], being an MeV-scale particle, can not be resonantly produced through an MSW effect [CITATION].', '1810.03626-2-32-12': 'Such effect is very relevant for keV-scale particles for which strong limits can be derived [CITATION].', '1810.03626-2-33-0': '# [MATH] and Neutrino Mass Generation', '1810.03626-2-34-0': 'In this section we will discuss the possible connection of the new fermion [MATH] to neutrino mass generation.', '1810.03626-2-34-1': 'Any fermion that couples to light active neutrinos must be investigated with regard to its contribution to neutrino mass.', '1810.03626-2-35-0': 'Let us first discuss the nature of the scalar [MATH] that appears in our framework.', '1810.03626-2-35-1': 'Given our preference for light [MATH], such construction is not achievable with representations higher than singlets.', '1810.03626-2-35-2': 'Namely, [MATH] can obviously not be the SM Higgs due to its tiny couplings with [MATH] and [MATH] quark as well as its heavy mass which would further suppress the strength of the CE[MATH]S process.', '1810.03626-2-35-3': 'If we replace the SM Higgs by a novel Higgs doublet [MATH] with possibly larger couplings to quarks (and hence nuclei), we face the problem of a neccessary huge mass splitting between the light neutral component and the charged ones, which have not been seen.', '1810.03626-2-35-4': 'An option would be to consider the following gauge invariant [MATH]-dimensional operator in the effective theory formalism [EQUATION] with singlets [MATH] and [MATH], where [MATH] represents the scale of new physics.', '1810.03626-2-35-5': 'After electroweak symmetry breaking this operator yields an interaction term [EQUATION]', '1810.03626-2-35-6': 'By assuming furthermore non-vanishing interactions between [MATH] and nuclei (or light quarks), the CE[MATH]S occurs through the process shown in [REF].', '1810.03626-2-35-7': 'We will now discuss a minimal model containing SM singlets only, which will generate the effective Lagrangian in Eq. ([REF]).', '1810.03626-2-36-0': '## The Model', '1810.03626-2-37-0': 'We supplement the SM particle content with [EQUATION] where [MATH] and [MATH] are Majorana fermions and [MATH] is a real scalar.', '1810.03626-2-37-1': 'The quantum numbers under the SM gauge group [MATH] are indicated in brackets, and clearly no charged degrees of freedom are introduced.', '1810.03626-2-37-2': 'One of the goals of this section is to demonstrate that the neutrino masses can be generated from this extended fermion sector via a modified type-I seesaw mechanism [CITATION].', '1810.03626-2-37-3': 'This means that we would require at least two generations of novel fermions which participate in this mechanism, such that at most one light neutrino is massless.', '1810.03626-2-37-4': 'Still, for simplicity, throughout this section we will focus on the [MATH]-generation case which can be straightforwardly extended.', '1810.03626-2-37-5': 'Similarly, we will also restrict our discussion to one active neutrino flavor state, namely for definitness the electron (anti)neutrino [MATH].', '1810.03626-2-38-0': 'The relevant part of the Lagrangian reads [EQUATION] where [MATH] are the Yukawa couplings, [MATH] and [MATH] are Majorana masses of [MATH] and [MATH] fields, respectively, and [MATH] is the Dirac mass which is allowed by gauge symmetries.', '1810.03626-2-39-0': 'This Lagrangian is a minimal UV complete realization of eq:5D,eq:eff_coupling with fermion singlet [MATH] interacting with the fields given in both brackets of eq:5D through Yukawa couplings [MATH] and [MATH] (see eq:lagr).', '1810.03626-2-39-1': 'We will show that the allowed values of [MATH] exceed the characteristic momentum exchange [MATH] in CE[MATH]S experiments, which justifies the analysis setup in sec:production.', '1810.03626-2-39-2': 'If [MATH], we can easily relate the parameters of the full theory with [MATH] and obtain [MATH].', '1810.03626-2-39-3': 'If that was not the case, the topology shown in [REF] including [MATH] as the dynamical degree of freedom should be considered.', '1810.03626-2-40-0': 'More importantly, within the presented model, we will demonstrate the existence of parameter space that can be probed by CE[MATH]S experiments, generates neutrino masses in the right ballpark, and is not excluded from the new physics searches at neutrino oscillation facilities, beam dump experiments, colliders, etc.', '1810.03626-2-40-1': 'This indicates the importance of the CE[MATH]S in future new physics searches as there are scenarios where it could yield the strongest limits or perhaps even lead to new discoveries.', '1810.03626-2-41-0': 'After electroweak symmetry breaking, the neutral fermion mass matrix reads [EQUATION] where [MATH] GeV and we assumed that [MATH] does not develop a non-vanishing vacuum expectation value.', '1810.03626-2-42-0': 'We furthermore assume for the mass matrix given in [REF] that [MATH].', '1810.03626-2-42-1': 'In this way, the mixing between [MATH] and [MATH] is suppressed and hence, the masses of heavy new fermions essentially match the parameters in the flavor basis, [MATH] and [MATH].', '1810.03626-2-42-2': 'Contrary, if [MATH], the two physical masses would be of similar size which is not wanted in our scenario.', '1810.03626-2-43-0': 'We start by performing a rotation in the [MATH]-[MATH] plane by an angle [MATH].', '1810.03626-2-43-1': 'As discussed above, [MATH] is the physical mass of a particle produced in CE[MATH]S experiments.', '1810.03626-2-43-2': 'We take [MATH] MeV as an illustrative number.', '1810.03626-2-43-3': 'The bounds on the mixing of active neutrinos with heavy fermions have been extensively studies in the literature.', '1810.03626-2-43-4': 'From Refs. [CITATION] we infer that the constraint on the mixing between [MATH] and [MATH] for [MATH] MeV reads [EQUATION] and is set by neutrinoless double beta decay experiments.', '1810.03626-2-43-5': 'Weaker limits apply for the other flavors, which therefore can be accommodated more easily.', '1810.03626-2-43-6': 'The mass matrix after the [MATH]-[MATH] rotation reads approximately [EQUATION] from where one can infer that [MATH] may serve as a potential source of neutrino mass.', '1810.03626-2-43-7': 'By taking the upper value of [MATH] in eq:limit we obtain [MATH] eV which matches the required order of magnitude for neutrino mass.', '1810.03626-2-44-0': 'It was demonstrated in sec:production that the numerical analysis of CE[MATH]S for [MATH] MeV yields the limit [EQUATION] where [MATH] was introduced in eq:eft-lagrangian and [MATH] roughly corresponds to the coupling strength to individual quarks.', '1810.03626-2-44-1': 'The values indicated in square brackets represent the range in which the bound, depending on specific values of [MATH] and [MATH], is set (see fig:constraints).', '1810.03626-2-44-2': 'We assume [MATH], i.e. similar size of the [MATH] coupling to quarks and fermions, such that [EQUATION] approximatively holds.', '1810.03626-2-44-3': 'Having now a feeling for the numbers in eq:neutral_mass2 , we continue the diagonalization.', '1810.03626-2-44-4': 'Performing a rotation in the [MATH]-[MATH] plane by an angle [MATH] and using this expression as well as eq:Y2 we can relate the mixing angle with the upper limit from CE[MATH]S experiments [EQUATION]', '1810.03626-2-44-5': 'Clearly, [MATH] must not be tiny as otherwise the large mixing would pose a problem for [MATH].', '1810.03626-2-44-6': 'We can safely assume [MATH] because it parametrizes the strength of the interaction between three hidden particles and moreover [MATH] does not mix with the SM Higgs.', '1810.03626-2-44-7': 'By inserting [MATH] GeV in [REF] we obtain the relation [EQUATION]', '1810.03626-2-44-8': 'The seesaw contribution to the neutrino mass from mixing between [MATH] and [MATH] is then [EQUATION] which gives the upper bound on [MATH] from neutrino mass considerations [EQUATION]', '1810.03626-2-44-9': 'From eq:y1-M1,eq:y_1 we infer that the [MATH] values which can contribute to this neutrino mass generation are in the [MATH] GeV mass range.', '1810.03626-2-44-10': 'Finally, we need to check if the mixing given in eq:mix_eN is compatible with such masses.', '1810.03626-2-44-11': 'To this end, we again employ the limits from Refs. [CITATION] and infer that [MATH] GeV is fully consistent, whereas the smaller values are marginally allowed, i.e. in tension with the constraints from neutrinoless double beta decay experiments, big bang nucleosynthesis as well as the PS191 [CITATION] beam dump experiment.', '1810.03626-2-44-12': 'Interestingly, GeV-scale [MATH] will be testable at some upcoming experiments such as DUNE [CITATION], SHiP [CITATION], FASER [CITATION], NA62 [CITATION] and MATHUSLA [CITATION].', '1810.03626-2-45-0': 'In summary, parameters that are compatible with all available laboratory constraints and give an observable signal in coherent scattering experiments, give a neutrino mass of order [EQUATION] which is compatible with observation.', '1810.03626-2-45-1': 'Here, [MATH] denotes relative contribution to the active neutrino mass from [MATH] and [MATH].', '1810.03626-2-46-0': '# [MATH] as Dark Matter Particle', '1810.03626-2-47-0': 'Limits on [MATH] from terrestrial experiments as well as astrophysics were discussed in [REF].', '1810.03626-2-47-1': 'This section is devoted to the evaluation of the cosmic abundance of [MATH].', '1810.03626-2-47-2': 'As a first observation, we note that the smallness of [MATH] relative to the electroweak scale implies that the production of DM from freeze-out might not yield desired results and signifies the preference for non-thermal production.', '1810.03626-2-47-3': 'Note further that within our framework there is no possibility for [MATH] to decay into a pair of electrons or neutrinos.', '1810.03626-2-47-4': 'However, there could be tree-level ([MATH]) and radiative decays ([MATH]).', '1810.03626-2-47-5': 'From the expressions for the decay rates of these processes [CITATION] we infer that in order to ensure [MATH] stability, [MATH] for [MATH] MeV.', '1810.03626-2-47-6': 'The implied tiny value of [MATH] does not jeopardize our new physics scenario at CE[MATH]S experiments because the rate for [MATH] depends on [MATH] and [MATH] couplings, but not on [MATH].', '1810.03626-2-47-7': 'Note however from Eq. ([REF]) that neutrino mass will be dominated by [MATH] in this case.', '1810.03626-2-48-0': 'If [MATH] is the DM particle, CE[MATH]S experiments would yield an entirely novel method for searching (MeV-scale) DM.', '1810.03626-2-48-1': 'Note that dark matter in CE[MATH]S experiments was discussed already in [CITATION], where kinetically mixed dark photons decay into DM pairs which subsequently scatters at CE[MATH]S experiments (note that the neutrino sources of those experiments also generate photons).', '1810.03626-2-48-2': 'Here, we will propose a novel framework by demonstrating that the DM particle can be produced directly via CE[MATH]S and the effects of such process are imprinted in the measurable recoils of the nuclei.', '1810.03626-2-48-3': 'Interestingly, this would resemble DM search in direct detection experiments because in both cases the observable signal is the nuclear recoil.', '1810.03626-2-48-4': 'While nuclear recoil in direct detection experiments stems from DM-nucleus interaction, in CE[MATH]S such effect is caused by neutrinos.', '1810.03626-2-48-5': 'Thus, the main goal would be to distinguish between the SM coherent neutrino scattering events and "new physics" events in which [MATH] is produced.', '1810.03626-2-48-6': 'We have shown in [REF] that the distributions of the recoil energy corresponding to these two cases can be vastly different and it is possible to discriminate between them.', '1810.03626-2-48-7': 'These arguments strongly motivate the search for DM at CE[MATH]S experiments if [MATH] can indeed play the role of DM.', '1810.03626-2-48-8': 'Hence, in what follows we will discuss if and how [MATH] can be produced in right amounts.', '1810.03626-2-48-9': 'We rely on the UV complete model introduced in sec:model which was already shown to be successful in generating non-vanishing neutrino masses.', '1810.03626-2-49-0': 'In order to realize the CE[MATH]S process shown in fig1 we require [MATH] to interact with quarks as well as with [MATH] and neutrinos.', '1810.03626-2-49-1': 'The Lagrangian for these interactions reads [EQUATION] where [MATH] and [MATH] are the couplings to quarks and leptons, respectively.', '1810.03626-2-49-2': 'We consider the case in which neither of these couplings is weaker than [MATH].', '1810.03626-2-49-3': 'For an UV complete model from which [REF] stems after electroweak symmetry breaking, we refer the reader to [REF].', '1810.03626-2-50-0': 'For [MATH] MeV and [MATH], the decay and inverse decay widths of the process [MATH] are much larger than the value of the Hubble parameter across the relevant temperatures.', '1810.03626-2-50-1': 'Thus, [MATH] is in thermal equilibrium with the SM bath.', '1810.03626-2-50-2': 'At the temperatures below the QCD phase transition ([MATH] MeV) quarks are no longer relevant degrees of freedom.', '1810.03626-2-50-3': 'The vast majority of the formed mesons and baryons are non-relativistic and hence effectively disappear from the SM thermal bath shortly below [MATH] (the only exception are pions which are somewhat lighter).', '1810.03626-2-50-4': 'The [MATH] interaction which was keeping [MATH] in thermal equilibrium is not relevant below [MATH].', '1810.03626-2-50-5': 'However, even below [MATH], [MATH] can be in thermal equilibrium through the process [MATH], provided there is a sufficiently large Yukawa coupling [MATH].', '1810.03626-2-50-6': 'Such process occurs via [MATH]-channel exchange of [MATH] and is governed exclusively by the second term in [REF].', '1810.03626-2-50-7': 'Note that [MATH] should decay shortly below [MATH] into [MATH] and neutrinos in order not to violate any of the BBN predictions.', '1810.03626-2-50-8': 'For the parameter values we will take below this is indeed the case.', '1810.03626-2-50-9': 'It is also worthwhile to mention that the [MATH], introduced in [REF] to UV-complete the model, decay well above [MATH] due to their large [MATH] coupling.', '1810.03626-2-50-10': 'On the other hand, the stability of [MATH] is ensured by assuming a small value of [MATH] which sufficiently suppresses [MATH] and [MATH] decays.', '1810.03626-2-51-0': 'Above [MATH], [MATH] is in thermal equilibrium with [MATH] and hence also with the SM bath, due to rapid [MATH] processes.', '1810.03626-2-51-1': 'Whether [MATH] remains in contact with the SM bath at lower temperatures (after [MATH] decays) depends on the strength of the [MATH] process which occurs via [MATH]-channel exchange of [MATH] (hence is proportional to [MATH]).', '1810.03626-2-52-0': 'In order to accurately determine the present-day abundance of [MATH] we solve the following Boltzmann equations [CITATION] [EQUATION]', '1810.03626-2-52-1': 'For a particle denoted with [MATH] we define the yield as [MATH] (superscript eq denotes equilibrium value), where [MATH] and [MATH] are number and entropy density, respectively.', '1810.03626-2-52-2': 'In addition, [MATH], [MATH] is the Hubble parameter, [MATH] is the thermally averaged cross section for a given process (evaluated following Ref. [CITATION]), [MATH], [MATH] and [MATH] are modified Bessel functions.', '1810.03626-2-52-3': 'The initial conditions are [MATH] and [MATH], since we argued above that the sizable coupling to free quarks leaves the particles in thermal equilibrium at temperatures above [MATH].', '1810.03626-2-53-0': 'The solution for [MATH] is shown in [REF].', '1810.03626-2-53-1': 'In both panels we set [MATH] MeV and show the results for [MATH] which are in the ballpark of testable values at CE[MATH]S experiments.', '1810.03626-2-53-2': 'In the (left) right panel, [MATH] is fixed to [MATH] MeV ([MATH] MeV).', '1810.03626-2-53-3': 'Much smaller values of [MATH] are not considered due to the requirement [MATH] which forbids [MATH] to decay and is thus essential to render our DM candidate stable.', '1810.03626-2-53-4': 'We discuss and interpret the results from the figure in what follows.', '1810.03626-2-54-0': 'In both panels we observe the standard thermal freeze-out of [MATH] for [MATH].', '1810.03626-2-54-1': 'The interaction which keeps [MATH] in thermal equilibrium with the SM is [MATH], and the strength of this process increases by reducing [MATH].', '1810.03626-2-54-2': 'Hence, for smaller [MATH], the stronger interaction indicates that [MATH] stays longer in the thermal equilibrium and undergoes freeze-out at later times (smaller temperatures) which implies smaller final abundance of [MATH].', '1810.03626-2-54-3': 'This is visible from fig:DM where the final value of [MATH] for [MATH] (blue line) in the left panel (lighter [MATH]) is much smaller that the one shown in the right panel.', '1810.03626-2-54-4': 'By using the relation between the yield and the relic abundance [EQUATION] it is clear from both panels of fig:DM that [MATH] is strongly overproduced with respect to the observed DM abundance [MATH] [CITATION].', '1810.03626-2-54-5': 'Moreover, the freeze-out temperatures are smaller than [MATH] MeV which means that any new physics contribution invoked to deplete [MATH] is strongly constrained by BBN considerations.', '1810.03626-2-55-0': 'The freeze-out also occurs for [MATH] MeV and [MATH] (red line in the left panel).', '1810.03626-2-55-1': 'In this case, initially, the interaction between [MATH] and the SM is not sufficiently strong to follow the sudden change of [MATH] which is induced by a rapid change of SM degrees of freedom present in the thermal bath at [MATH] MeV.', '1810.03626-2-55-2': 'However, [MATH] eventually reaches the equilibrium value and the freeze-out occurs leaving [MATH] with even larger abundance than in the previously discussed scenarios.', '1810.03626-2-55-3': 'For [MATH] MeV and [MATH] (red line in the right panel) as well as for both cases with [MATH], [MATH] does not reach equilibrium.', '1810.03626-2-55-4': 'This means that the production occurs via "freeze-in" [CITATION] where the weak interaction with the SM bath leads to a gradual accumulation of [MATH] abundance.', '1810.03626-2-55-5': 'From [REF] it is obvious that the final [MATH] in all "freeze-in" scenarios is chiefly set by the initial condition (non-vanishing [MATH] at QCD phase transition) and the freeze-in contribution yields only a subdominant effect.', '1810.03626-2-56-0': 'Even though the DM abundance is still too large, in the "freeze-in" scenarios where [MATH] is not in thermal equilibrium at [MATH] MeV temperatures, a late-time entropy injection episode below [MATH] can reduce the [MATH] abundance significantly.', '1810.03626-2-56-1': 'Such entropy injection can be achieved for instance via decays of heavy scalars into the states in the thermal bath (see Ref. [CITATION] and references therein).', '1810.03626-2-56-2': 'It is clear that [MATH] needs to be diluted by [MATH] in order to meet observation.', '1810.03626-2-56-3': 'Let us note that alternatives with respect to late-time entropy injection have also been considered.', '1810.03626-2-56-4': 'For instance, a heavier, overproduced particle could decay into lighter states which may be DM [CITATION].', '1810.03626-2-56-5': 'This helps because, as may be inferred from eq:rel, the abundance of DM is proportional to the mass of a DM particle.', '1810.03626-2-56-6': 'A detailed analysis of such a scenario is beyond the scope of this project as it would require a significant extension of the Boltzmann equations given in eq:Boltzmann.', '1810.03626-2-56-7': 'In such extension, the late-time entropy injection would be avoided.', '1810.03626-2-56-8': 'We also note that there may be more models in which MeV-scale particle is produced in right amounts, without a necessity for late-time entropy injection.', '1810.03626-2-57-0': 'In conclusion, for the couplings that can be probed in CONUS or COHERENT [MATH], we explored whether in our model [MATH] can be DM, i.e. produced in right amounts in the early Universe.', '1810.03626-2-57-1': 'It turns out (see again [REF]) that this is not achievable without extending the minimal model presented in [REF].', '1810.03626-2-57-2': 'Namely, we find that a late time entropy injection episode is necessary to sufficiently deplete the abundance of [MATH] and render it a viable DM candidate.', '1810.03626-2-58-0': 'Since [MATH] in our model does not scatter on nuclei and electrons at tree-level, the bounds from direct detection are not strong.', '1810.03626-2-58-1': 'Moreover, for the considered range of couplings and masses, the CE[MATH]S cross section is several orders of magnitude smaller than those that can be currently tested at direct DM detection facilities for MeV-scale DM [CITATION].', '1810.03626-2-58-2': 'Given the absence of the annihilation channel into [MATH] pairs, the constraints from CMB [CITATION] are also not probing the parameter space to which CE[MATH]S experiments are sensitive.', '1810.03626-2-59-0': '# Summary and Conclusions', '1810.03626-2-60-0': 'Coherent neutrino-nucleus scattering is a new window to probe physics within and beyond the Standard Model.', '1810.03626-2-60-1': 'We have noted here that the final state fermion does not necessarily have to be a light active neutrino.', '1810.03626-2-60-2': 'Instead, we have entertained the possibility that an MeV-scale fermion [MATH] is produced in the process, which will lead to a significant modification of the observable recoil spectrum.', '1810.03626-2-60-3': 'We have set limits on the parameters that are involved when the interaction of the neutrino-[MATH] pair with quarks is mediated by a light scalar.', '1810.03626-2-61-0': 'The measurable couplings are well compatible with neutrino mass generation via low-scale type-I seesaw mechanism where, interestingly, both [MATH] and the newly introduced GeV-scale fermion [MATH] can contribute.', '1810.03626-2-61-1': 'Furthermore, [MATH] may be the DM particle, which we have shown to be typically requiring an injection of entropy in the early Universe after the QCD phase transition.', '1810.03626-2-61-2': 'Such an entropy injection can be achieved by introducing a new scalar which decays to the states in the thermal bath at late times, diluting the dark matter to the abundance which is in accord with present observations.', '1810.03626-2-62-0': 'Thus, exotic physics in coherent neutrino-nucleus scattering has a variety of interesting implications in neutrino physics and cosmology.', '1810.03626-2-62-1': 'The present analysis is only one example of the exciting prospects that this new window to physics has given us the opportunity to probe.'} | [['1810.03626-1-26-0', '1810.03626-2-27-0'], ['1810.03626-1-26-1', '1810.03626-2-27-1'], ['1810.03626-1-26-2', '1810.03626-2-27-2'], 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'1810.03626-2-31-0'], ['1810.03626-1-30-1', '1810.03626-2-31-1'], ['1810.03626-1-30-2', '1810.03626-2-31-2'], ['1810.03626-1-30-3', '1810.03626-2-31-3'], ['1810.03626-1-30-4', '1810.03626-2-31-4'], ['1810.03626-1-30-5', '1810.03626-2-31-5'], ['1810.03626-1-30-6', '1810.03626-2-31-6'], ['1810.03626-1-0-0', '1810.03626-2-0-0'], ['1810.03626-1-0-1', '1810.03626-2-0-1'], ['1810.03626-1-0-2', '1810.03626-2-0-2'], ['1810.03626-1-0-3', '1810.03626-2-0-3'], ['1810.03626-1-0-4', '1810.03626-2-0-4'], ['1810.03626-1-53-0', '1810.03626-2-54-0'], ['1810.03626-1-53-1', '1810.03626-2-54-1'], ['1810.03626-1-53-2', '1810.03626-2-54-2'], ['1810.03626-1-53-3', '1810.03626-2-54-3'], ['1810.03626-1-53-4', '1810.03626-2-54-4'], ['1810.03626-1-53-5', '1810.03626-2-54-5'], ['1810.03626-1-5-0', '1810.03626-2-5-0'], ['1810.03626-1-5-1', '1810.03626-2-5-1'], ['1810.03626-1-39-0', '1810.03626-2-40-0'], ['1810.03626-1-39-1', '1810.03626-2-40-1'], ['1810.03626-1-48-0', '1810.03626-2-49-0'], ['1810.03626-1-48-1', 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'1810.03626-2-43-4'], ['1810.03626-1-42-5', '1810.03626-2-43-5'], ['1810.03626-1-42-6', '1810.03626-2-43-6'], ['1810.03626-1-42-7', '1810.03626-2-43-7'], ['1810.03626-1-41-0', '1810.03626-2-42-0'], ['1810.03626-1-41-1', '1810.03626-2-42-1'], ['1810.03626-1-41-2', '1810.03626-2-42-2'], ['1810.03626-1-13-0', '1810.03626-2-13-0'], ['1810.03626-1-13-1', '1810.03626-2-13-1'], ['1810.03626-1-13-2', '1810.03626-2-13-2'], ['1810.03626-1-13-3', '1810.03626-2-13-3'], ['1810.03626-1-28-0', '1810.03626-2-29-0'], ['1810.03626-1-28-1', '1810.03626-2-29-1'], ['1810.03626-1-28-2', '1810.03626-2-29-2'], ['1810.03626-1-28-3', '1810.03626-2-29-3'], ['1810.03626-1-28-4', '1810.03626-2-29-4'], ['1810.03626-1-28-5', '1810.03626-2-29-5'], ['1810.03626-1-28-6', '1810.03626-2-29-6'], ['1810.03626-1-28-7', '1810.03626-2-29-7'], ['1810.03626-1-28-8', '1810.03626-2-29-8'], ['1810.03626-1-28-9', '1810.03626-2-29-9'], ['1810.03626-1-11-0', '1810.03626-2-11-0'], ['1810.03626-1-29-0', '1810.03626-2-30-0'], ['1810.03626-1-29-1', '1810.03626-2-30-1'], ['1810.03626-1-29-2', '1810.03626-2-30-2'], ['1810.03626-1-29-3', '1810.03626-2-30-5'], ['1810.03626-1-29-4', '1810.03626-2-30-6'], ['1810.03626-1-29-5', '1810.03626-2-30-7'], ['1810.03626-1-29-6', '1810.03626-2-30-8'], ['1810.03626-1-29-7', '1810.03626-2-30-9'], ['1810.03626-1-29-8', '1810.03626-2-30-10'], ['1810.03626-1-29-9', '1810.03626-2-30-11'], ['1810.03626-1-17-0', '1810.03626-2-18-0'], ['1810.03626-1-17-1', '1810.03626-2-18-1'], ['1810.03626-1-17-2', '1810.03626-2-18-2'], ['1810.03626-1-46-0', '1810.03626-2-47-0'], ['1810.03626-1-46-1', '1810.03626-2-47-1'], ['1810.03626-1-46-2', '1810.03626-2-47-2'], ['1810.03626-1-46-3', '1810.03626-2-47-3'], ['1810.03626-1-46-4', '1810.03626-2-47-4'], ['1810.03626-1-46-5', '1810.03626-2-47-5'], ['1810.03626-1-46-6', '1810.03626-2-47-6'], ['1810.03626-1-46-7', '1810.03626-2-47-7'], ['1810.03626-1-14-0', '1810.03626-2-14-0'], ['1810.03626-1-14-1', '1810.03626-2-14-1'], ['1810.03626-1-14-2', '1810.03626-2-14-2'], ['1810.03626-1-14-3', '1810.03626-2-14-3'], ['1810.03626-1-14-4', '1810.03626-2-14-4'], ['1810.03626-1-59-0', '1810.03626-2-60-0'], ['1810.03626-1-59-1', '1810.03626-2-60-1'], ['1810.03626-1-59-2', '1810.03626-2-60-2']] | [['1810.03626-1-2-0', '1810.03626-2-2-0'], ['1810.03626-1-15-4', '1810.03626-2-15-4'], ['1810.03626-1-55-1', '1810.03626-2-56-1'], ['1810.03626-1-59-3', '1810.03626-2-60-3'], ['1810.03626-1-59-5', '1810.03626-2-61-1']] | [] | [['1810.03626-1-43-12', '1810.03626-2-44-12'], ['1810.03626-1-59-4', '1810.03626-2-61-0']] | [] | ['1810.03626-1-11-1', '1810.03626-2-11-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1810.03626 | null | null | null | null | null |
1710.08173 | {'1710.08173-1-0-0': 'Intra-unit cell magnetic order has been observed in four different families of high-temperature superconductors from polarized neutron diffraction experiments and supported by several other techniques.', '1710.08173-1-0-1': 'That order, which does not break translation symmetry, is consistent with the predicted spontaneous orbital moments generated by two microscopic loop currrents in each CuO[MATH] cell.', '1710.08173-1-0-2': 'In a recent report, Croft et al arXiv:1709.06128v1 claimed to find no evidence for such orbital loop currents in charge ordered [MATH] using detwinned samples at least 100 smaller than in our experiments.', '1710.08173-1-0-3': 'We show by a detailed quantitative analysis of their data that contrary to their conclusion, the observed magnetic signal falls below the threshold of detection for their experiment.', '1710.08173-1-0-4': 'This is shown by noting that Croft et al overestimate by about an order of magnitude the expected magnetic signal.', '1710.08173-1-0-5': 'A factor [MATH] 4-5 is due to improper comparison of samples doping level, not taking into account detwinning of the samples and incorrect data calibration plus another factor of 3 in not accounting for possible finite correlation length.', '1710.08173-1-0-6': 'Not determining the spin-flip reference line properly, not doing a polarization analysis and inadequate control of the flipping ratio of the neutron beam add to uncertainties in their measurements.', '1710.08173-1-1-0': '# Introduction', '1710.08173-1-2-0': 'In an extensive series of papers [CITATION], we demonstrated using polarized neutron diffraction (PND) that the pseudogap state of underdoped cuprate superconductors is characterized by a Q=0 magnetic order, also referred to as an intra unit cell (IUC) magnetic order[CITATION].', '1710.08173-1-2-1': 'That encompasses results in four different cuprates families with a large variety of dopings: [MATH] (YBCO) [CITATION], [MATH] [CITATION], [MATH] [CITATION] and [MATH] [CITATION].', '1710.08173-1-2-2': 'Two reviews were written to give more experimental and technical details and put the different neutron results in perspective with the other physical properties of high-temperature cuprates [CITATION].', '1710.08173-1-2-3': 'In a recent manuscript posted on the arXiv, Croft et al[CITATION] claimed that they "found no evidence for the appearance of magnetic order below 300 K" in two [MATH] samples.', '1710.08173-1-3-0': 'We here show that Croft et al [CITATION] could not observe the magnetic signal owing to the insufficient detection capability of their measurements.', '1710.08173-1-3-1': 'This is due to multiple limitations of their data and analysis, that are detailed below, spanning incorrect sample comparison, ignorance of the impact of detwinning, misleading data calibration and improper data collection strategy.', '1710.08173-1-3-2': 'We describe each of these shortcomings below and show that their estimation of the magnetic signal reported by us is wrong by an order of magnitude.', '1710.08173-1-3-3': 'Therefore, it is below the sensitivity of the experimental results of Croft et al [CITATION].', '1710.08173-1-3-4': 'The claimed upper bound for a possible magnetic moment is therefore not correct and should be disregarded.', '1710.08173-1-3-5': 'Further, the comparison with local probes results in [CITATION] is oversimplified as it overlooks all the recent literature about muon spin resonance results [CITATION] leading to the possibility of finite time-scales of the IUC order.', '1710.08173-1-3-6': 'Such short range correlations have been actually reported in a recent study in nearly optimally doped YBCO[CITATION] using PND.', '1710.08173-1-4-0': 'The polarized neutron diffraction experiment we discuss here is very challenging[CITATION].', '1710.08173-1-4-1': 'It is worth emphasizing that neutron polarization analysis has been employed to prove the magnetic nature of the reported signal[CITATION].', '1710.08173-1-4-2': 'The polarization sum-rule is systematically satisfied.', '1710.08173-1-4-3': 'We emphasize as well than the YBCO data are nearly indistinguishable from those for [MATH] [CITATION].', '1710.08173-1-4-4': 'There is no doubt left about the existence of a magnetic signal.', '1710.08173-1-5-0': 'Over the years, the data analysis has been improved (see for instance, the refine analysis on the sample C from our first report in Fauque et al[CITATION] compared to the recent one in Mangin-Thro et al [CITATION]).', '1710.08173-1-5-1': 'The earlier results [CITATION] have been upgraded for quantitative accuracy: it reveals for the first time the IUC magnetic signal in five different YBCO samples and its stricking evolution with hole doping, following the pseudogap physics.', '1710.08173-1-5-2': 'Gradually, the data analysis was more quantitative in the subsequent publications in YBCO [CITATION].', '1710.08173-1-5-3': 'This has important consequences on the magnetic signal amplitude and the confidence one can get from it.', '1710.08173-1-5-4': 'The neutron intensity is proportional to sample mass.', '1710.08173-1-5-5': 'By using samples [MATH] 100 times smaller than ours on a spectrometer with about 3 times larger neutron flux (at the used wavelength), Croft et al[CITATION] are thus facing more than 10 times more experimental issues.', '1710.08173-1-5-6': 'Even after a counting time of a few hours, their experiment does not reach the required sensitivity.', '1710.08173-1-6-0': 'We consider below different aspects which affect noticeably the comparison made by Croft et al[CITATION] with our data and analysis.', '1710.08173-1-6-1': 'We demonstrate that various mistakes invalidate their conclusion and that they did not have enough sensitivity to detect the IUC magnetic signal in their measurements.', '1710.08173-1-6-2': 'In section [REF], we first recall a few key results, already present in the literature, that, put together, show that Croft et al[CITATION] overestimated by about a factor [MATH] 4-5 the amplitude of the signal they were looking for.', '1710.08173-1-6-3': 'In section [REF], we compare the data collection stategy in the studies carried out by Fauque et al on the one hand and Croft et al on the other hand.', '1710.08173-1-6-4': 'This comparison shows that Croft et al[CITATION] overlooked important experimental features discussed previously [CITATION], that limit severely their accuracy.', '1710.08173-1-6-5': 'Next, in section [REF], we underline the recent evolution of muon spin resonance ([MATH]SR) data which suggests that the magnetic correlations are at short range with important consequences for the detection limit in PND, since instrumental resolution effects have to be considered very carefully.', '1710.08173-1-6-6': 'Finally, in section [REF], we present an improved analysis of our orignal paper Fauque et al[CITATION] by taking account of the instrument resolution, leading only to a small change, about 15% in the conclusion which were already reached.', '1710.08173-1-6-7': 'We can then assess a direct comparison with the data reported by Croft et al[CITATION] and demonstrate how the IUC magnetic signal falls below the threshold of detection of their measurements.', '1710.08173-1-7-0': '# Doping level and neutron structure factor', '1710.08173-1-8-0': '## Sample comparison', '1710.08173-1-9-0': 'The table [REF] shows all the underdoped YBCO samples utilized in PND experiments.', '1710.08173-1-9-1': 'Croft et al[CITATION] focussed exclusively on the pioneering work of Fauque et al[CITATION] neglecting further results.', '1710.08173-1-9-2': 'Their comparison with the previous results is also questionnable as they compare samples which are characterized by distinct hole doping levels.', '1710.08173-1-10-0': 'In four independant measurements on three different underdoped samples with superconducting transitions within the so-called 60K-plateau (samples YBCO[MATH]), very reproductible results were found: that corresponds to sample B and C for [CITATION] and the report of Mook et al [CITATION] (sample labelled here M) on a different sample.', '1710.08173-1-10-1': 'Two of these samples (samples B and M) are twinned and exhibit an oxygen ordering, ortho-II, corresponding to one Cu-O chain filled over two (see e.g. ref. [CITATION] for an extensive discussion of the oxygen ordering in YBCO).', '1710.08173-1-10-2': 'The third sample is detwinned and has a different ordering of extra oxygen in CuO-chains, ortho-VIII [CITATION].', '1710.08173-1-10-3': 'For all these reports, the magnetic cross-section at the Bragg position Q=(011) is systematically found to be 1.6[MATH]0.1 mbarn at a low temperature (typically at 70K, slightly above [MATH] and using the same calibration procedure) [CITATION].', '1710.08173-1-10-4': 'In constrast, an almost twice larger signal was reported in the sample A (see table [REF]) which had a lower [MATH] and was measured only during our first experiment[CITATION].', '1710.08173-1-11-0': 'Croft et al[CITATION] performed their measurements on two different YBCO samples with superconducting temperatures [MATH] K (doping p=0.104, sample labelled here H1 for clarity) and [MATH] K (p=0.123, sample labelled here H2).', '1710.08173-1-11-1': 'The first one (H1) exhibits ortho-II oxygen ordering and the second (H2) ortho-VIII oxygen ordering.', '1710.08173-1-12-0': 'Croft et al systematically compare their samples to samples of Fauque et al[CITATION] with smaller superconducting temperatures (say sample A with [MATH] K to H1 and sample C with [MATH] K for H2, respectively) meaning systematically lower doping.', '1710.08173-1-12-1': 'As the signal is found to increase with decreasing doping[CITATION], their comparison necessarily leads to an overestimation of the expected magnetic signal.', '1710.08173-1-12-2': 'In particular, Croft et al[CITATION] misleadingly compare in figure 8, sample A with doping level of p=0.091 with sample H1 with p=0.104.', '1710.08173-1-12-3': 'Later, in figure 9, they compare sample C with p=0.115 with sample H2 with p=0.123.', '1710.08173-1-12-4': 'For a fair discussion, their sample H1 should instead be better compared with sample B with [MATH] K [CITATION] and p=0.107[CITATION] or even with sample M [MATH] K[CITATION] with p= 0.112[CITATION].', '1710.08173-1-12-5': 'For all B,C and M samples, the magnetic cross-section at the Bragg position Q=(011) is systematically found to be 1.6 [MATH] 0.1 mbarn (see table [REF]).', '1710.08173-1-12-6': 'That is the correct order of magnitude which should be used to carry out a comparison with our data.', '1710.08173-1-12-7': 'The extrapolation of "Fauque et al" in Figures 8 and 11 (for sample H1) of Croft et al is then overestimated by a factor 2 by this first issue.', '1710.08173-1-12-8': 'As sample H1, both samples B and M show the ortho-II oxygen ordering whereas sample C exhibits the ortho-VIII oxyen ordering as sample H2.', '1710.08173-1-13-0': '## Impact of detwinning', '1710.08173-1-14-0': 'Next, both samples studied by Croft et al [CITATION] are detwinned samples.', '1710.08173-1-14-1': 'However, the possible impact of detwinning on the neutron structure factor is ignored.', '1710.08173-1-14-2': 'Even if the IUC structure factor is not expected to change upon an orthorhombic distortion, it was experimentally proved that it had an important impact in YBCO on the neutron structure factor of the IUC order[CITATION].', '1710.08173-1-14-3': 'Indeed, that study of detwinned YBCO reveals a L-dependent a-b anisotropy of the scattered magnetic intensity, pointing out that the bilayer mirror plane is lost.', '1710.08173-1-14-4': 'Such an effect can be accounted for by a stacking of criss-crossed magnetic pattern within a bilayer.', '1710.08173-1-14-5': 'For L=0, the magnetic intensity is predominantly along a[MATH] and weaker along b[MATH].', '1710.08173-1-14-6': 'However, that difference between a[MATH] and b[MATH] is negligible for L=1.', '1710.08173-1-14-7': 'In terms of loop current models[CITATION], it means that the sum of the toroidal moments of each plane of the CuO[MATH] bilayer is pointing along b[MATH], the direction of the Cu-O chains [CITATION].', '1710.08173-1-15-0': 'That study was made in the ortho-VIII (sample C), the same can be expected in ortho-II sample as what matters is the locking of the composite toroidal moment along the CuO chains, b[MATH].', '1710.08173-1-15-1': 'The magnetic scattering at the (010) Bragg reflection appears [MATH] 3 times weaker than that at the (100) reflection [CITATION].', '1710.08173-1-15-2': 'Croft et al [CITATION] ignores that result and compares their result on the (010) reflection with our sample A which was twinned where the magnetic intensity is predominantly controlled by the (100) reflection.', '1710.08173-1-15-3': 'As a result, an intensity of 2.0 mbarn at 70 K for the (010) reflection is expected for a detwinned sample instead of 9.0 mbarn (expected from twinned sample A) as is misleadingly reported in Fig 11.a', '1710.08173-1-15-4': 'This has a strong impact on the figure 8.a', '1710.08173-1-15-5': 'a and figure 11.a', '1710.08173-1-15-6': 'The expected amplitude from Fauque et al[CITATION] for a detwinned sample with an ortho-II structure is compatible within the statistical accuracy of the shown data for the (010) reflection (see below section [REF] and Fig. [REF]).', '1710.08173-1-16-0': '## Data calibration', '1710.08173-1-17-0': 'A third issue in the report of Croft et al [CITATION] is related to the use of a procedure of calibration of the data in absolute unit which differs substantially from the one systematically used in previous PND studies [CITATION].', '1710.08173-1-17-1': 'As explained in the figure 1 caption of ref. [CITATION] and in [CITATION], the magnetic cross sections were calibrated in mbarn using the nuclear Bragg cross section of the (004) nuclear Bragg reflection.', '1710.08173-1-17-2': 'One then needs to estimate the nuclear neutron structure factor [MATH] which reads: [EQUATION] where [MATH] and [MATH] correspond to the neutron scattering length and the position of the n-th atom in the unit cell.', '1710.08173-1-17-3': 'A calculation of that nuclear structure factor gives [MATH] barn [CITATION] using the measured atomic positions[CITATION].', '1710.08173-1-17-4': 'The Bragg peak (004) was chosen as we had to compare intensity of 5 different (twinned and detwinned) samples all together with different oxygen content.', '1710.08173-1-17-5': 'As discussed below, the calculated structure factors, [MATH], depend on the specific chain-oxygen ordering.', '1710.08173-1-17-6': 'However, this effect is limited for the (004) reflection; the estimate of 7 barns is an averaged value over the doping range given by table [REF].', '1710.08173-1-17-7': 'Actually, a collection of Bragg peaks intensity along (00L) direction has been measured and can be used to calibrate the magnetic intensity of our samples (see section [REF] and Fig. [REF] below).', '1710.08173-1-17-8': 'The Bragg peaks cross-sections along (00L) are typically less sensitive to specific oxygen ordering of oxygen atoms in the Cu-O chains.', '1710.08173-1-18-0': 'Next, we use the intensity ratio (or cross-section ratio) of the nuclear Bragg peak to magnetic one [MATH].', '1710.08173-1-18-1': 'As detailed in [CITATION], we found the smaller ratio [MATH] at 70K for the (011) reflection of our detwinned sample C.', '1710.08173-1-18-2': 'That gives [MATH] 0.5 barn for the (011) nuclear cross-section, about two times larger than the value quoted by Croft et al [CITATION] of 0.28 barn.', '1710.08173-1-18-3': 'That leads to an additional overestimation by a factor 2 of the magnetic intensity of Fauque et al[CITATION] in figures 8.', '1710.08173-1-18-4': 'e, 9.c and 11.', '1710.08173-1-18-5': 'b,c Croft et al [CITATION].', '1710.08173-1-19-0': 'Actually, Croft et al [CITATION] did not use the (00L) series of Bragg peak to calibrate but only two reflections with K=1 plus only two strong reflections (006) and (020) at high momentum transfer.', '1710.08173-1-19-1': 'That coarse calibration is presented in Fig. 10 of ref. [CITATION] in a log-log representation that we reproduced here in Fig. [REF].', '1710.08173-1-19-2': 'We note first the saturation of the measured intensities for (020) strong Bragg reflection due to extinction effect.', '1710.08173-1-19-3': 'Another issue with this calibration is the sensitivity of the calculated structure factors to the oxygen content and the specific oxygen ordering of a given orthorhombic phase.', '1710.08173-1-19-4': 'This is particularly important for the K=1 Bragg peaks cross-section but also for the (006) reflection.', '1710.08173-1-19-5': 'A straightforward calculation shows that Bragg peak intensity in the (10L) or (01L) Bragg peaks cross-section varies considerably with the amount of actual oxygen atoms from x=6 (tetragonal phase) to x=7 (orthorhombic phase with full Cu-O chain) [CITATION].', '1710.08173-1-19-6': 'That is assuming that all extra oxygen atoms contribute to the cross-section to the long range lattice (as in ref. [CITATION]), but this ideal description does not correspond to reality due to imperfect oxygen ordering.', '1710.08173-1-20-0': 'There is an abundant literature about the different stages of oxygen ordering in YBCO (see e.g. [CITATION]).', '1710.08173-1-20-1': 'Typically, the oxygen ordering is characterized by a difuse scattering with 2D ordering with large in-plane correlation lengh and short correlation range along [MATH].', '1710.08173-1-20-2': 'Only the ortho-II phase exhibits a three-dimensional ordering, the remaining ones are essentially two-dimensional[CITATION].', '1710.08173-1-20-3': 'The extra oxygens form in-plane chains that do not become coherent along the [MATH]-direction, perpendicular to the CuO[MATH] plane.', '1710.08173-1-20-4': 'Actually, even in ortho-II phase, the extra oxygen atoms do not fully contribute to the lattice, explaining why the (011) cross section is measured larger than the simple calculation is giving.', '1710.08173-1-21-0': 'More specifically, the nuclear structure factor for (01L) reflections can be written as, [EQUATION] where [MATH] is a sum of neutron coherent scattering length of the various atoms (Y,Ba,Cu,O) already present in the parent tetragonal compound, YBa[MATH]Cu[MATH]O[MATH].', '1710.08173-1-21-1': 'However, for these atoms, the structure factor is calculated using the atomic positions reported in the orthorhombic YBCO[MATH] phase [CITATION] to take into account the effect of the orthorhombic distortion.', '1710.08173-1-21-2': 'The contribution of the additional oxygen atoms in the Cu-O chains is not included in [MATH] but it is in the second term of Eq. [REF] where [MATH] (for sample H1) is the extra oxygen content, [MATH] is neutron coherent scattering length for oxygen, [MATH] is the prefactor of [MATH], the scattering function of the extra oxygen atoms typically described by a Lorentzian function[CITATION].', '1710.08173-1-21-3': 'By definition, [MATH] as the scattering function has to be normalized to 1 once integrated over the Brillouin zone (BZ).', '1710.08173-1-21-4': 'As shown in Eq. [REF], there is a destructive interference between both terms.', '1710.08173-1-21-5': 'If [MATH] is large enough, the interference with the extra oxygen scattering is not very large and [MATH] does not vary much with [MATH].', '1710.08173-1-21-6': 'In contrast, if [MATH] is similar to [MATH], the destructive interference cannot be neglected.', '1710.08173-1-21-7': 'However, if the order is 2D or at short-range, [MATH] is noticeably smaller than 1.', '1710.08173-1-21-8': 'For the larger correlation case considered in the ortho-II phase[CITATION], [MATH], that corresponds to [MATH] 1/3.', '1710.08173-1-21-9': '[MATH]=0.83 barn is found the smaller for [MATH].', '1710.08173-1-21-10': 'Putting all these numbers together, one finds that [MATH] barn for the (011) reflection not 0.28 barn as quoted in Croft et al [CITATION].', '1710.08173-1-22-0': 'In Fig. [REF], the measured intensities are reported versus the calculated structure factor, [MATH], for these two limits : (i) the long range oxygen ordered ortho-II phase considered by Croft et al, [MATH] and (ii) the short range oxygen ordered ortho-II phase with [MATH], [MATH].', '1710.08173-1-22-1': 'The figure shows that the expected linear behavior (dashed line) works as good for both models (only with three points as the (020) reflection is affected by extinction effect).', '1710.08173-1-22-2': 'Therefore, Fig. [REF] does not validate the structural model with [MATH], in contrast to the suggestion of Croft et al [CITATION].', '1710.08173-1-22-3': 'The H1 ortho-II sample has been studied with hard x-ray diffraction measurements [CITATION] where the oxygen-chain order is reported to have a finite coherence length along [MATH], [MATH] corresponding to [MATH].', '1710.08173-1-22-4': 'In ortho-VIII phase, the oxygen ordering is essentially 2D, so, [MATH] is even smaller and [MATH] larger.', '1710.08173-1-22-5': 'That underestimation of [MATH] of Croft et al [CITATION] yields an additional overestimation of the expected magnetic scattering by a factor 2.', '1710.08173-1-23-0': 'In this section [REF], we show an overestimation of the IUC intensity in Croft et al [CITATION] by factor 2 from the difference in doping (section [REF]).', '1710.08173-1-23-1': 'Another factor of more than 2 arises from twin-averaging vs fully detwinned sample (section [REF]) but that applies only for the (010) Bragg spot.', '1710.08173-1-23-2': 'Finally, a last factor of 2 is due to data calibration (section [REF]) which occurs for the (011) reflection where the value of [MATH] is smaller.', '1710.08173-1-23-3': 'Therefore, the overall conclusion of this section [REF] is a systematic overestimation by a factor of [MATH] 4-5 of the expected magnetic scattering (quoted "Fauque et al") in the figures 8,9 and 11, of Croft et al [CITATION], making the conclusion of lack of observation of our observed signal invalid.', '1710.08173-1-24-0': '# Data collection strategy', '1710.08173-1-25-0': 'Next we examine the different data collection strategies.', '1710.08173-1-25-1': 'As the expected magnetic signal is weak compared to the leakage of the nuclear intensity due to inperfect polarization, [MATH] 0.95-0.96, of the instruments, the only way to observe the signal is to perform a very accurate temperature dependence of the flipping ratio (R), ie.', '1710.08173-1-25-2': '[MATH] of a given Bragg spot ([MATH] and [MATH] stand for the non-spin-flip and spin-flip intensities, respectively).', '1710.08173-1-25-3': 'The expected signal can show up above a reference line (called baseline) which has to be determined at high temperature (above [MATH]).', '1710.08173-1-25-4': 'Unfortunately, only 1 or 2 points have been measured by Croft et al [CITATION] above the reported ordering temperature [MATH] (which coincides with the pseudogap temperature, [MATH]).', '1710.08173-1-25-5': 'This is unfortunately insufficient to properly characterize the shape of that baseline and distinguish it from a genuine magnetic signal.', '1710.08173-1-26-0': 'Further, Croft et al [CITATION] did not perform the same method to collect the data as the one used in previous PND studies.', '1710.08173-1-26-1': 'They used a more text-book way which consists of realigning the sample at each temperature, implying that the spectrometer has to move at every temperature.', '1710.08173-1-26-2': 'This method would be very reliable if the outcome of the measurements were not affected by erratic positioning errors of the spectrometer at each alignement.', '1710.08173-1-26-3': 'We actually discarded that method rapidly as it did not give reliable enough flipping ratios values compared to the required accuracy.', '1710.08173-1-26-4': 'This is due to the limit of mechanical stability of the instruments.', '1710.08173-1-26-5': 'It adds additionnal systematic and random errors in the countings.', '1710.08173-1-26-6': 'Although Croft et al [CITATION] do not report a very large distribution of countings between each given temperature, this can be nevertheless seen in their data where some of their points exhibit clear departure from the expected statistical errors of the counting.', '1710.08173-1-26-7': 'As the total number of points as a function of the temperature (6-8) for each curve is rather scarce, this is detrimental to get confident determination of both the shape of the baseline and possible magnetic signal on top of it, especially when the point at 300 K is the point which drifts away from the expected statistics.', '1710.08173-1-27-0': 'Croft et al [CITATION] argued that it is crucial to realign the sample at every temperature as the lattice parameters change with temperature and that the flipping ratio is sensisitve to A4 ("[MATH]"-scan) (Fig. 6d in [CITATION]).', '1710.08173-1-27-1': 'We are aware of this possible issue but we show that this effect is not crucial.', '1710.08173-1-27-2': 'This argument put forward by Croft et al leads to some inconsistency.', '1710.08173-1-27-3': 'Assuming that the argument were correct, then one should detect an artefact magnetic signal on any Bragg reflection.', '1710.08173-1-27-4': 'This is not what we have observed [CITATION].', '1710.08173-1-27-5': 'Furthermore, this signal should be stronger at large momentum transfer, such as for the (02L) reflections where no effect was reported [CITATION].', '1710.08173-1-27-6': 'Indeed, the effect should be about twice larger at large [MATH] such as the (020) reflection than at lower [MATH] for the (01L) reflections.', '1710.08173-1-27-7': 'Precisely, the effect is 2.4 less for (010) compared to (020) as it is in the ratio of [MATH] of both Bragg peaks indexes.', '1710.08173-1-27-8': 'This is not observed in any of our data (see e.g. [CITATION]).', '1710.08173-1-27-9': 'Furthermore, under this assumption, there is no way to undertstand how an artificial magnetic signal could fulfilled the magnetic sum rule obtained by the neutron polarization analysis [CITATION].', '1710.08173-1-27-10': 'Croft et al [CITATION] are incorrect when they wrote in their section III.C "(similar changes are expected for the (010) and (011) reflections)".', '1710.08173-1-27-11': 'It should be stressed that this effect is less important in configurations of broad momentum instrument resolution, for instance in the case of large samples with large mosaicities which broadens the instrument resolution.', '1710.08173-1-28-0': 'In order to keep a highly stable flipping ratio, previous PND studies rule out the full realignement of the sample at each temperature.', '1710.08173-1-28-1': 'This is why we have way better accurate flipping ratio, a keypoint of success to determine a genuine magnetic signal.', '1710.08173-1-28-2': 'However, we realign the rocking "A3" angle at each temperature if necessary.', '1710.08173-1-28-3': 'This has less impact of the flipping ratio stability.', '1710.08173-1-28-4': 'Our strategy to collect the data then consists of measuring a lot of points in temperature with high statistics with no spectrometer movement.', '1710.08173-1-28-5': 'It should be stressed that Croft et al [CITATION] do not report any data obtained this way although we have established that this is a proper technique to disentangle unwanted shift of the flipping ratio with temperature (the change of the "baseline") from a genuine magnetic signal[CITATION].', '1710.08173-1-29-0': 'Further, they should have considered the possibility of a drift of the baseline due to the fact that the sample move up and down as a function of temperature, following the constraction and dilatation of the cryostat sample stick on which the sample is attached.', '1710.08173-1-29-1': 'Therefore, the sample is displaced vertically in temperature and thus probes different flipping ratios due to inevitable polarization inhomogenities of the beam because of large monochromator and analyzer sizes[CITATION].', '1710.08173-1-29-2': 'This would be especially the case for a small sample compared to sample stick shriking as Croft et al [CITATION].', '1710.08173-1-29-3': 'Therefore, their claim that "the neutrons always emanate from the same part of the monochromator and strike the same part of the analyzer."', '1710.08173-1-29-4': 'cannot be correct when changing temperature.', '1710.08173-1-30-0': 'Incidently, another experimental issue in [CITATION] is related to the incoherent background subtraction, which should be measured away from the Bragg position.', '1710.08173-1-30-1': 'Again, a different strategy has been applied compared to the previous PND studies in [CITATION].', '1710.08173-1-30-2': 'We prove that this background subtraction to be important for a quantitative analysis of the data when the magnetic signal is weak [CITATION].', '1710.08173-1-30-3': 'No background subtraction was made in the original reports [CITATION].', '1710.08173-1-30-4': 'The choice in momentum space of the reference background wavevector is important as it has to be a location where no magnetic signal is present.', '1710.08173-1-30-5': 'We choose points by shifting the in-plane momentum transfer such as [MATH]=(0,0.9,1) for the reflection (011).', '1710.08173-1-30-6': 'Typically, on a triple axis instrument, the SF incoherent background intensity at [MATH]=(0,0.9,1) slightly increases upon cooling (negative slope with temperature), following the expected behaviour of a Debye-Waller factor.', '1710.08173-1-30-7': 'Using polarization analysis, we prove that magnetism at such a location is weaker by two order of magnitude (see e.g [CITATION]).', '1710.08173-1-31-0': 'Instead, Croft et al [CITATION] choose [MATH]=(0,1,0.9) along [MATH] for the background subtraction and did not mention any polarization analysis to address possible magnetism at this [MATH]-location.', '1710.08173-1-31-1': 'Near optimal doping, we show that magnetic signal is not negligible at this [MATH]-location [CITATION] due to short-ranged IUC correlations along [MATH].', '1710.08173-1-31-2': 'Therefore, that choice is particularly a bad idea as it would necessarily reduce possible intrinsic IUC signal.', '1710.08173-1-31-3': 'Actually, Croft et al [CITATION] report the intensity at [MATH]=(0,1,0.9) in Figs. 7b-c in a logarithm scale.', '1710.08173-1-31-4': 'Interestingly, it occurs that the SF intensity decreases upon cooling (positive slope with temperature) at this [MATH]-location (this is especially clear for the H1 sample, Fig. 7b).', '1710.08173-1-31-5': 'This implies one of two possibilities: either, this sloping background is the true incoherent background (and then it should be removed from the Bragg peak (011) SF intensity), or it suggests a vanishing at low temperature of a magnetic signal present at high temperature at this [MATH]-location.', '1710.08173-1-31-6': 'This latter possibility is related to the fact that the positive slope with temperature background behaviour contrasts with the known behaviour of the incoherent SF background (negative slope with temperature) discussed above.', '1710.08173-1-31-7': 'Both possibilities leads to interesting opportunuities.', '1710.08173-1-31-8': 'Croft et al [CITATION] made no attempt to analyse this data along these lines, only fitting it by a constant.', '1710.08173-1-32-0': 'Finally, no attempt of polarization analysis is reported in [CITATION] to search for the weak magnetic signal at the Bragg reflection (011) or to accurately determine an upper limit for a magnetic scattering intensity.', '1710.08173-1-32-1': 'It should be strongly emphasized that previous PND studies reported a magnetic signal at very specific Bragg reflections (the signal was absent on others) and complementary polarization analysis demonstrate unambigously the magnetic nature of the observed scattered intensity [CITATION].', '1710.08173-1-33-0': 'As a result of the various points discussed above, Croft et al [CITATION] cannot estimate the shape of the "baseline" as they can only rely on the point measured at 300K to re-scale their data (even if sometimes that point is not included in their fitting line).', '1710.08173-1-33-1': 'That necessarily alters the data accuracy and degrades the threshold of detection of a small signal.', '1710.08173-1-34-0': '# Effects of short-range magnetic correlations', '1710.08173-1-35-0': 'It has been already discussed by Fauque et al[CITATION] that local probe measurements had not observed a magnetic signal consistent with our measurements.', '1710.08173-1-35-1': 'Typically, zero-field muon spin resonance ([MATH]SR) [CITATION] and Nuclear Magnetic Resonance (NMR) [CITATION] experiments in YBCO found no evidence for a static and long range magnetic order.', '1710.08173-1-35-2': 'First, let us recall that the situation with [MATH]SR experiments has been contradictory (see [CITATION] for an extensive discussion).', '1710.08173-1-35-3': 'Second, it was envisaged[CITATION] that the magnetic moments could fluctuate slowly enough to appear static to neutrons, but too fast enough to be identified as a magnetic order in [MATH]SR and therefore even more for NMR which probes even slower timescales.', '1710.08173-1-35-4': 'Recently this picture has been confirmed as slow magnetic fluctuations have been discovered in YBCO samples [CITATION] using a longitudinal field [MATH]SR technique.', '1710.08173-1-35-5': 'These magnetic fluctuations are related to our magnetic order as a critical slowing down[CITATION] is also reported at the same temperature [MATH] where the neutron magnetic signal sets in.', '1710.08173-1-35-6': 'Beyond the YBCO bilayer system, new set of [MATH]SR data in the bilayer system, [MATH], is also found to support the existence of the magnetic signal[CITATION].', '1710.08173-1-36-0': 'Croft et al [CITATION] dismiss these recent developments and efforts made to refine the [MATH]SR studies.', '1710.08173-1-36-1': 'The conclusion of the PND experiment of Croft et al is no more supported by the most recent [MATH]SR studies.', '1710.08173-1-36-2': 'J. Zhang et al[CITATION] found that magnetic correlations fluctuate, [MATH] 10 ns timescale, even for strongly underdoped samples as the ones discussed here.', '1710.08173-1-36-3': 'These low frequency fluctuations are necessarily related to the formation of finite-size magnetic domains [CITATION], corresponding to finite correlation lengths of the magnetic order observed in neutron diffraction.', '1710.08173-1-36-4': 'Such a short range correlation length of the IUC order has been observed [CITATION] near optimally doped YBCO sample.', '1710.08173-1-36-5': 'Note that when the correlation length becomes too short, such as in [MATH] [CITATION], the magnetic signal cannot be observed anymore on the Bragg reflection.', '1710.08173-1-37-0': 'In underdoped samples, for a doping p[MATH]0.11 (samples A and M in Tab. [REF]), we observed resolution limited magnetic peaks[CITATION].', '1710.08173-1-37-1': 'However, the PND experiment was performed on large single crystals with large mosaicity, corresponding to rocking scans with a full width at half maxium (FWHM) of [MATH], implying that correlation length [MATH] AA[The rocking "A3" scan is similar than a scan along L. On Bragg reflections, (010) or (020), a rocking scan matches a transverse scan, i.e. a scan along the (001) direction].', '1710.08173-1-37-2': 'Instead, the measurements carried out by Croft et al [CITATION] are performed on small samples with high mosaic spread yielding an observed rocking scan FWHM of [MATH] (Fig 6 and 7).', '1710.08173-1-37-3': 'Therefore, if the magnetic order exhibited a finite correlation length [MATH] , the magnetic signal would be characterized by a FWHM of [MATH] in the A3-scan, with a peak intensity reduced by a factor [MATH]3 in comparison with a resolution limited signal, like a nuclear Bragg peak.', '1710.08173-1-37-4': 'A sketch of these two situations is reported in the top-right panel of Fig. [REF].', '1710.08173-1-37-5': 'The magnetic amplitude at the Bragg position would be accordingly [MATH]3 times reduced compared to the one of true long range ordered state.', '1710.08173-1-38-0': 'In conclusion of this section, in case of short range magnetic correlations, for example [MATH] , this factor 3 has to be added to the previous factor 4-5 discussed above in section [REF]; that amounts to a reduced factor 12 for a proper comparison with Fauque et al[CITATION] in this limit.', '1710.08173-1-39-0': '# Improved data analysis', '1710.08173-1-40-0': 'Finally, motivated by the fact that a detailed comparison is necessary to appreciate the possible differences between the study of Croft et al [CITATION] with earlier PND works, we refine the data calibration procedure of Fauque et al[CITATION].', '1710.08173-1-40-1': 'Originally, the data were calibrated onto the Bragg peak reflection (004) but the instrumental resolution was neglected as a subsidiary effect.', '1710.08173-1-40-2': 'We then re-assess below the complete data calibration of the results of the detwinned high-quality sample C[CITATION].', '1710.08173-1-41-0': 'To perform the calibration, one needs to estimate the instrumental resolution correction.', '1710.08173-1-41-1': 'The resolution of a triple-axis spectrometer is known to be described by a 4D (3 directions for momentum transfer and one for energy) Gaussian function [CITATION].', '1710.08173-1-41-2': 'The scattering intensity is the convolution of the neutron scattering cross section with this instrument resolution ellipsoid.', '1710.08173-1-41-3': 'A Bragg peak is a Dirac function in phase space of both the momentum transfer and the energy.', '1710.08173-1-41-4': 'For a Bragg scattering , the measured intensity at [MATH]=(HKL) then reduces to the product of the computed structure factor with the volume of the resolution ellipsoid.', '1710.08173-1-41-5': 'That reads: [EQUATION] [MATH] is determined by the normalization of the spectrometer 4D Gaussian resolution [CITATION].', '1710.08173-1-41-6': 'In general, for a triple-axis spectrometer[CITATION], [MATH] is never as simple as the Lorentz factor, [MATH], used for a diffractometer (two-axis instrument, no analysis of the scattered neutron energy).', '1710.08173-1-41-7': 'Croft et al [CITATION] are then not correct to apply the Lorentz factor in their calibration curve (fig. 10) even for A3-integrated intensities.', '1710.08173-1-41-8': 'To determine [MATH], we use a Cooper-Nathans[CITATION] formalism but with the use of mosaic spreads of the diffracting crystals and of the divergence angles given by the beam geometry (distances between the different elements composing the instrument, sizes of crystals,...).', '1710.08173-1-42-0': 'For the detwinned sample C[CITATION], one can compute the proper resolution factor [MATH].', '1710.08173-1-42-1': 'Fig. [REF] shows the comparison of the measured intensity of the nuclear Bragg peaks divided by the resolution factor [MATH].', '1710.08173-1-42-2': 'The figure shows canonical behaviour of a calibration curve (see an example, Fig. 1 in [CITATION], in the context of cuprates): i) the expected linear behavior given by Eq. [REF] at small structure factor and ii) a saturation at larger structure factor due to extinction effects.', '1710.08173-1-42-3': 'From this linear calibration at small enough [MATH] (black dashed line), one obtains a renormalization of about 15 % for the magnetic cross-section for sample C of 1.3 [MATH] 0.1 mbarn at (011) reflection compared to our previous estimate (red dashed line).', '1710.08173-1-42-4': 'As the IUC magnetic intensity at the reflection (011) does not depend much on the sample detwinning [CITATION], one can report the corrected intensity for different samples versus the doping level (Fig. [REF]) from the published data [CITATION].', '1710.08173-1-42-5': 'As the intensity is proportional to the square of the magnetic moments, that induces a renormalization of less than 10 % for the ordered IUC magnetic moment, still of the order of [MATH] 0.1 [MATH].', '1710.08173-1-42-6': 'Fig. [REF] shows as well the onset temperature of the IUC order, [MATH].', '1710.08173-1-42-7': 'As discussed in [CITATION], the measured magnetic intensity is not simply proportional to [MATH], possibly because of the short range nature of the IUC order.', '1710.08173-1-43-0': 'Lastly, we now turn to the comparison of the magnetic signal reported by Fauque et al[CITATION] to the experimental data of Croft et al [CITATION].', '1710.08173-1-43-1': 'In figs. [REF] and [REF], we present [MATH] data for both samples of Croft et al [CITATION]; the data points are exactly the same as those reported in Figs. 8-9 in [CITATION], but Figs. [REF]-[REF] zoom on the data.', '1710.08173-1-43-2': 'For instance, for a sake of clarity, we do not consider in Figs. [REF]-[REF] the possible issue with the background subtraction emphasized in section [REF].', '1710.08173-1-43-3': 'We stress that a background subtraction with a positive slope with temperature would lead to a more clearly noticeable IUC magnetic signal at low temperature.', '1710.08173-1-44-0': 'The inverse of flipping ratio at the reflection (020) is linear in temperature with a positive slope as it has been shown to exist [CITATION].', '1710.08173-1-44-1': 'As discussed in section [REF], that slope is inevitable as the sample is displaced in the neutron beam upon cooling.', '1710.08173-1-44-2': 'With the large grey shaded area, we represent the zone of the limit of detection.', '1710.08173-1-44-3': 'This area is due to combined effects of the statistical errors of each points, occurence of off-statistical points (possibly related to errors in positioning as discussed in section [REF]) and the scarce number of points.', '1710.08173-1-44-4': 'This area is simply deduced from the measurements at the Bragg (020) reflection where no magnetic signal is expected for the IUC order.', '1710.08173-1-44-5': 'For clarity, the same error of [MATH] is defined for both figures.', '1710.08173-1-44-6': 'That error is typically equivalent to an error of [MATH] on the flipping ratio or [MATH] of the spin-flip intensity.', '1710.08173-1-44-7': 'This uncertainty is not negligible as the highest ratio of the IUC intensity to the spin-flip total intensity, that we have reported, is only [MATH] for detwinned sample[CITATION] and about [MATH] otherwise[CITATION] for the corresponding doping levels.', '1710.08173-1-44-8': 'The error on the spin-flip intensity in previous PND experiments [CITATION] ranges between [MATH] %.', '1710.08173-1-44-9': '(see e.g. Figure S1 in Supplemental materials of Mangin-Thro et al [CITATION] for an estimate of [MATH]).', '1710.08173-1-45-0': 'On Figs. [REF]-[REF], the detection limit of [MATH] is next reported for the Bragg reflections (010) and (011) where the magnetic signal is expected.', '1710.08173-1-45-1': 'To compare that set of data, with those of Fauque et al[CITATION], we then consider three different scenarios which are plotted in Figs. [REF]-[REF]: A) no magnetic signal is present, B) a long range magnetic order is present with the amplitude reported in [CITATION] C) a magnetic order with short range correlation along the c-direction is present with the amplitude reported in [CITATION].', '1710.08173-1-45-2': 'For the sample H1, one compares with the amplitude measured for sample B (see table [REF]) assuming that this sample would be detwinned for a proper comparison.', '1710.08173-1-45-3': 'For the sample H2, one compares with the amplitude measured for sample C.', '1710.08173-1-45-4': 'The data have been normalized in absolute units using the structural model discussed in section [REF] of short range oxygen ordering, [MATH], which corresponds to the smaller ratio [MATH] ever reported [CITATION].', '1710.08173-1-45-5': 'That fully determines the amplitude of the expected signal from Fauque et al [CITATION] data.', '1710.08173-1-45-6': 'The top right panel of Fig. [REF] simulates the A3 rocking scan for the scenario B in blue and the scenario C in red where both hypothetical curves have been convoluted by the resolution function given by the Fig. 6.', '1710.08173-1-45-7': 'a in [CITATION].', '1710.08173-1-45-8': 'One clearly sees that the IUC signal is [MATH] 3 times weaker in scenario C compared to scenario B.', '1710.08173-1-46-0': 'First, Following the various points discussed above, one clearly sees that the expected signal from [CITATION] is much weaker than what Croft et al [CITATION] have estimated.', '1710.08173-1-46-1': 'For all the three scenarios presented above, only one parameter, the overall level of the background of the baseline, is fitted.', '1710.08173-1-46-2': 'All the other parameters are given from the literature [CITATION] and the discussion above.', '1710.08173-1-46-3': 'Clearly, for all cases, scenarios A (of Croft et al [CITATION]) and C (a short range along [MATH] IUC magnetic order compatible with the report of ref. [CITATION]) cannot be distinguished at all.', '1710.08173-1-46-4': 'Even for a true long range magnetic order along [MATH] (scenario B), the data are insufficient to eliminate with confidence the existence of IUC order.', '1710.08173-1-46-5': 'Clearly, for both samples, the experimental sensitivity is too limited to observe the IUC order contrary to the claim of Croft et al [CITATION].', '1710.08173-1-47-0': '# Conclusion', '1710.08173-1-48-0': 'In conclusion, Croft et al [CITATION] do not have the experimental accuracy to observe the IUC signal that have been reporting for the last decade in YBCO [CITATION].', '1710.08173-1-48-1': 'Different factors applied: first, the purported level of intensity of Fauque et al has to be systematically renormalized by a factor [MATH] 4-5 (see section [REF]).', '1710.08173-1-48-2': 'That corresponds to the scenario B of figs. [REF] and [REF], where a long range IUC order is assumed.', '1710.08173-1-48-3': 'Second, an additional factor 3 occurs if the signal is short ranged along L with [MATH] as discussed in section [REF] (scenario C of Figs. [REF] and [REF]).', '1710.08173-1-48-4': 'Third, one experimental issue is represented by the shaded areas which indicates the uncertainty of [MATH] on the thermal dependence of the baseline for the ratio [MATH] as underlined in section [REF].', '1710.08173-1-48-5': 'The most plausible scenario for the magnetic intensity lies in between scenarios B and C in Figs. [REF] and [REF].', '1710.08173-1-48-6': 'This is clearly below the detection limit of the data of Croft et al [CITATION].', '1710.08173-1-48-7': 'The claimed upper bound for a possible magnetic moment in [CITATION] is therefore baseless and should be disregarded.', '1710.08173-1-49-0': 'One positive conclusion that could be derived from the study of Croft et al [CITATION] is that the use of sample with a very good mosaic has raised question concerning the length of the magnetic correlations for the IUC order.', '1710.08173-1-49-1': 'Short ranged correlations were already established in YBCO around optimal doping [CITATION] but not for more underdoped samples.', '1710.08173-1-49-2': 'Put together with the recent [MATH]SR studies[CITATION], this could reinforce the idea according to which there could be finite size magnetic domain, with very slow dynamics.', '1710.08173-1-49-3': 'The role of the finite size correlations and related forward scattering was recently addressed by C.M. Varma[CITATION] within the loop current framework.', '1710.08173-1-50-0': 'Finally, it is worth pointing out that independent theoretical ideas have been developed to interpret our measurements.', '1710.08173-1-50-1': 'Firstly, the magnetic order is generally described by microscopic loop currents within each unit cell [CITATION] and is therefore usually associated with orbital magnetic moments.', '1710.08173-1-50-2': 'Further developments of the loop current theory have been made to account for additional experimental facts [CITATION].', '1710.08173-1-50-3': 'Recently, other types of loop currents have been discussed as well [CITATION] where translationally-invariant states with topological order co-existing with both Ising-nematic order and spontaneous charge currents.', '1710.08173-1-50-4': 'Following a different path, magnetic quadrupoles on the Cu ions have been also proposed to account our results [CITATION] with possible microscopic mechanism[CITATION].', '1710.08173-1-50-5': 'Further, we recently reported similar time-reversal broken symmetry in iridates [CITATION] implying the possible generalization of loop current-type electronic phase in other oxides.', '1710.08173-1-50-6': 'In both iridates [CITATION] and cuprates[CITATION], second harmonic generation optical measurements found an odd-parity magnetic order parameter exactly in the same temperature and doping ranges, fully consistent with the loop current-type phase.', '1710.08173-1-50-7': 'In cuprates, other experimental techniques confirm the same broken symmetry in the pseudogap phase, either time reversal symmetry broken symmetry[CITATION], or loss of both C4 rotation and mirror symmetry in optical birefrengence in YBCO[CITATION].', '1710.08173-1-51-0': 'acknowlegments', '1710.08173-1-52-0': 'We wish to thank Victor Baledent, Niels-Bech Christensen, Benoit Fauque, Martin Greven, Jaehong Jeong, Yuan Li and Chandra Varma for stimulating discussions on different aspects related to this work.', '1710.08173-1-52-1': 'We acknowledge financial supports from the project NirvAna (contract ANR-14-OHRI-0010) of the ANR French agency.'} | {'1710.08173-2-0-0': 'Intra-unit cell magnetic order has been observed in four different families of high-temperature superconductors from polarized neutron diffraction experiments and supported by several other techniques.', '1710.08173-2-0-1': 'That order, which does not break translation symmetry, is consistent with the predicted orbital moments generated by two microscopic loop currrents in each CuO[MATH] cell.', '1710.08173-2-0-2': 'Recently, Croft et al [Phys.', '1710.08173-2-0-3': 'Rev. B 96, 214504 (2017)] claim to find no evidence for such orbital loop currents in charge ordered [MATH] using detwinned samples at least 100 times smaller than in previous experiments.', '1710.08173-2-0-4': 'We show by a detailed quantitative analysis of their data that contrary to their conclusion, the observed magnetic signal falls below their threshold of detection.', '1710.08173-2-0-5': 'This is shown by noting that Croft et al could overestimate by about an order of magnitude the expected magnetic signal.', '1710.08173-2-0-6': 'A factor [MATH] 3 is simply due to incorrect comparison with the previous experiments.', '1710.08173-2-0-7': 'Another factor of 3 can be related to not accounting for possible finite correlation length of the magnetic signal.', '1710.08173-2-0-8': 'Additional major experimental uncertainties are also discussed.', '1710.08173-2-0-9': 'None of the data reported by Croft et al could suggest that the intra-unit cell order is not universal in cuprates.', '1710.08173-2-1-0': '# Introduction', '1710.08173-2-2-0': 'In an extensive series of papers [CITATION], we and our collaborators demonstrated using polarized neutron diffraction (PND) that the pseudogap state of underdoped cuprate superconductors is characterized by a Q=0 magnetic order, also referred to as an intra unit cell (IUC) magnetic order[CITATION].', '1710.08173-2-2-1': 'That encompasses results in four different cuprates families with a large variety of dopings: [MATH] (YBCO) [CITATION], [MATH] [CITATION], [MATH] [CITATION] and [MATH] [CITATION].', '1710.08173-2-2-2': 'Two reviews were written to give more experimental and technical details and put the different neutron results in perspective with the other physical properties of high-temperature cuprates [CITATION].', '1710.08173-2-3-0': 'In a recent paper, Croft et al[CITATION] claim that they "found no evidence for the appearance of magnetic order below 300 K" in two [MATH] samples.', '1710.08173-2-3-1': 'We here show that they could not observe the magnetic signal owing to the insufficient detection capability of their measurements.', '1710.08173-2-3-2': 'First, Croft et al[CITATION] erroneously overestimate by a factor [MATH] 3 the magnetic signal that Fauque et al[CITATION] have previously reported.', '1710.08173-2-3-3': 'This seems to be related to multiple simplifications of their analysis, spanning incorrect data calibration, misleading sample comparison, ignorance of the impact of detwinning.', '1710.08173-2-3-4': 'We describe each of these shortcomings below.', '1710.08173-2-3-5': 'Additional experimental limitations have been overlooked in [CITATION].', '1710.08173-2-3-6': 'Indeed, not determining the spin-flip reference line properly, not doing a polarization analysis and inadequate control of the flipping ratio of the neutron beam add to uncertainties in their measurements.', '1710.08173-2-4-0': 'Further, the comparison with local probes results in [CITATION] is outdated as it dismisses the recent literature about muon spin resonance results [CITATION] leading to the possibility of finite time-scales IUC order associated to short range correlations.', '1710.08173-2-4-1': 'Such finite correlations have been actually reported in a recent study in nearly optimally doped YBCO[CITATION] using PND.', '1710.08173-2-4-2': 'This would have the effect to reduce the magnetic signal by another factor of [MATH] 3 in the experiment of Croft et al [CITATION].', '1710.08173-2-4-3': 'The claimed upper bound for a possible magnetic moment is therefore not correct and should be disregarded.', '1710.08173-2-4-4': 'None of the data reported by Croft et al [CITATION] disprove that the IUC order is universal in all cuprates.', '1710.08173-2-5-0': 'The PND experiment we discuss here is very challenging[CITATION].', '1710.08173-2-5-1': 'It is worth emphasizing that neutron polarization analysis has been previoulsy employed[CITATION] but not attempted in [CITATION].', '1710.08173-2-5-2': 'The expected polarization sum-rule is systematically satisfied that proves the magnetic nature of the reported signal.', '1710.08173-2-5-3': 'We emphasize as well than the YBCO data are nearly indistinguishable from those for [MATH] [CITATION].', '1710.08173-2-5-4': 'There is no doubt left about the existence of a magnetic signal.', '1710.08173-2-6-0': 'The earlier results [CITATION] revealed for the first time the IUC magnetic signal in five different YBCO samples and its stricking evolution with hole doping, following the pseudogap physics.', '1710.08173-2-6-1': 'Gradually, the data analysis was more quantitative in the subsequent publications in YBCO [CITATION].', '1710.08173-2-6-2': 'Over the years, the data analysis has been improved for quantitative accuracy (see for instance, the refined analysis on the sample C in Mangin-Thro et al [CITATION] compare to our original report in Fauque et al[CITATION]).', '1710.08173-2-6-3': 'This has important consequences on the magnetic signal amplitude and the confidence one can get from it.', '1710.08173-2-6-4': 'The neutron intensity is proportional to sample mass.', '1710.08173-2-6-5': 'By using samples [MATH] 100 times smaller than ours on a spectrometer with about 3 times larger neutron flux (at the used wavelength), Croft et al[CITATION] face about 30 times more experimental limitations.', '1710.08173-2-6-6': 'The counting times in previous reports[CITATION] could reach 2 hours/point.', '1710.08173-2-6-7': 'Even after a counting time of 4 hours/point in some of their data, their experiment does not reach the required accuracy.', '1710.08173-2-7-0': 'We consider below different aspects which affect noticeably the comparison made by Croft et al[CITATION] with our data and analysis.', '1710.08173-2-7-1': 'We demonstrate that various mistakes invalidate their conclusion and that they did not have enough sensitivity to detect the IUC magnetic signal in their measurements.', '1710.08173-2-7-2': 'First, we report on the same graph the raw data obtained from the different measurements (section [REF]).', '1710.08173-2-7-3': 'Second, in section [REF], we recall a few key results, already present in the literature, that, put together, could explain why Croft et al[CITATION] overestimate the signal they were looking for.', '1710.08173-2-7-4': 'In section [REF], we compare the data collection stategy of the previous studies with the one of Croft et al. This comparison shows that Croft et al[CITATION] overlook important experimental features discussed previously [CITATION], that limit their accuracy.', '1710.08173-2-7-5': 'Next, in section [REF], we underline the recent outcome of muon spin resonance ([MATH]SR) data which suggests that the magnetic correlations could be at short range with important consequences for the detection limit in PND, since instrumental resolution effects have to be considered very carefully.', '1710.08173-2-7-6': 'Finally, in section [REF], we present an improved analysis of our orignal paper Fauque et al[CITATION] by taking account of the instrument resolution, leading only to a small reduction, about 20% of the observed magnetic signal.', '1710.08173-2-7-7': 'We can then assess a direct comparison with the data reported by Croft et al[CITATION] and demonstrate how the IUC magnetic signal falls below the threshold of detection of their measurements.', '1710.08173-2-8-0': '# Raw data comparison', '1710.08173-2-9-0': 'First, we compare the raw experimental data obtained by Fauque et al[CITATION] and Croft et al[CITATION].', '1710.08173-2-9-1': 'As the IUC magnetic order does not break the symmetry of the lattice, one should study the inverse flipping ratio [MATH] at a Bragg position when the signal is expected to highlight a possible magnetic intensity at low temperature.', '1710.08173-2-9-2': 'As a recall, the inverse flipping ratio [MATH] is given by the following equation: [EQUATION] where [MATH] and [MATH] stand for the non-spin-flip and spin-flip intensities, respectively.', '1710.08173-2-9-3': 'As regularly emphasized[CITATION], [MATH] is essentially a ratio of measured quantities and does not depend on any assumptions on nuclear structure factor or flipping ratio.', '1710.08173-2-9-4': 'Changes in [MATH] for the same Bragg peak should be comparable among the different studies as far as similar sample composition are compared.', '1710.08173-2-9-5': 'The second part of Eq. [REF] shows how the magnetic intensity [MATH] can be extracted by comparison with the bare inverse flipping ratio [MATH] as it has been shown in several reports [CITATION].', '1710.08173-2-10-0': 'We report in Fig. [REF] the raw [MATH] at a Bragg positions (010) and (011) for two detwinned samples: sample H1 with p=0.104 of Croft et al[CITATION] and sample C of Fauque et al[CITATION] (see table [REF] for samples description).', '1710.08173-2-10-1': 'Within error bars, there is no disagreement between both data for the (010) reflection in Fig [REF].a', '1710.08173-2-10-2': 'a. However, Fig [REF].b', '1710.08173-2-10-3': 'b shows a certain difference between both datasets for the (011) reflection whose possible origins can be understood and is discussed below in sections [REF]-[REF]-[REF].', '1710.08173-2-10-4': 'For both reflections, a more surprising discrepancy occurs between the actual data of Fauque et al[CITATION] and the alleged ones estimated by Croft et al[CITATION] (the full and dashed lines in their figures 8d and 8e).', '1710.08173-2-10-5': 'Obviously, both should exactly match but they do not.', '1710.08173-2-10-6': 'That underlines the erroneous analysis performed by Croft et al[CITATION], and this for both Bragg peaks.', '1710.08173-2-11-0': 'Before going into a discussion on possible origins of this discrepancy in the next section, two remarks are necessary.', '1710.08173-2-11-1': 'First, the ratio [MATH] in Eq. [REF] does not change appreciably versus doping for the Bragg peak (011) in the doping range of interest here (p=0.1-0.12) (see table [REF]).', '1710.08173-2-11-2': '[MATH] corresponds to the nuclear structure factor [MATH] where [MATH] is given for instance by Eq. (12) of Croft et al[CITATION] or Eq. [REF] below.', '1710.08173-2-11-3': 'Fig. [REF] depicts the magnetic intensity and nuclear structure factor for the Bragg reflection (011) versus oxygen content.', '1710.08173-2-11-4': 'Both quantities are decreasing with increasing oxygen content.', '1710.08173-2-11-5': 'Therefore, it cannot be objected that the ratio [MATH] should decrease upon doping as does the magnetic intensity as it is considered in [CITATION].', '1710.08173-2-11-6': 'As a result, similar [MATH] is expected for both samples in Fig. [REF] within a 20% difference.', '1710.08173-2-12-0': 'Second, it should be noticed that the data shown for the Bragg peak (011) of sample C is the best example of a IUC magnetic signal ever reported in YBCO[CITATION].', '1710.08173-2-12-1': 'That corresponds to the highest experimentally reported ratio of the magnetic intensity compared to the nuclear intensity in Eq. [REF]: [MATH]= 1/400 =0.25 % at 70K for the (011) reflection[CITATION].', '1710.08173-2-12-2': 'All other reports in twinned samples are lower [CITATION]; this is due to a larger nuclear intensity of the (101) Bragg intensity which is averaged with the (011) peak in twinned samples.', '1710.08173-2-12-3': 'As shown in Fig. [REF], Croft et al[CITATION] allegedly estimate [MATH]=0.75 % (see Fig. [REF]) at odds with our results.', '1710.08173-2-13-0': 'In conclusion of this section, based on raw data, we demonstrate that the analysis of Croft et al[CITATION] cannot be correct.', '1710.08173-2-13-1': 'In the next section, we list a number of flaws in their attempt to compare with the results of Fauque et al[CITATION].', '1710.08173-2-14-0': '# Doping level and neutron structure factor', '1710.08173-2-15-0': '## Sample comparison', '1710.08173-2-16-0': 'First, for clarity, we need to describe the various YBCO samples investigated in PND experiments, listed in table [REF] shows all the underdoped YBCO samples.', '1710.08173-2-16-1': 'In four independant measurements on three different underdoped samples with superconducting transitions within the so-called 60K-plateau (samples YBCO[MATH]), very reproductible results were found: that corresponds to sample B and C for [CITATION] and the report of Mook et al [CITATION] (sample labelled here M) on a different sample.', '1710.08173-2-16-2': 'Two of these samples (samples B and M) are twinned and exhibit an oxygen ordering, ortho-II, corresponding to one Cu-O chain filled over two (see e.g. ref. [CITATION] for an extensive discussion of the oxygen ordering in YBCO).', '1710.08173-2-16-3': 'The third sample is detwinned and has a different ordering of extra oxygen in CuO-chains, ortho-VIII [CITATION].', '1710.08173-2-16-4': 'For all these reports, the magnetic cross-section at the Bragg position Q=(011) is systematically found to be 1.3[MATH]0.1 mbarn at a low temperature (typically at 70K, slightly above [MATH] and using the same calibration procedure, see table [REF]) [CITATION].', '1710.08173-2-16-5': 'In constrast, an almost twice larger signal was reported in the sample A (see table [REF]) which had a lower [MATH] and was measured only during our first experiment[CITATION].', '1710.08173-2-16-6': 'Croft et al[CITATION] focus exclusively on the pioneering work of Fauque et al[CITATION] neglecting further improvements.', '1710.08173-2-17-0': 'Croft et al[CITATION] performed their measurements on two different YBCO samples with superconducting temperatures [MATH] K (doping p=0.104, sample labelled here H1 for clarity) and [MATH] K (p=0.123, sample labelled here H2).', '1710.08173-2-17-1': 'The first one (H1) exhibits ortho-II oxygen ordering and the second (H2) ortho-VIII oxygen ordering.', '1710.08173-2-17-2': 'Their comparison with the previous results is questionnable: the full lines in figs. 8, 9 and 11 in [CITATION] correspond to different samples of Fauque et al but do not correspond to the actual doping level.', '1710.08173-2-17-3': 'That typically overestimates the expected magnetic signal.', '1710.08173-2-17-4': 'As the signal is found to increase with decreasing doping[CITATION], their comparison necessarily leads to an overestimation of the expected magnetic signal.', '1710.08173-2-17-5': 'In particular, Croft et al[CITATION] misleadingly compare in figures 8d and 11a, sample A with doping level of p=0.091 with sample H1 with p=0.104.', '1710.08173-2-17-6': 'Later, in figures 9 and 11c, they innacurately compare sample C with p=0.115 with sample H2 with p=0.123.', '1710.08173-2-17-7': 'Next, they linearly extrapolated (dashed lines in figs. 8, 9 and 11) the expected magnetic signal for doping level where no sample was measured.', '1710.08173-2-17-8': 'The same blue lines are shown in Fig. [REF], they do not correspond to any of our measurements [CITATION].', '1710.08173-2-18-0': 'For a fair discussion, their sample H1 should instead be better compared with sample B with [MATH] K [CITATION] and p=0.107[CITATION] or even with sample M [MATH] K[CITATION] with p= 0.112[CITATION].', '1710.08173-2-18-1': 'For all B,C and M samples, the magnetic cross-section at the Bragg position Q=(011) is systematically found to be 1.3 [MATH] 0.1 mbarn (see table [REF]).', '1710.08173-2-18-2': 'That is the correct order of magnitude which should be used to carry out a comparison with our data.', '1710.08173-2-18-3': 'The extrapolation from sample A (blue lines in Figures 8 and 11 (for sample H1) of Croft et al is then misleading.', '1710.08173-2-18-4': 'As sample H1, both samples B and M show the ortho-II oxygen ordering whereas sample C exhibits the ortho-VIII oxyen ordering as sample H2.', '1710.08173-2-18-5': 'We will see below that both the oxygen content and ordering are important to estimate the proper data calibration.', '1710.08173-2-19-0': 'Further, Croft et al[CITATION] neglected the 20% reduction of the magnetic intensity that we found following the improved analysis discussed below in section [REF].', '1710.08173-2-19-1': 'In any of their figures [CITATION], they did consider I[MATH] that we give in table [REF].', '1710.08173-2-19-2': 'This reduction of intensity further systematically worsens the estimate of "Fauque et al" in Figures 8, 9 and 11 in [CITATION].', '1710.08173-2-20-0': '## Impact of detwinning', '1710.08173-2-21-0': 'Next, both samples studied by Croft et al [CITATION] are detwinned samples.', '1710.08173-2-21-1': 'Even if the IUC structure factor is not expected to change upon an orthorhombic distortion, it was experimentally proved that it has an important impact on the neutron structure factor of the IUC order[CITATION].', '1710.08173-2-21-2': 'Indeed, the study of detwinned YBCO reveals a L-dependent a-b anisotropy of the scattered magnetic intensity, pointing out that the bilayer mirror plane is lost.', '1710.08173-2-21-3': 'Such an effect can be accounted for by a stacking of criss-crossed magnetic pattern within a bilayer.', '1710.08173-2-21-4': 'For L=0, the magnetic intensity is predominantly along a[MATH] and weaker along b[MATH].', '1710.08173-2-21-5': 'However, that difference between a[MATH] and b[MATH] is negligible for L=1.', '1710.08173-2-21-6': 'In terms of loop current models[CITATION], it means that the sum of the toroidal moments of each plane of the CuO[MATH] bilayer points along b[MATH], the direction of the Cu-O chains [CITATION].', '1710.08173-2-22-0': 'That study was made in the ortho-VIII (sample C), the same can be expected in ortho-II sample as what matters is the locking of the composite toroidal moment along the CuO chains, b[MATH].', '1710.08173-2-22-1': 'The magnetic scattering at the (010) Bragg reflection appears [MATH] 3 times weaker than that at the (100) reflection [CITATION].', '1710.08173-2-22-2': 'Croft et al [CITATION] only elusively acknowledged the impact of detwinning on the magnetic structure factor.', '1710.08173-2-22-3': 'That corresponds to the dotted lines in figures 8d and 11a and the two other lines in these figures should then be disregarded as they do not correspond to any physical situation (all these lines are reproduced in Fig. [REF].', '1710.08173-2-22-4': 'a).', '1710.08173-2-22-5': 'The full line compares their result on the (010) reflection with our twinned sample A where the magnetic intensity is predominantly controlled by the (100) reflection.', '1710.08173-2-22-6': 'As a result, an intensity of 2.0 mbarn at 70 K for the (010) reflection is expected for a detwinned sample instead of 9.0 mbarn (expected from twinned sample A) as is misleadingly reported in Fig 11.a', '1710.08173-2-22-7': 'This has a strong impact on the figure 8.a', '1710.08173-2-22-8': 'a and figure 11.a', '1710.08173-2-22-9': 'The proper expected amplitude from Fauque et al[CITATION] for a detwinned sample is within the statistical accuracy of the data of Croft et al [CITATION] for the (010) reflection in agreement with the results shown in Fig. [REF].a', '1710.08173-2-22-10': 'a (see also below section [REF] and Fig. [REF]).', '1710.08173-2-23-0': '## Data calibration', '1710.08173-2-24-0': 'A third issue in [CITATION], which can give insight to explain the discrepancy of Fig. [REF], is related to the use of a procedure of calibration of the data in absolute unit which differs substantially from the one systematically used in the previous PND studies [CITATION].', '1710.08173-2-24-1': 'As explained in the figure 1 caption of ref. [CITATION] and in [CITATION], the magnetic cross sections were calibrated in mbarn using the nuclear Bragg cross section of the (004) nuclear Bragg reflection.', '1710.08173-2-24-2': 'One then needs to estimate the nuclear neutron structure factor [MATH] which reads: [EQUATION] where [MATH] and [MATH] correspond to the neutron scattering length and the position of the d-th atom in the unit cell.', '1710.08173-2-24-3': '[MATH] is the occupancy of each atomic site which is unity for all atoms except for the oxygen in the chains at (0,1/2,0) site (labelled O(1) or O(4) depending of the litterature) where it is found to correspond to the extra oxygen content, x[CITATION].', '1710.08173-2-24-4': 'Eq. [REF] differs only from Eq. 12 of Ref. [CITATION] by the Debye-Waller term which of common opinion [CITATION] has no strong influence on the structure factor for the considered Bragg peaks.', '1710.08173-2-24-5': 'It could be added to the calculation as all atomic mean squared atomic displacements have been measured in YBCO[MATH] by neutron diffraction [CITATION] without assuming any approximation.', '1710.08173-2-24-6': 'The most prominent effect is the anomalous static displacements in the (a,b) plane of the oxygen atom at the chain site.', '1710.08173-2-25-0': 'In Fauque et al [CITATION], we calibrate our data using the Bragg peak (004).', '1710.08173-2-25-1': 'The Bragg peak (004) was chosen as we had to compare intensity of 5 different (twinned and detwinned) samples all together with different oxygen content.', '1710.08173-2-25-2': 'As the neutron scattering length is similar to all atoms, the calculated neutron structure factors, [MATH], vary noticeably on the oxygen content, x, of [MATH].', '1710.08173-2-25-3': 'Further, as discussed below, [MATH], depend as well on the specific chain-oxygen ordering.', '1710.08173-2-25-4': 'However, these effects are limited for the (004) reflection.', '1710.08173-2-25-5': 'Using the measured atomic positions in the orthorhombic phase[CITATION], we made an estimate of [MATH] barns [CITATION], as an averaged value over the doping range given by table [REF].', '1710.08173-2-25-6': 'Actually, a collection of Bragg peaks along (00L) direction has been measured as well.', '1710.08173-2-25-7': 'This can be used to improve the calibration of the magnetic intensity of our samples (see below in section [REF] and Fig. [REF]).', '1710.08173-2-26-0': 'Using this calibration, one can next estimate the nuclear intensity of the (011) Bragg reflection in the detwinned sample C.', '1710.08173-2-26-1': 'That gives [MATH] 0.52 barn for the (011) nuclear cross-section.', '1710.08173-2-26-2': 'This estimation is coherent with the ratio of the nuclear Bragg peak to the magnetic one, [MATH] at 70K for the (011) reflection of our detwinned sample C as quoted above in section [REF] (the smaller ratio ever reported for the IUC order[CITATION]).', '1710.08173-2-26-3': 'Our consistent estimate is twice larger than the value quoted by Croft et al [CITATION] of 0.28 barn.', '1710.08173-2-26-4': 'That difference basically explains most of the discrepancy in Fig. [REF] between our results [CITATION] and those purportedly estimated by Croft et al [CITATION].', '1710.08173-2-26-5': 'That leads to a global overestimation by a factor 2 of the magnetic intensity of Fauque et al [CITATION] in figures 8.', '1710.08173-2-26-6': 'e, 9.c and 11.', '1710.08173-2-26-7': 'b,c Croft et al [CITATION].', '1710.08173-2-27-0': 'In contrast, Croft et al [CITATION] did not use the (00L) series of Bragg peak for calibratation but only two reflections with K=1 plus only two strong reflections (006) and (020) at high momentum transfer.', '1710.08173-2-27-1': 'That coarse calibration is presented in Fig. 10 of ref. [CITATION] in a log-log representation that we reproduced here in Fig. [REF].', '1710.08173-2-27-2': 'Their data calibration is only made using 4 Bragg peaks much less than standard data calibration usually performed in neutron diffraction.', '1710.08173-2-27-3': 'The (004) Bragg peak is even not reported in ref. [CITATION] altering the comparison.', '1710.08173-2-27-4': 'Another issue with this calibration is the sensitivity of the calculated structure factors to the oxygen content.', '1710.08173-2-27-5': 'This is particularly important for the K=1 Bragg peaks cross-section but also for the (006) reflection.', '1710.08173-2-27-6': 'A straightforward calculation using Eq. [REF] shows that Bragg peak intensiies in the (01L) and (006) Bragg peaks cross-section vary considerably with the amount of actual oxygen atoms from x=6 (tetragonal phase) to x=7 (orthorhombic phase with full Cu-O chain) [CITATION].', '1710.08173-2-27-7': 'In the table III and Fig. 10 of [CITATION], Croft et al erroneously compare the measurements for their H2 sample (x=0.67) with the calculation of the ortho-II sample H1 (x=0.54).', '1710.08173-2-28-0': '## Oxygen ordering', '1710.08173-2-29-0': 'Next, the structure factor of Eq. [REF] is not a correct description of the atomic structure of YBCO around x[MATH] 0.6 as it assumes that the oxygen chain site O(1) is randomly occupied.', '1710.08173-2-29-1': 'It has been extensively shown by electron, X-ray and single-crystal neutron diffraction [CITATION] that YBCO[MATH] exhibits superstructures for all [MATH] in the doping range of table [REF] .', '1710.08173-2-29-2': 'Depending on doping, there are different stages of oxygen ordering in YBCO on which there is an abundant literature (see e.g. [CITATION]).', '1710.08173-2-29-3': 'Typically, the oxygen ordering is characterized by a short range scattering with large in-plane correlation lengh but short correlation range along [MATH].', '1710.08173-2-29-4': 'Only the so-called ortho-II phase exhibits a three-dimensional ordering, the remaining ones are essentially two-dimensional[CITATION].', '1710.08173-2-29-5': 'The extra oxygens form in-plane chains that do not become coherent along the [MATH]-direction, perpendicular to the CuO[MATH] plane.', '1710.08173-2-29-6': 'Therefore, an orthorhombic crystal structure with random occupancy of the oxygen O(1) site as considered in [CITATION] cannot account for the crystal structure of their samples.', '1710.08173-2-29-7': 'For instance, the H1 ortho-II sample has been studied with hard x-ray diffraction measurements [CITATION] where the oxygen-chain order is reported to have a finite coherence length along [MATH], [MATH] in agreement with previous reports [CITATION].', '1710.08173-2-29-8': 'The sample H2 (ortho-VIII) has even lower coherence length, [MATH] AA[CITATION].', '1710.08173-2-29-9': 'Further, it should be noticed that the oxygen ordering occupies a large volume fraction [CITATION].', '1710.08173-2-29-10': 'Therefore, the oxygen atoms in the chains do not primarily contribute to the structure factors via random occupancies as in Eq. [REF].', '1710.08173-2-29-11': 'It should be also stressed that structural refinements using neutron diffraction in YBCO have all been perfomed in twinned single crystals [CITATION].', '1710.08173-2-30-0': 'Croft et al [CITATION] argued that the ordering of the chain oxygens has little effect on structure factors of the (010) or (011) reflections.', '1710.08173-2-30-1': 'This is not correct for short range order of oxygen atoms particurlarly for neutron diffraction where oxygen scattering lengh is not negligible compared to the other atoms in contrast of X-ray diffraction.', '1710.08173-2-30-2': 'The short-range ordering of oxygen is a fact checked and can be empirically described by the scattering function, [MATH][CITATION].', '1710.08173-2-30-3': 'Although there is no analytic expression of the neutron cross-section, [MATH] is basically given by the Q-dependence of the oxygen supertructure.', '1710.08173-2-30-4': 'This is typically described by the so-called [MATH]-function for a linear chain of partially ordered atoms which gives an excellent description of the neutron diffraction data[CITATION].', '1710.08173-2-30-5': 'When the correlation length is of the order of the lattice parameter, it displays broad L-dependence with smoothed maxima around integrer L[CITATION].', '1710.08173-2-30-6': 'In the long-range limit, it corresponds to a sum of Lorentzians around the same integer L values compatible with x-ray measurements where the Lorentzian peak shape observed at high temperature transforms to a Lorentzian-squared at low temperature[CITATION] due to the inhibition of long-range ortho-II order.', '1710.08173-2-30-7': 'The true long-range order is never attained, since the antiphase boundaries between Ortho-II domains do not anneal out [CITATION].', '1710.08173-2-30-8': 'As shown by Monte-Carlo calculations of the anisotropic next-nearest-neighbor lattice-gas model (so called ASYNNNI) model [CITATION], the scattering function [MATH] exhibits similar broad peaks at the superstructures positions as well as at the regular Bragg positions.', '1710.08173-2-30-9': 'Therefore, the short-range oxygen ordering is obviously affecting the structure factors of the (010) or (011) reflections.', '1710.08173-2-31-0': 'As an attempt to describe this experimental situation, the nuclear structure factor for the (0KL) reflections can be approximated by, [EQUATION] where [MATH] is a sum of neutron coherent scattering length of the various atoms (Y,Ba,Cu,O) already present in the parent compound, YBa[MATH]Cu[MATH]O[MATH].', '1710.08173-2-31-1': 'However, for these atoms, the structure factor is calculated using the atomic positions reported in the orthorhombic YBCO[MATH] phase [CITATION] to account for the effect of the orthorhombic distortion.', '1710.08173-2-31-2': 'The contribution of the additional oxygen atoms in the Cu-O chains is not included in [MATH] but it is in the second term of Eq. [REF] where [MATH] (for sample H2) is the extra oxygen content, [MATH] is neutron coherent scattering length for oxygen, [MATH] is the value of the scattering function [MATH] discussed above at the Bragg position.', '1710.08173-2-31-3': 'It is related to the amplitude of [MATH] at the superstructure position [CITATION].', '1710.08173-2-32-0': 'As shown in Eq. [REF], there is a destructive interference between both terms for K=1 when [MATH].', '1710.08173-2-32-1': 'If [MATH] is large enough, the interference with the extra oxygen scattering is not very large and [MATH] does not vary much with [MATH].', '1710.08173-2-32-2': 'In contrast, if [MATH] is similar to [MATH], the destructive interference cannot be neglected and gives a large effect, [MATH]=0.83 barn is found the smaller for [MATH].', '1710.08173-2-32-3': 'However, if the oxygen order is 2D or at short-range, it is noticeably smaller than 1 and the effect is reduced.', '1710.08173-2-32-4': 'It is not obvious to give an estimate of the amplitude of [MATH] as it depends on the correlations lengths in the three directions.', '1710.08173-2-33-0': 'Another aspect is related to the resolution effect of the instrument.', '1710.08173-2-33-1': 'Indeed, as the diffuse scattering and the regular contribution to the Bragg peak exhibit different q-dependences, the integration by the instrumental resolution of the intensity would differ considerably for both features.', '1710.08173-2-33-2': 'In their structural analysis, Croft et al [CITATION] do not care about that effect.', '1710.08173-2-33-3': 'Then, using an estimate of [MATH] 1/3, one finds that [MATH] barn and for the (011) reflection not 0.28 barn as quoted in Croft et al [CITATION] (sample H1).', '1710.08173-2-33-4': 'In sample H2 (ortho-VIII phase), the oxygen ordering is essentially 2D, so, [MATH] could be even smaller and [MATH] larger.', '1710.08173-2-34-0': 'In Fig. [REF], the measured intensities are reported versus the calculated structure factor, [MATH], for these two limits : (i) the random distribution of oxygen in an ortho-I configuration considered by Croft et al[CITATION] and (ii) a short range oxygen ordered phase with [MATH].', '1710.08173-2-34-1': 'The figure shows that the expected linear behavior works as good for both models.', '1710.08173-2-34-2': 'Therefore, Fig. [REF] does not validate the structural model of random occupancy of oxygen atoms, in contrast to the suggestion of Croft et al [CITATION].', '1710.08173-2-34-3': 'The finite coherence lengths reported by hard x-ray diffraction measurements [CITATION] necessarilly implies a reduced [MATH].', '1710.08173-2-34-4': 'That underestimation of [MATH] of Croft et al [CITATION] yields to an overestimation of the expected magnetic scattering by at least a factor 2 which occurs predominantly for the (011) reflection where the value of [MATH] is smaller.', '1710.08173-2-35-0': 'In this section [REF], we show various points which can explain the discrepancy of Fig. [REF] between the raw data of Fauque et al [CITATION] and the alledged estimate made by Croft et al [CITATION].', '1710.08173-2-35-1': 'For the Bragg peak (011), the main error in [CITATION] comes from improper data calibration (sections [REF] and [REF]).', '1710.08173-2-35-2': 'For the (010) Bragg posiition, the magnetic structure in detwinned sample (section [REF]) should be properly acknowledged.', '1710.08173-2-35-3': 'As a result, the full and dashed lines in the figures 8,9 and 11, of Croft et al [CITATION] are all misleading as they are erroneous.', '1710.08173-2-35-4': 'They should be disregarded.', '1710.08173-2-35-5': 'Another global overestimated factor of 20% has been overlooked (see section [REF]).', '1710.08173-2-36-0': 'In the next two sections [REF]-[REF], we discuss on the one hand limitations with polarized neutron experiments in [CITATION] and on the other hand the dynamical nature of IUC magnetic signal as shown by recent muon spin resonance data [CITATION] which have been overlookd in [CITATION].', '1710.08173-2-36-1': 'Both features are important to explain the difference for the (011) reflection in Fig. [REF].', '1710.08173-2-36-2': 'b between the data of Fauque et al [CITATION] and those of Croft et al[CITATION].', '1710.08173-2-37-0': '# Data collection strategy', '1710.08173-2-38-0': 'First, we examine the different data collection strategies.', '1710.08173-2-38-1': 'As the expected magnetic signal is weak compared to the leakage of the nuclear intensity due to imperfect polarization, [MATH] 0.95-0.96, of the instruments, the only way to observe the signal is to perform a very accurate temperature dependence of the flipping ratio (R), ie.', '1710.08173-2-38-2': '[MATH] of a given Bragg spot.', '1710.08173-2-38-3': 'The inverse of this quantity, [MATH], is reported in Fig. [REF].', '1710.08173-2-38-4': 'As given by Eq. [REF], the expected signal can show up above a reference line (called baseline) which has to be determined at high temperature (above [MATH]).', '1710.08173-2-38-5': 'Unfortunately, only 1 or 2 points are measured by Croft et al [CITATION] above the reported ordering temperature [MATH] (which coincides with the pseudogap temperature, [MATH]).', '1710.08173-2-38-6': 'This is unfortunately insufficient to properly characterize the shape of that baseline and distinguish it from a genuine magnetic signal.', '1710.08173-2-39-0': 'Further, Croft et al [CITATION] do not perform the same method to collect the data as the one used in previous PND studies.', '1710.08173-2-39-1': 'They realign the sample at each temperature, implying that the spectrometer has to move at every temperature.', '1710.08173-2-39-2': 'We actually discarded this method rapidly as it did not give reliable enough flipping ratios values compared to the required accuracy.', '1710.08173-2-39-3': 'This method would be very reliable if the outcome of the measurements were not affected by slight erratic positioning errors of the spectrometer at each alignement, which is due to the limit of mechanical stability of the instruments.', '1710.08173-2-39-4': 'It should be noted by all triple-axis instruments discussed here (4F1 at LLB or IN20 at ILL) have all the same nominal angular accuracy of 0.01[MATH] for the rocking angle (A3) and 0.02[MATH] for the scattering angle (A4).', '1710.08173-2-40-0': 'In general, this limit is usually passing unnoticed because the required accuracy on the flipping ratio is generally not so essential.', '1710.08173-2-40-1': 'However, for the present case where one needs a very high precision on [MATH] corresponding to an error of [MATH] 0.02% (see Fig. [REF]).', '1710.08173-2-40-2': 'Although Croft et al [CITATION] do not report a very large distribution of countings between each given temperature, this can be nevertheless seen in their data where some of their points exhibit clear departure from the expected statistical errors of the counting.', '1710.08173-2-40-3': 'As the total number of points (6-8) as a function of temperature for each curve is rather scarce, this is detrimental to get a confident determination of both the shape of the baseline and possible magnetic signal on top of it, especially when the point at 300 K is the point which drifts away from the expected statistics.', '1710.08173-2-40-4': 'It necessarilly adds additionnal systematic and random errors in the countings which yields, [MATH] 0.1%, as reported below in Figs. [REF] and [REF].', '1710.08173-2-41-0': 'Croft et al [CITATION] argue that it is crucial to realign the sample at every temperature as the lattice parameters change with temperature and that the flipping ratio is sensitive to A4 ("[MATH]"-scan) (Fig. 6d in [CITATION]).', '1710.08173-2-41-1': 'We are aware of this possible issue but we showed that this effect is not crucial.', '1710.08173-2-41-2': 'This argument put forward by Croft et al leads to some inconsistency.', '1710.08173-2-41-3': 'Assuming that the argument were correct, then one should detect an artefact magnetic signal on any Bragg reflection.', '1710.08173-2-41-4': 'This is not what we have observed [CITATION].', '1710.08173-2-41-5': 'Furthermore, this signal should be stronger at large momentum transfer, such as for the (02L) reflections where no effect was reported [CITATION].', '1710.08173-2-41-6': 'Indeed, the effect should be about twice larger at large [MATH] such as the (020) reflection than at lower [MATH] for the (01L) reflections.', '1710.08173-2-41-7': 'This is not observed in any of our data (see e.g. [CITATION]).', '1710.08173-2-41-8': 'Croft et al [CITATION] are incorrect when they wrote in their section III.C "(similar changes are expected for the (010) and (011) reflections)".', '1710.08173-2-41-9': 'Furthermore, under this false assumption, there is no way to understand how an artificial magnetic signal could fulfilled the magnetic sum rule obtained by the neutron polarization analysis [CITATION].', '1710.08173-2-41-10': 'It should be stressed that this effect is less important in configurations of broad momentum instrument resolution, for instance in the case of large samples with large mosaicities which broadens the instrument resolution.', '1710.08173-2-42-0': 'In order to keep a highly stable flipping ratio, previous PND studies rule out the full realignement of the sample at each temperature.', '1710.08173-2-42-1': 'This is why we have way better accurate flipping ratio, a keypoint of success to determine a genuine magnetic signal.', '1710.08173-2-42-2': 'However, we realign the rocking "A3" angle at each temperature if necessary.', '1710.08173-2-42-3': 'This has less impact on the flipping ratio stability.', '1710.08173-2-42-4': 'Our strategy to collect the data then consists of measuring a lot of points in temperature with high statistics with no spectrometer movement.', '1710.08173-2-42-5': 'It should be stressed that Croft et al [CITATION] do not report any data obtained this way although we have established that this is a proper technique to disentangle unwanted shift of the flipping ratio with temperature (the change of the "baseline") from a genuine magnetic signal[CITATION].', '1710.08173-2-43-0': 'Further, they neglect to consider the possibility of a drift of the baseline due to the fact that the sample move up and down as a function of temperature, following the contraction and dilatation of the cryostat sample stick on which the sample is attached.', '1710.08173-2-43-1': 'Therefore, the sample is displaced vertically as a function of temperature and thus probes different flipping ratios due to inevitable polarization inhomogenities of the beam because of large monochromator and analyzer sizes[CITATION].', '1710.08173-2-43-2': 'This would be especially the case for a small sample[CITATION] compared to the sample-stick shrinking.', '1710.08173-2-43-3': 'Therefore, their claim that "the neutrons always emanate from the same part of the monochromator and strike the same part of the analyzer" cannot be correct when changing temperature.', '1710.08173-2-43-4': 'If that was true, then why their [MATH] subtantially decreases upon cooling as shown in Figs. [REF] and [REF] with appropriate zooming ?', '1710.08173-2-44-0': 'Incidently, another experimental issue in [CITATION] is related to the incoherent background subtraction, which should be measured away from the Bragg position.', '1710.08173-2-44-1': 'Again, a different strategy has been applied compared to the previous PND studies in [CITATION].', '1710.08173-2-44-2': 'We prove that this background subtraction to be important for a quantitative analysis of the data when the magnetic signal is weak [CITATION].', '1710.08173-2-44-3': 'No background subtraction was made in the original reports [CITATION].', '1710.08173-2-44-4': 'The choice in momentum space of the reference background wavevector is important as it has to be a location where no magnetic signal is present.', '1710.08173-2-44-5': 'We choose points by shifting the in-plane momentum transfer such as [MATH]=(0,0.9,1) for the reflection (011).', '1710.08173-2-44-6': 'Typically, on a triple axis instrument, the SF incoherent background intensity at [MATH]=(0,0.9,1) slightly increases upon cooling (negative slope with temperature), following the expected behaviour of a Debye-Waller factor.', '1710.08173-2-44-7': 'Using polarization analysis, we prove that the magnetism at such a location is weaker by two order of magnitude (see e.g [CITATION]).', '1710.08173-2-45-0': 'Instead, Croft et al [CITATION] choose [MATH]=(0,1,0.9) along [MATH] for the background subtraction and do not mention any polarization analysis to address a possible magnetism at this [MATH]-location.', '1710.08173-2-45-1': 'Near optimal doping, we showed that magnetic signal is not negligible at this [MATH]-location [CITATION] due to short-ranged IUC correlations along [MATH].', '1710.08173-2-45-2': 'Therefore, that choice is unfortunate as it would necessarily reduce possible intrinsic IUC signal.', '1710.08173-2-45-3': 'Actually, Croft et al [CITATION] report the intensity at [MATH]=(0,1,0.9) in Figs. 7c.', '1710.08173-2-45-4': 'Interestingly, it occurs that the SF intensity decreases upon cooling (positive slope with temperature) at this [MATH]-location.', '1710.08173-2-45-5': 'This implies one of two possibilities: either, this sloping background is the true incoherent background (and then it should be removed from the Bragg peak (011) SF intensity point by point), or it suggests a vanishing at low temperature of a magnetic signal present at high temperature at this [MATH]-location.', '1710.08173-2-45-6': 'This latter possibility is related to the fact that the positive slope with temperature background behaviour contrasts with the known behaviour of the incoherent SF background (negative slope with temperature) discussed above.', '1710.08173-2-45-7': 'Both possibilities leads to interesting opportunuities.', '1710.08173-2-45-8': 'Croft et al [CITATION] made no attempt to analyse these data along these lines, only fitting them by a constant.', '1710.08173-2-46-0': 'Finally, no attempt of polarization analysis is reported in [CITATION] to search for the weak magnetic signal at the Bragg reflection (011) or to accurately determine an upper limit for a magnetic scattering intensity.', '1710.08173-2-46-1': 'It should be strongly emphasized that previous PND studies reported a magnetic signal at very specific Bragg reflections (the signal was absent on others) and complementary polarization analysis demonstrate unambigously the magnetic nature of the observed scattered intensity [CITATION].', '1710.08173-2-47-0': 'As a result of the various points discussed above, Croft et al [CITATION] cannot estimate the shape of the "baseline" as they can only rely on the point measured at 300K to re-scale their data (even if sometimes that point is not included in their fitting line).', '1710.08173-2-47-1': 'That necessarily alters the data accuracy with substantial error [MATH] 0.1% and degrades the threshold of detection of a small signal (see section [REF]).', '1710.08173-2-47-2': 'At all Bragg reflections, their [MATH], when appropriately plotted, is also not a constant but exhibits a slope in temperature (see Figs. [REF] and [REF]).', '1710.08173-2-48-0': '# Effects of short-range magnetic correlations', '1710.08173-2-49-0': 'It has been already discussed by Fauque et al[CITATION] that local probe measurements did not show a magnetic signal consistent with our measurements.', '1710.08173-2-49-1': 'Typically, zero-field muon spin resonance ([MATH]SR) [CITATION] and Nuclear Magnetic Resonance (NMR) [CITATION] experiments in YBCO found no evidence for a static and long range magnetic order.', '1710.08173-2-49-2': 'First, let us recall that the situation with [MATH]SR experiments has been contradictory (see [CITATION] for an extensive discussion).', '1710.08173-2-49-3': 'Second, it was envisaged[CITATION] that the magnetic moments could fluctuate slowly enough to appear static to neutrons, but too fast enough to be observed as a magnetic order in [MATH]SR and therefore even more for NMR which probes even slower timescales.', '1710.08173-2-49-4': 'Recently this picture has been confirmed as slow magnetic fluctuations have been discovered in YBCO samples [CITATION] using a longitudinal field [MATH]SR technique.', '1710.08173-2-49-5': 'These magnetic fluctuations are related to our magnetic order as a critical slowing down[CITATION] is also reported at the same temperature [MATH] where the neutron magnetic signal sets in.', '1710.08173-2-49-6': 'Beyond the YBCO bilayer system, a new set of [MATH]SR data in the bilayer system, [MATH], is also found to support the existence of the magnetic signal[CITATION].', '1710.08173-2-50-0': 'Croft et al [CITATION] do not mention these recent essential developments and efforts made to refine the [MATH]SR studies.', '1710.08173-2-50-1': 'Therefore, the conclusion of Croft et al is no more supported by the most recent [MATH]SR studies contrary to their claim.', '1710.08173-2-50-2': 'J. Zhang et al[CITATION] found that magnetic correlations fluctuate, [MATH] 10 ns timescale, even for strongly underdoped samples as the ones discussed here.', '1710.08173-2-50-3': 'These low frequency fluctuations are necessarily related to the formation of finite-size magnetic domains [CITATION], corresponding to finite correlation lengths of the magnetic order observed in neutron diffraction.', '1710.08173-2-50-4': 'Such a short range correlation length of the IUC order has been observed [CITATION] near optimally doped YBCO sample.', '1710.08173-2-50-5': 'Note that when the correlation length becomes too short, such as in [MATH] [CITATION], the magnetic signal cannot be observed on the Bragg reflection anymore[CITATION].', '1710.08173-2-51-0': 'In underdoped samples, for dopings p[MATH]0.1-0.11 (samples A and M in Tab. [REF]), we observed resolution limited magnetic peaks[CITATION].', '1710.08173-2-51-1': 'However, the PND experiment was performed on large single crystals with large mosaicity, corresponding to rocking scans with a full width at half maxium (FWHM) of [MATH], implying that correlation length [MATH] .', '1710.08173-2-51-2': 'As shown in Fig. [REF], the rocking "A3" scan is similar to a scan along L. On Bragg reflections, (010) or (020), a rocking scan matches a transverse scan, i.e. a scan along the (001) direction.', '1710.08173-2-51-3': 'Instead, the measurements carried out by Croft et al [CITATION] are performed on small samples with low mosaic spread yielding an observed rocking scan FWHM of [MATH] (Fig 6 and 7).', '1710.08173-2-51-4': 'Therefore, if the magnetic order exhibited a finite correlation length [MATH] , the magnetic signal would be characterized by a FWHM of [MATH] in the A3-scan, with a peak intensity reduced by a factor [MATH]3 in comparison with a resolution limited signal, like a nuclear Bragg peak.', '1710.08173-2-51-5': 'A sketch of these two situations is reported in the top-right panel of Fig. [REF].', '1710.08173-2-51-6': 'The magnetic amplitude at the Bragg position would be reduced accordingly [MATH]3 times compared to the one of true long range ordered state.', '1710.08173-2-52-0': 'In conclusion of this section, in case of short range magnetic correlations, for example [MATH] , this factor 3 can be invoked to explain the difference in Fig. [REF] between the data of Fauque et al[CITATION] and those of Croft et al[CITATION] due to the different mosaicity of the samples.', '1710.08173-2-52-1': 'This is what is illustrated in Figs [REF] and [REF] of the next section.', '1710.08173-2-53-0': '# Improved data analysis', '1710.08173-2-54-0': 'Finally, motivated by the fact that a detailed comparison is necessary to appreciate the possible differences between the study of Croft et al [CITATION] with earlier PND works, we refine the data calibration procedure of Fauque et al[CITATION].', '1710.08173-2-54-1': 'Originally, the data were calibrated onto the Bragg peak reflection (004) but the instrumental resolution was neglected as a secondary effect.', '1710.08173-2-54-2': 'We then re-assess below the complete data calibration of the results of the detwinned high-quality sample C[CITATION].', '1710.08173-2-55-0': 'To perform the calibration, one needs to estimate the instrumental resolution correction.', '1710.08173-2-55-1': 'The resolution of a triple-axis spectrometer is known to be described by a 4D (3 directions for momentum transfer and one for energy) Gaussian function [CITATION].', '1710.08173-2-55-2': 'The scattering intensity is the convolution of the neutron scattering cross section with this instrument resolution ellipsoid.', '1710.08173-2-55-3': 'A Bragg peak is a Dirac function in phase space.', '1710.08173-2-55-4': 'For a Bragg scattering, the measured intensity at [MATH]=(HKL) then reduces to the product of the computed structure factor with the volume of the resolution ellipsoid.', '1710.08173-2-55-5': 'That reads: [EQUATION] [MATH] is a normalization constant, [MATH] is determined by the normalization of the spectrometer 4D Gaussian resolution [CITATION].', '1710.08173-2-55-6': 'In general, for a triple-axis spectrometer[CITATION], [MATH] is not as simple as the Lorentz factor, [MATH], used for a diffractometer (two-axis instrument, no analysis of the scattered neutron energy); [MATH] only for a [MATH] scan and a perfect mosaicity sample[CITATION].', '1710.08173-2-55-7': 'That is actually the quantity from [MATH] scans, [MATH] from Croft et al [CITATION], which is reported in Fig. [REF] for sample H2.', '1710.08173-2-55-8': 'Otherwise, we need to estimate [MATH].', '1710.08173-2-55-9': 'To determine [MATH], we use a Cooper-Nathans[CITATION] formalism but with the use of mosaic spreads of the diffracting crystals and of the divergence angles given by the beam geometry (distances between the different elements composing the instrument, sizes of crystals,...).', '1710.08173-2-56-0': 'For the detwinned sample C[CITATION], one can compute the proper resolution factor [MATH].', '1710.08173-2-56-1': 'The structure factors are calculated with Eq. [REF] using crystallographic data of refs. [CITATION] and S(0)=1/3.', '1710.08173-2-56-2': 'Fig. [REF] shows the comparison of the measured intensity of the nuclear Bragg peaks divided by the resolution factor [MATH].', '1710.08173-2-56-3': 'The figure shows canonical behaviour of a calibration curve (see an example, Fig. 1 in [CITATION], in the context of cuprates): i) the expected linear behavior given by Eq. [REF] at small structure factor and ii) a saturation at larger structure factor due to extinction effects.', '1710.08173-2-56-4': 'From this linear calibration at small enough [MATH] (black dashed line), one obtains a renormalization of about 20 % for the magnetic cross-section for sample C of 1.3 [MATH] 0.1 mbarn at (011) reflection compared to our previous estimate (red dashed line).', '1710.08173-2-56-5': 'As the IUC magnetic intensity at the reflection (011) does not depend much on the sample detwinning [CITATION], one can report the corrected intensity for different samples versus the doping level (Fig. [REF]) from the published data [CITATION].', '1710.08173-2-56-6': 'As the intensity is proportional to the square of the magnetic moments, that induces a renormalization of less than 10 % for the ordered IUC magnetic moment, still of the order of [MATH] 0.1 [MATH].', '1710.08173-2-56-7': 'Fig. [REF] shows as well the onset temperature of the IUC order, [MATH].', '1710.08173-2-56-8': 'The measured magnetic intensity is not simply proportional to [MATH], possibly because of the short range nature of the IUC order[CITATION].', '1710.08173-2-57-0': 'Lastly, we now turn to the comparison of the magnetic signal reported by Fauque et al[CITATION] to the experimental data of Croft et al [CITATION].', '1710.08173-2-57-1': 'In figs. [REF] and [REF], we present [MATH] data for both samples of Croft et al [CITATION]; the data points are exactly the same as those reported in Figs. 8-9 in [CITATION], but Figs. [REF]-[REF] zoom on the data.', '1710.08173-2-57-2': 'For instance, for a sake of clarity, we do not consider in Figs. [REF]-[REF] the possible issue with the background subtraction emphasized in section [REF].', '1710.08173-2-57-3': 'We stress that a background subtraction with a positive slope with temperature would lead to a more clearly noticeable IUC magnetic signal at low temperature.', '1710.08173-2-58-0': 'The inverse of flipping ratio at the reflection (020) is linear in temperature with a positive slope as it has been shown to exist [CITATION].', '1710.08173-2-58-1': 'As discussed in section [REF], this slope is inevitable as the sample is displaced in the neutron beam upon cooling.', '1710.08173-2-58-2': 'Croft et al [CITATION] arbitrarily describe that systematic sloping line with an horizontal line only.', '1710.08173-2-58-3': 'At the accuracy required to observe the IUC magnetic signal, this is not a correct approximation.', '1710.08173-2-58-4': 'With the large grey shaded area, we next represent the zone of the limit of detection, corresponding to [MATH], on both sides of an average sloping line.', '1710.08173-2-58-5': 'This area is due to combined effects of the statistical errors of each points, occurence of off-statistical points (possibly related to errors in positioning as discussed in section [REF]) and the scarce number of points.', '1710.08173-2-58-6': 'This area is simply deduced from the measurements at the Bragg (020) reflection where no magnetic signal is expected for the IUC order.', '1710.08173-2-59-0': 'For clarity, the same error of [MATH] 0.1% is used for all plots in both figures.', '1710.08173-2-59-1': 'That error is typically equivalent to an error of [MATH] on the flipping ratio or [MATH] of the spin-flip intensity.', '1710.08173-2-59-2': 'This uncertainty is not negligible as the highest ratio of the IUC intensity to the spin-flip total intensity, that we have reported, is only [MATH] for detwinned sample[CITATION] and about [MATH]3-4 otherwise[CITATION] for the corresponding doping levels.', '1710.08173-2-59-3': 'The error on the spin-flip intensity in previous PND experiments [CITATION] ranges between [MATH] 0.5-1 .', '1710.08173-2-59-4': 'That leads to an estimate of [MATH] 0.01-0.02 (see e.g. Fig. S1 in the Supplemental materials of Mangin-Thro et al [CITATION] for an example of measured error bars) for a magnetic signal at best of [MATH] 0.25 (Fig. [REF]).', '1710.08173-2-60-0': 'On Figs. [REF]-[REF], the detection limit of [MATH] is next reported for the Bragg reflections (010) and (011) where the magnetic signal is expected.', '1710.08173-2-60-1': 'To compare that set of data, with those of Fauque et al[CITATION], we then consider three different scenarios which are plotted in Figs. [REF]-[REF]: A) no magnetic signal is present, B) a long range magnetic order is present with the amplitude reported in [CITATION] C) a magnetic order with short range correlation along the c-direction is present with the amplitude reported in [CITATION].', '1710.08173-2-60-2': 'For the sample H1, one compares with the amplitude measured for sample B (see table [REF]) assuming that this sample would be detwinned for a proper comparison.', '1710.08173-2-60-3': 'For the sample H2, one compares with the amplitude measured for sample C.', '1710.08173-2-60-4': 'The data have been normalized in absolute units using the structural model discussed in section [REF] of short range oxygen ordering, [MATH].', '1710.08173-2-60-5': 'In both cases, that basically corresponds to our best experimental evidence of a magnetic signal [MATH] 0.25% as plotted in Fig. [REF].', '1710.08173-2-60-6': 'Indeed, as discusssed in section [REF], this ratio is weakly dependent of doping within the doping range p[MATH]0.1-0.12 (Fig. [REF]).', '1710.08173-2-60-7': 'The top right panel of Fig. [REF] simulates the A3 rocking scan for the scenario B in blue and the scenario C in red where both hypothetical curves have been convoluted by the resolution function given by the Fig. 6.', '1710.08173-2-60-8': 'a in [CITATION].', '1710.08173-2-60-9': 'One clearly sees that the IUC signal is [MATH] 3 times weaker in scenario C compared to scenario B.', '1710.08173-2-61-0': 'First, following the various points discussed above and in line with the Fig. [REF], one clearly sees that the expected signal from [CITATION] is only marginally larger than the experimental uncertainty (error area).', '1710.08173-2-61-1': 'For all the three scenarios presented above, only one parameter, the overall level of the background of the baseline, is fitted.', '1710.08173-2-61-2': 'All the other parameters are given from the literature [CITATION] and the discussion above.', '1710.08173-2-61-3': 'Clearly, for all cases, scenarios A (of Croft et al [CITATION]) and C (a short range along [MATH] IUC magnetic order compatible with the report of ref. [CITATION]) cannot be distinguished at all.', '1710.08173-2-61-4': 'Even for a true long range magnetic order along [MATH] (scenario B), the data are insufficient to eliminate with confidence the existence of IUC order.', '1710.08173-2-61-5': 'Clearly, for both samples, the sensitivity of the experiment is insufficient to observe the IUC order contrary to the claim of Croft et al [CITATION].', '1710.08173-2-62-0': '# Conclusion', '1710.08173-2-63-0': 'In conclusion, Croft et al [CITATION] do not have the experimental accuracy to observe the IUC signal that we have been reporting for the last decade in YBCO [CITATION].', '1710.08173-2-63-1': 'Different factors applied: first, the accuracy limit of the experiment of Croft et al [CITATION] is represented by the shaded areas which indicate the uncertainty of [MATH] 0.1% on the thermal dependence of the baseline for the ratio [MATH].', '1710.08173-2-63-2': 'Second, the purported level of intensity of Fauque et al has been erroneously and systematically overestimated by a factor [MATH] 3 (see Fig. [REF]).', '1710.08173-2-63-3': 'That corresponds to the scenario B of figs. [REF] and [REF], where a long range IUC order is assumed.', '1710.08173-2-63-4': 'Third, an additional factor 3 occurs if the signal is short ranged along L with [MATH] as discussed in section [REF] (scenario C of Figs. [REF] and [REF]).', '1710.08173-2-63-5': 'The most plausible scenario for the magnetic intensity lies in between scenarios B and C in Figs. [REF] and [REF].', '1710.08173-2-63-6': 'This is clearly below the detection limit of the data of Croft et al [CITATION].', '1710.08173-2-63-7': 'The claimed upper bound for a possible magnetic moment in [CITATION] is therefore baseless and should be disregarded.', '1710.08173-2-64-0': 'One positive conclusion that could be derived from the study of Croft et al [CITATION] is that the use of sample with a very good mosaic has raised question concerning the length of the magnetic correlations for the IUC order.', '1710.08173-2-64-1': 'Short ranged correlations were already established in YBCO around optimal doping [CITATION] but not for more underdoped samples.', '1710.08173-2-64-2': 'Put together with the recent [MATH]SR studies[CITATION], this could reinforce the idea according to which there could be finite size magnetic domain, with very slow dynamics.', '1710.08173-2-64-3': 'The role of the finite size correlations and related forward scattering was recently addressed by C.M. Varma[CITATION] within the loop current framework.', '1710.08173-2-65-0': 'Finally, it is worth pointing out that independent theoretical ideas have been developed to interpret our measurements.', '1710.08173-2-65-1': 'Firstly, the magnetic order is generally described by microscopic loop currents within each unit cell [CITATION] and is therefore usually associated with orbital magnetic moments.', '1710.08173-2-65-2': 'Further developments of the loop current theory have been made to account for additional experimental facts [CITATION].', '1710.08173-2-65-3': 'Recently, other types of loop currents have been discussed as well [CITATION] where translationally-invariant states with topological order co-existing with both Ising-nematic order and spontaneous charge currents.', '1710.08173-2-65-4': 'Following a different path, magnetic quadrupoles on the Cu ions have been also proposed to account our results [CITATION] with possible microscopic mechanism[CITATION].', '1710.08173-2-65-5': 'Further, we recently reported similar time-reversal broken symmetry in iridates [CITATION] implying the possible generalization of loop current-type electronic phase in other oxides.', '1710.08173-2-65-6': 'In both iridates [CITATION] and cuprates[CITATION], second harmonic generation optical measurements found an odd-parity magnetic order parameter exactly in the same temperature and doping ranges, fully consistent with the loop current-type phase.', '1710.08173-2-65-7': 'In cuprates, other experimental techniques confirm the same broken symmetry in the pseudogap phase, either time reversal broken symmetry[CITATION], or loss of both C[MATH] rotation and mirror symmetry in optical birefrengence in YBCO[CITATION].', '1710.08173-2-66-0': 'The intra unit cell magnetic signal has been well documented for the last decade using polarized neutron diffraction[CITATION].', '1710.08173-2-66-1': 'The magnetic signal, observed in four cuprate familiees, is seen only at specific Bragg positions.', '1710.08173-2-66-2': 'Data in YBCO and in [MATH] are nearly indistinguishable with a systematic doping dependence.', '1710.08173-2-66-3': 'In all cuprates, important polarization analysis has been conducted and the polarization sum rule, which demonstrates the magnetic nature of the signal, always nicely obeyed.', '1710.08173-2-66-4': 'The experiment is highly non-trivial with many technical pitfalls to miss the genuine signal.', '1710.08173-2-66-5': 'Among other features, it requires sufficient data at high temperature for a proper knowledge of the background.', '1710.08173-2-66-6': 'The IUC magnetic signal falls below the experimental sensitivity of the experiment of Croft et al [CITATION] predominantly because too tiny YBCO samples were studied (100 times smaller than in previous reports).', '1710.08173-2-66-7': 'Further, we demonstrate a large number of flaws in their data analysis which invalidate their comparison with the previous data.', '1710.08173-2-66-8': 'Their estimate of the magnetic signal previously reported could be incorrect by an order of magnitude.', '1710.08173-2-66-9': 'None of the data shown invalidate that the intra unit cell magnetic order is an intrinsic property of the pseudogap state of cuprates.', '1710.08173-2-67-0': 'acknowlegments', '1710.08173-2-68-0': 'We wish to thank Gabe Aeppli, Victor Baledent, Dalila Bounoua, Johan Chang, Niels-Bech Christensen, Benoit Fauque, Martin Greven, Jaehong Jeong, Steve Kivelson, Yuan Li, Tomo Uemura and Chandra Varma for stimulating discussions on different aspects related to this work.', '1710.08173-2-68-1': 'We acknowledge financial supports from the project NirvAna (contract ANR-14-OHRI-0010) of the ANR French agency.'} | [['1710.08173-1-31-6', '1710.08173-2-45-6'], ['1710.08173-1-31-7', '1710.08173-2-45-7'], ['1710.08173-1-35-1', '1710.08173-2-49-1'], ['1710.08173-1-35-2', '1710.08173-2-49-2'], ['1710.08173-1-35-4', '1710.08173-2-49-4'], ['1710.08173-1-35-5', '1710.08173-2-49-5'], 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['1710.08173-2-58-5', '1710.08173-3-21-5'], ['1710.08173-2-59-0', '1710.08173-3-22-0'], ['1710.08173-2-59-3', '1710.08173-3-22-3'], ['1710.08173-1-3-1', '1710.08173-2-3-3'], ['1710.08173-1-3-2', '1710.08173-2-3-4'], ['1710.08173-1-3-3', '1710.08173-2-4-4'], ['1710.08173-1-26-1', '1710.08173-2-39-1'], ['1710.08173-1-26-2', '1710.08173-2-39-3'], ['1710.08173-1-26-4', '1710.08173-2-39-3'], ['1710.08173-1-26-5', '1710.08173-2-40-4'], ['1710.08173-1-44-2', '1710.08173-2-58-4'], ['1710.08173-1-44-5', '1710.08173-2-59-0'], ['1710.08173-1-17-0', '1710.08173-2-24-0'], ['1710.08173-1-17-5', '1710.08173-2-25-3'], ['1710.08173-1-17-6', '1710.08173-2-25-4'], ['1710.08173-1-17-7', '1710.08173-2-25-6'], ['1710.08173-1-17-7', '1710.08173-2-25-7'], ['1710.08173-1-22-1', '1710.08173-2-34-1'], ['1710.08173-1-22-4', '1710.08173-2-33-4'], ['1710.08173-1-22-5', '1710.08173-2-34-4'], ['1710.08173-1-21-0', '1710.08173-2-31-0'], ['1710.08173-1-21-2', '1710.08173-2-31-2'], ['1710.08173-1-21-4', '1710.08173-2-32-0'], ['1710.08173-1-21-6', '1710.08173-2-32-2'], ['1710.08173-1-21-7', '1710.08173-2-32-3'], ['1710.08173-1-21-9', '1710.08173-2-32-2'], ['1710.08173-2-66-7', '1710.08173-3-28-3'], ['1710.08173-1-9-0', '1710.08173-2-16-0'], ['1710.08173-2-36-0', '1710.08173-3-19-0'], ['1710.08173-2-36-2', '1710.08173-3-19-7'], ['1710.08173-2-51-3', '1710.08173-3-19-4'], ['1710.08173-2-52-0', '1710.08173-3-19-7'], ['1710.08173-1-12-3', '1710.08173-2-17-6'], ['1710.08173-1-12-8', '1710.08173-2-17-1'], ['1710.08173-1-12-7', '1710.08173-2-18-3']] | [['1710.08173-2-4-1', '1710.08173-3-7-3'], ['1710.08173-2-4-2', '1710.08173-3-7-4'], ['1710.08173-2-9-0', '1710.08173-3-10-0'], ['1710.08173-2-9-1', '1710.08173-3-11-0'], ['1710.08173-2-9-2', '1710.08173-3-11-1'], ['1710.08173-2-9-3', '1710.08173-3-11-2'], ['1710.08173-2-9-4', '1710.08173-3-11-3'], ['1710.08173-2-9-5', '1710.08173-3-11-4'], ['1710.08173-2-10-0', '1710.08173-3-12-0'], ['1710.08173-2-10-1', '1710.08173-3-12-1'], ['1710.08173-2-10-3', '1710.08173-3-12-4'], ['1710.08173-2-10-4', '1710.08173-3-12-5'], ['1710.08173-2-10-5', '1710.08173-3-12-6'], ['1710.08173-2-10-6', '1710.08173-3-12-7'], ['1710.08173-2-11-1', '1710.08173-3-14-0'], ['1710.08173-2-11-2', '1710.08173-3-14-1'], ['1710.08173-2-11-5', '1710.08173-3-14-3'], ['1710.08173-2-11-6', '1710.08173-3-14-4'], ['1710.08173-2-13-0', '1710.08173-3-16-0'], ['1710.08173-2-12-3', '1710.08173-3-16-1']] | ['1710.08173-1-18-4', '1710.08173-1-45-7', '1710.08173-1-51-0', '1710.08173-2-22-4', '1710.08173-2-26-6', '1710.08173-2-60-8', '1710.08173-2-67-0', '1710.08173-3-12-2', '1710.08173-3-12-3', '1710.08173-3-13-0', '1710.08173-3-29-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1710.08173 | {'1710.08173-3-0-0': 'Intra-unit cell magnetic order has been observed in four different families of high-temperature superconductors from polarized neutron diffraction experiments and supported by several other techniques.', '1710.08173-3-0-1': 'That order, which does not break translation symmetry, is consistent with the predicted orbital moments generated by two microscopic loop currrents in each CuO[MATH] cell.', '1710.08173-3-0-2': 'Recently, using polarized neutron diffraction, Croft et al [Phys.', '1710.08173-3-0-3': 'Rev. B 96, 214504 (2017)] claim to find no evidence for such orbital loop currents in charge ordered [MATH].', '1710.08173-3-0-4': 'Their experiment is done with detwinned samples at least 100 times smaller than in previous experiments without counting much longer.', '1710.08173-3-0-5': 'We show by a detailed quantitative analysis of their data that contrary to their conclusion, the magnetic signal falls below their threshold of detection.', '1710.08173-3-0-6': 'None of the data reported by Croft et al challenge the universality of the intra-unit cell order in cuprates.', '1710.08173-3-1-0': '# Introduction', '1710.08173-3-2-0': 'In an extensive series of papers [CITATION], we and our collaborators demonstrated using polarized neutron diffraction (PND)[CITATION] that the pseudogap state of underdoped cuprate superconductors is characterized by a Q=0 magnetic order, also referred to as an intra unit cell (IUC) magnetic order[CITATION].', '1710.08173-3-2-1': 'That encompasses results in four different cuprates families with a large variety of hole doping (p): [MATH] (YBCO) [CITATION], [MATH] [CITATION], [MATH] [CITATION] and [MATH] [CITATION].', '1710.08173-3-2-2': 'Two reviews were written to give more experimental and technical details and put the different neutron results in perspective with the other physical properties of high-temperature cuprates [CITATION].', '1710.08173-3-3-0': 'The PND experiment we discuss here is very challenging[CITATION].', '1710.08173-3-3-1': 'The earlier results [CITATION] revealed for the first time the IUC magnetic signal in five different YBCO samples and its stricking evolution with hole doping, following the pseudogap physics.', '1710.08173-3-3-2': 'Gradually, the data analysis was more quantitative in the subsequent publications in YBCO [CITATION].', '1710.08173-3-3-3': 'Over the years, the data analysis has been improved for quantitative accuracy (see for instance, the refined analysis on the sample C in Mangin-Thro et al [CITATION] compare to our original report in Fauque et al[CITATION]).', '1710.08173-3-3-4': 'This has important consequences on the magnetic signal amplitude and the confidence one can get from it.', '1710.08173-3-4-0': 'In a recent paper, Croft et al[CITATION] claim that they found no evidence for the appearance of magnetic order below 300 K in two YBCO samples.', '1710.08173-3-4-1': 'We show that they could not observe the magnetic signal owing to the insufficient detection capability of their measurements (see [CITATION] for details).', '1710.08173-3-5-0': '[MATH] First, the neutron intensity is proportional to sample mass.', '1710.08173-3-5-1': 'By using samples at least [MATH] 100 times smaller than ours on a spectrometer with about 3 times larger neutron flux (at the used wavelength), Croft et al[CITATION] face about 30 times more experimental limitations.', '1710.08173-3-5-2': 'The counting times in previous reports[CITATION] could reach 2 hours/point.', '1710.08173-3-5-3': 'Even after a counting time of 4 hours/point in some of their data, their experiment does not reach the required accuracy.', '1710.08173-3-6-0': '[MATH] Second, Croft et al[CITATION] erroneously overestimate by a factor [MATH] 3 the magnetic signal that Fauque et al[CITATION] have previously reported.', '1710.08173-3-6-1': 'This seems to be related to multiple simplifications of their analysis, spanning incorrect data calibration, misleading sample comparison, ignorance of the impact of detwinning.', '1710.08173-3-6-2': 'Additional experimental limitations have been overlooked in [CITATION].', '1710.08173-3-6-3': 'Indeed, not determining the spin-flip reference line properly, not doing a polarization analysis and inadequate control of the flipping ratio of the neutron beam add to uncertainties in their measurements [CITATION].', '1710.08173-3-7-0': '[MATH] Third, the comparison with local probes results in [CITATION] is outdated and partial as it dismisses the recent literature about muon spin resonance results [CITATION].', '1710.08173-3-7-1': 'Recent muon spin rotation measurements report magnetic correlations at T* with finite time-scales ([MATH] 10 ns), which are fluctuating slowly enough to appear static to neutrons.', '1710.08173-3-7-2': 'This observation is in disagrement with the conclusion of Croft et al[CITATION], but fully consistent with our observation of IUC order [CITATION].', '1710.08173-3-7-3': 'Finite time scale magnetic correlation can be associated with slowly fluctuating magnetic domains, such short range correlations have been actually reported in nearly optimally doped YBCO[CITATION] using PND.', '1710.08173-3-7-4': 'This would have the effect to reduce the magnetic signal by another factor of [MATH] 3 in the experiment of Croft et al [CITATION].', '1710.08173-3-8-0': 'The claimed upper bound for a possible magnetic moment is therefore not correct and should be disregarded.', '1710.08173-3-8-1': 'None of the data reported by Croft et al [CITATION] disprove that the IUC magnetic order is universal in all cuprates.', '1710.08173-3-9-0': '# Raw data comparison', '1710.08173-3-10-0': 'First, we compare the raw experimental data obtained by Fauque et al[CITATION] and Croft et al[CITATION].', '1710.08173-3-10-1': 'In [CITATION], we have studied an array of co-aligned high-quality twin-free single crystals.', '1710.08173-3-10-2': 'As the samples studied by Croft et al[CITATION], each single crystal was synthesized using the same self-flux method using a BaZrO[MATH] crucible as described in previous reports [CITATION] and on which charge density wave order has been observed as well[CITATION].', '1710.08173-3-11-0': 'As the IUC magnetic order does not break the symmetry of the lattice, one should study the inverse flipping ratio [MATH] at a Bragg position where the signal is expected to highlight a possible magnetic intensity at low temperature.', '1710.08173-3-11-1': 'The inverse flipping ratio [MATH] reads: [EQUATION] where [MATH] and [MATH] stand for the non-spin-flip and spin-flip intensities, respectively.', '1710.08173-3-11-2': 'As regularly emphasized[CITATION], [MATH] is essentially a ratio of measured quantities and does not depend on any assumptions on nuclear structure factor or flipping ratio.', '1710.08173-3-11-3': 'Changes in [MATH] for the same Bragg peak should be comparable among the different studies.', '1710.08173-3-11-4': 'The second part of Eq. [REF] shows how the magnetic intensity [MATH] can be extracted by comparison with the bare inverse flipping ratio [MATH] (background baseline) as it has been shown in several reports [CITATION].', '1710.08173-3-12-0': 'We report in Fig. [REF] the raw [MATH] at a Bragg positions (010) and (011) for two detwinned samples: sample H1 with p=0.104 of Croft et al[CITATION] and sample C of Fauque et al[CITATION] (see [CITATION] for samples description).', '1710.08173-3-12-1': 'Within error bars, there is no disagreement between both data for the (010) reflection (Fig [REF].', '1710.08173-3-12-2': 'a).', '1710.08173-3-12-3': 'However, Fig [REF].', '1710.08173-3-12-4': 'b shows a certain difference between both datasets for the (011) reflection whose possible origins can be understood as discussed below in the next section (see also [CITATION]).', '1710.08173-3-12-5': 'For both reflections, a more surprising discrepancy occurs between the actual data of Fauque et al[CITATION] and the alleged ones estimated by Croft et al[CITATION] (the dashed and dotted lines in their figures 8d and 8e).', '1710.08173-3-12-6': 'Obviously, both quantities should exactly match but they do not.', '1710.08173-3-12-7': 'That underlines the erroneous analysis performed by Croft et al[CITATION], and this for both Bragg peaks.', '1710.08173-3-12-8': 'We give in the supplemental material[CITATION] a discussion on possible origins of this discrepancy.', '1710.08173-3-13-0': 'Before showing alternative analyses of Croft et al[CITATION] data, two remarks are necessary:', '1710.08173-3-14-0': '[MATH] First, the ratio [MATH] in Eq. [REF] does not change appreciably versus doping for the Bragg peak (011) in the doping range of interest here (p=0.1-0.12).', '1710.08173-3-14-1': '[MATH] corresponds to the nuclear structure factor [MATH] where [MATH] is given for instance by Eq. (12) in [CITATION].', '1710.08173-3-14-2': 'A simple calculation shows that nuclear structure factor for the Bragg reflection (011) versus oxygen content are decreasing with increasing oxygen content similarly than the IUC magnetic intensity[CITATION].', '1710.08173-3-14-3': 'Therefore, it cannot be objected that the ratio [MATH] should decrease upon doping as does the magnetic intensity like it is considered in [CITATION].', '1710.08173-3-14-4': 'As a result, similar [MATH] is expected for both samples in Fig. [REF] within a 20% difference.', '1710.08173-3-15-0': '[MATH] Second, it should be noticed that the data shown for the Bragg peak (011) of sample C is the best example of a IUC magnetic signal ever reported in YBCO[CITATION].', '1710.08173-3-15-1': 'That corresponds to the highest experimentally reported ratio of the magnetic intensity compared to the nuclear intensity in Eq. [REF]: [MATH]=0.25 % at 70K for the (011) reflection[CITATION].', '1710.08173-3-15-2': 'All other reports in twinned samples are lower [CITATION]; this is due to a larger nuclear intensity of the (101) Bragg intensity which is averaged with the (011) peak in twinned samples.', '1710.08173-3-16-0': 'Based on comparison of raw data in Fig. [REF], we demonstrate that the analysis of Croft et al[CITATION] is highly questionable.', '1710.08173-3-16-1': 'As shown in Fig. [REF], Croft et al[CITATION] allegedly estimate [MATH]=0.75 % (see Fig. [REF]) at odds with our results.', '1710.08173-3-17-0': '# Alternative data analysis', '1710.08173-3-18-0': 'In [CITATION], we show various points which can explain the discrepancy of Fig. [REF] between the raw data of Fauque et al [CITATION] and the alledged estimate made by Croft et al [CITATION].', '1710.08173-3-18-1': 'For the Bragg peak (011), the main error in [CITATION] comes from improper data calibration.', '1710.08173-3-18-2': 'For the (010) Bragg position, the magnetic structure in twin-free sample should be properly acknowledged.', '1710.08173-3-18-3': 'As a result, the full and dashed lines in the figures 8,9 and 11, of Croft et al [CITATION] are all erroneous.', '1710.08173-3-18-4': 'Another global overestimated factor of 20% has been neglected.', '1710.08173-3-18-5': 'Finally, a close comparison [CITATION] of the data analysis of both measurements reveal numerous limitations with the PND experiments in [CITATION].', '1710.08173-3-19-0': 'Next, the dynamical nature of IUC magnetic signal as shown by recent muon spin resonance data [CITATION] have been ignored in [CITATION] (see [CITATION] for details).', '1710.08173-3-19-1': 'Such low frequency fluctuations are necessarily related to the formation of finite-size magnetic domains [CITATION], corresponding to finite correlation lengths of the magnetic order observed in neutron diffraction.', '1710.08173-3-19-2': 'Such a short range correlation length of the IUC order has been observed [CITATION] near optimally doped YBCO sample.', '1710.08173-3-19-3': 'In underdoped samples (for dopings p[MATH]0.1-0.11), one can simply give an upper limit of the correlation length along the c axis [MATH] AA[CITATION] as the observed magnetic peak is resolution limited due to large crystals mosaicity of [MATH] in [CITATION].', '1710.08173-3-19-4': 'In contrast, the measurements carried out by Croft et al [CITATION] are performed on tiny samples with mosaic spread at least 10 times smalller.', '1710.08173-3-19-5': 'As shown in the top-right panel of Fig. [REF], a magnetic signal with a correlation length [MATH] would exhibit a significantly broader rocking scan (A3) peak than for a long range magnetic order assumed in [CITATION].', '1710.08173-3-19-6': 'The magnetic amplitude at the Bragg position would be reduced accordingly [MATH]3 times compared to the one of true long range ordered state.', '1710.08173-3-19-7': 'Clearly, this factor 3 can be invoked to explain the difference in Fig. [REF] between the data of Fauque et al[CITATION] and those of Croft et al[CITATION] due to the different mosaicity of the samples.', '1710.08173-3-20-0': 'Further, we turn to a consistent comparison of the magnetic signal reported by Fauque et al[CITATION] to the experimental data of Croft et al [CITATION].', '1710.08173-3-20-1': 'In Fig. [REF], we zoom [MATH] data for the sample p=0.104 of Croft et al [CITATION].', '1710.08173-3-20-2': 'The same treatment can be found in [CITATION] for the other sample (p=0.123) of Croft et al with the same conclusion.', '1710.08173-3-21-0': 'The inverse of flipping ratio at the reflection (020) is linear in temperature with a positive slope as it has been shown to exist [CITATION].', '1710.08173-3-21-1': 'As discussed in [CITATION], this slope is inevitable as the sample drifts in the neutron beam upon changing temperature.', '1710.08173-3-21-2': 'Croft et al [CITATION] arbitrarily describe it with a flat horizontal line only.', '1710.08173-3-21-3': 'At the accuracy required to observe the IUC magnetic signal, this is not a correct approximation.', '1710.08173-3-21-4': 'With the large grey shaded area, we next represent the zone of the limit of detection, corresponding to [MATH] [CITATION], on both sides of an average sloping line.', '1710.08173-3-21-5': 'This area is due to combined effects of the statistical errors of each points, occurence of off-statistical points (possibly related to mechanical errors in positioning) and the scarce number of points (especially above T*).', '1710.08173-3-21-6': 'This area is simply deduced from the measurements at the Bragg (020) reflection where no magnetic signal is expected for the IUC order.', '1710.08173-3-22-0': 'For clarity, the same error of [MATH] 0.1% is used for all plots.', '1710.08173-3-22-1': 'That error is typically equivalent to an error of [MATH] on the flipping ratio or [MATH] of the spin-flip intensity.', '1710.08173-3-22-2': 'This uncertainty is not negligible as our best report of a magnetic signal is [MATH] 0.25 (Fig. [REF]).', '1710.08173-3-22-3': 'The error on the spin-flip intensity in previous PND experiments [CITATION] leads to an estimate of [MATH] 0.01-0.02 (see e.g. Fig. S1 in the Supplemental materials of Mangin-Thro et al [CITATION] for an example of measured error bars).', '1710.08173-3-23-0': 'In Fig. [REF], the detection limit of [MATH] is next reported for the Bragg reflections (010) and (011) where the magnetic signal is expected.', '1710.08173-3-23-1': 'To compare that set of data, with those of Fauque et al[CITATION], we then consider three different scenarios which are plotted in Figs. [REF]: A) no magnetic signal is present, B) a long range magnetic order is present C) a magnetic order with short range correlation along the c-direction is present (with the same integrated intensity of scenario (B)).', '1710.08173-3-23-2': 'The amplitude in scenario (B) corresponds to the amplitude expected for a detwinned sample with the appropriate doping level[CITATION].', '1710.08173-3-23-3': 'That basically corresponds to our best experimental evidence of a magnetic signal [MATH] 0.25 as plotted in Fig. [REF].', '1710.08173-3-23-4': 'The top right panel of Fig. [REF] simulates the A3 rocking scan for the scenario B in blue and the scenario C in red where both hypothetical curves have been convoluted by the resolution function given by the Fig. 6.a', '1710.08173-3-24-0': 'Following the various points discussed above and in line with the Fig. [REF], one clearly sees that the expected signal from [CITATION] is only marginally larger than the experimental uncertainty (error area).', '1710.08173-3-24-1': 'For all the three scenarios presented above, only one parameter, the overall level of the background of the baseline, is fitted.', '1710.08173-3-24-2': 'All the other parameters are given from the literature [CITATION] and the discussion above.', '1710.08173-3-24-3': 'Clearly, for all cases, scenarios A (of Croft et al [CITATION]) and C (a short range along [MATH] IUC magnetic order compatible with the report of ref. [CITATION]) cannot be distinguished at all.', '1710.08173-3-24-4': 'Even for a true long range magnetic order along [MATH] (scenario B), the data are insufficient to eliminate with confidence the existence of IUC order due to indetermination of the baseline.', '1710.08173-3-24-5': 'Clearly, the sensitivity of the experiment is insufficient to observe the IUC order contrary to the claim of Croft et al [CITATION].', '1710.08173-3-25-0': '# Conclusion', '1710.08173-3-26-0': 'In conclusion, Croft et al [CITATION] do not have the experimental accuracy to observe the IUC signal that we have been reporting for the last decade in YBCO [CITATION].', '1710.08173-3-26-1': 'Different factors applied: first, the accuracy limit of the experiment of Croft et al [CITATION] is represented by the shaded areas which indicate the uncertainty of [MATH] 0.1% on the thermal dependence of the baseline for the ratio [MATH] in Eq. [REF].', '1710.08173-3-26-2': 'Second, the purported level of intensity of Fauque et al has been erroneously and systematically overestimated by a factor [MATH] 3 (see Fig. [REF]).', '1710.08173-3-26-3': 'That corresponds to the scenario B of figs. [REF], where a long range IUC order is assumed.', '1710.08173-3-26-4': 'Third, an additional factor 3 occurs if the signal is short ranged along L with [MATH] (scenario C of Figs. [REF]).', '1710.08173-3-26-5': 'The most plausible scenario for the magnetic intensity lies in between scenarios B and C in Fig. [REF].', '1710.08173-3-26-6': 'This is clearly below the detection limit of the data of Croft et al [CITATION].', '1710.08173-3-26-7': 'The claimed upper bound for a possible magnetic moment in [CITATION] is therefore baseless and should be disregarded.', '1710.08173-3-27-0': 'The IUC magnetic signal has been well documented for the last decade using polarized neutron diffraction[CITATION].', '1710.08173-3-27-1': 'The magnetic signal, observed in four cuprate familiees, is seen only at specific Bragg positions.', '1710.08173-3-27-2': 'Data in YBCO and in [MATH] are nearly indistinguishable with a systematic doping dependence.', '1710.08173-3-27-3': 'In all cuprates, important polarization analysis has been conducted and the polarization sum rule, which demonstrates the magnetic nature of the signal, always nicely obeyed.', '1710.08173-3-28-0': 'The experiment is highly non-trivial with many technical pitfalls to miss the genuine signal.', '1710.08173-3-28-1': 'Among other features, it requires sufficient data at high temperature for a proper knowledge of the background.', '1710.08173-3-28-2': 'The IUC magnetic signal falls below the experimental sensitivity of the experiment of Croft et al [CITATION] predominantly because too tiny YBCO samples were studied (at least 100 times smaller than in previous reports).', '1710.08173-3-28-3': 'Further, a large number of flaws and inaccuracies in their data analysis invalidates their comparison with the previous data[CITATION].', '1710.08173-3-28-4': 'Their estimate of the magnetic signal previously reported could be incorrect up to an order of magnitude.', '1710.08173-3-28-5': 'None of the data shown invalidate that the IUC magnetic order is an intrinsic property of the pseudogap state of cuprates.', '1710.08173-3-29-0': 'acknowlegments', '1710.08173-3-30-0': 'We wish to thank Gabriel Aeppli, Victor Baledent, Dalila Bounoua, Johan Chang, Niels-Bech Christensen, Benoit Fauque, Martin Greven, Jaehong Jeong, Steve Kivelson, Yuan Li, Tasutomo Uemura and Chandra Varma for stimulating discussions on different aspects related to this work.', '1710.08173-3-30-1': 'We acknowledge financial supports from the project NirvAna (contract ANR-14-OHRI-0010) of the ANR French agency.'} | null | null | null | null |
math-0111036 | {'math-0111036-1-0-0': 'We continue to study a model of disordered interface growth in two dimensions.', 'math-0111036-1-0-1': 'The interface is given by a height function on the sites of the one-dimensional integer lattice and grows in discrete time: (1) the height above the site [MATH] adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above [MATH] increases by 1 with probability [MATH].', 'math-0111036-1-0-2': 'We assume that [MATH] are chosen independently at random with a common distribution [MATH], and that the initial state is such that the origin is far above the other sites.', 'math-0111036-1-0-3': 'Provided that the tails of the distribution [MATH] at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of [MATH].', 'math-0111036-1-0-4': 'In the quenched case, the said fluctuations are asymptotically normal, while the annealed case they satisfy the appropriate extremal limit law.', 'math-0111036-1-1-0': '2000 Mathematics Subject Classification.', 'math-0111036-1-1-1': 'Primary 60K35.', 'math-0111036-1-1-2': 'Secondary 05A16, 33E17, 60K37, 60G70, 82C44.', 'math-0111036-1-2-0': 'Keywords: growth model, fluctuations, Fredholm determinant, phase transition, saddle point analysis, extremal order statistics.'} | {'math-0111036-2-0-0': 'We continue to study a model of disordered interface growth in two dimensions.', 'math-0111036-2-0-1': 'The interface is given by a height function on the sites of the one-dimensional integer lattice and grows in discrete time: (1) the height above the site [MATH] adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above [MATH] increases by 1 with probability [MATH].', 'math-0111036-2-0-2': 'We assume that [MATH] are chosen independently at random with a common distribution [MATH], and that the initial state is such that the origin is far above the other sites.', 'math-0111036-2-0-3': 'Provided that the tails of the distribution [MATH] at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of [MATH].', 'math-0111036-2-0-4': 'In the quenched case, the said fluctuations are asymptotically normal, while in the annealed case they satisfy the appropriate extremal limit law.', 'math-0111036-2-1-0': '2000 Mathematics Subject Classification.', 'math-0111036-2-1-1': 'Primary 60K35.', 'math-0111036-2-1-2': 'Secondary 05A16, 33E17, 60K37, 60G70, 82C44.', 'math-0111036-2-2-0': 'Keywords: growth model, fluctuations, Fredholm determinant, phase transition, saddle point analysis, extremal order statistics.'} | [['math-0111036-1-2-0', 'math-0111036-2-2-0'], ['math-0111036-1-0-0', 'math-0111036-2-0-0'], ['math-0111036-1-0-1', 'math-0111036-2-0-1'], ['math-0111036-1-0-2', 'math-0111036-2-0-2'], ['math-0111036-1-0-3', 'math-0111036-2-0-3'], ['math-0111036-1-0-4', 'math-0111036-2-0-4']] | [['math-0111036-1-2-0', 'math-0111036-2-2-0'], ['math-0111036-1-0-0', 'math-0111036-2-0-0'], ['math-0111036-1-0-1', 'math-0111036-2-0-1'], ['math-0111036-1-0-2', 'math-0111036-2-0-2'], ['math-0111036-1-0-3', 'math-0111036-2-0-3']] | [['math-0111036-1-0-4', 'math-0111036-2-0-4']] | [] | [] | [] | ['math-0111036-1-1-0', 'math-0111036-1-1-1', 'math-0111036-1-1-2', 'math-0111036-2-1-0', 'math-0111036-2-1-1', 'math-0111036-2-1-2'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/math/0111036 | null | null | null | null | null |
0910.5150 | {'0910.5150-1-0-0': 'This technical note shows how Electrodynamics and a Yukawa model are dressed after integrating out perturbative brane fluctuations, and it is found that first order corrections in the inverse of the brane tension occur for the fermion and scalar wave functions, the couplings and the masses.', '0910.5150-1-0-1': 'Nevertheless, field redefinitions actually lead to effective actions where only masses are dressed to this first order, in the regime of low-energy brane fluctuations, and we compare our results with the literature.', '0910.5150-1-1-0': '# Introduction', '0910.5150-1-2-0': 'In the framework of higher-dimensional theories, one is naturally led to describe the induced metric on a four-dimensional brane, which can give indications on the bulk structure.', '0910.5150-1-2-1': 'In this context, an elegant effective description of a four-dimensional brane embedded in a five-dimensional Universe is provided by a scalar field living on the brane, representing fluctuations in the position of the brane, as measured in the fifth dimension [CITATION].', '0910.5150-1-2-2': 'For small fluctuations of the brane about its equilibrium position, quantum fluctuations of the brane can then be interpreted as particles, the branons, which couple to the Standard Model (SM), with a dimensionful coupling constant, proportional to the inverse of the brane tension [MATH].', '0910.5150-1-3-0': 'We study here the coupling of branons to Electrodynamics and a Yukawa model, and we will see that, although the interaction of branons with SM particles is suppressed by [MATH], the derivative interactions between branons and the SM can compensate this suppression.', '0910.5150-1-3-1': 'We will consider a flat extra dimension, such that branons are massless.', '0910.5150-1-3-2': 'Also, because of the low branon frequency approximation, the couplings of branons to SM particles are quadratic in branons, and thus their one-loop integration is exact.', '0910.5150-1-4-0': 'Similar studies were done in [CITATION], where the authors study the coupling of branons to the SM, and integrate the former to obtain the corresponding effective action for SM particles.', '0910.5150-1-4-1': 'We compare this to our approach, where the regime of low-energy branons proves to be essential for no discrepancies to appear between our results.', '0910.5150-1-5-0': 'Section 2 describes the construction of the low energy theory describing the interaction branons/Electrodynamics.', '0910.5150-1-5-1': 'We derive then, in section 3, the effective action for Electrodynamics, obtained after exact integration of branons, and we give technical comments on this derivation.', '0910.5150-1-5-2': 'Finally, we briefly show in section 4 how a similar argument holds for a Yukawa model.', '0910.5150-1-6-0': '# Branon-Electrodynamics interactions', '0910.5150-1-7-0': 'We consider a 5-dimensional flat Universe with generic coordinates [MATH], where [MATH] are the coordinates on the brane, which is defined by the equation [MATH].', '0910.5150-1-7-1': 'The brane coordinates are denoted with the indices [MATH], and the induced metric [MATH] on the brane is [EQUATION]', '0910.5150-1-7-2': 'If [MATH] is the brane tension, the brane action is then [EQUATION] where dots represent higher orders in derivatives of [MATH].', '0910.5150-1-7-3': 'Our dynamical variable is the canonically normalized branon field [MATH], with mass dimension 1, and the brane ground state is [MATH].', '0910.5150-1-7-4': 'Ignoring field-independent terms, since we are looking at an effective theory in flat space time, and taking into account the low-energy branon approximation, the resulting effective action for branons is then [EQUATION] and describes a free theory.', '0910.5150-1-7-5': 'Interactions will occur, though, with particles propagating in the brane, and we now derive the corresponding action for Electrodynamics.', '0910.5150-1-8-0': 'In what follows, Latin indices refer to the local inertial frame and are contracted with [MATH], whereas Greek indices are contracted with the induced metric [MATH].', '0910.5150-1-8-1': 'The action describing Electrodynamics in curved background is [CITATION] [EQUATION] where the spin connection is [EQUATION] and the gamma matrices [MATH] are defined in the local inertial frame, therefore satisfying [EQUATION]', '0910.5150-1-8-2': 'Neglecting higher orders in the branon derivatives [MATH], we find the following approximate vierbeins on the brane [EQUATION] which lead to the expected definition [MATH], up to higher orders in [MATH].', '0910.5150-1-8-3': 'The Christoffel symbols for the induced metric are then [EQUATION] and the spin connection is [EQUATION] such that the total action for the system branons-Electrodynamics is, after integrations by parts, [EQUATION] where the indices are raised and lowered with the Minkowski metric.', '0910.5150-1-8-4': 'We note that, although the coupling between branons and Electrodynamics is proportional to [MATH], it is actually not negligible, because of derivative interactions.', '0910.5150-1-8-5': 'Finally, the Greek indices appearing in the effective action ([REF]) all denote flat four-dimensional space time coordinates.', '0910.5150-1-9-0': '# Effective action for Electrodynamics', '0910.5150-1-10-0': '## Derivation', '0910.5150-1-11-0': 'We assume from now on that branons have energies up to some value [MATH], where [MATH] for the low-energy branon approximation ([REF]) to be valid.', '0910.5150-1-11-1': 'We integrate out their degrees of freedom from the theory described by the action ([REF]), which can be done exactly, since this action is quadratic in the branon field.', '0910.5150-1-11-2': 'The resulting effective action for Electrodynamics is then [EQUATION] where the trace is taken over the branon momentum [MATH].', '0910.5150-1-11-3': 'This regularization does not contradict gauge invariance, since the momenta of photons and fermions are not restricted, but only the branon momentum is, as can be seen from the Fourier transform ([REF]), when the trace is taken ([MATH]).', '0910.5150-1-11-4': 'The second functional derivative of [MATH] is [EQUATION] where [MATH], and has the following Fourier transform [EQUATION] where [EQUATION]', '0910.5150-1-11-5': 'We expand then the logarithm in the effective action ([REF]) around the diagonal part (proportional to [MATH]), and keep the first order in the inverse brane tension, to obtain [EQUATION] where field-independent terms were ignored.', '0910.5150-1-11-6': 'Note that the contributions for the correction to [MATH] cancel each other, and the terms arising from the spin connection (last line in eq. ([REF])) also cancel in the trace.', '0910.5150-1-11-7': 'We will discuss these two points in the next subsection.', '0910.5150-1-11-8': 'We can also note that, although branons are massless, no IR divergences appear in the branon loop integrals, because of the derivative interactions compensating the possible divergences at [MATH].', '0910.5150-1-11-9': 'The effective action for Electrodynamics is finally [EQUATION] and we find a correction to the fermion wave function, the coupling and the mass.', '0910.5150-1-11-10': 'Also, as expected, the corrections to the fermion wave function and to the coupling are the same, which is a consequence of gauge invariance.', '0910.5150-1-12-0': '## Comments', '0910.5150-1-13-0': 'We now explain how the effective action ([REF]) could be obtained differently, as was done in [CITATION].', '0910.5150-1-14-0': 'In this work, the Authors start from the action [EQUATION] where [MATH] is the flat space time Standard Model Lagrangian, and [MATH] is its energy-momentum tensor.', '0910.5150-1-14-1': 'After integration over branons, they find that the first order correction to the Standard Model action is proportional to [EQUATION] which would vanish if the Standard Model was massless.', '0910.5150-1-15-0': 'To put our results in a similar context, we bear in mind that not only the metric, but also its derivatives depend on the branon field [EQUATION] such that [MATH] and [MATH] cannot be considered independent variables.', '0910.5150-1-15-1': 'As a consequence, expanding the following action up to the first order in [MATH] [EQUATION] we obtain (neglecting constant terms), [EQUATION] where "flat" denotes the corresponding quantity for vanishing branon field, and [EQUATION]', '0910.5150-1-15-2': 'Our results can be understood as follows:', '0910.5150-1-16-0': '## Phenomenological implication', '0910.5150-1-17-0': 'The effective action ([REF]) implies a change in the parameters describing Electrodynamics.', '0910.5150-1-17-1': 'To see what the change is, we consider the following rescaled fermion field [EQUATION] which is our new dynamical variable and has a canonical kinetic term.', '0910.5150-1-17-2': 'The resulting effective action for Electrodynamics becomes [EQUATION] where the effective mass is, [EQUATION]', '0910.5150-1-17-3': 'As a consequence, after field redefinition, we find that only the fermion mass gets a correction from branon dressing, with a value agreeing with [CITATION], since [MATH], and which leads to hardly no measurable effect.', '0910.5150-1-18-0': '# Coupling to a Yukawa model', '0910.5150-1-19-0': 'We briefly present here similar arguments for branons coupled to a Yukawa model.', '0910.5150-1-20-0': 'We start with the action [EQUATION] where the induced metric [MATH] is given by eq.([REF]).', '0910.5150-1-20-1': 'We know from the experience with Electrodynamics that the spin connection does not play any role, and that the integration of branons to the order [MATH] involves the trace of the corresponding energy momentum tensor [EQUATION] such that the effective Lagrangian is, after an integration by parts in the effective flat-space-time action, [EQUATION] where [MATH] is given is eq.([REF]).', '0910.5150-1-20-2': 'In terms of the rescaled fields [EQUATION] we obtain [EQUATION] where the effective masses are [EQUATION] and the effective coupling is [EQUATION]', '0910.5150-1-20-3': 'We are therefore left with a similar conclusion: although the different operators of the model are dressed at the order [MATH], field redefinitions show that the Yukawa coupling is eventually dressed at the order [MATH] only.', '0910.5150-1-20-4': 'Note that, as for the Electrodynamics case, our result agrees with the conclusions of [CITATION] in the correct regime of validity of the low-energy branon approximation [MATH].', '0910.5150-1-21-0': '# Conclusion', '0910.5150-1-22-0': 'The concept of branon gives an elegant effective description of low energy brane fluctuations, but the feedback on the coupling constants of the Standard Model is of order [MATH] only, as explained in [CITATION], making this effective description hard to test experimentally, especially if the brane tension is expected to be larger than (100 GeV)[MATH] [CITATION].', '0910.5150-1-22-1': 'Also, it is not really clear how to interpret the cut off [MATH] in branon energy, making even more difficult the experimental tests.', '0910.5150-1-22-2': 'One possibility is to assume that [MATH] corresponds to a centre of mass energy in a collision, in which case the branon dressing can be compared to quantum corrections.', '0910.5150-1-22-3': 'But the corresponding dressing of coupling constants, of order [MATH], can be in the error bars of the experimental measurements if the brane tension is large.', '0910.5150-1-22-4': 'Bounds have been discussed in [CITATION] though.', '0910.5150-1-22-5': 'Another possibility is to assume a branon bath of typical temperature [MATH], in a finite temperature collision, but it is not clear how efficiently branons thermalize to Standard Model particles.', '0910.5150-1-22-6': 'One can also assume that branons are produced by an astrophysical catastrophic event, but in this case, the parameter [MATH] is then certainly so small, that no effect can be detected.', '0910.5150-1-23-0': 'Finally, we would like to make a remark, concerning the initial approximation where we take into account the first order only in [MATH], in order to get the action describing the interaction branon-Standard Model.', '0910.5150-1-23-1': 'This approximation is based on the occurrence of low energy branons, and contains two derivatives of the branon field.', '0910.5150-1-23-2': 'But the resulting coupling [MATH] contains four derivatives, such that, should we follow a strict gradient expansion scheme, we would have to take into account also terms of order [MATH] in the expansion of the induced metric [MATH].', '0910.5150-1-23-3': 'This was not done, and would spoil the exact integration over branon degrees of freedom.', '0910.5150-1-23-4': 'These additional terms, though, might be relevant in the limit of the parameter [MATH] goes to 1, where the low-energy branon approximation is not valid anymore.', '0910.5150-1-23-5': 'These ambiguities show that a proper study of brane fluctuations feedback on the Standard Model might need to be non-perturbative.', '0910.5150-1-23-6': 'The concept of branon degree of freedom would then be difficult to define, but phenomenological effects might be more realistically predictable.'} | {'0910.5150-2-0-0': 'This technical note shows how Electrodynamics and a Yukawa model are dressed after integrating out perturbative brane fluctuations, and it is found that first order corrections in the inverse of the brane tension occur for the fermion and scalar wave functions, the couplings and the masses.', '0910.5150-2-0-1': 'Nevertheless, field redefinitions actually lead to effective actions where only masses are dressed to this first order.', '0910.5150-2-0-2': 'We compare our results with the literature and find discrepancies at the next order, which, however, might not be measurable in the valid regime of low-energy brane fluctuations.', '0910.5150-2-1-0': '# Introduction', '0910.5150-2-2-0': 'In the context of braneworld scenarios and higher dimensional theories, branons are modes which correspond to perturbative quantum fluctuations of the brane, about its equilibrium position in extra dimensions.', '0910.5150-2-2-1': 'Using an effective description of a four-dimensional brane embedded in a five-dimensional universe, branons can be thought of as a scalar field living on the brane, representing fluctuations in the position of the brane, as measured in the fifth dimension [CITATION].', '0910.5150-2-2-2': 'Branons can then be thought of as particles which couple to the Standard Model (SM), with a dimensionful coupling constant, proportional to the inverse of the brane tension [MATH].', '0910.5150-2-3-0': 'Branons are interesting to study in the context of Cosmology and Collider Physics, since there could be phenomenological effects arising from branons, that may be detectable in colliders as the LHC .', '0910.5150-2-3-1': 'In this paper we study the coupling of branons to Electrodynamics and a Yukawa model, and the corresponding possible phenomenological effects.', '0910.5150-2-3-2': 'We will see that, although the interaction of branons with SM particles is suppressed by [MATH], the derivative interactions between branons and the SM can compensate this suppression.', '0910.5150-2-3-3': 'The idea is then to integrate out branon degrees of freedom, in order to obtain the effective theory for the SM degrees of freedom.', '0910.5150-2-3-4': 'In the low energy approximation for branons, the coupling to the SM is quadratic in branons, such that the integration of the latter can be done exactly.', '0910.5150-2-3-5': 'As a consequence, the one-loop result is exact, and will be expanded in powers of the dimensionful coupling constant [MATH].', '0910.5150-2-3-6': 'The resulting effective theory contains corrections to SM interactions, but also new interactions, such as four-fermion interactions, all of dimensionality larger than 4.', '0910.5150-2-3-7': 'We will concentrate here on the corrections to the SM, since the other interactions are suppressed by higher orders of the inverse brane tension, [MATH].', '0910.5150-2-4-0': 'It has been shown in previous studies [CITATION] that warped extra dimensions lead to massive branons, but we will consider a flat extra dimension, leading to massless branons: the dressing effects that we study here are mainly due to UV dynamics, where the branon mass does not play an important role.', '0910.5150-2-5-0': 'Similar studies were done in [CITATION], where the authors look at the coupling of branons to the SM, and integrate the former to obtain the corresponding effective action for SM particles.', '0910.5150-2-5-1': 'We compare this to our approach, and find the same conclusion to the order [MATH], but a discrepancy to the next order [MATH], for the Yukawa model.', '0910.5150-2-5-2': 'The reason of the discrepancy is that the authors of [CITATION] assume equations of motion to hold, without taking into account the dressing from branons.', '0910.5150-2-5-3': 'We do not assume that degrees of freedom of the SM satisfy equations of motion, and find that the Yukawa coupling gets a correction.', '0910.5150-2-5-4': 'Nevertheless, we argue that this [MATH] correction is certainly negligible in the low-energy approximation, where the branon model is valid.', '0910.5150-2-6-0': 'The paper is structured as follows.', '0910.5150-2-6-1': 'Section 2 describes the construction of the low energy theory describing the interaction branons/Electrodynamics.', '0910.5150-2-6-2': 'Starting from the general expression for the Electrodynamics action in curved space time, we expand the metric in terms of the branon degree of freedom, which leads us to the interactions branon/Electrodynamics.', '0910.5150-2-6-3': 'We derive then, in section 3, the effective action for Electrodynamics, obtained after exact integration of branons.', '0910.5150-2-6-4': 'We give technical comments on this derivation, by comparing our work to the one given in [CITATION].', '0910.5150-2-6-5': 'The agreement between our approaches, in this specific case, is a consequence of gauge symmetry.', '0910.5150-2-6-6': 'Indeed, the gauge coupling and the fermion wave function renormalization need to get the same corrections for gauge symmetry to hold, such that, after a redefinition of the fields, the correction to the gauge coupling exactly vanishes, and only the fermion mass gets a correction.', '0910.5150-2-6-7': 'Another important fact is that the Maxwell free theory is conformally invariant, independently of equations of motion, such that the gauge field does not get any correction.', '0910.5150-2-6-8': 'Section 4 shows how a Yukawa model is dressed by branons.', '0910.5150-2-6-9': 'In this case, because no gauge invariance is required, we find that the Yukawa coupling does get a correction, leading to the discrepancy with [CITATION].', '0910.5150-2-6-10': 'Our conclusion questions the concept of branon, which, in order to be well defined, needs a low energy approximation, where the branon dressing might not be measurable in experiments.', '0910.5150-2-7-0': '# Branon/Electrodynamics interactions', '0910.5150-2-8-0': 'We consider a 5-dimensional flat Universe with generic coordinates [MATH], where [MATH] are the coordinates on the brane, which is defined by the equation [MATH].', '0910.5150-2-8-1': 'The brane coordinates are denoted with the indices [MATH], and the induced metric [MATH] on the brane is [EQUATION]', '0910.5150-2-8-2': 'If [MATH] is the brane tension, the brane action is then [EQUATION] where dots represent higher orders in derivatives of [MATH].', '0910.5150-2-8-3': 'Our dynamical variable is the canonically normalized branon field [MATH], with mass dimension 1, and the brane ground state is [MATH].', '0910.5150-2-8-4': 'Ignoring field-independent terms, since we are looking at an effective theory in flat space time, and taking into account the low-energy branon approximation, the resulting effective action for branons is then [EQUATION] and describes a free theory.', '0910.5150-2-8-5': 'Interactions will occur, though, with particles propagating in the brane over Cosmological distances, and we now derive the corresponding action for Electrodynamics.', '0910.5150-2-9-0': 'In what follows, Latin indices refer to the local inertial frame and are contracted with [MATH], whereas Greek indices are contracted with the induced metric [MATH].', '0910.5150-2-9-1': 'The action describing Electrodynamics in curved background is [CITATION] [EQUATION] where the spin connection is [EQUATION] and the gamma matrices [MATH] are defined in the local inertial frame, therefore satisfying [EQUATION]', '0910.5150-2-9-2': 'Neglecting higher orders in the branon derivatives [MATH], we find the following approximate vierbeins on the brane [EQUATION] which lead to the expected definition [MATH], up to higher orders in [MATH].', '0910.5150-2-9-3': 'The Christoffel symbols for the induced metric are then [EQUATION] and the spin connection is [EQUATION] such that the total action for the system branons/Electrodynamics is, after integrations by parts, [EQUATION] where the indices are raised and lowered with the Minkowski metric.', '0910.5150-2-9-4': 'We note that, although the coupling between branons and Electrodynamics is proportional to [MATH], it is actually not negligible, because of derivative interactions.', '0910.5150-2-9-5': 'Finally, the Greek indices appearing in the effective action ([REF]) all denote flat four-dimensional space time coordinates.', '0910.5150-2-10-0': '# Effective action for Electrodynamics', '0910.5150-2-11-0': '## Derivation', '0910.5150-2-12-0': 'We assume from now on that branons have energies up to some value [MATH], where [MATH] for the low-energy branon approximation ([REF]) to be valid.', '0910.5150-2-12-1': 'We integrate out their degrees of freedom from the theory described by the action ([REF]), which can be done exactly, since this action is quadratic in the branon field.', '0910.5150-2-12-2': 'The resulting effective action for Electrodynamics is then [EQUATION] where the trace is taken over the branon momentum [MATH].', '0910.5150-2-12-3': 'This regularization does not contradict gauge invariance, since the momenta of photons and fermions are not restricted, but only the branon momentum is, as can be seen from the Fourier transform ([REF]), when the trace is taken ([MATH]).', '0910.5150-2-12-4': 'The second functional derivative of [MATH] is [EQUATION] where [MATH], and has the following Fourier transform [EQUATION] where [EQUATION]', '0910.5150-2-12-5': 'We expand then the logarithm in the effective action ([REF]) around the diagonal part (proportional to [MATH]), and keep the first order in the inverse brane tension, to obtain [EQUATION] where field-independent terms were ignored.', '0910.5150-2-12-6': 'Note that the contributions for the correction to [MATH] cancel each other after taking the trace, because [EQUATION] and the terms arising from the spin connection (last line in eq. ([REF])) also cancel in the trace.', '0910.5150-2-12-7': 'We will discuss these two points in the next subsection.', '0910.5150-2-12-8': 'We can also note that, although branons are massless, no IR divergences appear in the branon loop integrals, because of the derivative interactions compensating the possible divergences at [MATH].', '0910.5150-2-12-9': 'The effective action for Electrodynamics is finally [EQUATION] and we find a correction to the fermion wave function, the coupling and the mass.', '0910.5150-2-12-10': 'Also, as expected, the corrections to the fermion wave function and to the coupling are the same, which is a consequence of gauge invariance.', '0910.5150-2-13-0': '## Comments', '0910.5150-2-14-0': 'We now explain how the effective action ([REF]) could be obtained differently, as was done in [CITATION].', '0910.5150-2-15-0': 'In this work, the Authors start from the action [EQUATION] where [MATH] is the flat space time Standard Model Lagrangian, and [MATH] is its energy-momentum tensor.', '0910.5150-2-15-1': 'After integration over branons, they find that the first order correction to the Standard Model action is proportional to [EQUATION] which would vanish if the Standard Model was scale invariant and if one assumes the equations of motion to be satisfied.', '0910.5150-2-16-0': 'To put our results in a similar context, we bear in mind that not only the metric, but also its derivatives depend on the branon field [EQUATION] such that [MATH] and [MATH] cannot be considered independent variables.', '0910.5150-2-16-1': 'As a consequence, expanding the following action up to the first order in [MATH] [EQUATION] we obtain (neglecting constant terms), [EQUATION] where "flat" denotes the corresponding quantity for vanishing branon field, and [EQUATION]', '0910.5150-2-16-2': 'Our results can be understood as follows:', '0910.5150-2-17-0': '## Phenomenological implication', '0910.5150-2-18-0': 'The effective action ([REF]) implies a change in the parameters describing Electrodynamics.', '0910.5150-2-18-1': 'To see what the change is, we consider the following rescaled fermion field [EQUATION] which is our new dynamical variable and has a canonical kinetic term.', '0910.5150-2-18-2': 'The resulting effective action for Electrodynamics becomes [EQUATION] where the effective mass is, [EQUATION]', '0910.5150-2-18-3': 'As a consequence, after field redefinition, we find that only the fermion mass gets a correction from branon dressing, with a value agreeing with [CITATION], since [MATH], and which leads to hardly no measurable effect.', '0910.5150-2-19-0': 'It is worth remembering that in experiments we measure only the dressed parameters.', '0910.5150-2-19-1': 'Therefore, in order to distinguish the branon-dressed parameters from the bare ones, we would need to compare an experiment with branons with a similar experiment without branons, which is difficult to set up.', '0910.5150-2-20-0': '# Coupling to a Yukawa model', '0910.5150-2-21-0': 'In the previous example of Electrodynamics, we found no dressing for the coupling after field redefinition, since corrections to the coupling, before field redefinition, have to be the same as corrections to the fermion kinetic term, because of gauge invariance.', '0910.5150-2-21-1': 'We consider here a Yukawa model, not "protected" by a gauge symmetry, and we will see that the coupling does get a correction, to the order [MATH] though.', '0910.5150-2-22-0': '## Corrections to the bare Lagrangian', '0910.5150-2-23-0': 'We briefly present here similar arguments for branons coupled to a Yukawa model.', '0910.5150-2-24-0': 'We start with the action [EQUATION] where the induced metric [MATH] is given by eq. ([REF]).', '0910.5150-2-24-1': 'We use here directly the approach given in [CITATION], and write the action ([REF]) in the form (neglecting higher orders in branon derivatives) [EQUATION] where [MATH] is the Yukawa Lagrangian in flat space time, and [MATH] the corresponding energy momentum tensor.', '0910.5150-2-24-2': 'The second functional derivative of [MATH] with respect to the branon field has Fourier transform [EQUATION] where [MATH] is the Fourier transform of [MATH].', '0910.5150-2-24-3': 'The integration over branons, up to the energy [MATH], leads to the following effective action (ignoring field independent terms) [EQUATION]', '0910.5150-2-24-4': 'When expanding the logarithm, we obtain (again, ignoring field independent terms) [EQUATION] and the trace over Lorentz indices of [MATH] is [EQUATION]', '0910.5150-2-24-5': 'In the situation where [MATH] (scale invariant theory), the trace ([REF]) can also be written [EQUATION] and is therefore the sum of a divergence plus terms which vanish if one assumes the equations of motions to hold, before dressing.', '0910.5150-2-24-6': 'But because of branon dressing, these equations of motion hold only up to terms of order [MATH], such that, because of the overall additional factor [MATH] in eq.([REF]), we expect a difference of order [MATH] with [CITATION].', '0910.5150-2-24-7': 'Therefore we do not assume any cancellation of the trace ([REF]), and, from eq.([REF]) and by analogy with the calculations performed for the Electrodynamics case, we are led to the following effective Lagrangian [EQUATION] where [MATH] is given is eq.([REF]).', '0910.5150-2-25-0': '## Phenomenological implications', '0910.5150-2-26-0': 'In terms of the rescaled fields [EQUATION] we obtain from the effective Lagrangian ([REF]) [EQUATION] where the effective masses are [EQUATION] and the effective coupling is [EQUATION]', '0910.5150-2-26-1': 'Hence, although the correction ([REF]) occurs at the order [MATH] only, it does not vanish exactly, as one could conclude from [CITATION].', '0910.5150-2-26-2': 'Note that this [MATH] correction has nothing to do with higher orders in the expansion of the logarithm in eq.([REF]), since these higher orders lead to four-fermion interactions and other operators with dimension larger than 4.', '0910.5150-2-26-3': 'The Yukawa coupling gets corrections from the first order only, in the expansion of the logarithm appearing in eq.([REF]).', '0910.5150-2-27-0': "Nevertheless, if one considers a low-energy branon regime, consistent with the initial approximation ([REF]), one should assume that [MATH], such that the correction ([REF]) satisfies [EQUATION] and the discrepancy with [CITATION] actually hasn't got measurable consequences, since the Yukawa coupling of the SM is measured via fermion masses: the correction ([REF]) is smaller than the error bars on the Cabibbo-Kobayashi-Maskawa matrix.", '0910.5150-2-28-0': 'Again, as with the electrodynamics case it is worth remembering that in experiments we measure only the dressed parameters and the comments at the end of section [REF] apply here as well.', '0910.5150-2-29-0': '# Conclusion', '0910.5150-2-30-0': 'We discussed in this paper, in the framework of brane models, how two sectors of the Standard Model are dressed by brane fluctuations.', '0910.5150-2-30-1': 'This was done by integrating out a massless scalar degree of freedom, associated to perturbative brane fluctuations, the branon.', '0910.5150-2-30-2': 'In order to check results previously published, we performed this integration explicitly, in the case of Electrodynamics, without assuming any symmetry of the energy-momentum tensor, and find the expected result.', '0910.5150-2-30-3': 'But in the situation of the Yukawa interaction, we find that the coupling does get a correction, which differs from previous studies, but with a difference hardly measurable experimentally.', '0910.5150-2-31-0': 'More generally, the concept of branon gives an elegant effective description of low energy brane fluctuations, but the feedback on the coupling constants of the Standard Model is of order [MATH] only, making this effective description hard to test experimentally, especially if the brane tension is expected to be larger than (100 GeV)[MATH] [CITATION].', '0910.5150-2-31-1': 'Also, it is not really clear how to interpret the cut off [MATH] in branon energy, making even more difficult the experimental tests.', '0910.5150-2-31-2': 'One possibility is to assume that [MATH] corresponds to a centre of mass energy in a collision, in which case the branon dressing can be compared to quantum corrections.', '0910.5150-2-31-3': 'But the corresponding dressing of coupling constants, of order [MATH], can be in the error bars of the experimental measurements if the brane tension is large.', '0910.5150-2-31-4': 'Bounds have been discussed in [CITATION] though.', '0910.5150-2-31-5': 'Another possibility is to assume a branon bath of typical temperature [MATH], in a finite temperature collision, but it is not clear how efficiently branons thermalize to Standard Model particles.', '0910.5150-2-31-6': 'One can also assume that branons are produced by an astrophysical catastrophic event, but in this case, the parameter [MATH] is then certainly so small, that no effect can be detected.', '0910.5150-2-32-0': 'Furthermore, we note that, in order to distinguish the branon-dressed parameters from the bare ones, we would need to compare an experiment with branons with a similar experiment without branons, which is difficult to set up.', '0910.5150-2-32-1': 'It seems more realistic to check the effects of branons at the next order in the inverse brane tension, since new interactions appear when the logarithm in the effective action ([REF]) is developed further.', '0910.5150-2-32-2': 'These effects are of order [MATH] though, and have been calculated in [CITATION].', '0910.5150-2-32-3': 'The discrepancy we find here with this reference would not change their results substantially, since the corresponding correction occurs at the order [MATH].', '0910.5150-2-33-0': 'We also would like to remark that the initial approximation ([REF]) takes into account the first order only in [MATH], to get the action describing the interaction branon-Standard Model.', '0910.5150-2-33-1': 'This approximation is based on the occurrence of low energy branons, and contains two derivatives of the branon field.', '0910.5150-2-33-2': 'But the resulting coupling [MATH] contains four derivatives, such that, should we follow a strict gradient expansion scheme, we would have to take into account also terms of order [MATH] in the expansion of the induced metric [MATH].', '0910.5150-2-33-3': 'This was not done, and would spoil the exact integration over branon degrees of freedom.', '0910.5150-2-33-4': 'These additional terms, though, might be relevant in the limit of the parameter [MATH] goes to 1, where the low-energy branon approximation is not valid anymore.', '0910.5150-2-33-5': 'These ambiguities show that a proper study of brane fluctuations feedback on the Standard Model might need to be non-perturbative.', '0910.5150-2-33-6': 'The concept of branon degree of freedom would then be difficult to define, but phenomenological effects might be more realistically predictable.'} | [['0910.5150-1-19-0', '0910.5150-2-23-0'], ['0910.5150-1-8-0', '0910.5150-2-9-0'], ['0910.5150-1-8-1', '0910.5150-2-9-1'], ['0910.5150-1-8-2', '0910.5150-2-9-2'], ['0910.5150-1-8-4', '0910.5150-2-9-4'], ['0910.5150-1-8-5', '0910.5150-2-9-5'], ['0910.5150-1-22-1', '0910.5150-2-31-1'], ['0910.5150-1-22-2', '0910.5150-2-31-2'], ['0910.5150-1-22-3', '0910.5150-2-31-3'], ['0910.5150-1-22-4', '0910.5150-2-31-4'], ['0910.5150-1-22-5', '0910.5150-2-31-5'], ['0910.5150-1-22-6', '0910.5150-2-31-6'], ['0910.5150-1-14-0', '0910.5150-2-15-0'], ['0910.5150-1-15-0', '0910.5150-2-16-0'], ['0910.5150-1-15-1', '0910.5150-2-16-1'], ['0910.5150-1-7-0', '0910.5150-2-8-0'], 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'0910.5150-2-26-0']] | [] | [['0910.5150-1-2-2', '0910.5150-2-2-2'], ['0910.5150-1-14-1', '0910.5150-2-15-1'], ['0910.5150-1-3-2', '0910.5150-2-3-4'], ['0910.5150-1-23-0', '0910.5150-2-33-0'], ['0910.5150-1-0-1', '0910.5150-2-0-1'], ['0910.5150-1-5-1', '0910.5150-2-6-3'], ['0910.5150-1-5-1', '0910.5150-2-6-4'], ['0910.5150-1-5-2', '0910.5150-2-6-8'], ['0910.5150-1-20-1', '0910.5150-2-24-1'], ['0910.5150-1-20-1', '0910.5150-2-24-7']] | [] | ['0910.5150-1-15-2', '0910.5150-2-16-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0910.5150 | null | null | null | null | null |
astro-ph-0601562 | {'astro-ph-0601562-1-0-0': 'The last decade has seen applications of Adaptive Mesh Refinement (AMR) methods for a wide range of problems from space physics to cosmology.', 'astro-ph-0601562-1-0-1': 'With the advent of these methods, in which space is discretized into a mesh of many individual cubic elements, the contemporary analog of the extensively studied line radiative transfer (RT) in a semi-infinite slab is that of RT in a cube.', 'astro-ph-0601562-1-0-2': 'In this study we provide an approximate solution of the RT equation, as well as analytic expressions for the probability distribution functions (pdfs) of the properties of photons emerging from a cube, and compare them with the corresponding slab problem.', 'astro-ph-0601562-1-0-3': 'These pdfs can be used to perform fast resonant-line RT in optically thick AMR cells where, otherwise, it could take unrealistically long times to transfer even a handful of photons.', 'astro-ph-0601562-1-1-0': ']', 'astro-ph-0601562-1-2-0': '1Lyman Spitzer Fellow, Department of Astrophysical Sciences, Princeton University, Peyton Hall, Ivy Lane, Princeton, NJ 08544-1001; [email protected]', 'astro-ph-0601562-1-3-0': '# Introduction', 'astro-ph-0601562-1-4-0': 'The classic problem of resonant radiative transfer (RT) in a semi-infinite slab has been extensively studied in literature .', 'astro-ph-0601562-1-4-1': 'On the other hand, the last decade has seen applications of Adaptive Mesh Refinement (AMR) methods to problems as diverse as solar physics, supernovae and nucleosynthesis, interstellar medium physics, star formation, astrophysical jets, cosmology, etc. [CITATION].', 'astro-ph-0601562-1-4-2': 'In mesh-based methods the continuous domain of interest is discretized into a grid of many individual cubic elements.', 'astro-ph-0601562-1-4-3': 'With the advent of AMR codes, the contemporary analogue - at least in terms of usefulness and applicability range - of the extensively studied problem of resonant-line RT in a slab is the relatively unexplored problem of RT in a cube.', 'astro-ph-0601562-1-5-0': 'Understanding resonant RT in optically thick cubes is useful in particular because to perform RT in AMR simulations one must solve numerous cube RT problems, since each time a photon enters an AMR cell one has a new cube RT problem.', 'astro-ph-0601562-1-5-1': 'Furthermore, as is the case, e.g., for Ly-[MATH] line RT in cosmological simulations of galaxy formation, the high resolution achieved with AMR codes (along with the cooling of the gas), leads to very high scattering optical depths [CITATION].', 'astro-ph-0601562-1-5-2': 'To obtain results within realistic times, we need a very fast RT algorithm, faster than the standard direct Monte Carlo approach in which one follows the detailed scattering of photons in each one of the cells.', 'astro-ph-0601562-1-5-3': 'A way to obtain this very fast algorithm is to study a priori the RT in cubes of various physical conditions.', 'astro-ph-0601562-1-5-4': 'Using the results of such a study, we can avoid following the detailed photon scattering in each cell.', 'astro-ph-0601562-1-5-5': 'Instead, we can immediately take the photon out of the cell treating thus the problem on a per cell rather than on a per scattering basis, thus accelerating the RT scheme considerably.', 'astro-ph-0601562-1-6-0': 'In this letter we discuss results on the resonant RT in a cube, and in comparison with resonant RT in a slab of similar physical conditions.', 'astro-ph-0601562-1-7-0': '# Description of the problem and definitions', 'astro-ph-0601562-1-8-0': 'In what follows we assume slab and cube configurations for the scattering medium.', 'astro-ph-0601562-1-8-1': 'The slab is semi-infinite, namely it is finite in one spatial dimension and infinite in the other two.', 'astro-ph-0601562-1-8-2': 'The scattering medium is uniform in its properties and has a central source of center-of-line photons that scatter resonantly before escaping.', 'astro-ph-0601562-1-8-3': '[CITATION] has solved the slab RT equation in the limit of high optical depths and obtained a one parameter solution .', 'astro-ph-0601562-1-8-4': 'The parameter is [MATH], with [MATH] and [MATH] the line natural and thermal Doppler widths, respectively, and [MATH] the center-of-line optical depth from the center of the scattering medium to one of its edges.', 'astro-ph-0601562-1-8-5': 'More specifically, [MATH] is defined so that the optical depth at frequency shift [MATH] is [MATH], with [MATH] the normalized line profile.', 'astro-ph-0601562-1-8-6': 'The discussion that follows applies to optically thick media ([MATH]).', 'astro-ph-0601562-1-9-0': '# Results', 'astro-ph-0601562-1-10-0': '## Emerging frequency distribution', 'astro-ph-0601562-1-11-0': '### Analytic solution for resonant RT in a cube', 'astro-ph-0601562-1-12-0': 'Following [CITATION], one can show that the RT equation we need to solve is [EQUATION] where [MATH] is the zeroth moment of the intensity [MATH], [MATH] is defined through [MATH], [MATH] is defined through [MATH], and [MATH] is the emissivity.', 'astro-ph-0601562-1-12-1': 'This equation is identical to the equation used previously for a semi-infinite slab or a spherically symmetric distribution .', 'astro-ph-0601562-1-12-2': 'The only difference is that those previous cases were for one spatial dimension, hence [MATH] consisted of only one term, whereas in the case of a cube it has contributions from all three dimensions, i.e. [MATH].', 'astro-ph-0601562-1-13-0': 'To obtain an approximate solution for equation ([REF]), we focus first on the solution of the corresponding homogeneous equation .', 'astro-ph-0601562-1-13-1': 'We assume that the solution is separable, namely that [MATH].', 'astro-ph-0601562-1-13-2': 'Substituting in equation ([REF]) and after some rearrangement we get [EQUATION] where we have perfomed a first separation with separation constant [MATH].', 'astro-ph-0601562-1-13-3': 'Assuming furthermore that [MATH], after performing additional separations we end up with equations of the form [EQUATION] and similarly for [MATH] with separation constants [MATH] and [MATH], respectively, and with [MATH].', 'astro-ph-0601562-1-13-4': 'The general solution for each one of these equations consists of both sine and cosine terms.', 'astro-ph-0601562-1-13-5': 'Since we only consider central point sources, i.e. [MATH], each one of the functions [MATH],[MATH], and [MATH] must be separately even.', 'astro-ph-0601562-1-13-6': 'Thus, we set [MATH] and [MATH].', 'astro-ph-0601562-1-14-0': 'We calculate the constants [MATH] using boundary conditions that are generated assuming the Milne-Eddington approximation [CITATION].', 'astro-ph-0601562-1-14-1': 'In fact, we adapt the usual two stream approximation to the cube problem.', 'astro-ph-0601562-1-14-2': 'We take into account all 3 spatial directions and treat them equivalently.', 'astro-ph-0601562-1-14-3': 'That is, we use an eight-stream approximation, which nevertheless, leads to the same conditions as the two stream approximation.', 'astro-ph-0601562-1-14-4': 'Now, however, these conditions refer separately to each one of the three independent directions.', 'astro-ph-0601562-1-14-5': 'Thus, we get [MATH], with [MATH] the flux components.', 'astro-ph-0601562-1-14-6': 'This condition yields [EQUATION] with [MATH] the first moment of [MATH] (differing from the flux [MATH] by a factor of 1/4), and where, in order to obtain the last equality, we used the other Milne-Eddington condition, [MATH], with [MATH] the second moment of [MATH] with respect to [MATH], combined with that [MATH].', 'astro-ph-0601562-1-14-7': 'Using equation ([REF]) we get for [MATH] the condition [CITATION] [EQUATION] as well as similar conditions for [MATH] and [MATH].', 'astro-ph-0601562-1-15-0': 'Normalizing the cosine solutions in [MATH] in each direction separately, the solution to the homogeneous equation RT equation takes the form [EQUATION]', 'astro-ph-0601562-1-15-1': 'Substituting this in equation ([REF]), and after mutliplying by [MATH], and integrating over volume we get [EQUATION] with [EQUATION]', 'astro-ph-0601562-1-15-2': 'For a source of unit strength at the center of the cube we have [MATH], and thus [MATH].', 'astro-ph-0601562-1-15-3': 'In equation ([REF]) we set [MATH] [CITATION].', 'astro-ph-0601562-1-16-0': 'Away from [MATH] equation ([REF]) is homogeneous.', 'astro-ph-0601562-1-16-1': 'Imposing the boundary condition [MATH] that reflects the fact that we do not expect photons with infinite frequency shifts (thus, [MATH] must go to zero for high - postive or negative - [MATH] values) we get [EQUATION]', 'astro-ph-0601562-1-16-2': 'We obtain the value of the constant [MATH] exactly as in [CITATION].', 'astro-ph-0601562-1-16-3': 'Plugging all these in equation ([REF]) and, since we are interested in the overall spectrum of radiation emerging from one side of the cube, say along the z axis, after integrating over [MATH] and setting [MATH] we get [EQUATION]', 'astro-ph-0601562-1-16-4': 'At this last step we also substituted [MATH] with [MATH].', 'astro-ph-0601562-1-16-5': "For a comparison of the spectrum emerging from one side of a cube to that emerging from one of the two 'sides' of a slab we must multiply our cube solution by a factor of 3 so that both solutions have the same normalization.", 'astro-ph-0601562-1-16-6': "This is so because, for the same central source, we expect 1/6 of the photons to emerge from a certain cube side and 1/2 of the photons to emerge from a certain slab 'side'.", 'astro-ph-0601562-1-17-0': 'Each individual series term for both the cube and the slab solution depends only on [MATH].', 'astro-ph-0601562-1-17-1': 'This becomes obvious when one takes into account the definition of [MATH] and the approximation for [MATH] in the wings, as well as condition ([REF]).', 'astro-ph-0601562-1-17-2': 'The slab solution is an alternate series and can be written in closed form.', 'astro-ph-0601562-1-17-3': 'The cube solution cannot be written in closed form, but, using equations ([REF]) and ([REF]) we see that [MATH] and [MATH].', 'astro-ph-0601562-1-17-4': 'Thus, the [MATH] term in equation ([REF]) can be written as [MATH], indicating that the cube series may also be alternating.', 'astro-ph-0601562-1-17-5': 'Writing this three-sum series as one sum, i.e., [MATH], we find that indeed the cube series is alternating as well.', 'astro-ph-0601562-1-17-6': 'Some of the first finite sums of the alternating series for a [MATH] cube and slab are shown at the top two panels, and the left bottom panel of Figure [REF], respectively.', 'astro-ph-0601562-1-17-7': 'In the case of the cube the infinite number of terms result (solid histograms) is obtained by the Monte Carlo method described in detail in [CITATION], whereas for the slab we use the closed form slab analytical solution of [CITATION].', 'astro-ph-0601562-1-18-0': 'The cube series solution will be of some practical value, only if a few first terms contribute significantly to the sum.', 'astro-ph-0601562-1-18-1': "The exponential decay in [MATH] indicates that the terms should die off for 'high' [MATH] and [MATH] values-with the exact values where this happens dependent on the frequency (or [MATH]) one calculates the spectrum at.", 'astro-ph-0601562-1-18-2': 'The logarithm of the absolute ratio of the series terms, [MATH], in units of the first term, [MATH], for 3 different values of the frequency shift is shown in the bottom right panel of Figure [REF].', 'astro-ph-0601562-1-18-3': "As before, [MATH], but we find that these 'convergence curves' remain identical for all cube [MATH] at the extremely optically thick regime.", 'astro-ph-0601562-1-18-4': "The dashed-dotted line is the 'convergence curve' for a slab for frequency shift equal to the shift where the emergent spectrum is known to have a maximum )^1/3[MATH][MATH]S[MATH](110^-4)S_1[MATH]x_f=39[MATH](1 0.3) S_1[MATH]x=x_max[MATH]S_i[MATH]3%[MATH]i=4[MATH]S-S_4/S 0.03[MATH]i=30[MATH]x=2[MATH]i=30[MATH]x=2[MATH][MATH]x x_max[MATH]_0[MATH]_0[MATH]_0[MATH]2/3[MATH]_0[MATH]N_sc[MATH]N_sc[MATH]_0[MATH]N_sc[MATH]_0[MATH]_0[MATH]f[MATH]_0[MATH]f[MATH]0.66-0.77[MATH]_0[MATH]210^3-10^8[MATH]f[MATH]_0[MATH]f[MATH]_0[MATH]2/3[MATH]f[MATH]f=2/3[MATH]f[MATH]f[MATH][MATH]I()/I(0)=(1+2)/3[MATH][MATH]I() d[MATH][MATH]_0=2 10^3[MATH][MATH][0,2][MATH][MATH][MATH][MATH][MATH][MATH][MATH]z[MATH]x[MATH]y[MATH]x(y) 5[MATH]"} | {'astro-ph-0601562-2-0-0': 'The last decade has seen applications of Adaptive Mesh Refinement (AMR) methods for a wide range of problems from space physics to cosmology.', 'astro-ph-0601562-2-0-1': 'With the advent of these methods, in which space is discretized into a mesh of many individual cubic elements, the contemporary analog of the extensively studied line radiative transfer (RT) in a semi-infinite slab is that of RT in a cube.', 'astro-ph-0601562-2-0-2': 'In this study we provide an approximate solution of the RT equation, as well as analytic expressions for the probability distribution functions (pdfs) of the properties of photons emerging from a cube, and compare them with the corresponding slab problem.', 'astro-ph-0601562-2-0-3': 'These pdfs can be used to perform fast resonant-line RT in optically thick AMR cells where, otherwise, it could take unrealistically long times to transfer even a handful of photons.', 'astro-ph-0601562-2-1-0': ']', 'astro-ph-0601562-2-2-0': '1Lyman Spitzer Fellow, Department of Astrophysical Sciences, Princeton University, Peyton Hall, Ivy Lane, Princeton, NJ 08544-1001; [email protected]', 'astro-ph-0601562-2-3-0': '# Introduction', 'astro-ph-0601562-2-4-0': 'The classic problem of resonant radiative transfer (RT) in a semi-infinite slab has been extensively studied in literature .', 'astro-ph-0601562-2-4-1': 'On the other hand, the last decade has seen applications of Adaptive Mesh Refinement (AMR) methods to problems as diverse as solar physics, supernovae and nucleosynthesis, interstellar medium physics, star formation, astrophysical jets, cosmology, etc. [CITATION].', 'astro-ph-0601562-2-4-2': 'In mesh-based methods the continuous domain of interest is discretized into a grid of many individual cubic elements.', 'astro-ph-0601562-2-4-3': 'With the advent of AMR codes, the contemporary analogue - at least in terms of usefulness and applicability range - of the extensively studied problem of resonant-line RT in a slab is the relatively unexplored problem of RT in a cube.', 'astro-ph-0601562-2-5-0': 'Understanding resonant RT in optically thick cubes is useful in particular because to perform RT in AMR simulations one must solve numerous cube RT problems, since each time a photon enters an AMR cell one has a new cube RT problem.', 'astro-ph-0601562-2-5-1': 'Furthermore, as is the case, e.g., for Ly-[MATH] line RT in cosmological simulations of galaxy formation, the high resolution achieved with AMR codes (along with the cooling of the gas), leads to very high scattering optical depths [CITATION].', 'astro-ph-0601562-2-5-2': 'To obtain results within realistic times, we need a very fast RT algorithm, faster than the standard direct Monte Carlo approach in which one follows the detailed scattering of photons in each one of the cells.', 'astro-ph-0601562-2-5-3': 'A way to obtain this very fast algorithm is to study a priori the RT in cubes of various physical conditions.', 'astro-ph-0601562-2-5-4': 'Using the results of such a study, we can avoid following the detailed photon scattering in each cell.', 'astro-ph-0601562-2-5-5': 'Instead, we can immediately take the photon out of the cell treating thus the problem on a per cell rather than on a per scattering basis, thus accelerating the RT scheme considerably.', 'astro-ph-0601562-2-6-0': 'In this letter we discuss results on the resonant RT in a cube, and in comparison with resonant RT in a slab of similar physical conditions.', 'astro-ph-0601562-2-7-0': '# Description of the problem and definitions', 'astro-ph-0601562-2-8-0': 'In what follows we assume slab and cube configurations for the scattering medium.', 'astro-ph-0601562-2-8-1': 'The slab is semi-infinite, namely it is finite in one spatial dimension and infinite in the other two.', 'astro-ph-0601562-2-8-2': 'The scattering medium is uniform in its properties and has a central source of center-of-line photons that scatter resonantly before escaping.', 'astro-ph-0601562-2-8-3': '[CITATION] has solved the slab RT equation in the limit of high optical depths and obtained a one parameter solution .', 'astro-ph-0601562-2-8-4': 'The parameter is [MATH], with [MATH] and [MATH] the line natural and thermal Doppler widths, respectively, and [MATH] the center-of-line optical depth from the center of the scattering medium to one of its edges.', 'astro-ph-0601562-2-8-5': 'More specifically, [MATH] is defined so that the optical depth at frequency shift [MATH] is [MATH], with [MATH] the normalized line profile.', 'astro-ph-0601562-2-8-6': 'The discussion that follows applies to optically thick media ([MATH]).', 'astro-ph-0601562-2-9-0': '# Results', 'astro-ph-0601562-2-10-0': '## Emerging frequency distribution', 'astro-ph-0601562-2-11-0': '### Approximate analytic solution for resonant RT in a cube', 'astro-ph-0601562-2-12-0': 'Following [CITATION], one can show that the RT equation we need to solve is [EQUATION] where [MATH] is the zeroth moment of the intensity [MATH], [MATH] is defined through [MATH], [MATH] is defined through [MATH], [MATH] is the emissivity, and [MATH] is measured from the center of the cube.', 'astro-ph-0601562-2-12-1': 'This equation is identical to the equation used previously for a semi-infinite slab or a spherically symmetric distribution .', 'astro-ph-0601562-2-12-2': 'The only difference is that those previous cases were for one spatial dimension, hence [MATH] consisted of only one term, whereas in the case of a cube it has contributions from all three dimensions, i.e. [MATH], with [MATH] and [MATH] the components of [MATH] along the x, y and z-axis, respectively.', 'astro-ph-0601562-2-13-0': 'Equation ([REF]) is a linear, inhomogeneous, partial differential equation.', 'astro-ph-0601562-2-13-1': 'To solve it we use the eigenfunction expansion method.', 'astro-ph-0601562-2-13-2': 'Namely, motivated by the method of separation of variables (applicable in the case of the corresponding homogeneous equation), we assume that the solution can be written as [EQUATION]', 'astro-ph-0601562-2-13-3': 'When applying this expansion method in inhomogeneous problems, the idea is that [MATH] and [MATH] will be known (eigen)functions and [MATH] will be the unknown coefficients of the sum (here frequency-dependent) that are to be determined through the solution process.', 'astro-ph-0601562-2-13-4': 'In reality, we have to specify all four functions since we do not have any "known" eigenfunctions.', 'astro-ph-0601562-2-13-5': 'We take the "known" (eigen)functions of position to be the solutions to the "associated" homogeneous ordinary differential equations.', 'astro-ph-0601562-2-13-6': 'The term "associated" here implies that the best choice for the basis position (eigen)functions should be the solution sets from Sturm-Liouville problems that closely resemble the problem being addressed.', 'astro-ph-0601562-2-13-7': 'This will give a set of functions that are orthogonal over the domain defined by the problem.', 'astro-ph-0601562-2-13-8': 'Thus, to find the position (eigen)functions, we focus first on the solution of the corresponding homogeneous equation.', 'astro-ph-0601562-2-13-9': 'We assume that the solution is separable, namely that [MATH].', 'astro-ph-0601562-2-13-10': 'Substituting in equation ([REF]) and after some rearrangement we get [EQUATION] where we have performed a first separation with separation constant [MATH].', 'astro-ph-0601562-2-13-11': 'As already implied by equation ([REF]), we furthermore assume that [MATH] in Cartesian coordinates.', 'astro-ph-0601562-2-13-12': 'After performing additional separations we end up with equations of the form [EQUATION] and similarly for [MATH] with separation constants [MATH] and [MATH], respectively, and with [MATH].', 'astro-ph-0601562-2-13-13': 'The general solution for each one of these equations consists of both sine and cosine terms.', 'astro-ph-0601562-2-13-14': 'Since we only consider central point sources, i.e. [MATH], each one of the functions [MATH],[MATH], and [MATH] must be separately even.', 'astro-ph-0601562-2-13-15': 'Thus, we set [MATH] and [MATH].', 'astro-ph-0601562-2-14-0': 'We calculate the constants [MATH] using boundary conditions that are generated assuming the Eddington approximation [CITATION], where near isotropy is assumed.', 'astro-ph-0601562-2-14-1': 'Given the optical depths we are concern with, the near isotropy assumption should be fairly accurate.', 'astro-ph-0601562-2-14-2': 'In fact, we adapt the usual two stream approximation to the cube problem.', 'astro-ph-0601562-2-14-3': 'Instead of assuming mild anisotropy in the form of two streams (i.e., [MATH] for [MATH] and [MATH] for [MATH]), we take into account all 3 spatial directions and treat them equivalently.', 'astro-ph-0601562-2-14-4': 'That is, we use an eight-stream approximation, which nevertheless, leads to the same conditions as the two stream approximation.', 'astro-ph-0601562-2-14-5': 'Now, however, these conditions refer separately to each one of the three independent directions.', 'astro-ph-0601562-2-14-6': 'Thus, we get [EQUATION] with [MATH] the first moment of [MATH] and where, in order to obtain the last equality, we used the other Eddington condition, [MATH], with [MATH] the second moment of [MATH] with respect to [MATH], combined with that [MATH].', 'astro-ph-0601562-2-14-7': 'Using equation ([REF]) we get for [MATH] the condition [CITATION] [EQUATION] as well as similar conditions for [MATH] and [MATH].', 'astro-ph-0601562-2-15-0': 'Normalizing the cosine solutions in [MATH] in each direction separately, the solution to the homogeneous equation RT equation takes the form [EQUATION]', 'astro-ph-0601562-2-15-1': 'Now we proceed to calculate the "coefficients" of the sum given in equation ([REF]).', 'astro-ph-0601562-2-15-2': 'Substituting this in equation ([REF]), and after multiplying by [MATH], and integrating over volume we get [EQUATION] with [EQUATION]', 'astro-ph-0601562-2-15-3': 'For a source of unit strength at the center of the cube we have [MATH], and thus [MATH].', 'astro-ph-0601562-2-15-4': 'In equation ([REF]) we set [MATH] [CITATION].', 'astro-ph-0601562-2-16-0': 'Away from [MATH] equation ([REF]) is homogeneous.', 'astro-ph-0601562-2-16-1': 'Imposing the boundary condition [MATH] that reflects the fact that we do not expect photons with infinite frequency shifts (thus, [MATH] must go to zero for high - positive or negative - [MATH] values) we get [EQUATION]', 'astro-ph-0601562-2-16-2': 'We obtain the value of the constant [MATH] exactly as in [CITATION].', 'astro-ph-0601562-2-16-3': 'Plugging all these in equation ([REF]) and, since we are interested in the overall spectrum of radiation emerging from one side of the cube, say along the z axis, after integrating over [MATH] and setting [MATH] we get [EQUATION]', 'astro-ph-0601562-2-16-4': 'At this last step we also substituted [MATH] with [MATH].', 'astro-ph-0601562-2-16-5': "For a comparison of the spectrum emerging from one side of a cube to that emerging from one of the two 'sides' of a slab we must multiply our cube solution by a factor of 3 so that both solutions have the same normalization.", 'astro-ph-0601562-2-16-6': "This is so because, for the same central source, we expect 1/6 of the photons to emerge from a certain cube side and 1/2 of the photons to emerge from a certain slab 'side'.", 'astro-ph-0601562-2-17-0': 'Each individual series term for both the cube and the slab solution depends only on [MATH].', 'astro-ph-0601562-2-17-1': 'This becomes obvious when one takes into account the definition of [MATH] and the approximation for [MATH] in the wings, as well as condition ([REF]).', 'astro-ph-0601562-2-17-2': 'The slab solution is an alternate series and can be written in closed form.', 'astro-ph-0601562-2-17-3': 'The cube solution cannot be written in closed form, but, using equations ([REF]) and ([REF]) we see that [MATH] and [MATH].', 'astro-ph-0601562-2-17-4': 'Thus, the [MATH] term in equation ([REF]) can be written as [MATH], indicating that the cube series may also be alternating.', 'astro-ph-0601562-2-17-5': 'Writing this three-sum series as one sum, i.e., [MATH], we find that indeed the cube series is alternating as well.', 'astro-ph-0601562-2-17-6': 'Some of the first finite sums of the alternating series for a [MATH] cube and slab are shown at the top two panels, and the left bottom panel of Figure [REF], respectively.', 'astro-ph-0601562-2-17-7': 'In the case of the cube the infinite number of terms result (solid histograms) is obtained by the Monte Carlo method described in detail in [CITATION], whereas for the slab we use the closed form slab analytical solution of [CITATION].', 'astro-ph-0601562-2-18-0': 'The cube series solution will be of some practical value, only if a few first terms contribute significantly to the sum.', 'astro-ph-0601562-2-18-1': "The exponential decay in [MATH] indicates that the terms should die off for 'high' [MATH] and [MATH] values-with the exact values where this happens dependent on the frequency (or [MATH]) one calculates the spectrum at.", 'astro-ph-0601562-2-18-2': 'The logarithm of the absolute ratio of the series terms, [MATH], in units of the first term, [MATH], for 3 different values of the frequency shift is shown in the bottom right panel of Figure [REF].', 'astro-ph-0601562-2-18-3': "As before, [MATH], but we find that these 'convergence curves' remain identical for all cube [MATH] at the extremely optically thick regime.", 'astro-ph-0601562-2-18-4': "The dashed-dotted line is the 'convergence curve' for a slab for frequency shift equal to the shift where the emergent spectrum is known to have a maximum )^1/3[MATH][MATH]S[MATH](110^-4)S_1[MATH]x_f=39[MATH](1 0.3) S_1[MATH]x=x_max[MATH]S_i[MATH]3%[MATH]i=4[MATH]S-S_4/S 0.03[MATH]i=30[MATH]x=2[MATH]i=30[MATH]x=2[MATH][MATH]x x_max[MATH]_0[MATH]_0[MATH]_0[MATH]2/3[MATH]_0[MATH]N_sc[MATH]N_sc[MATH]_0[MATH]N_sc[MATH]_0[MATH]_0[MATH]f[MATH]_0[MATH]f[MATH]0.66-0.77[MATH]_0[MATH]210^3-10^8[MATH]f[MATH]_0[MATH]f[MATH]_0[MATH]2/3[MATH]f[MATH]f=2/3[MATH]f[MATH]f[MATH][MATH]I()/I(0)=(1+2)/3[MATH][MATH]I() d[MATH][MATH]_0=2 10^3[MATH][MATH][0,2][MATH][MATH][MATH][MATH][MATH][MATH][MATH]z[MATH]x[MATH]y[MATH]x(y) 5[MATH]"} | [['astro-ph-0601562-1-13-1', 'astro-ph-0601562-2-13-9'], ['astro-ph-0601562-1-13-4', 'astro-ph-0601562-2-13-13'], ['astro-ph-0601562-1-13-5', 'astro-ph-0601562-2-13-14'], ['astro-ph-0601562-1-13-6', 'astro-ph-0601562-2-13-15'], ['astro-ph-0601562-1-15-0', 'astro-ph-0601562-2-15-0'], ['astro-ph-0601562-1-15-2', 'astro-ph-0601562-2-15-3'], ['astro-ph-0601562-1-15-3', 'astro-ph-0601562-2-15-4'], ['astro-ph-0601562-1-4-0', 'astro-ph-0601562-2-4-0'], ['astro-ph-0601562-1-4-1', 'astro-ph-0601562-2-4-1'], ['astro-ph-0601562-1-4-2', 'astro-ph-0601562-2-4-2'], ['astro-ph-0601562-1-4-3', 'astro-ph-0601562-2-4-3'], ['astro-ph-0601562-1-5-0', 'astro-ph-0601562-2-5-0'], ['astro-ph-0601562-1-5-1', 'astro-ph-0601562-2-5-1'], ['astro-ph-0601562-1-5-2', 'astro-ph-0601562-2-5-2'], ['astro-ph-0601562-1-5-3', 'astro-ph-0601562-2-5-3'], ['astro-ph-0601562-1-5-4', 'astro-ph-0601562-2-5-4'], ['astro-ph-0601562-1-5-5', 'astro-ph-0601562-2-5-5'], ['astro-ph-0601562-1-8-0', 'astro-ph-0601562-2-8-0'], ['astro-ph-0601562-1-8-1', 'astro-ph-0601562-2-8-1'], ['astro-ph-0601562-1-8-2', 'astro-ph-0601562-2-8-2'], ['astro-ph-0601562-1-8-3', 'astro-ph-0601562-2-8-3'], ['astro-ph-0601562-1-8-4', 'astro-ph-0601562-2-8-4'], ['astro-ph-0601562-1-8-5', 'astro-ph-0601562-2-8-5'], ['astro-ph-0601562-1-8-6', 'astro-ph-0601562-2-8-6'], ['astro-ph-0601562-1-12-1', 'astro-ph-0601562-2-12-1'], ['astro-ph-0601562-1-16-0', 'astro-ph-0601562-2-16-0'], ['astro-ph-0601562-1-16-2', 'astro-ph-0601562-2-16-2'], ['astro-ph-0601562-1-16-3', 'astro-ph-0601562-2-16-3'], ['astro-ph-0601562-1-16-4', 'astro-ph-0601562-2-16-4'], ['astro-ph-0601562-1-16-5', 'astro-ph-0601562-2-16-5'], ['astro-ph-0601562-1-16-6', 'astro-ph-0601562-2-16-6'], ['astro-ph-0601562-1-17-0', 'astro-ph-0601562-2-17-0'], ['astro-ph-0601562-1-17-1', 'astro-ph-0601562-2-17-1'], ['astro-ph-0601562-1-17-2', 'astro-ph-0601562-2-17-2'], ['astro-ph-0601562-1-17-3', 'astro-ph-0601562-2-17-3'], ['astro-ph-0601562-1-17-4', 'astro-ph-0601562-2-17-4'], ['astro-ph-0601562-1-17-5', 'astro-ph-0601562-2-17-5'], ['astro-ph-0601562-1-17-6', 'astro-ph-0601562-2-17-6'], ['astro-ph-0601562-1-17-7', 'astro-ph-0601562-2-17-7'], ['astro-ph-0601562-1-2-0', 'astro-ph-0601562-2-2-0'], ['astro-ph-0601562-1-18-0', 'astro-ph-0601562-2-18-0'], ['astro-ph-0601562-1-18-1', 'astro-ph-0601562-2-18-1'], ['astro-ph-0601562-1-18-2', 'astro-ph-0601562-2-18-2'], ['astro-ph-0601562-1-18-3', 'astro-ph-0601562-2-18-3'], ['astro-ph-0601562-1-0-0', 'astro-ph-0601562-2-0-0'], ['astro-ph-0601562-1-0-1', 'astro-ph-0601562-2-0-1'], ['astro-ph-0601562-1-0-2', 'astro-ph-0601562-2-0-2'], ['astro-ph-0601562-1-0-3', 'astro-ph-0601562-2-0-3'], ['astro-ph-0601562-1-6-0', 'astro-ph-0601562-2-6-0'], ['astro-ph-0601562-1-14-1', 'astro-ph-0601562-2-14-2'], ['astro-ph-0601562-1-14-3', 'astro-ph-0601562-2-14-4'], ['astro-ph-0601562-1-14-4', 'astro-ph-0601562-2-14-5'], ['astro-ph-0601562-1-14-7', 'astro-ph-0601562-2-14-7'], ['astro-ph-0601562-1-13-2', 'astro-ph-0601562-2-13-10'], ['astro-ph-0601562-1-13-3', 'astro-ph-0601562-2-13-12'], ['astro-ph-0601562-1-15-1', 'astro-ph-0601562-2-15-2'], ['astro-ph-0601562-1-12-0', 'astro-ph-0601562-2-12-0'], ['astro-ph-0601562-1-12-2', 'astro-ph-0601562-2-12-2'], ['astro-ph-0601562-1-16-1', 'astro-ph-0601562-2-16-1'], ['astro-ph-0601562-1-13-0', 'astro-ph-0601562-2-13-8'], ['astro-ph-0601562-1-14-0', 'astro-ph-0601562-2-14-0'], ['astro-ph-0601562-1-14-2', 'astro-ph-0601562-2-14-3'], ['astro-ph-0601562-1-14-6', 'astro-ph-0601562-2-14-6']] | [['astro-ph-0601562-1-13-1', 'astro-ph-0601562-2-13-9'], ['astro-ph-0601562-1-13-4', 'astro-ph-0601562-2-13-13'], ['astro-ph-0601562-1-13-5', 'astro-ph-0601562-2-13-14'], ['astro-ph-0601562-1-13-6', 'astro-ph-0601562-2-13-15'], ['astro-ph-0601562-1-15-0', 'astro-ph-0601562-2-15-0'], ['astro-ph-0601562-1-15-2', 'astro-ph-0601562-2-15-3'], ['astro-ph-0601562-1-15-3', 'astro-ph-0601562-2-15-4'], ['astro-ph-0601562-1-4-0', 'astro-ph-0601562-2-4-0'], ['astro-ph-0601562-1-4-1', 'astro-ph-0601562-2-4-1'], ['astro-ph-0601562-1-4-2', 'astro-ph-0601562-2-4-2'], ['astro-ph-0601562-1-4-3', 'astro-ph-0601562-2-4-3'], ['astro-ph-0601562-1-5-0', 'astro-ph-0601562-2-5-0'], ['astro-ph-0601562-1-5-1', 'astro-ph-0601562-2-5-1'], ['astro-ph-0601562-1-5-2', 'astro-ph-0601562-2-5-2'], ['astro-ph-0601562-1-5-3', 'astro-ph-0601562-2-5-3'], ['astro-ph-0601562-1-5-4', 'astro-ph-0601562-2-5-4'], ['astro-ph-0601562-1-5-5', 'astro-ph-0601562-2-5-5'], ['astro-ph-0601562-1-8-0', 'astro-ph-0601562-2-8-0'], ['astro-ph-0601562-1-8-1', 'astro-ph-0601562-2-8-1'], ['astro-ph-0601562-1-8-2', 'astro-ph-0601562-2-8-2'], ['astro-ph-0601562-1-8-3', 'astro-ph-0601562-2-8-3'], ['astro-ph-0601562-1-8-4', 'astro-ph-0601562-2-8-4'], ['astro-ph-0601562-1-8-5', 'astro-ph-0601562-2-8-5'], ['astro-ph-0601562-1-8-6', 'astro-ph-0601562-2-8-6'], ['astro-ph-0601562-1-12-1', 'astro-ph-0601562-2-12-1'], ['astro-ph-0601562-1-16-0', 'astro-ph-0601562-2-16-0'], ['astro-ph-0601562-1-16-2', 'astro-ph-0601562-2-16-2'], ['astro-ph-0601562-1-16-3', 'astro-ph-0601562-2-16-3'], ['astro-ph-0601562-1-16-4', 'astro-ph-0601562-2-16-4'], ['astro-ph-0601562-1-16-5', 'astro-ph-0601562-2-16-5'], ['astro-ph-0601562-1-16-6', 'astro-ph-0601562-2-16-6'], ['astro-ph-0601562-1-17-0', 'astro-ph-0601562-2-17-0'], ['astro-ph-0601562-1-17-1', 'astro-ph-0601562-2-17-1'], ['astro-ph-0601562-1-17-2', 'astro-ph-0601562-2-17-2'], ['astro-ph-0601562-1-17-3', 'astro-ph-0601562-2-17-3'], ['astro-ph-0601562-1-17-4', 'astro-ph-0601562-2-17-4'], ['astro-ph-0601562-1-17-5', 'astro-ph-0601562-2-17-5'], ['astro-ph-0601562-1-17-6', 'astro-ph-0601562-2-17-6'], ['astro-ph-0601562-1-17-7', 'astro-ph-0601562-2-17-7'], ['astro-ph-0601562-1-2-0', 'astro-ph-0601562-2-2-0'], ['astro-ph-0601562-1-18-0', 'astro-ph-0601562-2-18-0'], ['astro-ph-0601562-1-18-1', 'astro-ph-0601562-2-18-1'], ['astro-ph-0601562-1-18-2', 'astro-ph-0601562-2-18-2'], ['astro-ph-0601562-1-18-3', 'astro-ph-0601562-2-18-3'], ['astro-ph-0601562-1-0-0', 'astro-ph-0601562-2-0-0'], ['astro-ph-0601562-1-0-1', 'astro-ph-0601562-2-0-1'], ['astro-ph-0601562-1-0-2', 'astro-ph-0601562-2-0-2'], ['astro-ph-0601562-1-0-3', 'astro-ph-0601562-2-0-3'], ['astro-ph-0601562-1-6-0', 'astro-ph-0601562-2-6-0'], ['astro-ph-0601562-1-14-1', 'astro-ph-0601562-2-14-2'], ['astro-ph-0601562-1-14-3', 'astro-ph-0601562-2-14-4'], ['astro-ph-0601562-1-14-4', 'astro-ph-0601562-2-14-5'], ['astro-ph-0601562-1-14-7', 'astro-ph-0601562-2-14-7']] | [['astro-ph-0601562-1-13-2', 'astro-ph-0601562-2-13-10'], ['astro-ph-0601562-1-13-3', 'astro-ph-0601562-2-13-12'], ['astro-ph-0601562-1-15-1', 'astro-ph-0601562-2-15-2'], ['astro-ph-0601562-1-12-0', 'astro-ph-0601562-2-12-0'], ['astro-ph-0601562-1-12-2', 'astro-ph-0601562-2-12-2'], ['astro-ph-0601562-1-16-1', 'astro-ph-0601562-2-16-1']] | [] | [['astro-ph-0601562-1-13-0', 'astro-ph-0601562-2-13-8'], ['astro-ph-0601562-1-14-0', 'astro-ph-0601562-2-14-0'], ['astro-ph-0601562-1-14-2', 'astro-ph-0601562-2-14-3'], ['astro-ph-0601562-1-14-6', 'astro-ph-0601562-2-14-6']] | [] | ['astro-ph-0601562-1-1-0', 'astro-ph-0601562-1-18-4', 'astro-ph-0601562-2-1-0', 'astro-ph-0601562-2-18-4'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/0601562 | null | null | null | null | null |
1901.08164 | {'1901.08164-1-0-0': 'A commonly cited inefficiency of neural network training by back-propagation is the update locking problem: each layer must wait for the signal to propagate through the network before updating.', '1901.08164-1-0-1': 'We consider and analyze a training procedure, Decoupled Greedy Learning (DGL), that addresses this problem more effectively and at scales beyond those of previous solutions.', '1901.08164-1-0-2': 'It is based on a greedy relaxation of the joint training objective, recently shown to be effective in the context of Convolutional Neural Networks (CNNs) on large-scale image classification.', '1901.08164-1-0-3': 'We consider an optimization of this objective that permits us to decouple the layer training, allowing for layers or modules in networks to be trained with a potentially linear parallelization in layers.', '1901.08164-1-0-4': 'We show theoretically and empirically that this approach converges.', '1901.08164-1-0-5': 'In addition, we empirically find that it can lead to better generalization than sequential greedy optimization and even standard end-to-end back-propagation.', '1901.08164-1-0-6': 'We show that an extension of this approach to asynchronous settings, where modules can operate with large communication delays, is possible with the use of a replay buffer.', '1901.08164-1-0-7': 'We demonstrate the effectiveness of DGL on the CIFAR-10 datasets against alternatives and on the large-scale ImageNet dataset, where we are able to effectively train VGG and ResNet-152 models.', '1901.08164-1-1-0': '# Introduction', '1901.08164-1-2-0': 'Training jointly all layers using back-propagation is the standard method for learning neural networks, including the computationally intensive vision models based on Convolutional Neural Networks (CNNs) [CITATION].', '1901.08164-1-2-1': 'Due to the sequential nature of the gradient processing, standard back-propagation has several well-known inefficiencies that do not permit the computations of the different constituent modules to be parallelized.', '1901.08164-1-2-2': '[CITATION] characterizes these in order of severity as the forward, update, and backward locking problems.', '1901.08164-1-2-3': 'Backward unlocking would permit updates of all modules once signals have propagated to all subsequent modules, update unlocking would permit updates of a module before a signal has reached all subsequent modules, and forward unlocking would permit a module to operate asynchronously from its predecessor and dependent modules.', '1901.08164-1-3-0': 'Multiple methods have been proposed, which can deal up to a certain degree with the backward unlocking challenge [CITATION].', '1901.08164-1-3-1': '[CITATION] propose and analyze DNI, a method that addresses the more challenging update locking.', '1901.08164-1-3-2': 'The DNI approach uses an auxiliary network to predict the gradient of the backward pass directly from the input.', '1901.08164-1-3-3': 'This method is not shown to scale well computationally or in terms of accuracy, especially in the case of CNNs [CITATION].', '1901.08164-1-3-4': 'Indeed, auxiliary networks must predict a weight gradient that can be very large in dimensionality, which can be inaccurate and challenging to scale when intermediate representations are large, as is the case for larger models and input image sizes.', '1901.08164-1-4-0': 'Recently, several authors have revisited the classic [CITATION] approach of supervised greedy layer-wise training of neural networks [CITATION].', '1901.08164-1-4-1': 'In [CITATION] it is shown that such an approach, which relaxes the joint learning objective, can lead to high performance deep CNNs on large-scale datasets.', '1901.08164-1-4-2': 'Some of these works also consider the use of auxiliary networks with hidden layers as part of the local auxiliary problems which has some analogs to the auxiliary networks of DNI and target propagation [CITATION].', '1901.08164-1-4-3': 'We will show that the greedy learning objective can be solved with an alternative optimization algorithm, which permits decoupling the computations and achieving update unlocking.', '1901.08164-1-4-4': 'This can also be augmented with replay buffers [CITATION] to permit forward unlocking.', '1901.08164-1-4-5': 'This strategy can be shown to be a state-of-the-art baseline for parallelizing the training across modules of a neural network.', '1901.08164-1-5-0': 'Our contributions in this work are as follows.', '1901.08164-1-5-1': 'We (a) propose an optimization procedure for a decoupled greedy learning objective that solves the update locking problem.', '1901.08164-1-5-2': '(b) Empirically, we show that it exhibits similar convergence rates and generalization as its non-decoupled counterpart.', '1901.08164-1-5-3': '(c) We show that it can be extended to an asynchronous setting by use of a replay buffer, providing a step towards addressing the forward locking problem.', '1901.08164-1-5-4': '(d) We motivate these observations theoretically, showing that the proposed optimization procedure converges and recovers standard rates of non-convex optimization.', '1901.08164-1-5-5': 'Experimentally we (e) design an improved auxiliary network structure for greedy layer-wise training of CNNs that permits to maintain accuracy while having negligible cost for the auxiliary task.', '1901.08164-1-5-6': 'We (f) show that the decoupled greedy learning can well outperform competing methods in terms of scalability to larger and deeper models and stability to optimization hyper-parameters, allowing it to be applied to large datasets.', '1901.08164-1-5-7': 'We then demonstrate on the ImageNet dataset that we can train the deep models VGG-19 and ResNet-152 with larger degrees of parallelism than other proposals and reduced memory consumption.', '1901.08164-1-5-8': 'Code for experiments will be made available.', '1901.08164-1-6-0': '# Related work', '1901.08164-1-7-0': 'To the best of our knowledge [CITATION] is the only work which directly addresses the update or forward locking problems in deep feed-forward networks.', '1901.08164-1-7-1': 'Other works [CITATION] consider the backward locking problem, furthermore a number of back-propagation alternatives such as [CITATION] are also able to address this problem.', '1901.08164-1-7-2': 'However, update locking is a more severe inefficiency.', '1901.08164-1-7-3': "Consider the case where each layer's forward processing time is [MATH] and is equal across a network of [MATH] layers.", '1901.08164-1-7-4': 'Given that the backward pass of back-propagation is a constant multiple in time of the forward pass, in the most ideal case the backward unlocking will still only scale as [MATH] with [MATH] parallel nodes, while update unlocking could scale as [MATH].', '1901.08164-1-8-0': 'One class of alternatives to standard back-propagation aims to avoid its biologically implausible aspects, most notably the weight transport problem [CITATION].', '1901.08164-1-8-1': 'Some of these methods [CITATION] can also achieve backward unlocking as they permit all parameters to be updated at the same time, but only once the signal has propagated to the top layer.', '1901.08164-1-8-2': 'None of them however solve the update locking problem or forward locking problem which we consider.', '1901.08164-1-8-3': 'Target propagation uses a local auxiliary network as in our approach, which is used to propagate backward the optimal activations computed from the layer above.', '1901.08164-1-8-4': 'Feedback alignment replaces the symmetric weights of the backward pass with random weights.', '1901.08164-1-8-5': 'Direct feedback alignment extends the idea of feedback alignment passing errors from the top to all layers, potentially permitting a simultaneous update.', '1901.08164-1-8-6': 'These approaches have also not been shown to be scalable to large datasets [CITATION], obtaining only [MATH] top-5 accuracy on ImageNet (for a reference model that achieves [MATH]).', '1901.08164-1-8-7': 'On the other hand a greedy learning strategy has been shown to work well on the same task [CITATION].', '1901.08164-1-9-0': 'Another line of related work inspired by optimization methods such as Alternating Direction Method of Multipliers (ADMM) [CITATION] considers approaches that use auxiliary variables to break optimization into sub-problems.', '1901.08164-1-9-1': "These approaches are fundamentally different from ours as they optimization for the joint training objective, the auxiliary variables providing a link between a layer and it's successive layers, whereas we consider a different objective where a layer has no dependence on its successors.", '1901.08164-1-9-2': 'None of these methods can achieve update or forward unlocking, however some [CITATION] are able to have a simultaneous weight updates (backward unlocked).', '1901.08164-1-9-3': 'Another issue with these methods is that most of the existing approaches except for [CITATION] require standard ("batch") gradient descent and are thus difficult to scale.', '1901.08164-1-9-4': 'They also often involve an inner minimization problem and have thus not been demonstrated to work on realistic large scale datasets.', '1901.08164-1-9-5': 'Furthermore, none of these have been combined with CNNs.', '1901.08164-1-10-0': 'Distributed optimization based on data parallelism is a popular area of research in machine learning beyond deep learning models and often studied in the convex setting [CITATION].', '1901.08164-1-10-1': 'In deep network optimization the predominant method is distributed synchronous SGD [CITATION] and variants, as well as asynchronous [CITATION] variants.', '1901.08164-1-10-2': 'Our approach on the other hand can be seen as closer to exploiting a type of model parallelism vs data parallelism and can be easily combined with many of these methods, particularly distributed synchronous SGD.', '1901.08164-1-11-0': '# Parallel Decoupled Greedy Learning', '1901.08164-1-12-0': 'In this section we formally define the greedy objective and parallel optimization which we study in both the synchronous and asynchronous setting.', '1901.08164-1-12-1': 'We mainly consider the online setting and assume a stream of samples or mini-batches denoted [MATH], that can be run during [MATH] iterations.', '1901.08164-1-13-0': '## Optimization for Greedy Objective', '1901.08164-1-14-0': 'Let [MATH] and [MATH] be the data matrix and labels for the training data.', '1901.08164-1-14-1': 'Let [MATH] be the output representation for module [MATH].', '1901.08164-1-14-2': 'We will denote the per-module objective function [MATH], where the parameters [MATH] correspond to the module parameter (i.e. [MATH]), [MATH] correspond to auxiliary parameters used to compute the objective.', '1901.08164-1-14-3': '[MATH] in our case will be the empirical risk with a cross-entropy loss.', '1901.08164-1-14-4': 'The greedy training objective is thus given recursively by defining [MATH]: [EQUATION] where [MATH] and [MATH] is the minimizer of Problem [MATH].', '1901.08164-1-14-5': 'A natural way to solve the optimization problem for [MATH] modules, [MATH], is thus by sequentially solving the problems [MATH] starting with [MATH].', '1901.08164-1-14-6': 'Here we consider an alternative procedure for optimizing this objective given in Alg. [REF]: individual updates of each set of parameters are performed in sequence across the different layers.', '1901.08164-1-14-7': 'Each layer processes a sample or mini-batch, then passes it to the next layer.', '1901.08164-1-14-8': 'Note that at line [MATH] the subsequent layer can already begin computing line [MATH].', '1901.08164-1-14-9': 'Therefore, this algorithm achieves update unlocking.', '1901.08164-1-14-10': 'An explicit version of an equivalent multi-worker pseudo-code is included in Appendix [REF].', '1901.08164-1-15-0': 'Fig. [REF] illustrates the decoupling compared to how samples are processed in standard back-propagation.', '1901.08164-1-15-1': 'We observe that once a [MATH] has been computed, processing by subsequent layers can begin.', '1901.08164-1-15-2': 'Sec. [REF] will also consider a version of the algorithm that can be made asynchronous by introducing a replay buffer.', '1901.08164-1-16-0': '## Asynchronous Decoupled Greedy Learning with Replay', '1901.08164-1-17-0': 'We consider an extension of this framework that addresses forward unlocking [CITATION].', '1901.08164-1-17-1': 'Since the computations of the modules in DGL are only loosely dependent, we can attempt an extension that allows to also remove some dependency of the computations on the previous modules such that this can operate asynchronously.', '1901.08164-1-17-2': 'This is achieved by the use of a replay buffer that is shared between adjacent modules and allows modules to reuse older samples.', '1901.08164-1-17-3': 'It can be beneficial in scenarios with communication delays or substantial variations in speeds between layers/modules.', '1901.08164-1-17-4': 'We will evaluate one instance of such an algorithm based on the use of a replay buffer of size [MATH], shown in Alg. [REF].', '1901.08164-1-17-5': 'Here each module maintains a buffer to which it writes its output representations, which is read by the module above.', '1901.08164-1-18-0': '[H] Stream [MATH]; Distribution of the delay [MATH]; Buffer size [MATH] Initialize: Buffers [MATH] with size [MATH]; params [MATH]', '1901.08164-1-19-0': 'trainingSample [MATH] in [MATH] following [MATH].', '1901.08164-1-20-0': '[MATH] Compute [MATH] Update parameters [MATH]', '1901.08164-1-21-0': 'Asynchronous DGL with Replay Buffer', '1901.08164-1-22-0': 'Our minimal distributed setting is as follows.', '1901.08164-1-22-1': 'Each worker [MATH] has a buffer that it writes to and that worker [MATH] can read from.', '1901.08164-1-22-2': 'The buffer uses a simple read and write protocol.', '1901.08164-1-22-3': 'A buffer [MATH] permits layer [MATH] to write new samples.', '1901.08164-1-22-4': 'When it reaches capacity it overwrites the oldest sample.', '1901.08164-1-22-5': 'Layer [MATH] requests samples from the buffer [MATH].', '1901.08164-1-22-6': 'The sample is selected by a simple last-in-first-out (LIFO) rule, with a precedence for the least-reused samples.', '1901.08164-1-22-7': 'The speed of the worker is constant, yet can be potentially different across workers and is also subject to small random fluctuations.', '1901.08164-1-22-8': 'Our algorithm does not require a shared buffer across all workers, but only across pairs of workers.', '1901.08164-1-22-9': 'Alg. [REF] simulates potential delays in such a setup by the use of a probability mass function (pmf) [MATH] over workers analogous to typical asynchronous settings such as [CITATION].', '1901.08164-1-22-10': 'At each iteration a layer is chosen at random according to [MATH] to perform computation.', '1901.08164-1-22-11': 'In our experiments we will limit ourselves to pmfs that are uniform over workers except for a single layer which is chosen to be selected less frequently on average.', '1901.08164-1-22-12': 'We note even in the case of a uniform pmf, asynchronous behavior will naturally arise requiring the reuse of samples.', '1901.08164-1-22-13': 'Alg. [REF] permits a controlled simulation of processing speed discrepancies and will be used over settings of [MATH] and [MATH] to demonstrate that training accuracy and testing accuracy remain robust in practical regimes.', '1901.08164-1-22-14': 'Appendix [REF] also provides a more intuitive pseudo-code for how this buffer-based algorithm would be implemented in a parallel environment.', '1901.08164-1-23-0': 'As will be demonstrated in our experiments, the DGL can potentially be robust to substantial asynchronous behavior.', '1901.08164-1-23-1': 'Unlike common data-parallel asynchronous algorithms [CITATION], the asynchronous DGL does not rely on a master node and requires only local communication similar to recent decentralized schemes [CITATION].', '1901.08164-1-23-2': 'Unlike decentralized SGD algorithms, nodes only need to maintain and update the parameters of their local model, potentially supporting much larger models.', '1901.08164-1-23-3': 'Combining DGL with data-parallel methods is also natural.', '1901.08164-1-23-4': 'For example a common issue of the popular distributed synchronous SGD in deep CNNs is the often limited maximum batch size [CITATION].', '1901.08164-1-23-5': 'This suggests that DGL can be used in combination with data parallelism to add an additional dimension of parallelization.', '1901.08164-1-23-6': 'Potentially combining asynchronous DGL with distributed synchronous SGD for sub-problem optimization is a promising direction.', '1901.08164-1-24-0': '## Auxiliary and Primary Network Design', '1901.08164-1-25-0': 'The DNI method requires an auxiliary network to predict the gradient.', '1901.08164-1-25-1': 'The greedy layerwise CNN training procedure of [CITATION], which we parallelize, similarly relies on an auxiliary network.', '1901.08164-1-25-2': 'This requires the design of an auxiliary network in addition to the CNN architecture design.', '1901.08164-1-25-3': '[CITATION] have shown that simple averaging operations can be used to construct a scalable auxiliary network.', '1901.08164-1-25-4': 'However, they did not directly consider the parallel training use case.', '1901.08164-1-25-5': 'Here care must be taken in the design, as will be discussed in the experimental section.', '1901.08164-1-25-6': 'The primary considerations in our case is the relative speed of the auxiliary network with respect to the associated module it is attached to in the primary network.', '1901.08164-1-25-7': 'We will use primarily FLOP count in our analysis and aim to restrict our auxiliary networks to be [MATH] of the primary network.', '1901.08164-1-26-0': 'Although auxiliary network design might seem like an additional layer of complexity in CNN design and might potentially require slightly different architecture principles for the primary network than standard end-to-end trained deep CNNs, this is not inherently prohibitive since architecture design is well known to be related to the training.', '1901.08164-1-26-1': 'As an example, consider the typical motivation for residual connections which are originally motivated by optimization issues inherent to end-to-end backpropagation of deep networks.', '1901.08164-1-27-0': 'We note although we focus on the distributed learning context, the proposed optimization algorithm and associated theory for greedy objectives is generic and has other potential applications.', '1901.08164-1-27-1': 'Greedy objectives have recently been used in several applications in reinforcement learning [CITATION] and in ensemble methods like Boosting [CITATION].', '1901.08164-1-27-2': 'Even with a single worker the synchronous DGL has a gain in terms of memory.', '1901.08164-1-27-3': 'Moreover it is easier to implement efficiently than sequential greedy training since in the naive sequential training scheme, a forward pass through old modules or caching of previous activations is needed for optimal performance.', '1901.08164-1-28-0': '# Theoretical Analysis', '1901.08164-1-29-0': 'In this section we analyze the convergence of Alg. [REF] when the update steps are obtained from stochastic gradient methods.', '1901.08164-1-29-1': 'We show that under the DGL optimization scheme a critical point can be reached .', '1901.08164-1-29-2': 'In standard stochastic optimization schemes, the input distribution fed to a model is fixed [CITATION].', '1901.08164-1-29-3': 'With the decoupled training procedure the input distribution to each module is time-varying and dependent on the convergence of the previous module.', '1901.08164-1-29-4': 'At time step [MATH], for simplicity we will denote all parameters of a module (including auxiliary) as [MATH], and samples as [MATH], which follow the density [MATH].', '1901.08164-1-29-5': 'We aim to prove that each auxiliary problem of the DGL approach will converge to a critical point despite the time varying inputs corresponding to sub-optimal outputs from prior modules.', '1901.08164-1-29-6': 'Proofs are given in Appendix [REF].', '1901.08164-1-30-0': 'Let us fix a depth [MATH], such that [MATH] and consider the target density of the previous layer, [MATH].', '1901.08164-1-30-1': 'We consider the following distance: [MATH].', '1901.08164-1-30-2': 'Denoting [MATH] the composition of the non-negative loss function and the network, we will study the expected risk [MATH].', '1901.08164-1-30-3': 'We will now state several standard assumptions we use.', '1901.08164-1-31-0': '[[MATH]-smoothness] [MATH] is differentiable and its gradient is [MATH]-Lipschitz.', '1901.08164-1-32-0': 'We consider the SGD scheme with learning rate [MATH]: [EQUATION] [Robbins-Monro conditions] The step sizes satisfy [MATH] yet [MATH].', '1901.08164-1-33-0': 'We also assume bounded gradient moments:', '1901.08164-1-34-0': '[Finite variance] There exists [MATH], for any [MATH] and [MATH], [MATH].', '1901.08164-1-35-0': 'The Assumptions 1, 2 and 3 are standard [CITATION], and we show in the following that our proof of convergence leads to similar rates, up to a multiplicative constant.', '1901.08164-1-35-1': 'The following assumption is specific to our setting where we consider a time-varying distribution: [Convergence of the previous layer] We assume that [MATH].', '1901.08164-1-36-0': 'Assumption 3 can be extend to [MATH], Under Assumption 3 and 4, one has: [MATH].', '1901.08164-1-37-0': 'We are now ready to prove the core statement for the convergence results in this setting: Under Assumptions 1, 3 and 4, we have: [EQUATION]', '1901.08164-1-38-0': 'The expectation is taken over each random variable.', '1901.08164-1-38-1': 'Also, note that without the temporal dependency (i.e. [MATH]), this becomes analogous to Lemma 4.4 in [CITATION].', '1901.08164-1-38-2': 'Naturally it follows, that', '1901.08164-1-39-0': 'Thus the DGL scheme converges in the sense of [CITATION].', '1901.08164-1-39-1': 'It is also possible to obtain the following rate:', '1901.08164-1-40-0': 'The sequence of expected gradient norm accumulates around 0 at the following rate: [EQUATION]', '1901.08164-1-40-1': 'Thus compared to the sequential case, the parallel setting adds a delay that is controlled by [MATH].', '1901.08164-1-40-2': 'We now evaluate DGL empirically.', '1901.08164-1-41-0': '# Experiments', '1901.08164-1-42-0': 'We conducted several experiments that empirically show that DGL optimizes the greedy objective well.', '1901.08164-1-42-1': 'We compare the DGL method to others, showing it is a state-of-the-art solution for decoupling training of deep network modules.', '1901.08164-1-42-2': 'We show that it can still work on a large-scale dataset (ImageNet) and that it can, in some cases, generalize better than standard back-propagation.', '1901.08164-1-42-3': 'We also demonstrate positive initial results for the asynchronous variant of the algorithm.', '1901.08164-1-43-0': '## Other Approaches and Auxiliary Network Designs', '1901.08164-1-44-0': 'This section presents experiments evaluating DGL with the CIFAR-10 dataset [CITATION] and standard data augmentation.', '1901.08164-1-44-1': 'We first use a setup that permits us to compare against the DNI method and which also highlights the generality and scalability of DGL.', '1901.08164-1-44-2': 'We then consider the design of a more efficient auxiliary network which we will subsequently use to permit scaling to the ImageNet dataset.', '1901.08164-1-44-3': 'We will also show that DGL is effective at optimizing the greedy objective compared to a naive sequential algorithm.', '1901.08164-1-45-0': 'Comparison to DNI We reproduce the CIFAR-10 CNN experiment described in [CITATION], Appendix C.1.', '1901.08164-1-45-1': 'This experiment utilizes a 3 layer network with auxiliary networks of 2 hidden CNN layers.', '1901.08164-1-45-2': 'We compare our reproduction to the DGL approach.', '1901.08164-1-45-3': 'Instead of the final synthetic gradient prediction for the DGL we apply a final projection to the target prediction space.', '1901.08164-1-45-4': 'We follow the prescribed optimization procedure from [CITATION] in this comparison, using Adam with a learning rate of 3e-5.', '1901.08164-1-45-5': 'We run training for 1500 epochs and compare standard backpropagation, DNI, cDNI [CITATION] and DGL.', '1901.08164-1-45-6': 'Results are shown in Fig. [REF].', '1901.08164-1-45-7': 'Further details are included in the Appendix.', '1901.08164-1-45-8': 'We find that the DGL method outperforms DNI and the context DNI by a substantial amount both in test accuracy and training loss.', '1901.08164-1-45-9': 'We also find in this setting that the DGL can generalize better than standard backpropagation and obtains a very close final training loss.', '1901.08164-1-46-0': 'We also attempted DNI with the more commonly used optimization settings for CNNs (SGD with momentum and step decay), but found that the DNI would diverge when larger learning rates were used, although DGL sub-problem optimization worked effectively with common CNN optimization strategies.', '1901.08164-1-46-1': 'We also note that the prescribed experiment uses a setting where the scalability of our method is not fully exploited.', '1901.08164-1-46-2': 'Each layer of the primary network of [CITATION] has a pooling operation, which permits the auxiliary network to be small for synthetic gradient prediction.', '1901.08164-1-46-3': 'This however severely restricts the architecture choices in the primary network to using a pooling operation at each layer.', '1901.08164-1-46-4': 'In DGL we can apply the pooling operations in the auxiliary network thus permitting the auxiliary network to be negligible in cost even for layers without pooling (whereas for synthetic gradient prediction they often have to be as costly as the base network).', '1901.08164-1-46-5': 'Overall we find that the DGL approach is not only far more scalable and accurate but also more stable and robust to changes in optimization hyper-parameters than DNI.', '1901.08164-1-47-0': 'Auxiliary Network Design We consider different auxiliary networks for CNNs.', '1901.08164-1-47-1': 'As a baseline we use convolutional auxiliary layers as in [CITATION] and [CITATION].', '1901.08164-1-47-2': 'For distributed training application this approach is sub-optimal as the auxiliary network can be substantial compared to the base network, leading to poorer parallelization gains.', '1901.08164-1-47-3': "We note however that even in those cases (that we don't study here) where the auxiliary network computation is potentially on the order of the the primary network, it can still give advantages for parallelization for very deep networks and many available workers.", '1901.08164-1-48-0': 'The primary network architecture we use for these experiments is a simple CNN similar to VGG family models [CITATION].', '1901.08164-1-48-1': 'It consists of 6 convolutional layers with [MATH] kernels, batchnorm and shape preserving padding, with [MATH] maxpooling operations at layers 1 and 3.', '1901.08164-1-48-2': 'The channel width of the first layer is 128 and is doubled at each downsampling operation.', '1901.08164-1-48-3': 'The final layer does not have an auxiliary model, it is learned with a linear spatial averaging followed by a 2-hidden layer constant depth fully connected network, for all experiments.', '1901.08164-1-48-4': 'Two alternatives to the CNN auxiliary of [CITATION] are explored, which exploit a spatial averaging operation.', '1901.08164-1-48-5': 'We re-iterate that this kind of approach and even the simple network structure we consider is not easily applicable in the case of DNI and synthetic gradient prediction.', '1901.08164-1-48-6': 'Optimization is done using a standard strategy for CIFAR CNN training.', '1901.08164-1-48-7': 'We apply SGD with momentum of [MATH] and weight decay 5e-4 [CITATION] and decaying step sizes.', '1901.08164-1-48-8': 'For these experiments we use a short schedule of 50 epochs and decay factor of [MATH] every 15 epochs [CITATION].', '1901.08164-1-48-9': 'Results of comparisons are given in Table [REF].', '1901.08164-1-49-0': 'The baseline auxiliary strategy based on [CITATION] and [CITATION] applies 2 CNN layers followed by a spatial averaging to [MATH] before a final projection.', '1901.08164-1-49-1': 'We denote this CNN-aux.', '1901.08164-1-49-2': 'The first alternative we explore is a direct application of the spatial averaging to [MATH] output shape (regardless of the input resolution) followed by a 3 layer MLP (of constant width).', '1901.08164-1-49-3': 'This is denoted MLP-aux and drastically reduces the FLOP count with minimal accuracy loss compared to CNN-aux.', '1901.08164-1-49-4': 'Finally we consider applying a staged spatial resolution, first reducing the spatial resolution by 4[MATH] (and total size 16[MATH]), then applying 3 [MATH] convolutions followed by a reduction to [MATH] and a 3 layer MLP.', '1901.08164-1-49-5': 'We denote this approach MLP-SR-aux.', '1901.08164-1-49-6': 'These latter two strategies that leverage the spatial averaging produce auxiliary networks that are less than [MATH] of the FLOP count of the primary network even for large spatial resolutions as in real world image datasets.', '1901.08164-1-49-7': 'We will show that MLP-SR-aux is still effective even for the large-scale ImageNet dataset.', '1901.08164-1-50-0': 'Sequential vs. Parallel Optimization of Greedy Objective', '1901.08164-1-51-0': 'We briefly compare the sequential optimization of the greedy objective [CITATION] to the DGL (Alg. [REF]).', '1901.08164-1-51-1': 'We use a 4 layer CIFAR-10 network with an MLP-SR-aux auxiliary model and a final layer attached to a 2 layer MLP.', '1901.08164-1-51-2': 'We use the same optimization settings with 50 epochs as in the last experiment.', '1901.08164-1-51-3': 'In the sequential training we train each layer for 50 epochs before moving to the subsequent one.', '1901.08164-1-51-4': 'Thus the difference to DGL lies only in the input received at each layer (fully converged previous layer versus not fully converged previous layer).', '1901.08164-1-51-5': 'The rest of the optimization settings are identical.', '1901.08164-1-51-6': 'Figure [REF] shows comparisons of the learning curves for sequential training and DGL at layer 4 (layer 1 is the same for both as the input representation is not varying over the training period).', '1901.08164-1-51-7': 'We observe that the DGL quickly catches up with the sequential training scheme and appears to sometimes generalize better.', '1901.08164-1-51-8': 'We additionally visualize the dynamics of training per layer in Fig. [REF], which demonstrates that after just a few epochs the individual layers build a dynamic of progressive improvement with depth.', '1901.08164-1-51-9': 'Additional visualizations are included in the supplementary materials.', '1901.08164-1-52-0': 'Multi-Layer modules Although we have so far considered the setting of layer-wise decoupling, this approach can easily be applied to generic modules.', '1901.08164-1-52-1': 'Indeed approaches such as DNI [CITATION] often consider decoupling entire multi-layer modules.', '1901.08164-1-52-2': 'Furthermore the propositions for backward unlocking [CITATION] also rely on and report they can often only decouple 100 layer networks into 2 or 4 blocks before observing optimization issues or performance losses and require that the number of parallel modules is much lower than the network depth for the theoretical guarantees to hold.', '1901.08164-1-52-3': 'As in those cases, using multi-layer decoupled modules can improve performance and is natural in the case of deeper networks.', '1901.08164-1-52-4': 'We now use such a multi-layer approach to directly compare to the backward unlocking of [CITATION] and then subsequently we will apply this on deep networks for ImageNet.', '1901.08164-1-52-5': 'We will denote from here-on the number of total modules a network is split into as [MATH].', '1901.08164-1-53-0': 'Comparison to DDG [CITATION] proposes a solution to the backward locking (less efficient than solving update-locking, see discussion above) We show that even in this situation the DGL method can provide a strong baseline for work on backward unlocking.', '1901.08164-1-53-1': 'We take the example from [CITATION], which considers a ResNet-110 parallelized into [MATH] blocks.', '1901.08164-1-53-2': 'We use the auxiliary network MLP-SR-aux which has less than [MATH] the FLOP count of the primary network.', '1901.08164-1-53-3': 'We use the exact optimization and network split points as in [CITATION].', '1901.08164-1-53-4': 'To assess variance in the accuracy for CIFAR-10 we perform 3 trials.', '1901.08164-1-53-5': 'We observe in Tab. [REF] that the accuracy is the same across the DDG method, backprop, and our approach.', '1901.08164-1-53-6': 'DDG achieves better parallelization because it also splits the forward pass.', '1901.08164-1-54-0': '## Large-scale Experiments', '1901.08164-1-55-0': 'Existing methods considering update or backward locking have not been evaluated on large image datasets as they are often unstable or already show large losses in accuracy on smaller datasets.', '1901.08164-1-55-1': 'Here we study the optimization of several well-known architectures, mainly the VGG family [CITATION] and the ResNet [CITATION].', '1901.08164-1-55-2': 'In all our experiments we use the MLP-SR-aux auxiliary network which scales well from the smaller CIFAR-10 images to the larger ImageNet ones.', '1901.08164-1-55-3': 'The final module does not have an auxiliary model.', '1901.08164-1-56-0': 'For all optimization of auxiliary problems and for end-to-end optimization of reference models we use the optimization schedule prescribed in [CITATION].', '1901.08164-1-56-1': 'It consists of training for 50 epochs with mini-batches size [MATH], uses SGD with momentum of 0.9, weight decay of 1e-4, and a learning rate of [MATH] reduced by a factor 10 every 10 epochs.', '1901.08164-1-56-2': 'Results are shown in Tab. [REF].', '1901.08164-1-56-3': 'For several of the models DGL can perform as well and sometimes better than the end-to-end trained model, while permitting parallel training.', '1901.08164-1-56-4': 'For the VGG-13 architecture we also evaluate the case where the model is trained layer by layer ([MATH]).', '1901.08164-1-56-5': 'Although performance is degraded by this split we find its performance surprising considering that no backward communication is performed.', '1901.08164-1-56-6': 'We conjecture that improved auxiliary models and combinations with methods such as [CITATION] to allow feedback on top of the local model, may further improve performance.', '1901.08164-1-56-7': 'Also as mentioned for the settings with larger potential parallelization, slower but more performant auxiliary models could potentially be considered as well.', '1901.08164-1-57-0': 'We also remark that the synchronous DGL has favorable memory usage compared to DDG and to the DNI method.', '1901.08164-1-57-1': 'DNI requiring to store larger activations and DDG having memory.', '1901.08164-1-57-2': 'Although not the focus of this work in the single worker version of DGL has favorable memory usage compared to standard end-to-end backpropagation training.', '1901.08164-1-57-3': 'For example the ResNet-152 DGL [MATH] setting we have considered can fit [MATH] more samples on a single 16GB GPU than the the standard end-to-end training.', '1901.08164-1-58-0': '## Asynchronous DGL with Replay', '1901.08164-1-59-0': 'We now study the stability of Alg. [REF] w.r.t. the buffers.', '1901.08164-1-59-1': 'We use a 5 layer CIFAR-10 network with the MLP-aux and with all other architecture and optimization settings as in the auxiliary network experiments of Sec. [REF].', '1901.08164-1-59-2': 'Each layer is equipped with a buffer of size [MATH].', '1901.08164-1-59-3': 'At each iteration, a layer is chosen according to the pmf [MATH], and a batch selected from buffer [MATH].', '1901.08164-1-59-4': 'One layer is slowed down by decreasing its selection probability in the pmf [MATH] by a factor [MATH].', '1901.08164-1-59-5': 'We evaluate different slowdown factors (up to [MATH]).', '1901.08164-1-59-6': 'Accuracy versus slowdown factor is shown in Fig. [REF].', '1901.08164-1-59-7': 'For this experiment we use a buffer of size [MATH].', '1901.08164-1-59-8': 'We run separate experiments with the slowdown applied at each layer of the network as well as 3 random seeds for each of these settings (for a total of 18 experiments per data point).', '1901.08164-1-59-9': 'We show the evaluations for 10 values of [MATH].', '1901.08164-1-59-10': 'To ensure a fair comparison we also stop updating layers once they have completed the iterations for 50 epochs, thus assuring identical number of gradient updates for all layers in all experiments compared.', '1901.08164-1-59-11': 'In practice one could continue updating until all layers have completed training.', '1901.08164-1-59-12': 'In Fig. [REF] and compare it to the synchronous case (standard DGL).', '1901.08164-1-59-13': 'We first observe that the accuracy of the synchronous algorithm is maintained in the setting where [MATH] and the pmf is uniform.', '1901.08164-1-59-14': 'Note that even this is a non-trivial case, as it will mean that layers inherently have random delays (as compared to the synchronous Alg. [REF]).', '1901.08164-1-59-15': 'We see that accuracy is maintained until approximately [MATH] and accuracy losses after that still remain small.', '1901.08164-1-59-16': 'Our maximum slowdown factor of [MATH] is somewhat drastic - it means that for the 50 epochs of training, the slowed-down layer is only on epoch 25 while those following it are at epoch 50.', '1901.08164-1-60-0': 'In a second experiment we evaluate performance with respect to the buffer size.', '1901.08164-1-60-1': 'Results are shown in Fig. [REF].', '1901.08164-1-60-2': 'For this experiment we fix the slowdown factor to [MATH].', '1901.08164-1-60-3': 'We observe that even when a very small buffer size can yield only a slight loss in performance accuracy.', '1901.08164-1-60-4': 'Indeed building on this demonstration there are multiple direction to improve Async DGL with replay.', '1901.08164-1-60-5': 'For example improving the efficiency of the buffer, by including data augmentation in feature space [CITATION], mixing samples in batches, and improved batch sampling among other directions.', '1901.08164-1-61-0': '# Conclusion', '1901.08164-1-62-0': 'We have analyzed and introduced a simple and strong baseline for parallelizing per layer and per module computations in CNN training.', '1901.08164-1-62-1': 'Our approach is shown to match or exceed state of the art approaches addressing these problems and shown able to scale to much large datasets than others.', '1901.08164-1-62-2': 'Future work can develop improved auxiliary problem objectives and combinations with delayed feedback.'} | {'1901.08164-2-0-0': 'A commonly cited inefficiency of neural network training by back-propagation is the update locking problem: each layer must wait for the signal to propagate through the network before updating.', '1901.08164-2-0-1': 'In recent years multiple authors have considered alternatives that can alleviate this issue.', '1901.08164-2-0-2': 'In this context, we consider a simpler, but more effective, substitute that uses minimal feedback, which we call Decoupled Greedy Learning (DGL).', '1901.08164-2-0-3': 'It is based on a greedy relaxation of the joint training objective, recently shown to be effective in the context of Convolutional Neural Networks (CNNs) on large-scale image classification.', '1901.08164-2-0-4': 'We consider an optimization of this objective that permits us to decouple the layer training, allowing for layers or modules in networks to be trained with a potentially linear parallelization in layers.', '1901.08164-2-0-5': 'We show theoretically and empirically that this approach converges.', '1901.08164-2-0-6': 'Then, we empirically find that it can lead to better generalization than sequential greedy optimization and sometimes end-to-end back-propagation.', '1901.08164-2-0-7': 'We show an extension of this approach to asynchronous settings, where modules can operate with large communication delays, is possible with the use of a replay buffer.', '1901.08164-2-0-8': 'We demonstrate the effectiveness of DGL on the CIFAR-10 dataset against alternatives and on the large-scale ImageNet dataset.', '1901.08164-2-1-0': '# Introduction', '1901.08164-2-2-0': 'Jointly training all layers using back-propagation is the standard method for learning neural networks, including the computationally intensive vision models based on Convolutional Neural Networks (CNNs) [CITATION].', '1901.08164-2-2-1': 'Due to the sequential nature of gradient processing, standard back-propagation has several well-known inefficiencies that prohibit parallelization of the computations of the different constituent modules.', '1901.08164-2-2-2': '[CITATION] characterizes these in order of severity as the forward, update, and backward locking problems.', '1901.08164-2-2-3': 'Backward unlocking would permit updates of all modules once forward signals have propagated to all subsequent modules, update unlocking would permit updates of a module before a signal has reached all subsequent modules, and forward unlocking would permit a module to operate asynchronously from its predecessor and dependent modules.', '1901.08164-2-3-0': 'Methods addressing backward locking to a certain degree have been proposed in [CITATION].', '1901.08164-2-3-1': 'However, update locking is a far more severe inefficiency.', '1901.08164-2-3-2': 'Thus [CITATION] propose and analyze Decoupled Neural Interfaces (DNI), a method that uses an auxiliary network to predict the gradient of the backward pass directly from the input.', '1901.08164-2-3-3': 'This method unfortunately does not scale well computationally or in terms of accuracy, especially in the case of CNNs [CITATION].', '1901.08164-2-3-4': 'Indeed, auxiliary networks must predict a weight gradient that can be very large in dimensionality for larger models and input sizes.', '1901.08164-2-4-0': 'A major obstacle to update unlocking is the heavy reliance on the upper modules for feedback.', '1901.08164-2-4-1': 'Several works have recently revisited the classic [CITATION] approach of supervised greedy layer-wise training of neural networks [CITATION].', '1901.08164-2-4-2': 'In [CITATION] it is shown that such an approach, which relaxes the joint learning objective, and does not require global feedback, can lead to high-performance deep CNNs on large-scale datasets.', '1901.08164-2-4-3': 'We will show that the greedy learning objective used in these papers can be solved with an alternative optimization algorithm, which permits decoupling the computations and achieves update unlocking.', '1901.08164-2-4-4': 'It can be augmented with replay buffers [CITATION] to permit forward unlocking.', '1901.08164-2-4-5': 'This simpler strategy can be shown to be a state-of-the-art baseline for parallelizing the training across modules of a neural network.', '1901.08164-2-5-0': 'Our contributions in this work are as follows.', '1901.08164-2-5-1': '(a) We propose an optimization procedure for a decoupled greedy learning objective that achieves update unlocking.', '1901.08164-2-5-2': '(b) Empirically, we show that it exhibits similar convergence rates and generalization as its non-decoupled counterpart.', '1901.08164-2-5-3': '(c) We show that it can be extended to an asynchronous setting using a replay buffer, providing a step towards addressing the forward locking problem.', '1901.08164-2-5-4': '(d) We motivate these observations theoretically, showing that the proposed optimization procedure converges and recovers standard rates of non-convex optimization.', '1901.08164-2-5-5': 'Experimentally we (e) design an improved auxiliary network structure for greedy layer-wise training of CNNs that maintains accuracy with negligible computational cost for the auxiliary task.', '1901.08164-2-5-6': '(f) We show that DGL can outperform competing methods in terms of scalability to larger and deeper models and stability to optimization hyperparameters, allowing it to be applied to large datasets.', '1901.08164-2-5-7': 'We then demonstrate on the ImageNet dataset that we can train the deep models VGG-19 and ResNet-152 with larger degrees of parallelism and reduced memory consumption compared to other works.', '1901.08164-2-6-0': '# Parallel Decoupled Greedy Learning', '1901.08164-2-7-0': 'In this section we formally define the greedy objective and parallel optimization which we study in both the synchronous and asynchronous setting.', '1901.08164-2-7-1': 'We mainly consider the online setting and assume a stream of samples or mini-batches denoted [MATH], that can be run during [MATH] iterations.', '1901.08164-2-8-0': '## Preliminaries', '1901.08164-2-9-0': 'For comparison purposes, we briefly review the update unlocking approach from DNI [CITATION].', '1901.08164-2-9-1': 'There, each network module has an associated auxiliary net which, given the output activation of the module, predicts the gradient signal from subsequent modules: the module can thus perform an update while modules above are still forward processing.', '1901.08164-2-9-2': 'The DNI auxiliary model is trained by using true gradients provided by upper modules when they become available, requiring activation caching.', '1901.08164-2-9-3': 'This also means that the auxiliary module can become out of sync with the changing output activation distribution, often requiring slow learning rates.', '1901.08164-2-9-4': 'Due to this and the high dimensionality of the predicted gradient which scales with module size, this estimate is challenging.', '1901.08164-2-9-5': 'One may ask how well a method that entirely avoids the use of feedback from upper modules would fare given similarly-sized auxiliary networks.', '1901.08164-2-9-6': 'We will show that adapting the objective in [CITATION] can also allow for update unlock and a degree of forward unlocking, with better properties.', '1901.08164-2-10-0': '## Optimization for Greedy Objective', '1901.08164-2-11-0': 'Let [MATH] and [MATH] be the data matrix and labels for the training data, [MATH] be the output representation for module [MATH].', '1901.08164-2-11-1': 'We will denote the per-module objective function [MATH], where the parameters [MATH] correspond to the module parameter (i.e. [MATH]) and [MATH] represents auxiliary parameters used to compute the objective.', '1901.08164-2-11-2': '[MATH] in our case will be the empirical risk with a cross-entropy loss.', '1901.08164-2-11-3': 'The greedy training objective is thus given recursively by defining [MATH]: [EQUATION] where [MATH] and [MATH] is the minimizer of Problem ( [MATH] ).', '1901.08164-2-11-4': 'A natural way to solve the optimization problem for [MATH] modules, [MATH], is thus by sequentially solving the problems [MATH] starting with [MATH] (e.g. [CITATION]).', '1901.08164-2-11-5': 'Here we consider an alternative procedure for optimizing this objective given in Alg. [REF]: individual updates of each set of parameters are performed in sequence across the different layers.', '1901.08164-2-11-6': 'Each layer processes a sample or mini-batch, then passes it to the next layer.', '1901.08164-2-11-7': 'Note that at line [MATH] the subsequent layer can already begin computing line [MATH].', '1901.08164-2-11-8': 'Therefore, this algorithm achieves update unlocking.', '1901.08164-2-11-9': 'Appendix [REF] gives an explicit version of an equivalent multi-worker pseudo-code.', '1901.08164-2-11-10': 'Fig. [REF] illustrates the decoupling compared to how samples are processed in standard back-propagation.', '1901.08164-2-11-11': 'Once [MATH] has been computed, processing by subsequent layers can begin.', '1901.08164-2-12-0': '## Asynchronous Decoupled Greedy Learning with Replay', '1901.08164-2-13-0': 'We extend this framework to address forward unlocking [CITATION].', '1901.08164-2-13-1': 'DGL modules already do not depend on their successors for updates.', '1901.08164-2-13-2': 'We can further reduce dependency on the previous modules such that they can operate asynchronously.', '1901.08164-2-13-3': 'This is achieved via a replay buffer that is shared between adjacent modules, enabling them to reuse older samples.', '1901.08164-2-13-4': 'Scenarios with communication delays or substantial variations in speed between layers/modules benefit from this.', '1901.08164-2-13-5': 'We study one instance of such an algorithm that uses a replay buffer of size [MATH], shown in Alg. [REF].', '1901.08164-2-13-6': 'L0.52', '1901.08164-2-14-0': '[H] Synchronous DGL Stream [MATH] of samples or mini-batches.', '1901.08164-2-14-1': 'Initialize Parameters [MATH].', '1901.08164-2-14-2': '[MATH].', '1901.08164-2-14-3': 'Compute [MATH].', '1901.08164-2-14-4': '[MATH]Update parameters [MATH].', '1901.08164-2-15-0': '[H] Stream [MATH]; Distribution of the delay [MATH]; Buffer size [MATH].', '1901.08164-2-15-1': 'Initialize: Buffers [MATH]; params [MATH].', '1901.08164-2-16-0': 'trainingSample [MATH] in [MATH] following [MATH].', '1901.08164-2-17-0': '[MATH].', '1901.08164-2-17-1': '[MATH].', '1901.08164-2-17-2': 'Compute [MATH].', '1901.08164-2-17-3': '[MATH] Update parameters [MATH].', '1901.08164-2-17-4': '[MATH].', '1901.08164-2-18-0': 'Asynchronous DGL with Replay', '1901.08164-2-19-0': 'Our minimal distributed setting is as follows.', '1901.08164-2-19-1': 'Each worker [MATH] has a buffer that it writes to and that worker [MATH] can read from.', '1901.08164-2-19-2': 'The buffer uses a simple read/write protocol.', '1901.08164-2-19-3': 'A buffer [MATH] lets layer [MATH] write new samples.', '1901.08164-2-19-4': 'When it reaches capacity it overwrites the oldest sample.', '1901.08164-2-19-5': 'Layer [MATH] requests samples from the buffer [MATH].', '1901.08164-2-19-6': 'They are selected by a last-in-first-out (LIFO) rule, with precedence for the least-reused samples.', '1901.08164-2-19-7': 'Alg. [REF] simulates potential delays in such a setup by the use of a probability mass function (pmf) [MATH] over workers, analogous to typical asynchronous settings such as [CITATION].', '1901.08164-2-19-8': 'At each iteration, a layer is chosen at random according to [MATH] to perform a computation.', '1901.08164-2-19-9': 'In our experiments we limit ourselves to pmfs that are uniform over workers except for a single layer which is chosen to be selected less frequently on average.', '1901.08164-2-19-10': 'Even in the case of a uniform pmf, asynchronous behavior will naturally arise requiring the reuse of samples.', '1901.08164-2-19-11': 'Alg. [REF] permits a controlled simulation of processing speed discrepancies and will be used over settings of [MATH] and [MATH] to demonstrate that training and testing accuracy remain robust in practical regimes.', '1901.08164-2-19-12': 'Appendix [REF] also provides pseudo-code for implementation in a parallel environment.', '1901.08164-2-20-0': 'Unlike common data-parallel asynchronous algorithms [CITATION], the asynchronous DGL does not rely on a master node and requires only local communication similar to recent decentralized schemes [CITATION].', '1901.08164-2-20-1': 'Contrary to decentralized SGD, DGL nodes only need to maintain and update the parameters of their local module, permitting much larger modules.', '1901.08164-2-20-2': 'Combining asynchronous DGL with distributed synchronous SGD for sub-problem optimization is a promising direction.', '1901.08164-2-20-3': 'For example it can alleviate a common issue of the popular distributed synchronous SGD in deep CNNs, which is the often limiting maximum batch size [CITATION].', '1901.08164-2-21-0': '## Auxiliary and Primary Network Design', '1901.08164-2-22-0': 'Like DNI our procedure relies on an auxiliary network to obtain update signal, both methods thus require auxiliary network design in addition to the main CNN architecture.', '1901.08164-2-22-1': '[CITATION] have shown that spatial averaging operations can be used to construct a scalable auxiliary network for the same objective as used in Sec [REF].', '1901.08164-2-22-2': 'However, they did not directly consider the parallel training use case, where additional care must be taken in the design: The primary consideration is the relative speed of the auxiliary network with respect to its associated main network module.', '1901.08164-2-22-3': 'We will use primarily FLOP count in our analysis and aim to restrict our auxiliary networks to be [MATH] of the main network.', '1901.08164-2-23-0': 'Although auxiliary network design might seem like an additional layer of complexity in CNN design and may require invoking slightly different architecture principles, this is not inherently prohibitive since architecture design is often related to the training (e.g., the use of residuals is originally motivated by optimization issues inherent to end-to-end backprop [CITATION]).', '1901.08164-2-24-0': 'Finally, we note although we focus on the distributed learning context, this algorithm and associated theory for greedy objectives is generic and has other potential applications.', '1901.08164-2-24-1': 'For example greedy objectives have recently been used in [CITATION] and even with a single worker DGL reduces memory.', '1901.08164-2-25-0': '# Theoretical Analysis', '1901.08164-2-26-0': 'We now analyze the convergence of Alg. [REF] when the update steps are obtained from stochastic gradient methods.', '1901.08164-2-26-1': 'We show convergence guarantees [CITATION] under reasonable assumptions.', '1901.08164-2-26-2': 'In standard stochastic optimization schemes, the input distribution fed to a model is fixed [CITATION].', '1901.08164-2-26-3': 'In this work, the input distribution to each module is time-varying and dependent on the convergence of the previous module.', '1901.08164-2-26-4': 'At time step [MATH], for simplicity we will denote all parameters of a module (including auxiliary) as [MATH], and samples as [MATH], which follow the density [MATH].', '1901.08164-2-26-5': 'For each auxiliary problem, we aim to prove the strongest existing guarantees [CITATION] for the non-convex setting despite time-varying input distributions from prior modules.', '1901.08164-2-26-6': 'Proofs are given in the Appendix.', '1901.08164-2-27-0': 'Let us fix a depth [MATH], such that [MATH] and consider the converged density of the previous layer, [MATH].', '1901.08164-2-27-1': 'We consider the following distance: [MATH].', '1901.08164-2-27-2': 'Denoting [MATH] the composition of the non-negative loss function and the network, we will study the expected risk [MATH].', '1901.08164-2-27-3': 'We will now state several standard assumptions we use.', '1901.08164-2-28-0': '[[MATH]-smoothness] [MATH] is differentiable and its gradient is [MATH]-Lipschitz.', '1901.08164-2-28-1': 'We consider the SGD scheme with learning rate [MATH]: [EQUATION] [Robbins-Monro conditions] The step sizes satisfy [MATH] yet [MATH].', '1901.08164-2-29-0': '[Finite variance][MATH].', '1901.08164-2-30-0': 'The Assumptions 1, 2 and 3 are standard [CITATION], and we show in the following that our proof of convergence leads to similar rates, up to a multiplicative constant.', '1901.08164-2-30-1': 'The following assumption is specific to our setting where we consider a time-varying distribution: [Convergence of the previous layer] We assume that [MATH].', '1901.08164-2-30-2': 'Under Assumption 3 and 4, one has: [MATH].', '1901.08164-2-31-0': 'We are now ready to prove the core statement for the convergence results in this setting: Under Assumptions 1, 3 and 4, we have: [EQUATION]', '1901.08164-2-32-0': 'The expectation is taken over each random variable.', '1901.08164-2-32-1': 'Also, note that without the temporal dependency (i.e. [MATH]), this becomes analogous to Lemma 4.4 in [CITATION].', '1901.08164-2-32-2': 'Naturally it follows, that', '1901.08164-2-33-0': 'Thus the DGL scheme converges in the sense of [CITATION].', '1901.08164-2-33-1': 'We can also obtain the following rate: The sequence of expected gradient norm accumulates around 0 at the following rate: [EQUATION]', '1901.08164-2-34-0': 'Thus compared to the sequential case, the parallel setting adds a delay that is controlled by [MATH].', '1901.08164-2-35-0': '# Experiments', '1901.08164-2-36-0': 'We conduct experiments that empirically show that DGL optimizes the greedy objective well, showing it is favorable against recent state-of-the-art proposals for decoupling training of deep network modules.', '1901.08164-2-36-1': 'We show that it can still work on a large-scale dataset (ImageNet) and that it can, in some cases, generalize better than standard back-propagation.', '1901.08164-2-36-2': 'We also demonstrate positive initial results for the asynchronous variant of the algorithm.', '1901.08164-2-36-3': 'For all experiments we use architectures taken from prior works and standard optimization settings.', '1901.08164-2-37-0': '## Comparisons to Other Approaches and Auxiliary Network Designs', '1901.08164-2-38-0': 'This section presents experiments evaluating DGL with the CIFAR-10 dataset [CITATION] and standard data augmentation.', '1901.08164-2-38-1': 'We first use a setup that permits us to compare against the DNI method and which also highlights the generality and scalability of DGL.', '1901.08164-2-38-2': 'We then consider the design of a more efficient auxiliary network which will help to scale to the ImageNet dataset.', '1901.08164-2-38-3': 'We will also show that DGL is effective at optimizing the greedy objective compared to a naive sequential algorithm.', '1901.08164-2-39-0': 'Comparison to DNI We reproduce the CIFAR-10 CNN experiment described in [CITATION], Appendix C.1.', '1901.08164-2-39-1': 'This experiment utilizes a 3 layer network with auxiliary networks of 2 hidden CNN layers.', '1901.08164-2-39-2': 'We compare our reproduction to the DGL approach.', '1901.08164-2-39-3': 'Instead of the final synthetic gradient prediction for the DGL we apply a final projection to the target prediction space.', '1901.08164-2-39-4': 'Here, we follow the prescribed optimization procedure from [CITATION], using Adam with a learning rate of [MATH].', '1901.08164-2-39-5': 'We run training for 1500 epochs and compare standard backprop, DNI, context DNI (cDNI) [CITATION] and DGL.', '1901.08164-2-39-6': 'Results are shown in Fig. [REF].', '1901.08164-2-39-7': 'Details are included in the Appendix.', '1901.08164-2-39-8': 'The DGL method outperforms DNI and the cDNI by a substantial amount both in test accuracy and training loss.', '1901.08164-2-39-9': 'Also in this setting, DGL can generalize better than standard backprop and obtains a very close final training loss.', '1901.08164-2-40-0': 'We also attempted DNI with the more commonly used optimization settings for CNNs (SGD with momentum and step decay), but found that DNI would diverge when larger learning rates were used, although DGL sub-problem optimization worked effectively with common CNN optimization strategies.', '1901.08164-2-40-1': 'We also note that the prescribed experiment uses a setting where the scalability of our method is not fully exploited.', '1901.08164-2-40-2': 'Each layer of the primary network of [CITATION] has a pooling operation, which permits the auxiliary network to be small for synthetic gradient prediction.', '1901.08164-2-40-3': 'This however severely restricts the architecture choices in the primary network to using a pooling operation at each layer.', '1901.08164-2-40-4': 'In DGL, we can apply the pooling operations in the auxiliary network, thus permitting the auxiliary network to be negligible in cost even for layers without pooling (whereas synthetic gradient predictions often have to be as costly as the base network).', '1901.08164-2-40-5': 'Overall, DGL is far more scalable, accurate, and more robust to changes in optimization hyper-parameters than DNI.', '1901.08164-2-41-0': 'Auxiliary Network Design We consider different auxiliary networks for CNNs.', '1901.08164-2-41-1': 'As a baseline we use convolutional auxiliary layers as in [CITATION] and [CITATION].', '1901.08164-2-41-2': 'For distributed training application this approach is sub-optimal as the auxiliary network can be substantial compared to the base network, leading to poorer parallelization gains.', '1901.08164-2-41-3': "We note however that even in those cases (that we don't study here) where the auxiliary network computation is potentially on the order of the the primary network, it can still give advantages for parallelization for very deep networks and many available workers.", '1901.08164-2-42-0': 'The primary network architecture we use for this study is a simple CNN similar to VGG family models [CITATION] and those used in [CITATION].', '1901.08164-2-42-1': 'It consists of 6 convolutions of size [MATH], batchnorm and shape preserving padding, with [MATH] maxpooling at layers 1 and 3.', '1901.08164-2-42-2': 'The width of the first layer is 128 and is doubled at each downsampling operation.', '1901.08164-2-42-3': 'The final layer does not have an auxiliary model- it is followed by a pooling and 2-hidden layer fully connected network, for all experiments.', '1901.08164-2-42-4': 'Two alternatives to the CNN auxiliary of [CITATION] are explored below.', '1901.08164-2-42-5': 'Results are given in Tab. [REF].', '1901.08164-2-42-6': 'R0.45', '1901.08164-2-43-0': 'Comparison of auxiliary networks on CIFAR.', '1901.08164-2-43-1': 'CNN-aux applied in previous work is inefficient w.r.t. the primary module.', '1901.08164-2-43-2': 'We report flop count of the aux net relative to the largest module.', '1901.08164-2-43-3': 'MLP-aux and MLP-SR-aux applied after spatial averaging operations are far more effective with min. acc.', '1901.08164-2-43-4': 'loss.', '1901.08164-2-44-0': 'The baseline auxiliary strategy based on [CITATION] and [CITATION] applies 2 CNN layers followed by a [MATH] averaging and projection, denoted as CNN-aux.', '1901.08164-2-44-1': 'First, we explore a direct application of the spatial averaging to [MATH] output shape (regardless of the resolution) followed by a 3-layer MLP (of constant width).', '1901.08164-2-44-2': 'This is denoted MLP-aux and drastically reduces the FLOP count with minimal accuracy loss compared to CNN-aux.', '1901.08164-2-44-3': 'Finally, we study a staged spatial resolution, first reducing the spatial resolution by 4[MATH] (and total size 16[MATH]), then applying 3 [MATH] convolutions followed by a reduction to [MATH] and a 3 layer MLP, that we denote as MLP-SR-aux.', '1901.08164-2-44-4': 'These latter two strategies that leverage the spatial averaging produce auxiliary networks that are less than [MATH] of the FLOP count of the primary network even for large spatial resolutions as in real world image datasets.', '1901.08164-2-44-5': 'We will show that MLP-SR-aux is still effective even for the large-scale ImageNet dataset.', '1901.08164-2-44-6': "We note that these more effective auxiliary models are not easily applicable in the case of DNI's gradient prediction.", '1901.08164-2-45-0': 'Sequential vs. Parallel Optimization of Greedy Objective We briefly compare the sequential optimization of the greedy objective [CITATION] to the DGL (Alg. [REF]).', '1901.08164-2-45-1': 'We use a 6 layer CIFAR-10 network with an MLP-SR-aux auxiliary model.', '1901.08164-2-45-2': 'In parallel we train the layers together for 50 epochs and in the sequential training we train each layer for 50 epochs before moving to the subsequent one.', '1901.08164-2-45-3': 'Thus the difference to DGL lies only in the input received at each layer (fully converged previous layer versus not fully converged previous layer).', '1901.08164-2-45-4': 'The rest of the optimization settings are identical.', '1901.08164-2-45-5': 'Fig. [REF] shows comparisons of the learning curves for sequential training and DGL at layer 4 (layer 1 is the same for both as the input representation is not varying over the training period).', '1901.08164-2-45-6': 'DGL quickly catches up with the sequential training scheme and appears to sometimes generalize better.', '1901.08164-2-45-7': 'We also visualize the dynamics of training per layer in Fig. [REF], which demonstrates that after just a few epochs the individual layers build a dynamic of progressive improvement with depth.', '1901.08164-2-46-0': 'Multi-Layer modules We have so far mainly considered the setting of layer-wise decoupling.', '1901.08164-2-46-1': 'This approach however can easily be applied to generic modules.', '1901.08164-2-46-2': 'Indeed, approaches such as DNI [CITATION] often consider decoupling entire multi-layer modules.', '1901.08164-2-46-3': 'Furthermore the propositions for backward unlocking [CITATION] also rely on and report they can often only decouple 100 layer networks into 2 or 4 blocks before observing optimization issues or performance losses and require that the number of parallel modules be much lower than the network depth for the theoretical guarantees to hold.', '1901.08164-2-46-4': 'As in those cases, using multi-layer decoupled modules can improve performance and is natural in the case of deeper networks.', '1901.08164-2-46-5': 'We now use such a multi-layer approach to directly compare to the backward unlocking of [CITATION] and then subsequently we will apply this on deep networks for ImageNet.', '1901.08164-2-46-6': 'From here on we will denote [MATH] the number of total modules a network is split into.', '1901.08164-2-47-0': 'Comparison to DDG [CITATION] proposes a solution to the backward locking (less efficient than solving update-locking, see discussion in Sec [REF]).', '1901.08164-2-47-1': 'R0.39 ResNet-110([MATH]) for Backprop and DDG method from [CITATION].', '1901.08164-2-47-2': 'DGL is run for 3 trials to compute variance.', '1901.08164-2-47-3': 'They give the same acc.', '1901.08164-2-47-4': 'with DGL being update unlocked, DDG only backward unlocked.', '1901.08164-2-47-5': 'DNI is reported to not work in this setting [CITATION].', '1901.08164-2-48-0': 'We show that even in this situation the DGL method can provide a strong baseline for work on backward unlocking.', '1901.08164-2-48-1': 'We take the experimental setup from [CITATION], which considers a ResNet-110 parallelized into [MATH] blocks.', '1901.08164-2-48-2': 'We use the auxiliary network MLP-SR-aux which has less than [MATH] the FLOP count of the primary network.', '1901.08164-2-48-3': 'We use the exact optimization and network split points as in [CITATION].', '1901.08164-2-49-0': 'To assess variance in CIFAR-10 accuracy, we perform 3 trials.', '1901.08164-2-49-1': 'Tab. [REF] shows that the accuracy is the same across the DDG method, backprop, and our approach.', '1901.08164-2-49-2': 'DGL achieves better parallelization because it is update unlocked.', '1901.08164-2-49-3': 'We use the parallel implementation provided by [CITATION] to obtain a direct wall clock time comparison.', '1901.08164-2-49-4': 'We note that there are multiple considerations for comparing speed across these methods (see Appendix [REF]).', '1901.08164-2-50-0': 'Wall Time Comparison We compare to the parallel implementation of [CITATION] using the same communication protocols and run on the same hardware.', '1901.08164-2-50-1': 'We find for [MATH] GPU gives a [MATH] respectively speedup over DDG.', '1901.08164-2-50-2': 'With DDG [MATH] giving approximately [MATH] speedup over standard backprop on same hardware (close to results from [CITATION]).', '1901.08164-2-51-0': '## Large-scale Experiments', '1901.08164-2-52-0': 'Existing methods considering update or backward locking have not been evaluated on large image datasets as they are often unstable or already show large losses in accuracy on smaller datasets.', '1901.08164-2-52-1': 'Here we study the optimization of several well-known architectures, mainly the VGG family [CITATION] and the ResNet [CITATION], with DGL.', '1901.08164-2-52-2': 'In all our experiments we use the MLP-SR-aux auxiliary network which scales well from the smaller CIFAR-10 to the larger ImageNet.', '1901.08164-2-52-3': 'The final module has no auxiliary network.', '1901.08164-2-52-4': 'For all optimization of auxiliary problems and for end-to-end optimization of reference models we use the shortened optimization schedule prescribed in [CITATION].', '1901.08164-2-52-5': 'Results are shown in Tab. [REF].', '1901.08164-2-52-6': 'We see that for all the models DGL can perform as well and sometimes better than the end-to-end trained models, while permitting parallel training.', '1901.08164-2-52-7': 'In all these cases the auxiliary networks are neglibile (see Appendix Table [REF] for more details).', '1901.08164-2-52-8': 'For the VGG-13 architecture we also evaluate the case where the model is trained layer by layer ([MATH]).', '1901.08164-2-52-9': 'Although here performance is slightly degraded, we find it is suprisingly high given that no backward communication is performed.', '1901.08164-2-52-10': 'We conjecture that improved auxiliary models and combinations with methods such as [CITATION] to allow feedback on top of the local model, may further improve performance.', '1901.08164-2-52-11': 'Also for the settings with larger potential parallelization, slower but more performant auxiliary models could potentially be considered as well.', '1901.08164-2-53-0': 'The synchronous DGL has also favorable memory usage compared to DDG and to the DNI method, DNI requiring to store larger activations and DDG having high memory compared to the base network even for few splits [CITATION].', '1901.08164-2-53-1': 'Although not our focus, the single worker version of DGL has favorable memory usage compared to standard backprop training.', '1901.08164-2-53-2': 'For example, the ResNet-152 DGL [MATH] setting can fit [MATH] more samples on a single 16GB GPU than the standard end-to-end training.', '1901.08164-2-54-0': '## Asynchronous DGL with Replay', '1901.08164-2-55-0': 'R0.35buffer_size_vs_mean_accuracy.', '1901.08164-2-55-1': 'pdfBuffer size vs. Acc.', '1901.08164-2-55-2': 'for Async DGL.', '1901.08164-2-55-3': 'Smaller buffers produce only small loss in acc.', '1901.08164-2-56-0': 'We now study the stability of Alg. [REF] w.r.t delays.', '1901.08164-2-56-1': 'We use a 5 layer CIFAR-10 network with the MLP-aux and with all other architecture and optimization settings as in the auxiliary network experiments of Sec. [REF].', '1901.08164-2-56-2': 'Each layer is equipped with a buffer of size [MATH].', '1901.08164-2-56-3': 'At each iteration, a layer is chosen according to the pmf [MATH], and a batch selected from buffer [MATH].', '1901.08164-2-56-4': 'One layer is slowed down by decreasing its selection probability in the pmf [MATH] by a factor [MATH].', '1901.08164-2-56-5': 'We evaluate different slowdown factors (up to [MATH]).', '1901.08164-2-56-6': 'Accuracy versus [MATH] is shown in Fig. [REF].', '1901.08164-2-56-7': 'For this experiment we use a buffer of size [MATH].', '1901.08164-2-56-8': 'We run separate experiments with the slowdown applied at each layer of the network as well as 3 random seeds for each of these settings (thus 18 experiments per data point).', '1901.08164-2-56-9': 'We show the evaluations for 10 values of [MATH].', '1901.08164-2-56-10': 'To ensure a fair comparison we stop updating layers once they have completed 50 epochs, assuring identical number of gradient updates for all layers in all experiments.', '1901.08164-2-57-0': 'In practice one could continue updating until all layers are trained.', '1901.08164-2-57-1': 'In Fig. [REF] we compare to the synchronous case.', '1901.08164-2-57-2': 'First, observe that the accuracy of the synchronous algorithm is maintained in the setting where [MATH] and the pmf is uniform.', '1901.08164-2-57-3': 'Note that even this is a non-trivial case, as it will mean that layers inherently have random delays (as compared to Alg. [REF]).', '1901.08164-2-57-4': 'Secondly, observe accuracy is maintained until approximately [MATH] and accuracy losses after that the difference remains small.', '1901.08164-2-57-5': 'Note that even case [MATH] is somewhat drastic: for 50 training epochs, the slowed-down layer is only on epoch 25 while those following it are at epoch 50.', '1901.08164-2-58-0': 'We now consider the performance with respect to the buffer size.', '1901.08164-2-58-1': 'Results are shown in Fig. [REF].', '1901.08164-2-58-2': 'For this experiment we set [MATH].', '1901.08164-2-58-3': 'Observe even a tiny buffer size can yield only a slight loss in performance accuracy.', '1901.08164-2-58-4': 'Building on this demonstration there are multiple directions to improve Async DGL with replay.', '1901.08164-2-58-5': 'For example improving the efficiency of the buffer, by including data augmentation in feature space [CITATION], mixing samples in batches, or improved batch sampling, among others.', '1901.08164-2-59-0': '# Related work', '1901.08164-2-60-0': 'To the best of our knowledge [CITATION] is the only work which directly addresses the update or forward locking problems in deep feed-forward networks.', '1901.08164-2-60-1': 'Other works [CITATION] study the backward locking problem.', '1901.08164-2-60-2': 'Furthermore, a number of backpropagation alternatives [CITATION] can address backward locking.', '1901.08164-2-60-3': 'However, update locking is a more severe inefficiency.', '1901.08164-2-60-4': "Consider the case where each layer's forward processing time is [MATH] and is equal across a network of [MATH] layers.", '1901.08164-2-60-5': 'Given that the backward pass is a constant multiple in time of the forward, in the most ideal case the backward unlocking will still only scale as [MATH] with [MATH] parallel nodes, while update unlocking could scale as [MATH].', '1901.08164-2-61-0': 'One class of alternatives to standard back-propagation aims to avoid its biologically implausible aspects, most notably the weight transport problem [CITATION].', '1901.08164-2-61-1': 'Some of these methods [CITATION] can also achieve backward unlocking as they permit all parameters to be updated at the same time, but only once the signal has propagated to the top layer.', '1901.08164-2-61-2': 'However, they do not solve the update or forward locking problems, which we consider.', '1901.08164-2-61-3': 'Target propagation uses a local auxiliary network as in our approach, for propagating backward optimal activations computed from the layer above.', '1901.08164-2-61-4': 'Feedback alignment replaces the symmetric weights of the backward pass with random weights.', '1901.08164-2-61-5': 'Direct feedback alignment extends the idea of feedback alignment passing errors from the top to all layers, potentially enabling simultaneous updates.', '1901.08164-2-61-6': 'These approaches have also not been shown to scale to large datasets [CITATION], obtaining only [MATH] top-5 accuracy on ImageNet (reference model achieving [MATH]).', '1901.08164-2-61-7': 'On the other hand, greedy learning has been shown to work well on this task [CITATION].', '1901.08164-2-61-8': 'We also note concurrent work of [CITATION] which shows an approach similar to the synchronous version of DGL in the context of biological plausibility.', '1901.08164-2-62-0': 'Another line of related work inspired by optimization methods such as Alternating Direction Method of Multipliers (ADMM) [CITATION] use auxiliary variables to break the optimization into sub-problems.', '1901.08164-2-62-1': 'These approaches are fundamentally different from ours as they optimize for the joint training objective, the auxiliary variables providing a link between a layer and its successive layers, whereas we consider a different objective where a layer has no dependence on its successors.', '1901.08164-2-62-2': 'None of these methods can achieve update or forward unlocking.', '1901.08164-2-62-3': 'However, some [CITATION] are able to have simultaneous weight updates (backward unlocked).', '1901.08164-2-62-4': 'Another issue with ADMM methods is that most of the existing approaches except for [CITATION] require standard ("batch") gradient descent and are thus difficult to scale.', '1901.08164-2-62-5': 'They also often involve an inner minimization problem and have thus not been demonstrated to work on large-scale datasets.', '1901.08164-2-62-6': 'Furthermore, none of these have been combined with CNNs.', '1901.08164-2-63-0': 'Distributed optimization based on data parallelism is a popular area in machine learning beyond deep learning models and often studied in the convex setting [CITATION].', '1901.08164-2-63-1': 'For deep network optimization the predominant method is distributed synchronous SGD [CITATION] and variants, as well as asynchronous [CITATION] variants.', '1901.08164-2-63-2': 'Our work is closer to a form of model parallelism rather than data parallelism, and can be easily combined with many data parallel methods (e.g. distributed synchronous SGD).', '1901.08164-2-63-3': 'Finally, recent proposals for "pipelining "[CITATION] consider system level approaches to optimize latency times.', '1901.08164-2-63-4': 'These methods do not address the update, forward, locking problems[CITATION] which are algorithmic constraints of backpropagation.', '1901.08164-2-63-5': 'Pipelining can be seen as a systems-level solution attempting to work around these restrictions, with the fundamental limitations remaining.', '1901.08164-2-63-6': 'Removing and reducing update, backward, forward locking would simplify the design and efficiency of such systems-level machinery.', '1901.08164-2-64-0': '# Conclusion', '1901.08164-2-65-0': 'We have analyzed and introduced a simple and strong baseline for parallelizing per layer and per module computations in CNN training.', '1901.08164-2-65-1': 'This work matches or exceeds state-of-the-art approaches addressing these problems and is able to scale to much large datasets than others.', '1901.08164-2-65-2': 'Future work can develop improved auxiliary problem objectives and combinations with delayed feedback.'} | [['1901.08164-1-12-0', '1901.08164-2-7-0'], ['1901.08164-1-12-1', '1901.08164-2-7-1'], ['1901.08164-1-22-0', '1901.08164-2-19-0'], ['1901.08164-1-22-1', '1901.08164-2-19-1'], ['1901.08164-1-22-4', '1901.08164-2-19-4'], ['1901.08164-1-22-5', '1901.08164-2-19-5'], ['1901.08164-1-37-0', '1901.08164-2-31-0'], ['1901.08164-1-47-0', '1901.08164-2-41-0'], ['1901.08164-1-47-1', '1901.08164-2-41-1'], ['1901.08164-1-47-2', '1901.08164-2-41-2'], ['1901.08164-1-47-3', '1901.08164-2-41-3'], ['1901.08164-1-9-5', '1901.08164-2-62-6'], ['1901.08164-1-60-1', '1901.08164-2-58-1'], ['1901.08164-1-0-0', '1901.08164-2-0-0'], ['1901.08164-1-0-2', '1901.08164-2-0-3'], ['1901.08164-1-0-3', '1901.08164-2-0-4'], ['1901.08164-1-0-4', '1901.08164-2-0-5'], ['1901.08164-1-2-2', '1901.08164-2-2-2'], ['1901.08164-1-40-1', 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'1901.08164-2-13-4'], ['1901.08164-1-17-4', '1901.08164-2-13-5'], ['1901.08164-1-51-1', '1901.08164-2-45-1'], ['1901.08164-1-23-2', '1901.08164-2-20-1'], ['1901.08164-1-26-0', '1901.08164-2-23-0'], ['1901.08164-1-48-0', '1901.08164-2-42-0'], ['1901.08164-1-48-3', '1901.08164-2-42-3'], ['1901.08164-1-48-4', '1901.08164-2-42-4'], ['1901.08164-1-49-0', '1901.08164-2-44-0'], ['1901.08164-1-49-4', '1901.08164-2-44-3'], ['1901.08164-1-45-4', '1901.08164-2-39-4'], ['1901.08164-1-45-5', '1901.08164-2-39-5'], ['1901.08164-1-45-9', '1901.08164-2-39-9'], ['1901.08164-1-3-1', '1901.08164-2-3-2'], ['1901.08164-1-3-2', '1901.08164-2-3-2'], ['1901.08164-1-3-4', '1901.08164-2-3-4'], ['1901.08164-1-52-0', '1901.08164-2-46-0'], ['1901.08164-1-52-0', '1901.08164-2-46-1'], ['1901.08164-1-42-0', '1901.08164-2-36-0'], ['1901.08164-1-42-1', '1901.08164-2-36-0'], ['1901.08164-1-29-3', '1901.08164-2-26-3'], ['1901.08164-1-29-6', '1901.08164-2-26-6'], ['1901.08164-1-57-0', '1901.08164-2-53-0'], ['1901.08164-1-57-0', '1901.08164-2-53-1'], ['1901.08164-1-57-1', '1901.08164-2-53-0'], ['1901.08164-1-57-2', '1901.08164-2-53-1'], ['1901.08164-1-46-5', '1901.08164-2-40-5'], ['1901.08164-1-59-6', '1901.08164-2-56-6'], ['1901.08164-1-59-11', '1901.08164-2-57-0'], ['1901.08164-1-59-12', '1901.08164-2-57-1'], ['1901.08164-1-59-15', '1901.08164-2-57-4'], ['1901.08164-1-59-16', '1901.08164-2-57-5'], ['1901.08164-1-53-0', '1901.08164-2-48-0'], ['1901.08164-1-53-6', '1901.08164-2-49-2'], ['1901.08164-1-36-0', '1901.08164-2-30-2'], ['1901.08164-1-55-3', '1901.08164-2-52-3'], ['1901.08164-1-56-5', '1901.08164-2-52-9'], ['1901.08164-1-14-0', '1901.08164-2-11-0'], ['1901.08164-1-14-1', '1901.08164-2-11-0'], ['1901.08164-1-14-10', '1901.08164-2-11-9']] | [['1901.08164-1-7-0', '1901.08164-2-60-0'], ['1901.08164-1-7-2', '1901.08164-2-60-3'], ['1901.08164-1-7-2', '1901.08164-2-3-1'], ['1901.08164-1-7-3', '1901.08164-2-60-4'], ['1901.08164-1-7-4', '1901.08164-2-60-5'], ['1901.08164-1-8-0', '1901.08164-2-61-0'], ['1901.08164-1-8-1', '1901.08164-2-61-1'], ['1901.08164-1-8-2', '1901.08164-2-61-2'], ['1901.08164-1-8-3', '1901.08164-2-61-3'], ['1901.08164-1-8-4', '1901.08164-2-61-4'], ['1901.08164-1-8-5', '1901.08164-2-61-5'], ['1901.08164-1-8-6', '1901.08164-2-61-6'], ['1901.08164-1-8-7', '1901.08164-2-61-7'], ['1901.08164-1-10-0', '1901.08164-2-63-0'], ['1901.08164-1-10-1', '1901.08164-2-63-1'], ['1901.08164-1-10-2', '1901.08164-2-63-2']] | ['1901.08164-1-18-0', '1901.08164-1-19-0', '1901.08164-1-20-0', '1901.08164-1-21-0', '1901.08164-1-31-0', '1901.08164-1-33-0', '1901.08164-1-34-0', '1901.08164-1-39-0', '1901.08164-1-39-1', '1901.08164-1-50-0', '1901.08164-2-13-6', '1901.08164-2-14-0', '1901.08164-2-14-1', '1901.08164-2-14-2', '1901.08164-2-14-3', '1901.08164-2-14-4', '1901.08164-2-15-0', '1901.08164-2-15-1', '1901.08164-2-16-0', '1901.08164-2-17-0', '1901.08164-2-17-1', '1901.08164-2-17-2', '1901.08164-2-17-3', '1901.08164-2-17-4', '1901.08164-2-18-0', '1901.08164-2-29-0', '1901.08164-2-42-6', '1901.08164-2-43-4', '1901.08164-2-55-0', '1901.08164-2-55-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1901.08164 | null | null | null | null | null |
1811.05028 | {'1811.05028-1-0-0': 'This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise.', '1811.05028-1-0-1': 'The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-side Lipschitz condition.', '1811.05028-1-0-2': 'These assumptions are the same ones as used in [CITATION] where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under "minimum assumptions" were studied.', '1811.05028-1-0-3': 'As a result, the semilinear SPDEs considered in this paper is a direct generalization of the SODEs considered in [CITATION].', '1811.05028-1-0-4': 'There are several difficulties which need to be overcome for this generalization.', '1811.05028-1-0-5': 'First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty.', '1811.05028-1-0-6': 'It turns out a special discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.', '1811.05028-1-0-7': 'In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.', '1811.05028-1-0-8': 'Second, In order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the [MATH]-seminorm of the numerical solution must be established, which are difficult and delicate.', '1811.05028-1-0-9': 'A judicious combination of the properties of the drift and diffusion terms and a nontrivial technique borrowed from [CITATION] is used in this paper to achieve the goal.', '1811.05028-1-0-10': 'Finally, stability estimates for the second and higher order moments of the [MATH]-norm of the numerical solution is also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.', '1811.05028-1-0-11': 'This is done by utilizing the interpolation theory and the higher moment estimates for the [MATH]-seminorm of the numerical solution.', '1811.05028-1-0-12': 'After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.', '1811.05028-1-0-13': 'Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.', '1811.05028-1-1-0': '# Introduction', '1811.05028-1-2-0': 'We consider the following initial-boundary value problem for general semilinear stochastic partial differential equations (SPDEs) with function-type multiplicative noise:', '1811.05028-1-3-0': 'Here [MATH] is a bounded domain, [MATH] denotes the standard Weiner process on the filtered probability space [MATH], and [MATH] are two given functions and [MATH] takes the form [EQUATION] where [MATH].', '1811.05028-1-3-1': 'For the sake of clarity, we only consider the case [MATH] in this paper, where [MATH] is an odd integer (it is trivial when [MATH]).', '1811.05028-1-3-2': 'We remark that similar results still hold for the general nonlinear function [MATH] in [REF], and when [MATH], [REF] is known as the stochastic Allen-Cahn equation with function-type multiplicative noise and interaction length [MATH] [CITATION].', '1811.05028-1-3-3': 'We also assume that [MATH] is globally Lipschitz, that is, there exists a constant [MATH] such that [EQUATION]', '1811.05028-1-3-4': 'Setting [MATH] in [REF], we get [EQUATION]', '1811.05028-1-3-5': 'Under the above assumptions for the drift term and the diffusion term, it can be proved that [CITATION] there exists a unique strong variational solution u such that [EQUATION] holds [MATH]-almost surely.', '1811.05028-1-3-6': 'Moreover, when the initial condition [MATH] is sufficiently smooth, the following stability estimate for the strong solution [MATH] holds: [EQUATION]', '1811.05028-1-3-7': 'Clearly, when the [MATH] term in [REF] is dropped, the PDE reduces to a stochastic ODE.', '1811.05028-1-3-8': 'A convergence theory for numerical approximations for this stochastic ODE was established long ago (cf. [CITATION]) under the global Lipschitz assumptions on [MATH] and [MATH].', '1811.05028-1-3-9': 'Later, the convergence was proved in [CITATION] under a weaker condition on [MATH] known as a one-side Lipschitz condition in the sense that there exists a constant [MATH] such that [EQUATION]', '1811.05028-1-3-10': 'The optimal rate of convergence was also obtained in [CITATION] under an extra assumption that [MATH] behaves like a polynomial.', '1811.05028-1-3-11': 'The one-side Lipschitz condition is widely used and it has broad applications [CITATION].', '1811.05028-1-4-0': 'We also note that numerical approximations of the SPDE [REF] with various special drift terms and/or diffusion terms have been extensively investigated in the literature, see [CITATION].', '1811.05028-1-4-1': 'In particular, we mention that the case that [MATH], [MATH] are bounded and [MATH] is global Lipschitz continuous was studied in [CITATION], the high moments of the [MATH]-norm of the numerical solution were proved to be stable, and a nearly optimal strong convergence rate was established.', '1811.05028-1-4-2': 'A specially designed discretization is used for [MATH], and it is not trivial to extend the idea to the case when [MATH] where [MATH].', '1811.05028-1-5-0': 'The goal of this paper is to generalize the numerical SODE theory of [CITATION] to the SPDE case.', '1811.05028-1-5-1': 'Specifically, we want to design a fully discrete finite element method for problem [REF]-[REF] which can be proved to be stable and convergent with optimal rates in strong norms under "minimum" assumptions on nonlinear functions [MATH] and [MATH] as those used in [CITATION].', '1811.05028-1-5-2': 'We recall that the "minimum" assumptions refer to that [MATH] is assumed to be global Lipschitz, and [MATH] satisfies the one-side Lipschitz condition [REF] and it behaves like a polynomial.', '1811.05028-1-5-3': 'To the best of our knowledge, such a goal has yet been achieved before in the literature.', '1811.05028-1-6-0': 'The remainder of this paper is organized as follows.', '1811.05028-1-6-1': 'In Section [REF], we establish several Holder continuity properties (in different norms) for the SPDE solution [MATH] and for the composite function [MATH].', '1811.05028-1-6-2': 'These properties play an important role in our error analysis.', '1811.05028-1-6-3': 'In Section [REF], we first present our fully discrete finite element method for problem [REF]-[REF], which consists of an Euler-type scheme for time discretization and a nonstandard finite element method for spatial discretization.', '1811.05028-1-6-4': 'The novelty of our spatial discretization is to approximate the nonlinear function [MATH] by its finite element interpolation in the scheme.', '1811.05028-1-6-5': 'We then establish several key properties for the numerical solution, among them are the stability of the second and higher order moments of its [MATH]-seminorm and the stability of the second and higher order moments of its [MATH]-norm.', '1811.05028-1-6-6': 'We note that the proofs of the stability of these higher order moments are quite involved, and they require some special techniques and rely on the structure of the proposed numerical method.', '1811.05028-1-6-7': 'For example, the diagonal dominance property of the stiffness matrix is needed to show the stability of the second and higher order moments of the [MATH]-seminorm of the numerical solution, however, the mass matrix may not be diagonally dominant.', '1811.05028-1-6-8': 'To circumvent this difficulty, we use the stability of the second and higher order moments of the [MATH]-seminorm of the numerical solution and the interpolation theory to get the desired [MATH]-norm stability.', '1811.05028-1-6-9': 'Finally, in this section we prove nearly optimal order error estimates for the numerical solution by utilizing the stability of higher order moments of the [MATH]-norm and [MATH]-seminorm of the numerical solution.', '1811.05028-1-6-10': 'We like to emphasize that only sub-optimal order error estimates could be obtained should the stability of higher order moments of the [MATH]-seminorm of the numerical solution were not known, see [CITATION] where the special case [MATH] was considered.', '1811.05028-1-6-11': 'In Section [REF], we present several numerical experiments to validate our theoretical results, especially to verify the stability of numerical solution using different initial conditions [MATH] and different functions [MATH] and [MATH].', '1811.05028-1-6-12': 'As a special case, the stochastic Allen-Cahn equation with function-type multiplicative noise is also tested.', '1811.05028-1-7-0': '# Preliminaries and properties of the SPDE solution', '1811.05028-1-8-0': 'Throughout this paper, we shall use [MATH] to denote a generic constant, and we take the standard Sobolev notations in [CITATION].', '1811.05028-1-8-1': 'When it is the whole domain [MATH], [MATH] and [MATH] are used to simplify [MATH] and [MATH] respectively, and [MATH] is used to denote the standard inner product of [MATH].', '1811.05028-1-8-2': '[MATH] denotes the expectation operator on the filtered probability space [MATH].', '1811.05028-1-9-0': 'In this section, we first derive the Holder continuity in time for the strong solution [MATH] with respect to the spatial [MATH]-seminorm and for the composite function [MATH] with respect to the spatial [MATH]-norm.', '1811.05028-1-9-1': 'Both results will play a key role in the error analysis (see Subsection [REF]).', '1811.05028-1-9-2': 'The time derivatives of [MATH] and the composite function [MATH] do not exist in the stochastic case, so these Holder continuity results will substitute for the differentiability of [MATH] and [MATH] with respect to time in the error analysis.', '1811.05028-1-10-0': 'Let [MATH] be the strong solution to problem [REF].', '1811.05028-1-10-1': 'Then for any [MATH] with [MATH], we have [EQUATION] where [EQUATION]', '1811.05028-1-10-2': "Applying Ito's formula to the functional [MATH] with fixed [MATH] and using integration by parts, we get [EQUATION]", '1811.05028-1-10-3': 'The expectation of the first term on the right-hand side of [REF] can be bounded by the Cauchy-Schwarz inequality as follows [EQUATION]', '1811.05028-1-10-4': 'The expectation of the second term on the right-hand side of [REF] can be bounded by [EQUATION]', '1811.05028-1-10-5': 'Next we bound the expectation of the fourth term on the right-hand side of [REF] as follows [EQUATION]', '1811.05028-1-10-6': 'Then Lemma [REF] follows from [REF]-[REF] and the fact that the expectation of the third term on the right-hand side of [REF] is zero.', '1811.05028-1-11-0': 'Next we prove the Holder continuity result for the nonlinear term [MATH] with respect to the spatial [MATH]-norm.', '1811.05028-1-12-0': 'Let [MATH] be the strong solution to problem [REF].', '1811.05028-1-12-1': 'Then for any [MATH] with [MATH], we have [EQUATION] where [EQUATION]', '1811.05028-1-12-2': "Applying Ito's formula to [MATH] with fixed [MATH], we obtain [EQUATION]", '1811.05028-1-12-3': "Taking the expectation on both sides, it follows from integration by parts and Young's inequality that [EQUATION]", '1811.05028-1-12-4': 'Finally, the desired Lemma [REF] follows from [REF].', '1811.05028-1-13-0': '(a) For the diffusion term, the global Lipschitz condition, which is stronger than the one-side Lipschitz condition, is needed as in the SODE case.', '1811.05028-1-13-1': 'Using the [MATH] assumption and the global Lipschitz assumption, we can derive that the derivative of the diffusion term is bounded by the Lipschitz constant [MATH], i.e., [MATH], but the diffusion term itself may not be bounded.', '1811.05028-1-13-2': 'For instance, [MATH], [MATH], etc.', '1811.05028-1-13-3': 'Notice these two assumptions are consistent with the SODE case in [CITATION], and they are also the conditions to guarantee the well-posedness [CITATION] of the strong SODE solution;', '1811.05028-1-14-0': '(b) We can verify [MATH] in [REF] satisfies a one-sided Lipschitz condition [REF].', '1811.05028-1-14-1': 'If the drift term [MATH] behaves polynomially, then for the one-sided Lipschitz condition [REF], we have the following conclusions:', '1811.05028-1-15-0': '(1).', '1811.05028-1-15-1': 'The power [MATH] of the highest order term must be odd.', '1811.05028-1-15-2': 'Because when the highest power [MATH] is even, dividing [MATH] by [MATH] yields the the quotient is odd so that it can be [MATH] and [MATH].', '1811.05028-1-15-3': 'When choosing [MATH] and [MATH] sufficiently large or small, the absolute value of this term is dominant and the left-hand side of [REF] is [MATH] where [MATH] can be [MATH], which is a contradiction;', '1811.05028-1-16-0': '(2).', '1811.05028-1-16-1': 'The sign of the highest odd order term must be negative.', '1811.05028-1-16-2': 'Because this term is dominant and the quotient of dividing [MATH] by [MATH] can be [MATH], which contradicts [REF].', '1811.05028-1-17-0': '# Fully discrete finite element approximation', '1811.05028-1-18-0': '## Formulation of the finite element method', '1811.05028-1-19-0': 'In this section, we first construct a fully discrete finite element method for problem [REF]-[REF].', '1811.05028-1-19-1': 'we then establish several stability properties for the numerical solution including the stability of higher order moments for its [MATH]-seminorm and [MATH]-norm.', '1811.05028-1-19-2': 'Finally, we derive optimal order error estimates in strong norms for the numerical solution using the stability estimates.', '1811.05028-1-20-0': 'Let [MATH] be a uniform partition of [MATH] and [MATH] be the triangulation of [MATH] satisfying the following assumption [CITATION]: [EQUATION] where [MATH] denotes the edge of simplex [MATH].', '1811.05028-1-20-1': 'It was proved in [CITATION] that the stiffness matrix for the Poisson equation with zero Dirichlet boundary is an [MATH]-matrix if and only if this assumption holds for all edges.', '1811.05028-1-20-2': 'The stiffness matrix is diagonally dominant if the Neumann boundary condition is considered.', '1811.05028-1-20-3': 'Notice this assumption is just the Delaunay triangulation when [MATH].', '1811.05028-1-20-4': 'In 3D, the notations in the assumption [REF] are as follows: [MATH] denote the vertices of [MATH], [MATH] the edge connecting two vertices [MATH] and [MATH], [MATH] the [MATH]-dimensional simplex opposite to the vertex [MATH], [MATH] or [MATH] the angle between the faces [MATH] and [MATH], [MATH] , the [MATH]-dimensional simplex opposite to the edge [MATH].', '1811.05028-1-20-5': 'See Figure [REF] below.', '1811.05028-1-21-0': 'Consider the [MATH]-Lagrangian finite element space [EQUATION] where [MATH] denotes the space of all linear polynomials.', '1811.05028-1-21-1': 'Then the finite element approximation of [REF] is to seek an [MATH] adapted [MATH]-valued process [MATH] such that it holds [MATH]-almost surely that [EQUATION] where [MATH], [MATH], and [MATH] is the standard nodal value interpolation operator [MATH], i.e., [EQUATION] where [MATH] denotes the number of vertices of [MATH], and [MATH] denotes the nodal basis function of [MATH] corresponding to the vertex [MATH].', '1811.05028-1-21-2': 'The initial condition is chosen by [MATH] where [MATH] is the [MATH]-projection operator defined by [EQUATION]', '1811.05028-1-22-0': 'For all [MATH], the following well-known error estimate results can be found in [CITATION]: [EQUATION]', '1811.05028-1-22-1': 'Finally, given [MATH], we define the discrete Laplace operator [MATH] by [EQUATION]', '1811.05028-1-23-0': '## Stability estimates for the [MATH]-th moment of the [MATH]-seminorm of [MATH]', '1811.05028-1-24-0': 'First we shall prove the second moment discrete [MATH]-seminorm stability result, which is necessary to establish the corresponding higher moment stability result.', '1811.05028-1-25-0': 'Suppose the mesh assumption in [REF] holds, then [EQUATION]', '1811.05028-1-25-1': 'Testing [REF] with [MATH], then [EQUATION]', '1811.05028-1-25-2': 'Using the definition of the discrete Laplace operator, we get [EQUATION] where the stability in the [MATH]-seminorm of the [MATH] projection [CITATION] is used in the inequality of [REF].', '1811.05028-1-26-0': 'The crucial part is to bound the first term on the right-hand side of [REF] since it cannot be treated as a bad term, which aligns with the continuous case.', '1811.05028-1-26-1': 'Denote [MATH], then [EQUATION] where [MATH].', '1811.05028-1-27-0': "Using Young's inequality when [MATH], we have [EQUATION]", '1811.05028-1-27-1': 'Besides, since the stiffness matrix is diagonally dominant, then [EQUATION]', '1811.05028-1-27-2': 'Then we have [EQUATION]', '1811.05028-1-27-3': 'Combining [REF]-[REF] and [REF], and taking the summation, we have [EQUATION]', '1811.05028-1-27-4': "Using Gronwall's inequality, we obtain [REF].", '1811.05028-1-28-0': 'Before we establish the error estimates, we need to prove the stability of the higher order moments for the [MATH]-seminorm of the numerical solution.', '1811.05028-1-29-0': 'Suppose the mesh assumption in [REF] holds, then for any [MATH], [EQUATION]', '1811.05028-1-29-1': 'The proof is divided into three steps.', '1811.05028-1-29-2': 'In Step 1, we establish the bound for [MATH].', '1811.05028-1-29-3': 'In Step 2, we give the bound for [MATH], where [MATH] and [MATH] is an arbitrary positive integer.', '1811.05028-1-29-4': 'In Step 3, we obtain the bound for [MATH], where [MATH] is an arbitrary real number and [MATH].', '1811.05028-1-30-0': 'Step 1.', '1811.05028-1-30-1': 'Based on [REF]-[REF], we have [EQUATION]', '1811.05028-1-30-2': 'Notice the following identity [EQUATION] and multiplying [REF] with [MATH], we obtain [EQUATION]', '1811.05028-1-30-3': 'The first term on the right-hand side of [REF] can be written as [EQUATION] where [MATH] will be determined later.', '1811.05028-1-31-0': 'The second term on the right-hand side of [REF] can be written as [EQUATION]', '1811.05028-1-31-1': 'For the right-hand side of [REF], using the Cauchy-Schwarz inequality, we get [EQUATION] where [MATH] will be determined later.', '1811.05028-1-31-2': 'Similarly, using the Cauchy-Schwarz inequality, we have [EQUATION] where [MATH] will be determined later.', '1811.05028-1-32-0': 'Choosing [MATH] such that [MATH], then taking the summation over [MATH] from [MATH] to [MATH] and taking the expectation on both sides of [REF], we obtain [EQUATION]', '1811.05028-1-32-1': 'When restricting [MATH], we have [EQUATION]', '1811.05028-1-32-2': "Using Gronwall's inequality, we obtain [EQUATION]", '1811.05028-1-32-3': 'Step 2.', '1811.05028-1-32-4': 'Similar to Step 1, using [REF]-[REF], we have [EQUATION]', '1811.05028-1-32-5': 'Proceed similarly as in Step 1, multiplying [REF] with [MATH], we can obtain the 8-th moment of the [MATH]-seminorm stability result of the numerical solution.', '1811.05028-1-32-6': 'Then repeating this process, the [MATH]-th moment of the [MATH]-seminorm stability result of the numerical solution can be obtained.', '1811.05028-1-33-0': 'Step 3.', '1811.05028-1-33-1': "Suppose [MATH], then using Young's inequality, we have [EQUATION] where the second inequality follows from the results of Step 2.", '1811.05028-1-33-2': 'The proof is complete.', '1811.05028-1-34-0': '## Stability estimates for the [MATH]-th moment of the [MATH]-norm of [MATH]', '1811.05028-1-35-0': 'Since the mass matrix may not be the diagonally dominated matrix, we cannot use the above idea to prove the [MATH] stability.', '1811.05028-1-35-1': 'Instead, we prove the stability results by utilizing the above established results.', '1811.05028-1-35-2': 'The following results hold when [MATH] is the odd integer in 2D case, and when [MATH] or [MATH] in 3D case.', '1811.05028-1-36-0': 'Suppose the mesh assumption in [REF] holds, then [EQUATION]', '1811.05028-1-36-1': 'Testing [REF] with [MATH], then [EQUATION]', '1811.05028-1-36-2': 'We can easily prove the following inequalities: [EQUATION] where [REF] is used in the inequality above.', '1811.05028-1-37-0': 'We have the following standard interpolation result and the inverse inequality [CITATION]: [EQUATION]', '1811.05028-1-37-1': "Using [REF]-[REF], and Young's inequality, we have [EQUATION]", '1811.05028-1-37-2': 'Notice when [MATH], [MATH] if [MATH], and when [MATH], [MATH] if [MATH].', '1811.05028-1-37-3': 'Using the above inequalities, Theorem [REF], taking summation over [MATH] from [MATH] to [MATH], and taking expectation on both sides of [REF], we obtain [EQUATION] where Theorem [REF] is used in the last inequality.', '1811.05028-1-38-0': "The conclusion is a direct result by using Gronwall's inequality.", '1811.05028-1-39-0': 'To obtain the error estimates results, we need to establish a higher moment discrete [MATH] stability result for the numerical solution [MATH].', '1811.05028-1-40-0': 'Suppose the mesh assumption in [REF] holds, then for any [MATH], [EQUATION]', '1811.05028-1-40-1': 'The proof is divided into three steps.', '1811.05028-1-40-2': 'In Step 1, we give the bound for [MATH].', '1811.05028-1-40-3': 'In Step 2, we give the bound for [MATH], where [MATH] and [MATH] is an arbitrary positive integer.', '1811.05028-1-40-4': 'In Step 3, we give the bound for [MATH], where [MATH] is an arbitrary real number and [MATH].', '1811.05028-1-41-0': 'Step 1.', '1811.05028-1-41-1': 'Based on [REF] and [REF], we have [EQUATION]', '1811.05028-1-41-2': 'Notice the following identity [EQUATION]', '1811.05028-1-41-3': 'Multiplying [REF] with [MATH], we obtain [EQUATION]', '1811.05028-1-41-4': 'The first term on the right-hand side of [REF] can be written as [EQUATION] where [MATH] will be determined later.', '1811.05028-1-42-0': 'The second term on the right-hand side of [REF] can be written as [EQUATION]', '1811.05028-1-42-1': 'For the second term on the right-hand side of [REF], using the Cauchy-Schwarz inequality, we get [EQUATION] where [MATH] will be determined later.', '1811.05028-1-42-2': 'Using [REF], the third term on the right-hand side of [REF] can be bounded by [EQUATION] where [MATH] will be determined later.', '1811.05028-1-43-0': 'Choosing [MATH] such that [MATH], then taking the summation over [MATH] from [MATH] to [MATH] and taking the expectation on both sides of [REF], we obtain [EQUATION]', '1811.05028-1-43-1': 'When [MATH], we have [EQUATION]', '1811.05028-1-43-2': "Using Gronwall's inequality, we obtain [EQUATION]", '1811.05028-1-43-3': 'Step 2.', '1811.05028-1-43-4': 'Similar to Step 1, using [REF]-[REF], we have [EQUATION]', '1811.05028-1-43-5': 'Similar to Step 1, multiplying [REF] with [MATH], we can obtain the 8-th moment of the [MATH] stability result of the discrete solution.', '1811.05028-1-43-6': 'Then repeating this process, the [MATH]-th moment of the [MATH] stability result of the discrete solution can be obtained.', '1811.05028-1-44-0': 'Step 3.', '1811.05028-1-44-1': "Suppose [MATH], then using Young's inequality, we have [EQUATION] where the second inequality uses Step 2.", '1811.05028-1-44-2': 'The proof is complete.', '1811.05028-1-45-0': '## Error estimates', '1811.05028-1-46-0': 'Let [MATH].', '1811.05028-1-46-1': 'In the following theorem, the [MATH] projection is used in the proof of the error estimates and the strong convergence rate is given.', '1811.05028-1-47-0': 'Let [MATH] and [MATH] denote respectively the solutions of problem [REF] and scheme [REF], then there holds [EQUATION]', '1811.05028-1-47-1': 'We write [MATH] where [EQUATION]', '1811.05028-1-47-2': 'It follows from [REF] that for all [MATH]) there holds [MATH]-almost surely [EQUATION]', '1811.05028-1-47-3': 'Subtracting [REF] from [REF] and setting [MATH], the following error equation holds [MATH]-almost surely, [EQUATION]', '1811.05028-1-47-4': 'The left-hand side of [REF] can be handled by [EQUATION]', '1811.05028-1-47-5': 'Next, we bound the right-hand side of [REF].', '1811.05028-1-47-6': 'First, since [MATH] is the [MATH]-projection operator, we have [MATH].', '1811.05028-1-48-0': 'For the second term on the right-hand side of [REF], using the Holder continuity in Lemma [REF], we have [EQUATION]', '1811.05028-1-48-1': 'In order to estimate the third term on the right-hand side of [REF], we write [EQUATION]', '1811.05028-1-48-2': 'Using the Holder continuity in Lemma [REF], we obtain [EQUATION]', '1811.05028-1-48-3': 'Next, using properties of the projection, we have [EQUATION]', '1811.05028-1-48-4': 'The third term on the right-hand side of [REF] can be bounded by [EQUATION]', '1811.05028-1-48-5': 'Using Theorem [REF], properties of the interpolation operator, the inverse inequality, and the fact that [MATH] is a piecewise linear polynomial, the fourth term on the right-hand side of [REF] can be handled by [EQUATION]', '1811.05028-1-48-6': 'Combining [REF]-[REF] to obtain [EQUATION]', '1811.05028-1-48-7': 'By the martingale property, the Ito isometry, the Holder continuity of [MATH] and the global Lipschitz condition [REF], we have [EQUATION]', '1811.05028-1-48-8': 'Taking the expectation on [REF] and combining estimates [REF]-[REF], summing over [MATH] with [MATH], and using the properties of the [MATH] projection and the regularity assumption, we obtain [EQUATION]', '1811.05028-1-48-9': "Finally, the assertion of the theorem follows from [REF], the discrete Gronwall's inequality, the [MATH]-projection properties, the fact that [MATH] and the triangle inequality.", '1811.05028-1-48-10': 'The proof is complete.', '1811.05028-1-49-0': 'The following strong stability result is a direct corollary of Theorem [REF].', '1811.05028-1-50-0': 'Suppose the mesh assumption in [REF] holds and [MATH], then [EQUATION]', '1811.05028-1-50-1': 'For each sample point, [EQUATION]', '1811.05028-1-50-2': 'When [MATH], taking the expectation on both sides of [REF], and using Theorem [REF], we obtain [EQUATION] (a) Notice the elliptic projection cannot be used due to the first term [MATH] in [REF].', '1811.05028-1-50-3': 'In reference [CITATION], it is [MATH] since [MATH] projection is used there.', '1811.05028-1-51-0': '(b) For the diffusion term, We need [MATH] and [MATH] to be Lipschitz continuous, which are the same assumptions as in stochastic ODE case [CITATION].', '1811.05028-1-51-1': 'The analysis in [CITATION] requires two extra conditions: [MATH] and [MATH] are bounded.', '1811.05028-1-51-2': 'Notice [MATH], [MATH] or some others satisfy the assumptions in this paper, but they do not satisfy the assumptions in [CITATION].', '1811.05028-1-52-0': '# Numerical experiments', '1811.05028-1-53-0': 'In this section, we present several two dimensional numerical examples to gauge the performance of the proposed stochastic finite element scheme for the stochastic partial differential equations satisfying the proposed assumptions for the nonlinear term and the diffusion term.', '1811.05028-1-53-1': 'Test 1 is designed to demonstrate the error orders with respect to mesh size [MATH] for small and big noises; Test 2 is designed to demonstrate the stability results and evolution of the stochastic Allen-Cahn equation, which is a special case of the SPDE in this paper; Test 3 is designed to demonstrate the stability results of the SPDE with a different initial condition; Test 4 is designed to demonstrate the stability results of the SPDE with a different nonlinear term; Test 5 is designed to demonstrate the stability results of the SPDE with a different diffusion term.', '1811.05028-1-53-2': 'The square domain [MATH], and 500 sample points are used in these tests.', '1811.05028-1-54-0': 'Test 1 Consider the following smooth initial condition [EQUATION] where [MATH].', '1811.05028-1-54-1': 'Time step size [MATH] is used in this Test 1.', '1811.05028-1-55-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-1-55-1': 'Table [REF] shows the following three types of errors [MATH], [MATH], and [MATH] respectively, and the rates of convergence.', '1811.05028-1-55-2': 'The noise intensity [MATH].', '1811.05028-1-55-3': 'In the table, we use [MATH], [MATH] and [MATH] to denote these three types of errors respectively.', '1811.05028-1-56-0': 'Table [REF] shows the errors [MATH], [MATH] and [MATH] respectively, and the rates of convergence at final time [MATH].', '1811.05028-1-56-1': 'The noise intensity [MATH].', '1811.05028-1-57-0': 'From these two tables, we observe that the error orders of [MATH] and [MATH] are 2, and the error order of [MATH] is 1.', '1811.05028-1-57-1': 'Besides, the error orders keep the same when the noise intensity increases.', '1811.05028-1-58-0': 'In the following tests, [MATH] and [MATH] are used to denote [MATH] and [MATH] respectively.', '1811.05028-1-59-0': 'Test 2 Consider the following initial condition [EQUATION]', '1811.05028-1-59-1': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH], which corresponds to the stochastic Allen-Cahn equation.', '1811.05028-1-59-2': 'More tests related to the Allen-Cahn equation can be found in [CITATION].', '1811.05028-1-59-3': 'Figure [REF] shows the evolution of the zero-level sets of the solutions under different intensity of the noise.', '1811.05028-1-59-4': 'We observe that although the circle may shrink or dilate (depending on the sign of the diffusion term), the average zero-level sets shrink for smaller and bigger noises.', '1811.05028-1-59-5': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-1-59-6': 'We also observe that they are both bounded.', '1811.05028-1-60-0': 'Test 3 Consider the following initial condition [EQUATION]', '1811.05028-1-60-1': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-1-60-2': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-1-61-0': 'Test 4 Consider the initial condition in [REF] with [MATH].', '1811.05028-1-62-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-1-62-1': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-1-63-0': 'Test 5 Consider the initial condition in [REF] with [MATH].', '1811.05028-1-64-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-1-64-1': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].'} | {'1811.05028-2-0-0': 'This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise.', '1811.05028-2-0-1': 'The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-side Lipschitz condition.', '1811.05028-2-0-2': 'These assumptions are the same ones as used in [CITATION] where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under "minimum assumptions" were studied.', '1811.05028-2-0-3': 'As a result, the semilinear SPDEs considered in this paper is a direct generalization of the SODEs considered in [CITATION].', '1811.05028-2-0-4': 'There are several difficulties which need to be overcome for this generalization.', '1811.05028-2-0-5': 'First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty.', '1811.05028-2-0-6': 'It turns out a special discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.', '1811.05028-2-0-7': 'In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.', '1811.05028-2-0-8': 'Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the [MATH]-seminorm of the numerical solution must be established, which are difficult and delicate.', '1811.05028-2-0-9': 'A judicious combination of the properties of the drift and diffusion terms and a nontrivial technique borrowed from [CITATION] is used in this paper to achieve the goal.', '1811.05028-2-0-10': 'Finally, stability estimates for the second and higher order moments of the [MATH]-norm of the numerical solution is also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.', '1811.05028-2-0-11': 'This is done by utilizing the interpolation theory and the higher moment estimates for the [MATH]-seminorm of the numerical solution.', '1811.05028-2-0-12': 'After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.', '1811.05028-2-0-13': 'Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.', '1811.05028-2-1-0': '# Introduction', '1811.05028-2-2-0': 'We consider the following initial-boundary value problem for general semilinear stochastic partial differential equations (SPDEs) with function-type multiplicative noise:', '1811.05028-2-3-0': 'Here [MATH] is a bounded domain, [MATH] denotes the standard Weiner process on the filtered probability space [MATH], and [MATH] are two given functions and [MATH] takes the form [EQUATION] where [MATH].', '1811.05028-2-3-1': 'For the sake of clarity, we only consider the case [MATH] in this paper, where [MATH] is an odd integer (it is trivial when [MATH]).', '1811.05028-2-3-2': 'We remark that similar results still hold for the general nonlinear function [MATH] in [REF], and when [MATH], [REF] is known as the stochastic Allen-Cahn equation with function-type multiplicative noise and interaction length [MATH] [CITATION].', '1811.05028-2-3-3': 'We also assume that [MATH] is globally Lipschitz, that is, there exists a constant [MATH] such that [EQUATION]', '1811.05028-2-3-4': 'Setting [MATH] in [REF], we get [EQUATION]', '1811.05028-2-3-5': 'Under the above assumptions for the drift term and the diffusion term, it can be proved that [CITATION] there exists a unique strong variational solution u such that [EQUATION] holds [MATH]-almost surely.', '1811.05028-2-3-6': 'Moreover, when the initial condition [MATH] is sufficiently smooth, the following stability estimate for the strong solution [MATH] holds: [EQUATION]', '1811.05028-2-3-7': 'Clearly, when the [MATH] term in [REF] is dropped, the PDE reduces to a stochastic ODE.', '1811.05028-2-3-8': 'A convergence theory for numerical approximations for this stochastic ODE was established long ago (cf. [CITATION]) under the global Lipschitz assumptions on [MATH] and [MATH].', '1811.05028-2-3-9': 'Later, the convergence was proved in [CITATION] under a weaker condition on [MATH] known as a one-side Lipschitz condition in the sense that there exists a constant [MATH] such that [EQUATION]', '1811.05028-2-3-10': 'The optimal rate of convergence was also obtained in [CITATION] under an extra assumption that [MATH] behaves like a polynomial.', '1811.05028-2-3-11': 'The one-side Lipschitz condition is widely used and it has broad applications [CITATION].', '1811.05028-2-4-0': 'We also note that numerical approximations of the SPDE [REF] with various special drift terms and/or diffusion terms have been extensively investigated in the literature, see [CITATION].', '1811.05028-2-4-1': 'In particular, we mention that the case that [MATH], [MATH] are bounded and [MATH] is global Lipschitz continuous was studied in [CITATION], the high moments of the [MATH]-norm of the numerical solution were proved to be stable, and a nearly optimal strong convergence rate was established.', '1811.05028-2-4-2': 'A specially designed discretization is used for [MATH], and it is not trivial to extend the idea to the case when [MATH] where [MATH].', '1811.05028-2-5-0': 'The goal of this paper is to generalize the numerical SODE theory of [CITATION] to the SPDE case.', '1811.05028-2-5-1': 'Specifically, we want to design a fully discrete finite element method for problem [REF]-[REF] which can be proved to be stable and convergent with optimal rates in strong norms under "minimum" assumptions on nonlinear functions [MATH] and [MATH] as those used in [CITATION].', '1811.05028-2-5-2': 'We recall that the "minimum" assumptions refer to that [MATH] is assumed to be global Lipschitz, and [MATH] satisfies the one-side Lipschitz condition [REF] and it behaves like a polynomial.', '1811.05028-2-5-3': 'To the best of our knowledge, such a goal has yet been achieved before in the literature.', '1811.05028-2-6-0': 'The remainder of this paper is organized as follows.', '1811.05028-2-6-1': 'In Section [REF], we establish several Holder continuity properties (in different norms) for the SPDE solution [MATH] and for the composite function [MATH].', '1811.05028-2-6-2': 'These properties play an important role in our error analysis.', '1811.05028-2-6-3': 'In Section [REF], we first present our fully discrete finite element method for problem [REF]-[REF], which consists of an Euler-type scheme for time discretization and a nonstandard finite element method for spatial discretization.', '1811.05028-2-6-4': 'The novelty of our spatial discretization is to approximate the nonlinear function [MATH] by its finite element interpolation in the scheme.', '1811.05028-2-6-5': 'We then establish several key properties for the numerical solution, among them are the stability of the second and higher order moments of its [MATH]-seminorm and the stability of the second and higher order moments of its [MATH]-norm.', '1811.05028-2-6-6': 'We note that the proofs of the stability of these higher order moments are quite involved, and they require some special techniques and rely on the structure of the proposed numerical method.', '1811.05028-2-6-7': 'For example, the diagonal dominance property of the stiffness matrix is needed to show the stability of the second and higher order moments of the [MATH]-seminorm of the numerical solution, however, the mass matrix may not be diagonally dominant.', '1811.05028-2-6-8': 'To circumvent this difficulty, we use the stability of the second and higher order moments of the [MATH]-seminorm of the numerical solution and the interpolation theory to get the desired [MATH]-norm stability.', '1811.05028-2-6-9': 'Finally, in this section we prove nearly optimal order error estimates for the numerical solution by utilizing the stability of higher order moments of the [MATH]-norm and [MATH]-seminorm of the numerical solution.', '1811.05028-2-6-10': 'We like to emphasize that only sub-optimal order error estimates could be obtained should the stability of higher order moments of the [MATH]-seminorm of the numerical solution were not known, see [CITATION] where the special case [MATH] was considered.', '1811.05028-2-6-11': 'In Section [REF], we present several numerical experiments to validate our theoretical results, especially to verify the stability of numerical solution using different initial conditions [MATH] and different functions [MATH] and [MATH].', '1811.05028-2-6-12': 'As a special case, the stochastic Allen-Cahn equation with function-type multiplicative noise is also tested.', '1811.05028-2-7-0': '# Preliminaries and properties of the SPDE solution', '1811.05028-2-8-0': 'Throughout this paper, we shall use [MATH] to denote a generic constant, and we take the standard Sobolev notations in [CITATION].', '1811.05028-2-8-1': 'When it is the whole domain [MATH], [MATH] and [MATH] are used to simplify [MATH] and [MATH] respectively, and [MATH] is used to denote the standard inner product of [MATH].', '1811.05028-2-8-2': '[MATH] denotes the expectation operator on the filtered probability space [MATH].', '1811.05028-2-9-0': 'In this section, we first derive the Holder continuity in time for the strong solution [MATH] with respect to the spatial [MATH]-seminorm and for the composite function [MATH] with respect to the spatial [MATH]-norm.', '1811.05028-2-9-1': 'Both results will play a key role in the error analysis (see Subsection [REF]).', '1811.05028-2-9-2': 'The time derivatives of [MATH] and the composite function [MATH] do not exist in the stochastic case, so these Holder continuity results will substitute for the differentiability of [MATH] and [MATH] with respect to time in the error analysis.', '1811.05028-2-10-0': 'Let [MATH] be the strong solution to problem [REF].', '1811.05028-2-10-1': 'Then for any [MATH] with [MATH], we have [EQUATION] where [EQUATION]', '1811.05028-2-10-2': "Applying Ito's formula to the functional [MATH] with fixed [MATH] and using integration by parts, we get [EQUATION]", '1811.05028-2-10-3': 'The expectation of the first term on the right-hand side of [REF] can be bounded by the Cauchy-Schwarz inequality as follows [EQUATION]', '1811.05028-2-10-4': 'The expectation of the second term on the right-hand side of [REF] can be bounded by [EQUATION]', '1811.05028-2-10-5': 'Next we bound the expectation of the fourth term on the right-hand side of [REF] as follows [EQUATION]', '1811.05028-2-10-6': 'Then Lemma [REF] follows from [REF]-[REF] and the fact that the expectation of the third term on the right-hand side of [REF] is zero.', '1811.05028-2-11-0': 'Next we prove the Holder continuity result for the nonlinear term [MATH] with respect to the spatial [MATH]-norm.', '1811.05028-2-12-0': 'Let [MATH] be the strong solution to problem [REF].', '1811.05028-2-12-1': 'Then for any [MATH] with [MATH], we have [EQUATION] where [EQUATION]', '1811.05028-2-12-2': "Applying Ito's formula to [MATH] with fixed [MATH], we obtain [EQUATION]", '1811.05028-2-12-3': "Taking the expectation on both sides, it follows from integration by parts and Young's inequality that [EQUATION]", '1811.05028-2-12-4': 'Finally, the desired Lemma [REF] follows from [REF].', '1811.05028-2-13-0': '(a) For the diffusion term, the global Lipschitz condition, which is stronger than the one-side Lipschitz condition, is needed as in the SODE case.', '1811.05028-2-13-1': 'Using the [MATH] assumption and the global Lipschitz assumption, we can derive that the derivative of the diffusion term is bounded by the Lipschitz constant [MATH], i.e., [MATH], but the diffusion term itself may not be bounded.', '1811.05028-2-13-2': 'For instance, [MATH], [MATH], etc.', '1811.05028-2-13-3': 'Notice these two assumptions are consistent with the SODE case in [CITATION], and they are also the conditions to guarantee the well-posedness [CITATION] of the strong SODE solution;', '1811.05028-2-14-0': '(b) We can verify [MATH] in [REF] satisfies a one-sided Lipschitz condition [REF].', '1811.05028-2-14-1': 'If the drift term [MATH] behaves polynomially, then for the one-sided Lipschitz condition [REF], we have the following conclusions:', '1811.05028-2-15-0': '(1).', '1811.05028-2-15-1': 'The power [MATH] of the highest order term must be odd.', '1811.05028-2-15-2': 'Because when the highest power [MATH] is even, dividing [MATH] by [MATH] yields the the quotient is odd so that it can be [MATH] and [MATH].', '1811.05028-2-15-3': 'When choosing [MATH] and [MATH] sufficiently large or small, the absolute value of this term is dominant and the left-hand side of [REF] is [MATH] where [MATH] can be [MATH], which is a contradiction;', '1811.05028-2-16-0': '(2).', '1811.05028-2-16-1': 'The sign of the highest odd order term must be negative.', '1811.05028-2-16-2': 'Because this term is dominant and the quotient of dividing [MATH] by [MATH] can be [MATH], which contradicts [REF].', '1811.05028-2-17-0': '# Fully discrete finite element approximation', '1811.05028-2-18-0': '## Formulation of the finite element method', '1811.05028-2-19-0': 'In this section, we first construct a fully discrete finite element method for problem [REF]-[REF].', '1811.05028-2-19-1': 'we then establish several stability properties for the numerical solution including the stability of higher order moments for its [MATH]-seminorm and [MATH]-norm.', '1811.05028-2-19-2': 'Finally, we derive optimal order error estimates in strong norms for the numerical solution using the stability estimates.', '1811.05028-2-20-0': 'Let [MATH] be a uniform partition of [MATH] and [MATH] be the triangulation of [MATH] satisfying the following assumption [CITATION]: [EQUATION] where [MATH] denotes the edge of simplex [MATH].', '1811.05028-2-20-1': 'It was proved in [CITATION] that the stiffness matrix for the Poisson equation with zero Dirichlet boundary is an [MATH]-matrix if and only if this assumption holds for all edges.', '1811.05028-2-20-2': 'The stiffness matrix is diagonally dominant if the Neumann boundary condition is considered.', '1811.05028-2-20-3': 'Notice this assumption is just the Delaunay triangulation when [MATH].', '1811.05028-2-20-4': 'In 3D, the notations in the assumption [REF] are as follows: [MATH] denote the vertices of [MATH], [MATH] the edge connecting two vertices [MATH] and [MATH], [MATH] the [MATH]-dimensional simplex opposite to the vertex [MATH], [MATH] or [MATH] the angle between the faces [MATH] and [MATH], [MATH] , the [MATH]-dimensional simplex opposite to the edge [MATH].', '1811.05028-2-20-5': 'See Figure [REF] below.', '1811.05028-2-21-0': 'Consider the [MATH]-Lagrangian finite element space [EQUATION] where [MATH] denotes the space of all linear polynomials.', '1811.05028-2-21-1': 'Then the finite element approximation of [REF] is to seek an [MATH] adapted [MATH]-valued process [MATH] such that it holds [MATH]-almost surely that [EQUATION] where [MATH], [MATH], and [MATH] is the standard nodal value interpolation operator [MATH], i.e., [EQUATION] where [MATH] denotes the number of vertices of [MATH], and [MATH] denotes the nodal basis function of [MATH] corresponding to the vertex [MATH].', '1811.05028-2-21-2': 'The initial condition is chosen by [MATH] where [MATH] is the [MATH]-projection operator defined by [EQUATION]', '1811.05028-2-22-0': 'For all [MATH], the following well-known error estimate results can be found in [CITATION]: [EQUATION]', '1811.05028-2-22-1': 'Finally, given [MATH], we define the discrete Laplace operator [MATH] by [EQUATION]', '1811.05028-2-23-0': '## Stability estimates for the [MATH]-th moment of the [MATH]-seminorm of [MATH]', '1811.05028-2-24-0': 'First we shall prove the second moment discrete [MATH]-seminorm stability result, which is necessary to establish the corresponding higher moment stability result.', '1811.05028-2-25-0': 'Suppose the mesh assumption in [REF] holds, then [EQUATION]', '1811.05028-2-25-1': 'Testing [REF] with [MATH], then [EQUATION]', '1811.05028-2-25-2': 'Using the definition of the discrete Laplace operator, we get [EQUATION] where the stability in the [MATH]-seminorm of the [MATH] projection [CITATION] is used in the inequality of [REF].', '1811.05028-2-26-0': 'The crucial part is to bound the first term on the right-hand side of [REF] since it cannot be treated as a bad term, which aligns with the continuous case.', '1811.05028-2-26-1': 'Denote [MATH], then [EQUATION] where [MATH].', '1811.05028-2-27-0': "Using Young's inequality when [MATH], we have [EQUATION]", '1811.05028-2-27-1': 'Besides, since the stiffness matrix is diagonally dominant, then [EQUATION]', '1811.05028-2-27-2': 'Then we have [EQUATION]', '1811.05028-2-27-3': 'Combining [REF]-[REF] and [REF], and taking the summation, we have [EQUATION]', '1811.05028-2-27-4': "Using Gronwall's inequality, we obtain [REF].", '1811.05028-2-28-0': 'Before we establish the error estimates, we need to prove the stability of the higher order moments for the [MATH]-seminorm of the numerical solution.', '1811.05028-2-29-0': 'Suppose the mesh assumption in [REF] holds, then for any [MATH], [EQUATION]', '1811.05028-2-29-1': 'The proof is divided into three steps.', '1811.05028-2-29-2': 'In Step 1, we establish the bound for [MATH].', '1811.05028-2-29-3': 'In Step 2, we give the bound for [MATH], where [MATH] and [MATH] is an arbitrary positive integer.', '1811.05028-2-29-4': 'In Step 3, we obtain the bound for [MATH], where [MATH] is an arbitrary real number and [MATH].', '1811.05028-2-30-0': 'Step 1.', '1811.05028-2-30-1': 'Based on [REF]-[REF], we have [EQUATION]', '1811.05028-2-30-2': 'Notice the following identity [EQUATION] and multiplying [REF] with [MATH], we obtain [EQUATION]', '1811.05028-2-30-3': 'The first term on the right-hand side of [REF] can be written as [EQUATION] where [MATH] will be determined later.', '1811.05028-2-31-0': 'The second term on the right-hand side of [REF] can be written as [EQUATION]', '1811.05028-2-31-1': 'For the right-hand side of [REF], using the Cauchy-Schwarz inequality, we get [EQUATION] where [MATH] will be determined later.', '1811.05028-2-31-2': 'Similarly, using the Cauchy-Schwarz inequality, we have [EQUATION] where [MATH] will be determined later.', '1811.05028-2-32-0': 'Choosing [MATH] such that [MATH], then taking the summation over [MATH] from [MATH] to [MATH] and taking the expectation on both sides of [REF], we obtain [EQUATION]', '1811.05028-2-32-1': 'When restricting [MATH], we have [EQUATION]', '1811.05028-2-32-2': "Using Gronwall's inequality, we obtain [EQUATION]", '1811.05028-2-32-3': 'Step 2.', '1811.05028-2-32-4': 'Similar to Step 1, using [REF]-[REF], we have [EQUATION]', '1811.05028-2-32-5': 'Proceed similarly as in Step 1, multiplying [REF] with [MATH], we can obtain the 8-th moment of the [MATH]-seminorm stability result of the numerical solution.', '1811.05028-2-32-6': 'Then repeating this process, the [MATH]-th moment of the [MATH]-seminorm stability result of the numerical solution can be obtained.', '1811.05028-2-33-0': 'Step 3.', '1811.05028-2-33-1': "Suppose [MATH], then using Young's inequality, we have [EQUATION] where the second inequality follows from the results of Step 2.", '1811.05028-2-33-2': 'The proof is complete.', '1811.05028-2-34-0': '## Stability estimates for the [MATH]-th moment of the [MATH]-norm of [MATH]', '1811.05028-2-35-0': 'Since the mass matrix may not be the diagonally dominated matrix, we cannot use the above idea to prove the [MATH] stability.', '1811.05028-2-35-1': 'Instead, we prove the stability results by utilizing the above established results.', '1811.05028-2-35-2': 'The following results hold when [MATH] is the odd integer in 2D case, and when [MATH] or [MATH] in 3D case.', '1811.05028-2-36-0': 'Suppose the mesh assumption in [REF] holds, then [EQUATION]', '1811.05028-2-36-1': 'Testing [REF] with [MATH], then [EQUATION]', '1811.05028-2-36-2': 'We can easily prove the following inequalities: [EQUATION] where [REF] is used in the inequality above.', '1811.05028-2-37-0': 'We have the following standard interpolation result and the inverse inequality [CITATION]: [EQUATION]', '1811.05028-2-37-1': "Using [REF]-[REF], and Young's inequality, we have [EQUATION]", '1811.05028-2-37-2': 'Notice when [MATH], [MATH] if [MATH], and when [MATH], [MATH] if [MATH].', '1811.05028-2-37-3': 'Using the above inequalities, Theorem [REF], taking summation over [MATH] from [MATH] to [MATH], and taking expectation on both sides of [REF], we obtain [EQUATION] where Theorem [REF] is used in the last inequality.', '1811.05028-2-38-0': "The conclusion is a direct result by using Gronwall's inequality.", '1811.05028-2-39-0': 'To obtain the error estimates results, we need to establish a higher moment discrete [MATH] stability result for the numerical solution [MATH].', '1811.05028-2-40-0': 'Suppose the mesh assumption in [REF] holds, then for any [MATH], [EQUATION]', '1811.05028-2-40-1': 'The proof is divided into three steps.', '1811.05028-2-40-2': 'In Step 1, we give the bound for [MATH].', '1811.05028-2-40-3': 'In Step 2, we give the bound for [MATH], where [MATH] and [MATH] is an arbitrary positive integer.', '1811.05028-2-40-4': 'In Step 3, we give the bound for [MATH], where [MATH] is an arbitrary real number and [MATH].', '1811.05028-2-41-0': 'Step 1.', '1811.05028-2-41-1': 'Based on [REF] and [REF], we have [EQUATION]', '1811.05028-2-41-2': 'Notice the following identity [EQUATION]', '1811.05028-2-41-3': 'Multiplying [REF] with [MATH], we obtain [EQUATION]', '1811.05028-2-41-4': 'The first term on the right-hand side of [REF] can be written as [EQUATION] where [MATH] will be determined later.', '1811.05028-2-42-0': 'The second term on the right-hand side of [REF] can be written as [EQUATION]', '1811.05028-2-42-1': 'For the second term on the right-hand side of [REF], using the Cauchy-Schwarz inequality, we get [EQUATION] where [MATH] will be determined later.', '1811.05028-2-42-2': 'Using [REF], the third term on the right-hand side of [REF] can be bounded by [EQUATION] where [MATH] will be determined later.', '1811.05028-2-43-0': 'Choosing [MATH] such that [MATH], then taking the summation over [MATH] from [MATH] to [MATH] and taking the expectation on both sides of [REF], we obtain [EQUATION]', '1811.05028-2-43-1': 'When [MATH], we have [EQUATION]', '1811.05028-2-43-2': "Using Gronwall's inequality, we obtain [EQUATION]", '1811.05028-2-43-3': 'Step 2.', '1811.05028-2-43-4': 'Similar to Step 1, using [REF]-[REF], we have [EQUATION]', '1811.05028-2-43-5': 'Similar to Step 1, multiplying [REF] with [MATH], we can obtain the 8-th moment of the [MATH] stability result of the discrete solution.', '1811.05028-2-43-6': 'Then repeating this process, the [MATH]-th moment of the [MATH] stability result of the discrete solution can be obtained.', '1811.05028-2-44-0': 'Step 3.', '1811.05028-2-44-1': "Suppose [MATH], then using Young's inequality, we have [EQUATION] where the second inequality uses Step 2.", '1811.05028-2-44-2': 'The proof is complete.', '1811.05028-2-45-0': '## Error estimates', '1811.05028-2-46-0': 'Let [MATH].', '1811.05028-2-46-1': 'In the following theorem, the [MATH] projection is used in the proof of the error estimates and the strong convergence rate is given.', '1811.05028-2-47-0': 'Let [MATH] and [MATH] denote respectively the solutions of problem [REF] and scheme [REF], then there holds [EQUATION]', '1811.05028-2-47-1': 'We write [MATH] where [EQUATION]', '1811.05028-2-47-2': 'It follows from [REF] that for all [MATH]) there holds [MATH]-almost surely [EQUATION]', '1811.05028-2-47-3': 'Subtracting [REF] from [REF] and setting [MATH], the following error equation holds [MATH]-almost surely, [EQUATION]', '1811.05028-2-47-4': 'The left-hand side of [REF] can be handled by [EQUATION]', '1811.05028-2-47-5': 'Next, we bound the right-hand side of [REF].', '1811.05028-2-47-6': 'First, since [MATH] is the [MATH]-projection operator, we have [MATH].', '1811.05028-2-48-0': 'For the second term on the right-hand side of [REF], using the Holder continuity in Lemma [REF], we have [EQUATION]', '1811.05028-2-48-1': 'In order to estimate the third term on the right-hand side of [REF], we write [EQUATION]', '1811.05028-2-48-2': 'Using the Holder continuity in Lemma [REF], we obtain [EQUATION]', '1811.05028-2-48-3': 'Next, using properties of the projection, we have [EQUATION]', '1811.05028-2-48-4': 'The third term on the right-hand side of [REF] can be bounded by [EQUATION]', '1811.05028-2-48-5': 'Using Theorem [REF], properties of the interpolation operator, the inverse inequality, and the fact that [MATH] is a piecewise linear polynomial, the fourth term on the right-hand side of [REF] can be handled by [EQUATION]', '1811.05028-2-48-6': 'Combining [REF]-[REF] to obtain [EQUATION]', '1811.05028-2-48-7': 'By the martingale property, the Ito isometry, the Holder continuity of [MATH] and the global Lipschitz condition [REF], we have [EQUATION]', '1811.05028-2-48-8': 'Taking the expectation on [REF] and combining estimates [REF]-[REF], summing over [MATH] with [MATH], and using the properties of the [MATH] projection and the regularity assumption, we obtain [EQUATION]', '1811.05028-2-48-9': "Finally, the assertion of the theorem follows from [REF], the discrete Gronwall's inequality, the [MATH]-projection properties, the fact that [MATH] and the triangle inequality.", '1811.05028-2-48-10': 'The proof is complete.', '1811.05028-2-49-0': 'The following strong stability result is a direct corollary of Theorem [REF].', '1811.05028-2-50-0': 'Suppose the mesh assumption in [REF] holds and [MATH], then [EQUATION]', '1811.05028-2-50-1': 'For each sample point, [EQUATION]', '1811.05028-2-50-2': 'When [MATH], taking the expectation on both sides of [REF], and using Theorem [REF], we obtain [EQUATION] (a) Notice the elliptic projection cannot be used due to the first term [MATH] in [REF].', '1811.05028-2-50-3': 'In reference [CITATION], it is [MATH] since [MATH] projection is used there.', '1811.05028-2-51-0': '(b) For the diffusion term, We need [MATH] and [MATH] to be Lipschitz continuous, which are the same assumptions as in stochastic ODE case [CITATION].', '1811.05028-2-51-1': 'The analysis in [CITATION] requires two extra conditions: [MATH] and [MATH] are bounded.', '1811.05028-2-51-2': 'Notice [MATH], [MATH] or some others satisfy the assumptions in this paper, but they do not satisfy the assumptions in [CITATION].', '1811.05028-2-52-0': '# Numerical experiments', '1811.05028-2-53-0': 'In this section, we present several two dimensional numerical examples to gauge the performance of the proposed stochastic finite element scheme for the stochastic partial differential equations satisfying the proposed assumptions for the nonlinear term and the diffusion term.', '1811.05028-2-53-1': 'Test 1 is designed to demonstrate the error orders with respect to mesh size [MATH] for small and big noises; Test 2 is designed to demonstrate the stability results and evolution of the stochastic Allen-Cahn equation, which is a special case of the SPDE in this paper; Test 3 is designed to demonstrate the stability results of the SPDE with a different initial condition; Test 4 is designed to demonstrate the stability results of the SPDE with a different nonlinear term; Test 5 is designed to demonstrate the stability results of the SPDE with a different diffusion term.', '1811.05028-2-53-2': 'The square domain [MATH], and 500 sample points are used in these tests.', '1811.05028-2-54-0': 'Test 1 Consider the following smooth initial condition [EQUATION] where [MATH].', '1811.05028-2-54-1': 'Time step size [MATH] is used in this Test 1.', '1811.05028-2-55-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-2-55-1': 'Table [REF] shows the following three types of errors [MATH], [MATH], and [MATH] respectively, and the rates of convergence.', '1811.05028-2-55-2': 'The noise intensity [MATH].', '1811.05028-2-55-3': 'In the table, we use [MATH], [MATH] and [MATH] to denote these three types of errors respectively.', '1811.05028-2-56-0': 'Table [REF] shows the errors [MATH], [MATH] and [MATH] respectively, and the rates of convergence at final time [MATH].', '1811.05028-2-56-1': 'The noise intensity [MATH].', '1811.05028-2-57-0': 'From these two tables, we observe that the error orders of [MATH] and [MATH] are 2, and the error order of [MATH] is 1.', '1811.05028-2-57-1': 'Besides, the error orders keep the same when the noise intensity increases.', '1811.05028-2-58-0': 'In the following tests, [MATH] and [MATH] are used to denote [MATH] and [MATH] respectively.', '1811.05028-2-59-0': 'Test 2 Consider the following initial condition [EQUATION]', '1811.05028-2-59-1': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH], which corresponds to the stochastic Allen-Cahn equation.', '1811.05028-2-59-2': 'More tests related to the Allen-Cahn equation can be found in [CITATION].', '1811.05028-2-59-3': 'Figure [REF] shows the evolution of the zero-level sets of the solutions under different intensity of the noise.', '1811.05028-2-59-4': 'We observe that although the circle may shrink or dilate (depending on the sign of the diffusion term), the average zero-level sets shrink for smaller and bigger noises.', '1811.05028-2-59-5': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-2-59-6': 'We also observe that they are both bounded.', '1811.05028-2-60-0': 'Test 3 Consider the following initial condition [EQUATION]', '1811.05028-2-60-1': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-2-60-2': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-2-61-0': 'Test 4 Consider the initial condition in [REF] with [MATH].', '1811.05028-2-62-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-2-62-1': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].', '1811.05028-2-63-0': 'Test 5 Consider the initial condition in [REF] with [MATH].', '1811.05028-2-64-0': 'In this test, the nonlinear term [MATH], and the diffusion term [MATH].', '1811.05028-2-64-1': 'Figure [REF] shows the [MATH] and [MATH] stability results at each time step, which verifies the results in Theorems [REF] and [REF].'} | [['1811.05028-1-50-0', '1811.05028-2-50-0'], ['1811.05028-1-50-1', '1811.05028-2-50-1'], ['1811.05028-1-50-2', '1811.05028-2-50-2'], ['1811.05028-1-50-3', '1811.05028-2-50-3'], ['1811.05028-1-14-0', '1811.05028-2-14-0'], ['1811.05028-1-40-0', '1811.05028-2-40-0'], 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'1811.05028-1-63-0', '1811.05028-2-2-0', '1811.05028-2-13-2', '1811.05028-2-14-1', '1811.05028-2-15-0', '1811.05028-2-16-0', '1811.05028-2-30-0', '1811.05028-2-32-3', '1811.05028-2-33-0', '1811.05028-2-37-2', '1811.05028-2-38-0', '1811.05028-2-41-0', '1811.05028-2-43-3', '1811.05028-2-44-0', '1811.05028-2-46-0', '1811.05028-2-48-6', '1811.05028-2-61-0', '1811.05028-2-63-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1811.05028 | null | null | null | null | null |
physics-0304012 | {'physics-0304012-1-0-0': 'The creation of a Fermi-degenerate gas molecules using either photoassociation or the Feshbach resonance is theoretically examined.', 'physics-0304012-1-0-1': 'This problem raises an interest because, unlike bosons, fermions in general do not behave cooperatively, so that the collective association of, say, two million atoms into one million molecules is not to be expected.', 'physics-0304012-1-0-2': 'Nevertheless, we find that the coupled Fermi system displays collective Rabi-like oscillations and adiabatic passage between atoms and molecules, thereby mimicking Bose-Einstein statistics.', 'physics-0304012-1-0-3': 'Cooperative association of a degenerate mixture of Bose and Fermi gases could therefore serve as a shortcut to a degenerate gas of Fermi molecules.', 'physics-0304012-1-1-0': 'Because of the large number of ro-vibrational degrees of freedom in a molecule, laser cooling is difficult if not impossible to apply in practice, so that a laser-cooling-based approach [CITATION] to a Fermi-degenerate gas of atoms will not work for molecules.', 'physics-0304012-1-1-1': 'Moreover, buffer-gas cooling [CITATION] and Stark-deceleration [CITATION] techniques could take years to reach quantum degeneracy.', 'physics-0304012-1-1-2': 'Here we report that a degenerate gas of Fermi molecules could be formed by applying either photoassociation [CITATION] or a Feshbach resonance [CITATION] to a degenerate mixture of Bose and Fermi atoms [CITATION].', 'physics-0304012-1-1-3': 'The collective Fermi-atom-molecule system is predicted to undergo Rabi-like oscillations, as well as adiabatic passage, from atoms to molecules, thereby mimicking cooperative behavior that was previously attributed to Bose statistics [CITATION].', 'physics-0304012-1-2-0': 'Photoassociation occurs when two atoms absorb a laser photon, thereby jumping from the free two-atom continuum to a bound molecular state.', 'physics-0304012-1-2-1': 'If the initial atoms form a Bose-Einstein condensate (BEC), then a photoassociation laser could conceivably be used to collectively convert a BEC of atoms into a BEC of molecules [CITATION].', 'physics-0304012-1-2-2': 'As it happens, collective free-bound photoassociation is theoretically identical to magnetoassociation, whereby a molecular condensate can be created when one atom from a condensed pair spin flips in the presence of a magnetic field tuned near a Feshbach resonance [CITATION].', 'physics-0304012-1-2-3': 'Intuition developed in one instance is therefore generally applicable to the other, and we will often refer simply to collective association.', 'physics-0304012-1-3-0': 'In analogy with coherent optical transients in few level atomic systems [CITATION], photoassociation of a BEC has been predicted to facilitate Rabi-like oscillations between atomic and molecular condensates [CITATION], whereby an entire gas of, say, two million Bose-condensed atoms are collectively converted into a million molecules that are, in turn, collectively converted back into (roughly) two million atoms, ad infinitum.', 'physics-0304012-1-3-1': 'Another interesting possibility arises because the ground state of the system is all atoms for large positive detunings and all molecules for extremely negative detunings, so that a slow sweep of the laser detuning from one extreme to the other will collectively convert a BEC of atoms into a BEC of molecules [CITATION].', 'physics-0304012-1-3-2': 'The Feshbach resonance was used in a similar manner to observe [CITATION] collective Ramsey fringes between an atomic condensate and a small fraction of molecular condensate dressed by dissociated atom pairs [CITATION].', 'physics-0304012-1-4-0': 'The statistics of neutral atoms is determined by the number of neutrons in the nucleus, which must be odd for fermionic atoms and even for bosonic atoms.', 'physics-0304012-1-4-1': 'The sum of the total number of neutrons in the nucleus of the constituent atoms similarly determines the statistics of neutral molecules.', 'physics-0304012-1-4-2': 'Accordingly, molecules formed by free-bound association of two fermions will necessarily result in a boson, whereas fermionic molecules are born from the union of a boson and a fermion.', 'physics-0304012-1-4-3': 'Given a degenerate mixture of Bose and Fermi gases [CITATION], is it possible that collective free-bound association could serve as a source of Fermi molecules?', 'physics-0304012-1-5-0': 'To address this question, we model a degenerate Bose-Fermi mixture of atoms [CITATION] coupled by either a Feshbach resonance or photoassociation to a Fermi-degenerate gas of molecules.', 'physics-0304012-1-5-1': 'The initial bosonic [fermionic] atoms are denoted by the field [MATH] [[MATH]], and the fermionic molecules by the field [MATH].', 'physics-0304012-1-5-2': 'Neglecting particle-particle interactions, the Hamiltonian density for such an untrapped system can be written [MATH], where [EQUATION]', 'physics-0304012-1-5-3': 'Here [MATH]) is the mass of a bosonic (fermionic) atom, [MATH] is the mass of a molecule, [MATH] is the so-called chemical potential that implicitly accounts for particle trapping, [MATH] is the binding energy (detuning) of the molecular state relative to the dissociation threshold, and [MATH] is the (real) atom-molecule coupling.', 'physics-0304012-1-6-0': 'Without loss of generality, we now make some simplifications.', 'physics-0304012-1-6-1': 'First, the number of atoms, and the traps that contain them, can be adjusted so that [MATH].', 'physics-0304012-1-6-2': 'Second is that atom-molecule conversion occurs on a timescale much faster than the motion of the atoms in the trap, allowing us to neglect the kinetic energies, and justifying our omission of an explicit trapping potential for the particles.', 'physics-0304012-1-6-3': 'Third, since the total number of particles, [MATH] is a conserved quantity, we may add the term [MATH] to the Hamiltonian without altering the physics; upon doing so, we also redefine the detuning as [MATH].', 'physics-0304012-1-6-4': 'Fourth is that, whatever the Fermi energy associated with the fermionic components, it lies within the Wigner threshold regime [CITATION], so that the coupling [MATH] can be taken as the same for all modes.', 'physics-0304012-1-6-5': 'Finally, we switch to momentum space, but retain only the [MATH] atomic condensate modes since these transitions are favored over [MATH] modes due to Bose stimulation.', 'physics-0304012-1-6-6': 'The simplified Hamiltonian reads [EQUATION] where [MATH] annihilates a bosonic atom with wavevector [MATH], [MATH]) annihilates an atom (molecule) with wavevector [MATH], and [MATH].', 'physics-0304012-1-7-0': 'The system described by the Hamiltonian ([REF]) is a particular nonlinear version of a two-level atom driven by an external field.', 'physics-0304012-1-7-1': 'Of course, the two-level atom is well known to undergo Rabi flopping [CITATION], and similar behavior observed for the present system would in principle answer question of whether collective association will provide a shortcut to the formation of a Fermi-degenerate gas of molecules.', 'physics-0304012-1-7-2': 'We therefore consider an on-resonance system, for which [MATH] and the Heisenberg equations of motion are [EQUATION]', 'physics-0304012-1-7-3': 'If the system starts out with [MATH] bosonic atoms, the operator [MATH] will have a characteristic size [MATH].', 'physics-0304012-1-7-4': 'On the other hand, while the fermionic operators [MATH] and [MATH] are characteristically of order unity, the sum over momentum states will ultimately introduce the initial number of fermionic atoms.', 'physics-0304012-1-7-5': 'Hence, for equal numbers of bosons and fermions, [MATH], it is intuitively obvious that the equations of motion ([REF]) yield a system that evolves with the characteristic frequency [MATH].', 'physics-0304012-1-8-0': 'Said intuition is confirmed by solving the time-dependent Schrodinger equation numerically, the results of which are shown in Fig. [REF].', 'physics-0304012-1-8-1': 'For the simplest case of [MATH], the system undergoes complete conversion from a doubly-degenerate Fermi-Bose system of atoms to a Fermi-degenerate gas of molecules in a time [MATH].', 'physics-0304012-1-8-2': 'However, considering a larger number of initial atoms, [MATH], we see that quantum many-body fluctuations not only frustrate complete conversion, but also adjust the oscillation frequency and lead to a series of collapse and revivals.', 'physics-0304012-1-8-3': 'Increasing the initial particle number to [MATH] brings the amplitude of the first half-oscillation closer to the ideal case.', 'physics-0304012-1-8-4': 'This behavior is exactly analogous to the single-component bosonic case [CITATION].', 'physics-0304012-1-8-5': 'Although limited computational resources preclude the explicit investigation of large particle number, based on the bosonic analogy [CITATION] we fully expect the first half oscillation to be complete for large [MATH], i.e., [MATH] initial Fermi atoms should be converted to [MATH] Fermi molecules over a timescale [MATH].', 'physics-0304012-1-9-0': 'Nevertheless, collisions between particles will shift the energy of a state with a given number of atoms and molecules in a manner that depends nonlinearly on the numbers, making it difficult to maintain exact resonance.', 'physics-0304012-1-9-1': 'Furthermore, once a the system has Rabi flopped to molecules, the depopulated atomic condensate makes Bose stimulation, and thus neglect of the [MATH] BEC modes, questionable.', 'physics-0304012-1-9-2': 'And so we investigate the more robust possibility of rapid adiabatic passage [CITATION].', 'physics-0304012-1-9-3': 'From the Hamiltonian ([REF]), it should be clear that the system will favor all atoms for large positive detunings, while favoring all molecules for large negative detunings.', 'physics-0304012-1-9-4': 'With [MATH] established as the characteristic frequency for collective atom-molecule conversion, then changes in the detuning that are of the order of [MATH], and occur over a time of order [MATH], should qualify as adiabatic.', 'physics-0304012-1-9-5': 'Hence, if the detuning is swept in a linear fashion according to [MATH], then dimensionless sweep rates [MATH] should enable rapid adiabatic passage to a degenerate gas of Fermi molecules.', 'physics-0304012-1-10-0': 'Our suspicions are again corroborated by a numerical solution to the Schrodinger equation, shown in Fig. [REF].', 'physics-0304012-1-10-1': 'While many-body effects appear to rather seriously affect the efficiency of a marginally adiabatic sweep ([MATH]) compared to the [MATH] case, the difference between [MATH] and [MATH] (not shown) is in fact small.', 'physics-0304012-1-10-2': 'Overall, many-body effects are expected to be weak for near-adiabatic sweeps ([MATH]), and vanishing for sweeps that are deep-adiabatic ([MATH]), again in agreement with our BEC experience [CITATION].', 'physics-0304012-1-10-3': 'On the matter of transitions to noncondensate modes, we note that, in the all-boson case [CITATION], such transitions can be neglected for a sweep directed as in Fig. [REF], i.e., for [MATH].', 'physics-0304012-1-10-4': 'Given the success of the analogy so far, similar results are expected for a Fermi system.', 'physics-0304012-1-11-0': 'Before closing, we estimate some explicit numbers.', 'physics-0304012-1-11-1': 'We eschew photoassociation because of the losses associated with the electronically-excited state [CITATION], and focus on the atom-molecule coupling provided by the [MATH]-[MATH]K Feshbach resonance located at [MATH]G [CITATION], which has a width [MATH]G and an associated zero-field Fermi-Bose-atom scattering length [MATH]nm.', 'physics-0304012-1-11-2': 'Accordingly, the atom-molecule coupling is [MATH], where we have estimated the difference between the fermionic-atom and -molecule magnetic moments to be equal to the Bohr magneton [MATH].', 'physics-0304012-1-11-3': 'Assuming [MATH] condensate atoms in a trap with respective radial and axial frequencies [MATH]Hz and [MATH]Hz [CITATION], the density of bosons is [MATH].', 'physics-0304012-1-11-4': 'As for the fermions, we assume a modest number, say, [MATH], which two consequences: (i) the atomic BEC will act as a reservoir, thus absorbing any heat created by holes in the Fermi sea [CITATION]; (ii) barring an unfortunately large scattering length for Bose-atom and Fermi-molecule collisions, we can neglect the possibility of any Fermi-Bose collapse instabilities [CITATION].', 'physics-0304012-1-11-5': 'For [MATH], the timescale for atom-molecule conversion is [MATH].', 'physics-0304012-1-11-6': 'This timescale is safely below the fastest timescale for trapped-atom motion [MATH], justifying our neglect of trap dynamics and the kinetic energy.', 'physics-0304012-1-11-7': 'Similarly, the timescale for Bose atom-atom and Fermi-Bose atom-molecule collisions are roughly equal, [MATH]ms, and are also reasonably neglected.', 'physics-0304012-1-12-0': 'Lastly, a problem with the Feshbach resonance is noted: the molecules it creates, while translationally and rotationally ultracold, are vibrationally very hot, and are therefore highly susceptible to vibrational relaxation induced by collisions with unconverted atoms (molecule-molecule collisions are Pauli blocked).', 'physics-0304012-1-12-1': 'There is no known handle on such transitions for the present system.', 'physics-0304012-1-12-2': 'Using the sodium numbers [CITATION] for comparison, Fermi-Bose [Fermi-Fermi] collisions are estimated to cause vibrational relaxation on a timescale [MATH] [[MATH]], which is also negligible.', 'physics-0304012-1-12-3': 'If collision-induced vibrational relaxation of the molecules does turn out to be a problem, the trap could be expanded to further reduce the rate of collisions with unconverted atoms.', 'physics-0304012-1-13-0': 'In conclusion, we highlight that the term [MATH] was previously referred to as the Bose-enhanced free-bound coupling, and the detuning sweep, e.g., referred to as Bose-stimulated rapid adiabatic passage from atoms to molecules.', 'physics-0304012-1-13-1': 'However, this behavior is now clearly independent of statistics, so that Bose stimulation of free-bound association has nothing whatsoever to do with Bose statistics, but is instead a many-body cooperative effect that applies equally well to Fermi-degenerate systems.', 'physics-0304012-1-13-2': 'To four-wave mixing in a Fermi gas of atoms [CITATION], we therefore add the above two examples of fermions mimicking an effect that was previously attributed to Bose-Einstein statistics.', 'physics-0304012-1-13-3': 'Such behavior should allow for a shortcut to a degenerate gas of Fermi molecules via photoassociation or the Feshbach resonance.', 'physics-0304012-1-14-0': 'The authors acknowledge Juha Javanainen for helpful discussions.', 'physics-0304012-1-14-1': 'This work supported by the Academy of Finland (MM and KAS, project 50314), and the Magnus Ehrnrooth foundation (OD).', 'physics-0304012-1-15-0': 'MAC00 For a discussion of the role of Bose enhancement in photoassociation, see M. Mackie and J. Javanainen, 47, 2645 (2000).', 'physics-0304012-1-16-0': 'JAV02 For a brief review of cooperative fermion behavior, see J. Javanainen, Nature 412, 689 (2002).', 'physics-0304012-1-17-0': 'ALL87 L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, (Dover, New York, 1987).', 'physics-0304012-1-18-0': 'TWO_CO A two-color photoassociation scheme, whereby electronically excited molecules are transferred to a stable molecule state, would of course avert such losses, and the work herein indicates that "Bose-stimulated" Raman adiabatic passage from atoms to molecules [M. Mackie et al., 84, 3803 (2000)] will in principle work for fermions.'} | {'physics-0304012-2-0-0': 'We theoretically examine the creation of a Fermi-degenerate gas of molecules by considering a photoassociation or Feshbach resonance applied to a degenerate Bose-Fermi mixture of atoms.', 'physics-0304012-2-0-1': 'This problem raises an interest because, unlike bosons, fermions in general do not behave cooperatively, so that the collective conversion of, say, two million atoms into one million molecules is not to be expected.', 'physics-0304012-2-0-2': 'Nevertheless, we find that the coupled Fermi system displays collective Rabi-like oscillations and adiabatic passage between atoms and molecules, thereby mimicking Bose-Einstein statistics.', 'physics-0304012-2-0-3': 'Cooperative association of a degenerate mixture of Bose and Fermi gases could therefore serve as a shortcut to a degenerate gas of Fermi molecules.', 'physics-0304012-2-1-0': 'Laser cooling of molecules is difficult, if not impossible, to apply in practice.', 'physics-0304012-2-1-1': 'This bottleneck occurs because the removal of energy from the translational degrees of freedom is most often accompanied by excitation of ro-vibrational modes which, upon de-excitation, couple back into the translational motion in a manner that heats the system.', 'physics-0304012-2-1-2': 'A laser-cooling-based approach[CITATION] to a Fermi-degenerate gas of molecules is therefore unlikely to succeed.', 'physics-0304012-2-1-3': 'Moreover, buffer-gas cooling [CITATION] and Stark-deceleration [CITATION] techniques, while no doubt of practical interest in their own right, currently view quantum degeneracy from the horizon.', 'physics-0304012-2-2-0': 'One possible route to ultracold molecules is photoassociation[CITATION], and the formation of nondegenerate Fermi molecules was indeed recently observed[CITATION].', 'physics-0304012-2-2-1': 'A shortcut to quantum degenerate molecules is in the early stages of development for Bose-Einstein condensates (BECs), so far delivering strongly enhanced molecule formation[CITATION], precise measurements of the light-induced shift of the binding energy[CITATION], and searches for a fundamental limit to the atom-molecule conversion rate[CITATION].', 'physics-0304012-2-2-2': 'The magnetic field counterpart to photoassociation is the Feshbach resonance[CITATION].', 'physics-0304012-2-2-3': 'Often seen as a tool for adjusting the scattering lengths, thereby enabling condensation in otherwise-incondensible systems[CITATION], the latest studies of a BEC tuned near a Feshbach resonance have culminated in evidence for atom-molecule coherence[CITATION].', 'physics-0304012-2-3-0': 'Motivated by these exciting experiments, we report that a degenerate gas of Fermi molecules could be formed by applying either a photoassociation [CITATION] or Feshbach resonance [CITATION] to a degenerate mixture of Bose and Fermi atoms [CITATION].', 'physics-0304012-2-3-1': 'The collective Fermi atom-molecule system is predicted to undergo Rabi-like oscillations, as well as adiabatic passage, from atoms to molecules, thereby mimicking cooperative behavior that was previously attributed to Bose statistics [CITATION].', 'physics-0304012-2-4-0': 'Photoassociation occurs when two atoms absorb a laser photon, thereby jumping from the free two-atom continuum to a bound molecular state.', 'physics-0304012-2-4-1': 'If the initial atoms form a Bose-Einstein condensate, then a photoassociation laser could conceivably be used to convert a BEC of atoms into a BEC of molecules [CITATION].', 'physics-0304012-2-4-2': 'Collective free-bound photoassociation is theoretically identical to magnetoassociation, whereby a molecular condensate can be created when one atom from a condensed pair spin flips in the presence of a magnetic field tuned near a Feshbach resonance [CITATION].', 'physics-0304012-2-4-3': 'Intuition developed in one instance is therefore generally applicable to the other, and we will often refer simply to collective association.', 'physics-0304012-2-5-0': 'Analogous with coherent optical transients in few level atomic systems [CITATION], photoassociation of a BEC has been predicted to induce Rabi-like oscillations between atomic and molecular condensates [CITATION], whereby an entire gas of, say, two million Bose-condensed atoms are collectively converted into a million molecules that are, in turn, collectively converted back into (roughly) two million atoms, ad infinitum.', 'physics-0304012-2-5-1': 'Another interesting possibility arises because the ground state of the system is all atoms for large positive detunings (far below threshold) and all molecules for large negative detunings (far above threshold), so that a slow sweep of the laser detuning from one extreme to the other will collectively convert a BEC of atoms into a BEC of molecules [CITATION].', 'physics-0304012-2-5-2': 'Incidentally, it was a particular combination of these two concepts, applied instead to magnetoassociation, that led to the observation[CITATION] of collective Ramsey fringes between an atomic condensate and a small fraction of molecular condensate dressed by dissociated atom pairs [CITATION].', 'physics-0304012-2-6-0': 'The statistics of neutral atoms is determined by the number of neutrons in the nucleus, which must be odd for fermionic atoms and even for bosonic atoms.', 'physics-0304012-2-6-1': 'The sum of the total number of neutrons in the nucleus of the constituent atoms similarly determines the statistics of neutral molecules.', 'physics-0304012-2-6-2': 'Accordingly, molecules formed by free-bound association of two fermions will necessarily result in a boson, whereas fermionic molecules are born from the union of a boson and a fermion.', 'physics-0304012-2-6-3': 'Given a degenerate mixture of Bose and Fermi gases [CITATION], is it possible that collective free-bound association could serve as a source of degenerate Fermi molecules?', 'physics-0304012-2-7-0': 'To address this question, we model a degenerate Bose-Fermi mixture of atoms [CITATION] coupled by either a Feshbach or photoassociation resonance to a Fermi-degenerate gas of molecules.', 'physics-0304012-2-7-1': 'The initial bosonic [fermionic] atoms are denoted by the field [MATH] [[MATH]], and the fermionic molecules by the field [MATH].', 'physics-0304012-2-7-2': 'Neglecting particle-particle interactions, the Hamiltonian density for such an untrapped system can be written [MATH], where [EQUATION]', 'physics-0304012-2-7-3': 'Here [MATH]) is the mass of a bosonic (fermionic) atom, [MATH] is the mass of a molecule, [MATH] is the so-called chemical potential that implicitly accounts for particle trapping, [MATH] is the binding energy (detuning) of the molecular state relative to the dissociation threshold, and [MATH] is the (real) atom-molecule coupling.', 'physics-0304012-2-8-0': 'We now make some simplifications to allow for ease of modeling.', 'physics-0304012-2-8-1': 'First, we scale the Fermi (Bose) atom fields as [MATH]), and the molecule field as [MATH], which yields [MATH]; in turn, [MATH] is absorbed into the detuning.', 'physics-0304012-2-8-2': 'Second is that atom-molecule conversion occurs on a timescale much faster than the motion of the atoms in the trap, allowing us to neglect the kinetic energies, and justifying our omission of an explicit trapping potential for the particles.', 'physics-0304012-2-8-3': 'Third is that, whatever the Fermi energy associated with the fermionic components, it lies within the Wigner threshold regime, so that the coupling [MATH] can be taken as the same for all modes.', 'physics-0304012-2-8-4': 'Finally, we switch to momentum space, but retain only the [MATH] atomic condensate modes since, due to Bose stimulation, these transitions are favored over [MATH] modes.', 'physics-0304012-2-8-5': 'The simplified Hamiltonian reads [EQUATION] where [MATH] annihilates a bosonic atom with wavevector [MATH], [MATH]) annihilates an atom (molecule) with wavevector [MATH], and [MATH].', 'physics-0304012-2-9-0': 'The system described by the Hamiltonian ([REF]) is a particular nonlinear version of a two-level atom driven by an external field.', 'physics-0304012-2-9-1': 'Of course, the two-level atom is well known to undergo Rabi flopping [CITATION], and similar behavior observed for the present system would in principle answer question of whether collective association will provide a shortcut to the formation of a Fermi-degenerate gas of molecules.', 'physics-0304012-2-9-2': 'We therefore consider an on-resonance system, for which [MATH] and the Heisenberg equations of motion are [EQUATION]', 'physics-0304012-2-9-3': 'If the system starts out with [MATH] bosonic atoms, the operator [MATH] will have a characteristic size [MATH].', 'physics-0304012-2-9-4': 'On the other hand, while the fermionic operators [MATH] and [MATH] are characteristically of order unity, the sum over momentum states will ultimately introduce the initial number of fermionic atoms.', 'physics-0304012-2-9-5': 'Hence, for equal numbers of bosons and fermions, [MATH], it is intuitively obvious that the equations of motion ([REF]) yield a system that evolves with the characteristic frequency [MATH].', 'physics-0304012-2-10-0': 'Using Fock states, said intuition is confirmed by solving the time-dependent Schrodinger equation numerically, the results of which are shown in Fig. [REF].', 'physics-0304012-2-10-1': 'For the simplest case of [MATH], the system undergoes complete conversion from a doubly-degenerate Fermi-Bose system of atoms to a Fermi-degenerate gas of molecules in a time [MATH].', 'physics-0304012-2-10-2': 'However, considering a larger number of initial atoms, [MATH], we see that quantum many-body fluctuations not only frustrate complete conversion, but also adjust the oscillation frequency and lead to collapse and revival.', 'physics-0304012-2-10-3': 'Increasing the initial particle number to [MATH] brings the amplitude of the first half-oscillation closer to the ideal case.', 'physics-0304012-2-10-4': 'This behavior is exactly analogous to the single-component bosonic case [CITATION].', 'physics-0304012-2-10-5': 'Although limited computational resources preclude the explicit investigation of large particle number, based on the bosonic analogy [CITATION] we fully expect the first half oscillation to be complete for large [MATH], i.e., [MATH] initial Fermi atoms should be converted to [MATH] Fermi molecules over a timescale [MATH].', 'physics-0304012-2-11-0': 'Nevertheless, collisions between particles will shift the energy of a state with a given number of atoms and molecules in a manner that depends nonlinearly on the numbers, making it difficult to maintain exact resonance.', 'physics-0304012-2-11-1': 'Furthermore, once a the system has Rabi flopped to molecules, the depopulated atomic condensate makes Bose stimulation, and thus neglect of the [MATH] BEC modes, questionable.', 'physics-0304012-2-11-2': 'And so we investigate the more robust possibility of rapid adiabatic passage [CITATION].', 'physics-0304012-2-11-3': 'From the Hamiltonian ([REF]), it should be clear that the system will favor all atoms for large positive detunings, while favoring all molecules for large negative detunings.', 'physics-0304012-2-11-4': 'With [MATH] established as the characteristic frequency for collective atom-molecule conversion, changes in the detuning that are of the order of [MATH], and occur over a time of order [MATH], should qualify as adiabatic.', 'physics-0304012-2-11-5': 'Hence, if the detuning is swept in a linear fashion according to [MATH], then dimensionless sweep rates [MATH] should enable rapid adiabatic passage to a degenerate gas of Fermi molecules.', 'physics-0304012-2-12-0': 'Our suspicions are again corroborated by a Fock-state-based numerical solution to the Schrodinger equation, shown in Fig. [REF].', 'physics-0304012-2-12-1': 'While many-body effects appear to rather seriously affect the efficiency of a marginally adiabatic sweep ([MATH]) compared to the [MATH] case, the difference between [MATH] and [MATH] (not shown) is in fact small.', 'physics-0304012-2-12-2': 'Overall, many-body effects are expected to be weak for near-adiabatic sweeps ([MATH]), and vanishing for sweeps that are deep-adiabatic ([MATH]), again in agreement with our BEC experience [CITATION].', 'physics-0304012-2-12-3': 'On the matter of photodissociation to noncondensate modes-and the related pair correlations[CITATION]-we note that, in the all-boson case [CITATION], such transitions can be neglected for a sweep directed as in Fig. [REF], i.e., for [MATH].', 'physics-0304012-2-12-4': 'Given the success of the analogy so far, similar results are expected for a Fermi system.', 'physics-0304012-2-13-0': 'Before closing, we estimate some explicit numbers.', 'physics-0304012-2-13-1': 'We eschew photoassociation because of the losses associated with the electronically-excited state [CITATION], and focus on the atom-molecule coupling provided by the [MATH]-[MATH]K Feshbach resonance located at [MATH]G [CITATION], which has a width [MATH]G and an associated zero-field Fermi-Bose-atom scattering length [MATH]nm.', 'physics-0304012-2-13-2': 'Accordingly, the atom-molecule coupling is [MATH], where we have estimated the difference between the Fermi-Bose atom pair and molecular magnetic moments to be equal to the Bohr magneton [MATH].', 'physics-0304012-2-13-3': 'Assuming [MATH] condensate atoms in a trap with respective radial and axial frequencies [MATH]Hz and [MATH]Hz [CITATION], the density of bosons is [MATH].', 'physics-0304012-2-13-4': 'As for the fermions, we assume a modest number, say, [MATH], which has three consequences: (i) the atomic BEC will act as a reservoir, thus absorbing any heat created by holes in the Fermi sea [CITATION]; (ii) barring an unfortunately large scattering length for Bose-atom and Fermi-molecule collisions, we can neglect the possibility of any Fermi-Bose collapse instabilities [CITATION]; (iii) the size of the Fermi cloud ([MATH]) is smaller than the BEC ([MATH]), so that overlap is not an issue.', 'physics-0304012-2-13-5': 'Moreover, for [MATH], the timescale for atom-molecule conversion is [MATH].', 'physics-0304012-2-13-6': 'This timescale is safely below the fastest timescale for trapped-atom motion [MATH], justifying our neglect of trap dynamics and the kinetic energy; physically put, this means that the Fermi energy is negligible compared to the atom-molecule coupling strength.', 'physics-0304012-2-14-0': 'Lastly we discuss the neglected role of particle-particle interactions, i.e., collisions.', 'physics-0304012-2-14-1': 'These are described in terms of a single parameter, the [MATH]-wave scattering length [MATH], where the value of [MATH] differs depending on whether one has in mind Bose-Bose atom, Fermi-Bose atom, Fermi-Bose atom-molecule, or Fermi-Fermi atom-molecule collisions ([MATH]-wave Fermi-Fermi atom and molecule collisions are of course Pauli blocked).', 'physics-0304012-2-14-2': 'Whatever the species involved, the real part of [MATH] describes elastic collisions between particles, and leads to so-called mean-field shifts.', 'physics-0304012-2-14-3': 'The imaginary part need only be considered for atom-molecule collisions terms, and arises because the molecules are vibrationally very hot, and therefore undergo vibrational relaxation induced by collisions with unconverted atoms.', 'physics-0304012-2-14-4': 'The collisional coupling strength is then [MATH], where [MATH] is the mass of the atom or the reduced mass of the atom-atom (atom-molecule) pair, and it is included in the theory as a density dependent detuning.', 'physics-0304012-2-14-5': 'The units of the collisional coupling are [MATH], so that an appropriate density factor is needed to arrive at a collisional rate.', 'physics-0304012-2-14-6': 'Estimating the real part of [MATH] from the [MATH]-[MATH]K collisions[CITATION], and the imaginary part from collisions [CITATION], a typical density ([MATH]) leads to elastic and inelastic collisions that occur on a timescale [MATH].', 'physics-0304012-2-14-7': 'For rapid adiabatic passage, the contribution from the collisional interaction is negligible compared to the detuning, except near-resonance; however, the system only spends about [MATH] in this region, which is short enough to expect collisions to be negligible.', 'physics-0304012-2-14-8': 'If, by chance, collisions do turn out to be a problem, the trap could be expanded to further reduce the rate of collisions with unconverted atoms.', 'physics-0304012-2-15-0': 'In conclusion, we highlight that the term [MATH] was previously referred to as the Bose-enhanced free-bound coupling, and the detuning sweep, for example, referred to as Bose-stimulated rapid adiabatic passage from atoms to molecules.', 'physics-0304012-2-15-1': 'However, this behavior is now clearly independent of statistics, so that Bose stimulation of free-bound association has nothing whatsoever to do with Bose statistics, but is instead a many-body cooperative effect that applies equally well to Fermi-degenerate systems.', 'physics-0304012-2-15-2': 'We expect this situation to arise whenever the system is addressed as a unit, i.e., when the atom-molecule coupling strength is larger than the Fermi energy.', 'physics-0304012-2-15-3': 'To four-wave mixing in a Fermi gas of atoms [CITATION], we therefore add a further example of fermions mimicking an effect that was previously attributed to Bose-Einstein statistics.', 'physics-0304012-2-15-4': 'Such behavior should allow for a shortcut to a degenerate gas of Fermi molecules via a collective photoassociation or Feshbach resonance.', 'physics-0304012-2-16-0': 'The authors acknowledge Juha Javanainen for helpful discussions.', 'physics-0304012-2-16-1': 'This work supported by the Academy of Finland (MM and KAS, project 50314), and the Magnus Ehrnrooth foundation (OD).', 'physics-0304012-2-17-0': 'SCH02 U. Schloder, C. Silber, T. Deuschle, and C. Zimmermann, Phys.', 'physics-0304012-2-18-0': 'TIM99a E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, Phys.', 'physics-0304012-2-19-0': 'MAC00 For a review of the role of Bose enhancement in photoassociation, see M. Mackie and J. Javanainen, 47, 2645 (2000).', 'physics-0304012-2-20-0': 'JAV02 For a brief review of cooperative fermion behavior in four-wave mixing, see J. Javanainen, Nature 412, 689 (2002).', 'physics-0304012-2-21-0': 'ALL87 L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, (Dover, New York, 1987).', 'physics-0304012-2-22-0': 'TWO_CO A two-color photoassociation scheme, whereby electronically excited molecules are transferred to a stable molecule state, could of course avert such losses, and the work herein indicates that "Bose-stimulated" Raman adiabatic passage [M. Mackie et al., 84, 3803 (2000)] will in principle create for Fermi molecules [M. Mackie et al., (physics/0305057)].'} | [['physics-0304012-1-11-0', 'physics-0304012-2-13-0'], ['physics-0304012-1-11-1', 'physics-0304012-2-13-1'], ['physics-0304012-1-11-3', 'physics-0304012-2-13-3'], ['physics-0304012-1-7-0', 'physics-0304012-2-9-0'], ['physics-0304012-1-7-1', 'physics-0304012-2-9-1'], ['physics-0304012-1-7-2', 'physics-0304012-2-9-2'], ['physics-0304012-1-7-3', 'physics-0304012-2-9-3'], ['physics-0304012-1-7-4', 'physics-0304012-2-9-4'], ['physics-0304012-1-7-5', 'physics-0304012-2-9-5'], ['physics-0304012-1-13-1', 'physics-0304012-2-15-1'], ['physics-0304012-1-0-2', 'physics-0304012-2-0-2'], ['physics-0304012-1-0-3', 'physics-0304012-2-0-3'], ['physics-0304012-1-10-1', 'physics-0304012-2-12-1'], ['physics-0304012-1-10-2', 'physics-0304012-2-12-2'], ['physics-0304012-1-10-4', 'physics-0304012-2-12-4'], ['physics-0304012-1-17-0', 'physics-0304012-2-21-0'], ['physics-0304012-1-8-1', 'physics-0304012-2-10-1'], ['physics-0304012-1-8-3', 'physics-0304012-2-10-3'], 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'physics-0304012-2-11-1'], ['physics-0304012-1-9-2', 'physics-0304012-2-11-2'], ['physics-0304012-1-9-3', 'physics-0304012-2-11-3'], ['physics-0304012-1-9-5', 'physics-0304012-2-11-5'], ['physics-0304012-1-4-0', 'physics-0304012-2-6-0'], ['physics-0304012-1-4-1', 'physics-0304012-2-6-1'], ['physics-0304012-1-4-2', 'physics-0304012-2-6-2'], ['physics-0304012-1-14-0', 'physics-0304012-2-16-0'], ['physics-0304012-1-14-1', 'physics-0304012-2-16-1'], ['physics-0304012-1-6-2', 'physics-0304012-2-8-2'], ['physics-0304012-1-6-6', 'physics-0304012-2-8-5'], ['physics-0304012-1-2-0', 'physics-0304012-2-4-0'], ['physics-0304012-1-2-3', 'physics-0304012-2-4-3']] | [['physics-0304012-1-11-2', 'physics-0304012-2-13-2'], ['physics-0304012-1-11-4', 'physics-0304012-2-13-4'], ['physics-0304012-1-11-5', 'physics-0304012-2-13-5'], ['physics-0304012-1-18-0', 'physics-0304012-2-22-0'], ['physics-0304012-1-13-0', 'physics-0304012-2-15-0'], ['physics-0304012-1-13-2', 'physics-0304012-2-15-3'], ['physics-0304012-1-13-3', 'physics-0304012-2-15-4'], ['physics-0304012-1-0-1', 'physics-0304012-2-0-1'], ['physics-0304012-1-10-0', 'physics-0304012-2-12-0'], ['physics-0304012-1-10-3', 'physics-0304012-2-12-3'], ['physics-0304012-1-15-0', 'physics-0304012-2-19-0'], ['physics-0304012-1-16-0', 'physics-0304012-2-20-0'], ['physics-0304012-1-8-0', 'physics-0304012-2-10-0'], ['physics-0304012-1-8-2', 'physics-0304012-2-10-2'], ['physics-0304012-1-5-0', 'physics-0304012-2-7-0'], ['physics-0304012-1-9-4', 'physics-0304012-2-11-4'], ['physics-0304012-1-4-3', 'physics-0304012-2-6-3'], ['physics-0304012-1-3-0', 'physics-0304012-2-5-0'], ['physics-0304012-1-3-1', 'physics-0304012-2-5-1'], ['physics-0304012-1-6-4', 'physics-0304012-2-8-3'], ['physics-0304012-1-6-5', 'physics-0304012-2-8-4'], ['physics-0304012-1-12-3', 'physics-0304012-2-14-8'], ['physics-0304012-1-1-2', 'physics-0304012-2-3-0'], ['physics-0304012-1-1-3', 'physics-0304012-2-3-1'], ['physics-0304012-1-2-1', 'physics-0304012-2-4-1'], ['physics-0304012-1-2-2', 'physics-0304012-2-4-2']] | [] | [['physics-0304012-1-11-6', 'physics-0304012-2-13-6'], ['physics-0304012-1-0-0', 'physics-0304012-2-0-0'], ['physics-0304012-1-3-2', 'physics-0304012-2-5-2'], ['physics-0304012-1-6-0', 'physics-0304012-2-8-0'], ['physics-0304012-1-12-0', 'physics-0304012-2-14-3']] | [] | [] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/physics/0304012 | null | null | null | null | null |
0709.1733 | {'0709.1733-1-0-0': 'We investigate the low-lying excited states of the spin [MATH] ferromagnetic XXZ chain with Ising anisotropy [MATH] and kink boundary conditions.', '0709.1733-1-0-1': 'Since the third component of the total magnetization, [MATH], is conserved, it is meaningful to study the spectrum for each fixed value of [MATH].', '0709.1733-1-0-2': 'We prove that for [MATH] the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain.', '0709.1733-1-0-3': 'In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.', '0709.1733-1-1-0': 'Keywords: Anisotropic Heisenberg Ferromagnet, XXZ Model, Perturbation Theory PACS numbers: 05.30.', '0709.1733-1-1-1': 'Ch, 05.50.', '0709.1733-1-1-2': '+q MCS numbers: 81Q15, 82B10, 82B24, 82D40 width2truein Copyright 2007 by the authors.', '0709.1733-1-1-3': 'Reproduction of this article in its entirety, by any means, is permitted for non-commercial purposes.', '0709.1733-1-2-0': '# Introduction', '0709.1733-1-3-0': 'In order to make the promise of quantum computation a reality we need to find or build physical systems that faithfully implement the quantum gates used in the algorithms of quantum computation [CITATION].', '0709.1733-1-3-1': 'The basic requirement is that the experimenter has access to two states of a quantum system that can be effectively decoupled from environmental noise for a sufficiently long time, and that transitions between these two states can be controlled to simulate a number of elementary quantum gates (unitary transformations).', '0709.1733-1-3-2': 'Systems that have been investigated intensively are single photon systems, cavity QED, nuclear spins (using NMR in suitable molecules), atomic levels in ion traps, and Josephson rings [CITATION].', '0709.1733-1-4-0': 'In this paper we are investigating the possibility of using excited states at the interfaces, called kinks, in one-dimensional magnetic systems.', '0709.1733-1-4-1': 'It turns out that if the spins are of magnitude [MATH] or more and their interactions have a suitable anisotropy, such as in the ferromagnetic XXZ Heisenberg model, isolated excited states are possible.', '0709.1733-1-4-2': 'Our main result is a mathematical demonstration that such states indeed exist for sufficiently large anisotropy.', '0709.1733-1-4-3': 'We believe that if one could build one-dimensional spin [MATH] systems with [MATH], which interact through an anisotropic interaction such as in the XXZ model, this would be a good starting point to encode qubits and unitary gates.', '0709.1733-1-4-4': 'The natural candidates for control parameters in such systems would be the components of a localized magnetic field.', '0709.1733-1-4-5': 'From the experimental point of view this is certainly a challenging problem.', '0709.1733-1-4-6': 'This work is a first step toward a mathematical model to study the optimal control of these systems such as has already been carried out for nuclear magnetic resonance (NMR) [CITATION] and superconducting Joshepson qubits [CITATION].', '0709.1733-1-5-0': 'Concretely, we study the one-dimensional spin [MATH] ferromagnetic XXZ model with kink boundary terms.', '0709.1733-1-5-1': 'The Hamiltonian is [EQUATION] where [MATH],[MATH] and [MATH] are the spin [MATH] matrices acting on the site [MATH].', '0709.1733-1-5-2': 'Apart from the magnitude of the spins, [MATH], the main parameter of the model is the anisotropy [MATH] and we will refer to the limit [MATH] as the Ising limit.', '0709.1733-1-5-3': 'In the case of [MATH] these boundary conditions were first introduced in [CITATION].', '0709.1733-1-6-0': 'The XXZ kink Hamiltonian commutes with the operator [MATH].', '0709.1733-1-6-1': 'We define [MATH] to be the eigenspace of [MATH] with eigenvalue [MATH].', '0709.1733-1-6-2': 'These subspaces are called "sectors", and they are invariant subspaces for [MATH].', '0709.1733-1-7-0': 'It was shown in [CITATION] that for each sector there is a unique ground state of [MATH] with eigenvalue 0.', '0709.1733-1-7-1': 'Moreover, this ground state, [MATH], is given by the following expression: [EQUATION] where the sum is over all configurations for which [MATH] and the relationship between [MATH] and [MATH] is given by [MATH].', '0709.1733-1-7-2': 'A straightforward calculation shows a sharp transition in the magnetization from fully polarized down at the left to fully polarized up at the right.', '0709.1733-1-7-3': 'For this reason they are called kink ground states.', '0709.1733-1-7-4': 'In [CITATION] Koma and Nachtergaele proved that the kink ground states (as well as their spin-flipped or reflected versions the antikinks) comprise the entire set of ground states for the infinite-volume model, aside from the 2 other ground states: the translation invariant maximally magnetized and minimally magnetized all +J and all -J groundstates.', '0709.1733-1-7-5': 'Since the infinite-volume Hamiltonian incorporates all possible limits for all possible boundary conditions, this is a strong a posteriori justification for choosing the kink boundary conditions.', '0709.1733-1-7-6': 'It is worth noting that it has been proved for the antiferromagnetic model that no such ground states exist [CITATION].', '0709.1733-1-8-0': 'In [CITATION], Koma, Nachtergaele, and Starr showed that there is a spectral gap above each of the ground states in this model for all values of [MATH].', '0709.1733-1-8-1': 'Based on numerical evidence, they also made a conjecture that for [MATH] the first excited state of the XXZ model is an isolated eigenvalue, and that the magnitude of the spectral gap is asymptotically given by [MATH], where [MATH] is an eigenvalue of a particular one-particle problem.', '0709.1733-1-8-2': 'Caputo and Martinelli [CITATION] showed that the gap is indeed of order [MATH].', '0709.1733-1-9-0': 'Our main result is a proof that for all sufficiently large [MATH] the first few excitations of the XXZ model are isolated eigenvalues.', '0709.1733-1-9-1': 'This is true for all [MATH] and for spin 1 with [MATH] even, which is illustrated in Figure [REF] for the spin 3/2 and spin 2 chains.', '0709.1733-1-9-2': 'See Section [REF] for the precise statements.', '0709.1733-1-9-3': 'It turns out that in the Ising limit the eigenvalues less than [MATH] are all of multiplicity at most [MATH] in each sector.', '0709.1733-1-9-4': 'Moreover, the first excited states are simple except in the case when [MATH] is an integer and [MATH] mod [MATH].', '0709.1733-1-9-5': 'In this case, they are doubly degenerate.', '0709.1733-1-9-6': 'This is discussed in Section [REF].', '0709.1733-1-9-7': 'In Section [REF], we write the XXZ Hamiltonian as an explicit perturbation of the Ising limit.', '0709.1733-1-9-8': 'Theorem [REF] verifies that the perturbation is relatively bounded with respect to the Ising limit, and we finish this section by demonstrating that our estimates suffice to guarantee analytic continuation of the limiting eigenvalues.', '0709.1733-1-9-9': 'It is clear that the same method of proof can be applied to other Hamiltonians.', '0709.1733-1-10-0': '# Set-up', '0709.1733-1-11-0': 'We study the spin [MATH] ferromagnetic XXZ model on the one-dimensional lattice [MATH].', '0709.1733-1-11-1': 'The local Hilbert space for a single site [MATH] is [MATH] with [MATH].', '0709.1733-1-11-2': 'We consider the Hilbert space for a finite chain on the sites [MATH].', '0709.1733-1-11-3': 'This is [MATH].', '0709.1733-1-11-4': 'The Hamiltonian of the spin-[MATH] XXZ model is [EQUATION] where [MATH],[MATH] and [MATH] are the spin-[MATH] matrices acting on the site [MATH], tensored with the identity operator acting on the other sites.', '0709.1733-1-11-5': 'The main parameter of the model is the anisotropy [MATH] and we get the Ising limit as [MATH].', '0709.1733-1-11-6': 'It is mathematically more convenient to work with the parameter [MATH], which we then assume is in the interval [MATH].', '0709.1733-1-11-7': 'As we said, [MATH] is the Ising limit, and [MATH] is the isotropic XXX Heisenberg model.', '0709.1733-1-11-8': 'It was shown [CITATION] that additional ground states emerge when we add particular boundary terms.', '0709.1733-1-11-9': 'Examples of this are the kink and antikink Hamiltonians [EQUATION]', '0709.1733-1-11-10': 'It is easy to see that the kink and anti-kink Hamiltonians are unitarily equivalent.', '0709.1733-1-11-11': 'We will be mainly interested in the kink Hamiltonian with [MATH].', '0709.1733-1-11-12': 'Note that, by a telescoping sum, we can absorb the boundary fields into the local interactions:', '0709.1733-1-12-0': 'H_L^k(^-1) = _L-1 h^k_,+1(^-1) ,', '0709.1733-1-13-0': 'h^k_,+1(^-1) = J^2-S_^3 S_+1^3 - ^-1(S_^1 S_+1^1 + S_^2 S_+1^2) + J 1-^-2 (S_^3 - S_+1^3)', '0709.1733-1-14-0': 'The Ising kink Hamiltonian is the result of taking [MATH], equivalently setting [MATH], namely it is [EQUATION]', '0709.1733-1-14-1': 'Each of the Hamiltonians introduced above commutes with the total magnetization [EQUATION].', '0709.1733-1-14-2': 'As indicated in the introduction, for each [MATH], the corresponding sector is defined to be the eigenspace of [MATH] with eigenvalue [MATH]; clearly, these are invariant subspaces for all the Hamiltonians introduced above.', '0709.1733-1-15-0': 'The Ising basis is a natural orthonormal basis for [MATH].', '0709.1733-1-15-1': 'At each site we have an orthonormal basis of the Hilbert space [MATH] given by the eigenvectors of [MATH] and labeled according to their eigenvalues.', '0709.1733-1-15-2': 'We will denote this by [MATH] for [MATH], and [MATH].', '0709.1733-1-15-3': 'Here, and throughout the remainder of the paper, we will use the notation [MATH] for the set [MATH] as we have done with [MATH].', '0709.1733-1-15-4': 'Finally, there is an orthonormal basis of the entire Hilbert space consisting of simple tensor product vectors: [MATH].', '0709.1733-1-16-0': 'Also recall that the raising and lowering operators are defined such that [EQUATION]', '0709.1733-1-16-1': 'A short calculation shows that [MATH] and [MATH] are given by [MATH] and [MATH], and [EQUATION]', '0709.1733-1-17-0': '# Main theorem', '0709.1733-1-18-0': 'Many of the results in this paper concern the kink Hamiltonian given by ([REF]).', '0709.1733-1-18-1': 'We will study it as a perturbation of the Ising Hamiltonian ([REF]) in the regime [MATH].', '0709.1733-1-18-2': 'We denote an Ising configuration as [MATH], where [MATH] for each [MATH], and the corresponding basis vector as [EQUATION].', '0709.1733-1-18-3': 'Observe that the Ising kink Hamiltonian is diagonal with respect to this basis, [EQUATION]', '0709.1733-1-18-4': 'Since each of the [MATH] are non-negative, it is easy to see that the ground states of [MATH] are all of the form [EQUATION] with some [MATH] and [MATH].', '0709.1733-1-18-5': 'Note that the total magnetization corresponding to [MATH] is [MATH].', '0709.1733-1-18-6': 'As we will verify in Proposition [REF], it is easy to check that these ground states are unique per sector.', '0709.1733-1-18-7': 'We do point out that there is a slight ambiguity in the above labeling scheme, however, since [MATH] and [MATH] coincide.', '0709.1733-1-18-8': 'Let us consider the following elementary result.', '0709.1733-1-19-0': 'For [MATH], [MATH] and [MATH], the eigenspace of [MATH] corresponding to energy [MATH] has dimension at least equal to [MATH].', '0709.1733-1-20-0': 'Because of this theorem, we see that, in the [MATH] limit, the energy [MATH] is essential spectrum; an eigenvalue with infinite multiplicity.', '0709.1733-1-20-1': 'As our interest is in perturbation theory, it is natural for us to restrict our attention to energies strictly between [MATH] and [MATH].', '0709.1733-1-20-2': 'We will call the corresponding eigenvectors "low energy excitations".', '0709.1733-1-21-0': 'We will now describe all the low energy excitations for the Ising kink Hamiltonian.', '0709.1733-1-21-1': 'In order to do so, it is convenient to introduce a family of eigenvectors which contains all possible low energy excitations.', '0709.1733-1-22-0': 'Let [MATH] be any site away from the boundary, i.e. take [MATH], and choose [MATH].', '0709.1733-1-22-1': 'These choices specify a sector [MATH] and a groundstate [MATH].', '0709.1733-1-22-2': 'If [MATH], we define the following vectors: For [MATH] set [EQUATION]', '0709.1733-1-22-3': 'If [MATH], we define the following vectors: For [MATH] set [EQUATION]', '0709.1733-1-22-4': 'We call these two sequences of Ising basis vectors the "localized kink excitations".', '0709.1733-1-22-5': 'Clearly, these vectors [MATH] have the same total magnetization [MATH] and moreover, [EQUATION] where [MATH] has been calculated using ([REF]); note that in each case there is only one non-zero term.', '0709.1733-1-22-6': 'In Figure [REF] we have shown two ground states, in the left and right graphs, as well as two low excitations.', '0709.1733-1-22-7': 'These are the schematic diagrams for [MATH], in solid in both pictures, and [MATH] and [MATH], in a dashed line, in the left and right, respectively', '0709.1733-1-23-0': 'We now define two sets of labels: [EQUATION]', '0709.1733-1-23-1': 'Depending on [MATH] and [MATH], neither, one, or both of these sets may be empty.', '0709.1733-1-23-2': 'We have the following theorem.', '0709.1733-1-24-0': '(1) The low energy excitations of [MATH] form a subset of the localized kink excitations introduced in ([REF]) and ([REF]) above.', '0709.1733-1-25-0': '(2) For [MATH] of the form [MATH] with some [MATH] and [MATH], the set of low energy excitations equals [EQUATION].', '0709.1733-1-25-1': 'This is a nonempty set except in the following two cases: [MATH], or [MATH], and [MATH].', '0709.1733-1-26-0': '(3) The low energy excitations of [MATH] are at most two-fold degenerate.', '0709.1733-1-26-1': 'The first excitation is simple, except for the case that [MATH] is an integer [MATH], and [MATH] mod [MATH].', '0709.1733-1-27-0': 'The two-fold degeneracy of the first excited state, guaranteed by part (3) of the theorem, occurs due to the spin flip and reflection symmetry.', '0709.1733-1-27-1': 'All other degeneracies with energy [MATH] occur as follows.', '0709.1733-1-27-2': 'Suppose [MATH] for integers [MATH].', '0709.1733-1-27-3': 'In this case, let [MATH].', '0709.1733-1-27-4': 'Then [MATH] is degenerate with [MATH], both with energy [MATH].', '0709.1733-1-27-5': 'Similarly, [MATH] is degenerate with [MATH] for the same energy.', '0709.1733-1-28-0': 'Next we consider the perturbed Hamiltonian [MATH].', '0709.1733-1-28-1': 'As is discussed in the appendix, each low-lying eigenvalue [MATH], associated to some [MATH], is isolated from the rest of the spectrum by an isolation distance [MATH], independent of [MATH], which is defined by [EQUATION].', '0709.1733-1-29-0': 'Let [MATH], and fix [MATH].', '0709.1733-1-29-1': 'For any [MATH], consider the interval [MATH] about the low lying energy [MATH].', '0709.1733-1-29-2': 'The spectral projection of [MATH] onto [MATH] is analytic for large enough values of [MATH].', '0709.1733-1-29-3': 'In particular, the dimension of the spectral projection onto [MATH] is constant for this range of [MATH].', '0709.1733-1-30-0': 'Our estimates yield a lower bound on [MATH] as is provided by ([REF]).', '0709.1733-1-30-1': 'A slightly worse bound demonstrates that taking [MATH] suffices, but we do not expect either estimate to be sharp.', '0709.1733-1-31-0': 'The above theorem confirms the structure of the spectrum shown in Figure [REF].', '0709.1733-1-31-1': 'Moreover, since our numerical calculations indicate that some of the eigenvalues enter the continous spectrum, we do not expect this type of perturbation theory to work for the entire range of [MATH].', '0709.1733-1-32-0': '# Proof of Theorems [REF] and [REF] (Excitations of the Ising Model)', '0709.1733-1-33-0': 'In this section, we will focus on the Ising kink Hamiltonian [MATH], as introduced in ([REF]), for a fixed [MATH].', '0709.1733-1-33-1': 'The ground states can be characterized as follows:', '0709.1733-1-34-0': 'The ground states of the Ising kink Hamiltonian are all of the form [EQUATION] for [MATH] and [MATH].', '0709.1733-1-34-1': 'Moreover, there is exactly one ground state in each sector; the total magnetization eigenvalue for [MATH] is [MATH].', '0709.1733-1-35-0': 'This proposition was already used, implicitly, in setting up Theorem [REF].', '0709.1733-1-35-1': 'Here we prove it.', '0709.1733-1-36-0': 'Given any Ising configuration [MATH], equation ([REF]) demonstrates that the energy associated to [MATH], [MATH], is the sum of [MATH] non-negative terms.', '0709.1733-1-36-1': 'Therefore, the only way to have [MATH] is if all of the summands are [MATH].', '0709.1733-1-36-2': 'This is the case only if either [MATH] or [MATH] for all [MATH].', '0709.1733-1-36-3': 'Clearly, this is satisfied for the Ising configurations with [MATH] for all [MATH], [MATH] for all [MATH], and [MATH] equal to any number in [MATH].', '0709.1733-1-36-4': 'It is equally easy to see that these are all of the ground state configurations: if [MATH] is a groundstate configuration and for some [MATH] we have [MATH], then [MATH], which, by induction, means that [MATH] for all [MATH].', '0709.1733-1-36-5': 'Similar reasoning yields that [MATH] for all [MATH].', '0709.1733-1-37-0': 'To show that the ground states are unique in each sector, consider the equation [MATH], subject to the constraints: [MATH], [MATH], and [MATH].', '0709.1733-1-37-1': 'If [MATH] mod [MATH], then there is a unique pair [MATH] which satisfies this equation.', '0709.1733-1-37-2': 'If [MATH] mod [MATH], then there are two possible solutions [MATH] and [MATH] for some [MATH].', '0709.1733-1-37-3': 'But then, it is trivial to see that [MATH] and [MATH] coincide.', '0709.1733-1-38-0': 'Thus in any sector [MATH], there is a unique ground state eigenvector [MATH] for some choice of [MATH] with [MATH].', '0709.1733-1-38-1': 'We next observe that there are many eigenvectors with eigenvalue [MATH]; recall this was the statement of Theorem [REF].', '0709.1733-1-39-0': 'Theorem [REF] First consider the case that [MATH] is not divisible by [MATH].', '0709.1733-1-39-1': 'Then the unique groundstate eigenvector is [MATH] for some [MATH] and [MATH].', '0709.1733-1-39-2': 'If [MATH], then for each [MATH] consider the Ising configuration [MATH] with components [EQUATION]', '0709.1733-1-39-3': 'The total magnetization for the vector [MATH] is still [MATH].', '0709.1733-1-39-4': 'Using the formula ([REF]) again, it is easy to check that [MATH], [MATH] for all [MATH], and therefore, [MATH].', '0709.1733-1-39-5': 'This constitutes [MATH] possible values of [MATH]; producing at least this many distinct eigenvectors with energy [MATH].', '0709.1733-1-39-6': 'Similarly, if [MATH] then there are [MATH] Ising configurations of the form [MATH] with components [EQUATION] corresponding to some [MATH].', '0709.1733-1-39-7': 'In total, this gives [MATH] orthonormal eigenvectors corresponding to eigenvalue [MATH], yielding a lower bound on the dimension of the eigenspace.', '0709.1733-1-39-8': 'If [MATH] or [MATH], then the dimension is increased by at least 1.', '0709.1733-1-39-9': 'In the special case where [MATH] is divisible by [MATH], the eigenvector can be written in two ways as [MATH] or [MATH].', '0709.1733-1-39-10': 'When constructing excitations for [MATH] to the left of the kink, use the first formula above relative to [MATH].', '0709.1733-1-39-11': 'When constructing excitations for [MATH] to the right of the kink, use the second formula above relative to [MATH].', '0709.1733-1-39-12': 'Once again, this results in [MATH] orthonormal eigenvectors corresponding to eigenvalue [MATH].', '0709.1733-1-40-0': 'We now claim that any Ising configuration which is neither a ground state nor a localized kink excitation corresponds to an energy that is at least [MATH].', '0709.1733-1-40-1': 'This is the content of the following lemma.', '0709.1733-1-41-0': 'Consider an Ising configuration [MATH].', '0709.1733-1-42-0': '(1) If there is any [MATH] such that [MATH], then [MATH].', '0709.1733-1-43-0': '(2) If there is any [MATH] such that [MATH] and [MATH], then [MATH].', '0709.1733-1-44-0': '(1) It is clear from ([REF]) that we need only prove [EQUATION]', '0709.1733-1-44-1': 'Since [MATH], we have that [MATH].', '0709.1733-1-44-2': 'We need only verify that [MATH] to establish the claim.', '0709.1733-1-44-3': 'The product of two integers [MATH] is at most equal to [MATH].', '0709.1733-1-44-4': 'But this is only attained by [MATH].', '0709.1733-1-44-5': 'Since [MATH], we can neither have [MATH] nor [MATH].', '0709.1733-1-44-6': 'The next largest possible value of [MATH] is [MATH], and we have verified the claim.', '0709.1733-1-45-0': '(2) Again, because all terms are nonnegative, it suffices to show that [EQUATION].', '0709.1733-1-45-1': 'Using the formula for [MATH] and simplifying gives [EQUATION].', '0709.1733-1-45-2': 'Since [MATH], we have [EQUATION]', '0709.1733-1-45-3': 'Therefore, [EQUATION]', '0709.1733-1-45-4': 'Since [MATH] and [MATH], in either case we have proven the claim.', '0709.1733-1-46-0': 'Now we can finish the proof of the first main result, Theorem [REF].', '0709.1733-1-47-0': 'Theorem [REF] Part (1) of the theorem is a direct consequence of Lemma [REF] because the only Ising configurations that do not satisfy condition (1) or (2) of that lemma are the ground state configurations and the localized kink excitations.', '0709.1733-1-48-0': 'To prove part (2), let [MATH] for some [MATH] and [MATH].', '0709.1733-1-48-1': 'We first consider the case that [MATH].', '0709.1733-1-49-0': 'If [MATH] is sufficiently large and [MATH], then [MATH] as [EQUATION].', '0709.1733-1-49-1': 'So, in this case, both [MATH] and [MATH] are nonempty.', '0709.1733-1-50-0': 'Similarly, if [MATH], then either [EQUATION] or [EQUATION].', '0709.1733-1-50-1': 'Hence, [MATH] if [MATH] and [MATH].', '0709.1733-1-51-0': 'Lastly, if [MATH], then either [EQUATION] or [EQUATION].', '0709.1733-1-51-1': 'Hence, [MATH] if [MATH] and [MATH].', '0709.1733-1-52-0': 'We have proved (2) in the case that [MATH].', '0709.1733-1-52-1': 'Actually, the last observation above also verifies (2) in the case that [MATH] and [MATH].', '0709.1733-1-53-0': 'Finally, if [MATH] and [MATH], then [MATH] and if [MATH], then [MATH].', '0709.1733-1-53-1': 'In both these cases, the set of kink excitations is empty.', '0709.1733-1-54-0': 'We now prove the first part of (3).', '0709.1733-1-54-1': 'First, observe that for any [MATH], [MATH] and therefore with [MATH] fixed, both [MATH] are increasing functions of [MATH] for [MATH].', '0709.1733-1-54-2': 'Therefore the only degeneracies that can occur for a particular energy [MATH] is if [MATH] and [MATH] for some integers [MATH] and [MATH].', '0709.1733-1-54-3': 'This is obviously at most a two-fold degeneracy.', '0709.1733-1-55-0': 'In order to prove the second part of (3), we will simply prove Remark [REF].', '0709.1733-1-55-1': 'Without loss of generality, suppose [MATH].', '0709.1733-1-55-2': 'Then [MATH].', '0709.1733-1-55-3': 'So the only possibility for [MATH] to be less than [MATH] is if [MATH], which gives the energy [MATH] and an auxiliary condition, namely [MATH].', '0709.1733-1-55-4': 'A degeneracy happens only if there is a [MATH] such that [MATH].', '0709.1733-1-55-5': 'This means [EQUATION].', '0709.1733-1-55-6': 'Setting [MATH] and [MATH] (which satisfies [MATH] because [MATH]) we have exactly the result claimed.', '0709.1733-1-55-7': 'Note that the second part of (3) refers to the first excitation.', '0709.1733-1-55-8': 'For [MATH], the lowest excitation is [MATH].', '0709.1733-1-55-9': 'This means [MATH], which implies [MATH] and therefore [MATH] is even with [MATH] (because [MATH]).', '0709.1733-1-56-0': '# Proof of the main theorem', '0709.1733-1-57-0': 'The goal of this section is to prove Theorem [REF].', '0709.1733-1-57-1': 'We will do so by analyzing the kink Hamiltonian [MATH] as a perturbation of the Ising limit [MATH].', '0709.1733-1-57-2': 'Within the first subsection below, specifically in Theorem [REF], we prove that the operators which arise in our expansion of [MATH] are relatively bounded with respect to [MATH].', '0709.1733-1-57-3': 'In the next subsection, we discuss how the explicit bounds on [MATH], those claimed in Theorem [REF], follow from relative boundedness and basic perturbation theory.', '0709.1733-1-58-0': '## Relative Boundedness', '0709.1733-1-59-0': 'In this subsection, we will anaylze the kink Hamiltonian introduced in ([REF]).', '0709.1733-1-59-1': 'Recall that this Hamiltonian is written as', '0709.1733-1-60-0': 'H_L^k(^-1) = _L-1 h^k_,+1(^-1) ,', '0709.1733-1-61-0': 'h^k_,+1(^-1) = J^2-S_^3 S_+1^3 - ^-1(S_^1 S_+1^1 + S_^2 S_+1^2) + J 1-^-2 (S_^3 - S_+1^3).', '0709.1733-1-62-0': 'By adding and subtracting terms of the form [MATH] to the local Hamiltonians, we find that [EQUATION] where [EQUATION] and [EQUATION]', '0709.1733-1-62-1': 'In Theorem [REF] below, we will show that both [MATH] and [MATH] are relatively bounded perturbations of [MATH].', '0709.1733-1-62-2': 'To prove such estimates, we will use the following lemma several times.', '0709.1733-1-63-0': 'Let [MATH] and [MATH] be self-adjoint [MATH] matrices.', '0709.1733-1-63-1': 'If [MATH] and [MATH], then there exists a constant [MATH] for which, [EQUATION]', '0709.1733-1-63-2': 'One may take [MATH] where [MATH] denotes the smallest positive eigenvalue of [MATH].', '0709.1733-1-64-0': 'Any vector [MATH] can be written as [MATH] where [MATH] and [MATH].', '0709.1733-1-64-1': 'Clearly then, [MATH] and therefore [EQUATION] as claimed.', '0709.1733-1-65-0': 'For our proof of Theorem [REF], we find it useful to introduce the Ising model without boundary conditions as an auxillary Hamiltonian, i.e., [EQUATION].', '0709.1733-1-65-1': 'It is easy to prove the next lemma.', '0709.1733-1-66-0': 'The Ising model without boundary terms is relatively bounded with respect to the Ising kink Hamiltonian.', '0709.1733-1-66-1': 'In particular, for any vector [MATH], [EQUATION].', '0709.1733-1-67-0': 'Consider the terms of the Ising kink Hamiltonian: [EQUATION]', '0709.1733-1-67-1': 'Summing on [MATH] then, we find that [EQUATION] and therefore, the bound [EQUATION] is clear for any vector [MATH].', '0709.1733-1-68-0': 'We now state the relative boundedness result.', '0709.1733-1-69-0': 'The linear term in the perturbation expansion of [MATH], see ([REF]), satisfies [EQUATION] for any vector [MATH].', '0709.1733-1-69-1': 'Moreover, we also have that [EQUATION]', '0709.1733-1-69-2': 'Using Lemma [REF], it is clear we need only prove that [EQUATION] to establish ([REF]).', '0709.1733-1-69-3': 'To this end, Lemma [REF] provides an immediate bound on the individual terms of these Hamiltonians.', '0709.1733-1-69-4': 'In fact, observe that for any fixed [MATH], both [MATH] and [MATH] are self-adjoint with [MATH] and [EQUATION]', '0709.1733-1-69-5': 'It is also easy to see that, for every [MATH], the first positive eigenvalue of [MATH] is [MATH], and we have that [EQUATION]', '0709.1733-1-69-6': 'An application of Lemma [REF] yields the operator inequality [EQUATION] valid for any [MATH].', '0709.1733-1-70-0': 'The norm bound we seek to prove will follow from considering products of these local Hamiltonians.', '0709.1733-1-70-1': 'For any vector [MATH], one has that [EQUATION] and [EQUATION]', '0709.1733-1-70-2': 'The arguments we provided above apply equally well to the diagonal terms of ([REF]) and ([REF]) in the sense that [EQUATION] is also valid for any [MATH].', '0709.1733-1-70-3': 'We find a similar bound by considering the terms on the right hand side of ([REF]) and ([REF]) for which [MATH].', '0709.1733-1-70-4': 'In this case, each of the operators [MATH] and [MATH] commute with both of the operators [MATH] and [MATH].', '0709.1733-1-70-5': 'Moreover, we conclude from ([REF]) that the operators [MATH] are non-negative for every [MATH].', '0709.1733-1-70-6': 'Since all the relevant quantities commute, it is clear that [EQUATION]', '0709.1733-1-70-7': 'Our observations above imply the following bound [EQUATION]', '0709.1733-1-70-8': 'In fact, the terms on the right hand side of ([REF]) for which either [MATH] or [MATH] are non-positive by ([REF]), respectively, ([REF]).', '0709.1733-1-70-9': 'In the case that [MATH], the operators [MATH] are non-negative (since they commute) and hence we may drop these terms; those terms that remain we group as the self-adjoint operators appearing on the right hand side of ([REF]) above.', '0709.1733-1-71-0': 'Our estimate is completed by applying Lemma [REF] one more time.', '0709.1733-1-71-1': 'Note that for any [MATH] the operator [MATH] is self-adjoint, non-negative, and [EQUATION] where the self-adjoint operator [MATH] appearing above is given by [EQUATION]', '0709.1733-1-71-2': 'For each [MATH], the first positive eigenvalue of [MATH] is [MATH], and it is also easy to see that [MATH].', '0709.1733-1-71-3': 'Thus, term by term Lemma [REF] implies that [EQUATION] from which we conclude that [EQUATION] as claimed in ([REF]).', '0709.1733-1-71-4': 'We have proved ([REF]).', '0709.1733-1-72-0': 'Equation ([REF]) follows directly from the easy observation that [MATH] is equal to [MATH].', '0709.1733-1-73-0': '## Perturbation theory', '0709.1733-1-74-0': 'In Section [REF], we verified that, in any given sector, the spectrum of the Ising kink Hamiltonian, [MATH], when restricted to the interval [MATH] consists of only isolated eigenvalues whose multiplicity is at most two.', '0709.1733-1-74-1': 'In fact, for the sector [MATH] these eigenvalues are determined by [EQUATION] for those values of [MATH] with [MATH].', '0709.1733-1-74-2': 'It is clear from ([REF]) that each of these eigenvalues have an isolation distance [MATH] from the rest of the spectrum and that this distance is independent of the length scale [MATH].', '0709.1733-1-75-0': 'For our proof of the relative boundedness result in Theorem [REF], we expanded the Hamiltonian as [EQUATION]', '0709.1733-1-75-1': 'Using the first resolvent formula, it is easy to see that [EQUATION] where we have denoted the resolvent by [MATH], and it is assumed that [MATH] has been chosen small enough so that [EQUATION]', '0709.1733-1-75-2': 'It is clear from sections II.1.3-4 of [CITATION] and chapter I of [CITATION] that the spectral projections corresponding to [MATH] can be written as a power series in [MATH], the coefficients of which being integrals of the resolvent over a fixed contour [MATH].', '0709.1733-1-75-3': 'Proving an estimate of the form ([REF]) for [MATH] large enough, uniformly with respect to [MATH], is sufficient to guarantee analyticity of the spectral projections.', '0709.1733-1-75-4': 'We verify such a uniform estimate below.', '0709.1733-1-76-0': 'Let [MATH] be an eigenvalue of [MATH] with isolation distance [MATH] as specified above.', '0709.1733-1-76-1': 'Denote by [MATH] the circle in the complex plane centered at [MATH] with radius [MATH].', '0709.1733-1-76-2': 'We claim that if [EQUATION] then ([REF]) is satisfied uniformly for [MATH].', '0709.1733-1-77-0': 'We proved in Theorem [REF] that for any vector [MATH], [EQUATION]', '0709.1733-1-77-1': 'Applying this bound to vectors [MATH] of the form [MATH] yields a norm estimate on [MATH], i.e., [EQUATION]', '0709.1733-1-77-2': 'Moreover, since [EQUATION] we have proved that [EQUATION]', '0709.1733-1-77-3': 'Similar arguments, again using Theorem [REF], imply that [EQUATION]', '0709.1733-1-77-4': 'For [MATH], the circular contour described above, we have that [EQUATION] and [EQUATION]', '0709.1733-1-77-5': 'We derive a bound of the form ([REF]), uniform for [MATH], by ensuring [MATH] large enough so that [EQUATION] where [EQUATION]', '0709.1733-1-77-6': 'Explicitly, one finds that the inequality ([REF]) is satisfied for all [EQUATION]', '0709.1733-1-77-7': 'Equation ([REF]) is a simple sufficient condition for [MATH] to satisfy this inequality.', '0709.1733-1-77-8': 'This is easy to verify if one first replaces [MATH] by [MATH] in ([REF]).'} | {'0709.1733-2-0-0': 'We investigate the low-lying excited states of the spin [MATH] ferromagnetic XXZ chain with Ising anisotropy [MATH] and kink boundary conditions.', '0709.1733-2-0-1': 'Since the third component of the total magnetization, [MATH], is conserved, it is meaningful to study the spectrum for each fixed value of [MATH].', '0709.1733-2-0-2': 'We prove that for [MATH] the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain.', '0709.1733-2-0-3': 'In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.', '0709.1733-2-1-0': 'Keywords: Anisotropic Heisenberg Ferromagnet, XXZ Model, Perturbation Theory PACS numbers: 05.30.', '0709.1733-2-1-1': 'Ch, 05.50.', '0709.1733-2-1-2': '+q MCS numbers: 81Q15, 82B10, 82B24, 82D40 width2truein Copyright 2007 by the authors.', '0709.1733-2-1-3': 'Reproduction of this article in its entirety, by any means, is permitted for non-commercial purposes.', '0709.1733-2-2-0': '# Introduction', '0709.1733-2-3-0': 'In this paper, we are investigating the existence of isolated excited states in certain one-dimensional, quantum spin models of magnetic systems.', '0709.1733-2-3-1': 'It turns out that if the spins are of magnitude [MATH] or more and their interactions have a suitable anisotropy, such as in the ferromagnetic XXZ Heisenberg model, isolated excited states are possible.', '0709.1733-2-3-2': 'For the spin [MATH] chain the ground states are separated by a gap to the rest of the spectrum, and there are no isolated eigenvalues below the continuum.', '0709.1733-2-4-0': 'Our main result is a mathematical demonstration that such states indeed exist for sufficiently large anisotropy.', '0709.1733-2-4-1': 'Concretely, we study the one-dimensional spin [MATH] ferromagnetic XXZ model with the following boundary terms.', '0709.1733-2-4-2': 'The Hamiltonian is [EQUATION] where [MATH],[MATH] and [MATH] are the spin [MATH] matrices acting on the site [MATH].', '0709.1733-2-4-3': 'Apart from the magnitude of the spins, [MATH], the main parameter of the model is the anisotropy [MATH], and we will refer to the limit [MATH] as the Ising limit.', '0709.1733-2-4-4': 'In the case of [MATH] these boundary conditions were first introduced in [CITATION].', '0709.1733-2-4-5': 'They lead to ground states with a domain wall between down spins on the left portion of the chain and up spins on the right.', '0709.1733-2-4-6': 'The domain wall is exponentially localized [CITATION].', '0709.1733-2-4-7': 'The third component of the magnetization, [MATH], is conserved, and there is exactly one ground state for each value of [MATH].', '0709.1733-2-4-8': 'Different values of [MATH] correspond to different positions of the domain walls, which in one dimension are sometimes referred to as kinks.', '0709.1733-2-4-9': 'In [CITATION] and [CITATION] the ground states for this type boundary conditions were further analyzed and generalized to higher spin.', '0709.1733-2-4-10': 'A careful analysis of the Ising limit, see Section [REF], reveals that for [MATH] one or more low-lying excitations, each with a finite degeneracy, closely resemble the domain wall, i.e. kinked, ground states, and therefore one should expect them to be resolute under perturbations.', '0709.1733-2-4-11': 'In this paper, we first show that these states exists and correspond to isolated eigenvalues of the finite volume XXZ chain with sufficiently strong anisotropy.', '0709.1733-2-4-12': 'We illustrate this feature in Figure [REF].', '0709.1733-2-4-13': 'Moreover, as consequence of the strong localization near the position of the ground state kink, these eigenvalues only weakly depend on the distance of the domain wall to the edges of the chain, and for this reason, we are next able to demonstrate that they persits even after the thermodynamic limit.', '0709.1733-2-4-14': 'The main difficulty we must overcome corresponds to the fact that, in the thermodynamic limit, the perturbation of the entire chain is an unbounded operator, and therefore, the standard, finite-order perturbation theory is inadequate for a rigorous argument.', '0709.1733-2-5-0': 'The XXZ kink Hamiltonian commutes with the operator [MATH].', '0709.1733-2-5-1': 'We define [MATH] to be the eigenspace of [MATH] with eigenvalue [MATH].', '0709.1733-2-5-2': 'These subspaces are called "sectors", and they are invariant subspaces for [MATH].', '0709.1733-2-6-0': 'It was shown in [CITATION] that for each sector there is a unique ground state of [MATH] with eigenvalue 0.', '0709.1733-2-6-1': 'Moreover, this ground state, [MATH], is given by the following expression: [EQUATION] where the sum is over all configurations for which [MATH] and the relationship between [MATH] and [MATH] is given by [MATH].', '0709.1733-2-6-2': 'A straightforward calculation shows a sharp transition in the magnetization from fully polarized down at the left to fully polarized up at the right.', '0709.1733-2-6-3': 'For this reason they are called kink ground states.', '0709.1733-2-6-4': 'In [CITATION] Koma and Nachtergaele proved that the kink ground states (as well as their spin-flipped or reflected versions the antikinks) comprise the entire set of ground states for the infinite-volume model, aside from the 2 other ground states: the translation invariant maximally magnetized and minimally magnetized all +J and all -J groundstates.', '0709.1733-2-6-5': 'Since the infinite-volume Hamiltonian incorporates all possible limits for all possible boundary conditions, this is a strong a posteriori justification for choosing the kink boundary conditions.', '0709.1733-2-6-6': 'It is worth noting that it has been proved for the antiferromagnetic model that no such ground states exist [CITATION].', '0709.1733-2-7-0': 'In [CITATION], Koma, Nachtergaele, and Starr showed that there is a spectral gap above each of the ground states in this model for all values of [MATH].', '0709.1733-2-7-1': 'Based on numerical evidence, they also made a conjecture that for [MATH] the first excited state of the XXZ model is an isolated eigenvalue, and that the magnitude of the spectral gap is asymptotically given by [MATH], where [MATH] is an eigenvalue of a particular one-particle problem.', '0709.1733-2-7-2': 'Caputo and Martinelli [CITATION] showed that the gap is indeed of order [MATH].', '0709.1733-2-8-0': 'Our main result is a proof that for all sufficiently large [MATH] the first few excitations of the XXZ model are isolated eigenvalues.', '0709.1733-2-8-1': 'This is true for all [MATH] and for spin 1 with [MATH] even, which is illustrated in Figure [REF] for the spin 3/2 and spin 2 chains.', '0709.1733-2-8-2': 'See Section [REF] for the precise statements.', '0709.1733-2-8-3': 'It turns out that in the Ising limit the eigenvalues less than [MATH] are all of multiplicity at most [MATH] in each sector.', '0709.1733-2-8-4': 'Moreover, the first excited states are simple except in the case when [MATH] is an integer and [MATH] mod [MATH].', '0709.1733-2-8-5': 'In this case, they are doubly degenerate.', '0709.1733-2-8-6': 'This is discussed in Section [REF].', '0709.1733-2-8-7': 'In Section [REF], we write the XXZ Hamiltonian as an explicit perturbation of the Ising limit.', '0709.1733-2-8-8': 'Theorem [REF] verifies that the perturbation is relatively bounded with respect to the Ising limit, and we finish this section by demonstrating that our estimates suffice to guarantee analytic continuation of the limiting eigenvalues.', '0709.1733-2-8-9': 'It is clear that the same method of proof can be applied to other Hamiltonians.', '0709.1733-2-9-0': 'While the question of low-lying excitations is generally interesting, it is of particular importance in the context of quantum computation.', '0709.1733-2-9-1': 'For quantum computers to become a reality we need to find or build physical systems that faithfully implement the quantum gates used in the algorithms of quantum computation [CITATION].', '0709.1733-2-9-2': 'The basic requirement is that the experimenter has access to two states of a quantum system that can be effectively decoupled from environmental noise for a sufficiently long time, and that transitions between these two states can be controlled to simulate a number of elementary quantum gates (unitary transformations).', '0709.1733-2-9-3': 'Systems that have been investigated intensively are single photon systems, cavity QED, nuclear spins (using NMR in suitable molecules), atomic levels in ion traps, and Josephson rings [CITATION].', '0709.1733-2-9-4': 'We believe that if one could build one-dimensional spin [MATH] systems with [MATH], which interact through an anisotropic interaction such as in the XXZ model, this would be a good starting point to encode qubits and unitary gates.', '0709.1733-2-9-5': 'The natural candidates for control parameters in such systems would be the components of a localized magnetic field.', '0709.1733-2-9-6': 'From the experimental point of view this is certainly a challenging problem.', '0709.1733-2-9-7': 'This work is a first step toward developing a mathematical model useful in the study of optimal control for these systems such as has already been carried out for nuclear magnetic resonance (NMR) [CITATION] and superconducting Joshepson qubits [CITATION].', '0709.1733-2-10-0': '# Set-up', '0709.1733-2-11-0': 'We study the spin [MATH] ferromagnetic XXZ model on the one-dimensional lattice [MATH].', '0709.1733-2-11-1': 'The local Hilbert space for a single site [MATH] is [MATH] with [MATH].', '0709.1733-2-11-2': 'We consider the Hilbert space for a finite chain on the sites [MATH].', '0709.1733-2-11-3': 'This is [MATH].', '0709.1733-2-11-4': 'The Hamiltonian of the spin-[MATH] XXZ model is [EQUATION] where [MATH],[MATH] and [MATH] are the spin-[MATH] matrices acting on the site [MATH], tensored with the identity operator acting on the other sites.', '0709.1733-2-11-5': 'The main parameter of the model is the anisotropy [MATH] and we get the Ising limit as [MATH].', '0709.1733-2-11-6': 'It is mathematically more convenient to work with the parameter [MATH], which we then assume is in the interval [MATH].', '0709.1733-2-11-7': 'As we said, [MATH] is the Ising limit, and [MATH] is the isotropic XXX Heisenberg model.', '0709.1733-2-11-8': 'It was shown [CITATION] that additional ground states emerge when we add particular boundary terms.', '0709.1733-2-11-9': 'Examples of this are the kink and antikink Hamiltonians [EQUATION]', '0709.1733-2-11-10': 'It is easy to see that the kink and anti-kink Hamiltonians are unitarily equivalent.', '0709.1733-2-11-11': 'We will be mainly interested in the kink Hamiltonian with [MATH].', '0709.1733-2-11-12': 'Note that, by a telescoping sum, we can absorb the boundary fields into the local interactions:', '0709.1733-2-12-0': 'H_L^k(^-1) = _L-1 h^k_,+1(^-1) ,', '0709.1733-2-13-0': 'h^k_,+1(^-1) = J^2-S_^3 S_+1^3 - ^-1(S_^1 S_+1^1 + S_^2 S_+1^2) + J 1-^-2 (S_^3 - S_+1^3)', '0709.1733-2-14-0': 'The Ising kink Hamiltonian is the result of taking [MATH], equivalently setting [MATH], namely it is [EQUATION]', '0709.1733-2-14-1': 'Each of the Hamiltonians introduced above commutes with the total magnetization [EQUATION].', '0709.1733-2-14-2': 'As indicated in the introduction, for each [MATH], the corresponding sector is defined to be the eigenspace of [MATH] with eigenvalue [MATH]; clearly, these are invariant subspaces for all the Hamiltonians introduced above.', '0709.1733-2-15-0': 'The Ising basis is a natural orthonormal basis for [MATH].', '0709.1733-2-15-1': 'At each site we have an orthonormal basis of the Hilbert space [MATH] given by the eigenvectors of [MATH] and labeled according to their eigenvalues.', '0709.1733-2-15-2': 'We will denote this by [MATH] for [MATH], and [MATH].', '0709.1733-2-15-3': 'Here, and throughout the remainder of the paper, we will use the notation [MATH] for the set [MATH] as we have done with [MATH].', '0709.1733-2-15-4': 'Finally, there is an orthonormal basis of the entire Hilbert space consisting of simple tensor product vectors: [MATH].', '0709.1733-2-16-0': 'Also recall that the raising and lowering operators are defined such that [EQUATION]', '0709.1733-2-16-1': 'A short calculation shows that [MATH] and [MATH] are given by [MATH] and [MATH], and [EQUATION]', '0709.1733-2-17-0': '# Main theorem', '0709.1733-2-18-0': 'Many of the results in this paper concern the kink Hamiltonian given by ([REF]).', '0709.1733-2-18-1': 'We will study it as a perturbation of the Ising Hamiltonian ([REF]) in the regime [MATH].', '0709.1733-2-18-2': 'We denote an Ising configuration as [MATH], where [MATH] for each [MATH], and the corresponding basis vector as [EQUATION].', '0709.1733-2-18-3': 'Observe that the Ising kink Hamiltonian is diagonal with respect to this basis, [EQUATION]', '0709.1733-2-18-4': 'Since each of the [MATH] are non-negative, it is easy to see that the ground states of [MATH] are all of the form [EQUATION] with some [MATH] and [MATH].', '0709.1733-2-18-5': 'Note that the total magnetization corresponding to [MATH] is [MATH].', '0709.1733-2-18-6': 'As we will verify in Proposition [REF], it is easy to check that these ground states are unique per sector.', '0709.1733-2-18-7': 'We do point out that there is a slight ambiguity in the above labeling scheme, however, since [MATH] and [MATH] coincide.', '0709.1733-2-18-8': 'Let us consider the following elementary result.', '0709.1733-2-19-0': 'For [MATH], [MATH] and [MATH], the eigenspace of [MATH] corresponding to energy [MATH] has dimension at least equal to [MATH].', '0709.1733-2-20-0': 'Because of this theorem, we see that, in the [MATH] limit, the energy [MATH] is essential spectrum; an eigenvalue with infinite multiplicity.', '0709.1733-2-20-1': 'As our interest is in perturbation theory, it is natural for us to restrict our attention to energies strictly between [MATH] and [MATH].', '0709.1733-2-20-2': 'We will call the corresponding eigenvectors "low energy excitations".', '0709.1733-2-21-0': 'We will now describe all the low energy excitations for the Ising kink Hamiltonian.', '0709.1733-2-21-1': 'In order to do so, it is convenient to introduce a family of eigenvectors which contains all possible low energy excitations.', '0709.1733-2-22-0': 'Let [MATH] be any site away from the boundary, i.e. take [MATH], and choose [MATH].', '0709.1733-2-22-1': 'These choices specify a sector [MATH] and a groundstate [MATH].', '0709.1733-2-22-2': 'If [MATH], we define the following vectors: For [MATH] set [EQUATION]', '0709.1733-2-22-3': 'If [MATH], we define the following vectors: For [MATH] set [EQUATION]', '0709.1733-2-22-4': 'We call these two sequences of Ising basis vectors the "localized kink excitations".', '0709.1733-2-22-5': 'Clearly, these vectors [MATH] have the same total magnetization [MATH] and moreover, [EQUATION] where [MATH] has been calculated using ([REF]); note that in each case there is only one non-zero term.', '0709.1733-2-22-6': 'In Figure [REF] we have shown two ground states, in the left and right graphs, as well as two low excitations.', '0709.1733-2-22-7': 'These are the schematic diagrams for [MATH], in solid in both pictures, and [MATH] and [MATH], in a dashed line, in the left and right, respectively', '0709.1733-2-23-0': 'We now define two sets of labels: [EQUATION]', '0709.1733-2-23-1': 'Depending on [MATH] and [MATH], neither, one, or both of these sets may be empty.', '0709.1733-2-23-2': 'We have the following theorem.', '0709.1733-2-24-0': '(1) The low energy excitations of [MATH] form a subset of the localized kink excitations introduced in ([REF]) and ([REF]) above.', '0709.1733-2-25-0': '(2) For [MATH] of the form [MATH] with some [MATH] and [MATH], the set of low energy excitations equals [EQUATION].', '0709.1733-2-25-1': 'This is a nonempty set except in the following two cases: [MATH], or [MATH], and [MATH].', '0709.1733-2-26-0': '(3) The low energy excitations of [MATH] are at most two-fold degenerate.', '0709.1733-2-26-1': 'The first excitation is simple, except for the case that [MATH] is an integer [MATH], and [MATH] mod [MATH].', '0709.1733-2-27-0': 'The two-fold degeneracy of the first excited state, guaranteed by part (3) of the theorem, occurs due to the spin flip and reflection symmetry.', '0709.1733-2-27-1': 'All other degeneracies with energy [MATH] occur as follows.', '0709.1733-2-27-2': 'Suppose [MATH] for integers [MATH].', '0709.1733-2-27-3': 'In this case, let [MATH].', '0709.1733-2-27-4': 'Then [MATH] is degenerate with [MATH], both with energy [MATH].', '0709.1733-2-27-5': 'Similarly, [MATH] is degenerate with [MATH] for the same energy.', '0709.1733-2-28-0': 'Next we consider the perturbed Hamiltonian [MATH].', '0709.1733-2-28-1': 'As is discussed in Section [REF], each low-lying eigenvalue [MATH], associated to some [MATH], is isolated from the rest of the spectrum by an isolation distance [MATH], independent of [MATH], which is defined by [EQUATION].', '0709.1733-2-29-0': 'Let [MATH], and fix [MATH].', '0709.1733-2-29-1': 'For any [MATH], consider the interval [MATH] about the low lying energy [MATH].', '0709.1733-2-29-2': 'The spectral projection of [MATH] onto [MATH] is analytic for large enough values of [MATH].', '0709.1733-2-29-3': 'In particular, the dimension of the spectral projection onto [MATH] is constant for this range of [MATH].', '0709.1733-2-30-0': 'Our estimates yield a lower bound on [MATH] as is provided by ([REF]).', '0709.1733-2-30-1': 'A slightly worse bound demonstrates that taking [MATH] suffices, but we do not expect either estimate to be sharp.', '0709.1733-2-31-0': 'The above theorem confirms the structure of the spectrum shown in Figure [REF].', '0709.1733-2-31-1': 'Moreover, since our numerical calculations indicate that some of the eigenvalues enter the continous spectrum, we do not expect this type of perturbation theory to work for the entire range of [MATH].', '0709.1733-2-32-0': '# Proof of Theorems [REF] and [REF] (Excitations of the Ising Model)', '0709.1733-2-33-0': 'In this section, we will focus on the Ising kink Hamiltonian [MATH], as introduced in ([REF]), for a fixed [MATH].', '0709.1733-2-33-1': 'The ground states can be characterized as follows:', '0709.1733-2-34-0': 'The ground states of the Ising kink Hamiltonian are all of the form [EQUATION] for [MATH] and [MATH].', '0709.1733-2-34-1': 'Moreover, there is exactly one ground state in each sector; the total magnetization eigenvalue for [MATH] is [MATH].', '0709.1733-2-35-0': 'This proposition was already used, implicitly, in setting up Theorem [REF].', '0709.1733-2-35-1': 'Here we prove it.', '0709.1733-2-36-0': 'Given any Ising configuration [MATH], equation ([REF]) demonstrates that the energy associated to [MATH], [MATH], is the sum of [MATH] non-negative terms.', '0709.1733-2-36-1': 'Therefore, the only way to have [MATH] is if all of the summands are [MATH].', '0709.1733-2-36-2': 'This is the case only if either [MATH] or [MATH] for all [MATH].', '0709.1733-2-36-3': 'Clearly, this is satisfied for the Ising configurations with [MATH] for all [MATH], [MATH] for all [MATH], and [MATH] equal to any number in [MATH].', '0709.1733-2-36-4': 'It is equally easy to see that these are all of the ground state configurations: if [MATH] is a groundstate configuration and for some [MATH] we have [MATH], then [MATH], which, by induction, means that [MATH] for all [MATH].', '0709.1733-2-36-5': 'Similar reasoning yields that [MATH] for all [MATH].', '0709.1733-2-37-0': 'To show that the ground states are unique in each sector, consider the equation [MATH], subject to the constraints: [MATH], [MATH], and [MATH].', '0709.1733-2-37-1': 'If [MATH] mod [MATH], then there is a unique pair [MATH] which satisfies this equation.', '0709.1733-2-37-2': 'If [MATH] mod [MATH], then there are two possible solutions [MATH] and [MATH] for some [MATH].', '0709.1733-2-37-3': 'But then, it is trivial to see that [MATH] and [MATH] coincide.', '0709.1733-2-38-0': 'Thus in any sector [MATH], there is a unique ground state eigenvector [MATH] for some choice of [MATH] with [MATH].', '0709.1733-2-38-1': 'We next observe that there are many eigenvectors with eigenvalue [MATH]; recall this was the statement of Theorem [REF].', '0709.1733-2-39-0': 'Theorem [REF] First consider the case that [MATH] is not divisible by [MATH].', '0709.1733-2-39-1': 'Then the unique groundstate eigenvector is [MATH] for some [MATH] and [MATH].', '0709.1733-2-39-2': 'If [MATH], then for each [MATH] consider the Ising configuration [MATH] with components [EQUATION]', '0709.1733-2-39-3': 'The total magnetization for the vector [MATH] is still [MATH].', '0709.1733-2-39-4': 'Using the formula ([REF]) again, it is easy to check that [MATH], [MATH] for all [MATH], and therefore, [MATH].', '0709.1733-2-39-5': 'This constitutes [MATH] possible values of [MATH]; producing at least this many distinct eigenvectors with energy [MATH].', '0709.1733-2-39-6': 'Similarly, if [MATH] then there are [MATH] Ising configurations of the form [MATH] with components [EQUATION] corresponding to some [MATH].', '0709.1733-2-39-7': 'In total, this gives [MATH] orthonormal eigenvectors corresponding to eigenvalue [MATH], yielding a lower bound on the dimension of the eigenspace.', '0709.1733-2-39-8': 'If [MATH] or [MATH], then the dimension is increased by at least 1.', '0709.1733-2-39-9': 'In the special case where [MATH] is divisible by [MATH], the eigenvector can be written in two ways as [MATH] or [MATH].', '0709.1733-2-39-10': 'When constructing excitations for [MATH] to the left of the kink, use the first formula above relative to [MATH].', '0709.1733-2-39-11': 'When constructing excitations for [MATH] to the right of the kink, use the second formula above relative to [MATH].', '0709.1733-2-39-12': 'Once again, this results in [MATH] orthonormal eigenvectors corresponding to eigenvalue [MATH].', '0709.1733-2-40-0': 'We now claim that any Ising configuration which is neither a ground state nor a localized kink excitation corresponds to an energy that is at least [MATH].', '0709.1733-2-40-1': 'This is the content of the following lemma.', '0709.1733-2-41-0': 'Consider an Ising configuration [MATH].', '0709.1733-2-42-0': '(1) If there is any [MATH] such that [MATH], then [MATH].', '0709.1733-2-43-0': '(2) If there is any [MATH] such that [MATH] and [MATH], then [MATH].', '0709.1733-2-44-0': '(1) It is clear from ([REF]) that we need only prove [EQUATION]', '0709.1733-2-44-1': 'Since [MATH], we have that [MATH].', '0709.1733-2-44-2': 'We need only verify that [MATH] to establish the claim.', '0709.1733-2-44-3': 'The product of two integers [MATH] is at most equal to [MATH].', '0709.1733-2-44-4': 'But this is only attained by [MATH].', '0709.1733-2-44-5': 'Since [MATH], we can neither have [MATH] nor [MATH].', '0709.1733-2-44-6': 'The next largest possible value of [MATH] is [MATH], and we have verified the claim.', '0709.1733-2-45-0': '(2) Again, because all terms are nonnegative, it suffices to show that [EQUATION].', '0709.1733-2-45-1': 'Using the formula for [MATH] and simplifying gives [EQUATION].', '0709.1733-2-45-2': 'Since [MATH], we have [EQUATION]', '0709.1733-2-45-3': 'Therefore, [EQUATION]', '0709.1733-2-45-4': 'Since [MATH] and [MATH], in either case we have proven the claim.', '0709.1733-2-46-0': 'Now we can finish the proof of the first main result, Theorem [REF].', '0709.1733-2-47-0': 'Theorem [REF] Part (1) of the theorem is a direct consequence of Lemma [REF] because the only Ising configurations that do not satisfy condition (1) or (2) of that lemma are the ground state configurations and the localized kink excitations.', '0709.1733-2-48-0': 'To prove part (2), let [MATH] for some [MATH] and [MATH].', '0709.1733-2-48-1': 'We first consider the case that [MATH].', '0709.1733-2-49-0': 'If [MATH] is sufficiently large and [MATH], then [MATH] as [EQUATION].', '0709.1733-2-49-1': 'So, in this case, both [MATH] and [MATH] are nonempty.', '0709.1733-2-50-0': 'Similarly, if [MATH], then either [EQUATION] or [EQUATION].', '0709.1733-2-50-1': 'Hence, [MATH] if [MATH] and [MATH].', '0709.1733-2-51-0': 'Lastly, if [MATH], then either [EQUATION] or [EQUATION].', '0709.1733-2-51-1': 'Hence, [MATH] if [MATH] and [MATH].', '0709.1733-2-52-0': 'We have proved (2) in the case that [MATH].', '0709.1733-2-52-1': 'Actually, the last observation above also verifies (2) in the case that [MATH] and [MATH].', '0709.1733-2-53-0': 'Finally, if [MATH] and [MATH], then [MATH] and if [MATH], then [MATH].', '0709.1733-2-53-1': 'In both these cases, the set of kink excitations is empty.', '0709.1733-2-54-0': 'We now prove the first part of (3).', '0709.1733-2-54-1': 'First, observe that for any [MATH], [MATH] and therefore with [MATH] fixed, both [MATH] are increasing functions of [MATH] for [MATH].', '0709.1733-2-54-2': 'Therefore the only degeneracies that can occur for a particular energy [MATH] is if [MATH] and [MATH] for some integers [MATH] and [MATH].', '0709.1733-2-54-3': 'This is obviously at most a two-fold degeneracy.', '0709.1733-2-55-0': 'In order to prove the second part of (3), we will simply prove Remark [REF].', '0709.1733-2-55-1': 'Without loss of generality, suppose [MATH].', '0709.1733-2-55-2': 'Then [MATH].', '0709.1733-2-55-3': 'So the only possibility for [MATH] to be less than [MATH] is if [MATH], which gives the energy [MATH] and an auxiliary condition, namely [MATH].', '0709.1733-2-55-4': 'A degeneracy happens only if there is a [MATH] such that [MATH].', '0709.1733-2-55-5': 'This means [EQUATION].', '0709.1733-2-55-6': 'Setting [MATH] and [MATH] (which satisfies [MATH] because [MATH]) we have exactly the result claimed.', '0709.1733-2-55-7': 'Note that the second part of (3) refers to the first excitation.', '0709.1733-2-55-8': 'For [MATH], the lowest excitation is [MATH].', '0709.1733-2-55-9': 'This means [MATH], which implies [MATH] and therefore [MATH] is even with [MATH] (because [MATH]).', '0709.1733-2-56-0': '# Proof of the main theorem', '0709.1733-2-57-0': 'The goal of this section is to prove Theorem [REF].', '0709.1733-2-57-1': 'We will do so by analyzing the kink Hamiltonian [MATH] as a perturbation of the Ising limit [MATH].', '0709.1733-2-57-2': 'Within the first subsection below, specifically in Theorem [REF], we prove that the operators which arise in our expansion of [MATH] are relatively bounded with respect to [MATH].', '0709.1733-2-57-3': 'In the next subsection, we discuss how the explicit bounds on [MATH], those claimed in Theorem [REF], follow from relative boundedness and basic perturbation theory.', '0709.1733-2-58-0': '## Relative Boundedness', '0709.1733-2-59-0': 'In this subsection, we will analyze the kink Hamiltonian introduced in ([REF]).', '0709.1733-2-59-1': 'Recall that this Hamiltonian is written as', '0709.1733-2-60-0': 'H_L^k(^-1) = _L-1 h^k_,+1(^-1) ,', '0709.1733-2-61-0': 'h^k_,+1(^-1) = J^2-S_^3 S_+1^3 - ^-1(S_^1 S_+1^1 + S_^2 S_+1^2) + J 1-^-2 (S_^3 - S_+1^3).', '0709.1733-2-62-0': 'By adding and subtracting terms of the form [MATH] to the local Hamiltonians, we find that [EQUATION] where [EQUATION] and [EQUATION]', '0709.1733-2-62-1': 'In Theorem [REF] below, we will show that both [MATH] and [MATH] are relatively bounded perturbations of [MATH].', '0709.1733-2-62-2': 'To prove such estimates, we will use the following lemma several times.', '0709.1733-2-63-0': 'Let [MATH] and [MATH] be self-adjoint [MATH] matrices.', '0709.1733-2-63-1': 'If [MATH] and [MATH], then there exists a constant [MATH] for which, [EQUATION]', '0709.1733-2-63-2': 'One may take [MATH] where [MATH] denotes the smallest positive eigenvalue of [MATH].', '0709.1733-2-64-0': 'Any vector [MATH] can be written as [MATH] where [MATH] and [MATH].', '0709.1733-2-64-1': 'Clearly then, [MATH] and therefore [EQUATION] as claimed.', '0709.1733-2-65-0': 'For our proof of Theorem [REF], we find it useful to introduce the Ising model without boundary conditions as an auxillary Hamiltonian, i.e., [EQUATION].', '0709.1733-2-65-1': 'It is easy to prove the next lemma.', '0709.1733-2-66-0': 'The Ising model without boundary terms is relatively bounded with respect to the Ising kink Hamiltonian.', '0709.1733-2-66-1': 'In particular, for any vector [MATH], [EQUATION].', '0709.1733-2-67-0': 'Consider the terms of the Ising kink Hamiltonian: [EQUATION]', '0709.1733-2-67-1': 'Summing on [MATH] then, we find that [EQUATION] and therefore, the bound [EQUATION] is clear for any vector [MATH].', '0709.1733-2-68-0': 'We now state the relative boundedness result.', '0709.1733-2-69-0': 'The linear term in the perturbation expansion of [MATH], see ([REF]), satisfies [EQUATION] for any vector [MATH].', '0709.1733-2-69-1': 'Moreover, we also have that [EQUATION]', '0709.1733-2-69-2': 'Using Lemma [REF], it is clear we need only prove that [EQUATION] to establish ([REF]).', '0709.1733-2-69-3': 'To this end, Lemma [REF] provides an immediate bound on the individual terms of these Hamiltonians.', '0709.1733-2-69-4': 'In fact, observe that for any fixed [MATH], both [MATH] and [MATH] are self-adjoint with [MATH] and [EQUATION]', '0709.1733-2-69-5': 'It is also easy to see that, for every [MATH], the first positive eigenvalue of [MATH] is [MATH], and we have that [EQUATION]', '0709.1733-2-69-6': 'An application of Lemma [REF] yields the operator inequality [EQUATION] valid for any [MATH].', '0709.1733-2-70-0': 'The norm bound we seek to prove will follow from considering products of these local Hamiltonians.', '0709.1733-2-70-1': 'For any vector [MATH], one has that [EQUATION] and [EQUATION]', '0709.1733-2-70-2': 'The arguments we provided above apply equally well to the diagonal terms of ([REF]) and ([REF]) in the sense that [EQUATION] is also valid for any [MATH].', '0709.1733-2-70-3': 'We find a similar bound by considering the terms on the right hand side of ([REF]) and ([REF]) for which [MATH].', '0709.1733-2-70-4': 'In this case, each of the operators [MATH] and [MATH] commute with both of the operators [MATH] and [MATH].', '0709.1733-2-70-5': 'Moreover, we conclude from ([REF]) that the operators [MATH] are non-negative for every [MATH].', '0709.1733-2-70-6': 'Since all the relevant quantities commute, it is clear that [EQUATION]', '0709.1733-2-70-7': 'Our observations above imply the following bound [EQUATION]', '0709.1733-2-70-8': 'In fact, the terms on the right hand side of ([REF]) for which either [MATH] or [MATH] are non-positive by ([REF]), respectively, ([REF]).', '0709.1733-2-70-9': 'In the case that [MATH], the operators [MATH] are non-negative (since they commute) and hence we may drop these terms; those terms that remain we group as the self-adjoint operators appearing on the right hand side of ([REF]) above.', '0709.1733-2-71-0': 'Our estimate is completed by applying Lemma [REF] one more time.', '0709.1733-2-71-1': 'Note that for any [MATH] the operator [MATH] is self-adjoint, non-negative, and [EQUATION] where the self-adjoint operator [MATH] appearing above is given by [EQUATION]', '0709.1733-2-71-2': 'For each [MATH], the first positive eigenvalue of [MATH] is [MATH], and it is also easy to see that [MATH].', '0709.1733-2-71-3': 'Thus, term by term Lemma [REF] implies that [EQUATION] from which we conclude that [EQUATION] as claimed in ([REF]).', '0709.1733-2-71-4': 'We have proved ([REF]).', '0709.1733-2-72-0': 'Equation ([REF]) follows directly from the easy observation that [MATH] is equal to [MATH].', '0709.1733-2-73-0': '## Perturbation theory', '0709.1733-2-74-0': 'In Section [REF], we verified that, in any given sector, the spectrum of the Ising kink Hamiltonian, [MATH], when restricted to the interval [MATH] consists of only isolated eigenvalues whose multiplicity is at most two.', '0709.1733-2-74-1': 'In fact, for the sector [MATH] these eigenvalues are determined by [EQUATION] for those values of [MATH] with [MATH].', '0709.1733-2-74-2': 'It is clear from ([REF]) that each of these eigenvalues have an isolation distance [MATH] from the rest of the spectrum and that this distance is independent of the length scale [MATH].', '0709.1733-2-75-0': 'For our proof of the relative boundedness result in Theorem [REF], we expanded the Hamiltonian as [EQUATION]', '0709.1733-2-75-1': 'Using the first resolvent formula, it is easy to see that [EQUATION] where we have denoted the resolvent by [MATH], and it is assumed that [MATH] has been chosen small enough so that [EQUATION]', '0709.1733-2-75-2': 'It is clear from sections II.1.3-4 of [CITATION] and chapter I of [CITATION] that the spectral projections corresponding to [MATH] can be written as a power series in [MATH], the coefficients of which being integrals of the resolvent over a fixed contour [MATH].', '0709.1733-2-75-3': 'Proving an estimate of the form ([REF]) for [MATH] large enough, uniformly with respect to [MATH], is sufficient to guarantee analyticity of the spectral projections.', '0709.1733-2-75-4': 'We verify such a uniform estimate below.', '0709.1733-2-76-0': 'Let [MATH] be an eigenvalue of [MATH] with isolation distance [MATH] as specified above.', '0709.1733-2-76-1': 'Denote by [MATH] the circle in the complex plane centered at [MATH] with radius [MATH].', '0709.1733-2-76-2': 'We claim that if [EQUATION] then ([REF]) is satisfied uniformly for [MATH].', '0709.1733-2-77-0': 'We proved in Theorem [REF] that for any vector [MATH], [EQUATION]', '0709.1733-2-77-1': 'Applying this bound to vectors [MATH] of the form [MATH] yields a norm estimate on [MATH], i.e., [EQUATION]', '0709.1733-2-77-2': 'Moreover, since [EQUATION] we have proved that [EQUATION]', '0709.1733-2-77-3': 'Similar arguments, again using Theorem [REF], imply that [EQUATION]', '0709.1733-2-77-4': 'For [MATH], the circular contour described above, we have that [EQUATION] and [EQUATION]', '0709.1733-2-77-5': 'We derive a bound of the form ([REF]), uniform for [MATH], by ensuring [MATH] large enough so that [EQUATION] where [EQUATION]', '0709.1733-2-77-6': 'Explicitly, one finds that the inequality ([REF]) is satisfied for all [EQUATION]', '0709.1733-2-77-7': 'Equation ([REF]) is a simple sufficient condition for [MATH] to satisfy this inequality.', '0709.1733-2-77-8': 'This is easy to verify if one first replaces [MATH] by [MATH] in ([REF]).'} | [['0709.1733-1-7-0', '0709.1733-2-6-0'], ['0709.1733-1-7-1', '0709.1733-2-6-1'], ['0709.1733-1-7-2', '0709.1733-2-6-2'], ['0709.1733-1-7-3', '0709.1733-2-6-3'], ['0709.1733-1-7-4', '0709.1733-2-6-4'], ['0709.1733-1-7-5', '0709.1733-2-6-5'], ['0709.1733-1-7-6', '0709.1733-2-6-6'], ['0709.1733-1-48-0', '0709.1733-2-48-0'], ['0709.1733-1-48-1', '0709.1733-2-48-1'], ['0709.1733-1-49-0', '0709.1733-2-49-0'], ['0709.1733-1-49-1', '0709.1733-2-49-1'], ['0709.1733-1-70-0', '0709.1733-2-70-0'], ['0709.1733-1-70-1', '0709.1733-2-70-1'], ['0709.1733-1-70-2', '0709.1733-2-70-2'], ['0709.1733-1-70-3', '0709.1733-2-70-3'], ['0709.1733-1-70-4', '0709.1733-2-70-4'], ['0709.1733-1-70-5', '0709.1733-2-70-5'], ['0709.1733-1-70-6', '0709.1733-2-70-6'], ['0709.1733-1-70-7', '0709.1733-2-70-7'], ['0709.1733-1-70-8', '0709.1733-2-70-8'], ['0709.1733-1-70-9', '0709.1733-2-70-9'], ['0709.1733-1-34-0', '0709.1733-2-34-0'], ['0709.1733-1-34-1', '0709.1733-2-34-1'], ['0709.1733-1-38-0', '0709.1733-2-38-0'], ['0709.1733-1-38-1', '0709.1733-2-38-1'], ['0709.1733-1-23-0', '0709.1733-2-23-0'], ['0709.1733-1-23-1', '0709.1733-2-23-1'], ['0709.1733-1-23-2', '0709.1733-2-23-2'], ['0709.1733-1-14-0', '0709.1733-2-14-0'], ['0709.1733-1-14-1', '0709.1733-2-14-1'], ['0709.1733-1-14-2', '0709.1733-2-14-2'], ['0709.1733-1-37-0', '0709.1733-2-37-0'], ['0709.1733-1-37-1', '0709.1733-2-37-1'], ['0709.1733-1-37-2', '0709.1733-2-37-2'], ['0709.1733-1-37-3', '0709.1733-2-37-3'], ['0709.1733-1-27-0', '0709.1733-2-27-0'], ['0709.1733-1-27-1', '0709.1733-2-27-1'], ['0709.1733-1-27-2', '0709.1733-2-27-2'], ['0709.1733-1-27-3', '0709.1733-2-27-3'], ['0709.1733-1-27-4', '0709.1733-2-27-4'], ['0709.1733-1-27-5', '0709.1733-2-27-5'], ['0709.1733-1-64-0', '0709.1733-2-64-0'], ['0709.1733-1-64-1', '0709.1733-2-64-1'], ['0709.1733-1-33-0', '0709.1733-2-33-0'], ['0709.1733-1-25-0', '0709.1733-2-25-0'], ['0709.1733-1-25-1', '0709.1733-2-25-1'], ['0709.1733-1-35-0', '0709.1733-2-35-0'], ['0709.1733-1-35-1', '0709.1733-2-35-1'], ['0709.1733-1-18-0', '0709.1733-2-18-0'], ['0709.1733-1-18-1', '0709.1733-2-18-1'], ['0709.1733-1-18-2', '0709.1733-2-18-2'], ['0709.1733-1-18-3', '0709.1733-2-18-3'], ['0709.1733-1-18-4', '0709.1733-2-18-4'], ['0709.1733-1-18-5', '0709.1733-2-18-5'], ['0709.1733-1-18-6', '0709.1733-2-18-6'], ['0709.1733-1-18-7', '0709.1733-2-18-7'], ['0709.1733-1-18-8', '0709.1733-2-18-8'], ['0709.1733-1-72-0', '0709.1733-2-72-0'], ['0709.1733-1-8-0', '0709.1733-2-7-0'], ['0709.1733-1-8-1', '0709.1733-2-7-1'], ['0709.1733-1-8-2', '0709.1733-2-7-2'], ['0709.1733-1-24-0', '0709.1733-2-24-0'], ['0709.1733-1-44-0', '0709.1733-2-44-0'], ['0709.1733-1-44-1', '0709.1733-2-44-1'], ['0709.1733-1-44-2', '0709.1733-2-44-2'], ['0709.1733-1-44-3', '0709.1733-2-44-3'], ['0709.1733-1-44-4', '0709.1733-2-44-4'], ['0709.1733-1-44-5', '0709.1733-2-44-5'], ['0709.1733-1-44-6', '0709.1733-2-44-6'], ['0709.1733-1-55-0', '0709.1733-2-55-0'], ['0709.1733-1-55-1', '0709.1733-2-55-1'], ['0709.1733-1-55-3', '0709.1733-2-55-3'], ['0709.1733-1-55-4', '0709.1733-2-55-4'], ['0709.1733-1-55-6', '0709.1733-2-55-6'], ['0709.1733-1-55-7', '0709.1733-2-55-7'], ['0709.1733-1-55-8', '0709.1733-2-55-8'], ['0709.1733-1-55-9', '0709.1733-2-55-9'], ['0709.1733-1-6-0', '0709.1733-2-5-0'], ['0709.1733-1-6-1', '0709.1733-2-5-1'], ['0709.1733-1-6-2', '0709.1733-2-5-2'], ['0709.1733-1-59-1', '0709.1733-2-59-1'], ['0709.1733-1-28-0', '0709.1733-2-28-0'], ['0709.1733-1-5-1', '0709.1733-2-4-2'], ['0709.1733-1-5-3', '0709.1733-2-4-4'], ['0709.1733-1-40-0', '0709.1733-2-40-0'], ['0709.1733-1-40-1', '0709.1733-2-40-1'], ['0709.1733-1-31-0', '0709.1733-2-31-0'], ['0709.1733-1-31-1', '0709.1733-2-31-1'], ['0709.1733-1-46-0', '0709.1733-2-46-0'], ['0709.1733-1-74-0', '0709.1733-2-74-0'], ['0709.1733-1-74-1', '0709.1733-2-74-1'], ['0709.1733-1-74-2', '0709.1733-2-74-2'], ['0709.1733-1-15-0', '0709.1733-2-15-0'], ['0709.1733-1-15-1', '0709.1733-2-15-1'], ['0709.1733-1-15-2', '0709.1733-2-15-2'], ['0709.1733-1-15-3', '0709.1733-2-15-3'], ['0709.1733-1-15-4', '0709.1733-2-15-4'], ['0709.1733-1-9-0', '0709.1733-2-8-0'], ['0709.1733-1-9-1', '0709.1733-2-8-1'], ['0709.1733-1-9-2', '0709.1733-2-8-2'], ['0709.1733-1-9-3', '0709.1733-2-8-3'], ['0709.1733-1-9-4', '0709.1733-2-8-4'], ['0709.1733-1-9-5', '0709.1733-2-8-5'], ['0709.1733-1-9-6', '0709.1733-2-8-6'], ['0709.1733-1-9-7', '0709.1733-2-8-7'], ['0709.1733-1-9-8', '0709.1733-2-8-8'], ['0709.1733-1-9-9', '0709.1733-2-8-9'], ['0709.1733-1-77-0', '0709.1733-2-77-0'], ['0709.1733-1-77-1', '0709.1733-2-77-1'], ['0709.1733-1-77-2', '0709.1733-2-77-2'], ['0709.1733-1-77-3', '0709.1733-2-77-3'], ['0709.1733-1-77-4', '0709.1733-2-77-4'], ['0709.1733-1-77-5', '0709.1733-2-77-5'], ['0709.1733-1-77-6', '0709.1733-2-77-6'], ['0709.1733-1-77-7', '0709.1733-2-77-7'], ['0709.1733-1-77-8', '0709.1733-2-77-8'], ['0709.1733-1-66-0', '0709.1733-2-66-0'], ['0709.1733-1-66-1', '0709.1733-2-66-1'], ['0709.1733-1-21-0', '0709.1733-2-21-0'], ['0709.1733-1-21-1', '0709.1733-2-21-1'], ['0709.1733-1-54-0', '0709.1733-2-54-0'], ['0709.1733-1-54-1', '0709.1733-2-54-1'], ['0709.1733-1-54-2', '0709.1733-2-54-2'], ['0709.1733-1-54-3', '0709.1733-2-54-3'], ['0709.1733-1-71-0', '0709.1733-2-71-0'], ['0709.1733-1-71-1', '0709.1733-2-71-1'], ['0709.1733-1-71-2', '0709.1733-2-71-2'], ['0709.1733-1-71-3', '0709.1733-2-71-3'], ['0709.1733-1-71-4', '0709.1733-2-71-4'], ['0709.1733-1-76-0', '0709.1733-2-76-0'], ['0709.1733-1-76-1', '0709.1733-2-76-1'], ['0709.1733-1-76-2', '0709.1733-2-76-2'], ['0709.1733-1-75-0', '0709.1733-2-75-0'], ['0709.1733-1-75-1', '0709.1733-2-75-1'], ['0709.1733-1-75-2', '0709.1733-2-75-2'], ['0709.1733-1-75-3', '0709.1733-2-75-3'], ['0709.1733-1-75-4', '0709.1733-2-75-4'], ['0709.1733-1-63-0', '0709.1733-2-63-0'], ['0709.1733-1-63-1', '0709.1733-2-63-1'], ['0709.1733-1-63-2', '0709.1733-2-63-2'], ['0709.1733-1-36-0', '0709.1733-2-36-0'], ['0709.1733-1-36-1', '0709.1733-2-36-1'], ['0709.1733-1-36-2', '0709.1733-2-36-2'], ['0709.1733-1-36-3', '0709.1733-2-36-3'], ['0709.1733-1-36-4', '0709.1733-2-36-4'], ['0709.1733-1-36-5', '0709.1733-2-36-5'], ['0709.1733-1-26-0', '0709.1733-2-26-0'], ['0709.1733-1-26-1', '0709.1733-2-26-1'], ['0709.1733-1-69-0', '0709.1733-2-69-0'], ['0709.1733-1-69-1', '0709.1733-2-69-1'], ['0709.1733-1-69-2', '0709.1733-2-69-2'], ['0709.1733-1-69-3', '0709.1733-2-69-3'], ['0709.1733-1-69-4', '0709.1733-2-69-4'], ['0709.1733-1-69-5', '0709.1733-2-69-5'], ['0709.1733-1-69-6', '0709.1733-2-69-6'], ['0709.1733-1-30-0', '0709.1733-2-30-0'], ['0709.1733-1-30-1', '0709.1733-2-30-1'], ['0709.1733-1-47-0', '0709.1733-2-47-0'], ['0709.1733-1-53-0', '0709.1733-2-53-0'], ['0709.1733-1-53-1', '0709.1733-2-53-1'], ['0709.1733-1-45-0', '0709.1733-2-45-0'], ['0709.1733-1-45-1', '0709.1733-2-45-1'], ['0709.1733-1-45-2', '0709.1733-2-45-2'], ['0709.1733-1-45-4', '0709.1733-2-45-4'], ['0709.1733-1-22-0', '0709.1733-2-22-0'], ['0709.1733-1-22-1', '0709.1733-2-22-1'], ['0709.1733-1-22-2', '0709.1733-2-22-2'], ['0709.1733-1-22-3', '0709.1733-2-22-3'], ['0709.1733-1-22-4', '0709.1733-2-22-4'], ['0709.1733-1-22-5', '0709.1733-2-22-5'], ['0709.1733-1-22-6', '0709.1733-2-22-6'], ['0709.1733-1-22-7', '0709.1733-2-22-7'], ['0709.1733-1-19-0', '0709.1733-2-19-0'], ['0709.1733-1-1-0', '0709.1733-2-1-0'], ['0709.1733-1-1-3', '0709.1733-2-1-3'], ['0709.1733-1-62-0', '0709.1733-2-62-0'], ['0709.1733-1-62-1', '0709.1733-2-62-1'], ['0709.1733-1-62-2', '0709.1733-2-62-2'], ['0709.1733-1-29-1', '0709.1733-2-29-1'], ['0709.1733-1-29-2', '0709.1733-2-29-2'], ['0709.1733-1-29-3', '0709.1733-2-29-3'], ['0709.1733-1-65-0', '0709.1733-2-65-0'], ['0709.1733-1-65-1', '0709.1733-2-65-1'], ['0709.1733-1-39-0', '0709.1733-2-39-0'], ['0709.1733-1-39-1', '0709.1733-2-39-1'], ['0709.1733-1-39-2', '0709.1733-2-39-2'], ['0709.1733-1-39-3', '0709.1733-2-39-3'], ['0709.1733-1-39-4', '0709.1733-2-39-4'], ['0709.1733-1-39-5', '0709.1733-2-39-5'], ['0709.1733-1-39-6', 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['0709.1733-1-3-0', '0709.1733-2-9-1']] | [] | [['0709.1733-1-4-0', '0709.1733-2-3-0']] | [] | ['0709.1733-1-1-1', '0709.1733-1-1-2', '0709.1733-1-11-3', '0709.1733-1-11-12', '0709.1733-1-12-0', '0709.1733-1-13-0', '0709.1733-1-29-0', '0709.1733-1-33-1', '0709.1733-1-41-0', '0709.1733-1-43-0', '0709.1733-1-45-3', '0709.1733-1-50-0', '0709.1733-1-50-1', '0709.1733-1-51-0', '0709.1733-1-51-1', '0709.1733-1-55-2', '0709.1733-1-55-5', '0709.1733-1-60-0', '0709.1733-1-61-0', '0709.1733-1-68-0', '0709.1733-2-1-1', '0709.1733-2-1-2', '0709.1733-2-11-3', '0709.1733-2-11-12', '0709.1733-2-12-0', '0709.1733-2-13-0', '0709.1733-2-29-0', '0709.1733-2-33-1', '0709.1733-2-41-0', '0709.1733-2-43-0', '0709.1733-2-45-3', '0709.1733-2-50-0', '0709.1733-2-50-1', '0709.1733-2-51-0', '0709.1733-2-51-1', '0709.1733-2-55-2', '0709.1733-2-55-5', '0709.1733-2-60-0', '0709.1733-2-61-0', '0709.1733-2-68-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/0709.1733 | null | null | null | null | null |
1503.01735 | {'1503.01735-1-0-0': 'In this review we discuss spin and charge transport properties in graphene-based single-layer and few-layer spin-valve devices.', '1503.01735-1-0-1': 'We give an overview of challenges and recent advances in the field of device fabrication and discuss two of our fabrication methods in more detail which result in distinctly different device performances.', '1503.01735-1-0-2': 'In the first class of devices, Co/MgO electrodes are directly deposited onto graphene which results in rough MgO-to-Co interfaces and favor the formation of conducting pinholes throughout the MgO layer.', '1503.01735-1-0-3': 'We show that the contact resistance area product ([MATH]) is a benchmark for spin transport properties as it scales with the measured spin lifetime in these devices indicating that contact-induced spin dephasing is the bottleneck for spin transport even in devices with large [MATH] values.', '1503.01735-1-0-4': 'In a second class of devices, Co/MgO electrodes are first patterned onto a silicon substrate.', '1503.01735-1-0-5': 'Subsequently, a graphene-hBN heterostructure is directly transferred onto these prepatterned electrodes which provides improved interface properties.', '1503.01735-1-0-6': 'This is seen by a strong enhancement of both charge and spin transport properties yielding charge carrier mobilities exceeding [MATH] and spin lifetimes up to 3.7 ns at room temperature.', '1503.01735-1-0-7': 'We discuss several shortcomings in the determination of both quantities which complicates the analysis of both extrinsic and intrinsic spin scattering mechanisms.', '1503.01735-1-0-8': 'Furthermore, we show that contacts can be the origin of a second charge neutrality point in gate dependent resistance measurements which is influenced by the quantum capacitance of the underlying graphene layer .', '1503.01735-1-1-0': '# Introduction', '1503.01735-1-2-0': 'In spin-based electronics, three main aspects have to be considered when exploring suitable spin transport materials and material combinations: (1) electrical injection and detection of spins, (2) their manipulation, and (3) the transport of spins in the material [CITATION].', '1503.01735-1-2-1': 'What makes graphene a promising material in the field of spintronics, is its unique spin transport performance in particular at room temperature [CITATION] where spin lifetimes of up to [MATH] [CITATION] and spin diffusion length of [MATH] [CITATION] have been measured by means of electrical Hanle spin precession measurements in non-local spin-valve devices.', '1503.01735-1-2-2': 'The corresponding charge carrier mobilities in these devices are above [MATH] [CITATION].', '1503.01735-1-2-3': 'Other interesting materials in the field of spintronics, e.g. Si, also exhibit nanosecond spin lifetimes at room temperature but fail short to graphene in respect to the spin diffusion lengths and charge carrier mobilities [CITATION].', '1503.01735-1-2-4': 'We note that we only compare device and material properties from electrical spin precession measurements at room temperature.', '1503.01735-1-2-5': 'Less invasive spin sensitive methods such as electron spin resonance or optical pump-probe methods can yield much longer spin lifetimes, especially at low temperatures [CITATION].', '1503.01735-1-3-0': "Form a theoretical point of view the long spin lifetimes are attributed to graphene's extraordinary band structure [CITATION] which exhibits weak spin-orbit coupling at energies close to the Dirac point [CITATION].", '1503.01735-1-3-1': 'Because for the most prominent spin relaxation mechanisms the spin relaxation rate scales with the spin-orbit coupling strength [CITATION].', '1503.01735-1-3-2': 'But the weak spin-orbit coupling in graphene is a mixed blessing because for spin manipulation, e.g. via the Bychkov-Rashba effect, a strong spin-orbit coupling is needed [CITATION].', '1503.01735-1-3-3': 'While there are many publications by different groups about electrical injection and detection of spins in graphene (e.g. [CITATION]) the manipulation of spins in these experiments is therefore solely done by an external magnetic field.', '1503.01735-1-4-0': 'Therefore, one emerging topic in graphene-based spintronic research is the partial functionalization of graphene with the goal to achieve other ways for spin manipulation, e.g. by electrostatic gating.', '1503.01735-1-4-1': 'The aim is to use high quality graphene parts as leads for efficient spin transport whereas the spin manipulation is realized in a functionalized part of the graphene device with enhanced spin-orbit coupling.', '1503.01735-1-4-2': 'Routes to increase the spin-orbit coupling include the spin-orbit proximity effect, in which additional materials such as the two dimensional transition metal dichalcogenides (e.g. tungsten disulfide [CITATION]) or ferromagnetic insulators (e.g. EuO [CITATION]) are put in direct contact to graphene.', '1503.01735-1-4-3': 'Also hydrogenation of graphene [CITATION] or the deposition of heavy adatoms [CITATION] can yield enhanced spin-orbit coupling in graphene, but as a drawback, these approaches are already known to strongly change its band structure [CITATION].', '1503.01735-1-5-0': 'Very recently, an excellent review article about graphene spintronics compiled a significant part of both theoretical and experimental work which has been carried out on spin phenomena in graphene [CITATION].', '1503.01735-1-5-1': 'In particular, this review focuses on how to measure spin transport in graphene, the spin-orbit coupling in pristine and modified graphene, magnetic moments from defects and adatoms, and the open question about which spin relaxation mechanisms limit spin transport in graphene.', '1503.01735-1-5-2': 'In our article, we expand the review to open questions and challenges in the experimental field of work.', '1503.01735-1-5-3': 'In section [REF] we discuss new routes in device fabrication by the usage of novel transfer techniques and outline the issue of device contaminations during fabrication.', '1503.01735-1-5-4': 'Then, we cover the influence of metallic contacts on the underlying graphene in section [REF] and present new results on the appearance of a contact-induced second charge neutrality point in gate dependent resistance measurements.', '1503.01735-1-5-5': 'Next, the reliability of extracted values for both spin lifetimes and charge carrier mobilities is discussed in sections [REF] and [REF], respectively.', '1503.01735-1-5-6': 'In section [REF], we briefly comment on spin relaxation mechanisms in graphene.', '1503.01735-1-6-0': '# New methods of device fabrication', '1503.01735-1-7-0': 'A completely new road of device fabrication was paved by the introduction of mechanical transfer techniques for 2d materials (see e.g. [CITATION]).', '1503.01735-1-7-1': 'These transfer techniques provide a controlled deposition of flakes with high spatial precision, which results in two important advantages: (1) The device fabrication becomes now independent of the random position of exfoliated flakes on a substrate and (2) the possibility to deposit different 2d materials on top of each other in order to build so-called van der Waals heterostructures (review about these heterostructures in reference [CITATION]).', '1503.01735-1-8-0': 'To highlight the first advantage, we first have to discuss the shortcomings when growing complicated structures directly on top of graphene.', '1503.01735-1-8-1': 'Without the novel transfer techniques, the starting point of a graphene-based spin transport device is the randomly exfoliated graphene flake which is typically deposited onto Si[MATH]/SiO[MATH].', '1503.01735-1-8-2': 'The next fabrication step is usually a lithography process on top of graphene which is needed for the deposition of ferromagnetic electrodes.', '1503.01735-1-8-3': 'But as soon as graphene gets into contact with an organic resist, it is extremely difficult to completely remove the contamination of hydrocarbons.', '1503.01735-1-8-4': 'This is illustrated in the atomic force microscopy (AFM) images of figures [REF](a)-(c), which show one of our earlier but not optimized dosage tests.', '1503.01735-1-8-5': 'As a resist, we used PMMA (950K) which was dissolved in ethyl lactate and n-butyl acetate with a thickness of 250 nm after spin-coating and baking.', '1503.01735-1-8-6': 'For developing we used isopropyl alcohol and methyl isobutyl ketone with a developing time of 105 s. E-beam writing was performed with an acceleration voltage of 10 kV.', '1503.01735-1-8-7': 'The dosages in figures [REF](a)-(c) are given in fractions of [MATH].', '1503.01735-1-8-8': 'The scale in figure [REF](a) was chosen to depicted the flake in high contrast.', '1503.01735-1-8-9': 'The resist is underdeveloped for a dosage below [MATH].', '1503.01735-1-8-10': 'But also for dosages larger than [MATH] an increasing contamination of the graphene flake can be observed, which can be easier be visualized in the phase signal of the AFM image in tipping mode (figure [REF](b)).', '1503.01735-1-8-11': 'The difference in contrast can be understood by different attenuations of the AFM cantilever for SiO[MATH], graphene on SiO[MATH], and PMMA on graphene.', '1503.01735-1-8-12': 'The increasing contamination for higher dosages can be explained by crosslinking of PMMA [CITATION] or e-beam-induced defects in the graphene flake [CITATION] which enables a stronger binding with hydrocarbons.', '1503.01735-1-8-13': 'We observe the cleanest graphene surface at a dosage of [MATH] and were able to reduce the amount of leftover resist residues even further by increasing the developing time to 210 s.', '1503.01735-1-9-0': 'But even for optimized lithography conditions the graphene flake will be contaminated with hydrocarbons on an atomic scale.', '1503.01735-1-9-1': 'Such contaminations are clearly seen by transmission electron microscopy and annealing temperatures of up to [MATH] are needed to remove the hydrocarbons [CITATION].', '1503.01735-1-9-2': 'But such high temperatures are far beyond the glass transition temperature of the resist and therefore will destroy the patterning mask.', '1503.01735-1-9-3': 'Next to thermal annealing also other methods are applied to remove resist residues after the development step of the resist, e.g. low-density inductively coupled Ar plasma [CITATION], [MATH] cluster cleaning [CITATION], or ultraviolet-ozone treatment [CITATION].', '1503.01735-1-9-4': 'But it still has to be seen if these cleaning methods are able to remove even the last atomic layer of hydrocarbons from the graphene flake.', '1503.01735-1-9-5': 'While in most of these publications only rather macroscopic analysis methods such as Raman spectroscopy or even qualitative electrical measurements are used, only the high-resolution X-ray photoelectron spectroscopy in case of the ultraviolet-ozone treatment of reference [CITATION] may really demonstrate the complete removal of all contaminations.', '1503.01735-1-9-6': 'There is, however, also a drawback to this method as long treatments can also create defects in the graphene flake.', '1503.01735-1-10-0': 'The next issue arises because of the chemically inert nature of graphene and its [MATH] hybridization, which leads to unfavorable growth conditions of many materials on top of graphene.', '1503.01735-1-10-1': 'This can already be seen in the stronger accumulation of the PMMA on the graphene flake compared to the area of SiO[MATH] for dosages lower than [MATH] in figure [REF](b) (in this figure the scale is optimized to the height of the underdeveloped PMMA).', '1503.01735-1-10-2': 'The reason for this accumulation is the low wettability of graphene [CITATION] and again it is transmission electron microscopy which reveals the clustering of almost every metal deposited on graphene [CITATION].', '1503.01735-1-10-3': 'The interaction between metals and graphene is so weak, that in sub-monolayer growth of metal layers on graphene, the metal atoms solely resides on the hydrocarbon contamination and not on the clean graphene parts [CITATION].', '1503.01735-1-10-4': 'By now, there are only few studies reporting epitaxial-like growth of materials on graphene, e.g. EuO [CITATION] or Ti [CITATION].', '1503.01735-1-11-0': 'The hydrocarbon contaminations from the lithography step and the clustering of many materials are important issues for graphene-based spin transport devices because both can reduce the quality of insulating oxide barriers between graphene and ferromagnetic electrodes which are needed for spin injection and spin detection.', '1503.01735-1-11-1': 'But an insufficient barrier quality can be the bottleneck for the overall spin transport properties of the device [CITATION].', '1503.01735-1-11-2': 'Presently, there are two widely used oxide materials: MgO and Al[MATH]O[MATH].', '1503.01735-1-11-3': 'The direct growth of MgO on graphene is governed by the Volmer-Weber island growth mechanism [CITATION] resulting in inhomogeneous oxide layers with presumably pinholes [CITATION].', '1503.01735-1-11-4': 'This is illustrated in the AFM image of a 3 nm thick MgO layer grown on top of graphene in figure [REF](d), which exhibits peak-to-peak values of up to 2 nm.', '1503.01735-1-11-5': 'The use of a Ti/TiO[MATH] wetting layer yields more homogeneous MgO layers [CITATION] but the impact of Ti on the properties of graphene is still an open question [CITATION].', '1503.01735-1-11-6': 'For Al[MATH]O[MATH] the evaporation of Al and its subsequent oxidation to Al[MATH]O[MATH] can also yield rough layers with pinholes [CITATION].', '1503.01735-1-11-7': 'On the other hand, there is a report on homogeneous and pinhole free Al[MATH]O[MATH] oxide barriers on graphene by argon sputter deposition [CITATION].', '1503.01735-1-11-8': 'But the same group also demonstrated by Raman spectroscopy that this technique may also induces defects in graphene [CITATION].', '1503.01735-1-12-0': 'It has been shown that the new transfer techniques can overcome the described problems by avoiding the direct growth of electrode material onto graphene.', '1503.01735-1-12-1': 'In a first step the electrode structure is fabricated by means of e-beam lithography and metallization.', '1503.01735-1-12-2': 'Only afterwards the graphene flake is deposited on top of the prepatterned structure (figure [REF](e)).', '1503.01735-1-12-3': 'Therefore, the graphene flake is not exposed to the e-beam lithography step, hence resist residues and e-beam induced defects can be avoided.', '1503.01735-1-12-4': 'Furthermore, the growth conditions of the ferromagnetic electrodes and the oxide barrier are far more suitable and can be adapted to an adequate substrate.', '1503.01735-1-12-5': 'Especially, the whole expertise already achieved in the fabrication of high quality magnetic tunnel junction devices (e.g. [CITATION]) may now also be applied to graphene spin transport devices.', '1503.01735-1-12-6': 'In this respect, we especially refer to the improvement of magnetic tunnel junctions by annealing which results in crystallization of the interface between ferromagnetic metals and oxide barriers [CITATION].', '1503.01735-1-12-7': 'The approach to first fabricate spin injection and detection electrodes on a wafer and then to deposit a stack of graphene on hexagonal boron nitride on top of it was applied in reference [CITATION], where spin lifetimes of 3.7 ns were measured at room temperature in trilayer graphene, which is the longest room temperature spin lifetime reported so far.', '1503.01735-1-13-0': 'As already mentioned in the first paragraph of this section, the possibility to deposit and stack various 2d materials on top of each other in order to fabricate van der Waals heterostructures is the second advantage of the transfer techniques.', '1503.01735-1-13-1': 'So far, many possible 2d materials beyond graphene have emerged (overview e.g. in [CITATION]) and combining these 2d materials with graphene has expanded the field of graphene research dramatically.', '1503.01735-1-13-2': 'One of the first applications was the fabrication of stacks consisting of hexagonal boron nitride (hBN) and graphene.', '1503.01735-1-13-3': 'Due to the atomically smooth surface and similar lattice constant of hBN to graphene, these devices exhibit significantly improved electrical properties compared to graphene deposited on SiO[MATH] [CITATION].', '1503.01735-1-13-4': 'The incorporation of such hBN-graphene stacks in graphene spintronic devices led to a significant enhancement of the measured spin diffusion lengths as well as the charge carrier mobility [CITATION].', '1503.01735-1-13-5': 'Another application of hBN which became feasible with the transfer techniques is its usage as an oxygen-free injection barrier when placed between graphene and the ferromagnetic electrodes [CITATION].', '1503.01735-1-13-6': 'By transferring exfoliated hBN flakes, the above-mentioned problems in oxide barrier growth may be avoided.', '1503.01735-1-14-0': 'Finally, the transfer techniques offer a new route to functionalize graphene by putting different 2d materials in direct contact to graphene.', '1503.01735-1-14-1': 'One important application of such a heterostructure is the enhancement of the spin-orbit coupling (e.g. by the spin-orbit proximity effect of tungsten disulfide on graphene [CITATION]) to allow for spin manipulation via the Bychkov-Rashba effect.', '1503.01735-1-14-2': 'We emphasize that many effects like the spin-orbit proximity effect greatly rely on contamination-free interfaces.', '1503.01735-1-14-3': 'However, depending on the exact kind of transfer technique different amounts of hydrocarbon contamination between the layers of the heterostructures can be observed [CITATION].', '1503.01735-1-14-4': 'Interestingly, there seems to be "self-cleansing" effects in some heterostructures (e.g. graphene on hBN, MoS[MATH], and WS[MATH]), in which the surface contamination of the 2d materials automatically aggregates into bubbles, leaving behind rather clean interfaces throughout [MATH]m-sized graphene/hBN areas [CITATION].', '1503.01735-1-15-0': '# Impact of metals on graphene', '1503.01735-1-16-0': 'In this section, we discuss the influence of metallic electrodes on both spin and charge properties of the underlying graphene layer.', '1503.01735-1-16-1': 'One important tool to investigate the electronic properties is angle-resolved photoemission spectroscopy (ARPES), which can directly probe the band structure of graphene which is in direct contact to the metal (see e.g. references [CITATION] for ARPES measurements of graphene in contact with Au, Ag, Fe, Ni, Co, Al, and Cu).', '1503.01735-1-16-2': 'Of special interest in the area of graphene spintronics are the 3d transition metals Fe, Ni, and Co, which are used as ferromagnetic electrodes to generate spin polarized currents.', '1503.01735-1-16-3': "ARPES measurements have revealed that all three metals significantly alter the band structure of graphene: The Dirac point of graphene shifts around 2.8 eV below the Fermi energy and the graphene's [MATH] band hybridizes with 3d bands of Co near the Fermi level [CITATION].", '1503.01735-1-17-0': 'As a results, as soon as the ferromagnetic electrode gets into direct contact with graphene (which can be the case for pinholes in the separating oxide layer), the spin current is initially, i.e. right after electrical spin injection, no longer carried by pure graphene states near the Dirac point but rather by the hybridized states near the Fermi level.', '1503.01735-1-17-1': 'A significant higher spin scattering rate in such states in comparison to the states near the Dirac point of unmodified graphene may explain the overall short spin lifetime measured in devices with ohmic contacts exhibiting small contact resistance area products [MATH] [CITATION] (see also figure [REF]).', '1503.01735-1-17-2': 'Furthermore, the large density of states at the Fermi level due to hybridization results in a pinning of the Fermi level.', '1503.01735-1-17-3': 'Considering the n-doping of graphene underneath the contacts caused by the Co, the gate voltage dependent doping profiles of figure [REF](c) can be deduced for ohmic contacts, which in turn can explain the gate dependent resistance curve of such a device in figure [REF](a).', '1503.01735-1-18-0': 'A sufficiently thick oxide barrier without pinholes, which is preferable for spin transport devices, is expected to suppress hybridization between the metal and graphene.', '1503.01735-1-18-1': 'Nevertheless, a field effect doping of graphene in such metal/dielectric/graphene heterostructures is still expected because of different respective work functions [CITATION].', '1503.01735-1-18-2': 'The thicker the insulating oxide barrier between graphene and metallic electrode (which results in larger [MATH] products) the lower the corresponding doping [CITATION].', '1503.01735-1-18-3': 'At some point the doping is small enough to be also tuned by the back gate voltage (see corresponding doping profile in figure [REF](d)).', '1503.01735-1-18-4': 'The different doping of graphene in between and underneath the electrodes can be seen by two distinct charge neutrality points (CNP) in gate dependent resistance measurements (figure [REF](b)): the right peak (blue arrow) corresponds to the graphene part between the electrodes (CNP mostly around a gate voltage of [MATH] for untreated devices or at positive gate voltages because of p-doping during oxygen treatments, see reference [CITATION]) while the left one (red arrow) results from the graphene underneath the electrodes (CNP at large negative gate voltages because of the n-doping of Co).', '1503.01735-1-18-5': 'For our devices, the contact-induced CNP at negative gate voltage (see red arrow in figure [REF](b)) typically appears for [MATH] [CITATION].', '1503.01735-1-19-0': 'In the following, we discuss the origin of the contact-induced second CNP in more detail.', '1503.01735-1-19-1': "Although this review article is about spin transport, we consider this discussion as crucial as it highlights two important aspects which are often neglected: (1) The significant modification of graphene's transport properties right underneath the contacts and (2) the existence of lateral pn-junctions which the spin polarized charge carriers have to overcome.", '1503.01735-1-20-0': 'To confirm that the left CNP in our devices is indeed contact-induced, we now focus on a non-local spin-valve device with Co/MgO electrodes in more detail.', '1503.01735-1-20-1': 'This device exhibit a large electrode-to-electrode variation of the respective [MATH] values.', '1503.01735-1-20-2': 'Figure [REF](a) depicts the schematic layout.', '1503.01735-1-20-3': 'The measured contact resistance area products of the inner contacts 2 to 4 are: [MATH], [MATH], and [MATH].', '1503.01735-1-20-4': 'Consistent to the above explanations, we assign higher doping levels to lower [MATH] values which yields the potential profile of figure [REF](a) for which we assume that the graphene between the contacts is hole doped by an applied gate voltage.', '1503.01735-1-20-5': "The depicted black curve corresponds to the position of graphene's CNP relative to the Fermi level [MATH] (also see depicted Dirac cone).", '1503.01735-1-20-6': 'Gate dependent 4-terminal resistance measurements were performed as a function of dc-current.', '1503.01735-1-20-7': 'For this, we apply a total current of [MATH] over the outermost electrodes 1 and 5 and measure the voltage drop between contacts 2 and 3 for region A and contacts 3 and 4 for region B by standard low frequency lock-in technique (see also reference [CITATION]).', '1503.01735-1-21-0': 'As it can be seen for both regions in figures [REF](b) and (d) both the position and magnitude of the left CNP significantly depends on the applied dc-current whereas for the right CNP of the bare graphene there is only a small decrease in resistance for both positive and negative dc current which can be explained by the larger local temperatures from current-induced Joule heating.', '1503.01735-1-21-1': 'Figure [REF](c) depicts the dependence of the gate voltage position of the contact-induced left CNP as a function of dc-current.', '1503.01735-1-21-2': 'We show this data not only for regions A and B but also between contacts 2 and 4, which we call region AB.', '1503.01735-1-21-3': 'Interestingly, the gate voltage shift of the CNP has different sign and amplitude in regions A and B. Both, the different sign and amplitude can also be seen in the doping profile of figure [REF](a) (indicated by the arrows).', '1503.01735-1-21-4': 'Consistently, the dc-current dependent shift of the contact-induced CNP measured between contacts 2 and 4 (green curve in figure [REF](c)) is in very good agreement the arithmetic mean of the respective curves of region A and B.', '1503.01735-1-22-0': 'With the help of theoretical considerations about graphene field effect transistors [CITATION] we are able to qualitatively understand the gate voltage shift of the contact-induced CNP.', '1503.01735-1-22-1': 'For this we assume that high quality tunnel barriers induce a field effect similar to top gates, which changes the Fermi level in graphene underneath the contacts.', '1503.01735-1-22-2': 'In fact, a reverse argumentation might be more intuitive (we refer to reference [CITATION]): Because of the different work functions between graphene and a metal a charge transfer will occur during the alignment of the Fermi levels.', '1503.01735-1-22-3': 'The charge transfer yields a carrier doping of graphene, but it also creates a voltage drop and therefore a local electric field over the separating dielectric MgO barrier as the barrier acts as a capacitor with the graphene and the metallic electrodes as its plates.', '1503.01735-1-22-4': 'These voltages [MATH] (TG as in top gate) are depicted in the schematic layout of a graphene field effect transistor in figure [REF](a).', '1503.01735-1-22-5': 'In the following, we assume that these voltages remain unaffected if the two inner contacts in figure [REF](a) are used as voltage probes while the outer ones are source and drain contacts (this wiring applies to the measurement of figure [REF]).', '1503.01735-1-23-0': 'Next, we determine the charge carrier density in the graphene flake.', '1503.01735-1-23-1': 'For the graphene underneath an inner electrode (marked by "X" in figure [REF](a)) we can apply the equivalent circuit in figure [REF](b).', '1503.01735-1-23-2': 'The MgO layer, which acts as a barrier for spin injection and spin detection, is approximated to be a top gate with capacitance [MATH], whereas the back gate voltage [MATH] is applied over the capacitance [MATH], which represents the SiO[MATH] layer.', '1503.01735-1-23-3': 'Far more complicated is the incorporation of graphene in this picture.', '1503.01735-1-23-4': 'For this we first discuss the Fermi level shift by applying a bias voltage over the graphene flake.', '1503.01735-1-23-5': 'In figure [REF](c) this effect is illustrated in a very simplified system, in which e.g. the doping effects of the contacts are neglected.', '1503.01735-1-23-6': 'We first consider ideal, undoped graphene, in which case the Fermi level of the graphene underneath the grounded contact (left contact in figure [REF](c)) is at the Dirac point.', '1503.01735-1-23-7': 'A bias voltage applied to the right contact results in a lateral voltage gradient along the graphene channel with corresponding shifts of the Fermi level [CITATION].', '1503.01735-1-23-8': "Along the whole graphene flake a gradual nn'-junction is generated, which can be verified by photocurrent measurements [CITATION].", '1503.01735-1-24-0': 'The Fermi level shift underneath the contacts with the applied source-drain-voltage [MATH] can be written as [MATH] where the energy scale is set to zero at the CNP of graphene.', '1503.01735-1-24-1': 'In this simplified system the voltage drop [MATH] along the graphene channel only depends on the electrostatic potential by the applied bias.', '1503.01735-1-24-2': 'But in general, the Fermi level also depends on the electrochemical potential.', '1503.01735-1-24-3': 'Accordingly, the position of the Fermi level [MATH] also depends on other factors such as the applied gate voltage or changes in the electrochemical potential by adsorbates.', '1503.01735-1-24-4': 'This is also illustrated in figure [REF](c) in the case that graphene is p-doped by an applied gate voltage or adsorbates.', '1503.01735-1-24-5': 'Apparently, the local area with vanishing charge carrier density, i.e. where the Fermi level is at the Dirac point, moves along the graphene channel (from left to right in the lower panel of figure [REF](c)) with increasing p-doping concentration.', '1503.01735-1-24-6': 'Such a shift of the CNP by an applied gate voltage is e.g. observed in photocurrent and thermal infrared microscopy [CITATION].', '1503.01735-1-25-0': 'The capacitance of the graphene part in the equivalent circuit over which the voltage [MATH] drops is the quantum capacitance [MATH] of graphene [CITATION].', '1503.01735-1-25-1': 'The quantum capacitance directly results from the Pauli principle and becomes relevant in materials with a small density of states (DOS) near the Fermi level.', '1503.01735-1-25-2': 'To explain the quantum capacitance, we first consider ideal graphene with the Fermi level at the Dirac point, where the DOS vanishes (figure [REF](d), left band structure).', '1503.01735-1-25-3': 'If now the charge carrier density in graphene is changed by [MATH] the Fermi level shifts by [MATH] as the electrons have to occupy higher energy states.', '1503.01735-1-25-4': 'For single layer graphene, the Fermi level [MATH] and the DOS at the Fermi level [MATH] are given by [CITATION]: [EQUATION] where [MATH] is the Fermi velocity.', '1503.01735-1-25-5': 'If now the charge carrier density is doubled by adding the same amount of charges [MATH] a second time, the increase of the Fermi level becomes less strong because of the previous increase in DOS ([MATH]) (see figure [REF](d)).', '1503.01735-1-25-6': 'As mentioned above, the electrons at the Fermi level have the energy [MATH].', '1503.01735-1-25-7': 'Combining this expression with equation [REF] and the general definition of a capacitance [MATH] (with the charge [MATH] and area [MATH]) results in the quantum capacitance of graphene: [EQUATION]', '1503.01735-1-25-8': 'In figure [REF](e) we plot this quantum capacitance of graphene as a function of the applied gate voltage (black solid line).', '1503.01735-1-25-9': 'Due to broadening of the DOS by both thermal excitation at room temperature and by structural inhomogeneities the experimentally determined graphene quantum capacitance is also broadened [CITATION] which is depicted as a black dashed line.', '1503.01735-1-25-10': 'Furthermore, we include the capacitance of both oxide layers (the MgO of the top electrode with thicknesses of 2 and 3 nm and the SiO[MATH] of the back gate).', '1503.01735-1-26-0': 'Now we go back to the equivalent circuit in figure [REF](b).', '1503.01735-1-26-1': 'First, we start with the graphene part between the contacts (marked by a circle in figure [REF](a)).', '1503.01735-1-26-2': 'As there is no top gate capacitance in this region, back gate and quantum capacitances are put in series and give a total capacitance [MATH].', '1503.01735-1-26-3': 'As the back gate capacitance is much smaller than the quantum capacitance (compare to figure [REF](e)), it dominates the total capacitance: [MATH].', '1503.01735-1-26-4': 'Accordingly, the quantum capacitance does not play any significant role in the back gate induced field effect (see also [CITATION]).', '1503.01735-1-26-5': 'This is an important conclusion as the bias voltage can only change the charge carrier density at the node of the equivalent circuit in figure [REF](b) with help of the quantum capacitance.', '1503.01735-1-26-6': 'The fact that the quantum capacitance can be neglected now explains why the CNP of the bare graphene part between the contacts does not shift with dc currents (figures [REF](b) and (d)).', '1503.01735-1-26-7': 'In contrast, the quantum capacitance plays a significant role for graphene parts underneath the electrodes as it is in the same order of magnitude as the electrode capacitance (figure [REF](e)).', '1503.01735-1-26-8': 'If we assume a fixed charge carrier density at the node of the equivalent circuit of figure [REF](b) for the contact region, now the shift of the contact-induced CNP in figures [REF](b) and (d) becomes clear: By changing the dc current (or accordingly the bias voltage) the voltage [MATH] will change according to the discussion in figure [REF](c).', '1503.01735-1-26-9': 'If we want to keep the charge carrier density at the node of the equivalent circuit at [MATH] (i.e. to the contact-induced CNP) the voltages [MATH] (i.e. the actual doping of the graphene by the contacts) and [MATH] also have to change.', '1503.01735-1-26-10': 'To simulate this, all changes of [MATH], [MATH], and [MATH] have to be calculated self-consistently (e.g. a change in the back gate voltage will shift the Fermi level and therefore the value of the graphene quantum capacitance), which goes beyond the scope of this paper.', '1503.01735-1-26-11': 'For a far more detailed description we refer to the references [CITATION].', '1503.01735-1-27-0': 'Finally, we again point to the fact that the different charge transport properties of graphene underneath the contacts and graphene in between the contacts are so far widely neglected.', '1503.01735-1-27-1': 'It has to be seen how the change in the doping underneath the contacts with different dc-biases or the pn-junctions along the graphene channel have an impact on the performance of graphene-based spintronic devices.', '1503.01735-1-28-0': '# Determination of spin lifetimes', '1503.01735-1-29-0': 'In this section we discuss on how to extract the spin lifetime out of spin precession measurements and address how several limitations to this approach have recently been recognized.', '1503.01735-1-29-1': 'We restrict ourselves to non-local spin transport measurements, so we can neglect the drift term in the Bloch-Torrey equation and only consider changes in the net spin vector [MATH] by spin precession about a perpendicular magnetic field [MATH] (see measurement configuration in figure [REF](a)), spin diffusion, and spin dephasing and relaxation [CITATION]: [EQUATION]', '1503.01735-1-29-2': 'Here [MATH] is the Larmor frequency, [MATH] is the gyromagnetic ratio, [MATH] the spin diffusion constant, and [MATH] the spin lifetime.', '1503.01735-1-30-0': 'The advantage of non-local measurements is that the spin signal can be significantly decoupled from spurious charge signals [CITATION].', '1503.01735-1-30-1': 'To achieve such a non-local configuration, dc measurements or lock-in techniques with low frequencies are needed, because electric pulses in case of RF measurements propagate through the whole device, which means that no non-local part can exist.', '1503.01735-1-30-2': 'The dc or low frequency measurements lead to a stationary or quasi-stationary condition and therefore the time derivative of equation [REF] is set to zero.', '1503.01735-1-30-3': 'Hence a solution to [EQUATION] has to be found to extract the spin lifetime by means of a fit to the dc spin precession curve (so-called Hanle curve).', '1503.01735-1-30-4': 'Typical Hanle curves for a spin transport device are shown in figure [REF](b) with both parallel and antiparallel alignments of the inner Co electrodes in figure [REF](a).', '1503.01735-1-30-5': 'The non-local spin resistance [MATH] can be determined at [MATH].', '1503.01735-1-30-6': 'As expected, both Hanle curves merge at larger magnetic fields.', '1503.01735-1-30-7': 'However, they do not become constant but rather increase above [MATH] T.', '1503.01735-1-30-8': 'We note that this magnetic field dependent background signal is typical for most studies and it can even be much more pronounced.', '1503.01735-1-30-9': 'Recently, it has been shown that the background can be caused by an inhomogeneous current flow through the oxide barriers.', '1503.01735-1-30-10': 'As a result there is a charge accumulation signal next to the actual spin accumulation signal in the non-local voltage which can be be explained by a redistribution of charge carriers by a perpendicular magnetic field similar to the classical Hall effect [CITATION].', '1503.01735-1-31-0': 'One fundamental drawback of every solution to equation [REF] is that the three parameters [MATH], [MATH], and [MATH] cannot be determined independently.', '1503.01735-1-31-1': 'This is because of the steady state condition, which allows the multiplication of equation [REF] by any factor [MATH] without changing the overall result.', '1503.01735-1-31-2': 'Therefore a fit using a solution to equation [REF] is invariant with respect to the transition ([MATH].', '1503.01735-1-31-3': 'Hence one of the three parameters has to be assumed or determined by other measurements.', '1503.01735-1-32-0': 'Often [MATH] is assumed for the analysis of spin precession measurements.', '1503.01735-1-32-1': 'For pristine or moderately modified graphene flakes this assumption is confirmed by electron spin resonance (ESR) measurements [CITATION].', '1503.01735-1-32-2': 'It remains to be seen if this value also holds for functionalized graphene.', '1503.01735-1-32-3': 'By now, spin precession measurements of hydrogenated graphene suggest larger [MATH]-factors [CITATION].', '1503.01735-1-32-4': 'Of course, the most elegant way to circumvent this uncertainty is to give up on the non-local, steady state condition and apply time-resolved RF measurements which offer the possibility to directly determine the [MATH]-factor from time dependent spin precession.', '1503.01735-1-33-0': 'In the following, we discuss some assumptions and simplifications, which have to be made in order to find useable solutions to equation [REF].', '1503.01735-1-33-1': 'One of the first and still widely used analytical solution was developed by Johnson and Silsbee [CITATION].', '1503.01735-1-33-2': 'But in recent years it was found that the measured spin lifetime of a graphene device scales with the [MATH] product at least up to values of several tens of k[MATH]m[MATH] (see figure [REF]) [CITATION]) suggesting that the measured spin lifetime is not the intrinsic spin lifetime of graphene but rather has extrinsic origin.', '1503.01735-1-33-3': 'One route to explain this dependence is that the measured spin lifetime can be significantly underestimated if spin relaxation by the contacts is not included in the fit model and hence some work was done in the direction of more elaborated models [CITATION].', '1503.01735-1-33-4': 'But even these models make two crucial assumptions: (1) The injection barriers are homogeneous and can be characterized by the [MATH] value only and (2) the spin lifetime in graphene underneath the contacts is the same as the spin lifetime in the bare graphene part between the contacts.', '1503.01735-1-34-0': 'As it was already discussed in section [REF] the growth of homogeneous oxide barriers on graphene is quite challenging.', '1503.01735-1-34-1': 'But as long as the contact is not spatially homogenous, the measured contact resistance can only be an averaged value.', '1503.01735-1-34-2': 'Especially if there are pinholes within the barrier the contact resistance area product cannot precisely be determined, because the exact number and sizes of the pinholes cannot be deduced from electrical measurements.', '1503.01735-1-34-3': 'Therefore, the uncertainty in the contact resistance will also lead to an uncertainty in the extracted spin lifetimes by the aforementioned models.', '1503.01735-1-35-0': 'And also the second assumption of a single spin lifetime seems to be oversimplified considering section [REF].', '1503.01735-1-35-1': 'As it was demonstrated by ARPES measurements [CITATION] the interaction between graphene and ferromagnetic metals can significantly change the band structure of graphene due to hybridization.', '1503.01735-1-35-2': 'The hybridized states are directly at the Fermi level and therefore both charge and spin transport will occur through these states, which may have a strong impact on the spin lifetime.', '1503.01735-1-35-3': 'But even if we assume a sufficiently thick oxide barrier, which inhibits this hybridization, the assumption of a single spin lifetime may still be too simple if the spin lifetime depends on the Fermi level or, correspondingly, the charge carrier density.', '1503.01735-1-35-4': 'Because the field effect doping of the contacts [CITATION] can e.g. lead to the case at which the graphene underneath the contacts is highly n-doped whereas the graphene between the contacts is at its CNP.', '1503.01735-1-35-5': 'By now, only some research was done on the topic how two spatially different graphene parts with different spin relaxation times influence the overall measured spin lifetime (e.g. in reference [CITATION]).', '1503.01735-1-36-0': 'Finally, we note that a rough MgO layer like the one in figure [REF](d) not only can cause pinholes but also leads to an equally rough surface of the Co electrode which is deposited on top.', '1503.01735-1-36-1': 'But such a rough ferromagnetic interface yield stray fields, which may result in a broadening of the spin precession curve and thereby an apparent reduction of the extracted spin lifetime [CITATION].', '1503.01735-1-37-0': '# Determination of charge carrier mobilities', '1503.01735-1-38-0': 'In this section we discuss the determination of the charge carrier mobility [MATH] from gate dependent resistance measurements.', '1503.01735-1-38-1': 'There are two reasons making this an important topic for graphene-based spin transport studies.', '1503.01735-1-38-2': 'Firstly, the dependence of the spin lifetime and the charge carrier mobility is often used to identify the dominating spin relaxation mechanism [CITATION].', '1503.01735-1-38-3': 'Secondly, the use of oxide barriers for spin injection and detection can lead to the appearance of the contact-induced CNP as it has been discussed in section [REF].', '1503.01735-1-38-4': 'We will demonstrate that this 2nd CNP has significant impact on the determination of the charge carrier mobility.', '1503.01735-1-39-0': 'Firstly, we analyze the electron mobility in one of our devices by different methods which are currently used in literature showing that the determination of the mobility is not unambiguous and yield values which vary by more than a factor of 2.', '1503.01735-1-39-1': 'The simplest model considers a graphene field effect transistor, for which a linear dependence between the charge carrier density [MATH] (or accordingly back gate voltage) and the conductivity [MATH] is often observed (see e.g. [CITATION]).', '1503.01735-1-39-2': 'For this case several groups (e.g. [CITATION]) assume a simple Drude model and define the Drude mobility as: [EQUATION]', '1503.01735-1-39-3': 'We emphasize that this Drude mobility is extracted at a single point of the conductivity curve.', '1503.01735-1-39-4': 'To account for the residual conductivity [MATH] even in the case of [MATH] the conductivity can be written as [MATH].', '1503.01735-1-39-5': 'Starting from this equation the extracted field effect mobility [MATH] from fitting the slope of the conductivity curves is given by (e.g. in references [CITATION]): [EQUATION]', '1503.01735-1-39-6': 'We applied both methods to a conductivity curve from one of our actual spin valve devices in figure [REF](b).', '1503.01735-1-39-7': 'The red circle marks the point at which [MATH] and [MATH] have been determined.', '1503.01735-1-39-8': 'Remarkably, [MATH] exceeds [MATH] by almost 60[MATH].', '1503.01735-1-39-9': 'In the as-fabricated state of this device the conductivity drops only slightly at large negative gate voltages, which indicates that the contact induced CNP is close but nevertheless out of range of the applicable gate voltage.', '1503.01735-1-39-10': 'After an oxygen treatment which primarily reduces the doping effect of the electrodes [CITATION] the contact-induced CNP shifts into the accessible gate voltage range (figure [REF](e)) and furthermore complicates the analysis of the carrier mobilities.', '1503.01735-1-40-0': 'This can be seen in the resistance curve of another spin valve device in figure [REF](a) that also shows two distinct CNPs.', '1503.01735-1-40-1': 'We can decompose resistance contributions of both graphene regions (underneath and in between the contacts; model for fitting is discussed further below), which is shown by red and blue dashed lines, respectively.', '1503.01735-1-40-2': 'The overlapping of both curves clearly demonstrates one major issue of the two mobility fits discussed so far: As soon as the doping underneath the contact approaches the one of the bare graphene part, both the absolute value and the slope of the measured gate dependent resistance curve significantly changes in the overlapping part.', '1503.01735-1-40-3': 'And so does the extracted mobility.', '1503.01735-1-40-4': 'The reason for this is quite clear: Both equation [REF] and [REF] are based on one well-defined transport regime, whereas the contact-induced CNP points to the fact that the charge transport in graphene underneath and in between the contacts can differ significantly.', '1503.01735-1-41-0': 'But before we discuss more detailed fitting models, which include different transport regimes, we first want to discuss another issue when using equation [REF] which arises as soon as the conductivity curves are not linear as the ones in figure [REF].', '1503.01735-1-41-1': 'In this case, the extracted mobility [MATH], which is determined by the slope, depend on the charge carrier density [MATH].', '1503.01735-1-41-2': 'But this is a mathematical contradiction which is often neglected.', '1503.01735-1-41-3': 'The derivative of [MATH] with respect to [MATH] is now given by: [EQUATION]', '1503.01735-1-41-4': 'The term [MATH]/[MATH] is neglected by the approach of equation [REF] and only vanishes for strictly linear conductivity curves.', '1503.01735-1-42-0': 'But the question arises if the mobility indeed depends on the charge carrier density in case of a non-linear conductivity curve as the non-linearity can also be explained by a constant mobility ([MATH]/[MATH]) if there is a charge carrier independent contribution [MATH] to the overall resistivity.', '1503.01735-1-42-1': 'Such a contribution can result from both short- and correlated long-range disorder [CITATION].', '1503.01735-1-42-2': 'Hence, the corresponding conductivity is given by (see e.g. references [CITATION]):', '1503.01735-1-43-0': '[EQUATION]', '1503.01735-1-43-1': 'Figure [REF](c) demonstrates that a significant part of the electron branch for [MATH] 0 of the non-linear conductivity curve can be fitted under the assumption of the constant mobility [MATH]cm[MATH]/Vs. However, the fitting fails completely in the hole branch (not shown), because here the impact of the contact-induced CNP has to be considered.', '1503.01735-1-43-2': 'Furthermore, all three approaches that are discussed so far break down near the CNP.', '1503.01735-1-43-3': 'But there are also models trying to fit the whole conductivity curve including the two CNPs (e.g. in reference [CITATION]).', '1503.01735-1-43-4': 'We also use the following model which assumes two differently doped graphene parts with different mobilities [MATH]: [EQUATION] where [MATH] is the gate voltage [MATH] dependent charge carrier density, [MATH] the capacitive coupling for a [MATH] thick SiO[MATH] layer, [MATH] the residual charge carrier densities due to the presence of electron-hole puddles [CITATION] and thermally excited carriers, which prevent a divergence at the CNPs [CITATION], and a gate independent resistivity [MATH].', '1503.01735-1-43-5': 'This model can be fitted quite well to the conductivity curve of figure [REF](d) (the given mobility [MATH] cm[MATH]/Vs corresponds to the Dirac curve with its CNP near zero gate voltage).', '1503.01735-1-44-0': 'But now there are two issues: At first, the values of the charge carrier mobilities as extracted from the four models spread over a broad range between [MATH] and [MATH].', '1503.01735-1-44-1': 'Secondly, there is a serious issue when extracting the mobility from equation [REF], which already contains seven parameters, as it cannot be used to adequately fit the conductivity curve of the same device after oxygen treatment (figure [REF](e)).', '1503.01735-1-44-2': 'We tried several other models to fit the curve in figure [REF](e), but for a reasonable fit we need at least eight free parameters.', '1503.01735-1-44-3': 'But as we already briefly discussed in one of our previous publications [CITATION] and will discuss in this section in more detail, such a large number of free parameters make such a model useless.', '1503.01735-1-44-4': 'Because as long as the Dirac curves have symmetric shapes, which they have in figure [REF](a) and in our previous publications [CITATION], only seven parameters can reliably be extracted out of a gate dependent resistivity curve with two visible CNPs: The position, magnitude and width for each of the two peaks and the background.', '1503.01735-1-44-5': 'Therefore, a model of eight parameters is overdetermined and accordingly we are able to change the mobility values in such a model over an unreasonable wide range without seriously worsening the fit results.', '1503.01735-1-45-0': 'To approach reliable mobility fits, we now deduce several parameters which have to be included into a conductivity model to adequately fit a gate dependent resistivity curve with two CNPs.', '1503.01735-1-45-1': 'First, there is the carrier mobility as the actual quantity of interest.', '1503.01735-1-45-2': 'As already discussed in section [REF], the interaction with the electrodes can significantly change the electronic properties of graphene.', '1503.01735-1-45-3': 'We therefore have to consider two mobilities for each device: one for the graphene underneath the contacts and one for the graphene area in between the contacts.', '1503.01735-1-45-4': 'The same argument holds for the electrochemical doping of the respective graphene parts, which is obvious as we observe two CNPs.', '1503.01735-1-45-5': 'Two additional parameters are needed to include spatial variations of the electrochemical potentials within each graphene region.', '1503.01735-1-45-6': 'These two parameters can also account for the minimum conductivities as the variations in the potential lead to the electron-hole puddles at the charge neutrality point [CITATION].', '1503.01735-1-45-7': 'So far this yields six parameters.', '1503.01735-1-46-0': 'In the next step, we have to critically review the gate voltage as the tuning parameter for the resistivity measurement.', '1503.01735-1-46-1': 'As discussed e.g. in [CITATION] the interaction between contact and graphene may lead to a screening of the gate electric field underneath the electrodes.', '1503.01735-1-46-2': 'We assume that there is a gradual transition from pinning to depinning of the Fermi level underneath the contacts with increasing oxide barrier thickness and quality.', '1503.01735-1-46-3': 'Therefore, we have to consider a screening factor as a seventh parameter which accounts for the effective Fermi level shift in the graphene underneath the contacts as a function of applied gate voltage.', '1503.01735-1-47-0': 'Furthermore, there is the unknown transition of the electrochemical potentials between contact covered and bare graphene part, i.e. the exact shape and lateral extension of the pn-junctions, which are known to exist near the edges of the contacts [CITATION].', '1503.01735-1-47-1': 'In the most simple approximation, at least one more parameter has to be considered which describes the decaying length of the pn-junction (see e.g. references [CITATION]).', '1503.01735-1-47-2': 'And finally, there might be a gate voltage independent contribution [MATH] to the overall resistance as already discussed in equation [REF].', '1503.01735-1-47-3': 'Hence, at least nine parameters are necessary to simulate the gate dependent resistivity, whereas only seven independent parameters may be experimentally determined by a gate dependent resistance measurement.', '1503.01735-1-48-0': 'The whole problem with the analysis of the carrier mobility from gate dependent resistance measurements is that both the contact-covered and bare graphene parts are measured simultaneously in series.', '1503.01735-1-48-1': 'The lack of independently probing both regions is bound to the very nature of a pure electrical transport measurement.', '1503.01735-1-48-2': 'Therefore, independent measurements have to be conducted to determine some of the aforementioned parameters separately.', '1503.01735-1-48-3': 'Especially, information about the spatial change in the electrochemical potential along the contact induced pn-junctions may be helpful.', '1503.01735-1-48-4': 'These can be obtained from scanning photocurrent microscopy [CITATION].', '1503.01735-1-49-0': 'Finally, we note that most other graphene-based nanoelectronic devices are fabricated with low-ohmic, metallic contacts.', '1503.01735-1-49-1': 'This might be the reason why the aforementioned issues concerning the determination of the charge carrier mobility is not discussed in recent publications in more detail.', '1503.01735-1-49-2': 'For metallic contacts, a complete pinning of the Fermi level underneath the contacts can be assumed (see section [REF]).', '1503.01735-1-49-3': 'Therefore, the complete contact area of the device only contributes as a gate independent constant resistance to the overall gate dependent resistance (e.g. [MATH] in equation [REF]).', '1503.01735-1-49-4': 'Additionally, the impact of both the contact area and the pn-junctions on the transport measurement decreases with increasing separation of the electrodes.', '1503.01735-1-49-5': 'We note that in many charge-based electrical transport studies the contact separation is much larger than in graphene-based spin transport devices.', '1503.01735-1-50-0': '# Spin relaxation mechanism and perspectives', '1503.01735-1-51-0': 'In this final section we review several studies which were used to investigate the relevant spin relaxation mechanism in graphene.', '1503.01735-1-51-1': 'Maybe the most direct way to identify the dominating spin relaxation mechanism is to evaluate the dependence of the spin lifetime on the charge carrier mobility [CITATION].', '1503.01735-1-51-2': "A linear dependence of [MATH] on [MATH] or [MATH], which is the momentum scattering time, is a priori suggestive of an Elliott-Yafet (EY) spin scattering mechanism, while the inverse relation [MATH] indicate the dominance of a D'yakonov-Perel'-like (DP) spin scattering mechanism.", '1503.01735-1-51-3': 'In our previous studies we found an inverse dependence of spin lifetime on the mobility for both single-layer [CITATION] and bilayer [CITATION] graphene which is thus indicative for DP-like spin scattering.', '1503.01735-1-51-4': 'This dependence was only found in devices with contact resistance area products larger than [MATH] (figure [REF](a)).', '1503.01735-1-51-5': 'In contrast, all devices with [MATH] (figure [REF](a)) and devices where the contact resistances were enhanced by subsequent oxygen treatments (figure [REF](b)) do not show this 1/[MATH]-dependence.', '1503.01735-1-51-6': 'All these devices have been prepared by the conventional top-down method for which the MgO injection and detection barrier was directly deposited onto graphene.', '1503.01735-1-52-0': 'Most strikingly, the new generation of devices where we apply the bottom-up approach by transferring a hBN-graphene-stack on prepatterned electrodes (see figure [REF](e) and section [REF]) exhibit significantly enhanced charge and spin transport properties (see full symbols in figure [REF](c)).', '1503.01735-1-52-1': 'We attribute the increase in mobility to the hBN substrate while we relate the increase in spin lifetime to improved contact properties according to our advanced transfer technique which has several advantages over the previous fabrication methods.', '1503.01735-1-52-2': 'At first, the contact region has not been exposed to an electron beam which reduces the number of defects in graphene.', '1503.01735-1-52-3': 'Secondly, the interface between MgO and graphene is expected to be of higher quality yielding more homogeneous barriers which can be seen by the larger [MATH] values for devices with thinner MgO layer thicknesses [CITATION].', '1503.01735-1-53-0': 'Our results indicate that the overall improvement of the spin properties primarily result from the improvement of the contact properties suggesting that the observed 1/[MATH] dependence in the initial work is of extrinsic origin.', '1503.01735-1-53-1': 'In this context, we again want to emphasize (sections [REF] and [REF]) that the determination of both the charge carrier mobility and the spin lifetime is by no means unambiguous and may thus result in contradicting [MATH] dependencies.', '1503.01735-1-54-0': 'Furthermore, we demonstrated in [CITATION] and [CITATION] that the spin lifetimes in [CITATION] are most likely limited by contact properties (also see figure [REF]) in all devices.', '1503.01735-1-54-1': 'In fact, there are indications that even in our newest bottom-up devices the contacts may still limit spin transport properties (paper in preparation).', '1503.01735-1-54-2': 'If in current devices extrinsic parameters limit spin transport properties, it will be interesting to see how far more advanced fabrication methods will yield devices with even longer spin lifetimes and larger carrier mobilities which ultimately allows to unveil intrinsic spin scattering mechanisms.'} | {'1503.01735-2-0-0': 'In this review we discuss spin and charge transport properties in graphene-based single-layer and few-layer spin-valve devices.', '1503.01735-2-0-1': 'We give an overview of challenges and recent advances in the field of device fabrication and discuss two of our fabrication methods in more detail which result in distinctly different device performances.', '1503.01735-2-0-2': 'In the first class of devices, Co/MgO electrodes are directly deposited onto graphene which results in rough MgO-to-Co interfaces and favor the formation of conducting pinholes throughout the MgO layer.', '1503.01735-2-0-3': 'We show that the contact resistance area product ([MATH]) is a benchmark for spin transport properties as it scales with the measured spin lifetime in these devices indicating that contact-induced spin dephasing is the bottleneck for spin transport even in devices with large [MATH] values.', '1503.01735-2-0-4': 'In a second class of devices, Co/MgO electrodes are first patterned onto a silicon substrate.', '1503.01735-2-0-5': 'Subsequently, a graphene-hBN heterostructure is directly transferred onto these prepatterned electrodes which provides improved interface properties.', '1503.01735-2-0-6': 'This is seen by a strong enhancement of both charge and spin transport properties yielding charge carrier mobilities exceeding [MATH] and spin lifetimes up to 3.7 ns at room temperature.', '1503.01735-2-0-7': 'We discuss several shortcomings in the determination of both quantities which complicates the analysis of both extrinsic and intrinsic spin scattering mechanisms.', '1503.01735-2-0-8': 'Furthermore, we show that contacts can be the origin of a second charge neutrality point in gate dependent resistance measurements which is influenced by the quantum capacitance of the underlying graphene layer.', '1503.01735-2-1-0': '# Introduction', '1503.01735-2-2-0': 'In spin-based electronics, three main aspects have to be considered when exploring suitable spin transport materials and material combinations: (1) electrical injection and detection of spins, (2) their manipulation, and (3) the transport of spins in the material [CITATION].', '1503.01735-2-2-1': 'What makes graphene a promising material in the field of spintronics, is its unique spin transport performance in particular at room temperature [CITATION] where spin lifetimes of up to [MATH] [CITATION] and spin diffusion length of [MATH] [CITATION] have been measured by means of electrical Hanle spin precession measurements in non-local spin-valve devices.', '1503.01735-2-2-2': 'The corresponding charge carrier mobilities in these devices are above [MATH] [CITATION].', '1503.01735-2-2-3': 'Other interesting materials in the field of spintronics, e.g. Si, also exhibit nanosecond spin lifetimes at room temperature but fail short to graphene in respect to the spin diffusion lengths and charge carrier mobilities [CITATION].', '1503.01735-2-2-4': 'We note that we only compare device and material properties from electrical spin precession measurements at room temperature.', '1503.01735-2-2-5': 'Less invasive spin sensitive methods such as electron spin resonance or optical pump-probe methods can yield much longer spin lifetimes, especially at low temperatures [CITATION].', '1503.01735-2-3-0': 'Graphene has an extraordinary band structure [CITATION] and a weak intrinsic spin-orbit coupling at energies close to the Dirac point [CITATION].', '1503.01735-2-3-1': 'Often, this property is mentioned as a favorable aspect for graphene spintronics, because for the most prominent spin relaxation mechanisms the spin relaxation rate scales with the spin-orbit coupling strength [CITATION].', '1503.01735-2-3-2': 'Accordingly, initial calculations promised quite long spin lifetimes in pristine graphene flakes up to the ms regime [CITATION].', '1503.01735-2-3-3': 'But the experimental values are orders of magnitude smaller than these predictions and only exhibit spin lifetimes in the range of [MATH] to [MATH] at room temperature [CITATION].', '1503.01735-2-3-4': 'More elaborated calculations which included novel spin scattering mechanisms such as resonant scattering by magnetic impurities [CITATION] or entanglement between spin and pseudospin by random spin orbit coupling [CITATION] can explain these short spin lifetimes.', '1503.01735-2-3-5': 'Nevertheless, final answers about both the limiting spin relaxation mechanism and the maximal achievable spin lifetime in graphene are still missing.', '1503.01735-2-3-6': 'Considering the latter, it is interesting to mention that electron spin resonance (ESR) experiments in synthesized graphene flakes yield spin lifetimes of conduction electrons in the range of [MATH], despite of a significant defect density in the studied samples [CITATION].', '1503.01735-2-3-7': 'It has been suggested that the much longer spin lifetimes in ESR is due to the fact that the graphene sheets are free from substrate effects and metallic electrodes [CITATION].', '1503.01735-2-4-0': 'There is a large number of publication from different groups about electrical injection and detection of spins in graphene (e.g. [CITATION]).', '1503.01735-2-4-1': 'But although electric fields from back and/or top gates or biases along the graphene channel can strongly modify spin and charge transport properties, the actual spin precession in these experiments is always triggered by external magnetic fields.', '1503.01735-2-4-2': 'The reason for this is that the before-mentioned weak spin-orbit coupling in graphene is also a mixed blessing because for spin manipulation, e.g. via the Bychkov-Rashba effect, a strong spin-orbit coupling is needed [CITATION], which, on the other hand favors spin dephasing and spin scattering as it has been explored in III-V semiconductors [CITATION].', '1503.01735-2-4-3': 'Therefore, one emerging topic in graphene-based spintronic research is the partial functionalization of graphene with the goal to achieve other ways for spin manipulation, e.g. by electrostatic gating.', '1503.01735-2-4-4': 'The aim is to use high quality graphene parts as leads for efficient spin transport whereas the spin manipulation is realized in a functionalized part of the graphene device with enhanced spin-orbit coupling.', '1503.01735-2-4-5': 'Routes to increase the spin-orbit coupling include the spin-orbit proximity effect, in which additional materials such as the two dimensional transition metal dichalcogenides (e.g. tungsten disulfide [CITATION]) or ferromagnetic insulators (e.g. EuO [CITATION]) are put in direct contact to graphene.', '1503.01735-2-4-6': 'Also hydrogenation of graphene [CITATION] or the deposition of heavy adatoms [CITATION] can yield enhanced spin-orbit coupling in graphene, but as a drawback, these approaches are already known to strongly change its band structure [CITATION].', '1503.01735-2-5-0': 'Very recently, an excellent review article about graphene spintronics compiled a significant part of both theoretical and experimental work which has been carried out on spin phenomena in graphene [CITATION].', '1503.01735-2-5-1': 'In particular, this review focuses on how to measure spin transport in graphene, the spin-orbit coupling in pristine and modified graphene, magnetic moments from defects and adatoms, and the open question about which spin relaxation mechanisms limit spin transport in graphene.', '1503.01735-2-5-2': 'In our article, we expand the review to open questions and challenges in the experimental field of work.', '1503.01735-2-5-3': 'In section [REF] we discuss new routes in device fabrication by the usage of novel transfer techniques and outline the issue of device contaminations during fabrication.', '1503.01735-2-5-4': 'Then, we cover the influence of metallic contacts on the underlying graphene in section [REF] and present new results on the appearance of a contact-induced second charge neutrality point in gate dependent resistance measurements.', '1503.01735-2-5-5': 'Next, the reliability of extracted values for both spin lifetimes and charge carrier mobilities is discussed in sections [REF] and [REF], respectively.', '1503.01735-2-5-6': 'In section [REF], we briefly comment on some experimental studies which were used to investigate the relevant spin relaxation mechanism in graphene.', '1503.01735-2-6-0': '# Methods of device fabrication', '1503.01735-2-7-0': 'In recent years, it has been demonstrated that electrical injection and detection of spins in graphene can be accomplished by a variety of different electrode materials, such as Al[MATH]O[MATH]/Co [CITATION], MgO/Co [CITATION], submonolayer TiO[MATH]/MgO/Co [CITATION], TiO[MATH]/Co [CITATION], Cu/NiFe [CITATION], amorphous carbon/Co [CITATION], fluorinated graphene/NiFe [CITATION], hydrogenated graphene/NiFe [CITATION], PTCA/ALD Al[MATH]O[MATH]/NiFe [CITATION], only Co [CITATION], h-BN/NiFe [CITATION], h-BN/Co [CITATION], and YO/Co [CITATION].', '1503.01735-2-7-1': 'However, as mentioned in the introductory part, the measured spin lifetimes are only in the range of [MATH] to [MATH] at room temperature.', '1503.01735-2-7-2': 'This is well below the lifetimes of [MATH] measured by ESR experiments in graphene flakes without any contacts [CITATION].', '1503.01735-2-7-3': 'Next to these ESR experiments there are also other studies which indicates that an insufficient barrier quality can be the bottleneck for the overall spin transport [CITATION].', '1503.01735-2-7-4': 'In the following, we therefore address the challenges and summarize the progress in the field of device fabrication.', '1503.01735-2-8-0': 'First, we discuss several shortcomings when growing electrode material directly onto the graphene surface.', '1503.01735-2-8-1': 'Because often the starting point of a graphene-based spin transport device is a randomly exfoliated graphene flake which is typically deposited onto Si[MATH]/SiO[MATH].', '1503.01735-2-8-2': 'The next fabrication step is usually a lithography process on top of graphene which is needed for the deposition of ferromagnetic electrodes.', '1503.01735-2-8-3': 'But as soon as graphene gets into contact with an organic resist, it is extremely difficult to completely remove the contamination of hydrocarbons.', '1503.01735-2-8-4': 'This is illustrated in the atomic force microscopy (AFM) images of figures [REF](a)-(c), which show one of our earlier but not optimized dosage tests.', '1503.01735-2-8-5': 'As a resist, we used PMMA (950K) which was dissolved in ethyl lactate and n-butyl acetate with a thickness of 250 nm after spin-coating and baking.', '1503.01735-2-8-6': 'For developing we used isopropyl alcohol and methyl isobutyl ketone with a developing time of 105 s. E-beam writing was performed with an acceleration voltage of 10 kV.', '1503.01735-2-8-7': 'The dosages in figures [REF](a)-(c) are given in fractions of [MATH].', '1503.01735-2-8-8': 'The scale in figure [REF](a) was chosen to depicted the flake in high contrast.', '1503.01735-2-8-9': 'The resist is underdeveloped for a dosage below [MATH].', '1503.01735-2-8-10': 'But also for dosages larger than [MATH] an increasing contamination of the graphene flake can be observed, which can be easier be visualized in the phase signal of the AFM image in tipping mode (figure [REF](b)).', '1503.01735-2-8-11': 'The difference in contrast can be understood by different attenuations of the AFM cantilever for SiO[MATH], graphene on SiO[MATH], and PMMA on graphene.', '1503.01735-2-8-12': 'The increasing contamination for higher dosages can be explained by crosslinking of PMMA [CITATION] or e-beam-induced defects in the graphene flake [CITATION] which enables a stronger binding with hydrocarbons.', '1503.01735-2-8-13': 'We observe the cleanest graphene surface at a dosage of [MATH] and were able to reduce the amount of leftover resist residues even further by increasing the developing time to 210 s.', '1503.01735-2-9-0': 'But even for optimized lithography conditions the graphene flake will be contaminated with hydrocarbons on an atomic scale.', '1503.01735-2-9-1': 'Such contaminations are clearly seen by transmission electron microscopy and annealing temperatures of up to [MATH] are needed to remove the hydrocarbons [CITATION].', '1503.01735-2-9-2': 'But such high temperatures are far beyond the glass transition temperature of the resist and therefore will destroy the patterning mask.', '1503.01735-2-9-3': 'Next to thermal annealing also other methods are applied to remove resist residues after the development step of the resist, e.g. low-density inductively coupled Ar plasma [CITATION], [MATH] cluster cleaning [CITATION], or ultraviolet-ozone treatment [CITATION].', '1503.01735-2-9-4': 'But it still has to be seen if these cleaning methods are able to remove even the last atomic layer of hydrocarbons from the graphene flake.', '1503.01735-2-9-5': 'While in most of these publications only rather macroscopic analysis methods such as Raman spectroscopy or even qualitative electrical measurements are used, only the high-resolution X-ray photoelectron spectroscopy in case of the ultraviolet-ozone treatment of reference [CITATION] may really demonstrate the complete removal of all contaminations.', '1503.01735-2-9-6': 'There is, however, also a drawback to this method as long treatments can also create defects in the graphene flake.', '1503.01735-2-10-0': 'The next issue arises because of the chemically inert nature of graphene and its [MATH] hybridization, which leads to unfavorable growth conditions of many materials on top of graphene.', '1503.01735-2-10-1': 'This can already be seen in the stronger accumulation of the PMMA on the graphene flake compared to the area of SiO[MATH] for dosages lower than [MATH] in figure [REF](b) (in this figure the scale is optimized to the height of the underdeveloped PMMA).', '1503.01735-2-10-2': 'The reason for this accumulation is the low wettability of graphene [CITATION] and again it is transmission electron microscopy which reveals the clustering of almost every metal deposited on graphene [CITATION].', '1503.01735-2-10-3': 'The interaction between metals and graphene is so weak, that in sub-monolayer growth of metal layers on graphene, the metal atoms solely resides on the hydrocarbon contamination and not on the clean graphene parts [CITATION].', '1503.01735-2-10-4': 'For example, the direct growth of MgO on graphene is governed by the Volmer-Weber island growth mechanism [CITATION] resulting in inhomogeneous oxide layers with presumably pinholes [CITATION].', '1503.01735-2-10-5': 'This is illustrated in the AFM image of a 3 nm thick MgO layer grown on top of graphene in figure [REF](d), which exhibits peak-to-peak values of up to 2 nm.', '1503.01735-2-10-6': 'The use of a Ti/TiO[MATH] wetting layer yields more homogeneous MgO layers [CITATION] but the impact of Ti on the properties of graphene is still an open question [CITATION].', '1503.01735-2-10-7': 'For Al[MATH]O[MATH] the evaporation of Al and its subsequent oxidation to Al[MATH]O[MATH] can also yield rough layers with pinholes [CITATION].', '1503.01735-2-10-8': 'On the other hand, there is a report on homogeneous and pinhole free Al[MATH]O[MATH] oxide barriers on graphene by argon sputter deposition [CITATION].', '1503.01735-2-10-9': 'But the same group also demonstrated by Raman spectroscopy that this technique may also induces defects in graphene [CITATION].', '1503.01735-2-10-10': 'By now, there are only few studies reporting epitaxial-like growth of materials on graphene, e.g. EuO [CITATION] or Ti [CITATION].', '1503.01735-2-11-0': 'The hydrocarbon contaminations from the lithography step and the clustering of many materials on top of graphene are important issues for graphene-based spin transport devices because both can reduce the quality of the insulating oxide barrier between graphene and the ferromagnetic electrodes.', '1503.01735-2-11-1': 'But as mentioned above, an insufficient barrier quality can be the bottleneck for the overall spin transport [CITATION].', '1503.01735-2-11-2': 'A possible way to circumvent some of these problems was paved by the introduction of mechanical transfer techniques for 2d materials (see e.g. [CITATION]).', '1503.01735-2-11-3': 'These transfer techniques provide a controlled deposition of flakes with high spatial precision, which results in two important advantages: (1) The device fabrication becomes now independent of the random position of exfoliated flakes on a substrate and (2) the possibility to deposit different 2d materials on top of each other in order to build so-called van der Waals heterostructures (review about these heterostructures in reference [CITATION]).', '1503.01735-2-12-0': 'It has been shown that these new transfer techniques can be used to overcome the problem of direct growth of electrode material onto graphene.', '1503.01735-2-12-1': 'For this the electrode structure is fabricated by means of e-beam lithography and metallization in a first step.', '1503.01735-2-12-2': 'Only afterwards the graphene flake is deposited on top of the prepatterned structure (figure [REF](e)) [CITATION].', '1503.01735-2-12-3': 'Therefore, the graphene flake is not exposed to the e-beam lithography step, hence resist residues and e-beam induced defects can be avoided.', '1503.01735-2-12-4': 'Furthermore, the growth conditions of the ferromagnetic electrodes and the oxide barrier are far more suitable and can be adapted to an adequate substrate.', '1503.01735-2-12-5': 'Especially, the whole expertise already achieved in the fabrication of high quality magnetic tunnel junction devices (e.g. [CITATION]) may now also be applied to graphene spin transport devices.', '1503.01735-2-12-6': 'In this respect, we especially refer to the improvement of magnetic tunnel junctions by annealing which results in crystallization of the interface between ferromagnetic metals and oxide barriers [CITATION].', '1503.01735-2-12-7': 'The approach to first fabricate spin injection and detection electrodes on a wafer and then to deposit a stack of graphene on hexagonal boron nitride on top of it was e.g. applied in reference [CITATION], where spin lifetimes of 3.7 ns were measured at room temperature in trilayer graphene, which is the longest room temperature spin lifetime reported so far.', '1503.01735-2-12-8': 'Nevertheless, there might still be some issues with this new fabrication process because the insulating barrier between graphene and ferromagnetic electrode is exposed to air at some point during the process.', '1503.01735-2-12-9': 'It is well known that many oxides [CITATION] and particularly MgO [CITATION] are hygroscopic.', '1503.01735-2-12-10': 'Therefore, it cannot be excluded that a partial hydroxylation of the MgO barrier may still limit the barrier quality in reference [CITATION].', '1503.01735-2-13-0': 'As already mentioned, the possibility to deposit and stack various 2d materials on top of each other in order to fabricate van der Waals heterostructures is the second advantage of the transfer techniques.', '1503.01735-2-13-1': 'So far, many possible 2d materials beyond graphene have emerged (overview e.g. in [CITATION]) and combining these 2d materials with graphene has expanded the field of graphene research dramatically.', '1503.01735-2-13-2': 'One of the first applications was the fabrication of stacks consisting of hexagonal boron nitride (hBN) and graphene.', '1503.01735-2-13-3': 'Due to the atomically smooth surface and similar lattice constant of hBN to graphene, these devices exhibit significantly improved electrical properties compared to graphene deposited on SiO[MATH] [CITATION].', '1503.01735-2-13-4': 'The incorporation of such hBN-graphene stacks in graphene spintronic devices led to a significant enhancement of the measured spin diffusion lengths as well as the charge carrier mobility [CITATION].', '1503.01735-2-13-5': 'Another application of hBN which became feasible with the transfer techniques is its usage as an oxygen-free injection barrier when placed between graphene and the ferromagnetic electrodes [CITATION].', '1503.01735-2-13-6': 'By transferring exfoliated hBN flakes, the above-mentioned problems in oxide barrier growth may be avoided.', '1503.01735-2-14-0': 'Finally, the transfer techniques offer a new route to functionalize graphene by putting different 2d materials in direct contact to graphene.', '1503.01735-2-14-1': 'One important application of such a heterostructure is the enhancement of the spin-orbit coupling (e.g. by the spin-orbit proximity effect of tungsten disulfide on graphene [CITATION]) to allow for spin manipulation via the Bychkov-Rashba effect.', '1503.01735-2-14-2': 'We emphasize that many effects like the spin-orbit proximity effect greatly rely on contamination-free interfaces.', '1503.01735-2-14-3': 'However, depending on the exact kind of transfer technique different amounts of hydrocarbon contamination between the layers of the heterostructures can be observed [CITATION].', '1503.01735-2-14-4': 'Interestingly, there seems to be "self-cleansing" effects in some heterostructures (e.g. graphene on hBN, MoS[MATH], and WS[MATH]), in which the surface contamination of the 2d materials automatically aggregates into bubbles, leaving behind rather clean interfaces throughout [MATH]m-sized graphene/hBN areas [CITATION].', '1503.01735-2-15-0': '# Impact of metals on graphene', '1503.01735-2-16-0': 'In this section, we discuss the influence of metallic electrodes on both spin and charge properties of the underlying graphene layer.', '1503.01735-2-16-1': 'One important tool to investigate the electronic properties is angle-resolved photoemission spectroscopy (ARPES), which can directly probe the band structure of graphene which is in direct contact to the metal (see e.g. references [CITATION] for ARPES measurements of graphene in contact with Au, Ag, Fe, Ni, Co, Al, and Cu).', '1503.01735-2-16-2': 'Of special interest in the area of graphene spintronics are the 3d transition metals Fe, Ni, and Co, which are used as ferromagnetic electrodes to generate spin polarized currents.', '1503.01735-2-16-3': "ARPES measurements have revealed that all three metals significantly alter the band structure of graphene: The Dirac point of graphene shifts around 2.8 eV below the Fermi energy and the graphene's [MATH] band hybridizes with 3d bands of Co near the Fermi level [CITATION].", '1503.01735-2-17-0': 'As a results, as soon as the ferromagnetic electrode gets into direct contact with graphene (which can be the case for pinholes in the separating oxide layer), the spin current is initially, i.e. right after electrical spin injection, no longer carried by pure graphene states near the Dirac point but rather by the hybridized states near the Fermi level.', '1503.01735-2-17-1': 'A significant higher spin scattering rate in such states in comparison to the states near the Dirac point of unmodified graphene may explain the overall short spin lifetime measured in devices with ohmic contacts exhibiting small contact resistance area products [MATH] [CITATION] (see also figure [REF]).', '1503.01735-2-17-2': 'Furthermore, the large density of states at the Fermi level due to hybridization results in a pinning of the Fermi level, which is equivalent to a screening of a gate electric field underneath the electrodes [CITATION].', '1503.01735-2-17-3': 'Considering both the n-doping and the screening underneath the contacts caused by the Co, the gate voltage dependent doping profiles of figure [REF](c) can be deduced for ohmic contacts, which in turn can explain the gate dependent resistance curve of such a device in figure [REF](a).', '1503.01735-2-18-0': 'A sufficiently thick oxide barrier without pinholes, which is preferable for spin transport devices, is expected to suppress hybridization between the metal and graphene.', '1503.01735-2-18-1': 'Nevertheless, a field effect doping of graphene in such metal/dielectric/graphene heterostructures is still expected because of different respective work functions [CITATION].', '1503.01735-2-18-2': 'The thicker the insulating oxide barrier between graphene and metallic electrode (which results in larger [MATH] products) the lower the corresponding doping [CITATION].', '1503.01735-2-18-3': 'At some point the doping is small enough to be also tuned by the back gate voltage (see corresponding doping profile in figure [REF](d)).', '1503.01735-2-18-4': 'The different doping of graphene in between and underneath the electrodes can be seen by two distinct charge neutrality points (CNP) in gate dependent resistance measurements (figure [REF](b)): the right peak (blue arrow) corresponds to the graphene part between the electrodes (CNP mostly around a gate voltage of [MATH] for untreated devices or at positive gate voltages because of p-doping during oxygen treatments, see reference [CITATION]) while the left one (red arrow) results from the graphene underneath the electrodes (CNP at large negative gate voltages because of the n-doping of Co).', '1503.01735-2-18-5': 'For our Co/MgO/graphene devices, the contact-induced CNP at negative gate voltage (see red arrow in figure [REF](b)) typically appears for [MATH] [CITATION], which is also the value at which the [MATH]-[MATH]-curves of the contacts exhibit non-linear behavior [CITATION].', '1503.01735-2-18-6': 'At this point, we want to emphasize that only a sufficiently thick insulating barrier diminishes the metal-induced doping density of graphene strong enough that the contact-induced 2nd charge neutrality point becomes accessible by large back gate voltages.', '1503.01735-2-18-7': 'The required thickness depends on the used contact materials [CITATION].', '1503.01735-2-18-8': 'Therefore, the value of [MATH] is most likely only holds for our Co/MgO/graphene devices and may strongly differ for other material combinations.', '1503.01735-2-19-0': 'In the following, we discuss the origin of the contact-induced second CNP in more detail.', '1503.01735-2-19-1': "Although this review article is about spin transport, we consider this discussion as crucial as it highlights two important aspects which are often neglected: (1) The significant modification of graphene's transport properties right underneath the contacts and (2) the existence of lateral pn-junctions which the spin polarized charge carriers have to overcome.", '1503.01735-2-20-0': 'To confirm that the left CNP in our devices is indeed contact-induced, we now focus on a non-local spin-valve device with Co/MgO electrodes in more detail.', '1503.01735-2-20-1': 'This device exhibit a large electrode-to-electrode variation of the respective [MATH] values.', '1503.01735-2-20-2': 'Figure [REF](a) depicts the schematic layout.', '1503.01735-2-20-3': 'The measured contact resistance area products of the inner contacts 2 to 4 are: [MATH], [MATH], and [MATH].', '1503.01735-2-20-4': 'According to our previous studies [CITATION], we now assume that the interaction between Co and graphene is the dominating source of doping, whereas doping from fabrication-induced contaminations (like resist residues) or from the MgO itself are subordinated.', '1503.01735-2-20-5': 'Following the above explanation that thicker barriers with larger [MATH] values diminishes Co-induced doping of graphene [CITATION], we can now assign larger doping levels to lower [MATH] values.', '1503.01735-2-20-6': 'This yields the potential profile of figure [REF](a) for which we assume that the graphene between the contacts is hole doped by an applied gate voltage.', '1503.01735-2-20-7': "The depicted black curve corresponds to the position of graphene's CNP relative to the Fermi level [MATH] (also see depicted Dirac cone).", '1503.01735-2-20-8': 'Gate dependent 4-terminal resistance measurements were performed as a function of dc-current.', '1503.01735-2-20-9': 'For this, we apply a total current of [MATH] over the outermost electrodes 1 and 5 and measure the voltage drop between contacts 2 and 3 for region A and contacts 3 and 4 for region B by standard low frequency lock-in technique.', '1503.01735-2-21-0': 'As it can be seen for both regions in figures [REF](b) and (d) both the position and magnitude of the left CNP significantly depends on the applied dc-current whereas for the right CNP of the bare graphene there is only a small decrease in resistance for both positive and negative dc current which can be explained by the larger local temperatures from current-induced Joule heating.', '1503.01735-2-21-1': 'Figure [REF](c) depicts the dependence of the gate voltage position of the contact-induced left CNP as a function of dc-current.', '1503.01735-2-21-2': 'We show this data not only for regions A and B but also between contacts 2 and 4, which we call region AB.', '1503.01735-2-21-3': 'Interestingly, the gate voltage shift of the CNP has different sign and amplitude in regions A and B. Both, the different sign and amplitude can also be seen in the doping profile of figure [REF](a) (indicated by the arrows).', '1503.01735-2-21-4': 'Consistently, the dc-current dependent shift of the contact-induced CNP measured between contacts 2 and 4 (green curve in figure [REF](c)) is in good agreement the arithmetic mean of the respective curves of regions A and B (blue curve in figure [REF](c)).', '1503.01735-2-22-0': 'With the help of theoretical considerations about graphene field effect transistors [CITATION] we are able to qualitatively understand the gate voltage shift of the contact-induced CNP.', '1503.01735-2-22-1': 'For this we assume that high quality tunnel barriers induce a field effect similar to top gates, which changes the Fermi level in graphene underneath the contacts.', '1503.01735-2-22-2': 'In fact, a reverse argumentation might be more intuitive (we refer to reference [CITATION]): Because of the different work functions between graphene and a metal a charge transfer will occur during the alignment of the Fermi levels.', '1503.01735-2-22-3': 'The charge transfer yields a carrier doping of graphene, but it also creates a voltage drop and therefore a local electric field over the separating dielectric MgO barrier as the barrier acts as a capacitor with the graphene and the metallic electrodes as its plates.', '1503.01735-2-22-4': 'These voltages [MATH] (TG as in top gate) are depicted in the schematic layout of a graphene field effect transistor in figure [REF](a).', '1503.01735-2-22-5': 'In the following, we assume that these voltages remain unaffected if the two inner contacts in figure [REF](a) are used as voltage probes while the outer ones are source and drain contacts (this wiring applies to the measurement of figure [REF]).', '1503.01735-2-23-0': 'Next, we determine the charge carrier density in the graphene flake.', '1503.01735-2-23-1': 'For the graphene underneath an inner electrode (marked by "X" in figure [REF](a)) we can apply the equivalent circuit in figure [REF](b).', '1503.01735-2-23-2': 'The MgO layer, which acts as a barrier for spin injection and spin detection, is approximated to be a top gate with capacitance [MATH], whereas the back gate voltage [MATH] is applied over the capacitance [MATH], which represents the SiO[MATH] layer.', '1503.01735-2-23-3': 'Far more complicated is the incorporation of graphene in this picture.', '1503.01735-2-23-4': 'For this we first discuss the Fermi level shift by applying a bias voltage over the graphene flake.', '1503.01735-2-23-5': 'In figure [REF](c) this effect is illustrated in a very simplified system, in which e.g. the doping effects of the contacts are neglected.', '1503.01735-2-23-6': 'We first consider ideal, undoped graphene, in which case the Fermi level of the graphene underneath the grounded contact (left contact in figure [REF](c)) is at the Dirac point.', '1503.01735-2-23-7': 'A bias voltage applied to the right contact results in a lateral voltage gradient along the graphene channel with corresponding shifts of the Fermi level [CITATION].', '1503.01735-2-23-8': "Along the whole graphene flake a gradual nn'-junction is generated, which can be verified by photocurrent measurements [CITATION].", '1503.01735-2-24-0': 'The Fermi level shift underneath the contacts with the applied source-drain-voltage [MATH] can be written as [MATH] where the energy scale is set to zero at the CNP of graphene.', '1503.01735-2-24-1': 'In this simplified system the voltage drop [MATH] along the graphene channel only depends on the electrostatic potential by the applied bias.', '1503.01735-2-24-2': 'But in general, the Fermi level also depends on the electrochemical potential.', '1503.01735-2-24-3': 'Accordingly, the position of the Fermi level [MATH] also depends on other factors such as the applied gate voltage or changes in the electrochemical potential by adsorbates.', '1503.01735-2-24-4': 'This is also illustrated in figure [REF](c) in the case that graphene is p-doped by an applied gate voltage or adsorbates.', '1503.01735-2-24-5': 'Apparently, the local area with vanishing charge carrier density, i.e. where the Fermi level is at the Dirac point, moves along the graphene channel (from left to right in the lower panel of figure [REF](c)) with increasing p-doping concentration.', '1503.01735-2-24-6': 'Such a shift of the CNP by an applied gate voltage is e.g. observed in photocurrent and thermal infrared microscopy [CITATION].', '1503.01735-2-25-0': 'The capacitance of the graphene part in the equivalent circuit over which the voltage [MATH] drops is the quantum capacitance [MATH] of graphene [CITATION].', '1503.01735-2-25-1': 'The quantum capacitance directly results from the Pauli principle and becomes relevant in materials with a small density of states (DOS) near the Fermi level.', '1503.01735-2-25-2': 'To explain the quantum capacitance, we first consider ideal graphene with the Fermi level at the Dirac point, where the DOS vanishes (figure [REF](d), left band structure).', '1503.01735-2-25-3': 'If now the charge carrier density in graphene is changed by [MATH] the Fermi level shifts by [MATH] as the electrons have to occupy higher energy states.', '1503.01735-2-25-4': 'For single layer graphene, the Fermi level [MATH] and the DOS at the Fermi level [MATH] are given by [CITATION]: [EQUATION] where [MATH] is the Fermi velocity.', '1503.01735-2-25-5': 'If now the charge carrier density is doubled by adding the same amount of charges [MATH] a second time, the increase of the Fermi level becomes less strong because of the previous increase in DOS ([MATH]) (see figure [REF](d)).', '1503.01735-2-25-6': 'As mentioned above, the electrons at the Fermi level have the energy [MATH].', '1503.01735-2-25-7': 'Combining this expression with equation [REF] and the general definition of a capacitance [MATH] (with the charge [MATH] and area [MATH]) results in the quantum capacitance of graphene: [EQUATION]', '1503.01735-2-25-8': 'In figure [REF](e) we plot this quantum capacitance of graphene as a function of the applied gate voltage (black solid line).', '1503.01735-2-25-9': 'Due to broadening of the DOS by both thermal excitation at room temperature and by structural inhomogeneities the experimentally determined graphene quantum capacitance is also broadened [CITATION] which is depicted as a black dashed line.', '1503.01735-2-25-10': 'Furthermore, we include the capacitance of both oxide layers (the MgO of the top electrode with thicknesses of 2 and 3 nm and the SiO[MATH] of the back gate).', '1503.01735-2-26-0': 'Now we go back to the equivalent circuit in figure [REF](b).', '1503.01735-2-26-1': 'First, we start with the graphene part between the contacts (marked by a circle in figure [REF](a)).', '1503.01735-2-26-2': 'As there is no top gate capacitance in this region, back gate and quantum capacitances are put in series and give a total capacitance [MATH].', '1503.01735-2-26-3': 'As the back gate capacitance is much smaller than the quantum capacitance (compare to figure [REF](e)), it dominates the total capacitance: [MATH].', '1503.01735-2-26-4': 'Accordingly, the quantum capacitance does not play any significant role in the back gate induced field effect (see also [CITATION]).', '1503.01735-2-26-5': 'This is an important conclusion as the bias voltage can only change the charge carrier density at the node of the equivalent circuit in figure [REF](b) with help of the quantum capacitance.', '1503.01735-2-26-6': 'The fact that the quantum capacitance can be neglected now explains why the CNP of the bare graphene part between the contacts does not shift with dc currents (figures [REF](b) and (d)).', '1503.01735-2-26-7': 'In contrast, the quantum capacitance plays a significant role for graphene parts underneath the electrodes as it is in the same order of magnitude as the electrode capacitance (figure [REF](e)).', '1503.01735-2-26-8': 'If we assume a fixed charge carrier density at the node of the equivalent circuit of figure [REF](b) for the contact region, now the shift of the contact-induced CNP in figures [REF](b) and (d) becomes clear: By changing the dc current (or accordingly the bias voltage) the voltage [MATH] will change according to the discussion in figure [REF](c).', '1503.01735-2-26-9': 'If we want to keep the charge carrier density at the node of the equivalent circuit at [MATH] (i.e. to the contact-induced CNP) the voltages [MATH] (i.e. the actual doping of the graphene by the contacts) and [MATH] also have to change.', '1503.01735-2-26-10': 'To simulate this, all changes of [MATH], [MATH], and [MATH] have to be calculated self-consistently (e.g. a change in the back gate voltage will shift the Fermi level and therefore the value of the graphene quantum capacitance), which goes beyond the scope of this paper.', '1503.01735-2-26-11': 'For a far more detailed description we refer to the references [CITATION].', '1503.01735-2-27-0': 'Finally, we again point to the fact that the different charge transport properties of graphene underneath the contacts and graphene in between the contacts are so far widely neglected.', '1503.01735-2-27-1': 'It has to be seen how the change in the doping underneath the contacts with different dc-biases or the pn-junctions along the graphene channel have an impact on the performance of graphene-based spintronic devices [CITATION].', '1503.01735-2-28-0': '# Determination of spin lifetimes', '1503.01735-2-29-0': 'In this section we discuss on how to extract the spin lifetime out of spin precession measurements and address how several limitations to this approach have recently been recognized.', '1503.01735-2-29-1': 'We restrict ourselves to non-local spin transport measurements, so we can neglect the drift term in the Bloch-Torrey equation and only consider changes in the net spin vector [MATH] by spin precession about a perpendicular magnetic field [MATH] (see measurement configuration in figure [REF](a)), spin diffusion, and spin dephasing and relaxation [CITATION]: [EQUATION]', '1503.01735-2-29-2': 'Here [MATH] is the Larmor frequency, [MATH] is the gyromagnetic ratio, [MATH] the spin diffusion constant, and [MATH] the spin lifetime.', '1503.01735-2-30-0': 'The advantage of non-local measurements is that the spin signal can be significantly decoupled from spurious charge signals [CITATION].', '1503.01735-2-30-1': 'To achieve such a non-local configuration, dc measurements or lock-in techniques with low frequencies are needed, because electric pulses in case of RF measurements propagate through the whole device, which means that no non-local part can exist.', '1503.01735-2-30-2': 'The dc or low frequency measurements lead to a stationary or quasi-stationary condition and therefore the time derivative of equation [REF] is set to zero.', '1503.01735-2-30-3': 'Hence a solution to [EQUATION] has to be found to extract the spin lifetime by means of a fit to the dc spin precession curve (so-called Hanle curve).', '1503.01735-2-30-4': 'Typical Hanle curves for a spin transport device are shown in figure [REF](b) with both parallel and antiparallel alignments of the inner Co electrodes in figure [REF](a).', '1503.01735-2-30-5': 'The non-local spin resistance [MATH] can be determined at [MATH].', '1503.01735-2-30-6': 'As expected, both Hanle curves merge at larger magnetic fields.', '1503.01735-2-30-7': 'However, they do not become constant but rather increase above [MATH] T.', '1503.01735-2-30-8': 'We note that this magnetic field dependent background signal is typical for most studies and it can even be much more pronounced.', '1503.01735-2-30-9': 'One explanation for this background is that the magnetization of the ferromagnetic electrodes can rotate out-of-plane with increasing perpendicular magnetic field [CITATION].', '1503.01735-2-30-10': 'In principle, this rotation of the magnetization can account for a background signal which is symmetric in magnetic field.', '1503.01735-2-30-11': 'But we often observe an antisymmetric background signal with a linear term in [MATH], which also dependents on the charge carrier density and the wiring of the device [CITATION].', '1503.01735-2-30-12': "These findings cannot be explained by the rotation of the electrode's magnetization alone.", '1503.01735-2-30-13': 'We have recently shown that such a background of second polynomial order can be caused by an inhomogeneous current flow through the oxide barriers.', '1503.01735-2-30-14': 'As a result there is a charge accumulation signal next to the actual spin accumulation signal in the non-local voltage which can be explained by a redistribution of charge carriers by a perpendicular magnetic field similar to the classical Hall effect [CITATION].', '1503.01735-2-31-0': 'One fundamental drawback of every solution to equation [REF] is that the three parameters [MATH], [MATH], and [MATH] cannot be determined independently.', '1503.01735-2-31-1': 'This is because of the steady state condition, which allows the multiplication of equation [REF] by any factor [MATH] without changing the overall result.', '1503.01735-2-31-2': 'Therefore a fit using a solution to equation [REF] is invariant with respect to the transition ([MATH].', '1503.01735-2-31-3': 'Hence one of the three parameters has to be assumed or determined by other measurements.', '1503.01735-2-32-0': 'Often [MATH] is assumed for the analysis of spin precession measurements.', '1503.01735-2-32-1': 'For pristine or moderately modified graphene flakes this assumption is confirmed by electron spin resonance (ESR) measurements [CITATION].', '1503.01735-2-32-2': 'It remains to be seen if this value also holds for functionalized graphene.', '1503.01735-2-32-3': 'By now, spin precession measurements of hydrogenated graphene suggest larger [MATH]-factors [CITATION].', '1503.01735-2-32-4': 'Of course, the most elegant way to circumvent this uncertainty is to give up on the non-local, steady state condition and apply time-resolved RF measurements which offer the possibility to directly determine the [MATH]-factor from time dependent spin precession.', '1503.01735-2-33-0': 'In the following, we discuss some assumptions and simplifications, which have to be made in order to find useable solutions to equation [REF].', '1503.01735-2-33-1': 'One of the first and still widely used analytical solution was developed by Johnson and Silsbee [CITATION].', '1503.01735-2-33-2': 'But in recent years it was found that the measured spin lifetime of a graphene device scales with the [MATH] product at least up to values of several tens of k[MATH]m[MATH] (see figure [REF]) [CITATION]) suggesting that the measured spin lifetime is not the intrinsic spin lifetime of graphene but rather has extrinsic origin.', '1503.01735-2-33-3': 'One route to explain this dependence is that the measured spin lifetime can be significantly underestimated if spin relaxation by the contacts is not included in the fit model and hence some work was done in the direction of more elaborated models [CITATION].', '1503.01735-2-33-4': 'But even these models make two crucial assumptions: (1) The injection barriers are homogeneous and can be characterized by the [MATH] value only and (2) the spin lifetime in graphene underneath the contacts is the same as the spin lifetime in the bare graphene part between the contacts.', '1503.01735-2-34-0': 'As it was already discussed in section [REF] the growth of homogeneous oxide barriers on graphene is quite challenging.', '1503.01735-2-34-1': 'But as long as the contact is not spatially homogenous, the measured contact resistance can only be an averaged value.', '1503.01735-2-34-2': 'Especially if there are pinholes within the barrier the contact resistance area product cannot precisely be determined, because the exact number and sizes of the pinholes cannot be deduced from electrical measurements.', '1503.01735-2-34-3': 'Therefore, the uncertainty in the contact resistance will also lead to an uncertainty in the extracted spin lifetimes by the aforementioned models.', '1503.01735-2-35-0': 'And also the second assumption of a single spin lifetime seems to be oversimplified considering section [REF].', '1503.01735-2-35-1': 'As it was demonstrated by ARPES measurements [CITATION] the interaction between graphene and ferromagnetic metals can significantly change the band structure of graphene due to hybridization.', '1503.01735-2-35-2': 'The hybridized states are directly at the Fermi level and therefore both charge and spin transport will occur through these states, which may have a strong impact on the spin lifetime.', '1503.01735-2-35-3': 'But even if we assume a sufficiently thick oxide barrier, which inhibits this hybridization, the assumption of a single spin lifetime may still be too simple if the spin lifetime depends on the Fermi level or, correspondingly, the charge carrier density.', '1503.01735-2-35-4': 'Because the field effect doping of the contacts [CITATION] can e.g. lead to the case at which the graphene underneath the contacts is highly n-doped whereas the graphene between the contacts is at its CNP.', '1503.01735-2-35-5': 'By now, only some research was done on the topic how two spatially different graphene parts with different spin relaxation times influence the overall measured spin lifetime (e.g. in reference [CITATION]).', '1503.01735-2-36-0': 'Finally, we note that a rough MgO layer like the one in figure [REF](d) not only can cause pinholes but also leads to an equally rough surface of the Co electrode which is deposited on top.', '1503.01735-2-36-1': 'But such a rough ferromagnetic interface can yield stray fields, which may result in a broadening of the spin precession curve and thereby an apparent reduction of the extracted spin lifetime [CITATION].', '1503.01735-2-37-0': '# Determination of charge carrier mobilities', '1503.01735-2-38-0': 'In this section we discuss the determination of the charge carrier mobility [MATH] from gate dependent resistance measurements.', '1503.01735-2-38-1': 'There are two reasons making this an important topic for graphene-based spin transport studies.', '1503.01735-2-38-2': 'Firstly, the dependence of the spin lifetime and the charge carrier mobility is often used to identify the dominating spin relaxation mechanism [CITATION].', '1503.01735-2-38-3': 'Secondly, the use of oxide barriers for spin injection and detection can lead to the appearance of the contact-induced CNP as it has been discussed in section [REF].', '1503.01735-2-38-4': 'We will demonstrate that this 2nd CNP has significant impact on the determination of the charge carrier mobility.', '1503.01735-2-39-0': 'Firstly, we analyze the electron mobility in one of our devices by different methods which are currently used in literature showing that the determination of the mobility is not unambiguous and yield values which vary by more than a factor of 2.', '1503.01735-2-39-1': 'The simplest model considers a graphene field effect transistor, for which a linear dependence between the charge carrier density [MATH] (or accordingly back gate voltage) and the conductivity [MATH] is often observed (see e.g. [CITATION]).', '1503.01735-2-39-2': 'For this case several groups (e.g. [CITATION]) assume a simple Drude model and define the Drude mobility as: [EQUATION]', '1503.01735-2-39-3': 'We emphasize that this Drude mobility is extracted at a single point of the conductivity curve.', '1503.01735-2-39-4': 'To account for the residual conductivity [MATH] even in the case of [MATH] the conductivity can be written as [MATH].', '1503.01735-2-39-5': 'Starting from this equation the extracted field effect mobility [MATH] from fitting the slope of the conductivity curves is given by (e.g. in references [CITATION]): [EQUATION]', '1503.01735-2-39-6': 'We applied both methods to a conductivity curve from one of our actual spin valve devices in figure [REF](b).', '1503.01735-2-39-7': 'The red circle marks the point at which [MATH] and [MATH] have been determined.', '1503.01735-2-39-8': 'Remarkably, [MATH] exceeds [MATH] by almost 60[MATH].', '1503.01735-2-39-9': 'In the as-fabricated state of this device the conductivity drops only slightly at large negative gate voltages, which indicates that the contact induced CNP is close but nevertheless out of range of the applicable gate voltage.', '1503.01735-2-39-10': 'After an oxygen treatment which primarily reduces the doping effect of the electrodes [CITATION] the contact-induced CNP shifts into the accessible gate voltage range (figure [REF](e)) and furthermore complicates the analysis of the carrier mobilities.', '1503.01735-2-40-0': 'This can be seen in the resistance curve of another spin valve device in figure [REF](a) that also shows two distinct CNPs.', '1503.01735-2-40-1': 'We can decompose resistance contributions of both graphene regions (underneath and in between the contacts; model for fitting is discussed further below), which is shown by red and blue dashed lines, respectively.', '1503.01735-2-40-2': 'The overlapping of both curves clearly demonstrates one major issue of the two mobility fits discussed so far: As soon as the doping underneath the contact approaches the one of the bare graphene part, both the absolute value and the slope of the measured gate dependent resistance curve significantly changes in the overlapping part.', '1503.01735-2-40-3': 'And so does the extracted mobility.', '1503.01735-2-40-4': 'The reason for this is quite clear: Both equation [REF] and [REF] are based on one well-defined transport regime, whereas the contact-induced CNP points to the fact that the charge transport in graphene underneath and in between the contacts can differ significantly.', '1503.01735-2-41-0': 'But before we discuss more detailed fitting models, which include different transport regimes, we first want to discuss another issue when using equation [REF] which arises as soon as the conductivity curves are not linear as the ones in figure [REF].', '1503.01735-2-41-1': 'In this case, the extracted mobility [MATH], which is determined by the slope, depend on the charge carrier density [MATH].', '1503.01735-2-41-2': 'But this is a mathematical contradiction which is often neglected.', '1503.01735-2-41-3': 'The derivative of [MATH] with respect to [MATH] is now given by: [EQUATION]', '1503.01735-2-41-4': 'The term [MATH]/[MATH] is neglected by the approach of equation [REF] and only vanishes for strictly linear conductivity curves.', '1503.01735-2-42-0': 'But the question arises if the mobility indeed depends on the charge carrier density in case of a non-linear conductivity curve as the non-linearity can also be explained by a constant mobility ([MATH]/[MATH]) if there is a charge carrier independent contribution [MATH] to the overall resistivity.', '1503.01735-2-42-1': 'Such a contribution can result from both short- and correlated long-range disorder [CITATION].', '1503.01735-2-42-2': 'Hence, the corresponding conductivity is given by (see e.g. references [CITATION]):', '1503.01735-2-43-0': '[EQUATION]', '1503.01735-2-43-1': 'Figure [REF](c) demonstrates that a significant part of the electron branch for [MATH] 0 of the non-linear conductivity curve can be fitted under the assumption of the constant mobility [MATH]cm[MATH]/Vs. However, the fitting fails completely in the hole branch (not shown), because here the impact of the contact-induced CNP has to be considered.', '1503.01735-2-43-2': 'Furthermore, all three approaches that are discussed so far break down near the CNP.', '1503.01735-2-43-3': 'But there are also models trying to fit the whole conductivity curve including the two CNPs (e.g. in reference [CITATION]).', '1503.01735-2-43-4': 'We also use the following model which assumes two differently doped graphene parts with different mobilities [MATH]: [EQUATION] where [MATH] is the gate voltage [MATH] dependent charge carrier density, [MATH] the capacitive coupling for a [MATH] thick SiO[MATH] layer, [MATH] the residual charge carrier densities due to the presence of electron-hole puddles [CITATION] and thermally excited carriers, which prevent a divergence at the CNPs [CITATION], and a gate independent resistivity [MATH].', '1503.01735-2-43-5': 'This model can be fitted quite well to the conductivity curve of figure [REF](d) (the given mobility [MATH] cm[MATH]/Vs corresponds to the Dirac curve with its CNP near zero gate voltage).', '1503.01735-2-44-0': 'But now there are two issues: At first, the values of the charge carrier mobilities as extracted from the four models spread over a broad range between [MATH] and [MATH].', '1503.01735-2-44-1': 'Secondly, there is a serious issue when extracting the mobility from equation [REF], which already contains seven parameters, as it cannot be used to adequately fit the conductivity curve of the same device after oxygen treatment (figure [REF](e)).', '1503.01735-2-44-2': 'We tried several other models to fit the curve in figure [REF](e), but for a reasonable fit we need at least eight free parameters.', '1503.01735-2-44-3': 'But as we already briefly discussed in one of our previous publications [CITATION] and will discuss in this section in more detail, such a large number of free parameters make such a model useless.', '1503.01735-2-44-4': 'Because as long as the Dirac curves have symmetric shapes, which they have in figure [REF](a) and in our previous publications [CITATION], only seven parameters can reliably be extracted out of a gate dependent resistivity curve with two visible CNPs: The position, magnitude and width for each of the two peaks and the background.', '1503.01735-2-44-5': 'Therefore, a model of eight parameters is overdetermined and accordingly we are able to change the mobility values in such a model over an unreasonable wide range without seriously worsening the fit results.', '1503.01735-2-45-0': 'To approach reliable mobility fits, we now deduce several parameters which have to be included into a conductivity model to adequately fit a gate dependent resistivity curve with two CNPs.', '1503.01735-2-45-1': 'First, there is the carrier mobility as the actual quantity of interest.', '1503.01735-2-45-2': 'As already discussed in section [REF], the interaction with the electrodes can significantly change the electronic properties of graphene.', '1503.01735-2-45-3': 'We therefore have to consider two mobilities for each device: one for the graphene underneath the contacts and one for the graphene area in between the contacts.', '1503.01735-2-45-4': 'The same argument holds for the electrochemical doping of the respective graphene parts, which is obvious as we observe two CNPs.', '1503.01735-2-45-5': 'Two additional parameters are needed to include spatial variations of the electrochemical potentials within each graphene region.', '1503.01735-2-45-6': 'These two parameters can also account for the minimum conductivities as the variations in the potential lead to the electron-hole puddles at the charge neutrality point [CITATION].', '1503.01735-2-45-7': 'So far this yields six parameters.', '1503.01735-2-46-0': 'In the next step, we have to critically review the gate voltage as the tuning parameter for the resistivity measurement.', '1503.01735-2-46-1': 'As discussed e.g. in [CITATION] the interaction between contact and graphene may lead to a screening of the gate electric field underneath the electrodes.', '1503.01735-2-46-2': 'We assume that there is a gradual transition from pinning to depinning of the Fermi level underneath the contacts with increasing oxide barrier thickness and quality.', '1503.01735-2-46-3': 'Therefore, we have to consider a screening factor as a seventh parameter which accounts for the effective Fermi level shift in the graphene underneath the contacts as a function of applied gate voltage.', '1503.01735-2-47-0': 'Furthermore, there is the unknown transition of the electrochemical potentials between contact covered and bare graphene part, i.e. the exact shape and lateral extension of the pn-junctions, which are known to exist near the edges of the contacts [CITATION].', '1503.01735-2-47-1': 'In the most simple approximation, at least one more parameter has to be considered which describes the decaying length of the pn-junction (see e.g. references [CITATION]).', '1503.01735-2-47-2': 'And finally, there might be a gate voltage independent contribution [MATH] to the overall resistance as already discussed in equation [REF].', '1503.01735-2-47-3': 'Hence, at least nine parameters are necessary to simulate the gate dependent resistivity, whereas only seven independent parameters may be experimentally determined by a gate dependent resistance measurement.', '1503.01735-2-48-0': 'The whole problem with the analysis of the carrier mobility from gate dependent resistance measurements is that both the contact-covered and bare graphene parts are measured simultaneously in series.', '1503.01735-2-48-1': 'The lack of independently probing both regions is bound to the very nature of a pure electrical transport measurement.', '1503.01735-2-48-2': 'Therefore, independent measurements have to be conducted to determine some of the aforementioned parameters separately.', '1503.01735-2-48-3': 'Especially, information about the spatial change in the electrochemical potential along the contact induced pn-junctions may be helpful.', '1503.01735-2-48-4': 'These can be obtained from scanning photocurrent microscopy [CITATION].', '1503.01735-2-49-0': 'Finally, we note that most other graphene-based nanoelectronic devices are fabricated with low-ohmic, metallic contacts.', '1503.01735-2-49-1': 'This might be the reason why the aforementioned issues concerning the determination of the charge carrier mobility is not discussed in recent publications in more detail.', '1503.01735-2-49-2': 'For metallic contacts, a complete pinning of the Fermi level underneath the contacts can be assumed (see section [REF]).', '1503.01735-2-49-3': 'Therefore, the complete contact area of the device only contributes as a gate independent constant resistance to the overall gate dependent resistance (e.g. [MATH] in equation [REF]).', '1503.01735-2-49-4': 'Additionally, the impact of both the contact area and the pn-junctions on the transport measurement decreases with increasing separation of the electrodes.', '1503.01735-2-49-5': 'We note that in many charge-based electrical transport studies the contact separation is much larger than in graphene-based spin transport devices.', '1503.01735-2-50-0': '# Critical review and perspectives', '1503.01735-2-51-0': 'In this final section, we critically review some experimental studies which were used to investigate the dominant spin relaxation mechanism in graphene.', '1503.01735-2-51-1': 'Maybe the most direct way to identify the dominating spin relaxation mechanism is to evaluate the dependence of the spin lifetime on the charge carrier mobility [CITATION].', '1503.01735-2-51-2': "A linear dependence of [MATH] on [MATH] or [MATH], which is the momentum scattering time, is a priori suggestive of an Elliott-Yafet (EY) spin scattering mechanism, while the inverse relation [MATH] indicate the dominance of a D'yakonov-Perel'-like (DP) spin scattering mechanism.", '1503.01735-2-51-3': 'In our previous studies we found an inverse dependence of spin lifetime on the mobility for both single-layer [CITATION] and bilayer [CITATION] graphene which is thus indicative for DP-like spin scattering.', '1503.01735-2-51-4': 'This dependence was only found in devices with contact resistance area products larger than [MATH] (figure [REF](a)).', '1503.01735-2-51-5': 'In contrast, all devices with [MATH] (figure [REF](a)) and devices where the contact resistances were enhanced by subsequent oxygen treatments (figure [REF](b)) do not show this 1/[MATH]-dependence.', '1503.01735-2-51-6': 'All these devices have been prepared by the conventional top-down method for which the MgO injection and detection barrier was directly deposited onto graphene.', '1503.01735-2-52-0': 'Most strikingly, the new generation of devices where we apply the bottom-up approach by transferring a hBN-graphene-stack on prepatterned electrodes (see figure [REF](e) and section [REF]) exhibit significantly enhanced charge and spin transport properties (see full symbols in figure [REF](c)).', '1503.01735-2-52-1': 'We attribute the increase in mobility to the hBN substrate while we relate the increase in spin lifetime to improved contact properties according to our advanced transfer technique which has several advantages over the previous fabrication methods.', '1503.01735-2-52-2': 'At first, the contact region has not been exposed to an electron beam which may reduce the number of defects in graphene.', '1503.01735-2-52-3': '[CITATION] Secondly, the interface between MgO and graphene is expected to be of higher quality yielding more homogeneous barriers which can be seen by the larger [MATH] values for devices with thinner MgO layer thicknesses [CITATION].', '1503.01735-2-53-0': 'Our results indicate that the overall improvement of the spin properties primarily result from the improvement of the contact properties suggesting that the observed 1/[MATH] dependence in the initial work is of extrinsic origin.', '1503.01735-2-53-1': 'In this context, we again want to emphasize (sections [REF] and [REF]) that the determination of both the charge carrier mobility and the spin lifetime is by no means unambiguous and may thus result in contradicting [MATH] dependencies.', '1503.01735-2-54-0': 'Furthermore, we demonstrated in [CITATION] and [CITATION] that the spin lifetimes in [CITATION] are most likely limited by contact properties (also see figure [REF]) in all devices.', '1503.01735-2-54-1': 'In fact, there are indications that even in our newest bottom-up devices the contacts may still limit spin transport properties (paper in preparation).', '1503.01735-2-54-2': 'If in current devices extrinsic parameters limit spin transport properties, it will be interesting to see how far more advanced fabrication methods will yield devices with even longer spin lifetimes and larger carrier mobilities which ultimately allows to unveil intrinsic spin scattering mechanisms.'} | [['1503.01735-1-20-0', '1503.01735-2-20-0'], ['1503.01735-1-20-1', '1503.01735-2-20-1'], ['1503.01735-1-20-2', '1503.01735-2-20-2'], ['1503.01735-1-20-3', '1503.01735-2-20-3'], ['1503.01735-1-20-5', '1503.01735-2-20-7'], ['1503.01735-1-20-6', '1503.01735-2-20-8'], ['1503.01735-1-48-0', '1503.01735-2-48-0'], ['1503.01735-1-48-1', '1503.01735-2-48-1'], ['1503.01735-1-48-2', '1503.01735-2-48-2'], ['1503.01735-1-48-3', '1503.01735-2-48-3'], ['1503.01735-1-48-4', '1503.01735-2-48-4'], ['1503.01735-1-0-0', '1503.01735-2-0-0'], ['1503.01735-1-0-1', '1503.01735-2-0-1'], ['1503.01735-1-0-2', '1503.01735-2-0-2'], ['1503.01735-1-0-3', '1503.01735-2-0-3'], ['1503.01735-1-0-4', '1503.01735-2-0-4'], ['1503.01735-1-0-5', '1503.01735-2-0-5'], ['1503.01735-1-0-6', '1503.01735-2-0-6'], ['1503.01735-1-0-7', '1503.01735-2-0-7'], ['1503.01735-1-16-0', '1503.01735-2-16-0'], ['1503.01735-1-16-1', '1503.01735-2-16-1'], ['1503.01735-1-16-2', '1503.01735-2-16-2'], 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'1503.01735-2-38-0'], ['1503.01735-1-38-1', '1503.01735-2-38-1'], ['1503.01735-1-38-2', '1503.01735-2-38-2'], ['1503.01735-1-38-3', '1503.01735-2-38-3'], ['1503.01735-1-38-4', '1503.01735-2-38-4'], ['1503.01735-1-41-0', '1503.01735-2-41-0'], ['1503.01735-1-41-1', '1503.01735-2-41-1'], ['1503.01735-1-41-2', '1503.01735-2-41-2'], ['1503.01735-1-41-3', '1503.01735-2-41-3'], ['1503.01735-1-41-4', '1503.01735-2-41-4'], ['1503.01735-1-47-0', '1503.01735-2-47-0'], ['1503.01735-1-47-1', '1503.01735-2-47-1'], ['1503.01735-1-47-2', '1503.01735-2-47-2'], ['1503.01735-1-47-3', '1503.01735-2-47-3'], ['1503.01735-1-35-0', '1503.01735-2-35-0'], ['1503.01735-1-35-1', '1503.01735-2-35-1'], ['1503.01735-1-35-2', '1503.01735-2-35-2'], ['1503.01735-1-35-3', '1503.01735-2-35-3'], ['1503.01735-1-35-4', '1503.01735-2-35-4'], ['1503.01735-1-35-5', '1503.01735-2-35-5'], ['1503.01735-1-44-0', '1503.01735-2-44-0'], ['1503.01735-1-44-1', '1503.01735-2-44-1'], ['1503.01735-1-44-2', '1503.01735-2-44-2'], ['1503.01735-1-44-3', '1503.01735-2-44-3'], ['1503.01735-1-44-4', '1503.01735-2-44-4'], ['1503.01735-1-44-5', '1503.01735-2-44-5'], ['1503.01735-1-10-0', '1503.01735-2-10-0'], ['1503.01735-1-10-1', '1503.01735-2-10-1'], ['1503.01735-1-10-2', '1503.01735-2-10-2'], ['1503.01735-1-10-3', '1503.01735-2-10-3'], ['1503.01735-1-10-4', '1503.01735-2-10-10'], ['1503.01735-1-11-4', '1503.01735-2-10-5'], ['1503.01735-1-11-5', '1503.01735-2-10-6'], ['1503.01735-1-11-6', '1503.01735-2-10-7'], ['1503.01735-1-11-7', '1503.01735-2-10-8'], ['1503.01735-1-11-8', '1503.01735-2-10-9']] | [['1503.01735-1-20-7', '1503.01735-2-20-9'], ['1503.01735-1-0-8', '1503.01735-2-0-8'], ['1503.01735-1-21-4', '1503.01735-2-21-4'], ['1503.01735-1-13-0', '1503.01735-2-13-0'], ['1503.01735-1-17-3', '1503.01735-2-17-3'], ['1503.01735-1-52-2', '1503.01735-2-52-2'], ['1503.01735-1-52-3', '1503.01735-2-52-3'], ['1503.01735-1-27-1', '1503.01735-2-27-1'], ['1503.01735-1-36-1', '1503.01735-2-36-1'], ['1503.01735-1-8-1', '1503.01735-2-8-1'], ['1503.01735-1-12-0', '1503.01735-2-12-0'], ['1503.01735-1-12-1', '1503.01735-2-12-1'], ['1503.01735-1-12-2', '1503.01735-2-12-2'], ['1503.01735-1-12-7', '1503.01735-2-12-7'], ['1503.01735-1-30-10', '1503.01735-2-30-14'], ['1503.01735-1-11-3', '1503.01735-2-10-4']] | [] | [['1503.01735-1-20-4', '1503.01735-2-20-6'], ['1503.01735-1-7-0', '1503.01735-2-11-2'], ['1503.01735-1-5-6', '1503.01735-2-5-6'], ['1503.01735-1-17-2', '1503.01735-2-17-2'], ['1503.01735-1-18-5', '1503.01735-2-18-5'], ['1503.01735-1-51-0', '1503.01735-2-51-0'], ['1503.01735-1-8-0', '1503.01735-2-8-0'], ['1503.01735-1-3-1', '1503.01735-2-3-1'], ['1503.01735-1-30-9', '1503.01735-2-30-13']] | [] | ['1503.01735-1-42-2', '1503.01735-1-43-0', '1503.01735-2-42-2', '1503.01735-2-43-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1503.01735 | null | null | null | null | null |
1903.09202 | {'1903.09202-1-0-0': 'We report high spatial resolution ([MATH]0076, 410pc) ALMA imaging of the dust continuum and the ionised carbon line [C2] in a luminous quasar host galaxy at [MATH], 800 million years after the Big Bang.', '1903.09202-1-0-1': 'Based on previous studies, this galaxy hosts a [MATH][MATH] and with the gas having a velocity dispersion of [MATH]110kms[MATH] (based on the FWHM of 255kms[MATH]), the region where the black hole dominates the gravitational potential, the so-called black hole sphere of influence, has a radius of [MATH]355pc.', '1903.09202-1-0-2': 'Thus the achieved spatial resolution of the observations in principle allows us to probe scales similar to the black hole sphere of influence in order to search for kinematic signatures of the central black hole.', '1903.09202-1-1-0': '# Results', '1903.09202-1-2-0': '## Dust and [C 2] emission', '1903.09202-1-3-0': 'In Figure [REF] we show the results of the observations.', '1903.09202-1-3-1': 'The first two panels in the figure show the [C2] and continuum maps.', '1903.09202-1-3-2': 'The bottom two panels in Figure [REF] show the kinematics of the [C2] emission.', '1903.09202-1-3-3': 'To match the position of the [C2] and FIR dust emission with that of the accreting black hole, we recomputed the near-infrared location of the quasar by correcting the positions of nearby stars with a match in the GAIA DR2 catalog .', '1903.09202-1-3-4': 'The resulting near-infrared location of the quasar is: 03[MATH]16.919[MATH], -31[MATH]55.86[MATH] with an uncertainty of 002, which is consistent with the previously published coordinates of the quasar .', '1903.09202-1-3-5': 'The distribution of the far-infrared continuum emission has a pronounced peak that is coincident with the near-infrared location of the quasar.', '1903.09202-1-4-0': 'We measured the total flux of the quasar host galaxy in an aperture with radius of 075.', '1903.09202-1-4-1': 'The total continuum flux density is 5.34[MATH]0.19mJy around the [C2] line, making this quasar host one of the most luminous, unlensed sources known at high redshift .', '1903.09202-1-4-2': 'The total brightness of the [C2] line is 5.43[MATH]0.33Jykms[MATH], which corresponds to a luminosity of [MATH],=[MATH][MATH].', '1903.09202-1-4-3': 'We derive a systemic redshift of the quasar host by fitting a Gaussian to the [C2] emission line, resulting in [MATH], consistent with earlier measurements .', '1903.09202-1-5-0': 'Both the continuum and the [C2] emission are spatially resolved, and the emission is extended over [MATH]5kpc.', '1903.09202-1-5-1': 'The gas distribution and kinematics, as traced by the [C2] emission, are highly complex.', '1903.09202-1-5-2': 'There is a pronounced lack of [C2] emission in towards the east of the quasar.', '1903.09202-1-5-3': 'This cavity in the [C2] emission is also seen in the dust continuum observations.', '1903.09202-1-5-4': 'In general, the continuum and [C2] emission trace similar structures, the main difference being the bright peak in the continuum.', '1903.09202-1-6-0': 'In Figure [REF] we compare the [C2] emission to that of the dust.', '1903.09202-1-6-1': 'The [MATH]/[MATH] ratio is lowest at the location of the quasar where the dust continuum peaks ([MATH]/[MATH],[MATH]).', '1903.09202-1-6-2': 'This is reminiscent of the centers of local star-bursts and ULIRGs, where low [C2]-to-FIR luminosity ratios are observed in regions with high FIR surface brightness .', '1903.09202-1-6-3': 'Away from the central regions, the [C2]-to-FIR luminosity ratio is in the range [MATH]/[MATH],[MATH].', '1903.09202-1-6-4': 'More centrally concentrated continuum emission has been observed in some other high redshift galaxies and quasar hosts, albeit at lower spatial resolution .', '1903.09202-1-7-0': '## Gas Dynamics', '1903.09202-1-8-0': 'From the mean velocity map of the [C2] emission (third panel in Figure [REF] it is evident that the position of the accreting black hole coincides with the kinematic centre of the [C2] emission.', '1903.09202-1-8-1': 'It is also located at the centre of the global [C2] emission.', '1903.09202-1-8-2': 'Generally, the gas exhibits some ordered motion along the line of sight, with the gas having positive line-of-sight velocities towards the east and negative velocities towards the west.', '1903.09202-1-8-3': 'The highest velocity component, which was already discovered in low resolution data , is now clearly separated, both spatially and in frequency.', '1903.09202-1-8-4': 'It is therefore likely that this is a companion galaxy close to the quasar host (see also Figures [REF] and [REF]).', '1903.09202-1-8-5': 'This will be further discussed in Section [REF].', '1903.09202-1-9-0': 'The velocity dispersion in the quasar host is roughly uniform (in the range 50-100kms[MATH]) throughout the galaxy.', '1903.09202-1-9-1': 'The dispersion is almost equal to the projected line-of-sight velocity.', '1903.09202-1-9-2': 'At the position of the quasar, there is no sharp increase in the velocity dispersion; the dispersion of 110kms[MATH] at that location is not found to be higher compared to the remainder of the host galaxy.', '1903.09202-1-9-3': 'This indicates that, with the current resolution, the mass budget within the central [MATH]400pc is not dominated by the black hole.', '1903.09202-1-9-4': 'Indeed, assuming typical dust properties and a gas-to-dust ratio of 70 , the inferred gas mass within our central resolution element is [MATH][MATH] , higher than the [MATH][MATH] of the black hole .', '1903.09202-1-10-0': 'To explore how the distribution of [C2] emission changes with line-of-sight velocity, we have averaged the emission into channels with a width of 30MHz (36kms[MATH]), see Figure [REF].', '1903.09202-1-10-1': 'There are two striking features in the channels centred around 0kms[MATH]; i) the [C2] emission covers the whole spatial extent seen in the integrated emission map (Figure [REF]), and ii) there are two depressions/cavities in the [C2] emission with diameters of [MATH]0.5kpc on either side of the black hole.', '1903.09202-1-10-2': 'The gas with the highest velocities - that was already seen at positive velocities towards the North-East in the low resolution data - is clearly offset from the quasar host.', '1903.09202-1-11-0': 'We can capture some of the complexity of the kinematics, by generating a position-velocity diagram of the [C2] emission (Figure [REF]).', '1903.09202-1-11-1': 'A position-velocity diagram is a planar slice through the datacube and a useful way to visualise the complex kinematics as it shows the line-of-sight velocities of the gas as a function of distance from the centre.', '1903.09202-1-11-2': "This 'pseudo-longslit' spectrum, oriented along the major kinematical axis through the position of the quasar and the [C2] cavity to the east, shows an 'S-like' shape in the velocity curve.", '1903.09202-1-11-3': 'This implies that, in addition to the cavity that is clearly seen towards the east, there is a second cavity to the west, which is also seen in the channel maps at zero velocity (Figure [REF]).', '1903.09202-1-11-4': 'Interestingly, the line-of-sight velocities approach zero at the largest radii, consistent with the finding that the channel map at zero velocity shows the full extent of the integrated line emission.', '1903.09202-1-12-0': 'In the next Section, we will fit several models to explain the distribution and velocity of the gas.', '1903.09202-1-13-0': '# Kinematic Modeling of the [C 2] emission', '1903.09202-1-14-0': 'lcccc Model Parameters.', '1903.09202-1-14-1': '0pt & Disk A & Disk B & Disk C & Rot + Jet D Velocity profile & constant & Keplerian & truncated & constant', '1903.09202-1-15-0': 'Right ascension[MATH] & 03[MATH]16.9225(1)[MATH] & 03[MATH]16.9219(1)[MATH] & 03[MATH]16.9219(1)[MATH] & 03[MATH]16.9227(2)[MATH]', '1903.09202-1-16-0': 'Declination[MATH] & -31[MATH]55.938(2)[MATH] & -31[MATH]55.939(2)[MATH] & -31[MATH]55.940(1)[MATH] & -31[MATH]55.939(3)[MATH]', '1903.09202-1-17-0': 'Redshift[MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-18-0': 'Inclination (deg.)', '1903.09202-1-18-1': '& [MATH] & [MATH] & [MATH] & -', '1903.09202-1-19-0': 'Position angle (deg.)', '1903.09202-1-19-1': '& [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-20-0': 'Central intensity (mJybeam[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-21-0': 'Intensity scale radius (kpc) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-22-0': 'Circular velocity (kms[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-23-0': 'Velocity dispersion (kms[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-24-0': 'Velocity scale radius (kpc) & - & [MATH] & [MATH] & -', '1903.09202-1-25-0': 'Jet opening angle (deg.)', '1903.09202-1-25-1': '& - & - & - & [MATH]', '1903.09202-1-26-0': 'Jet z-axis angle (deg.)', '1903.09202-1-26-1': '[MATH] & - & - & - & [MATH]', '1903.09202-1-27-0': 'Jet x-axis angle (deg.)', '1903.09202-1-27-1': '[MATH] & - & - & - & [MATH]', '1903.09202-1-28-0': 'aPosition of the kinematic centre of the model.', '1903.09202-1-28-1': 'bNumber in parentheses is the uncertainty in the last number.', '1903.09202-1-28-2': 'cAngle of the jet compared to the axis of rotation.', '1903.09202-1-28-3': 'dAngle of the jet compared to the position angle of the rotation.', '1903.09202-1-29-0': 'To better understand the spatial extent and kinematic signature of the [C2] emission, we have fitted a range of simple models to the data.', '1903.09202-1-29-1': 'These models are discussed below and are summarised in Table [REF].', '1903.09202-1-29-2': 'To fit the models, we generate a cube from the given model parameters, which we will refer to as the model cube.', '1903.09202-1-29-3': 'This model cube is then convolved with the ALMA primary beam to account for potential beam smearing effects.', '1903.09202-1-29-4': 'We compare this convolved model data cube, [MATH]), with the data cube, [MATH], using a standard chi-squared algorithm, [EQUATION] where [MATH] is the uncertainty (i.e., the root mean square (rms) uncertainty of the pixels showing no emission in the same channel as pixel [MATH]).', '1903.09202-1-29-5': 'The model cube has velocities that range from -216 to +216kms[MATH].', '1903.09202-1-29-6': 'This range covers the velocities where the data show [C2] emission but excludes the channels affected by the close companion (see Figures [REF] and [REF]).', '1903.09202-1-29-7': 'Since the adjacent pixels are highly correlated in interferometric data, we do not sum up all pixels in our data, but instead use a bootstrap method, whereby we randomly select [MATH] pixels from the data cube and calculate the [MATH] value of these pixels.', '1903.09202-1-29-8': 'We repeat this process until the median [MATH] value remains unchanged.', '1903.09202-1-29-9': 'Here the number of pixels, [MATH], is chosen so that on average each beam only contains a single pixel per single [MATH] calculation.', '1903.09202-1-30-0': 'To fully sample the parameter space of the model and provide more realistic constraints on the possible values of each parameter, we have used a Markov Chain Monte Carlo (MCMC) method to sample the parameter space.', '1903.09202-1-30-1': 'In particular, we have used the affine invariant MCMC ensemble sampler code, emcee .', '1903.09202-1-30-2': 'For all parameters, we assume flat priors, and we initiate the parameters with rough estimates from visual inspection of the data.', '1903.09202-1-30-3': 'We have verified that these initial guesses do not affect the results of the MCMC analysis.', '1903.09202-1-30-4': 'Results of the different models are shown in Figure [REF] and Table [REF].', '1903.09202-1-31-0': '## Modeling Results', '1903.09202-1-32-0': 'We have modeled the [C2] kinematics in 3D using several simple models.', '1903.09202-1-32-1': 'The following models were considered in this analysis: a) A thin rotating disk with a constant velocity profile ("Disk A").', '1903.09202-1-32-2': 'This is the fiducial profile of a dark matter dominated disk galaxy, which normally is the model employed to explain the gas kinematics in marginally resolved [MATH] quasar host galaxies .', '1903.09202-1-32-3': 'b) A thin rotating disk with a decreasing velocity profile ("Disk B").', '1903.09202-1-32-4': 'This profile would arise if the mass of the galaxy is centrally located.', '1903.09202-1-32-5': 'c) A thin rotating disk with a constant velocity profile up to a certain radius, after which it decreases to zero ("Disk C").', '1903.09202-1-32-6': 'This is the velocity profile of a disk that is truncated at a certain radius, or the velocity profile of a disk that is warped into the plane of the sky at this radius.', '1903.09202-1-32-7': 'd) An inclined biconical jet embedded in a uniform rotating spherical gas ("Rot+Jet D").', '1903.09202-1-32-8': 'This is a simple model of an active galactic nucleus (AGN) jet accelerating and/or ionising the [C2] emitting ISM.', '1903.09202-1-32-9': 'This possibility will be further discussed in Section [REF].', '1903.09202-1-32-10': 'All of the models have a decreasing intensity profile dependent only on the distance from the centre.', '1903.09202-1-32-11': 'This clearly is an oversimplification, as the moment-zero map shows obvious evidence of non-exponential emission (see Figure [REF]).', '1903.09202-1-33-0': 'The number of parameters in each of the models varies from 9 to 12 (Table [REF]).', '1903.09202-1-33-1': "Each model contains the central position of the galaxy (in two spatial and one frequency dimension), the position angle of the maximum rotation, the central intensity and 'scale height' of the emission, the maximum velocity of rotation, and the dispersion along the line of sight.", '1903.09202-1-33-2': 'Besides these 8 parameters, the disk models also need an inclination, and disk models "B" and "C" require a scale radius for the velocity.', '1903.09202-1-33-3': 'Finally the jet model "D" requires three additional angles to describe the jet angle and opening angle of the jet, as well as a terminal radius at which the jet stops accelerating/ionising the ISM.', '1903.09202-1-34-0': 'From the fits shown in Figure [REF], it is clear that the kinematics are inconsistent with a simple rotating disk with a constant velocity ("Disk A"), which is typically assumed in order to derived dynamical masses of marginally resolved [MATH] quasar host galaxies .', '1903.09202-1-34-1': 'To account for the extended emission detected at the systemic velocity, we need a truncated disk model ("Disk C") whereby the systemic velocity goes to zero beyond a certain radius.', '1903.09202-1-34-2': 'Such a scenario could arise from either a gravitationally-dominated disk that has formed in the centre of a dispersion-dominated source, or alternatively, a warp of the outer disk into the plane of the sky.', '1903.09202-1-34-3': 'Both scenarios, however, do not provide adequate fits to the observed [C2] distribution and kinematics (Figures [REF] and [REF]).', '1903.09202-1-35-0': 'As can be seen in Figure 3, from the position of the black hole, the velocities increase to a radius of 0.5kpc (0.1[MATH]) and reaches a peak line-of-sight velocity of 150-200kms[MATH].', '1903.09202-1-35-1': 'Assuming these kinematics can be described by a rotating disk, the dynamical mass within this radius is [MATH]/sin[MATH])M[MATH], with [MATH] the inclination angle of the disk.', '1903.09202-1-35-2': 'For comparison, the mass of the black hole is [MATH]M[MATH], and the inferred gas mass within this radius is [MATH]M[MATH].', '1903.09202-1-35-3': 'For an inclination angle of [MATH] (see Table [REF]), the dynamical mass and molecular gas mass within the central 1kpc are consistent with each other within the large uncertainties.', '1903.09202-1-35-4': 'Beyond this radius, the line-of-sight velocity is decreasing faster than Keplerian (shown by the "Disk B" model in Figure 3), and reaches approximately systemic velocity at the last measured points at 1.5-2.0kpc.', '1903.09202-1-35-5': 'To summarise, the kinematics appear to be dispersion dominated with some overall rotation (i.e. net angular momentum) in the central kpc.', '1903.09202-1-35-6': 'This implies that most of the gas has not yet settled in a disk.', '1903.09202-1-36-0': '# Origin of the [C 2] cavities', '1903.09202-1-37-0': 'Instead of a kinematic origin, the presence of the two kpc-scale [C2] cavities could be the result of energy injection into the interstellar medium (ISM).', '1903.09202-1-37-1': 'Similar shell-like structure are seen in the neutral ISM of local galaxies , and are often interpreted as the results of supernova or active galactic nucleus (AGN) feedback.', '1903.09202-1-37-2': 'We can calculate the energy needed to created such structures seen here using : [EQUATION] with [MATH] being the density in cm[MATH], [MATH] being the diameter of the cavity in parsecs and [MATH] the expansion velocity in kms[MATH].', '1903.09202-1-37-3': 'The cavities have a diameter of approximately [MATH]0.5kpc (see Figure [REF]), and the luminosity-weighted, average density of the ISM in the quasar host has been estimated to be 10[MATH]cm[MATH] based on a multi-line analysis in this source.', '1903.09202-1-37-4': "Under the assumption that the 'S-like' structure is due to energy input in the plane of the galaxy, the expansion velocity can be approximated by the amplitude of the derivation from systemic velocity .", '1903.09202-1-37-5': 'We estimate an amplitude of [MATH]80kms[MATH] at the centre of the cavities (Figure 3).', '1903.09202-1-37-6': 'Taken together, we derive a required energy of [MATH]b (kms[MATH]) & - & +361 & -137 & -289 & +595', '1903.09202-1-38-0': '[MATH]a (Jykms[MATH]) & 5.43[MATH]0.33 & 0.43[MATH]0.05 & 0.31[MATH]0.08 & 1.13[MATH]0.17 & [MATH]0.23', '1903.09202-1-39-0': 'FWHM[MATH] (kms[MATH]) & 225[MATH]15 & 180[MATH]25 & 135[MATH]40 & 330[MATH]55 & -', '1903.09202-1-40-0': '[MATH]c (mJy) & 5.34[MATH]0.19 & -d & [MATH]0.23 & 0.58[MATH]0.12 & [MATH]0.33', '1903.09202-1-41-0': '[MATH]) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]) & [MATH] & - & [MATH]', '1903.09202-1-42-0': 'SFR[MATH]e ([MATH]yr[MATH]) & 1016[MATH]73 & 51[MATH]9 & 35[MATH]10 & 159[MATH]28 & [MATH]f ([MATH]) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-1-43-0': 'aThe redshift and [C2] line flux ([MATH]) are measured using a Gaussian fit to the [C2] line for the quasar and the companions C1, C2 and C3 (as shown in Figure [REF]).', '1903.09202-1-43-1': 'The redshift of the Lyman alpha emitter (LAE) is taken from [CITATION].', '1903.09202-1-43-2': 'bThe line-of-sight velocity [MATH] is computed using [MATH].', '1903.09202-1-43-3': 'c The continuum flux density at a rest-frame wavelength of 158[MATH]m ([MATH]) is measured by averaging the line-free channels in the spectrum around the [C2] line.', '1903.09202-1-43-4': 'dThe continuum flux density of C1 could not be accurately determined due to contamination by the quasar host galaxy.', '1903.09202-1-43-5': 'eThe star-formation rate SFR[MATH] is derived using SFR[MATH] .', '1903.09202-1-43-6': 'fMolecular gas mass, derived from the [C2] luminosity using a [MATH]-to-H[MATH] conversion factor of [MATH][MATH]/[MATH] .', '1903.09202-1-44-0': 'The environment of J0305-3150 has been the subject of various studies.', '1903.09202-1-44-1': 'A study searching for Lyman alpha emission from galaxies at the redshift of the quasar revealed the presence of a Lyman alpha halo around the quasar, and a faint Ly[MATH] emitting (LAE) companion at a distance of 12.5kpc .', '1903.09202-1-44-2': 'In addition, a larger field-of-view narrowband search for Lyman alpha emitters around the quasar indicated that the quasar is not located in a dense environment on Mpc scales .', '1903.09202-1-44-3': 'Our previous, shallow ALMA data of the field contained two faint (rest-frame 158[MATH]m flux densities of [MATH]mJy) continuum sources .', '1903.09202-1-45-0': 'In addition to high-resolution imaging of the quasar host galaxy, the improved sensitivity of the new ALMA data allowed us to search for additional sources in the field.', '1903.09202-1-45-1': 'Previously, bright companion sources have been found in the vicinity of some high redshift quasars .', '1903.09202-1-45-2': 'Our new data reveal the presence of 3 [C2] emitters at the same redshift as the quasar (Figure [REF]; physical parameters in Table [REF]).', '1903.09202-1-45-3': 'One companion, C1, could already be seen in Figure [REF].', '1903.09202-1-45-4': 'The brightest companion, C3, was already identified as a continuum source in the field in [CITATION].', '1903.09202-1-46-0': 'The FIR luminosity of companion C3 is comparable to that of the [C2] companions near several [MATH] quasars presented in [CITATION].', '1903.09202-1-46-1': 'However, in contrast to the companions found by [CITATION], all companions identified near J0305-3150 have FIR luminosities that are a factor [MATH]10 smaller than that of the quasar host.', '1903.09202-1-46-2': 'Also, the estimated molecular gas masses of the companions are at least a factor of [MATH]5 smaller than that of the quasar host.', '1903.09202-1-46-3': 'Despite the small masses, the interaction of companions with the quasar host could provide an alternative explanation of the gas morphology and kinematics in the quasar host galaxy.', '1903.09202-1-46-4': 'Furthermore, it is a possibility that such interactions triggered the high star-formation rate in the quasar host and the AGN activity.', '1903.09202-1-47-0': '# Summary', '1903.09202-1-48-0': 'The high spatial resolution ([MATH]400pc) imaging of a quasar host galaxy at [MATH] shows that its formation is a complex and chaotic process.', '1903.09202-1-48-1': 'We find that the ISM in the quasar host has not yet settled in a simple disk.', '1903.09202-1-48-2': 'While there are signs that the AGN is affecting the ISM, this feedback is not suppressing the formation of stars in the quasar host: In fact, the star-formation rate inferred by the far-infrared luminosity of [MATH]yr[MATH] is high among galaxies and quasar hosts at similar redshifts .', '1903.09202-1-48-3': 'The observed high star-formation rate and rapid black hole growth could be triggered by interactions with the newly detected nearby companion galaxies.', '1903.09202-1-48-4': 'The observations presented here show the unique role that high-angular resolution observations with ALMA can play in studies of the ISM in some of the most distant massive galaxies.', '1903.09202-1-49-0': 'B.P.V., M.N., and F.W. acknowledge funding through the ERC grant "Cosmic Gas".', '1903.09202-1-49-1': 'This paper makes use of the following ALMA data: ADS/JAO.ALMA2017.1.01532.', '1903.09202-1-49-2': 'S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile.', '1903.09202-1-49-3': 'The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.', '1903.09202-1-50-0': 'ALMA'} | {'1903.09202-2-0-0': 'We report high spatial resolution ([MATH]0076, 410pc) Atacama Large Millimeter/submillimeter Array imaging of the dust continuum and the ionized carbon line [C2] in a luminous quasar host galaxy at [MATH], 800 million years after the big bang.', '1903.09202-2-0-1': 'Based on previous studies, this galaxy hosts a [MATH][MATH] and with the gas having a velocity dispersion of [MATH]110kms[MATH] (based on the FWHM of 255kms[MATH]), the region where the black hole dominates the gravitational potential, the so-called black hole sphere of influence, has a radius of [MATH]355pc.', '1903.09202-2-0-2': 'Thus the achieved spatial resolution of the observations in principle allow us to probe scales similar to the black hole sphere of influence in order to search for kinematic signatures of the central black hole.', '1903.09202-2-1-0': '# Results', '1903.09202-2-2-0': '## Dust and [C 2] emission', '1903.09202-2-3-0': 'In Figure [REF] we show the results of the observations.', '1903.09202-2-3-1': 'The first two panels in the figure show the [C2] and continuum maps.', '1903.09202-2-3-2': 'The bottom two panels in Figure [REF] show the kinematics of the [C2] emission.', '1903.09202-2-3-3': 'To match the position of the [C2] and FIR dust emission with that of the accreting black hole, we recomputed the near-infrared (NIR) location of the quasar by correcting the positions of nearby stars with a match in the GAIA DR2 catalog .', '1903.09202-2-3-4': 'The resulting NIR location of the quasar is 03[MATH].919, -31[MATH]5586 with an uncertainty of 002, which is consistent with the previously published coordinates of the quasar .', '1903.09202-2-3-5': 'The distribution of the FIR continuum emission has a pronounced peak that is coincident with the NIR location of the quasar.', '1903.09202-2-4-0': 'We measured the total flux of the quasar host galaxy in an aperture with a radius of 075.', '1903.09202-2-4-1': 'The total continuum flux density is 5.34[MATH]0.19mJy around the [C2] line, making this quasar host one of the most luminous, unlensed sources known at high redshift .', '1903.09202-2-4-2': 'The total brightness of the [C2] line is 5.43[MATH]0.33Jykms[MATH], which corresponds to a luminosity of [MATH],=[MATH][MATH].', '1903.09202-2-4-3': 'We derive a systemic redshift of the quasar host by fitting a Gaussian to the [C2] emission line, resulting in [MATH], consistent with earlier measurements .', '1903.09202-2-5-0': 'Both the continuum and the [C2] emission are spatially resolved, and the emission is extended over [MATH]5kpc.', '1903.09202-2-5-1': 'The gas distribution and kinematics, as traced by the [C2] emission, are highly complex.', '1903.09202-2-5-2': 'There is a pronounced lack of [C2] emission toward the east of the quasar.', '1903.09202-2-5-3': 'This cavity in the [C2] emission is also seen in the dust continuum observations.', '1903.09202-2-5-4': 'In general, the continuum and [C2] emission trace similar structures, the main difference being the bright peak in the continuum.', '1903.09202-2-6-0': 'In Figure [REF] we compare the [C2] emission to that of the dust.', '1903.09202-2-6-1': 'The [MATH]/[MATH] ratio is lowest at the location of the quasar where the dust continuum peaks ([MATH]/[MATH],[MATH]).', '1903.09202-2-6-2': 'This is reminiscent of the centers of local star bursts and ultraluminous infrared galaxies (ULIRGs), where low [C2]-to-FIR luminosity ratios are observed in regions with high FIR surface brightness .', '1903.09202-2-6-3': 'Away from the central regions, the [C2]-to-FIR luminosity ratio is in the range of [MATH]/[MATH],[MATH].', '1903.09202-2-6-4': 'More centrally concentrated continuum emission has been observed in some other high-redshift galaxies and quasar hosts, albeit at lower spatial resolution .', '1903.09202-2-7-0': '## Gas Dynamics', '1903.09202-2-8-0': 'From the mean velocity map of the [C2] emission (third panel in Figure [REF]) it is evident that the position of the accreting black hole coincides with the kinematic center of the [C2] emission.', '1903.09202-2-8-1': 'It is also located at the center of the global [C2] emission.', '1903.09202-2-8-2': 'Generally, the gas exhibits some ordered motion along the line of sight, with the gas having positive line-of-sight velocities toward the east and negative velocities toward the west.', '1903.09202-2-8-3': 'The highest velocity component, which was already discovered in low-resolution data , is now clearly separated, both spatially and in frequency.', '1903.09202-2-8-4': 'It is therefore likely that this is a companion galaxy close to the quasar host (see also Figures [REF] and [REF]).', '1903.09202-2-8-5': 'This will be further discussed in Section [REF].', '1903.09202-2-9-0': 'The velocity dispersion in the quasar host is roughly uniform (in the range 50-100kms[MATH]) throughout the galaxy.', '1903.09202-2-9-1': 'The dispersion is almost equal to the projected line-of-sight velocity.', '1903.09202-2-9-2': 'At the position of the quasar, there is no sharp increase in the velocity dispersion; the dispersion of 110kms[MATH] at that location is not found to be higher compared to the remainder of the host galaxy.', '1903.09202-2-9-3': 'This indicates that, with the current resolution, the mass budget within the central [MATH]400pc is not dominated by the black hole.', '1903.09202-2-9-4': 'Indeed, assuming typical dust properties and a gas-to-dust ratio of 70 , the inferred gas mass within our central resolution element is [MATH][MATH] , higher than the [MATH][MATH] of the black hole .', '1903.09202-2-10-0': 'To explore how the distribution of [C2] emission changes with line-of-sight velocity, we have averaged the emission into channels with a width of 30MHz (36kms[MATH]; see Figure [REF]).', '1903.09202-2-10-1': 'There are two striking features in the channels centered around 0kms[MATH]: (i) the [C2] emission covers the whole spatial extent seen in the integrated emission map (Figure [REF]), and (ii) there are two depressions/cavities in the [C2] emission with diameters of [MATH]0.5kpc on either side of the black hole.', '1903.09202-2-10-2': 'The gas with the highest velocities - that was already seen at positive velocities toward the northeast in the low-resolution data - is clearly offset from the quasar host.', '1903.09202-2-11-0': 'We can capture some of the complexity of the kinematics by generating a position-velocity diagram of the [C2] emission (Figure [REF]).', '1903.09202-2-11-1': 'A position-velocity diagram is a planar slice through the data cube and a useful way to visualize the complex kinematics as it shows the line-of-sight velocities of the gas as a function of distance from the center.', '1903.09202-2-11-2': 'This pseudo-longslit spectrum, oriented along the major kinematical axis through the position of the quasar and the [C2] cavity to the east, shows an S-like shape in the velocity curve.', '1903.09202-2-11-3': 'This implies that, in addition to the cavity that is clearly seen toward the east, there is a second cavity to the west, which is also seen in the channel maps at zero velocity (Figure [REF]).', '1903.09202-2-11-4': 'Interestingly, the line-of-sight velocities approach zero at the largest radii, consistent with the finding that the channel map at zero velocity shows the full extent of the integrated line emission.', '1903.09202-2-12-0': 'In the next Section, we will fit several models to explain the distribution and velocity of the gas.', '1903.09202-2-13-0': '# Kinematic Modeling of the [C 2] emission', '1903.09202-2-14-0': 'lcccc Model Parameters.', '1903.09202-2-14-1': '0pt & Disk A & Disk B & Disk C & Rot + Jet D Velocity profile & Constant & Keplerian & Truncated & Constant', '1903.09202-2-15-0': 'R.A.[MATH] & 03[MATH].9225(1) & 03[MATH].9219(1) & 03[MATH].9219(1) & 03[MATH].9227(2)', '1903.09202-2-16-0': 'decl.', '1903.09202-2-16-1': '[MATH] & -31[MATH]55938(2) & -31[MATH]55939(2) & -31[MATH]55940(1) & -31[MATH]55939(3)', '1903.09202-2-17-0': 'Redshift[MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-18-0': 'Inclination (deg.)', '1903.09202-2-18-1': '& [MATH] & [MATH] & [MATH] & -', '1903.09202-2-19-0': 'Position angle (deg.)', '1903.09202-2-19-1': '& [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-20-0': 'Central intensity (mJybeam[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-21-0': 'Intensity scale radius (kpc) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-22-0': 'Circular velocity (kms[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-23-0': 'Velocity dispersion (kms[MATH]) & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-24-0': 'Velocity scale radius (kpc) & - & [MATH] & [MATH] & -', '1903.09202-2-25-0': 'Jet opening angle (deg.)', '1903.09202-2-25-1': '& - & - & - & [MATH]', '1903.09202-2-26-0': 'Jet [MATH]-axis angle (deg.)', '1903.09202-2-26-1': '[MATH] & - & - & - & [MATH]', '1903.09202-2-27-0': 'Jet [MATH]-axis angle (deg.)', '1903.09202-2-27-1': '[MATH] & - & - & - & [MATH]', '1903.09202-2-28-0': 'aPosition of the kinematic center of the model.', '1903.09202-2-28-1': 'bNumber in parentheses is the uncertainty in the last number.', '1903.09202-2-28-2': 'cAngle of the jet compared to the axis of rotation.', '1903.09202-2-28-3': 'dAngle of the jet compared to the position angle of the rotation.', '1903.09202-2-29-0': 'To better understand the spatial extent and kinematic signature of the [C2] emission, we have fitted a range of simple models to the data.', '1903.09202-2-29-1': 'These models are discussed below and are summarized in Table [REF].', '1903.09202-2-29-2': 'To fit the models, we generate a cube from the given model parameters, which we will refer to as the model cube.', '1903.09202-2-29-3': 'This model cube is then convolved with the ALMA primary beam to account for potential beam smearing effects.', '1903.09202-2-29-4': 'We compare this convolved model data cube, [MATH], with the data cube, [MATH], using a standard chi-squared algorithm, [EQUATION] where [MATH] is the uncertainty (i.e., rms uncertainty of the pixels showing no emission in the same channel as pixel [MATH]).', '1903.09202-2-29-5': 'The model cube has velocities that range from -216 to +216kms[MATH].', '1903.09202-2-29-6': 'This range covers the velocities where the data show [C2] emission but excludes the channels affected by the close companion (see Figures [REF] and [REF]).', '1903.09202-2-29-7': 'Since the adjacent pixels are highly correlated in interferometric data, we do not sum up all of the pixels in our data, but instead use a bootstrap method, whereby we randomly select [MATH] pixels from the data cube and calculate the [MATH] value of these pixels.', '1903.09202-2-29-8': 'We repeat this process until the median [MATH] value remains unchanged.', '1903.09202-2-29-9': 'Here the number of pixels, [MATH], is chosen so that on average each beam only contains a single pixel per single [MATH] calculation.', '1903.09202-2-30-0': 'To fully sample the parameter space of the model and provide more realistic constraints on the possible values of each parameter, we have used a Markov chain Monte Carlo (MCMC) method to sample the parameter space.', '1903.09202-2-30-1': 'In particular, we have used the affine invariant MCMC ensemble sampler code, emcee .', '1903.09202-2-30-2': 'For all parameters, we assume flat priors, and we initiate the parameters with rough estimates from visual inspection of the data.', '1903.09202-2-30-3': 'We have verified that these initial guesses do not affect the results of the MCMC analysis.', '1903.09202-2-30-4': 'Results of the different models are shown in Figure [REF] and Table [REF].', '1903.09202-2-31-0': '## Modeling Results', '1903.09202-2-32-0': 'We have modeled the [C2] kinematics in 3D using several simple models.', '1903.09202-2-32-1': 'The following models were considered in this analysis: (a) a thin rotating disk with a constant velocity profile (Disk A).', '1903.09202-2-32-2': 'This is the fiducial profile of a dark-matter-dominated disk galaxy, which normally is the model employed to explain the gas kinematics in marginally resolved [MATH] quasar host galaxies .', '1903.09202-2-32-3': 'Model (b) is a thin rotating disk with a decreasing velocity profile (Disk B).', '1903.09202-2-32-4': 'This profile would arise if the mass of the galaxy is centrally located.', '1903.09202-2-32-5': 'Model (c) is a thin rotating disk with a constant velocity profile up to a certain radius, after which it decreases to zero (Disk C).', '1903.09202-2-32-6': 'This is the velocity profile of a disk that is truncated at a certain radius, or the velocity profile of a disk that is warped into the plane of the sky at this radius.', '1903.09202-2-32-7': 'Model (d) is an inclined biconical jet embedded in a uniform rotating spherical gas (Rot+Jet D).', '1903.09202-2-32-8': 'This is a simple model of an active galactic nucleus (AGN) jet accelerating and/or ionizing the [C2] emitting ISM.', '1903.09202-2-32-9': 'This possibility will be further discussed in Section [REF].', '1903.09202-2-32-10': 'All of the models have a decreasing intensity profile dependent only on the distance from the center.', '1903.09202-2-32-11': 'This clearly is an oversimplification, as the moment 0 map shows obvious evidence of nonexponential emission (see Figure [REF]).', '1903.09202-2-33-0': 'The number of parameters in each of the models varies from 9 to 12 (Table [REF]).', '1903.09202-2-33-1': 'Each model contains the central position of the galaxy (in two spatial and one frequency dimension), the position angle of the maximum rotation, the central intensity and scale height of the emission, the maximum velocity of rotation, and the dispersion along the line of sight.', '1903.09202-2-33-2': 'Besides these eight parameters, the disk models also need an inclination, and disk models B and C require a scale radius for the velocity.', '1903.09202-2-33-3': 'Finally, the jet model D requires three additional angles to describe the jet angle and opening angle of the jet, as well as a terminal radius at which the jet stops accelerating/ionizing the ISM.', '1903.09202-2-34-0': 'From the fits shown in Figure [REF], it is clear that the kinematics are inconsistent with a simple rotating disk with a constant velocity (Disk A), which is typically assumed in order to derive the dynamical masses of marginally resolved [MATH] quasar host galaxies .', '1903.09202-2-34-1': 'To account for the extended emission detected at the systemic velocity, we need a truncated disk model (Disk C) whereby the systemic velocity goes to zero beyond a certain radius.', '1903.09202-2-34-2': 'Such a scenario could arise from either a gravitationally-dominated disk that has formed in the center of a dispersion-dominated source, or alternatively, a warp of the outer disk into the plane of the sky.', '1903.09202-2-34-3': 'Both scenarios, however, do not provide adequate fits to the observed [C2] distribution and kinematics (Figures [REF] and [REF]).', '1903.09202-2-35-0': 'As can be seen in Figure [REF], from the position of the black hole, the velocity increases to a radius of 0.5kpc (01) and reaches a peak line-of-sight velocity of 150-200kms[MATH].', '1903.09202-2-35-1': 'Assuming these kinematics can be described by a rotating disk, and the dynamical mass within this radius is [MATH]/sin[MATH], with [MATH] the inclination angle of the disk.', '1903.09202-2-35-2': 'For comparison, the mass of the black hole is [MATH][MATH], and the inferred gas mass within this radius is [MATH][MATH].', '1903.09202-2-35-3': 'For an inclination angle of [MATH] (see Table [REF]), the dynamical mass and molecular gas mass within the central 1kpc are consistent with each other within the large uncertainties.', '1903.09202-2-35-4': 'Beyond this radius, the line-of-sight velocity is decreasing faster than Keplerian (shown by the Disk B model in Figure [REF]), and reaches approximately systemic velocity at the last measured points at 1.5-2.0kpc.', '1903.09202-2-35-5': 'To summarize, the kinematics appear to be dispersion-dominated with some overall rotation (i.e. net angular momentum) in the central kiloparsec. This implies that most of the gas has not yet settled in a disk.', '1903.09202-2-36-0': '# Origin of the [C 2] cavities', '1903.09202-2-37-0': 'Instead of a kinematic origin, the presence of the two-kiloparsec-scale [C2] cavities could be the result of energy injection into the ISM.', '1903.09202-2-37-1': 'Similar shell-like structures are seen in the neutral ISM of local galaxies , and are often interpreted as the results of supernova or AGN feedback.', '1903.09202-2-37-2': 'We can calculate the energy needed to created such structures seen here using [EQUATION] with [MATH] being the density in cm[MATH], [MATH] being the diameter of the cavity in parsec, and [MATH] the expansion velocity in kms[MATH].', '1903.09202-2-37-3': 'The cavities have a diameter of approximately [MATH]0.5kpc (see Figure [REF]), and the luminosity-weighted, average density of the ISM in the quasar host has been estimated to be 10[MATH]cm[MATH] based on a multiline analysis in this source.', '1903.09202-2-37-4': 'Under the assumption that the S-like structure is due to energy input in the plane of the galaxy, the expansion velocity can be approximated by the amplitude of the derivation from systemic velocity .', '1903.09202-2-37-5': 'We estimate an amplitude of [MATH]80kms[MATH] at the center of the cavities (Figure [REF]).', '1903.09202-2-37-6': 'Taken together, we derive a required energy of [MATH]b (kms[MATH]) & ... & +361 & -137 & -289 & +595', '1903.09202-2-38-0': '[MATH]a (Jykms[MATH]) & 5.43[MATH]0.33 & 0.43[MATH]0.05 & 0.31[MATH]0.08 & 1.13[MATH]0.17 & [MATH]0.23', '1903.09202-2-39-0': 'FWHM[MATH] (kms[MATH]) & 225[MATH]15 & 180[MATH]25 & 135[MATH]40 & 330[MATH]55 & -', '1903.09202-2-40-0': '[MATH]c (mJy) & 5.34[MATH]0.19 & ...d & [MATH]0.23 & 0.58[MATH]0.12 & [MATH]0.33', '1903.09202-2-41-0': '[MATH]) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]) & [MATH] & ... & [MATH]', '1903.09202-2-42-0': 'SFR[MATH]e ([MATH]yr[MATH]) & 1016[MATH]73 & 51[MATH]9 & 35[MATH]10 & 159[MATH]28 & [MATH]f ([MATH]) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1903.09202-2-43-0': 'Notes.', '1903.09202-2-43-1': 'All the quoted errors are [MATH] and the upper limits are [MATH].', '1903.09202-2-43-2': 'aThe redshift and [C2] line flux ([MATH]) are measured using a Gaussian fit to the [C2] line for the quasar and the companions C1, C2, and C3 (as shown in Figure [REF]).', '1903.09202-2-43-3': 'The redshift of the Ly[MATH] emitter (LAE) is taken from [CITATION].', '1903.09202-2-43-4': 'bThe line-of-sight velocity [MATH] is computed using [MATH].', '1903.09202-2-43-5': 'c The continuum flux density at a rest-frame wavelength of 158[MATH]m ([MATH]) is measured by averaging the line-free channels in the spectrum around the [C2] line.', '1903.09202-2-43-6': 'dThe continuum flux density of C1 could not be accurately determined due to contamination by the quasar host galaxy.', '1903.09202-2-43-7': 'eThe star-formation rate SFR[MATH] is derived using SFR[MATH] .', '1903.09202-2-43-8': 'fMolecular gas mass, derived from the [C2] luminosity using a [MATH]-to-H[MATH] conversion factor of [MATH][MATH]/[MATH] .', '1903.09202-2-44-0': 'The environment of J0305-3150 has been the subject of various studies.', '1903.09202-2-44-1': 'A study searching for Ly[MATH] emission from galaxies at the redshift of the quasar revealed the presence of a Ly[MATH] halo around the quasar, and a faint Ly[MATH] emitting (LAE) companion at a distance of 12.5kpc .', '1903.09202-2-44-2': 'In addition, a larger field-of-view narrowband search for Ly[MATH] emitters around the quasar indicated that the quasar is not located in a dense environment on megaparsec scales .', '1903.09202-2-44-3': 'Our previous, shallow ALMA data of the field contained two faint (rest frame 158[MATH]m flux densities of [MATH]mJy) continuum sources .', '1903.09202-2-45-0': 'In addition to high-resolution imaging of the quasar host galaxy, the improved sensitivity of the new ALMA data allowed us to search for additional sources in the field.', '1903.09202-2-45-1': 'Previously, bright companion sources have been found in the vicinity of some high-redshift quasars .', '1903.09202-2-45-2': 'Our new data reveal the presence of three [C2] emitters at the same redshift as the quasar (Figure [REF]; physical parameters in Table [REF]).', '1903.09202-2-45-3': 'One companion, C1, could already be seen in Figure [REF].', '1903.09202-2-45-4': 'The brightest companion, C3, was already identified as a continuum source in the field in [CITATION].', '1903.09202-2-46-0': 'The FIR luminosity of companion C3 is comparable to that of the [C2] companions near several [MATH] quasars presented in [CITATION].', '1903.09202-2-46-1': 'However, in contrast to the companions found by [CITATION], all companions identified near J0305-3150 have FIR luminosities that are a factor [MATH]10 smaller than that of the quasar host.', '1903.09202-2-46-2': 'Also, the estimated molecular gas masses of the companions are at least a factor of [MATH]5 smaller than that of the quasar host.', '1903.09202-2-46-3': 'Despite the small masses, the interaction of companions with the quasar host could provide an alternative explanation of the gas morphology and kinematics in the quasar host galaxy.', '1903.09202-2-46-4': 'Furthermore, it is a possibility that such interactions triggered the high star-formation rate in the quasar host and the AGN activity.', '1903.09202-2-47-0': '# Summary', '1903.09202-2-48-0': 'The high spatial resolution ([MATH]400pc) imaging of a quasar host galaxy at [MATH] shows that its formation is a complex and chaotic process.', '1903.09202-2-48-1': 'We find that the ISM in the quasar host has not yet settled in a simple disk.', '1903.09202-2-48-2': 'While there are signs that the AGN is affecting the ISM, this feedback is not suppressing the formation of stars in the quasar host.', '1903.09202-2-48-3': 'In fact, the star-formation rate inferred by the FIR luminosity of [MATH]yr[MATH] is high among galaxies and quasar hosts at similar redshifts .', '1903.09202-2-48-4': 'The observed high star-formation rate and rapid black hole growth could be triggered by interactions with the newly detected nearby companion galaxies.', '1903.09202-2-48-5': 'The observations presented here show the unique role that high-angular resolution observations with ALMA can play in studies of the ISM in some of the most distant massive galaxies.', '1903.09202-2-49-0': 'B.P.V., M.N., and F.W. acknowledge funding through the ERC grant "Cosmic Gas."', '1903.09202-2-49-1': 'This paper makes use of the following ALMA data: ADS/JAO.ALMA2017.1.01532.', '1903.09202-2-49-2': 'S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile.', '1903.09202-2-49-3': 'The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ.', '1903.09202-2-50-0': 'ALMA'} | [['1903.09202-1-29-0', '1903.09202-2-29-0'], ['1903.09202-1-29-2', '1903.09202-2-29-2'], ['1903.09202-1-29-3', '1903.09202-2-29-3'], ['1903.09202-1-29-5', '1903.09202-2-29-5'], ['1903.09202-1-29-6', '1903.09202-2-29-6'], ['1903.09202-1-29-8', '1903.09202-2-29-8'], ['1903.09202-1-29-9', '1903.09202-2-29-9'], ['1903.09202-1-45-0', '1903.09202-2-45-0'], ['1903.09202-1-45-3', '1903.09202-2-45-3'], ['1903.09202-1-45-4', '1903.09202-2-45-4'], ['1903.09202-1-4-1', '1903.09202-2-4-1'], ['1903.09202-1-4-2', '1903.09202-2-4-2'], ['1903.09202-1-4-3', '1903.09202-2-4-3'], ['1903.09202-1-28-1', '1903.09202-2-28-1'], ['1903.09202-1-28-2', '1903.09202-2-28-2'], ['1903.09202-1-28-3', 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['1903.09202-1-6-2', '1903.09202-2-6-2'], ['1903.09202-1-6-3', '1903.09202-2-6-3'], ['1903.09202-1-6-4', '1903.09202-2-6-4'], ['1903.09202-1-34-0', '1903.09202-2-34-0'], ['1903.09202-1-34-1', '1903.09202-2-34-1'], ['1903.09202-1-34-2', '1903.09202-2-34-2'], ['1903.09202-1-3-3', '1903.09202-2-3-3'], ['1903.09202-1-3-5', '1903.09202-2-3-5'], ['1903.09202-1-14-1', '1903.09202-2-14-1'], ['1903.09202-1-5-2', '1903.09202-2-5-2'], ['1903.09202-1-30-0', '1903.09202-2-30-0'], ['1903.09202-1-37-0', '1903.09202-2-37-0'], ['1903.09202-1-37-1', '1903.09202-2-37-1'], ['1903.09202-1-37-2', '1903.09202-2-37-2'], ['1903.09202-1-37-3', '1903.09202-2-37-3'], ['1903.09202-1-37-4', '1903.09202-2-37-4'], ['1903.09202-1-37-5', '1903.09202-2-37-5'], ['1903.09202-1-49-0', '1903.09202-2-49-0'], ['1903.09202-1-49-2', '1903.09202-2-49-2'], ['1903.09202-1-49-3', '1903.09202-2-49-3'], ['1903.09202-1-11-0', '1903.09202-2-11-0'], ['1903.09202-1-11-1', '1903.09202-2-11-1'], ['1903.09202-1-11-2', '1903.09202-2-11-2'], ['1903.09202-1-11-3', '1903.09202-2-11-3'], ['1903.09202-1-8-0', '1903.09202-2-8-0'], ['1903.09202-1-8-1', '1903.09202-2-8-1'], ['1903.09202-1-8-2', '1903.09202-2-8-2'], ['1903.09202-1-8-3', '1903.09202-2-8-3'], ['1903.09202-1-43-0', '1903.09202-2-43-2'], ['1903.09202-1-43-1', '1903.09202-2-43-3'], ['1903.09202-1-0-0', '1903.09202-2-0-0'], ['1903.09202-1-0-2', '1903.09202-2-0-2'], ['1903.09202-1-32-1', '1903.09202-2-32-1'], ['1903.09202-1-32-2', '1903.09202-2-32-2'], ['1903.09202-1-32-8', '1903.09202-2-32-8'], ['1903.09202-1-32-10', '1903.09202-2-32-10'], ['1903.09202-1-32-11', '1903.09202-2-32-11'], ['1903.09202-1-35-0', '1903.09202-2-35-0'], ['1903.09202-1-35-1', '1903.09202-2-35-1'], ['1903.09202-1-35-2', '1903.09202-2-35-2'], ['1903.09202-1-35-4', '1903.09202-2-35-4'], ['1903.09202-1-10-0', '1903.09202-2-10-0'], ['1903.09202-1-10-1', '1903.09202-2-10-1'], ['1903.09202-1-10-2', '1903.09202-2-10-2'], ['1903.09202-1-33-1', '1903.09202-2-33-1'], ['1903.09202-1-33-2', '1903.09202-2-33-2'], ['1903.09202-1-33-3', '1903.09202-2-33-3'], ['1903.09202-1-44-1', '1903.09202-2-44-1'], ['1903.09202-1-44-2', '1903.09202-2-44-2'], ['1903.09202-1-44-3', '1903.09202-2-44-3']] | [] | [['1903.09202-1-3-4', '1903.09202-2-3-4'], ['1903.09202-1-48-2', '1903.09202-2-48-2'], ['1903.09202-1-48-2', '1903.09202-2-48-3'], ['1903.09202-1-32-3', '1903.09202-2-32-3'], ['1903.09202-1-32-5', '1903.09202-2-32-3'], ['1903.09202-1-32-5', '1903.09202-2-32-5'], ['1903.09202-1-32-7', '1903.09202-2-32-7'], ['1903.09202-1-35-5', '1903.09202-2-35-5'], ['1903.09202-1-35-6', '1903.09202-2-35-5']] | [] | ['1903.09202-1-14-0', '1903.09202-1-15-0', '1903.09202-1-16-0', '1903.09202-1-17-0', '1903.09202-1-18-0', '1903.09202-1-18-1', '1903.09202-1-19-0', '1903.09202-1-19-1', '1903.09202-1-20-0', '1903.09202-1-21-0', '1903.09202-1-22-0', '1903.09202-1-23-0', '1903.09202-1-24-0', '1903.09202-1-25-0', '1903.09202-1-25-1', '1903.09202-1-26-0', '1903.09202-1-26-1', '1903.09202-1-27-0', '1903.09202-1-27-1', '1903.09202-1-37-6', '1903.09202-1-38-0', '1903.09202-1-39-0', '1903.09202-1-40-0', '1903.09202-1-41-0', '1903.09202-1-42-0', '1903.09202-1-50-0', '1903.09202-2-14-0', '1903.09202-2-15-0', '1903.09202-2-16-0', '1903.09202-2-16-1', '1903.09202-2-17-0', '1903.09202-2-18-0', '1903.09202-2-18-1', '1903.09202-2-19-0', '1903.09202-2-19-1', '1903.09202-2-20-0', '1903.09202-2-21-0', '1903.09202-2-22-0', '1903.09202-2-23-0', '1903.09202-2-24-0', '1903.09202-2-25-0', '1903.09202-2-25-1', '1903.09202-2-26-0', '1903.09202-2-26-1', '1903.09202-2-27-0', '1903.09202-2-27-1', '1903.09202-2-37-6', '1903.09202-2-38-0', '1903.09202-2-39-0', '1903.09202-2-40-0', '1903.09202-2-41-0', '1903.09202-2-42-0', '1903.09202-2-43-0', '1903.09202-2-50-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1903.09202 | null | null | null | null | null |
1412.4415 | {'1412.4415-1-0-0': 'A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced.', '1412.4415-1-0-1': 'A classification of Lie symmetries and low-order conservation laws, along with peaked travelling wave solutions, is presented for this family.', '1412.4415-1-0-2': 'These classifications pick out a 1-parameter equation that shares the same low-order Hamiltonian conservation laws and peakon solutions as the Camassa-Holm and Novikov equations.', '1412.4415-1-1-0': '# Introduction', '1412.4415-1-2-0': 'There is considerable interest in the study of equations of the form [MATH] that describe breaking waves.', '1412.4415-1-2-1': 'In this paper we consider the equation [EQUATION] with parameters [MATH] (not all zero) and [MATH].', '1412.4415-1-2-2': 'This 4-parameter family contains several integrable equations.', '1412.4415-1-2-3': 'For [MATH] and [MATH], equation [REF] reduces respectively to the Camassa-Holm equation [CITATION] [EQUATION] and the Degasperis-Procesi equation [CITATION] [EQUATION] while for [MATH], equation [REF] becomes the Novikov equation [CITATION] [EQUATION]', '1412.4415-1-2-4': 'The three equations [REF], [REF], [REF] are integrable in the sense of having a Lax pair, a bi-Hamiltonian structure, as well as hierarchies of local symmetries and local conservation laws, and they also possess peaked travelling wave solutions.', '1412.4415-1-3-0': 'In addition to these integrable equations, a considerable number of other non-integrable equations that admit breaking waves are included in the 4-parameter family [REF].', '1412.4415-1-3-1': 'For instance, there is the [MATH]-equation [EQUATION] which unifies the Camassa-Holm and Degasperis-Procesi equations [CITATION].', '1412.4415-1-3-2': 'There is also a modified version of the [MATH]-equation [CITATION] [EQUATION] which includes the Novikov equation.', '1412.4415-1-3-3': 'No other cases of the two equations [REF] and [REF] are integrable [CITATION].', '1412.4415-1-4-0': 'An equivalent form of the 4-parameter equation [REF] is given by [EQUATION] in terms of the momentum variable [EQUATION] with parameters [EQUATION]', '1412.4415-1-4-1': 'This parametric equation [REF] is invariant under the group of scaling transformations [MATH], [MATH], [MATH] with [MATH].', '1412.4415-1-5-0': 'In section [REF], we classify the Lie symmetry group of equation [REF].', '1412.4415-1-5-1': 'In section [REF], we classify the low-order conservation laws of equation [REF] and show that the Hamiltonians of the Camassa-Holm and Novikov equations are admitted as local conservation laws by equation [REF] if and only if [MATH] and [MATH].', '1412.4415-1-5-2': 'We consider peaked travelling waves in section [REF] and show that equation [REF] admits a peakon solution if and only if [MATH] and [MATH].', '1412.4415-1-5-3': 'Finally, in section [REF], we combine the previous results to obtain a natural subfamily of equations given by [MATH], [MATH], [MATH], [MATH] which unifies the Camassa-Holm and Novikov equations into a 1-parameter equation with a rich structure of local conservation laws and peakon solutions.', '1412.4415-1-6-0': '# Lie symmetries', '1412.4415-1-7-0': 'The Lie symmetry group of equation [REF] comprises point symmetries as well as contact symmetries, since the equation involves only a single dependent variable [MATH].', '1412.4415-1-8-0': 'A point symmetry [CITATION] of equation [REF] is a group of transformations on [MATH] given by an infinitesimal generator [EQUATION] whose prolongation satisfies [EQUATION] for all formal solutions [MATH] of equation [REF].', '1412.4415-1-8-1': 'When acting on solutions, any point symmetry [REF] is equivalent to an infinitesimal generator with the characteristic form [EQUATION] where the characteristic functions [MATH], [MATH], [MATH] are determined by [EQUATION] holding for all formal solutions [MATH] of equation [REF].', '1412.4415-1-8-2': 'Thus, equation [REF] constitutes the determining equation for point symmetries of equation [REF].', '1412.4415-1-8-3': 'This formulation is useful for doing computations and for considering extensions to contact symmetries, as well as for making a connection with conserved densities.', '1412.4415-1-9-0': 'A contact symmetry [CITATION] extends the definition of invariance [REF] by allowing the transformations to depend essentially on first order derivatives of [MATH], as given by an infinitesimal generator with the characteristic form [EQUATION]', '1412.4415-1-9-1': 'The corresponding transformations on [MATH] are given by [EQUATION] where [EQUATION] which follows from preservation of the contact condition [MATH].', '1412.4415-1-9-2': 'Note that a contact symmetry reduces to a (prolonged) point symmetry if and only if [MATH] is a linear function of [MATH] and [MATH].', '1412.4415-1-10-0': 'The set of all Lie symmetries admitted by equation [REF] inherits the structure of a Lie algebra under commutation of the operators [MATH].', '1412.4415-1-10-1': 'For a given (sub)algebra of point or contact symmetries, the corresponding group of transformations has a natural action [CITATION] on the set of all solutions [MATH].', '1412.4415-1-11-0': 'To classify all of the Lie symmetries admitted by equation [REF], we first substitute a general characteristic function [MATH] into the symmetry determining equation [REF].', '1412.4415-1-11-1': 'Next we eliminate [MATH], [MATH], [MATH] through writing the equation in the solved form [EQUATION] and doing the same for its differential consequences.', '1412.4415-1-11-2': 'The determining equation [REF] then splits with respect to [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] into a linear overdetermined system of equations on [MATH], which can be straightforwardly solved by use of Maple (see the Appendix).', '1412.4415-1-11-3': 'This leads to the following classification result.', '1412.4415-1-12-0': 'A symmetry analysis of particular equations in the family [REF] can be found in Refs. [CITATION].', '1412.4415-1-13-0': '# Conservation laws', '1412.4415-1-14-0': 'A local conservation law [CITATION] for equation [REF] is a space-time divergence [EQUATION] holding for all formal solutions [MATH] of equation [REF], where the conserved density [MATH] and the spatial flux [MATH] are functions of [MATH], [MATH], [MATH] and derivatives of [MATH].', '1412.4415-1-14-1': 'The spatial integral of the conserved density [MATH] satisfies [EQUATION] and so if the flux [MATH] vanishes at spatial infinity, then [EQUATION] formally yields a conserved quantity for equation [REF].', '1412.4415-1-14-2': 'Conversely, any such conserved quantity arises from a local conservation law [REF].', '1412.4415-1-15-0': 'If the conserved quantity [REF] is purely a boundary term, then the local conservation law is called trivial.', '1412.4415-1-15-1': 'This occurs when (and only when) the conserved density is a total [MATH]-derivative and the flux is a total [MATH]-derivative, related by [EQUATION] for all formal solutions [MATH] of equation [REF], where [MATH] is some function of [MATH], [MATH], [MATH] and derivatives of [MATH].', '1412.4415-1-15-2': 'Two local conservation laws are equivalent if they differ by a trivial conservation law, thereby giving the same conserved quantity up to boundary terms.', '1412.4415-1-16-0': 'The set of all conservation laws (up to equivalence) admitted by equation [REF] forms a vector space on which there is a natural action [CITATION] by the group of all Lie symmetries of the equation.', '1412.4415-1-17-0': 'For conserved densities and fluxes depending on at most [MATH], a conservation law can be expressed in an equivalent form by a divergence identity [EQUATION] holding off solutions, where [EQUATION] is called the multiplier.', '1412.4415-1-17-1': 'This identity [REF]-[REF] is called the characteristic equation [CITATION] for the conserved density and flux.', '1412.4415-1-17-2': 'By balancing the highest order [MATH]-derivative terms [MATH] on both sides of the equation, we directly find that [MATH] and [MATH].', '1412.4415-1-17-3': 'Then balancing the terms [MATH], we see that [MATH].', '1412.4415-1-17-4': 'Hence the conserved density and the flux of such a conservation law must have the form [EQUATION]', '1412.4415-1-17-5': 'Consequently, the multiplier [REF] in the characteristic equation [REF] has the form [EQUATION]', '1412.4415-1-17-6': 'In general, the differential order of a local conservation law is defined to be the smallest differential order among all equivalent conserved densities.', '1412.4415-1-17-7': 'A local conservation law is said to be of low order if the differential order of its multiplier is strictly less than the differential order of the equation.', '1412.4415-1-18-0': 'All low-order multipliers [REF] are determined by the condition [EQUATION] since space-time divergences are characterized by the fact that their variational derivative with respect to [MATH] vanishes identically, where [EQUATION] denotes the variational derivative (Euler operator) [CITATION].', '1412.4415-1-18-1': 'The determining equation [REF] has a direct splitting with respect to [MATH] and [MATH]-derivatives of [MATH], yielding an equivalent overdetermined system of equations on [MATH].', '1412.4415-1-18-2': 'One equation in this system is given by the adjoint of the symmetry determining equation [REF], [EQUATION] holding for all formal solutions [MATH] of equation [REF].', '1412.4415-1-18-3': 'Solutions [MATH] of this equation [REF] are called adjoint-symmetries (or cosymmetries) [CITATION].', '1412.4415-1-18-4': 'The remaining equations in the system comprise Helmholtz conditions which are necessary and sufficient for [MATH] to have the form [REF].', '1412.4415-1-18-5': 'As a consequence, multipliers [REF] are simply adjoint-symmetries that have a certain variational form, and the determination of local conservation laws via multipliers is a kind of adjoint problem of the determination of symmetries [CITATION].', '1412.4415-1-19-0': 'For any solution [REF] of the determining equation [REF], a corresponding conserved density and flux of the form [REF] can be recovered either through integration [CITATION] of the characteristic equation [REF], which splits with respect to [MATH], [MATH], [MATH], [MATH] into a system of equations for [MATH] and [MATH], or through a homotopy integral formula [CITATION] , which expresses [MATH] and [MATH] directly in terms of [MATH].', '1412.4415-1-19-1': 'It is straightforward to show that [MATH] and [MATH] have the form [REF] of a trivial conservation law iff [MATH].', '1412.4415-1-19-2': 'Thus there is a one-to-one correspondence between equivalence classes of non-trivial low-order conservation laws [REF] and non-zero low-order multipliers [REF].', '1412.4415-1-20-0': 'Both the Camassa-Holm equation [REF] and Novikov equation [REF] possess low-order local conservations law given by the conserved densities [CITATION]', '1412.4415-1-21-0': 'T=mu = u^2 + u_x^2 +D_x(-uu_x),', '1412.4415-1-22-0': 'T=m^p,', '1412.4415-1-23-0': 'where [MATH] and [MATH], respectively, for the two equations.', '1412.4415-1-23-1': 'In addition, the Camassa-Holm equation [REF] itself is a low-order local conservation law having the conserved density [EQUATION]', '1412.4415-1-23-2': 'All of these conserved densities are related to Hamiltonian structures for the two equations [CITATION].', '1412.4415-1-23-3': 'The corresponding multipliers are respectively given by [EQUATION]', '1412.4415-1-23-4': 'To look for conserved densities of the same form for equation [REF], we now classify all multipliers up to 1st-order [EQUATION] as well as all 2nd-order multipliers with the specific form [EQUATION]', '1412.4415-1-23-5': 'In each case it is straightforward to solve the determining equation [REF] by use of Maple (see the Appendix), which leads to the following classification result.', '1412.4415-1-24-0': 'In light of the adjoint relationship between multipliers and symmetries, the classification of 0th- and 1st- order multipliers in Proposition [REF] is a counterpart of the classification of point and contact symmetries in Proposition [REF].', '1412.4415-1-25-0': 'Next we obtain the corresponding conserved densities and fluxes for each multiplier [REF]-[REF] by first splitting the characteristic equation [REF] with respect to [MATH], [MATH], [MATH], [MATH] where [MATH] and [MATH] have the form [REF], and then integrating the resulting system of equations.', '1412.4415-1-25-1': 'This yields the following low-order local conservation laws for equation [REF].', '1412.4415-1-26-0': '# Peakon solutions', '1412.4415-1-27-0': 'Both the Camassa-Holm and Novikov equations possess peaked travelling wave solutions, called peakons [CITATION], [EQUATION] where [MATH] and [MATH], respectively, for the two equations.', '1412.4415-1-27-1': 'Peakons have attracted much attention in the study of breaking wave equations (see, e.g. Refs. [CITATION] and references therein).', '1412.4415-1-28-0': 'In general, a peakon is a weak travelling wave solution satisfying an integral (i.e. weak) formulation of a breaking wave equation.', '1412.4415-1-28-1': 'Such a formulation is essential for deriving multi-peakon solutions.', '1412.4415-1-28-2': 'However, single peakons can be derived directly from the travelling wave reduction of a breaking wave equation, which will be the approach we use here.', '1412.4415-1-29-0': 'Invariance of the 4-parameter equation [REF] under time-translation and space-translation point symmetries implies the existence of travelling wave solutions [EQUATION] where [MATH] satisfies the ODE [EQUATION]', '1412.4415-1-29-1': 'For the travelling wave ODE [REF], an integral formulation is obtained through multiplying this ODE by a test function [MATH] (which is smooth and has compact support) and integrating over [MATH], leaving at most first derivatives of [MATH] in the integral, which yields [EQUATION]', '1412.4415-1-29-2': 'A weak solution of ODE [REF] is a function [MATH] that belongs to the Sobolev space [MATH] and that satisfies the integral equation [REF] for all smooth test functions [MATH] with compact support on [MATH].', '1412.4415-1-30-0': 'To proceed we substitute a peaked travelling wave expression [EQUATION] into equation [REF] and split up the integral into the intervals [MATH] and [MATH].', '1412.4415-1-30-1': 'The first term in equation [REF] yields, after integration by parts, [EQUATION]', '1412.4415-1-30-2': 'Similarly, the second term in equation [REF] gives [EQUATION] provided [EQUATION] so that the boundary terms at [MATH] vanish.', '1412.4415-1-30-3': 'The third and fourth terms in equation [REF] together yield [EQUATION]', '1412.4415-1-30-4': 'When the terms [REF]-[REF] are combined, we find that equation [REF] reduces to [EQUATION]', '1412.4415-1-30-5': 'This equation is satisfied for all test functions [MATH] iff [EQUATION] which determines the amplitude [MATH] in the peakon expression [REF].', '1412.4415-1-30-6': 'Thus we obtain the following result.', '1412.4415-1-31-0': 'The resulting peakon solution of equation [REF] is given by [EQUATION]', '1412.4415-1-31-1': 'When the nonlinearity power [MATH] is a positive integer, then the wave speed is necessarily positive, [MATH], if [MATH] is even, as in the case ([MATH]) of the Novikov equation [REF], while if [MATH] is odd, the wave speed can be either positive or negative, [MATH], as in the case ([MATH]) of the Camassa-Holm equation [REF].', '1412.4415-1-32-0': 'We remark that the peakon solution satisfies equation [REF] in the sense of a weak solution.', '1412.4415-1-32-1': 'This means [MATH] is a distribution in [MATH] with respect to [MATH] for some [MATH] and in [MATH] with respect to [MATH] such that it satisfies the integral equation [EQUATION] for all test functions [MATH] in [MATH].', '1412.4415-1-33-0': '# Unified family of Camassa-Holm-Novikov equations', '1412.4415-1-34-0': 'From Propositions [REF] and [REF], the low-order conservation laws [REF]-[REF] as well as the peakon solutions [REF] of the Camassa-Holm and Novikov equations are admitted simultaneously by the 4-parameter wave-breaking equation [REF] iff its parameters [MATH] satisfy [EQUATION]', '1412.4415-1-34-1': 'In this case, equation [REF] reduces to the 1-parameter equation [EQUATION] where the parameter [MATH] has been absorbed by the scaling transformation [MATH].', '1412.4415-1-35-0': 'We will refer to equation [REF] as the gCHN equation, since it is the simplest 1-parameter polynomial family of wave-breaking equations unifying both the Camassa-Holm and Novikov equations (given by [MATH] and [MATH], respectively) as well as their low-order conservation laws and peakon solutions.', '1412.4415-1-35-1': 'This equation [REF] was first deduced in Ref. [CITATION] using a different approach.', '1412.4415-1-36-0': 'At first sight, the gCHN equation [REF] seems closely analogous to the [MATH]-equation [REF]: both equations unify two integrable equations, possess peakon solutions, and exhibit wave breaking phenomena.', '1412.4415-1-36-1': 'However, there are important differences.', '1412.4415-1-36-2': 'Firstly, the nonlinearities in the [MATH]-equation are purely quadratic, whereas the gCHN equation has nonlinearities of degree [MATH] and thus connects two integrable equations with different nonlinearities.', '1412.4415-1-36-3': 'Secondly, from Proposition [REF], the [MATH]-equation admits the conserved density [MATH] only for [MATH], when the [MATH]-equation reduces to the Camassa-Holm equation.', '1412.4415-1-36-4': 'In contrast, the gCHN equation admits this conserved density for all [MATH].', '1412.4415-1-36-5': 'This implies that the [MATH] norm of solutions [MATH] is conserved for the gCHN equation but not for the [MATH]-equation if [MATH].', '1412.4415-1-37-0': 'In a subsequent paper [CITATION], we will explore the integrability properties of the gCHN equation [REF].', '1412.4415-1-37-1': 'Interesting questions are whether, for any [MATH] other than the two known integrable cases [MATH] and [MATH], this equation [REF] admits multi-peakon solutions with a Hamiltonian structure; a Lax pair; a bi-Hamiltonian formulation; and a hierarchy of higher order symmetries and conservation laws.'} | {'1412.4415-2-0-0': 'A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced.', '1412.4415-2-0-1': 'A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family.', '1412.4415-2-0-2': 'These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved [MATH] norm and it possesses [MATH]-peakon solutions, when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case [MATH].', '1412.4415-2-1-0': '# Introduction', '1412.4415-2-2-0': 'There is considerable interest in the study of equations of the form [MATH] that describe breaking waves.', '1412.4415-2-2-1': 'In this paper we consider the equation [EQUATION] with parameters [MATH] (not all zero) and [MATH].', '1412.4415-2-2-2': 'This 4-parameter family contains several integrable equations.', '1412.4415-2-2-3': 'For [MATH] and [MATH], equation [REF] reduces respectively to the Camassa-Holm equation [CITATION] [EQUATION] and the Degasperis-Procesi equation [CITATION] [EQUATION] while for [MATH], equation [REF] becomes the Novikov equation [CITATION] [EQUATION]', '1412.4415-2-2-4': 'The three equations [REF], [REF], [REF] are integrable in the sense of having a Lax pair, a bi-Hamiltonian structure, as well as hierarchies of local symmetries and local conservation laws, and they also possess peaked travelling wave solutions.', '1412.4415-2-3-0': 'In addition to these integrable equations, many other non-integrable equations that admit breaking waves are included in the 4-parameter family [REF].', '1412.4415-2-3-1': 'For instance, there is the [MATH]-equation [EQUATION] which unifies the Camassa-Holm and Degasperis-Procesi equations [CITATION].', '1412.4415-2-3-2': 'There is also a modified version of the [MATH]-equation [CITATION] [EQUATION] which includes the Novikov equation.', '1412.4415-2-3-3': 'No other cases of the two equations [REF] and [REF] are known to be integrable [CITATION].', '1412.4415-2-4-0': 'An equivalent form of the 4-parameter equation [REF] is given by [EQUATION] in terms of the momentum variable [EQUATION] with parameters [EQUATION]', '1412.4415-2-4-1': 'This parametric equation [REF] is invariant under the group of scaling transformations [MATH], [MATH], [MATH] with [MATH].', '1412.4415-2-5-0': 'In section [REF], we classify the low-order conservation laws of equation [REF] and show that the Hamiltonians of the Camassa-Holm and Novikov equations are admitted as local conservation laws by equation [REF] if and only if [MATH] and [MATH].', '1412.4415-2-5-1': 'We consider peaked travelling waves in section [REF] and use a weak formulation of equation [REF] to show that single peakon and multi-peakon solutions are admitted if and only if [MATH] and [MATH] when [MATH].', '1412.4415-2-5-2': 'We derive the explicit equations of motion for [MATH] peakon/anti-peakon solutions and also obtain the constants of motion inherited from the local conservation laws of equation [REF].', '1412.4415-2-6-0': 'In section [REF], we combine the previous results to obtain a natural 1-parameter family of equations [EQUATION] given by [MATH], [MATH], [MATH], [MATH], where a scaling transformation [MATH] is used to put [MATH].', '1412.4415-2-6-1': 'Since this 1-parameter family [REF] unifies the Camassa-Holm and Novikov equations, we will refer to it as the gCHN equation.', '1412.4415-2-6-2': '(Similar unified equations have been considered previously from related perspectives [CITATION].)', '1412.4415-2-6-3': 'We then discuss some general features of the dynamics of its [MATH] peakon/anti-peakon solutions and we show that wave-breaking occurs for certain solutions describing collisions between peakons and anti-peakons in the case [MATH].', '1412.4415-2-7-0': 'Finally, in section [REF], we make some concluding remarks including a possible scenario for wave-breaking in the Cauchy problem for weak solutions.', '1412.4415-2-8-0': '# Conservation laws', '1412.4415-2-9-0': 'For the 4-parameter equation [REF], a local conservation law [CITATION] is a space-time divergence [EQUATION] holding for all solutions [MATH] of equation [REF], where the conserved density [MATH] and the spatial flux [MATH] are functions of [MATH], [MATH], [MATH] and derivatives of [MATH].', '1412.4415-2-9-1': 'The spatial integral of the conserved density [MATH] satisfies [EQUATION] and so if the flux [MATH] vanishes at spatial infinity, then [EQUATION] formally yields a conserved quantity for equation [REF].', '1412.4415-2-9-2': 'Conversely, any such conserved quantity arises from a local conservation law [REF].', '1412.4415-2-10-0': 'If the conserved quantity [REF] is purely a boundary term, then the local conservation law is called trivial.', '1412.4415-2-10-1': 'This occurs when (and only when) the conserved density is a total [MATH]-derivative and the flux is a total [MATH]-derivative, related by [EQUATION] for all solutions [MATH] of equation [REF], where [MATH] is some function of [MATH], [MATH], [MATH] and derivatives of [MATH].', '1412.4415-2-10-2': 'Two local conservation laws are equivalent if they differ by a trivial conservation law, thereby giving the same conserved quantity up to boundary terms.', '1412.4415-2-11-0': 'The set of all conservation laws (up to equivalence) admitted by equation [REF] forms a vector space on which there is a natural action [CITATION] by the group of all Lie symmetries of the equation.', '1412.4415-2-12-0': 'For conserved densities and fluxes depending on at most [MATH], a conservation law can be expressed in an equivalent form by a divergence identity [EQUATION] where [EQUATION] is called the multiplier.', '1412.4415-2-12-1': 'This identity [REF]-[REF] is called the characteristic equation [CITATION] for the conserved density and flux.', '1412.4415-2-12-2': 'By balancing the highest order [MATH]-derivative terms [MATH] on both sides of the equation, we directly find that [MATH] and [MATH].', '1412.4415-2-12-3': 'Then balancing the terms [MATH], we see that [MATH].', '1412.4415-2-12-4': 'Hence the conserved density and the flux in the divergence identity must have the form [EQUATION]', '1412.4415-2-12-5': 'Its multiplier [REF] thus has the form [EQUATION]', '1412.4415-2-12-6': 'In general, the differential order of a local conservation law is defined to be the smallest differential order among all equivalent conserved densities.', '1412.4415-2-12-7': 'A local conservation law is said to be of low order if the differential orders of [MATH] and [MATH] are both strictly less than the differential order of the equation.', '1412.4415-2-13-0': 'Consequently, conserved densities and fluxes of the form [REF] comprise all possible low-order conservation laws of equation [REF].', '1412.4415-2-13-1': 'The problem of finding all low-order conservations then reduces to the simpler problem of finding all low-order multipliers [REF].', '1412.4415-2-13-2': 'Since equation [REF] is an evolution equation, it has no Lagrangian formulation in terms of the variable [MATH].', '1412.4415-2-13-3': 'In this situation, the problem of finding multipliers can be understood as a kind of adjoint [CITATION] of the problem of finding symmetries.', '1412.4415-2-14-0': 'An infinitesimal symmetry [CITATION] of equation [REF] is a generator [EQUATION] whose coefficient [MATH] is given by a function of [MATH], [MATH], [MATH] and derivatives of [MATH], such that the prolonged generator satisfies the invariance condition [EQUATION] holding for all solutions [MATH] of equation [REF].', '1412.4415-2-14-1': 'The Lie symmetry group of equation [REF] is generated by infinitesimal symmetries [REF] with coefficients of the form [EQUATION]', '1412.4415-2-14-2': 'If [MATH] is at most linear in [MATH] and [MATH], then the resulting generator [REF] will yield a group of point transformations [CITATION], whereas if [MATH] is nonlinear in [MATH] or [MATH], then a group of contact transformations [CITATION] will be generated.', '1412.4415-2-14-3': 'Hence, all generators of Lie symmetries admitted by equation [REF] are determined by the solutions of condition [REF] for [MATH].', '1412.4415-2-14-4': '(It is straightforward to solve this determining equation by Maple to classify the Lie symmetry group of equation [REF], as shown in the Appendix.)', '1412.4415-2-15-0': 'The condition for determining all multipliers [MATH] of low-order conservation laws [REF] admitted by equation [REF] consists of [EQUATION] which arises from the property that the variational derivative (Euler operator) [EQUATION] annihilates an expression identically iff it is a space-time divergence [CITATION].', '1412.4415-2-15-1': 'This condition [REF] can be split with respect to [MATH] and [MATH]-derivatives of [MATH], yielding an equivalent overdetermined system of equations on [MATH].', '1412.4415-2-15-2': 'One equation in this system is given by the adjoint of the symmetry determining equation [REF], [EQUATION] holding for all solutions [MATH] of equation [REF].', '1412.4415-2-15-3': 'Solutions [MATH] of this equation [REF] are called adjoint-symmetries (or cosymmetries) [CITATION].', '1412.4415-2-15-4': 'The remaining equations in the system comprise Helmholtz conditions which are necessary and sufficient for [MATH] to have the form [REF].', '1412.4415-2-15-5': 'As a consequence, multipliers [REF] are simply adjoint-symmetries that have a certain variational form.', '1412.4415-2-16-0': 'For any solution [REF] of the multiplier determining equation [REF], a corresponding conserved density and flux of the form [REF] can be recovered either through integration [CITATION] of the characteristic equation [REF], which splits with respect to [MATH], [MATH], [MATH], [MATH] into a system of equations for [MATH] and [MATH], or through a homotopy integral formula [CITATION] , which expresses [MATH] and [MATH] directly in terms of [MATH].', '1412.4415-2-16-1': 'It is straightforward to show that [MATH] and [MATH] have the form [REF] of a trivial conservation law iff [MATH].', '1412.4415-2-16-2': 'Thus there is a one-to-one correspondence between equivalence classes of non-trivial low-order conservation laws [REF] and non-zero low-order multipliers [REF].', '1412.4415-2-17-0': '## Classification results', '1412.4415-2-18-0': 'Both the Camassa-Holm equation [REF] and Novikov equation [REF] possess low-order local conservations law given by the conserved densities [CITATION]', '1412.4415-2-19-0': 'T=mu = u^2 + u_x^2 +D_x(-uu_x),', '1412.4415-2-20-0': 'T=m^q,', '1412.4415-2-21-0': 'where [MATH] and [MATH], respectively, for the two equations.', '1412.4415-2-21-1': 'In addition, the Camassa-Holm equation [REF] itself is a low-order local conservation law having the conserved density [EQUATION]', '1412.4415-2-21-2': 'All of these conserved densities are related to Hamiltonian structures for the two equations [CITATION].', '1412.4415-2-21-3': 'The corresponding multipliers are respectively given by [EQUATION]', '1412.4415-2-21-4': 'To look for conserved densities of the same form for equation [REF], we now classify all multipliers up to 1st-order [EQUATION] as well as all 2nd-order multipliers with the specific form [EQUATION]', '1412.4415-2-21-5': 'In each case it is straightforward to solve the determining equation [REF] by use of Maple (as shown in the Appendix), which leads to the following classification result.', '1412.4415-2-22-0': 'In light of the adjoint relationship between multipliers and symmetries, the classification of 0th- and 1st- order multipliers in Proposition [REF] is a counterpart of the classification of Lie symmetries (cf. Proposition [REF]).', '1412.4415-2-23-0': 'Next we obtain the corresponding conserved densities and fluxes for each multiplier [REF]-[REF] by first splitting the characteristic equation [REF] with respect to [MATH], [MATH], [MATH], [MATH] where [MATH] and [MATH] have the form [REF], and then integrating the resulting system of equations.', '1412.4415-2-23-1': 'This yields the following low-order local conservation laws for equation [REF].', '1412.4415-2-24-0': '(i) The local conservation laws admitted by the wave-breaking equation [REF] with multipliers of at most 1st-order consist of three 0th-order conservation laws [EQUATION] and two 1st-order conservation laws [EQUATION] (ii) The local conservation laws admitted by the wave-breaking equation [REF] with 2nd-order multipliers of the form [REF] consist of two 2nd-order conservation laws [EQUATION]', '1412.4415-2-24-1': 'In these conservation laws [REF]-[REF], any terms of the form [MATH] in the case [MATH] should be replaced by [MATH].', '1412.4415-2-25-0': 'These conservation laws yield the following conserved integrals.', '1412.4415-2-25-1': 'We start with the conservation laws at 0th order.', '1412.4415-2-25-2': 'From [MATH], we have [EQUATION] which is the conserved mass for equation [REF].', '1412.4415-2-25-3': 'The conserved integral arising from [MATH] is a weighted mass, [EQUATION]', '1412.4415-2-25-4': 'Interestingly, from [MATH] we get a conserved integral which vanishes, but has a non-zero spatial flux.', '1412.4415-2-25-5': 'This type of conservation law arises because the multiplier [REF] converts equation [REF] into the form of a total [MATH]-derivative.', '1412.4415-2-26-0': 'Next we look at the conservation laws at 1st order.', '1412.4415-2-26-1': 'From [MATH], the [MATH] norm of [MATH] is conserved, [EQUATION]', '1412.4415-2-26-2': 'From [MATH], we have [EQUATION] where [MATH] is the center of mass of [MATH].', '1412.4415-2-26-3': 'Since [MATH] is conserved, it can be evaluated at [MATH], which yields the relation [MATH].', '1412.4415-2-26-4': 'This shows that the center of mass moves at a constant speed controlled by the [MATH] norm of [MATH].', '1412.4415-2-27-0': 'Finally, we consider the conservation laws at 2nd order.', '1412.4415-2-27-1': 'From [MATH], we get [EQUATION]', '1412.4415-2-27-2': 'This shows that the [MATH] norm of [MATH] is conserved if [MATH] does not change sign or if [MATH] is an even integer.', '1412.4415-2-27-3': 'The conserved integral arising from [MATH] is a linear combination of the [MATH] norms of [MATH] as given by [EQUATION]', '1412.4415-2-27-4': 'This can be written alternatively as a weighted [MATH] norm when [MATH].', '1412.4415-2-27-5': 'It is interesting to note that simultaneous conservation of both the [MATH] and the weighted [MATH] norms requires the condition [MATH] which holds iff [MATH] and [MATH], but in this case [MATH].', '1412.4415-2-28-0': '# Peakon solutions', '1412.4415-2-29-0': 'Both the Camassa-Holm and Novikov equations possess peaked travelling wave solutions [CITATION], called peakons, [EQUATION] where [MATH] and [MATH], respectively, for the two equations.', '1412.4415-2-29-1': 'Peakons have attracted much attention in the study of breaking wave equations.', '1412.4415-2-30-0': 'In general, on [MATH], a peakon is a weak travelling wave solution satisfying an integral (i.e. weak) formulation of a breaking wave equation.', '1412.4415-2-30-1': 'Such a formulation is essential for deriving multi-peakon solutions.', '1412.4415-2-30-2': 'However, single peakons can be derived directly from the travelling wave reduction of a breaking wave equation, which will be the approach we use here.', '1412.4415-2-31-0': '## Single peakon solution', '1412.4415-2-32-0': 'The manifest invariance of the 4-parameter equation [REF] under time-translation and space-translation symmetries implies the existence of travelling wave solutions [EQUATION] where [MATH] satisfies the ODE [EQUATION]', '1412.4415-2-32-1': 'For the travelling wave ODE [REF], an integral formulation is obtained through multiplying this ODE by a test function [MATH] (which is smooth and has compact support) and integrating over [MATH], leaving at most first derivatives of [MATH] in the integral, which yields [EQUATION]', '1412.4415-2-32-2': 'A weak solution of ODE [REF] is a function [MATH] that belongs to the Sobolev space [MATH] and that satisfies the integral equation [REF] for all smooth test functions [MATH] with compact support on [MATH].', '1412.4415-2-33-0': 'To proceed we substitute a peaked travelling wave expression [EQUATION] into equation [REF] and split up the integral into the intervals [MATH] and [MATH].', '1412.4415-2-33-1': 'The first term in equation [REF] yields, after integration by parts, [EQUATION]', '1412.4415-2-33-2': 'Similarly, the second term in equation [REF] gives [EQUATION] provided [MATH] so that the boundary terms at [MATH] vanish.', '1412.4415-2-33-3': 'The third and fourth terms in equation [REF] together yield [EQUATION]', '1412.4415-2-33-4': 'When the terms [REF]-[REF] are combined, we find that equation [REF] reduces to [EQUATION]', '1412.4415-2-33-5': 'This equation is satisfied for all test functions [MATH] iff [EQUATION] which determines the amplitude [MATH] in the peakon expression [REF].', '1412.4415-2-33-6': 'Thus we obtain the following result.', '1412.4415-2-34-0': 'The resulting peakon solution of equation [REF] is given by [EQUATION]', '1412.4415-2-34-1': 'When the nonlinearity power [MATH] is a positive integer, then the wave speed is necessarily positive, [MATH], if [MATH] is even, as in the case ([MATH]) of the Novikov equation [REF], while if [MATH] is odd, the wave speed can be either positive or negative, [MATH], as in the case ([MATH]) of the Camassa-Holm equation [REF].', '1412.4415-2-35-0': 'The peakon solution [REF] satisfies equation [REF] only in the sense of a weak solution.', '1412.4415-2-35-1': 'This means [MATH] is a distribution in [MATH] with respect to [MATH] for some [MATH] and in [MATH] with respect to [MATH] such that it satisfies the integral equation [EQUATION] for all test functions [MATH] in [MATH].', '1412.4415-2-36-0': '## Multi-peakon solution', '1412.4415-2-37-0': 'Both the Camassa-Holm and Novikov equations possess multi-peakon solutions [CITATION] which are a linear superposition of peaked travelling waves with time-dependent amplitudes and positions.', '1412.4415-2-37-1': 'The form of these solutions is given by [EQUATION] where the amplitudes [MATH] and positions [MATH] satisfy a Hamiltonian system of ODEs [EQUATION] given in terms of the Hamiltonian function [EQUATION]', '1412.4415-2-37-2': 'The Poisson bracket [MATH] in this system [REF] arises from the respective Hamiltonian operator formulations [CITATION] of these two equations and has the standard canonical form in the case of Camassa-Holm equation and a certain non-canonical form in the case of the Novikov equation.', '1412.4415-2-38-0': 'We now investigate whether equation [REF] also admits multi-peakon solutions.', '1412.4415-2-38-1': 'It will be convenient to use the notation [EQUATION] where the summation is understood to go from [MATH] to [MATH].', '1412.4415-2-38-2': 'Note that the [MATH]-derivatives of [MATH] are given by [EQUATION] and [EQUATION] in terms of the sign function [EQUATION] and the Dirac delta distribution [EQUATION] which has the properties [MATH] for [MATH], and [MATH] for all [MATH].', '1412.4415-2-39-0': 'To begin, we substitute the general multi-peakon expression [REF] into the integral equation [REF].', '1412.4415-2-39-1': 'There are two ways we can then proceed.', '1412.4415-2-39-2': 'One way is to assume [MATH] at a fixed [MATH], split up the integral over [MATH] into corresponding intervals, and integrate by parts, similarly to the derivation of the single peakon solution.', '1412.4415-2-39-3': 'Another way, which is simpler, is to employ the following result from distribution theory [CITATION].', '1412.4415-2-40-0': 'Let [MATH] be a piecewise [MATH] function having at most jump discontinuities at a finite number of points [MATH] in [MATH].', '1412.4415-2-40-1': 'Then, for any test function [MATH], [EQUATION] where [EQUATION] is the jump in [MATH] at the point [MATH], and [EQUATION] is the non-singular part of the distributional derivative of [MATH].', '1412.4415-2-40-2': 'We will now use this integration by parts relation [REF] to evaluate each term in the integral equation [REF].', '1412.4415-2-41-0': 'The first term in equation [REF] yields [EQUATION]', '1412.4415-2-41-1': 'From expression [REF], we see [EQUATION] so thus [MATH] holds a. e. in [MATH].', '1412.4415-2-41-2': 'Hence [EQUATION] and thus we get [EQUATION]', '1412.4415-2-41-3': 'Next, the second term in equation [REF] gives [EQUATION]', '1412.4415-2-41-4': 'We now simplify the two parts of the integral involving [MATH].', '1412.4415-2-41-5': 'For the first part, we have [EQUATION] after using relation [REF] and then integrating by parts.', '1412.4415-2-41-6': 'For the second part, since [MATH] holds a. e. in [MATH], we have [EQUATION] from applying the integration by parts relation [REF].', '1412.4415-2-41-7': 'By simplifying [MATH] a. e. with the use of relation [REF], we see [EQUATION]', '1412.4415-2-41-8': 'Hence the integral [REF] becomes [EQUATION]', '1412.4415-2-41-9': 'Then we have [EQUATION]', '1412.4415-2-41-10': 'Similarly, the third term in equation [REF] gives [EQUATION]', '1412.4415-2-41-11': 'We can simplify the integral involving [MATH] by the same steps used for the previous integral.', '1412.4415-2-41-12': 'This yields [EQUATION]', '1412.4415-2-41-13': 'Hence we then have [EQUATION]', '1412.4415-2-41-14': 'Finally, by combining the three terms [REF], [REF], [REF] with the fourth term in equation [REF], we obtain [EQUATION]', '1412.4415-2-41-15': 'The jump terms are evaluated by', '1412.4415-2-42-0': "[u_t]__i = 2_i_i', [u_x]__i = -2_i", '1412.4415-2-43-0': "[u_tx]__i = d[u_x]__idt = -2_i'", '1412.4415-2-44-0': '[u_x^2]__i = 2u_x(_i)[u_x]__i = -4_iu_x(_i)', '1412.4415-2-45-0': 'which all follow directly from the expressions [REF] and [REF].', '1412.4415-2-45-1': 'Thus we get [EQUATION]', '1412.4415-2-45-2': 'This equation is satisfied for all test functions [MATH] iff [EQUATION] which determines the amplitudes [MATH] and the positions [MATH] in the multi-peakon expression [REF].', '1412.4415-2-45-3': 'Thus we have established the following result.', '1412.4415-2-46-0': 'From Propositions [REF] and [REF], we have a classification of all cases for which the 4-parameter equation [REF] possesses both single peakon and multi-peakon solutions.', '1412.4415-2-47-0': 'The 4-parameter equation [REF] admits single peakon and multi-peakon solutions iff [EQUATION]', '1412.4415-2-47-1': 'In this case, a general [MATH]-peakon solution has the form [REF], where the amplitudes [MATH] and positions [MATH] satisfy the system of ODEs [EQUATION] in terms of the separations [EQUATION]', '1412.4415-2-47-2': 'This result generalizes related work in Ref.[CITATION] which established the existence of single and multi-peakon solutions for a 2-parameter equation defined by the case [MATH], [MATH] of equation [REF].', '1412.4415-2-47-3': '(In particular, the derivation in Ref.[CITATION] was completely formal, whereas the steps here provide a rigorous proof applied to the more general 4-parameter equation [REF].)', '1412.4415-2-48-0': 'It is easy to check that the general [MATH]-peakon ODE system [REF]-[REF] reduces to the well-known multi-peakon systems for the [MATH]-equation [REF] when [MATH], which includes the the Camassa-Holm equation and the Degasperis-Procesi equation when [MATH] and [MATH], respectively, as well as for the Novikov equation when [MATH].', '1412.4415-2-49-0': '## Constants of motion', '1412.4415-2-50-0': 'The ODE system [REF]-[REF] for the amplitudes and positions of the [MATH] peakons in the expression [REF] inherits constants of motion (i.e. time-independent quantities) given by the conserved integrals that are admitted by equation [REF] in the case [REF].', '1412.4415-2-50-1': 'From Theorem [REF], there are six conserved integrals [REF]-[REF] which we can consider.', '1412.4415-2-51-0': 'The first conserved integral [REF] yields [EQUATION]', '1412.4415-2-51-1': 'This quantity [MATH] is the total mass for the [MATH]-peakon solution.', '1412.4415-2-51-2': 'A weighted mass arises from the second conserved integral [REF], [EQUATION]', '1412.4415-2-51-3': 'The next conserved integral [REF] gives [EQUATION] which is the [MATH] norm of the [MATH]-peakon solution.', '1412.4415-2-52-0': 'The fourth conserved integral [REF] does not exist in the case [REF], since [MATH] and [MATH] together imply that [MATH].', '1412.4415-2-52-1': 'Last, the two conserved integrals [REF] and [REF] are nonlinear in [MATH] which is a distribution.', '1412.4415-2-52-2': 'As a consequence, both these integrals are ill-defined for the the [MATH]-peakon solution.', '1412.4415-2-53-0': '# Unified family of Camassa-Holm-Novikov equations', '1412.4415-2-54-0': 'From Theorems [REF] and [REF], the low-order conservation laws [REF]-[REF] as well as the [MATH]-peakon solution expression [REF] of the Camassa-Holm and Novikov equations are admitted simultaneously by the 4-parameter equation [REF] iff its parameters [MATH] satisfy [EQUATION]', '1412.4415-2-54-1': 'After a scaling transformation [MATH] is used to put [MATH], equation [REF] reduces to the 1-parameter gCHN equation [REF] presented in section [REF].', '1412.4415-2-55-0': '## Dynamics of multi-peakon solutions', '1412.4415-2-56-0': 'The explicit system describing [MATH]-peakon solutions of the gCHN equation [REF] for all [MATH] is given by [EQUATION] where [MATH] and [MATH] are, respectively, the amplitudes and positions appearing in the general [MATH]-peakon expression [REF].', '1412.4415-2-56-1': 'The [MATH] norm [REF] of the [MATH]-peakon solution provides a constant of motion [EQUATION] which is determined by the initial amplitudes and initial separations.', '1412.4415-2-57-0': 'When all of the amplitudes are positive, [MATH], for all [MATH], the solution expression [REF] is a superposition of [MATH] peakons, each of which is right moving.', '1412.4415-2-57-1': 'In this case, the constant of motion [REF] directly gives the inequality [MATH], [MATH], which implies that any collisions among the [MATH] peakons are elastic.', '1412.4415-2-58-0': 'When all of the amplitudes are negative, [MATH], for all [MATH], the solution expression [REF] is instead a superposition of [MATH] anti-peakons, each of which is either right moving if [MATH] is even or left moving if [MATH] is odd.', '1412.4415-2-58-1': 'Similarly to the previous case, the constant of motion [REF] yields [MATH], [MATH], implying that any collisions among the [MATH] anti-peakons are elastic.', '1412.4415-2-59-0': 'In the case when some amplitudes have opposite signs, or an amplitude changes its sign at some [MATH], the solution expression [REF] then describes a superposition of both peakons and anti-peakons.', '1412.4415-2-59-1': 'Although the constant of motion is still non-negative, the amplitudes are no longer bounded by [MATH].', '1412.4415-2-59-2': 'As a consequence, wave breaking can occur in collisions, which we will now show for the case [MATH].', '1412.4415-2-60-0': '## Wave breaking in collisions between peakons and anti-peakons', '1412.4415-2-61-0': 'For [MATH], the system [REF]-[REF] describing [MATH]-peakon solutions [EQUATION] takes a simple form.', '1412.4415-2-61-1': 'First, the constant of motion [REF] can be used to express the relative separation [MATH] in terms of the two amplitudes [MATH] and [MATH] through the relation [EQUATION]', '1412.4415-2-61-2': 'Then, the equations of motion for the two positions [MATH] and [MATH] and the two amplitudes [MATH] and [MATH] are given by', '1412.4415-2-62-0': "_1' = A_1^p,", '1412.4415-2-63-0': "_2' = A_2^p,", '1412.4415-2-64-0': "_1' = 12 sgn(_1,2) A_1^p-1(H -_1^2 -_2^2),", '1412.4415-2-65-0': "_2' = -12 sgn(_1,2) A_2^p-1(H -_1^2 -_2^2),", '1412.4415-2-66-0': 'with [EQUATION] and [EQUATION]', '1412.4415-2-66-1': 'If another constant of motion could be found for this system, then the system could be reduced to two separated ODEs for the two amplitudes, plus two quadratures for the two positions, which would allow the general solution to be obtained.', '1412.4415-2-66-2': 'Even without another constant of motion, it is still possible to do a qualitative analysis of all solutions by studying the phase plane [MATH] of the coupled ODEs [REF]-[REF] for the amplitudes.', '1412.4415-2-67-0': 'We start from the relation [REF], which imposes inequalities on the amplitudes, [EQUATION]', '1412.4415-2-67-1': 'For a given value of [MATH], these two inequalities define the domain for all [MATH]-peakon solutions in the phase plane [MATH].', '1412.4415-2-67-2': 'The boundary of the domain corresponds to the two equalities [EQUATION] and [EQUATION] which consist of a circle and two parallel lines.', '1412.4415-2-67-3': 'The circle comprises the equilibrium points of the amplitude ODEs [REF]-[REF] in the phase plane.', '1412.4415-2-67-4': 'Each point on the circle is a limit of a [MATH]-peakon solution describing an asymptotic superposition of two [MATH]-peakon solutions, in which the amplitudes are constant and the positions are infinitely separated.', '1412.4415-2-67-5': 'The lines each constitute a degenerate [MATH]-peakon solution in which the two positions coincide and the sum of the two amplitudes is constant, describing a peakon solution [EQUATION] in the case of the upper line, and an anti-peakon solution [EQUATION] in the case of the lower line.', '1412.4415-2-68-0': 'The entire solution domain divides into four parts which are related by a reflection symmetry [MATH].', '1412.4415-2-68-1': 'One part of the domain is given by the points lying between the circle [REF] and the upper line [REF] in the first quadrant, which comprises all solutions describing two peakons.', '1412.4415-2-68-2': 'There is a counterpart given by the points lying between the circle [REF] and the lower line [REF] in the third quadrant, which comprises all solutions describing two anti-peakons.', '1412.4415-2-68-3': 'The two other parts of the domain comprise all solutions describing a peakon and an anti-peakon.', '1412.4415-2-68-4': 'These parts are given by the points between the segments of the upper and lower lines that lie outside of the circle.', '1412.4415-2-69-0': 'Within this solution domain in the phase plane, the flow defined by the amplitude ODEs [REF]-[REF] depends on the nonlinearity power [MATH] and the sign of the separation [MATH].', '1412.4415-2-69-1': 'We are interested in flows that describe a collision between a peakon and an anti-peakon.', '1412.4415-2-69-2': 'This condition can be used to determine [MATH] at [MATH] at each point in the phase plane by considering the ODE [EQUATION] for the separation.', '1412.4415-2-69-3': 'If [MATH], then the relative separation [MATH] between the peakon and anti-peakon will be decreasing only if [MATH].', '1412.4415-2-69-4': 'Similarly, if [MATH], then the relative separation [MATH] between the peakon and anti-peakon will be decreasing only if [MATH].', '1412.4415-2-69-5': 'Hence, a necessary condition for a collision to occur is that [MATH] and [MATH] have opposite signs during the flow.', '1412.4415-2-69-6': 'Since [MATH] can occur only on the upper and lower lines [REF], which are boundaries of the domain in which solutions describe a collision between a peakon and an anti-peakon, we can impose [EQUATION] at each point in the phase plane.', '1412.4415-2-69-7': 'Note [MATH] holds iff [MATH] when [MATH] is odd, and [MATH] when [MATH] is even.', '1412.4415-2-69-8': 'The points given by [MATH] in the phase plane consist of the lines [REF] and [MATH], while the points given by [MATH] consist of the lines that are perpendicular to each of those three lines.', '1412.4415-2-69-9': 'Consequently, hereafter we will consider initial conditions [EQUATION] and [EQUATION] without loss of generality.', '1412.4415-2-69-10': '(Note that reversing the sign in the initial condition [REF] will correspond to reflecting the flow about the line [MATH] in the phase plane.)', '1412.4415-2-70-0': 'Under the collision condition [REF] and initial conditions [REF]-[REF], the flow then depends only on the nonlinearity power [MATH].', '1412.4415-2-70-1': 'The case [MATH], which represents the Camassa-Holm equation, is special, since there is another constant of motion [MATH] which is given by the total mass [REF].', '1412.4415-2-70-2': 'This implies that the flow simply consists of parallel lines in the phase plane.', '1412.4415-2-70-3': 'In all other cases [MATH], the flow is no longer given by straight lines and has a much richer structure.', '1412.4415-2-71-0': 'The flows for all even powers [MATH] are qualitatively similar to the case [MATH], which represents the Novikov equation.', '1412.4415-2-71-1': 'A picture of the phase plane for [MATH] is shown in Fig. [REF].', '1412.4415-2-71-2': 'Clearly, in the second quadrant, the upper line [MATH] is a stable asymptotic attractor for solutions describing a peakon ([MATH]) and an anti-peakon ([MATH]), while the lower line [MATH] is an unstable asymptotic attractor.', '1412.4415-2-71-3': 'In the fourth quadrant, these behaviours are reversed.', '1412.4415-2-72-0': 'The flows for all other odd powers [MATH] are qualitatively similar to the case [MATH] which is shown in Fig. [REF].', '1412.4415-2-72-1': 'In the second quadrant, both the upper and lower lines [MATH] are stable asymptotic attractors for solutions describing a peakon ([MATH]) and an anti-peakon ([MATH]).', '1412.4415-2-72-2': 'The line [MATH] is an unstable asymptotic attractor.', '1412.4415-2-72-3': 'In the fourth quadrant, the behaviour is the same.', '1412.4415-2-73-0': 'In all cases [MATH], the flow will evolve the initial amplitudes toward a stable attractor line.', '1412.4415-2-73-1': 'This evolution is shown in Figs. [REF] and [REF] for the cases [MATH] and [MATH], respectively, where the initial positions of the peakon and anti-peakon are chosen to be distinctly separated.', '1412.4415-2-73-2': 'We see that the peakon and anti-peakon collide such that their peak amplitudes become closer while the slope at locations [MATH] in between the two peaks rapidly increases (without bound) as relative separation between their positions decreases to zero in a finite time.', '1412.4415-2-73-3': 'This blow-up in the slope seen in Figs. [REF] and [REF] is an example of wave breaking.', '1412.4415-2-74-0': 'There is a qualitative explanation of why the blow-up in the slope [MATH] between the two peaks in a collision solution [REF] occurs in a finite time.', '1412.4415-2-74-1': 'Consider the asymptotic attractor solution [MATH] corresponding to the upper line [REF].', '1412.4415-2-74-2': 'This solution arises from the initial condition [MATH] and [MATH].', '1412.4415-2-74-3': 'The amplitude ODEs [REF]-[REF] yield [EQUATION] whereby [MATH] and [MATH] as [MATH] such that [MATH] is constant for all [MATH].', '1412.4415-2-74-4': 'Any solution having an initial condition close to [MATH] and [MATH] will exhibit a similar long-time behaviour for [MATH] and [MATH], as a consequence of continuous dependence of solutions on initial data for the ODEs [REF]-[REF].', '1412.4415-2-74-5': 'Since [MATH], the solution [REF] remains continuous and bounded at all [MATH] for [MATH], whereas the slope [EQUATION] has jump discontinuities at [MATH] and [MATH] and becomes unbounded at [MATH] (with [MATH]) as [MATH].', '1412.4415-2-75-0': 'The same kind of wave-breaking behaviour can be expected to occur in collisions between peakons and anti-peakons when [MATH].', '1412.4415-2-76-0': '# Concluding remarks', '1412.4415-2-77-0': 'At first sight, the gCHN equation [REF] seems closely analogous to the [MATH]-equation [REF]: both equations unify two integrable equations, possess [MATH]-peakon solutions, and exhibit wave breaking phenomena.', '1412.4415-2-77-1': 'However, there are important differences.', '1412.4415-2-77-2': 'Firstly, the nonlinearities in the [MATH]-equation are purely quadratic, whereas the gCHN equation has nonlinearities of degree [MATH] and thereby connects two integrable equations with different nonlinearities.', '1412.4415-2-77-3': 'Secondly, the [MATH] norm of solutions [MATH] is conserved for the [MATH]-equation only if [MATH], when the [MATH]-equation reduces to the Camassa-Holm equation.', '1412.4415-2-77-4': 'In contrast, the [MATH] norm is conserved for the gCHN equation for all [MATH].', '1412.4415-2-78-0': 'In a subsequent work, we will explore further properties of the gCHN equation [REF] and its multi-peakon solutions.', '1412.4415-2-78-1': 'There are numerous interesting questions.', '1412.4415-2-78-2': 'Can a wave-breaking result similar to those for the Camassa-Holm and Novikov equations be established for classical solutions?', '1412.4415-2-78-3': 'How will the wave-breaking behaviour depend on [MATH]?', '1412.4415-2-78-4': 'In particular, a plausible criteria for wave-breaking is [MATH] which generalizes the criteria known [CITATION] in the Camassa-Holm case [MATH] and the Novikov case [MATH].', '1412.4415-2-78-5': 'In another direction, for any [MATH] other than these two known integrable cases [MATH] and [MATH], does the equation have a Hamiltonian formulation or perhaps integrability properties?'} | [['1412.4415-1-32-1', '1412.4415-2-35-1'], 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['1412.4415-1-30-3', '1412.4415-2-33-3'], ['1412.4415-1-30-4', '1412.4415-2-33-4'], ['1412.4415-1-30-5', '1412.4415-2-33-5'], ['1412.4415-1-30-6', '1412.4415-2-33-6'], ['1412.4415-1-3-1', '1412.4415-2-3-1'], ['1412.4415-1-3-2', '1412.4415-2-3-2'], ['1412.4415-1-29-1', '1412.4415-2-32-1'], ['1412.4415-1-29-2', '1412.4415-2-32-2'], ['1412.4415-1-14-1', '1412.4415-2-9-1'], ['1412.4415-1-14-2', '1412.4415-2-9-2'], ['1412.4415-1-15-0', '1412.4415-2-10-0'], ['1412.4415-1-15-2', '1412.4415-2-10-2'], ['1412.4415-1-0-0', '1412.4415-2-0-0'], ['1412.4415-1-31-0', '1412.4415-2-34-0'], ['1412.4415-1-31-1', '1412.4415-2-34-1'], ['1412.4415-1-32-0', '1412.4415-2-35-0'], ['1412.4415-1-19-0', '1412.4415-2-16-0'], ['1412.4415-1-23-5', '1412.4415-2-21-5'], ['1412.4415-1-34-0', '1412.4415-2-54-0'], ['1412.4415-1-28-0', '1412.4415-2-30-0'], ['1412.4415-1-36-0', '1412.4415-2-77-0'], ['1412.4415-1-36-2', '1412.4415-2-77-2'], ['1412.4415-1-30-2', '1412.4415-2-33-2'], ['1412.4415-1-3-0', '1412.4415-2-3-0'], ['1412.4415-1-3-3', '1412.4415-2-3-3'], ['1412.4415-1-29-0', '1412.4415-2-32-0'], ['1412.4415-1-14-0', '1412.4415-2-9-0'], ['1412.4415-1-15-1', '1412.4415-2-10-1'], ['1412.4415-1-0-1', '1412.4415-2-0-1'], ['1412.4415-1-5-2', '1412.4415-2-5-1'], ['1412.4415-1-37-0', '1412.4415-2-78-0'], ['1412.4415-1-37-1', '1412.4415-2-78-5'], ['1412.4415-1-36-3', '1412.4415-2-77-3'], ['1412.4415-1-36-4', '1412.4415-2-77-4'], ['1412.4415-1-36-5', '1412.4415-2-77-4'], ['1412.4415-1-0-2', '1412.4415-2-0-2'], ['1412.4415-1-17-0', '1412.4415-2-12-0'], ['1412.4415-1-17-1', '1412.4415-2-12-1'], ['1412.4415-1-17-2', '1412.4415-2-12-2'], ['1412.4415-1-17-3', '1412.4415-2-12-3'], ['1412.4415-1-17-4', '1412.4415-2-12-4'], ['1412.4415-1-17-5', '1412.4415-2-12-5'], ['1412.4415-1-17-6', '1412.4415-2-12-6'], ['1412.4415-1-17-7', '1412.4415-2-12-7'], ['1412.4415-1-18-1', '1412.4415-2-15-1'], ['1412.4415-1-18-2', '1412.4415-2-15-2'], ['1412.4415-1-18-3', '1412.4415-2-15-3'], ['1412.4415-1-18-4', '1412.4415-2-15-4'], ['1412.4415-1-18-5', '1412.4415-2-15-5'], ['1412.4415-1-24-0', '1412.4415-2-22-0'], ['1412.4415-1-27-0', '1412.4415-2-29-0'], ['1412.4415-1-27-1', '1412.4415-2-29-1'], ['1412.4415-1-35-0', '1412.4415-2-6-1']] | [['1412.4415-1-32-1', '1412.4415-2-35-1'], ['1412.4415-1-25-0', '1412.4415-2-23-0'], ['1412.4415-1-25-1', '1412.4415-2-23-1'], ['1412.4415-1-5-1', '1412.4415-2-5-0'], ['1412.4415-1-4-0', '1412.4415-2-4-0'], ['1412.4415-1-4-1', '1412.4415-2-4-1'], ['1412.4415-1-16-0', '1412.4415-2-11-0'], ['1412.4415-1-19-1', '1412.4415-2-16-1'], ['1412.4415-1-19-2', '1412.4415-2-16-2'], ['1412.4415-1-23-0', '1412.4415-2-21-0'], ['1412.4415-1-23-1', '1412.4415-2-21-1'], ['1412.4415-1-23-2', '1412.4415-2-21-2'], ['1412.4415-1-23-3', '1412.4415-2-21-3'], ['1412.4415-1-23-4', '1412.4415-2-21-4'], ['1412.4415-1-20-0', '1412.4415-2-18-0'], ['1412.4415-1-28-1', '1412.4415-2-30-1'], ['1412.4415-1-28-2', '1412.4415-2-30-2'], ['1412.4415-1-36-1', '1412.4415-2-77-1'], ['1412.4415-1-2-0', '1412.4415-2-2-0'], ['1412.4415-1-2-1', '1412.4415-2-2-1'], ['1412.4415-1-2-2', '1412.4415-2-2-2'], ['1412.4415-1-2-3', '1412.4415-2-2-3'], ['1412.4415-1-2-4', '1412.4415-2-2-4'], ['1412.4415-1-30-0', '1412.4415-2-33-0'], ['1412.4415-1-30-1', '1412.4415-2-33-1'], ['1412.4415-1-30-3', '1412.4415-2-33-3'], ['1412.4415-1-30-4', '1412.4415-2-33-4'], ['1412.4415-1-30-5', '1412.4415-2-33-5'], ['1412.4415-1-30-6', '1412.4415-2-33-6'], ['1412.4415-1-3-1', '1412.4415-2-3-1'], ['1412.4415-1-3-2', '1412.4415-2-3-2'], ['1412.4415-1-29-1', '1412.4415-2-32-1'], ['1412.4415-1-29-2', '1412.4415-2-32-2'], ['1412.4415-1-14-1', '1412.4415-2-9-1'], ['1412.4415-1-14-2', '1412.4415-2-9-2'], ['1412.4415-1-15-0', '1412.4415-2-10-0'], ['1412.4415-1-15-2', '1412.4415-2-10-2'], ['1412.4415-1-0-0', '1412.4415-2-0-0'], ['1412.4415-1-31-0', '1412.4415-2-34-0'], ['1412.4415-1-31-1', '1412.4415-2-34-1']] | [['1412.4415-1-32-0', '1412.4415-2-35-0'], ['1412.4415-1-19-0', '1412.4415-2-16-0'], ['1412.4415-1-23-5', '1412.4415-2-21-5'], ['1412.4415-1-34-0', '1412.4415-2-54-0'], ['1412.4415-1-28-0', '1412.4415-2-30-0'], ['1412.4415-1-36-0', '1412.4415-2-77-0'], ['1412.4415-1-36-2', '1412.4415-2-77-2'], ['1412.4415-1-30-2', '1412.4415-2-33-2'], ['1412.4415-1-3-0', '1412.4415-2-3-0'], ['1412.4415-1-3-3', '1412.4415-2-3-3'], ['1412.4415-1-29-0', '1412.4415-2-32-0'], ['1412.4415-1-14-0', '1412.4415-2-9-0'], ['1412.4415-1-15-1', '1412.4415-2-10-1'], ['1412.4415-1-0-1', '1412.4415-2-0-1']] | [] | [['1412.4415-1-5-2', '1412.4415-2-5-1'], ['1412.4415-1-37-0', '1412.4415-2-78-0'], ['1412.4415-1-37-1', '1412.4415-2-78-5'], ['1412.4415-1-36-3', '1412.4415-2-77-3'], ['1412.4415-1-36-4', '1412.4415-2-77-4'], ['1412.4415-1-36-5', '1412.4415-2-77-4'], ['1412.4415-1-0-2', '1412.4415-2-0-2']] | [['1412.4415-1-17-0', '1412.4415-2-12-0'], ['1412.4415-1-17-1', '1412.4415-2-12-1'], ['1412.4415-1-17-2', '1412.4415-2-12-2'], ['1412.4415-1-17-3', '1412.4415-2-12-3'], ['1412.4415-1-17-4', '1412.4415-2-12-4'], ['1412.4415-1-17-5', '1412.4415-2-12-5'], ['1412.4415-1-17-6', '1412.4415-2-12-6'], ['1412.4415-1-17-7', '1412.4415-2-12-7'], ['1412.4415-1-18-1', '1412.4415-2-15-1'], ['1412.4415-1-18-2', '1412.4415-2-15-2'], ['1412.4415-1-18-3', '1412.4415-2-15-3'], ['1412.4415-1-18-4', '1412.4415-2-15-4'], ['1412.4415-1-18-5', '1412.4415-2-15-5'], ['1412.4415-1-24-0', '1412.4415-2-22-0'], ['1412.4415-1-27-0', '1412.4415-2-29-0'], ['1412.4415-1-27-1', '1412.4415-2-29-1'], ['1412.4415-1-35-0', '1412.4415-2-6-1']] | ['1412.4415-1-21-0', '1412.4415-1-22-0', '1412.4415-2-19-0', '1412.4415-2-20-0', '1412.4415-2-41-12', '1412.4415-2-42-0', '1412.4415-2-43-0', '1412.4415-2-44-0', '1412.4415-2-62-0', '1412.4415-2-63-0', '1412.4415-2-64-0', '1412.4415-2-65-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1412.4415 | null | null | null | null | null |
1804.08343 | {'1804.08343-1-0-0': 'The orbifold construction [MATH] for a finite group [MATH] is fundamental in rational conformal field theory.', '1804.08343-1-0-1': 'The construction of [MATH] from [MATH] on the categorical level, often called gauging, is also prominent in the study of topological phases of matter.', '1804.08343-1-0-2': 'The key step in this construction is to find a braided [MATH]-crossed extension of [MATH] compatible with a given action of [MATH] on a non-degenerate braided fusion category [MATH].', '1804.08343-1-0-3': 'The extension theory of Etingof-Nikshych-Ostrik gives two obstructions for this problem, [MATH] and [MATH] for certain coefficients, the latter of which depends on a categorical lifting of the action and is notoriously difficult to compute.', '1804.08343-1-0-4': 'We show that in the case where [MATH] acts by permutations on [MATH], both of these obstructions vanish.', '1804.08343-1-0-5': 'This verifies a conjecture of Muger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.', '1804.08343-1-1-0': '# Introduction.', '1804.08343-1-2-0': 'Modular tensor categories play an important role in low dimensional physics, both in 2-dimensional conformal field theories and topological phases of matter.', '1804.08343-1-2-1': 'In conformal field theory, modular tensor categories arise as local representations of fields of observables.', '1804.08343-1-2-2': 'An important construction in this context is the orbifold construction, which takes the fixed point fields of observables by a global action of a finite group.', '1804.08343-1-2-3': 'In the completely-rational conformal net axiomatization, Muger has shown that the resulting modular tensor category can be constructed from the original by a categorical procedure known as gauging [CITATION].', '1804.08343-1-2-4': 'Gauging of modular categories also appears in the context of topological phases of matter, describing the process of promoting a group of global symmetries to local symmetries [CITATION].', '1804.08343-1-3-0': 'In the setting of abstract modular tensor categories, gauging is a multi-step process [CITATION].', '1804.08343-1-3-1': 'The starting point is an action of [MATH] on [MATH] compatible with the braiding, i.e. a homomorphism [MATH], where [MATH] is the group of equivalence classes of braided autoequivalences of [MATH].', '1804.08343-1-3-2': 'To gauge [MATH] by [MATH], we first need to find a compatible braided G-crossed extension of [MATH].', '1804.08343-1-3-3': 'We then equivariantize by the associated action of [MATH], which produces a new modular tensor category called a gauging of [MATH] by [MATH].', '1804.08343-1-3-4': 'Gaugings may not exist at all, and if they do they are usually not unique.', '1804.08343-1-4-0': 'The problem is that associated to the initial homomorphism [MATH] are two cohomological obstructions to finding a compatible braided [MATH]-crossed extension [CITATION].', '1804.08343-1-4-1': 'First one asks if [MATH] can be lifted to a categorical action [MATH].', '1804.08343-1-4-2': 'There is an obstruction to obtaining a lifting, given by a cohomology class [MATH], where [MATH] is the group of invertible objects in [MATH], also known as the simple currents of [MATH].', '1804.08343-1-4-3': 'If [MATH] vanishes, the liftings form a torsor over [MATH].', '1804.08343-1-4-4': 'Once we have a categorical action [MATH], in order to obtain a braided [MATH]-crossed extension, we now need to lift this to a tri-functor [MATH] [CITATION], Theorem 7.12.', '1804.08343-1-4-5': 'The obstruction for this lifting is a cohomology class [MATH].', '1804.08343-1-4-6': 'If this vanishes, the liftings form a torsor over [MATH].', '1804.08343-1-4-7': 'Thus starting with [MATH], there are two obstructions to the existence of a desired braided [MATH]-crossed extension of [MATH], [MATH] and [MATH], where the latter depends on a choice of lifting of [MATH].', '1804.08343-1-5-0': 'This obstruction theory has direct relevance to any (chiral) rational conformal field theory [MATH], which we can axiomatize either as a strongly-rational vertex operator algebra or a completely-rational conformal net of von Neumann algebras.', '1804.08343-1-5-1': 'The following statements are theorems for conformal nets, though many are still only conjectures for vertex operator algebras.', '1804.08343-1-5-2': 'The category of representations [MATH] is a modular tensor category.', '1804.08343-1-5-3': 'Suppose a finite group [MATH] acts faithfully on [MATH].', '1804.08343-1-5-4': 'Then [MATH] acts as automorphisms on [MATH] which respect the braiding.', '1804.08343-1-5-5': 'The orbifold [MATH] (the subtheory of [MATH] consisting of [MATH]-fixed-points) will also be rational, so [MATH] will also be a modular tensor category.', '1804.08343-1-5-6': 'This category contains [MATH] as a subcategory - this fact goes back to Doplicher-Haag-Roberts, but also follows because extending [MATH] by the objects in that subcategory recovers [MATH].', '1804.08343-1-5-7': 'The [MATH]-twisted (solotonic) representations of [MATH] form the braided [MATH]-crossed category [MATH], called the de-equivariantization of [MATH] [CITATION].', '1804.08343-1-5-8': 'The equivariantization [MATH] recovers [MATH].', '1804.08343-1-5-9': 'In particular, the obstruction [MATH] must vanish, and for some resulting choice of [MATH] so must the obstruction [MATH].', '1804.08343-1-5-10': 'This means that if we can find some [MATH]-action [MATH] on a modular tensor category [MATH], for which either the obstruction [MATH] or all [MATH] fail to vanish, then no completely-rational conformal net can have a [MATH]-action which acts like [MATH] (respectively [MATH]) on its representations.', '1804.08343-1-6-0': 'An obvious class of orbifolds are the permutation orbifolds.', '1804.08343-1-6-1': 'We start with a rational conformal field theory [MATH], say a conformal net.', '1804.08343-1-6-2': 'Write [MATH].', '1804.08343-1-6-3': 'Then any subgroup [MATH] acts by global automorphisms on the tensor power theory [MATH]; the subtheory [MATH] is called the permutation orbifold.', '1804.08343-1-6-4': 'Then [MATH] acts on the objects and morphisms of the Deligne product [MATH] in the obvious way by permuting factors, as given explicitly in Section [REF].', '1804.08343-1-6-5': 'As shown in Proposition [REF] below, this always lifts to the standard permutation categorical action [MATH] of [MATH] by permutation functors, which is a strict action whose tensorators are all identities.', '1804.08343-1-6-6': 'Hence the obstruction [MATH] must vanish.', '1804.08343-1-6-7': 'Even in this simple setting, however, the obstruction [MATH] is difficult to compute explicitly.', '1804.08343-1-6-8': 'If [MATH] did not vanish for the standard permutation categorical action [MATH], then there could be no conformal net [MATH] realizing [MATH].', '1804.08343-1-6-9': 'But it has been conjectured that any modular tensor category is [MATH] for some rational conformal field theory [MATH].', '1804.08343-1-6-10': 'For this reason, Muger conjectured that [MATH] always vanishes (Conjecture 5.5, Appendix 5, [CITATION]).', '1804.08343-1-6-11': 'This same reason makes the question of whether all [MATH] vanish, both natural and compelling.', '1804.08343-1-6-12': "The main result of our paper answers Muger's conjecture affirmatively.", '1804.08343-1-7-0': 'Let [MATH] be a finite group and [MATH] a non-degenerate braided fusion category.', '1804.08343-1-7-1': 'Let [MATH] be the standard permutation categorical action.', '1804.08343-1-8-0': "Note that we are able to work in the a-priori greater generality of non-degenerate braided fusion categories, since our arguments don't require modular data.", '1804.08343-1-8-1': 'In Question [REF] we ask whether [MATH] vanishes for the 2-cocycle twistings of [MATH].', '1804.08343-1-8-2': 'We wish to point out two immediate corollaries of our main result.', '1804.08343-1-8-3': "By [CITATION], Theorem 7.12, we obtain the following which explicitly addresses Muger's conjecture.", '1804.08343-1-9-0': 'Let [MATH] be a non-degenerate braided fusion category and [MATH].', '1804.08343-1-9-1': 'Then there exist braided [MATH]-crossed fusion categories whose trivially graded component is [MATH], and whose categorical action restricted to the trivial component is the standard permutation braided categorical action of [MATH] on [MATH].', '1804.08343-1-9-2': 'The equivalence classes of such extensions form a torsor over [MATH].', '1804.08343-1-10-0': 'We point out Muger actually conjectures that there is a distinguished such category.', '1804.08343-1-10-1': 'Although we have a distinguished categorical action [MATH], our result does not pick out a distinguished extension, and indeed we have several: the equivalence classes form a torsor as stated in the corollary, with no special class distinguished.', '1804.08343-1-10-2': "Muger's Conjecture 6.3 discusses the permutation orbifolds of conformal field theory, but here we now know there again is not a distinguished category: e.g. in the special case where [MATH] and 3 divides the order of [MATH], exactly 3 different categories can arise (the resulting category depends on the central charge mod 24) [CITATION].", '1804.08343-1-10-3': 'Thus the "distinguished" part of Muger\'s conjecture fails, though for conformal field theories the resulting categories certainly are severely limited.', '1804.08343-1-11-0': 'If [MATH] is a braided fusion category with Muger center [MATH], a minimal non-degenerate extension is a non-degenerate braided fusion category [MATH] containing [MATH] as a full braided subcategory such that [MATH].', '1804.08343-1-11-1': 'These do not always exist (for example, see Proposition 4.11 of [CITATION]).', '1804.08343-1-11-2': 'However, the existence of such an extension for a category with [MATH] for some finite group [MATH] is closely tied with the vanishing of the [MATH] obstruction associated with the canonical categorical action of [MATH] on the de-equivariantization of [MATH].', '1804.08343-1-11-3': 'From either the arguments of Appendix 5, Theorem 5.4 [CITATION], or more directly from Theorem 4.8 (i) of [CITATION], we have the following immediate corollary of Theorem [REF].', '1804.08343-1-12-0': 'Let [MATH] be a non-degenerate braided fusion category, [MATH], and [MATH] the equivariantization of [MATH] by the associated categorical permutation action.', '1804.08343-1-12-1': 'Then there exists a minimal non-degenerate extension of [MATH].', '1804.08343-1-13-0': 'We hope these results clear the path to understanding the categories resulting from permutation gauging.', '1804.08343-1-13-1': 'Recent progress has been made in the case of [MATH], [CITATION].', '1804.08343-1-13-2': 'The results there show that basic descriptions in the general case promises to be very complicated, or at best, difficult to prove using known techniques.', '1804.08343-1-13-3': 'On the other hand, a great deal is known about permutation orbifolds of conformal field theories [CITATION], [CITATION], [CITATION], [CITATION].', '1804.08343-1-13-4': 'We believe that comparing permutation orbifolds in conformal field theory and permutation gauging of modular categories could provide new clues for answering the important question: does every modular category arise from conformal field theory?', '1804.08343-1-14-0': 'The outline of the paper is as follows.', '1804.08343-1-14-1': 'In Section [MATH], we review some basics of braided [MATH]-crossed extension theory, and discuss the behavior of the [MATH] obstruction under tensor product splitting.', '1804.08343-1-14-2': 'In Section 3 we discuss some lemmas regarding the cohomological restriction maps for subgroups of symmetric groups.', '1804.08343-1-14-3': 'In Section 4, we define the standard permutation categorical actions, and investigate possible twistings.', '1804.08343-1-14-4': 'Finally, we apply the group cohomology lemmas to prove Theorem [REF].', '1804.08343-1-14-5': 'We include an appendix with GAP computations.'} | {'1804.08343-2-0-0': 'The orbifold construction [MATH] for a finite group [MATH] is fundamental in rational conformal field theory.', '1804.08343-2-0-1': 'The construction of [MATH] from [MATH] on the categorical level, often called gauging, is also prominent in the study of topological phases of matter.', '1804.08343-2-0-2': 'Given a non-degenerate braided fusion category [MATH] with a [MATH]-action, the key step in this construction is to find a braided [MATH]-crossed extension compatible with the action.', '1804.08343-2-0-3': 'The extension theory of Etingof-Nikshych-Ostrik gives two obstructions for this problem, [MATH] and [MATH] for certain coefficients, the latter depending on a categorical lifting of the action and is notoriously difficult to compute.', '1804.08343-2-0-4': 'We show that in the case where [MATH] acts by permutations on [MATH], both of these obstructions vanish.', '1804.08343-2-0-5': 'This verifies a conjecture of Muger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.', '1804.08343-2-1-0': '# Introduction.', '1804.08343-2-2-0': 'Modular tensor categories play an important role in low dimensional physics, both in 2-dimensional conformal field theories and topological phases of matter.', '1804.08343-2-2-1': 'In conformal field theory, modular tensor categories arise as local representations of fields of observables.', '1804.08343-2-2-2': 'An important construction in this context is the orbifold construction, which takes the fixed-point fields of observables by a global action of a finite group.', '1804.08343-2-2-3': 'In the completely-rational conformal net axiomatization, Muger has shown that the resulting modular tensor category can be constructed from the original by a categorical procedure known as gauging [CITATION].', '1804.08343-2-2-4': 'Gauging of modular categories also appears in the context of topological phases of matter, describing the process of promoting a group of global symmetries to local symmetries [CITATION].', '1804.08343-2-3-0': 'In the setting of abstract modular tensor categories, gauging is a multi-step process [CITATION].', '1804.08343-2-3-1': 'The starting point is an action of [MATH] on [MATH] compatible with the braiding, i.e. a homomorphism [MATH], where [MATH] is the group of equivalence classes of braided autoequivalences of [MATH].', '1804.08343-2-3-2': 'To gauge [MATH] by [MATH], we first need to find a compatible braided G-crossed extension of [MATH].', '1804.08343-2-3-3': 'We then equivariantize by the associated action of [MATH], which produces a new modular tensor category called a gauging of [MATH] by [MATH].', '1804.08343-2-3-4': 'Gaugings may not exist at all, and if they do they are usually not unique.', '1804.08343-2-4-0': 'The problem is that associated to the initial homomorphism [MATH] are two cohomological obstructions to finding a compatible braided [MATH]-crossed extension [CITATION].', '1804.08343-2-4-1': 'First one asks if [MATH] can be lifted to a categorical action [MATH] (we define [MATH] etc in Section 2 below).', '1804.08343-2-4-2': 'There is an obstruction to obtaining a lifting, given by a cohomology class [MATH], where [MATH] is the group of invertible objects in [MATH], also known as the simple currents of [MATH].', '1804.08343-2-4-3': 'If [MATH] vanishes, the liftings form a torsor over [MATH].', '1804.08343-2-4-4': 'Once we have a categorical action [MATH], in order to obtain a braided [MATH]-crossed extension, we now need to lift this to a tri-functor [MATH] [CITATION], Theorem 7.12.', '1804.08343-2-4-5': 'The obstruction for this lifting is a cohomology class [MATH].', '1804.08343-2-4-6': 'If this vanishes, the liftings form a torsor over [MATH].', '1804.08343-2-4-7': 'Thus starting with [MATH], there are two obstructions to the existence of a desired braided [MATH]-crossed extension of [MATH], [MATH] and [MATH], where the latter depends on a choice of lifting of [MATH].', '1804.08343-2-5-0': 'This obstruction theory has direct relevance to any (chiral) rational conformal field theory [MATH], which we can axiomatize either as a strongly-rational vertex operator algebra or a completely-rational conformal net of von Neumann algebras.', '1804.08343-2-5-1': 'The following statements are theorems for conformal nets, though many are still only conjectures for vertex operator algebras.', '1804.08343-2-5-2': 'The category of representations [MATH] is a modular tensor category.', '1804.08343-2-5-3': 'Suppose a finite group [MATH] acts faithfully on [MATH].', '1804.08343-2-5-4': 'Then [MATH] acts as automorphisms on [MATH] which respect the braiding.', '1804.08343-2-5-5': 'The orbifold [MATH] (the subtheory of [MATH] consisting of [MATH]-fixed-points) will also be rational, so [MATH] will also be a modular tensor category.', '1804.08343-2-5-6': 'This category contains [MATH] as a fusion subcategory - this fact goes back to Doplicher-Haag-Roberts, but also follows because extending [MATH] by the objects in that subcategory recovers [MATH].', '1804.08343-2-5-7': 'The [MATH]-twisted (solotonic) representations of [MATH] form the braided [MATH]-crossed category [MATH], called the de-equivariantization of [MATH] [CITATION].', '1804.08343-2-5-8': 'The equivariantization [MATH] recovers [MATH].', '1804.08343-2-5-9': 'In particular, the obstruction [MATH] must vanish, and for some resulting choice of [MATH] so must the obstruction [MATH].', '1804.08343-2-5-10': 'This means that if we can find some [MATH]-action [MATH] on a modular tensor category [MATH], for which either the obstruction [MATH] or all [MATH] fail to vanish, then no completely-rational conformal net can have a [MATH]-action which acts like [MATH] (respectively [MATH]) on its representations.', '1804.08343-2-6-0': 'An obvious class of orbifolds are the permutation orbifolds.', '1804.08343-2-6-1': 'Start with a rational conformal field theory [MATH], say a conformal net.', '1804.08343-2-6-2': 'Write [MATH].', '1804.08343-2-6-3': 'Then any subgroup [MATH] acts by global automorphisms on the tensor power theory [MATH]; the subtheory [MATH] is called the permutation orbifold.', '1804.08343-2-6-4': 'Then [MATH] acts on the objects and morphisms of the Deligne product [MATH] in the obvious way by permuting factors, as given explicitly in Section [REF].', '1804.08343-2-6-5': 'As shown in Proposition [REF] below, this always lifts to the standard permutation categorical action [MATH] of [MATH] by permutation functors, which is a strict action whose tensorators are all identities.', '1804.08343-2-6-6': 'Hence the obstruction [MATH] must vanish.', '1804.08343-2-6-7': 'Even in this simple setting, however, the obstruction [MATH] is difficult to compute explicitly.', '1804.08343-2-6-8': 'If [MATH] did not vanish for the standard permutation categorical action [MATH], then there could be no conformal net [MATH] realizing [MATH].', '1804.08343-2-6-9': 'But it has been conjectured that any modular tensor category is [MATH] for some rational conformal field theory [MATH].', '1804.08343-2-6-10': 'For this reason, Muger conjectured that [MATH] always vanishes (Conjecture 5.5, Appendix 5, [CITATION]).', '1804.08343-2-6-11': 'This same reason makes the question of whether all [MATH] vanish, both natural and compelling.', '1804.08343-2-6-12': "The main result of our paper answers Muger's conjecture affirmatively.", '1804.08343-2-7-0': 'Let [MATH] be a finite group and [MATH] a non-degenerate braided fusion category.', '1804.08343-2-7-1': 'Let [MATH] be the standard permutation categorical action.', '1804.08343-2-8-0': "Note that we are able to work in the a priori greater generality of non-degenerate braided fusion categories, since our arguments don't require modular data.", '1804.08343-2-8-1': 'In Question [REF] we ask whether [MATH] vanishes for the 2-cocycle twistings of [MATH].', '1804.08343-2-8-2': 'We wish to point out two immediate corollaries of our main result.', '1804.08343-2-8-3': "By [CITATION], Theorem 7.12, we obtain the following statement which explicitly addresses Muger's conjecture.", '1804.08343-2-9-0': 'Let [MATH] be a non-degenerate braided fusion category and [MATH].', '1804.08343-2-9-1': 'Then there exist braided [MATH]-crossed fusion categories whose trivially-graded component is [MATH], and whose categorical action restricted to the trivial component is the standard permutation braided categorical action of [MATH] on [MATH].', '1804.08343-2-9-2': 'The equivalence classes (up to braided equivalence) of such extensions form a torsor over [MATH].', '1804.08343-2-10-0': 'We point out Muger actually conjectures that there is a distinguished such category.', '1804.08343-2-10-1': 'Although we have a distinguished categorical action [MATH], our result does not pick out a distinguished extension, and indeed we have several: the equivalence classes form a torsor as stated in the corollary, with no special class distinguished.', '1804.08343-2-10-2': "Muger's Conjecture 6.3 discusses the permutation orbifolds of conformal field theory, but even in this context there is not a distinguished category: e.g. in the special case where [MATH] and 3 divides the order of [MATH], exactly 3 different categories can arise (the resulting category depends on the central charge modulo 24 - this has been observed by Marcel Bischoff, as well as Example 2.1.1 in [CITATION] and Theorem 2 in [CITATION].", '1804.08343-2-10-3': 'Thus the "distinguished" part of Muger\'s conjecture fails, though for conformal field theories the resulting categories certainly are severely limited.', '1804.08343-2-11-0': 'Given a finite [MATH]-set, [CITATION] construct a weak [MATH]-equivariant fusion category.', '1804.08343-2-11-1': 'It is tempting to guess that this matches a braided [MATH]-crossed extension of Corollary 1.2 but we cannot say for sure: it is only weakly rigid, and not necessarily rigid, because it is constructed using modular functors, though it does give a fusion category in the [MATH] case.', '1804.08343-2-12-0': "Bantay's Orbifold Covariance Principle [CITATION] is the requirement that all properties of a rational conformal field theory have to be compatible with the fact that permutation orbifolds of a rational theory is rational.", '1804.08343-2-12-1': 'For example, you get fairly easily from this the congruence property of the SL[MATH] representation (modular data) of these theories.', '1804.08343-2-12-2': 'Modulo one subtlety, Corollary 1.2 requires that a modular tensor category must be compatible with the fact that its gaugings by permutation actions, which all must exist, are themselves all healthy modular tensor categories.', '1804.08343-2-12-3': 'This should have plenty of consequences, including a faster proof of the congruence property.', '1804.08343-2-12-4': 'The subtlety though is that [CITATION] ignores spherical structures for extensions, and in this setting one expects the theory to be slightly different, though we expect the obstructions for permutation functors to be the same in any case.', '1804.08343-2-13-0': 'If [MATH] is a braided fusion category with Muger center [MATH], a minimal non-degenerate extension is a non-degenerate braided fusion category [MATH] containing [MATH] as a full fusion subcategory such that [MATH].', '1804.08343-2-13-1': 'These do not always exist (for example, see Proposition 4.11 of [CITATION]).', '1804.08343-2-13-2': 'However, the existence of such an extension for a category with [MATH] for some finite group [MATH] is closely tied with the vanishing of the [MATH] obstruction associated with the canonical categorical action of [MATH] on the de-equivariantization of [MATH].', '1804.08343-2-13-3': 'From the arguments of Appendix 5, Theorem 5.4 of [CITATION], or more directly from Theorem 4.8 (i) of [CITATION], we have the following immediate corollary of Theorem [REF].', '1804.08343-2-14-0': 'Let [MATH] be a non-degenerate braided fusion category, [MATH], and [MATH] the equivariantization of [MATH] by the associated categorical permutation action.', '1804.08343-2-14-1': 'Then there exists a minimal non-degenerate extension of [MATH].', '1804.08343-2-15-0': 'We hope these results clear the path to understanding the categories resulting from permutation gauging.', '1804.08343-2-15-1': 'Recent progress has been made in the case of [MATH], [CITATION].', '1804.08343-2-15-2': 'The results there show that basic descriptions in the general case promises to be very complicated, or at best, difficult to prove using known techniques.', '1804.08343-2-15-3': 'On the other hand, a great deal is known about permutation orbifolds of conformal field theories [CITATION], [CITATION], [CITATION], [CITATION].', '1804.08343-2-15-4': 'We believe that comparing permutation orbifolds in conformal field theory and permutation gauging of modular categories could provide new clues for answering the important question: does every modular category arise from conformal field theory?', '1804.08343-2-16-0': 'The outline of the paper is as follows.', '1804.08343-2-16-1': 'In Section [MATH], we review some basics of braided [MATH]-crossed extension theory, and discuss the behavior of the [MATH] obstruction under tensor product splitting.', '1804.08343-2-16-2': 'In Section 3 we discuss some lemmas regarding the cohomological restriction maps for subgroups of symmetric groups.', '1804.08343-2-16-3': 'In Section 4, we define the standard permutation categorical actions, and investigate possible twistings.', '1804.08343-2-16-4': 'Finally, we apply the group cohomology lemmas to prove Theorem [REF].', '1804.08343-2-16-5': 'We include an appendix with GAP computations.'} | [['1804.08343-1-3-0', '1804.08343-2-3-0'], ['1804.08343-1-3-1', '1804.08343-2-3-1'], 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0811.3360 | {'0811.3360-1-0-0': 'M.S., Department of Mathematics and Statistics, Wright State University, 2007.', '0811.3360-1-0-1': 'Estimates in the Generalized Morrey Space for Parabolic Systems', '0811.3360-1-1-0': 'The purpose of this paper is to study the parabolic system [MATH] in the generalized Morrey Space [MATH] .', '0811.3360-1-1-1': 'We would like to understand the regularity of the solutions of this system.', '0811.3360-1-1-2': 'It will be shown that 1: if [MATH] then [MATH] , and 2: if [MATH] then [MATH].', '0811.3360-1-1-3': 'Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions.', '0811.3360-1-2-0': 'roman', '0811.3360-1-3-0': 'arabic', '0811.3360-1-4-0': '# Introduction', '0811.3360-1-5-0': 'In this paper we will be investing the following linear parabolic systems of the form', '0811.3360-1-6-0': '[MATH]', '0811.3360-1-7-0': 'where [MATH] and the repeated indices denote summation such as', '0811.3360-1-8-0': '[MATH]', '0811.3360-1-9-0': 'Throughout the paper we assume an uniform ellipticity condition, namely:', '0811.3360-1-10-0': '[MATH] where [MATH] , [MATH]', '0811.3360-1-11-0': 'The main purpose of this paper is to demonstrate that one can obtain the gradient estimates in generalized Morrey spaces [MATH] for weak solutions of (1-1).', '0811.3360-1-12-0': '# Preliminaries', '0811.3360-1-13-0': 'Notations:', '0811.3360-1-14-0': '[MATH]-ball in [MATH] centered at [MATH] with radius [MATH] and [MATH] for [MATH] and [MATH] - parabolic cylinder in [MATH] which has a vertex at [MATH]', '0811.3360-1-15-0': 'Boundary Terms of the Parabolic Cylinder:', '0811.3360-1-16-0': 'The boundary of the parabolic cylinder consists of the lateral walls, the lower boundary, and the lower corners, however we will use [MATH] to denote the parabolic boundary of the parabolic cylinder', '0811.3360-1-17-0': 'Morrey Space for Parabolic Setting:', '0811.3360-1-18-0': '[MATH] with [MATH] is a continuous function on [MATH] on [MATH] is the diameter of [MATH] .', '0811.3360-1-19-0': 'Lemma 2.1:', '0811.3360-1-20-0': '[MATH] is a Banach Space under the norm', '0811.3360-1-21-0': '[MATH]', '0811.3360-1-22-0': 'Proof:', '0811.3360-1-23-0': 'So the "norm" must satisfy the three properties to classify as a norm, then it must be shown that the space is complete.', '0811.3360-1-23-1': '[MATH] is trival similarly [MATH] is quite obvious, though the triangle inequality must be shown since it is not very simple.', '0811.3360-1-24-0': "Consider [MATH] via Minkowski's inequality.", '0811.3360-1-24-1': 'The one applies the [MATH] function to both sides which yields,', '0811.3360-1-25-0': '[MATH] .', '0811.3360-1-25-1': 'This implies that [MATH] Therefore the triangle inequality is satisfied, and [MATH] is a norm on [MATH].', '0811.3360-1-26-0': 'Next it must be shown that [MATH] is complete under the norm [MATH] .', '0811.3360-1-26-1': 'Let [MATH] be a Cauchy sequence in [MATH].', '0811.3360-1-26-2': "Tschebyshev's inequality implies that [MATH] .", '0811.3360-1-26-3': 'Therefore, there exists a subsequence [MATH] and [MATH] such that [MATH].', '0811.3360-1-26-4': 'For every [MATH] there exists [MATH] such that [MATH] if [MATH].', '0811.3360-1-26-5': "Let [MATH] then by Fatou's lemma, one obtains, [MATH] for [MATH].", '0811.3360-1-26-6': 'Thus [MATH] by [MATH] and [MATH] .', '0811.3360-1-26-7': 'Therefore [MATH] is complete and hence it is a Banach space.', '0811.3360-1-26-8': '[MATH]', '0811.3360-1-27-0': 'Morrey Space for p=2:', '0811.3360-1-28-0': 'We consider the case of [MATH] for [MATH].', '0811.3360-1-28-1': 'Define the Morrey space for [MATH] by:', '0811.3360-1-29-0': '[MATH]', '0811.3360-1-30-0': 'Definition 2.1:', '0811.3360-1-31-0': 'A function [MATH] is said to be almost increasing if there exists [MATH] such that [MATH] for [MATH]', '0811.3360-1-32-0': 'The next proposition is due to [Hu] which will be useful for the main results of this paper.', '0811.3360-1-33-0': 'Proposition 2.1:', '0811.3360-1-34-0': 'Let [MATH] be a non-negative almost increasing function in [MATH] and [MATH] on [MATH].', '0811.3360-1-34-1': 'Suppose that', '0811.3360-1-35-0': '(a) There exists [MATH] such that [MATH] for [MATH]', '0811.3360-1-36-0': '(b) There exists [MATH] such that [MATH] is almost increasing in [MATH]', '0811.3360-1-37-0': 'Then there exists [MATH] and [MATH] such that if [MATH] then [MATH]', '0811.3360-1-38-0': 'BMO and VMO Spaces:', '0811.3360-1-39-0': 'Definition 2.2:', '0811.3360-1-40-0': 'Let [MATH] and [MATH] on [MATH] then [MATH] is defined by:', '0811.3360-1-41-0': '[MATH]', '0811.3360-1-42-0': 'where [MATH] and [MATH]', '0811.3360-1-43-0': 'Definition 2.3:', '0811.3360-1-44-0': 'Letting [MATH] one defines [MATH] by:', '0811.3360-1-45-0': '[MATH]', '0811.3360-1-46-0': 'Weak Solutions:', '0811.3360-1-47-0': 'We would like to discuss the energy estimates for the system (2-3) [MATH] in [MATH] and [MATH] is constant.', '0811.3360-1-47-1': 'For [MATH], let [MATH] be a test function with [MATH] and [MATH] with [MATH] and [MATH] .', '0811.3360-1-47-2': 'Next we multiply the test function by (2-3) and use integration by parts.', '0811.3360-1-48-0': '[MATH] The boundary term is zero by definition of [MATH] and [MATH].', '0811.3360-1-49-0': '[MATH] Therefore by uniform ellipticity and Cauchy-Schwartz inequality:', '0811.3360-1-50-0': '[MATH] Then since [MATH] and [MATH] This implies', '0811.3360-1-51-0': '[MATH] Then this implies', '0811.3360-1-52-0': '[MATH] We call this last line the energy estimate for (2-3).', '0811.3360-1-52-1': 'Then define [MATH] This space is the Sobolov space counterpart for parabolic equations.', '0811.3360-1-53-0': 'Using energy estimates and Sobolev embedding theorem, one can get the Morrey estimate for (2-3) with constant coefficients.', '0811.3360-1-54-0': 'Lemma 2.2:', '0811.3360-1-55-0': 'Let [MATH] solve the following,', '0811.3360-1-56-0': '[MATH] Then for [MATH] and [MATH]', '0811.3360-1-57-0': 'Proof:', '0811.3360-1-58-0': 'By [Sc] one has [MATH] .', '0811.3360-1-58-1': 'Since [MATH] is constant, by differentiating (2-3) we obtain that [MATH] is still a solution.', '0811.3360-1-58-2': 'Hence [MATH]', '0811.3360-1-59-0': 'We will now turn our attention to the main results of this paper.', '0811.3360-1-60-0': '# Main Results', '0811.3360-1-61-0': 'In the preliminaries section the Morrey estimate we are interested in was shown for [MATH] by [Sc].', '0811.3360-1-61-1': 'In this section we will extend this result to system (1-1).', '0811.3360-1-61-2': 'We establish the Morrey estimate first for the case [MATH] and second for [MATH] .', '0811.3360-1-62-0': 'Theorem 3.1:', '0811.3360-1-63-0': 'Let [MATH] be a weak solution to [MATH].', '0811.3360-1-63-1': 'Let [MATH] and assume the uniform ellipticity condition and [MATH] also assume there exists [MATH] such that [MATH] and [MATH] is almost increasing, then [MATH] for any [MATH], for [MATH] and [MATH] .', '0811.3360-1-63-2': 'Moreover', '0811.3360-1-64-0': '[MATH] such that for [MATH].', '0811.3360-1-65-0': 'Proof:', '0811.3360-1-66-0': 'Let [MATH] .', '0811.3360-1-66-1': 'As in Theorem 3.1 let [MATH] and [MATH] satisfy respectively[MATH]', '0811.3360-1-67-0': '(3-3) [MATH] and', '0811.3360-1-68-0': '(3-4) [MATH]', '0811.3360-1-69-0': 'Similar to the proof of Theorem 3.1 one obtains', '0811.3360-1-70-0': '[MATH] .', '0811.3360-1-70-1': 'We multiply (3-4) by [MATH] and perform an integration by parts to obtain', '0811.3360-1-71-0': "[MATH] Then by Holder's inequality, along with [MATH] and by proposition 3.1 one has", '0811.3360-1-72-0': '[MATH] and therefore one gets', '0811.3360-1-73-0': '[MATH] .', '0811.3360-1-73-1': 'It follows that', '0811.3360-1-74-0': '[MATH] .', '0811.3360-1-74-1': 'Then by using proposition 2.1 one can achieve the desired result.', '0811.3360-1-75-0': 'Our final theorem will establish the generalized Morrey estimate for system (1-1).', '0811.3360-1-76-0': 'Theorem 3.2:', '0811.3360-1-77-0': 'Let [MATH] be a weak solution to [MATH] with the uniform ellipticity condition.', '0811.3360-1-77-1': 'Suppose there exist [MATH] such that [MATH] and [MATH] is almost increasing.', '0811.3360-1-77-2': 'Assume [MATH] and [MATH], then [MATH] for any [MATH] and for [MATH] and [MATH] .', '0811.3360-1-77-3': 'Moreover one obtains the following interior estimate', '0811.3360-1-78-0': '[MATH] .', '0811.3360-1-79-0': 'Proof:', '0811.3360-1-80-0': 'Again like in theorem 3.1, let [MATH] and [MATH] satisfy respectively', '0811.3360-1-81-0': '(3-5) [MATH] and', '0811.3360-1-82-0': '(3-6) [MATH] Apply Lemma 3.1 to [MATH] one obtains the following', '0811.3360-1-83-0': '(3-7)[MATH] .', '0811.3360-1-83-1': 'Also multiplying (3-6) to [MATH] and performing integration by parts, just as in the proof of Theorem 3.1 one has', '0811.3360-1-84-0': '[MATH] .', '0811.3360-1-84-1': "Applying Cauchy-Schwartz's inequality one has", '0811.3360-1-85-0': '(3-8)[MATH] .', '0811.3360-1-85-1': 'By combining (3-7), (3-8) and the fact that [MATH] one obtains', '0811.3360-1-86-0': '[MATH] .', '0811.3360-1-86-1': 'Therefore the result will follow applying Proposition 2.1 [MATH]', '0811.3360-1-87-0': '# References', '0811.3360-1-88-0': '[Hu] Q. Huang, Estimates on the Generalized Morrey Spaces [MATH] and [MATH] for Linear Elliptic Systems.', '0811.3360-1-88-1': 'Indiana University Mathematics Journal Vol. 45, No. 2 (1996)', '0811.3360-1-89-0': '[Sc] W. Schlag, Schauder and [MATH] Estimates for Parabolic Systems via Campanato Spaces.', '0811.3360-1-89-1': 'Commun.', '0811.3360-1-89-2': 'in Partial Differential Equations, 21(78), 1141-1175 (1996)', '0811.3360-1-90-0': '[St-Gi] M. Struwe M. Giaquinta, On the partial regularity of weak solutions of Nonlinear Parabolic Systems, Mathematische Zeitschrift Vol. 179'} | {'0811.3360-2-0-0': 'The purpose of this paper is to study the parabolic system [MATH] in the generalized Morrey Space [MATH].', '0811.3360-2-0-1': 'We would like to understand the regularity of the solutions of this system.', '0811.3360-2-0-2': 'It will be shown that 1: if [MATH] then [MATH] , and 2: if [MATH] then [MATH].', '0811.3360-2-0-3': 'Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions.', '0811.3360-2-1-0': '# Introduction', '0811.3360-2-2-0': 'In this paper we will be investigating the following linear parabolic systems of the form [EQUATION] where [MATH] and the repeated indices denote summation such as [EQUATION]', '0811.3360-2-2-1': 'Throughout the paper we assume an uniform ellipticity condition, namely: [EQUATION] where [MATH], [MATH], [MATH], [MATH], and [MATH].', '0811.3360-2-2-2': 'The main purpose of this paper is to demonstrate that one can obtain the gradient estimates in generalized Morrey spaces [MATH] for weak solutions of ([REF]).', '0811.3360-2-2-3': 'In the next section we discuss some definitions and preliminaries.', '0811.3360-2-3-0': '# Preliminaries and Weak Solutions', '0811.3360-2-4-0': 'In this section we discuss some needed theorems and lemmas for the main results of this paper.', '0811.3360-2-4-1': 'More over we state some notation that will be used throughout this paper.', '0811.3360-2-4-2': 'We denote the [MATH]-dimensional ball centered at [MATH] with radius [MATH] as [EQUATION]', '0811.3360-2-4-3': 'We will also let [MATH] represent a [MATH] dimensional coordinate, i.e. [MATH], where [MATH], [MATH] and [MATH].', '0811.3360-2-4-4': 'Similarly [MATH].', '0811.3360-2-4-5': 'We will denote the parabolic cylinder in [MATH] with vertex at [MATH] by [EQUATION]', '0811.3360-2-4-6': 'The boundary of the parabolic cylinder consists of the lateral walls, the lower boundary, and the lower corners, however we will use [MATH] to denote the parabolic boundary of the parabolic cylinder.', '0811.3360-2-4-7': 'Next we define what the Morrey space is for the parabolic setting.', '0811.3360-2-5-0': 'The parabolic Morrey space is defined to be the following [EQUATION] with [MATH], [MATH], and [MATH] is a continuous function on [MATH], [MATH] on [MATH], and [MATH] is the diameter of [MATH] with [MATH].', '0811.3360-2-6-0': 'We now come to the first lemma which states that the parabolic Morrey space is a Banach space.', '0811.3360-2-7-0': 'The space [MATH] is a Banach space under the following norm [EQUATION]', '0811.3360-2-7-1': 'First it must be shown that [MATH] is indeed a norm.', '0811.3360-2-7-2': 'Then it must be shown that the space is complete.', '0811.3360-2-7-3': 'Showing the conditions [MATH] is positive definite and the homogeneity condition are trivial.', '0811.3360-2-7-4': 'However the triangle inequality is not obvious.', '0811.3360-2-7-5': "Using Minkowski's inequality and the face that [MATH] on [MATH] one gets the following, [EQUATION]", '0811.3360-2-7-6': 'This string of inequalities shows the triangle inequality is satisfied and thus [MATH] defines a norm.', '0811.3360-2-7-7': 'All that is left to show is that the space is complete under the norm.', '0811.3360-2-7-8': 'To do this one must show that every Cauchy sequence from [MATH] converges to an element in [MATH].', '0811.3360-2-7-9': 'Let [MATH] be a Cauchy sequence in [MATH].', '0811.3360-2-7-10': "Tschebyshev's inequality implies that [EQUATION] where [MATH] is standard Lebesgue measure.", '0811.3360-2-7-11': 'Therefore, there exists a subsequence [MATH] and a [MATH] such that [MATH] converges to [MATH] a.e. in [MATH].', '0811.3360-2-7-12': 'Then for every [MATH] there exists [MATH] such that [MATH] if [MATH].', '0811.3360-2-7-13': "Letting [MATH], Fatou's lemma implies that [MATH] for [MATH].", '0811.3360-2-7-14': 'Thus [MATH] by [MATH] and [MATH] as [MATH] .', '0811.3360-2-7-15': 'Therefore [MATH] is complete and hence it is a Banach space.', '0811.3360-2-7-16': 'This completes the proof.', '0811.3360-2-8-0': 'For the rest this paper we will set [MATH] in the Morrey space [MATH].', '0811.3360-2-8-1': 'We state this next for convenience.', '0811.3360-2-8-2': '[EQUATION]', '0811.3360-2-8-3': 'The next definition was coined in [CITATION] and the subsidiary lemma was also proven in the same paper.', '0811.3360-2-8-4': 'The following definition defines when a function is said to be "almost" increasing.', '0811.3360-2-8-5': 'We say "almost" since the natural thought with almost would be that the function is increasing everywhere except on a set of measure zero, however, here is not the case.', '0811.3360-2-9-0': 'A function [MATH] is said to be almost increasing if there exists [MATH] such that [MATH] for [MATH].', '0811.3360-2-10-0': 'Now we state the lemma that was proven in [CITATION].', '0811.3360-2-11-0': 'Let [MATH] be a non-negative almost increasing function in [MATH] and [MATH] a positive function on [MATH].', '0811.3360-2-11-1': 'Suppose that', '0811.3360-2-12-0': 'There exists [MATH] such that [MATH] for [MATH]', '0811.3360-2-13-0': 'There exists [MATH] such that [MATH] is almost increasing in [MATH]', '0811.3360-2-14-0': 'Then there exists [MATH] and [MATH] such that if [MATH] then [MATH].', '0811.3360-2-15-0': 'Next we define the bounded mean oscillation and vanishing mean oscillation spaces in the parabolic setting.', '0811.3360-2-15-1': 'This is already well understood in say a [MATH]-dimensional ball or on some bounded domain [MATH] in [MATH].', '0811.3360-2-15-2': 'We define the bounded mean oscillation, [MATH], in the following way.', '0811.3360-2-16-0': 'Let [MATH] and [MATH] on [MATH]; so [MATH] is a positive continuous function on the interval [MATH].', '0811.3360-2-16-1': 'The [MATH] is defined by [EQUATION] where [MATH] and [MATH].', '0811.3360-2-17-0': 'Next we define the vanishing mean oscillation space for the parabolic setup in a similar way.', '0811.3360-2-18-0': 'If [MATH], where [MATH] is the continuous function defined in the bounded mean oscillation space definition, then the [MATH] is defined by [EQUATION] where [EQUATION]', '0811.3360-2-18-1': 'For the rest of this section discuss the weak solutions to variation of the system of parabolic parabolic partial differential equations that this paper is concerned with.', '0811.3360-2-18-2': 'Let [MATH] be constant and consider the following system in [MATH] : [EQUATION]', '0811.3360-2-18-3': 'For [MATH], let [MATH] be a test function with [MATH], the space of smooth functions vanishing at infinity, [MATH], and [MATH] with [MATH] and [MATH] defined in the following way [EQUATION]', '0811.3360-2-18-4': 'Multiplying ([REF]) by the test function, using integration by parts and noticing that the boundary term is zero by the definition of [MATH] and [MATH] one gets [EQUATION]', '0811.3360-2-18-5': 'Then by the uniform ellipticity condition, ([REF]), and the Cauchy-Schwartz inequality one has [EQUATION]', '0811.3360-2-18-6': 'Then since [MATH] and, by a simple computation, [MATH] one gets [EQUATION]', '0811.3360-2-18-7': 'This last inequality in turn implies the following inequality [EQUATION]', '0811.3360-2-18-8': 'The inequality, ([REF]), is called the energy estimate for the system of partial differential equations stated in ([REF]).', '0811.3360-2-18-9': 'We derived this energy estimate as it will have applications to the proofs of the main results it also lets us define a Sobolev space counterpart for parabolic equations.', '0811.3360-2-18-10': 'Consider the following definition of a space [EQUATION] [MATH] is said to be the Solobev space counterpart for parabolic equations.', '0811.3360-2-18-11': 'We call it this since our definition of it looks very similar to the definition of the Solobev space with [MATH].', '0811.3360-2-18-12': 'Using these energy estimates and the Sobolev embedding theorem, one can get the Morrey estimate for the system of partial differential equations stated in ([REF]) with constant coefficients.', '0811.3360-2-18-13': 'The Morrey estimate for a system of homogeneous parabolic partial differential equations with constant coefficients is [EQUATION] for [MATH] and [MATH].', '0811.3360-2-18-14': 'We end this section with the formal statement of this and a proof.', '0811.3360-2-19-0': 'Let [MATH] be a solution to the system of partial differential equations defined in ([REF]) in [MATH].', '0811.3360-2-19-1': 'Then for [MATH] and [MATH] the following inequality holds [EQUATION]', '0811.3360-2-19-2': 'Recall the system of partial differential equations stated in ([REF]): [EQUATION]', '0811.3360-2-19-3': 'Since the coefficients, [MATH], are constant, differentiating the above equation with respect to [MATH] shows that [MATH] is still a solution to the system of differential equations.', '0811.3360-2-19-4': 'By [CITATION] one has [EQUATION] where [MATH] is a solution to the above system of equations.', '0811.3360-2-19-5': 'Using this inequality and the fact that [MATH] is still a solution one arrives at [EQUATION] and the result follows.', '0811.3360-2-19-6': 'Thus this completes the proof.', '0811.3360-2-20-0': 'We now close this section and turn the main results of this paper.', '0811.3360-2-21-0': '# Main Results', '0811.3360-2-22-0': 'In the previous section the Morrey estimate we are interested in was shown when the coefficients, [MATH], were constant and mainly done by [CITATION].', '0811.3360-2-22-1': 'In this section we will extend the result to the system of partial differential equations defined in ([REF]).', '0811.3360-2-22-2': 'We first establish the Morrey estimate for the case [MATH], the space of continuous functions on the closure of [MATH], and second the case [MATH], the space of all bounded functions with vanishing mean oscillation.', '0811.3360-2-23-0': 'Let [MATH] be a weak solution, in [MATH], to the following system of partial differential equations [EQUATION] for [MATH].', '0811.3360-2-23-1': 'Let [MATH] and suppose they satisfy the uniform ellipticity condition with [MATH].', '0811.3360-2-23-2': 'Suppose there exists [MATH] and [MATH] such that [MATH] and that the function [MATH] is almost increasing, then [MATH] for any [MATH], for [MATH] and [MATH].', '0811.3360-2-23-3': 'Moreover the following inequality holds [EQUATION]', '0811.3360-2-23-4': 'Let [MATH] satisfy the following system [EQUATION] where [MATH] is a fixed point.', '0811.3360-2-23-5': 'Then [MATH] will satisfy this system [EQUATION]', '0811.3360-2-23-6': 'Clearly by Lemma ([REF]) one obtains the following inequality [EQUATION]', '0811.3360-2-23-7': 'Multiplying equation ([REF]) by [MATH], integrating and performing integration by parts, one obtains the following [EQUATION]', '0811.3360-2-23-8': 'Since [MATH], for small enough [MATH] one has [MATH] for some [MATH].', '0811.3360-2-23-9': 'Therefore one gets [EQUATION] using the Schwartz inequality.', '0811.3360-2-23-10': 'This last inequality yields [EQUATION]', '0811.3360-2-23-11': 'Therefore one obtains the following [EQUATION]', '0811.3360-2-23-12': 'Then the desired result follows immediately from lemma ([REF]).', '0811.3360-2-23-13': 'Thus the theorem has been proved.', '0811.3360-2-24-0': 'It has just been shown that the Morrey estimate is valid for continuous functions on the closure of the parabolic domain, which can be thought of as a finite cylinder.', '0811.3360-2-24-1': 'Before the desired estimate can be proven, we will need two additional lemmas.', '0811.3360-2-24-2': 'The first lemma, dubbed the "reverse" Holder inequality can be found in [CITATION].', '0811.3360-2-24-3': 'We will use this in proving the second lemma that we need before the final estimate is shown.', '0811.3360-2-25-0': 'Let [MATH] be a weak solution to the following system [EQUATION] in [MATH] with [MATH].', '0811.3360-2-25-1': 'Assume that the [MATH] satisfy the uniform ellipticity condition, then there exists some [MATH] such that [MATH] and for every [MATH] the following inequality holds [EQUATION]', '0811.3360-2-25-2': 'We leave without proof as this was done in [CITATION].', '0811.3360-2-25-3': 'We now move to the second lemma needed for the last Morrey estimate that we desire.', '0811.3360-2-26-0': 'Let [MATH] be a weak solution to the following system [EQUATION] in [MATH] with [MATH].', '0811.3360-2-26-1': 'Assume that the [MATH] and that they satisfy the uniform ellipticity condition.', '0811.3360-2-26-2': 'Then for any [MATH] there exists [MATH] and [MATH] depending only on [MATH], [MATH], [MATH], [MATH], and [MATH] such that for [MATH] the following inequality holds [EQUATION]', '0811.3360-2-26-3': 'First define the following [EQUATION]', '0811.3360-2-26-4': 'As in theorem ([REF]), let [MATH] and [MATH] satisty the following systems of partial differential equations respectively [EQUATION] and [EQUATION]', '0811.3360-2-26-5': 'Similar to the proof of theorem ([REF]) one can obtain the following estimate [EQUATION]', '0811.3360-2-26-6': 'Multiplying [MATH] to equation ([REF]), integrating and performing an integration by parts, one gets the following inequality [EQUATION]', '0811.3360-2-26-7': "Using Holder's inequality, lemma ([REF]) and with the fact that the [MATH] one obtains the following inequality [EQUATION]", '0811.3360-2-26-8': 'Therefore one gets the following inequality [EQUATION]', '0811.3360-2-26-9': 'Then by using lemma ([REF]) one can achieve the desired result.', '0811.3360-2-26-10': 'Thus this completes the proof.', '0811.3360-2-27-0': 'We are now in the position to state and prove the final theorem in this paper.', '0811.3360-2-27-1': 'We have all the tools necessary to get the final Morrey estimate which extends the result in the system of linear elliptic partial differential equations case.', '0811.3360-2-28-0': 'Let [MATH] be a weak solution to the following system of parabolic partial differential equations [EQUATION] in [MATH] with [MATH] and let the [MATH] satisfy the uniform ellipticity condition.', '0811.3360-2-28-1': 'Suppose there exists [MATH] and [MATH] such that [MATH] and that the function [MATH] is almost increasing.', '0811.3360-2-28-2': 'If the [MATH] and [MATH], then [MATH] for any [MATH] and for [MATH] and [MATH].', '0811.3360-2-28-3': 'Moreover the following interior integral estimate holds [EQUATION]', '0811.3360-2-28-4': 'Again as in theorem ([REF]), let [MATH] and [MATH] satisfy the following systems of partial differential equations respectively [EQUATION] and [EQUATION]', '0811.3360-2-28-5': 'Applying lemma ([REF]) to [MATH] yields the following inequality [EQUATION]', '0811.3360-2-28-6': 'Then multiplying [MATH] to equation ([REF]), integrating and performing an integration by parts, just as in theorem ([REF]), one has the following [EQUATION]', '0811.3360-2-28-7': 'Using the Cauchy-Schwartz inequality one this last inequality one gets [EQUATION]', '0811.3360-2-28-8': 'Since one has [MATH], combining equations ([REF]) and ([REF]) one obtains the following inequality [EQUATION]', '0811.3360-2-28-9': 'Therefore the result will follow by applying lemma ([REF]).', '0811.3360-2-28-10': 'Thus this completes the proof.'} | [['0811.3360-1-1-1', '0811.3360-2-0-1'], ['0811.3360-1-1-2', '0811.3360-2-0-2'], ['0811.3360-1-1-3', '0811.3360-2-0-3'], ['0811.3360-1-26-1', '0811.3360-2-7-9'], ['0811.3360-1-26-7', '0811.3360-2-7-15'], ['0811.3360-1-31-0', '0811.3360-2-9-0'], ['0811.3360-1-16-0', '0811.3360-2-4-6'], ['0811.3360-1-36-0', '0811.3360-2-13-0'], ['0811.3360-1-1-0', '0811.3360-2-0-0'], ['0811.3360-1-34-0', '0811.3360-2-11-0'], ['0811.3360-1-26-3', '0811.3360-2-7-11'], ['0811.3360-1-26-4', '0811.3360-2-7-12'], ['0811.3360-1-77-1', '0811.3360-2-28-1'], ['0811.3360-1-11-0', '0811.3360-2-2-2'], ['0811.3360-1-18-0', '0811.3360-2-5-0'], ['0811.3360-1-61-0', '0811.3360-2-22-0'], ['0811.3360-1-61-1', '0811.3360-2-22-1'], ['0811.3360-1-61-2', '0811.3360-2-22-2'], ['0811.3360-1-59-0', '0811.3360-2-20-0'], ['0811.3360-1-47-1', '0811.3360-2-18-3'], ['0811.3360-1-47-2', '0811.3360-2-18-4'], ['0811.3360-1-52-1', '0811.3360-2-18-9'], ['0811.3360-1-52-1', '0811.3360-2-18-10'], ['0811.3360-1-53-0', '0811.3360-2-18-12'], ['0811.3360-1-53-0', '0811.3360-2-18-13'], ['0811.3360-1-23-0', '0811.3360-2-7-2'], ['0811.3360-1-23-0', '0811.3360-2-7-7'], ['0811.3360-1-25-1', '0811.3360-2-7-6'], ['0811.3360-1-26-0', '0811.3360-2-7-1'], ['0811.3360-1-26-0', '0811.3360-2-7-2'], ['0811.3360-1-26-0', '0811.3360-2-7-7'], ['0811.3360-1-26-2', '0811.3360-2-7-10'], ['0811.3360-1-26-5', '0811.3360-2-7-13'], ['0811.3360-1-77-0', '0811.3360-2-28-0'], ['0811.3360-1-77-2', '0811.3360-2-28-2'], ['0811.3360-1-77-3', '0811.3360-2-28-3'], ['0811.3360-1-80-0', '0811.3360-2-28-4'], ['0811.3360-1-82-0', '0811.3360-2-28-5'], ['0811.3360-1-5-0', '0811.3360-2-2-0'], ['0811.3360-1-74-1', '0811.3360-2-26-9'], ['0811.3360-1-63-0', '0811.3360-2-25-0'], ['0811.3360-1-63-0', '0811.3360-2-26-0'], ['0811.3360-1-63-1', '0811.3360-2-23-2'], ['0811.3360-1-63-1', '0811.3360-2-25-1']] | [['0811.3360-1-1-1', '0811.3360-2-0-1'], ['0811.3360-1-1-2', '0811.3360-2-0-2'], ['0811.3360-1-1-3', '0811.3360-2-0-3'], ['0811.3360-1-26-1', '0811.3360-2-7-9'], ['0811.3360-1-26-7', '0811.3360-2-7-15']] | [['0811.3360-1-31-0', '0811.3360-2-9-0'], ['0811.3360-1-16-0', '0811.3360-2-4-6'], ['0811.3360-1-36-0', '0811.3360-2-13-0'], ['0811.3360-1-1-0', '0811.3360-2-0-0'], ['0811.3360-1-34-0', '0811.3360-2-11-0'], ['0811.3360-1-26-3', '0811.3360-2-7-11'], ['0811.3360-1-26-4', '0811.3360-2-7-12'], ['0811.3360-1-77-1', '0811.3360-2-28-1'], ['0811.3360-1-11-0', '0811.3360-2-2-2']] | [] | [['0811.3360-1-18-0', '0811.3360-2-5-0'], ['0811.3360-1-61-0', '0811.3360-2-22-0'], ['0811.3360-1-61-1', '0811.3360-2-22-1'], ['0811.3360-1-61-2', '0811.3360-2-22-2'], ['0811.3360-1-59-0', '0811.3360-2-20-0'], ['0811.3360-1-47-1', '0811.3360-2-18-3'], ['0811.3360-1-47-2', '0811.3360-2-18-4'], ['0811.3360-1-52-1', '0811.3360-2-18-9'], ['0811.3360-1-52-1', '0811.3360-2-18-10'], ['0811.3360-1-53-0', '0811.3360-2-18-12'], ['0811.3360-1-53-0', '0811.3360-2-18-13'], ['0811.3360-1-23-0', '0811.3360-2-7-2'], ['0811.3360-1-23-0', '0811.3360-2-7-7'], ['0811.3360-1-25-1', '0811.3360-2-7-6'], ['0811.3360-1-26-0', '0811.3360-2-7-1'], ['0811.3360-1-26-0', '0811.3360-2-7-2'], ['0811.3360-1-26-0', '0811.3360-2-7-7'], ['0811.3360-1-26-2', '0811.3360-2-7-10'], ['0811.3360-1-26-5', '0811.3360-2-7-13'], ['0811.3360-1-77-0', '0811.3360-2-28-0'], ['0811.3360-1-77-2', '0811.3360-2-28-2'], ['0811.3360-1-77-3', '0811.3360-2-28-3'], ['0811.3360-1-80-0', '0811.3360-2-28-4'], ['0811.3360-1-82-0', '0811.3360-2-28-5'], ['0811.3360-1-5-0', '0811.3360-2-2-0'], ['0811.3360-1-74-1', '0811.3360-2-26-9']] | [['0811.3360-1-63-0', '0811.3360-2-25-0'], ['0811.3360-1-63-0', '0811.3360-2-26-0'], ['0811.3360-1-63-1', '0811.3360-2-23-2'], ['0811.3360-1-63-1', '0811.3360-2-25-1']] | ['0811.3360-1-2-0', '0811.3360-1-3-0', '0811.3360-1-6-0', '0811.3360-1-7-0', '0811.3360-1-8-0', '0811.3360-1-9-0', '0811.3360-1-10-0', '0811.3360-1-13-0', '0811.3360-1-14-0', '0811.3360-1-15-0', '0811.3360-1-17-0', '0811.3360-1-19-0', '0811.3360-1-20-0', '0811.3360-1-21-0', '0811.3360-1-22-0', '0811.3360-1-24-0', '0811.3360-1-24-1', '0811.3360-1-25-0', '0811.3360-1-26-6', '0811.3360-1-26-8', '0811.3360-1-27-0', '0811.3360-1-28-0', '0811.3360-1-28-1', '0811.3360-1-29-0', '0811.3360-1-30-0', '0811.3360-1-33-0', '0811.3360-1-34-1', '0811.3360-1-35-0', '0811.3360-1-37-0', '0811.3360-1-38-0', '0811.3360-1-39-0', '0811.3360-1-40-0', '0811.3360-1-41-0', '0811.3360-1-42-0', '0811.3360-1-43-0', '0811.3360-1-44-0', '0811.3360-1-45-0', '0811.3360-1-46-0', '0811.3360-1-49-0', '0811.3360-1-50-0', '0811.3360-1-51-0', '0811.3360-1-54-0', '0811.3360-1-55-0', '0811.3360-1-56-0', '0811.3360-1-57-0', '0811.3360-1-58-2', '0811.3360-1-62-0', '0811.3360-1-63-2', '0811.3360-1-64-0', '0811.3360-1-65-0', '0811.3360-1-66-0', '0811.3360-1-66-1', '0811.3360-1-67-0', '0811.3360-1-68-0', '0811.3360-1-69-0', '0811.3360-1-70-0', '0811.3360-1-72-0', '0811.3360-1-73-0', '0811.3360-1-73-1', '0811.3360-1-74-0', '0811.3360-1-76-0', '0811.3360-1-78-0', '0811.3360-1-79-0', '0811.3360-1-81-0', '0811.3360-1-83-0', '0811.3360-1-84-0', '0811.3360-1-84-1', '0811.3360-1-85-0', '0811.3360-1-86-0', '0811.3360-1-86-1', '0811.3360-1-89-1', '0811.3360-1-89-2', '0811.3360-2-4-4', '0811.3360-2-7-14', '0811.3360-2-8-2', '0811.3360-2-10-0', '0811.3360-2-11-1', '0811.3360-2-12-0', '0811.3360-2-14-0', '0811.3360-2-16-0', '0811.3360-2-16-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0811.3360 | null | null | null | null | null |
1512.07107 | {'1512.07107-1-0-0': 'The direct detection of a stellar system that explodes as a Type Ia supernova (SN Ia) has not yet been successful.', '1512.07107-1-0-1': 'Various indirect methods have been used to investigate SN Ia progenitor systems but none have produced conclusive results.', '1512.07107-1-0-2': 'A prediction of single-degenerate models is that H- (or He-) rich material from the envelope of the companion star should be swept up by the SN ejecta in the explosion.', '1512.07107-1-0-3': 'Seven SNe Ia have been analysed to date looking for signs of H-rich material in their late-time spectra and none were detected.', '1512.07107-1-0-4': 'We present results from new late-time spectra of 11 SNe Ia obtained at the VLT using XShooter and FORS2.', '1512.07107-1-0-5': 'We present the tentative detection of H[MATH] emission for SN 2013ct, corresponding to [MATH]0.007 M[MATH] of stripped/ablated companion star material (under the assumptions of the spectral modelling).', '1512.07107-1-0-6': 'This mass is significantly lower than expected for single-degenerate scenarios, suggesting that [MATH]0.1 M[MATH] of H-rich is present but not observed.', '1512.07107-1-0-7': 'We do not detect H[MATH] emission in the other 10 SNe Ia.', '1512.07107-1-0-8': 'This brings the total sample of normal SNe Ia with non-detections ([MATH]0.001-0.074 M[MATH]) of H-rich material to 17 events.', '1512.07107-1-0-9': 'The simplest explanation for these non-detections is that these objects did not result from the explosion of a CO white dwarf accreting matter from a H-rich companion star via Roche-lobe overflow or symbiotic channels.', '1512.07107-1-0-10': 'However, further spectral modelling is needed to confirm this.', '1512.07107-1-0-11': 'We also find no evidence of He-emission features, but models with He-rich companion stars are not available to place mass limits.', '1512.07107-1-1-0': '# Introduction', '1512.07107-1-2-0': 'The stellar systems and explosion mechanisms that produce Type Ia supernovae (SNe Ia) are still under debate.', '1512.07107-1-2-1': 'There are two main competing scenarios: the double-degenerate scenario [CITATION] where the companion star to the exploding white dwarf is another white dwarf, and the single-degenerate scenario [CITATION] where the companion star is a non-degenerate star such as a main-sequence (MS), giant or sub-giant.', '1512.07107-1-2-2': "The 'double-detonation' explosion mechanism has also seen a recent resurgence in popularity.", '1512.07107-1-2-3': 'In this scenario, the explosion results from the detonation of He on the white dwarf surface (accreted from either a degenerate or non-degenerate companion star), which triggers a subsequent C detonation in the core that unbinds the star .', '1512.07107-1-3-0': "Recent work has highlighted that it is likely that more than one of these channels contributes to produce 'normal' SNe Ia .", '1512.07107-1-3-1': 'However, clear observational diagnostics of the progenitor channel for individual objects have still not been determined.', '1512.07107-1-4-0': 'The presence of circumstellar material (CSM) around SNe Ia was originally suggested as a key observational diagnostic of the SD scenario (expected not to be present in DD scenarios).', '1512.07107-1-4-1': 'CSM signatures have been detected in a number of SNe Ia through studies of time-varying Nai D absorption lines in maximum-light spectra .', '1512.07107-1-4-2': 'The presence of CSM has also been inferred through statistical studies that have shown that [MATH]20 per cent of SNe Ia show signs of CSM and its presence is more common in more luminous SNe Ia .', '1512.07107-1-4-3': 'However, recent results have also shown at least qualitatively that the observed Nai D absorption profiles could also be produced in merger scenarios and/or where the companion star is a second white dwarf .', '1512.07107-1-5-0': "A unusual class of thermonuclear SNe, dubbed 'SNe Ia-CSM' , display strong signatures of H (H[MATH] emission with typical widths of [MATH]) in their spectra at early times, consistent with interaction with a H-rich CSM: SN2002ic , SN2005gj , SN2008J , and PTF11kx .", '1512.07107-1-5-1': 'A sample including new objects discovered by the Palomar Transient Factory is discussed in [CITATION].', '1512.07107-1-6-0': 'SNe Ia-CSM are suggested to result from a progenitor channel involving a non-degenerate companion star such as a symbiotic system .', '1512.07107-1-6-1': 'Some of these objects, such as PTF11kx, have been observed at late times ([MATH]200 d post maximum), where their H emission due to interaction with CSM is found to be still strong .', '1512.07107-1-6-2': "Since these SNe appear very unusual also at early times and perhaps represent a distinct class of SNe Ia, the astrophysical community does not consider them to be 'normal' SNe Ia, i.e. SNe Ia that would be used in cosmological studies.", '1512.07107-1-7-0': 'Another observational signature of a non-degenerate companion star would be shock interaction from the SN ejecta impacting on the companion star, which was calculated by [CITATION].', '1512.07107-1-7-1': 'This interaction would take the form of an increased luminosity in the very early-time (a few days after explosion) light curves of SNe Ia but would be dependent on the SN viewing angle.', '1512.07107-1-7-2': 'Signatures of this were searched for in a number of optical-wavelength studies but no clear detection was made .', '1512.07107-1-7-3': 'Recent early photometric observations of a SN Ia, iPTF14atg, have shown evidence of ultra-violet emission soon after explosion, consistent with the predictions of interaction with a non-degenerate companion star .', '1512.07107-1-7-4': "However, iPTF14atg was not a normal SN Ia; it displayed unusual photometric and spectroscopic properties being most similar to the peculiar class of sub-luminous 'SN 2002es-like' SNe Ia", '1512.07107-1-8-0': "Given these inconclusive results on the progenitor configuration of 'normal' SNe Ia from CSM and shock-interaction studies to date, complementary tracers of progenitor scenarios are very important.", '1512.07107-1-8-1': 'One particular area of interest is the detection of hydrogen in late-time ([MATH]200 d) SN Ia spectra - H features are expected to be present only in the SD scenario, where H-rich material is removed from a non-degenerate companion star.', '1512.07107-1-8-2': 'A key benefit of this nebular-phase probe is that the SN ejecta is expected to be optically thin at these late phases, and therefore the detection of H-rich material does not depend on orientation angle.', '1512.07107-1-8-3': 'This H emission is distinct from that seen in SNe Ia-CSM at late times, where the broader H emission is due to interaction with H-rich CSM.', '1512.07107-1-9-0': 'The ablation (heating) or stripping (momentum transfer) of H-rich material from a non-degenerate companion star through the impact of the SN Ia ejecta on the companion was discussed by [CITATION] and [CITATION].', '1512.07107-1-9-1': 'A number of early numerical studies were performed looking at the interaction between the SN ejecta and a non-degenerate companion star .', '1512.07107-1-9-2': 'The predicted properties of the H- and He-rich material have since been constrained in greater detail in a number of independent studies .', '1512.07107-1-10-0': 'The amount of unbound material present in the system after explosion, as well as its velocity distribution, is found to depend on the properties of the companion star.', '1512.07107-1-10-1': 'Assuming the case of Roche-lobe overflow (RLOF), where the binary separation is typically [MATH]3R[MATH] (where R[MATH] is the radius of the companion star), the latest three-dimensional simulations of MS, red giant (RG) and He-star companions find unbound masses of [MATH]0.1-0.2, 0.6, and 0.02-0.06 M[MATH], respectively .', '1512.07107-1-10-2': 'The smallest amount of material is removed in the WD+He star scenario since although the initial binary separation is small (a key parameter determining the amount of mass removed through stripping and ablation), a He star is more compact and has a higher binding energy than a MS or RG companion.', '1512.07107-1-10-3': 'This makes the removal of its envelope in the explosion more difficult.', '1512.07107-1-10-4': 'In the case of the WD+He star scenario, the material is expected to be He- instead of H-rich.', '1512.07107-1-10-5': 'The peak velocities of the unbound material are also predicted to depend on the companion-star properties.', '1512.07107-1-10-6': 'In the simulations of [CITATION], the peak velocities of the unbound material are [MATH]550, 660, and 955 [MATH] for WD+MS, WD+RG and WD+He star systems, respectively.', '1512.07107-1-10-7': 'The [CITATION] velocity predictions are in good agreement with these values.', '1512.07107-1-11-0': "Alternative mass-transfer mechanisms such as WD+RG systems that transfer mass to the white dwarf through a stellar wind ('symbiotic systems') can have binary separations larger than those permitted for RLOF systems.", '1512.07107-1-11-1': 'Some recurrent nova systems such as RS Ophiuchi and T Coronae Borealis can have separations of up to [MATH]5R[MATH] , greater than the allowed separation for RLOF.', '1512.07107-1-11-2': 'However, [MATH]0.5 M[MATH] of material is still expected to be removed from the companion star in this scenario .', '1512.07107-1-12-0': "Observational studies of seven nearby 'normal' SNe Ia have been performed to look for low-velocity ([MATH]) H-features in their late-time spectra .", '1512.07107-1-12-1': 'No positive detections have been made.', '1512.07107-1-12-2': 'Three of the late-time spectra studied were of low-resolution ([MATH]), comparable to the width of the expected H features: SN 2001el , SN 1998bu, SN 2000cx .', '1512.07107-1-12-3': 'The spectra used by [CITATION] for SN 2005am and SN 2005cf, and by [CITATION] for SN 2011fe, were significantly higher resolution ([MATH]).', '1512.07107-1-12-4': 'A spectrum obtained for SN 2014J, the closest SN Ia for several decades, had a resolution of [MATH] .', '1512.07107-1-13-0': 'The tightest limits on the H flux present have been obtained for the nearby SN 2011fe [CITATION], corresponding to an upper limit of 0.001 M[MATH] of swept-up material if a linear extrapolation of the H fraction of the [CITATION] spectral synthesis model holds .', '1512.07107-1-13-1': 'Possible signatures of H at near-infrared (NIR) wavelengths were investigated for SN 1998bu using the Pa[MATH] and Pa[MATH] lines, but less stringent mass limits of [MATH]0.5 M[MATH] were placed .', '1512.07107-1-14-0': 'A search for signatures of a He-rich companion star in the late-time spectra of SNe 2011fe and 2014J was carried out .', '1512.07107-1-14-1': 'Based on the results of [CITATION], it is expected that line fluxes of O and Ca produced in He-rich material would be stronger than those of He.', '1512.07107-1-14-2': 'Therefore, the presence of [Caii] and [Oi] lines with widths of [MATH] was investigated by [CITATION] - they detected no features at the expected wavelengths and placed upper He-mass limits from the [Oi] 6300 line of [MATH]0.002 and [MATH]0.005 M[MATH] for SNe 2011fe and 2014J, respectively.', '1512.07107-1-14-3': 'For SN 2011fe, they suggested that this limit is sufficient to rule out all He-rich companion star models from [CITATION] and [CITATION], including for larger-than-RLOF binary separations.', '1512.07107-1-15-0': 'If the model predictions are correct, then the non-detection mass limits of the literature sample are sufficient to rule out MS or RG companion stars in RLOF or accreting mass via a stellar wind for these seven SNe Ia.', '1512.07107-1-15-1': 'However, the sample size is small and it is unclear if these SNe Ia are representative of the full SN Ia population.', '1512.07107-1-15-2': 'Therefore, a study looking at a significantly larger sample of nebular phase spectra of SNe Ia is needed.', '1512.07107-1-16-0': 'In this paper, we present low- and medium-resolution nebular-phase spectroscopic data of 11 nearby SNe Ia obtained using the ESO Very Large Telescope (VLT) with the XShooter spectrograph and with the FOcal Reducer and low dispersion Spectrograph .', '1512.07107-1-16-1': 'In Section [REF], we detail the observations and data reduction steps employed.', '1512.07107-1-16-2': 'The analysis and search for the predicted H[MATH] lines, as well as for Pa[MATH] and Pa[MATH] lines at NIR wavelengths are described in Section [REF].', '1512.07107-1-16-3': 'In the same section, we also discuss the results of a search for signatures of He-rich material measured using the available Hei, as well as [Caii] and [Oi], lines.', '1512.07107-1-16-4': 'The discussion and conclusions are presented in Section [REF].', '1512.07107-1-16-5': 'Throughout this paper we assume a Hubble constant, [MATH]kms[MATH]Mpc[MATH].', '1512.07107-1-17-0': '# Observations and Data Reduction', '1512.07107-1-18-0': 'Nebular-phase spectra ([MATH]230 d) of 11 SNe Ia are used in this study: six obtained using VLT+XShooter and six obtained using VLT+FORS2.', '1512.07107-1-18-1': 'One object, SN 2012cg was observed with both instruments and serves as a check of the analysis and methods.', '1512.07107-1-18-2': 'We discuss the data reduction for the two instruments separately.', '1512.07107-1-19-0': '## XShooter spectra', '1512.07107-1-20-0': 'XShooter is an echelle spectrograph with three arms (UVB, VIS and NIR) covering the wavelength range of [MATH]3000-25000 .', '1512.07107-1-20-1': 'The spectral format of XShooter is fixed but the resolution can be adjusted using different slit widths.', '1512.07107-1-20-2': 'For our data, we used slit widths of 0.8[MATH], 0.9[MATH] and 0.9[MATH] in the UVB, VIS and NIR arms, corresponding to resolutions of [MATH] 6200, 8800 and 5300, respectively.', '1512.07107-1-20-3': 'Details of the spectral observations are given in Table [REF].', '1512.07107-1-21-0': 'The spectra were reduced using the REFLEX pipeline (version 2.6) with the XShooter module (version 2.5.0), producing flux-calibrated one-dimensional spectra in each of the three arms .', '1512.07107-1-21-1': 'Photometric conditions and seeing of [MATH] were required before execution of the observations could begin.', '1512.07107-1-21-2': 'Due to the relatively high spectral resolution of XShooter ([MATH]35 kms[MATH]), host galaxy features were easily identified and removed in the reduction process.', '1512.07107-1-21-3': 'For one SN in our sample, SN 2013ct, due to the potential detection of a feature consistent with H[MATH] emission (see Section [REF] for more details), we also performed an additional extraction of the spectra using a custom-written pipeline.', '1512.07107-1-21-4': 'The two-dimensional data products from the REFLEX pipeline were used as input, but with sky-subtraction, extraction and flux-calibration performed order-by-order on the unresampled REFLEX images.', '1512.07107-1-21-5': 'This was to check carefully the host galaxy feature removal and relative flux calibration, and confirm that this was not causing the tentative emission feature that was seen.', '1512.07107-1-22-0': 'We scale the flux of our spectra to photometry taken close in time to the spectroscopic observations to obtain absolute flux-calibrated spectra.', '1512.07107-1-22-1': "We obtained acquisition images using r' and i' band filters, similar to the filters used by the Sloan Digital Sky Survey , and in the Johnson I band filter.", '1512.07107-1-22-2': 'The SN magnitudes were determined using point spread function (PSF) photometry.', '1512.07107-1-22-3': 'The zero points of the images were obtained using aperture photometry of comparison stars in the images, which were then calibrated by comparison to either catalogue magnitudes from the SDSS Data Release 10 for SNe 2012cg and 2012ht, or using the XShooter acquisition and guide camera zero points for SNe 2012fr, 2013ct, 2013cs, 2013aa.', '1512.07107-1-22-4': "XShooter zero points were not available for r'.", '1512.07107-1-22-5': "Therefore, for the three SNe (2013ct, 2013cs, and 2013aa) that were observed in the r' band, a transformation, using the relative flux throughputs of the SN spectra in the relevant filters, was applied to convert r' to Johnson R for which an XShooter zero point was available.", '1512.07107-1-23-0': 'The flux-calibrated spectra were corrected to rest-frame wavelengths using the heliocentric velocities given in Table [REF].', '1512.07107-1-23-1': 'These values were obtained from the NASA/IPAC Extragalactic Database (NED) galaxy spectra, apart from SN 2012cg, which following [CITATION] used the stellar velocity of [CITATION] at the SN position.', '1512.07107-1-24-0': 'Following [CITATION], the optical regions of the XShooter spectra were rebinned to 3 .', '1512.07107-1-24-1': 'The NIR portions of the spectra were rebinned to 6 to increase the signal-to-noise ratio.', '1512.07107-1-24-2': 'A host-galaxy origin for any potential [MATH] H[MATH] emission can be confidently ruled out since it would only contribute to the central spectral bin at the binned XSH dispersion.', '1512.07107-1-25-0': '## FORS2 spectra', '1512.07107-1-26-0': 'For the FORS2 observations, a combination of grisms 300V and 300I (+OG590) and a 1[MATH] slit were used, yielding a resolution [MATH]-[MATH] over a wavelength range of 3500-10500.', '1512.07107-1-26-1': 'After bias subtraction and flat-fielding an optimal, variance-weighted extraction of the spectra was performed with the iraf task apall.', '1512.07107-1-26-2': 'The dispersion solution was established using arc-lamp exposures and cross-checked against night-sky emission lines.', '1512.07107-1-26-3': 'Spectrophotometric standard stars, usually observed during the same nights as the SNe, were used to perform a relative flux calibration of the spectra and to correct for telluric absorptions.', '1512.07107-1-27-0': 'To calibrate to an absolute flux scale and correct for slit losses, the spectra were scaled to match FORS2 [MATH] photometry of the SNe obtained close in time to the spectral observations (see Table 1).', '1512.07107-1-27-1': 'The SN magnitudes in the [MATH] frames were measured using PSF photometry, and calibrated using the FORS2 zero points and extinction coefficients provided by the ESO data-quality monitoring.', '1512.07107-1-28-0': 'As a result of the significantly lower spectral resolution of the FORS2 spectra ([MATH]450 kms[MATH]) compared to the XShooter spectra, the host-galaxy subtraction was more complex.', '1512.07107-1-28-1': 'For nine of the SNe Ia observed with FORS2, emission from a Hii-region was visible in the two-dimensional spectra either at the SN location or very nearby.', '1512.07107-1-28-2': 'Therefore, a clean background subtraction could not be guaranteed for these objects and we remove these SNe from further discussion since the uncertain background subtraction could mask H[MATH] emission from the SN itself.', '1512.07107-1-28-3': 'However, we note that these SNe Ia could have displayed narrow H features that went undetected.', '1512.07107-1-28-4': 'Therefore, the number of SNe Ia with narrow H features could be higher than measured in our sample.', '1512.07107-1-28-5': 'The six remaining SNe Ia observed with FORS2, and that have a clean background subtraction, are detailed in Table [REF].', '1512.07107-1-28-6': 'For these spectra, the dispersion was left at its native value of 3.25 .', '1512.07107-1-29-0': '# Analysis', '1512.07107-1-30-0': 'We have searched for the presence of material that could have been removed from a non-degenerate companion star (via ablation and/or stripping) in a new late-time SN Ia spectral sample, as well as used model predictions to constrain quantitatively the presence or absence of solar abundance material.', '1512.07107-1-31-0': '## Searching for signatures of H', '1512.07107-1-32-0': 'We searched for the presence of narrow ([MATH]) lines of H (H[MATH], Pa[MATH], Pa[MATH]) in the spectra.', '1512.07107-1-32-1': 'To do this an underlying continuum must first be defined.', '1512.07107-1-32-2': 'The underlying continuum of the spectra in these regions was fit using a second-order Savitzky-Golay smoothing polynomial , with the search wavelength region of [MATH]22 of the H features excluded from the smoothing.', '1512.07107-1-32-3': 'This was to avoid biasing the continuum towards higher values if a H-emission feature was present.', '1512.07107-1-32-4': 'The smoothing scales used were significantly larger than the predicted widths of the narrow features, with widths of 80-140 in the optical and [MATH]200 in the NIR.', '1512.07107-1-32-5': 'Further analysis and the associated uncertainties on estimating the continuum for the H[MATH] region is discussed in Section [REF].', '1512.07107-1-33-0': 'The rebinned spectra and continuum fits at the position of H[MATH] for our SN Ia sample are shown in Fig. [REF].', '1512.07107-1-33-1': 'The Pa[MATH]) regions of the XShooter spectra are shown in Fig. [REF].', '1512.07107-1-33-2': 'The position of any potential Pa[MATH]) feature falls in the middle of a telluric band and detection of SN flux is very difficult.', '1512.07107-1-33-3': 'Therefore, the only NIR H line we discuss is the Pa[MATH] feature.', '1512.07107-1-33-4': 'We did not detect any strong unambiguous H[MATH] or Pa[MATH] features in our sample when inspecting the appropriate wavelength regions.', '1512.07107-1-33-5': 'However, this does not exclude less prominent emission features being present.', '1512.07107-1-33-6': 'A quantitative estimate of the presence of H[MATH] and associated non-detection limits is described in Section [REF].', '1512.07107-1-34-0': 'We have estimated the potential effect of telluric features on detection of narrow features in the H[MATH] and Pa[MATH] spectral regions.', '1512.07107-1-34-1': 'The closest significant telluric feature (transmission [MATH]0.9) to the wavelength of H[MATH] is a feature at [MATH]6519 identified using the ESO SKYCALC Sky Model Calculator.', '1512.07107-1-34-2': 'For the SN in our sample with the lowest heliocentric redshift, SN 2012cg, this feature would lie at a rest frame wavelength of [MATH]6508 .', '1512.07107-1-34-3': 'This is well outside the region of interest of [MATH]22 of the H[MATH] feature at 6563 and the feature is weak enough not to be visible in our spectra.', '1512.07107-1-34-4': 'Therefore, we conclude that this has negligible effect on our results.', '1512.07107-1-34-5': 'We have also investigated potential telluric features in the Pa[MATH] wavelength region and find no significant telluric features within the observed wavelength region.', '1512.07107-1-35-0': '### Model predictions of stripped/ablated material', '1512.07107-1-36-0': 'Based on the calculations of [CITATION], [CITATION] used a one-dimensional nebular-phase spectral code to give a quantitative estimate of the H[MATH] emission expected from different amounts of solar-abundance material stripped or ablated from a companion star.', '1512.07107-1-36-1': 'This solar abundance material was located in the inner 1000 [MATH] of the ejecta using the W7 explosion model .', '1512.07107-1-36-2': 'The peak luminosity of the H[MATH] line for 0.05 M[MATH] of solar-abundance material was found to be [MATH]3.36 [MATH] erg s[MATH] .', '1512.07107-1-37-0': 'These model H[MATH] luminosities were calculated at +380 d and the H[MATH] emission is expected to be time-dependent .', '1512.07107-1-37-1': 'However, [CITATION] noted that since the optical depth to gamma rays should be higher at earlier times, the H[MATH] emission should be stronger at epochs earlier than +380 d. Therefore, the estimated H[MATH] flux at +380 d can be considered a lower limit for earlier spectra as long as the condition that the ejecta are transparent is fulfilled ([MATH]200 d) and we can observe this low-velocity material.', '1512.07107-1-37-2': 'Sources of additional uncertainty in the modelling are discussed in Section [REF].', '1512.07107-1-38-0': 'For three SNe Ia in our sample (SNe 2009ig, 2011ek, and 2012ht), the spectra were obtained at epochs greater than +380 d (between +405 and +433 d).', '1512.07107-1-38-1': 'Given the small difference in the epoch studied in the model and the observed data, the model fluxes are assumed to be still applicable at these slightly later phases.', '1512.07107-1-39-0': 'We have calculated the expected H[MATH] emission for this luminosity for each SN in our sample, using the distance and Galactic extinction values (corrected to the wavelength of H[MATH]) in Table [REF].', '1512.07107-1-39-1': 'Host galaxy extinction corrections were not applied for most SNe Ia in our sample since we found no evidence for significant host galaxy extinction.', '1512.07107-1-39-2': 'This was based on an analysis of their maximum-light spectra, absolute magnitudes at peak, and locations within their host galaxy.', '1512.07107-1-39-3': 'However, for SN 2012cu, the optical spectra at maximum showed signs of significant additional extinction, although the exact amount is unknown .', '1512.07107-1-39-4': 'Therefore, we computed two H[MATH] luminosities for SN 2012cu, one with no additional extinction and one with a conservative estimate of one magnitude of E(B-V) extinction.', '1512.07107-1-40-0': 'The expected integrated H[MATH] fluxes for 0.05 M[MATH] of H-rich material, assuming a Gaussian profile with a full width at half-maximum (FWHM) of 1000 [MATH], and a central wavelength of 6563 are given in Table [REF].', '1512.07107-1-40-1': 'The chosen value of the FWHM is consistent with the previous studies, as well as the model constraint of [CITATION] that the H-rich material is located at velocities below [MATH].', '1512.07107-1-40-2': 'Both values (with/without additional extinction) for SN 2012cu are given but the more conservative limit assuming one magnitude of extinction is quoted for the final mass limit range.', '1512.07107-1-40-3': 'The expected H[MATH] line profiles for 0.05 M[MATH] of solar abundance material are plotted in Fig. [REF] for each SN in our sample.', '1512.07107-1-41-0': '### Estimation of contributions from narrow H[MATH]', '1512.07107-1-42-0': 'To estimate the amount of H[MATH] that could be present, as well as set limits on any non-detections, we made model spectra combining the continuum fit with a Gaussian to represent a potential H[MATH] emission feature.', '1512.07107-1-42-1': 'The Gaussian was set to have a fixed FWHM of 1000 [MATH] and fixed central wavelength of 6563 , but a varying peak flux.', '1512.07107-1-42-2': 'A likelihood function was calculated as a function of varying peak flux in steps of 0.001 M[MATH] within the range corresponding to [MATH]0.05 M[MATH] (negative values are included to sample a complete peak-flux distribution).', '1512.07107-1-42-3': '[CITATION] and [CITATION] calculated the expected flux at the position of H[MATH] for differing input masses of stripped/ablated H-rich companion star material (0.01, 0.05, 0.1, 0.5 M[MATH]), and found a linear relation between H[MATH] flux and mass.', '1512.07107-1-42-4': 'Therefore, we use a linear scaling between H[MATH] flux and mass in our analysis.', '1512.07107-1-43-0': 'The largest uncertainty in the likelihood calculation is the estimation of the continuum fit.', '1512.07107-1-43-1': 'Therefore, to estimate the associated uncertainty we have calculated a number of continuum fits using the Savitzky-Golay smoothing polynomial with widths varying from 80 to 140 .', '1512.07107-1-43-2': 'This range was chosen so has to be significantly larger than the width of the feature we are searching for ([MATH]22 ) but not so big that the underlying continuum features are not well fit.', '1512.07107-1-43-3': 'A value of 100 was used in the analysis of [CITATION].', '1512.07107-1-43-4': 'The mean of these continuum fits is then used as input to the likelihood calculation.', '1512.07107-1-44-0': 'The sigma entering the likelihood was calculated from the r.m.s. scatter per wavelength bin from the continuum fits (an estimation of the uncertainty in the continuum definition) combined with the scatter around the normalised continuum (an estimation of the spectral noise).', '1512.07107-1-45-0': '### Tentative detection of H[MATH] in SN 2013ct', '1512.07107-1-46-0': 'For one SN in our sample, SN 2013ct, we have identified a weak but broad feature at the position of H[MATH], consistent with the stripping/ablation model predictions of H[MATH] emission with velocities of [MATH]600-1000 [MATH].', '1512.07107-1-46-1': 'By varying the width of the feature, we found the strongest significance feature for a FWHM of 850 [MATH].', '1512.07107-1-46-2': 'Using this FWHM and the method detailed in Section [REF], we found that the H[MATH] flux of SN 2013ct corresponded to [MATH]0.005 M[MATH] of stripped or ablated H-rich material.', '1512.07107-1-47-0': 'Due to the strong underlying continuum for SN 2013ct and the potential detection of a narrow H[MATH] feature, we investigated further fits to the underlying continuum.', '1512.07107-1-47-1': 'We expanded the continuum fitting models to include a Gaussian fit to the underlying continuum.', '1512.07107-1-47-2': 'The parameters of the Gaussian were set to be variable.', '1512.07107-1-47-3': 'The region of the spectrum included in the Gaussian was varied between [MATH]70 and [MATH]130 of the centre of the underlying profile (6540 ).', '1512.07107-1-47-4': 'The region [MATH] from the rest wavelength of H[MATH] was again excluded from the continuum fitting.', '1512.07107-1-48-0': 'The mean continuum of the combined Savitzky-Golay and Gaussian fitting was used as the continuum in the likelihood calculation.', '1512.07107-1-48-1': 'The best fit for the H[MATH] feature for SN 2013ct using this expanded continuum fitting is shown in Fig. [REF].', '1512.07107-1-48-2': 'This H[MATH] flux corresponds to [MATH]0.007[MATH]0.001 M[MATH] of stripped or ablated H-rich material at a 3.7[MATH] significance.', '1512.07107-1-49-0': 'To help determine the robustness of our H[MATH] detection for SN 2013ct we performed two additional tests.', '1512.07107-1-49-1': 'Firstly, we searched for similar features at different rest wavelengths by varying the wavelength of the expected feature within [MATH]120 of the detected H[MATH] position.', '1512.07107-1-49-2': 'We used the same analysis method as for the search at the position of H[MATH], including excluding the search region from the continuum fitting.', '1512.07107-1-49-3': 'Secondly, we tested if a feature of a similar significance was detected on top of another broad nebular emission feature (the 5800 feature) in the SN 2013ct spectrum.', '1512.07107-1-49-4': 'This was to determine if residuals in the fitting of the continuum for the broad emission features would result in a similar feature to that seen at the position of H[MATH].', '1512.07107-1-49-5': 'In both cases, no significant detection ([MATH]) is associated within any feature outside of the H[MATH] search region.', '1512.07107-1-49-6': 'However, we caution that the significance of the H[MATH] detection is under the assumptions of the fitting to an unknown underlying continuum, as well as the predictions of the models of the position and velocity distribution of H-rich material.', '1512.07107-1-50-0': 'If our measurements of the SNe are assumed to be independent, then as the sample size increases, there is an increased probability of finding a 3[MATH] detection by chance.', '1512.07107-1-50-1': 'We used the correction of [CITATION] to estimate the increased significance needed to claim a 3[MATH] detection for one object out of the 8 SNe Ia in our combined new data and literature sample for which the mass detection limit at the position of H[MATH] is [MATH]0.007 M[MATH] (equivalent to the SN 2013ct detection).', '1512.07107-1-50-2': 'We found that a 3.7[MATH] detection in any one spectrum is equivalent to a 3.1[MATH] detection if we make 8 comparisons, which we use as our significance value for this detection.', '1512.07107-1-51-0': '### H[MATH] detection limits', '1512.07107-1-52-0': 'The rest of the SNe Ia in our sample showed no obvious H[MATH] emission.', '1512.07107-1-52-1': 'Using the same method as for SN 2013ct, we determined that the Gaussian fits at the position of H[MATH] for the rest of the sample are consistent with zero within the 3[MATH] uncertainties.', '1512.07107-1-52-2': 'We placed 3[MATH] upper limits on the strength of features that could have remained undetected at the position of H[MATH].', '1512.07107-1-53-0': 'The line profiles representing these flux limits are shown in Fig. [REF], and values for the integrated H[MATH] flux and corresponding mass limits are given in Table [REF].', '1512.07107-1-53-1': 'The 3[MATH] limiting mass range for the SN sample is 0.001-0.074 M[MATH].', '1512.07107-1-53-2': 'The bright, nearby SN 2012cg was observed with both XShooter and FORS2 and we obtain a consistent mass limit ([MATH]0.010 M[MATH]) for both spectra.', '1512.07107-1-54-0': 'Since these limits are strictly statistical, [CITATION] estimated how systematic uncertainties from our lack of knowledge of the underlying spectral continuum could affect the measured F(3[MATH]) values of H[MATH] emission for SNe 2011fe and 2014J.', '1512.07107-1-54-1': 'They suggested, using an inspection by eye, that the limits on the masses of stripped/ablated material could be 2-3 greater than their measured values.', '1512.07107-1-55-0': 'As discussed in Section [REF], to minimise the uncertainty in the selection of the continuum in our sample, we have excluded the region of [MATH]22 with respect to H[MATH] when smoothing the spectra to avoid biasing the continuum fit of the search wavelength region.', '1512.07107-1-55-1': 'We have also tested our continuum fits by checking for similar strength features outside the H[MATH] search region, as well as on top of the broad 5800 nebular feature, and no detection was made.', '1512.07107-1-55-2': 'Therefore, we have chosen to use our original F(3[MATH]) values calculated from the probability distributions to measure M(3[MATH]) and these are the values we quote in Table [REF].', '1512.07107-1-56-0': '## Searching for signatures of He', '1512.07107-1-57-0': 'The companion star to the exploding white dwarf could also potentially be a He-rich star instead of a H-rich star, which could result in the presence of narrow ([MATH]) He lines in late-time SN Ia spectra.', '1512.07107-1-57-1': 'Less material is expected to become unbound than in the H-rich case; just 0.02-0.06 M[MATH] is predicted by the models .', '1512.07107-1-58-0': 'To investigate the presence of He emission at late times, we focus first on the Hei 5876, 10830, and 20587 features in the XShooter VIS and NIR spectra.', '1512.07107-1-58-1': 'The wavelength regions for the Hei lines are shown in Fig. [REF].', '1512.07107-1-58-2': 'The Hei 5876 and 10830 spectral regions are found to be relatively free of telluric absorption features.', '1512.07107-1-58-3': 'However, the Hei 20590 region is moderately affected.', '1512.07107-1-58-4': 'For completeness, we show this region in Fig. [REF] but do not base our results on this region alone.', '1512.07107-1-58-5': 'We did not have model predictions of the peak luminosity for these He features but from Fig. [REF] it is clear that there are no strong Hei signatures detected in our SN Ia sample within the velocity range predicted by [CITATION].', '1512.07107-1-59-0': '[CITATION] detailed how [Oi] and [Caii] emission features are promising probes of stripped/ablated He-rich material.', '1512.07107-1-59-1': 'We have also searched for [Oi] 6300, 6364 and [Caii] 7291, 7324 emission features with velocities of [MATH] of the rest wavelength in our XShooter and FORS2 spectra but did not find any features consistent with the qualitative model predictions.', '1512.07107-1-59-2': 'However, further analysis, when robust spectral modelling predictions of peak luminosities are available, will be necessary to put flux (and mass) limits on these He, Ca and O non-detections.', '1512.07107-1-60-0': '# Discussion', '1512.07107-1-61-0': 'Using the expected H[MATH] luminosities from a time-dependent nebular synthesis calculation for solar abundance material confined within 1000 [MATH] , we have quantified the presence or absence of H-rich companion star material in 11 SNe Ia for which late-time spectra were obtained.', '1512.07107-1-61-1': 'For 10 SNe Ia in the sample, we did not find evidence of H-rich companion star material in their late-time spectra.', '1512.07107-1-61-2': 'Under the assumptions of the modelling, we can place limits of [MATH]0.001-0.074 M[MATH] on the mass of H-rich material that could remain undetected.', '1512.07107-1-61-3': 'For one SN in our sample, SN 2013ct, we have made a tentative detection (3.1[MATH]) of emission at the position of H[MATH], corresponding to 0.007[MATH]0.001 M[MATH] of H-rich material.', '1512.07107-1-61-4': "Combining these new data with previous samples , this means one potential detection of H[MATH] emission out of 18 'normal' SNe Ia that have been studied at late times to look for signatures of H-rich material.", '1512.07107-1-62-0': 'Emission at the wavelength of Pa[MATH] was also investigated for five SNe Ia in the sample with suitable NIR spectral coverage and no clear emission was identified.', '1512.07107-1-62-1': 'However, as noted in [CITATION], the mass limits obtained from Pa[MATH] are less constraining than for H[MATH].', '1512.07107-1-63-0': 'We do not detect strong signatures of Hei 5876, 10830, 20587 features within a velocity range of [MATH].', '1512.07107-1-63-1': 'We also do not detect [Oi] or [Caii] emission features, which were suggested by [CITATION] to be promising tracers of He-rich material.', '1512.07107-1-63-2': '[CITATION] estimated very rough limits on the He mass present for SNe 2011fe and 2014J by adjusting the H mass limits for the He to H number density ratio assuming solar metallicity.', '1512.07107-1-63-3': 'For the tightest mass limits for SN 2011fe in [CITATION], all He-rich companion stars from the models of [CITATION] and [CITATION] can be ruled out based on the non-detection of [Oi] emission.', '1512.07107-1-63-4': 'However, since the He-mass predictions obtained are simple extrapolations from models of H-rich solar abundance material placed in the inner [MATH] of the SN ejecta and the He-rich material is expected to be at higher velocity than this, they should be taken as very rough upper limits.', '1512.07107-1-64-0': '## First detection of H[MATH] in the late-time spectrum of a normal SN Ia?', '1512.07107-1-65-0': 'The detection of a potential H[MATH] feature consistent in velocity ([MATH]) with the models of [CITATION] and [CITATION] is suggested for SN 2013ct.', '1512.07107-1-65-1': 'Using the models of [CITATION], this is estimated to be equivalent to 0.007[MATH]0.001 M[MATH] of solar abundance material being present at velocities below 1000 [MATH].', '1512.07107-1-65-2': 'However, the models of [CITATION] and [CITATION] have shown that if the companion star to the exploding white dwarf was a MS star in any realistic binary scenario, at least 0.1-0.2 M[MATH] of material should be removed from the companion star after explosion.', '1512.07107-1-65-3': 'The companion star can not be artificially moved to greater separations to reduce the amount of stripped/ablated material because then the necessary condition of RLOF would not be fulfilled.', '1512.07107-1-65-4': 'For a RG companion, [MATH]0.6 M[MATH] of material is predicted to be stripped/ablated from the companion in RLOF.', '1512.07107-1-65-5': 'For systems transferring mass via a stellar wind, [MATH]0.5 M[MATH] of material would be removed from the companion star', '1512.07107-1-66-0': 'Therefore, in order to explain an emission feature corresponding to [MATH]0.007 M[MATH] of H-rich material, an additional [MATH]0.1-0.6 M[MATH] of material would have to be present but not visible as a narrow H[MATH] emission within 1000 [MATH].', '1512.07107-1-66-1': 'A tail of material extending to higher velocities could hide some material that would not contribute significantly to the observed narrow H[MATH] feature.', '1512.07107-1-66-2': 'However, the models predict that the amount of stripped or ablated material present in a high-velocity tail is expected to be low .', '1512.07107-1-67-0': 'Another possibility is that the majority of the H-rich companion star material is not sufficiently powered by radioactive heating to produce H[MATH] emission.', '1512.07107-1-67-1': 'This may be because there is little SN ejecta remaining below 1000 [MATH] at these times .', '1512.07107-1-67-2': 'Therefore, while the presence of a narrow H[MATH] feature suggests a H-rich non-degenerate companion star to the white dwarf for SN 2013ct, further constraints on the companion star properties can not currently be placed.', '1512.07107-1-67-3': 'The uncertainties of the model and observations are detailed further in Section [REF].', '1512.07107-1-68-0': '## Are H-rich SD systems sub-dominant for producing normal SNe Ia?', '1512.07107-1-69-0': 'Unfortunately, little early-time data are available for SN 2013ct to determine its detailed properties.', '1512.07107-1-69-1': 'The SN was discovered on 2013 May 10 but not spectroscopically classified until 2013 May 22, where the best near-infrared spectroscopic match was a normal SN Ia approximately 20 d after maximum light .', '1512.07107-1-69-2': 'Therefore, while this spectrum appears similar to other normal SNe Ia, subtle differences that may have been apparent with a larger dataset are not quantifiable.', '1512.07107-1-70-0': "Although the first tentative detection of H[MATH] emission in a late-time spectrum of a 'normal' SN Ia is very interesting, the absence of H[MATH] emission in the late-time spectra of 17 other SNe Ia is also worthy of discussion.", '1512.07107-1-70-1': 'Using the models of [CITATION], stripped/ablated mass limits in the range 0.001-0.074 M[MATH] were determined for these 17 SNe Ia.', '1512.07107-1-70-2': 'If we assume that the stripping/ablation and spectral model calculations are correct, then these limits are sufficient to rule out all MS and RG companions transferring mass via RLOF or a stellar wind at all plausible separations.', '1512.07107-1-71-0': 'However, the H[MATH] detection equivalent to [MATH]0.007 M[MATH] of H-rich material for SN 2013ct potentially weakens these constraints, since it suggests that [MATH]0.1-0.6 M[MATH] of H-rich companion star material was hidden for this SN.', '1512.07107-1-71-1': 'Therefore, H-rich material may have also been present in the other 17 SNe Ia but also hidden.', '1512.07107-1-72-0': 'The spin-up/spin-down scenario of [CITATION] and [CITATION] could reduce the amount of H-rich material stripped from a non-degenerate companion star at the time of explosion.', '1512.07107-1-72-1': 'This involves a white dwarf becoming spun up by the mass accreted from its companion star, resulting in a stable white dwarf above the Chandrasekhar mass.', '1512.07107-1-72-2': 'This gives time for the companion star to evolve and contract before the SN explosion.', '1512.07107-1-72-3': 'Therefore, at the time of explosion, the companion star is much smaller and more tightly bound, significantly reducing the amount of material removed by the impact of the SN ejecta.', '1512.07107-1-72-4': 'However, some fine-tuning is necessary to produce this scenario and as the sample of SNe Ia without H emission grows, it is becoming increasingly unlikely that this can explain all the observed non-detections.', '1512.07107-1-73-0': 'It has also been suggested that the broad underlying emission feature at the position of H[MATH] in our sample (usually attributed to [Feii] 6559 emission) may have a contribution from a broad H[MATH] emission component .', '1512.07107-1-73-1': 'However, in this case, the H-rich material would have to be present at much higher velocities than those currently predicted by modelling efforts .', '1512.07107-1-73-2': 'Further studies are necessary to determine if this is physically plausible.', '1512.07107-1-74-0': 'However, it is likely that the sub-class of SNe Ia, SNe Ia-CSM, do result from a SD progenitor channel.', '1512.07107-1-74-1': 'The most likely scenario is that of a symbiotic system involving a RG or asymptotic giant branch star .', '1512.07107-1-74-2': 'This scenario would produce the necessary CSM to explain the interaction features seen in their spectra such as broad H emission extending till late times .', '1512.07107-1-74-3': "If SNe Ia-CSM and at least some so-called 'normal' SNe Ia originate from a SD scenario, then it may not be surprising to identify weaker H features, such as that detected for SN 2013ct, in some SNe Ia (coming from stripped/ablated companion material instead of pre-explosion mass loss).", '1512.07107-1-75-0': '## He-rich companion stars', '1512.07107-1-76-0': "Another solution for producing 'normal' SNe Ia through the SD channel and avoiding H contamination is to invoke a He- instead of H-rich companion star.", '1512.07107-1-76-1': 'The latest simulations of the interaction between a He-rich companion star and the SN ejecta predict stripped/ablated masses of 0.02-0.06 M[MATH] .', '1512.07107-1-76-2': '[CITATION] put limits on the mass of He-rich material present in SNe 2011fe and 2014J of [MATH]0.002 and [MATH]0.005 M[MATH], respectively.', '1512.07107-1-76-3': 'Under the assumption of RLOF, these limits rule out the He-rich companion star models of [CITATION] and [CITATION] for both SNe Ia.', '1512.07107-1-77-0': 'We have presented a qualitative discussion of the non-detection of He emission in our late-time spectral sample of 11 SNe Ia - we do not find obvious features that could be attributed to Hei, [Caii] or [Oi] emission.', '1512.07107-1-77-1': 'However, as cautioned in [CITATION], these limits are based on a rough correction from H- to He-rich material, and have not been modelled explicitly.', '1512.07107-1-77-2': 'Therefore, we await more detailed spectral modelling to place limits on the presence of He-rich material from a companion star swept-up in the SN ejecta.', '1512.07107-1-78-0': "Companion stars with He-rich outer layers are also present in the 'double-detonation' scenario, where a thin layer of He on the surface of the primary white dwarf is responsible for the first detonation .", '1512.07107-1-78-1': 'This He material is accreted from a He star, He WD, a CO white dwarf with a thin layer of He on its surface, or it may be already present on the primary white dwarf surface.', '1512.07107-1-78-2': 'At the time of the subsequent detonation of the core, the He on the surface is expected to have velocities of [MATH].', '1512.07107-1-78-3': "Therefore, this material is not expected to be visible at the low velocities studied here, and the non-detection of He features in the late-time spectra does not place constraints on the presence of high-velocity He-rich material in the 'double-detonation' scenario.", '1512.07107-1-78-4': 'Detection of this high-velocity He is also unlikely in early-time observations because of insufficient heating of this material so far out in the ejecta to cause He emission lines to be observed.', '1512.07107-1-79-0': '## Sample selection', '1512.07107-1-80-0': 'The SNe Ia in our late-time spectral sample were selected for observation based on their proximity ([MATH]) and visibility at Paranal at [MATH]200 d after maximum light.', '1512.07107-1-80-1': 'If a SN Ia was significantly sub-luminous at maximum light then it would not have been scheduled for observations because it would be deemed too faint at [MATH]200 d after maximum light.', '1512.07107-1-80-2': 'This would bias our sample towards more luminous SNe Ia.', '1512.07107-1-80-3': 'However, the SNe Ia in our sample span a wide range of host galaxy types from early- to late-type galaxies, suggesting a spread in SN luminosity .', '1512.07107-1-80-4': 'SN 2011iv which occurred in an elliptical galaxy was a low-luminosity SN Ia .', '1512.07107-1-80-5': "Therefore, we conclude that our sample is not significantly biased towards more luminous events, and covers a range of 'normal' SN Ia luminosities.", '1512.07107-1-80-6': 'There is also no reason to expect that less-luminous SNe Ia are more likely to have H or He features present in their spectra.', '1512.07107-1-80-7': 'In fact, recent work suggested the opposite - that it is likely that more luminous SNe Ia occur preferentially through SD channels .', '1512.07107-1-81-0': '## Observational and model uncertainties', '1512.07107-1-82-0': 'In this section, we discuss the uncertainties and limitations of the observations and modelling that could cause the results and interpretation of our analysis to be less constraining.', '1512.07107-1-82-1': 'A number of independent simulations have been made of the amount of stripped/ablated material, and its velocity distribution, that is expected to be removed from a non-degenerate companion star after explosion .', '1512.07107-1-82-2': 'There is now reasonable agreement among the different groups for these masses and velocity distributions for different companion star setups.', '1512.07107-1-82-3': 'However, only one analysis has been done to determine the spectral line strengths associated with different mass and velocity values .', '1512.07107-1-82-4': 'This study was based on the earliest impact simulations of [CITATION].', '1512.07107-1-82-5': 'However, the amount and velocity distributions of the stripped/ablated material have not been updated drastically in more recent modelling, and this should not significantly affect the spectral modelling results.', '1512.07107-1-83-0': 'The largest uncertainty in the estimation of the presence of H- or He-emission features in the observed spectra is the calculation of the underlying continuum.', '1512.07107-1-83-1': 'We have used a second-degree Savitzky-Golay smoothing polynomial to fit the underlying continuum, as was used in previous late-time narrow H[MATH] searches.', '1512.07107-1-83-2': 'This gives a good fit to the underlying continua in our sample, even in the case of broad underlying emission features.', '1512.07107-1-83-3': 'For SN 2013ct, we expanded our analysis of the continuum fitting to include also broad Gaussian fits to the underlying spectral feature, and again found a significant detection.', '1512.07107-1-83-4': 'However, given the tentative nature of the detection of H[MATH] emission in SN 2013ct, we could not completely exclude the possibility that the continuum fits results in a residual consistent with the detected H[MATH] emission.', '1512.07107-1-83-5': 'To determine how likely this is, we performed tests, detailed in Section [REF], looking for similar strength features at different wavelengths around H[MATH] and also on top of a different broad emission feature at [MATH]5800 .', '1512.07107-1-83-6': 'In neither case, was a similar feature found.', '1512.07107-1-84-0': 'With regard to the spectral synthesis models, the addition of the H-rich material potentially swept-up from a non-degenerate companion star in the models of [CITATION] and [CITATION] is somewhat ad-hoc - it is made by adding varying amounts of solar metallicity material with velocities [MATH] in a W7 density model, artificially increasing the density in the innermost region.', '1512.07107-1-84-1': 'It is uncertain if this is consistent with the density structure obtained from the three-dimensional modelling of the impact of the SN ejecta on the companion star.', '1512.07107-1-85-0': 'In particular, a major source of uncertainty in the spectral synthesis modelling is whether the low-velocity H-rich material is sufficiently powered by radioactive heating to produce H[MATH] emission.', '1512.07107-1-85-1': 'This depends on the location of the H-rich material relative to the radioactive material; if they are not co-located then H-rich companion star material may be present in the ejecta but not observable.', '1512.07107-1-85-2': '[CITATION] performed the spectral line strength calculations at an epoch of +380 d.', '1512.07107-1-85-3': 'In the range of 150-300 d, the SN envelope is expected to become transparent to [MATH]-rays and enter a positron-dominated phase .', '1512.07107-1-85-4': 'However, this is not expected to be the case for the central high-density regions, where the H is located.', '1512.07107-1-85-5': 'In these high-density regions, the optical depth to [MATH]-rays is likely to be high enough to power the H lines , but more detailed modelling need to be carried out to confirm this.', '1512.07107-1-85-6': 'Indeed, the tentative detection of low-velocity H emission in SN 2013ct, corresponding to [MATH]0.007 M[MATH] of H-rich material, suggests that [MATH]0.1-0.6 M[MATH] of stripped/ablated material must be present but not observable.', '1512.07107-1-86-0': 'The one-dimensional models of [CITATION] and [CITATION] also assume spherical symmetry.', '1512.07107-1-86-1': 'However, the impact simulations show that the stripping and ablation of material from the companion star is not symmetric; the material is predominantly confined to the downstream region behind the companion star .', '1512.07107-1-86-2': 'Since the ejecta are assumed to be optically thin at the late phases studied here ([MATH]200 d), the detection of the swept-up material is not viewing angle dependent.', '1512.07107-1-86-3': 'However, the predicted asymmetry of the unbound material could affect the shape and wavelength of the observed line profiles.', '1512.07107-1-86-4': 'As discussed in [CITATION], the model used by [CITATION] and [CITATION] also included only a limited number of elements, ionisation states, and atomic levels.', '1512.07107-1-86-5': 'No macroscopic mixing of the companion material was included in the spectral modelling either, which could affect the predicted shape and flux of the emission lines but is not expected to be a dominant source of uncertainty.', '1512.07107-1-87-0': 'Therefore, given the discussed uncertainties and limitations, future modelling, using the three-dimensional simulated ejecta structure of [CITATION] and [CITATION] as input to a multi-dimensional radiative transfer calculation for computing expected spectral fluxes, is of vital importance to confirm the model predictions, and hence observational mass limits.', '1512.07107-1-88-0': '# Conclusions', '1512.07107-1-89-0': 'We have presented a search for the presence of H- and He-rich material stripped or ablated from a non-degenerate companion star in new late-time spectra of 11 SNe Ia, obtained at the VLT+XShooter and the VLT+FORS2.', '1512.07107-1-89-1': 'The observed fluxes (or limits) at the position of H[MATH] have been converted to masses using the spectral synthesis modelling described in [CITATION] and [CITATION].', '1512.07107-1-89-2': 'Our main results are:', '1512.07107-1-90-0': 'We find evidence at the 3.1[MATH] level of H[MATH] emission with a best fit FWHM of [MATH] for one SN Ia in our sample, SN 2013ct.', '1512.07107-1-90-1': 'This corresponds to 0.007[MATH]0.001 M[MATH] of H-rich material stripped/ablated from a non-degenerate companion star.', '1512.07107-1-90-2': 'This mass is much lower than expected for MS+WD or RG+WD progenitor systems, suggesting at least 0.1 M[MATH] of H-rich material is present in SN 2013ct but not observed as narrow H[MATH] emission.', '1512.07107-1-90-3': 'We find no evidence of H emission (H[MATH], Pa[MATH]) in the late-time spectra of 10 other SNe Ia, bringing the total sample with no H emission detected to 17 SNe Ia.', '1512.07107-1-90-4': 'Upper limits on the stripped/ablated mass of solar abundance material of 0.001-0.074 M[MATH] are placed for these SNe Ia.', '1512.07107-1-90-5': 'These upper mass limits of H-rich solar abundance material are inconsistent with MS or RG companion stars transferring mass via RLOF or wind-driven accretion (under the assumptions of current modelling).', '1512.07107-1-90-6': 'No signatures of He-rich material in the form of Hei, [Oii] or [Caii] emission lines are identified.', '1512.07107-1-90-7': 'However, spectral modelling of the expected flux of lines from He-rich material is not available.', '1512.07107-1-91-0': 'While future observational studies will increase the sample size of SNe Ia with the necessary late-time observations, and perhaps identify narrow H[MATH] emission in more objects, major future improvements are also likely to come from the next generation of spectral-synthesis modelling, allowing us to confirm (or adjust) these mass limits and determine if the SD scenario for producing the majority of SNe Ia is really in jeopardy.'} | {'1512.07107-2-0-0': 'The direct detection of a stellar system that explodes as a Type Ia supernova (SN Ia) has not yet been successful.', '1512.07107-2-0-1': 'Various indirect methods have been used to investigate SN Ia progenitor systems but none have produced conclusive results.', '1512.07107-2-0-2': 'A prediction of single-degenerate models is that H- (or He-) rich material from the envelope of the companion star should be swept up by the SN ejecta in the explosion.', '1512.07107-2-0-3': 'Seven SNe Ia have been analysed to date looking for signs of H-rich material in their late-time spectra and none were detected.', '1512.07107-2-0-4': 'We present results from new late-time spectra of 11 SNe Ia obtained at the Very Large Telescope using XShooter and FORS2.', '1512.07107-2-0-5': 'We present the tentative detection of H[MATH] emission for SN 2013ct, corresponding to [MATH]0.007 M[MATH] of stripped/ablated companion star material (under the assumptions of the spectral modelling).', '1512.07107-2-0-6': 'This mass is significantly lower than expected for single-degenerate scenarios, suggesting that [MATH]0.1 M[MATH] of H-rich is present but not observed.', '1512.07107-2-0-7': 'We do not detect H[MATH] emission in the other 10 SNe Ia.', '1512.07107-2-0-8': 'This brings the total sample of normal SNe Ia with non-detections ([MATH]0.001-0.058 M[MATH]) of H-rich material to 17 events.', '1512.07107-2-0-9': 'The simplest explanation for these non-detections is that these objects did not result from the explosion of a CO white dwarf accreting matter from a H-rich companion star via Roche lobe overflow or symbiotic channels.', '1512.07107-2-0-10': 'However, further spectral modelling is needed to confirm this.', '1512.07107-2-0-11': 'We also find no evidence of He-emission features, but models with He-rich companion stars are not available to place mass limits.', '1512.07107-2-1-0': '# Introduction', '1512.07107-2-2-0': 'The stellar systems and explosion mechanisms that produce Type Ia supernovae (SNe Ia) are still under debate.', '1512.07107-2-2-1': 'There are two main competing scenarios: the double-degenerate scenario [CITATION] where the companion star to the exploding white dwarf is another white dwarf, and the single-degenerate scenario [CITATION] where the companion star is a non-degenerate star such as a main-sequence (MS), giant or sub-giant.', '1512.07107-2-2-2': "The 'double-detonation' explosion mechanism has also seen a recent resurgence in popularity.", '1512.07107-2-2-3': 'In this scenario, the explosion results from the detonation of He on the white dwarf surface (accreted from either a degenerate or non-degenerate companion star), which triggers a subsequent C detonation in the core that unbinds the star .', '1512.07107-2-3-0': "Recent work has highlighted that it is likely that more than one of these channels contributes to produce 'normal' SNe Ia .", '1512.07107-2-3-1': 'However, clear observational diagnostics of the progenitor channel for individual objects have still not been determined.', '1512.07107-2-4-0': 'The presence of circumstellar material (CSM) around SNe Ia was originally suggested as a key observational diagnostic of the SD scenario (expected not to be present in DD scenarios).', '1512.07107-2-4-1': 'CSM signatures have been detected in a number of SNe Ia through studies of time-varying Nai D absorption lines in maximum-light spectra .', '1512.07107-2-4-2': 'The presence of CSM has also been inferred through statistical studies that have shown that [MATH]20 per cent of SNe Ia show signs of CSM and its presence is more common in more luminous SNe Ia .', '1512.07107-2-4-3': 'However, recent results have also shown at least qualitatively that the observed Nai D absorption profiles could also be produced in merger scenarios and/or where the companion star is a second white dwarf .', '1512.07107-2-5-0': "A unusual class of thermonuclear SNe, dubbed 'SNe Ia-CSM' , display strong signatures of H (H[MATH] emission with typical widths of [MATH]) in their spectra at early times, consistent with interaction with a H-rich CSM: SN2002ic , SN2005gj , SN2008J , and PTF11kx .", '1512.07107-2-5-1': 'A sample including new objects discovered by the Palomar Transient Factory is discussed in [CITATION].', '1512.07107-2-6-0': 'SNe Ia-CSM are suggested to result from a progenitor channel involving a non-degenerate companion star such as a symbiotic system .', '1512.07107-2-6-1': 'Some of these objects, such as PTF11kx, have been observed at late times ([MATH]200 d post maximum), where their H emission due to interaction with CSM is found to be still strong .', '1512.07107-2-6-2': "Since these SNe appear very unusual also at early times and perhaps represent a distinct class of SNe Ia, the astrophysical community does not consider them to be 'normal' SNe Ia, i.e. SNe Ia that would be used in cosmological studies.", '1512.07107-2-7-0': 'Another observational signature of a non-degenerate companion star would be shock interaction from the SN ejecta impacting on the companion star, which was calculated by [CITATION].', '1512.07107-2-7-1': 'This interaction would take the form of an increased luminosity in the very early-time (a few days after explosion) light curves of SNe Ia but would be dependent on the SN viewing angle.', '1512.07107-2-7-2': 'Signatures of this were searched for in a number of optical-wavelength studies but no clear detection was made .', '1512.07107-2-7-3': 'Recent early photometric observations of a SN Ia, iPTF14atg, have shown evidence of ultra-violet emission soon after explosion, consistent with the predictions of interaction with a non-degenerate companion star .', '1512.07107-2-7-4': "However, iPTF14atg was not a normal SN Ia; it displayed unusual photometric and spectroscopic properties being most similar to the peculiar class of sub-luminous 'SN 2002es-like' SNe Ia", '1512.07107-2-8-0': "Given these inconclusive results on the progenitor configuration of 'normal' SNe Ia from CSM and shock-interaction studies to date, complementary tracers of progenitor scenarios are very important.", '1512.07107-2-8-1': 'One particular area of interest is the detection of hydrogen in late-time ([MATH]200 d) SN Ia spectra - H features are expected to be present only in the SD scenario, where H-rich material is removed from a non-degenerate companion star.', '1512.07107-2-8-2': 'A key benefit of this nebular-phase probe is that the SN ejecta is expected to be optically thin at these late phases, and therefore the detection of H-rich material does not depend on orientation angle.', '1512.07107-2-8-3': 'This H emission is distinct from that seen in SNe Ia-CSM at late times, where the broader H emission is due to interaction with H-rich CSM.', '1512.07107-2-9-0': 'The ablation (heating) or stripping (momentum transfer) of H-rich material from a non-degenerate companion star through the impact of the SN Ia ejecta on the companion was discussed by [CITATION] and [CITATION].', '1512.07107-2-9-1': 'A number of early numerical studies were performed looking at the interaction between the SN ejecta and a non-degenerate companion star .', '1512.07107-2-9-2': 'The predicted properties of the H- and He-rich material have since been constrained in greater detail in a number of independent studies .', '1512.07107-2-10-0': 'The amount of unbound material present in the system after explosion, as well as its velocity distribution, is found to depend on the properties of the companion star.', '1512.07107-2-10-1': 'Assuming the case of Roche lobe overflow (RLOF), where the binary separation is typically [MATH]3R[MATH] (where R[MATH] is the radius of the companion star), the latest three-dimensional simulations of MS, red giant (RG) and He-star companions find unbound masses of [MATH]0.1-0.2, 0.6 and 0.02-0.06 M[MATH], respectively .', '1512.07107-2-10-2': 'The smallest amount of material is removed in the WD+He star scenario since although the initial binary separation is small (a key parameter determining the amount of mass removed through stripping and ablation), a He star is more compact and has a higher binding energy than an MS or RG companion.', '1512.07107-2-10-3': 'This makes the removal of its envelope in the explosion more difficult.', '1512.07107-2-10-4': 'In the case of the WD+He star scenario, the material is expected to be He- instead of H-rich.', '1512.07107-2-10-5': 'The peak velocities of the unbound material are also predicted to depend on the companion-star properties.', '1512.07107-2-10-6': 'In the simulations of [CITATION], the peak velocities of the unbound material are [MATH]550, 660, and 955 [MATH] for WD+MS, WD+RG and WD+He star systems, respectively.', '1512.07107-2-10-7': 'The [CITATION] velocity predictions are in good agreement with these values.', '1512.07107-2-11-0': 'Alternative mass-transfer mechanisms such as WD+RG systems that transfer mass to the white dwarf through a stellar wind (symbiotic systems) can have binary separations larger than those permitted for RLOF systems.', '1512.07107-2-11-1': 'Some recurrent nova systems such as RS Ophiuchi and T Coronae Borealis can have separations of up to [MATH]5R[MATH] , greater than the allowed separation for RLOF.', '1512.07107-2-11-2': 'However, [MATH]0.5 M[MATH] of material is still expected to be removed from the companion star in this scenario .', '1512.07107-2-12-0': "Observational studies of seven nearby 'normal' SNe Ia have been performed to look for low-velocity ([MATH]) H-features in their late-time spectra .", '1512.07107-2-12-1': 'No positive detections have been made.', '1512.07107-2-12-2': 'Three of the late-time spectra studied were of low-resolution ([MATH]), comparable to the width of the expected H features: SN 2001el , SN 1998bu, SN 2000cx .', '1512.07107-2-12-3': 'The spectra used by [CITATION] for SN 2005am and SN 2005cf, and by [CITATION] for SN 2011fe, were significantly higher resolution ([MATH]).', '1512.07107-2-12-4': 'A spectrum obtained for SN 2014J, the closest SN Ia for several decades, had a resolution of [MATH] .', '1512.07107-2-13-0': 'The tightest limits on the H flux present have been obtained for the nearby SN 2011fe [CITATION], corresponding to an upper limit of 0.001 M[MATH] of swept-up material if a linear extrapolation of the H fraction of the [CITATION] spectral synthesis model holds .', '1512.07107-2-13-1': 'Possible signatures of H at near-infrared (NIR) wavelengths were investigated for SN 1998bu using the Pa[MATH] and Pa[MATH] lines, but less stringent mass limits of [MATH]0.5 M[MATH] were placed .', '1512.07107-2-14-0': 'A search for signatures of a He-rich companion star in the late-time spectra of SNe 2011fe and 2014J was carried out .', '1512.07107-2-14-1': 'Based on the results of [CITATION], it is expected that line fluxes of O and Ca produced in He-rich material would be stronger than those of He.', '1512.07107-2-14-2': 'Therefore, the presence of [Caii] and [Oi] lines with widths of [MATH] was investigated by [CITATION] - they detected no features at the expected wavelengths and placed upper He-mass limits from the [Oi] 6300 line of [MATH]0.002 and [MATH]0.005 M[MATH] for SNe 2011fe and 2014J, respectively.', '1512.07107-2-14-3': 'For SN 2011fe, they suggested that this limit is sufficient to rule out all He-rich companion star models from [CITATION] and [CITATION], including for larger-than-RLOF binary separations.', '1512.07107-2-15-0': 'If the model predictions are correct, then the non-detection mass limits of the literature sample are sufficient to rule out MS or RG companion stars in RLOF or accreting mass via a stellar wind for these seven SNe Ia.', '1512.07107-2-15-1': 'However, the sample size is small and it is unclear if these SNe Ia are representative of the full SN Ia population.', '1512.07107-2-15-2': 'Therefore, a study looking at a significantly larger sample of nebular phase spectra of SNe Ia is needed.', '1512.07107-2-16-0': 'In this paper, we present low- and medium-resolution nebular-phase spectroscopic data of 11 nearby SNe Ia obtained using the European Southern Observatory (ESO) Very Large Telescope (VLT) with the XShooter spectrograph and with the FOcal Reducer and low dispersion Spectrograph .', '1512.07107-2-16-1': 'In Section [REF], we detail the observations and data reduction steps employed.', '1512.07107-2-16-2': 'The analysis and search for the predicted H[MATH] lines, as well as for Pa[MATH] and Pa[MATH] lines at NIR wavelengths are described in Section [REF].', '1512.07107-2-16-3': 'In the same section, we also discuss the results of a search for signatures of He-rich material measured using the available Hei, as well as [Caii] and [Oi], lines.', '1512.07107-2-16-4': 'The discussion and conclusions are presented in Section [REF].', '1512.07107-2-16-5': 'Throughout this paper we assume a Hubble constant, [MATH]kms[MATH]Mpc[MATH].', '1512.07107-2-17-0': '# Observations and Data Reduction', '1512.07107-2-18-0': 'Nebular-phase spectra ([MATH]230 d) of 11 SNe Ia are used in this study: six obtained using VLT+XShooter and six obtained using VLT+FORS2.', '1512.07107-2-18-1': 'One object, SN 2012cg was observed with both instruments and serves as a check of the analysis and methods.', '1512.07107-2-18-2': 'We discuss the data reduction for the two instruments separately.', '1512.07107-2-19-0': '## XShooter spectra', '1512.07107-2-20-0': 'XShooter is an echelle spectrograph with three arms (UVB, VIS and NIR) covering the wavelength range of [MATH]3000-25000 .', '1512.07107-2-20-1': 'The spectral format of XShooter is fixed but the resolution can be adjusted using different slit widths.', '1512.07107-2-20-2': 'For our data, we used slit widths of 0.8, 0.9 and 0.9 arcsec in the UVB, VIS and NIR arms, corresponding to resolutions of [MATH] 6200, 8800 and 5300, respectively.', '1512.07107-2-20-3': 'Details of the spectral observations are given in Table [REF].', '1512.07107-2-21-0': 'The spectra were reduced using the REFLEX pipeline (version 2.6) with the XShooter module (version 2.5.0), producing flux-calibrated one-dimensional spectra in each of the three arms .', '1512.07107-2-21-1': 'Photometric conditions and seeing of [MATH]0.8 arcsec were required before execution of the observations could begin.', '1512.07107-2-21-2': 'Due to the relatively high spectral resolution of XShooter ([MATH]35 kms[MATH]), host galaxy features were easily identified and removed in the reduction process.', '1512.07107-2-21-3': 'For one SN in our sample, SN 2013ct, due to the potential detection of a feature consistent with H[MATH] emission (see Section [REF] for more details), we also performed an additional extraction of the spectra using a custom-written pipeline.', '1512.07107-2-21-4': 'The two-dimensional data products from the REFLEX pipeline were used as input, but with sky-subtraction, extraction and flux-calibration performed order-by-order on the unresampled REFLEX images.', '1512.07107-2-21-5': 'This was to check carefully the host galaxy feature removal and relative flux calibration, and confirm that this was not causing the tentative emission feature that was seen.', '1512.07107-2-22-0': 'We scale the flux of our spectra to photometry taken close in time to the spectroscopic observations to obtain absolute flux-calibrated spectra.', '1512.07107-2-22-1': "We obtained acquisition images using r' and i' band filters, similar to the filters used by the Sloan Digital Sky Survey , and in the Johnson I band filter.", '1512.07107-2-22-2': 'The SN magnitudes were determined using point spread function (PSF) photometry.', '1512.07107-2-22-3': 'The zero-points of the images were obtained using aperture photometry of comparison stars in the images, which were then calibrated by comparison to either catalogue magnitudes from the SDSS Data Release 10 for SNe 2012cg and 2012ht, or using the XShooter acquisition and guide camera zero-points for SNe 2012fr, 2013ct, 2013cs, 2013aa.', '1512.07107-2-22-4': "XShooter zero-points were not available for r'.", '1512.07107-2-22-5': "Therefore, for the three SNe (2013ct, 2013cs, and 2013aa) that were observed in the r' band, a transformation, using the relative flux throughputs of the SN spectra in the relevant filters, was applied to convert r' to Johnson R for which an XShooter zero-point was available.", '1512.07107-2-23-0': 'The flux-calibrated spectra were corrected to rest-frame wavelengths using the heliocentric velocities given in Table [REF].', '1512.07107-2-23-1': 'These values were obtained from the NASA/IPAC Extragalactic Database (NED) galaxy spectra, apart from SN 2012cg, which following [CITATION] used the stellar velocity of [CITATION] at the SN position.', '1512.07107-2-24-0': 'Following [CITATION], the optical regions of the XShooter spectra were rebinned to 3 .', '1512.07107-2-24-1': 'The NIR portions of the spectra were rebinned to 6 to increase the signal-to-noise ratio.', '1512.07107-2-24-2': 'A host-galaxy origin for any potential [MATH] H[MATH] emission can be confidently ruled out since it would only contribute to the central spectral bin at the binned XSH dispersion.', '1512.07107-2-25-0': '## FORS2 spectra', '1512.07107-2-26-0': 'For the FORS2 observations, a combination of grisms 300V and 300I (+OG590) and a 1 arcsec slit were used, yielding a resolution [MATH]-[MATH] over a wavelength range of 3500-10500.', '1512.07107-2-26-1': 'After bias subtraction and flat-fielding an optimal, variance-weighted extraction of the spectra was performed wite iraf task apall.', '1512.07107-2-26-2': 'The dispersion solution was established using arc-lamp exposures and cross-checked against night-sky emission lines.', '1512.07107-2-26-3': 'Spectrophotometric standard stars, usually observed during the same nights as the SNe, were used to perform a relative flux calibration of the spectra and to correct for telluric absorptions.', '1512.07107-2-27-0': 'To calibrate to an absolute flux scale and correct for slit losses, the spectra were scaled to match FORS2 [MATH] photometry of the SNe obtained close in time to the spectral observations (see Table 1).', '1512.07107-2-27-1': 'The SN magnitudes in the [MATH] frames were measured using PSF photometry, and calibrated using the FORS2 zero-points and extinction coefficients provided by the ESO data-quality monitoring.', '1512.07107-2-28-0': 'As a result of the significantly lower spectral resolution of the FORS2 spectra ([MATH]450 kms[MATH]) compared to the XShooter spectra, the host-galaxy subtraction was more complex.', '1512.07107-2-28-1': 'For nine of the SNe Ia observed with FORS2, emission from a Hii-region was visible in the two-dimensional spectra either at the SN location or very nearby.', '1512.07107-2-28-2': 'Therefore, a clean background subtraction could not be guaranteed for these objects and we remove these SNe from further discussion since the uncertain background subtraction could mask H[MATH] emission from the SN itself.', '1512.07107-2-28-3': 'However, we note that these SNe Ia could have displayed narrow H features that went undetected.', '1512.07107-2-28-4': 'Therefore, the number of SNe Ia with narrow H features could be higher than measured in our sample.', '1512.07107-2-28-5': 'The six remaining SNe Ia observed with FORS2, and that have a clean background subtraction, are detailed in Table [REF].', '1512.07107-2-28-6': 'For these spectra, the dispersion was left at its native value of 3.25 .', '1512.07107-2-29-0': '# Analysis', '1512.07107-2-30-0': 'We have searched for the presence of material that could have been removed from a non-degenerate companion star (via ablation and/or stripping) in a new late-time SN Ia spectral sample, as well as used model predictions to constrain quantitatively the presence or absence of solar abundance material.', '1512.07107-2-31-0': '## Searching for signatures of H', '1512.07107-2-32-0': 'We searched for the presence of narrow ([MATH]) lines of H (H[MATH], Pa[MATH], Pa[MATH]) in the spectra.', '1512.07107-2-32-1': 'To do this an underlying continuum must first be defined.', '1512.07107-2-32-2': 'The underlying continuum of the spectra in these regions was fit using a second-order Savitzky-Golay smoothing polynomial , with the search wavelength region of [MATH]22 of the H features excluded from the smoothing.', '1512.07107-2-32-3': 'This was to avoid biasing the continuum towards higher values if a H-emission feature was present.', '1512.07107-2-32-4': 'The smoothing scales used were significantly larger than the predicted widths of the narrow features, with widths of 80-140 in the optical and [MATH]200 in the NIR.', '1512.07107-2-32-5': 'Further analysis and the associated uncertainties on estimating the continuum for the H[MATH] region is discussed in Section [REF].', '1512.07107-2-33-0': 'The rebinned spectra and continuum fits at the position of H[MATH] for our SN Ia sample are shown in Fig. [REF].', '1512.07107-2-33-1': 'The Pa[MATH]) regions of the XShooter spectra are shown in Fig. [REF].', '1512.07107-2-33-2': 'The position of any potential Pa[MATH]) feature falls in the middle of a telluric band and detection of SN flux is very difficult.', '1512.07107-2-33-3': 'Therefore, the only NIR H line we discuss is the Pa[MATH] feature.', '1512.07107-2-33-4': 'We did not detect any strong unambiguous H[MATH] or Pa[MATH] features in our sample when inspecting the appropriate wavelength regions.', '1512.07107-2-33-5': 'However, this does not exclude less prominent emission features being present.', '1512.07107-2-33-6': 'A quantitative estimate of the presence of H[MATH] and associated non-detection limits is described in Section [REF].', '1512.07107-2-34-0': 'We have estimated the potential effect of telluric features on detection of narrow features in the H[MATH] and Pa[MATH] spectral regions.', '1512.07107-2-34-1': 'The closest significant telluric feature (transmission [MATH]0.9) to the wavelength of H[MATH] is a feature at [MATH]6519 identified using the ESO SKYCALC Sky Model Calculator.', '1512.07107-2-34-2': 'For the SN in our sample with the lowest heliocentric redshift, SN 2012cg, this feature would lie at a rest frame wavelength of [MATH]6508 .', '1512.07107-2-34-3': 'This is well outside the region of interest of [MATH]22 of the H[MATH] feature at 6563 and the feature is weak enough not to be visible in our spectra.', '1512.07107-2-34-4': 'Therefore, we conclude that this has negligible effect on our results.', '1512.07107-2-34-5': 'We have also investigated potential telluric features in the Pa[MATH] wavelength region and find no significant telluric features within the observed wavelength region.', '1512.07107-2-35-0': '### Model predictions of stripped/ablated material', '1512.07107-2-36-0': 'Based on the calculations of [CITATION], [CITATION] used a one-dimensional nebular-phase spectral code to give a quantitative estimate of the H[MATH] emission expected from different amounts of solar-abundance material stripped or ablated from a companion star.', '1512.07107-2-36-1': 'This solar abundance material was located in the inner 1000 [MATH] of the ejecta using the W7 explosion model .', '1512.07107-2-36-2': 'The peak luminosity of the H[MATH] line for 0.05 M[MATH] of solar-abundance material was found to be [MATH]3.36 [MATH] erg s[MATH] .', '1512.07107-2-37-0': 'These model H[MATH] luminosities were calculated at +380 d and the H[MATH] emission is expected to be time-dependent .', '1512.07107-2-37-1': 'However, [CITATION] noted that since the optical depth to gamma rays should be higher at earlier times, the H[MATH] emission should be stronger at epochs earlier than +380 d. Therefore, the estimated H[MATH] flux at +380 d can be considered a lower limit for earlier spectra as long as the condition that the ejecta are transparent is fulfilled ([MATH]200 d) and we can observe this low-velocity material.', '1512.07107-2-37-2': 'Sources of additional uncertainty in the modelling are discussed in Section [REF].', '1512.07107-2-38-0': 'For three SNe Ia in our sample (SNe 2009ig, 2011ek, and 2012ht), the spectra were obtained at epochs greater than +380 d (between +405 and +433 d).', '1512.07107-2-38-1': 'Given the small difference in the epoch studied in the model and the observed data, the model fluxes are assumed to be still applicable at these slightly later phases.', '1512.07107-2-39-0': 'We have calculated the expected H[MATH] emission for this luminosity for each SN in our sample, using the distance and Galactic extinction values [corrected to the wavelength of H[MATH] using the [CITATION] extinction curve with R[MATH]] in Table [REF].', '1512.07107-2-39-1': 'Host galaxy extinction corrections were not applied for most SNe Ia in our sample since we found no evidence for significant host galaxy extinction.', '1512.07107-2-39-2': 'This was based on an analysis of their maximum-light spectra, absolute magnitudes at peak, and locations within their host galaxy.', '1512.07107-2-40-0': 'The expected integrated H[MATH] fluxes for 0.05 M[MATH] of H-rich material, assuming a Gaussian profile with a full width at half-maximum (FWHM) of 1000 [MATH], and a central wavelength of 6563 are given in Table [REF].', '1512.07107-2-40-1': 'The chosen value of the FWHM is consistent with the previous studies, as well as the model constraint of [CITATION] that the H-rich material is located at velocities below [MATH].', '1512.07107-2-40-2': 'The expected H[MATH] line profiles for 0.05 M[MATH] of solar abundance material are plotted in Fig. [REF] for each SN in our sample.', '1512.07107-2-41-0': '### Estimation of contributions from narrow H[MATH]', '1512.07107-2-42-0': 'To estimate the amount of H[MATH] that could be present, as well as set limits on any non-detections, we made model spectra combining the continuum fit with a Gaussian to represent a potential H[MATH] emission feature.', '1512.07107-2-42-1': 'The Gaussian was set to have a fixed FWHM of 1000 [MATH] and fixed central wavelength of 6563 , but a varying peak flux.', '1512.07107-2-42-2': 'A likelihood function was calculated as a function of varying peak flux in steps of 0.001 M[MATH] within the range corresponding to [MATH]0.05 M[MATH] (negative values are included to sample a complete peak-flux distribution).', '1512.07107-2-42-3': '[CITATION] and [CITATION] calculated the expected flux at the position of H[MATH] for differing input masses of stripped/ablated H-rich companion star material (0.01, 0.05, 0.1, 0.5 M[MATH]), and found a linear relation between H[MATH] flux and mass.', '1512.07107-2-42-4': 'Therefore, we use a linear scaling between H[MATH] flux and mass in our analysis.', '1512.07107-2-43-0': 'The largest uncertainty in the likelihood calculation is the estimation of the continuum fit.', '1512.07107-2-43-1': 'Therefore, to estimate the associated uncertainty we have calculated a number of continuum fits using the Savitzky-Golay smoothing polynomial with widths varying from 80 to 140 .', '1512.07107-2-43-2': 'This range was chosen so has to be significantly larger than the width of the feature we are searching for ([MATH]22 ) but not so big that the underlying continuum features are not well fit.', '1512.07107-2-43-3': 'A value of 100 was used in the analysis of [CITATION].', '1512.07107-2-43-4': 'The mean of these continuum fits is then used as input to the likelihood calculation.', '1512.07107-2-44-0': 'The sigma entering the likelihood was calculated from the rms scatter per wavelength bin from the continuum fits (an estimation of the uncertainty in the continuum definition) combined with the scatter around the normalized continuum (an estimation of the spectral noise).', '1512.07107-2-45-0': '### Tentative detection of H[MATH] in SN 2013ct', '1512.07107-2-46-0': 'For one SN in our sample, SN 2013ct, we have identified a weak but broad feature at the position of H[MATH], consistent with the stripping/ablation model predictions of H[MATH] emission with velocities of [MATH]600-1000 [MATH].', '1512.07107-2-46-1': 'By varying the width of the feature, we found the strongest significance feature for an FWHM of 850 [MATH].', '1512.07107-2-46-2': 'Using this FWHM and the method detailed in Section [REF], we found that the H[MATH] flux of SN 2013ct corresponded to [MATH]0.005 M[MATH] of stripped or ablated H-rich material.', '1512.07107-2-47-0': 'Due to the strong underlying continuum for SN 2013ct and the potential detection of a narrow H[MATH] feature, we investigated further fits to the underlying continuum.', '1512.07107-2-47-1': 'We expanded the continuum fitting models to include a Gaussian fit to the underlying continuum.', '1512.07107-2-47-2': 'The parameters of the Gaussian were set to be variable.', '1512.07107-2-47-3': 'The region of the spectrum included in the Gaussian was varied between [MATH]70 and [MATH]130 of the centre of the underlying profile (6540 ).', '1512.07107-2-47-4': 'The region [MATH] from the rest wavelength of H[MATH] was again excluded from the continuum fitting.', '1512.07107-2-48-0': 'The mean continuum of the combined Savitzky-Golay and Gaussian fitting was used as the continuum in the likelihood calculation.', '1512.07107-2-48-1': 'The best-fitting for the H[MATH] feature for SN 2013ct using this expanded continuum fitting is shown in Fig. [REF].', '1512.07107-2-48-2': 'This H[MATH] flux corresponds to [MATH]0.007[MATH]0.001 M[MATH] of stripped or ablated H-rich material at a 3.7[MATH] significance.', '1512.07107-2-49-0': 'To help determine the robustness of our H[MATH] detection for SN 2013ct, we performed two additional tests.', '1512.07107-2-49-1': 'First, we searched for similar features at different rest wavelengths by varying the wavelength of the expected feature within [MATH]120 of the detected H[MATH] position.', '1512.07107-2-49-2': 'We used the same analysis method as for the search at the position of H[MATH], including excluding the search region from the continuum fitting.', '1512.07107-2-49-3': 'Secondly, we tested if a feature of a similar significance was detected on top of another broad nebular emission feature (the 5800 feature) in the SN 2013ct spectrum.', '1512.07107-2-49-4': 'This was to determine if residuals in the fitting of the continuum for the broad emission features would result in a similar feature to that seen at the position of H[MATH].', '1512.07107-2-49-5': 'In both cases, no significant detection ([MATH]) is associated within any feature outside of the H[MATH] search region.', '1512.07107-2-49-6': 'However, we caution that the significance of the H[MATH] detection is under the assumptions of the fitting to an unknown underlying continuum, as well as the predictions of the models of the position and velocity distribution of H-rich material.', '1512.07107-2-50-0': 'If our measurements of the SNe are assumed to be independent, then as the sample size increases, there is an increased probability of finding a 3[MATH] detection by chance.', '1512.07107-2-50-1': 'We used the correction of [CITATION] to estimate the increased significance needed to claim a 3[MATH] detection for one object out of the eight SNe Ia in our combined new data and literature sample for which the mass detection limit at the position of H[MATH] is [MATH]0.007 M[MATH] (equivalent to the SN 2013ct detection).', '1512.07107-2-50-2': 'We found that a 3.7[MATH] detection in any one spectrum is equivalent to a 3.1[MATH] detection if we make eight comparisons, which we use as our significance value for this detection.', '1512.07107-2-51-0': '### H[MATH] detection limits', '1512.07107-2-52-0': 'The rest of the SNe Ia in our sample showed no obvious H[MATH] emission.', '1512.07107-2-52-1': 'Using the same method as for SN 2013ct, we determined that the Gaussian fits at the position of H[MATH] for the rest of the sample are consistent with zero within the 3[MATH] uncertainties.', '1512.07107-2-52-2': 'We placed 3[MATH] upper limits on the strength of features that could have remained undetected at the position of H[MATH].', '1512.07107-2-53-0': 'The line profiles representing these flux limits are shown in Fig. [REF], and values for the integrated H[MATH] flux and corresponding mass limits are given in Table [REF].', '1512.07107-2-53-1': 'The 3[MATH] limiting mass range for the SN sample is 0.001-0.058 M[MATH].', '1512.07107-2-53-2': 'The bright, nearby SN 2012cg was observed with both XShooter and FORS2 and we obtain a consistent mass limit ([MATH]0.010 M[MATH]) for both spectra.', '1512.07107-2-54-0': 'Since these limits are strictly statistical, [CITATION] estimated how systematic uncertainties from our lack of knowledge of the underlying spectral continuum could affect the measured F(3[MATH]) values of H[MATH] emission for SNe 2011fe and 2014J.', '1512.07107-2-54-1': 'They suggested, using an inspection by eye, that the limits on the masses of stripped/ablated material could be 2-3 greater than their measured values.', '1512.07107-2-55-0': 'As discussed in Section [REF], to minimize the uncertainty in the selection of the continuum in our sample, we have excluded the region of [MATH]22 with respect to H[MATH] when smoothing the spectra to avoid biasing the continuum fit of the search wavelength region.', '1512.07107-2-55-1': 'We have also tested our continuum fits by checking for similar strength features outside the H[MATH] search region, as well as on top of the broad 5800 nebular feature, and no detection was made.', '1512.07107-2-55-2': 'Therefore, we have chosen to use our original F(3[MATH]) values calculated from the probability distributions to measure M(3[MATH]) and these are the values we quote in Table [REF].', '1512.07107-2-56-0': '## Searching for signatures of He', '1512.07107-2-57-0': 'The companion star to the exploding white dwarf could also potentially be a He-rich star instead of a H-rich star, which could result in the presence of narrow ([MATH]) He lines in late-time SN Ia spectra.', '1512.07107-2-57-1': 'Less material is expected to become unbound than in the H-rich case; just 0.02-0.06 M[MATH] is predicted by the models .', '1512.07107-2-58-0': 'To investigate the presence of He emission at late times, we focus first on the Hei 5876, 10830, and 20587 features in the XShooter VIS and NIR spectra.', '1512.07107-2-58-1': 'The wavelength regions for the Hei lines are shown in Fig. [REF].', '1512.07107-2-58-2': 'The Hei 5876 and 10830 spectral regions are found to be relatively free of telluric absorption features.', '1512.07107-2-58-3': 'However, the Hei 20590 region is moderately affected.', '1512.07107-2-58-4': 'For completeness, we show this region in Fig. [REF] but do not base our results on this region alone.', '1512.07107-2-58-5': 'We did not have model predictions of the peak luminosity for these He features but from Fig. [REF] it is clear that there are no strong Hei signatures detected in our SN Ia sample within the velocity range predicted by [CITATION].', '1512.07107-2-59-0': '[CITATION] detailed how [Oi] and [Caii] emission features are promising probes of stripped/ablated He-rich material.', '1512.07107-2-59-1': 'We have also searched for [Oi] 6300, 6364 and [Caii] 7291, 7324 emission features with velocities of [MATH] of the rest wavelength in our XShooter and FORS2 spectra but did not find any features consistent with the qualitative model predictions.', '1512.07107-2-59-2': 'However, further analysis, when robust spectral modelling predictions of peak luminosities are available, will be necessary to put flux (and mass) limits on these He, Ca and O non-detections.', '1512.07107-2-60-0': '# Discussion', '1512.07107-2-61-0': 'Using the expected H[MATH] luminosities from a time-dependent nebular synthesis calculation for solar abundance material confined within 1000 [MATH] , we have quantified the presence or absence of H-rich companion star material in 11 SNe Ia for which late-time spectra were obtained.', '1512.07107-2-61-1': 'For 10 SNe Ia in the sample, we did not find evidence of H-rich companion star material in their late-time spectra.', '1512.07107-2-61-2': 'Under the assumptions of the modelling, we can place limits of [MATH]0.001-0.058 M[MATH] on the mass of H-rich material that could remain undetected.', '1512.07107-2-61-3': 'For one SN in our sample, SN 2013ct, we have made a tentative detection (3.1[MATH]) of emission at the position of H[MATH], corresponding to 0.007[MATH]0.001 M[MATH] of H-rich material.', '1512.07107-2-61-4': "Combining these new data with previous samples , this means one potential detection of H[MATH] emission out of 18 'normal' SNe Ia that have been studied at late times to look for signatures of H-rich material.", '1512.07107-2-62-0': 'Emission at the wavelength of Pa[MATH] was also investigated for five SNe Ia in the sample with suitable NIR spectral coverage and no clear emission was identified.', '1512.07107-2-62-1': 'However, as noted in [CITATION], the mass limits obtained from Pa[MATH] are less constraining than for H[MATH].', '1512.07107-2-63-0': 'We do not detect strong signatures of Hei 5876, 10830, 20587 features within a velocity range of [MATH].', '1512.07107-2-63-1': 'We also do not detect [Oi] or [Caii] emission features, which were suggested by [CITATION] to be promising tracers of He-rich material.', '1512.07107-2-63-2': '[CITATION] estimated very rough limits on the He mass present for SNe 2011fe and 2014J by adjusting the H mass limits for the He to H number density ratio assuming solar metallicity.', '1512.07107-2-63-3': 'For the tightest mass limits for SN 2011fe in [CITATION], all He-rich companion stars from the models of [CITATION] and [CITATION] can be ruled out based on the non-detection of [Oi] emission.', '1512.07107-2-63-4': 'However, since the He-mass predictions obtained are simple extrapolations from models of H-rich solar abundance material placed in the inner [MATH] of the SN ejecta and the He-rich material is expected to be at higher velocity than this, they should be taken as very rough upper limits.', '1512.07107-2-64-0': '## First detection of H[MATH] in the late-time spectrum of a normal SN Ia?', '1512.07107-2-65-0': 'The detection of a potential H[MATH] feature consistent in velocity ([MATH]) with the models of [CITATION] and [CITATION] is suggested for SN 2013ct.', '1512.07107-2-65-1': 'Using the models of [CITATION], this is estimated to be equivalent to 0.007[MATH]0.001 M[MATH] of solar abundance material being present at velocities below 1000 [MATH].', '1512.07107-2-65-2': 'However, the models of [CITATION] and [CITATION] have shown that if the companion star to the exploding white dwarf was an MS star in any realistic binary scenario, at least 0.1-0.2 M[MATH] of material should be removed from the companion star after explosion.', '1512.07107-2-65-3': 'The companion star can not be artificially moved to greater separations to reduce the amount of stripped/ablated material because then the necessary condition of RLOF would not be fulfilled.', '1512.07107-2-65-4': 'For an RG companion, [MATH]0.6 M[MATH] of material is predicted to be stripped/ablated from the companion in RLOF.', '1512.07107-2-65-5': 'For systems transferring mass via a stellar wind, [MATH]0.5 M[MATH] of material would be removed from the companion star', '1512.07107-2-66-0': 'Therefore, in order to explain an emission feature corresponding to [MATH]0.007 M[MATH] of H-rich material, an additional [MATH]0.1-0.6 M[MATH] of material would have to be present but not visible as a narrow H[MATH] emission within 1000 [MATH].', '1512.07107-2-66-1': 'A tail of material extending to higher velocities could hide some material that would not contribute significantly to the observed narrow H[MATH] feature.', '1512.07107-2-66-2': 'However, the models predict that the amount of stripped or ablated material present in a high-velocity tail is expected to be low .', '1512.07107-2-67-0': 'Another possibility is that the majority of the H-rich companion star material is not sufficiently powered by radioactive heating to produce H[MATH] emission.', '1512.07107-2-67-1': 'This may be because there is little SN ejecta remaining below 1000 [MATH] at these times .', '1512.07107-2-67-2': 'Therefore, while the presence of a narrow H[MATH] feature suggests a H-rich non-degenerate companion star to the white dwarf for SN 2013ct, further constraints on the companion star properties can not currently be placed.', '1512.07107-2-67-3': 'The uncertainties of the model and observations are detailed further in Section [REF].', '1512.07107-2-68-0': '## Are H-rich SD systems sub-dominant for producing normal SNe Ia?', '1512.07107-2-69-0': 'Unfortunately, little early-time data are available for SN 2013ct to determine its detailed properties.', '1512.07107-2-69-1': 'The SN was discovered on 2013 May 10 but not spectroscopically classified until 2013 May 22, where the best near-infrared spectroscopic match was a normal SN Ia approximately 20 d after maximum light .', '1512.07107-2-69-2': 'Therefore, while this spectrum appears similar to other normal SNe Ia, subtle differences that may have been apparent with a larger data set are not quantifiable.', '1512.07107-2-70-0': "Although the first tentative detection of H[MATH] emission in a late-time spectrum of a 'normal' SN Ia is very interesting, the absence of H[MATH] emission in the late-time spectra of 17 other SNe Ia is also worthy of discussion.", '1512.07107-2-70-1': 'Using the models of [CITATION], stripped/ablated mass limits in the range 0.001-0.058 M[MATH] were determined for these 17 SNe Ia.', '1512.07107-2-70-2': 'If we assume that the stripping/ablation and spectral model calculations are correct, then these limits are sufficient to rule out all MS and RG companions transferring mass via RLOF or a stellar wind at all plausible separations.', '1512.07107-2-71-0': 'However, the H[MATH] detection equivalent to [MATH]0.007 M[MATH] of H-rich material for SN 2013ct potentially weakens these constraints, since it suggests that [MATH]0.1-0.6 M[MATH] of H-rich companion star material was hidden for this SN.', '1512.07107-2-71-1': 'Therefore, H-rich material may have also been present in the other 17 SNe Ia but also hidden.', '1512.07107-2-72-0': 'The spin-up/spin-down scenario of [CITATION] and [CITATION] could reduce the amount of H-rich material stripped from a non-degenerate companion star at the time of explosion.', '1512.07107-2-72-1': 'This involves a white dwarf becoming spun-up by the mass accreted from its companion star, resulting in a stable white dwarf above the Chandrasekhar mass.', '1512.07107-2-72-2': 'This gives time for the companion star to evolve and contract before the SN explosion.', '1512.07107-2-72-3': 'Therefore, at the time of explosion, the companion star is much smaller and more tightly bound, significantly reducing the amount of material removed by the impact of the SN ejecta.', '1512.07107-2-72-4': 'However, some fine-tuning is necessary to produce this scenario and as the sample of SNe Ia without H emission grows, it is becoming increasingly unlikely that this can explain all the observed non-detections.', '1512.07107-2-73-0': 'It has also been suggested that the broad underlying emission feature at the position of H[MATH] in our sample (usually attributed to [Feii] 6559 emission) may have a contribution from a broad H[MATH] emission component .', '1512.07107-2-73-1': 'However, in this case, the H-rich material would have to be present at much higher velocities than those currently predicted by modelling efforts .', '1512.07107-2-73-2': 'Further studies are necessary to determine if this is physically plausible.', '1512.07107-2-74-0': 'However, it is likely that the sub-class of SNe Ia, SNe Ia-CSM, do result from an SD progenitor channel.', '1512.07107-2-74-1': 'The most likely scenario is that of a symbiotic system involving an RG or asymptotic giant branch star .', '1512.07107-2-74-2': 'This scenario would produce the necessary CSM to explain the interaction features seen in their spectra such as broad H emission extending till late times .', '1512.07107-2-74-3': "If SNe Ia-CSM and at least some so-called 'normal' SNe Ia originate from a SD scenario, then it may not be surprising to identify weaker H features, such as that detected for SN 2013ct, in some SNe Ia (coming from stripped/ablated companion material instead of pre-explosion mass loss).", '1512.07107-2-75-0': '## He-rich companion stars', '1512.07107-2-76-0': "Another solution for producing 'normal' SNe Ia through the SD channel and avoiding H contamination is to invoke a He- instead of H-rich companion star.", '1512.07107-2-76-1': 'The latest simulations of the interaction between a He-rich companion star and the SN ejecta predict stripped/ablated masses of 0.02-0.06 M[MATH] .', '1512.07107-2-76-2': '[CITATION] put limits on the mass of He-rich material present in SNe 2011fe and 2014J of [MATH]0.002 and [MATH]0.005 M[MATH], respectively.', '1512.07107-2-76-3': 'Under the assumption of RLOF, these limits rule out the He-rich companion star models of [CITATION] and [CITATION] for both SNe Ia.', '1512.07107-2-77-0': 'We have presented a qualitative discussion of the non-detection of He emission in our late-time spectral sample of 11 SNe Ia - we do not find obvious features that could be attributed to Hei, [Caii] or [Oi] emission.', '1512.07107-2-77-1': 'However, as cautioned in [CITATION], these limits are based on a rough correction from H- to He-rich material, and have not been modelled explicitly.', '1512.07107-2-77-2': 'Therefore, we await more detailed spectral modelling to place limits on the presence of He-rich material from a companion star swept-up in the SN ejecta.', '1512.07107-2-78-0': "Companion stars with He-rich outer layers are also present in the 'double-detonation' scenario, where a thin layer of He on the surface of the primary white dwarf is responsible for the first detonation .", '1512.07107-2-78-1': 'This He material is accreted from a He star, He WD, a CO white dwarf with a thin layer of He on its surface, or it may be already present on the primary white dwarf surface.', '1512.07107-2-78-2': 'At the time of the subsequent detonation of the core, the He on the surface is expected to have velocities of [MATH].', '1512.07107-2-78-3': "Therefore, this material is not expected to be visible at the low velocities studied here, and the non-detection of He features in the late-time spectra does not place constraints on the presence of high-velocity He-rich material in the 'double-detonation' scenario.", '1512.07107-2-78-4': 'Detection of this high-velocity He is also unlikely in early-time observations because of insufficient heating of this material so far out in the ejecta to cause He emission lines to be observed.', '1512.07107-2-79-0': '## Sample selection', '1512.07107-2-80-0': 'The SNe Ia in our late-time spectral sample were selected for observation based on their proximity ([MATH]) and visibility at Paranal at [MATH]200 d after maximum light.', '1512.07107-2-80-1': 'If an SN Ia was significantly sub-luminous at maximum light then it would not have been scheduled for observations because it would be deemed too faint at [MATH]200 d after maximum light.', '1512.07107-2-80-2': 'This would bias our sample towards more luminous SNe Ia.', '1512.07107-2-80-3': 'However, the SNe Ia in our sample span a wide range of host galaxy types from early- to late-type galaxies, suggesting a spread in SN luminosity .', '1512.07107-2-80-4': 'SN 2011iv which occurred in an elliptical galaxy was a low-luminosity SN Ia .', '1512.07107-2-80-5': "Therefore, we conclude that our sample is not significantly biased towards more luminous events, and covers a range of 'normal' SN Ia luminosities.", '1512.07107-2-80-6': 'There is also no reason to expect that less-luminous SNe Ia are more likely to have H or He features present in their spectra.', '1512.07107-2-80-7': 'In fact, recent work suggested the opposite - that it is likely that more luminous SNe Ia occur preferentially through SD channels .', '1512.07107-2-81-0': '## Observational and model uncertainties', '1512.07107-2-82-0': 'In this section, we discuss the uncertainties and limitations of the observations and modelling that could cause the results and interpretation of our analysis to be less constraining.', '1512.07107-2-82-1': 'A number of independent simulations have been made of the amount of stripped/ablated material, and its velocity distribution, that is expected to be removed from a non-degenerate companion star after explosion .', '1512.07107-2-82-2': 'There is now reasonable agreement among the different groups for these masses and velocity distributions for different companion star setups.', '1512.07107-2-82-3': 'However, only one analysis has been done to determine the spectral line strengths associated with different mass and velocity values .', '1512.07107-2-82-4': 'This study was based on the earliest impact simulations of [CITATION].', '1512.07107-2-82-5': 'However, the amount and velocity distributions of the stripped/ablated material have not been updated drastically in more recent modelling, and this should not significantly affect the spectral modelling results.', '1512.07107-2-83-0': 'The largest uncertainty in the estimation of the presence of H- or He-emission features in the observed spectra is the calculation of the underlying continuum.', '1512.07107-2-83-1': 'We have used a second-degree Savitzky-Golay smoothing polynomial to fit the underlying continuum, as was used in previous late-time narrow H[MATH] searches.', '1512.07107-2-83-2': 'This gives a good fit to the underlying continua in our sample, even in the case of broad underlying emission features.', '1512.07107-2-83-3': 'For SN 2013ct, we expanded our analysis of the continuum fitting to include also broad Gaussian fits to the underlying spectral feature, and again found a significant detection.', '1512.07107-2-83-4': 'However, given the tentative nature of the detection of H[MATH] emission in SN 2013ct, we could not completely exclude the possibility that the continuum fits results in a residual consistent with the detected H[MATH] emission.', '1512.07107-2-83-5': 'To determine how likely this is, we performed tests, detailed in Section [REF], looking for similar strength features at different wavelengths around H[MATH] and also on top of a different broad emission feature at [MATH]5800 .', '1512.07107-2-83-6': 'In neither case, was a similar feature found.', '1512.07107-2-84-0': 'With regard to the spectral synthesis models, the addition of the H-rich material potentially swept-up from a non-degenerate companion star in the models of [CITATION] and [CITATION] is somewhat ad hoc - it is made by adding varying amounts of solar metallicity material with velocities [MATH] in a W7 density model, artificially increasing the density in the innermost region.', '1512.07107-2-84-1': 'It is uncertain if this is consistent with the density structure obtained from the three-dimensional modelling of the impact of the SN ejecta on the companion star.', '1512.07107-2-85-0': 'In particular, a major source of uncertainty in the spectral synthesis modelling is whether the low-velocity H-rich material is sufficiently powered by radioactive heating to produce H[MATH] emission.', '1512.07107-2-85-1': 'This depends on the location of the H-rich material relative to the radioactive material; if they are not co-located then H-rich companion star material may be present in the ejecta but not observable.', '1512.07107-2-85-2': '[CITATION] performed the spectral line strength calculations at an epoch of +380 d.', '1512.07107-2-85-3': 'In the range of 150-300 d, the SN envelope is expected to become transparent to gamma-rays and enter a positron-dominated phase .', '1512.07107-2-85-4': 'However, this is not expected to be the case for the central high-density regions, where the H is located.', '1512.07107-2-85-5': 'In these high-density regions, the optical depth to gamma-rays is likely to be high enough to power the H lines , but more detailed modelling need to be carried out to confirm this.', '1512.07107-2-85-6': 'Indeed, the tentative detection of low-velocity H emission in SN 2013ct, corresponding to [MATH]0.007 M[MATH] of H-rich material, suggests that [MATH]0.1-0.6 M[MATH] of stripped/ablated material must be present but not observable.', '1512.07107-2-86-0': 'The one-dimensional models of [CITATION] and [CITATION] also assume spherical symmetry.', '1512.07107-2-86-1': 'However, the impact simulations show that the stripping and ablation of material from the companion star is not symmetric; the material is predominantly confined to the downstream region behind the companion star .', '1512.07107-2-86-2': 'Since the ejecta are assumed to be optically thin at the late phases studied here ([MATH]200 d), the detection of the swept-up material is not viewing angle dependent.', '1512.07107-2-86-3': 'However, the predicted asymmetry of the unbound material could affect the shape and wavelength of the observed line profiles.', '1512.07107-2-86-4': 'As discussed in [CITATION], the model used by [CITATION] and [CITATION] also included only a limited number of elements, ionization states and atomic levels.', '1512.07107-2-86-5': 'No macroscopic mixing of the companion material was included in the spectral modelling either, which could affect the predicted shape and flux of the emission lines but is not expected to be a dominant source of uncertainty.', '1512.07107-2-87-0': 'Therefore, given the discussed uncertainties and limitations, future modelling, using the three-dimensional simulated ejecta structure of [CITATION] and [CITATION] as input to a multi-dimensional radiative transfer calculation for computing expected spectral fluxes, is of vital importance to confirm the model predictions, and hence observational mass limits.', '1512.07107-2-88-0': '# Conclusions', '1512.07107-2-89-0': 'We have presented a search for the presence of H- and He-rich material stripped or ablated from a non-degenerate companion star in new late-time spectra of 11 SNe Ia, obtained at the VLT+XShooter and the VLT+FORS2.', '1512.07107-2-89-1': 'The observed fluxes (or limits) at the position of H[MATH] have been converted to masses using the spectral synthesis modelling described in [CITATION] and [CITATION].', '1512.07107-2-89-2': 'Our main results are as follows.', '1512.07107-2-90-0': 'We find evidence at the 3.1[MATH] level of H[MATH] emission with a best-fitting FWHM of [MATH] for one SN Ia in our sample, SN 2013ct.', '1512.07107-2-90-1': 'This corresponds to 0.007[MATH]0.001 M[MATH] of H-rich material stripped/ablated from a non-degenerate companion star.', '1512.07107-2-90-2': 'This mass is much lower than expected for MS+WD or RG+WD progenitor systems, suggesting at least 0.1 M[MATH] of H-rich material is present in SN 2013ct but not observed as narrow H[MATH] emission.', '1512.07107-2-90-3': 'We find no evidence of H emission (H[MATH], Pa[MATH]) in the late-time spectra of 10 other SNe Ia, bringing the total sample with no H emission detected to 17 SNe Ia.', '1512.07107-2-90-4': 'Upper limits on the stripped/ablated mass of solar abundance material of 0.001-0.058 M[MATH] are placed for these SNe Ia.', '1512.07107-2-90-5': 'These upper mass limits of H-rich solar abundance material are inconsistent with MS or RG companion stars transferring mass via RLOF or wind-driven accretion (under the assumptions of current modelling).', '1512.07107-2-90-6': 'No signatures of He-rich material in the form of Hei, [Oii] or [Caii] emission lines are identified.', '1512.07107-2-90-7': 'However, spectral modelling of the expected flux of lines from He-rich material is not available.', '1512.07107-2-91-0': 'While future observational studies will increase the sample size of SNe Ia with the necessary late-time observations, and perhaps identify narrow H[MATH] emission in more objects, major future improvements are also likely to come from the next generation of spectral-synthesis modelling, allowing us to confirm (or adjust) these mass limits and determine if the SD scenario for producing the majority of SNe Ia is really in jeopardy.'} | [['1512.07107-1-26-2', '1512.07107-2-26-2'], ['1512.07107-1-26-3', '1512.07107-2-26-3'], ['1512.07107-1-69-0', '1512.07107-2-69-0'], ['1512.07107-1-69-1', '1512.07107-2-69-1'], ['1512.07107-1-8-0', 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'1512.07107-2-86-5'], ['1512.07107-1-90-1', '1512.07107-2-90-1'], ['1512.07107-1-90-2', '1512.07107-2-90-2'], ['1512.07107-1-90-3', '1512.07107-2-90-3'], ['1512.07107-1-90-5', '1512.07107-2-90-5'], ['1512.07107-1-90-6', '1512.07107-2-90-6'], ['1512.07107-1-90-7', '1512.07107-2-90-7'], ['1512.07107-1-46-0', '1512.07107-2-46-0'], ['1512.07107-1-46-2', '1512.07107-2-46-2'], ['1512.07107-1-91-0', '1512.07107-2-91-0'], ['1512.07107-1-59-0', '1512.07107-2-59-0'], ['1512.07107-1-59-1', '1512.07107-2-59-1'], ['1512.07107-1-59-2', '1512.07107-2-59-2']] | [['1512.07107-1-26-0', '1512.07107-2-26-0'], ['1512.07107-1-26-1', '1512.07107-2-26-1'], ['1512.07107-1-69-2', '1512.07107-2-69-2'], ['1512.07107-1-49-0', '1512.07107-2-49-0'], ['1512.07107-1-49-1', '1512.07107-2-49-1'], ['1512.07107-1-72-1', '1512.07107-2-72-1'], ['1512.07107-1-10-1', '1512.07107-2-10-1'], ['1512.07107-1-10-2', '1512.07107-2-10-2'], ['1512.07107-1-20-2', '1512.07107-2-20-2'], ['1512.07107-1-21-1', '1512.07107-2-21-1'], ['1512.07107-1-55-0', '1512.07107-2-55-0'], ['1512.07107-1-11-0', '1512.07107-2-11-0'], ['1512.07107-1-80-1', '1512.07107-2-80-1'], ['1512.07107-1-85-3', '1512.07107-2-85-3'], ['1512.07107-1-85-5', '1512.07107-2-85-5'], ['1512.07107-1-65-2', '1512.07107-2-65-2'], ['1512.07107-1-65-4', '1512.07107-2-65-4'], ['1512.07107-1-48-1', '1512.07107-2-48-1'], ['1512.07107-1-0-4', '1512.07107-2-0-4'], ['1512.07107-1-0-8', '1512.07107-2-0-8'], ['1512.07107-1-0-9', '1512.07107-2-0-9'], ['1512.07107-1-39-0', '1512.07107-2-39-0'], ['1512.07107-1-61-2', '1512.07107-2-61-2'], ['1512.07107-1-53-1', '1512.07107-2-53-1'], ['1512.07107-1-16-0', '1512.07107-2-16-0'], ['1512.07107-1-74-0', '1512.07107-2-74-0'], ['1512.07107-1-74-1', '1512.07107-2-74-1'], ['1512.07107-1-27-1', '1512.07107-2-27-1'], ['1512.07107-1-70-1', '1512.07107-2-70-1'], ['1512.07107-1-84-0', '1512.07107-2-84-0'], ['1512.07107-1-22-3', '1512.07107-2-22-3'], ['1512.07107-1-22-4', '1512.07107-2-22-4'], ['1512.07107-1-22-5', '1512.07107-2-22-5'], ['1512.07107-1-44-0', '1512.07107-2-44-0'], ['1512.07107-1-50-1', '1512.07107-2-50-1'], ['1512.07107-1-50-2', '1512.07107-2-50-2'], ['1512.07107-1-86-4', '1512.07107-2-86-4'], ['1512.07107-1-90-0', '1512.07107-2-90-0'], ['1512.07107-1-90-4', '1512.07107-2-90-4'], ['1512.07107-1-46-1', '1512.07107-2-46-1']] | [] | [] | [] | ['1512.07107-1-89-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1512.07107 | null | null | null | null | null |
1703.09021 | {'1703.09021-1-0-0': 'Following our earlier finding based on RHIC data on the dominant jet production from nucleus corona region, we reconsider this effect in nucleus-nucleus collisions at LHC energy.', '1703.09021-1-0-1': 'Our hypothesis was based on experimental data, which raised the idea of a finite formation time for the produced medium.', '1703.09021-1-0-2': 'At RHIC energy and in low density corona region this time reaches about 2 fm/c.', '1703.09021-1-0-3': 'All observed high [MATH] particles are produced in the corona region and have a chance to escape during this 2 fm/c.', '1703.09021-1-0-4': 'After that, the formed high density matter absorbs all jets.', '1703.09021-1-0-5': 'Following this hypothesis, the nuclear modification factor [MATH] should be independent on particle momentum and be flat versus [MATH].', '1703.09021-1-0-6': 'At the same time, we describe the finite azimuthal anisotropy of high [MATH] particles, [MATH].', '1703.09021-1-0-7': 'A separate prediction held that, at LHC energy, the formation time in the corona region should be two times smaller, about 1 fm/c. New data at LHC show that [MATH] is not flat and is rising with [MATH].', '1703.09021-1-0-8': 'We add to our original hypothesis an assumption that a fast parton traversing the produced medium loses the fixed portion of its energy.', '1703.09021-1-0-9': 'A shift of about 7 GeV from the original power law [MATH] production cross section in [MATH] explains well all the observed [MATH] dependences.', '1703.09021-1-0-10': 'The shift of about 7 GeV is also valid at RHIC energy, where the cross section follows a power law with about [MATH] and this shift explains a very slow rise of [MATH] seen for neutral pions with [MATH] above 15 GeV/c.', '1703.09021-1-0-11': 'We also show that the observed at LHC dependence of [MATH] at high [MATH] and our previous predictions agree.', '1703.09021-1-1-0': 'Over the last 17 years of relativistic nucleus-nucleus collisions at RHIC and LHC, a set of observables was found which confirms the formation of high energy and high density matter.', '1703.09021-1-1-1': 'Among these features are the strong jet suppression manifested in particle suppression at high transverse momentum, [MATH], and large particle anisotropy.', '1703.09021-1-1-2': 'There is also a long list of models and theoretical assumptions to explain these effects.', '1703.09021-1-1-3': 'In our view, when one talks about jet suppression, a significant effect of particle production from the nucleus corona region is often ignored or underestimated.', '1703.09021-1-1-4': 'In a previous publication based purely on experimental data at RHIC, a simple model was proposed [CITATION] to explain the angular dependence in the reaction plane of the nuclear modification factor [MATH].', '1703.09021-1-1-5': 'The model nicely described the centrality and azimuthal dependence (or factor [MATH] for high [MATH]) of [MATH] at RHIC energy.', '1703.09021-1-1-6': 'In the model, there is one free parameter of about 2.3 fm/c which was interpreted as plasma formation time at the low density corona region.', '1703.09021-1-1-7': 'The physical meaning of this parameter is that fast partons have roughly this time to escape from the produced medium and, theafter, they are absorbed by the absolutely opaque central region.', '1703.09021-1-1-8': 'This value of [MATH]=2.3 fm/c is not "crazy large" because the number of nuclear collisions, [MATH], near corona region is rather small, but it should be less than 0.8 fm/c in the core region of the produced matter [CITATION].', '1703.09021-1-1-9': 'Time, necessary to form the strongly interacting colored matter, should be proportional to the mean distance between the interaction or collision points with a color exchange.', '1703.09021-1-1-10': 'This distance, itself, should be inversely proportional to the square root of the density of such interactions.', '1703.09021-1-1-11': 'The picture in some sense is similar to the percolation scenario.', '1703.09021-1-1-12': 'If the density of [MATH] in [MATH] plane of colliding nuclei near the corona region is [MATH] then the formation time [MATH] versus density [MATH] will be: [EQUATION] where [MATH] is the distance from the center.', '1703.09021-1-1-13': 'In Fig. [REF] we plot the evolution of the formation time versus the distance from the center of the region for the colliding Au+Au nuclei in the 0-5% centrality bin.', '1703.09021-1-1-14': 'For the [MATH] density distribution of colliding nucleons we used density profiles generated for our first publication [CITATION].', '1703.09021-1-1-15': 'To demonstrate how formation time works we show in Fig. [REF] two extreme cases: when the fast parton is produced in the center of the colliding region, arrow 1, and near the surface at a depth of about 2 fm from the Woods-Saxon radius, arrow 2.', '1703.09021-1-1-16': 'The first parton moves with the speed of light along its world line 1 only for about 0.8 fm and then is stopped by the produced matter.', '1703.09021-1-1-17': 'The second parton will survive.', '1703.09021-1-1-18': 'The proposed model in [CITATION] works well at RHIC energy.', '1703.09021-1-2-0': 'In "The last call for prediction" published prior to the start of LHC we also proposed some features which should be observed at LHC if a similar picture with formation time is valid [CITATION] (see pages 119-121 and figures 99-100 in the e-print version).', '1703.09021-1-2-1': 'As we already mentioned, the formation time should be proportional to the mean distance between interactions with the color exchange.', '1703.09021-1-2-2': 'It means also that only part of the nucleon-nucleon inelastic cross section will contribute to the process: single- and double diffractive and soft process with meson exchange will not be relevant here.', '1703.09021-1-2-3': 'If a relativistic rise of the total nucleon-nucleon, NN, cross section comes purely from the contribution from the colored parton hard scatterings, we can estimate the relative value of hard scatterings to the total nucleon-nucleon cross section.', '1703.09021-1-2-4': 'At [MATH]=20 GeV the NN total cross section is at its minimum of 30 millibars - there is no hard scattering, but mostly soft nucleon-nucleon interactions with meson exchange.', '1703.09021-1-2-5': 'At 200 GeV the cross section rises by 13 mb, at 5500 GeV - by 49 mb.', '1703.09021-1-2-6': 'The formation time of the colored matter should be proportional to one over the square root of these numbers because the density of [MATH] is proportional to the cross section .', '1703.09021-1-2-7': 'If we get [MATH]=2.3 fm/c at 200 GeV, then from the rise of the NN total cross section, we estimate T=1.2 fm/c near the corona region at 5500 GeV of LHC energy.', '1703.09021-1-2-8': 'In the center of the collision zone it will be about three times shorter.', '1703.09021-1-2-9': 'Calculations show that such a value of [MATH] should give a constant [MATH]=0.1 for high [MATH] particles in the most central collisions.', '1703.09021-1-2-10': 'Predictions made in [CITATION] assume that the core of the produced matter is opaque, but experimental data for Pb+Pb collisions obtained by CMS, ATLAS and ALICE show that [MATH] is continuously rising at high [MATH].', '1703.09021-1-2-11': 'It means that the core of the collision zone becomes more transparent for fast particles.', '1703.09021-1-2-12': 'It is natural to assume that the parton loses some portion of its energy.', '1703.09021-1-2-13': 'We found that a constant energy loss of 7 GeV describes well the data for [MATH] versus [MATH].', '1703.09021-1-2-14': 'Particle, namely pion, production cross section at LHC energy follows a simple power law [MATH] [CITATION] at high [MATH].', '1703.09021-1-2-15': 'Thus, the energy drop by 7 GeV becomes less significant with increasing parton [MATH].', '1703.09021-1-2-16': 'In Fig. [REF] we present results for the [MATH] versus [MATH] from CMS data [CITATION] and our calculations for most central collisions.', '1703.09021-1-2-17': 'There are two contributions: a constant value of 0.1 for a particle from the corona region, as was predicted in ref. [CITATION], and a new momentum dependent component when matter becomes more transparent for fast parton, which loses 7 GeV.', '1703.09021-1-2-18': 'This provides excellent agreement with the data.', '1703.09021-1-2-19': 'In Fig. [REF] we show a similar plot for mid-central collisions.', '1703.09021-1-2-20': 'In this case the contribution to [MATH] from the corona region reaches 0.35 [CITATION], but the penetrating parton contribution is about the same.', '1703.09021-1-3-0': 'Out of curiosity we checked how this 7 GeV energy loss works at RHIC and added this component to the previous calculation with the corona region and absolutely black core, Fig. [REF].', '1703.09021-1-3-1': 'The only difference here is that the production cross section at RHIC follows a more steep power law [MATH] [CITATION].', '1703.09021-1-3-2': 'Within the error bars our line follows the experimental points.', '1703.09021-1-3-3': 'Such a large energy loss (7 GeV) at RHIC explains why the assumption about the complete black core with some corona contribution worked so well - the loss is too big for produced particles at RHIC.', '1703.09021-1-4-0': 'Our model worked well at RHIC for the observed large azimuthal of high [MATH] particles or parameter [MATH].', '1703.09021-1-4-1': 'Nearly 10 years ago we did a prediction for [MATH] at LHC [CITATION].', '1703.09021-1-4-2': 'It seems that the prediction is valid.', '1703.09021-1-4-3': 'In Fig. [REF] we compare our estimations with CMS results at [MATH]=15 GeV/c [CITATION].', '1703.09021-1-4-4': 'The prediction of a large [MATH] even at LHC is confirmed, the sensitivity to the collision geometry persists up to high [MATH].', '1703.09021-1-4-5': 'There is a deviation at small [MATH] but this is due to the well known effect of distortion by the initial geometry fluctuations.', '1703.09021-1-4-6': 'We also can explain the observed drop of [MATH] with particle or jet momentum above 15 GeV/c.', '1703.09021-1-4-7': 'The corona effect for in- and out-of-plane particle production is diluted by penetrating partons with energy loss.', '1703.09021-1-4-8': 'For example, looking at Fig. [REF] and Fig. [REF], one can see that at [MATH]=40 GeV/c particles from the corona region count for about one half of the total yield at this momentum.', '1703.09021-1-4-9': 'Thus, [MATH] should drop to about a factor of 2.', '1703.09021-1-4-10': 'This what is qualitatively seen by the three experiments [CITATION].', '1703.09021-1-5-0': 'In conclusion, we demonstrate that the contribution from the corona region and the assumption of a finite formation time for the colored strongly interacting matter are the reasons for the observed centrality and momentum dependence of particle [MATH].', '1703.09021-1-5-1': 'At LHC energy, a fast parton escapes the interaction zone by losing about 7 GeV.', '1703.09021-1-5-2': 'This value does not depend on momentum, centrality, energy density, and, probably, on beam energy.', '1703.09021-1-5-3': 'The observed azimuthal angular asymmetry at a high transverse momentum is well described at RHIC and LHC energies.', '1703.09021-1-6-0': 'This work was partially supported by the RFBR grant numbers 14-22-03069-ofi-m and 14-02-00570-a.', '1703.09021-1-6-1': 'We would like to thank Keith Guzik for help with the text.'} | {'1703.09021-2-0-0': 'Following our earlier finding based on RHIC data on the dominant jet production from nucleus corona region, we reconsider this effect in nucleus-nucleus collisions at LHC energies.', '1703.09021-2-0-1': 'Our hypothesis was based on experimental data, which raised the idea of a finite formation time for the produced medium.', '1703.09021-2-0-2': 'At RHIC energy and in low density corona region this time reaches about 2 fm/[MATH].', '1703.09021-2-0-3': 'Following this hypothesis, the nuclear modification factor [MATH] at high [MATH] should be independent on particle momentum, and the azimuthal anisotropy of high [MATH] particles, [MATH], should be finite.', '1703.09021-2-0-4': 'A separate prediction held that, at LHC energy, the formation time in the corona region should be about 1 fm/[MATH].', '1703.09021-2-0-5': 'New data at LHC show that [MATH] is not flat and is rising with [MATH].', '1703.09021-2-0-6': 'We add to our original hypothesis an assumption that a fast parton traversing the produced medium loses the fixed portion of its energy.', '1703.09021-2-0-7': 'A shift of about 7 GeV from the original power law [MATH] production cross section in [MATH] explains well all the observed [MATH] dependencies.', '1703.09021-2-0-8': 'The shift of about 7 GeV is also valid at RHIC energy.', '1703.09021-2-0-9': 'We also show that the observed at LHC dependence of [MATH] at high [MATH] and our previous predictions agree.', '1703.09021-2-1-0': 'Over the last 17 years of relativistic nucleus-nucleus collisions at RHIC and LHC, a set of observables was found which confirms the formation of high energy and high density matter.', '1703.09021-2-1-1': 'Among these features are the strong jet suppression manifested in particle suppression at high transverse momentum, [MATH], and large particle anisotropy.', '1703.09021-2-1-2': 'There is also a long list of models and theoretical assumptions to explain these effects.', '1703.09021-2-1-3': 'In our view, when one talks about jet suppression, a significant effect of particle production from the nucleus corona region is often ignored or underestimated.', '1703.09021-2-1-4': 'In a previous publication based purely on experimental data at RHIC, a simple model was proposed [CITATION] to explain the angular dependence in the reaction plane of the nuclear modification factor [MATH].', '1703.09021-2-1-5': 'The model nicely described the centrality and azimuthal dependence (or factor [MATH] for high [MATH]) of [MATH] at RHIC energy.', '1703.09021-2-1-6': 'In the model, there is one free parameter of about 2.3 fm/[MATH] which was interpreted as plasma formation time at the low density corona region.', '1703.09021-2-1-7': 'The physical meaning of this parameter is that fast partons have roughly this time to escape from the produced medium and, theafter, they are absorbed by the absolutely opaque central region.', '1703.09021-2-1-8': 'This value of [MATH]=2.3 fm/[MATH] is not "crazy large" because the number of nuclear collisions, [MATH], near corona region is rather small, but it should be less than 0.8 fm/c in the core region of the produced matter [CITATION].', '1703.09021-2-1-9': 'Time, necessary to form the strongly interacting colored matter, should be proportional to the mean distance between the interaction or collision points with a color exchange.', '1703.09021-2-1-10': 'This distance, itself, should be inversely proportional to the square root of the density of such interactions.', '1703.09021-2-1-11': 'The picture in some sense is similar to the percolation scenario [CITATION].', '1703.09021-2-1-12': 'If the density of [MATH] in [MATH] plane of colliding nuclei near the corona region is [MATH] then the formation time [MATH] versus density [MATH] will be: [EQUATION] where [MATH] is the distance from the center.', '1703.09021-2-1-13': 'In Fig. [REF] we plot the evolution of the formation time versus the distance from the center of the region for the colliding Au+Au nuclei in the 0-5% centrality bin.', '1703.09021-2-1-14': 'For the [MATH] density distribution of colliding nucleons we used density profiles generated for our first publication [CITATION].', '1703.09021-2-1-15': 'To demonstrate how formation time works we show in Fig. [REF] two extreme cases: when the fast parton is produced in the center of the colliding region, arrow 1, and near the surface at a depth of about 2 fm from the Woods-Saxon radius, arrow 2.', '1703.09021-2-1-16': 'The first parton moves with the speed of light along its world line 1 only for about 0.8 fm and then is stopped by the produced matter.', '1703.09021-2-1-17': 'The second parton will survive.', '1703.09021-2-1-18': 'The proposed model in [CITATION] works well at RHIC energy.', '1703.09021-2-2-0': 'In "The last call for prediction" published prior to the start of LHC we also proposed some features which should be observed at LHC if a similar picture with formation time is valid [CITATION] (see pages 119-121 and figures 99-100 in the e-print version).', '1703.09021-2-2-1': 'As we already mentioned, the formation time should be proportional to the mean distance between interactions with the color exchange.', '1703.09021-2-2-2': 'It means also that only part of the nucleon-nucleon inelastic cross section will contribute to the process: single- and double diffractive and soft process with meson exchange will not be relevant here.', '1703.09021-2-2-3': 'If a relativistic rise of the total nucleon-nucleon, NN, cross section comes purely from the contribution from the colored parton hard scatterings, we can estimate the relative value of hard scatterings to the total nucleon-nucleon cross section.', '1703.09021-2-2-4': 'At [MATH]=20 GeV the NN total cross section is at its minimum of 30 millibars - there is almost no hard scattering, but mostly soft nucleon-nucleon interactions with meson exchange.', '1703.09021-2-2-5': 'At 200 GeV the cross section rises by 13 mb, at 5500 GeV - by 49 mb.', '1703.09021-2-2-6': 'The formation time of the colored matter should be proportional to one over the square root of these numbers because the density of [MATH] is proportional to the cross section.', '1703.09021-2-2-7': 'If we get [MATH]=2.3 fm/[MATH] at 200 GeV, then from the rise of the NN total cross section, we estimate T=1.2 fm/[MATH] near the corona region at around 5 TeV of LHC energy.', '1703.09021-2-2-8': 'In the center of the collision zone it will be about three times shorter.', '1703.09021-2-2-9': 'Calculations show that such a value of [MATH] should give a constant [MATH]=0.1 for high [MATH] particles in the most central collisions.', '1703.09021-2-2-10': 'We have to emphasize that the value of T around 1.2 fm/[MATH] is valid within uncertainty of 5% in the LHC energy range of 2.7-5 TeV.', '1703.09021-2-2-11': 'It comes from a little change of [MATH] total cross section between 85 mb and 90 mb if one interpolates the existing [MATH] data [CITATION], thus, and relative change of hard scattering contribution is on the level of 5 mb.', '1703.09021-2-3-0': 'Predictions made in [CITATION] assume that the core of the produced matter is opaque, but experimental data for Pb+Pb collisions obtained by ALICE, CMS and ATLAS [CITATION] show that [MATH] is continuously rising at high [MATH].', '1703.09021-2-3-1': 'It means that the core of the collision zone becomes more transparent for fast particles.', '1703.09021-2-3-2': 'It is natural to assume that the parton loses some portion of its energy.', '1703.09021-2-3-3': 'We found that a constant energy loss of 7 GeV describes well the data for [MATH] versus [MATH].', '1703.09021-2-3-4': 'Particle, namely pion, production cross section at LHC energy follows a simple power law [MATH] [CITATION] at high [MATH].', '1703.09021-2-3-5': 'Thus, the energy drop by 7 GeV becomes less significant with increasing parton [MATH].', '1703.09021-2-3-6': 'In Fig. [REF] we present results for the [MATH] versus [MATH] from CMS data [CITATION] and our calculations for most central collisions.', '1703.09021-2-3-7': 'There are two contributions: a constant value of 0.1 for a particle from the corona region, as was predicted in ref. [CITATION], and a new momentum dependent component when matter becomes more transparent for fast parton, which loses 7 GeV.', '1703.09021-2-3-8': 'This provides excellent agreement with the data.', '1703.09021-2-3-9': 'In Fig. [REF] we show a similar plot for mid-central collisions.', '1703.09021-2-3-10': 'In this case the contribution to [MATH] from the corona region reaches 0.35 [CITATION], but the penetrating parton contribution is about the same.', '1703.09021-2-4-0': 'Out of curiosity we checked how this 7 GeV energy loss works at RHIC and added this component to the previous calculation with the corona region and absolutely black core, Fig. [REF].', '1703.09021-2-4-1': 'The only difference here is that the production cross section at RHIC follows a more steep power law [MATH] [CITATION].', '1703.09021-2-4-2': 'Within the error bars our line follows the experimental points.', '1703.09021-2-4-3': 'Such a large energy loss (7 GeV) at RHIC explains why the assumption about the complete black core with some corona contribution worked so well - the loss is too big for produced particles at RHIC.', '1703.09021-2-5-0': 'Our model worked well at RHIC for the observed large azimuthal of high [MATH] particles or parameter [MATH].', '1703.09021-2-5-1': 'Nearly 10 years ago we did a prediction for [MATH] at LHC [CITATION].', '1703.09021-2-5-2': 'It seems that the prediction is valid.', '1703.09021-2-5-3': 'In Fig. [REF] we compare our estimations with CMS results at [MATH]=15 GeV/[MATH] [CITATION].', '1703.09021-2-5-4': 'The prediction of a large [MATH] even at LHC is confirmed, the sensitivity to the collision geometry persists up to high [MATH].', '1703.09021-2-5-5': 'There is a deviation at small [MATH] but this is due to the well known effect of distortion by the initial geometry fluctuations (see for example, PHOBOS paper [CITATION]).', '1703.09021-2-5-6': 'We also can explain the observed drop of [MATH] with particle or jet momentum above 15 GeV/[MATH].', '1703.09021-2-5-7': 'The corona effect for in- and out-of-plane particle production is diluted by penetrating partons with energy loss.', '1703.09021-2-5-8': 'For example, looking at Fig. [REF] and Fig. [REF], one can see that at [MATH]=40 GeV/[MATH] particles from the corona region count for about one half of the total yield at this momentum.', '1703.09021-2-5-9': 'Thus, [MATH] should drop to about a factor of 2.', '1703.09021-2-5-10': 'This what is qualitatively seen by the three experiments [CITATION].', '1703.09021-2-6-0': 'In conclusion, we demonstrate that in PbPb collisions at LHC the contribution from the corona region and the assumption of a finite formation time for the colored strongly interacting matter are the reasons for the observed centrality and momentum dependence of particle [MATH].', '1703.09021-2-6-1': 'At LHC energies, a fast parton escapes the interaction zone by losing about 7 GeV.', '1703.09021-2-6-2': 'Within our model this value does not depend on momentum, centrality, energy density, and, probably, on beam energy.', '1703.09021-2-6-3': 'The observed azimuthal angular asymmetry at a high transverse momentum is well described at RHIC and LHC energies.', '1703.09021-2-7-0': 'This work was partially supported by the RFBR grant numbers 14-22-03069-ofi-m and 14-02-00570-a.', '1703.09021-2-7-1': 'We would like to thank Keith Guzik for help with the text.'} | [['1703.09021-1-1-0', '1703.09021-2-1-0'], ['1703.09021-1-1-1', '1703.09021-2-1-1'], ['1703.09021-1-1-2', '1703.09021-2-1-2'], ['1703.09021-1-1-3', '1703.09021-2-1-3'], ['1703.09021-1-1-4', '1703.09021-2-1-4'], ['1703.09021-1-1-5', '1703.09021-2-1-5'], ['1703.09021-1-1-7', '1703.09021-2-1-7'], ['1703.09021-1-1-9', '1703.09021-2-1-9'], ['1703.09021-1-1-10', '1703.09021-2-1-10'], ['1703.09021-1-1-12', '1703.09021-2-1-12'], ['1703.09021-1-1-13', '1703.09021-2-1-13'], ['1703.09021-1-1-14', '1703.09021-2-1-14'], ['1703.09021-1-1-15', '1703.09021-2-1-15'], ['1703.09021-1-1-16', 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'1703.09021-2-2-0'], ['1703.09021-1-2-1', '1703.09021-2-2-1'], ['1703.09021-1-2-2', '1703.09021-2-2-2'], ['1703.09021-1-2-3', '1703.09021-2-2-3'], ['1703.09021-1-2-5', '1703.09021-2-2-5'], ['1703.09021-1-2-8', '1703.09021-2-2-8'], ['1703.09021-1-2-9', '1703.09021-2-2-9'], ['1703.09021-1-2-11', '1703.09021-2-3-1'], ['1703.09021-1-2-12', '1703.09021-2-3-2'], ['1703.09021-1-2-13', '1703.09021-2-3-3'], ['1703.09021-1-2-14', '1703.09021-2-3-4'], ['1703.09021-1-2-15', '1703.09021-2-3-5'], ['1703.09021-1-2-16', '1703.09021-2-3-6'], ['1703.09021-1-2-17', '1703.09021-2-3-7'], ['1703.09021-1-2-18', '1703.09021-2-3-8'], ['1703.09021-1-2-19', '1703.09021-2-3-9'], ['1703.09021-1-2-20', '1703.09021-2-3-10']] | [['1703.09021-1-1-6', '1703.09021-2-1-6'], ['1703.09021-1-1-8', '1703.09021-2-1-8'], ['1703.09021-1-1-11', '1703.09021-2-1-11'], ['1703.09021-1-0-0', '1703.09021-2-0-0'], ['1703.09021-1-0-2', '1703.09021-2-0-2'], ['1703.09021-1-0-9', '1703.09021-2-0-7'], ['1703.09021-1-5-0', '1703.09021-2-6-0'], ['1703.09021-1-5-1', '1703.09021-2-6-1'], ['1703.09021-1-5-2', '1703.09021-2-6-2'], ['1703.09021-1-4-3', '1703.09021-2-5-3'], ['1703.09021-1-4-5', '1703.09021-2-5-5'], ['1703.09021-1-4-6', '1703.09021-2-5-6'], ['1703.09021-1-4-8', '1703.09021-2-5-8'], ['1703.09021-1-2-4', '1703.09021-2-2-4'], ['1703.09021-1-2-6', '1703.09021-2-2-6'], ['1703.09021-1-2-7', '1703.09021-2-2-7'], ['1703.09021-1-2-10', '1703.09021-2-3-0']] | [] | [['1703.09021-1-0-7', '1703.09021-2-0-5'], ['1703.09021-1-0-7', '1703.09021-2-0-4'], ['1703.09021-1-0-5', '1703.09021-2-0-3'], ['1703.09021-1-0-6', '1703.09021-2-0-3'], ['1703.09021-1-0-10', '1703.09021-2-0-8']] | [] | ['1703.09021-1-6-0', '1703.09021-2-7-0', '1703.09021-3-7-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1703.09021 | {'1703.09021-3-0-0': 'Following our earlier finding based on RHIC data on the dominant jet production from nucleus corona region, we reconsider this effect in nucleus-nucleus collisions at LHC energies.', '1703.09021-3-0-1': 'Our hypothesis was based on experimental data, which raised the idea of a finite formation time for the produced medium.', '1703.09021-3-0-2': 'At RHIC energy and in low density corona region this time reaches about 2 fm/[MATH].', '1703.09021-3-0-3': 'Following this hypothesis, the nuclear modification factor [MATH] at high [MATH] should be independent on particle momentum, and the azimuthal anisotropy of high [MATH] particles, [MATH], should be finite.', '1703.09021-3-0-4': 'A separate prediction held that, at LHC energy, the formation time in the corona region should be about 1 fm/[MATH].', '1703.09021-3-0-5': 'New data at LHC show that [MATH] is not flat and is rising with [MATH].', '1703.09021-3-0-6': 'We add to our original hypothesis an assumption that a fast parton traversing the produced medium loses the fixed portion of its energy.', '1703.09021-3-0-7': 'A shift of about 7 GeV from the original power law [MATH] production cross section in [MATH] explains well all the observed [MATH] dependencies.', '1703.09021-3-0-8': 'The shift of about 7 GeV is also valid at RHIC energy.', '1703.09021-3-0-9': 'We also show that the observed at LHC dependence of [MATH] at high [MATH] and our previous predictions agree.', '1703.09021-3-1-0': 'Over the last 17 years of relativistic nucleus-nucleus collisions at RHIC and LHC, a set of observables was found which confirms the formation of high energy and high density matter.', '1703.09021-3-1-1': 'Among these features are the strong jet suppression manifested in particle suppression at high transverse momentum, [MATH], and large particle anisotropy.', '1703.09021-3-1-2': 'There is also a long list of models and theoretical assumptions to explain these effects.', '1703.09021-3-1-3': 'In our view, when one talks about jet suppression, a significant effect of particle production from the nucleus corona region is often ignored or underestimated.', '1703.09021-3-1-4': 'In a previous publication based purely on experimental data at RHIC, a simple model was proposed [CITATION] to explain the angular dependence in the reaction plane of the nuclear modification factor [MATH].', '1703.09021-3-1-5': 'The model nicely described the centrality and azimuthal dependence (or factor [MATH] for high [MATH]) of [MATH] at RHIC energy.', '1703.09021-3-1-6': 'In the model, there is one free parameter of about 2.3 fm/[MATH] which was interpreted as plasma formation time at the low density corona region.', '1703.09021-3-1-7': 'The physical meaning of this parameter is that fast partons have roughly this time to escape from the produced medium and, theafter, they are absorbed by the absolutely opaque central region.', '1703.09021-3-1-8': 'This value of [MATH]=2.3 fm/[MATH] is not "crazy large" because the number of nuclear collisions, [MATH], near corona region is rather small, but it should be less than 0.8 fm/c in the core region of the produced matter [CITATION].', '1703.09021-3-1-9': 'Time, necessary to form the strongly interacting colored matter, should be proportional to the mean distance between the interaction or collision points with a color exchange.', '1703.09021-3-1-10': 'This distance, itself, should be inversely proportional to the square root of the density of such interactions.', '1703.09021-3-1-11': 'The picture in some sense is similar to the percolation scenario [CITATION].', '1703.09021-3-1-12': 'If the density of [MATH] in [MATH] plane of colliding nuclei near the corona region is [MATH] then the formation time [MATH] versus density [MATH] will be: [EQUATION] where [MATH] is the distance from the center.', '1703.09021-3-1-13': 'In Fig. [REF] we plot the evolution of the formation time versus the distance from the center of the region for the colliding Au+Au nuclei in the 0-5% centrality bin.', '1703.09021-3-1-14': 'For the [MATH] density distribution of colliding nucleons we used density profiles generated for our first publication [CITATION].', '1703.09021-3-1-15': 'To demonstrate how formation time works we show in Fig. [REF] two extreme cases: when the fast parton is produced in the center of the colliding region, arrow 1, and near the surface at a depth of about 2 fm from the Woods-Saxon radius, arrow 2.', '1703.09021-3-1-16': 'The first parton moves with the speed of light along its world line 1 only for about 0.8 fm and then is stopped by the produced matter.', '1703.09021-3-1-17': 'The second parton will survive.', '1703.09021-3-1-18': 'The proposed model in [CITATION] works well at RHIC energy.', '1703.09021-3-2-0': 'In "The last call for prediction" published prior to the start of LHC we also proposed some features which should be observed at LHC if a similar picture with formation time is valid [CITATION] (see pages 119-121 and figures 99-100 in the e-print version).', '1703.09021-3-2-1': 'As we already mentioned, the formation time should be proportional to the mean distance between interactions with the color exchange.', '1703.09021-3-2-2': 'It means also that only part of the nucleon-nucleon inelastic cross section will contribute to the process: single- and double diffractive and soft process with meson exchange will not be relevant here.', '1703.09021-3-2-3': 'If a relativistic rise of the total nucleon-nucleon, NN, cross section comes purely from the contribution from the colored parton hard scatterings, we can estimate the relative value of hard scatterings to the total nucleon-nucleon cross section.', '1703.09021-3-2-4': 'At [MATH]=20 GeV the NN total cross section is at its minimum of 30 millibars - there is almost no hard scattering, but mostly soft nucleon-nucleon interactions with meson exchange.', '1703.09021-3-2-5': 'At 200 GeV the cross section rises by 13 mb, at 5500 GeV - by 49 mb.', '1703.09021-3-2-6': 'The formation time of the colored matter should be proportional to one over the square root of these numbers because the density of [MATH] is proportional to the cross section.', '1703.09021-3-2-7': 'If we get [MATH]=2.3 fm/[MATH] at 200 GeV, then from the rise of the NN total cross section, we estimate T=1.2 fm/[MATH] near the corona region at around 5 TeV of LHC energy.', '1703.09021-3-2-8': 'In the center of the collision zone it will be about three times shorter.', '1703.09021-3-2-9': 'Calculations show that such a value of [MATH] should give a constant [MATH]=0.1 for high [MATH] particles in the most central collisions.', '1703.09021-3-2-10': 'We have to emphasize that the value of T around 1.2 fm/[MATH] is valid within uncertainty of 5% in the LHC energy range of 2.7-5 TeV.', '1703.09021-3-2-11': 'It comes from a little change of [MATH] total cross section between 85 mb and 90 mb if one interpolates the existing [MATH] data [CITATION], thus, and relative change of hard scattering contribution is on the level of 5 mb.', '1703.09021-3-3-0': 'Predictions made in [CITATION] assume that the core of the produced matter is opaque, but experimental data for Pb+Pb collisions obtained by ALICE, CMS and ATLAS [CITATION] show that [MATH] is continuously rising at high [MATH].', '1703.09021-3-3-1': 'It means that the core of the collision zone becomes more transparent for fast particles.', '1703.09021-3-3-2': 'It is natural to assume that the parton loses some portion of its energy.', '1703.09021-3-3-3': 'We found that a constant energy loss of 7 GeV describes well the data for [MATH] versus [MATH].', '1703.09021-3-3-4': 'Particle, namely pion, production cross section at LHC energy follows a simple power law [MATH] [CITATION] at high [MATH].', '1703.09021-3-3-5': 'Thus, the energy drop by 7 GeV becomes less significant with increasing parton [MATH].', '1703.09021-3-3-6': 'In Fig. [REF] we present results for the [MATH] versus [MATH] from CMS data [CITATION] and our calculations for most central collisions.', '1703.09021-3-3-7': 'There are two contributions: a constant value of 0.1 for a particle from the corona region, as was predicted in ref. [CITATION], and a new momentum dependent component when matter becomes more transparent for fast parton, which loses 7 GeV.', '1703.09021-3-3-8': 'This provides excellent agreement with the data.', '1703.09021-3-3-9': 'In Fig. [REF] we show a similar plot for mid-central collisions.', '1703.09021-3-3-10': 'In this case the contribution to [MATH] from the corona region reaches 0.35 [CITATION], but the penetrating parton contribution is about the same.', '1703.09021-3-4-0': 'Out of curiosity we checked how this 7 GeV energy loss works at RHIC and added this component to the previous calculation with the corona region and absolutely black core, Fig. [REF].', '1703.09021-3-4-1': 'The only difference here is that the production cross section at RHIC follows a more steep power law [MATH] [CITATION].', '1703.09021-3-4-2': 'Within the error bars our line follows the experimental points.', '1703.09021-3-4-3': 'Such a large energy loss (7 GeV) at RHIC explains why the assumption about the complete black core with some corona contribution worked so well - the loss is too big for produced particles at RHIC.', '1703.09021-3-5-0': 'Our model worked well at RHIC for the observed large azimuthal of high [MATH] particles or parameter [MATH].', '1703.09021-3-5-1': 'Nearly 10 years ago we did a prediction for [MATH] at LHC [CITATION].', '1703.09021-3-5-2': 'It seems that the prediction is valid.', '1703.09021-3-5-3': 'In Fig. [REF] we compare our estimations with CMS results at [MATH]=15 GeV/[MATH] [CITATION].', '1703.09021-3-5-4': 'The prediction of a large [MATH] even at LHC is confirmed, the sensitivity to the collision geometry persists up to high [MATH].', '1703.09021-3-5-5': 'There is a deviation at small [MATH] but this is due to the well known effect of distortion by the initial geometry fluctuations (see for example, PHOBOS paper [CITATION]).', '1703.09021-3-5-6': 'We also can explain the observed drop of [MATH] with particle or jet momentum above 15 GeV/[MATH].', '1703.09021-3-5-7': 'The corona effect for in- and out-of-plane particle production is diluted by penetrating partons with energy loss.', '1703.09021-3-5-8': 'For example, looking at Fig. [REF] and Fig. [REF], one can see that at [MATH]=40 GeV/[MATH] particles from the corona region count for about one half of the total yield at this momentum.', '1703.09021-3-5-9': 'Thus, [MATH] should drop to about a factor of 2.', '1703.09021-3-5-10': 'This what is qualitatively seen by the three experiments [CITATION].', '1703.09021-3-6-0': 'In conclusion, we demonstrate that in PbPb collisions at LHC the contribution from the corona region and the assumption of a finite formation time for the colored strongly interacting matter are the reasons for the observed centrality and momentum dependence of particle [MATH].', '1703.09021-3-6-1': 'At LHC energies, a fast parton escapes the interaction zone by losing about 7 GeV.', '1703.09021-3-6-2': 'Within our model this value does not depend on momentum, centrality, energy density, and, probably, on beam energy.', '1703.09021-3-6-3': 'The observed azimuthal angular asymmetry at a high transverse momentum is well described at RHIC and LHC energies.', '1703.09021-3-7-0': 'This work was partially supported by the RFBR grant numbers 14-22-03069-ofi-m and 14-02-00570-a.', '1703.09021-3-7-1': 'We would like to thank Keith Guzik for help with the text.'} | null | null | null | null |
1208.5012 | {'1208.5012-1-0-0': 'The spectrum sharing has recently passed into a mainstream Cognitive Radio (CR) strategy.', '1208.5012-1-0-1': 'We investigate the core issue in this strategy: interference mitigation at Primary Receiver (PR).', '1208.5012-1-0-2': 'We propose a linear precoder design which aims at alleviating the interference caused by Secondary User (SU) from the source for Orthogonal Space-Time Block Coding (OSTBC) based CR.', '1208.5012-1-0-3': 'We resort to Minimum Variance (MV) approach to contrive the precoding matrix at Secondary Transmitter (ST) in order to maximize the Signal to Noise Ratio (SNR) at Secondary Receiver (SR) on the premise that the orthogonality of OSTBC is kept, the interference introduced to Primary Link (PL) by Secondary Link (SL) is maintained under a tolerable level and the total transmitted power constraint at ST is satisfied.', '1208.5012-1-0-4': 'Moreover, the selection of polarization mode for SL is incorporated in the precoder design.', '1208.5012-1-0-5': 'In order to provide an analytic solution with low computational cost, we put forward an original precoder design algorithm which exploits an auxiliary variable to treat the optimization problem with a mixture of linear and quadratic constraints.', '1208.5012-1-0-6': 'Numerical results demonstrate that our proposed precoder design enable SR to have an agreeable SNR on the prerequisite that the interference at PR is maintained below the threshold.', '1208.5012-1-1-0': 'Cognitive radio, precoder design, orthogonal space-time block coding, polarized antennas.', '1208.5012-1-2-0': '# Introduction', '1208.5012-1-3-0': 'Cognitive Radio (CR) is an encouraging technology to combat the spectrum scarcity.', '1208.5012-1-3-1': 'In order to further enhance the spectrum utilization, the spectrum sharing strategy that Primary Users (PUs) and Secondary Users (SUs) coexist in licensed bands as long as PUs are preserved from the interference caused by SUs attracts much research efforts.', '1208.5012-1-3-2': 'Such a strategy is tantamount to a multi-user system in which the inter-user interference mitigation is the core.', '1208.5012-1-3-3': 'Various inter-user interference mitigation techniques for spectrum sharing CR systems have been put forward.', '1208.5012-1-3-4': 'They can be roughly grouped into two categories: power allocation [CITATION]-[CITATION] and precoding in Multiple-Input Multiple-Output (MIMO) CR systems [CITATION]-[CITATION].', '1208.5012-1-4-0': 'Space Time Block Coding (STBC) exploits time and space diversity in MIMO systems so as to heighten the reliability of the message signal.', '1208.5012-1-4-1': 'Orthogonal STBC (OSTBC) are contrived in such a fashion that the vectors of coding matrix are orthogonal in both time and space dimensions.', '1208.5012-1-4-2': 'This feature yields a simple linear decoding at the receiver side so that no complex matrix manipulation-Singular Value Decomposition (SVD), for instance, is required for recovering the information bit from the gathered received symbols.', '1208.5012-1-4-3': 'Numerous precoding techniques have been mooted for unstructured codes.', '1208.5012-1-4-4': 'However, these techniques cannot be applied to OSTBC which should forcibly preserve a special space-time structure.', '1208.5012-1-4-5': 'The precoding design for OSTBC CR systems attracts less attention in previous work.', '1208.5012-1-4-6': 'Such previous work in [CITATION] was based on the Maximum Likelihood (ML) space-time decoder, whereas the ML decoder is a nonlinear method.', '1208.5012-1-4-7': 'Inspired by Minimum Variance (MV) receiver applied for OSTBC multi-access systems [CITATION] which used a weight matrix at the receiver side to quell the inter-user interference, we make use of MV approach to design a precoding matrix at Secondary Transmitter (ST).', '1208.5012-1-5-0': 'The precoding matrix at ST is designed to comply with the needs in our CR system: maximizing the Signal to Noise Ratio (SNR) at Secondary Receiver (SR) on the premise that the orthogonality of OSTBC is kept, the interference introduced to Primary Link (PL) by Secondary Link (SL) is maintained under a tolerable level and the total transmitted power constraint at ST is satisfied.', '1208.5012-1-6-0': 'The classic MV beamforming [CITATION], [CITATION] built an optimization problem which includes only one linear constraint, that cannot administer to the needs in our CR system.', '1208.5012-1-6-1': 'On the other hand, some precoder designs for CR systems [CITATION] introduced a mixture of linear and quadratic constraints to the optimization problem which leads to iterative solutions with high computational complexity.', '1208.5012-1-6-2': 'For the purpose of contriving a precoder that applies to our CR system and provides an analytic solution with low computational cost, we moot an original precoder design algorithm: we first take advantage of an optimization problem which includes one linear constraint with the objective of preserving the orthogonality of OSTBC and making SL introduce minimal interference to PL for different combinations of the polarization mode at ST and SR.', '1208.5012-1-6-3': 'This optimization problem provides an analytic solution in terms of an auxiliary variable which is the system gain on SL.', '1208.5012-1-6-4': 'Then we regulate this auxiliary variable using the quadratic constraints evoked by the transmitted power budget at ST and the maximum tolerable interference at Primary Receiver (PR).', '1208.5012-1-6-5': 'The polarization mode at ST and SR are conclusively settled on based upon the maximization criteria of SNR at SR.', '1208.5012-1-7-0': 'The rest of the paper is organized as follows.', '1208.5012-1-7-1': 'The system model and OSTBC are presented in Section II.', '1208.5012-1-7-2': 'In Section III, we introduce the proposed precoder design for OSTBC based CR with polarized antennas.', '1208.5012-1-7-3': 'We report the numerical results and provide insights on the expected performance in Section IV.', '1208.5012-1-7-4': 'Finally, we give the conclusion in Section V.', '1208.5012-1-8-0': '# System Descriptions', '1208.5012-1-9-0': 'We consider a CR system that consists of one SL which exploits OSTBC and one PL.', '1208.5012-1-9-1': 'ST and PT are only allowed to communicate with their peers.', '1208.5012-1-9-2': 'ST or PT is equipped with [MATH] antennas and SR or PR is equipped with [MATH] antennas.', '1208.5012-1-9-3': 'The antennas in the same array have identical polarization mode.', '1208.5012-1-9-4': 'On each link, the transmit antenna array or the receive antenna array is able to switch its polarization mode between vertical mode [MATH] and horizontal mode [MATH].', '1208.5012-1-9-5': "We denote by [MATH] and [MATH], respectively, the transmit antenna array's polarization mode and the receive antenna array's polarization mode.", '1208.5012-1-10-0': '## System Model', '1208.5012-1-11-0': 'In this paper, we exploit 3GPP Spatial Channel Model (SCM) [CITATION].', '1208.5012-1-11-1': 'The space channel impulse response between a pair of antennas [MATH] and [MATH] of path [MATH] can be expressed as a function in terms of the polarization channel response and the geometric configuration of the antennas at both sides of the link: [EQUATION] where [MATH] is the BS antenna complex response for the V-pol component, [MATH] is the BS antenna complex response for the H-pol component, [MATH] is the MS antenna complex response for the V-pol component, [MATH] is the MS antenna complex response for the H-pol component, [MATH] is the Angle of Departure (AOD) for the [MATH]th subpath of the [MATH]th path and [MATH] is the Angle of Arrival (AOA) for the [MATH]th subpath of the [MATH]th path.', '1208.5012-1-12-0': 'We assume that the system is operated over a frequency-flat channel with [MATH] paths and each path contains only one subpath.', '1208.5012-1-12-1': 'For a point to point communication link, the baseband input-output relationship at time-slot [MATH] is expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna, [MATH] is a [MATH] size transmitted signal vector which satisfies [MATH], [MATH] is a [MATH] size complex Gaussian noise vector at receiver with zero-mean and unit-variance and [MATH] is the [MATH] channel matrix for the specified [MATH] and [MATH] with the entry [EQUATION] where [MATH].', '1208.5012-1-12-2': '[MATH] has unit variance and satisfies [MATH].', '1208.5012-1-13-0': 'Assuming that the channel is constant from [MATH] to [MATH], then Equation ([REF]) can be extended into: [EQUATION] where [MATH], [MATH] and [MATH].', '1208.5012-1-14-0': '## Orthogonal Space-Time Block Coding', '1208.5012-1-15-0': 'If [MATH] is OSTBC matrix, then [MATH] has a linear representation in terms of complex information symbols prior to space-time encoding [MATH] [CITATION]: [EQUATION] where [MATH] and [MATH] are [MATH] code matrices [CITATION].', '1208.5012-1-16-0': 'OSTBC matrix has the following unitary property: [EQUATION]', '1208.5012-1-16-1': 'In order to represent the relationship between the original symbols and the received signal by multiplication of matrices, we introduce the "underline" operator [CITATION] to rewrite Equation ([REF]) as: [EQUATION] where [MATH] is the data stream which is QPSK modulated in this paper, [MATH] is the equivalent channel matrix with the specified polarization mode, [MATH] is the OSTBC compact dispersion matrix and the "underline" operator for any matrix [MATH] is defined as: [EQUATION] where [MATH] is the vectorization operator stacking all columns of a matrix on top of each other.', '1208.5012-1-17-0': "The earliest OSTBC scheme which is well known as Alamouti's code was proposed in [CITATION].", '1208.5012-1-17-1': "Alamouti's code gives full diversity in the spatial dimension without data rate loss.", '1208.5012-1-17-2': "The transmission matrix of Alamouti's code [MATH] is given as: [EQUATION]", '1208.5012-1-17-3': "In [CITATION], Alamouti's code was extended for more antennas.", '1208.5012-1-17-4': 'For instance, four antennas, the transmission matrix of the half rate code [MATH] is given as: [EQUATION]', '1208.5012-1-18-0': '# Precoder for OSTBC based CR with Polarized Antennas', '1208.5012-1-19-0': "We design a precoding matrix at ST which acts on the entry of the OSTBC compact dispersion matrix and has no influence on the codes' structure.", '1208.5012-1-19-1': 'Our precoder design relies on the equivalent transmit correlation matrix on the link between ST and PR (SPL).', '1208.5012-1-19-2': 'This matrix can be estimated easily by SU in the sensing step and enables our precoder design to regulate the interference introduced by SL to PL.', '1208.5012-1-20-0': '## Constraints from SL', '1208.5012-1-21-0': 'With the precoding operation, the received signal at SR for the specified polarization mode at ST and SR can be expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna of SR, [MATH] is the SL equivalent channel matrix with the specified polarization mode at ST and SR, [MATH] is the precoding matrix for the specified polarization mode at ST and SR.', '1208.5012-1-22-0': 'A straightforward approach to estimate the transmitted signal from ST is using the following soft output detector: [EQUATION]', '1208.5012-1-22-1': 'The OSTBC structure conservation puts forward the following constraint: [EQUATION] where [MATH] is the system gain on SL for the specified polarization mode at ST and SR which will be adjusted to satisfy the other constraints.', '1208.5012-1-23-0': 'Additionally, the transmitted power budget at ST induces another constraint: [EQUATION] where [MATH] and [MATH] are, respectively, the transmitted power for the specified polarization mode at ST and SR and the maximum transmitted power at ST.', '1208.5012-1-24-0': '## Constraints from PL', '1208.5012-1-25-0': 'The received signal at PR from ST is deemed as baleful signal by PL and can be expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna of PR and [MATH] is the equivalent channel matrix for the specified polarization mode at ST and PR.', '1208.5012-1-26-0': 'The interference power introduced by SL to PL for the specified polarization mode at ST and PR can be calculated as: [EQUATION] where [MATH] is the equivalent transmit correlation matrix on SPL for the specified polarization mode at ST and PR.', '1208.5012-1-26-1': 'The maximum tolerable interference power [MATH] at PR evokes the following constraint: [EQUATION]', '1208.5012-1-27-0': '## Minimum Variance Algorithm', '1208.5012-1-28-0': 'SU can dominate the configuration of the precoding matrix and the polarization mode on SL, while SU has no eligibility to select the polarization mode on PL.', '1208.5012-1-28-1': 'Our algorithm is based on an optimization problem which includes one linear constraint with the objective of preserving the orthogonality of OSTBC and making SL introduce minimal interference to PL for different combinations of the polarization mode at ST and SR.', '1208.5012-1-28-2': 'This optimization problem provides an analytic solution in terms of an auxiliary variable which is the system gain on SL.', '1208.5012-1-28-3': 'Then this auxiliary variable is regulated by using the quadratic constraints evoked by the transmitted power budget at ST and the maximum tolerable interference at PR.', '1208.5012-1-28-4': 'The polarization mode at ST and SR are conclusively settled on based upon the maximization criteria of SNR at SR.', '1208.5012-1-29-0': 'Such an optimization problem that includes one linear constraint is described as follow: [EQUATION]', '1208.5012-1-29-1': 'We exploit the method of Lagrange multipliers to find [MATH] for each combination of the polarization mode at ST and SR.', '1208.5012-1-29-2': 'The Lagrangian function can be written as: [EQUATION] where [MATH] and [MATH] is a [MATH] size matrix of Lagrange multipliers.', '1208.5012-1-30-0': 'By differentiating the Lagrange function with respect to [MATH] and equating it to zero, we obtain an analytic solution in terms of [MATH] which is expressed as: [EQUATION] where [MATH].', '1208.5012-1-31-0': 'The estimated interference power at PR can be expressed in terms of [MATH] as: [EQUATION]', '1208.5012-1-31-1': 'The estimated SNR at SR can be written in terms of [MATH] as: [EQUATION] where [EQUATION] .', '1208.5012-1-32-0': 'The estimated transmit power at ST in terms of [MATH] is given by: [EQUATION] where [EQUATION]', '1208.5012-1-32-1': 'We derive [MATH] by substituting [MATH] and [MATH] into Equation ([REF]) and Equation ([REF]) which indicate the transmitted power budget constraint and the maximum tolerable interference constraint: [EQUATION]', '1208.5012-1-32-2': 'Therefore the estimated SNR at SR can be determined as: [EQUATION]', '1208.5012-1-32-3': 'Based upon the maximization criteria of SNR at SR, Finally, we destine the estimated polarization mode of ST and SR as: [EQUATION]', '1208.5012-1-33-0': '# Numerical Results', '1208.5012-1-34-0': 'For the purpose of validating our proposed precoding design algorithm, we simulated our CR system using the proposed precoder design algorithm and measure the SNR at SR by using varying maximum transmitted power at ST and a reasonable interference threshold at PR.', '1208.5012-1-35-0': "We firstly carried out our simulation with Alamouti's code at ST for different combinations of [MATH] and [MATH] on SL under different multipath scenarios.", '1208.5012-1-35-1': 'Then, we executed our simulation with different codes for different number of transmit antennas at ST based upon a determinate combination of [MATH] and [MATH] on SL and multipath scenario.', '1208.5012-1-35-2': 'In both simulation scenarios, the Signal to Interference plus Noise Ratio (SINR) threshold to perceive the received signal at PR was chosen equal to [MATH] and the Cross-polar Discrimination (XPD) was set to [MATH].', '1208.5012-1-35-3': 'The channel matrix on each link was modeled according to 3GPP SCM.', '1208.5012-1-35-4': 'Since the status of polarization at PR [MATH] is normally unidentified for SU, the equivalent transmit correlation matrix on SPL becomes random.', '1208.5012-1-35-5': 'This thereby results in a random SNR at SR.', '1208.5012-1-35-6': 'In our simulation, we calculated the SNR at SR in terms of the polarization tilt angle at PR by introducing a rotation matrix to the equivalent transmit correlation matrix on SPL.', '1208.5012-1-35-7': 'We assumed that the polarization tilt angle at PR follows a continuous uniform distribution between [MATH] and [MATH].', '1208.5012-1-35-8': 'Then we sampled uniformly over the range of the polarization tilt angle at PR and calculated the SNR at SR for each sample of tilt angle.', '1208.5012-1-35-9': 'Finally, we worked out an average the SNR at SR to evaluate the system performance.', '1208.5012-1-36-0': '## Performance Analysis of Polarization Diversity', '1208.5012-1-37-0': 'We simulated a CR system, where ST is equipped with 2 antennas, SR is equipped with 1 antenna and PR is equipped with 2 antennas.', '1208.5012-1-37-1': 'We observe the variation of the average SNR at SR for different combinations of [MATH] and [MATH] on SL as the transmit power at ST increases.', '1208.5012-1-37-2': 'First, we set SL channel as a 2-path frequency flat fading channel and SPL channel as a single path frequency flat fading channel.', '1208.5012-1-37-3': 'The variation tendencies in this scenario were depicted in Fig.[REF].', '1208.5012-1-37-4': 'Then we reset SPL channel as a 4-path frequency flat fading channel and the corresponding variation tendencies were shown in Fig.[REF].', '1208.5012-1-37-5': 'The average SNR at SR for a large number of samples leads to the smooth curves.', '1208.5012-1-37-6': 'As the transmit power at ST increases, the average SNR at SR of all different combinations of [MATH] and [MATH] on SL exhibit uptrend in both scenarios and linear increase is obtained when [MATH] are below [MATH] in both scenarios, where [MATH] denotes the noise power at SR.', '1208.5012-1-37-7': 'The mismatch of [MATH] and [MATH] on SL induces a [MATH] gap between the matched modes and the mismatched modes when the average SNR at SR has linear increase in the first scenario.', '1208.5012-1-37-8': 'When we enhanced the number of paths in SPL channel, the average SNR at SR for the mismatched modes was declined by [MATH] and the gap was enlarged in the second scenario.', '1208.5012-1-38-0': '## Performance Analysis of Transmit Antennas Diversity', '1208.5012-1-39-0': 'In the second simulation, we aimed to observe the average SNR at SR by using different number of transmit antennas.', '1208.5012-1-39-1': "In the first circumstance, 2 transmit antennas and Alamouti's code [MATH] were utilized at ST. In the second circumstance, 4 transmit antennas and the half rate code [MATH] were utilized at ST. In both circumstances, we set [MATH] and [MATH].", '1208.5012-1-39-2': 'SR is equipped with 1 antenna and PR is equipped with 4 antennas.', '1208.5012-1-39-3': 'The number of paths is chosen equal to 2 on SL and 6 on SPL.', '1208.5012-1-40-0': 'For the case of 2 transmit antennas at ST, the SNR at SR reaches the saturation point at [MATH] when [MATH] achieves [MATH].', '1208.5012-1-40-1': 'Compare to the previous results in Fig. [REF] and [REF], the SNR at SR reaches the saturation point faster due to the increase in number of paths on the SPL.', '1208.5012-1-40-2': 'However, the increase in number of antennas will significantly delay the arrival of the saturation point even the number of paths on the SPL is also increased.', '1208.5012-1-40-3': 'For the case of 4 transmit antennas at ST, the SNR at SR reaches the saturation point at [MATH] when [MATH] achieves [MATH].', '1208.5012-1-41-0': '# Conclusions', '1208.5012-1-42-0': 'A linear precoder design which aims at alleviating the interference at PR for OSTBC based CR has been introduced.', '1208.5012-1-42-1': 'One of the principal contributions is to endow the conventional prefiltering technique with the excellent features of OSTBC in the context of CR.', '1208.5012-1-42-2': 'The prefiltering technique has been optimized for the purpose of maximizing the SNR at SR on the premise that the orthogonality of OSTBC is kept, the interference introduced to PL by SL is maintained under a tolerable level and the total transmitted power constraint is satisfied.', '1208.5012-1-42-3': 'Numeral Results have shown that polarization diversity contributes to achieve better SNR at SR, moreover, the increase in number of antennas will significantly delay the arrival of the saturation point for the SNR at SR.'} | {'1208.5012-2-0-0': 'The spectrum sharing has recently passed into a mainstream Cognitive Radio (CR) strategy.', '1208.5012-2-0-1': 'We investigate the core issue in this strategy: interference mitigation at Primary Receiver (PR).', '1208.5012-2-0-2': 'We propose a linear precoder design which aims at alleviating the interference caused by Secondary User (SU) from the source for Orthogonal Space-Time Block Coding (OSTBC) based CR.', '1208.5012-2-0-3': 'We resort to Minimum Variance (MV) approach to contrive the precoding matrix at Secondary Transmitter (ST) in order to maximize the Signal to Noise Ratio (SNR) at Secondary Receiver (SR) on the premise that the orthogonality of OSTBC is kept, the interference introduced to Primary Link (PL) by Secondary Link (SL) is maintained under a tolerable level and the total transmitted power constraint at ST is satisfied.', '1208.5012-2-0-4': 'Moreover, the selection of polarization mode for SL is incorporated in the precoder design.', '1208.5012-2-0-5': 'In order to provide an analytic solution with low computational cost, we put forward an original precoder design algorithm which exploits an auxiliary variable to treat the optimization problem with a mixture of linear and quadratic constraints.', '1208.5012-2-0-6': 'Numerical results demonstrate that our proposed precoder design enable SR to have an agreeable SNR on the prerequisite that the interference at PR is maintained below the threshold.', '1208.5012-2-1-0': 'Cognitive radio, precoder design, orthogonal space-time block coding, polarized antennas.', '1208.5012-2-2-0': '# Introduction', '1208.5012-2-3-0': 'Cognitive Radio (CR) is an encouraging technology to combat the spectrum scarcity.', '1208.5012-2-3-1': 'In order to further enhance the spectrum utilization, the spectrum sharing strategy that Primary Users (PUs) and Secondary Users (SUs) coexist in licensed bands as long as PUs are preserved from the interference caused by SUs attracts much research efforts.', '1208.5012-2-3-2': 'Such a strategy is tantamount to a multi-user system in which the inter-user interference mitigation is the core.', '1208.5012-2-3-3': 'Various inter-user interference mitigation techniques for spectrum sharing CR systems have been put forward.', '1208.5012-2-3-4': 'They can be roughly grouped into two categories: power allocation [CITATION]-[CITATION] and precoding in Multiple-Input Multiple-Output (MIMO) CR systems [CITATION]-[CITATION].', '1208.5012-2-4-0': 'Space Time Block Coding (STBC) exploits time and space diversity in MIMO systems so as to heighten the reliability of the message signal.', '1208.5012-2-4-1': 'Orthogonal STBC (OSTBC) are contrived in such a fashion that the vectors of coding matrix are orthogonal in both time and space dimensions.', '1208.5012-2-4-2': 'This feature yields a simple linear decoding at the receiver side so that no complex matrix manipulation-Singular Value Decomposition (SVD), for instance, is required for recovering the information bit from the gathered received symbols.', '1208.5012-2-4-3': 'Numerous precoding techniques have been mooted for unstructured codes.', '1208.5012-2-4-4': 'However, these techniques cannot be applied to OSTBC which should forcibly preserve a special space-time structure.', '1208.5012-2-4-5': 'The precoding design for OSTBC CR systems attracts less attention in previous work.', '1208.5012-2-4-6': 'Such previous work in [CITATION] was based on the Maximum Likelihood (ML) space-time decoder, whereas the ML decoder is a nonlinear method.', '1208.5012-2-4-7': 'Inspired by Minimum Variance (MV) receiver applied for OSTBC multi-access systems [CITATION] which used a weight matrix at the receiver side to quell the inter-user interference, we make use of MV approach to design a precoding matrix at Secondary Transmitter (ST).', '1208.5012-2-5-0': 'The precoding matrix at ST is designed to comply with the needs in our CR system: maximizing the Signal to Noise Ratio (SNR) at Secondary Receiver (SR) on the premise that the orthogonality of OSTBC is kept, the interference introduced to Primary Link (PL) by Secondary Link (SL) is maintained under a tolerable level and the total transmitted power constraint at ST is satisfied.', '1208.5012-2-6-0': 'The classic MV beamforming [CITATION], [CITATION] built an optimization problem which includes only one linear constraint, that cannot administer to the needs in our CR system.', '1208.5012-2-6-1': 'On the other hand, some precoder designs for CR systems [CITATION] introduced a mixture of linear and quadratic constraints to the optimization problem which leads to iterative solutions with high computational complexity.', '1208.5012-2-6-2': 'For the purpose of contriving a precoder that applies to our CR system and provides an analytic solution with low computational cost, we moot an original precoder design algorithm: we first take advantage of an optimization problem which includes one linear constraint with the objective of preserving the orthogonality of OSTBC and making SL introduce minimal interference to PL for different combinations of the polarization mode at ST and SR.', '1208.5012-2-6-3': 'This optimization problem provides an analytic solution in terms of an auxiliary variable which is the system gain on SL.', '1208.5012-2-6-4': 'Then we regulate this auxiliary variable using the quadratic constraints evoked by the transmitted power budget at ST and the maximum tolerable interference at Primary Receiver (PR).', '1208.5012-2-6-5': 'The polarization mode at ST and SR are conclusively settled on based upon the maximization criteria of SNR at SR.', '1208.5012-2-7-0': 'The rest of the paper is organized as follows.', '1208.5012-2-7-1': 'The system model and OSTBC are presented in Section II.', '1208.5012-2-7-2': 'In Section III, we introduce the proposed precoder design for OSTBC based CR with polarized antennas.', '1208.5012-2-7-3': 'We report the numerical results and provide insights on the expected performance in Section IV.', '1208.5012-2-7-4': 'Finally, we give the conclusion in Section V.', '1208.5012-2-8-0': '# System Descriptions', '1208.5012-2-9-0': 'We consider a CR system that consists of one SL which exploits OSTBC and one PL.', '1208.5012-2-9-1': 'ST and PT are only allowed to communicate with their peers.', '1208.5012-2-9-2': 'ST or PT is equipped with [MATH] antennas and SR or PR is equipped with [MATH] antennas.', '1208.5012-2-9-3': 'The antennas in the same array have identical polarization mode.', '1208.5012-2-9-4': 'On each link, the transmit antenna array or the receive antenna array is able to switch its polarization mode between vertical mode [MATH] and horizontal mode [MATH].', '1208.5012-2-9-5': "We denote by [MATH] and [MATH], respectively, the transmit antenna array's polarization mode and the receive antenna array's polarization mode.", '1208.5012-2-10-0': '## System Model', '1208.5012-2-11-0': 'In this paper, we exploit 3GPP Spatial Channel Model (SCM) [CITATION].', '1208.5012-2-11-1': 'The space channel impulse response between a pair of antennas [MATH] and [MATH] of path [MATH] can be expressed as a function in terms of the polarization channel response and the geometric configuration of the antennas at both sides of the link: [EQUATION] where [MATH] is the BS antenna complex response for the V-pol component, [MATH] is the BS antenna complex response for the H-pol component, [MATH] is the MS antenna complex response for the V-pol component, [MATH] is the MS antenna complex response for the H-pol component, [MATH] is the Angle of Departure (AOD) for the [MATH]th subpath of the [MATH]th path and [MATH] is the Angle of Arrival (AOA) for the [MATH]th subpath of the [MATH]th path.', '1208.5012-2-12-0': 'We assume that the system is operated over a frequency-flat channel with [MATH] paths and each path contains only one subpath.', '1208.5012-2-12-1': 'For a point to point communication link, the baseband input-output relationship at time-slot [MATH] is expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna, [MATH] is a [MATH] size transmitted signal vector which satisfies [MATH], [MATH] is a [MATH] size complex Gaussian noise vector at receiver with zero-mean and unit-variance and [MATH] is the [MATH] channel matrix for the specified [MATH] and [MATH] with the entry [EQUATION] where [MATH].', '1208.5012-2-12-2': '[MATH] has unit variance and satisfies [MATH].', '1208.5012-2-13-0': 'Assuming that the channel is constant from [MATH] to [MATH], then Equation ([REF]) can be extended into: [EQUATION] where [MATH], [MATH] and [MATH].', '1208.5012-2-14-0': '## Orthogonal Space-Time Block Coding', '1208.5012-2-15-0': 'If [MATH] is OSTBC matrix, then [MATH] has a linear representation in terms of complex information symbols prior to space-time encoding [MATH] [CITATION]: [EQUATION] where [MATH] and [MATH] are [MATH] code matrices [CITATION].', '1208.5012-2-16-0': 'OSTBC matrix has the following unitary property: [EQUATION]', '1208.5012-2-16-1': 'In order to represent the relationship between the original symbols and the received signal by multiplication of matrices, we introduce the "underline" operator [CITATION] to rewrite Equation ([REF]) as: [EQUATION] where [MATH] is the data stream which is QPSK modulated in this paper, [MATH] is the equivalent channel matrix with the specified polarization mode, [MATH] is the OSTBC compact dispersion matrix and the "underline" operator for any matrix [MATH] is defined as: [EQUATION] where [MATH] is the vectorization operator stacking all columns of a matrix on top of each other.', '1208.5012-2-17-0': "The earliest OSTBC scheme which is well known as Alamouti's code was proposed in [CITATION].", '1208.5012-2-17-1': "Alamouti's code gives full diversity in the spatial dimension without data rate loss.", '1208.5012-2-17-2': "The transmission matrix of Alamouti's code [MATH] is given as: [EQUATION]", '1208.5012-2-17-3': "In [CITATION], Alamouti's code was extended for more antennas.", '1208.5012-2-17-4': 'For instance, four antennas, the transmission matrix of the half rate code [MATH] is given as: [EQUATION]', '1208.5012-2-18-0': '# Precoder for OSTBC based CR with Polarized Antennas', '1208.5012-2-19-0': "We design a precoding matrix at ST which acts on the entry of the OSTBC compact dispersion matrix and has no influence on the codes' structure.", '1208.5012-2-19-1': 'Our precoder design relies on the equivalent transmit correlation matrix on the link between ST and PR (SPL).', '1208.5012-2-19-2': 'This matrix can be estimated easily by SU in the sensing step and enables our precoder design to regulate the interference introduced by SL to PL.', '1208.5012-2-20-0': '## Constraints from SL', '1208.5012-2-21-0': 'With the precoding operation, the received signal at SR for the specified polarization mode at ST and SR can be expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna of SR, [MATH] is the SL equivalent channel matrix with the specified polarization mode at ST and SR, [MATH] is the precoding matrix for the specified polarization mode at ST and SR.', '1208.5012-2-22-0': 'A straightforward approach to estimate the transmitted signal from ST is using the following soft output detector: [EQUATION]', '1208.5012-2-22-1': 'The OSTBC structure conservation puts forward the following constraint: [EQUATION] where [MATH] is the system gain on SL for the specified polarization mode at ST and SR which will be adjusted to satisfy the other constraints.', '1208.5012-2-23-0': 'Additionally, the transmitted power budget at ST induces another constraint: [EQUATION] where [MATH] and [MATH] are, respectively, the transmitted power for the specified polarization mode at ST and SR and the maximum transmitted power at ST.', '1208.5012-2-24-0': '## Constraints from PL', '1208.5012-2-25-0': 'The received signal at PR from ST is deemed as baleful signal by PL and can be expressed as: [EQUATION] where [MATH] is the SNR at each receive antenna of PR and [MATH] is the equivalent channel matrix for the specified polarization mode at ST and PR.', '1208.5012-2-26-0': 'The interference power introduced by SL to PL for the specified polarization mode at ST and PR can be calculated as: [EQUATION] where [MATH] is the equivalent transmit correlation matrix on SPL for the specified polarization mode at ST and PR.', '1208.5012-2-26-1': 'The maximum tolerable interference power [MATH] at PR evokes the following constraint: [EQUATION]', '1208.5012-2-27-0': '## Minimum Variance Algorithm', '1208.5012-2-28-0': 'SU can dominate the configuration of the precoding matrix and the polarization mode on SL, while SU has no eligibility to select the polarization mode on PL.', '1208.5012-2-28-1': 'Our algorithm is based on an optimization problem which includes one linear constraint with the objective of preserving the orthogonality of OSTBC and making SL introduce minimal interference to PL for different combinations of the polarization mode at ST and SR.', '1208.5012-2-28-2': 'This optimization problem provides an analytic solution in terms of an auxiliary variable which is the system gain on SL.', '1208.5012-2-28-3': 'Then this auxiliary variable is regulated by using the quadratic constraints evoked by the transmitted power budget at ST and the maximum tolerable interference at PR.', '1208.5012-2-28-4': 'The polarization mode at ST and SR are conclusively settled on based upon the maximization criteria of SNR at SR.', '1208.5012-2-29-0': 'Such an optimization problem that includes one linear constraint is described as follow: [EQUATION]', '1208.5012-2-29-1': 'We exploit the method of Lagrange multipliers to find [MATH] for each combination of the polarization mode at ST and SR.', '1208.5012-2-29-2': 'The Lagrangian function can be written as: [EQUATION] where [MATH] and [MATH] is a [MATH] size matrix of Lagrange multipliers.', '1208.5012-2-30-0': 'By differentiating the Lagrange function with respect to [MATH] and equating it to zero, we obtain an analytic solution in terms of [MATH] which is expressed as: [EQUATION] where [MATH].', '1208.5012-2-31-0': 'The estimated interference power at PR can be expressed in terms of [MATH] as: [EQUATION]', '1208.5012-2-31-1': 'The estimated SNR at SR can be written in terms of [MATH] as: [EQUATION] where [EQUATION] .', '1208.5012-2-32-0': 'The estimated transmit power at ST in terms of [MATH] is given by: [EQUATION] where [EQUATION]', '1208.5012-2-32-1': 'We derive [MATH] by substituting [MATH] and [MATH] into Equation ([REF]) and Equation ([REF]) which indicate the transmitted power budget constraint and the maximum tolerable interference constraint: [EQUATION]', '1208.5012-2-32-2': 'Therefore the estimated SNR at SR can be determined as: [EQUATION]', '1208.5012-2-32-3': 'Based upon the maximization criteria of SNR at SR, Finally, we destine the estimated polarization mode of ST and SR as: [EQUATION]', '1208.5012-2-33-0': '# Numerical Results', '1208.5012-2-34-0': 'For the purpose of validating our proposed precoding design algorithm, we simulated our CR system using the proposed precoder design algorithm and measure the SNR at SR by using varying maximum transmitted power at ST and a reasonable interference threshold at PR.', '1208.5012-2-35-0': "We firstly carried out our simulation with Alamouti's code at ST for different combinations of [MATH] and [MATH] on SL under different multipath scenarios.", '1208.5012-2-35-1': 'Then, we executed our simulation with different codes for different number of transmit antennas at ST based upon a determinate combination of [MATH] and [MATH] on SL and multipath scenario.', '1208.5012-2-35-2': 'In both simulation scenarios, the Signal to Interference plus Noise Ratio (SINR) threshold to perceive the received signal at PR was chosen equal to [MATH] and the Cross-polar Discrimination (XPD) was set to [MATH].', '1208.5012-2-35-3': 'The channel matrix on each link was modeled according to 3GPP SCM.', '1208.5012-2-35-4': 'Since the status of polarization at PR [MATH] is normally unidentified for SU, the equivalent transmit correlation matrix on SPL becomes random.', '1208.5012-2-35-5': 'This thereby results in a random SNR at SR.', '1208.5012-2-35-6': 'In our simulation, we calculated the SNR at SR in terms of the polarization tilt angle at PR by introducing a rotation matrix to the equivalent transmit correlation matrix on SPL.', '1208.5012-2-35-7': 'We assumed that the polarization tilt angle at PR follows a continuous uniform distribution between [MATH] and [MATH].', '1208.5012-2-35-8': 'Then we sampled uniformly over the range of the polarization tilt angle at PR and calculated the SNR at SR for each sample of tilt angle.', '1208.5012-2-35-9': 'Finally, we worked out an average the SNR at SR to evaluate the system performance.', '1208.5012-2-36-0': '## Performance Analysis of Polarization Diversity', '1208.5012-2-37-0': 'We simulated a CR system, where ST is equipped with 2 antennas, SR is equipped with 1 antenna and PR is equipped with 2 antennas.', '1208.5012-2-37-1': 'We observe the variation of the average SNR at SR for different combinations of [MATH] and [MATH] on SL as the transmit power at ST increases.', '1208.5012-2-37-2': 'First, we set SL channel as a 2-path frequency flat fading channel and SPL channel as a single path frequency flat fading channel.', '1208.5012-2-37-3': 'The variation tendencies in this scenario were depicted in Fig.[REF].', '1208.5012-2-37-4': 'Then we reset SPL channel as a 4-path frequency flat fading channel and the corresponding variation tendencies were shown in Fig.[REF].', '1208.5012-2-37-5': 'The average SNR at SR for a large number of samples leads to the smooth curves.', '1208.5012-2-37-6': 'As the transmit power at ST increases, the average SNR at SR of all different combinations of [MATH] and [MATH] on SL exhibit uptrend in both scenarios and linear increase is obtained when [MATH] are below [MATH] in both scenarios, where [MATH] denotes the noise power at SR.', '1208.5012-2-37-7': 'The mismatch of [MATH] and [MATH] on SL induces a [MATH] gap between the matched modes and the mismatched modes when the average SNR at SR has linear increase in the first scenario.', '1208.5012-2-37-8': 'When we enhanced the number of paths in SPL channel, the average SNR at SR for the mismatched modes was declined by [MATH] and the gap was enlarged in the second scenario.', '1208.5012-2-38-0': '## Performance Analysis of Transmit Antennas Diversity', '1208.5012-2-39-0': 'In the second simulation, we aimed to observe the average SNR at SR by using different number of transmit antennas.', '1208.5012-2-39-1': "In the first circumstance, 2 transmit antennas and Alamouti's code [MATH] were utilized at ST. In the second circumstance, 4 transmit antennas and the half rate code [MATH] were utilized at ST. In both circumstances, we set [MATH] and [MATH].", '1208.5012-2-39-2': 'SR is equipped with 1 antenna and PR is equipped with 4 antennas.', '1208.5012-2-39-3': 'The number of paths is chosen equal to 2 on SL and 6 on SPL.', '1208.5012-2-40-0': 'For the case of 2 transmit antennas at ST, the SNR at SR reaches the saturation point at [MATH] when [MATH] achieves [MATH].', '1208.5012-2-40-1': 'Compare to the previous results in Fig. [REF] and [REF], the SNR at SR reaches the saturation point faster due to the increase in number of paths on the SPL.', '1208.5012-2-40-2': 'However, the increase in number of antennas will significantly delay the arrival of the saturation point even the number of paths on the SPL is also increased.', '1208.5012-2-40-3': 'For the case of 4 transmit antennas at ST, the SNR at SR reaches the saturation point at [MATH] when [MATH] achieves [MATH].', '1208.5012-2-41-0': '# Conclusions', '1208.5012-2-42-0': 'A linear precoder design which aims at alleviating the interference at PR for OSTBC based CR has been introduced.', '1208.5012-2-42-1': 'One of the principal contributions is to endow the conventional prefiltering technique with the excellent features of OSTBC in the context of CR.', '1208.5012-2-42-2': 'The prefiltering technique has been optimized for the purpose of maximizing the SNR at SR on the premise that the orthogonality of OSTBC is kept, the interference introduced to PL by SL is maintained under a tolerable level and the total transmitted power constraint is satisfied.', '1208.5012-2-42-3': 'Numeral Results have shown that polarization diversity contributes to achieve better SNR at SR, moreover, the increase in number of antennas will significantly delay the arrival of the saturation point for the SNR at SR.'} | [['1208.5012-1-15-0', '1208.5012-2-15-0'], ['1208.5012-1-3-0', '1208.5012-2-3-0'], ['1208.5012-1-3-1', '1208.5012-2-3-1'], ['1208.5012-1-3-2', '1208.5012-2-3-2'], ['1208.5012-1-3-3', '1208.5012-2-3-3'], ['1208.5012-1-3-4', '1208.5012-2-3-4'], ['1208.5012-1-9-0', '1208.5012-2-9-0'], ['1208.5012-1-9-1', '1208.5012-2-9-1'], ['1208.5012-1-9-2', '1208.5012-2-9-2'], ['1208.5012-1-9-3', '1208.5012-2-9-3'], ['1208.5012-1-9-4', '1208.5012-2-9-4'], ['1208.5012-1-9-5', '1208.5012-2-9-5'], ['1208.5012-1-35-0', '1208.5012-2-35-0'], ['1208.5012-1-35-1', '1208.5012-2-35-1'], ['1208.5012-1-35-2', '1208.5012-2-35-2'], ['1208.5012-1-35-3', '1208.5012-2-35-3'], ['1208.5012-1-35-4', '1208.5012-2-35-4'], ['1208.5012-1-35-5', '1208.5012-2-35-5'], ['1208.5012-1-35-6', '1208.5012-2-35-6'], ['1208.5012-1-35-7', '1208.5012-2-35-7'], ['1208.5012-1-35-8', '1208.5012-2-35-8'], ['1208.5012-1-35-9', '1208.5012-2-35-9'], ['1208.5012-1-39-0', '1208.5012-2-39-0'], ['1208.5012-1-39-1', '1208.5012-2-39-1'], ['1208.5012-1-39-2', '1208.5012-2-39-2'], ['1208.5012-1-39-3', '1208.5012-2-39-3'], ['1208.5012-1-25-0', '1208.5012-2-25-0'], ['1208.5012-1-22-0', '1208.5012-2-22-0'], ['1208.5012-1-22-1', '1208.5012-2-22-1'], ['1208.5012-1-23-0', '1208.5012-2-23-0'], ['1208.5012-1-26-0', '1208.5012-2-26-0'], ['1208.5012-1-26-1', '1208.5012-2-26-1'], ['1208.5012-1-31-0', '1208.5012-2-31-0'], ['1208.5012-1-31-1', '1208.5012-2-31-1'], ['1208.5012-1-16-0', '1208.5012-2-16-0'], ['1208.5012-1-16-1', '1208.5012-2-16-1'], ['1208.5012-1-21-0', '1208.5012-2-21-0'], ['1208.5012-1-42-0', '1208.5012-2-42-0'], ['1208.5012-1-42-1', '1208.5012-2-42-1'], ['1208.5012-1-42-2', '1208.5012-2-42-2'], ['1208.5012-1-42-3', '1208.5012-2-42-3'], ['1208.5012-1-7-0', '1208.5012-2-7-0'], ['1208.5012-1-7-1', '1208.5012-2-7-1'], ['1208.5012-1-7-2', '1208.5012-2-7-2'], ['1208.5012-1-7-3', '1208.5012-2-7-3'], ['1208.5012-1-7-4', '1208.5012-2-7-4'], ['1208.5012-1-11-0', '1208.5012-2-11-0'], ['1208.5012-1-11-1', '1208.5012-2-11-1'], ['1208.5012-1-19-0', '1208.5012-2-19-0'], ['1208.5012-1-19-1', '1208.5012-2-19-1'], ['1208.5012-1-19-2', '1208.5012-2-19-2'], ['1208.5012-1-0-0', '1208.5012-2-0-0'], ['1208.5012-1-0-1', '1208.5012-2-0-1'], 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['1208.5012-1-35-5', '1208.5012-2-35-5'], ['1208.5012-1-35-6', '1208.5012-2-35-6'], ['1208.5012-1-35-7', '1208.5012-2-35-7'], ['1208.5012-1-35-8', '1208.5012-2-35-8'], ['1208.5012-1-35-9', '1208.5012-2-35-9'], ['1208.5012-1-39-0', '1208.5012-2-39-0'], ['1208.5012-1-39-1', '1208.5012-2-39-1'], ['1208.5012-1-39-2', '1208.5012-2-39-2'], ['1208.5012-1-39-3', '1208.5012-2-39-3'], ['1208.5012-1-25-0', '1208.5012-2-25-0'], ['1208.5012-1-22-0', '1208.5012-2-22-0'], ['1208.5012-1-22-1', '1208.5012-2-22-1'], ['1208.5012-1-23-0', '1208.5012-2-23-0'], ['1208.5012-1-26-0', '1208.5012-2-26-0'], ['1208.5012-1-26-1', '1208.5012-2-26-1'], ['1208.5012-1-31-0', '1208.5012-2-31-0'], ['1208.5012-1-31-1', '1208.5012-2-31-1'], ['1208.5012-1-16-0', '1208.5012-2-16-0'], ['1208.5012-1-16-1', '1208.5012-2-16-1'], ['1208.5012-1-21-0', '1208.5012-2-21-0'], ['1208.5012-1-42-0', '1208.5012-2-42-0'], ['1208.5012-1-42-1', '1208.5012-2-42-1'], ['1208.5012-1-42-2', '1208.5012-2-42-2'], ['1208.5012-1-42-3', '1208.5012-2-42-3'], ['1208.5012-1-7-0', '1208.5012-2-7-0'], ['1208.5012-1-7-1', '1208.5012-2-7-1'], ['1208.5012-1-7-2', '1208.5012-2-7-2'], ['1208.5012-1-7-3', '1208.5012-2-7-3'], ['1208.5012-1-7-4', '1208.5012-2-7-4'], ['1208.5012-1-11-0', '1208.5012-2-11-0'], ['1208.5012-1-11-1', '1208.5012-2-11-1'], ['1208.5012-1-19-0', '1208.5012-2-19-0'], ['1208.5012-1-19-1', '1208.5012-2-19-1'], ['1208.5012-1-19-2', '1208.5012-2-19-2'], ['1208.5012-1-0-0', '1208.5012-2-0-0'], ['1208.5012-1-0-1', '1208.5012-2-0-1'], ['1208.5012-1-0-2', '1208.5012-2-0-2'], ['1208.5012-1-0-3', '1208.5012-2-0-3'], ['1208.5012-1-0-4', '1208.5012-2-0-4'], ['1208.5012-1-0-5', '1208.5012-2-0-5'], ['1208.5012-1-0-6', '1208.5012-2-0-6'], ['1208.5012-1-30-0', '1208.5012-2-30-0'], ['1208.5012-1-32-0', '1208.5012-2-32-0'], ['1208.5012-1-32-1', '1208.5012-2-32-1'], ['1208.5012-1-32-2', '1208.5012-2-32-2'], ['1208.5012-1-32-3', '1208.5012-2-32-3'], ['1208.5012-1-5-0', '1208.5012-2-5-0'], ['1208.5012-1-12-0', '1208.5012-2-12-0'], ['1208.5012-1-12-1', '1208.5012-2-12-1'], ['1208.5012-1-12-2', '1208.5012-2-12-2'], ['1208.5012-1-37-0', '1208.5012-2-37-0'], ['1208.5012-1-37-1', '1208.5012-2-37-1'], ['1208.5012-1-37-2', '1208.5012-2-37-2'], ['1208.5012-1-37-3', '1208.5012-2-37-3'], ['1208.5012-1-37-4', '1208.5012-2-37-4'], ['1208.5012-1-37-5', '1208.5012-2-37-5'], ['1208.5012-1-37-6', '1208.5012-2-37-6'], ['1208.5012-1-37-7', '1208.5012-2-37-7'], ['1208.5012-1-37-8', '1208.5012-2-37-8'], ['1208.5012-1-6-0', '1208.5012-2-6-0'], ['1208.5012-1-6-1', '1208.5012-2-6-1'], ['1208.5012-1-6-2', '1208.5012-2-6-2'], ['1208.5012-1-6-3', '1208.5012-2-6-3'], ['1208.5012-1-6-4', '1208.5012-2-6-4'], ['1208.5012-1-6-5', '1208.5012-2-6-5'], ['1208.5012-1-17-0', '1208.5012-2-17-0'], ['1208.5012-1-17-1', '1208.5012-2-17-1'], ['1208.5012-1-17-2', '1208.5012-2-17-2'], ['1208.5012-1-17-3', '1208.5012-2-17-3'], ['1208.5012-1-17-4', '1208.5012-2-17-4'], ['1208.5012-1-28-0', '1208.5012-2-28-0'], ['1208.5012-1-28-1', '1208.5012-2-28-1'], ['1208.5012-1-28-2', '1208.5012-2-28-2'], ['1208.5012-1-28-3', '1208.5012-2-28-3'], ['1208.5012-1-28-4', '1208.5012-2-28-4'], ['1208.5012-1-34-0', '1208.5012-2-34-0'], ['1208.5012-1-4-0', '1208.5012-2-4-0'], ['1208.5012-1-4-1', '1208.5012-2-4-1'], ['1208.5012-1-4-2', '1208.5012-2-4-2'], ['1208.5012-1-4-3', '1208.5012-2-4-3'], ['1208.5012-1-4-4', '1208.5012-2-4-4'], ['1208.5012-1-4-5', '1208.5012-2-4-5'], ['1208.5012-1-4-6', '1208.5012-2-4-6'], ['1208.5012-1-4-7', '1208.5012-2-4-7'], ['1208.5012-1-29-0', '1208.5012-2-29-0'], ['1208.5012-1-29-1', '1208.5012-2-29-1'], ['1208.5012-1-29-2', '1208.5012-2-29-2'], ['1208.5012-1-40-0', '1208.5012-2-40-0'], ['1208.5012-1-40-1', '1208.5012-2-40-1'], ['1208.5012-1-40-2', '1208.5012-2-40-2'], ['1208.5012-1-40-3', '1208.5012-2-40-3']] | [] | [] | [] | [] | ['1208.5012-1-1-0', '1208.5012-1-13-0', '1208.5012-2-1-0', '1208.5012-2-13-0'] | {'1': 'http://creativecommons.org/licenses/publicdomain/', '2': 'http://creativecommons.org/licenses/publicdomain/'} | https://arxiv.org/abs/1208.5012 | null | null | null | null | null |
1611.03044 | {'1611.03044-1-0-0': 'Control reserves are power generation or consumption entities that ensure balance of supply and demand of electricity in real-time.', '1611.03044-1-0-1': 'In many countries, they are operated through a market mechanism in which entities provide bids.', '1611.03044-1-0-2': 'The system operator determines the accepted bids based on an optimization algorithm.', '1611.03044-1-0-3': 'We develop the Vickery-Clarke-Groves (VCG) mechanism for these electricity markets.', '1611.03044-1-0-4': 'We show that all advantages of the VCG mechanism including incentive compatibility of the equilibria and efficiency of the outcome can be guaranteed in these markets.', '1611.03044-1-0-5': 'Furthermore, we derive conditions to ensure collusion and shill bidding are not profitable.', '1611.03044-1-0-6': 'Our results are verified with numerical examples.', '1611.03044-1-1-0': '# Introduction', '1611.03044-1-2-0': 'The liberalization of electricity markets leads to opportunities and challenges for ensuring stability and efficiency of the power grid.', '1611.03044-1-2-1': 'For a stable grid, the supply and demand of electricity at all times need to be balanced.', '1611.03044-1-2-2': 'This instantaneous balance is reflected in the grid frequency.', '1611.03044-1-2-3': 'Whereas scheduling (yearly, day-ahead) is based on forecast supply and demand of power, the control reserves (also referred to as ancillary services) provide additional controllability to balance supply and demand of power in real-time.', '1611.03044-1-2-4': 'With increasing volatile renewable sources of energy, the need for control reserves also has increased.', '1611.03044-1-2-5': 'This motivates analysis and design of optimization algorithms and market mechanisms that procure these reserves.', '1611.03044-1-3-0': 'The objective of this paper is a game theoretic exploration of an alternative market mechanism for the control reserves with potential improvements.', '1611.03044-1-3-1': 'To further discuss this, we briefly discuss relevant features of the existing market mechanism.', '1611.03044-1-3-2': 'Control reserves are categorized as primary, secondary, and tertiary.', '1611.03044-1-3-3': 'Primary reserves balance frequency deviations in timescale of seconds.', '1611.03044-1-3-4': 'Secondary reserves balance the deviations on a timescale of seconds to minutes not resolved by primary control.', '1611.03044-1-3-5': 'Tertiary reserves restore secondary reserves and typically act 15 minutes after a disturbance to frequency.', '1611.03044-1-3-6': 'The secondary and tertiary control reserves in several countries are procured in a market.', '1611.03044-1-3-7': 'In the Swiss market for example, the auction mechanism implemented by the Transmission System Operator (TSO) minimizes the cost of procurement of required amounts of power, given bids .', '1611.03044-1-4-0': 'In a pay-as-bid mechanism, since payments to winners are equal to their bid prices, a rational player may over-bid to ensure profit.', '1611.03044-1-4-1': 'As an alternative to pay-as-bid, we explore the Vickery Clarke Groves (VCG) mechanism.', '1611.03044-1-4-2': 'This is one of the most prominent auction mechanisms.', '1611.03044-1-4-3': 'The first analysis of the VCG mechanism was carried out by for the sale of a single item.', '1611.03044-1-4-4': 'This work was subsequently generalized to multiple items by and .', '1611.03044-1-5-0': 'It has been shown that the VCG mechanism is the only mechanism that possesses efficiency and incentive compatibility.', '1611.03044-1-5-1': 'Efficiency implies that goods are exchanged between buyers and sellers in a way that creates maximal social value.', '1611.03044-1-5-2': 'Incentive compatibility means that it is optimal for each participant to bid their true value.', '1611.03044-1-5-3': 'Variants of the VCG mechanism have been successfully deployed generating billions of dollars in Spectrum auctions, for instance, in the 2012 UK spectrum auction and in advertising, for instance, by Facebook .', '1611.03044-1-5-4': 'For further discussion on the VCG mechanism and its application to real auctions we recommend .', '1611.03044-1-6-0': 'Investigation must be performed before applying the VCG mechanism.', '1611.03044-1-6-1': 'As outlined in the paper of Ausubel and Milgrom , coalitions of participants can influence the auction in order to obtain higher collective profit.', '1611.03044-1-6-2': 'These peculiarities occur when the outcome of the auction is not in the core.', '1611.03044-1-6-3': 'The core is a solution concept in coalition game theory where prices are distributed so that there is no incentive for participants to leave the coalition .', '1611.03044-1-6-4': 'This has recently motivated the study and application of VCG auctions where the outcome is projected to the core .', '1611.03044-1-7-0': 'The electricity market can be thought of as a reverse auction.', '1611.03044-1-7-1': 'In contrast to an auction with multiple goods, in an electricity market, each participant can bid for continuum values of power.', '1611.03044-1-7-2': 'Furthermore, to clear this market, certain constraints, such as balance of supply and demand and network constraints need to be guaranteed.', '1611.03044-1-7-3': 'Due to the differences between an electricity market and an auction mechanism for multiple items (such as spectrum or adverts), there are conceptual and theoretical advances in VCG mechanism that need to be analyzed.', '1611.03044-1-8-0': 'In this paper, we apply the VCG mechanism to control reserve markets and provide a mathematically rigorous analysis of it.', '1611.03044-1-8-1': 'We show that efficiency and incentive compatibility of the VCG mechanism will hold even in the case of stochastic markets, see Theorem [REF].', '1611.03044-1-8-2': 'On the other hand, we provide examples where shill bidding might occur.', '1611.03044-1-8-3': 'The remainder of the paper develops ways to resolve this issue.', '1611.03044-1-8-4': 'In particular, building upon a series of results based on coalitional game theory, in Theorem [REF] we show how a simple pay-off monotonicity condition removes incentives for shill bidding and other collusions.', '1611.03044-1-8-5': 'The proofs developed significantly simplify the arguments of Ausubel and Milgrom .', '1611.03044-1-9-0': 'The paper is organized as follows.', '1611.03044-1-9-1': 'In Section [REF] we introduce the VCG mechanism for control reserve markets, analyzing its positive and negative aspects.', '1611.03044-1-9-2': 'Throughout Section [REF] we investigate conditions that can mitigate these problems making the mechanism competitive.', '1611.03044-1-9-3': 'We conclude with specific simulations based on data available from Swissgrid (the Swiss TSO) showing the applicability of VCG mechanism to the Swiss ancillary service market.', '1611.03044-1-10-0': '# Electricity auction market setup', '1611.03044-1-11-0': 'We briefly describe the control reserve market of Switzerland.', '1611.03044-1-11-1': 'The formulation and results derived are generalizable to alternative markets, with similar features as will be discussed.', '1611.03044-1-11-2': 'The Swiss system operator (TSO), Swissgrid, procures secondary and tertiary reserves in its reserves markets.', '1611.03044-1-11-3': 'These consist of a weekly market where secondary reserves are procured and daily markets where both secondary and tertiary reserves are procured.', '1611.03044-1-11-4': 'Each market participant submits a bid that consists of a price per unit of power (CHF/MW, swiss franc per megawatt) and a volume of power which it can supply (MW).', '1611.03044-1-11-5': 'Offers are indivisible and thus, must be accepted entirely or rejected.', '1611.03044-1-11-6': 'Moreover, conditional offers are accepted.', '1611.03044-1-11-7': 'This means that a participant can offer a set of bids, of which only one can be accepted.', '1611.03044-1-11-8': 'If an offer is accepted, the participant is paid for its availability irrespective of whether these reserves are deployed (an additional payment is made in case of deployment).', '1611.03044-1-11-9': 'This availability payment, under the current swiss reserve market, is pay-as-bid.', '1611.03044-1-11-10': 'An extensive description of the Swiss Ancillary market is given in .', '1611.03044-1-12-0': 'We abstract the control reserve market summarized above as follows.', '1611.03044-1-12-1': 'Let [MATH] denote the set of auction participants and [MATH].', '1611.03044-1-12-2': 'Let [MATH] be all the bids placed by participant [MATH], where [MATH] is the vector of power supplies offered (MW) and [MATH] are their corresponding requested costs (or prices).', '1611.03044-1-12-3': 'Here [MATH] is the number of bids from participant [MATH].', '1611.03044-1-12-4': 'Let [MATH] be the set of all bids and [MATH].', '1611.03044-1-12-5': 'Given a set [MATH], a mechanism defines which bids are accepted with a choice function, [MATH] and a payment to each participant, payment rule [MATH].', '1611.03044-1-12-6': "The utility of participant [MATH] is hence [EQUATION] where [MATH] is participant [MATH]'s true cost of providing the offered power [MATH] and [MATH] is the binary vector indicating his accepted bids.", '1611.03044-1-13-0': "The transmission system operator's objective function is [EQUATION].", '1611.03044-1-13-1': "The variable [MATH] selects the accepted bids, [MATH] can be any additional variables entering the TSO's optimization and [MATH] is a general function.", '1611.03044-1-13-2': 'In most electricity market, the objective is to minimize the cost of procurement subject to some constraints: [EQUATION]', '1611.03044-1-13-3': 'The above constraints correspond to procurement of the required amounts of power, e.g. in the Swiss reserve markets accepted reserves must have a deficit probability of less than 0.2%.', '1611.03044-1-13-4': 'We let [MATH] be the feasible values of [MATH] for this optimization.', '1611.03044-1-13-5': 'The optimization defines a general class of models, where the cost function is affine in [MATH] and the prices of bids do not enter the constraints.', '1611.03044-1-14-0': '## The pay-as-bid mechanism', '1611.03044-1-15-0': 'In the current pay-as-bid mechanism we recognize: [EQUATION]', '1611.03044-1-15-1': "It follows that each participant's utility is [MATH].", '1611.03044-1-15-2': 'As such, rational participants would bid more than their true values to make profit.', '1611.03044-1-15-3': 'Consequently, under pay-as-bid, the TSO attempts to minimize inflated bids rather than true costs.', '1611.03044-1-15-4': 'Thus, pay-as-bid cannot guarantee power reserves are procured cost effectively.', '1611.03044-1-16-0': '## The VCG mechanism', '1611.03044-1-17-0': 'The VCG mechanism is characterized with the same choice function as the pay-as-bid mechanism but a different payment rule.', '1611.03044-1-18-0': 'The Vickrey-Clarke-Groves (VCG) choice function and payment rule are defined as: [EQUATION] where [MATH] denotes the vector of bids placed by all participants excluding [MATH].', '1611.03044-1-18-1': 'The function [MATH] must be carefully chosen to make the mechanism meaningful.', '1611.03044-1-18-2': 'Namely, we require that payments go from the TSO to power plants, positive transfers, and that power plants will not face negative utilities participating to such auctions, individual rationality.', '1611.03044-1-18-3': "A particular choice of [MATH] is the Clarke pivot-rule, which minimizes the procurement cost given all bids excluding [MATH]'s: [EQUATION].", '1611.03044-1-19-0': 'A set of bids [MATH] is a dominant-strategy Nash equilibrium if for each participant [MATH], [EQUATION].', '1611.03044-1-19-1': 'Moreover, a dominant-strategy equilibrium is incentive compatible if [MATH] where [MATH] is the true cost of power [MATH], as given in [REF].', '1611.03044-1-19-2': "That is, each participant finds it more profitable to bid truthfully [MATH], rather than any other vector [MATH], regardless of other participants' bids.", '1611.03044-1-19-3': 'Hence, all the bidding strategies are dominated by strategy [MATH].', '1611.03044-1-20-0': 'topQ: can you comment on how this result differs from existing proofs in VCG - in particular, some note about the fact that such results were derived before for single/multi-item but not in terms of a optimization variable with continuum of bids, correct?', '1611.03044-1-20-1': 'red NW: an attempt is here', '1611.03044-1-21-0': 'The following theorem summarizes the contributions of , and in designing the VCG mechanism.', '1611.03044-1-21-1': 'In our proof, we are mindful of the slightly non-standard setting of the electrical markets: that auctions are "reverse-auctions", i.e. with a single buyer and many sellers, and that constraints in the optimization problem may be non-standard.', '1611.03044-1-22-0': 'Given the clearing model of [REF].', '1611.03044-1-22-1': '[a)] The energy procurement auction under VCG choice function and payment rule is a Dominant-Strategy Incentive-Compatible (D.S.I.C) mechanism.', '1611.03044-1-22-2': 'The VCG outcomes are efficient, that is, the sum of all the utilities is maximized.', '1611.03044-1-22-3': 'The Clarke pivot rule ensures positive transfers and individual rationality.', '1611.03044-1-23-0': 'a) We distinguish between the participant [MATH] placing a generic bid [MATH] and biding truthfully [MATH].', '1611.03044-1-23-1': 'For [MATH], substituting the VCG choice function and payment rule with [MATH] as in [REF]: -0.3em [EQUATION] where the term in brackets is the cost [MATH] of [MATH] but evaluated at [MATH].', '1611.03044-1-23-2': 'For [MATH], however, [MATH].', '1611.03044-1-23-3': 'We then have [MATH] because [MATH] is a feasible suboptimal allocation for the available bids [MATH].', '1611.03044-1-24-0': 'b) Let [MATH] denote the utility gained by the TSO, that is, [MATH].', '1611.03044-1-24-1': 'By Definition [REF] and incentive compatibility, [MATH].', '1611.03044-1-24-2': 'We then have: [MATH].', '1611.03044-1-24-3': 'Hence, [MATH], which is maximized by the clearing model [REF].', '1611.03044-1-25-0': 'c) This can be easily verified substituting [MATH]: [EQUATION]', '1611.03044-1-26-0': 'In summary, all producers have incentive to reveal their true values for price of power in a VCG market.', '1611.03044-1-26-1': 'Thus, it becomes easier for entities to enter the auction, without spending resources in computing optimal bidding strategies.', '1611.03044-1-26-2': 'This can help in achieving market liberalization objectives.', '1611.03044-1-26-3': 'Moreover, from the above theorem it follows that the winners of the auctions are the producers with the lowest true values.', '1611.03044-1-26-4': 'This is because participants bid truthfully and the VCG choice function minimizes the cost of the accepted bids.', '1611.03044-1-27-0': 'So, there are persuasive arguments for considering VCG market for control reserves.', '1611.03044-1-27-1': 'However, there are potential disadvantages that must be eliminated.', '1611.03044-1-28-0': 'Suppose the TSO has to procure 800 MW from PowerPlant1, [MATH], who bids [MATH] CHF for 800 MW, and PowerPlant2, [MATH], who bids [MATH] CHF for 800 MW.', '1611.03044-1-28-1': 'Under the VCG mechanism, PowerPlant1 wins the auction receiving a payment of [MATH] CHF.', '1611.03044-1-28-2': 'Suppose now that power plants [MATH] and [MATH] entered the auction each bidding 0 CHF for 200 MW.', '1611.03044-1-28-3': 'Clearly, the new entrants become winners and each of them would receive a VCG payment of [MATH] CHF.', '1611.03044-1-29-0': "This example shows that: (a) producers with very low prices (in this case 0 CHF) could receive very high payments; (b) collusion or shill bidding can increase participants' profits.", '1611.03044-1-29-1': "In fact, [MATH] and [MATH] could be a group of losers who jointly lowered their bids to win the auction, or they could represent multiple identities of the same losing participant (i.e. a power plant with true value greater than 40'000 CHF for 800 MW).", '1611.03044-1-29-2': "Entering the auction with four shills, however, this participant would have received a payment of 4[MATH]40'000 CHF.", '1611.03044-1-30-0': 'Our goal is now to derive conditions that make VCG outcomes competitive and prevent shill bidding or collusion.', '1611.03044-1-31-0': '# Solution approach for VCG market', '1611.03044-1-32-0': 'In coalition game theory, the core is the set of allocations of goods that cannot be improved upon by the formation of coalitions.', '1611.03044-1-32-1': 'identify conditions for a VCG outcome to lie in the core.', '1611.03044-1-32-2': 'Following their analysis we derive conditions for core outcomes in our setting and provide new simpler proofs relevant to our problem formulation that show that shill bidding and collusion can be eliminated from certain class of electricity markets under the VCG mechanism.', '1611.03044-1-33-0': 'Given a game where [MATH] is the set of participants, let [MATH] denote the coalitional value function [EQUATION]', '1611.03044-1-33-1': 'This function provides the optimal objective function, for any subset of participants [MATH] that includes the TSO.', '1611.03044-1-33-2': 'Here, [MATH] is the cost the TSO incurs for the VCG outcome with participants [MATH].', '1611.03044-1-33-3': 'That is, [MATH] is the solution to optimization [REF] with [MATH] for all [MATH], and with additional constraints that [MATH] for all [MATH].', '1611.03044-1-33-4': 'Clearly [MATH] for [MATH] since increasing participation reduces costs.', '1611.03044-1-33-5': 'We thus let [MATH] represent the coalition game associated with the auction.', '1611.03044-1-34-0': 'The [MATH] is defined as follows [EQUATION]', '1611.03044-1-34-1': 'The core is thus the set of all the feasible outcomes, coming from an efficient mechanism (first equality above), that are unblocked by any coalition (the inequality).', '1611.03044-1-34-2': 'We say that an outcome is competitive if it lies in the core; that is, there is no incentive for forming coalitions.', '1611.03044-1-34-3': 'In the previous example, the outcome was not competitive because it was blocked by coalition [MATH].', '1611.03044-1-34-4': "PowerPlant1 was offering only 40'000 CHF for the total amount of 800 MW.", '1611.03044-1-34-5': 'It will be also shown in Theorem [REF] that core outcomes eliminate any incentives for collusions and shill bidding.', '1611.03044-1-35-0': '## Ensuring core payments', '1611.03044-1-36-0': 'Since core outcome is a competitive outcome, we investigate under which conditions the outcomes of the VCG mechanism applied to the control reserve market will be in the core.', '1611.03044-1-36-1': 'Note that there are [MATH] constraints that define a core outcome.', '1611.03044-1-36-2': 'Our first Lemma provides an equivalent characterization of the core with significantly lower number of constraints.', '1611.03044-1-37-0': 'Given a VCG auction [MATH], let [MATH] be its outcome and [MATH] the corresponding winners.', '1611.03044-1-37-1': 'Assuming participants revealed their true values, [MATH] if and only if, [MATH], [EQUATION]', '1611.03044-1-37-2': 'Core constraints with [MATH] are immediately satisfied as [MATH] (individual rationality, Theorem 1c).', '1611.03044-1-37-3': 'Now, [MATH] is unblocked by any [MATH] (since [MATH] is the outcome with [MATH] participants).', '1611.03044-1-37-4': 'Thus, [MATH].', '1611.03044-1-37-5': 'Moreover, fixing [MATH], the dominant constraints are those corresponding to minimal [MATH], in particular, when [MATH] (this being maximal set with [MATH] not taking part in the coalition [MATH]).', '1611.03044-1-37-6': 'Finally recall from [REF] that, under the VCG mechanism, [MATH].', '1611.03044-1-37-7': 'The following definition and theorem act over subsets of participants.', '1611.03044-1-37-8': 'Here, we imagine that there is a set of potential participants [MATH] and, for each subset [MATH] of [MATH], we consider whether the outcome of the auction with L participants lies in the core.', '1611.03044-1-38-0': 'Participant [MATH] displays payoff monotonicity if [MATH], [EQUATION]', '1611.03044-1-38-1': 'The outcome of the VCG auction [MATH] lies in the core for all [MATH] if and only if payoff monotonicity holds for each participant in [MATH].', '1611.03044-1-39-0': 'To prove that payoff monotonicity is sufficient for [MATH] to lie in the core, we prove that [REF] holds.', '1611.03044-1-39-1': 'Let [MATH].', '1611.03044-1-39-2': 'Considering [MATH], we notice that: [MATH] since [MATH] displays payoff monotonicity over [MATH]; we also have [MATH] since [MATH] displays payoff monotonicity over [MATH].', '1611.03044-1-39-3': 'We can continue with the same considerations up to [MATH] since [MATH] displays payoff monotonicity over [MATH].', '1611.03044-1-39-4': 'Therefore, [MATH].', '1611.03044-1-39-5': 'This same argument holds for any subset of participants [MATH] and [MATH].', '1611.03044-1-39-6': 'Thus [REF] holds and so, by Lemma [REF], the VCG outcome belongs to the core.', '1611.03044-1-40-0': 'To prove that payoff monotonicity is also necessary for outcomes to lie in the core, suppose that [MATH] does not display payoff monotonicity.', '1611.03044-1-40-1': 'Then, there exist sets [MATH],[MATH] where [REF] does not hold.', '1611.03044-1-40-2': 'We may chose [MATH] for some [MATH].', '1611.03044-1-40-3': 'To see this, take [MATH] and [MATH] with [MATH], then, since payoff monotonicity does not hold, [EQUATION]', '1611.03044-1-40-4': 'The strict inequality above must hold for one of the summands [MATH].', '1611.03044-1-40-5': 'So we may consider sets [MATH] that differ by one participant, say [MATH].', '1611.03044-1-40-6': 'Let [MATH] and [MATH] be the minimal such sets.', '1611.03044-1-40-7': 'By minimality, [MATH] for [MATH].', '1611.03044-1-40-8': 'Further, after rearranging the above inequality we see that [MATH].', '1611.03044-1-40-9': 'That is both participant [MATH] and [MATH] are winners of the VCG auction with participants [MATH] (instead of [MATH]).', '1611.03044-1-40-10': 'Then, considering Lemma [REF] for the auction [MATH] with [MATH], and [MATH], we have: [MATH].', '1611.03044-1-40-11': 'Thus [REF] does not hold since the outcome [MATH] is blocked by coalition [MATH].', '1611.03044-1-40-12': 'Whether a VCG outcome is competitive hence depends on a particular property of the optimal cost [MATH].', '1611.03044-1-40-13': 'Namely, [MATH] has to make [REF] hold for each [MATH].', '1611.03044-1-40-14': 'Note that a similar result was proven in , for a sale auction of a finite number of objects, without any constraints.', '1611.03044-1-40-15': 'Our result generalizes this to markets with continuous goods and arbitrary social planner objectives of the form [REF].', '1611.03044-1-41-0': '## Single stage electricity procurement auction', '1611.03044-1-42-0': 'The class of auctions cleared by [REF] is very general and suitable for mechanisms with multiple stages of decisions.', '1611.03044-1-42-1': 'We will see, in fact, how the two-stages Swiss clearing model described in can be abstracted as in [REF].', '1611.03044-1-42-2': 'But first, we start considering simpler auctions, characterized by single-stage decisions.', '1611.03044-1-42-3': 'More specifically, energy procurement auctions where the TSO has to procure a fixed amount of [MATH] MW, subject to conditional offer constraints.', '1611.03044-1-42-4': 'Hence, we consider auctions cleared by: [EQUATION]', '1611.03044-1-42-5': 'The model above is a simple clearing model within class [REF].', '1611.03044-1-42-6': "We can now derive conditions on participants' bids to ensure pay-off monotonicity, condition [REF], is satisfied.", '1611.03044-1-42-7': 'Thus, we derive conditions under which the outcome of auctions cleared by [REF], [REF] would lie in the core.', '1611.03044-1-43-0': 'Given [REF], if [MATH] with [MATH] implies that [EQUATION] for each [MATH], then bidders satisfy payoff monotonicity condition [REF] under the VCG payment rule.', '1611.03044-1-44-0': 'In words, marginally increasing cost condition [REF] implies core outcomes, and thus eliminates incentives for collusions.', '1611.03044-1-44-1': 'Condition [REF] is visualized in Figure [REF].', '1611.03044-1-45-0': 'To prove that condition [REF] is sufficient for payoff monotonicity, the following Lemma is needed.', '1611.03044-1-46-0': 'Under clearing model [REF], for an auction with participants [MATH] and [MATH] with corresponding optimal power allocations [MATH] and [MATH], condition [REF] implies that [EQUATION].', '1611.03044-1-47-0': 'In the following proofs we apply the notation that if [MATH] is the accepted power allocation from bidder [MATH], then [MATH] is the associated cost bid from [MATH].', '1611.03044-1-47-1': 'If the accepted allocation is zero, we define [MATH].', '1611.03044-1-48-0': 'The proof follows by contradiction.', '1611.03044-1-48-1': 'That is, we will show that when [MATH] is such that [MATH], for some [MATH] then [MATH] can be modified to provide a lower cost allocation, [MATH], for participants [MATH] (thus contradicting optimality of [MATH]).', '1611.03044-1-48-2': 'First, we notice that since bids are equally spaced by [MATH] MW [REF] and satisfy condition [REF], it is never optimal to accept more than [MATH] MW ([MATH]).', '1611.03044-1-49-0': 'Now, assume [MATH] is such that [MATH], for some [MATH].', '1611.03044-1-49-1': "In order to procure exactly [MATH] MW, some participants' accepted MWs must decrease, that is, the set [MATH] is non-empty.", '1611.03044-1-49-2': 'Consider a feasible allocation [MATH] for the auction with [MATH] participants where [MATH] units of power are procured and [EQUATION]', '1611.03044-1-49-3': 'So, [MATH] is constructed from [MATH] by transferring [MATH] units of power from participants in [MATH] to participants in [MATH].', '1611.03044-1-49-4': 'In doing so, the inequality [MATH] can be maintained: [EQUATION].', '1611.03044-1-49-5': "The above inequality holds because when summing over [MATH], [MATH]'s sums to [MATH] and [MATH]'s sums to [MATH].", '1611.03044-1-50-0': 'Since [MATH] is optimal for participants [MATH] and [MATH] is not: [EQUATION] where we used [MATH] as a short-hand-notation for the cost corresponding to choosing [MATH] bids.', '1611.03044-1-51-0': 'Now, we use [REF] to replace the summations over [MATH] in [REF].', '1611.03044-1-51-1': 'In particular, define [MATH] so that [EQUATION].', '1611.03044-1-51-2': 'Note that [MATH] is feasible since [MATH] and [MATH] have the same sum over [MATH] (and thus cancel) and [MATH] is feasible.', '1611.03044-1-51-3': 'Further, since [MATH], by condition [REF]: [EQUATION]', '1611.03044-1-51-4': 'Adding [REF] to either side of [REF] (and canceling [MATH]) gives [EQUATION] which contradicts the optimality of [MATH].', '1611.03044-1-51-5': 'magentaMK: why is this in the footnote?', '1611.03044-1-51-6': 'if it is an assumption, it should be stated.normalsizeRed PGS: I was thinking more about it.', '1611.03044-1-51-7': 'What happens is that, if none in K has strictly increasing marginal cost, an allocation where [MATH] can be as optimal as the one where the accepted amounts are all decreased.', '1611.03044-1-52-0': 'If [MATH] has constant marginal cost (over such interval of MWs), in [REF] we have an equality.', '1611.03044-1-52-1': 'Then, we also have equality in the equation below.', '1611.03044-1-52-2': 'If we define as tie-breaking rule to accept always the lowest amount of MWs from participants with constant marginal cost (in the interval over which exist more than one optimal solution), we get the contradiction.', '1611.03044-1-52-3': 'Because we found an equivalent allocation that leads to the same cost but where for each [MATH] MWs are accepted.', '1611.03044-1-52-4': 'where for each [MATH], [MATH] is such that [MATH].', '1611.03044-1-52-5': 'Then, adding either side of [MATH] to either side of [REF], gives, [EQUATION]', '1611.03044-1-52-6': 'Being the left hand side a feasible allocation (in fact [MATH]), we ended up in a contradiction.', '1611.03044-1-53-0': 'A consequence of the above Lemma is that given the optimal allocation to procure [MATH] MW, for any lower amount of MWs (while being multiple of [MATH]) to be procured, the total MWs accepted from each participant never increases.', '1611.03044-1-53-1': 'Now, we are ready to prove Theorem [REF].', '1611.03044-1-54-0': 'We prove that under condition [REF], inequality [REF] holds for each [MATH], for any [MATH] with [MATH] being a generic new entered participant.', '1611.03044-1-54-1': 'Since [MATH] is a generic set and [MATH] can be any new participant, this is sufficient to prove that the payoffs are monotonic over all the possible couple of sets (the generalization to arbitrary sets [MATH] can then be achieved by the interpolating sums, as was done in [REF]).', '1611.03044-1-54-2': 'We adopt the same notation used in Lemma [REF] and we identify with [MATH] the set of winners.', '1611.03044-1-55-0': 'For each [MATH] we have by definition [MATH].', '1611.03044-1-55-1': 'Thus [MATH], (since the optimal solution is unchanged when [MATH] is removed from S).', '1611.03044-1-55-2': 'By Lemma [REF], [MATH] and so [MATH] also.', '1611.03044-1-55-3': 'Thus, payoff monotonicity holds for [MATH]: [MATH] for [MATH].', '1611.03044-1-55-4': 'This says that a loser of the auction cannot become a winner as more participants enter.', '1611.03044-1-56-0': 'For each winning participant, [MATH], recall that [MATH].', '1611.03044-1-56-1': 'Adopting the same notation of Lemma [REF], we can indicate it as: [EQUATION] where [MATH] are the optimal amounts to be accepted from [MATH], when [MATH] exits the auction.', '1611.03044-1-56-2': 'By Lemma [REF], in fact, [MATH].', '1611.03044-1-56-3': 'Similarly, after participant [MATH] enters the auction, [MATH].', '1611.03044-1-56-4': 'That is, [EQUATION] where [MATH] are the amounts accepted from [MATH] when [MATH] exits the new auction.', '1611.03044-1-56-5': 'By Lemma [REF], we again have [MATH].', '1611.03044-1-57-0': 'Notice that so far we applied Lemma [REF] to justify the increase of the accepted amounts, first, from each [MATH] and now from [MATH], due to the exit of [MATH] from the auctions.', '1611.03044-1-57-1': 'We can apply Lemma [REF] again and affirm that [MATH], and in particular [MATH], because of the entrance of [MATH].', '1611.03044-1-58-0': 'We now find suitable lower and upper bounds to ensure inequality [MATH].', '1611.03044-1-58-1': 'First, note that [MATH], where [MATH] are the cheapest allocation to procure [MATH] MW among [MATH].', '1611.03044-1-58-2': "By Lemma [REF] we have [MATH] , since [MATH]'s sum to [MATH] with [MATH], and [MATH]'s sum to [MATH].", '1611.03044-1-58-3': "Moreover, since [REF] holds and every [MATH] satisfies [REF], [MATH]'s are such that [MATH], because exactly [MATH] MW are purchased.", '1611.03044-1-58-4': 'Using the above suboptimal allocation, we have a lower bound for [MATH]: [EQUATION]', '1611.03044-1-58-5': 'Defining now [MATH] we must have [MATH] and [MATH] , since the right hand side is a feasible cost to procure [MATH] MW among the participants [MATH].', '1611.03044-1-58-6': 'Indeed, [MATH] and [MATH].', '1611.03044-1-58-7': 'Hence, we have: [EQUATION]', '1611.03044-1-58-8': "Moreover, since [MATH]'s satisfy [REF], we have: [EQUATION]", '1611.03044-1-58-9': 'The above holds because [MATH] , [MATH] and [MATH].', '1611.03044-1-58-10': 'In particular, [MATH] are the amounts accepted to procure [MATH] MW among [MATH], while [MATH] are to procure the same MWs among [MATH].', '1611.03044-1-58-11': 'Then, combining [REF] , [REF] and [REF], we finally obtain [MATH]', '1611.03044-1-59-0': "Condition [REF] on every participant's bids hence is sufficient to ensure that our VCG procurement auctions will always have core outcomes.", '1611.03044-1-59-1': 'While we do not show here that the condition is necessary, we illustrate that there are certainly auctions where condition [REF] is violated and for which payoff monotonicity does not hold.', '1611.03044-1-60-0': 'Consider Example [REF] where power plants [MATH] and [MATH] placed just one bid for 800 MW hence violating condition [REF].', '1611.03044-1-60-1': 'It is easy to see that the payoffs of each of the four winners are not monotonic.', '1611.03044-1-60-2': 'In fact, if just one of them (e.g. [MATH]) was participating, he would receive no payment; when [MATH],[MATH],[MATH] enter the auction, however, he becomes a winner hence making positive profit.', '1611.03044-1-60-3': 'Suppose now that [MATH] and [MATH] bid accordingly to [REF], but the bids [MATH] have a decreasing marginal cost: [MATH], [MATH].', '1611.03044-1-60-4': 'In this case, when [MATH] participates alone, he receives a VCG payment of [MATH] CHF; when [MATH],[MATH],[MATH] enter the auction, however, he receives [MATH] CHF.', '1611.03044-1-61-0': 'As previously anticipated, we are now able to prove that the condition derived also makes collusions and shill bidding unprofitable.', '1611.03044-1-61-1': 'Therefore, the participants are better off with their dominant strategies, which is truthful bidding.', '1611.03044-1-61-2': 'Although the result is well-known in literature , and motivates the choice of the core as a competitive standard, we can now prove it using the tools developed so far for the problem at hand.', '1611.03044-1-62-0': 'Consider a generic VCG auction [MATH] cleared by [REF].', '1611.03044-1-62-1': 'If [MATH] , [MATH] satisfies condition [REF].', '1611.03044-1-62-2': 'Then,', '1611.03044-1-63-0': 'Recall that under condition [REF] the participants display payoff monotonicity (Theorem [REF]).', '1611.03044-1-64-0': '(i) Let [MATH] be a set of colluders who would lose the auction when bidding their true values [MATH], while bidding [MATH] they become winners.', '1611.03044-1-64-1': 'Defining [MATH] and [MATH], the VCG payment that each player [MATH] in [MATH] receives is [EQUATION] where the first equality comes from definition of VCG payment, the first inequality comes from the fact that [MATH] since [MATH] displays payoff monotonicity and [MATH] because, when [MATH] decrease their bids, less MWs are being accepted from [MATH] (Lemma [REF]) and [MATH] is bidding with increasing marginal cost.', '1611.03044-1-64-2': 'The last equality comes from the fact that [MATH] originally was a group of non-winners.', '1611.03044-1-64-3': 'Then, [MATH], [MATH] is bounded by the payment that [MATH] would receive when he is the only one lowering its bid.', '1611.03044-1-64-4': 'By Theorem [REF].', '1611.03044-1-64-5': 'a he will not face any benefit in doing so.', '1611.03044-1-65-0': '(ii) We denote with [MATH] multiple identities of the same participant [MATH].', '1611.03044-1-65-1': 'Since every participant bids accordingly to [REF], the outcome is guaranteed to lie in the core.', '1611.03044-1-65-2': 'Hence, by Lemma [REF] and substituting [MATH], we have: [EQUATION] where [MATH] is the cost when [MATH], or equivalently [MATH], is removed from the auction.', '1611.03044-1-65-3': 'Therefore, the total payment that [MATH] would receive is bounded by the one he would receive bidding as a single participant.', '1611.03044-1-65-4': 'Making use of shills, hence, is not profitable.', '1611.03044-1-66-0': 'To confirm the previous theoretical results, we come back to Example 1, where the TSO had to procure a fixed amount of 800 MW.', '1611.03044-1-66-1': 'That is a simple auction cleared by [REF].', '1611.03044-1-66-2': 'Suppose that now power plants [MATH] and [MATH] bid according to [REF] (with [MATH] = 200 MW) and condition [REF].', '1611.03044-1-66-3': 'The available bids are now: [MATH], [MATH], [MATH].', '1611.03044-1-66-4': 'The winners of the auction are still power plants [MATH],[MATH],[MATH],[MATH] but now the VCG payments that they receive is [MATH] CHF [MATH].', '1611.03044-1-66-5': 'The total cost incurred by the TSO is much lower than before and no coalition of players now blocks the outcome.', '1611.03044-1-66-6': "If [MATH],[MATH],[MATH] and [MATH] were multiple identities of the same losing participant (i.e. a power plant with true value greater than 40'000 CHF for 800 MW), shill bidding would become unprofitable (as expected).", '1611.03044-1-66-7': "If, moreover, they were losing participants who jointly lowered their bids, the payments of 8'000 CHF surely made at least one of them to have negative profit.", '1611.03044-1-67-0': 'The diagram in Fig. [REF] summarizes and links the concepts we developed so far.', '1611.03044-1-67-1': 'Notice that Lemma [REF], Theorem [REF] and Theorem [REF] are specific for the class of auctions [REF].', '1611.03044-1-68-0': '## Application to two-stage stochastic market', '1611.03044-1-69-0': 'As we anticipated, the Swiss reserve market as described in can be modeled abstractly according to the optimization problem [REF].', '1611.03044-1-69-1': 'There are two stages of decision variables corresponding to weekly ([MATH]) and daily ([MATH]) bids.', '1611.03044-1-69-2': 'Weekly bids are available at the instance of optimization, whereas daily bids are unknown.', '1611.03044-1-69-3': 'A number of stochastic scenarios corresponding to likely possibilities of daily bids based on their past values is used in the optimization ([MATH]).', '1611.03044-1-69-4': 'The cost function corresponds to the cost of weekly bids and the expected cost of daily bids.', '1611.03044-1-69-5': 'Thus, the cost can be written as [MATH].', '1611.03044-1-69-6': 'The choice function determines the accepted weekly bids.', '1611.03044-1-70-0': 'The function [MATH] captures three types of constraints: (a) those corresponding to procurement of certain amount of tertiary reserves; (b) probabilistic constraints, which ensure that with sufficiently high probabilities, the supply and demand of power is balanced; (c) those corresponding to conditional bids.', '1611.03044-1-70-1': 'Constraint (b) links the daily and weekly variables.', '1611.03044-1-70-2': 'Constraints (a) and (c) correspond to those present in the optimization formulation [REF].', '1611.03044-1-71-0': 'It follows from the analysis of Section [REF], that the VCG mechanism applied to the two-stage stochastic market is an incentive compatible dominant strategy mechanism with socially efficient outcome.', '1611.03044-1-71-1': 'Due to coupling of the two stage decision variables, the analysis of the core payment is significantly more difficult.', '1611.03044-1-71-2': 'In particular, the result derived in Theorem [REF] do not readily apply.', '1611.03044-1-71-3': 'The amount of procured MWs is not anymore fixed and thus [REF] is not well defined.', '1611.03044-1-71-4': 'Selecting [MATH] infinitely small (forcing participants to provide continuous bid curves) and linearizing the probabilistic constraints (b), however, we could show that under condition [REF] this clearing model follows the same regularity property of Lemma [REF].', '1611.03044-1-71-5': 'Whether this makes all the participants displaying payoff monotonicity is a subject of our current study.', '1611.03044-1-71-6': 'Nevertheless, in the numerical example section, we evaluate the performance of the VCG mechanism and compare it to the pay-as-bid mechanism.', '1611.03044-1-72-0': '# Simulations and Analysis', '1611.03044-1-73-0': 'The following simulations are based on the bids placed in the 46th Swiss weekly procurement auction of 2014, where 21 power plants bid for secondary reserves, 25 for tertiary positive and 21 for tertiary negative reserves.', '1611.03044-1-73-1': 'Note that the secondary reserves are symmetric, that is, participants need to provide same amount of positive and negative power.', '1611.03044-1-73-2': 'Tertiary reserves are on the other hand asymmetric.', '1611.03044-1-73-3': 'Thus, participants bid for tertiary negative [MATH], and tertiary positive [MATH].', '1611.03044-1-73-4': 'As in , probabilistic scenarios for future daily auctions are assumed.', '1611.03044-1-73-5': 'The amount of daily reserves is based on the data of the previous week.', '1611.03044-1-73-6': 'Three scenarios are considered corresponding to nominal, high (20% higher) and low prices (20% lower) compared to the previous week.', '1611.03044-1-74-0': 'The corresponding outcome of the pay-as-bid mechanism and the VCG mechanism is shown in Table [REF].', '1611.03044-1-74-1': 'Note that in reality, in a repeated bidding process, the VCG mechanism would lead to different bidding behaviors, which we have not modeled here.', '1611.03044-1-75-0': 'Recall that in a pay-as-bid mechanism, a rational participant will overbid to ensure positive profit.', '1611.03044-1-75-1': 'Unfortunately, it is hard to know the true values of the bids for each participant.', '1611.03044-1-75-2': 'So, it is hard to have an accurate comparison between the VCG and pay-as-bid based on past data.', '1611.03044-1-75-3': "We now scale all the bid prices down by 90%, assuming that those were participants' true values and hence the bids that they would have placed under the VCG mechanism.", '1611.03044-1-75-4': 'The outcome of both mechanisms is shown in Table [REF].', '1611.03044-1-76-0': 'All the results are proportional (as it could be expected) and the sum of VCG payments is lower than the sum of pay-as-bid payments we had in the first scenario.', '1611.03044-1-76-1': "This means that assuming such scaled bids were participants' true values, the VCG mechanism would have led to a lower procurement cost than the implemented pay-as-bid mechanism.", '1611.03044-1-76-2': 'Hence, the VCG mechanism, apart from leading to a dominant strategy equilibrium with an efficient allocation, would have been beneficial also in terms of costs, for this particular case study based on the past data.', '1611.03044-1-77-0': 'This does not happen in generic VCG auctions.', '1611.03044-1-77-1': 'In particular, the cost incurred by the auctioneer in a VCG auction is usually higher than the cost under a pay-as-bid mechanism, considering the same set of bids.', '1611.03044-1-77-2': 'To see this, recall that the VCG payments are [MATH].', '1611.03044-1-77-3': 'These payments measure the benefit that each participant brings to the auction.', '1611.03044-1-77-4': 'When the VCG payments are computed through the two-stage stochastic optimization algorithm of we observed that the costs are not significantly different from the pay-as-bid mechanism.', '1611.03044-1-77-5': 'Intuitively, the two-stage market softens the benefit that every participant brings to the weekly auction: his accepted bids can always be replaced by amounts of MWs allocated to the future.', '1611.03044-1-77-6': 'In fact, the amounts of MWs bought in the weekly auction are not fixed and they are flexible depending on the future daily bids available in that week.', '1611.03044-1-78-0': 'To confirm the intuition above, we now assume that we had perfect information about the future daily auctions.', '1611.03044-1-78-1': 'As such, we run a deterministic auction assuming that the TSO already knew that the optimal amounts to be purchased were 409 MW for SRL , 114 MW for TRL- and 100 MW for TRL+ as predicted in Table [REF] .', '1611.03044-1-78-2': 'Given fixed MWs to be procured, the auction is cleared by the simplified model [REF].', '1611.03044-1-78-3': 'In this case, naturally, we have the same winners of the auction as the previous case for both VCG and pay-as-bid mechanism.', '1611.03044-1-78-4': 'The VCG payments however are significantly higher than the pay-as-bid payments.', '1611.03044-1-78-5': 'The results are shown in Table [REF].', '1611.03044-1-79-0': 'The result can be explained as follows.', '1611.03044-1-79-1': 'When a winner [MATH] is removed from the auction (to compute the term [MATH]) the amounts of MWs to be accepted among the other participants originally were subject to flexibility due to two stage decision variables and lack of a fixed total amount for each type of reserve SRL, TRL-, TRL+.', '1611.03044-1-79-2': 'If these total reserves are fixed for each type, the benefit that every participant brings to the Swiss weekly auction is much higher, and this results in higher VCG payments.', '1611.03044-1-80-0': 'The mixed integer optimization problems were solved with GUROBI, on a quad-core computer with processing speed 1.7 GHz and memory 4 Gb.', '1611.03044-1-80-1': 'The first two simulations had a computation time of 9 min, with an average of 17 s for each optimal cost [MATH].', '1611.03044-1-80-2': 'The last simulation took 7 min, with an average of 14 s for each [MATH].', '1611.03044-1-81-0': '# Conclusion', '1611.03044-1-82-0': 'We developed a VCG market mechanism for electricity markets, motivated by the set-up of the control reserves (ancillary services) market.', '1611.03044-1-82-1': 'We showed that the mechanism results in an incentive compatible dominant strategy Nash equilibrium.', '1611.03044-1-82-2': 'Furthermore, this mechanism is socially efficient.', '1611.03044-1-82-3': 'Through examples, we showed that shill bidding can occur.', '1611.03044-1-82-4': 'We thus, derived conditions under which a deterministic procurement mechanism can guarantee no shill bidding and thus competitive outcomes.', '1611.03044-1-82-5': 'These findings, both theoretical and empirical, act to support the application of VCG mechanism for the electricity markets under consideration.', '1611.03044-1-82-6': 'By removing incentives for collusion and by providing a simple truthful mechanism, we expect that the implementation simplifies the biding process, increases markets efficiency and encourages participation from increasing number of entities.', '1611.03044-1-82-7': 'We verified our results based on market data available from Swissgrid.', '1611.03044-1-82-8': 'Future work consists of deriving conditions under which Theorem [REF] could be extended to the stochastic market.', '1611.03044-1-83-0': 'We thank Farzaneh Abbaspour and Marek Zima from Swissgrid for helpful discussions.', '1611.03044-1-83-1': '2em'} | {'1611.03044-2-0-0': 'Control reserves are power generation or consumption entities that ensure balance of supply and demand of electricity in real-time.', '1611.03044-2-0-1': 'In many countries, they are operated through a market mechanism in which entities provide bids.', '1611.03044-2-0-2': 'The system operator determines the accepted bids based on an optimization algorithm.', '1611.03044-2-0-3': 'We develop the Vickrey-Clarke-Groves (VCG) mechanism for these electricity markets.', '1611.03044-2-0-4': 'We show that all advantages of the VCG mechanism including incentive compatibility of the equilibria and efficiency of the outcome can be guaranteed in these markets.', '1611.03044-2-0-5': 'Furthermore, we derive conditions to ensure collusion and shill bidding are not profitable.', '1611.03044-2-0-6': 'Our results are verified with numerical examples.', '1611.03044-2-1-0': '# Introduction', '1611.03044-2-2-0': 'The liberalization of electricity markets leads to opportunities and challenges for ensuring stability and efficiency of the power grid.', '1611.03044-2-2-1': 'For a stable grid, the supply and demand of electricity at all times need to be balanced.', '1611.03044-2-2-2': 'This instantaneous balance is reflected in the grid frequency.', '1611.03044-2-2-3': 'Whereas scheduling (yearly, day-ahead) is based on forecast supply and demand of power, the control reserves (also referred to as ancillary services) provide additional controllability to balance supply and demand of power in real-time.', '1611.03044-2-2-4': 'With increasing volatile renewable sources of energy, the need for control reserves also has increased.', '1611.03044-2-2-5': 'This motivates analysis and design of optimization algorithms and market mechanisms that procure these reserves.', '1611.03044-2-3-0': 'The objective of this paper is a game theoretic exploration of an alternative market mechanism for the control reserves with potential improvements.', '1611.03044-2-3-1': 'To further discuss this, we briefly discuss relevant features of the existing market mechanism.', '1611.03044-2-3-2': 'Control reserves are categorized as primary, secondary, and tertiary.', '1611.03044-2-3-3': 'Primary reserves balance frequency deviations in timescale of seconds.', '1611.03044-2-3-4': 'Secondary reserves balance the deviations on a timescale of seconds to minutes not resolved by primary control.', '1611.03044-2-3-5': 'Tertiary reserves restore secondary reserves and typically act 15 minutes after a disturbance to frequency.', '1611.03044-2-3-6': 'The secondary and tertiary control reserves in several countries are procured in a market.', '1611.03044-2-3-7': 'In the Swiss market for example, the auction mechanism implemented by the Transmission System Operator (TSO) minimizes the cost of procurement of required amounts of power, given bids .', '1611.03044-2-4-0': 'In a pay-as-bid mechanism, since payments to winners are equal to their bid prices, a rational player may over-bid to ensure profit.', '1611.03044-2-4-1': 'As an alternative to pay-as-bid, we explore the Vickrey Clarke Groves (VCG) mechanism.', '1611.03044-2-4-2': 'This is one of the most prominent auction mechanisms.', '1611.03044-2-4-3': 'The first analysis of the VCG mechanism was carried out by for the sale of a single item.', '1611.03044-2-4-4': 'This work was subsequently generalized to multiple items by and .', '1611.03044-2-5-0': 'It has been shown that the VCG mechanism is the only mechanism that possesses efficiency and incentive compatibility.', '1611.03044-2-5-1': 'Efficiency implies that goods are exchanged between buyers and sellers in a way that creates maximal social value.', '1611.03044-2-5-2': 'Incentive compatibility means that it is optimal for each participant to bid their true value.', '1611.03044-2-5-3': 'Variants of the VCG mechanism have been successfully deployed generating billions of dollars in Spectrum auctions, for instance, in the 2012 UK spectrum auction and in advertising, for instance, by Facebook .', '1611.03044-2-5-4': 'For further discussion on the VCG mechanism and its application to real auctions we recommend .', '1611.03044-2-6-0': 'Investigation must be performed before applying the VCG mechanism.', '1611.03044-2-6-1': 'As outlined in the paper of Ausubel and Milgrom , coalitions of participants can influence the auction in order to obtain higher collective profit.', '1611.03044-2-6-2': 'These peculiarities occur when the outcome of the auction is not in the core.', '1611.03044-2-6-3': 'The core is a solution concept in coalition game theory where prices are distributed so that there is no incentive for participants to leave the coalition .', '1611.03044-2-6-4': 'This has recently motivated the study and application of VCG auctions where the outcome is projected to the core .', '1611.03044-2-7-0': 'The electricity market can be thought of as a reverse auction.', '1611.03044-2-7-1': 'In contrast to an auction with multiple goods, in an electricity market, each participant can bid for continuum values of power.', '1611.03044-2-7-2': 'Furthermore, to clear this market, certain constraints, such as balance of supply and demand and network constraints need to be guaranteed.', '1611.03044-2-7-3': 'Due to the differences between an electricity market and an auction mechanism for multiple items (such as spectrum or adverts), there are conceptual and theoretical advances in VCG mechanism that need to be analyzed.', '1611.03044-2-8-0': 'In this paper, we apply the VCG mechanism to control reserve markets and provide a mathematically rigorous analysis of it.', '1611.03044-2-8-1': 'We show that efficiency and incentive compatibility of the VCG mechanism will hold even in the case of stochastic markets, see Theorem [REF].', '1611.03044-2-8-2': 'On the other hand, we provide examples where shill bidding might occur.', '1611.03044-2-8-3': 'The remainder of the paper develops ways to resolve this issue.', '1611.03044-2-8-4': 'In particular, building upon a series of results based on coalitional game theory, in Theorem [REF] we show how a simple pay-off monotonicity condition removes incentives for shill bidding and other collusions.', '1611.03044-2-8-5': 'The proofs developed significantly simplify the arguments of Ausubel and Milgrom .', '1611.03044-2-9-0': 'The paper is organized as follows.', '1611.03044-2-9-1': 'In Section [REF] we introduce the VCG mechanism for control reserve markets, analyzing its positive and negative aspects.', '1611.03044-2-9-2': 'Throughout Section [REF] we investigate conditions that can mitigate these problems making the mechanism competitive.', '1611.03044-2-9-3': 'We conclude with specific simulations based on data available from Swissgrid (the Swiss TSO) showing the applicability of VCG mechanism to the Swiss ancillary service market.', '1611.03044-2-10-0': '# Electricity auction market setup', '1611.03044-2-11-0': 'We briefly describe the control reserve market of Switzerland.', '1611.03044-2-11-1': 'The formulation and results derived are generalizable to alternative markets, with similar features as will be discussed.', '1611.03044-2-11-2': 'The Swiss system operator (TSO), Swissgrid, procures secondary and tertiary reserves in its reserves markets.', '1611.03044-2-11-3': 'These consist of a weekly market where secondary reserves are procured and daily markets where both secondary and tertiary reserves are procured.', '1611.03044-2-11-4': 'Each market participant submits a bid that consists of a price per unit of power (CHF/MW, swiss franc per megawatt) and a volume of power which it can supply (MW).', '1611.03044-2-11-5': 'Offers are indivisible and thus, must be accepted entirely or rejected.', '1611.03044-2-11-6': 'Moreover, conditional offers are accepted.', '1611.03044-2-11-7': 'This means that a participant can offer a set of bids, of which only one can be accepted.', '1611.03044-2-11-8': 'If an offer is accepted, the participant is paid for its availability irrespective of whether these reserves are deployed (an additional payment is made in case of deployment).', '1611.03044-2-11-9': 'This availability payment, under the current swiss reserve market, is pay-as-bid.', '1611.03044-2-11-10': 'An extensive description of the Swiss Ancillary market is given in .', '1611.03044-2-12-0': 'We abstract the control reserve market summarized above as follows.', '1611.03044-2-12-1': 'Let [MATH] denote the set of auction participants and [MATH].', '1611.03044-2-12-2': 'Let [MATH] be all the bids placed by participant [MATH], where [MATH] is the vector of power supplies offered (MW) and [MATH] are their corresponding requested costs (or prices).', '1611.03044-2-12-3': 'Here [MATH] is the number of bids from participant [MATH].', '1611.03044-2-12-4': 'Let [MATH] be the set of all bids and [MATH].', '1611.03044-2-12-5': 'Given a set [MATH], a mechanism defines which bids are accepted with a choice function, [MATH] and a payment to each participant, payment rule [MATH].', '1611.03044-2-12-6': "The utility of participant [MATH] is hence [EQUATION] where [MATH] is participant [MATH]'s true cost of providing the offered power [MATH] and [MATH] is the binary vector indicating his accepted bids.", '1611.03044-2-13-0': "The transmission system operator's objective function is [EQUATION].", '1611.03044-2-13-1': "The variable [MATH] selects the accepted bids, [MATH] can be any additional variables entering the TSO's optimization and [MATH] is a general function.", '1611.03044-2-13-2': 'In most electricity market, the objective is to minimize the cost of procurement subject to some constraints: [EQUATION]', '1611.03044-2-13-3': 'The above constraints correspond to procurement of the required amounts of power, e.g. in the Swiss reserve markets accepted reserves must have a deficit probability of less than 0.2%.', '1611.03044-2-13-4': 'We let [MATH] be the feasible values of [MATH] for this optimization.', '1611.03044-2-13-5': 'The optimization defines a general class of models, where the cost function is affine in [MATH] and the prices of bids do not enter the constraints.', '1611.03044-2-14-0': '## The pay-as-bid mechanism', '1611.03044-2-15-0': 'In the current pay-as-bid mechanism we recognize: [EQUATION]', '1611.03044-2-15-1': "It follows that each participant's utility is [MATH].", '1611.03044-2-15-2': 'As such, rational participants would bid more than their true values to make profit.', '1611.03044-2-15-3': 'Consequently, under pay-as-bid, the TSO attempts to minimize inflated bids rather than true costs.', '1611.03044-2-15-4': 'Thus, pay-as-bid cannot guarantee power reserves are procured cost effectively.', '1611.03044-2-16-0': '## The VCG mechanism', '1611.03044-2-17-0': 'The VCG mechanism is characterized with the same choice function as the pay-as-bid mechanism but a different payment rule.', '1611.03044-2-18-0': 'The Vickrey-Clarke-Groves (VCG) choice function and payment rule are defined as: [EQUATION] where [MATH] denotes the vector of bids placed by all participants excluding [MATH].', '1611.03044-2-18-1': 'The function [MATH] must be carefully chosen to make the mechanism meaningful.', '1611.03044-2-18-2': 'Namely, we require that payments go from the TSO to power plants, positive transfers, and that power plants will not face negative utilities participating to such auctions, individual rationality.', '1611.03044-2-18-3': "A particular choice of [MATH] is the Clarke pivot-rule, which minimizes the procurement cost given all bids excluding [MATH]'s: [EQUATION].", '1611.03044-2-19-0': 'A set of bids [MATH] is a dominant-strategy Nash equilibrium if for each participant [MATH], [EQUATION].', '1611.03044-2-19-1': 'Moreover, a dominant-strategy equilibrium is incentive compatible if [MATH] where [MATH] is the true cost of power [MATH], as given in [REF].', '1611.03044-2-19-2': "That is, each participant finds it more profitable to bid truthfully [MATH], rather than any other vector [MATH], regardless of other participants' bids.", '1611.03044-2-19-3': 'Hence, all the bidding strategies are dominated by strategy [MATH].', '1611.03044-2-20-0': 'topQ: can you comment on how this result differs from existing proofs in VCG - in particular, some note about the fact that such results were derived before for single/multi-item but not in terms of a optimization variable with continuum of bids, correct?', '1611.03044-2-20-1': 'red NW: an attempt is here', '1611.03044-2-21-0': 'The following theorem summarizes the contributions of , and in designing the VCG mechanism.', '1611.03044-2-21-1': 'In our proof, we are mindful of the slightly non-standard setting of the electrical markets: that auctions are "reverse-auctions", i.e. with a single buyer and many sellers, and that constraints in the optimization problem may be non-standard.', '1611.03044-2-22-0': 'Given the clearing model of [REF].', '1611.03044-2-22-1': '[a)] The energy procurement auction under VCG choice function and payment rule is a Dominant-Strategy Incentive-Compatible (D.S.I.C) mechanism.', '1611.03044-2-22-2': 'The VCG outcomes are efficient, that is, the sum of all the utilities is maximized.', '1611.03044-2-22-3': 'The Clarke pivot rule ensures positive transfers and individual rationality.', '1611.03044-2-23-0': 'a) We distinguish between the participant [MATH] placing a generic bid [MATH] and biding truthfully [MATH].', '1611.03044-2-23-1': 'For [MATH], substituting the VCG choice function and payment rule with [MATH] as in [REF]: -0.3em [EQUATION] where the term in brackets is the cost [MATH] of [MATH] but evaluated at [MATH].', '1611.03044-2-23-2': 'For [MATH], however, [MATH].', '1611.03044-2-23-3': 'Note that [EQUATION]', '1611.03044-2-23-4': 'We then have [MATH] because [MATH] is a feasible suboptimal allocation for the available bids [MATH].', '1611.03044-2-24-0': 'b) Let [MATH] denote the utility gained by the TSO, that is, [MATH].', '1611.03044-2-24-1': 'By Definition [REF] and incentive compatibility, [MATH].', '1611.03044-2-24-2': 'We then have: [MATH].', '1611.03044-2-24-3': 'Hence, [MATH], which is maximized by the clearing model [REF].', '1611.03044-2-25-0': 'c) This can be easily verified substituting [MATH]: [EQUATION]', '1611.03044-2-26-0': 'In summary, all producers have incentive to reveal their true values for price of power in a VCG market.', '1611.03044-2-26-1': 'Thus, it becomes easier for entities to enter the auction, without spending resources in computing optimal bidding strategies.', '1611.03044-2-26-2': 'This can help in achieving market liberalization objectives.', '1611.03044-2-26-3': 'Moreover, from the above theorem it follows that the winners of the auctions are the producers with the lowest true values.', '1611.03044-2-26-4': 'This is because participants bid truthfully and the VCG choice function minimizes the cost of the accepted bids.', '1611.03044-2-27-0': 'Note that the result above are very general.', '1611.03044-2-27-1': 'We do not need to assume any particular form for the term [MATH] and the constraints [MATH].', '1611.03044-2-27-2': 'Furthermore, using the exact same approach in the proof, we can state the exact same theorem for the case in which the participants provide continuous bid curves: [MATH], where [MATH].', '1611.03044-2-27-3': 'The difference is only in notation; we use [MATH] for the optimal bid accepted from player [MATH] instead of the binary variables [MATH].', '1611.03044-2-28-0': 'So, there are persuasive arguments for considering VCG market for control reserves.', '1611.03044-2-28-1': 'However, there are potential disadvantages that must be eliminated.', '1611.03044-2-29-0': 'Suppose the TSO has to procure 800 MW from PowerPlant1, [MATH], who bids [MATH] CHF for 800 MW, and PowerPlant2, [MATH], who bids [MATH] CHF for 800 MW.', '1611.03044-2-29-1': 'Under the VCG mechanism, PowerPlant1 wins the auction receiving a payment of [MATH] CHF.', '1611.03044-2-29-2': 'Suppose now that power plants [MATH] and [MATH] entered the auction each bidding 0 CHF for 200 MW.', '1611.03044-2-29-3': 'Clearly, the new entrants become winners and each of them would receive a VCG payment of [MATH] CHF.', '1611.03044-2-30-0': "This example shows that: (a) producers with very low prices (in this case 0 CHF) could receive very high payments; (b) collusion or shill bidding can increase participants' profits.", '1611.03044-2-30-1': "In fact, [MATH] and [MATH] could be a group of losers who jointly lowered their bids to win the auction, or they could represent multiple identities of the same losing participant (i.e. a power plant with true value greater than 40'000 CHF for 800 MW).", '1611.03044-2-30-2': "Entering the auction with four shills, however, this participant would have received a payment of 4[MATH]40'000 CHF.", '1611.03044-2-31-0': 'Our goal is now to derive conditions that make VCG outcomes competitive and prevent shill bidding or collusion.', '1611.03044-2-32-0': '# Solution approach for VCG market', '1611.03044-2-33-0': 'In coalition game theory, the core is the set of allocations of goods that cannot be improved upon by the formation of coalitions.', '1611.03044-2-33-1': 'identify conditions for a VCG outcome to lie in the core.', '1611.03044-2-33-2': 'Following their analysis we derive conditions for core outcomes in our setting and provide new simpler proofs relevant to our problem formulation that show that shill bidding and collusion can be eliminated from certain class of electricity markets under the VCG mechanism.', '1611.03044-2-34-0': 'Given a game where [MATH] is the set of participants, let [MATH] denote the coalitional value function [EQUATION]', '1611.03044-2-34-1': 'This function provides the optimal objective function, for any subset of participants [MATH] that includes the TSO.', '1611.03044-2-34-2': 'Here, [MATH] is the cost the TSO incurs for the VCG outcome with participants [MATH].', '1611.03044-2-34-3': 'That is, [MATH] is the solution to optimization [REF] with [MATH] for all [MATH], and with additional constraints that [MATH] for all [MATH].', '1611.03044-2-34-4': 'Clearly [MATH] for [MATH] since increasing participation reduces costs.', '1611.03044-2-34-5': 'We thus let [MATH] represent the coalition game associated with the auction.', '1611.03044-2-35-0': 'The [MATH] is defined as follows [EQUATION]', '1611.03044-2-35-1': 'The core is thus the set of all the feasible outcomes, coming from an efficient mechanism (first equality above), that are unblocked by any coalition (the inequality).', '1611.03044-2-35-2': 'We say that an outcome is competitive if it lies in the core; that is, there is no incentive for forming coalitions.', '1611.03044-2-35-3': 'In the previous example, the outcome was not competitive because it was blocked by coalition [MATH].', '1611.03044-2-35-4': "PowerPlant1 was offering only 40'000 CHF for the total amount of 800 MW.", '1611.03044-2-35-5': 'It will be also shown in Theorem [REF] that core outcomes eliminate any incentives for collusions and shill bidding.', '1611.03044-2-36-0': '## Ensuring core payments', '1611.03044-2-37-0': 'Since core outcome is a competitive outcome, we investigate under which conditions the outcomes of the VCG mechanism applied to the control reserve market will be in the core.', '1611.03044-2-37-1': 'Note that there are [MATH] constraints that define a core outcome.', '1611.03044-2-37-2': 'Our first Lemma provides an equivalent characterization of the core with significantly lower number of constraints.', '1611.03044-2-38-0': 'Given a VCG auction [MATH], let [MATH] be its outcome and [MATH] the corresponding winners.', '1611.03044-2-38-1': 'Assuming participants revealed their true values, [MATH] if and only if, [MATH], [EQUATION]', '1611.03044-2-38-2': 'Core constraints with [MATH] are immediately satisfied as [MATH] (individual rationality, Theorem 1c).', '1611.03044-2-38-3': 'Now, [MATH] is unblocked by any [MATH] (since [MATH] is the outcome with [MATH] participants).', '1611.03044-2-38-4': 'Thus, [MATH].', '1611.03044-2-38-5': 'Moreover, fixing [MATH], the dominant constraints are those corresponding to minimal [MATH], in particular, when [MATH] (this being maximal set with [MATH] not taking part in the coalition [MATH]).', '1611.03044-2-38-6': 'Finally recall from [REF] that, under the VCG mechanism, [MATH].', '1611.03044-2-38-7': 'The following definition and theorem act over subsets of participants.', '1611.03044-2-38-8': 'Here, we imagine that there is a set of potential participants [MATH] and, for each subset [MATH] of [MATH], we consider whether the outcome of the auction with L participants lies in the core.', '1611.03044-2-39-0': 'Participant [MATH] displays payoff monotonicity if [MATH], [EQUATION]', '1611.03044-2-39-1': 'The outcome of the VCG auction [MATH] lies in the core for all [MATH] if and only if payoff monotonicity holds for each participant in [MATH].', '1611.03044-2-40-0': 'To prove that payoff monotonicity is sufficient for [MATH] to lie in the core, we prove that [REF] holds.', '1611.03044-2-40-1': 'Let [MATH].', '1611.03044-2-40-2': 'Considering [MATH], we notice that: [MATH] since [MATH] displays payoff monotonicity over [MATH]; we also have [MATH] since [MATH] displays payoff monotonicity over [MATH].', '1611.03044-2-40-3': 'We can continue with the same considerations up to [MATH] since [MATH] displays payoff monotonicity over [MATH].', '1611.03044-2-40-4': 'Therefore, [MATH].', '1611.03044-2-40-5': 'This same argument holds for any subset of participants [MATH] and [MATH].', '1611.03044-2-40-6': 'Thus [REF] holds and so, by Lemma [REF], the VCG outcome belongs to the core.', '1611.03044-2-41-0': 'To prove that payoff monotonicity is also necessary for outcomes to lie in the core, suppose that [MATH] does not display payoff monotonicity.', '1611.03044-2-41-1': 'Then, there exist sets [MATH],[MATH] where [REF] does not hold.', '1611.03044-2-41-2': 'We may chose [MATH] for some [MATH].', '1611.03044-2-41-3': 'To see this, take [MATH] and [MATH] with [MATH], then, since payoff monotonicity does not hold, [EQUATION]', '1611.03044-2-41-4': 'The strict inequality above must hold for one of the summands [MATH].', '1611.03044-2-41-5': 'So we may consider sets [MATH] that differ by one participant, say [MATH].', '1611.03044-2-41-6': 'Let [MATH] and [MATH] be the minimal such sets.', '1611.03044-2-41-7': 'By minimality, [MATH] for [MATH].', '1611.03044-2-41-8': 'Further, after rearranging the above inequality we see that [MATH].', '1611.03044-2-41-9': 'That is both participant [MATH] and [MATH] are winners of the VCG auction with participants [MATH] (instead of [MATH]).', '1611.03044-2-41-10': 'Then, considering Lemma [REF] for the auction [MATH] with [MATH], and [MATH], we have: [MATH].', '1611.03044-2-41-11': 'Thus [REF] does not hold since the outcome [MATH] is blocked by coalition [MATH].', '1611.03044-2-41-12': 'Whether a VCG outcome is competitive hence depends on a particular property of the optimal cost [MATH].', '1611.03044-2-41-13': 'Namely, [MATH] has to make [REF] hold for each [MATH].', '1611.03044-2-41-14': 'Note that a similar result was proven in , for a sale auction of a finite number of objects, without any constraints.', '1611.03044-2-41-15': 'Our result generalizes this to markets with continuous goods and arbitrary social planner objectives of the form [REF].', '1611.03044-2-42-0': '## Single stage electricity procurement auction', '1611.03044-2-43-0': 'The class of auctions cleared by [REF] is very general and suitable for mechanisms with multiple stages of decisions.', '1611.03044-2-43-1': 'We will see, in fact, how the two-stages Swiss clearing model described in can be abstracted as in [REF].', '1611.03044-2-43-2': 'But first, we start considering simpler auctions, characterized by single-stage decisions.', '1611.03044-2-43-3': 'More specifically, energy procurement auctions where the TSO has to procure a fixed amount of [MATH] MW, subject to conditional offer constraints.', '1611.03044-2-43-4': 'Hence, we consider auctions cleared by: [EQUATION]', '1611.03044-2-43-5': 'The model above is a simple clearing model within class [REF].', '1611.03044-2-43-6': "We can now derive conditions on participants' bids to ensure pay-off monotonicity, condition [REF], is satisfied.", '1611.03044-2-43-7': 'Thus, we derive conditions under which the outcome of auctions cleared by [REF], [REF] would lie in the core.', '1611.03044-2-44-0': 'Given [REF], [REF] if [MATH] with [MATH] implies that [EQUATION] for each [MATH], then bidders satisfy payoff monotonicity condition [REF] under the VCG payment rule.', '1611.03044-2-45-0': 'In words, marginally increasing cost condition [REF] implies core outcomes, and thus eliminates incentives for collusions.', '1611.03044-2-45-1': 'Condition [REF] is visualized in Figure [REF].', '1611.03044-2-46-0': 'To prove that condition [REF] is sufficient for payoff monotonicity, the following Lemma is needed.', '1611.03044-2-47-0': 'Under clearing model [REF], for an auction with participants [MATH] and [MATH] with corresponding optimal power allocations [MATH] and [MATH], condition [REF] implies that [EQUATION].', '1611.03044-2-48-0': 'In the following proofs we apply the notation that if [MATH] is the accepted power allocation from bidder [MATH], then [MATH] is the associated cost bid from [MATH].', '1611.03044-2-48-1': 'If the accepted allocation is zero, we define [MATH].', '1611.03044-2-49-0': 'The proof follows by contradiction.', '1611.03044-2-49-1': 'That is, we will show that when [MATH] is such that [MATH], for some [MATH] then [MATH] can be modified to provide a lower cost allocation, [MATH], for participants [MATH] (thus contradicting optimality of [MATH]).', '1611.03044-2-49-2': 'First, we notice that since bids are equally spaced by [MATH] MW [REF] and satisfy condition [REF], it is never optimal to accept more than [MATH] MW ([MATH]).', '1611.03044-2-50-0': 'Now, assume [MATH] is such that [MATH], for some [MATH].', '1611.03044-2-50-1': "In order to procure exactly [MATH] MW, some participants' accepted MWs must decrease, that is, the set [MATH] is non-empty.", '1611.03044-2-50-2': 'Consider a feasible allocation [MATH] for the auction with [MATH] participants where [MATH] units of power are procured and [EQUATION]', '1611.03044-2-50-3': 'So, [MATH] is constructed from [MATH] by transferring [MATH] units of power from participants in [MATH] to participants in [MATH].', '1611.03044-2-50-4': 'In doing so, the inequality [MATH] can be maintained: [EQUATION].', '1611.03044-2-50-5': "The above inequality holds because when summing over [MATH], [MATH]'s sums to [MATH] and [MATH]'s sums to [MATH].", '1611.03044-2-51-0': 'Since [MATH] is optimal for participants [MATH] and [MATH] is not: [EQUATION] where we used [MATH] as a short-hand-notation for the cost corresponding to choosing [MATH] bids.', '1611.03044-2-52-0': 'Now, we use [REF] to replace the summations over [MATH] in [REF].', '1611.03044-2-52-1': 'In particular, define [MATH] so that [EQUATION].', '1611.03044-2-52-2': 'Note that [MATH] is feasible since [MATH] and [MATH] have the same sum over [MATH] (and thus cancel) and [MATH] is feasible.', '1611.03044-2-52-3': 'Further, since [MATH], by condition [REF]: [EQUATION]', '1611.03044-2-52-4': 'Adding [REF] to both side of [REF] (and canceling [MATH]) gives [EQUATION] which contradicts the optimality of [MATH].', '1611.03044-2-52-5': 'magentaMK: why is this in the footnote?', '1611.03044-2-52-6': 'if it is an assumption, it should be stated.normalsizeRed PGS: I was thinking more about it.', '1611.03044-2-52-7': 'What happens is that, if none in K has strictly increasing marginal cost, an allocation where [MATH] can be as optimal as the one where the accepted amounts are all decreased.', '1611.03044-2-53-0': 'If [MATH] has constant marginal cost (over such interval of MWs), in [REF] we have an equality.', '1611.03044-2-53-1': 'Then, we also have equality in the equation below.', '1611.03044-2-53-2': 'If we define as tie-breaking rule to accept always the lowest amount of MWs from participants with constant marginal cost (in the interval over which exist more than one optimal solution), we get the contradiction.', '1611.03044-2-53-3': 'Because we found an equivalent allocation that leads to the same cost but where for each [MATH] MWs are accepted.', '1611.03044-2-53-4': 'where for each [MATH], [MATH] is such that [MATH].', '1611.03044-2-53-5': 'Then, adding either side of [MATH] to either side of [REF], gives, [EQUATION]', '1611.03044-2-53-6': 'Being the left hand side a feasible allocation (in fact [MATH]), we ended up in a contradiction.', '1611.03044-2-54-0': 'A consequence of the above Lemma is that given the optimal allocation to procure [MATH] MW, for any lower amount of MWs (while being multiple of [MATH]) to be procured, the total MWs accepted from each participant never increases.', '1611.03044-2-54-1': 'Now, we are ready to prove Theorem [REF].', '1611.03044-2-55-0': 'We prove that under condition [REF], inequality [REF] holds for each [MATH], for any [MATH] with [MATH] being a generic new entered participant.', '1611.03044-2-55-1': 'Since [MATH] is a generic set and [MATH] can be any new participant, this is sufficient to prove that the payoffs are monotonic over all the possible couple of sets (the generalization to arbitrary sets [MATH] can then be achieved by the interpolating sums, as was done in [REF]).', '1611.03044-2-55-2': 'We adopt the same notation used in Lemma [REF] and we identify with [MATH] the set of winners.', '1611.03044-2-56-0': 'For each [MATH] we have by definition [MATH].', '1611.03044-2-56-1': 'Thus [MATH], (since the optimal solution is unchanged when [MATH] is removed from S).', '1611.03044-2-56-2': 'By Lemma [REF], [MATH] and so [MATH] also.', '1611.03044-2-56-3': 'Thus, payoff monotonicity holds for [MATH]: [MATH] for [MATH].', '1611.03044-2-56-4': 'This says that a loser of the auction cannot become a winner as more participants enter.', '1611.03044-2-57-0': 'For each winning participant, [MATH], recall that [MATH].', '1611.03044-2-57-1': 'Adopting the same notation of Lemma [REF], we can indicate it as: [EQUATION] where [MATH] are the optimal amounts to be accepted from [MATH], when [MATH] exits the auction.', '1611.03044-2-57-2': 'By Lemma [REF], in fact, [MATH].', '1611.03044-2-57-3': 'Similarly, after participant [MATH] enters the auction, [MATH].', '1611.03044-2-57-4': 'That is, [EQUATION] where [MATH] are the amounts accepted from [MATH] when [MATH] exits the new auction.', '1611.03044-2-57-5': 'By Lemma [REF], we again have [MATH].', '1611.03044-2-58-0': 'Notice that so far we applied Lemma [REF] to justify the increase of the accepted amounts, first, from each [MATH] and now from [MATH], due to the exit of [MATH] from the auctions.', '1611.03044-2-58-1': 'We can apply Lemma [REF] again and affirm that [MATH], and in particular [MATH], because of the entrance of [MATH].', '1611.03044-2-59-0': 'We now find suitable lower and upper bounds to ensure inequality [MATH].', '1611.03044-2-59-1': 'First, note that [MATH], where [MATH] are the cheapest allocation to procure [MATH] MW among [MATH].', '1611.03044-2-59-2': "By Lemma [REF] we have [MATH] , since [MATH]'s sum to [MATH] (due to [MATH]), and [MATH]'s sum to [MATH].", '1611.03044-2-59-3': "Moreover, since [REF] holds and every [MATH] satisfies [REF], [MATH]'s are such that [MATH], because exactly [MATH] MW are purchased.", '1611.03044-2-59-4': 'Using the above suboptimal allocation, we have a lower bound for [MATH]: [EQUATION]', '1611.03044-2-59-5': 'Defining now [MATH] we must have [MATH] and [MATH] , since the right hand side is a feasible cost to procure [MATH] MW among the participants [MATH].', '1611.03044-2-59-6': 'Indeed, [MATH] and [MATH].', '1611.03044-2-59-7': 'Hence, we have: [EQUATION]', '1611.03044-2-59-8': "Moreover, since [MATH]'s satisfy [REF], we have: [EQUATION]", '1611.03044-2-59-9': 'The above holds because [MATH] , [MATH] and [MATH].', '1611.03044-2-59-10': 'In particular, [MATH] are the amounts accepted to procure [MATH] MW among [MATH], while [MATH] are to procure the same MWs among [MATH].', '1611.03044-2-59-11': 'Then, combining [REF] , [REF] and [REF], we finally obtain [MATH]', '1611.03044-2-60-0': "Condition [REF] on every participant's bids hence is sufficient to ensure that our VCG procurement auctions will always have core outcomes.", '1611.03044-2-60-1': 'While we do not show here that the condition is necessary, we illustrate that there are certainly auctions where condition [REF] is violated and for which payoff monotonicity does not hold.', '1611.03044-2-61-0': 'Consider Example [REF] where power plants [MATH] and [MATH] placed just one bid for 800 MW hence violating condition [REF].', '1611.03044-2-61-1': 'It is easy to see that the payoffs of each of the four winners are not monotonic.', '1611.03044-2-61-2': 'In fact, if just one of them (e.g. [MATH]) was participating, he would receive no payment; when [MATH],[MATH],[MATH] enter the auction, however, he becomes a winner hence making positive profit.', '1611.03044-2-61-3': 'Suppose now that [MATH] and [MATH] bid accordingly to [REF], but the bids [MATH] have a decreasing marginal cost: [MATH], [MATH].', '1611.03044-2-61-4': 'In this case, when [MATH] participates alone, he receives a VCG payment of [MATH] CHF; when [MATH],[MATH],[MATH] enter the auction, however, he receives [MATH] CHF.', '1611.03044-2-62-0': 'As previously anticipated, we are now able to prove that the condition derived also makes collusions and shill bidding unprofitable.', '1611.03044-2-62-1': 'Therefore, the participants are better off with their dominant strategies, which is truthful bidding.', '1611.03044-2-62-2': 'Although the result is well-known in literature , and motivates the choice of the core as a competitive standard, we can now prove it using the tools developed so far for the problem at hand.', '1611.03044-2-63-0': 'Consider a generic VCG auction [MATH] cleared by [REF].', '1611.03044-2-63-1': 'If [MATH] , [MATH] satisfies condition [REF].', '1611.03044-2-63-2': 'Then,', '1611.03044-2-64-0': 'Recall that under condition [REF] the participants display payoff monotonicity (Theorem [REF]).', '1611.03044-2-65-0': '(i) Let [MATH] be a set of colluders who would lose the auction when bidding their true values [MATH], while bidding [MATH] they become winners.', '1611.03044-2-65-1': 'Defining [MATH] and [MATH], the VCG payment that each player [MATH] in [MATH] receives is [EQUATION] where the first equality comes from definition of VCG payment, the first inequality comes from the fact that [MATH] since [MATH] displays payoff monotonicity and [MATH] because, when [MATH] decrease their bids, less MWs are being accepted from [MATH] (Lemma [REF]) and [MATH] is bidding with increasing marginal cost.', '1611.03044-2-65-2': 'The last equality comes from the fact that [MATH] originally was a group of non-winners.', '1611.03044-2-65-3': 'Then, [MATH], [MATH] is bounded by the payment that [MATH] would receive when he is the only one lowering its bid.', '1611.03044-2-65-4': 'By Theorem [REF].', '1611.03044-2-65-5': 'a he will not face any benefit in doing so.', '1611.03044-2-66-0': '(ii) We denote with [MATH] multiple identities of the same participant [MATH].', '1611.03044-2-66-1': 'Since every participant bids accordingly to [REF], the outcome is guaranteed to lie in the core.', '1611.03044-2-66-2': 'Hence, by Lemma [REF] and substituting [MATH], we have: [EQUATION] where [MATH] is the cost when [MATH], or equivalently [MATH], is removed from the auction.', '1611.03044-2-66-3': 'Therefore, the total payment that [MATH] would receive is bounded by the one he would receive bidding as a single participant.', '1611.03044-2-66-4': 'Making use of shills, hence, is not profitable.', '1611.03044-2-67-0': 'To confirm the previous theoretical results, we come back to Example 1, where the TSO had to procure a fixed amount of 800 MW.', '1611.03044-2-67-1': 'That is a simple auction cleared by [REF].', '1611.03044-2-67-2': 'Suppose that now power plants [MATH] and [MATH] bid according to [REF] (with [MATH] = 200 MW) and condition [REF].', '1611.03044-2-67-3': 'The available bids are now: [MATH], [MATH], [MATH].', '1611.03044-2-67-4': 'The winners of the auction are still power plants [MATH],[MATH],[MATH],[MATH] but now the VCG payments that they receive is [MATH] CHF [MATH].', '1611.03044-2-67-5': 'The total cost incurred by the TSO is much lower than before and no coalition of players now blocks the outcome.', '1611.03044-2-67-6': "If [MATH],[MATH],[MATH] and [MATH] were multiple identities of the same losing participant (i.e. a power plant with true value greater than 40'000 CHF for 800 MW), shill bidding would become unprofitable (as expected).", '1611.03044-2-67-7': "If, moreover, they were losing participants who jointly lowered their bids, the payments of 8'000 CHF surely made at least one of them to have negative profit.", '1611.03044-2-68-0': 'The diagram in Fig. [REF] summarizes and links the concepts we developed so far.', '1611.03044-2-68-1': 'Notice that Lemma [REF], Theorem [REF] and Theorem [REF] are specific for the class of auctions [REF].', '1611.03044-2-69-0': '## Application to two-stage stochastic market', '1611.03044-2-70-0': 'As we anticipated, the Swiss reserve market as described in can be modeled abstractly according to the optimization problem [REF].', '1611.03044-2-70-1': 'There are two stages of decision variables corresponding to weekly ([MATH]) and daily ([MATH]) bids.', '1611.03044-2-70-2': 'Weekly bids are available at the instance of optimization, whereas daily bids are unknown.', '1611.03044-2-70-3': 'A number of stochastic scenarios corresponding to likely possibilities of daily bids based on their past values is used in the optimization ([MATH]).', '1611.03044-2-70-4': 'The cost function corresponds to the cost of weekly bids and the expected cost of daily bids.', '1611.03044-2-70-5': 'Thus, the cost can be written as [MATH].', '1611.03044-2-70-6': 'The choice function determines the accepted weekly bids.', '1611.03044-2-71-0': 'The function [MATH] captures three types of constraints: (a) those corresponding to procurement of certain amount of tertiary reserves; (b) probabilistic constraints, which ensure that with sufficiently high probabilities, the supply and demand of power is balanced; (c) those corresponding to conditional bids.', '1611.03044-2-71-1': 'Constraint (b) links the daily and weekly variables.', '1611.03044-2-71-2': 'Constraints (a) and (c) correspond to those present in the optimization formulation [REF].', '1611.03044-2-72-0': 'It follows from the analysis of Section [REF], that the VCG mechanism applied to the two-stage stochastic market is an incentive compatible dominant strategy mechanism with socially efficient outcome.', '1611.03044-2-72-1': 'Due to coupling of the two stage decision variables, the analysis of the core payment is significantly more difficult.', '1611.03044-2-72-2': 'In particular, the result derived in Theorem [REF] do not readily apply.', '1611.03044-2-72-3': 'The amount of procured MWs is not anymore fixed and thus [REF] is not well defined.', '1611.03044-2-72-4': 'Selecting [MATH] infinitely small (forcing participants to provide continuous bid curves) and linearizing the probabilistic constraints (b), however, we could show that under condition [REF] this clearing model follows the same regularity property of Lemma [REF].', '1611.03044-2-72-5': 'Whether this makes all the participants display payoff monotonicity is a subject of our current study.', '1611.03044-2-72-6': 'Nevertheless, in the numerical example section, we evaluate the performance of the VCG mechanism and compare it to the pay-as-bid mechanism.', '1611.03044-2-73-0': '# Simulations and Analysis', '1611.03044-2-74-0': 'The following simulations are based on the bids placed in the 46th Swiss weekly procurement auction of 2014, where 21 power plants bid for secondary reserves, 25 for tertiary positive and 21 for tertiary negative reserves.', '1611.03044-2-74-1': 'Note that the secondary reserves are symmetric, that is, participants need to provide same amount of positive and negative power.', '1611.03044-2-74-2': 'Tertiary reserves are on the other hand asymmetric.', '1611.03044-2-74-3': 'Thus, participants bid for tertiary negative [MATH], and tertiary positive [MATH].', '1611.03044-2-74-4': 'As in , probabilistic scenarios for future daily auctions are assumed.', '1611.03044-2-74-5': 'The amount of daily reserves is based on the data of the previous week.', '1611.03044-2-74-6': 'Three scenarios are considered corresponding to nominal, high (20% higher) and low prices (20% lower) compared to the previous week.', '1611.03044-2-75-0': 'The corresponding outcome of the pay-as-bid mechanism and the VCG mechanism is shown in Table [REF].', '1611.03044-2-75-1': 'Note that in reality, in a repeated bidding process, the VCG mechanism would lead to different bidding behaviors, which we have not modeled here.', '1611.03044-2-76-0': 'Recall that in a pay-as-bid mechanism, a rational participant will overbid to ensure positive profit.', '1611.03044-2-76-1': 'Unfortunately, it is hard to know the true values of the bids for each participant.', '1611.03044-2-76-2': 'So, it is hard to have an accurate comparison between the VCG and pay-as-bid based on past data.', '1611.03044-2-76-3': "We now scale all the bid prices down by 90%, assuming that those were participants' true values and hence the bids that they would have placed under the VCG mechanism.", '1611.03044-2-76-4': 'The outcome of both mechanisms is shown in Table [REF].', '1611.03044-2-77-0': 'All the results are proportional (as it could be expected) and the sum of VCG payments is lower than the sum of pay-as-bid payments we had in the first scenario.', '1611.03044-2-77-1': "This means that assuming such scaled bids were participants' true values, the VCG mechanism would have led to a lower procurement cost than the implemented pay-as-bid mechanism.", '1611.03044-2-77-2': 'Hence, the VCG mechanism, apart from leading to a dominant strategy equilibrium with an efficient allocation, would have been beneficial also in terms of costs, for this particular case study based on the past data.', '1611.03044-2-78-0': 'This does not happen in generic VCG auctions.', '1611.03044-2-78-1': 'In particular, the cost incurred by the auctioneer in a VCG auction is usually higher than the cost under a pay-as-bid mechanism, considering the same set of bids.', '1611.03044-2-78-2': 'To see this, recall that the VCG payments are [MATH].', '1611.03044-2-78-3': 'These payments measure the benefit that each participant brings to the auction.', '1611.03044-2-78-4': 'When the VCG payments are computed through the two-stage stochastic optimization algorithm of we observed that the costs are not significantly different from the pay-as-bid mechanism.', '1611.03044-2-78-5': 'Intuitively, the two-stage market softens the benefit that every participant brings to the weekly auction: his accepted bids can always be replaced by amounts of MWs allocated to the future.', '1611.03044-2-78-6': 'In fact, the amounts of MWs bought in the weekly auction are not fixed and they are flexible depending on the future daily bids available in that week.', '1611.03044-2-79-0': 'To confirm the intuition above, we now assume that we had perfect information about the future daily auctions.', '1611.03044-2-79-1': 'As such, we run a deterministic auction assuming that the TSO already knew that the optimal amounts to be purchased were 409 MW for SRL , 114 MW for TRL- and 100 MW for TRL+ as predicted in Table [REF] .', '1611.03044-2-79-2': 'Given fixed MWs to be procured, the auction is cleared by the simplified model [REF].', '1611.03044-2-79-3': 'In this case, naturally, we have the same winners of the auction as the previous case for both VCG and pay-as-bid mechanism.', '1611.03044-2-79-4': 'The VCG payments however are significantly higher than the pay-as-bid payments.', '1611.03044-2-79-5': 'The results are shown in Table [REF].', '1611.03044-2-80-0': 'The result can be explained as follows.', '1611.03044-2-80-1': 'When a winner [MATH] is removed from the auction (to compute the term [MATH]) the amounts of MWs to be accepted among the other participants originally were subject to flexibility due to two stage decision variables and lack of a fixed total amount for each type of reserve SRL, TRL-, TRL+.', '1611.03044-2-80-2': 'If these total reserves are fixed for each type, the benefit that every participant brings to the Swiss weekly auction is much higher, and this results in higher VCG payments.', '1611.03044-2-81-0': 'The mixed integer optimization problems were solved with GUROBI, on a quad-core computer with processing speed 1.7 GHz and memory 4 Gb.', '1611.03044-2-81-1': 'The first two simulations had a computation time of 9 min, with an average of 17 s for each optimal cost [MATH].', '1611.03044-2-81-2': 'The last simulation took 7 min, with an average of 14 s for each [MATH].', '1611.03044-2-82-0': '# Conclusion', '1611.03044-2-83-0': 'We developed a VCG market mechanism for electricity markets, motivated by the set-up of the control reserves (ancillary services) market.', '1611.03044-2-83-1': 'We showed that the mechanism results in an incentive compatible dominant strategy Nash equilibrium.', '1611.03044-2-83-2': 'Furthermore, this mechanism is socially efficient.', '1611.03044-2-83-3': 'Through examples, we showed that shill bidding can occur.', '1611.03044-2-83-4': 'We thus, derived conditions under which a deterministic procurement mechanism can guarantee no shill bidding and thus competitive outcomes.', '1611.03044-2-83-5': 'These findings, both theoretical and empirical, act to support the application of VCG mechanism for the electricity markets under consideration.', '1611.03044-2-83-6': 'By removing incentives for collusion and by providing a simple truthful mechanism, we expect that the implementation simplifies the biding process, increases markets efficiency and encourages participation from increasing number of entities.', '1611.03044-2-83-7': 'We verified our results based on market data available from Swissgrid.', '1611.03044-2-83-8': 'Future work consists of deriving conditions under which Theorem [REF] could be extended to the stochastic market.', '1611.03044-2-84-0': 'We thank Farzaneh Abbaspour and Marek Zima from Swissgrid for helpful discussions.', '1611.03044-2-84-1': '2em'} | [['1611.03044-1-3-0', '1611.03044-2-3-0'], ['1611.03044-1-3-1', '1611.03044-2-3-1'], ['1611.03044-1-3-2', '1611.03044-2-3-2'], ['1611.03044-1-3-3', '1611.03044-2-3-3'], ['1611.03044-1-3-4', '1611.03044-2-3-4'], ['1611.03044-1-3-5', '1611.03044-2-3-5'], ['1611.03044-1-3-6', '1611.03044-2-3-6'], ['1611.03044-1-3-7', '1611.03044-2-3-7'], ['1611.03044-1-82-0', '1611.03044-2-83-0'], ['1611.03044-1-82-1', '1611.03044-2-83-1'], ['1611.03044-1-82-2', '1611.03044-2-83-2'], ['1611.03044-1-82-3', '1611.03044-2-83-3'], ['1611.03044-1-82-4', '1611.03044-2-83-4'], ['1611.03044-1-82-5', '1611.03044-2-83-5'], ['1611.03044-1-82-6', '1611.03044-2-83-6'], ['1611.03044-1-82-7', '1611.03044-2-83-7'], ['1611.03044-1-82-8', '1611.03044-2-83-8'], ['1611.03044-1-54-0', '1611.03044-2-55-0'], ['1611.03044-1-54-1', '1611.03044-2-55-1'], ['1611.03044-1-54-2', '1611.03044-2-55-2'], ['1611.03044-1-6-0', 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['1611.03044-1-22-0', '1611.03044-2-22-0'], ['1611.03044-1-22-1', '1611.03044-2-22-1'], ['1611.03044-1-22-2', '1611.03044-2-22-2'], ['1611.03044-1-22-3', '1611.03044-2-22-3'], ['1611.03044-1-37-0', '1611.03044-2-38-0'], ['1611.03044-1-37-1', '1611.03044-2-38-1'], ['1611.03044-1-37-2', '1611.03044-2-38-2'], ['1611.03044-1-37-3', '1611.03044-2-38-3'], ['1611.03044-1-37-5', '1611.03044-2-38-5'], ['1611.03044-1-37-6', '1611.03044-2-38-6'], ['1611.03044-1-37-7', '1611.03044-2-38-7'], ['1611.03044-1-37-8', '1611.03044-2-38-8'], ['1611.03044-1-63-0', '1611.03044-2-64-0'], ['1611.03044-1-65-0', '1611.03044-2-66-0'], ['1611.03044-1-65-1', '1611.03044-2-66-1'], ['1611.03044-1-65-2', '1611.03044-2-66-2'], ['1611.03044-1-65-3', '1611.03044-2-66-3'], ['1611.03044-1-65-4', '1611.03044-2-66-4'], ['1611.03044-1-12-0', '1611.03044-2-12-0'], ['1611.03044-1-12-1', '1611.03044-2-12-1'], ['1611.03044-1-12-2', '1611.03044-2-12-2'], ['1611.03044-1-12-3', '1611.03044-2-12-3'], ['1611.03044-1-12-4', '1611.03044-2-12-4'], ['1611.03044-1-12-5', '1611.03044-2-12-5'], ['1611.03044-1-12-6', '1611.03044-2-12-6'], ['1611.03044-1-50-0', '1611.03044-2-51-0'], ['1611.03044-1-21-0', '1611.03044-2-21-0'], ['1611.03044-1-21-1', '1611.03044-2-21-1'], ['1611.03044-1-66-0', '1611.03044-2-67-0'], ['1611.03044-1-66-1', '1611.03044-2-67-1'], ['1611.03044-1-66-2', '1611.03044-2-67-2'], ['1611.03044-1-66-3', '1611.03044-2-67-3'], ['1611.03044-1-66-4', '1611.03044-2-67-4'], ['1611.03044-1-66-5', '1611.03044-2-67-5'], ['1611.03044-1-66-6', '1611.03044-2-67-6'], ['1611.03044-1-66-7', '1611.03044-2-67-7'], ['1611.03044-1-2-0', '1611.03044-2-2-0'], ['1611.03044-1-2-1', '1611.03044-2-2-1'], ['1611.03044-1-2-2', '1611.03044-2-2-2'], ['1611.03044-1-2-3', '1611.03044-2-2-3'], ['1611.03044-1-2-4', '1611.03044-2-2-4'], ['1611.03044-1-2-5', '1611.03044-2-2-5'], ['1611.03044-1-13-0', '1611.03044-2-13-0'], ['1611.03044-1-13-1', '1611.03044-2-13-1'], ['1611.03044-1-13-2', '1611.03044-2-13-2'], ['1611.03044-1-13-3', '1611.03044-2-13-3'], ['1611.03044-1-13-4', '1611.03044-2-13-4'], ['1611.03044-1-13-5', '1611.03044-2-13-5'], ['1611.03044-1-5-0', '1611.03044-2-5-0'], ['1611.03044-1-5-1', '1611.03044-2-5-1'], ['1611.03044-1-5-2', '1611.03044-2-5-2'], ['1611.03044-1-5-3', '1611.03044-2-5-3'], ['1611.03044-1-5-4', '1611.03044-2-5-4'], ['1611.03044-1-56-0', '1611.03044-2-57-0'], ['1611.03044-1-56-1', '1611.03044-2-57-1'], ['1611.03044-1-56-2', '1611.03044-2-57-2'], ['1611.03044-1-56-3', '1611.03044-2-57-3'], ['1611.03044-1-56-4', '1611.03044-2-57-4'], ['1611.03044-1-56-5', '1611.03044-2-57-5'], ['1611.03044-1-28-0', '1611.03044-2-29-0'], ['1611.03044-1-28-1', '1611.03044-2-29-1'], ['1611.03044-1-28-2', '1611.03044-2-29-2'], ['1611.03044-1-28-3', '1611.03044-2-29-3'], ['1611.03044-1-71-0', '1611.03044-2-72-0'], ['1611.03044-1-71-1', '1611.03044-2-72-1'], ['1611.03044-1-71-2', '1611.03044-2-72-2'], ['1611.03044-1-71-3', '1611.03044-2-72-3'], ['1611.03044-1-71-4', '1611.03044-2-72-4'], ['1611.03044-1-71-6', '1611.03044-2-72-6']] | [['1611.03044-1-51-4', '1611.03044-2-52-4'], ['1611.03044-1-4-1', '1611.03044-2-4-1'], ['1611.03044-1-0-3', '1611.03044-2-0-3'], ['1611.03044-1-58-2', '1611.03044-2-59-2'], ['1611.03044-1-43-0', '1611.03044-2-44-0'], ['1611.03044-1-71-5', '1611.03044-2-72-5']] | [] | [] | [] | ['1611.03044-1-23-2', '1611.03044-1-25-0', '1611.03044-1-37-4', '1611.03044-1-39-1', '1611.03044-1-39-4', '1611.03044-1-58-6', '1611.03044-1-62-0', '1611.03044-1-62-1', '1611.03044-1-62-2', '1611.03044-1-64-4', '1611.03044-1-83-1', '1611.03044-2-23-2', '1611.03044-2-23-3', '1611.03044-2-25-0', '1611.03044-2-38-4', '1611.03044-2-40-1', '1611.03044-2-40-4', '1611.03044-2-59-6', '1611.03044-2-63-0', '1611.03044-2-63-1', '1611.03044-2-63-2', '1611.03044-2-65-4', '1611.03044-2-84-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1611.03044 | null | null | null | null | null |
1308.5420 | {'1308.5420-1-0-0': "Dissections of a square into smaller squares, with the smaller squares having relatively prime sizes, are known as Mrs Perkins's quilts.", '1308.5420-1-0-1': 'A representation of these dissections using graphs is presented.', '1308.5420-1-0-2': 'The edges are directed and coloured North-South or West-East, and the graph corresponds naturally to the dissection.', '1308.5420-1-0-3': 'This representation allowed the exhaustive generation of all dissections up to order 18, using the plantri software.', '1308.5420-1-0-4': 'The results were cross-checked by generating all dissections of small sizes using a direct approach.', '1308.5420-1-0-5': 'The results confirm and extend several integer sequences.', '1308.5420-1-1-0': '# Introduction', '1308.5420-1-2-0': 'We are interested in dissecting squares into smaller squares; the order of a dissection is the number of smaller squares.', '1308.5420-1-2-1': 'We do not require that the dissection be perfect (that is, that the smaller squares are all of distinct sizes), but we do require that it is prime (that is, that the greatest divisor of the sizes is 1).', '1308.5420-1-2-2': "Such dissections are known as Mrs Perkins's quilts, after the name of a puzzle [CITATION].", '1308.5420-1-3-0': 'We define the function [MATH] to be the least possible order of a prime dissection of a square of side [MATH].', '1308.5420-1-3-1': 'It is known that [MATH]; the lower bound is due to Conway [CITATION] and the upper bound is due to Trustrum [CITATION].', '1308.5420-1-3-2': "Determination of [MATH] is problem C3 in [CITATION], where values were supplied 'with some slightly increasing lack of confidence' for [MATH].", '1308.5420-1-3-3': "The values came from [CITATION], where it was admitted: 'Strictly speaking, these are only upper bounds for [MATH].", '1308.5420-1-3-4': "Readers may care to accept the implied challenge.'", '1308.5420-1-3-5': 'The lack of confidence was justified: lower orders were subsequently found for several values of [MATH], starting with 51.', '1308.5420-1-3-6': 'The best values known to date are listed in the On-Line Encyclopedia of Integer Sequences [CITATION]equence A005670]OEIS.', '1308.5420-1-3-7': "However, some of the uncertainty persists: in the OEIS entry, it is stated that '[MATH] (and possibly 16) proved minimal by J. H. Conway'.", '1308.5420-1-4-0': 'The current article reports an exhaustive generation of prime dissections of squares with orders up to 18.', '1308.5420-1-4-1': 'This is sufficient to confirm the currently conjectured values of [MATH] for [MATH].', '1308.5420-1-4-2': 'Several other sequences are extended.', '1308.5420-1-5-0': 'The current approach is to define a way in which dissections can be represented by graphs, as discussed in the next section.', '1308.5420-1-5-1': 'The representation used here can be compared to the use of graphs in similar areas - most notably, the analogy with electrical networks.', '1308.5420-1-5-2': 'One of the inventors of that analogy (in 1940) has more recently written an entertaining account of that work [CITATION].', '1308.5420-1-5-3': 'The analogy was successfully applied to discovery of perfect dissections [CITATION].', '1308.5420-1-5-4': 'The graph in that approach is a representation of a scan in either the horizontal or the vertical direction.', '1308.5420-1-5-5': 'The vertices correspond to boundaries between squares, and an edge connecting two vertices corresponds to a square that spans those boundaries.', '1308.5420-1-5-6': 'The network can be efficiently solved to give sizes of squares.', '1308.5420-1-5-7': 'Planar imbeddings of these graphs give possible layouts of dissections, but these are not guaranteed to fit together in the other direction, or (if they do fit together) to form squares.', '1308.5420-1-5-8': 'For perfect dissections, the graphs have minimum degree 3, but this simplification would not be appropriate if they were used to investigate imperfect dissections.', '1308.5420-1-5-9': 'Reviews in this area, concentrating on perfect dissections, are by Federico [CITATION] and Anderson [CITATION].', '1308.5420-1-6-0': 'The new approach described here uses entirely different graphs; these require considerably more computation for each graph, including a search for ways to direct and colour it, but the requirements of triangulation and minimum degree 4 reduce the number of candidate graphs.', '1308.5420-1-6-1': 'This graph representation is conceptually straightforward: vertices represent squares, and edges represent connections.', '1308.5420-1-6-2': 'Therefore, similar graphs have been used by several authors, and applied to architectural floor plans [CITATION], for example.', '1308.5420-1-6-3': 'Several variations are discussed by Felsner [CITATION].', '1308.5420-1-6-4': 'In particular, Fusy [CITATION] investigates irreducible triangulations of the 4-gon with transversal structures (which are defined in very similar terms to the direction and colouring of the edges, described in the next section).', '1308.5420-1-6-5': 'Several properties of these graphs are proved, and the results are applied to straight-line drawings of planar graphs.', '1308.5420-1-6-6': 'However, it has been shown [CITATION] that the results are also relevant to square or rectangle dissections.', '1308.5420-1-7-0': 'In contrast to the methods developed to represent and solve specific problems, it is interesting to note an alternative approach [CITATION], where the problem is stated in terms of integer linear programming.', '1308.5420-1-7-1': 'This can then be solved using more general solvers.', '1308.5420-1-7-2': 'This approach may reduce the need for ingenuity and thought; the extra expense in computer time, if any, has not yet been fully investigated.', '1308.5420-1-8-0': '# Representation of dissections by two-coloured, directed graphs', '1308.5420-1-9-0': 'The basis for the generation of the dissections is their representation by graphs.', '1308.5420-1-9-1': 'An example is shown in Figure [REF].', '1308.5420-1-9-2': 'Let [MATH] be a dissection, and [MATH] be a graph that represents [MATH] in the following way:', '1308.5420-1-10-0': 'Properties of [MATH] can be deduced:', '1308.5420-1-11-0': 'The current work used the plantri software [CITATION] to generate an exhaustive set of candidate graphs; in the terminology of this software, we must consider one member of each isomorphism class of imbedded triangulations of a 4-sided disk.', '1308.5420-1-11-1': 'Plantri is well suited to this task.', '1308.5420-1-11-2': 'The plantri software was adapted to restrict the search to graphs with minimum degree 4; this restriction was not available by default for imbeddings of a disk.', '1308.5420-1-11-3': 'Some candidates could be rejected on the basis of the required properties, discussed above.', '1308.5420-1-11-4': 'For all other candidates, all possible directions and colourings were considered, using an exhaustive backtracking search.', '1308.5420-1-11-5': 'Each new direction and colouring was tested for conformance with the required properties discussed above; unsuitable choices were rejected.', '1308.5420-1-11-6': 'The remaining directed and coloured graphs were analysed to deduce, if possible, lengths of squares that would produce a corresponding dissection of the square.', '1308.5420-1-11-7': 'This analysis is described in the next section; a graph can be shown to correspond to a single prime dissection of a square, or shown not to correspond to any.', '1308.5420-1-11-8': 'Thus, an exhaustive search was conducted for dissections of small orders; results are presented in Section [REF].', '1308.5420-1-12-0': '# Deducing sizes', '1308.5420-1-13-0': 'From now on, we will consider only coloured, directed graphs, in specified imbeddings, that satisfy the properties mentioned in the previous section.', '1308.5420-1-13-1': 'For a dissection of order [MATH], the lengths of the squares, [MATH], can be regarded as unknowns, and any such graph can be used to deduce equations that these lengths obey.', '1308.5420-1-13-2': 'For example, consider every path of North-South edges from [MATH] to [MATH], and similarly every West-East path.', '1308.5420-1-13-3': 'The sum of the lengths associated with the non-cardinal vertices in such a path equals the length [MATH] of the dissected square.', '1308.5420-1-13-4': 'Kurz [CITATION] gives an example of these equations (not using the graph representation).', '1308.5420-1-14-0': 'At this stage, [MATH] is also unknown.', '1308.5420-1-14-1': 'However, we may equate the sum of the lengths on each path to unity.', '1308.5420-1-14-2': 'In a successful solution, the resulting lengths will then be rational numbers (since all the equations have integer, indeed unity, coefficients) and will satisfy [MATH].', '1308.5420-1-14-3': 'If all sizes are multiplied by the least common multiple of all their denominators, the result will be a prime dissection with integer lengths.', '1308.5420-1-14-4': 'For some graphs, the equations have a solution with lengths not satisfying [MATH]; this indicates that the graph does not correspond to a dissection of the square.', '1308.5420-1-15-0': 'The number of paths, and hence the number of equations, will depend on the graph.', '1308.5420-1-15-1': 'However, even the horizontal or vertical constraints are sufficient to define a minimal integer solution.', '1308.5420-1-15-2': '(This is noted by Kurz [CITATION].', '1308.5420-1-15-3': 'Acton [CITATION] discusses the solution of similar sets of equations.)', '1308.5420-1-15-4': 'The current equations contain both horizontal and vertical constraints; either these are inconsistent, so that the graph does not correspond to a square dissection, or there are redundant equations in the set.', '1308.5420-1-16-0': 'Some dissections (specifically, those containing crosses) have several graphs.', '1308.5420-1-16-1': 'All graphs were generated, and duplicate solutions were removed.', '1308.5420-1-17-0': 'Some solutions were generated that satisfied the equations, but did not match the graph that defined the equations: for example, solutions contained pairs of squares that were connected in the graph and were correctly aligned in the deduced dissection (for example, in that the lower edge of one had the same vertical coordinate as the top edge of the other) but did not actually touch, even at a cross.', '1308.5420-1-17-1': 'These solutions were discarded; if the dissections were valid, they would also be generated from their correct graph(s).', '1308.5420-1-18-0': 'In fact, the search did not use the approach of building up equations from cardinal-to-cardinal paths, because the full collection of paths is not available until the graph has been fully coloured and directed.', '1308.5420-1-18-1': 'Instead, a greater number of unknowns was used, so that each new direction and colouring of an edge would generate a new equation.', '1308.5420-1-18-2': 'The new equation could then be tested for compatibility with the existing set of equations, and impossible graphs could be rejected at a higher level of the search tree.', '1308.5420-1-18-3': "This approach used [MATH] unknowns in total: the [MATH] lengths of the squares, and also the [MATH] and [MATH] coordinates of the squares' North-West corners.", '1308.5420-1-18-4': 'Some of the unknowns were known immediately: for example, with [MATH] pointing East, we stipulated that [MATH] for all squares on the Western extreme, and these squares were known from their connection to [MATH].', '1308.5420-1-18-5': 'Solving for fractional dissections of a unit square, then a square [MATH] on the Eastern extreme satisfied [MATH].', '1308.5420-1-18-6': 'During the direction/ colouring search, the assignment of a direction and a colour to an edge produced a new equation immediately.', '1308.5420-1-18-7': 'For example, if a West-East edge pointed from square [MATH] to square [MATH], then [MATH].', '1308.5420-1-18-8': 'The additional complication of [MATH] equations was found to be justified by the earlier detection of graphs without solutions as dissections.', '1308.5420-1-19-0': 'Equations were combined by forming them into an upper-triangular matrix, for a specified ordering of the unknowns.', '1308.5420-1-19-1': 'This was effectively an incremental process of Gaussian elimination.', '1308.5420-1-19-2': 'Each new equation was considered in terms of its earliest unknown.', '1308.5420-1-19-3': 'If no equation had previously been noted with that unknown as the earliest, then the new equation contributed new information and could not be inconsistent with the previous equations; the new equation would be duly noted.', '1308.5420-1-19-4': 'Alternatively, if a previous equation already had that unknown as its earliest, then the two equations would be combined to eliminate that unknown.', '1308.5420-1-19-5': 'If the combined equation contained no unknowns, then there were two possible conclusions: if the combined equation contained a non-zero constant term, then the new equation was not consistent with the existing equations; or if the combined equation effectively stated [MATH], then the new equation was redundant, since it was a linear combination of previous equations.', '1308.5420-1-19-6': 'If the combined equation had non-zero coefficients of unknowns, then the process would be repeated in terms of the new earliest unknown.', '1308.5420-1-20-0': 'Integer arithmetic was used to combine equations.', '1308.5420-1-20-1': 'It was verified that overflow did not occur.', '1308.5420-1-20-2': 'The magnitudes of coefficients often increased when equations were combined; to keep the coefficients within manageable magnitudes, it was necessary to divide every newly-combined equation by the greatest common divisor of its coefficients.', '1308.5420-1-20-3': "For order [MATH], the largest considered in this study, all terms had magnitudes less than [MATH], and so it was sufficient to use 64-bit 'long long' integers in the C programming language.", '1308.5420-1-21-0': '# Cross-check', '1308.5420-1-22-0': 'As a check on the correctness of the program, an entirely different method was used to generate all dissections for small lengths.', '1308.5420-1-22-1': 'This method simply considered all possible ways of dissecting squares of length [MATH].', '1308.5420-1-22-2': 'An appropriate method for this was to regard it as an example of the exact cover problem: covering each of the [MATH] unit subsquares by selecting from all possible smaller squares without overlap.', '1308.5420-1-22-3': "Knuth's Dancing Links X algorithm [CITATION] was used.", '1308.5420-1-23-0': 'This alternative generation was used to give a complete generation of solutions of all orders, for sizes [MATH].', '1308.5420-1-23-1': 'This itself enabled a cross-check, because the results agreed with sequences in the OEIS: A045846, A221845 and A224239.', '1308.5420-1-23-2': 'These results also agreed with the five terms of A221844 given in the OEIS, and added four new terms; the details of these and other sequences are listed in the Appendix.', '1308.5420-1-24-0': "The alternative generation was also used to give all solutions with both [MATH] and [MATH], and its results agreed with the main method's.", '1308.5420-1-24-1': 'The numbers of solutions are shown in Table [REF].', '1308.5420-1-25-0': 'The alternative method was not intensively optimized.', '1308.5420-1-25-1': 'The complete generation for [MATH] required 8.5 hours on an Intel Core i7-3770 machine, as a single process.', '1308.5420-1-25-2': 'The equivalent time for [MATH] was 4 minutes, which suggested that optimisation would be justified before trying [MATH].', '1308.5420-1-26-0': "An efficient way to extend A224239 ('Number of inequivalent ways to cut an [MATH] square into squares with integer sides') was to count the number of symmetric solutions.", '1308.5420-1-26-1': 'This was similarly regarded as an exact cover problem, selecting from symmetric collections of smaller squares.', '1308.5420-1-26-2': "These counts were combined with the published values of A045846 ('Number of distinct ways to cut an [MATH] square into squares with integer sides') to deduce the number of asymmetric dissections.", '1308.5420-1-26-3': 'Each asymmetric dissection in A224239 corresponds to 8 dissections in A045846; symmetric dissections correspond to only 4, 2 or 1 dissections, depending on the symmetries.', '1308.5420-1-26-4': 'In this method, the counts for [MATH] took less than 10 seconds.', '1308.5420-1-26-5': 'Results are given in the Appendix.', '1308.5420-1-27-0': '# Results', '1308.5420-1-28-0': 'The exhaustive generation was conducted for orders up to 18.', '1308.5420-1-28-1': 'The computer time increased by a factor close to 10 for each successive order.', '1308.5420-1-28-2': 'For example, orders 15, 16 and 17 required 0.8, 8.0 and 77 hours respectively on a four-core Intel Core i7-3770 machine, running eight processes simultaneously.', '1308.5420-1-28-3': 'Order 18 required 18.4 days on an eight-core Intel Xeon E5-2680 machine, running sixteen processes simultaneously.', '1308.5420-1-29-0': 'Up to symmetry, the numbers of prime dissections of order [MATH] are 1, 0, 0, 1, 0, 1, 2, 6, 16, 56, 183, 657, 2277, 8813, 34178, 137578, 558734, 2285694.', '1308.5420-1-29-1': 'This sequence (A221841) and others are listed in the Appendix.', '1308.5420-1-30-0': 'The results confirm that the known dissections for orders up to 18 (available at, for example, [CITATION]) are the largest possible.', '1308.5420-1-30-1': 'Table [REF] shows that, for example, the single known solution for [MATH], [MATH] is uniquely, up to symmetry, the largest possible at that order.', '1308.5420-1-30-2': 'The largest size, defined as [MATH] in Table [REF], is Sequence A089047.', '1308.5420-1-30-3': 'Since dissections of order 19 are known [CITATION] up to [MATH], then the uncertain, incorrect list in [CITATION] can be definitely replaced by the one in Table [REF].', '1308.5420-1-30-4': 'Thus the uncertainty in Sequence A005670 has been removed - or in fact, of course, displaced to higher [MATH].', '1308.5420-1-31-0': '# Confirmed and extended sequences in OEIS', '1308.5420-1-32-0': 'This appendix lists values from several sequences in the OEIS [CITATION] with previously known values that have been confirmed in the current work (or, conversely, have been used to check the current work).', '1308.5420-1-32-1': 'Also, new (or newly definite) values are shown in italics.', '1308.5420-1-33-0': "These sequences, as quoted in [CITATION], include the trivial dissection of a square into itself (except where 'smaller' is specified, in A018835 and A211302).", '1308.5420-1-34-0': 'Sequences A045846 and A221845 have known values beyond those listed here.', '1308.5420-1-34-1': 'In particular, A045846 is known for [MATH] - and the values for 10, 11 and 12 (1500957422222, 790347882174804 and 781621363452405930) were used to confirm and extend A224239, as described in Section [REF].', '1308.5420-1-35-0': 'A sequence not currently in the OEIS is the number of size collections in prime dissections of integer-sided squares into [MATH] squares, for [MATH]: 1, 0, 0, 1, 0, 1, 1, 2, 4, 7, 18, 40, 119, 323, 1100, 3594, 13068, 47444.'} | {'1308.5420-2-0-0': "Dissections of a square into smaller squares, with the smaller squares having relatively prime sizes, are known as Mrs Perkins's quilts.", '1308.5420-2-0-1': 'A representation of these dissections using graphs is presented.', '1308.5420-2-0-2': 'The edges are directed and coloured North-South or West-East, and the graph corresponds naturally to the dissection.', '1308.5420-2-0-3': 'This representation allowed the exhaustive generation of all dissections up to order 18, using the plantri software.', '1308.5420-2-0-4': 'The results were cross-checked by generating all dissections of small sizes using a direct approach.', '1308.5420-2-0-5': 'The results confirm, extend and introduce several integer sequences.', '1308.5420-2-1-0': '# Introduction', '1308.5420-2-2-0': 'We are interested in dissecting squares into smaller squares; the order of a dissection is the number of smaller squares, which are here called subsquares.', '1308.5420-2-2-1': 'We do not require that the dissection be perfect (that is, that the subsquares are all of distinct sizes), but we do require that it is prime (that is, that the greatest divisor of their sizes is 1).', '1308.5420-2-2-2': "Such dissections are known as Mrs Perkins's quilts, after the name of a puzzle [CITATION].", '1308.5420-2-3-0': 'We define the function [MATH] to be the least possible order of a prime dissection of a square of side [MATH].', '1308.5420-2-3-1': 'It is known that [MATH] for [MATH]; the lower bound is due to Conway [CITATION] and the upper bound is due to Trustrum [CITATION].', '1308.5420-2-3-2': "Determination of [MATH] is problem C3 in [CITATION], where values were supplied 'with some slightly increasing lack of confidence' for [MATH].", '1308.5420-2-3-3': "The values came from [CITATION], where it was admitted: 'Strictly speaking, these are only upper bounds for [MATH].", '1308.5420-2-3-4': "Readers may care to accept the implied challenge.'", '1308.5420-2-3-5': 'The lack of confidence was justified: lower orders were subsequently found for several values of [MATH], starting with 51.', '1308.5420-2-3-6': 'The best values known to date are listed in the On-Line Encyclopedia of Integer Sequences [CITATION]equence A005670]OEIS.', '1308.5420-2-3-7': "However, some of the uncertainty persists: in the OEIS entry, it is stated that '[MATH] (and possibly 16) proved minimal by J. H. Conway'.", '1308.5420-2-4-0': 'The current article reports an exhaustive generation of prime dissections of squares with orders up to 18.', '1308.5420-2-4-1': 'This confirms the currently conjectured values of [MATH] for [MATH].', '1308.5420-2-4-2': 'Several other sequences are extended.', '1308.5420-2-5-0': 'The current approach is to define a way in which dissections can be represented by graphs, as discussed in the next section.', '1308.5420-2-5-1': 'The representation used here can be compared to the use of graphs in similar areas - most notably, the analogy with electrical networks.', '1308.5420-2-5-2': 'One of the inventors of that analogy (in 1940) has more recently written an entertaining account of that work [CITATION].', '1308.5420-2-5-3': 'The electrical-network analogy (here abbreviated to ENA) was successfully applied to discovery of perfect dissections [CITATION].', '1308.5420-2-5-4': 'The graph in that approach is a representation of a scan in either the horizontal or the vertical direction.', '1308.5420-2-5-5': 'The vertices correspond to boundaries between subsquares, and an edge connecting two vertices corresponds to a subsquare that spans those boundaries.', '1308.5420-2-5-6': 'The network can be efficiently solved to give sizes of subsquares.', '1308.5420-2-5-7': 'Planar imbeddings of these graphs give possible layouts of dissections, but these are not guaranteed to fit together in the other direction, or (if they do fit together) to form squares.', '1308.5420-2-5-8': 'For perfect dissections, the graphs have minimum degree 3, but this simplification would not be appropriate if they were used to investigate imperfect dissections.', '1308.5420-2-5-9': 'Reviews in this area, concentrating on perfect dissections, are by Federico [CITATION] and Anderson [CITATION].', '1308.5420-2-6-0': 'The alternative approach described here also uses planar imbeddings of graphs to represent the layout of dissections.', '1308.5420-2-6-1': 'One difference in the graphs is that, as described in Section [REF], vertices here represent subsquares, rather than boundaries between subsquares.', '1308.5420-2-6-2': 'Another difference is that here a graph includes both horizontal and vertical connections, so a set of sizes consistent with a graph will fit together to a full square.', '1308.5420-2-6-3': 'Considerably more computation is required for each graph, including a search for ways to direct and colour it, but the requirements of triangulation and minimum degree 4 reduce the number of candidate graphs.', '1308.5420-2-6-4': 'Despite the differences between the graphs, there are considerable similarities between the equations generated; these are discussed in Section [REF].', '1308.5420-2-7-0': 'The graph representation used here is conceptually straightforward: vertices represent subsquares, and edges represent connections.', '1308.5420-2-7-1': 'Therefore, similar graphs have been used by several authors, and applied to architectural floor plans [CITATION], for example.', '1308.5420-2-7-2': 'Several variations are discussed by Felsner [CITATION].', '1308.5420-2-7-3': 'In particular, Fusy [CITATION] investigates irreducible triangulations of the 4-gon with transversal structures (which are defined in very similar terms to the direction and colouring of the edges, described in the next section).', '1308.5420-2-7-4': 'Fusy proved several properties of these graphs, and applied the results to straight-line drawings of planar graphs.', '1308.5420-2-7-5': 'However, it has been shown [CITATION] that the results are also relevant to square or rectangle dissections.', '1308.5420-2-8-0': 'In contrast to the methods developed to solve specific problems, it is interesting to note an alternative approach [CITATION], where the problem is stated in terms of integer linear programming.', '1308.5420-2-8-1': 'This can then be solved using general solvers.', '1308.5420-2-8-2': 'This approach may reduce the need for ingenuity and thought; the extra expense in computer time, if any, has not yet been fully investigated.', '1308.5420-2-9-0': '# Representation of dissections by two-coloured, directed graphs', '1308.5420-2-10-0': 'The basis for the generation of the dissections is their representation by graphs.', '1308.5420-2-10-1': 'An example is shown in Figure [REF].', '1308.5420-2-10-2': 'Let [MATH] be a dissection, and [MATH] be a graph that represents [MATH] in the following way:', '1308.5420-2-11-0': 'Each cardinal vertex is connected to the two neighbouring cardinals in the cycle.', '1308.5420-2-11-1': 'The direction and colour of these edges is arbitrary; to be consistent with the properties discussed below, they can be North-South edges, directed away from North or towards South.', '1308.5420-2-12-0': 'Properties of [MATH] can be deduced:', '1308.5420-2-13-0': 'The current work used the plantri software [CITATION] to generate an exhaustive set of candidate graphs; in the terminology of this software, we must consider one member of each isomorphism class of imbedded triangulations of a 4-sided disk.', '1308.5420-2-13-1': 'Plantri is well suited to this task.', '1308.5420-2-13-2': 'The plantri software was adapted to restrict the search to graphs with minimum degree 4; this restriction was not available by default for imbeddings of a disk.', '1308.5420-2-13-3': 'Some candidates could be rejected on the basis of the required properties, discussed above.', '1308.5420-2-13-4': 'For all other candidates, all possible directions and colourings were considered, using an exhaustive backtracking search.', '1308.5420-2-13-5': 'Each new direction and colouring was tested for conformance with the required properties discussed above; unsuitable choices were rejected.', '1308.5420-2-13-6': 'The remaining directed and coloured graphs were analysed to deduce, if possible, lengths of subsquares that would produce a corresponding dissection of the square.', '1308.5420-2-13-7': 'This analysis is described in the next section; each graph considered was shown to correspond to a single prime dissection of a square, or shown not to correspond to any.', '1308.5420-2-13-8': 'Thus, an exhaustive search was conducted for dissections of small orders; results are presented in Section [REF].', '1308.5420-2-14-0': '# Deducing sizes', '1308.5420-2-15-0': 'From now on, we will consider only coloured, directed graphs, in specified imbeddings, that satisfy the properties mentioned in the previous section.', '1308.5420-2-15-1': 'For a dissection of order [MATH], the lengths of the subsquares, [MATH], can be regarded as unknowns, and any such graph can be used to deduce equations that these lengths obey.', '1308.5420-2-15-2': 'For example, define a traverse to be a path of North-South edges from [MATH] to [MATH], or an equivalent West-East path.', '1308.5420-2-15-3': 'The sum of the lengths associated with the non-cardinal vertices in a traverse equals the length [MATH] of the dissected square, so each traverse provides an equation.', '1308.5420-2-15-4': 'Kurz [CITATION] gives an example of these equations (not using the graph representation).', '1308.5420-2-16-0': 'At this stage, [MATH] is also unknown.', '1308.5420-2-16-1': "Instead of equating sums of lengths to [MATH], we may define a subsquare's normalised length to be its length divided by [MATH].", '1308.5420-2-16-2': 'Each sum of these normalised lengths is then equated to 1.', '1308.5420-2-16-3': 'In a successful solution, the resulting normalised lengths will then be rational numbers (since all the equations have integer, indeed unity, coefficients); they must also satisfy [MATH].', '1308.5420-2-16-4': 'If all normalised lengths are multiplied by the least common multiple of all their denominators, the result will be a prime dissection with integer lengths.', '1308.5420-2-16-5': 'For some graphs, the equations have a solution with normalised lengths not satisfying [MATH]; this indicates that the graph does not correspond to a dissection of the square.', '1308.5420-2-16-6': 'Whether a solution uses lengths or normalised lengths, a multiplicative constant can be found to give a prime dissection; therefore, it is not always necessary to distinguish between lengths and normalised lengths.', '1308.5420-2-16-7': 'It has been noted previously [CITATION] that solutions to dissections typically provide lengths fixed to within a multiplicative constant.', '1308.5420-2-16-8': 'As discussed in Section [REF], none of the systems of linear equations had more than one solution in the current work.', '1308.5420-2-17-0': 'The number of traverses, and hence the number of equations, will depend on the graph.', '1308.5420-2-17-1': 'However, in the electrical-network analogy (ENA), even the horizontal or vertical constraints are sufficient to define a minimal integer solution.', '1308.5420-2-17-2': '(This is noted by Kurz [CITATION].', '1308.5420-2-17-3': 'Acton [CITATION] discusses the solution of similar sets of equations.)', '1308.5420-2-17-4': 'The current equations contain both horizontal and vertical constraints.', '1308.5420-2-17-5': 'Section [REF] gives an example of the specific equations used and shows that the different approaches are closely related.', '1308.5420-2-18-0': 'Some dissections (specifically, those containing crosses) have several graphs.', '1308.5420-2-18-1': 'All graphs were generated, and duplicate solutions were removed.', '1308.5420-2-19-0': 'Some solutions were generated that satisfied the equations, but did not match the graph that defined the equations: for example, solutions contained pairs of subsquares that were connected in the graph and were correctly aligned in the deduced dissection (for example, in that the lower edge of one had the same vertical coordinate as the top edge of the other) but did not actually touch, even at a cross.', '1308.5420-2-19-1': 'These solutions were discarded; if the dissections were valid, they would also be generated from their correct graphs.', '1308.5420-2-20-0': 'In fact, the search did not use the approach of building up equations from traverses, because the full collection of traverses is not available until the graph has been fully coloured and directed.', '1308.5420-2-20-1': 'Instead, a greater number of unknowns was used, so that each new direction and colouring of an edge would generate a new equation.', '1308.5420-2-20-2': 'We refer to this as the local equation approach.', '1308.5420-2-20-3': 'Each new local equation could then be tested for compatibility with the existing set of equations, and impossible graphs could be rejected at a higher level of the search tree.', '1308.5420-2-20-4': "This approach used [MATH] unknowns in total: the [MATH] lengths of the subsquares, and also the [MATH] and [MATH] coordinates of the subsquares' North-West corners.", '1308.5420-2-20-5': 'Some of the unknowns were known immediately: for example, with [MATH] pointing East, [MATH] for all subsquares on the Western extreme; these subsquares were known from their connection to [MATH].', '1308.5420-2-20-6': 'Solving for fractional dissections of a unit square (in other words, using normalised sizes), then a subsquare [MATH] on the Eastern extreme satisfied [MATH].', '1308.5420-2-20-7': 'During the direction/colouring search, the assignment of a direction and a colour to an edge produced a new equation immediately.', '1308.5420-2-20-8': 'For example, if a West-East edge pointed from subsquare [MATH] to subsquare [MATH], then [MATH].', '1308.5420-2-20-9': 'The additional complication of [MATH] unknowns was found to be justified by the earlier detection of graphs without solutions as dissections.', '1308.5420-2-21-0': 'Equations were combined by forming them into an upper-triangular matrix, for a specified ordering of the unknowns.', '1308.5420-2-21-1': 'This was effectively an incremental process of Gaussian elimination.', '1308.5420-2-21-2': 'Each new equation was considered in terms of its earliest unknown.', '1308.5420-2-21-3': 'If no equation had previously been noted with that unknown as the earliest, then the new equation contributed new information and could not be inconsistent with the previous equations; the new equation would be duly noted.', '1308.5420-2-21-4': 'Alternatively, if a previous equation already had that unknown as its earliest, then the two equations would be combined to eliminate that unknown.', '1308.5420-2-21-5': 'If the combined equation contained no unknowns, then there were two possible conclusions: if the combined equation contained a non-zero constant term, then the new equation was not consistent with the existing equations; alternatively, if the combined equation effectively stated [MATH], then the new equation was redundant, since it was a linear combination of previous equations.', '1308.5420-2-21-6': 'If the combined equation had non-zero coefficients of unknowns, then the process would be repeated in terms of the new earliest unknown.', '1308.5420-2-22-0': 'Integer arithmetic was used to combine equations.', '1308.5420-2-22-1': 'It was verified that overflow did not occur.', '1308.5420-2-22-2': 'The magnitudes of coefficients often increased when equations were combined; to keep the coefficients within manageable magnitudes, it was necessary to divide every newly-combined equation by the greatest common divisor of its coefficients.', '1308.5420-2-22-3': "For order [MATH], all terms had magnitudes less than [MATH], and so 64-bit 'long long' integers in the C programming language were adequate.", '1308.5420-2-23-0': '# Relationship between different approaches', '1308.5420-2-24-0': 'This section presents the equations that apply to the dissection of Figure [REF] using three approaches: the electrical-network analogy (ENA) of [CITATION]; the new approach using traverses; and the new approach using local equations.', '1308.5420-2-24-1': 'By reference to the example, it is shown that the approaches are closely related.', '1308.5420-2-25-0': '## Electrical-network analogy (ENA)', '1308.5420-2-26-0': 'We start with the ENA and consider only the vertical (North-South) scan, whose graph is shown in Figure [REF].', '1308.5420-2-26-1': "In a formulation by the ENA's original authors [CITATION], a matrix [MATH] is defined as follows: [EQUATION]", '1308.5420-2-26-2': 'In general let the number of nodes be [MATH].', '1308.5420-2-26-3': 'For the graph in Figure [REF], with 5 nodes and 8 edges, the matrix is [EQUATION]', '1308.5420-2-26-4': 'A matrix [MATH] is defined: [EQUATION]', '1308.5420-2-26-5': 'The equations to be solved are then: [EQUATION] for all rows of [MATH] except the top and bottom rows: [MATH] (using the original formulation where indices start at 1).', '1308.5420-2-27-0': "The original formulation refers to the unknowns [MATH] as 'potentials', but here we will explore the geometric basis for the equations, so we note that [MATH] is the vertical coordinate of node [MATH]; for example [MATH].", '1308.5420-2-27-1': 'It is observed [CITATION] that one of the coordinates, such as [MATH], can be fixed to be 0 without loss of generality.', '1308.5420-2-27-2': 'Therefore ([REF]) represents [MATH] equations in the remaining [MATH] unknowns.', '1308.5420-2-27-3': 'It has been shown [CITATION] that these equations are linearly independent, and therefore they fix the remaining coordinates to within a multiplicative constant.', '1308.5420-2-28-0': 'In the example, we will take several steps: form [MATH] from [MATH]; remove the top and bottom rows; ignore [MATH] since it is fixed to be 0; take [MATH] to the right of the equation; and divide through by [MATH].', '1308.5420-2-28-1': 'The result is: [EQUATION]', '1308.5420-2-28-2': 'Noting that [MATH] equals [MATH], the length of the square, we have [MATH] linearly independent equations in [MATH] normalised coordinates, giving the solutions [MATH], [MATH] and [MATH].', '1308.5420-2-28-3': 'Figure [REF] confirms that these are correct.', '1308.5420-2-29-0': 'For geometrical insight, it may be preferable to expand [MATH] in terms of [MATH].', '1308.5420-2-29-1': "Each edge in the ENA's graph represents a subsquare of the dissection, and is represented by a column of [MATH].", '1308.5420-2-29-2': 'This column contains a 1 and a [MATH] for the starting and ending nodes.', '1308.5420-2-29-3': "Therefore, [MATH] equals [MATH], where [MATH] is the vector of subsquares' lengths, possibly normalised.", '1308.5420-2-29-4': 'For example, [EQUATION]', '1308.5420-2-29-5': 'Applying the expansion of [MATH] to equation ([REF]) in our example, we obtain: [EQUATION]', '1308.5420-2-29-6': 'The geometric origin of these equations can be found, perhaps surprisingly, from the horizontal traverses in the new approach.', '1308.5420-2-29-7': 'This is discussed in the next section.', '1308.5420-2-30-0': '## Traverse equations', '1308.5420-2-31-0': 'By reference to Figure [REF], the traverses in the example can be found.', '1308.5420-2-31-1': 'From North-South traverses, the equations are: [EQUATION]', '1308.5420-2-31-2': 'From West-East traverses, the equations are: [EQUATION]', '1308.5420-2-31-3': 'Since the right sides of these equations are all [MATH], the same left sides can be expressed in terms of normalised lengths and equated to 1.', '1308.5420-2-31-4': 'This is computationally much more convenient, because [MATH] is not known at the start.', '1308.5420-2-31-5': 'In fact, [MATH] is fixed by the specification of a prime dissection rather than by the geometrical constraints.', '1308.5420-2-32-0': 'The connection with the ENA can be observed by taking the differences of successive equations in ([REF]): the resulting equations are identical to ([REF]).', '1308.5420-2-32-1': 'This is despite the fact that ([REF]) was generated by the vertical scan in the ENA, and ([REF]) from horizontal traverses.', '1308.5420-2-32-2': 'The reason is that both these equations represent constraints on horizontal distances, in equivalent situations, as will now be discussed.', '1308.5420-2-33-0': 'An example of the correspondence is shown in Figure [REF].', '1308.5420-2-33-1': "In the ENA's graph, only edges e and c enter the node, and only edges f, g and h leave the node.", '1308.5420-2-33-2': 'It can be deduced that this part of the dissection must be similar to the part shown in the figure: the West edges of e and f are aligned, and so are the East edges of c and h.', '1308.5420-2-33-3': 'We deduce that [EQUATION] which is identical to the last equation in ([REF]).', '1308.5420-2-33-4': "Thus we have a geometric interpretation of the nodes in the ENA's graph: each node corresponds to a horizontal line in the dissection, extending as far as possible to West and East.", '1308.5420-2-33-5': '(This interpretation is, of course, not novel, being consistent with the original expositions of this method.)', '1308.5420-2-33-6': 'The equations in ([REF]) follow: [MATH] is expanded using ([REF]); [MATH] equals [MATH]; and then each inner row of [MATH] contains [MATH] for each subsquare entering that vertex, and [MATH] for each subsquare leaving that vertex.', '1308.5420-2-34-0': "In the current work's graph, the corresponding feature to an ENA node is a horizontal pocket: two West-East paths that have the same starting vertex and ending vertex as each other, and that enclose no other West-East edges.", '1308.5420-2-34-1': 'An example is shown in Figure [REF].', '1308.5420-2-34-2': 'There is at least one West-East path from the West cardinal vertex to the starting vertex, and at least one West-East path from the ending vertex to the East cardinal vertex.', '1308.5420-2-34-3': 'Therefore there are two traverses that differ only in the vertices in the horizontal pocket.', '1308.5420-2-34-4': 'Equation ([REF]) is then the difference between the corresponding traverse equations.', '1308.5420-2-34-5': 'For the example, the relevant traverse equations are the last two in ([REF]).', '1308.5420-2-35-0': 'The four linear equations in ([REF]) are not sufficient to fix all lengths.', '1308.5420-2-35-1': 'For a full square dissection, the equations in ([REF]) also apply.', '1308.5420-2-35-2': 'We offer no proof that the combined traverse equations are sufficient for all graphs, where sufficient means either fixing all lengths to values consistent with all equations and with a dissection, or forming an inconsistent set of equations.', '1308.5420-2-35-3': 'For all cases where solution has been attempted for a set of traverse equations, they have been found to be sufficient.', '1308.5420-2-36-0': '## Local equations', '1308.5420-2-37-0': 'Here we consider the local equations, which use coordinates [MATH] and [MATH] as well as lengths [MATH].', '1308.5420-2-37-1': 'Whenever a West-East edge connects subsquare a to subsquare b, there is a local equation: [MATH].', '1308.5420-2-37-2': 'Similarly, a North-South edge connecting a to e gives [MATH].', '1308.5420-2-38-0': "Here we show that the local equations are equivalent to the ENA's equations, as conventionally stated in terms of [MATH].", '1308.5420-2-38-1': "We have shown that the analogy's equations can be expressed in terms of subsquare lengths, and that these equations can also be deduced from the traverse equations.", '1308.5420-2-38-2': "The analogy's equations have a single solution for a valid dissection, but this demonstration relies on the equations being stated in terms of [MATH] as in ([REF]).", '1308.5420-2-38-3': 'This ensures that there are only [MATH] unknowns in the [MATH] equations.', '1308.5420-2-38-4': 'This can be regarded as starting from the length-based equations and substituting ([REF]) etc.', '1308.5420-2-39-0': 'The local equations can clearly be combined to form the traverse equations: for example, the West-East traverse through a and b can be assembled from [MATH] and [MATH].', '1308.5420-2-39-1': '(Again we note that uppercase E and lowercase e represent cardinal and subsquare vertices respectively.)', '1308.5420-2-39-2': 'In the ENA, [MATH] was arbitrarily set to 0 and [MATH] was set to either [MATH] or 1.', '1308.5420-2-39-3': 'Similarly, all subsquares connected to the West cardinal vertex have zero [MATH]-coordinate, and [MATH] is [MATH].', '1308.5420-2-39-4': 'Thus we can combine these local equations to give the first traverse equation in ([REF]): [MATH].', '1308.5420-2-39-5': 'In practice, it was easier to produce equations in terms of normalised variables.', '1308.5420-2-40-0': 'When the local equations from North-South edges inside a horizontal pocket are combined, it is clear that there is a single [MATH]-coordinate that applies to all the lower subsquares.', '1308.5420-2-40-1': 'In the example of Figure [REF], [MATH], [MATH] and [MATH] are each equated to [MATH], so they must equal each other.', '1308.5420-2-40-2': 'This is implicit in the ENA, where edges leaving the corresponding vertex are all associated with the same [MATH].', '1308.5420-2-40-3': 'Finally, the expansion of lengths into [MATH]- or [MATH]-coordinates, as in ([REF]), is enforced by the North-South local equations.', '1308.5420-2-40-4': 'Therefore, we put forward the proposition that the local equations can be rearranged to imply the equations from the vertical scan of the ENA, and similarly the horizontal scan.', '1308.5420-2-40-5': 'It has been shown [CITATION] that the ENA equations, from either a horizontal or a vertical scan alone, are sufficient.', '1308.5420-2-40-6': 'Therefore, it is conjectured that the local equations are sufficient.', '1308.5420-2-40-7': 'More concretely, the solution procedure described in Section [REF] can detect whether a set of local equations is underdetermined, and no such sets were found in the current work.', '1308.5420-2-41-0': '# Cross-check', '1308.5420-2-42-0': 'As a check on the results, an entirely different method was used to generate all dissections for small lengths.', '1308.5420-2-42-1': 'This simply considered all possible ways of dissecting squares of length [MATH].', '1308.5420-2-42-2': "This was tackled as an exact cover problem: covering each of the [MATH] unit subsquares by selecting from all possible smaller subsquares without overlap, using Knuth's Dancing Links X algorithm [CITATION].", '1308.5420-2-43-0': 'This alternative generation was used to give a complete generation of solutions of all orders, for sizes [MATH].', '1308.5420-2-43-1': 'This itself enabled a cross-check, because the results agreed with sequences in the OEIS: A045846, A221845 and A224239.', '1308.5420-2-43-2': 'These results also agreed with the five terms of A221844 given in the OEIS, and added four new terms; details are in the Appendix.', '1308.5420-2-44-0': "The alternative generation was also used to give all solutions with both [MATH] and [MATH], and its results agreed with the main method's.", '1308.5420-2-44-1': 'The numbers of solutions are shown in Table [REF].', '1308.5420-2-45-0': 'The alternative method was not intensively optimised.', '1308.5420-2-45-1': 'The complete generation for [MATH] required 8.5 hours on an Intel Core i7-3770 machine, as a single process.', '1308.5420-2-45-2': 'The equivalent time for [MATH] was 4 minutes, which suggested that optimisation would be justified before trying [MATH].', '1308.5420-2-46-0': "An efficient way to extend A224239 ('Number of inequivalent ways to cut an [MATH] square into squares with integer sides') was to count symmetric solutions.", '1308.5420-2-46-1': 'This was similarly regarded as an exact cover problem, selecting from symmetric collections of smaller subsquares.', '1308.5420-2-46-2': "These counts were combined with the published values of A045846 ('Number of distinct ways to cut an [MATH] square into squares with integer sides') to deduce the number of asymmetric dissections.", '1308.5420-2-46-3': 'Each asymmetric dissection in A224239 corresponds to 8 dissections in A045846; symmetric dissections correspond to only 4, 2 or 1 dissections, depending on the symmetries.', '1308.5420-2-46-4': 'In this method, the counts for only the symmetric solutions took less than 10 seconds for [MATH], 1 minute for [MATH], and 2 hours for [MATH].', '1308.5420-2-46-5': 'Results are given in the Appendix.', '1308.5420-2-47-0': '# Results', '1308.5420-2-48-0': 'The exhaustive generation was conducted for orders up to 18.', '1308.5420-2-48-1': 'The computer time increased by a factor close to 10 for each successive order.', '1308.5420-2-48-2': 'For example, orders 15, 16 and 17 required 0.8, 8.0 and 77 hours respectively on a four-core Intel Core i7-3770 machine, running eight processes simultaneously.', '1308.5420-2-48-3': 'Order 18 required 18.4 days on an eight-core Intel Xeon E5-2680 machine, running sixteen processes simultaneously.', '1308.5420-2-49-0': 'Up to symmetry, the numbers of prime dissections of order [MATH] are 1, 0, 0, 1, 0, 1, 2, 6, 16, 56, 183, 657, 2277, 8813, 34178, 137578, 558734, 2285694.', '1308.5420-2-49-1': 'This sequence (A221841) and others are listed in the Appendix.', '1308.5420-2-49-2': 'The numbers of graphs and solutions are given in Table [REF].', '1308.5420-2-50-0': 'The results confirm that the known dissections for orders up to 18 (available at, for example, [CITATION]) include the largest possible.', '1308.5420-2-50-1': 'Table [REF] shows that, for example, the single known solution for [MATH], [MATH] is uniquely, up to symmetry, the largest possible at that order.', '1308.5420-2-50-2': 'The largest size, defined as [MATH] in Table [REF], is Sequence A089047.', '1308.5420-2-50-3': 'Since dissections of order 19 are known [CITATION] up to [MATH], then the uncertain, incorrect list in [CITATION] can be definitely replaced by the one in Table [REF].', '1308.5420-2-50-4': 'Thus the uncertainty in Sequence A005670 has been removed - or in fact, of course, displaced to higher [MATH].', '1308.5420-2-51-0': '# Confirmed and extended sequences in OEIS', '1308.5420-2-52-0': 'This appendix lists values from several sequences in the OEIS [CITATION] with previously known values that have been confirmed in the current work (or, conversely, have been used to check the current work).', '1308.5420-2-52-1': 'Also, new (or newly definite) values are shown in italics.', '1308.5420-2-53-0': "These sequences, as quoted in [CITATION], include the trivial dissection of a square into itself (except where 'smaller' is specified, in A018835 and A211302).", '1308.5420-2-54-0': 'Sequences A045846 and A221845 have known values beyond those listed here.', '1308.5420-2-54-1': 'In particular, the values of A045846[MATH] for [MATH] (1500957422222, 790347882174804 and 781621363452405930) were used to confirm and extend A224239, as described in Section [REF].', '1308.5420-2-55-0': 'The following new sequences have been added to the OEIS as results of the current work:'} | [['1308.5420-1-6-2', '1308.5420-2-7-1'], ['1308.5420-1-6-3', '1308.5420-2-7-2'], ['1308.5420-1-6-4', '1308.5420-2-7-3'], ['1308.5420-1-6-6', '1308.5420-2-7-5'], ['1308.5420-1-0-0', '1308.5420-2-0-0'], ['1308.5420-1-0-1', '1308.5420-2-0-1'], ['1308.5420-1-0-2', '1308.5420-2-0-2'], ['1308.5420-1-0-3', '1308.5420-2-0-3'], ['1308.5420-1-0-4', '1308.5420-2-0-4'], ['1308.5420-1-33-0', '1308.5420-2-53-0'], ['1308.5420-1-26-2', '1308.5420-2-46-2'], ['1308.5420-1-26-3', '1308.5420-2-46-3'], ['1308.5420-1-26-5', '1308.5420-2-46-5'], ['1308.5420-1-19-0', '1308.5420-2-21-0'], ['1308.5420-1-19-1', '1308.5420-2-21-1'], ['1308.5420-1-19-2', '1308.5420-2-21-2'], ['1308.5420-1-19-3', '1308.5420-2-21-3'], ['1308.5420-1-19-4', '1308.5420-2-21-4'], ['1308.5420-1-19-6', '1308.5420-2-21-6'], ['1308.5420-1-4-0', '1308.5420-2-4-0'], ['1308.5420-1-4-2', '1308.5420-2-4-2'], ['1308.5420-1-3-0', '1308.5420-2-3-0'], ['1308.5420-1-3-2', '1308.5420-2-3-2'], ['1308.5420-1-3-3', '1308.5420-2-3-3'], ['1308.5420-1-3-4', '1308.5420-2-3-4'], ['1308.5420-1-3-5', '1308.5420-2-3-5'], ['1308.5420-1-3-6', '1308.5420-2-3-6'], ['1308.5420-1-3-7', '1308.5420-2-3-7'], ['1308.5420-1-34-0', '1308.5420-2-54-0'], ['1308.5420-1-25-1', '1308.5420-2-45-1'], ['1308.5420-1-25-2', '1308.5420-2-45-2'], ['1308.5420-1-15-2', '1308.5420-2-17-2'], ['1308.5420-1-15-3', '1308.5420-2-17-3'], ['1308.5420-1-9-0', '1308.5420-2-10-0'], ['1308.5420-1-9-1', '1308.5420-2-10-1'], ['1308.5420-1-18-1', '1308.5420-2-20-1'], ['1308.5420-1-20-0', '1308.5420-2-22-0'], ['1308.5420-1-20-1', '1308.5420-2-22-1'], ['1308.5420-1-20-2', '1308.5420-2-22-2'], ['1308.5420-1-14-0', '1308.5420-2-16-0'], ['1308.5420-1-30-1', '1308.5420-2-50-1'], ['1308.5420-1-30-2', '1308.5420-2-50-2'], ['1308.5420-1-30-3', '1308.5420-2-50-3'], ['1308.5420-1-30-4', '1308.5420-2-50-4'], ['1308.5420-1-13-0', '1308.5420-2-15-0'], ['1308.5420-1-13-4', '1308.5420-2-15-4'], ['1308.5420-1-24-0', '1308.5420-2-44-0'], ['1308.5420-1-24-1', '1308.5420-2-44-1'], ['1308.5420-1-2-2', '1308.5420-2-2-2'], ['1308.5420-1-23-0', '1308.5420-2-43-0'], ['1308.5420-1-23-1', '1308.5420-2-43-1'], ['1308.5420-1-7-2', '1308.5420-2-8-2'], ['1308.5420-1-32-0', '1308.5420-2-52-0'], ['1308.5420-1-32-1', '1308.5420-2-52-1'], ['1308.5420-1-16-0', '1308.5420-2-18-0'], ['1308.5420-1-16-1', '1308.5420-2-18-1'], ['1308.5420-1-28-0', '1308.5420-2-48-0'], ['1308.5420-1-28-1', '1308.5420-2-48-1'], ['1308.5420-1-28-2', '1308.5420-2-48-2'], ['1308.5420-1-28-3', '1308.5420-2-48-3'], ['1308.5420-1-5-0', '1308.5420-2-5-0'], ['1308.5420-1-5-1', '1308.5420-2-5-1'], ['1308.5420-1-5-2', '1308.5420-2-5-2'], ['1308.5420-1-5-4', '1308.5420-2-5-4'], ['1308.5420-1-5-7', '1308.5420-2-5-7'], ['1308.5420-1-5-8', '1308.5420-2-5-8'], ['1308.5420-1-5-9', '1308.5420-2-5-9'], ['1308.5420-1-11-0', '1308.5420-2-13-0'], ['1308.5420-1-11-1', '1308.5420-2-13-1'], ['1308.5420-1-11-2', '1308.5420-2-13-2'], ['1308.5420-1-11-3', '1308.5420-2-13-3'], ['1308.5420-1-11-4', '1308.5420-2-13-4'], ['1308.5420-1-11-5', '1308.5420-2-13-5'], ['1308.5420-1-11-8', '1308.5420-2-13-8'], ['1308.5420-1-22-0', '1308.5420-2-42-0'], ['1308.5420-1-22-1', '1308.5420-2-42-1'], ['1308.5420-1-17-0', '1308.5420-2-19-0'], ['1308.5420-1-17-1', '1308.5420-2-19-1'], ['1308.5420-1-26-0', '1308.5420-2-46-0'], ['1308.5420-1-26-1', '1308.5420-2-46-1'], ['1308.5420-1-19-5', '1308.5420-2-21-5'], ['1308.5420-1-4-1', '1308.5420-2-4-1'], ['1308.5420-1-3-1', '1308.5420-2-3-1'], ['1308.5420-1-25-0', '1308.5420-2-45-0'], ['1308.5420-1-15-0', '1308.5420-2-17-0'], ['1308.5420-1-15-1', '1308.5420-2-17-1'], ['1308.5420-1-18-0', '1308.5420-2-20-0'], ['1308.5420-1-18-2', '1308.5420-2-20-3'], ['1308.5420-1-18-3', '1308.5420-2-20-4'], ['1308.5420-1-18-4', '1308.5420-2-20-5'], ['1308.5420-1-18-6', '1308.5420-2-20-7'], ['1308.5420-1-18-7', '1308.5420-2-20-8'], ['1308.5420-1-18-8', '1308.5420-2-20-9'], ['1308.5420-1-14-2', '1308.5420-2-16-3'], ['1308.5420-1-14-3', '1308.5420-2-16-4'], ['1308.5420-1-14-4', '1308.5420-2-16-5'], ['1308.5420-1-30-0', '1308.5420-2-50-0'], ['1308.5420-1-13-1', '1308.5420-2-15-1'], ['1308.5420-1-2-0', '1308.5420-2-2-0'], ['1308.5420-1-2-1', '1308.5420-2-2-1'], ['1308.5420-1-23-2', '1308.5420-2-43-2'], ['1308.5420-1-7-0', '1308.5420-2-8-0'], ['1308.5420-1-7-1', '1308.5420-2-8-1'], ['1308.5420-1-5-5', '1308.5420-2-5-5'], ['1308.5420-1-5-6', '1308.5420-2-5-6'], ['1308.5420-1-11-6', '1308.5420-2-13-6'], ['1308.5420-1-11-7', '1308.5420-2-13-7'], ['1308.5420-1-6-1', '1308.5420-2-7-0'], ['1308.5420-1-6-5', '1308.5420-2-7-4'], ['1308.5420-1-0-5', '1308.5420-2-0-5'], ['1308.5420-1-22-2', '1308.5420-2-42-2'], ['1308.5420-1-26-4', '1308.5420-2-46-4'], ['1308.5420-1-15-4', '1308.5420-2-17-4'], ['1308.5420-1-18-5', '1308.5420-2-20-6'], ['1308.5420-1-20-3', '1308.5420-2-22-3'], ['1308.5420-1-14-1', '1308.5420-2-16-2'], ['1308.5420-1-13-2', '1308.5420-2-15-2'], ['1308.5420-1-13-3', '1308.5420-2-15-3'], ['1308.5420-1-5-3', '1308.5420-2-5-3']] | [['1308.5420-1-6-2', '1308.5420-2-7-1'], ['1308.5420-1-6-3', '1308.5420-2-7-2'], ['1308.5420-1-6-4', '1308.5420-2-7-3'], ['1308.5420-1-6-6', '1308.5420-2-7-5'], ['1308.5420-1-0-0', '1308.5420-2-0-0'], ['1308.5420-1-0-1', '1308.5420-2-0-1'], ['1308.5420-1-0-2', '1308.5420-2-0-2'], ['1308.5420-1-0-3', '1308.5420-2-0-3'], ['1308.5420-1-0-4', '1308.5420-2-0-4'], ['1308.5420-1-33-0', '1308.5420-2-53-0'], ['1308.5420-1-26-2', '1308.5420-2-46-2'], ['1308.5420-1-26-3', '1308.5420-2-46-3'], ['1308.5420-1-26-5', '1308.5420-2-46-5'], ['1308.5420-1-19-0', '1308.5420-2-21-0'], ['1308.5420-1-19-1', '1308.5420-2-21-1'], ['1308.5420-1-19-2', '1308.5420-2-21-2'], ['1308.5420-1-19-3', '1308.5420-2-21-3'], ['1308.5420-1-19-4', '1308.5420-2-21-4'], ['1308.5420-1-19-6', '1308.5420-2-21-6'], ['1308.5420-1-4-0', '1308.5420-2-4-0'], ['1308.5420-1-4-2', '1308.5420-2-4-2'], ['1308.5420-1-3-0', '1308.5420-2-3-0'], ['1308.5420-1-3-2', '1308.5420-2-3-2'], ['1308.5420-1-3-3', '1308.5420-2-3-3'], ['1308.5420-1-3-4', '1308.5420-2-3-4'], ['1308.5420-1-3-5', '1308.5420-2-3-5'], ['1308.5420-1-3-6', '1308.5420-2-3-6'], ['1308.5420-1-3-7', '1308.5420-2-3-7'], ['1308.5420-1-34-0', '1308.5420-2-54-0'], ['1308.5420-1-25-1', '1308.5420-2-45-1'], ['1308.5420-1-25-2', '1308.5420-2-45-2'], ['1308.5420-1-15-2', '1308.5420-2-17-2'], ['1308.5420-1-15-3', '1308.5420-2-17-3'], ['1308.5420-1-9-0', '1308.5420-2-10-0'], ['1308.5420-1-9-1', '1308.5420-2-10-1'], ['1308.5420-1-18-1', '1308.5420-2-20-1'], ['1308.5420-1-20-0', '1308.5420-2-22-0'], ['1308.5420-1-20-1', '1308.5420-2-22-1'], ['1308.5420-1-20-2', '1308.5420-2-22-2'], ['1308.5420-1-14-0', '1308.5420-2-16-0'], ['1308.5420-1-30-1', '1308.5420-2-50-1'], ['1308.5420-1-30-2', '1308.5420-2-50-2'], ['1308.5420-1-30-3', '1308.5420-2-50-3'], ['1308.5420-1-30-4', '1308.5420-2-50-4'], ['1308.5420-1-13-0', '1308.5420-2-15-0'], ['1308.5420-1-13-4', '1308.5420-2-15-4'], ['1308.5420-1-24-0', '1308.5420-2-44-0'], ['1308.5420-1-24-1', '1308.5420-2-44-1'], ['1308.5420-1-2-2', '1308.5420-2-2-2'], ['1308.5420-1-23-0', '1308.5420-2-43-0'], ['1308.5420-1-23-1', '1308.5420-2-43-1'], ['1308.5420-1-7-2', '1308.5420-2-8-2'], ['1308.5420-1-32-0', '1308.5420-2-52-0'], ['1308.5420-1-32-1', '1308.5420-2-52-1'], ['1308.5420-1-16-0', '1308.5420-2-18-0'], ['1308.5420-1-16-1', '1308.5420-2-18-1'], ['1308.5420-1-28-0', '1308.5420-2-48-0'], ['1308.5420-1-28-1', '1308.5420-2-48-1'], ['1308.5420-1-28-2', '1308.5420-2-48-2'], ['1308.5420-1-28-3', '1308.5420-2-48-3'], ['1308.5420-1-5-0', '1308.5420-2-5-0'], ['1308.5420-1-5-1', '1308.5420-2-5-1'], ['1308.5420-1-5-2', '1308.5420-2-5-2'], ['1308.5420-1-5-4', '1308.5420-2-5-4'], ['1308.5420-1-5-7', '1308.5420-2-5-7'], ['1308.5420-1-5-8', '1308.5420-2-5-8'], ['1308.5420-1-5-9', '1308.5420-2-5-9'], ['1308.5420-1-11-0', '1308.5420-2-13-0'], ['1308.5420-1-11-1', '1308.5420-2-13-1'], ['1308.5420-1-11-2', '1308.5420-2-13-2'], ['1308.5420-1-11-3', '1308.5420-2-13-3'], ['1308.5420-1-11-4', '1308.5420-2-13-4'], ['1308.5420-1-11-5', '1308.5420-2-13-5'], ['1308.5420-1-11-8', '1308.5420-2-13-8']] | [['1308.5420-1-22-0', '1308.5420-2-42-0'], ['1308.5420-1-22-1', '1308.5420-2-42-1'], ['1308.5420-1-17-0', '1308.5420-2-19-0'], ['1308.5420-1-17-1', '1308.5420-2-19-1'], ['1308.5420-1-26-0', '1308.5420-2-46-0'], ['1308.5420-1-26-1', '1308.5420-2-46-1'], ['1308.5420-1-19-5', '1308.5420-2-21-5'], ['1308.5420-1-4-1', '1308.5420-2-4-1'], ['1308.5420-1-3-1', '1308.5420-2-3-1'], ['1308.5420-1-25-0', '1308.5420-2-45-0'], ['1308.5420-1-15-0', '1308.5420-2-17-0'], ['1308.5420-1-15-1', '1308.5420-2-17-1'], ['1308.5420-1-18-0', '1308.5420-2-20-0'], ['1308.5420-1-18-2', '1308.5420-2-20-3'], ['1308.5420-1-18-3', '1308.5420-2-20-4'], ['1308.5420-1-18-4', '1308.5420-2-20-5'], ['1308.5420-1-18-6', '1308.5420-2-20-7'], ['1308.5420-1-18-7', '1308.5420-2-20-8'], ['1308.5420-1-18-8', '1308.5420-2-20-9'], ['1308.5420-1-14-2', '1308.5420-2-16-3'], ['1308.5420-1-14-3', '1308.5420-2-16-4'], ['1308.5420-1-14-4', '1308.5420-2-16-5'], ['1308.5420-1-30-0', '1308.5420-2-50-0'], ['1308.5420-1-13-1', '1308.5420-2-15-1'], ['1308.5420-1-2-0', '1308.5420-2-2-0'], ['1308.5420-1-2-1', '1308.5420-2-2-1'], ['1308.5420-1-23-2', '1308.5420-2-43-2'], ['1308.5420-1-7-0', '1308.5420-2-8-0'], ['1308.5420-1-7-1', '1308.5420-2-8-1'], ['1308.5420-1-5-5', '1308.5420-2-5-5'], ['1308.5420-1-5-6', '1308.5420-2-5-6'], ['1308.5420-1-11-6', '1308.5420-2-13-6'], ['1308.5420-1-11-7', '1308.5420-2-13-7']] | [] | [['1308.5420-1-6-1', '1308.5420-2-7-0'], ['1308.5420-1-6-5', '1308.5420-2-7-4'], ['1308.5420-1-0-5', '1308.5420-2-0-5'], ['1308.5420-1-22-2', '1308.5420-2-42-2'], ['1308.5420-1-26-4', '1308.5420-2-46-4'], ['1308.5420-1-15-4', '1308.5420-2-17-4'], ['1308.5420-1-18-5', '1308.5420-2-20-6'], ['1308.5420-1-20-3', '1308.5420-2-22-3'], ['1308.5420-1-14-1', '1308.5420-2-16-2'], ['1308.5420-1-13-2', '1308.5420-2-15-2'], ['1308.5420-1-13-3', '1308.5420-2-15-3'], ['1308.5420-1-5-3', '1308.5420-2-5-3']] | [] | ['1308.5420-1-9-2', '1308.5420-1-10-0', '1308.5420-1-29-0', '1308.5420-1-29-1', '1308.5420-1-34-1', '1308.5420-1-35-0', '1308.5420-2-10-2', '1308.5420-2-12-0', '1308.5420-2-29-4', '1308.5420-2-49-0', '1308.5420-2-54-1', '1308.5420-2-55-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1308.5420 | null | null | null | null | null |
1802.02556 | {'1802.02556-1-0-0': 'Current flow closeness centrality (CFCC) has a better discriminating ability than the ordinary closeness centrality based on shortest paths.', '1802.02556-1-0-1': 'In this paper, we extend the notion of CFCC to a group of vertices in a weighted graph.', '1802.02556-1-0-2': 'For a graph with [MATH] vertices and [MATH] edges, the CFCC [MATH] for a vertex group [MATH] is equal to the ratio of [MATH] to the sum of effective resistances from [MATH] to all other vertices.', '1802.02556-1-0-3': 'We then study the problem of finding a group [MATH] of [MATH] vertices, so that the CFCC [MATH] is maximized.', '1802.02556-1-0-4': 'We alternatively solve this problem by minimizing the reciprocal of [MATH].', '1802.02556-1-0-5': 'We show that the problem is NP-hard, and prove that the objective function is monotone and supermodular.', '1802.02556-1-0-6': 'We propose two greedy algorithms with provable approximation guarantees.', '1802.02556-1-0-7': 'The first is a deterministic algorithm with an approximation factor [MATH] and [MATH] running time; while the second is a randomized algorithm with a [MATH]-approximation and [MATH] running time for any small [MATH], where the [MATH] notation hides the [MATH] factors.', '1802.02556-1-0-8': 'Extensive experiments on models and real networks demonstrate that our algorithms are effective and efficient, with the second algorithm being scalable to massive networks with more than a million vertices.', '1802.02556-1-1-0': '# Introduction', '1802.02556-1-2-0': 'A fundamental problem in network science and graph mining is to identify crucial vertices [CITATION], which has become an important tool in network analysis and found numerous applications in various areas [CITATION].', '1802.02556-1-2-1': 'The first step of finding central vertices is to define suitable indices measuring relative importance of vertices.', '1802.02556-1-2-2': 'Over the past decades, a lot of centrality metrics have been introduced to characterize and analyze the roles of vertices in networks [CITATION].', '1802.02556-1-2-3': 'Among different centrality measures, closeness centrality [CITATION] is a popular one.', '1802.02556-1-2-4': 'The closeness of a vertex is the harmonic mean of shortest path distances between the vertex and all other vertices.', '1802.02556-1-2-5': 'However, this metric considers only the shortest paths, neglecting the contributions from other paths, and thus can produce some odd effects, or even counterintuitive results [CITATION].', '1802.02556-1-2-6': 'To avoid this shortcoming, Brandes and Fleischer presented current flow closeness centrality [CITATION] based on electrical networks [CITATION] , which counts the contributions from all paths between vertices.', '1802.02556-1-2-7': 'It has been shown that the current flow based closeness [CITATION] can better discriminate vertices than its traditional counterpart.', '1802.02556-1-3-0': 'Most of previous works focused on the measures and algorithms for importance of individual vertices in a network [CITATION].', '1802.02556-1-3-1': 'They are not suitable to the problem of determining a group of [MATH] vertices that is the most important among all vertex groups, each containing exactly [MATH] vertices.', '1802.02556-1-3-2': 'Such a problem is encountered in various data mining and graph applications.', '1802.02556-1-3-3': 'For example, in social networks, retailers may want to choose [MATH] vertices as promoters of product, such that the number of the potentially influenced customers is maximized [CITATION].', '1802.02556-1-3-4': 'Again for instance, in P2P networks, how to place resources on a fixed number of [MATH] peers such that these resources are easily accessible to other peers [CITATION].', '1802.02556-1-3-5': 'In order to measure the importance of a group of vertices, Everett and Borgatti [CITATION] extended the idea of individual centrality to group centrality, and introduced the concepts of group centrality, for example, group closeness.', '1802.02556-1-3-6': 'Recently, some algorithms have been developed to compute or estimate group closeness [CITATION].', '1802.02556-1-3-7': 'However, similar to the individual vertex closeness, the group closeness also disregards the contributions from the non-geodesic paths.', '1802.02556-1-4-0': 'In this paper, we propose current flow closeness centrality (CFCC) of a group of vertices, which is an extension of current flow closeness of an individual vertex [CITATION].', '1802.02556-1-4-1': 'We prove that for a graph with [MATH] vertices and [MATH] edges, the CFCC [MATH] for a vertex group [MATH] is equal to the ratio of [MATH] to the sum of effective resistances from vertices in [MATH] to all other vertices.', '1802.02556-1-4-2': 'We then consider the optimization problem: how can we find a group [MATH] of [MATH] vertices so as to maximize CFCC [MATH].', '1802.02556-1-4-3': 'We solve this problem by minimizing the reciprocal of [MATH], instead of maximizing [MATH].', '1802.02556-1-4-4': 'We show that the problem is NP-hard, and prove that the problem is an instance of supermodular set function optimization with cardinality constraint.', '1802.02556-1-4-5': 'We thus resort to devise greedy algorithms to solve the problem.', '1802.02556-1-4-6': 'For this purpose, we develop two greedy algorithms with provable approximation guarantees.', '1802.02556-1-4-7': 'The first one is a deterministic algorithm with a [MATH] approximation factor and [MATH] running time; while the second one is randomized algorithm with a [MATH]-approximation factor and [MATH] running time for small [MATH], where the [MATH] notation suppresses the [MATH] factors.', '1802.02556-1-4-8': 'Finally, we perform extensive experiments on some networks to evaluate our algorithm.', '1802.02556-1-4-9': 'The results show that both algorithms are effective.', '1802.02556-1-4-10': 'Moreover, the second algorithm is efficient and is scalable to large networks with more than a million vertices.', '1802.02556-1-5-0': '## Related Works', '1802.02556-1-6-0': 'There exist various measures for centrality of a group of vertices, based on graph structure or dynamic processes, such as betweenness [CITATION], absorbing random-walk centrality [CITATION], and grounding centrality [CITATION].', '1802.02556-1-6-1': 'Since the criterion for importance of a vertex group is application dependent [CITATION], many previous works focused on choosing (or deleting) a group of a fixed number of [MATH] vertices in order to optimize related quantities/performances as measures of vertex group importance for different application considerations, including minimizing the leading eigenvalue of adjacency matrix for vertex immunization [CITATION], minimizing the mean steady-state variance for first-order leader-follower noisy consensus dynamics [CITATION], maximizing average distance for identifying structural hole spanners [CITATION], among others.', '1802.02556-1-7-0': 'Since we focus on group closeness centrality in this paper, we now introduce in detail the notation of closeness centrality and related algorithms.', '1802.02556-1-7-1': 'The closeness centrality for an individual vertex was proposed [CITATION] and formalized [CITATION] by Bavelas.', '1802.02556-1-7-2': 'For a given vertex, its closeness centrality is defined as the harmonic average of shortest path distances of the vertex to all the other vertices.', '1802.02556-1-7-3': 'Everett and Borgatti [CITATION] extended the individual closeness centrality to group closeness centrality, which measures how close a vertex group is to all other vertices.', '1802.02556-1-7-4': 'For a graph with [MATH] vertices and [MATH] edges, exactly computing the closeness centrality of a group of vertices involves calculating all-pairwise shortest path length, the time complexity of the state-of-the-art algorithm [CITATION] for which is [MATH].', '1802.02556-1-7-5': 'To reduce the computation complexity, various approximate algorithms were developed.', '1802.02556-1-7-6': 'For the disk-resident graphs with unit edge weight, a [MATH] algorithm was proposed for estimating the group closeness with high accuracy [CITATION].', '1802.02556-1-7-7': 'For weighted graphs, a greedy [MATH] algorithm with approximation ratio [MATH] was devised [CITATION], and a sampling algorithm without approximation guarantee was also proposed in the same paper to scale to large networks.', '1802.02556-1-7-8': 'Very recently, new techniques have been developed to speed up the greedy algorithm in [CITATION], but preserving its theoretical guarantee.', '1802.02556-1-8-0': 'Conventional closeness centrality is based on the shortest paths, omitting the contributions from other paths.', '1802.02556-1-8-1': 'In order to overcome this drawback, Brandes and Fleischer introduced current flow closeness centrality for an individual vertex [CITATION], which essentially considers all paths between vertices, but still gives large weight to short paths.', '1802.02556-1-8-2': 'In this paper, we will extend the current flow closeness centrality to a group of vertices.', '1802.02556-1-8-3': 'Moreover, we propose two algorithms that provide large approximation ratios with high probabilities.', '1802.02556-1-8-4': 'Particularly, one of our algorithms is scalable with nearly-linear time complexity [MATH] for small [MATH], where notation [MATH] hides the [MATH] factors.', '1802.02556-1-9-0': '# Preliminaries', '1802.02556-1-10-0': 'In this section, we briefly introduce some useful notations and tools for the convenience of description of our problem and algorithms.', '1802.02556-1-11-0': '## Notations', '1802.02556-1-12-0': 'We use normal lowercase letters like [MATH] to denote scalars in [MATH], normal uppercase letters like [MATH] to denote sets, bold lowercase letters like [MATH] to denote vectors, and bold uppercase letters like [MATH] to denote matrices.', '1802.02556-1-12-1': 'We write [MATH] to denote the [MATH] entry of vector [MATH] and [MATH] to denote entry [MATH] of matrix [MATH].', '1802.02556-1-12-2': 'We also write [MATH] to denote the [MATH] row of [MATH] and [MATH] to denote the [MATH] column of [MATH].', '1802.02556-1-13-0': 'We write sets in matrix subscripts to denote submatrices.', '1802.02556-1-13-1': 'For example, [MATH] denotes the submatrix of [MATH] with row indices in [MATH] and column indices in [MATH].', '1802.02556-1-13-2': 'To simplify notation, we also write [MATH] to denote the submatrix of [MATH] obtained by removing the [MATH] row and [MATH] column of [MATH].', '1802.02556-1-13-3': 'For example, for an [MATH] matrix [MATH], [MATH] denotes the submatrix [MATH].', '1802.02556-1-14-0': 'Note that the precedence of matrix subscripts is the lowest.', '1802.02556-1-14-1': 'Thus, [MATH] denotes the inverse of [MATH] instead of a submatrix of [MATH].', '1802.02556-1-15-0': 'For two matrices [MATH] and [MATH], we write [MATH] to denote that [MATH] is positive semidefinite, i.e., [MATH] holds for every real vector [MATH].', '1802.02556-1-16-0': 'We use [MATH] to denote the [MATH] standard basis vector of appropriate dimension, and [MATH] to denote the indicator vector of [MATH].', '1802.02556-1-17-0': '## Graphs, Laplacians, and Effective Resistances', '1802.02556-1-18-0': 'We write [MATH] to denote a positively weighted undirected graph with [MATH] vertices, [MATH] edges, and edge weight function [MATH].', '1802.02556-1-18-1': 'The Laplacian matrix [MATH] of [MATH] is defined as [EQUATION] where [MATH] is the weighted degree of [MATH] and [MATH] means [MATH].', '1802.02556-1-18-2': 'Let [MATH] and [MATH] denote, respectively, the maximum weight and minimum weight among all edges.', '1802.02556-1-18-3': "If we orient each edge of [MATH] arbitrarily, we can also write it's Laplacian as [MATH], where [MATH] is the signed edge-vertex incidence matrix defined by [EQUATION] and [MATH] is a diagonal matrix with [MATH].", '1802.02556-1-18-4': 'It is not hard to show that quadratic forms of [MATH] can be written as [EQUATION] which immediately implies that [MATH] is positive semidefinite, and [MATH] only has one zero eigenvalue if [MATH] is a connected graph.', '1802.02556-1-19-0': 'The following fact shows that submatrices of Laplacians are always positive definite and inverse-positive.', '1802.02556-1-20-0': 'Let [MATH] be the Laplacian of a connected graph and let [MATH] be a nonnegative, diagonal matrix with at least one nonzero entry.', '1802.02556-1-20-1': 'Then, [MATH] is positive definite, and every entry of [MATH] is positive.', '1802.02556-1-21-0': 'Let [MATH] be eigenvalues of [MATH] of a connected graph [MATH], and [MATH] be the corresponding orthonormal unit eigenvectors.', '1802.02556-1-21-1': 'Then we can decompose [MATH] as [MATH] and define its pseudoinverse as [MATH].', '1802.02556-1-22-0': 'It is not hard to verify that if [MATH] and [MATH] are Laplacians of connected graphs supported on the same vertex set, then [MATH] implies [MATH].', '1802.02556-1-23-0': 'We now give the definition of effective resistance between two vertices [CITATION].', '1802.02556-1-24-0': 'For a connected graph [MATH] with Laplacian matrix [MATH], the effective resistance between vertices [MATH] and [MATH] is defined as [EQUATION]', '1802.02556-1-24-1': 'One can also write the effective resistance as a diagonal element of the inverse of a submatrix of [MATH].', '1802.02556-1-25-0': 'The effective resistance can also be defined between a vertex and a vertex set.', '1802.02556-1-26-0': '[Effective Resistance Between a Vertex and a Vertex Set [CITATION]] For a connected graph [MATH] with Laplacian matrix [MATH], the effective resistance between vertices [MATH] and [MATH] is defined as [EQUATION] [MATH] can be interpreted as the electrical resistance when the graph is treated as a resistor network in which vertices in [MATH] are grounded.', '1802.02556-1-26-1': 'Namely, if we treat every edge [MATH] as a resistor with resistance [MATH] and ground all vertices in [MATH], then [MATH] equals the voltage of [MATH] when an electrical flow sends one unit of current into [MATH] and removes one unit of current from [MATH] (i.e. the ground).', '1802.02556-1-27-0': '## Current Flow Closeness Centrality', '1802.02556-1-28-0': 'The current flow closeness centrality was proposed in [CITATION].', '1802.02556-1-28-1': 'It is based on the assumption that information spreads efficiently like an electrical current.', '1802.02556-1-29-0': 'To define current-flow closeness, we treat the graph [MATH] as a resistor network by treating every edge [MATH] as a resistor with resistance [MATH].', '1802.02556-1-29-1': 'Let [MATH] denote the voltage of [MATH] when a unit current enters the network at [MATH] and leaves it at [MATH].', '1802.02556-1-30-0': 'The current flow closeness [MATH] of a vertex [MATH] is defined as [EQUATION]', '1802.02556-1-30-1': 'It has been proved [CITATION] that the current flow closeness of vertex [MATH] equals the ratio of [MATH] to the sum of effective resistances between [MATH] and other vertices.', '1802.02556-1-31-0': '[MATH].', '1802.02556-1-32-0': 'Actually, current flow closeness centrality is equivalent to information centrality [CITATION].', '1802.02556-1-33-0': '## Supermodular Functions', '1802.02556-1-34-0': 'We now give the definitions for monotone and supermodular set functions.', '1802.02556-1-34-1': 'For simplicity, we write [MATH] to denote [MATH] and [MATH] to denote [MATH].', '1802.02556-1-35-0': '[Monotonicity] A set function [MATH] is monotone if [MATH] holds for all [MATH].', '1802.02556-1-36-0': '[Supermodularity] A set function [MATH] is supermodular if [MATH] holds for all [MATH] and [MATH].', '1802.02556-1-37-0': '# Current Flow Closeness of a Group of Vertices', '1802.02556-1-38-0': 'We follow the idea of [CITATION] to define current flow closeness centrality (CFCC) of a group of vertices.', '1802.02556-1-39-0': 'To define current flow closeness centrality for a vertex set [MATH], we treat the graph [MATH] as a resistor network in which all vertices in [MATH] are grounded.', '1802.02556-1-39-1': 'Thus, vertices in [MATH] always have voltage [MATH].', '1802.02556-1-39-2': 'For a vertex [MATH], let [MATH] be the voltage of [MATH] when a unit current enters the network at [MATH] and leaves it at [MATH] (i.e. the ground).', '1802.02556-1-39-3': 'Then, we define the current-flow closeness of [MATH] as follows.', '1802.02556-1-40-0': 'Let [MATH] be a connected weighted graph.', '1802.02556-1-40-1': 'The current flow closeness centrality [MATH] of a vertex group [MATH] is defined as [EQUATION]', '1802.02556-1-40-2': 'Note that there are different variants of the definition of CFCC for a vertex group.', '1802.02556-1-40-3': 'For example, we can use [MATH] as the measure of CFCC for a vertex set [MATH].', '1802.02556-1-40-4': 'Definition [REF] adopts the standard form as the classic closeness centrality [CITATION].', '1802.02556-1-41-0': 'We now write [MATH] into a ratio of [MATH] to a sum of effective resistances as in Fact [REF].', '1802.02556-1-42-0': 'Let [MATH] be a fixed vertex.', '1802.02556-1-42-1': 'Suppose there is a unit current enters the network at [MATH] and leaves it at [MATH].', '1802.02556-1-42-2': 'Let [MATH] be a vector of voltages at vertices.', '1802.02556-1-42-3': "By Kirchhoff's Current Law and Ohm's Law, we have [EQUATION] where [MATH] denotes the amount of current flowing out of [MATH].", '1802.02556-1-42-4': 'Writing the above equation in block form gives [EQUATION]', '1802.02556-1-42-5': 'Since vertices in [MATH] all have voltage [MATH], we can write the above linear equation into [EQUATION] where ([REF]) leads to [EQUATION]', '1802.02556-1-42-6': 'The above equation results in the expression of voltage at [MATH] as [EQUATION].', '1802.02556-1-42-7': 'Now we can write the CFCC of [MATH] as [EQUATION]', '1802.02556-1-42-8': 'If we define the effective resistance between a vertex [MATH] and a vertex set [MATH] as [CITATION] [EQUATION] we have the following relation between [MATH] and [MATH].', '1802.02556-1-43-0': '[MATH].', '1802.02556-1-44-0': 'Being able to define CFCC of a vertex set raises the problem of maximizing current flow closeness subject to a cardinality constraint, which we state below.', '1802.02556-1-45-0': '[Current Flow Closeness Maximization, CFCM]', '1802.02556-1-46-0': 'Given a connected graph [MATH] with [MATH] vertices, [MATH] edges, and edge weight function [MATH] and an integer [MATH], find a vertex group [MATH] such that the CFCC [MATH] is maximized, that is [EQUATION]', '1802.02556-1-47-0': '# Hardness of Current Flow Closeness Maximization', '1802.02556-1-48-0': 'In this section, we prove that Problem [REF] is NP-hard.', '1802.02556-1-48-1': 'We will give a reduction from vertex cover on 3-regular graphs (graphs whose vertices all have degree 3), which is an NP-complete problem [CITATION].', '1802.02556-1-48-2': 'The decision version of this problem is stated below.', '1802.02556-1-48-3': '[Vertex Cover on 3-regular graphs, VC3] Given a connected 3-regular graph [MATH] and an integer [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH] is a vertex cover of [MATH] (i.e. every edge in [MATH] is incident with at least one vertex in [MATH]).', '1802.02556-1-49-0': 'An instance of this problem is denoted by VC3[MATH].', '1802.02556-1-50-0': 'We then give the decision version of Problem [REF].', '1802.02556-1-51-0': '[Current Flow Closeness Maximization, Decision Version, CFCMD] Given a connected graph [MATH], an integer [MATH], and a real number [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH].', '1802.02556-1-52-0': 'An instance of this problem is denoted by CFCMD[MATH].', '1802.02556-1-53-0': 'To show that VC3[MATH]CFCMD, we will need the following lemma.', '1802.02556-1-54-0': 'Let [MATH] be a connected 3-regular graph with all edge weights being [MATH] (i.e. [MATH] for all [MATH]).', '1802.02556-1-54-1': 'Let [MATH] be a nonempty vertex set, and [MATH].', '1802.02556-1-54-2': 'Then, [MATH] and the equality holds if and only if [MATH] is a vertex cover of [MATH].', '1802.02556-1-55-0': 'We first show that if [MATH] is a vertex cover of [MATH] then [MATH].', '1802.02556-1-55-1': 'When [MATH] is a vertex cover, [MATH] is an independent set.', '1802.02556-1-55-2': 'Thus, [MATH] is a diagonal matrix with all diagonal entries being [MATH].', '1802.02556-1-55-3': 'So we have [EQUATION].', '1802.02556-1-56-0': 'We then show that if [MATH] is not a vertex cover of [MATH] then [MATH].', '1802.02556-1-56-1': 'When [MATH] is not a vertex cover, [MATH] is not an independent set.', '1802.02556-1-56-2': 'Thus, [MATH] is a block diagonal matrix, where each block corresponding to a connected component of [MATH], the induced graph of [MATH] on [MATH].', '1802.02556-1-56-3': 'Let [MATH] be a connected component of [MATH] such that [MATH].', '1802.02556-1-56-4': 'Then, the block of [MATH] corresponding to [MATH] is [MATH].', '1802.02556-1-56-5': 'For a vertex [MATH], let the [MATH] column of [MATH] be [MATH].', '1802.02556-1-56-6': 'Then, we can write [MATH] into block form as [EQUATION] where [MATH].', '1802.02556-1-56-7': 'By blockwise matrix inversion we have [EQUATION]', '1802.02556-1-56-8': 'Since [MATH] is positive definite, we have [MATH] and hence [MATH].', '1802.02556-1-56-9': 'Since [MATH] is a connected component, [MATH] is not a zero vector, which coupled with the fact that [MATH] is positive definite gives [MATH].', '1802.02556-1-56-10': 'Thus, [MATH].', '1802.02556-1-56-11': 'Since this holds for all [MATH], we have [MATH].', '1802.02556-1-56-12': 'Also, since [MATH] can be any connected component of [MATH] with at least two vertices, and a block of an isolate vertex in [MATH] contributes a [MATH] to [MATH], we have for any [MATH] which is not a vertex cover of [MATH] [EQUATION] which implies [EQUATION].', '1802.02556-1-56-13': 'This completes the proof.', '1802.02556-1-57-0': 'Maximizing current flow closeness subject to a cardinality constraint is NP-hard.', '1802.02556-1-58-0': 'We give a polynomial reduction [EQUATION] from instances of VC3 to instances of CFCMD.', '1802.02556-1-58-1': 'For a connected 3-regular graph [MATH] with [MATH] vertices, we construct a weighted graph [MATH] with the same vertex set and edge set and an edge weight function [MATH] mapping all edges to [MATH].', '1802.02556-1-58-2': 'Then, we construct a reduction [MATH] as [EQUATION]', '1802.02556-1-58-3': 'Since by Lemma [REF] [MATH] holds for any [MATH], and [MATH] if and only if [MATH] is a vertex cover of [MATH], we can deduce that there is a vertex cover with at most [MATH] vertices of [MATH] if and only if there is a vertex set with at most [MATH] vertices with information centrality [MATH].', '1802.02556-1-58-4': 'Thus, [MATH] is a polynomial reduction from VC3 to CFCMD.', '1802.02556-1-58-5': 'Since VC3 is NP-complete and CFCMD is the decision version of CFCM, we have that CFCM is NP-hard.', '1802.02556-1-59-0': '# Supermodularity of the Reciprocal of Current Flow Group Closeness', '1802.02556-1-60-0': 'In this section, we give a fully algebraic proof for the supermodularity of reciprocal of current flow group closeness, i.e., [MATH].', '1802.02556-1-60-1': 'The supermodularity of [MATH] is a consequence of a more general result which shows that [MATH] is entrywise supermodular.', '1802.02556-1-61-0': 'Let [MATH] be an arbitrary pair of vertices.', '1802.02556-1-61-1': 'Then, the entry [MATH] is a monotone supermodular function.', '1802.02556-1-62-0': 'To prove Lemma [REF], we first define a linear relaxation [MATH] of [MATH] as [EQUATION]', '1802.02556-1-62-1': 'We remark the intuition behind this relaxation [MATH].', '1802.02556-1-62-2': 'Let [MATH] denote the indices of entries of [MATH] equal to one, and let [MATH] denote the indices of entries of [MATH] less than one.', '1802.02556-1-62-3': 'Then, by the definition in ([REF]), we can write [MATH] into a block diagonal matrix as [EQUATION] where [MATH] is itself a diagonal matrix.', '1802.02556-1-62-4': 'This means that if [MATH] for some nonempty vertex set [MATH], the following statement holds: [EQUATION].', '1802.02556-1-62-5': 'The condition that every entry of [MATH] is in [MATH] coupled with Fact [REF] also implies that all submatrices of [MATH] are positive definite and inverse-positive.', '1802.02556-1-63-0': 'Now for vertices [MATH] and nonempty vertex set [MATH] such that [MATH], we can write the marginal gain of a vertex [MATH] as [EQUATION]', '1802.02556-1-63-1': 'We can further write the matrix on the rhs of ([REF]) as an integral by [EQUATION] where the second equality follows by the identity [EQUATION] for any invertible matrix [MATH].', '1802.02556-1-64-0': 'To prove Lemma [REF], we will also need the following lemma, which shows the entrywise monoticity of [MATH].', '1802.02556-1-65-0': 'For [MATH], the following statement holds for any vertices [MATH] and nonempty vertex sets [MATH] such that [MATH]: [EQUATION]', '1802.02556-1-65-1': 'For simplicity, we let [MATH] and [MATH].', '1802.02556-1-65-2': 'We also write [MATH], [MATH], and [MATH].', '1802.02556-1-65-3': 'Due to the block diagonal structures of [MATH] and [MATH], we have [EQUATION] and [EQUATION]', '1802.02556-1-65-4': 'Since [MATH] and [MATH] agree on entries with indices in [MATH], we can write the submatrix [MATH] of [MATH] in block form as [EQUATION]', '1802.02556-1-65-5': 'By blockwise matrix inversion, we have [EQUATION] where the second equality follows by negating both [MATH] and [MATH].', '1802.02556-1-65-6': 'By definition the matrix [MATH] is entrywise nonnegative.', '1802.02556-1-65-7': 'By Fact [REF], every entry of [MATH] and [MATH] is also nonnegative.', '1802.02556-1-65-8': 'Thus, the matrix [EQUATION] is entrywise nonnegative, which coupled with ([REF]) and ([REF]), implies [MATH].', '1802.02556-1-66-0': '[Proof of Lemma [REF]]', '1802.02556-1-67-0': 'By definition the matrix [MATH] is entrywise nonnegative when [MATH].', '1802.02556-1-67-1': 'By Fact [REF], the matrix [MATH] is also entrywise nonnegative when [MATH].', '1802.02556-1-67-2': 'Thus, the derivative in ([REF]) is entrywise nonnegative, which implies the the monotonicity of [MATH] for any pair of vertices [MATH].', '1802.02556-1-68-0': 'We then prove the supermodularity, i.e., [EQUATION] for any [MATH] and [MATH].', '1802.02556-1-68-1': 'Lemma [REF], coupled with the fact that [MATH] and [MATH] are both entrywise nonnegative, gives the entrywise monotonicity of the derivative in ([REF]) as [EQUATION]', '1802.02556-1-68-2': 'Integrating both sides of the above inequality with respect to [MATH] on the interval [MATH] gives ([REF]).', '1802.02556-1-69-0': 'We now prove the supermodularity of [MATH].', '1802.02556-1-70-0': 'The reciprocal of current flow group centrality, i.e., [MATH], is a monotone supermodular function.', '1802.02556-1-71-0': 'Let [MATH] be vertex sets and [MATH] be a vertex.', '1802.02556-1-72-0': 'For monotonicity, we have [EQUATION] where the first inequality follows by the fact that [MATH] is entrywise nonnegative, and the second inequality follows from the entrywise monotinicity of [MATH].', '1802.02556-1-73-0': 'For supermodularity, we have [EQUATION] where the first inequality follows from the entrywise monotonicity of [MATH], and the second inequality follows from the entrywise supermodularity of [MATH].', '1802.02556-1-74-0': 'Theorem [REF] indicates that one can obtain a [MATH]-approximation to the optimum [MATH] by a simple greedy algorithm, by picking the vertex with the maximum marginal gain each time [CITATION].', '1802.02556-1-74-1': 'However, since computing [MATH] involves matrix inversions, a naive implementation of this greedy algorithm will take [MATH] time, assuming that one matrix inversion runs in [MATH] time.', '1802.02556-1-74-2': 'We will show in the next section how to implement this greedy algorithm in [MATH] time using blockwise matrix inversion.', '1802.02556-1-75-0': '# A Deterministic Greedy Algorithm', '1802.02556-1-76-0': 'We now consider how to accelerate the naive greedy algorithm.', '1802.02556-1-76-1': 'Suppose that after the [MATH] step, the algorithm has selected a set [MATH] containing [MATH] vertices.', '1802.02556-1-76-2': 'We next compute the marginal gain [MATH] of each vertex [MATH].', '1802.02556-1-77-0': 'For a vertex [MATH], let [MATH] denote the [MATH] column of the submatrix [MATH].', '1802.02556-1-77-1': 'Then we write [MATH] in block form as [EQUATION] where [MATH].', '1802.02556-1-77-2': 'By blockwise matrix inversion, we have [EQUATION] where [MATH].', '1802.02556-1-77-3': 'Then the marginal gain of [MATH] can be written as [EQUATION] where the second equality and the fourth equality follow by ([REF]), while the third equality follows by the cyclicity of trace.', '1802.02556-1-78-0': 'By ([REF]), we can also update the inverse [MATH] upon a vertex [MATH] by [EQUATION]', '1802.02556-1-78-1': 'At the first step, we need to pick a vertex [MATH] with minimum resistance [MATH], which can be done by computing [MATH] for all [MATH] using the relation [CITATION] [EQUATION]', '1802.02556-1-78-2': 'We give the [MATH]-time algorithm as follows.', '1802.02556-1-79-0': '[MATH]', '1802.02556-1-80-0': 'Compute [MATH] by inverting [MATH] in [MATH] time.', '1802.02556-1-80-1': '[MATH] where [EQUATION]', '1802.02556-1-80-2': 'Compute [MATH] in [MATH] time.', '1802.02556-1-80-3': 'Repeat the following steps for [MATH]:', '1802.02556-1-81-0': '[MATH] where [EQUATION]', '1802.02556-1-81-1': 'Compute [MATH] in [MATH] time by [EQUATION]', '1802.02556-1-82-0': 'Return [MATH].', '1802.02556-1-83-0': 'The performance of [MATH] is characterized in the following theorem.', '1802.02556-1-84-0': 'The algorithm [MATH] takes an undirected positive weighted graph [MATH] with associated Laplacian [MATH] and an integer [MATH], and returns a vertex set [MATH] with [MATH].', '1802.02556-1-84-1': 'The algorithm runs in time [MATH].', '1802.02556-1-84-2': 'The vertex set [MATH] satisfies', '1802.02556-1-85-0': '[EQUATION] where [MATH] and [MATH].', '1802.02556-1-86-0': 'The running time is easy to verify.', '1802.02556-1-86-1': 'We only need to prove the approximation ratio.', '1802.02556-1-87-0': 'By supermodularity, for any [MATH] [EQUATION] which implies [EQUATION]', '1802.02556-1-87-1': 'Then, we have [EQUATION] which coupled with [MATH] implies ([REF]).', '1802.02556-1-88-0': '# A Randomized Greedy Algorithm', '1802.02556-1-89-0': 'The deterministic greedy algorithm [MATH] has a computation complexity [MATH], which is still not acceptable for large networks.', '1802.02556-1-89-1': 'In this section, we provide efficient randomized algorithm, which achieves a [MATH] approximation factor in time [MATH].', '1802.02556-1-90-0': 'To further accelerate algorithm [MATH] we need to compute the marginal gains [EQUATION] for all [MATH] and a vertex set [MATH] more quickly.', '1802.02556-1-90-1': 'We also need a faster way to compute [EQUATION] for all [MATH] at the [MATH] step.', '1802.02556-1-90-2': 'We will show how to solve both problems in nearly linear time using Johnson-Lindenstrauss Lemma and Fast SDDM Solvers.', '1802.02556-1-90-3': 'The idea is similar to that of section 7 of [CITATION] and section 5 of [CITATION].', '1802.02556-1-91-0': '[Johnson-Lindenstrauss Lemma [CITATION]]', '1802.02556-1-92-0': 'Let [MATH] vertices very quickly.', '1802.02556-1-92-1': 'In future work, we plan to introduce and study the betweenness group centrality based on current flow [CITATION], which takes into account all possible paths.'} | {'1802.02556-2-0-0': 'Current flow closeness centrality (CFCC) has a better discriminating ability than the ordinary closeness centrality based on shortest paths.', '1802.02556-2-0-1': 'In this paper, we extend this notion to a group of vertices in a weighted graph, and then study the problem of finding a subset [MATH] of [MATH] vertices to maximize its CFCC [MATH], both theoretically and experimentally.', '1802.02556-2-0-2': 'We show that the problem is NP-hard, but propose two greedy algorithms for minimizing the reciprocal of [MATH] with provable guarantees using the monotoncity and supermodularity.', '1802.02556-2-0-3': 'The first is a deterministic algorithm with an approximation factor [MATH] and cubic running time; while the second is a randomized algorithm with a [MATH]-approximation and nearly-linear running time for any [MATH].', '1802.02556-2-0-4': 'Extensive experiments on model and real networks demonstrate that our algorithms are effective and efficient, with the second algorithm being scalable to massive networks with more than a million vertices.', '1802.02556-2-1-0': '# Introduction', '1802.02556-2-2-0': 'A fundamental problem in network science and graph mining is to identify crucial vertices [CITATION].', '1802.02556-2-2-1': 'It is an important tool in network analysis and found numerous applications in various areas [CITATION].', '1802.02556-2-2-2': 'The first step of finding central vertices is to define suitable indices measuring relative importance of vertices.', '1802.02556-2-2-3': 'Over the past decades, many centrality measures were introduced to characterize and analyze the roles of vertices in networks [CITATION].', '1802.02556-2-2-4': 'Among them, a popular one is closeness centrality [CITATION]: the closeness of a vertex is the reciprocal of the sum of shortest path distances between it and all other vertices.', '1802.02556-2-2-5': 'However, this metric considers only the shortest paths, and more importantly, neglects contributions from other paths.', '1802.02556-2-2-6': 'Therefore it can produce some odd effects, or even counterintuitive results [CITATION].', '1802.02556-2-2-7': 'To avoid this shortcoming, Brandes and Fleischer presented current flow closeness centrality [CITATION] based on electrical networks [CITATION] , which takes into account contributions from all paths between vertices.', '1802.02556-2-2-8': 'Current flow based closeness has been shown to better discriminate vertices than its traditional counterparts [CITATION].', '1802.02556-2-3-0': 'While most previous works focus on measures and algorithms for the importance of individual vertices in networks [CITATION], the problem of determining a group of [MATH] most important vertices arises frequently in data mining and graph applications.', '1802.02556-2-3-1': 'For example, in social networks, retailers may want to choose [MATH] vertices as promoters of product, such that the number of the potentially influenced customers is maximized [CITATION].', '1802.02556-2-3-2': 'Another example is P2P networks, where one wants to place resources on a fixed number of [MATH] peers so they are easily accessed by others [CITATION].', '1802.02556-2-3-3': 'In order to measure the importance of a group of vertices, Everett and Borgatti [CITATION] extended the idea of individual centrality to group centrality, and introduced the concepts of group centrality, for example, group closeness.', '1802.02556-2-3-4': 'Recently, some algorithms have been developed to compute or estimate group closeness [CITATION].', '1802.02556-2-3-5': 'However, similar to the case of individual vertices, these notions of group centrality also disregard contributions from paths that are not shortest.', '1802.02556-2-4-0': 'In this paper, we extend current flow closeness of individual vertices [CITATION] by proposing current flow closeness centrality (CFCC) for group of vertices.', '1802.02556-2-4-1': 'In a graph with [MATH] vertices and [MATH] edges, the CFCC [MATH] of a vertex group [MATH] is equal to the ratio of [MATH] to the sum of effective resistances between [MATH] and all vertices [MATH] in [MATH].', '1802.02556-2-4-2': 'We then consider the optimization problem: how can we find a group [MATH] of [MATH] vertices so as to maximize [MATH].', '1802.02556-2-4-3': 'We solve this problem by considering an equivalent problem of minimizing the reciprocal of [MATH].', '1802.02556-2-4-4': 'We show that the problem is NP-hard in Section [REF], but also prove that the problem is an instance of supermodular set function optimization with cardinality constraint in Section [REF].', '1802.02556-2-4-5': 'The latter allows us to devise greedy algorithms to solve this problem, leading to two greedy algorithms with provable approximation guarantees:', '1802.02556-2-5-0': 'A deterministic algorithm with a [MATH] approximation factor and [MATH] running time (Section [REF]); A randomized algorithm with a [MATH]-approximation factor and [MATH] running time for any small [MATH] (Section [REF]).', '1802.02556-2-6-0': 'A key ingredient of our second algorithm is nearly linear time solvers for Laplacians and symmetric, diagonally dominant, M-matrices (SDDM) [CITATION], which has been used in various optimization problems on graphs [CITATION].', '1802.02556-2-7-0': 'We perform extensive experiments on some networks to evaluate our algorithm, and some of their results are in Section [REF].', '1802.02556-2-7-1': 'Our code is available on GitHub at https://github.com/lchc/CFCC-maximization.', '1802.02556-2-7-2': 'These results show that both algorithms are effective.', '1802.02556-2-7-3': 'Moreover, the second algorithm is efficient and is scalable to large networks with more than a million vertices.', '1802.02556-2-8-0': '## Related Works', '1802.02556-2-9-0': 'There exist various measures for centrality of a group of vertices, based on graph structure or dynamic processes, such as betweenness [CITATION], absorbing random-walk centrality [CITATION], and grounding centrality [CITATION].', '1802.02556-2-9-1': 'Since the criterion for importance of a vertex group is application dependent [CITATION], many previous works focus on selecting (or deleting) a group of [MATH] vertices (for some given [MATH]) in order to optimize related quantities.', '1802.02556-2-9-2': 'These quantities are often measures of vertex group importance motivated by the applications, including minimizing the leading eigenvalue of adjacency matrix for vertex immunization [CITATION], minimizing the mean steady-state variance for first-order leader-follower noisy consensus dynamics [CITATION], maximizing average distance for identifying structural hole spanners [CITATION], and others.', '1802.02556-2-10-0': 'Previous works on closeness centrality and related algorithms are most directly related to our focus on the group closeness centrality in this paper.', '1802.02556-2-10-1': 'The closeness centrality for an individual vertex was proposed [CITATION] and formalized [CITATION] by Bavelas.', '1802.02556-2-10-2': 'For a given vertex, its closeness centrality is defined as the reciprocal of the sum of shortest path distances of the vertex to all the other vertices.', '1802.02556-2-10-3': 'Everett and Borgatti [CITATION] extended the individual closeness centrality to group closeness centrality, which measures how close a vertex group is to all other vertices.', '1802.02556-2-10-4': 'For a graph with [MATH] vertices and [MATH] edges, exactly computing the closeness centrality of a group of vertices involves calculating all-pairwise shortest path length, the time complexity of the state-of-the-art algorithm [CITATION] for which is [MATH].', '1802.02556-2-10-5': 'To reduce the computation complexity, various approximation algorithms were developed.', '1802.02556-2-10-6': 'A greedy [MATH] algorithm with approximation ratio [MATH] was devised [CITATION], and a sampling algorithm that scales better to large networks, but without approximation guarantee was also proposed in the same paper.', '1802.02556-2-10-7': 'Very recently, new techniques [CITATION] have been developed to speed up the greedy algorithm in [CITATION] while preserving its theoretical guarantees.', '1802.02556-2-11-0': 'Conventional closeness centrality is based on the shortest paths, omitting the contributions from other paths.', '1802.02556-2-11-1': 'In order to overcome this drawback, Brandes and Fleischer introduced current flow closeness centrality for an individual vertex [CITATION], which essentially considers all paths between vertices, but still gives large weight to short paths.', '1802.02556-2-11-2': 'Our investigation can be viewed as combining this line of current based centrality measures with the study of selecting groups of [MATH] vertices.', '1802.02556-2-11-3': 'For the former, a subset of the authors of this paper (Li and Zhang) recently demonstrated that current flow centrality measures for single edges can be computed provably efficiently [CITATION].', '1802.02556-2-11-4': 'Our approximation algorithm in Section [REF] is directly motivated by that routine.', '1802.02556-2-12-0': '# Preliminaries', '1802.02556-2-13-0': 'In this section, we briefly introduce some useful notations and tools for the convenience of description of our problem and algorithms.', '1802.02556-2-14-0': '## Notations', '1802.02556-2-15-0': 'We use normal lowercase letters like [MATH] to denote scalars in [MATH], normal uppercase letters like [MATH] to denote sets, bold lowercase letters like [MATH] to denote vectors, and bold uppercase letters like [MATH] to denote matrices.', '1802.02556-2-15-1': 'We write [MATH] to denote the [MATH] entry of vector [MATH] and [MATH] to denote entry [MATH] of matrix [MATH].', '1802.02556-2-15-2': 'We also write [MATH] to denote the [MATH] row of [MATH] and [MATH] to denote the [MATH] column of [MATH].', '1802.02556-2-16-0': 'We write sets in matrix subscripts to denote submatrices.', '1802.02556-2-16-1': 'For example, [MATH] denotes the submatrix of [MATH] with row indices in [MATH] and column indices in [MATH].', '1802.02556-2-16-2': 'To simplify notation, we also write [MATH] to denote the submatrix of [MATH] obtained by removing the [MATH] row and [MATH] column of [MATH].', '1802.02556-2-16-3': 'For example, for an [MATH] matrix [MATH], [MATH] denotes the submatrix [MATH].', '1802.02556-2-17-0': 'Note that the precedence of matrix subscripts is the lowest.', '1802.02556-2-17-1': 'Thus, [MATH] denotes the inverse of [MATH] instead of a submatrix of [MATH].', '1802.02556-2-18-0': 'For two matrices [MATH] and [MATH], we write [MATH] to denote that [MATH] is positive semidefinite, i.e., [MATH] holds for every real vector [MATH].', '1802.02556-2-19-0': 'We use [MATH] to denote the [MATH] standard basis vector of appropriate dimension, and [MATH] to denote the indicator vector of [MATH].', '1802.02556-2-20-0': '## Graphs, Laplacians, and Effective Resistances', '1802.02556-2-21-0': 'We write [MATH] to denote a positively weighted undirected graph with [MATH] vertices, [MATH] edges, and edge weight function [MATH].', '1802.02556-2-21-1': 'The Laplacian matrix [MATH] of [MATH] is defined as [MATH] if [MATH], [MATH] if [MATH], and [MATH] otherwise, where [MATH] is the weighted degree of [MATH] and [MATH] means [MATH].', '1802.02556-2-21-2': 'Let [MATH] and [MATH] denote, respectively, the maximum weight and minimum weight among all edges.', '1802.02556-2-21-3': "If we orient each edge of [MATH] arbitrarily, we can also write it's Laplacian as [MATH], where [MATH] is the signed edge-vertex incidence matrix defined by [MATH] if [MATH] is [MATH]'s head, [MATH] if [MATH] is [MATH]'s tail, and [MATH] otherwise, and [MATH] is a diagonal matrix with [MATH].", '1802.02556-2-21-4': 'It is not hard to show that quadratic forms of [MATH] can be written as [MATH] which immediately implies that [MATH] is positive semidefinite, and [MATH] only has one zero eigenvalue if [MATH] is a connected graph.', '1802.02556-2-22-0': 'The following fact shows that submatrices of Laplacians are always positive definite and inverse-positive.', '1802.02556-2-23-0': 'Let [MATH] be the Laplacian of a connected graph and let [MATH] be a nonnegative, diagonal matrix with at least one nonzero entry.', '1802.02556-2-23-1': 'Then, [MATH] is positive definite, and every entry of [MATH] is positive.', '1802.02556-2-24-0': 'Let [MATH] be eigenvalues of [MATH] of a connected graph [MATH], and [MATH] be the corresponding orthonormal eigenvectors.', '1802.02556-2-24-1': 'Then we can decompose [MATH] as [MATH] and define its pseudoinverse as [MATH].', '1802.02556-2-25-0': 'It is not hard to verify that if [MATH] and [MATH] are Laplacians of connected graphs supported on the same vertex set, then [MATH] implies [MATH].', '1802.02556-2-26-0': 'The pseudoinverse of Laplacian matrix can be used to define effective resistance between any pair of vertices [CITATION].', '1802.02556-2-27-0': 'For a connected graph [MATH] with Laplacian matrix [MATH], the effective resistance between vertices [MATH] and [MATH] is defined as [MATH].', '1802.02556-2-28-0': 'The effective resistance between two vertices can also be expressed in term of the diagonal elements of the inverse for submatrices of [MATH].', '1802.02556-2-29-0': 'The effective resistance can also be defined between a vertex and a vertex set.', '1802.02556-2-30-0': '[Effective Resistance Between a Vertex and a Vertex Set [CITATION]] For a connected graph [MATH] with Laplacian matrix [MATH], the effective resistance between vertices [MATH] and [MATH] is defined as [EQUATION] [MATH] can be interpreted as the electrical resistance when the graph is treated as a resistor network in which vertices in [MATH] are grounded.', '1802.02556-2-30-1': 'Namely, if we treat every edge [MATH] as a resistor with resistance [MATH] and ground all vertices in [MATH], then [MATH] equals the voltage of [MATH] when an electrical flow sends one unit of current into [MATH] and removes one unit of current from [MATH] (i.e. the ground).', '1802.02556-2-31-0': '## Current Flow Closeness Centrality', '1802.02556-2-32-0': 'The current flow closeness centrality was proposed in [CITATION].', '1802.02556-2-32-1': 'It is based on the assumption that information spreads efficiently like an electrical current.', '1802.02556-2-33-0': 'To define current flow closeness, we treat the graph [MATH] as a resistor network via replacing every edge [MATH] by a resistor with resistance [MATH].', '1802.02556-2-33-1': 'Let [MATH] denote the voltage of [MATH] when a unit current enters the network at [MATH] and leaves it at [MATH].', '1802.02556-2-34-0': 'The current flow closeness [MATH] of a vertex [MATH] is defined as [MATH].', '1802.02556-2-35-0': 'It has been proved [CITATION] that the current flow closeness of vertex [MATH] equals the ratio of [MATH] to the sum of effective resistances between [MATH] and other vertices.', '1802.02556-2-36-0': '[MATH].', '1802.02556-2-37-0': 'Actually, current flow closeness centrality is equivalent to information centrality [CITATION].', '1802.02556-2-38-0': '## Supermodular Functions', '1802.02556-2-39-0': 'We now give the definitions for monotone and supermodular set functions.', '1802.02556-2-39-1': 'For simplicity, we write [MATH] to denote [MATH] and [MATH] to denote [MATH].', '1802.02556-2-40-0': '[Monotonicity] A set function [MATH] is monotone if [MATH] holds for all [MATH].', '1802.02556-2-41-0': '[Supermodularity] A set function [MATH] is supermodular if [MATH] holds for all [MATH] and [MATH].', '1802.02556-2-42-0': '# Current Flow Closeness of a Group of Vertices', '1802.02556-2-43-0': 'We follow the idea of [CITATION] to define current flow closeness centrality (CFCC) of a group of vertices.', '1802.02556-2-44-0': 'To define current flow closeness centrality for a vertex set [MATH], we treat the graph [MATH] as a resistor network in which all vertices in [MATH] are grounded.', '1802.02556-2-44-1': 'Thus, vertices in [MATH] always have voltage [MATH].', '1802.02556-2-44-2': 'For a vertex [MATH], let [MATH] be the voltage of [MATH] when a unit current enters the network at [MATH] and leaves it at [MATH] (i.e. the ground).', '1802.02556-2-44-3': 'Then, we define the current flow closeness of [MATH] as follows.', '1802.02556-2-45-0': 'Let [MATH] be a connected weighted graph.', '1802.02556-2-45-1': 'The current flow closeness centrality [MATH] of a vertex group [MATH] is defined as [MATH].', '1802.02556-2-46-0': 'Note that there are different variants of the definition of CFCC for a vertex group.', '1802.02556-2-46-1': 'For example, we can use [MATH] as the measure of CFCC for a vertex set [MATH].', '1802.02556-2-46-2': 'Definition [REF] adopts the standard form as the classic closeness centrality [CITATION].', '1802.02556-2-47-0': 'We next show that [MATH] is in fact equal to the ratio of [MATH] to a sum of effective resistances as in Fact [REF].', '1802.02556-2-48-0': 'Let [MATH] be a fixed vertex.', '1802.02556-2-48-1': 'Suppose there is a unit current enters the network at [MATH] and leaves it at [MATH].', '1802.02556-2-48-2': 'Let [MATH] be a vector of voltages at vertices.', '1802.02556-2-48-3': "By Kirchhoff's Current Law and Ohm's Law, we have [MATH] where [MATH] denotes the amount of current flowing out of [MATH].", '1802.02556-2-48-4': 'Since vertices in [MATH] all have voltage [MATH], we can restrict this equation to vertices in [MATH] as [MATH], which leads to [MATH].', '1802.02556-2-48-5': 'This gives the expression of voltage at [MATH] as [MATH].', '1802.02556-2-48-6': 'Now we can write the CFCC of [MATH] as [EQUATION]', '1802.02556-2-48-7': 'Note that the diagonal entry [MATH] of [MATH] is exactly the effective resistance [MATH] between vertex [MATH] and vertex set [MATH] [CITATION], with [MATH] for any [MATH].', '1802.02556-2-48-8': 'Then we have the following relation governing [MATH] and [MATH].', '1802.02556-2-49-0': '[MATH].', '1802.02556-2-50-0': 'Being able to define CFCC of a vertex set raises the problem of maximizing current flow closeness subject to a cardinality constraint, which we state below.', '1802.02556-2-51-0': '[Current Flow Closeness Maximization, CFCM]', '1802.02556-2-52-0': 'Given a connected graph [MATH] with [MATH] vertices, [MATH] edges, and edge weight function [MATH] and an integer [MATH], find a vertex group [MATH] such that the CFCC [MATH] is maximized, that is [MATH]', '1802.02556-2-53-0': '# Hardness of Current Flow Closeness Maximization', '1802.02556-2-54-0': 'In this section, we prove that Problem [REF] is NP-hard.', '1802.02556-2-54-1': 'We will give a reduction from vertex cover on 3-regular graphs (graphs whose vertices all have degree 3), which is an NP-complete problem [CITATION].', '1802.02556-2-54-2': 'The decision version of this problem is stated below.', '1802.02556-2-54-3': '[Vertex Cover on 3-regular graphs, VC3] Given a connected 3-regular graph [MATH] and an integer [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH] is a vertex cover of [MATH] (i.e. every edge in [MATH] is incident with at least one vertex in [MATH]).', '1802.02556-2-55-0': 'An instance of this problem is denoted by VC3[MATH].', '1802.02556-2-56-0': 'We then give the decision version of Problem [REF].', '1802.02556-2-57-0': '[Current Flow Closeness Maximization, Decision Version, CFCMD] Given a connected graph [MATH], an integer [MATH], and a real number [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH].', '1802.02556-2-58-0': 'An instance of this problem is denoted by CFCMD[MATH].', '1802.02556-2-59-0': 'To give the reduction, we will need the following lemma.', '1802.02556-2-60-0': 'Let [MATH] be a connected 3-regular graph with all edge weights being [MATH] (i.e. [MATH] for all [MATH]).', '1802.02556-2-60-1': 'Let [MATH] be a nonempty vertex set, and [MATH].', '1802.02556-2-60-2': 'Then, [MATH] and the equality holds if and only if [MATH] is a vertex cover of [MATH].', '1802.02556-2-61-0': 'We first show that if [MATH] is a vertex cover of [MATH] then [MATH].', '1802.02556-2-61-1': 'When [MATH] is a vertex cover, [MATH] is an independent set.', '1802.02556-2-61-2': 'Thus, [MATH] is a diagonal matrix with all diagonal entries being [MATH].', '1802.02556-2-61-3': 'So we have [MATH].', '1802.02556-2-62-0': 'We then show that if [MATH] is not a vertex cover of [MATH] then [MATH].', '1802.02556-2-62-1': 'When [MATH] is not a vertex cover, [MATH] is not an independent set.', '1802.02556-2-62-2': 'Thus, [MATH] is a block diagonal matrix, with each block corresponding to a connected component of [MATH], the induced graph of [MATH] on [MATH].', '1802.02556-2-62-3': 'Let [MATH] be a connected component of [MATH] such that [MATH].', '1802.02556-2-62-4': 'Then, the block of [MATH] corresponding to [MATH] is [MATH].', '1802.02556-2-62-5': 'For a vertex [MATH], let the [MATH] column of [MATH] be [MATH].', '1802.02556-2-62-6': 'Then, we can write [MATH] into block form as [MATH] where [MATH].', '1802.02556-2-62-7': 'By blockwise matrix inversion we have [MATH].', '1802.02556-2-62-8': 'Since [MATH] is positive definite, we have [MATH] and hence [MATH].', '1802.02556-2-62-9': 'Since [MATH] is a connected component, [MATH] is not a zero vector, which coupled with the fact that [MATH] is positive definite gives [MATH].', '1802.02556-2-62-10': 'Thus, [MATH].', '1802.02556-2-62-11': 'Since this holds for all [MATH], we have [MATH].', '1802.02556-2-62-12': 'Also, since [MATH] can be any connected component of [MATH] with at least two vertices, and a block of an isolate vertex in [MATH] contributes a [MATH] to [MATH], we have for any [MATH] which is not a vertex cover of [MATH] which implies [MATH].', '1802.02556-2-63-0': 'The following theorem then follows by Lemma [REF].', '1802.02556-2-64-0': 'Maximizing current flow closeness subject to a cardinality constraint is NP-hard.', '1802.02556-2-65-0': 'We give a polynomial reduction [MATH] from instances of VC3 to instances of CFCMD.', '1802.02556-2-65-1': 'For a connected 3-regular graph [MATH] with [MATH] vertices, we construct a weighted graph [MATH] with the same vertex set and edge set and an edge weight function [MATH] mapping all edges to weight [MATH].', '1802.02556-2-65-2': 'Then, we construct a reduction [MATH] as [EQUATION]', '1802.02556-2-65-3': 'By Lemma [REF], [MATH] is a polynomial reduction from VC3 to CFCMD, which implies that CFCM is NP-hard.', '1802.02556-2-66-0': 'In this section, we prove that Problem [REF] is NP-hard.', '1802.02556-2-66-1': 'We will give a reduction from vertex cover on 3-regular graphs (graphs whose vertices all have degree 3), which is an NP-complete problem [CITATION].', '1802.02556-2-66-2': 'The decision version of this problem is stated below.', '1802.02556-2-66-3': '[Vertex Cover on 3-regular graphs, VC3] Given a connected 3-regular graph [MATH] and an integer [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH] is a vertex cover of [MATH] (i.e. every edge in [MATH] is incident with at least one vertex in [MATH]).', '1802.02556-2-67-0': '[Current Flow Closeness Maximization, Decision Version, CFCMD] Given a connected graph [MATH], an integer [MATH], and a real number [MATH], decide whether or not there is a vertex set [MATH] such that [MATH] and [MATH].', '1802.02556-2-68-0': 'Let [MATH] be a connected 3-regular graph with all edge weights being [MATH] (i.e. [MATH] for all [MATH]).', '1802.02556-2-68-1': 'Let [MATH] be a nonempty vertex set, and [MATH].', '1802.02556-2-68-2': 'Then, [MATH] and the equality holds if and only if [MATH] is a vertex cover of [MATH].', '1802.02556-2-69-0': 'We first show that if [MATH] is a vertex cover of [MATH] then [MATH].', '1802.02556-2-69-1': 'When [MATH] is a vertex cover, [MATH] is an independent set.', '1802.02556-2-69-2': 'Thus, [MATH] is a diagonal matrix with all diagonal entries being [MATH].', '1802.02556-2-69-3': 'So we have [EQUATION].', '1802.02556-2-70-0': 'We then show that if [MATH] is not a vertex cover of [MATH] then [MATH].', '1802.02556-2-70-1': 'When [MATH] is not a vertex cover, [MATH] is not an independent set.', '1802.02556-2-70-2': 'Thus, [MATH] is a block diagonal matrix, where each block corresponding to a connected component of [MATH], the induced graph of [MATH] on [MATH].', '1802.02556-2-70-3': 'Let [MATH] be a connected component of [MATH] such that [MATH].', '1802.02556-2-70-4': 'Then, the block of [MATH] corresponding to [MATH] is [MATH].', '1802.02556-2-70-5': 'For a vertex [MATH], let the [MATH] column of [MATH] be [MATH].', '1802.02556-2-70-6': 'Then, we can write [MATH] into block form as [EQUATION] where [MATH].', '1802.02556-2-70-7': 'By blockwise matrix inversion we have [EQUATION]', '1802.02556-2-70-8': 'Since [MATH] is positive definite, we have [MATH] and hence [MATH].', '1802.02556-2-70-9': 'Since [MATH] is a connected component, [MATH] is not a zero vector, which coupled with the fact that [MATH] is positive definite gives [MATH].', '1802.02556-2-70-10': 'Since this holds for all [MATH], we have [MATH].', '1802.02556-2-70-11': 'Also, since [MATH] can be any connected component of [MATH] with at least two vertices, and a block of an isolate vertex in [MATH] contributes a [MATH] to [MATH], we have for any [MATH] which is not a vertex cover of [MATH] [EQUATION] which implies [EQUATION].', '1802.02556-2-70-12': 'This completes the proof.', '1802.02556-2-71-0': '# Supermodularity of the Reciprocal of Current Flow Group Closeness', '1802.02556-2-72-0': 'In this section, we prove that the reciprocal of current flow group closeness, i.e., [MATH], is a monotone supermodular function.', '1802.02556-2-72-1': 'Our proof uses the following lemma, which shows that [MATH] is entrywise supermodular.', '1802.02556-2-73-0': 'Let [MATH] be an arbitrary pair of vertices.', '1802.02556-2-73-1': 'Then, the entry [MATH] is a monotone supermodular function.', '1802.02556-2-73-2': 'Namely, for vertices [MATH] and nonempty vertex sets [MATH] such that [MATH], [MATH] and [EQUATION]', '1802.02556-2-73-3': 'To prove Lemma [REF], we first define a linear relaxation [MATH] of [MATH] as [EQUATION]', '1802.02556-2-73-4': 'We remark the intuition behind this relaxation [MATH].', '1802.02556-2-73-5': 'Let [MATH] denote the indices of entries of [MATH] equal to one, and let [MATH] denote the indices of entries of [MATH] less than one.', '1802.02556-2-73-6': 'Then, by the definition in ([REF]), we can write [MATH] into a block diagonal matrix as [EQUATION] where [MATH] is itself a diagonal matrix.', '1802.02556-2-73-7': 'This means that if [MATH] for some nonempty vertex set [MATH], the following statement holds: [EQUATION].', '1802.02556-2-73-8': 'The condition that every entry of [MATH] is in [MATH] coupled with Fact [REF] also implies that all submatrices of [MATH] are positive definite and inverse-positive.', '1802.02556-2-74-0': 'Now for vertices [MATH] and nonempty vertex set [MATH] such that [MATH], we can write the marginal gain of a vertex [MATH] as [EQUATION]', '1802.02556-2-74-1': 'We can further write the matrix on the rhs of ([REF]) as an integral by [EQUATION] where the second equality follows by the identity [EQUATION] for any invertible matrix [MATH].', '1802.02556-2-75-0': 'To prove Lemma [REF], we will also need the following lemma, which shows the entrywise monoticity of [MATH].', '1802.02556-2-76-0': 'For [MATH], the following statement holds for any vertices [MATH] and nonempty vertex sets [MATH] such that [MATH]: [EQUATION]', '1802.02556-2-76-1': 'For simplicity, we let [MATH] and [MATH].', '1802.02556-2-76-2': 'We also write [MATH], [MATH], and [MATH].', '1802.02556-2-76-3': 'Due to the block diagonal structures of [MATH] and [MATH], we have [EQUATION] and [EQUATION]', '1802.02556-2-76-4': 'Since [MATH] and [MATH] agree on entries with indices in [MATH], we can write the submatrix [MATH] of [MATH] in block form as [EQUATION]', '1802.02556-2-76-5': 'By blockwise matrix inversion, we have [EQUATION] where the second equality follows by negating both [MATH] and [MATH].', '1802.02556-2-76-6': 'By definition the matrix [MATH] is entrywise nonnegative.', '1802.02556-2-76-7': 'By Fact [REF], every entry of [MATH] and [MATH] is also nonnegative.', '1802.02556-2-76-8': 'Thus, the matrix [EQUATION] is entrywise nonnegative, which coupled with ([REF]) and ([REF]), implies [MATH].', '1802.02556-2-77-0': '[Proof of Lemma [REF]]', '1802.02556-2-78-0': 'By definition the matrix [MATH] is entrywise nonnegative when [MATH].', '1802.02556-2-78-1': 'By Fact [REF], the matrix [MATH] is also entrywise nonnegative when [MATH].', '1802.02556-2-78-2': 'Thus, the derivative in ([REF]) is entrywise nonnegative, which implies the the monotonicity of [MATH] for any pair of vertices [MATH].', '1802.02556-2-79-0': 'We then prove the supermodularity, i.e., [EQUATION] for any [MATH] and [MATH].', '1802.02556-2-79-1': 'Lemma [REF], coupled with the fact that [MATH] and [MATH] are both entrywise nonnegative, gives the entrywise monotonicity of the derivative in ([REF]) as [EQUATION]', '1802.02556-2-79-2': 'Integrating both sides of the above inequality with respect to [MATH] on the interval [MATH] gives ([REF]).', '1802.02556-2-80-0': 'The following theorem follows by Lemma [REF].', '1802.02556-2-81-0': 'The reciprocal of current flow group centrality, i.e., [MATH], is a monotone supermodular function.', '1802.02556-2-82-0': 'Let [MATH] be vertex sets and [MATH] be a vertex.', '1802.02556-2-83-0': 'For monotonicity, we have [EQUATION] where the first inequality follows by the fact that [MATH] is entrywise nonnegative, and the second inequality follows from the entrywise monotinicity of [MATH].', '1802.02556-2-84-0': 'For supermodularity, we have [EQUATION] where the first inequality follows from the entrywise monotonicity of [MATH], and the second inequality follows from the entrywise supermodularity of [MATH].', '1802.02556-2-85-0': 'We note that [CITATION] has previously proved that [MATH] is monotone and supermodular by using the connection between effective resistance and commute time for random walks.', '1802.02556-2-85-1': 'However, our proof is fully algebraic.', '1802.02556-2-85-2': 'Moreover, we present a more general result that [MATH] is entrywise supermodular.', '1802.02556-2-86-0': 'Theorem [REF] indicates that one can obtain a [MATH]-approximation to the optimum [MATH] by a simple greedy algorithm, by picking the vertex with the maximum marginal gain each time [CITATION].', '1802.02556-2-86-1': 'However, since computing [MATH] involves matrix inversions, a naive implementation of this greedy algorithm will take [MATH] time, assuming that one matrix inversion runs in [MATH] time.', '1802.02556-2-86-2': 'We will show in the next section how to implement this greedy algorithm in [MATH] time using blockwise matrix inversion.', '1802.02556-2-87-0': 'In this section, we give a fully algebraic proof for the supermodularity of reciprocal of current flow group closeness, i.e., [MATH].', '1802.02556-2-87-1': 'The supermodularity of [MATH] is a consequence of a more general result which shows that [MATH] is entrywise supermodular.', '1802.02556-2-88-0': 'Let [MATH] be an arbitrary pair of vertices.', '1802.02556-2-88-1': 'Then, the entry [MATH] is a monotone supermodular function.', '1802.02556-2-89-0': 'To prove Lemma [REF], we first define a linear relaxation [MATH] of [MATH] as [EQUATION]', '1802.02556-2-89-1': 'We remark the intuition behind this relaxation [MATH].', '1802.02556-2-89-2': 'Let [MATH] denote the indices of entries of [MATH] equal to one, and let [MATH] denote the indices of entries of [MATH] less than one.', '1802.02556-2-89-3': 'Then, by the definition in ([REF]), we can write [MATH] into a block diagonal matrix as [EQUATION] where [MATH] is itself a diagonal matrix.', '1802.02556-2-89-4': 'This means that if [MATH] for some nonempty vertex set [MATH], the following statement holds: [EQUATION].', '1802.02556-2-89-5': 'The condition that every entry of [MATH] is in [MATH] coupled with Fact [REF] also implies that all submatrices of [MATH] are positive definite and inverse-positive.', '1802.02556-2-90-0': 'Now for vertices [MATH] and nonempty vertex set [MATH] such that [MATH], we can write the marginal gain of a vertex [MATH] as [EQUATION]', '1802.02556-2-90-1': 'We can further write the matrix on the rhs of ([REF]) as an integral by [EQUATION] where the second equality follows by the identity [EQUATION] for any invertible matrix [MATH].', '1802.02556-2-91-0': 'To prove Lemma [REF], we will also need the following lemma, which shows the entrywise monoticity of [MATH].', '1802.02556-2-92-0': 'For [MATH], the following statement holds for any vertices [MATH] and nonempty vertex sets [MATH] such that [MATH]: [EQUATION]', '1802.02556-2-92-1': 'For simplicity, we let [MATH] and [MATH].', '1802.02556-2-92-2': 'Due to the block diagonal structures of [MATH] and [MATH], we have [EQUATION] and [EQUATION]', '1802.02556-2-92-3': 'Since [MATH] and [MATH] agree on entries with indices in [MATH], we can write the submatrix [MATH] of [MATH] in block form as [EQUATION]', '1802.02556-2-92-4': 'By blockwise matrix inversion, we have [EQUATION] where the second equality follows by negating both [MATH] and [MATH].', '1802.02556-2-92-5': 'By definition the matrix [MATH] is entrywise nonnegative.', '1802.02556-2-92-6': 'By Fact [REF], every entry of [MATH] and [MATH] is also nonnegative.', '1802.02556-2-92-7': 'Thus, the matrix [EQUATION] is entrywise nonnegative, which coupled with ([REF]) and ([REF]), implies [MATH].', '1802.02556-2-93-0': 'By definition the matrix [MATH] is entrywise nonnegative when [MATH].', '1802.02556-2-93-1': 'By Fact [REF], the matrix [MATH] is also entrywise nonnegative when [MATH].', '1802.02556-2-93-2': 'Thus, the derivative in ([REF]) is entrywise nonnegative, which implies the the monotonicity of [MATH] for any pair of vertices [MATH].', '1802.02556-2-94-0': 'We then prove the supermodularity, i.e., [EQUATION] for any [MATH] and [MATH].', '1802.02556-2-94-1': 'Lemma [REF], coupled with the fact that [MATH] and [MATH] are both entrywise nonnegative, gives the entrywise monotonicity of the derivative in ([REF]) as [EQUATION]', '1802.02556-2-94-2': 'Integrating both sides of the above inequality with respect to [MATH] on the interval [MATH] gives ([REF]).', '1802.02556-2-95-0': 'The reciprocal of current flow group centrality, i.e., [MATH], is a monotone supermodular function.', '1802.02556-2-96-0': 'For monotonicity, we have [EQUATION] where the first inequality follows by the fact that [MATH] is entrywise nonnegative, and the second inequality follows from the entrywise monotinicity of [MATH].', '1802.02556-2-97-0': 'For supermodularity, we have [EQUATION] where the first inequality follows from the entrywise monotonicity of [MATH], and the second inequality follows from the entrywise supermodularity of [MATH].', '1802.02556-2-98-0': 'Theorem [REF] indicates that one can obtain a [MATH]-approximation to the optimum [MATH] by a simple greedy algorithm, by picking the vertex with the maximum marginal gain each time [CITATION].', '1802.02556-2-98-1': 'However, since computing [MATH] involves matrix inversions, a naive implementation of this greedy algorithm will take [MATH] time, assuming that one matrix inversion runs in [MATH] time.', '1802.02556-2-98-2': 'We will show in the next section how to implement this greedy algorithm in [MATH] time using blockwise matrix inversion.', '1802.02556-2-99-0': '# A Deterministic Greedy Algorithm', '1802.02556-2-100-0': 'We now consider how to accelerate the naive greedy algorithm.', '1802.02556-2-100-1': 'Suppose that after the [MATH] step, the algorithm has selected a set [MATH] containing [MATH] vertices.', '1802.02556-2-100-2': 'We next compute the marginal gain [MATH] of each vertex [MATH].', '1802.02556-2-101-0': 'For a vertex [MATH], let [MATH] denote the [MATH] column of the submatrix [MATH].', '1802.02556-2-101-1': 'Then we write [MATH] in block form as [MATH] where [MATH].', '1802.02556-2-101-2': 'By blockwise matrix inversion, we have [EQUATION] where [MATH].', '1802.02556-2-101-3': 'Then the marginal gain of [MATH] can be further expressed as [EQUATION] where the second equality and the fourth equality follow by ([REF]), while the third equality follows by the cyclicity of trace.', '1802.02556-2-102-0': 'By ([REF]), we can also update the inverse [MATH] upon a vertex [MATH] by [EQUATION]', '1802.02556-2-102-1': 'At the first step, we need to pick a vertex [MATH] with minimum [MATH], which can be done by computing [MATH] for all [MATH] using the relation [CITATION] [MATH].', '1802.02556-2-103-0': 'We give the [MATH]-time algorithm as follows.', '1802.02556-2-104-0': '[MATH]', '1802.02556-2-105-0': 'Compute [MATH] by inverting [MATH] in [MATH] time.', '1802.02556-2-105-1': '[MATH] where [MATH] Compute [MATH] in [MATH] time.', '1802.02556-2-105-2': 'Repeat the following steps for [MATH]:', '1802.02556-2-106-0': '[MATH], [MATH] Compute [MATH] in [MATH] time by [EQUATION]', '1802.02556-2-107-0': 'Return [MATH].', '1802.02556-2-108-0': 'The performance of [MATH] is characterized in the following theorem.', '1802.02556-2-109-0': 'The algorithm [MATH] takes an undirected positive weighted graph [MATH] with associated Laplacian [MATH] and an integer [MATH], and returns a vertex set [MATH] with [MATH].', '1802.02556-2-109-1': 'The algorithm runs in time [MATH].', '1802.02556-2-109-2': 'The vertex set [MATH] satisfies [EQUATION] where [MATH] and [MATH].', '1802.02556-2-110-0': 'The running time is easy to verify.', '1802.02556-2-110-1': 'We only need to prove the approximation ratio.', '1802.02556-2-111-0': 'By supermodularity, for any [MATH] [EQUATION] which implies [EQUATION]', '1802.02556-2-111-1': 'Then, we have [EQUATION] which coupled with [MATH] completes the proof.', '1802.02556-2-112-0': '# A Randomized Greedy Algorithm', '1802.02556-2-113-0': 'The deterministic greedy algorithm [MATH] has a time complexity [MATH], which is still not acceptable for large networks.', '1802.02556-2-113-1': 'In this section, we provide an efficient randomized algorithm, which achieves a [MATH] approximation factor in time [MATH].', '1802.02556-2-114-0': 'To further accelerate algorithm [MATH] we need to compute the marginal gains [EQUATION] for all [MATH] and a vertex set [MATH] more quickly.', '1802.02556-2-114-1': 'We also need a faster way to compute [EQUATION] for all [MATH] at the [MATH] step.', '1802.02556-2-114-2': 'We will show how to solve both problems in nearly linear time using Johnson-Lindenstrauss Lemma and Fast SDDM Solvers.', '1802.02556-2-114-3': 'Our routines are motivated by the effective resistance estimation routine in [CITATION].', '1802.02556-2-115-0': '[Johnson-Lindenstrauss Lemma [CITATION]]', '1802.02556-2-116-0': 'Let [MATH] vertices.', '1802.02556-2-116-1': 'In future works, we plan to introduce and study the betweenness group centrality based on current flow [CITATION], which takes into account all possible paths.'} | [['1802.02556-1-12-0', '1802.02556-2-15-0'], ['1802.02556-1-12-1', '1802.02556-2-15-1'], ['1802.02556-1-12-2', '1802.02556-2-15-2'], ['1802.02556-1-56-0', '1802.02556-2-70-0'], ['1802.02556-1-56-1', '1802.02556-2-70-1'], ['1802.02556-1-56-2', '1802.02556-2-70-2'], ['1802.02556-1-56-3', '1802.02556-2-70-3'], ['1802.02556-1-56-4', '1802.02556-2-70-4'], ['1802.02556-1-56-5', '1802.02556-2-70-5'], ['1802.02556-1-56-6', '1802.02556-2-70-6'], ['1802.02556-1-56-7', '1802.02556-2-70-7'], ['1802.02556-1-56-8', '1802.02556-2-70-8'], ['1802.02556-1-56-9', '1802.02556-2-70-9'], ['1802.02556-1-56-11', '1802.02556-2-70-10'], ['1802.02556-1-56-12', '1802.02556-2-70-11'], ['1802.02556-1-56-13', '1802.02556-2-70-12'], ['1802.02556-1-63-0', '1802.02556-2-90-0'], ['1802.02556-1-63-1', '1802.02556-2-90-1'], ['1802.02556-1-51-0', '1802.02556-2-67-0'], ['1802.02556-1-70-0', '1802.02556-2-95-0'], 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'1802.02556-1-81-0', '1802.02556-1-81-1', '1802.02556-1-82-0', '1802.02556-1-83-0', '1802.02556-1-85-0', '1802.02556-1-87-0', '1802.02556-1-87-1', '1802.02556-1-91-0', '1802.02556-2-4-5', '1802.02556-2-36-0', '1802.02556-2-49-0', '1802.02556-2-51-0', '1802.02556-2-55-0', '1802.02556-2-56-0', '1802.02556-2-58-0', '1802.02556-2-59-0', '1802.02556-2-62-10', '1802.02556-2-63-0', '1802.02556-2-76-2', '1802.02556-2-77-0', '1802.02556-2-80-0', '1802.02556-2-82-0', '1802.02556-2-103-0', '1802.02556-2-104-0', '1802.02556-2-105-0', '1802.02556-2-105-1', '1802.02556-2-105-2', '1802.02556-2-106-0', '1802.02556-2-107-0', '1802.02556-2-108-0', '1802.02556-2-115-0', '1802.02556-2-116-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1802.02556 | null | null | null | null | null |
1509.00021 | {'1509.00021-1-0-0': 'Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHsSH).', '1509.00021-1-0-1': 'KBHsSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs.', '1509.00021-1-0-2': 'Moving from the weak to the strong gravity region, BSs - which by themselves have no shadows - are classified, according to the lensing produced, as: [MATH] non-compact, which yield no multiple images; [MATH] compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere; [MATH] ultra-compact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure.', '1509.00021-1-0-3': 'The shadows of KBHsSH, for Kerr-like horizons and non-compact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs.', '1509.00021-1-0-4': 'But for non-Kerr-like horizons and ultra-compact BS-like hair, the shadows of KBHsSH are drastically different: novel shapes arise, sizes are considerably smaller and multiple shadows of a single BH become possible.', '1509.00021-1-0-5': 'Thus, KBHsSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry (VLBI) observations of BH shadows, such as those of the Event Horizon Telescope.', '1509.00021-1-1-0': 'Introduction.', '1509.00021-1-1-1': '100 years after General Relativity (GR) was formulated, we finally face a realistic prospect of testing one of its most dramatic consequences: black holes (BHs).', '1509.00021-1-1-2': 'The evidence for astrophysical BHs, gathered for over half a century, has built a strong case [CITATION], but it could not confirm the existence of event horizons, the defining property of BHs.', '1509.00021-1-1-3': 'The near future promises to open up a new channel of observation - gravitational waves - and deliver electromagnetic measurements of unprecedented precision, hopefully clarifying this central issue [CITATION].', '1509.00021-1-2-0': 'A particularly exciting prospect is the use of VLBI techniques to resolve the angular scale of the event horizon for some supermassive BH candidates and determine the corresponding BH shadow [CITATION].', '1509.00021-1-2-1': 'Its observation would probe the spacetime geometry in the vicinity of the horizon and consequently test the existence and properties of the latter [CITATION].', '1509.00021-1-2-2': 'It is therefore timely to study BH models that yield phenomenological deviations from the paradigmatic GR BH, described by the Kerr metric.', '1509.00021-1-3-0': 'One approach is to parameterize families of metric deviations from Kerr [CITATION].', '1509.00021-1-3-1': 'Another approach is to use exact solutions of GR (or generalizations thereof) yielding deviations from Kerr ([MATH] [CITATION]).', '1509.00021-1-3-2': 'Exact solutions with physically reasonable and astrophysically plausible matter sources, however, are scarce; but Kerr BHs with scalar hair (KBHsSH) [CITATION] are argueably one such model.', '1509.00021-1-3-3': "These are exact solutions of Einstein's gravity minimally coupled to a massive complex scalar field, and interpolate between Kerr BHs and gravitating solitons - boson stars (BSs) [CITATION] - which have been suggested as dark matter candidates and BH mimickers [CITATION].", '1509.00021-1-4-0': 'In this letter we show that the shadows of KBHsSH are distinguishable, or even drastically different, from those of Kerr BHs, and can thus yield new templates for the ongoing and future VLBI searches [CITATION] of BH shadows.', '1509.00021-1-5-0': 'The solutions.', '1509.00021-1-5-1': 'KBHsSH can be expressed by a stationary and axi-symmetric line element, in spheroidal coordinates [MATH] [CITATION], together with the (mass [MATH]) scalar field [MATH], where [MATH] is the frequency and [MATH] is the azimuthal harmonic index.', '1509.00021-1-5-2': 'The metric functions and [MATH] are determined numerically by solving five coupled, non-linear PDEs [CITATION].', '1509.00021-1-5-3': 'For [MATH], both BSs and KBHsSH solutions can be obtained.', '1509.00021-1-5-4': 'For [MATH], spherical BSs exist.', '1509.00021-1-5-5': 'The space of solutions for [MATH] is summarized in Fig. [REF].', '1509.00021-1-5-6': 'BSs exist for a limited range of [MATH] along spiraling curves.', '1509.00021-1-5-7': 'KBHsSH exist inside an open set, bounded by the [MATH] BS curve, a set of Kerr BHs and the set of extremal KBHsSH.', '1509.00021-1-6-0': 'Setup.', '1509.00021-1-6-1': 'Our goal is to compute photon geodesics in the above geometries and obtain, at a given observation point, the distorted apparent sky, due to the gravitational lensing of BSs, as well as the shadow, when a horizon is present.', '1509.00021-1-6-2': 'To interpret the patterns obtained, we divide the "celestial sphere" light source into four quadrants, each painted with a different color.', '1509.00021-1-6-3': 'On top of these, a grid of constant longitude and latitude (black) lines is introduced, with adjacent lines separated by [MATH] - Fig. [REF] (left).', '1509.00021-1-6-4': 'This setup mimicks closely the one in [CITATION].', '1509.00021-1-7-0': 'The observer, henceforth taken to be a camera, is placed inside the celestial sphere at an off-centered position to be specified below, and its viewing angle captures [MATH] of that sphere - Fig. [REF] (right).', '1509.00021-1-7-1': 'The point on the celestial sphere immediately in front of the observer (dubbed [MATH]) lies at the intersection of the four colored quadrants and it is marked with a white dot .', '1509.00021-1-8-0': "From the camera's position we span its viewing angle by performing, numerically, backwards ray tracing of [MATH] photon trajectories .", '1509.00021-1-8-1': 'We integrate these null geodesics until they either reach a point on the celestial sphere or they hit the BH horizon (when it is present).', '1509.00021-1-8-2': 'The latter directions form the shadow.', '1509.00021-1-8-3': "The camera's image is obtained upon a projection onto its local frame, by a method similar to that in [CITATION].", '1509.00021-1-9-0': 'Quantitative shadow parameters.', '1509.00021-1-9-1': 'To analyse the shadows obtained below we introduce six parameters, [MATH], mostly following [CITATION].', '1509.00021-1-9-2': "Let the camera's image be parametrized by the Cartesian coordinates [MATH], obtained from multiplying the observation angles by the circumferential radius [MATH] at the camera's position.", '1509.00021-1-9-3': 'The origin of this coordinate system, [MATH], points at the (unlensed) white dot on the celestial sphere.', '1509.00021-1-9-4': "The centre of the shadow, [MATH], has [MATH] where [MATH] and [MATH] and [MATH]) are respectively the maximum and minimum abscissae (ordinates) of the shadow's edge.", '1509.00021-1-9-5': 'Due to reflection symmetry for observations on the equatorial plane, [MATH].', '1509.00021-1-9-6': '[MATH] and [MATH] need not coincide; the displacement, [MATH], measures their difference.', '1509.00021-1-9-7': 'The width and height of the shadow are, respectively, [MATH] and [MATH].', '1509.00021-1-10-0': "A generic point [MATH] on the shadow's edge is at a distance [MATH] from [MATH].", '1509.00021-1-10-1': 'Let [MATH] be the angle between the line [MATH] and the [MATH] axis; the average radius is [MATH] and the deviation from sphericity is [MATH].', '1509.00021-1-10-2': 'Finally, [EQUATION] is the relative deviation from a comparable Kerr BH, either with the same ADM mass and angular momentum, [MATH], [MATH], or with the same horizon quantities, [MATH] (as long as the Kerr bound is not violated for the comparable Kerr BH).', '1509.00021-1-10-3': 'These comparable Kerr BHs are denoted below as Kerr[MATH] and Kerr[MATH], respectively.', '1509.00021-1-10-4': '[MATH] is made to coincide for both BHs in ([REF]).', '1509.00021-1-11-0': 'Lensing by spherical BSs.', '1509.00021-1-11-1': 'We first look at the lensing due to spherically symmetric BSs - Fig. [REF].', '1509.00021-1-11-2': 'We set the camera on the equator and always at [MATH], for the different BS solutions.', '1509.00021-1-11-3': 'Throughout we take [MATH] and in the following, unless otherwise stated, [MATH].', '1509.00021-1-12-0': 'Starting from vacuum ([MATH]) we find a set of non-compact BSs, for which any meaningful effective radius is large as compared to the corresponding Schwarzschild radius.', '1509.00021-1-12-1': 'These are "weak gravity" solutions and the corresponding lensing is illustrated by the BS with [MATH] in Fig. [REF], where only a small distortion of the background is observed.', '1509.00021-1-12-2': 'Moving further along the spiral, an Einstein ring appears at [MATH].', '1509.00021-1-12-3': 'The Einstein ring is formed by the lensing of [MATH], and it encloses two inverted copies of a region around [MATH].', '1509.00021-1-12-4': 'This is illustrated in Fig. [REF] using a BS with [MATH].', '1509.00021-1-12-5': 'The appearance of the first Einstein ring defines the transition from non-compact to compact BSs.', '1509.00021-1-13-0': 'Moving further along the spiral the region duplicated inside the Einstein ring becomes an increasingly larger part of the celestial sphere, as can be observed for the BS with [MATH] in Fig. [REF].', '1509.00021-1-13-1': "Shortly after the spiral's first backbending, at [MATH].", '1509.00021-1-13-2': 'Fig. [REF]), the full celestial sphere becomes duplicated, starting at the BS with [MATH].', '1509.00021-1-13-3': 'At, and beyond, this point, two further Einstein rings emerge - now corresponding to lensings of the point immediately behind the observer (dubbed [MATH]).', '1509.00021-1-13-4': 'This is illustrated by the bottom left panel in Fig. [REF], for a BS with [MATH].', '1509.00021-1-13-5': 'In between these two new Einstein rings, further pairs of Einstein rings can be seen to appear, progressively, further inside the spiral.', '1509.00021-1-13-6': 'The first such pair is illustrated in Fig. [REF] for [MATH].', '1509.00021-1-13-7': 'Each new pair of Einstein rings corresponds to images of either [MATH] or [MATH], in an alternating fashion, and to a further complete copy of the full celestial sphere.', '1509.00021-1-14-0': 'An infinite number of copies, and a corresponding self-similar structure, is expected to arise when a light-ring - corresponding in this case to a photon sphere - appears , marking the transition from compact to ultra-compact [CITATION] BSs.', '1509.00021-1-14-1': "This occurs well inside the spiral, on the third branch (after the second backbending), starting at the BS solution with [MATH], marked as the blue star '[MATH]' point on the [MATH] spiral of Fig. [REF].", '1509.00021-1-15-0': 'Lensing by rotating BSs.', '1509.00021-1-15-1': 'We now turn to rotating BSs (spin axis pointing up).', '1509.00021-1-15-2': 'Starting from vacuum, we again find a region of non-compact BSs, [MATH] without multiple images.', '1509.00021-1-15-3': 'Two differences, however, with respect to the top left panel of Fig. [REF] are: an asymmetric lensing, with an amplification of the side rotating away from the camera and the slight shift of point [MATH] to the left, due to frame dragging; these are illustrated in Fig. [REF] for a BS with [MATH].', '1509.00021-1-15-4': 'At [MATH] an Einstein ring appears, starting the set of compact rotating BSs.', '1509.00021-1-15-5': 'This is the case, in Fig. [REF], for the BSs with [MATH].', '1509.00021-1-15-6': 'The ring encloses again two inverted copies of part of the celestial sphere, but it is now elliptic and the duplicated image of the side rotating towards the observer is suppressed.', '1509.00021-1-15-7': 'The inversion shifts point [MATH] to the right.', '1509.00021-1-16-0': "Further following the spiral, new Einstein 'rings' appear, just as in the case of spherical BSs, except that instead of an 'O-shape', they have a squashed 'D-shape'.", '1509.00021-1-16-1': 'An example, for a BS with [MATH] is shown in Fig. [REF].', '1509.00021-1-16-2': 'Then, a light ring on the equatorial plane appears at [MATH].', '1509.00021-1-16-3': 'Beyond this point multiple (presumably infinitely many) images of the celestial sphere arise, with, we conjecture, a fractal structure.', '1509.00021-1-16-4': 'This is illustrated by the BSs with [MATH] and [MATH] in Fig. [REF].', '1509.00021-1-17-0': 'Shadows of KBHsSH.', '1509.00021-1-17-1': 'A KBHSH may be regarded as a BS around (and co-rotating with) a central horizon.', '1509.00021-1-17-2': 'The latter may be non-Kerr-like, [MATH] violate the Kerr bound in terms of horizon quantities [CITATION] and the former may have strong lensing effects, as seen above.', '1509.00021-1-17-3': 'Consequently, it is expectable that KBHsSH with ultra-compact BS-like hair and with a non-Kerr-like central horizon will have unfamiliar shadows.', '1509.00021-1-17-4': "This expectation is confirmed in Fig. [REF], where we exhibit the shadows and lensing for three examples of KBHsSH (together with four 'transition' examples) and for comparable Kerr BHs.", '1509.00021-1-17-5': 'The three main examples are configuration I-III in Fig. [REF] , and their physical quantities, in units of [MATH] when dimensionful, are summarized as follows:', '1509.00021-1-18-0': 'In Fig. [REF], we have scaled [MATH] in each case so that [MATH] is the same for all KBHsSH presented, and the camera sits at [MATH].', '1509.00021-1-18-1': 'Comparing the shadows in this way bears more significance for real observations, where [MATH] of the BH is fixed by the data.', '1509.00021-1-19-0': 'The top left panel of Fig. [REF] shows the shadow of configuration I (shadow I for short).', '1509.00021-1-19-1': "For this KBHSH the 'hair' contains only [MATH] of the total mass; the 'nearby' BSs in Fig. [REF] are non-compact; and the central horizon is Kerr-like ([MATH]).", '1509.00021-1-19-2': 'The shadow, albeit qualitatively familiar, is nevertheless distinguishable from that of the Kerr[MATH] BH, which is exhibited in the top middle panel.', '1509.00021-1-19-3': 'The latter is slightly larger and more [MATH]-like - a characteristic of extremal Kerr BHs.', '1509.00021-1-19-4': 'Shadow I turns out to be closer to the one of the Kerr[MATH] BH, exhibited in the top right panel.', '1509.00021-1-19-5': 'This observation can be quantitatively checked: [MATH], taking Kerr[MATH]/Kerr[MATH] for the comparable BH, [MATH].', '1509.00021-1-19-6': 'Table I .', '1509.00021-1-20-0': 'New types of BH shadows, quite distinct from those of Kerr[MATH] BHs, appear on the left 2[MATH] and 4[MATH] row panels of Fig. [REF], corresponding to shadow II and III.', '1509.00021-1-20-1': 'In both cases, the central BH is non-Kerr-like ([MATH]).', '1509.00021-1-20-2': 'Shadow II is smaller (average radius of [MATH]75%) than that of the Kerr[MATH] BH.', '1509.00021-1-20-3': "It is also more 'square', with a larger normalized deviation from sphericity.", '1509.00021-1-20-4': 'Shadow III is remarkably distinct.', '1509.00021-1-20-5': "Its central BH has [MATH], allowed by the 'heavy' hair that it is dragging ([MATH] the discussion in [CITATION]).", '1509.00021-1-20-6': 'The lensing of this hair resembles closely that of the ultra-compact BS on the bottom left panel of Fig. [REF].', '1509.00021-1-20-7': "Interestingly, multiple disconnected shadows of the (single) BH occur: the largest ones (besides the main 'hammer-like' shadow) are two eyebrows [CITATION], at symmetric positions above and below the main shadow; but we have detected many other smaller shadows, hinting again at a self-similar structure.", '1509.00021-1-20-8': 'On the 3[MATH] row of Fig. [REF], the shadows of four solutions in between configurations II and III illustrate the transition between them.', '1509.00021-1-20-9': 'Finally, we remark that the shadows of KBHsSH can have arbitrarily small sizes by considering solutions arbitrarily close to the BS curve in Fig. [REF].', '1509.00021-1-21-0': 'Remarks.', '1509.00021-1-21-1': 'KBHsSH can lead to qualitatively novel types of shadows in GR, as shown by shadows II and III.', '1509.00021-1-21-2': 'Even for KBHsSH close to Kerr, their shadows are distinguishable from the latter, with the same asymptotic quantities, as illustrated by shadow I. Regardless of the astrophysical relevance of these solutions - which is unclear - they can yield new templates with small or large deviations from the Kerr shadows, hopefully of use for VLBI observations.', '1509.00021-1-21-3': 'An exhaustive analysis of KBHsSH shadows spanning the space of solutions in Fig. [REF], and at different observation angles, will be presented elsewhere, for producing such templates [CITATION].', '1509.00021-1-21-4': 'But the examples herein already raise a challenge to the parameterizations of deviations from Kerr suggested in the literature [CITATION]: can they describe shadows with such large deviations?', '1509.00021-1-22-0': 'Besides the peculiar shape of some of the shadows exhibited, this model has one general prediction: smaller observed shadows than those expected for Kerr BHs with the same asymptotic charges.', '1509.00021-1-22-1': "Indeed, a 'smaller' central BH seems a natural consequence of the existence of hair, carrying part of the total energy.", '1509.00021-1-23-0': "Finally, for the setup herein, the redshift, which depends only on the source's and camera's positions, is constant throughout the image and has been neglected."} | {'1509.00021-2-0-0': 'Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHsSH).', '1509.00021-2-0-1': 'KBHsSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs.', '1509.00021-2-0-2': 'Moving from the weak to the strong gravity region, BSs - which by themselves have no shadows - are classified, according to the lensing produced, as: [MATH] non-compact, which yield no multiple images; [MATH] compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere; [MATH] ultra-compact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure.', '1509.00021-2-0-3': 'The shadows of KBHsSH, for Kerr-like horizons and non-compact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs.', '1509.00021-2-0-4': 'But for non-Kerr-like horizons and ultra-compact BS-like hair, the shadows of KBHsSH are drastically different: novel shapes arise, sizes are considerably smaller and multiple shadows of a single BH become possible.', '1509.00021-2-0-5': 'Thus, KBHsSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry (VLBI) observations of BH shadows, such as those of the Event Horizon Telescope.', '1509.00021-2-1-0': 'Introduction.', '1509.00021-2-1-1': '100 years after General Relativity (GR) was formulated, we finally face a realistic prospect of testing one of its most dramatic consequences: black holes (BHs).', '1509.00021-2-1-2': 'The evidence for astrophysical BHs, gathered for over half a century, has built a strong case [CITATION], but it could not confirm the existence of event horizons, the defining property of BHs.', '1509.00021-2-1-3': 'The near future promises to open up a new channel of observation - gravitational waves - and deliver electromagnetic measurements of unprecedented precision, hopefully clarifying this central issue [CITATION].', '1509.00021-2-2-0': 'A particularly exciting prospect is the use of VLBI techniques to resolve the angular scale of the event horizon for some supermassive BH candidates and determine the corresponding BH shadow [CITATION].', '1509.00021-2-2-1': 'Its observation would probe the spacetime geometry in the vicinity of the horizon and consequently test the existence and properties of the latter [CITATION].', '1509.00021-2-2-2': 'It is therefore timely to study BH models that yield phenomenological deviations from the paradigmatic GR BH, described by the Kerr metric.', '1509.00021-2-3-0': 'One approach is to parameterize families of metric deviations from Kerr [CITATION].', '1509.00021-2-3-1': 'Another approach is to use exact solutions of GR (or generalizations thereof) yielding deviations from Kerr ([MATH] [CITATION]).', '1509.00021-2-3-2': 'Exact solutions with physically reasonable and astrophysically plausible matter sources, however, are scarce; but Kerr BHs with scalar hair (KBHsSH) [CITATION] are arguably one such model.', '1509.00021-2-3-3': "These are exact solutions of Einstein's gravity minimally coupled to a massive complex scalar field, and interpolate between Kerr BHs and gravitating solitons - boson stars (BSs) [CITATION] - suggested as dark matter candidates (in the Newtonian limit) and BH mimickers [CITATION].", '1509.00021-2-4-0': 'In this letter we show that the shadows of KBHsSH are distinguishable, or even drastically different, from those of Kerr BHs, and can thus yield new templates for the ongoing and future VLBI searches [CITATION] of BH shadows.', '1509.00021-2-5-0': 'The solutions.', '1509.00021-2-5-1': 'KBHsSH can be expressed by a stationary and axi-symmetric line element, in spheroidal coordinates [MATH] [CITATION], together with the (mass [MATH]) scalar field [MATH], where [MATH] is the frequency and [MATH] is the azimuthal harmonic index.', '1509.00021-2-5-2': 'The metric functions and [MATH] are determined numerically by solving five coupled, non-linear PDEs [CITATION].', '1509.00021-2-5-3': 'For [MATH], both BSs and KBHsSH solutions can be obtained.', '1509.00021-2-5-4': 'For [MATH], spherical BSs exist.', '1509.00021-2-5-5': 'The space of solutions for [MATH] is summarized in Fig. [REF].', '1509.00021-2-5-6': 'BSs exist for a limited range of [MATH] along spiraling curves.', '1509.00021-2-5-7': 'KBHsSH exist inside an open set, bounded by the [MATH] BS curve, a set of Kerr BHs and the set of extremal KBHsSH.', '1509.00021-2-6-0': 'Setup.', '1509.00021-2-6-1': 'Our goal is to compute photon geodesics in the above geometries and obtain, at a given observation point, the distorted apparent sky, due to the gravitational lensing of BSs, as well as the shadow, when a horizon is present.', '1509.00021-2-6-2': 'To interpret the patterns obtained, we divide the "celestial sphere" light source into four quadrants, each painted with a different color.', '1509.00021-2-6-3': 'On top of these, a grid of constant longitude and latitude (black) lines is introduced, with adjacent lines separated by [MATH] - Fig. [REF] (left).', '1509.00021-2-6-4': 'This setup mimicks closely the one in [CITATION].', '1509.00021-2-7-0': 'The observer, henceforth denoted as [MATH], is placed inside the celestial sphere at an off-centered position to be specified below, and its viewing angle captures [MATH] of that sphere - Fig. [REF] (right).', '1509.00021-2-7-1': 'The point on the celestial sphere immediately in front of [MATH] (dubbed [MATH]) lies at the intersection of the four colored quadrants and it is marked with a white dot.', '1509.00021-2-8-0': "From the [MATH]'s position we span its viewing angle by performing, numerically, backwards ray tracing of [MATH] photon trajectories.", '1509.00021-2-8-1': 'We integrate these null geodesics until they either reach a point on the celestial sphere or they hit the BH horizon (when it is present).', '1509.00021-2-8-2': 'The latter directions form the shadow .', '1509.00021-2-8-3': "[MATH]'s image is obtained upon a projection onto its local frame, by a method similar to that in [CITATION].", '1509.00021-2-9-0': 'Quantitative shadow parameters.', '1509.00021-2-9-1': 'To analyse the shadows obtained below we introduce six parameters, [MATH], mostly following [CITATION].', '1509.00021-2-9-2': "Let [MATH]'s image be parametrized by the Cartesian coordinates [MATH], obtained from multiplying the observation angles by the circumferential radius [MATH] at [MATH]'s position.", '1509.00021-2-9-3': 'The origin of this coordinate system, [MATH], points at the (unlensed) white dot on the celestial sphere.', '1509.00021-2-9-4': "The centre of the shadow, [MATH], has [MATH] where [MATH] and [MATH] and [MATH]) are respectively the maximum and minimum abscissae (ordinates) of the shadow's edge.", '1509.00021-2-9-5': 'Due to reflection symmetry for observations on the equatorial plane, [MATH].', '1509.00021-2-9-6': '[MATH] and [MATH] need not coincide; the displacement, [MATH], measures their difference.', '1509.00021-2-9-7': 'The width and height of the shadow are, respectively, [MATH] and [MATH].', '1509.00021-2-10-0': "A generic point [MATH] on the shadow's edge is at a distance [MATH] from [MATH].", '1509.00021-2-10-1': 'Let [MATH] be the angle between the line [MATH] and the [MATH] axis; the average radius is [MATH] and the deviation from sphericity is [MATH].', '1509.00021-2-10-2': 'Finally, [EQUATION] is the relative deviation from a comparable Kerr BH, either with the same ADM mass and angular momentum, [MATH], [MATH], or with the same horizon quantities, [MATH] (as long as the Kerr bound is not violated for the comparable Kerr BH).', '1509.00021-2-10-3': 'These comparable Kerr BHs are denoted below as Kerr[MATH] and Kerr[MATH], respectively.', '1509.00021-2-10-4': '[MATH] is made to coincide for both BHs in ([REF]).', '1509.00021-2-11-0': 'Lensing by spherical BSs.', '1509.00021-2-11-1': 'We first look at the lensing due to spherically symmetric BSs - Fig. [REF].', '1509.00021-2-11-2': 'We set [MATH] on the equator and always at [MATH], for the different BS solutions.', '1509.00021-2-11-3': 'Throughout we take [MATH] and in the following, unless otherwise stated, [MATH].', '1509.00021-2-12-0': 'Starting from vacuum ([MATH]) we find a set of non-compact BSs, for which any meaningful effective radius is large as compared to the corresponding Schwarzschild radius.', '1509.00021-2-12-1': 'These are "weak gravity" solutions and the corresponding lensing is illustrated by the BS with [MATH] in Fig. [REF], where only a small distortion of the background is observed.', '1509.00021-2-12-2': 'Moving further along the spiral, an Einstein ring appears at [MATH].', '1509.00021-2-12-3': 'The Einstein ring is formed by the lensing of [MATH], and it encloses two inverted copies of a region around [MATH].', '1509.00021-2-12-4': 'This is illustrated in Fig. [REF] using a BS with [MATH].', '1509.00021-2-12-5': 'The appearance of the first Einstein ring defines the transition from non-compact to compact BSs.', '1509.00021-2-13-0': 'Moving further along the spiral the region duplicated inside the Einstein ring becomes an increasingly larger part of the celestial sphere, as can be observed for the BS with [MATH] in Fig. [REF].', '1509.00021-2-13-1': "Shortly after the spiral's first backbending, at [MATH].", '1509.00021-2-13-2': 'Fig. [REF]), the full celestial sphere becomes duplicated, starting at the BS with [MATH].', '1509.00021-2-13-3': 'At, and beyond, this point, two further Einstein rings emerge - now corresponding to lensings of the point immediately behind the observer (dubbed [MATH]).', '1509.00021-2-13-4': 'This is illustrated by the bottom left panel in Fig. [REF], for a BS with [MATH].', '1509.00021-2-13-5': 'In between these two new Einstein rings, further pairs of Einstein rings can be seen to appear, progressively, further inside the spiral.', '1509.00021-2-13-6': 'The first such pair is illustrated in Fig. [REF] for [MATH].', '1509.00021-2-13-7': 'Each new pair of Einstein rings corresponds to images of either [MATH] or [MATH], in an alternating fashion, and to a further complete copy of the full celestial sphere.', '1509.00021-2-14-0': 'An infinite number of copies, and a corresponding self-similar structure, is expected to arise when a light-ring - corresponding in this case to a photon sphere - appears , marking the transition from compact to ultra-compact [CITATION] BSs.', '1509.00021-2-14-1': "This occurs well inside the spiral, on the third branch (after the second backbending), starting at the BS solution with [MATH], marked as the blue star '[MATH]' point on the [MATH] spiral of Fig. [REF].", '1509.00021-2-15-0': 'Lensing by rotating BSs.', '1509.00021-2-15-1': 'We now turn to rotating BSs (spin axis pointing up).', '1509.00021-2-15-2': 'Starting from vacuum, we again find a region of non-compact BSs, [MATH] without multiple images.', '1509.00021-2-15-3': 'Two differences, however, with respect to the top left panel of Fig. [REF] are: an asymmetric lensing, with an amplification of the side rotating away from [MATH] and the slight shift of point [MATH] to the left, due to frame dragging; these are illustrated in Fig. [REF] for a BS with [MATH].', '1509.00021-2-15-4': 'At [MATH] an Einstein ring appears, starting the set of compact rotating BSs.', '1509.00021-2-15-5': 'This is the case, in Fig. [REF], for the BSs with [MATH].', '1509.00021-2-15-6': 'The ring encloses again two inverted copies of part of the celestial sphere, but it is now elliptic and the duplicated image of the side rotating towards the observer is suppressed.', '1509.00021-2-15-7': 'The inversion shifts point [MATH] to the right.', '1509.00021-2-16-0': "Further following the spiral, new Einstein 'rings' appear, just as in the case of spherical BSs, except that instead of an 'O-shape', they have a squashed 'D-shape'.", '1509.00021-2-16-1': 'An example, for a BS with [MATH] is shown in Fig. [REF].', '1509.00021-2-16-2': 'Then, a light ring on the equatorial plane appears at [MATH].', '1509.00021-2-16-3': 'Beyond this point multiple (presumably infinitely many) images of the celestial sphere arise, with, we conjecture, a fractal structure.', '1509.00021-2-16-4': 'This is illustrated by the BSs with [MATH] and [MATH] in Fig. [REF].', '1509.00021-2-17-0': 'Shadows of KBHsSH.', '1509.00021-2-17-1': 'A KBHSH may be regarded as a BS around (and co-rotating with) a central horizon.', '1509.00021-2-17-2': 'The latter may be non-Kerr-like, [MATH] violate the Kerr bound in terms of horizon quantities [CITATION] and the former may have strong lensing effects, as seen above.', '1509.00021-2-17-3': 'Consequently, it is expectable that KBHsSH with ultra-compact BS-like hair and with a non-Kerr-like central horizon will have unfamiliar shadows.', '1509.00021-2-17-4': "This expectation is confirmed in Fig. [REF], where we exhibit the shadows and lensing for three examples of KBHsSH (together with four 'transition' examples) and for comparable Kerr BHs.", '1509.00021-2-17-5': 'The three main examples are configuration I-III in Fig. [REF], and their physical quantities, in units of [MATH] when dimensionful, are summarized as follows:', '1509.00021-2-18-0': 'In Fig. [REF], we have scaled [MATH] in each case so that [MATH] is the same for all KBHsSH presented, and [MATH] sits at [MATH].', '1509.00021-2-18-1': 'Comparing the shadows in this way bears more significance for real observations, where [MATH] of the BH is fixed by the data.', '1509.00021-2-19-0': 'The top left panel of Fig. [REF] shows the shadow of configuration I (shadow I for short).', '1509.00021-2-19-1': "For this KBHSH the 'hair' contains only [MATH] of the total mass; the 'nearby' BSs in Fig. [REF] are non-compact; and the central horizon is Kerr-like ([MATH]).", '1509.00021-2-19-2': 'The shadow, albeit qualitatively familiar, is nevertheless distinguishable from that of the Kerr[MATH] BH, which is exhibited in the top middle panel.', '1509.00021-2-19-3': 'The latter is slightly larger and more [MATH]-like - a characteristic of extremal Kerr BHs.', '1509.00021-2-19-4': 'Shadow I turns out to be closer to the one of the Kerr[MATH] BH, exhibited in the top right panel.', '1509.00021-2-19-5': 'This observation can be quantitatively checked: [MATH], taking Kerr[MATH]/Kerr[MATH] for the comparable BH, [MATH].', '1509.00021-2-19-6': 'Table I.', '1509.00021-2-20-0': 'New types of BH shadows, quite distinct from those of Kerr[MATH] BHs, appear on the left 2[MATH] and 4[MATH] row panels of Fig. [REF], corresponding to shadow II and III.', '1509.00021-2-20-1': 'In both cases, the central BH is non-Kerr-like ([MATH]).', '1509.00021-2-20-2': 'Shadow II is smaller (average radius of [MATH]75%) than that of the Kerr[MATH] BH.', '1509.00021-2-20-3': "It is also more 'square', with a larger normalized deviation from sphericity.", '1509.00021-2-20-4': 'Shadow III is remarkably distinct.', '1509.00021-2-20-5': "Its central BH has [MATH], allowed by the 'heavy' hair that it is dragging ([MATH] the discussion in [CITATION]).", '1509.00021-2-20-6': 'The lensing of this hair resembles closely that of the ultra-compact BS on the bottom left panel of Fig. [REF].', '1509.00021-2-20-7': "Interestingly, multiple disconnected shadows of the (single) BH occur: the largest ones (besides the main 'hammer-like' shadow) are two eyebrows [CITATION], at symmetric positions above and below the main shadow; but we have detected many other smaller shadows, hinting again at a self-similar structure.", '1509.00021-2-20-8': 'On the 3[MATH] row of Fig. [REF], the shadows of four solutions in between configurations II and III illustrate the transition between them.', '1509.00021-2-20-9': 'Finally, we remark that the shadows of KBHsSH can have arbitrarily small sizes by considering solutions arbitrarily close to the BS curve in Fig. [REF].', '1509.00021-2-21-0': 'Remarks.', '1509.00021-2-21-1': 'KBHsSH can lead to qualitatively novel types of shadows in GR, as shown by shadows II and III.', '1509.00021-2-21-2': 'Even for KBHsSH close to Kerr, their shadows are distinguishable from the latter, with the same asymptotic quantities, as illustrated by shadow I. Regardless of the astrophysical relevance of these solutions - which is unclear - they can yield new templates with small or large deviations from the Kerr shadows, hopefully of use for VLBI observations.', '1509.00021-2-21-3': 'An exhaustive analysis of KBHsSH shadows spanning the space of solutions in Fig. [REF], and at different observation angles, will be presented elsewhere, for producing such templates [CITATION].', '1509.00021-2-21-4': 'But the examples herein already raise a challenge to the parameterizations of deviations from Kerr suggested in the literature [CITATION]: can they describe shadows with such large deviations?', '1509.00021-2-22-0': 'Besides the peculiar shape of some of the shadows exhibited, this model has one general prediction: smaller observed shadows than those expected for Kerr BHs with the same asymptotic charges.', '1509.00021-2-22-1': "Indeed, a 'smaller' central BH seems a natural consequence of the existence of hair, carrying part of the total energy.", '1509.00021-2-23-0': "Finally, for the setup herein, the redshift, which depends only on the source's and [MATH]'s positions, is constant throughout the image and has been neglected."} | [['1509.00021-1-1-1', '1509.00021-2-1-1'], ['1509.00021-1-1-2', '1509.00021-2-1-2'], ['1509.00021-1-1-3', '1509.00021-2-1-3'], ['1509.00021-1-8-1', '1509.00021-2-8-1'], ['1509.00021-1-20-0', '1509.00021-2-20-0'], ['1509.00021-1-20-1', '1509.00021-2-20-1'], ['1509.00021-1-20-2', '1509.00021-2-20-2'], ['1509.00021-1-20-3', '1509.00021-2-20-3'], ['1509.00021-1-20-4', '1509.00021-2-20-4'], ['1509.00021-1-20-5', '1509.00021-2-20-5'], ['1509.00021-1-20-6', '1509.00021-2-20-6'], ['1509.00021-1-20-7', '1509.00021-2-20-7'], ['1509.00021-1-20-8', '1509.00021-2-20-8'], ['1509.00021-1-20-9', '1509.00021-2-20-9'], ['1509.00021-1-18-1', '1509.00021-2-18-1'], ['1509.00021-1-16-0', '1509.00021-2-16-0'], ['1509.00021-1-16-1', '1509.00021-2-16-1'], ['1509.00021-1-16-2', '1509.00021-2-16-2'], 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'1509.00021-2-15-5'], ['1509.00021-1-15-6', '1509.00021-2-15-6'], ['1509.00021-1-15-7', '1509.00021-2-15-7'], ['1509.00021-1-13-0', '1509.00021-2-13-0'], ['1509.00021-1-13-1', '1509.00021-2-13-1'], ['1509.00021-1-13-2', '1509.00021-2-13-2'], ['1509.00021-1-13-3', '1509.00021-2-13-3'], ['1509.00021-1-13-4', '1509.00021-2-13-4'], ['1509.00021-1-13-5', '1509.00021-2-13-5'], ['1509.00021-1-13-6', '1509.00021-2-13-6'], ['1509.00021-1-13-7', '1509.00021-2-13-7'], ['1509.00021-1-21-1', '1509.00021-2-21-1'], ['1509.00021-1-21-2', '1509.00021-2-21-2'], ['1509.00021-1-21-3', '1509.00021-2-21-3'], ['1509.00021-1-21-4', '1509.00021-2-21-4'], ['1509.00021-1-0-0', '1509.00021-2-0-0'], ['1509.00021-1-0-1', '1509.00021-2-0-1'], ['1509.00021-1-0-2', '1509.00021-2-0-2'], ['1509.00021-1-0-3', '1509.00021-2-0-3'], ['1509.00021-1-0-4', '1509.00021-2-0-4'], ['1509.00021-1-0-5', '1509.00021-2-0-5'], ['1509.00021-1-5-1', '1509.00021-2-5-1'], ['1509.00021-1-5-2', '1509.00021-2-5-2'], ['1509.00021-1-5-3', '1509.00021-2-5-3'], ['1509.00021-1-5-4', '1509.00021-2-5-4'], ['1509.00021-1-5-5', '1509.00021-2-5-5'], ['1509.00021-1-5-6', '1509.00021-2-5-6'], ['1509.00021-1-5-7', '1509.00021-2-5-7'], ['1509.00021-1-11-0', '1509.00021-2-11-0'], ['1509.00021-1-11-1', '1509.00021-2-11-1'], ['1509.00021-1-11-3', '1509.00021-2-11-3'], ['1509.00021-1-22-0', '1509.00021-2-22-0'], ['1509.00021-1-22-1', '1509.00021-2-22-1'], ['1509.00021-1-6-1', '1509.00021-2-6-1'], ['1509.00021-1-6-2', '1509.00021-2-6-2'], ['1509.00021-1-6-3', '1509.00021-2-6-3'], ['1509.00021-1-6-4', '1509.00021-2-6-4'], ['1509.00021-1-2-0', '1509.00021-2-2-0'], ['1509.00021-1-2-1', '1509.00021-2-2-1'], ['1509.00021-1-2-2', '1509.00021-2-2-2'], ['1509.00021-1-17-1', '1509.00021-2-17-1'], ['1509.00021-1-17-2', '1509.00021-2-17-2'], ['1509.00021-1-17-3', '1509.00021-2-17-3'], ['1509.00021-1-17-4', '1509.00021-2-17-4'], ['1509.00021-1-9-1', '1509.00021-2-9-1'], ['1509.00021-1-9-3', '1509.00021-2-9-3'], ['1509.00021-1-9-4', '1509.00021-2-9-4'], ['1509.00021-1-9-5', '1509.00021-2-9-5'], ['1509.00021-1-9-6', '1509.00021-2-9-6'], ['1509.00021-1-9-7', '1509.00021-2-9-7'], ['1509.00021-1-10-0', '1509.00021-2-10-0'], ['1509.00021-1-10-1', '1509.00021-2-10-1'], ['1509.00021-1-10-2', '1509.00021-2-10-2'], ['1509.00021-1-10-3', '1509.00021-2-10-3'], ['1509.00021-1-10-4', '1509.00021-2-10-4'], ['1509.00021-1-19-0', '1509.00021-2-19-0'], ['1509.00021-1-19-1', '1509.00021-2-19-1'], ['1509.00021-1-19-2', '1509.00021-2-19-2'], ['1509.00021-1-19-3', '1509.00021-2-19-3'], ['1509.00021-1-19-4', '1509.00021-2-19-4'], ['1509.00021-1-19-5', '1509.00021-2-19-5'], ['1509.00021-1-4-0', '1509.00021-2-4-0'], ['1509.00021-1-14-0', '1509.00021-2-14-0'], ['1509.00021-1-14-1', '1509.00021-2-14-1'], ['1509.00021-1-3-0', '1509.00021-2-3-0'], ['1509.00021-1-3-1', '1509.00021-2-3-1'], ['1509.00021-1-12-0', '1509.00021-2-12-0'], ['1509.00021-1-12-1', '1509.00021-2-12-1'], ['1509.00021-1-12-2', '1509.00021-2-12-2'], ['1509.00021-1-12-3', '1509.00021-2-12-3'], ['1509.00021-1-12-4', '1509.00021-2-12-4'], ['1509.00021-1-12-5', '1509.00021-2-12-5']] | [['1509.00021-1-8-0', '1509.00021-2-8-0'], ['1509.00021-1-8-3', '1509.00021-2-8-3'], ['1509.00021-1-23-0', '1509.00021-2-23-0'], ['1509.00021-1-18-0', '1509.00021-2-18-0'], ['1509.00021-1-15-3', '1509.00021-2-15-3'], ['1509.00021-1-11-2', '1509.00021-2-11-2'], ['1509.00021-1-7-0', '1509.00021-2-7-0'], ['1509.00021-1-7-1', '1509.00021-2-7-1'], ['1509.00021-1-9-2', '1509.00021-2-9-2'], ['1509.00021-1-3-2', '1509.00021-2-3-2'], ['1509.00021-1-3-3', '1509.00021-2-3-3']] | [] | [['1509.00021-1-8-2', '1509.00021-2-8-2']] | [] | ['1509.00021-1-1-0', '1509.00021-1-5-0', '1509.00021-1-6-0', '1509.00021-1-9-0', '1509.00021-1-17-0', '1509.00021-1-17-5', '1509.00021-1-19-6', '1509.00021-1-21-0', '1509.00021-2-1-0', '1509.00021-2-5-0', '1509.00021-2-6-0', '1509.00021-2-9-0', '1509.00021-2-17-0', '1509.00021-2-17-5', '1509.00021-2-19-6', '1509.00021-2-21-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1509.00021 | null | null | null | null | null |
math-0403455 | {'math-0403455-1-0-0': 'We study the Gassner representation of the pure braid group [MATH] by considering its restriction to a free subgroup [MATH].', 'math-0403455-1-0-1': 'The kernel of the restriction is shown to lie in the subgroup [MATH], sharpening a result of Lipschutz.', 'math-0403455-1-1-0': '# Introduction', 'math-0403455-1-2-0': 'Denote by [MATH] the unreduced Gassner representation of the pure braid group [MATH] (a formula is given in Section [REF] below).', 'math-0403455-1-2-1': 'The faithfulness of [MATH] for [MATH] is a long-standing open question.', 'math-0403455-1-2-2': 'In this note, we investigate this by considering the restriction of [MATH] to a certain free subgroup [MATH] of [MATH]: [EQUATION].', 'math-0403455-1-2-3': 'The faithfulness of [MATH] would follow from that of [MATH].', 'math-0403455-1-3-0': 'For a group [MATH], denote by [MATH] the lower central series of [MATH].', 'math-0403455-1-3-1': 'The main result of this paper is the following.', 'math-0403455-1-4-0': 'Theorem [REF].', 'math-0403455-1-4-1': 'The kernel of [MATH] lies in the subgroup [MATH].', 'math-0403455-1-5-0': 'This is proved by passing to the graded Lie algebras associated to the lower central series of [MATH] and the filtration of [MATH] by powers of the augmentation ideal [MATH].', 'math-0403455-1-5-1': 'Unfortunately, our method does not extend to show that the kernel of [MATH] lies in [MATH] for any [MATH].', 'math-0403455-1-5-2': 'We explain this in Section [REF].', 'math-0403455-1-5-3': 'Moreover, the breakdown of our method does not imply that [MATH] is not injective.', 'math-0403455-1-6-0': 'In [CITATION], S. Lipschutz proved that the kernel of [MATH] lies in [MATH] using different techniques.', 'math-0403455-1-6-1': 'Our result reproves this and also sharpens it a bit.'} | {'math-0403455-2-0-0': 'We study the Gassner representation of the pure braid group [MATH] by considering its restriction to a free subgroup [MATH].', 'math-0403455-2-0-1': 'The kernel of the restriction is shown to lie in the subgroup [MATH], sharpening a result of Lipschutz.', 'math-0403455-2-1-0': '# Introduction', 'math-0403455-2-2-0': 'Denote by [MATH] the unreduced Gassner representation of the pure braid group [MATH] (a formula is given in Section [REF] below).', 'math-0403455-2-2-1': 'The faithfulness of [MATH] for [MATH] is a long-standing open question.', 'math-0403455-2-2-2': 'In this note, we investigate this by considering the restriction of [MATH] to a certain free subgroup [MATH] of [MATH]: [EQUATION].', 'math-0403455-2-2-3': 'The faithfulness of [MATH] would follow from that of [MATH] (see Proposition [REF] below, or [CITATION] for a more general result).', 'math-0403455-2-3-0': 'For a group [MATH], denote by [MATH] the lower central series of [MATH].', 'math-0403455-2-3-1': 'The main result of this paper is the following.', 'math-0403455-2-4-0': 'Theorem [REF].', 'math-0403455-2-4-1': 'The kernel of [MATH] lies in the subgroup [MATH].', 'math-0403455-2-5-0': 'This is proved by passing to the graded quotients associated to the lower central series of [MATH] and the filtration of [MATH] by powers of the augmentation ideal [MATH].', 'math-0403455-2-5-1': 'This allows us to show that the kernel of [MATH] lies in [MATH].', 'math-0403455-2-5-2': 'In [CITATION], S. Lipschutz proved that the kernel of [MATH] lies in [MATH] using different techniques (see also [CITATION] for another proof).', 'math-0403455-2-5-3': 'These two facts together allow us to prove Theorem [REF].', 'math-0403455-2-6-0': 'We also show (Theorem [REF]) that the intersection of the kernel of [MATH] with [MATH] lies in the subgroup [MATH].'} | [['math-0403455-1-0-0', 'math-0403455-2-0-0'], ['math-0403455-1-0-1', 'math-0403455-2-0-1'], ['math-0403455-1-2-0', 'math-0403455-2-2-0'], ['math-0403455-1-2-1', 'math-0403455-2-2-1'], ['math-0403455-1-2-2', 'math-0403455-2-2-2'], ['math-0403455-1-3-0', 'math-0403455-2-3-0'], ['math-0403455-1-3-1', 'math-0403455-2-3-1'], ['math-0403455-1-5-0', 'math-0403455-2-5-0'], ['math-0403455-1-5-1', 'math-0403455-2-5-1'], ['math-0403455-1-2-3', 'math-0403455-2-2-3']] | [['math-0403455-1-0-0', 'math-0403455-2-0-0'], ['math-0403455-1-0-1', 'math-0403455-2-0-1'], ['math-0403455-1-2-0', 'math-0403455-2-2-0'], ['math-0403455-1-2-1', 'math-0403455-2-2-1'], ['math-0403455-1-2-2', 'math-0403455-2-2-2'], ['math-0403455-1-3-0', 'math-0403455-2-3-0'], ['math-0403455-1-3-1', 'math-0403455-2-3-1']] | [['math-0403455-1-5-0', 'math-0403455-2-5-0']] | [] | [['math-0403455-1-5-1', 'math-0403455-2-5-1'], ['math-0403455-1-2-3', 'math-0403455-2-2-3']] | [] | ['math-0403455-1-4-0', 'math-0403455-1-4-1', 'math-0403455-2-4-0', 'math-0403455-2-4-1'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/math/0403455 | null | null | null | null | null |
quant-ph-0501138 | {'quant-ph-0501138-1-0-0': 'We study the application of self-induced decoherence to a spin bath model that has previously been used to analyze environment-induced decoherence.', 'quant-ph-0501138-1-0-1': 'We investigate whether, as claimed, decoherence in energy for systems endowed with a continuous energy spectrum is a universal process that does not require observational ignorance of the degrees of freedom of an environment.', 'quant-ph-0501138-1-0-2': 'To this end, we examine the time evolution of a general global observable in our model, analytically and numerically, and compare the results with those obtained for local observables in environment-induced decoherence.', 'quant-ph-0501138-1-0-3': 'We find that even for very large environments, corresponding to an approximately continuous energy spectrum, global decoherence may fail to occur, and that no direct correlation exists between the occurrence and strength of global decoherence and the size of the spin bath.', 'quant-ph-0501138-1-0-4': 'We show that the presence and degree of global decoherence is instead crucially dependent on both the observable and the state of the environment.', 'quant-ph-0501138-1-0-5': 'If suitable restrictions are imposed on the observable and the environment, we demonstrate that, and explain why, global decoherence can occur even for very small spin baths with a rate and strength similar to that of local decoherence.', 'quant-ph-0501138-1-0-6': 'Our study supports the idea of self-induced decoherence, that a tracing-out of the environment is not always necessary to obtain decoherence.', 'quant-ph-0501138-1-0-7': 'However, it also shows that the conditions for and the dynamics of decoherence differ significantly for global and local observables and in a way that may have experimentally verifiable consequences.', 'quant-ph-0501138-1-1-0': '# Introduction', 'quant-ph-0501138-1-2-0': 'The study of decoherence in quantum systems has led to important insights into the question of how classicality emerges from quantum mechanics and has been employed in proposals that suggest solutions to the quantum measurement problem [CITATION].', 'quant-ph-0501138-1-2-1': 'The main line of research has been centered around the approach of environment-induced decoherence(EID), based on pioneering work by Zeh [CITATION] and developed further by Zurek [CITATION] and others (for reviews, see [CITATION]).', 'quant-ph-0501138-1-2-2': 'Characteristic of this viewpoint is the emphasis of the openness of quantum systems: The interaction of a system [MATH] with its environment [MATH], consisting of a large number of degrees of freedom, is shown to lead to a typically rapid diagonalization of the reduced density matrix [MATH] (which describes the probability distribution of outcomes of measurements on [MATH]) in a preferred basis.', 'quant-ph-0501138-1-2-3': 'It follows that this form of decoherence is restricted to local observables that pertain only to the degrees of freedom of [MATH] but "ignore" [MATH].', 'quant-ph-0501138-1-2-4': '(If not stated otherwise, the term "decoherence" shall henceforth be understood as the decay of crossterms, representing interference, in the expectation value of an observable expressed in some particular basis.)', 'quant-ph-0501138-1-3-0': 'This has frequently raised the question of what role this "observational ignorance of the environment [MATH]" plays in obtaining decoherence (see, for example, Ref. [CITATION]).', 'quant-ph-0501138-1-3-1': 'Interestingly, the need for a tracing-over of degrees of freedom of [MATH] to demonstrate decoherence has been challenged by a series of papers published by the group of Castagnino et al. [CITATION].', 'quant-ph-0501138-1-3-2': 'The main claim of their new viewpoint on decoherence, termed self-induced decoherence (SID), is to suggest that decoherence in energy, i.e., in the eigenbasis of the total Hamiltonian of [MATH], is a universal effect that occurs for any observable, be it local or global, provided [MATH] is endowed with a continuous energy spectrum.', 'quant-ph-0501138-1-3-3': 'If true, this would imply that decoherence in energy is neither dependent on the particular form of interaction between [MATH] and [MATH] nor on the observational ignorance of [MATH], therefore explaining decoherence in a very general way and making the split into a "system of interest" and an "unobserved environment" superfluous.', 'quant-ph-0501138-1-4-0': 'The purpose of this paper is to evaluate the above claim in the context of an explicit model.', 'quant-ph-0501138-1-4-1': 'To this end, after introducing the basic ideas and formalism of SID in Sec. [REF], we study the time evolution of the expectation value of general local and global observables in a spin bath system previously analyzed in the context of EID (Sec. [REF]).', 'quant-ph-0501138-1-4-2': 'This allows us to evaluate under which conditions global decoherence in energy may occur in our model, and how the dynamics of such decoherence compare with that of EID-type local decoherence.', 'quant-ph-0501138-1-4-3': 'We discuss and summarize our results in Secs. [REF] and [REF].', 'quant-ph-0501138-1-5-0': '# Basics of self-induced decoherence', 'quant-ph-0501138-1-6-0': 'Consider a complete set of discrete eigenkets [MATH] of the Hamiltonian [MATH] with discrete spectrum [MATH].', 'quant-ph-0501138-1-6-1': 'Then, we can expand an arbitary observable [MATH] in terms of the [MATH], [EQUATION]', 'quant-ph-0501138-1-6-2': 'In the case of a continuous spectrum, i.e., [MATH], this generalizes to [EQUATION]', 'quant-ph-0501138-1-6-3': 'We can decompose the right-hand side of this equation into a "diagonal" part where the integration runs over [MATH] alone, and an "off-diagonal" part which contains the double integral of the original expression, [EQUATION]', 'quant-ph-0501138-1-6-4': 'The reason for this decomposition is that is we would like to use only expansion coefficients [MATH] that are sufficiently regular functions (in particular, that do not contain Dirac [MATH]-functions) but also account for observables such as the Hamiltonian, [MATH].', 'quant-ph-0501138-1-6-5': 'This decomposition can of course always be done, without restriction of generality.', 'quant-ph-0501138-1-7-0': 'The time evolution of the expectation value [MATH] of [MATH] in the pure state [EQUATION] is given by [EQUATION]', 'quant-ph-0501138-1-8-0': 'Now considering the limit [MATH], we can apply the Riemann-Lebesgue theorem [CITATION], which prescribes that [EQUATION] if [MATH] is a regular function and [MATH] integrable (i.e., [MATH]).', 'quant-ph-0501138-1-8-1': 'Provided these conditions are fulfilled by the coefficients [MATH], we obtain [EQUATION] that is, the off-diagonal terms [MATH] have vanished in the limit [MATH], yielding decoherence in the expectation value.', 'quant-ph-0501138-1-9-0': 'Formally, we can therefore write down a "diagonal-equivalent" density matrix [MATH], [EQUATION] that represents the density matrix corresponding to the "decohered" expectation value, i.e., [MATH] fulfills [MATH].', 'quant-ph-0501138-1-9-1': 'Note that [MATH] is only a formal equivalent and is not obtained through any dynamical process.', 'quant-ph-0501138-1-10-0': 'Thus, the main claim of SID is to suggest that the terms in the expectation value of any global observable corresponding to interference between different eigenstates of the total Hamiltonian [MATH] will decay as [MATH], provided [MATH] is endowed with a continuous spectrum (such that [MATH] is a regular function) and [MATH] is also a sufficiently regular function, in order for the Riemann-Lebesgue theorem to be applicable.', 'quant-ph-0501138-1-10-1': 'In contrast with EID, no explicit separation into a "system of interest" and its "environment" is made, and no tracing-out of the "unobserved degrees of freedom of the environment" is performed.', 'quant-ph-0501138-1-11-0': 'To evaluate the claim that global decoherence in energy is indeed an essentially universal phenomenon, we must investigate how the mathematical requirements and implications of the Riemann-Lebesgue theorem, the cornerstone of the program of SID, translate into the possibility of a physically relevant global decoherence effect in the context of realistic cases.', 'quant-ph-0501138-1-12-0': 'To this end, we can view the requirement for a continuous energy spectrum as an implicit inclusion of an "environment" with an essentially infinite number [MATH] of degrees of freedom.', 'quant-ph-0501138-1-12-1': 'However, for any realistic physical system, [MATH] will be finite, and therefore the energy spacing will be discrete.', 'quant-ph-0501138-1-12-2': 'If global decoherence in energy is indeed to be regarded as a physically relevant phenomenon, it is therefore reasonable to demand that it ought to occur, at least in an approximate manner, also in systems with a quasicontinuous energy spectrum, i.e., with a small but discrete energy spacing.', 'quant-ph-0501138-1-13-0': 'Similarly, although the Riemann-Lebesgue theorem gives only the infinite-time limit of the expectation value, Eq. [REF], we should be able to obtain approximate global decoherence also for the realistic case of finite times, thus posing the question of the explicit dynamics of SID.', 'quant-ph-0501138-1-13-1': 'Moreover, the properties of the product coefficients [MATH], see Eq. [REF], are crucial to the applicability of the Riemann-Lebesgue theorem.', 'quant-ph-0501138-1-13-2': 'It is therefore important to understand how these properties influence the time dependence of the expectation value and thus the potential occurence of global decoherence.', 'quant-ph-0501138-1-14-0': 'We shall investigate these questions in the following by studying the explicit time evolution of the expectation value of general local and global observables in a spin bath model.', 'quant-ph-0501138-1-15-0': '# Analyzing the spin bath model', 'quant-ph-0501138-1-16-0': '## The model and its time evolution', 'quant-ph-0501138-1-17-0': 'The probably most simple exactly solvable model for decoherence was introduced some years ago by Zurek .', 'quant-ph-0501138-1-17-1': 'Here, the system [MATH] consists of a spin-1/2 particle with two possible states [MATH] (representing spin up) and [MATH] (corresponding to spin down), interacting with a collection of [MATH] environmental spin-1/2 particles (described by the states [MATH] and [MATH]) via the total Hamiltonian [EQUATION]', 'quant-ph-0501138-1-17-2': 'Here, the [MATH] are coupling constants, and [MATH] is the identity operators for the [MATH]-th environmental spin.', 'quant-ph-0501138-1-17-3': 'The self-Hamiltonians of [MATH] and [MATH] are taken to be equal to zero.', 'quant-ph-0501138-1-17-4': 'Note that [MATH] has a particularly simple form since it contains only terms diagonal in the [MATH] and [MATH] bases.', 'quant-ph-0501138-1-18-0': 'It follows that the eigenstates of [MATH] are product states of the form [MATH], etc.', 'quant-ph-0501138-1-18-1': 'A general state [MATH] can then be written as a linear combination of product eigenstates [EQUATION]', 'quant-ph-0501138-1-18-2': 'This state evolves under the action of [MATH] into [EQUATION] where [EQUATION]', 'quant-ph-0501138-1-18-3': 'The density matrix is [EQUATION] and its part diagonal in energy (i.e., diagonal in eigenstates of the total Hamiltonian [MATH]) is [EQUATION]', 'quant-ph-0501138-1-19-0': 'In the following Secs. [REF] and [REF], we shall analyze the time evolution of the expectation value of two different kinds of observables: Local observables that take into account only the degrees of freedom of [MATH] (the situation encountered in EID), and global observables that pertain to both [MATH] and [MATH] (the case treated by SID).', 'quant-ph-0501138-1-20-0': '## Treatment within environment-induced decoherence: Local observables', 'quant-ph-0501138-1-21-0': 'Focusing, in the spirit of EID, on the system [MATH] alone, we trace out the degrees of freedom of the [MATH]-spin bath in the density operator [MATH].', 'quant-ph-0501138-1-21-1': 'This yields the reduced density operator [EQUATION] where the time dependence of the off-diagonal terms [MATH] and [MATH] is given by the decoherence factor [EQUATION]', 'quant-ph-0501138-1-22-0': 'As pointed out before, the motivation for tracing over the degrees of freedom of the environment lies in the fact that the reduced density matrix suffices to compute expectation values for any [MATH]-observable [EQUATION] where [MATH] and [MATH] are real numbers and [MATH] (since [MATH] must be Hermitian).', 'quant-ph-0501138-1-22-1': 'This is so because [EQUATION]', 'quant-ph-0501138-1-22-2': 'Fig. [REF] shows a plot of the logarithm of the magnitude of the decoherence factor [MATH], Eq. [REF], for two different bath sizes [MATH].', 'quant-ph-0501138-1-22-3': 'We see that [MATH] decays quickly by several orders of magnitude and then continues to oscillate about a very small mean value.', 'quant-ph-0501138-1-22-4': 'Thus, for local observables, terms corresponding to interference between the two [MATH]-spin states [MATH] and [MATH] become quickly and strongly supressed.', 'quant-ph-0501138-1-23-0': 'Following Ref. [CITATION], we can formally rewrite [MATH] as an integral, [EQUATION] with [MATH] given by [EQUATION] where the sum runs over all eigenstates [MATH] of the total Hamiltonian [MATH], with eigenvalues [MATH].', 'quant-ph-0501138-1-23-1': 'Thus, [MATH] is the characteristic function of the state density function [MATH].', 'quant-ph-0501138-1-24-0': '## Treatment within self-induced decoherence: Global observables', 'quant-ph-0501138-1-25-0': '### Computing the expectation value of global observables', 'quant-ph-0501138-1-26-0': 'Any global observable [MATH] can be written as a linear combination of the form [MATH], where the [MATH] are product eigenstates of the total Hamiltonian [MATH], Eq. [REF].', 'quant-ph-0501138-1-27-0': 'Since [MATH] must be Hermitian, [MATH], [MATH], [MATH], and [MATH] are real numbers, and [MATH], [MATH].', 'quant-ph-0501138-1-27-1': 'To keep the notation simple, we shall omit the sum over [MATH] (and thus the index [MATH]) in the following.', 'quant-ph-0501138-1-28-0': 'The part diagonal in energy, corresponding to the term [MATH] in Eq. [REF], is [EQUATION]', 'quant-ph-0501138-1-28-1': 'The expectation value of [MATH] in the state [MATH], Eq. [REF], is [EQUATION]', 'quant-ph-0501138-1-29-0': 'The special case of the expectation value of local observables as treated in EID can easily be recovered by remembering that tracing out the degrees of freedom of [MATH] is equivalent to choosing all coefficients [MATH] and [MATH], which yields [MATH] and [MATH], see Eq. [REF], in agreement with Eq. [REF].', 'quant-ph-0501138-1-30-0': 'Global decoherence in energy would be represented by the vanishing of all time-dependent terms in the above expression, i.e., [EQUATION] because we can easily show that [MATH], where [MATH], Eq. [REF], is the part of the density matrix that is diagonal in the eigenstates of the total Hamiltonian.', 'quant-ph-0501138-1-30-1': 'We also see that, as expected, [MATH], so decoherence in energy is characterized by the presence of only those product expansion coefficients that are contained in [MATH], Eq. [REF], i.e., in the part of [MATH] that is diagonal in energy.', 'quant-ph-0501138-1-31-0': 'The form of the two product terms [MATH] and [MATH] is similar: They only differ in the order of the pairing of the product expansion coefficients with the exponential factors.', 'quant-ph-0501138-1-31-1': 'Also, since the coefficients [MATH] are independent, global decoherence requires that individually [MATH] and [MATH] as [MATH].', 'quant-ph-0501138-1-31-2': 'We can therefore restrict our following analysis to [MATH] alone.', 'quant-ph-0501138-1-32-0': '### Analyzing the structure of the expectation value', 'quant-ph-0501138-1-33-0': 'First of all, let us rewrite [MATH] as a sum of [MATH] terms, [EQUATION]', 'quant-ph-0501138-1-33-1': 'Here the sets [MATH] specify over which indices [MATH] each product runs, namely, they are subsets of the set [MATH] of all integers between 1 and [MATH] such that [MATH] and [MATH].', 'quant-ph-0501138-1-33-2': 'The total energy [MATH] associated with each term in the sum, Eq. [REF], is [EQUATION]', 'quant-ph-0501138-1-33-3': 'We choose the index [MATH] such that [MATH] for all [MATH].', 'quant-ph-0501138-1-33-4': 'Clearly, [MATH] whenever [MATH] (i.e., if [MATH]), canceling out the time dependence of the associated product term in the expression for [MATH].', 'quant-ph-0501138-1-33-5': 'Thus, we can split [MATH] into a time-independent and a time-dependent part, [EQUATION] where the first sum runs over all [MATH] for which [MATH], while the second sum runs over all [MATH] for which [MATH].', 'quant-ph-0501138-1-33-6': 'We see that global decoherence in energy would require [MATH] to approach zero as [MATH].', 'quant-ph-0501138-1-34-0': 'Finally, we can formally rewrite [MATH] as an integral, [EQUATION] which shows that [MATH] is simply the Fourier transform of [MATH], where [EQUATION] with the [MATH] representing products of expansion coefficients, [EQUATION]', 'quant-ph-0501138-1-34-1': 'Written this way, we see that [MATH], Eq. [REF], has the same form as [MATH] derived in the context of EID, Eq. [REF].', 'quant-ph-0501138-1-34-2': 'This might not come as a surprise, since also the expression for [MATH] can be derived from the calculation of an expectation value of an observable, namely, that of the local observable [MATH] that measures the degree of local interference between the [MATH]-states [MATH] and [MATH] [see also Eq. [REF]].', 'quant-ph-0501138-1-34-3': 'However, in the case of EID, [MATH] is a product of [MATH] real and non-negative coefficients [MATH] and [MATH], while the [MATH] appearing in Eq. [REF] contain crossterms of the form [MATH] and [MATH], arbitrary real coefficients [MATH] and [MATH], and arbitrary complex coefficients [MATH] [see Eq. [REF]].', 'quant-ph-0501138-1-34-4': 'We shall investigate in the following the influence of this difference on the time evolution of [MATH].', 'quant-ph-0501138-1-35-0': '### Time dependence of the expectation value', 'quant-ph-0501138-1-36-0': 'Following the general formalism of SID, one might be tempted to apply the Riemann-Lebesgue theorem to the integral in Eq. [REF] to infer the vanishing of [MATH] in the limit [MATH].', 'quant-ph-0501138-1-36-1': 'However, the applicability of the theorem hinges on the properties of [MATH], Eq. [REF].', 'quant-ph-0501138-1-37-0': 'First, the [MATH]-functions appearing in the expression for [MATH] are not regular functions and will therefore in general prevent the direct application of the Riemann-Lebesgue theorem.', 'quant-ph-0501138-1-37-1': 'However, as indicated before, by increasing the number of environmental spins we can obtain an at least quasicontinuous energy spectrum, in which case we should be able to expect the Riemann-Lebesgue theorem to be applicable in an "approximate" sense.', 'quant-ph-0501138-1-38-0': 'Second, the behavior of [MATH] depends crucially on the [MATH], Eq. [REF].', 'quant-ph-0501138-1-38-1': 'The basic idea underlying the Riemann-Lebesgue theorem is that when a function [MATH] is multiplied by a phase factor [MATH] whose variation with [MATH] is much faster than the variation of [MATH], neighboring values [MATH] and [MATH] will have similiar magnitude and phases but will be weighted with two strongly different phase factors, leading to an averaging-out effect in the integral [MATH].', 'quant-ph-0501138-1-39-0': 'In our case, the phases of the [MATH] [which play here the role of [MATH]] will in general vary very rapidly with [MATH] and thus with [MATH].', 'quant-ph-0501138-1-39-1': 'This is a consequence of the fact that the [MATH] are composed of products of coefficients, so changing a single term in the product will in general result in a drastic change in the overall phase associated with the [MATH].', 'quant-ph-0501138-1-39-2': '(The variation in magnitude among the [MATH] can be expected to be comparably insignificant for larger [MATH]).', 'quant-ph-0501138-1-39-3': 'Such phase fluctuations are not present in the formally similiar EID decoherence factor [MATH], Eq. [REF], since there only the absolute value of the coefficients [MATH] and [MATH] enters.', 'quant-ph-0501138-1-40-0': 'We anticipate this phase-variation effect to counteract the averaging-out influence of the multiplying phase factor [MATH] and to thus make it more difficult for [MATH], Eq. [REF], to converge to zero.', 'quant-ph-0501138-1-40-1': 'On the other hand, if the average difference between the phases associated with the individual coefficients is decreased, we would expect that the rate and degree of decay of [MATH] will be improved, since in this case a small variation in the composition of a product term [MATH] will result in a smaller fluctuation in the overall phase of this term.', 'quant-ph-0501138-1-41-0': 'To check this prediction and to generally gain more insight into the behavior of [MATH], Eq. [REF], we studied numerically the time evolution of [MATH], Eq. [REF], by evaluating the normalized function [EQUATION]', 'quant-ph-0501138-1-41-1': 'The occurence of global decoherence in energy would then be represented by a decay of [MATH] from its initial value of one, with [MATH] approaching zero for sufficiently large [MATH].', 'quant-ph-0501138-1-42-0': 'To investigate the influence of phase fluctuations of the [MATH], Eq. [REF], on the time evolution of [MATH], we shall distinguish three different cases for selecting the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH]:', 'quant-ph-0501138-1-43-0': 'The resulting time evolution of [MATH] for these cases is plotted in Fig. [REF] for [MATH].', 'quant-ph-0501138-1-43-1': 'We chose [MATH] for the phase-restricted case (B), and drew the couplings [MATH] from a uniform distribution of random real numbers over the interval [MATH].', 'quant-ph-0501138-1-43-2': 'We show the results for ten consecutive runs, where in each run a new set of random numbers was chosen for the coefficients.', 'quant-ph-0501138-1-44-0': 'We observe a drastic influence of the range of phases and signs associated with the individual coefficients [MATH], [MATH], [MATH], [MATH], and [MATH], on the evolution of [MATH].', 'quant-ph-0501138-1-44-1': 'In the special case (C) of all coefficients being real non-negative numbers, [MATH] exhibits a strong and fast decay behavior, similiar to the decay of the factor [MATH] in EID [see Eq. [REF] and Fig. [REF]].', 'quant-ph-0501138-1-44-2': 'In the intermediate case (B), with restricted phases and signs, the degree of decay of [MATH] is decreased, while the decay rate stays roughly the same.', 'quant-ph-0501138-1-44-3': 'In the general case (A), in which no restriction on the spread of phases and on the signs of the coefficients is imposed, the time evolution of [MATH] is observed to be sensitive to the particular set of random numbers used for the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH] in each run.', 'quant-ph-0501138-1-44-4': 'For some of the chosen sets of random numbers, [MATH] is seen to lack any decay behavior at all.', 'quant-ph-0501138-1-44-5': 'In other runs, the baseline of oscillation lies below zero, indicating a very weak decoherence effect, albeit with the peaks of oscillation frequently reaching values greater than zero.', 'quant-ph-0501138-1-45-0': 'These results show that, for the value of [MATH] studied here, a consistent decay behavior of [MATH] hinges on the phase restrictions imposed on the coefficients.', 'quant-ph-0501138-1-45-1': 'If these restrictions are given up, the time evolution of [MATH] and the occurence of a (comparably weak) decay will depend on the particular set of values chosen for the coefficients.', 'quant-ph-0501138-1-46-0': 'To further illustrate the impact of phase restrictions on the occurence and the strength of decoherence, we shall study the normalized function [EQUATION]', 'quant-ph-0501138-1-46-1': 'This function is similiar to the EID decoherence factor [MATH], Eq. [REF], except for the substitution of the coefficients [MATH] and [MATH] appearing in [MATH] by [MATH] and [MATH], in order to allow for non-zero phase spreads.', 'quant-ph-0501138-1-46-2': 'Fig. [REF] shows a plot of [MATH] for both the case in which the phases of the [MATH] and [MATH] are completely unrestricted, and in which only the absolute value of these coefficients is used.', 'quant-ph-0501138-1-46-3': 'Similiar to the results obtained in the context of SID, we observe that when the coefficients multiplying the phase factors [MATH] have arbitrary random phases, decay either becomes entirely absent (as in the example shown) or at least decreases drastically in strength.', 'quant-ph-0501138-1-47-0': 'In any realistic situation, the coefficients [MATH] and [MATH] are determined by the physical state of the environment and should therefore typically be considered as not controllable; only in selecting the measured observable [MATH], and thus in choosing the [MATH]-coefficients [MATH], [MATH], and [MATH], we have flexibility.', 'quant-ph-0501138-1-47-1': 'Accordingly, one should in general allow the [MATH] and [MATH] to have arbitrary phases.', 'quant-ph-0501138-1-47-2': 'Since the [MATH]-coefficients [MATH], [MATH], and [MATH] are always paired with the [MATH]-coefficients [MATH] and [MATH] in the expression for the [MATH] that make up [MATH] and therefore [MATH] [see Eq. [REF]], giving up phase spread restrictions on the [MATH]-coefficients can be expected to render the restrictions imposed on the [MATH]-coefficients less effective, if not entirely irrelevant, in bringing about a decay of [MATH].', 'quant-ph-0501138-1-48-0': 'To study this case, in Fig. [REF] we show a representative plot of [MATH] using only the absolute values of the [MATH]-coefficients [MATH], [MATH], and [MATH] [as in cases (B) and (C)], while the [MATH]-coefficients [MATH] and [MATH] have random phases between 0 and [MATH] [as in case (A)].', 'quant-ph-0501138-1-48-1': 'While typically the amplitude of oscillation of [MATH] is greatly reduced in comparison with the typical amplitudes observed in the general case (A) (see Fig. [REF]), we find that decay either becomes entirely absent or is strongly diminished in strength, despite the fact that the strongest possible restriction on the phases and signs of the [MATH]-coefficients is imposed.', 'quant-ph-0501138-1-48-2': 'The particular behavior of [MATH] will again depend on the set of random numbers chosen for the coefficients, as in the case (A).', 'quant-ph-0501138-1-48-3': 'This result confirms our prediction above.', 'quant-ph-0501138-1-49-0': 'Of course, the main claim of SID, that global decoherence in energy is universal, is based on the assumption of a continuous energy spectrum, corresponding to [MATH], and on the consideration of the infinite-time limit [MATH], while so far we have only considered relatively modest values for [MATH] and [MATH].', 'quant-ph-0501138-1-49-1': 'However, since we know from Fig. [REF] that in the case of EID, strong and fast decoherence is obtained for the value of [MATH] and over the timescale used in the plots displayed in Fig. [REF], we can already now conclude that if global decoherence in energy does indeed occur in our model in the general case of no phase restrictions, it will require a much larger number of environmental spins and/or will be of significantly slower rate than the local decoherence described by EID.', 'quant-ph-0501138-1-49-2': 'In the following, we shall therefore study the time evolution of [MATH] for larger values of [MATH] and longer timescales.', 'quant-ph-0501138-1-49-3': 'We will not impose any phase or sign restrictions on the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH].', 'quant-ph-0501138-1-50-0': 'Fig. [REF] shows the time evolution of [MATH] for [MATH], for five different values of [MATH] between [MATH] and [MATH] (corresponding to the different rows of the plot), and for three different sets of random numbers for the coefficients (represented by the columns of the plot).', 'quant-ph-0501138-1-50-1': 'We observe that even for the comparably large [MATH] investigated, no consistent occurence of decoherence becomes apparent.', 'quant-ph-0501138-1-50-2': 'In particular, no generally valid direct correlation between the value of [MATH] and the time evolution of [MATH] is visible.', 'quant-ph-0501138-1-50-3': 'For example, in Run 1 (first column of Fig. [REF]) decay of [MATH] is present for small [MATH]) and large [MATH]), but not for the intermediate values of [MATH] studied.', 'quant-ph-0501138-1-50-4': 'Similiarly, in Run 2 (second column of Fig. [REF]), [MATH] represents decoherence for [MATH], but not for [MATH].', 'quant-ph-0501138-1-50-5': 'In Run 3 (third column of Fig. [REF]), although [MATH] exhibits decay for [MATH] and [MATH], the baseline of oscillation is shifted towards larger positive values as [MATH] is increased from [MATH] to [MATH].', 'quant-ph-0501138-1-51-0': 'These results indicate that increasing [MATH] over the range studied, and thus improving the degree of approximate continuity of the energy spectrum, does not generally translate into an onset of decoherence or an improvement of the strength of existing decoherence.', 'quant-ph-0501138-1-51-1': 'We also see that the choice of a longer timescale is irrelevant, since neither the baseline nor the amplitude of oscillation change significantly, over the investigated time interval, after a comparably short initial period.', 'quant-ph-0501138-1-51-2': 'Furthermore we observe that even if [MATH] decays, the function sustains a large-amplitude oscillation whose peaks often attain values much larger than the initial value of [MATH].', 'quant-ph-0501138-1-51-3': 'Our findings are in contrast with the small oscillation amplitude, as well as with the direct correlation between the strength of decoherence and the size [MATH] of the spin bath, found in the case of the EID decoherence factor [MATH] (Figs. [REF] and [REF]) and of [MATH] with phase-restricted coefficients (Fig. [REF]).', 'quant-ph-0501138-1-52-0': '# Discussion', 'quant-ph-0501138-1-53-0': 'The main claim of SID, that global decoherence in energy is universal provided the system is endowed with a continuous energy spectrum, is derived from an application of the Riemann-Lebesgue theorem to the expression for the expectation value of a general global observable, prescribing the vanishing of terms that are not diagonal in energy in the limit [MATH].', 'quant-ph-0501138-1-53-1': 'In the existing papers on SID that we are aware of, it is assumed from the outset that the energy spectrum is truly continuous, and that the expansion coefficients appearing in the off-diagonal part of the general expression for the expectation value are sufficiently regular functions.', 'quant-ph-0501138-1-53-2': 'This has allowed the authors to apply the Riemann-Lebesgue theorem to infer the occurrence of global decoherence directly and in a very general manner, without having to explicitly study the specific dynamical properties of a particular system.', 'quant-ph-0501138-1-54-0': 'However, for this approach to have physical significance, it must lead to approximate decoherence also for quasicontinuous spectra and for finite times, since [MATH] and [MATH] for any physically realizable situation.', 'quant-ph-0501138-1-54-1': 'Our study of the explicit time dependence of the expectation value of a general global observable in the context of a basic spin bath model of finite size demonstrates that the question of whether global decoherence is obtained for a certain value of [MATH] depends crucially on the set of random numbers chosen for the coefficients describing the measured observable and the state of the environment.', 'quant-ph-0501138-1-54-2': 'On the other hand, for a given set of random coefficients, global decoherence may appear and disappear for different [MATH], without an indication for a particular correlation between the value of [MATH] and the occurence of decoherence.', 'quant-ph-0501138-1-54-3': 'In essence, we deal with two independent variables in our model, namely, the size [MATH] of the spin bath, and the particular set of random numbers chosen for the coefficients describing the observable and the state of the environment.', 'quant-ph-0501138-1-54-4': 'Both variables influence the presence, or absence, of a decay of [MATH], however in a way that is not seen to follow a universal pattern in the examples studied in this paper.', 'quant-ph-0501138-1-55-0': 'This finding seems to be contrary to what would one expect based on the general formalism of SID.', 'quant-ph-0501138-1-55-1': 'Although SID claims the occurence of global decoherence only for truly continuous energy spectra, one would have anticipated that a possibly weaker, yet universal, decoherence effect would appear also for only approximately continuous spectra, and that there would exist a direct correlation between the degree of quasicontinuity and decoherence.', 'quant-ph-0501138-1-55-2': 'Although we cannot exclude the possibility for such behavior to appear for values of [MATH] larger than those studied here, this might be considered as rather unlikely, since our results have shown that increasing [MATH] often diminishes the degree of existing decay or even completely destroys the presence of any decay.', 'quant-ph-0501138-1-56-0': 'We have seen that only restrictions on the expansion coefficients describing the state of the environment and the part of the observable pertaining to the environment can ensure a consistent and general decay behavior of [MATH] and therefore the occurence of decoherence.', 'quant-ph-0501138-1-56-1': 'We found that global decoherence with a decay strength and rate similiar to that encountered in EID, for identical values of [MATH], can be obtained if the expansion coefficients possess similiar phases and signs.', 'quant-ph-0501138-1-56-2': 'The local observables treated in the framework of EID can then be viewed as a special case of this condition since the tracing-out of the environment amounts to the coefficients of both the observable and the environment entering only through their absolute values into the term relevant to decoherence, namely, the factor [MATH], Eq. [REF].', 'quant-ph-0501138-1-56-3': 'On the other hand, the less similiar the phases associated with the coefficients are, the worse the decay behavior becomes.', 'quant-ph-0501138-1-57-0': 'One might interpret this finding as a quantification of the connection between ignorance and the occurence of decoherence: As the phases associated with the observable become more dissimiliar, corresponding to an increased degree of observational distinction between the invididual spin states of the environment, the degree of global decoherence is diminished.', 'quant-ph-0501138-1-57-1': 'However, this argument requires the physically unrealistic assumption of the phases of the environmental states being restricted as well, since decay behavior that is present for a particular [MATH] will typically disappear if the constraint on the environmental phases is given up, regardless of the restriction imposed on the observable.', 'quant-ph-0501138-1-58-0': 'The numerical analysis has therefore confirmed our prediction that the phase fluctuations of the [MATH], Eq. [REF], determined by the phases of the individual coefficients describing the observable and the environment, have a drastic influence on the time evolution of [MATH].', 'quant-ph-0501138-1-58-1': 'In fact, our results show that the range of phases associated with the coefficients is the crucial factor in the strongly differing dynamics of SID and EID in our model.', 'quant-ph-0501138-1-58-2': 'This also poses the interesting question of how the differing strengths of decoherence for different choices of observables and environmental states might translate into experimentally observable effects.', 'quant-ph-0501138-1-58-3': 'Additionally, it would be worth exploring whether the connection between global decoherence and phase spreads also holds for other model systems.', 'quant-ph-0501138-1-59-0': 'From a conceptual point, the approach of SID is intriguing as it avoids the need for an observational ignorance of the environmental degrees of freedom in obtaining decoherence.', 'quant-ph-0501138-1-59-1': 'This concept of ignorance and its implications for a proper interpretation of EID have been a subject of ongoing critical discussions (see, for example, Ref. [CITATION] and references therein).', 'quant-ph-0501138-1-59-2': 'EID also requires the decomposition of the total Hilbert space into subsystems, but no general rule is available that would indicate where the split is to be placed, a conceptual difficulty admitted also by proponents of EID [CITATION].', 'quant-ph-0501138-1-60-0': 'One might argue that decoherence for local observables as considered in EID could be understood as a special case of global decoherence.', 'quant-ph-0501138-1-60-1': 'For the spin model studied in this paper, our results lend some support to this hypothesis, since we have shown that the requirement for a complete ignorance of the environment, as spelled out in EID, can be replaced by the more general condition that certain restrictions must be imposed both on the part of the observable pertaining to the environment and on the state of the environment.', 'quant-ph-0501138-1-60-2': 'As pointed out before, decoherence in the expectation value of local observables emerges from this condition as a special case in which particularly strong (value) constraints are imposed on the environmental part of the observable; in turn, these restrictions make constraints on the environmental states unnecessary.', 'quant-ph-0501138-1-61-0': 'However, it should be explored further to what extent the advantage of obtaining global decoherence might be outweighed by the requirement for restrictions on the state of the environment, which in realistic cases is typically not controllable.', 'quant-ph-0501138-1-61-1': 'Thus the dependence of global decoherence on both the observable and the state of the environment in our model could also be considered as diminishing the physical relevance of the concept, rather than being viewed as providing a more general framework for decoherence.', 'quant-ph-0501138-1-62-0': '# Summary and conclusions', 'quant-ph-0501138-1-63-0': 'We investigated, analytically and numerically, the time evolution of the expectation value of a general global observable in a simple spin bath model of finite size.', 'quant-ph-0501138-1-63-1': 'While the results of our study provide support for the idea of the SID program that ignorance of the degrees of freedom of the environment, represented by a restriction to local observables, is not necessarily required to obtain decoherence, we also found that the conditions for such global decoherence and the resulting decoherence dynamics differ strongly from those of EID.', 'quant-ph-0501138-1-64-0': 'We demonstrated that the occurrence and strength of global decoherence depend crucially on both the part of the observable pertaining to the degrees of freedom of the environment and the state of the environment.', 'quant-ph-0501138-1-64-1': 'We argued that, contrary to what one might expect from a natural extension of the concepts of SID to the realistic case of finite sizes of the environment and thus to only approximately continuous energy spectra, there exists no direct correlation between the size of the environment and the presence and degree of decoherence over the wide range of bath sizes investigated in our study.', 'quant-ph-0501138-1-65-0': 'We predicted and confirmed, however, that global decoherence in energy that is similiarly strong and rapid as the local decoherence treated in EID is possible if both the phase differences between individual environmental spin states and the degrees of distinction between the different possible states of the environment in a measurement are appropriately restricted.', 'quant-ph-0501138-1-65-1': 'We used this result to provide an explanation for the different dynamical properties of global and local decoherence in our model.', 'quant-ph-0501138-1-66-0': 'We suggested that the dependence of global decoherence on both the observable and the state of the environment, as observed for the case of our model, may lead to experimentally verifiable consequences.', 'quant-ph-0501138-1-66-1': 'It would be interesting to further investigate the relevance of this dependence by studying the time dependence of global observables in other quantum systems.', 'quant-ph-0501138-1-66-2': 'This should also provide additional insights into the general role of SID and the conditions under which global decoherence in energy may occur, and improve our understanding of the much-discussed relevance and interpretation of observational ignorance of the environment in EID.', 'quant-ph-0501138-1-67-0': 'The author would like to thank Arthur Fine for many thoughtful comments and helpful discussions.'} | {'quant-ph-0501138-2-0-0': 'The "self-induced decoherence" (SID) approach suggests that (1) the expectation value of any observable becomes diagonal in the eigenstates of the total Hamiltonian for systems endowed with a continuous energy spectrum, and (2), that this process can be interpreted as decoherence.', 'quant-ph-0501138-2-0-1': 'We evaluate the first claim in the context of a simple spin bath model.', 'quant-ph-0501138-2-0-2': 'We find that even for large environments, corresponding to an approximately continuous energy spectrum, diagonalization of the expectation value of random observables does in general not occur.', 'quant-ph-0501138-2-0-3': 'We explain this result and conjecture that SID is likely to fail also in other systems composed of discrete subsystems.', 'quant-ph-0501138-2-0-4': 'Regarding the second claim, we emphasize that SID does not describe a physically meaningful decoherence process for individual measurements, but only involves destructive interference that occurs collectively within an ensemble of presupposed "values" of measurements.', 'quant-ph-0501138-2-0-5': 'This leads us to question the relevance of SID for treating observed decoherence effects.', 'quant-ph-0501138-2-1-0': '# Introduction', 'quant-ph-0501138-2-2-0': 'In a series of papers [CITATION], the authors claim to present a "new approach to decoherence" [CITATION], termed "self-induced decoherence" (SID).', 'quant-ph-0501138-2-2-1': 'Their main assertion is that, for systems endowed with a continuous energy spectrum, the expectation value of an observable will become diagonal in the eigenbasis of the Hamiltonian of the system, and that this effect can be viewed as decoherence.', 'quant-ph-0501138-2-3-0': 'The basic idea underlying SID goes back to well-known arguments in the context of quantum measurement and the theory of irreversible processes [CITATION].', 'quant-ph-0501138-2-3-1': 'It rests on the observation that a superposition of a large number of terms with random phases in the expression for the expectation value of a typical observable, or for the matrix elements of the density operator, leads to destructive interference.', 'quant-ph-0501138-2-3-2': 'The phase differences are either due to a random-phase assumption [CITATION], or, as in SID, are created dynamically through the time evolution factor [MATH] associated with each energy eigenstate in the superposition.', 'quant-ph-0501138-2-3-3': 'These destructive interference effects are then responsible for the diagonalization of the expectation value in the energy eigenbasis as described by SID.', 'quant-ph-0501138-2-4-0': 'However, this process differs strongly from the mechanism of environment-induced decoherence (EID) [CITATION].', 'quant-ph-0501138-2-4-1': 'EID understands decoherence as the practically irreversible dislocalization of local phase relations between environment-selected preferred basis states due to entanglement with an environment.', 'quant-ph-0501138-2-4-2': 'The approximate diagonality of the expectation value of local observables expressed in the preferred basis is only a formal phenomenological consequence of the relative states of the environment becoming rapidly orthogonal during the decoherence process.', 'quant-ph-0501138-2-4-3': 'The fact that SID does not require an explicit environment interacting with the system motivated the term "self-induced" and was suggested [CITATION] to circumvent the question of a proper interpretation of the concept of "observational ignorance of the environment" in EID [CITATION].', 'quant-ph-0501138-2-5-0': 'This paper pursues two main goals.', 'quant-ph-0501138-2-5-1': 'First, after formalizing the basic idea of SID (Sec. [REF]), we shall discuss the question to what extent SID can claim to describe a physically relevant decoherence process (Sec. [REF]).', 'quant-ph-0501138-2-5-2': 'In particular, we will argue that, contrary to the claim of its proponents [CITATION], SID does not constitute a "new viewpoint" on decoherence in the usual definition of EID.', 'quant-ph-0501138-2-5-3': 'Second, we shall study whether diagonalization of the expectation value of random observables in the energy eigenbasis is obtained in the context of an explicit spin bath model (Sec. [REF]).', 'quant-ph-0501138-2-5-4': 'Deliberately, we have chosen a discrete model to investigate the required degree of "quasicontinuity" for SID to work as claimed.', 'quant-ph-0501138-2-5-5': 'To anticipate, we find that even for bath sizes large compared to what is typically considered in EID, no general decay of off-diagonal terms is found, unless both the observable and the initial state of the bath are appropriately restricted.', 'quant-ph-0501138-2-5-6': 'We explain and discuss this result in Sec. [REF], and present our conclusions in Sec. [REF].', 'quant-ph-0501138-2-6-0': '# Self-induced decoherence', 'quant-ph-0501138-2-7-0': 'The basic formalism of SID as developed in Refs. [CITATION] considers an arbitrary observable [EQUATION] expanded in the eigenstates [MATH] of the Hamiltonian [MATH] with continuous spectrum.', 'quant-ph-0501138-2-7-1': 'In the general treatment, only observables with [EQUATION] are considered, where [MATH] and the [MATH] are assumed to be regular functions.', 'quant-ph-0501138-2-7-2': 'The time evolution of the expectation value [MATH] of [MATH] in the pure state [MATH] (setting [MATH]) is then given by [EQUATION] where [MATH].', 'quant-ph-0501138-2-8-0': 'For large [MATH], the phase factor [MATH] fluctuates rapidly with [MATH], which leads to destructive interference in the double integral if the multiplying function [MATH], varies comparably slowly.', 'quant-ph-0501138-2-8-1': 'To formalize this argument, SID employs the Riemann-Lebesgue theorem [CITATION], which prescribes that [EQUATION] if [MATH] is a regular function and [MATH] integrable (i.e., [MATH]).', 'quant-ph-0501138-2-8-2': 'Provided these conditions are fulfilled by [MATH], it is concluded that [EQUATION] for large [MATH].', 'quant-ph-0501138-2-8-3': 'Thus, the off-diagonal terms [MATH] have collectively disappeared, which in SID is interpreted as "decoherence in the expectation value."', 'quant-ph-0501138-2-8-4': 'Formally, the SID program introduces a "diagonal-equivalent" density matrix [MATH], [EQUATION] which fulfills [MATH].', 'quant-ph-0501138-2-8-5': 'Note that [MATH] is only a formal equivalent and is not obtained through any dynamical process.', 'quant-ph-0501138-2-8-6': 'Also, expectation values of a nonexhaustive set of observables [see Eq. [REF]] do not uniquely determine the density matrix.', 'quant-ph-0501138-2-8-7': 'Therefore, one must not derive any conclusions about the possibility for certain states of the system from [MATH].', 'quant-ph-0501138-2-9-0': 'To summarize, the main result, Eq. [REF], has been obtained from two key assumptions:', 'quant-ph-0501138-2-10-0': 'The energy spectrum of the system is continuous.', 'quant-ph-0501138-2-11-0': 'The coefficients used in expanding the initial state and the observable in the energy eigenbasis form regular (and integrable) functions of the energy variable.', 'quant-ph-0501138-2-12-0': 'The first requirement of a continuous energy spectrum can be viewed as an implicit inclusion of an internal "environment" with an infinite number of degrees of freedom.', 'quant-ph-0501138-2-12-1': 'However, any realistic physical system is of finite size, and therefore the energy spacing will be discrete.', 'quant-ph-0501138-2-12-2': 'An approximate suppression of off-diagonal terms as given by Eq. [REF] should therefore occur also for quasicontinuous energy spectra, i.e., for small but discrete energy spacings.', 'quant-ph-0501138-2-13-0': 'The regularity assumption (2) is crucial, since it ensures that the phase factors [MATH] are able to lead to the required destructive interference of the expansion coefficients for large times.', 'quant-ph-0501138-2-13-1': 'However, especially in the realistic case of systems of finite size where the expansion coefficients will be a finite set of discrete values, this condition will not hold.', 'quant-ph-0501138-2-13-2': 'It is therefore important to understand the physical meaning and the consequences of a violation of this assumption.', 'quant-ph-0501138-2-14-0': 'Note also that the strict mathematical limit [MATH] employed in the Riemann-Lebesgue theorem, Eq. [REF], is not physically meaningful, and approximate suppression must therefore occur already over finite time scales, as indicated in Eq. [REF].', 'quant-ph-0501138-2-14-1': 'Also, for the realistic case of only quasicontinuous (i.e., essentially discrete) energy spectra, no conclusions about an "irreversibility" of the decay should be derived from the limit [MATH] (as it is done, for example, in Ref. [CITATION]), since the off-diagonal terms will return to their initial values within a finite recurrence time scale.', 'quant-ph-0501138-2-15-0': 'The issues outlined above will be illustrated and investigated in the context of a particular model system in Sec. [REF].', 'quant-ph-0501138-2-16-0': '# Does SID describe decoherence?', 'quant-ph-0501138-2-17-0': 'Despite the fact that SID and EID share the term "decoherence" in their name, we shall demonstrate in this section that their foundations, scope, and physical implications are fundamentally different.', 'quant-ph-0501138-2-17-1': 'Keeping these differences in mind is very important for a proper interpretation of the study of the bath model described in the following Sec. [REF].', 'quant-ph-0501138-2-18-0': 'As already briefly outlined in the Introduction, the standard approach of environmental decoherence [CITATION] describes the consequences of the ubiquitous interaction of any system with its environment.', 'quant-ph-0501138-2-18-1': 'This leads to entanglement between the system and the environment and singles out a preferred basis of the system that is dynamically determined by the Hamiltonian governing the interaction.', 'quant-ph-0501138-2-18-2': 'The relative environmental states associated with these preferred states rapidly approach orthogonality (i.e., macroscopic distinguishability).', 'quant-ph-0501138-2-18-3': 'Phase relations between the preferred states that were initially associated with the system alone are now "dislocalized" into the system-environment combination due to the entanglement, which constitutes the decoherence process.', 'quant-ph-0501138-2-18-4': 'In this sense, interference between the preferred states becomes locally suppressed, i.e., decoherence leads locally to a transition from a superposition to an apparent ("improper" [CITATION]) ensemble.', 'quant-ph-0501138-2-18-5': 'This can be used to define dynamically independent relative local wave-function components that can be related to local quasiclassical properties, thereby mimicking an apparent "collapse" of the wave function [CITATION].', 'quant-ph-0501138-2-19-0': 'The interaction between the system and its environment, often referred to as a "continuous measurement by the environment," is observer-independent and can be formulated entirely in terms of wave functions, without reference to presumed (classical) concepts such as "values of observables" and expectation values (see, for example, Chap.', 'quant-ph-0501138-2-19-1': '2 of Ref. [CITATION]).', 'quant-ph-0501138-2-19-2': 'As it has been emphasized frequently [CITATION], the formalism of local ("reduced") density matrices and expectation values presupposes the probabilistic interpretation of the wave function and ultimately relies on the occurence of a "collapse" of the wave function at some stage (or on the description of an observationally equivalent "branching" process in a relative-state framework [CITATION]).', 'quant-ph-0501138-2-19-3': 'The approximate diagonalization of the reduced density matrix [MATH] (describing the probability distribution of outcomes of measurements on the "system [MATH] of interest" immersed into an environment [MATH]) in the environment-selected basis should therefore be considered only as a phenomenological consequence of EID, but not as its essence (see also Ref. [CITATION]).', 'quant-ph-0501138-2-19-4': 'Given an ensemble of results of measurements of a local observable [MATH], the suppression of off-diagonal terms in [MATH] can then be related to the approximate diagonality of the expectation value of [MATH] in the preferred basis, since [MATH].', 'quant-ph-0501138-2-20-0': 'In contrast with EID, SID focuses solely on the derivation of a suppression of off-diagonal terms (in the energy eigenbasis only) in the expectation value of observables pertaining to a single undivided closed system; entanglement through interactions between subsystems plays no role in SID.', 'quant-ph-0501138-2-20-1': 'As indicated earlier, the damping effect is due to destructive interference between a large number of terms with dynamically induced phase differences.', 'quant-ph-0501138-2-20-2': 'Thus it is only the averaging process contained in the concept of expectation values that leads to a disappearance of interference terms.', 'quant-ph-0501138-2-20-3': 'Individually, each term remains present at all times and is not suppressed independently of the other terms.', 'quant-ph-0501138-2-20-4': 'The fact that collectively the off-diagonal terms may lead to a mutual canceling-out must not be misinterpreted as implying that the measurement "outcomes" corresponding to these terms do not occur.', 'quant-ph-0501138-2-20-5': 'Thus SID cannot pertain to the relevant problem of a loss of interference in individual measurements.', 'quant-ph-0501138-2-20-6': 'In view of this argument, the concept of the "diagonal-equivalent density matrix" [MATH], as introduced by the SID program [see Eq. [REF]], is rather misleading, since it gives the incorrect impression of an absence of interference terms [MATH], while the corresponding terms in the expression for the expectation value are individually present at all times.', 'quant-ph-0501138-2-20-7': 'Derivations of a "classical limit" based on [MATH] [CITATION] appear to have overlooked this issue.', 'quant-ph-0501138-2-21-0': 'While SID rests on the concept of expectation values, i.e., of weighted averages over an ensemble of measurement outcomes, it does not explain the physical origin of these outcomes and their ensembles.', 'quant-ph-0501138-2-21-1': 'In contrast with EID, SID does not contain a dynamical account of the measurement process itself that could motivate explanations for how measurement outcomes arise (if only, as in EID, in an "apparent," relative-state sense).', 'quant-ph-0501138-2-21-2': 'Consequently, the assumption of an a priori existence of an ensemble of measurement outcomes, as it is inherent in SID, could be viewed as a particular application of the Copenhagen interpretation.', 'quant-ph-0501138-2-21-3': 'One might then argue that in this case decoherence would not even be necessary in explaining the observed absense of (macrosopic) interference effects.', 'quant-ph-0501138-2-22-0': 'Note that EID makes crucial use of the concept of locality in deriving a loss of interference, since globally the quantum-mechanical superposition remains unchanged, as required by the unitarity of the time evolution of the total wave function.', 'quant-ph-0501138-2-22-1': 'As frequently emphasized by Zeh (e.g., in Refs. [CITATION]) and others (see, for example, Ref. [CITATION]), this locality can be grounded in the (nontrivial) empirical insight that all observers and interactions are intrinsically local.', 'quant-ph-0501138-2-22-2': 'On the other hand, the decomposition into a "system of interest" and an environment that is ignored from an observational point of view, as required in EID, and the resulting implication that the relevance of environmental decoherence is restricted to local subsystems of the total (nonlocal) quantum Universe, has been a subject of ongoing critical discussions (see, for example, Refs. [CITATION]).', 'quant-ph-0501138-2-22-3': 'Furthermore, no general rule is available that would indicate where the split between system and environment is to be placed, a conceptual difficulty admitted also by proponents of EID [CITATION].', 'quant-ph-0501138-2-22-4': 'These issues seem to have motivated the attempt of the SID program to derive decoherence for closed, undivided systems.', 'quant-ph-0501138-2-23-0': 'However, it is important to note that EID has clearly demonstrated that the assumption of the existence of closed system is unrealistic in essentially all cases [CITATION].', 'quant-ph-0501138-2-23-1': 'Enlarging the system by including parts of its environment, as it is implicitly done in SID in order to arrive at a quasicontinuous spectrum, will render the closed-system assumption even less physically viable: The combined system will in turn interact with its surroundings, and the degree of environmental interaction will increase with the number of degrees of freedom in the system.', 'quant-ph-0501138-2-23-2': 'Also, since some interaction with the external measuring device will be required, the assumption of a closed system simply bypasses the question of how the information contained in the ensemble is acquired in the first place.', 'quant-ph-0501138-2-23-3': 'Ultimately, the only truly closed "system" is the Universe in its entirety, and one can therefore question the physical relevance and motivation for a derivation of decoherence for subsystems that are presumed to be closed.', 'quant-ph-0501138-2-24-0': 'Furthermore, a general measurement in SID would pertain also to the environment implicitly contained in the "closed system," posing the question of how this could translate into an experimentally realizable situation.', 'quant-ph-0501138-2-24-1': 'And even if such a measurement can be carried out, its result would usually be of rather little physical interest in the typical situation of observing decoherence for a particular object due to its largely unobserved environment.', 'quant-ph-0501138-2-25-0': 'Finally, in SID, suppression of off-diagonal terms always occurs in the energy eigenbasis, which can therefore be viewed as the universal "preferred basis" in this approach.', 'quant-ph-0501138-2-25-1': 'However, this basis will generally not be useful in accounting for our observation of different preferred bases for the relevant local systems of interest (e.g., spatial localization of macroscopic bodies [CITATION], chirality eigenstates for molecules such as sugar [CITATION], and energy eigenstates in atoms [CITATION]).', 'quant-ph-0501138-2-25-2': 'Furthermore, the energy eigenbasis cannot be used to describe the emergence of time-dependent, quasiclassical properties.', 'quant-ph-0501138-2-26-0': 'In conclusion, not only is the scope of SID more limited than that of EID, but the two approaches also rest on different foundations.', 'quant-ph-0501138-2-26-1': 'The interpretation of the processes described by these theories is fundamentally different, even though phenomenological effects of EID can manifest themselves in a manner formally similiar to that of SID, i.e., as a disappearance of off-diagonal terms in expectation values.', 'quant-ph-0501138-2-26-2': 'Any proposed derivations of an "equivalence" between SID and EID [CITATION] can therefore at most claim to describe coincidental formal similiarities in the context of very particular models, and only if the scope of EID is reduced to its influence on expectation values.', 'quant-ph-0501138-2-26-3': 'On the basis of our arguments, we question the justification for labeling the process referred to by SID as "decoherence."', 'quant-ph-0501138-2-27-0': '# Analysis of the spin bath model', 'quant-ph-0501138-2-28-0': 'By studying an explicit model, we shall now directly investigate the claim of SID, that terms not diagonal in energy in the expectation value of arbitrary observables of the system decay if the system is endowed with a continuous energy spectrum.', 'quant-ph-0501138-2-28-1': 'We shall also illustrate formal and numerical differences in the time evolution of the expectation value of local observables that take into account only the degrees of freedom of the system [MATH] while ignoring the environment [MATH] (the situation encountered in EID), and global observables that pertain to both [MATH] and [MATH] (the case treated by SID).', 'quant-ph-0501138-2-28-2': 'However, in view of our arguments in the preceding Sec. [REF], this should not be misunderstood as a side-by-side comparison of SID and EID.', 'quant-ph-0501138-2-28-3': 'While expectation values may share formal similiarities in both approaches, they also obliterate fundamental differences between SID and EID that lead to very different implications of these expectation values for the question of decoherence.', 'quant-ph-0501138-2-29-0': '## The model and its time evolution', 'quant-ph-0501138-2-30-0': 'The probably most simple exactly solvable model for decoherence was introduced some years ago by Zurek .', 'quant-ph-0501138-2-30-1': 'Here, the system [MATH] consists of a spin-1/2 particle (a single qubit) with two possible states [MATH] (representing spin up) and [MATH] (corresponding to spin down), interacting with a collection of [MATH] environmental qubits (described by the states [MATH] and [MATH]) via the total Hamiltonian [EQUATION]', 'quant-ph-0501138-2-30-2': 'Here, the [MATH] are coupling constants, and [MATH] is the identity operator for the [MATH]-th environmental qubit.', 'quant-ph-0501138-2-30-3': 'The self-Hamiltonians of [MATH] and [MATH] are taken to be equal to zero.', 'quant-ph-0501138-2-30-4': 'Note that [MATH] has a particularly simple form, since it contains only terms diagonal in the [MATH] and [MATH] bases.', 'quant-ph-0501138-2-31-0': 'It follows that the eigenstates of [MATH] are product states of the form [MATH], etc.', 'quant-ph-0501138-2-31-1': 'A general state [MATH] can then be written as a linear combination of product eigenstates, [EQUATION]', 'quant-ph-0501138-2-31-2': 'This state evolves under the action of [MATH] into [EQUATION] where [EQUATION]', 'quant-ph-0501138-2-31-3': 'The density matrix is [EQUATION] and its part diagonal in energy (i.e., diagonal in the eigenstates [MATH] of [MATH]) is [EQUATION]', 'quant-ph-0501138-2-32-0': '## Expectation values of local observables', 'quant-ph-0501138-2-33-0': 'Focusing, in the spirit of EID, on the system [MATH] alone, we trace out the degrees of freedom of the spin bath in the density operator [MATH].', 'quant-ph-0501138-2-33-1': 'This yields the reduced density operator [EQUATION] where the time dependence of the off-diagonal terms [MATH] and [MATH] is given by the decoherence factor [EQUATION]', 'quant-ph-0501138-2-34-0': 'The expectation values of any local [MATH]-observable [EQUATION] is then given by [EQUATION]', 'quant-ph-0501138-2-34-1': 'We can formally rewrite [MATH] as a sum, [EQUATION] where the sum runs over all eigenstates [MATH] of the total Hamiltonian [MATH], with eigenvalues [MATH].', 'quant-ph-0501138-2-35-0': 'A concrete illustration for the time dependence of [MATH], Eq. [REF], for two different bath sizes is shown in Fig. [REF].', 'quant-ph-0501138-2-35-1': 'We see that [MATH] decays quickly by several orders of magnitude and then continues to oscillate about a very small mean value.', 'quant-ph-0501138-2-35-2': 'Thus, for local observables, terms corresponding to interference between the two [MATH]-states [MATH] and [MATH] become quickly and strongly supressed.', 'quant-ph-0501138-2-36-0': '## Expectation values of global observables', 'quant-ph-0501138-2-37-0': 'An arbitrary global observable [MATH] can be written as a linear combination of the form [MATH], where the [MATH] are product eigenstates of the total Hamiltonian [MATH], Eq. [REF].', 'quant-ph-0501138-2-38-0': 'Since [MATH] must be Hermitian, [MATH], [MATH], [MATH], and [MATH] are real numbers, and [MATH], [MATH].', 'quant-ph-0501138-2-38-1': 'To keep the notation simple, we shall omit the sum over [MATH] (and thus the index [MATH]) in the following.', 'quant-ph-0501138-2-39-0': 'The expectation value of [MATH] in the state [MATH], Eq. [REF], is [EQUATION]', 'quant-ph-0501138-2-39-1': 'The special case of the expectation value of local observables, as considered in the preceding Sec. [REF], can easily be recovered by remembering that tracing out the degrees of freedom of [MATH] is equivalent to choosing all coefficients [MATH] and [MATH], which yields [MATH] and [MATH], see Eq. [REF], in agreement with Eq. [REF].', 'quant-ph-0501138-2-40-0': 'Suppression of terms in [MATH] that are not diagonal in the energy eigenbasis would be represented by the vanishing of all time-dependent terms in the above expression, i.e., [EQUATION] because we can easily show that [MATH], where [MATH], Eq. [REF], is the part of the density matrix that is diagonal in the eigenstates of the total Hamiltonian.', 'quant-ph-0501138-2-40-1': 'We also see that [MATH], where [EQUATION] is the part of [MATH] diagonal in energy.', 'quant-ph-0501138-2-40-2': 'Thus, as expected, diagonality of [MATH] in energy can also be characterized by the presence of only those product expansion coefficients that are contained in [MATH].', 'quant-ph-0501138-2-41-0': 'The form of the two product terms [MATH] and [MATH] is similar: They only differ in the order of the pairing of the product expansion coefficients with the exponential factors.', 'quant-ph-0501138-2-41-1': 'Also, since the coefficients [MATH] are independent, diagonalization in energy will in general require that individually [MATH] and [MATH] for large [MATH].', 'quant-ph-0501138-2-41-2': 'We can therefore restrict our following analysis to [MATH] alone.', 'quant-ph-0501138-2-41-3': '(We shall also omit the subscript "0" in the following.)', 'quant-ph-0501138-2-42-0': 'First of all, let us rewrite [MATH] as a sum of [MATH] terms, [EQUATION] where the [MATH] represent products of expansion coefficients, [EQUATION]', 'quant-ph-0501138-2-42-1': 'Here the sets [MATH] specify over which indices [MATH] each product runs, namely, they are subsets of the set [MATH] of all integers between 1 and [MATH] such that [MATH] and [MATH].', 'quant-ph-0501138-2-42-2': 'The total energy [MATH] associated with each term in the sum, Eq. [REF], is [EQUATION]', 'quant-ph-0501138-2-42-3': 'We choose the index [MATH] such that [MATH] for all [MATH].', 'quant-ph-0501138-2-42-4': 'Clearly, [MATH] whenever [MATH] (i.e., if [MATH]), canceling out the time dependence of the associated product term in the expression for [MATH].', 'quant-ph-0501138-2-42-5': 'Thus, we can split [MATH] into a time-independent and a time-dependent part, [EQUATION] where now the first sum runs over all [MATH] for which [MATH], while the second sum runs over all [MATH] for which [MATH].', 'quant-ph-0501138-2-43-0': 'Diagonality in energy would require [MATH] as [MATH].', 'quant-ph-0501138-2-43-1': 'Written this way, we see that [MATH] is formally similiar to the function [MATH] derived for local observables, Eq. [REF].', 'quant-ph-0501138-2-43-2': 'This might not come as a surprise, since also the expression for [MATH] can be derived from the calculation of an expectation value of an observable, namely, that of the local observable [MATH] that measures the degree of local interference between the [MATH]-states [MATH] and [MATH].', 'quant-ph-0501138-2-43-3': 'However, in the case of [MATH], [MATH] is a product of [MATH] real and non-negative coefficients [MATH] and [MATH], while the [MATH] of Eq. [REF] contain crossterms of the form [MATH] and [MATH], arbitrary real coefficients [MATH] and [MATH], and arbitrary complex coefficients [MATH].', 'quant-ph-0501138-2-44-0': 'We expect this difference to have strong influence on the time evolution of [MATH] vs. that of [MATH].', 'quant-ph-0501138-2-44-1': 'The destructive interference needed to obtain suppression of the off-diagonal part of the expectation value relies on the idea that, when a function [MATH] is multiplied by a phase factor [MATH] whose variation with [MATH] is much faster than that of [MATH], neighboring values [MATH] and [MATH] will have similiar magnitude and phases, but will be weighted with two strongly different phase factors, which leads to an averaging-out effect in the sum [MATH].', 'quant-ph-0501138-2-45-0': 'In our case, writing [EQUATION] with [MATH], the phases [MATH] will in general vary very rapidly with [MATH] and, thus, with [MATH].', 'quant-ph-0501138-2-45-1': 'This is a consequence of the fact that the [MATH] are composed of products of coefficients, such that changing a single term in the product will in general result in a drastic change in the overall phase associated with the [MATH].', 'quant-ph-0501138-2-45-2': '(The variation in magnitude among the [MATH] can be expected to be comparably insignificant for larger [MATH]).', 'quant-ph-0501138-2-45-3': 'Such discontinuous phase fluctuations are absent in the formally similiar function [MATH], Eq. [REF], since there only the absolute value of the coefficients [MATH] and [MATH] enters.', 'quant-ph-0501138-2-45-4': 'Note that the impact of the phase fluctuations cannot be diminished by going to larger [MATH], since the [MATH]-periodicity of phases implies that the effect of a phase difference between terms [MATH] and [MATH] induced by [MATH] will in average be similiar to that induced by [MATH] for all (larger) values of [MATH].', 'quant-ph-0501138-2-46-0': 'We anticipate the described phase-variation effect to counteract the averaging-out influence of the multiplying phase factor [MATH], and to thus make it more difficult, if not entirely impossible, for [MATH], Eq. [REF], to converge to zero.', 'quant-ph-0501138-2-46-1': 'On the other hand, if the average difference between the phases associated with the individual coefficients is decreased, we would expect that the rate and degree of decay of [MATH] will be improved.', 'quant-ph-0501138-2-47-0': '## Numerical results for the expectation value of random global observables', 'quant-ph-0501138-2-48-0': 'To check this prediction and to generally gain more insight into the behavior of [MATH], Eq. [REF], we studied numerically the time evolution of [MATH], Eq. [REF], normalized by its initial value at [MATH], for sets of random observables [MATH].', 'quant-ph-0501138-2-48-1': 'Diagonalization of [MATH] in energy would then be represented by a decay of [MATH] from its initial value of one.', 'quant-ph-0501138-2-49-0': 'Fig. [REF] shows three typical examples for the time evolution of [MATH] for a fixed bath size of [MATH].', 'quant-ph-0501138-2-49-1': 'All couplings [MATH] were taken to be random real numbers between [MATH] and [MATH].', 'quant-ph-0501138-2-49-2': 'To investigate the influence of phase fluctuations of the [MATH], Eq. [REF], we considered three different cases for selecting the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH], i.e., for choosing the initial state of the environment and the observable.', 'quant-ph-0501138-2-49-3': 'In the completely random case (A), the coefficients [MATH], [MATH], and [MATH] were taken to be random complex numbers, with magnitudes and phases drawn from a uniform distribution over the intervals [MATH] and [MATH], respectively (and such that [MATH]).', 'quant-ph-0501138-2-49-4': 'Similiarly, the coefficients [MATH] and [MATH] were random real numbers drawn from a uniform distribution over the interval [MATH].', 'quant-ph-0501138-2-49-5': 'In the second case (B), the initial state of the environment was prepared such that the phases of the [MATH] and [MATH] were restricted to the interval [MATH].', 'quant-ph-0501138-2-49-6': 'Also, only observables with non-negative values of [MATH] and [MATH] were considered, such that sign reversals of [MATH] due to a change of product terms containing these coefficients were prevented.', 'quant-ph-0501138-2-49-7': 'Finally, in the third case (C), only the absolute values of the [MATH], [MATH], [MATH], [MATH], and [MATH] were used, which implies that the [MATH] fluctuated only in magnitude.', 'quant-ph-0501138-2-50-0': 'We observed a drastic influence of the range of phases and signs associated with the individual coefficients [MATH], [MATH], [MATH], [MATH], and [MATH], on the evolution of [MATH].', 'quant-ph-0501138-2-50-1': 'In the special case (C) of all coefficients being real non-negative numbers, [MATH] exhibited a consistently strong and fast decay behavior, similiar to the decay of the function [MATH], Eq. [REF], describing suppression of off-diagonal terms for local observables (see Fig. [REF]).', 'quant-ph-0501138-2-50-2': 'In the intermediate case (B), with restricted phases and signs, the degree of decay of [MATH] was decreased, while the decay rate stayed roughly the same.', 'quant-ph-0501138-2-50-3': 'In the general random case (A), in which no restriction on the spread of phases and on the signs of the coefficients was imposed, the time evolution of [MATH] was observed to be sensitive to the particular set of random numbers used for the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH] in each run.', 'quant-ph-0501138-2-50-4': 'For some of the sets, [MATH] was seen to lack any decay behavior at all.', 'quant-ph-0501138-2-50-5': 'In other cases, the baseline of oscillation was located below zero, indicating a very weak damping effect, albeit with the peaks of the large-amplitude oscillation frequently reaching values greater than zero.', 'quant-ph-0501138-2-51-0': 'These results show that, for the bath size studied here, a consistent occurence of a decay of [MATH] hinges on the phase restrictions imposed on the coefficients describing the observable and the initial state of the environment.', 'quant-ph-0501138-2-51-1': 'If these restrictions are given up, the time evolution of [MATH] and any occurence of a (comparably weak) decay will exhibit strong dependence on the particular set of values chosen for the coefficients.', 'quant-ph-0501138-2-52-0': 'However, it is important to realize that the assumption of a restricted initial state of [MATH] is not only unrealistic, since the environment is typically uncontrollable, but it will also lead to a circular argument when aiming at a derivation of a universal decay effect.', 'quant-ph-0501138-2-52-1': 'This is so because any restriction would require an appropriate preparation of the initial state through a measurement on the entire [MATH], which implies that suppression of off-diagonal terms would then in general be absent for the observable corresponding to this measurement, if the restriction of the initial state of [MATH] is relevant to the occurence of the suppression.', 'quant-ph-0501138-2-52-2': 'Consequently, the [MATH] and [MATH] must be allowed to possess arbitrary phases.', 'quant-ph-0501138-2-52-3': 'Then, since the [MATH], [MATH], and [MATH] are always paired with the [MATH] and [MATH] in the expression for the [MATH] that make up [MATH] [see Eq. [REF]], we anticipate that giving up phase restrictions on the [MATH] and [MATH] will render the restrictions imposed on the [MATH]-coefficients less effective, if not entirely irrelevant, in bringing about a decay of [MATH].', 'quant-ph-0501138-2-53-0': 'To study this prediction, in Fig. [REF] we show a representative plot of [MATH] using only the absolute values of the [MATH]-coefficients [MATH], [MATH], and [MATH], but with the [MATH]-coefficients [MATH] and [MATH] possessing random phases between 0 and [MATH].', 'quant-ph-0501138-2-53-1': 'We found that decay is either entirely absent or strongly diminished in strength, despite the fact that the strongest possible restriction on the phases and signs of the [MATH]-coefficients is imposed.', 'quant-ph-0501138-2-53-2': 'Similiar to the case of completely random coefficients, the behavior of [MATH] was observed to depend crucially on the particular set of random numbers chosen for the coefficients.', 'quant-ph-0501138-2-53-3': 'These results lead us to conclude that a universal decay of off-diagonal terms does not occur for the studied bath size and time scale.', 'quant-ph-0501138-2-54-0': 'To be sure, SID is based on the assumption of a quasicontinuous energy spectrum and very long timescales, corresponding to "sufficiently large" [MATH] and [MATH] (the existing derivations of SID [CITATION] even assume the strict limits [MATH] and [MATH], in order to allow for a direct application of the Riemann-Lebesgue theorem), while so far we have only considered relatively modest values for these parameters.', 'quant-ph-0501138-2-54-1': 'However, since we know from Fig. [REF] that for expectation values of local observables, strong and fast decay of off-diagonal terms is obtained for the value of [MATH] and over the timescale used in the plots shown in Fig. [REF], it is clear that, if a general global disappearance of interference terms is to occur in our model, it will require a much larger number of environmental qubits and/or longer time scales than typically considered for local observables.', 'quant-ph-0501138-2-55-0': 'Accordingly, in Fig. [REF] we show a typical example for the time evolution of [MATH] over the timescale [MATH] for the case of a completely random observable and initial state of [MATH], using comparably large bath sizes [MATH] between [MATH] and [MATH].', 'quant-ph-0501138-2-55-1': 'We observed that even for these values of [MATH], no consistent occurence of a decay became apparent.', 'quant-ph-0501138-2-55-2': 'In particular, no generally valid direct correlation between the value of [MATH] and the time evolution of [MATH] was visible.', 'quant-ph-0501138-2-55-3': 'Instead, it was again the particular set of random numbers included in the computation of [MATH] for a given value of [MATH] (but not to the size [MATH] of the set itself) that determined whether the baseline of oscillation of [MATH] was located above or below the zero line.', 'quant-ph-0501138-2-55-4': 'In agreement with analytical predictions in the preceding section, we also found that the choice of a longer timescale is irrelevant, since neither the baseline nor the amplitude of oscillation changed significantly over the investigated time interval after a comparably short initial period.', 'quant-ph-0501138-2-55-5': 'Furthermore, we observed that even if [MATH] "decayed" for a particular set of random numbers, the function sustained a large-amplitude oscillation whose peaks often attained values much larger than the initial value of [MATH].', 'quant-ph-0501138-2-56-0': 'Our results show that, in general, for the bath sizes and time scales studied, destructive interference of off-diagonal terms in the expectation value expressed in the energy eigenbasis [as quantified by [MATH], see Eq. [REF]] does not occur in our model.', 'quant-ph-0501138-2-56-1': 'Instead, the time evolution of [MATH] is simply determined by the particular random numbers used to describe the observable and the initial state of the environment.', 'quant-ph-0501138-2-56-2': 'Therefore, no general suppression of interference terms can be inferred.', 'quant-ph-0501138-2-57-0': '# Discussion', 'quant-ph-0501138-2-58-0': 'The process described by SID appears to be neither formally nor conceptually nor physically related to the decoherence mechanism in the standard sense of environmental decoherence.', 'quant-ph-0501138-2-58-1': 'EID accounts for the absence of interference from the perspective of the local (open) system by describing interactions with an environment in quantum-mechanical terms of wave-function entanglement.', 'quant-ph-0501138-2-58-2': 'In contrast, SID describes dynamically induced destructive interference between time-dependent terms in the expression for expectation values.', 'quant-ph-0501138-2-58-3': 'SID does not, however, explain the physical origin of the measurement outcomes and their probability-weighted ensembles needed to define the expectation values.', 'quant-ph-0501138-2-58-4': 'Even if this purely phenomenological basis of SID is accepted, the described process has no bearing on a loss of coherence in individual measurements, since it is only a consequence of averaging over a large number of measurement results.', 'quant-ph-0501138-2-58-5': 'This is in fundamental contrast to EID, where each measurement-like interaction leads to a dislocalization of interference and thus, locally, to a disappearance of interference.', 'quant-ph-0501138-2-59-0': 'The main result of our study of the spin bath model is the finding that the destructive interference predicted by SID will in general fail to occur in our model even for bath sizes and over time scales much larger than typically considered in treatments of the same model in environmental decoherence.', 'quant-ph-0501138-2-59-1': 'The source of this failure lies in the random relative phases associated with the individual initial bath spin states and the expansion coefficients of the observable.', 'quant-ph-0501138-2-59-2': 'The resulting discontinuous phase fluctuations in the coefficient function [MATH], as defined in Eq. [REF], counteract the supposed averaging-out effect of the dynamical phase factors [MATH] in a way that is, due to the [MATH]-periodicity of the phase, effectively independent of the value of [MATH].', 'quant-ph-0501138-2-60-0': 'Even when the bath size is increased, the function [MATH] remains a set of discrete values with discontinuously varying phases.', 'quant-ph-0501138-2-60-1': 'This can be explained by noting that, while the total energy is a sum of the energies of each subsystem, such that enlarging the number of contributing subsystems will in general lead to an improved quasicontinuity of the energy spectrum, the [MATH]-periodicity of the phases implies that the degree of phase discontinuity of the [MATH] will not be diminished by increasing the number of subsystems.', 'quant-ph-0501138-2-60-2': 'It is therefore unlikely that a consistent decay behavior could become apparent for spin baths much larger than those considered here.', 'quant-ph-0501138-2-61-0': 'This indicates that it is not the degree of continuity of the energy spectrum that represents the determining factor for obtaining destructive interference.', 'quant-ph-0501138-2-61-1': 'Rather, it is the discrete nature of the model itself that seems to lead to difficulties.', 'quant-ph-0501138-2-61-2': 'Only if restrictions are imposed on both the measured observable and the initial state of the environment, a consistent and general suppression of off-diagonal terms can occur.', 'quant-ph-0501138-2-61-3': 'But, as we have argued, the corresponding preparation of the initial state of the environment is physically unrealistic and renders the derivation of a universal decay effect circular.', 'quant-ph-0501138-2-62-0': 'We conjecture that the diagonalization of the expectation value, as described by SID, is likely to fail also in other systems composed of discrete individual subentities.', 'quant-ph-0501138-2-62-1': 'For, in such models, the relevant function will typically be represented by a large product of discrete expansion coefficients, similiar to the [MATH] of our model, whose discontinuous phase fluctuations will again be likely to counteract the averaging-out influence of the dynamical phases.', 'quant-ph-0501138-2-62-2': 'It is therefore clear that the seemingly innocuous mathematical requirement of regularity and integrability of the coefficient functions (see Sec. [REF]) is far from "valid in all relevant cases" where the condition of a sufficiently continuous energy spectrum holds.', 'quant-ph-0501138-2-62-3': 'The suggestion to approximate such discrete functions by a continuous function through interpolation [CITATION] does not appear to be viable, since the interpolated function would describe a physically different situation.', 'quant-ph-0501138-2-63-0': 'On a general note, it is also important to realize that dynamical phases are correlated.', 'quant-ph-0501138-2-63-1': 'Thus one could always construct an observable for which the initial phases of the coefficients seem completely random, but are in fact chosen such that recurrence of coherence will show up within a finite time interval, thus disproving the claimed universality of SID without any further argument.', 'quant-ph-0501138-2-64-0': '# Summary and conclusions', 'quant-ph-0501138-2-65-0': 'We have investigated the two main claims of the "self-induced decoherence" (SID) approach, namely, (1) that expectation values of observables pertaining to a closed system become diagonal in the eigenbasis of the Hamiltonian, provided the system is endowed with a continuous energy spectrum; and (2), that this process represents a new way of describing quantum decoherence, and that it leads to results equivalent to the standard approach of environment-induced decoherence.', 'quant-ph-0501138-2-66-0': 'We have evaluated the first claim in the context of a simple spin bath model of finite size by studying, analytically and numerically, the time evolution of expectation values of random global observables.', 'quant-ph-0501138-2-66-1': 'We have found that, in general, collective decay of terms off-diagonal in the energy eigenbasis does not occur over the large range of bath sizes and time scales considered.', 'quant-ph-0501138-2-66-2': 'This result is not due to an insufficient quasicontinuity of the energy spectrum, but is rather rooted in the randomness of the phases associated with the observable and the initial state of the environment.', 'quant-ph-0501138-2-66-3': 'Even in the limit of large bath sizes, the discrete functions for which destructive interference is to be derived do not approach their sufficiently smoothly varying interpolated approximations required for the dynamical phase averaging to have an effect.', 'quant-ph-0501138-2-67-0': 'These results represent an example for a simple model system that, although endowed with a quasicontinuous energy spectrum, fails to exhibit the decay of off-diagonal terms that would be expected from an extrapolation of SID to discrete models in the limit of comparably large sizes of the system.', 'quant-ph-0501138-2-67-1': 'Such an extrapolation should be possible if the approach is to have general physical relevance.', 'quant-ph-0501138-2-67-2': 'We have also anticipated that the decay effect described by SID will likely be absent also in other similiar models that are composed of discrete subsystems.', 'quant-ph-0501138-2-68-0': 'With respect to the second claim of the SID program, we have questioned the suggestion that SID represents a "new viewpoint" [CITATION] on the theory of environment-induced decoherence, since the two approaches are based on conceptually, formally, and physically unrelated mechanisms.', 'quant-ph-0501138-2-68-1': 'In particular, we have pointed out the following key differences and objections:', 'quant-ph-0501138-2-69-0': 'SID does not describe the suppression of interference for individual measurements, since interference terms in the expectation value are not damped individually.', 'quant-ph-0501138-2-70-0': 'SID simply presupposes the existence of an ensemble of measurement outcomes, without giving an account of its origin in terms of a physical description of measurement.', 'quant-ph-0501138-2-71-0': 'The assumption of closed systems is unrealistic, especially for systems containing the many degrees of freedom needed to obtain the required quasicontinuous energy spectrum.', 'quant-ph-0501138-2-72-0': 'The physical feasibility and relevance of measurements pertaining to the total system-environment combination is doubtful.', 'quant-ph-0501138-2-73-0': 'Energy as the universal preferred basis of the global closed system can usually not account for the different observed preferred bases for the local system of interest.', 'quant-ph-0501138-2-74-0': 'Our study leads us to two main conclusions.', 'quant-ph-0501138-2-74-1': 'First, it points to the need for more precise, physically motivated criteria for the occurence of the destructive interference effect described by SID.', 'quant-ph-0501138-2-74-2': 'Most importantly, however, the physical interpretation and relevance of this effect need to be explained.', 'quant-ph-0501138-2-74-3': 'We suspect that the SID approach may have mistakenly interpreted and labeled an unrelated process as "decoherence."', 'quant-ph-0501138-2-75-0': 'The author would like to thank A. Fine, M. Castagnino, E. Joos, O. Lombardi, and H.-D. Zeh for many thoughtful comments and helpful discussions.'} | [['quant-ph-0501138-1-33-1', 'quant-ph-0501138-2-42-1'], ['quant-ph-0501138-1-33-2', 'quant-ph-0501138-2-42-2'], ['quant-ph-0501138-1-33-3', 'quant-ph-0501138-2-42-3'], ['quant-ph-0501138-1-33-4', 'quant-ph-0501138-2-42-4'], ['quant-ph-0501138-1-27-0', 'quant-ph-0501138-2-38-0'], ['quant-ph-0501138-1-27-1', 'quant-ph-0501138-2-38-1'], ['quant-ph-0501138-1-21-1', 'quant-ph-0501138-2-33-1'], ['quant-ph-0501138-1-18-0', 'quant-ph-0501138-2-31-0'], ['quant-ph-0501138-1-18-2', 'quant-ph-0501138-2-31-2'], ['quant-ph-0501138-1-18-3', 'quant-ph-0501138-2-31-3'], ['quant-ph-0501138-1-31-0', 'quant-ph-0501138-2-41-0'], ['quant-ph-0501138-1-31-2', 'quant-ph-0501138-2-41-2'], ['quant-ph-0501138-1-39-2', 'quant-ph-0501138-2-45-2'], ['quant-ph-0501138-1-17-0', 'quant-ph-0501138-2-30-0'], ['quant-ph-0501138-1-17-3', 'quant-ph-0501138-2-30-3'], ['quant-ph-0501138-2-40-0', 'quant-ph-0501138-3-38-0'], ['quant-ph-0501138-2-40-1', 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'quant-ph-0501138-3-9-0'], ['quant-ph-0501138-1-8-0', 'quant-ph-0501138-2-8-1'], ['quant-ph-0501138-1-8-1', 'quant-ph-0501138-2-8-2'], ['quant-ph-0501138-1-8-1', 'quant-ph-0501138-2-8-3'], ['quant-ph-0501138-1-9-0', 'quant-ph-0501138-2-8-4'], ['quant-ph-0501138-1-50-0', 'quant-ph-0501138-2-55-0'], ['quant-ph-0501138-1-50-1', 'quant-ph-0501138-2-55-1'], ['quant-ph-0501138-1-51-1', 'quant-ph-0501138-2-55-4']] | [['quant-ph-0501138-1-4-3', 'quant-ph-0501138-2-5-6'], ['quant-ph-0501138-1-59-2', 'quant-ph-0501138-2-22-3']] | ['quant-ph-0501138-1-42-0', 'quant-ph-0501138-2-9-0', 'quant-ph-0501138-2-10-0', 'quant-ph-0501138-2-19-1', 'quant-ph-0501138-2-68-1', 'quant-ph-0501138-2-75-0', 'quant-ph-0501138-3-17-1'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/quant-ph/0501138 | {'quant-ph-0501138-3-0-0': 'The "self-induced decoherence" (SID) approach suggests that (1) the expectation value of any observable becomes diagonal in the eigenstates of the total Hamiltonian for systems endowed with a continuous energy spectrum, and that (2) this process can be interpreted as decoherence.', 'quant-ph-0501138-3-0-1': 'We evaluate the first claim in the context of a simple spin bath model.', 'quant-ph-0501138-3-0-2': 'We find that even for large environments, corresponding to an approximately continuous energy spectrum, diagonalization of the expectation value of random observables does in general not occur.', 'quant-ph-0501138-3-0-3': 'We explain this result and conjecture that SID is likely to fail also in other systems composed of discrete subsystems.', 'quant-ph-0501138-3-0-4': 'Regarding the second claim, we emphasize that SID does not describe a physically meaningful decoherence process for individual measurements, but only involves destructive interference that occurs collectively within an ensemble of presupposed "values" of measurements.', 'quant-ph-0501138-3-0-5': 'This leads us to question the relevance of SID for treating observed decoherence effects.', 'quant-ph-0501138-3-1-0': '# Introduction', 'quant-ph-0501138-3-2-0': 'In a series of papers [CITATION], the authors claim to present a "new approach to decoherence" [CITATION], termed "self-induced decoherence" (SID).', 'quant-ph-0501138-3-2-1': 'Their main assertion is that, for systems endowed with a continuous energy spectrum, the expectation value of an observable will become diagonal in the eigenbasis of the Hamiltonian of the system, and that this effect can be viewed as decoherence.', 'quant-ph-0501138-3-3-0': 'The basic idea underlying SID goes back to well-known arguments in the context of quantum measurement and the theory of irreversible processes [CITATION].', 'quant-ph-0501138-3-3-1': 'It rests on the observation that a superposition of a large number of terms with random phases in the expression for the expectation value of a typical observable, or for the matrix elements of the density operator, leads to destructive interference.', 'quant-ph-0501138-3-3-2': 'The phase differences are either due to a random-phase assumption [CITATION], or, as in SID, are created dynamically through the time evolution factor [MATH] associated with each energy eigenstate in the superposition.', 'quant-ph-0501138-3-3-3': 'These destructive interference effects are then responsible for the diagonalization of the expectation value in the energy eigenbasis as described by SID.', 'quant-ph-0501138-3-4-0': 'However, this process differs strongly from the mechanism of environment-induced decoherence (EID) [CITATION].', 'quant-ph-0501138-3-4-1': 'EID understands decoherence as the practically irreversible dislocalization of local phase relations between environment-selected preferred basis states due to entanglement with an environment.', 'quant-ph-0501138-3-4-2': 'The approximate diagonality of the expectation value of local observables expressed in the preferred basis is only a formal phenomenological consequence of the relative states of the environment becoming rapidly orthogonal during the decoherence process.', 'quant-ph-0501138-3-4-3': 'The fact that SID does not require an explicit environment interacting with the system motivated the term "self-induced" and was suggested [CITATION] to circumvent the question of a proper interpretation of the concept of "observational ignorance of the environment" in EID [CITATION].', 'quant-ph-0501138-3-5-0': 'This paper pursues two main goals.', 'quant-ph-0501138-3-5-1': 'First, after formalizing the basic idea of SID (Sec. [REF]), we shall discuss the question to what extent SID can claim to describe a physically relevant decoherence process (Sec. [REF]).', 'quant-ph-0501138-3-5-2': 'In particular, we will argue that, contrary to the claim of its proponents [CITATION], SID does not constitute a "new viewpoint" on decoherence in the usual definition of EID.', 'quant-ph-0501138-3-5-3': 'Second, we shall study whether diagonalization of the expectation value of random observables in the energy eigenbasis is obtained in the context of an explicit spin bath model (Sec. [REF]).', 'quant-ph-0501138-3-5-4': 'Deliberately, we have chosen a discrete model to investigate the required degree of "quasicontinuity" for SID to work as claimed.', 'quant-ph-0501138-3-5-5': 'To anticipate, we find that even for bath sizes large compared to what is typically considered in EID, no general decay of off-diagonal terms is found, unless both the observable and the initial state of the bath are appropriately restricted.', 'quant-ph-0501138-3-5-6': 'We explain and discuss this result in Sec. [REF], and present our conclusions in Sec. [REF].', 'quant-ph-0501138-3-6-0': '# Self-induced decoherence', 'quant-ph-0501138-3-7-0': 'The basic formalism of SID as developed in Refs. [CITATION] considers an arbitrary observable [EQUATION] expanded in the eigenstates [MATH] of the Hamiltonian [MATH] with continuous spectrum.', 'quant-ph-0501138-3-7-1': 'In the general treatment, only observables with [EQUATION] are considered, where [MATH] and the [MATH] are assumed to be regular functions.', 'quant-ph-0501138-3-7-2': 'The time evolution of the expectation value [MATH] of [MATH] in the pure state [MATH] (setting [MATH]) is then given by [EQUATION] where [MATH].', 'quant-ph-0501138-3-8-0': 'For large [MATH], the phase factor [MATH] fluctuates rapidly with [MATH], which leads to destructive interference in the double integral if the multiplying function [MATH] varies comparably slowly.', 'quant-ph-0501138-3-8-1': 'To formalize this argument, SID employs the Riemann-Lebesgue theorem [CITATION], which prescribes that [EQUATION] if [MATH] is a regular function and [MATH] integrable (i.e., [MATH]).', 'quant-ph-0501138-3-8-2': 'Provided these conditions are satisfied by [MATH], it is concluded that [EQUATION] for large [MATH].', 'quant-ph-0501138-3-8-3': 'Thus, the off-diagonal terms [MATH] have collectively disappeared, which in SID is interpreted as "decoherence in the expectation value."', 'quant-ph-0501138-3-8-4': 'Formally, the SID program introduces a "diagonal-equivalent" density matrix [MATH], [EQUATION] which satisfies [MATH].', 'quant-ph-0501138-3-8-5': 'Note that [MATH] is only a formal equivalent and is not obtained through any dynamical process.', 'quant-ph-0501138-3-8-6': 'Also, expectation values of a nonexhaustive set of observables [see Eq. [REF]] do not uniquely determine the density matrix.', 'quant-ph-0501138-3-8-7': 'Therefore, one must not derive any conclusions about the possibility for certain states of the system from [MATH].', 'quant-ph-0501138-3-9-0': 'To summarize, the main result Eq. [REF] has been obtained from two key assumptions: (1) The energy spectrum of the system is continuous; and (2) the coefficients used in expanding the initial state and the observable in the energy eigenbasis form regular (and integrable) functions of the energy variable.', 'quant-ph-0501138-3-10-0': 'The first requirement of a continuous energy spectrum can be viewed as an implicit inclusion of an internal "environment" with an infinite number of degrees of freedom.', 'quant-ph-0501138-3-10-1': 'However, any realistic physical system is of finite size, and therefore the energy spacing will be discrete.', 'quant-ph-0501138-3-10-2': 'An approximate suppression of off-diagonal terms as given by Eq. [REF] should therefore occur also for quasicontinuous energy spectra, i.e., for small but discrete energy spacings.', 'quant-ph-0501138-3-11-0': 'The regularity assumption (2) is crucial, since it ensures that the phase factors [MATH] are able to lead to the required destructive interference of the expansion coefficients for large times.', 'quant-ph-0501138-3-11-1': 'However, especially in the realistic case of systems of finite size where the expansion coefficients will be a finite set of discrete values, this condition will not hold.', 'quant-ph-0501138-3-11-2': 'It is therefore important to understand the physical meaning and the consequences of a violation of this assumption.', 'quant-ph-0501138-3-12-0': 'Note also that the strict mathematical limit [MATH] employed in the Riemann-Lebesgue theorem, Eq. [REF], is not physically meaningful, and approximate suppression must therefore occur already over finite time scales, as indicated in Eq. [REF].', 'quant-ph-0501138-3-12-1': 'Also, for the realistic case of only quasicontinuous (i.e., essentially discrete) energy spectra, no conclusions about an "irreversibility" of the decay should be derived from the limit [MATH] (as it is done, for example, in Ref. [CITATION]), since the off-diagonal terms will return to their initial values within a finite recurrence time scale.', 'quant-ph-0501138-3-13-0': 'The issues outlined above will be illustrated and investigated in the context of a particular model system in Sec. [REF].', 'quant-ph-0501138-3-14-0': '# Does SID describe decoherence?', 'quant-ph-0501138-3-15-0': 'Despite the fact that SID and EID share the term "decoherence" in their name, we shall demonstrate in this section that their foundations, scope, and physical implications are fundamentally different.', 'quant-ph-0501138-3-15-1': 'Keeping these differences in mind is very important for a proper interpretation of the study of the bath model described in the following Sec. [REF].', 'quant-ph-0501138-3-16-0': 'As already briefly outlined in the Introduction, the standard approach of environmental decoherence [CITATION] describes the consequences of the ubiquitous interaction of any system with its environment.', 'quant-ph-0501138-3-16-1': 'This leads to entanglement between the system and the environment and singles out a preferred basis of the system that is dynamically determined by the Hamiltonian governing the interaction.', 'quant-ph-0501138-3-16-2': 'The relative environmental states associated with these preferred states rapidly approach orthogonality (i.e., macroscopic distinguishability).', 'quant-ph-0501138-3-16-3': 'Phase relations between the preferred states that were initially associated with the system alone are now "dislocalized" into the system-environment combination due to the entanglement, which constitutes the decoherence process.', 'quant-ph-0501138-3-16-4': 'In this sense, interference between the preferred states becomes locally suppressed, i.e., decoherence leads locally to a transition from a superposition to an apparent ("improper" [CITATION]) ensemble.', 'quant-ph-0501138-3-16-5': 'This can be used to define dynamically independent relative local wave-function components that can be related to local quasiclassical properties, thereby mimicking an apparent "collapse" of the wave function [CITATION].', 'quant-ph-0501138-3-17-0': 'The interaction between the system and its environment, often referred to as a "continuous measurement by the environment," is observer independent and can be formulated entirely in terms of wave functions, without reference to presumed (classical) concepts such as "values of observables" and expectation values (see, for example, Chap.', 'quant-ph-0501138-3-17-1': '2 of Ref. [CITATION]).', 'quant-ph-0501138-3-17-2': 'As it has been emphasized frequently [CITATION], the formalism of local ("reduced") density matrices and expectation values presupposes the probabilistic interpretation of the wave function and ultimately relies on the occurence of a "collapse" of the wave function at some stage (or on the description of an observationally equivalent "branching" process in a relative-state framework [CITATION]).', 'quant-ph-0501138-3-17-3': 'The approximate diagonalization of the reduced density matrix [MATH] (describing the probability distribution of outcomes of measurements on the "system [MATH] of interest" immersed into an environment [MATH]) in the environment-selected basis should therefore be considered only as a phenomenological consequence of EID, but not as its essence (see also Ref. [CITATION]).', 'quant-ph-0501138-3-17-4': 'Given an ensemble of results of measurements of a local observable [MATH], the suppression of off-diagonal terms in [MATH] can then be related to the approximate diagonality of the expectation value of [MATH] in the preferred basis, since [MATH].', 'quant-ph-0501138-3-18-0': 'In contrast with EID, SID focuses solely on the derivation of a suppression of off-diagonal terms (in the energy eigenbasis only) in the expectation value of observables pertaining to a single undivided closed system; entanglement through interactions between subsystems plays no role in SID.', 'quant-ph-0501138-3-18-1': 'As indicated earlier, the damping effect is due to destructive interference between a large number of terms with dynamically induced phase differences.', 'quant-ph-0501138-3-18-2': 'Thus it is only the averaging process contained in the concept of expectation values that leads to a disappearance of interference terms.', 'quant-ph-0501138-3-18-3': 'Individually, each term remains present at all times and is not suppressed independently of the other terms.', 'quant-ph-0501138-3-18-4': 'The fact that collectively the off-diagonal terms may lead to a mutual canceling-out must not be misinterpreted as implying that the measurement "outcomes" corresponding to these terms do not occur.', 'quant-ph-0501138-3-18-5': 'Thus SID cannot pertain to the relevant problem of a loss of interference in individual measurements.', 'quant-ph-0501138-3-18-6': 'In view of this argument, the concept of the "diagonal-equivalent density matrix" [MATH], as introduced by the SID program [see Eq. [REF]], is rather misleading, since it gives the incorrect impression of an absence of interference terms [MATH], while the corresponding terms in the expression for the expectation value are individually present at all times.', 'quant-ph-0501138-3-18-7': 'Derivations of a "classical limit" based on [MATH] [CITATION] appear to have overlooked this issue.', 'quant-ph-0501138-3-19-0': 'While SID rests on the concept of expectation values, i.e., of weighted averages over an ensemble of measurement outcomes, it does not explain the physical origin of these outcomes and their ensembles.', 'quant-ph-0501138-3-19-1': 'In contrast with EID, SID does not contain a dynamical account of the measurement process itself that could motivate explanations for how measurement outcomes arise (if only, as in EID, in an "apparent," relative-state sense).', 'quant-ph-0501138-3-19-2': 'Consequently, the assumption of an a priori existence of an ensemble of measurement outcomes, as it is inherent in SID, could be viewed as a particular application of the Copenhagen interpretation.', 'quant-ph-0501138-3-19-3': 'One might then argue that in this case decoherence would not even be necessary in explaining the observed absense of (macrosopic) interference effects.', 'quant-ph-0501138-3-20-0': 'Note that EID makes crucial use of the concept of locality in deriving a loss of interference, since globally the quantum-mechanical superposition remains unchanged, as required by the unitarity of the time evolution of the total wave function.', 'quant-ph-0501138-3-20-1': 'As frequently emphasized by Zeh (e.g., in Refs. [CITATION]) and others (see, for example, Ref. [CITATION]), this locality can be grounded in the (nontrivial) empirical insight that all observers and interactions are intrinsically local.', 'quant-ph-0501138-3-20-2': 'On the other hand, the decomposition into a "system of interest" and an environment that is ignored from an observational point of view, as required in EID, and the resulting implication that the relevance of environmental decoherence is restricted to local subsystems of the total (nonlocal) quantum Universe, has been a subject of ongoing critical discussions (see, for example, Refs. [CITATION]).', 'quant-ph-0501138-3-20-3': 'Furthermore, no general rule is available that would indicate where the split between system and environment is to be placed, a conceptual difficulty admitted also by proponents of EID [CITATION].', 'quant-ph-0501138-3-20-4': 'These issues seem to have motivated the attempt of the SID program to derive decoherence for closed, undivided systems.', 'quant-ph-0501138-3-21-0': 'However, it is important to note that EID has clearly demonstrated that the assumption of the existence of closed system is unrealistic in essentially all cases [CITATION].', 'quant-ph-0501138-3-21-1': 'Enlarging the system by including parts of its environment, as it is implicitly done in SID in order to arrive at a quasicontinuous spectrum, will render the closed-system assumption even less physically viable: The combined system will in turn interact with its surroundings, and the degree of environmental interaction will increase with the number of degrees of freedom in the system.', 'quant-ph-0501138-3-21-2': 'Also, since some interaction with the external measuring device will be required, the assumption of a closed system simply bypasses the question of how the information contained in the ensemble is acquired in the first place.', 'quant-ph-0501138-3-21-3': 'Ultimately, the only truly closed "system" is the Universe in its entirety, and one can therefore question the physical relevance and motivation for a derivation of decoherence for subsystems that are presumed to be closed.', 'quant-ph-0501138-3-22-0': 'Furthermore, a general measurement in SID would pertain also to the environment implicitly contained in the "closed system," posing the question of how this could translate into an experimentally realizable situation.', 'quant-ph-0501138-3-22-1': 'And even if such a measurement can be carried out, its result would usually be of rather little physical interest in the typical situation of observing decoherence for a particular object due to its largely unobserved environment.', 'quant-ph-0501138-3-23-0': 'Finally, in SID, suppression of off-diagonal terms always occurs in the energy eigenbasis, which can therefore be viewed as the universal "preferred basis" in this approach.', 'quant-ph-0501138-3-23-1': 'However, this basis will generally not be useful in accounting for our observation of different preferred bases for the relevant local systems of interest (e.g., spatial localization of macroscopic bodies [CITATION], chirality eigenstates for molecules such as sugar [CITATION], and energy eigenstates in atoms [CITATION]).', 'quant-ph-0501138-3-23-2': 'Furthermore, the energy eigenbasis cannot be used to describe the emergence of time-dependent, quasiclassical properties.', 'quant-ph-0501138-3-24-0': 'In conclusion, not only is the scope of SID more limited than that of EID, but the two approaches also rest on different foundations.', 'quant-ph-0501138-3-24-1': 'The interpretation of the processes described by these theories is fundamentally different, even though phenomenological effects of EID can manifest themselves in a manner formally similiar to that of SID, i.e., as a disappearance of off-diagonal terms in expectation values.', 'quant-ph-0501138-3-24-2': 'Any proposed derivations of an "equivalence" between SID and EID [CITATION] can therefore at most claim to describe coincidental formal similiarities in the context of very particular models, and only if the scope of EID is reduced to the influence on expectation values.', 'quant-ph-0501138-3-24-3': 'On the basis of our arguments, we question the justification for labeling the process referred to by SID as "decoherence."', 'quant-ph-0501138-3-25-0': '# Analysis of the spin bath model', 'quant-ph-0501138-3-26-0': 'By studying an explicit model, we shall now directly investigate the claim of SID, that terms not diagonal in energy in the expectation value of arbitrary observables of the system decay if the system is endowed with a continuous energy spectrum.', 'quant-ph-0501138-3-26-1': 'We shall also illustrate formal and numerical differences in the time evolution of the expectation value of local observables that take into account only the degrees of freedom of the system [MATH] while ignoring the environment [MATH] (the situation encountered in EID), and global observables that pertain to both [MATH] and [MATH] (the case treated by SID).', 'quant-ph-0501138-3-26-2': 'However, in view of our arguments in the preceding Sec. [REF], this should not be misunderstood as a side-by-side comparison of SID and EID.', 'quant-ph-0501138-3-26-3': 'While expectation values may share formal similiarities in both approaches, they also obliterate fundamental differences between SID and EID that lead to very different implications of these expectation values for the question of decoherence.', 'quant-ph-0501138-3-27-0': '## The model and its time evolution', 'quant-ph-0501138-3-28-0': 'The probably most simple exactly solvable model for decoherence was introduced some years ago by Zurek [CITATION].', 'quant-ph-0501138-3-28-1': 'Here, the system [MATH] consists of a spin-1/2 particle (a single qubit) with two possible states [MATH] (representing spin up) and [MATH] (corresponding to spin down), interacting with a collection of [MATH] environmental qubits (described by the states [MATH] and [MATH]) via the total Hamiltonian [EQUATION]', 'quant-ph-0501138-3-28-2': 'Here, the [MATH] are coupling constants, and [MATH] is the identity operator for the [MATH]th environmental qubit.', 'quant-ph-0501138-3-28-3': 'The self-Hamiltonians of [MATH] and [MATH] are taken to be equal to zero.', 'quant-ph-0501138-3-28-4': 'Note that [MATH] has a particularly simple form, since it contains only terms diagonal in the [MATH] and [MATH] bases.', 'quant-ph-0501138-3-29-0': 'It follows that the eigenstates of [MATH] are product states of the form [MATH], etc.', 'quant-ph-0501138-3-29-1': 'A general state [MATH] can then be written as a linear combination of product eigenstates, [EQUATION]', 'quant-ph-0501138-3-29-2': 'This state evolves under the action of [MATH] into [EQUATION] where [EQUATION]', 'quant-ph-0501138-3-29-3': 'The density matrix is [EQUATION] and its part diagonal in energy (i.e., diagonal in the eigenstates [MATH] of [MATH]) is [EQUATION]', 'quant-ph-0501138-3-30-0': '## Expectation values of local observables', 'quant-ph-0501138-3-31-0': 'Focusing, in the spirit of EID, on the system [MATH] alone, we trace out the degrees of freedom of the spin bath in the density operator [MATH].', 'quant-ph-0501138-3-31-1': 'This yields the reduced density operator [EQUATION] where the time dependence of the off-diagonal terms [MATH] and [MATH] is given by the decoherence factor [EQUATION]', 'quant-ph-0501138-3-32-0': 'The expectation value of any local [MATH] observable [EQUATION] is then given by [EQUATION]', 'quant-ph-0501138-3-32-1': 'We can formally rewrite [MATH] as a sum, [EQUATION] where the sum runs over all eigenstates [MATH] of the total Hamiltonian [MATH], with eigenvalues [MATH].', 'quant-ph-0501138-3-33-0': 'A concrete illustration for the time dependence of [MATH], Eq. [REF], for two different bath sizes is shown in Fig. [REF].', 'quant-ph-0501138-3-33-1': 'We see that [MATH] decays quickly by several orders of magnitude and then continues to oscillate about a very small mean value.', 'quant-ph-0501138-3-33-2': 'Thus, for local observables, terms corresponding to interference between the two [MATH] states [MATH] and [MATH] become quickly and strongly suppressed.', 'quant-ph-0501138-3-34-0': '## Expectation values of global observables', 'quant-ph-0501138-3-35-0': 'An arbitrary global observable [MATH] can be written as a linear combination of the form [MATH], where the [MATH] are product eigenstates of the total Hamiltonian [MATH], Eq. [REF].', 'quant-ph-0501138-3-36-0': 'Since [MATH] must be Hermitian, [MATH], [MATH], [MATH], and [MATH] are real numbers, and [MATH], [MATH].', 'quant-ph-0501138-3-36-1': 'To keep the notation simple, we shall omit the sum over [MATH] (and thus the index [MATH]) in the following.', 'quant-ph-0501138-3-37-0': 'The expectation value of [MATH] in the state [MATH], Eq. [REF], is [EQUATION]', 'quant-ph-0501138-3-37-1': 'The special case of the expectation value of local observables, as considered in the preceding Sec. [REF], can easily be recovered by remembering that tracing out the degrees of freedom of [MATH] is equivalent to choosing all coefficients [MATH] and [MATH], which yields [MATH] and [MATH] [see Eq. [REF]], in agreement with Eq. [REF].', 'quant-ph-0501138-3-38-0': 'Suppression of terms in [MATH] that are not diagonal in the energy eigenbasis would be represented by the vanishing of all time-dependent terms in the above expression, i.e., [EQUATION] because we can easily show that [MATH], where [MATH], Eq. [REF], is the part of the density matrix that is diagonal in the eigenstates of the total Hamiltonian.', 'quant-ph-0501138-3-38-1': 'We also see that [MATH], where [EQUATION] is the part of [MATH] diagonal in energy.', 'quant-ph-0501138-3-38-2': 'Thus, as expected, diagonality of [MATH] in energy can also be characterized by the presence of only those product expansion coefficients that are contained in [MATH].', 'quant-ph-0501138-3-39-0': 'The form of the two product terms [MATH] and [MATH] is similar: They only differ in the order of the pairing of the product expansion coefficients with the exponential factors.', 'quant-ph-0501138-3-39-1': 'Also, since the coefficients [MATH] are independent, diagonalization in energy will in general require that individually [MATH] and [MATH] for large [MATH].', 'quant-ph-0501138-3-39-2': 'We can therefore restrict our following analysis to [MATH] alone.', 'quant-ph-0501138-3-39-3': '(We shall also omit the subscript "0" in the following.)', 'quant-ph-0501138-3-40-0': 'First of all, let us rewrite [MATH] as a sum of [MATH] terms, [EQUATION] where the [MATH] represent products of expansion coefficients, [EQUATION]', 'quant-ph-0501138-3-40-1': 'Here the sets [MATH] specify over which indices [MATH] each product runs, namely, they are subsets of the set [MATH] of all integers between 1 and [MATH] such that [MATH] and [MATH].', 'quant-ph-0501138-3-40-2': 'The total energy [MATH] associated with each term in the sum, Eq. [REF], is [EQUATION]', 'quant-ph-0501138-3-40-3': 'We choose the index [MATH] such that [MATH] for all [MATH].', 'quant-ph-0501138-3-40-4': 'Clearly, [MATH] whenever [MATH] [i.e., if [MATH]], canceling out the time dependence of the associated product term in the expression for [MATH].', 'quant-ph-0501138-3-40-5': 'Thus, we can split [MATH] into a time-independent and a time-dependent part, [EQUATION] where now the first sum runs over all [MATH] for which [MATH], while the second sum runs over all [MATH] for which [MATH].', 'quant-ph-0501138-3-41-0': 'Diagonality in energy would require [MATH] as [MATH].', 'quant-ph-0501138-3-41-1': 'Written this way, we see that [MATH] is formally similiar to the function [MATH] derived for local observables, Eq. [REF].', 'quant-ph-0501138-3-41-2': 'This might not come as a surprise, since also the expression for [MATH] can be derived from the calculation of an expectation value of an observable, namely, that of the local observable [MATH] that measures the degree of local interference between the [MATH] states [MATH] and [MATH].', 'quant-ph-0501138-3-41-3': 'However, in the case of [MATH], [MATH] is a product of [MATH] real and non-negative coefficients [MATH] and [MATH], while the [MATH] of Eq. [REF] contain cross terms of the form [MATH] and [MATH], arbitrary real coefficients [MATH] and [MATH], and arbitrary complex coefficients [MATH].', 'quant-ph-0501138-3-42-0': 'We expect this difference to have strong influence on the time evolution of [MATH] vs that of [MATH].', 'quant-ph-0501138-3-42-1': 'The destructive interference needed to obtain suppression of the off-diagonal part of the expectation value relies on the idea that, when a function [MATH] is multiplied by a phase factor [MATH] whose variation with [MATH] is much faster than that of [MATH], neighboring values [MATH] and [MATH] will have similiar magnitude and phases, but will be weighted with two strongly different phase factors, which leads to an averaging-out effect in the sum [MATH].', 'quant-ph-0501138-3-43-0': 'In our case, writing [EQUATION] with [MATH], the phases [MATH] will in general vary very rapidly with [MATH] and, thus, with [MATH].', 'quant-ph-0501138-3-43-1': 'This is a consequence of the fact that the [MATH] are composed of products of coefficients, such that changing a single term in the product will in general result in a drastic change in the overall phase associated with the [MATH].', 'quant-ph-0501138-3-43-2': '(The variation in magnitude among the [MATH] can be expected to be comparably insignificant for larger [MATH].)', 'quant-ph-0501138-3-43-3': 'Such discontinuous phase fluctuations are absent in the formally similiar function [MATH], Eq. [REF], since there only the absolute value of the coefficients [MATH] and [MATH] enters.', 'quant-ph-0501138-3-43-4': 'Note that the impact of the phase fluctuations cannot be diminished by going to larger [MATH], since the [MATH] periodicity of phases implies that the effect of a phase difference between terms [MATH] and [MATH] induced by [MATH] will in average be similiar to that induced by [MATH] for all (larger) values of [MATH].', 'quant-ph-0501138-3-44-0': 'We anticipate the described phase-variation effect to counteract the averaging-out influence of the multiplying phase factor [MATH], and to thus make it more difficult, if not entirely impossible, for [MATH], Eq. [REF], to converge to zero.', 'quant-ph-0501138-3-44-1': 'On the other hand, if the average difference between the phases associated with the individual coefficients is decreased, we would expect that the rate and degree of decay of [MATH] will be improved.', 'quant-ph-0501138-3-45-0': '## Numerical results for the expectation value of random global observables', 'quant-ph-0501138-3-46-0': 'To check this prediction and to generally gain more insight into the behavior of [MATH], Eq. [REF], we studied numerically the time evolution of [MATH], Eq. [REF], normalized by its initial value at [MATH], for sets of random observables [MATH].', 'quant-ph-0501138-3-46-1': 'Diagonalization of [MATH] in energy would then be represented by a decay of [MATH] from its initial value of one.', 'quant-ph-0501138-3-47-0': 'Figure [REF] shows three typical examples for the time evolution of [MATH] for a fixed bath size of [MATH].', 'quant-ph-0501138-3-47-1': 'All couplings [MATH] were taken to be random real numbers between [MATH] and [MATH].', 'quant-ph-0501138-3-47-2': 'To investigate the influence of phase fluctuations of the [MATH], Eq. [REF], we considered three different cases for selecting the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH], i.e., for choosing the initial state of the environment and the observable.', 'quant-ph-0501138-3-47-3': 'In the completely random case (A), the coefficients [MATH], [MATH], and [MATH] were taken to be random complex numbers, with magnitudes and phases drawn from a uniform distribution over the intervals [MATH] and [MATH], respectively (and such that [MATH]).', 'quant-ph-0501138-3-47-4': 'Similiarly, the coefficients [MATH] and [MATH] were random real numbers drawn from a uniform distribution over the interval [MATH].', 'quant-ph-0501138-3-47-5': 'In the second case (B), the initial state of the environment was prepared such that the phases of the [MATH] and [MATH] were restricted to the interval [MATH].', 'quant-ph-0501138-3-47-6': 'Also, only observables with non-negative values of [MATH] and [MATH] were considered, such that sign reversals of [MATH] due to a change of product terms containing these coefficients were prevented.', 'quant-ph-0501138-3-47-7': 'Finally, in the third case (C), only the absolute values of the [MATH], [MATH], [MATH], [MATH], and [MATH] were used, which implies that the [MATH] fluctuated only in magnitude.', 'quant-ph-0501138-3-48-0': 'We observed a drastic influence of the range of phases and signs associated with the individual coefficients [MATH], [MATH], [MATH], [MATH], and [MATH], on the evolution of [MATH].', 'quant-ph-0501138-3-48-1': 'In the special case (C) of all coefficients being real non-negative numbers, [MATH] exhibited a consistently strong and fast decay behavior, similiar to the decay of the function [MATH], Eq. [REF], describing suppression of off-diagonal terms for local observables (see Fig. [REF]).', 'quant-ph-0501138-3-48-2': 'In the intermediate case (B), with restricted phases and signs, the degree of decay of [MATH] was decreased, while the decay rate stayed roughly the same.', 'quant-ph-0501138-3-48-3': 'In the general random case (A), in which no restriction on the spread of phases and on the signs of the coefficients was imposed, the time evolution of [MATH] was observed to be sensitive to the particular set of random numbers used for the coefficients [MATH], [MATH], [MATH], [MATH], and [MATH] in each run.', 'quant-ph-0501138-3-48-4': 'For some of the sets, [MATH] was seen to lack any decay behavior at all.', 'quant-ph-0501138-3-48-5': 'In other cases, the baseline of oscillation was located below zero, indicating a very weak damping effect, albeit with the peaks of the large-amplitude oscillation frequently reaching values greater than zero.', 'quant-ph-0501138-3-49-0': 'These results show that, for the bath size studied here, a consistent occurence of a decay of [MATH] hinges on the phase restrictions imposed on the coefficients describing the observable and the initial state of the environment.', 'quant-ph-0501138-3-49-1': 'If these restrictions are given up, the time evolution of [MATH] and any occurence of a (comparably weak) decay will exhibit strong dependence on the particular set of values chosen for the coefficients.', 'quant-ph-0501138-3-50-0': 'However, it is important to realize that the assumption of a restricted initial state of [MATH] is not only unrealistic, since the environment is typically uncontrollable, but it will also lead to a circular argument when aiming at a derivation of a universal decay effect.', 'quant-ph-0501138-3-50-1': 'This is so because any restriction would require an appropriate preparation of the initial state through a measurement on the entire [MATH], which implies that suppression of off-diagonal terms would then in general be absent for the observable corresponding to this measurement, if the restriction of the initial state of [MATH] is relevant to the occurence of the suppression.', 'quant-ph-0501138-3-50-2': 'Consequently, the [MATH] and [MATH] must be allowed to possess arbitrary phases.', 'quant-ph-0501138-3-50-3': 'Then, since the [MATH], [MATH], and [MATH] are always paired with the [MATH] and [MATH] in the expression for the [MATH] that make up [MATH] [see Eq. [REF]], we anticipate that giving up phase restrictions on the [MATH] and [MATH] will render the restrictions imposed on the [MATH]-coefficients less effective, if not entirely irrelevant, in bringing about a decay of [MATH].', 'quant-ph-0501138-3-51-0': 'To study this prediction, in Fig. [REF] we show a representative plot of [MATH] using only the absolute values of the [MATH] coefficients [MATH], [MATH], and [MATH], but with the [MATH] coefficients [MATH] and [MATH] possessing random phases between 0 and [MATH].', 'quant-ph-0501138-3-51-1': 'We found that decay is either entirely absent or strongly diminished in strength, despite the fact that the strongest possible restriction on the phases and signs of the [MATH] coefficients is imposed.', 'quant-ph-0501138-3-51-2': 'Similiar to the case of completely random coefficients, the behavior of [MATH] was observed to depend crucially on the particular set of random numbers chosen for the coefficients.', 'quant-ph-0501138-3-51-3': 'These results lead us to conclude that a universal decay of off-diagonal terms does not occur for the studied bath size and time scale.', 'quant-ph-0501138-3-52-0': 'To be sure, SID is based on the assumption of a quasicontinuous energy spectrum and very long time scales, corresponding to "sufficiently large" [MATH] and [MATH] (the existing derivations of SID [CITATION] even assume the strict limits [MATH] and [MATH], in order to allow for a direct application of the Riemann-Lebesgue theorem), while so far we have only considered relatively modest values for these parameters.', 'quant-ph-0501138-3-52-1': 'However, since we know from Fig. [REF] that for expectation values of local observables, strong and fast decay of off-diagonal terms is obtained for the value of [MATH] and over the time scale used in the plots shown in Fig. [REF], it is clear that, if a general global disappearance of interference terms is to occur in our model, it will require a much larger number of environmental qubits and/or longer time scales than typically considered for local observables.', 'quant-ph-0501138-3-53-0': 'Accordingly, in Fig. [REF] we show a typical example for the time evolution of [MATH] over the time scale [MATH]-[MATH] for the case of a completely random observable and initial state of [MATH], using comparably large bath sizes [MATH] between [MATH] and [MATH].', 'quant-ph-0501138-3-53-1': 'We observed that even for these values of [MATH], no consistent occurrence of a decay became apparent.', 'quant-ph-0501138-3-53-2': 'In particular, no generally valid direct correlation between the value of [MATH] and the time evolution of [MATH] was visible.', 'quant-ph-0501138-3-53-3': 'Instead, it was again the particular set of random numbers included in the computation of [MATH] for a given value of [MATH] (but not to the size [MATH] of the set itself) that determined whether the baseline of oscillation of [MATH] was located above or below the zero line.', 'quant-ph-0501138-3-53-4': 'In agreement with analytical predictions in the preceding section, we also found that the choice of a longer timescale is irrelevant, since neither the baseline nor the amplitude of oscillation changed significantly over the investigated time interval after a comparably short initial period.', 'quant-ph-0501138-3-53-5': 'Furthermore, we observed that even if [MATH] "decayed" for a particular set of random numbers, the function sustained a large-amplitude oscillation whose peaks often attained values much larger than the initial value of [MATH].', 'quant-ph-0501138-3-54-0': 'Our results show that, in general, for the bath sizes and time scales studied, destructive interference of off-diagonal terms in the expectation value expressed in the energy eigenbasis [as quantified by [MATH], see Eq. [REF]] does not occur in our model.', 'quant-ph-0501138-3-54-1': 'Instead, the time evolution of [MATH] is simply determined by the particular random numbers used to describe the observable and the initial state of the environment.', 'quant-ph-0501138-3-54-2': 'Therefore, no general suppression of interference terms can be inferred.', 'quant-ph-0501138-3-55-0': '# Discussion', 'quant-ph-0501138-3-56-0': 'The process described by SID appears to be neither formally nor conceptually nor physically related to the decoherence mechanism in the standard sense of environmental decoherence.', 'quant-ph-0501138-3-56-1': 'EID accounts for the absence of interference from the perspective of the local (open) system by describing interactions with an environment in quantum-mechanical terms of wave-function entanglement.', 'quant-ph-0501138-3-56-2': 'In contrast, SID describes dynamically induced destructive interference between time-dependent terms in the expression for expectation values.', 'quant-ph-0501138-3-56-3': 'SID does not, however, explain the physical origin of the measurement outcomes and their probability-weighted ensembles needed to define the expectation values.', 'quant-ph-0501138-3-56-4': 'Even if this purely phenomenological basis of SID is accepted, the described process has no bearing on a loss of coherence in individual measurements, since it is only a consequence of averaging over a large number of measurement results.', 'quant-ph-0501138-3-56-5': 'This is in fundamental contrast to EID, where each measurementlike interaction leads to a dislocalization of interference and thus, locally, to a disappearance of interference.', 'quant-ph-0501138-3-57-0': 'The main result of our study of the spin bath model is the finding that the destructive interference predicted by SID will in general fail to occur in our model even for bath sizes and over time scales much larger than typically considered in treatments of the same model in environmental decoherence.', 'quant-ph-0501138-3-57-1': 'The source of this failure lies in the random relative phases associated with the individual initial bath spin states and the expansion coefficients of the observable.', 'quant-ph-0501138-3-57-2': 'The resulting discontinuous phase fluctuations in the coefficient function [MATH], as defined in Eq. [REF], counteract the supposed averaging-out effect of the dynamical phase factors [MATH] in a way that is, due to the [MATH] periodicity of the phase, effectively independent of the value of [MATH].', 'quant-ph-0501138-3-58-0': 'Even when the bath size is increased, the function [MATH] remains a set of discrete values with discontinuously varying phases.', 'quant-ph-0501138-3-58-1': 'This can be explained by noting that, while the total energy is a sum of the energies of each subsystem, such that enlarging the number of contributing subsystems will in general lead to an improved quasicontinuity of the energy spectrum, the [MATH] periodicity of the phases implies that the degree of phase discontinuity of the [MATH] will not be diminished by increasing the number of subsystems.', 'quant-ph-0501138-3-58-2': 'It is therefore unlikely that a consistent decay behavior could become apparent for spin baths much larger than those considered here.', 'quant-ph-0501138-3-59-0': 'This indicates that it is not the degree of continuity of the energy spectrum that represents the determining factor for obtaining destructive interference.', 'quant-ph-0501138-3-59-1': 'Rather, it is the discrete nature of the model itself that seems to lead to difficulties.', 'quant-ph-0501138-3-59-2': 'Only if restrictions are imposed on both the measured observable and the initial state of the environment, a consistent and general suppression of off-diagonal terms can occur.', 'quant-ph-0501138-3-59-3': 'But, as we have argued, the corresponding preparation of the initial state of the environment is physically unrealistic and renders the derivation of a universal decay effect circular.', 'quant-ph-0501138-3-60-0': 'We conjecture that the diagonalization of the expectation value, as described by SID, is likely to fail also in other systems composed of discrete individual subentities.', 'quant-ph-0501138-3-60-1': 'For, in such models, the relevant function will typically be represented by a large product of discrete expansion coefficients, similiar to the [MATH] of our model, whose discontinuous phase fluctuations will again be likely to counteract the averaging-out influence of the dynamical phases.', 'quant-ph-0501138-3-60-2': 'It is therefore clear that the seemingly innocuous mathematical requirement of regularity and integrability of the coefficient functions (see Sec. [REF]) is far from "valid in all relevant cases" where the condition of a sufficiently continuous energy spectrum holds.', 'quant-ph-0501138-3-60-3': 'The suggestion to approximate such discrete functions by a continuous function through interpolation [CITATION] does not appear to be viable, since the interpolated function would describe a physically different situation.', 'quant-ph-0501138-3-61-0': 'On a general note, it is also important to realize that dynamical phases are correlated.', 'quant-ph-0501138-3-61-1': 'Thus one could always construct an observable for which the initial phases of the coefficients seem completely random, but are in fact chosen such that recurrence of coherence will show up within a finite time interval, thus disproving the claimed universality of SID without any further argument.', 'quant-ph-0501138-3-62-0': '# Summary and conclusions', 'quant-ph-0501138-3-63-0': 'We have investigated the two main claims of the "self-induced decoherence" approach, namely, (1) that expectation values of observables pertaining to a closed system become diagonal in the eigenbasis of the Hamiltonian, provided the system is endowed with a continuous energy spectrum; and (2) that this process represents a new way of describing quantum decoherence, and that it leads to results equivalent to the standard approach of environment-induced decoherence.', 'quant-ph-0501138-3-64-0': 'We have evaluated the first claim in the context of a simple spin bath model of finite size by studying, analytically and numerically, the time evolution of expectation values of random global observables.', 'quant-ph-0501138-3-64-1': 'We have found that, in general, collective decay of terms off-diagonal in the energy eigenbasis does not occur over the large range of bath sizes and time scales considered.', 'quant-ph-0501138-3-64-2': 'This result is not due to an insufficient quasicontinuity of the energy spectrum, but is rather rooted in the randomness of the phases associated with the observable and the initial state of the environment.', 'quant-ph-0501138-3-64-3': 'Even in the limit of large bath sizes, the discrete functions for which destructive interference is to be derived do not approach their sufficiently smoothly varying interpolated approximations required for the dynamical phase averaging to have an effect.', 'quant-ph-0501138-3-65-0': 'These results represent an example for a simple model system that, although endowed with a quasicontinuous energy spectrum, fails to exhibit the decay of off-diagonal terms that would be expected from an extrapolation of SID to discrete models in the limit of comparably large sizes of the system.', 'quant-ph-0501138-3-65-1': 'Such an extrapolation should be possible if the approach is to have general physical relevance.', 'quant-ph-0501138-3-65-2': 'We have also anticipated that the decay effect described by SID will likely be absent also in other similiar models that are composed of discrete subsystems.', 'quant-ph-0501138-3-66-0': 'With respect to the second claim of the SID program, we have questioned the suggestion that SID represents a "new viewpoint" [CITATION] on the theory of environment-induced decoherence, since the two approaches are based on conceptually, formally, and physically unrelated mechanisms.', 'quant-ph-0501138-3-66-1': 'In particular, we have pointed out the following key differences and objections.', 'quant-ph-0501138-3-67-0': '(i) SID does not describe the suppression of interference for individual measurements, since interference terms in the expectation value are not damped individually.', 'quant-ph-0501138-3-68-0': '(ii) SID simply presupposes the existence of an ensemble of measurement outcomes, without giving an account of its origin in terms of a physical description of measurement.', 'quant-ph-0501138-3-69-0': '(iii) The assumption of closed systems is unrealistic, especially for systems containing the many degrees of freedom needed to obtain the required quasicontinuous energy spectrum.', 'quant-ph-0501138-3-70-0': '(iv) The physical feasibility and relevance of measurements pertaining to the total system-environment combination is doubtful.', 'quant-ph-0501138-3-71-0': '(v) Energy as the universal preferred basis of the global closed system can usually not account for the different observed preferred bases for the local system of interest.', 'quant-ph-0501138-3-72-0': 'Our study leads us to two main conclusions.', 'quant-ph-0501138-3-72-1': 'First, it points to the need for more precise, physically motivated criteria for the occurrence of the destructive interference effect described by SID.', 'quant-ph-0501138-3-72-2': 'Most importantly, however, the physical interpretation and relevance of this effect need to be explained.', 'quant-ph-0501138-3-72-3': 'We suspect that the SID approach may have mistakenly interpreted and labeled an unrelated process as "decoherence."', 'quant-ph-0501138-3-73-0': 'The author would like to thank A. Fine, M. Castagnino, E. Joos, O. Lombardi, and H. D. Zeh for many thoughtful comments and helpful discussions.'} | null | null | null | null |
1406.0676 | {'1406.0676-1-0-0': 'The evolution and lifetimes of thermally pulsating asymptotic giant branch (TP-AGB) stars suffer from significant uncertainties.', '1406.0676-1-0-1': 'In this work, we analyze the numbers and luminosity functions of TP-AGB stars in six quiescent, low metallicity ([Fe/H] [MATH]) galaxies taken from the ANGST sample, using HST photometry in both optical and near-infrared filters.', '1406.0676-1-0-2': 'The galaxies contain over 1000 TP-AGB stars (at least 60 per field).', '1406.0676-1-0-3': 'We compare the observed TP-AGB luminosity functions and relative numbers of TP-AGB and RGB stars, [MATH], to models generated from different suites of TP-AGB evolutionary tracks after adopting star formation histories (SFH) derived from the HST deep optical observations.', '1406.0676-1-0-4': 'We test various mass-loss prescriptions that differ in their treatments of mass-loss before the onset of dust-driven winds (pre-dust).', '1406.0676-1-0-5': 'These comparisons confirm that pre-dust mass-loss is important, since models that neglect pre-dust mass-loss fail to explain the observed [MATH] ratio or the luminosity functions.', '1406.0676-1-0-6': 'In contrast, models with more efficient pre-dust mass-loss produce results consistent with observations.', '1406.0676-1-0-7': 'We find that for [Fe/H][MATH], lower mass TP-AGB stars ([MATH]M_[MATH]) must have lifetimes of [MATH] Myr and higher masses ([MATH]M_[MATH]) must have lifetimes [MATH] Myr.', '1406.0676-1-0-8': 'In addition, assuming our best-fitting mass-loss prescription, we show that the third dredge up has no significant effect on TP-AGB lifetimes in this mass and metallicity range.', '1406.0676-1-1-0': '# Introduction', '1406.0676-1-2-0': 'Understanding the asymptotic giant branch (AGB) phase of stellar evolution is critical for various aspects of galaxy evolution.', '1406.0676-1-2-1': 'It affects the interpretation of the integrated light of distant galaxies, as well the origin of cosmic dust and the chemical enrichment of the interstellar medium.', '1406.0676-1-2-2': 'Although AGB lifetimes are generally very short (less than a few Myr), their high luminosities may contribute significantly to the integrated spectral energy distribution of galaxies , particularly at redder wavelengths.', '1406.0676-1-3-0': 'Despite its importance, the AGB phase is the most uncertain evolutionary phase of low and intermediate mass ([MATH]M_[MATH]) stars, primarily due its complexity .', '1406.0676-1-3-1': 'AGB stars have two burning shells around an electron-degenerate core.', '1406.0676-1-3-2': 'As they evolve, AGB stars undergo a series of He-shell flashes (called thermal pulses, TP), which cause the base of the convective envelope to deepen within the star.', '1406.0676-1-3-3': 'Depending on the stellar mass and metallicity, the base of the convective envelope may reach past the intershell region, bringing nuclearly-processed material from the interior up to the surface (the so-called third dredge-up), with consequent enrichment in [MATH]He, [MATH]C, [MATH]F, [MATH]Ne, [MATH]Mg, [MATH]Al and other isotopes, plus the heavy elements produced by the slow-neutron capture nucleosynthesis .', '1406.0676-1-3-4': 'Additionally, the most massive and luminous AGB stars (with initial masses [MATH] depending on metallicity) experience a process known as hot-bottom burning (HBB) during the quiescent interpulse periods.', '1406.0676-1-3-5': 'The innermost layers of the convective envelope become hot enough for the hydrogen-burning reactions to be activated, which results in a further modification of the surface chemical composition (mainly via the CNO, NeNa, MgAl cycles) and a sizable steepening of the brightening rate along the AGB .', '1406.0676-1-3-6': 'Eventually, the chemically enriched envelope is expelled into the interstellar medium through stellar winds .', '1406.0676-1-4-0': 'At present, though a general picture of the evolution during the AGB phase is reasonably clear, the quantitative details of all key physical processes at work (third dredge-up, strength and nucleosynthesis of HBB, mass-loss, pulsation, and dust formation) are still out of focus.', '1406.0676-1-4-1': 'The problem can be tackled from various perspectives and with different strategies.', '1406.0676-1-4-2': 'One approach is more linked to the underlying physics, involving detailed analysis of the physical processes and their control parameters (e.g. the dependence of HBB on the convection theory and nuclear network adopted, the role of overshooting for the efficiency of the third dredge-up, the sensitivity of pulsation models to the treatment of the subphotospheric convection, etc.).', '1406.0676-1-4-3': 'The nature of the other approach is more phenomenological, wherein one constrains uncertain parameters through statistical studies that compare observational data of AGB samples with population synthesis simulations.', '1406.0676-1-5-0': 'These two approaches are complementary.', '1406.0676-1-5-1': 'On one side, the results of population synthesis analysis can be extremely useful to the physics-based studies, as they provide the correct analysis for the development of improved theory.', '1406.0676-1-5-2': 'The calibration of the efficiency of the third dredge-up based on the observed luminosity functions of carbon stars is a fitting example .', '1406.0676-1-5-3': 'On the other side, any relevant improvement in the physical description of the AGB should be incorporated among the ingredients within population synthesis studies to enhance their predicting power.', '1406.0676-1-5-4': 'The adoption of variable molecular opacities to correctly describe the Hayashi lines of carbon stars, and hence interpret their position in the observed infrared color-magnitude diagrams, nicely illustrates this point .', '1406.0676-1-6-0': 'As part of the ambitious goal to reach a satisfactory understanding of the AGB phase, in this study we opt for the statistical population synthesis approach.', '1406.0676-1-6-1': 'The major aim is to obtain a calibration of the TP-AGB lifetimes as a function of the initial stellar mass in the low-metallicity regime.', '1406.0676-1-6-2': 'This calibration is important not only for the practical purposes of generating improved stellar tools useful to the community, but also for unveiling the behavior of uncertain processes, such as the mass-loss at low metallicity, outside of the regime probed in Milky Way (MW) and Magellanic Clouds (MCs) studies.', '1406.0676-1-7-0': '## Recent efforts to calibrate TP-AGB lifetimes', '1406.0676-1-8-0': 'The most popular method to constrain TP-AGB lifetimes is through determining the relative number densities of evolutionary phases on a color-magnitude diagram (CMD).', '1406.0676-1-8-1': 'However, small numbers of observed TP-AGB stars can hamper the usefulness of the derived model constraints , particularly in stellar clusters.', '1406.0676-1-9-0': 'Several studies have carried out detailed analysis of individual clusters in the Large and Small Magellanic Clouds (LMC, SMC) and the MW .', '1406.0676-1-9-1': '[CITATION] combined star counts from LMC and SMC clusters in age and metallicity bins to measure TP-AGB lifetimes with reduced uncertainties due to small number statistics.', '1406.0676-1-9-2': 'These studies have proven very useful for TP-AGB model calibrations in the specific ages and metallicities of the MW and MCs cluster populations.', '1406.0676-1-9-3': 'In the intermediate-metallicity regime ([MATH]) [CITATION] calibrated TP-AGB lifetimes based on the observed C-star and M-star luminosity functions in the MC fields and C- and M-star star counts in MC clusters .', '1406.0676-1-10-0': 'Recently, [CITATION] presented a substantial issue regarding the calibration of the lifetimes of AGB stars in MC clusters.', '1406.0676-1-10-1': 'It is known that 1) the lifetimes of core He-burning stars as a function of mass have a discontinuity and sharp increase at and above [MATH]M_[MATH] where stars become able to burn Helium in a non-degenerate core and 2) the lifetimes of AGB stars peak for masses of [MATH]M_[MATH] .', '1406.0676-1-10-2': 'These two effects conspire together to "boost" the number of TP-AGB stars observed in MC clusters of ages [MATH]1.5 Gyr, providing numbers far above those predicted by the fuel consumption theorem .', '1406.0676-1-10-3': 'In other words, the number of TP-AGB stars found on the CMD are not equally proportional to the lifetime of stars in that phase at all ages.', '1406.0676-1-10-4': 'The issue becomes particularly serious given that most of the TP-AGB stars in MC clusters are found in those clusters with ages close to 1.5 Gyr.', '1406.0676-1-11-0': 'Perhaps related to this issue, the derived lifetimes from TP-AGB stars in MC clusters were found to be overestimated when extrapolated to metallicities found in dwarf galaxies .', '1406.0676-1-11-1': 'In other words, TP-AGB models calibrated only with MC observations do not necessarily describe AGB star populations at low metallicities.', '1406.0676-1-11-2': 'Not only must the MC calibration be reassessed, but model constraints are needed at even lower metallicities than the MCs.', '1406.0676-1-12-0': 'In an attempt to extend the TP-AGB calibration to low metallicities, [CITATION] further constrained the [CITATION] TP-AGB models using optical observations of low-metallicity nearby, non-Local Group galaxies.', '1406.0676-1-12-1': 'These samples were from the ACS Nearby Galaxy Survey Treasury and the Archival Nearby Galaxies: Reduce, Reuse, Recycle database.', '1406.0676-1-12-2': 'ANGST is a volume limited sample of [MATH] non-Local Group galaxies to [MATH] Mpc, in which deep optical photometry allowed the measurement of their star formation histories (SFH).', '1406.0676-1-12-3': 'Using a subset of 12 dwarf galaxies from the ANGST sample that showed little to no recent star formation,taliasGirardi2010 compared the luminosity function (LF) of each galaxy with those of simulations using the measured SFHs, and updated the mass-loss prescriptions for low-mass, low-metallicity, TP-AGB stars before the onset of dust driven winds (pre-dust winds).', '1406.0676-1-12-4': 'This correction resulted in good data-model agreement.', '1406.0676-1-12-5': 'Indeed, the revised TP-AGB models reproduced the initial-to-final mass relationship of the Galactic globular cluster M4 .', '1406.0676-1-13-0': 'Using a completely different approach, [CITATION] put constraints on the lifetimes and the core mass growth of intermediate-age TP-AGB stars (with initial masses in the range 1.6 [MATH]) with slightly super-solar initial metallicity ([MATH]), combining recent accurate measurements of white dwarf masses in the Hyades and Praesepe star clusters and new TP-AGB models .', '1406.0676-1-14-0': '## This work in context', '1406.0676-1-15-0': 'The method applied bytaliasGirardi2010 represents a promising approach given the increasing availability of deep photometric data from which the detailed SFHs of nearby galaxies can be derived.', '1406.0676-1-15-1': 'Its range of applicability, however, was initially limited by the optical filter-set available in ANGST.', '1406.0676-1-15-2': 'The reddest filter in the ANGST survey is [MATH], which is more affected by circumstellar dust than redder filters and which also requires higher bolometric corrections for cooler TP-AGB stars .', '1406.0676-1-15-3': 'This limitation allows for a complete AGB sample only for relatively hot ([MATH]T_eff[MATH] K) TP-AGB stars.', '1406.0676-1-15-4': 'Therefore, the method was applied only to a few very metal-poor dwarf galaxies without any sign of recent star formation.', '1406.0676-1-16-0': 'To generate a complete sample of TP-AGB stars in galaxies with more recent star formation and higher metallicities (and hence cooler [MATH]T_eff[MATH]), near-infrared (NIR) filters are necessary.', '1406.0676-1-16-1': 'In a series of two papers, we build upon the method oftaliasGirardi2010 by calculating the LFs of modeled TP-AGB stars using the near-infrared [MATH] "snapshot" follow-up survey to ANGST .', '1406.0676-1-16-2': "This sample includes 26 NIR fields of 23 ANGST galaxies, with 10,000's of TP-AGB stars.", '1406.0676-1-16-3': 'In principle, this filter set allows us to probe TP-AGB masses from 1 to 5 [MATH] with [MATH]T_eff[MATH]K. However, we are still limited by the NIR filterset for which it is not possible to robustly separate C-rich from O-rich AGB stars .', '1406.0676-1-17-0': 'We divided the AGB SNAP sample into galaxies with and without evidence of recent star formation, which effectively limits the masses of the AGB star populations to [MATH]M_[MATH] for the former and includes all possible AGB masses for the latter.', '1406.0676-1-17-1': 'For this paper, we focus on six galaxies, DDO 71, DDO 78, NGC 2976, SCL DE1, HS 117, and KKH 37 shown in optical and NIR CMD and their LFs in Figure [REF].', '1406.0676-1-17-2': 'These galaxies are suitable to be modeled with the isochrones derived from the Padova and TRieste Stellar Evolution Code , which in its present version (v1.1) include masses up to [MATH]M_[MATH] and typical metallicities ranging from [MATH] ([Fe/H][MATH] -2.18 [MATH] -0.60.', '1406.0676-1-17-3': 'Notice that half of these galaxies (DDO 71, DDO 78, and SCL DE1), where already analyzed intaliasGirardi2010 but using the optical data only.', '1406.0676-1-17-4': 'In the next paper, we will extend the galaxy sample to those that show higher amounts of recent star formation with more massive stellar evolution models from PARSEC v2 (Bressan et al. in prep).', '1406.0676-1-18-0': 'The paper is organized as follows.', '1406.0676-1-18-1': 'In Section [REF], we briefly summarize the data reduction and photometry of the AGB-SNAP sub-sample.', '1406.0676-1-18-2': 'Next, we describe the process of recovering the SFH from the deeper optical photometry and associated artificial star tests and our method of isolating TP-AGB and red giant branch (RGB) stars.', '1406.0676-1-18-3': 'We discuss the TP-AGB stellar evolutionary models and the pre-dust mass-loss prescriptions in Section [REF].', '1406.0676-1-18-4': 'Our method of using star counts to robustly model the TP-AGB stars in data is presented in Section [REF].', '1406.0676-1-18-5': 'We compare the observational constraints from the TP-AGB models to those of data in Section [REF], and conclude in Section [REF].', '1406.0676-1-19-0': '# Data', '1406.0676-1-20-0': '## Reduction and Photometry', '1406.0676-1-21-0': 'We now briefly summarize the data reduction and photometry of the AGB-SNAP sample.', '1406.0676-1-21-1': 'For full details, we refer to the ANGST and AGB-SNAP survey papers .', '1406.0676-1-22-0': 'Optical data from the STScI ACS pipeline data were photometered using DOLPHOT2.0 including the ACS module.', '1406.0676-1-22-1': 'Cosmic rays were rejected after combining all images into a single drizzled image using the multidrizzle task within PyRAF .', '1406.0676-1-22-2': 'We use the conservative ANGST photometric catalog (*gst), which only includes objects with DOLPHOT parameters SNR < 4, ((sharp[MATH] + sharp[MATH]), and crowding ((crowd[MATH] + crowd[MATH]) in both filters.', '1406.0676-1-22-3': 'WFPC2 data from ANGRRR and ANGST (following the ACS camera failure) used the WFPC2 pipeline of [CITATION], which processes STScI baseline output through HSTphot, a WFPC2 optimized predecessor of DOLPHOT but updated to July 2008 CTE corrections (derived by A. Dolphin).', '1406.0676-1-22-4': 'To assess the photometric uncertainties and completeness, at least [MATH] artificial star tests were calculated for each galaxy.', '1406.0676-1-23-0': 'The NIR AGB-SNAP data were also reduced using the DOLPHOT package, using a significant update of the WFC3 module (among other enhancements).', '1406.0676-1-23-1': 'We again use the conservative photometry (*gst), in this case with DOLPHOT parameters S/N < 4, ((sharp[MATH] + sharp[MATH]) and ((crowd[MATH] +crowd[MATH]).', '1406.0676-1-24-0': '## Star Formation Histories of the Galaxy Sample', '1406.0676-1-25-0': 'We use the optical CMDs to derive the SFH.', '1406.0676-1-25-1': 'ANGST observations are deep enough to robustly measure the color and magnitude of the red clump (RC) and the shape of the RGB.', '1406.0676-1-25-2': 'Both place strong constraints on the metallicity evolution and past star formation rate .', '1406.0676-1-25-3': 'However, we exclude the AGB from contributing to the SFH fit by masking stars brighter than the TRGB (as was done in G10).', '1406.0676-1-26-0': 'Star formation histories were derived using the CMD-fitting MATCH package .', '1406.0676-1-26-1': 'MATCH finds the most likely SFH and metallicity evolution that fits the observed CMD based on a given IMF, binary fraction, and stellar isochrones while taking into account photometric uncertainties and completeness.', '1406.0676-1-26-2': 'We adopt a [CITATION] IMF, a binary fraction of 0.35, and the PARSEC isochrones .', '1406.0676-1-26-3': 'Photometric uncertainties and completeness were obtained from the artificial star tests described in [CITATION] and [CITATION].', '1406.0676-1-26-4': 'In addition, we allow MATCH to search distance and [MATH] parameter space, in finding the most likely SFH.', '1406.0676-1-27-0': 'Following the discussion in G10, we used MATCH with the zinc flag, which determines the best-fit CMD as a product of the most probable star formation history given an enrichment history in which the metallicity increases with galaxy age .', '1406.0676-1-27-1': 'MATCH derived SFR-weighted metallicities at ages older than 300 Myr range from Z=[MATH] ([Fe/H]=-1.54[MATH]-0.86; see Table [REF]) .', '1406.0676-1-27-2': 'Random uncertainties in the SFH were determined with the Hybrid Monte Carlo tests described in [CITATION].', '1406.0676-1-27-3': 'We use both the SFHs and their uncertainties to produce the model LFs (see Section [REF]).', '1406.0676-1-28-0': 'A summary of the cumulative SFHs of the galaxy sample are shown in Figure [REF].', '1406.0676-1-28-1': 'Derived SFHs all agree within uncertainties to those derived in [CITATION] or for NGC2976 [CITATION] (both studies used [CITATION] isochrones).', '1406.0676-1-28-2': 'In general, the CMD-fitting using PARSEC v1.1 compared to the previous Padova models produce better matches to the shape of the RGB found in the data.', '1406.0676-1-29-0': 'Our method to constrain TP-AGB models is ultimately limited to how well we can model the rest of the stellar populations in the data.', '1406.0676-1-29-1': 'One measure of how well the SFH is recovered from the data is the effective [MATH] .', '1406.0676-1-29-2': 'For the purposes of this study, we state the effective [MATH] values derived from the SFH fitting as a means to compare one SFH fit to the next in Table [REF].', '1406.0676-1-30-0': '## Selecting AGB and RGB Stars', '1406.0676-1-31-0': 'A successful TP-AGB model must be able to reproduce the lifetimes of TP-AGB stars.', '1406.0676-1-31-1': 'One classic observational comparison in stellar evolution modeling is to measure the relative lifetimes of an uncertain stellar evolution phase to a more certain stellar evolutionary phase by taking the ratio of stars found in different regions of a CMD.', '1406.0676-1-31-2': 'Therefore, a first test of the TP-AGB models is to calculate the number ratio of TP-AGB stars to RGB stars, [MATH].', '1406.0676-1-31-3': 'As we identify the TP-AGB and RGB stars, we must minimize the contamination from stars of other phases in the RGB region and the TP-AGB region.', '1406.0676-1-32-0': 'We selected RGB and TP-AGB stars in two regions of the CMDs that were defined to minimize any possible contamination from stars in other evolutionary phases.', '1406.0676-1-32-1': 'Figure [REF] shows example optical and NIR LFs of several evolutionary stages of a simulation massive enough to ensure all phases were well populated and based on the SFH of DDO71.', '1406.0676-1-33-0': 'We exclude any star within a magnitude offset around the TRGB to minimize the number of misclassified TP-AGB and/or RGB stars that scatter up or down in luminosity.', '1406.0676-1-33-1': 'Our simulations show that outside a region around the TRGB of [MATH] mag in [MATH] or [MATH] mag in [MATH] there is never more than 5% contamination of TP-AGB stars in the RGB.', '1406.0676-1-33-2': 'These "excluded regions" are shown in grey in all LF figures with the TRGB also indicated.', '1406.0676-1-34-0': 'With the optical and NIR filters used in our observations, there is no way of completely separating RGB stars from Early AGB (EAGB) stars, Red Helium Burning stars (RHeB), or RC stars.', '1406.0676-1-34-1': 'However, the RHeB and EAGB do not contribute more than [MATH] and [MATH] of the number of stars in the RGB region respectively, a percentage that greatly diminishes with decreasing brightness, based on our population synthesis of the galaxy sample.', '1406.0676-1-34-2': 'If we had included galaxies with more active recent SF, then the contamination from RHeB stars would have been higher.', '1406.0676-1-35-0': 'Unlike the EAGB and RHeB, possible contamination from the RC increases with fainter magnitudes.', '1406.0676-1-35-1': 'RC stars are several magnitudes fainter than the TRGB, and have higher photometric uncertainties, potentially causing RC stars to blend with faint RGB stars.', '1406.0676-1-35-2': 'The possible contamination is only in the optical, since the NIR data is never deep enough to reach the RC.', '1406.0676-1-35-3': 'Regardless, we mitigate this contamination of the RGB by only including stars above the 90 completeness limit in our analysis.', '1406.0676-1-35-4': 'In addition, high completeness assures precise counting of stars in each magnitude bin while being a bright enough limit to avoid stars from the RC (in the optical; the 90 completeness magnitudes for each galaxy in [MATH] and [MATH] are listed in Table [REF]).', '1406.0676-1-35-5': 'RC stars never make up more than 10 of stars in the RGB region included in our analysis.', '1406.0676-1-36-0': 'To exclude main sequence (MS) and blue He-burning (BHeB) stars, we restrict our analysis to stars redder than [MATH], [MATH] and [MATH].', '1406.0676-1-36-1': 'Though the color cuts are easily drawn by eye, they were based on population synthesis combined with photometric uncertainties measured in the data.', '1406.0676-1-36-2': 'We take the color cut as the reddest MS star that is brighter than the 90% completeness limit in a massive simulation calculated with constant SF ([MATH] range from [MATH]) and constant, high metallicity ([MATH] for the reddest MS available in PARSEC).', '1406.0676-1-36-3': 'We then add to the color cut the typical 1[MATH] color uncertainty found in the data (typical uncertainties are shown in Figure [REF].', '1406.0676-1-36-4': 'From our simulations of each galaxy, the analysis regions contain on average 15% MS stars and 1% BHeB stars.', '1406.0676-1-37-0': 'Finally, to robustly compare our models to data, we correct the data for completeness using the same artificial star tests derived for the SFH recovery.', '1406.0676-1-37-1': 'For our analysis, these corrections are minor as our faint limit is already at the 90 completeness.', '1406.0676-1-37-2': 'In effect, due to the completeness corrections, the number of RGB stars increases at most by 10 for optical data and 7 for NIR data.', '1406.0676-1-38-0': 'In summary, the TP-AGB stars are defined as those (optical/NIR) stars (0.1/0.2) magnitudes brighter than the TRGB and the RGB stars are defined as the stars from the 90 completeness limit to (0.1/0.2) magnitudes below the (optical/NIR) TRGB, excluding stars that are bluer in color than 0.2 for [MATH], 0.3 for [MATH], and 0.1 for [MATH].', '1406.0676-1-38-1': 'The numbers of TP-AGB and RGB stars as well as their ratio and Poisson uncertainties are listed in Table [REF].', '1406.0676-1-39-0': 'As eluded to in Section [REF], [MATH] observations can detect cooler TP-AGB stars than those using [MATH] as the latter is more affected by circumstellar dust.', '1406.0676-1-39-1': 'However, with the exception of NGC 2976, we detect fewer TP-AGB stars in NIR than in the optical.', '1406.0676-1-39-2': 'The reason for the discrepancy is simply due to the observations.', '1406.0676-1-39-3': 'While the WFC3/IR fields overlap the ACS/WFC3 fields , the ACS/WFC field of view (FOV; [MATH]) is larger than the WFC3/IR FOV ([MATH]) by a factor of 2.4.', '1406.0676-1-39-4': 'Indeed, the number densities of NIR TP-AGB stars are between 1.9-2.2 times higher than that of the optical TP-AGB stars (excluding NGC 2976 which is 4 times higher).', '1406.0676-1-40-0': 'The ratio of TP-AGB stars to RGB stars ranges from 0.017 to 0.253 and random errors in the measured ratio are never above 16%.', '1406.0676-1-40-1': 'When all galaxies are combined, the measured optical and NIR [MATH] ratio is [MATH] and [MATH], respectively.', '1406.0676-1-40-2': 'Therefore, random errors will have little effect when comparing the measured [MATH] ratio to that predicted by a grid of TP-AGB models.', '1406.0676-1-41-0': 'lcccccc', '1406.0676-1-42-0': 'Observational Data', '1406.0676-1-43-0': 'Target & [MATH] & [MATH] & 2cF814W & 2cF160W', '1406.0676-1-44-0': '& & & 90% Comp.', '1406.0676-1-44-1': '& [MATH] & 90% Comp.', '1406.0676-1-44-2': '& [MATH]', '1406.0676-1-45-0': 'DDO71 & 0.30 & 27.74 & 26.11 & 23.71 & 23.51 & 22.14', '1406.0676-1-46-0': 'HS117 & 0.36 & 27.91 & 24.76 & 23.86 & 23.85 & 22.34', '1406.0676-1-47-0': 'KKH37 & 0.23 & 27.57 & 24.41 & 23.54 & 23.26 & 21.96', '1406.0676-1-48-0': 'NGC2976 & 0.22 & 27.76 & 25.52 & 23.74 & 22.67 & 21.85', '1406.0676-1-49-0': 'DDO78 & 0.07 & 27.82 & 24.76 & 23.80 & 23.63 & 22.05', '1406.0676-1-50-0': 'SCL-DE1 & 0.05 & 28.11 & 25.65 & 24.09 & 24.27 & 22.57 Columns 2, 3, and 5 from [CITATION] and column 7 is from [CITATION], columns 4 and 6 are the 90% completeness magnitudes in [MATH] and [MATH], respectively.', '1406.0676-1-51-0': 'lrrcrrccc', '1406.0676-1-52-0': 'Star Counts and MATCH data', '1406.0676-1-53-0': 'Target & 3cF814W & 3cF160W & 2cResults from MATCH', '1406.0676-1-54-0': '& [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & SFR-weighted [Fe/H] & [MATH] of CMD-fit', '1406.0676-1-55-0': 'DDO71 & 149 & 8739 & [MATH] & 136 & 1730 & [MATH] & -1.13 & 1.48', '1406.0676-1-56-0': 'HS117 & 70 & 498 & [MATH] & 62 & 1003 & [MATH] & -1.20 & 1.07', '1406.0676-1-57-0': 'KKH37 & 135 & 669 & [MATH] & 122 & 923 & [MATH] & -1.38 & 1.14', '1406.0676-1-58-0': 'NGC2976 & 290 & 7695 & [MATH] & 490 & 1933 & [MATH] & -0.86 & 1.55', '1406.0676-1-59-0': 'DDO78 & 273 & 2783 & [MATH] & 215 & 2987 & [MATH] & -0.86 & 1.27', '1406.0676-1-60-0': 'SCL-DE1 & 83 & 1144 & [MATH] & 66 & 867 & [MATH] & -1.54 & 0.89', '1406.0676-1-61-0': 'Total & 1000 & 21528 & [MATH] & 1091 & 9443 & [MATH] & ... & ... The number of stars in the TP-AGB and RGB regions defined in Section [REF] and their ratio in [MATH] (columns 2-4) and [MATH] (columns 5-7).', '1406.0676-1-61-1': 'The final two columns come from the MATCH derived star formation histories, the sfr-weighted averaged metallicity older than 300 Myr (column 8) and the effective [MATH] of the MATCH solution .', '1406.0676-1-62-0': '# TP-AGB evolutionary models', '1406.0676-1-63-0': 'PARSEC is a thoroughly revised version of the popular Padova stellar evolution code used to compute stellar evolution tracks.', '1406.0676-1-63-1': 'We use its v1.1 release , which offers stellar tracks spanning the ranges [MATH], [MATH]M_[MATH]M_[MATH], and from the Pre-MS phase to the beginning of either the TP-AGB or the core Carbon ignition phase.', '1406.0676-1-63-2': 'PARSEC is used in this work both to derive the SFH through the MATCH package (see Section [REF]) as well as the input isochrones to the population synthesis code TRILEGAL (see Section [REF]), which simulates the LF under different AGB model assumptions.', '1406.0676-1-64-0': 'Following the first thermal pulse on the AGB, COLIBRI takes over the stellar evolution calculations from PARSEC.', '1406.0676-1-64-1': 'COLIBRI is an "almost-full" TP-AGB modeling code, that is, it relaxes many of the analytic forms of other synthetic TP-AGB models.', '1406.0676-1-64-2': 'All details are found in [CITATION].', '1406.0676-1-64-3': 'The most relevant to this study include: 1) a complete envelope model from the bottom of the quiescent H-burning shell up to the atmosphere; 2) Rosseland mean opacities calculated on-the-fly with the Opacity Project tools in the high temperature regime for [MATH] K and with the AESOPUS code at lower temperatures for [MATH] K [MATH] K, including the equation of state of [MATH] atomic and molecular species, assuring complete consistency with current chemical abundances; 3) complete nuclear network to follow the nucleosynthesis occurring at the base of the convective envelope (hot-bottom burning; HBB) in more massive AGB stars ([MATH]M_[MATH]) and in the pulse-driven convective zone at thermal pulses; and 4) efficiency of mass-loss treated as a free parameter to be calibrated by observations.', '1406.0676-1-65-0': 'In this paper, we focus on mass-loss and test three prescriptions by how well they reproduce the [MATH] ratio and the shape of the observed LFs.', '1406.0676-1-65-1': 'Although we also wish to constrain the efficiency of the third dredge up, such a test would require measuring the C/O ratio of the stellar population, and the NIR filters available are not red enough to robustly separate C-rich and O-rich stars .', '1406.0676-1-65-2': 'As we show below, the variation of the magnitude of the efficiency of the third dredge up does not change the TP-AGB lifetimes in mass-loss prescriptions with a high [MATH] dependence.', '1406.0676-1-66-0': '## Mass-Loss Prescriptions on the TP-AGB', '1406.0676-1-67-0': 'For a star to have a wind, there must be an outward force that provides momentum and energy input, accelerating the surface layers to velocities larger than the escape velocity.', '1406.0676-1-67-1': 'This may be realized in various ways, including the scattering of UV radiation by resonance line opacity in hot stars, the generation of magneto-acoustic waves above the photosphere in red giants, or the absorption of photons by dust grains in the outer atmospheres of the coolest and most luminous stars .', '1406.0676-1-68-0': "Mass-loss dominates an AGB star's evolution and fate.", '1406.0676-1-68-1': 'It is clear from observations of Mira and OH/IR stars that mass-loss rates increase exponentially along the AGB until they reach super-wind values of [MATH]M_[MATH] yr[MATH] .', '1406.0676-1-68-2': 'Despite the recent progress in the theory of AGB mass-loss , we still lack complete understanding of all the factors and their complex interplay which control the stellar winds on the AGB .', '1406.0676-1-69-0': 'Combining theoretical efforts and empirical evidence, a reasonable scenario takes form in which mass-loss on the AGB can be divided into three regimes: an initial period before the onset of the dust-driven wind (designated as "pre-dust mass-loss"); a subsequent phase characterised by an exponential increase of mass-loss driven by the combined action of dust and pulsation (designated as "dust-driven mass-loss"); and a final brief regime with high mass-loss (designated as "super-wind mass-loss").', '1406.0676-1-70-0': 'In our scheme, the phase of pre-dust mass-loss (with rate [MATH]) is thought to apply to the early stages on the AGB in which either dust has not yet formed in the outermost atmospheric layers, or if present in some small amount, is unable to generate an outflow.', '1406.0676-1-70-1': 'In these conditions a likely wind mechanism could be related to a strong flux of pressure waves or Alfven waves able to cause the spillover of the extended and highly turbulent chromospheres typical of red giants.', '1406.0676-1-70-2': 'The same mechanism might be at work during both the ascent along the RGB and the early stages of the AGB .', '1406.0676-1-71-0': 'In stellar evolutionary calculations a frequent choice to describe mass-loss during the early phases is the classical [CITATION] law, a simple scaling relation of stellar parameters based on observations of few red giants and supergiants.', '1406.0676-1-71-1': 'The Reimers relation is commonly multiplied by an efficiency parameter [MATH], whose value is calibrated such that it recovers the observed morphology of horizontal branch stars in Galactic Globular clusters.', '1406.0676-1-71-2': 'The calibration however, still depends on the residual envelope mass left over from the RGB .', '1406.0676-1-72-0': 'More recently [CITATION] proposed a modified version of the [CITATION] law, in which additional dependencies on the effective temperature and surface gravity follow from a physically-motivated consideration of the mechanical flux responsible for the wind.', '1406.0676-1-72-1': 'The role of the chromosphere in driving mass-loss in late-K to early-M giants is supported by the analysis [CITATION] of the H[MATH] and infrared calcium triplet lines in a sample of red giant stars hosted in Galactic globular clusters.', '1406.0676-1-72-2': 'Similarly to the Reimers relation, the [CITATION] formula also needs an efficiency parameter [MATH] to be specified.', '1406.0676-1-73-0': 'Novel efforts to model stellar winds from red giants were carried out by [CITATION].', '1406.0676-1-73-1': 'A self-consistent and more detailed theoretical approach is developed to follow the generation of energy flux due to magnetohydrodynamic turbulence from subsurface convection zones to its eventual dissipation and escape through the stellar wind.', '1406.0676-1-73-2': 'One major difference is that, while in [CITATION] the mass-loss rate is assumed to scale linearly with the photospheric mechanical energy flux ([MATH]) of Alfven waves ([MATH]), the analysis of [CITATION] yields a higher dependence ([MATH]).', '1406.0676-1-73-3': 'Analytic models for magnetic wave generation indicate that the mechanical energy flux scales as [MATH] .', '1406.0676-1-73-4': 'Hence, considering that the mass-loss rate is proportional to the surface-integrated mechanical energy flux, [MATH], and expressing the stellar radius [MATH] with the Stefan-Boltzmann law for a black body, we eventually obtain a significantly steeper dependence of the mass-loss rate on the effective temperature, i.e. [MATH] for [CITATION] and [MATH] following the results of [CITATION].', '1406.0676-1-74-0': 'Following the pre-dust phase of mass-loss, as the star climbs the AGB at increasing luminosity, suitable conditions can be met in the cool atmosphere for stellar winds to be generated through a different intervening mechanism.', '1406.0676-1-74-1': 'The most plausible hypothesis resides in the momentum input when the stellar radiation field is absorbed (or scattered) by dust grains and transferred to the gas via collisions.', '1406.0676-1-74-2': 'This wind is enhanced by pulsations that shock the envelope and periodically levitate matter up to regions where dust can more efficiently condense .', '1406.0676-1-74-3': 'Observationally there is a clear correlation, though with a large scatter, between the mass-loss rate (here designated with [MATH]) and the pulsation period [MATH] of AGB variables, such that [MATH] is seen to increase exponentially with the period .', '1406.0676-1-75-0': 'Finally, close to tip of the TP-AGB, the mass-loss rates almost level out to [MATH] yr[MATH] of the so-called super-wind phase ([MATH]), corresponding to the condition in which the maximum momentum of the radiation field is transferred to the stellar atmosphere.', '1406.0676-1-76-0': 'Within this framework, the mass-loss prescriptions adopted in the TP-AGB stellar models computed for this study are as follows.', '1406.0676-1-76-1': 'For the dust driven wind phase we adopt a formula similar to [CITATION], which predicts an exponential increase of mass-loss [MATH] dependent on stellar parameters derived from models of periodic shocked atmospheres.', '1406.0676-1-76-2': 'Coefficients [MATH] and [MATH] are calibrated on a sample of Galactic Mira stars.', '1406.0676-1-76-3': 'This prescription is also discussed in [CITATION], [CITATION], and [CITATION].', '1406.0676-1-77-0': 'For the super-wind phase we adopt the formalism of [CITATION], in which the mass-loss rate, [MATH], is proportional to the ratio of the stellar luminosity to the terminal velocity of the gas, which itself scales linearly with the pulsation period.', '1406.0676-1-77-1': 'In practice as soon as [MATH] days the super-wind regime is expected to set in.', '1406.0676-1-78-0': 'We keep the same prescriptions for [MATH] and [MATH] and vary only the [MATH].', '1406.0676-1-78-1': 'For the mass-loss rates [MATH] before the onset of dust-driven winds we consider four options:', '1406.0676-1-79-0': '[[MATH]] no mass-loss before the possible onset of the dust-driven wind, [MATH];', '1406.0676-1-80-0': '[[MATH]] the traditional [CITATION] mass-loss [EQUATION] with the efficiency parameter [MATH];', '1406.0676-1-81-0': '[[MATH]] the original [CITATION] law [EQUATION] with the efficiency parameter [MATH];', '1406.0676-1-82-0': '[[MATH]] a modified version of the [CITATION] scaling relation [EQUATION] in which, for the reasons explained above, the power-law dependence on the effective temperature is steepened; here the efficiency parameter is set to [MATH].', '1406.0676-1-83-0': 'In all formulas the mass-loss rate is given in [MATH], the effective temperature [MATH] is in Kelvin, the stellar radius [MATH], luminosity [MATH], the mass [MATH], and surface gravity [MATH] are expressed in solar units.', '1406.0676-1-83-1': 'Following [CITATION], at each time during the TP-AGB evolution, the current mass-loss rate is taken as [MATH].', '1406.0676-1-84-0': 'Finally, we caution the reader that our modified Schroder Cuntz relation, with [MATH], set for the early stages of the TP-AGB, may be too efficient to be extended to lower luminosities of RGB stars based on a quick comparison to the measured mass-loss rates of the sample, collected by [CITATION], that includes metal poor RGB stars with and effective temperatures in the range 3800 - 5800 K, as a function of the luminosity.', '1406.0676-1-84-1': 'In this context, the role of RGB mass-loss and its possible influence on the subsequent AGB evolution of low-mass stars is postponed to a future work.', '1406.0676-1-85-0': '## Main processes affecting the TP-AGB lifetimes', '1406.0676-1-86-0': 'Many physical processes and events are at work during the TP-AGB phase.', '1406.0676-1-86-1': 'However, there is no doubt that mass-loss is the principal mechanism that controls the duration of this phase, which ends when almost all the stellar mantle is ejected into the interstellar medium.', '1406.0676-1-87-0': 'In this study we opt to analyze the significance of the early stages of AGB mass-loss, since this regime may be particularly important for low-mass stars whose small envelopes may already be removed before the onset of the dust-driven wind (c.f., G10).', '1406.0676-1-87-1': 'This choice seems appropriate given the sample of galaxies under consideration, which are all characterized by a significant fraction of old stellar populations and thus will have TP-AGB populations dominated by lower mass stars (Figure [REF]).', '1406.0676-1-87-2': 'While analysing the impact of different laws for [MATH] and [MATH] is postponed to future works, it is worth mentioning that the prescriptions adopted here have already successfully passed a few observational tests, including the recovery of the expansion velocities of AGB circumstellar envelopes , and the Galactic initial-final mass relation .', '1406.0676-1-88-0': 'It turns out that, indeed, the efficiency of [MATH] plays a major role in determining the lifetimes of TP-AGB stars of lower TP-AGB mass.', '1406.0676-1-88-1': 'In Figure [REF] we compare the four mass-loss options applied to compute the TP-AGB evolution of two stars with initial masses of 1[MATH] and 2[MATH].', '1406.0676-1-88-2': 'The panels are organized from top to bottom following a sequence of progressively more efficient mass-loss.', '1406.0676-1-89-0': 'As expected, the larger the mass-loss rates, the shorter the TP-AGB lifetimes.', '1406.0676-1-89-1': 'From the case of no mass-loss [MATH] to the [MATH] case, the TP-AGB lifetimes are reduced by a factor that depends on the stellar mass, being roughly a factor of 4 for the 1[MATH] star and a factor of 2 for the 2[MATH] star.', '1406.0676-1-90-0': 'The [MATH] case corresponds to the longest duration of the TP-AGB phase, and also to the reddest excursion on the HR diagram, while the TP-AGB tracks computed with [MATH] have the shortest lifetimes and exhibit a smaller displacement towards the coolest [MATH] region.', '1406.0676-1-90-1': 'In general, comparing the panels from top to bottom we obtain a sequence of decreasing lifetimes and cooler HR tracks, a trend which is more pronounced at lower masses.', '1406.0676-1-90-2': 'The differences among models for the predicted TP-AGB lifetimes and effective temperatures become less pronounced at larger masses, as shown by the results for the 2[MATH] star (right panels of Figure [REF]).', '1406.0676-1-91-0': 'The correlation between the TP-AGB lifetime and the [MATH] redward excursion is explained as an effect related to C-star formation (when the surface C/O ratio increases from below to above unity): as more third dredge-up events are allowed to occur during the TP-AGB, a higher C-O excess is built in the atmosphere, leading to a stronger C-bearing molecular opacity, hence to a cooler Hayashi line.', '1406.0676-1-91-1': 'This latter aspect is particularly evident for the 1[MATH] star, which is able to become a carbon star with [MATH] (marked in red in Figure [REF]), whereas it remains oxygen-rich with the other mass-loss prescriptions.', '1406.0676-1-92-0': 'The third dredge-up is expected to affect the TP-AGB lifetimes, essentially due to its impact on the surface chemical composition (mainly in terms of the C/O ratio), which in turn controls both the atmospheric molecular opacity , and the mineralogy of the dust grains that grow in the expanding circumstellar envelope .', '1406.0676-1-92-1': 'Both factors combine to influence the mass-loss rates .', '1406.0676-1-92-2': 'In general, at the transition to the C-star regime, TP-AGB models with variable molecular opacities predict a sudden cooling of the track that makes the mass-loss rates increase (provided the adopted prescription is a sensitive function of [MATH]) with consequent reduction of the lifetimes.', '1406.0676-1-92-3': 'This point is fully discussed in [CITATION].', '1406.0676-1-93-0': 'The TP-AGB lifetime may also be sensitive to changes of the third dredge-up efficiency after the transition C-star phase.', '1406.0676-1-93-1': 'To test this possibility, we computed additional sets of TP-AGB tracks by varying the efficiency of the third dredge-up, while keeping the same [MATH] mass-loss formalism.', '1406.0676-1-93-2': 'Expressing the efficiency of the third dredge-up with the classical parameter [MATH], we adopt the original formalism for [MATH] proposed by [CITATION] ([MATH]), and test the two additional choices of doubling (2 [MATH]) and halving (0.5 [MATH]) the reference efficiency.', '1406.0676-1-93-3': 'We find that the predicted TP-AGB lifetimes barely change (see Fig. [REF], right panel), even when the total amount of dredged-up material varies by a factor of [MATH].', '1406.0676-1-93-4': 'The limited effect of the third dredge-up can be explained as a combination of two main factors.', '1406.0676-1-93-5': 'The first is due to the efficiency of the mass-loss prescription adopted here.', '1406.0676-1-93-6': 'The [MATH] model is quite efficient and the TP-AGB phase terminates quickly (for instance, at [MATH], the total number of thermal pulses is [MATH] in the relevant initial mass range).', '1406.0676-1-93-7': 'Therefore, there simply is not enough time for the third dredge-up to produce dramatic effects, no matter how it is varied.', '1406.0676-1-93-8': 'In addition, as the star reaches the super-wind phase, the mass-loss rates settle to typical values that are little affected by variations of other stellar parameters.', '1406.0676-1-94-0': 'The second factor is related to the sensitivity of the effective temperature as a function of the carbon excess in the atmosphere of carbon stars.', '1406.0676-1-94-1': 'In general, more carbon excess corresponds to lower effective temperature which causes higher rates of mass-loss.', '1406.0676-1-94-2': 'This response of the effective temperature to the increase of carbon can be described by the cooling rate (the derivative [MATH]) which is expected to progressively decrease at increasing C/O ratio .', '1406.0676-1-94-3': 'In other words, after the initial sizeable drop of [MATH] once C/O exceeds unity, as more third dredge-up events continue to take place, the atmospheric structure becomes less and less sensitive to further increase of carbon.', '1406.0676-1-94-4': 'As a consequence, the impact of the third dredge-up on the effective temperature becomes progressively weaker as more mixing events occur.', '1406.0676-1-95-0': 'In short, the duration of the TP-AGB phase is mainly controlled by mass-loss-at least for the mass/metallicity interval being considered here-while the effect of the third dredge-up is limited (see Fig. [REF], right panel).', '1406.0676-1-95-1': 'This result is important, as it strengthens the robustness of the analysis described next, whose primary aim is to obtain a quantitative estimation of the TP-AGB lifetimes as a function of the initial stellar mass, in the low-metallicity regime.', '1406.0676-1-96-0': 'However, we must also emphasize that although the TP-AGB lifetimes are found to be little influenced by the third dredge-up, the chemical composition of the ejecta is much affected by the properties of the mixing events.', '1406.0676-1-96-1': 'In particular, for the same amount of mass lost, the quantity of primary carbon, and hence of carbonaceous dust, that is injected in the interstellar medium does depend strongly on the efficiency of the third dredge-up.', '1406.0676-1-96-2': 'Therefore, although it is outside the scope of this paper, future calibration of the third dredge-up process is an essential step towards a comprehensive description of the TP-AGB phase that includes not only the spectro-photometric but also the chemical role of TP-AGB stars in the context of galaxy evolution.', '1406.0676-1-97-0': 'On a final note, we exclude the original [CITATION] from the subsequent discussion and analysis.', '1406.0676-1-97-1': 'As is shown in Figure [REF], [CITATION] mass-loss rate predicts a longer lifetime than the [MATH] model.', '1406.0676-1-97-2': 'There are two main reasons that the [CITATION] prescription is no longer satisfactory for the present work.', '1406.0676-1-97-3': 'First, the computations discussed here are based on a completely new release of stellar evolution models in which major modification and update of the input physics were introduced.', '1406.0676-1-97-4': 'This leads, for instance, to produce Hayashi lines that are on average somewhat cooler than in G10.', '1406.0676-1-97-5': 'Second, we found that if we vary the mass-loss efficiency ([MATH]) enough to account for the number of AGB stars in our predominantly old and metal-poor galaxies (increasing the efficiency parameter [MATH]), the effect on higher mass AGB stars (i.e. [MATH]) would be dramatic.', '1406.0676-1-97-6': 'In fact, that would lead to an extreme shortening of their lifetimes producing a deficit in contrast with the observations of AGB stars at typical metallities of the SMC and LMC ([MATH]).', '1406.0676-1-97-7': 'Finally, we underline that the modification to the original [CITATION] arises from replacing a simple assumption with a more physically-sound consideration of the dependence of the mass-loss rate on the mechanical magnetic flux .', '1406.0676-1-97-8': 'For these reasons, as discussed above, a new [MATH] scaling was in order, so we proceed with the [MATH] model in lieu of [MATH].', '1406.0676-1-98-0': '# Modeling the Data', '1406.0676-1-99-0': 'With optically derived SFHs of each galaxy and three sets of TP-AGB models, we now turn to our method of using star counts to robustly constrain TP-AGB lifetimes.', '1406.0676-1-99-1': 'We now discuss how we apply the derived SFHs and use TRILEGAL to create model LFs.', '1406.0676-1-100-0': '## Population Synthesis with TRILEGAL', '1406.0676-1-101-0': 'FollowingtaliasGirardi2010, we use TRILEGAL to synthesize the stellar populations for direct comparison to observations.', '1406.0676-1-101-1': 'TRILEGAL takes as input the PARSEC and COLIBRI stellar evolution libraries, a specified initial mass function (IMF), binary fraction, and the SFH.', '1406.0676-1-101-2': 'Importantly, TRILEGAL also simulates the [MATH]-[MATH] variations due to the thermal pulse cycles on the TP-AGB , and the reprocessing of radiation by their circumstellar dust-shells .', '1406.0676-1-101-3': 'For previous evolutionary phases, TRILEGAL provides simulations which are essentially identical to those performed by MATCH.', '1406.0676-1-102-0': 'The TRILEGAL input parameters are set to remain consistent with the parameters used in the SFH recovery (see Section [REF]).', '1406.0676-1-102-1': 'The stars produced by TRILEGAL are converted into absolute magnitudes in [MATH] filters using the set of bolometric corrections and extinction coefficients described in [CITATION], which are mostly based on ATLAS9 synthetic spectra, but with two important updates for cool giants: M giants now come from an extended database from Aringer et al. (in prep.)', '1406.0676-1-102-2': 'that covers the all relevant space of parameters ([MATH], [MATH], and [Fe/H]).', '1406.0676-1-102-3': 'For C-type stars, we adopt the [CITATION] library of C star models, interpolating inside the grids as a function of [MATH], [MATH], [Fe/H] and C/O ratio.', '1406.0676-1-102-4': "Radiation reprocessing by circumstellar dust shells in mass-losing stars are taken into account as in [CITATION], using the results of [CITATION]'s radiation transfer models for mixtures of 60% AlOx and 40% silicate, and of 85% amorphous Carbon and 15% silicon carbide (for M and C stars, respectively).", '1406.0676-1-102-5': 'Finally, the synthetic CMDs are corrected for distance and extinction, [MATH] using extinction coefficients from [CITATION], as listed in Table [REF].', '1406.0676-1-103-0': '## Accounting for Uncertainties in SFH', '1406.0676-1-104-0': 'To obtain a robust number ratio of TP-AGB stars to RGB stars as well as the range of probable LFs expected from a given TP-AGB model, one must account for the random uncertainties in SFH.', '1406.0676-1-104-1': 'We synthesize at least 50 stellar populations with SFHs that are randomly sampled within the uncertainties of the best fit SFH.', '1406.0676-1-104-2': 'Specifically, for each SFH sample, we take the value of SF in each time bin as a random draw of a Gaussian distribution whose mean is the best fit SFR in that time bin and whose [MATH] is the uncertainty associated with that time bin.', '1406.0676-1-104-3': 'If the SFR is zero in the time bin, we adopt only positive uncertainties.', '1406.0676-1-105-0': 'Figure [REF] shows an example of 50 SFH realizations based on the MATCH-derived SFH and hybrid Monte Carlo uncertainties .', '1406.0676-1-105-1': 'The effect of the randomly sampled SFH can be seen in the Figures [REF]-[REF] as a spread in LF at bright magnitudes.', '1406.0676-1-105-2': 'Accounting for uncertainties in the SFH shows a clearer picture of the model predictions on the LF by introducing a spread in the number of stars that are expected to be found in each magnitude bin on a LF.', '1406.0676-1-105-3': 'The effect on the mean [MATH] ratio is to consistently produce a standard deviation of [MATH], independent of the TP-AGB model.', '1406.0676-1-106-0': '## Creating LFs of the Galaxy Sample', '1406.0676-1-107-0': 'For each SFH of each galaxy and each TP-AGB model, we generate a model stellar population with TRILEGAL of sufficient size to both to completely sample the IMF and to have at least twice the number of RGB stars in the sample as there are in the data.', '1406.0676-1-107-1': 'We then correct for the discrepancy in total stellar mass by scaling the model LF by the number of stars in the RGB region of the data (defined in Section [REF]).', '1406.0676-1-107-2': 'In other words, we multiply the simulated LF by a factor [MATH], such that, [EQUATION]', '1406.0676-1-107-3': 'Example LFs of an un-scaled simulation are shown in Figure [REF], all simulated LFs in the figures following are scaled.', '1406.0676-1-108-0': 'To test the adequacy of the TP-AGB model, we compare the amplitude and the shape of the model LF to the observations.', '1406.0676-1-108-1': 'The number of stars in the model LF are compared to that in the data by calculating the [MATH] ratio as described in Section [REF].', '1406.0676-1-108-2': 'The [MATH] ratio is related to the average TP-AGB lifetime of the observed population, as it combines all the star counts into one data point.', '1406.0676-1-109-0': 'The [MATH] ratio is a useful first comparison to make between models, as well as comparisons to other studies (e.g., G10).', '1406.0676-1-109-1': 'A successful TP-AGB model must also match the shape of the observed LF.', '1406.0676-1-109-2': 'Therefore, we also compare the predicted LF with those of the observations.', '1406.0676-1-110-0': 'We calculate the Poisson-equivalent of the Gaussian [MATH] statistic in two regions of the LF to compare the model LF shape to that in the data.', '1406.0676-1-110-1': 'The first region is the "full LF" that is, from the 90% completeness magnitude and brighter (including the regions excluded from the [MATH] ratio calculation).', '1406.0676-1-110-2': 'The second region is only the TP-AGB region, defined as brighter than a small offset above the TRGB (0.1 mag in [MATH] and 0.2 mag in [MATH]; see Section [REF]).', '1406.0676-1-111-0': '# Analysis', '1406.0676-1-112-0': '## Ratio of TP-AGB to RGB stars', '1406.0676-1-113-0': 'Tables [REF] and [REF] list the mean [MATH] and standard deviation of their Poisson uncertainties calculated from 50 simulations for each galaxy for each TP-AGB model (columns 2, 4, and 6).', '1406.0676-1-113-1': 'Next to each mean [MATH] ratio are the fractional difference between the mean model ratio and that found in the data (columns 3, 5, and 7).', '1406.0676-1-113-2': 'We define the fractional difference to be [MATH]N_TP-AGB/N_RGB[MATH]N_TP-AGB/N_RGB[MATH].', '1406.0676-1-113-3': 'Therefore, a value of [MATH] would be perfect agreement between data and model while [MATH] would mean the model is overpredicting the number, and thus, the lifetimes of TP-AGB stars.', '1406.0676-1-114-0': 'For each individual galaxy, the [MATH] and [MATH] models consistently overpredict the [MATH] ratio.', '1406.0676-1-114-1': 'On average, the [MATH] mass-loss prescription overpredicts the number of optical TP-AGB stars by nearly a factor of 3 and the [MATH] mass-loss law overpredicts them by more than a factor of 2.', '1406.0676-1-114-2': "In the NIR, the overpredictions of TP-AGB stars are similar or higher, nearly a factor of 3 when not accounting for pre-dust mass-loss, and a factor of 2.5 when only including Reimers' mass-loss.", '1406.0676-1-114-3': 'The [MATH] model, however, is on average consistent or slightly lower within uncertainties to the observed [MATH] ratio in both filters.', '1406.0676-1-115-0': "The conclusion from comparing the [MATH] ratio from model to model, is that there are too many TP-AGB stars predicted by the mass-loss prescriptions that neglect pre-dust mass-loss, or assume only Reimers' relation.", '1406.0676-1-115-1': 'In contrast, pre-dust mass-loss in the [MATH] model is most consistent with the average lifetimes of observed TP-AGB stars.', '1406.0676-1-116-0': '## Luminosity Functions in the Optical and NIR', '1406.0676-1-117-0': 'Figures [REF], [REF], and [REF] show a set of panels with optical (left) and IR (right) LFs for each galaxy in the sample.', '1406.0676-1-117-1': 'Observations (corrected for completeness; red) are shown with Poisson uncertainties for each of the 50 model LFs per panel overplotted (grey).', '1406.0676-1-117-2': 'The regions around the TRGB which were excluded in calculating the [MATH] ratio are shaded (see Section [REF]).', '1406.0676-1-117-3': 'The fainter magnitudes between 90%-50% completeness are also shaded; they are never included in the analysis, though the RGB and RC are important constraints used to derive the SFH and best fit metallicity enrichment law.', '1406.0676-1-118-0': 'To propagate the completeness and photometric uncertainties from the observations to the model LFs, we make use of the uncertainties reported by MATCH in the derived SFHs (which include uncertainties from the artificial star tests).', '1406.0676-1-118-1': 'An example set of SFHs is shown in Figure [REF].', '1406.0676-1-119-0': 'Qualitatively, the model and data LF agree in the RGB region (between the shaded regions) and diverge at fainter magnitudes.', '1406.0676-1-119-1': 'The agreement around the RGB is by design, as discussed in Section [REF], because the model LFs are scaled to match the total number of observed RGB stars in this region.', '1406.0676-1-119-2': 'However, there are two observational sources that give rise to the disagreement between data and scaled models fainter than [MATH] completeness.', '1406.0676-1-119-3': 'The first observational source is decreasing completeness with increasing magnitude, and the second cause is due to photometric uncertainties (median uncertainties in each CMD are shown in each panel of Figure [REF]).', '1406.0676-1-120-0': 'In general, the model LFs follow the results in Section [REF] for the [MATH].', '1406.0676-1-120-1': 'The [MATH] and [MATH] LFs predict far more TP-AGB stars than are observed in each magnitude bin (the only exception is for bright TP-AGB stars in NGC2976 in the NIR).', '1406.0676-1-120-2': 'The [MATH] model however, shows excellent agreement in the TP-AGB region of LF compared to the observed LF in many cases.', '1406.0676-1-121-0': 'To robustly compare one TP-AGB model to another, we calculate the Poission-like [MATH] likelihood that the observational LF is randomly drawn from the model LF.', '1406.0676-1-121-1': 'Figure [REF] shows the results of this calculation for only the TP-AGB region compared to the same region in the data (top panels) and the LF brighter than the 90% completeness limit (bottom panels) for the optical (left panels) and the NIR (right panels).', '1406.0676-1-121-2': 'Each point of Figure [REF] represents the mean value of the [MATH] calculated individually for each of the 50 model LFs, with uncertainties corresponding to the standard deviation of the mean.', '1406.0676-1-122-0': 'The absolute placement of the [MATH] values for each galaxy are likely dominated by the goodness of the SFH recovery mentioned in Section [REF].', '1406.0676-1-122-1': 'For example, NGC 2976 has the largest effective [MATH], corresponding to the least-good SFH fit, while SCL-DE1 is the opposite.', '1406.0676-1-122-2': 'The same trend is seen in Figure [REF] but using TRILEGAL simulated LFs.', '1406.0676-1-122-3': 'Therefore, it is more meaningful to consider the relative [MATH] values in Figure [REF] to compare TP-AGB models.', '1406.0676-1-122-4': 'The [MATH] values follow the qualitative picture from the LF agreement described above.', '1406.0676-1-122-5': 'The [MATH] mass-loss produces the most consistent LF compared to that observed.', '1406.0676-1-123-0': 'With the success in the [MATH] mass-loss prescription we can now constrain the expected lifetimes of low metallicity, low mass TP-AGB stars.', '1406.0676-1-123-1': 'Figure [REF] shows the corresponding TP-AGB lifetimes for low and intermediate metallicities, for the [MATH] model.', '1406.0676-1-123-2': 'The left panel shows the entire lifetime of the TP-AGB, including the amount of time spent below the TRGB.', '1406.0676-1-123-3': 'The right panel shows the expected lifetimes of TP-AGB stars above the TRGB.', '1406.0676-1-123-4': 'Figure [REF] shows the distribution of TP-AGB masses from each of the best fitting LFs for each galaxy.', '1406.0676-1-123-5': 'Therefore, based on the number of observed TP-AGB stars in six galaxies that are low metallicity and have little recent SFH, the lifetimes of the typical TP-AGB star in our sample ([MATH]M_[MATH] or [MATH]) will be less than 1.2 Myr.', '1406.0676-1-123-6': 'For stars of mass [MATH]M_[MATH], we expect the TP-AGB lifetime to be less than half a Myr, as the star will expel much of its atmosphere during the pre-dust driven phase of the TP-AGB.', '1406.0676-1-124-0': '## The Initial and Final Mass Relationship', '1406.0676-1-125-0': 'As mentioned in G10, one of the clearest constraints on the evolution of low-mass low-metallicity TP-AGB stars is given by the (few) measured masses of white dwarfs (WDs) in globular clusters, and in particular those in M4, for which [CITATION] derived a mean mass of [MATH]M_[MATH].', '1406.0676-1-125-1': 'Assuming that M4 has a [Fe/H]=-1.07 and [[MATH]/Fe]=0.39 dex , it should be well represented by PARSEC tracks of [MATH].', '1406.0676-1-125-2': 'For this metallicity, the track which lifetime best fits the 12 Gyr age expected for globular clusters is the one with an initial mass of 0.85 [MATH], which takes 11.7 Gyr to evolve from the zero-age MS to the WD cooling ages of the observed M4 white dwarfs .', '1406.0676-1-125-3': 'This track finishes the TP-AGB with a remnant mass of 0.547 [MATH], which is just slightly larger than the mean value determined by [CITATION].', '1406.0676-1-125-4': 'The small difference in final mass can be easily explained by invoking a small additional amount of mass-loss on the RGB.', '1406.0676-1-125-5': 'We emphasize however that calibrating the RGB mass-loss to a precision of a few hundredths of solar masses, is certainly beyond the scope of this paper.', '1406.0676-1-126-0': 'lrrrrrrr', '1406.0676-1-127-0': 'Mean Optical [MATH] Ratios', '1406.0676-1-128-0': 'Target & [MATH] & Frac.', '1406.0676-1-128-1': 'Difference & [MATH] & Frac.', '1406.0676-1-128-2': 'Difference & [MATH] & Frac.', '1406.0676-1-128-3': 'Difference', '1406.0676-1-129-0': 'DDO71 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-130-0': 'DDO78 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-131-0': 'HS117 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-132-0': 'KKH37 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-133-0': 'NGC2976 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-134-0': 'SCL-DE1 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-135-0': 'Mean & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] Mean optical TP-AGB to RGB model ratios and their fractional differences compared to the data.', '1406.0676-1-135-1': 'For each galaxy for each TP-AGB model, 50 TRILEGAL simulations produced a model stellar population from the best fit SFH and its uncertainties (see Section [REF]).', '1406.0676-1-135-2': 'The only change from one set of simulations to the other is the TP-AGB model.', '1406.0676-1-135-3': 'The total fractional differences are calculated compared to the total [MATH] ratio in the data listed in Table [REF].', '1406.0676-1-136-0': 'lrrrrrrr', '1406.0676-1-137-0': 'Mean NIR [MATH] Ratios', '1406.0676-1-138-0': 'Target & [MATH] & Frac.', '1406.0676-1-138-1': 'Difference & [MATH] & Frac.', '1406.0676-1-138-2': 'Difference & [MATH] & Frac.', '1406.0676-1-138-3': 'Difference', '1406.0676-1-139-0': 'DDO71 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-140-0': 'DDO78 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-141-0': 'HS117 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-142-0': 'KKH37 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-143-0': 'NGC2976 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-144-0': 'SCL-DE1 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-1-145-0': 'Mean & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] Same as Table [REF] for WFC3/IR data.', '1406.0676-1-146-0': '# Conclusions', '1406.0676-1-147-0': 'We have extended the analysis from G10, and confirmed that pre-dust mass-loss plays an important role in the TP-AGB evolution of low-mass metal-poor stars.', '1406.0676-1-147-1': 'We have shown that neglecting this phase of mass-loss altogether will overpredict the numbers of TP-AGB stars by a factor of [MATH].', '1406.0676-1-147-2': "We also showed that assuming Reimers' scaling relation to describe the pre-dust mass-loss phase will also overpredict the number to TP-AGB stars but by a factor of [MATH].", '1406.0676-1-147-3': 'Following recent results of detailed models that describe the chromospheric winds of red giants, we implemented a revised mass-loss prescription with a stronger dependence on the effective temperature.', '1406.0676-1-147-4': 'Using the [MATH] ratio and comparing LFs, we showed this implementation reaches good to excellent agreement with the data.', '1406.0676-1-147-5': 'Moreover, the [MATH] mass-loss prescription further lowers the expected TP-AGB lifetime for low-mass, low-metallicity TP-AGB stars.', '1406.0676-1-148-0': 'An interesting outcome of the our analysis is the dependence on the initial chemical composition: TP-AGB lifetimes of low-mass stars are expected to become shorter at decreasing metallicity.', '1406.0676-1-148-1': 'We also show that, given our calibrated mass-loss relation, the efficiency of the third dredge-up has little effect on TP-AGB lifetimes in this regime.', '1406.0676-1-149-0': 'This paper represents the first step in a major calibration of the COLIBRI and PARSEC codes which will aid in the understanding of the physics involved in TP-AGB evolution.', '1406.0676-1-149-1': 'Upon the availability of PARSEC v2, which will include higher mass stellar evolution tracks, this study will be extended to include the 17 other galaxies in the AGB-SNAP sample.', '1406.0676-1-149-2': 'This addition will expand the metallicities available to test and increase the mass range of our sample to the complete sample of TP-AGB stars.', '1406.0676-1-150-0': 'These improvements to the TP-AGB models will be included in sets of isochrones and tools to synthesize stellar populations available on the CMD and TRILEGAL websites.'} | {'1406.0676-2-0-0': 'The evolution and lifetimes of thermally pulsating asymptotic giant branch (TP-AGB) stars suffer from significant uncertainties.', '1406.0676-2-0-1': 'In this work, we analyze the numbers and luminosity functions of TP-AGB stars in six quiescent, low metallicity ([Fe/H] [MATH]) galaxies taken from the ANGST sample, using HST photometry in both optical and near-infrared filters.', '1406.0676-2-0-2': 'The galaxies contain over 1000 TP-AGB stars (at least 60 per field).', '1406.0676-2-0-3': 'We compare the observed TP-AGB luminosity functions and relative numbers of TP-AGB and RGB stars, [MATH], to models generated from different suites of TP-AGB evolutionary tracks after adopting star formation histories (SFH) derived from the HST deep optical observations.', '1406.0676-2-0-4': 'We test various mass-loss prescriptions that differ in their treatments of mass-loss before the onset of dust-driven winds (pre-dust).', '1406.0676-2-0-5': 'These comparisons confirm that pre-dust mass-loss is important, since models that neglect pre-dust mass-loss fail to explain the observed [MATH] ratio or the luminosity functions.', '1406.0676-2-0-6': 'In contrast, models with more efficient pre-dust mass-loss produce results consistent with observations.', '1406.0676-2-0-7': 'We find that for [Fe/H][MATH], lower mass TP-AGB stars ([MATH]M_[MATH]) must have lifetimes of [MATH] Myr and higher masses ([MATH]M_[MATH]) must have lifetimes [MATH] Myr.', '1406.0676-2-0-8': 'In addition, assuming our best-fitting mass-loss prescription, we show that the third dredge up has no significant effect on TP-AGB lifetimes in this mass and metallicity range.', '1406.0676-2-1-0': '# Introduction', '1406.0676-2-2-0': 'Understanding the asymptotic giant branch (AGB) phase of stellar evolution is critical for various aspects of galaxy evolution.', '1406.0676-2-2-1': 'It affects the interpretation of the integrated light of distant galaxies, as well the origin of cosmic dust and the chemical enrichment of the interstellar medium.', '1406.0676-2-2-2': 'Although AGB lifetimes are generally very short (less than a few Myr), their high luminosities may contribute significantly to the integrated spectral energy distribution of galaxies , particularly at redder wavelengths.', '1406.0676-2-3-0': 'Despite its importance, the AGB phase is the most uncertain evolutionary phase of low and intermediate mass ([MATH]M_[MATH]) stars, primarily due its complexity .', '1406.0676-2-3-1': 'AGB stars have two burning shells around an electron-degenerate core.', '1406.0676-2-3-2': 'As they evolve, AGB stars undergo a series of He-shell flashes (called thermal pulses, TP), which cause the base of the convective envelope to deepen within the star.', '1406.0676-2-3-3': 'Depending on the stellar mass and metallicity, the base of the convective envelope may reach past the intershell region, bringing nuclearly-processed material from the interior up to the surface (the so-called third dredge-up), with consequent enrichment in [MATH]He, [MATH]C, [MATH]F, [MATH]Ne, [MATH]Mg, [MATH]Al and other isotopes, plus the heavy elements produced by the slow-neutron capture nucleosynthesis .', '1406.0676-2-3-4': 'Additionally, the most massive and luminous AGB stars (with initial masses [MATH] depending on metallicity) experience a process known as hot-bottom burning (HBB) during the quiescent interpulse periods.', '1406.0676-2-3-5': 'The innermost layers of the convective envelope become hot enough for the hydrogen-burning reactions to be activated, which results in a further modification of the surface chemical composition (mainly via the CNO, NeNa, MgAl cycles) and a sizable steepening of the brightening rate along the AGB .', '1406.0676-2-3-6': 'Eventually, the chemically enriched envelope is expelled into the interstellar medium through stellar winds .', '1406.0676-2-4-0': 'At present, though a general picture of the evolution during the AGB phase is reasonably clear, the quantitative details of all key physical processes at work (third dredge-up, strength and nucleosynthesis of HBB, mass-loss, pulsation, and dust formation) are still out of focus.', '1406.0676-2-4-1': 'The problem can be tackled from various perspectives and with different strategies.', '1406.0676-2-4-2': 'One approach is more linked to the underlying physics, involving detailed analysis of the physical processes and their control parameters (e.g. the dependence of HBB on the convection theory and nuclear network adopted, the role of overshooting for the efficiency of the third dredge-up, the sensitivity of pulsation models to the treatment of the subphotospheric convection, etc.).', '1406.0676-2-4-3': 'The nature of the other approach is more phenomenological, wherein one constrains uncertain parameters through statistical studies that compare observational data of AGB samples with population synthesis simulations.', '1406.0676-2-5-0': 'These two approaches are complementary.', '1406.0676-2-5-1': 'On one side, the results of population synthesis analysis can be extremely useful to the physics-based studies, as they provide the correct analysis for the development of improved theory.', '1406.0676-2-5-2': 'The calibration of the efficiency of the third dredge-up based on the observed luminosity functions of carbon stars is a fitting example .', '1406.0676-2-5-3': 'On the other side, any relevant improvement in the physical description of the AGB should be incorporated among the ingredients within population synthesis studies to enhance their predicting power.', '1406.0676-2-5-4': 'The adoption of variable molecular opacities to correctly describe the Hayashi lines of carbon stars, and hence interpret their position in the observed infrared color-magnitude diagrams, nicely illustrates this point .', '1406.0676-2-6-0': 'As part of the ambitious goal to reach a satisfactory understanding of the AGB phase, in this study we opt for the statistical population synthesis approach.', '1406.0676-2-6-1': 'The major aim is to obtain a calibration of the TP-AGB lifetimes as a function of the initial stellar mass in the low-metallicity regime.', '1406.0676-2-6-2': 'This calibration is important not only for the practical purposes of generating improved stellar tools useful to the community, but also for unveiling the behavior of uncertain processes, such as the mass-loss at low metallicity, outside of the regime probed in Milky Way (MW) and Magellanic Clouds (MCs) studies.', '1406.0676-2-7-0': '## Recent efforts to calibrate TP-AGB lifetimes', '1406.0676-2-8-0': 'The most popular method to constrain TP-AGB lifetimes is through determining the relative number densities of evolutionary phases on a color-magnitude diagram (CMD).', '1406.0676-2-8-1': 'However, small numbers of observed TP-AGB stars can hamper the usefulness of the derived model constraints , particularly in stellar clusters.', '1406.0676-2-9-0': 'Several studies have carried out detailed analysis of individual clusters in the Large and Small Magellanic Clouds (LMC, SMC) and the MW .', '1406.0676-2-9-1': '[CITATION] combined star counts from LMC and SMC clusters in age and metallicity bins to measure TP-AGB lifetimes with reduced uncertainties due to small number statistics.', '1406.0676-2-9-2': 'These studies have proven very useful for TP-AGB model calibrations in the specific ages and metallicities of the MW and MCs cluster populations.', '1406.0676-2-9-3': 'In the intermediate-metallicity regime ([MATH]) [CITATION] calibrated TP-AGB lifetimes based on the observed C-star and M-star luminosity functions in the MC fields and C- and M-star star counts in MC clusters .', '1406.0676-2-10-0': 'Recently, [CITATION] presented a substantial issue regarding the calibration of the lifetimes of AGB stars in MC clusters.', '1406.0676-2-10-1': 'It is known that 1) the lifetimes of core He-burning stars as a function of mass have a discontinuity and sharp increase at and above [MATH]M_[MATH] where stars become able to burn Helium in a non-degenerate core and 2) the lifetimes of AGB stars peak for masses of [MATH]M_[MATH] .', '1406.0676-2-10-2': 'These two effects conspire together to "boost" the number of TP-AGB stars observed in MC clusters of ages [MATH]1.5 Gyr, providing numbers far above those predicted by the fuel consumption theorem .', '1406.0676-2-10-3': 'In other words, the number of TP-AGB stars found on the CMD are not equally proportional to the lifetime of stars in that phase at all ages.', '1406.0676-2-10-4': 'The issue becomes particularly serious given that most of the TP-AGB stars in MC clusters are found in those clusters with ages close to 1.5 Gyr.', '1406.0676-2-11-0': 'Perhaps related to this issue, the derived lifetimes from TP-AGB stars in MC clusters were found to be overestimated when extrapolated to metallicities found in dwarf galaxies .', '1406.0676-2-11-1': 'In other words, TP-AGB models calibrated only with MC observations do not necessarily describe AGB star populations at low metallicities.', '1406.0676-2-11-2': 'Not only must the MC calibration be reassessed, but model constraints are needed at even lower metallicities than the MCs.', '1406.0676-2-12-0': 'In an attempt to extend the TP-AGB calibration to low metallicities, [CITATION] further constrained the [CITATION] TP-AGB models using optical observations of low-metallicity nearby, non-Local Group galaxies.', '1406.0676-2-12-1': 'These samples were from the ACS Nearby Galaxy Survey Treasury and the Archival Nearby Galaxies: Reduce, Reuse, Recycle database.', '1406.0676-2-12-2': 'ANGST is a volume limited sample of [MATH] non-Local Group galaxies to [MATH] Mpc, in which deep optical photometry allowed the measurement of their star formation histories (SFH).', '1406.0676-2-12-3': 'Using a subset of 12 dwarf galaxies from the ANGST sample that showed little to no recent star formation,taliasGirardi2010 compared the luminosity function (LF) of each galaxy with those of simulations using the measured SFHs, and updated the mass-loss prescriptions for low-mass, low-metallicity, TP-AGB stars before the onset of dust driven winds (pre-dust winds).', '1406.0676-2-12-4': 'This correction resulted in good data-model agreement.', '1406.0676-2-12-5': 'Indeed, the revised TP-AGB models reproduced the initial-to-final mass relationship of the Galactic globular cluster M4 .', '1406.0676-2-13-0': 'Using a completely different approach, [CITATION] put constraints on the lifetimes and the core mass growth of intermediate-age TP-AGB stars (with initial masses in the range 1.6 [MATH]) with slightly super-solar initial metallicity ([MATH]), combining recent accurate measurements of white dwarf masses in the Hyades and Praesepe star clusters and new TP-AGB models .', '1406.0676-2-14-0': '## This work in context', '1406.0676-2-15-0': 'The method applied bytaliasGirardi2010 represents a promising approach given the increasing availability of deep photometric data from which the detailed SFHs of nearby galaxies can be derived.', '1406.0676-2-15-1': 'Its range of applicability, however, was initially limited by the optical filter-set available in ANGST.', '1406.0676-2-15-2': 'The reddest filter in the ANGST survey is [MATH], which is more affected by circumstellar dust than redder filters and which also requires higher bolometric corrections for cooler TP-AGB stars .', '1406.0676-2-15-3': 'This limitation allows for a complete AGB sample only for relatively hot ([MATH]T_eff[MATH] K) TP-AGB stars.', '1406.0676-2-15-4': 'Therefore, the method was applied only to a few very metal-poor dwarf galaxies without any sign of recent star formation.', '1406.0676-2-16-0': 'To generate a complete sample of TP-AGB stars in galaxies with more recent star formation and higher metallicities (and hence cooler [MATH]T_eff[MATH]), near-infrared (NIR) filters are necessary.', '1406.0676-2-16-1': 'In a series of two papers, we build upon the method oftaliasGirardi2010 by calculating the LFs of modeled TP-AGB stars using the near-infrared [MATH] "snapshot" follow-up survey to ANGST .', '1406.0676-2-16-2': "This sample includes 26 NIR fields of 23 ANGST galaxies, with 10,000's of TP-AGB stars.", '1406.0676-2-16-3': 'In principle, this filter set allows us to probe TP-AGB masses from 1 to 5 [MATH] with [MATH]T_eff[MATH]K. However, we are still limited by the NIR filterset for which it is not possible to robustly separate C-rich from O-rich AGB stars .', '1406.0676-2-17-0': 'We divided the AGB SNAP sample into galaxies with and without evidence of recent star formation, which effectively limits the masses of the AGB star populations to [MATH]M_[MATH] for the former and includes all possible AGB masses for the latter.', '1406.0676-2-17-1': 'For this paper, we focus on six galaxies, DDO 71, DDO 78, NGC 2976, SCL DE1, HS 117, and KKH 37 shown in optical and NIR CMD and their LFs in Figure [REF].', '1406.0676-2-17-2': 'These galaxies are suitable to be modeled with the isochrones derived from the Padova and TRieste Stellar Evolution Code , which in its present version (v1.1) include masses up to [MATH]M_[MATH] and typical metallicities ranging from [MATH] ([Fe/H][MATH] -2.18 [MATH] -0.60.', '1406.0676-2-17-3': 'Notice that half of these galaxies (DDO 71, DDO 78, and SCL DE1), where already analyzed intaliasGirardi2010 but using the optical data only.', '1406.0676-2-17-4': 'In the next paper, we will extend the galaxy sample to those that show higher amounts of recent star formation with more massive stellar evolution models from PARSEC v2 (Bressan et al. in prep).', '1406.0676-2-18-0': 'The paper is organized as follows.', '1406.0676-2-18-1': 'In Section [REF], we briefly summarize the data reduction and photometry of the AGB-SNAP sub-sample.', '1406.0676-2-18-2': 'Next, we describe the process of recovering the SFH from the deeper optical photometry and associated artificial star tests and our method of isolating TP-AGB and red giant branch (RGB) stars.', '1406.0676-2-18-3': 'We discuss the TP-AGB stellar evolutionary models and the pre-dust mass-loss prescriptions in Section [REF].', '1406.0676-2-18-4': 'Our method of using star counts to robustly model the TP-AGB stars in data is presented in Section [REF].', '1406.0676-2-18-5': 'We compare the observational constraints from the TP-AGB models to those of data in Section [REF], and conclude in Section [REF].', '1406.0676-2-19-0': '# Data', '1406.0676-2-20-0': '## Reduction and Photometry', '1406.0676-2-21-0': 'We now briefly summarize the data reduction and photometry of the AGB-SNAP sample.', '1406.0676-2-21-1': 'For full details, we refer to the ANGST and AGB-SNAP survey papers .', '1406.0676-2-22-0': 'Optical data from the STScI ACS pipeline data were photometered using DOLPHOT2.0 including the ACS module.', '1406.0676-2-22-1': 'Cosmic rays were rejected after combining all images into a single drizzled image using the multidrizzle task within PyRAF .', '1406.0676-2-22-2': 'We use the conservative ANGST photometric catalog (*gst), which only includes objects with DOLPHOT parameters SNR < 4, ((sharp[MATH] + sharp[MATH]), and crowding ((crowd[MATH] + crowd[MATH]) in both filters.', '1406.0676-2-22-3': 'WFPC2 data from ANGRRR and ANGST (following the ACS camera failure) used the WFPC2 pipeline of [CITATION], which processes STScI baseline output through HSTphot, a WFPC2 optimized predecessor of DOLPHOT but updated to July 2008 CTE corrections (derived by A. Dolphin).', '1406.0676-2-22-4': 'To assess the photometric uncertainties and completeness, at least [MATH] artificial star tests were calculated for each galaxy.', '1406.0676-2-23-0': 'The NIR AGB-SNAP data were also reduced using the DOLPHOT package, using a significant update of the WFC3 module (among other enhancements).', '1406.0676-2-23-1': 'We again use the conservative photometry (*gst), in this case with DOLPHOT parameters S/N < 4, ((sharp[MATH] + sharp[MATH]) and ((crowd[MATH] +crowd[MATH]).', '1406.0676-2-24-0': '## Star Formation Histories of the Galaxy Sample', '1406.0676-2-25-0': 'We use the optical CMDs to derive the SFH.', '1406.0676-2-25-1': 'ANGST observations are deep enough to robustly measure the color and magnitude of the red clump (RC) and the shape of the RGB.', '1406.0676-2-25-2': 'Both place strong constraints on the metallicity evolution and past star formation rate .', '1406.0676-2-25-3': 'However, we exclude the AGB from contributing to the SFH fit by masking stars brighter than the TRGB (as was done in G10).', '1406.0676-2-26-0': 'Star formation histories were derived using the CMD-fitting MATCH package .', '1406.0676-2-26-1': 'MATCH finds the most likely SFH and metallicity evolution that fits the observed CMD based on a given IMF, binary fraction, and stellar isochrones while taking into account photometric uncertainties and completeness.', '1406.0676-2-26-2': 'We adopt a [CITATION] IMF, a binary fraction of 0.35, and the PARSEC isochrones .', '1406.0676-2-26-3': 'Photometric uncertainties and completeness were obtained from the artificial star tests described in [CITATION] and [CITATION].', '1406.0676-2-26-4': 'In addition, we allow MATCH to search distance and [MATH] parameter space, in finding the most likely SFH.', '1406.0676-2-27-0': 'Following the discussion in G10, we used MATCH with the zinc flag, which determines the best-fit CMD as a product of the most probable star formation history given an enrichment history in which the metallicity increases with galaxy age .', '1406.0676-2-27-1': 'MATCH derived SFR-weighted metallicities at ages older than 300 Myr range from Z=[MATH] ([Fe/H]=-1.54[MATH]-0.86; see Table [REF]) .', '1406.0676-2-27-2': 'Random uncertainties in the SFH were determined with the Hybrid Monte Carlo tests described in [CITATION].', '1406.0676-2-27-3': 'We use both the SFHs and their uncertainties to produce the model LFs (see Section [REF]).', '1406.0676-2-28-0': 'A summary of the cumulative SFHs of the galaxy sample are shown in Figure [REF].', '1406.0676-2-28-1': 'Derived SFHs all agree within uncertainties to those derived in [CITATION] or for NGC2976 [CITATION] (both studies used [CITATION] isochrones).', '1406.0676-2-28-2': 'In general, the CMD-fitting using PARSEC v1.1 compared to the previous Padova models produce better matches to the shape of the RGB found in the data.', '1406.0676-2-29-0': 'Our method to constrain TP-AGB models is ultimately limited to how well we can model the rest of the stellar populations in the data.', '1406.0676-2-29-1': 'One measure of how well the SFH is recovered from the data is the effective [MATH] .', '1406.0676-2-29-2': 'For the purposes of this study, we state the effective [MATH] values derived from the SFH fitting as a means to compare one SFH fit to the next in Table [REF].', '1406.0676-2-30-0': '## Selecting AGB and RGB Stars', '1406.0676-2-31-0': 'A successful TP-AGB model must be able to reproduce the lifetimes of TP-AGB stars.', '1406.0676-2-31-1': 'One classic observational comparison in stellar evolution modeling is to measure the relative lifetimes of an uncertain stellar evolution phase to a more certain stellar evolutionary phase by taking the ratio of stars found in different regions of a CMD.', '1406.0676-2-31-2': 'Therefore, a first test of the TP-AGB models is to calculate the number ratio of TP-AGB stars to RGB stars, [MATH].', '1406.0676-2-31-3': 'As we identify the TP-AGB and RGB stars, we must minimize the contamination from stars of other phases in the RGB region and the TP-AGB region.', '1406.0676-2-32-0': 'We selected RGB and TP-AGB stars in two regions of the CMDs that were defined to minimize any possible contamination from stars in other evolutionary phases.', '1406.0676-2-32-1': 'Figure [REF] shows example optical and NIR LFs of several evolutionary stages of a simulation massive enough to ensure all phases were well populated and based on the SFH of DDO71.', '1406.0676-2-33-0': 'We exclude any star within a magnitude offset around the TRGB to minimize the number of misclassified TP-AGB and/or RGB stars that scatter up or down in luminosity.', '1406.0676-2-33-1': 'Our simulations show that outside a region around the TRGB of [MATH] mag in [MATH] or [MATH] mag in [MATH] there is never more than 5% contamination of TP-AGB stars in the RGB.', '1406.0676-2-33-2': 'These "excluded regions" are shown in grey in all LF figures with the TRGB also indicated.', '1406.0676-2-34-0': 'With the optical and NIR filters used in our observations, there is no way of completely separating RGB stars from Early AGB (EAGB) stars, Red Helium Burning stars (RHeB), or RC stars.', '1406.0676-2-34-1': 'However, the RHeB and EAGB do not contribute more than [MATH] and [MATH] of the number of stars in the RGB region respectively, a percentage that greatly diminishes with decreasing brightness, based on our population synthesis of the galaxy sample.', '1406.0676-2-34-2': 'If we had included galaxies with more active recent SF, then the contamination from RHeB stars would have been higher.', '1406.0676-2-35-0': 'Unlike the EAGB and RHeB, possible contamination from the RC increases with fainter magnitudes.', '1406.0676-2-35-1': 'RC stars are several magnitudes fainter than the TRGB, and have higher photometric uncertainties, potentially causing RC stars to blend with faint RGB stars.', '1406.0676-2-35-2': 'The possible contamination is only in the optical, since the NIR data is never deep enough to reach the RC.', '1406.0676-2-35-3': 'Regardless, we mitigate this contamination of the RGB by only including stars above the 90 completeness limit in our analysis.', '1406.0676-2-35-4': 'In addition, high completeness assures precise counting of stars in each magnitude bin while being a bright enough limit to avoid stars from the RC (in the optical; the 90 completeness magnitudes for each galaxy in [MATH] and [MATH] are listed in Table [REF]).', '1406.0676-2-35-5': 'RC stars never make up more than 10 of stars in the RGB region included in our analysis.', '1406.0676-2-36-0': 'To exclude main sequence (MS) and blue He-burning (BHeB) stars, we restrict our analysis to stars redder than [MATH], [MATH] and [MATH].', '1406.0676-2-36-1': 'Though the color cuts are easily drawn by eye, they were based on population synthesis combined with photometric uncertainties measured in the data.', '1406.0676-2-36-2': 'We take the color cut as the reddest MS star that is brighter than the 90% completeness limit in a massive simulation calculated with constant SF ([MATH] range from [MATH]) and constant, high metallicity ([MATH] for the reddest MS available in PARSEC).', '1406.0676-2-36-3': 'We then add to the color cut the typical 1[MATH] color uncertainty found in the data (typical uncertainties are shown in Figure [REF].', '1406.0676-2-36-4': 'From our simulations of each galaxy, the analysis regions contain on average 15% MS stars and 1% BHeB stars.', '1406.0676-2-37-0': 'Finally, to robustly compare our models to data, we correct the data for completeness using the same artificial star tests derived for the SFH recovery.', '1406.0676-2-37-1': 'For our analysis, these corrections are minor as our faint limit is already at the 90 completeness.', '1406.0676-2-37-2': 'In effect, due to the completeness corrections, the number of RGB stars increases at most by 10 for optical data and 7 for NIR data.', '1406.0676-2-38-0': 'In summary, the TP-AGB stars are defined as those (optical/NIR) stars (0.1/0.2) magnitudes brighter than the TRGB and the RGB stars are defined as the stars from the 90 completeness limit to (0.1/0.2) magnitudes below the (optical/NIR) TRGB, excluding stars that are bluer in color than 0.2 for [MATH], 0.3 for [MATH], and 0.1 for [MATH].', '1406.0676-2-38-1': 'The numbers of TP-AGB and RGB stars as well as their ratio and Poisson uncertainties are listed in Table [REF].', '1406.0676-2-39-0': 'As eluded to in Section [REF], [MATH] observations can detect cooler TP-AGB stars than those using [MATH] as the latter is more affected by circumstellar dust.', '1406.0676-2-39-1': 'However, with the exception of NGC 2976, we detect fewer TP-AGB stars in NIR than in the optical.', '1406.0676-2-39-2': 'The reason for the discrepancy is simply due to the observations.', '1406.0676-2-39-3': 'While the WFC3/IR fields overlap the ACS/WFC3 fields , the ACS/WFC field of view (FOV; [MATH]) is larger than the WFC3/IR FOV ([MATH]) by a factor of 2.4.', '1406.0676-2-39-4': 'Indeed, the number densities of NIR TP-AGB stars are between 1.9-2.2 times higher than that of the optical TP-AGB stars (excluding NGC 2976 which is 4 times higher).', '1406.0676-2-40-0': 'The ratio of TP-AGB stars to RGB stars ranges from 0.017 to 0.253 and random errors in the measured ratio are never above 16%.', '1406.0676-2-40-1': 'When all galaxies are combined, the measured optical and NIR [MATH] ratio is [MATH] and [MATH], respectively.', '1406.0676-2-40-2': 'Therefore, random errors will have little effect when comparing the measured [MATH] ratio to that predicted by a grid of TP-AGB models.', '1406.0676-2-41-0': 'lcccccc', '1406.0676-2-42-0': 'Observational Data', '1406.0676-2-43-0': 'Target & [MATH] & [MATH] & 2cF814W & 2cF160W', '1406.0676-2-44-0': '& & & 90% Comp.', '1406.0676-2-44-1': '& [MATH] & 90% Comp.', '1406.0676-2-44-2': '& [MATH]', '1406.0676-2-45-0': 'DDO71 & 0.30 & 27.74 & 26.11 & 23.71 & 23.51 & 22.14', '1406.0676-2-46-0': 'HS117 & 0.36 & 27.91 & 24.76 & 23.86 & 23.85 & 22.34', '1406.0676-2-47-0': 'KKH37 & 0.23 & 27.57 & 24.41 & 23.54 & 23.26 & 21.96', '1406.0676-2-48-0': 'NGC2976 & 0.22 & 27.76 & 25.52 & 23.74 & 22.67 & 21.85', '1406.0676-2-49-0': 'DDO78 & 0.07 & 27.82 & 24.76 & 23.80 & 23.63 & 22.05', '1406.0676-2-50-0': 'SCL-DE1 & 0.05 & 28.11 & 25.65 & 24.09 & 24.27 & 22.57 Columns 2, 3, and 5 from [CITATION] and column 7 is from [CITATION], columns 4 and 6 are the 90% completeness magnitudes in [MATH] and [MATH], respectively.', '1406.0676-2-51-0': 'lrrcrrccc', '1406.0676-2-52-0': 'Star Counts and MATCH data', '1406.0676-2-53-0': 'Target & 3cF814W & 3cF160W & 2cResults from MATCH', '1406.0676-2-54-0': '& [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & SFR-weighted [Fe/H] & [MATH] of CMD-fit', '1406.0676-2-55-0': 'DDO71 & 149 & 8739 & [MATH] & 136 & 1730 & [MATH] & -1.13 & 1.48', '1406.0676-2-56-0': 'HS117 & 70 & 498 & [MATH] & 62 & 1003 & [MATH] & -1.20 & 1.07', '1406.0676-2-57-0': 'KKH37 & 135 & 669 & [MATH] & 122 & 923 & [MATH] & -1.38 & 1.14', '1406.0676-2-58-0': 'NGC2976 & 290 & 7695 & [MATH] & 490 & 1933 & [MATH] & -0.86 & 1.55', '1406.0676-2-59-0': 'DDO78 & 273 & 2783 & [MATH] & 215 & 2987 & [MATH] & -0.86 & 1.27', '1406.0676-2-60-0': 'SCL-DE1 & 83 & 1144 & [MATH] & 66 & 867 & [MATH] & -1.54 & 0.89', '1406.0676-2-61-0': 'Total & 1000 & 21528 & [MATH] & 1091 & 9443 & [MATH] & ... & ... The number of stars in the TP-AGB and RGB regions defined in Section [REF] and their ratio in [MATH] (columns 2-4) and [MATH] (columns 5-7).', '1406.0676-2-61-1': 'The final two columns come from the MATCH derived star formation histories, the sfr-weighted averaged metallicity older than 300 Myr (column 8) and the effective [MATH] of the MATCH solution .', '1406.0676-2-62-0': '# TP-AGB evolutionary models', '1406.0676-2-63-0': 'PARSEC is a thoroughly revised version of the popular Padova stellar evolution code used to compute stellar evolution tracks.', '1406.0676-2-63-1': 'We use its v1.1 release , which offers stellar tracks spanning the ranges [MATH], [MATH]M_[MATH]M_[MATH], and from the Pre-MS phase to the beginning of either the TP-AGB or the core Carbon ignition phase.', '1406.0676-2-63-2': 'PARSEC is used in this work both to derive the SFH through the MATCH package (see Section [REF]) as well as the input isochrones to the population synthesis code TRILEGAL (see Section [REF]), which simulates the LF under different AGB model assumptions.', '1406.0676-2-64-0': 'Following the first thermal pulse on the AGB, COLIBRI takes over the stellar evolution calculations from PARSEC.', '1406.0676-2-64-1': 'COLIBRI is an "almost-full" TP-AGB modeling code, that is, it relaxes many of the analytic forms of other synthetic TP-AGB models.', '1406.0676-2-64-2': 'All details are found in [CITATION].', '1406.0676-2-64-3': 'The most relevant to this study include: 1) a complete envelope model from the bottom of the quiescent H-burning shell up to the atmosphere; 2) Rosseland mean opacities calculated on-the-fly with the Opacity Project tools in the high temperature regime for [MATH] K and with the AESOPUS code at lower temperatures for [MATH] K [MATH] K, including the equation of state of [MATH] atomic and molecular species, assuring complete consistency with current chemical abundances; 3) complete nuclear network to follow the nucleosynthesis occurring at the base of the convective envelope (hot-bottom burning; HBB) in more massive AGB stars ([MATH]M_[MATH]) and in the pulse-driven convective zone at thermal pulses; and 4) efficiency of mass-loss treated as a free parameter to be calibrated by observations.', '1406.0676-2-65-0': 'In this paper, we focus on mass-loss and test three prescriptions by how well they reproduce the [MATH] ratio and the shape of the observed LFs.', '1406.0676-2-65-1': 'Although we also wish to constrain the efficiency of the third dredge up, such a test would require measuring the C/O ratio of the stellar population, and the NIR filters available are not red enough to robustly separate C-rich and O-rich stars .', '1406.0676-2-65-2': 'As we show below, the variation of the magnitude of the efficiency of the third dredge up does not change the TP-AGB lifetimes in mass-loss prescriptions with a high [MATH] dependence.', '1406.0676-2-66-0': '## Mass-Loss Prescriptions on the TP-AGB', '1406.0676-2-67-0': 'For a star to have a wind, there must be an outward force that provides momentum and energy input, accelerating the surface layers to velocities larger than the escape velocity.', '1406.0676-2-67-1': 'This may be realized in various ways, including the scattering of UV radiation by resonance line opacity in hot stars, the generation of magneto-acoustic waves above the photosphere in red giants, or the absorption of photons by dust grains in the outer atmospheres of the coolest and most luminous stars .', '1406.0676-2-68-0': "Mass-loss dominates an AGB star's evolution and fate.", '1406.0676-2-68-1': 'It is clear from observations of Mira and OH/IR stars that mass-loss rates increase exponentially along the AGB until they reach super-wind values of [MATH]M_[MATH] yr[MATH] .', '1406.0676-2-68-2': 'Despite the recent progress in the theory of AGB mass-loss , we still lack complete understanding of all the factors and their complex interplay which control the stellar winds on the AGB .', '1406.0676-2-69-0': 'Combining theoretical efforts and empirical evidence, a reasonable scenario takes form in which mass-loss on the AGB can be divided into three regimes: an initial period before the onset of the dust-driven wind (designated as "pre-dust mass-loss"); a subsequent phase characterised by an exponential increase of mass-loss driven by the combined action of dust and pulsation (designated as "dust-driven mass-loss"); and a final brief regime with high mass-loss (designated as "super-wind mass-loss").', '1406.0676-2-70-0': 'In our scheme, the phase of pre-dust mass-loss (with rate [MATH]) is thought to apply to the early stages on the AGB in which either dust has not yet formed in the outermost atmospheric layers, or if present in some small amount, is unable to generate an outflow.', '1406.0676-2-70-1': 'In these conditions a likely wind mechanism could be related to a strong flux of pressure waves or Alfven waves able to cause the spillover of the extended and highly turbulent chromospheres typical of red giants.', '1406.0676-2-70-2': 'The same mechanism might be at work during both the ascent along the RGB and the early stages of the AGB .', '1406.0676-2-71-0': 'In stellar evolutionary calculations a frequent choice to describe mass-loss during the early phases is the classical [CITATION] law, a simple scaling relation of stellar parameters based on observations of few red giants and supergiants.', '1406.0676-2-71-1': 'The Reimers relation is commonly multiplied by an efficiency parameter [MATH], whose value is calibrated such that it recovers the observed morphology of horizontal branch stars in Galactic Globular clusters.', '1406.0676-2-71-2': 'The calibration however, still depends on the residual envelope mass left over from the RGB .', '1406.0676-2-72-0': 'More recently [CITATION] proposed a modified version of the [CITATION] law, in which additional dependencies on the effective temperature and surface gravity follow from a physically-motivated consideration of the mechanical flux responsible for the wind.', '1406.0676-2-72-1': 'The role of the chromosphere in driving mass-loss in late-K to early-M giants is supported by the analysis [CITATION] of the H[MATH] and infrared calcium triplet lines in a sample of red giant stars hosted in Galactic globular clusters.', '1406.0676-2-72-2': 'Similarly to the Reimers relation, the [CITATION] formula also needs an efficiency parameter [MATH] to be specified.', '1406.0676-2-73-0': 'Novel efforts to model stellar winds from red giants were carried out by [CITATION].', '1406.0676-2-73-1': 'A self-consistent and more detailed theoretical approach is developed to follow the generation of energy flux due to magnetohydrodynamic turbulence from subsurface convection zones to its eventual dissipation and escape through the stellar wind.', '1406.0676-2-73-2': 'One major difference is that, while in [CITATION] the mass-loss rate is assumed to scale linearly with the photospheric mechanical energy flux ([MATH]) of Alfven waves ([MATH]), the analysis of [CITATION] yields a higher dependence ([MATH]).', '1406.0676-2-73-3': 'Analytic models for magnetic wave generation indicate that the mechanical energy flux scales as [MATH] .', '1406.0676-2-73-4': 'Hence, considering that the mass-loss rate is proportional to the surface-integrated mechanical energy flux, [MATH], and expressing the stellar radius [MATH] with the Stefan-Boltzmann law for a black body, we eventually obtain a significantly steeper dependence of the mass-loss rate on the effective temperature, i.e. [MATH] for [CITATION] and [MATH] following the results of [CITATION].', '1406.0676-2-74-0': 'Following the pre-dust phase of mass-loss, as the star climbs the AGB at increasing luminosity, suitable conditions can be met in the cool atmosphere for stellar winds to be generated through a different intervening mechanism.', '1406.0676-2-74-1': 'The most plausible hypothesis resides in the momentum input when the stellar radiation field is absorbed (or scattered) by dust grains and transferred to the gas via collisions.', '1406.0676-2-74-2': 'This wind is enhanced by pulsations that shock the envelope and periodically levitate matter up to regions where dust can more efficiently condense .', '1406.0676-2-74-3': 'Observationally there is a clear correlation, though with a large scatter, between the mass-loss rate (here designated with [MATH]) and the pulsation period [MATH] of AGB variables, such that [MATH] is seen to increase exponentially with the period .', '1406.0676-2-75-0': 'Finally, close to tip of the TP-AGB, the mass-loss rates almost level out to [MATH] yr[MATH] of the so-called super-wind phase ([MATH]), corresponding to the condition in which the maximum momentum of the radiation field is transferred to the stellar atmosphere.', '1406.0676-2-76-0': 'Within this framework, the mass-loss prescriptions adopted in the TP-AGB stellar models computed for this study are as follows.', '1406.0676-2-76-1': 'For the dust driven wind phase we adopt a formula similar to [CITATION], which predicts an exponential increase of mass-loss [MATH] dependent on stellar parameters derived from models of periodic shocked atmospheres.', '1406.0676-2-76-2': 'Coefficients [MATH] and [MATH] are calibrated on a sample of Galactic Mira stars.', '1406.0676-2-76-3': 'This prescription is also discussed in [CITATION], [CITATION], and [CITATION].', '1406.0676-2-77-0': 'For the super-wind phase we adopt the formalism of [CITATION], in which the mass-loss rate, [MATH], is proportional to the ratio of the stellar luminosity to the terminal velocity of the gas, which itself scales linearly with the pulsation period.', '1406.0676-2-77-1': 'In practice as soon as [MATH] days the super-wind regime is expected to set in.', '1406.0676-2-78-0': 'We keep the same prescriptions for [MATH] and [MATH] and vary only the [MATH].', '1406.0676-2-78-1': 'For the mass-loss rates [MATH] before the onset of dust-driven winds we consider four options:', '1406.0676-2-79-0': '[[MATH]] no mass-loss before the possible onset of the dust-driven wind, [MATH];', '1406.0676-2-80-0': '[[MATH]] the traditional [CITATION] mass-loss [EQUATION] with the efficiency parameter [MATH];', '1406.0676-2-81-0': '[[MATH]] the original [CITATION] law [EQUATION] with the efficiency parameter [MATH];', '1406.0676-2-82-0': '[[MATH]] a modified version of the [CITATION] scaling relation [EQUATION] in which, for the reasons explained above, the power-law dependence on the effective temperature is steepened; here the efficiency parameter is set to [MATH].', '1406.0676-2-83-0': 'In all formulas the mass-loss rate is given in [MATH], the effective temperature [MATH] is in Kelvin, the stellar radius [MATH], luminosity [MATH], the mass [MATH], and surface gravity [MATH] are expressed in solar units.', '1406.0676-2-83-1': 'Following [CITATION], at each time during the TP-AGB evolution, the current mass-loss rate is taken as [MATH].', '1406.0676-2-84-0': 'Finally, we caution the reader that our modified Schroder Cuntz relation, with [MATH], set for the early stages of the TP-AGB, may be too efficient to be extended to lower luminosities of RGB stars based on a quick comparison to the measured mass-loss rates of the sample, collected by [CITATION], that includes metal poor RGB stars with and effective temperatures in the range 3800 - 5800 K, as a function of the luminosity.', '1406.0676-2-84-1': 'In this context, the role of RGB mass-loss and its possible influence on the subsequent AGB evolution of low-mass stars is postponed to a future work.', '1406.0676-2-85-0': '## Main processes affecting the TP-AGB lifetimes', '1406.0676-2-86-0': 'Many physical processes and events are at work during the TP-AGB phase.', '1406.0676-2-86-1': 'However, there is no doubt that mass-loss is the principal mechanism that controls the duration of this phase, which ends when almost all the stellar mantle is ejected into the interstellar medium.', '1406.0676-2-87-0': 'In this study we opt to analyze the significance of the early stages of AGB mass-loss, since this regime may be particularly important for low-mass stars whose small envelopes may already be removed before the onset of the dust-driven wind (c.f., G10).', '1406.0676-2-87-1': 'This choice seems appropriate given the sample of galaxies under consideration, which are all characterized by a significant fraction of old stellar populations and thus will have TP-AGB populations dominated by lower mass stars (Figure [REF]).', '1406.0676-2-87-2': 'While analysing the impact of different laws for [MATH] and [MATH] is postponed to future works, it is worth mentioning that the prescriptions adopted here have already successfully passed a few observational tests, including the recovery of the expansion velocities of AGB circumstellar envelopes , and the Galactic initial-final mass relation .', '1406.0676-2-88-0': 'It turns out that, indeed, the efficiency of [MATH] plays a major role in determining the lifetimes of TP-AGB stars of lower TP-AGB mass.', '1406.0676-2-88-1': 'In Figure [REF] we compare the four mass-loss options applied to compute the TP-AGB evolution of two stars with initial masses of 1[MATH] and 2[MATH].', '1406.0676-2-88-2': 'The panels are organized from top to bottom following a sequence of progressively more efficient mass-loss.', '1406.0676-2-89-0': 'As expected, the larger the mass-loss rates, the shorter the TP-AGB lifetimes.', '1406.0676-2-89-1': 'From the case of no mass-loss [MATH] to the [MATH] case, the TP-AGB lifetimes are reduced by a factor that depends on the stellar mass, being roughly a factor of 4 for the 1[MATH] star and a factor of 2 for the 2[MATH] star.', '1406.0676-2-90-0': 'The [MATH] case corresponds to the longest duration of the TP-AGB phase, and also to the reddest excursion on the HR diagram, while the TP-AGB tracks computed with [MATH] have the shortest lifetimes and exhibit a smaller displacement towards the coolest [MATH] region.', '1406.0676-2-90-1': 'In general, comparing the panels from top to bottom we obtain a sequence of decreasing lifetimes and cooler HR tracks, a trend which is more pronounced at lower masses.', '1406.0676-2-90-2': 'The differences among models for the predicted TP-AGB lifetimes and effective temperatures become less pronounced at larger masses, as shown by the results for the 2[MATH] star (right panels of Figure [REF]).', '1406.0676-2-91-0': 'The correlation between the TP-AGB lifetime and the [MATH] redward excursion is explained as an effect related to C-star formation (when the surface C/O ratio increases from below to above unity): as more third dredge-up events are allowed to occur during the TP-AGB, a higher C-O excess is built in the atmosphere, leading to a stronger C-bearing molecular opacity, hence to a cooler Hayashi line.', '1406.0676-2-91-1': 'This latter aspect is particularly evident for the 1[MATH] star, which is able to become a carbon star with [MATH] (marked in red in Figure [REF]), whereas it remains oxygen-rich with the other mass-loss prescriptions.', '1406.0676-2-92-0': 'The third dredge-up is expected to affect the TP-AGB lifetimes, essentially due to its impact on the surface chemical composition (mainly in terms of the C/O ratio), which in turn controls both the atmospheric molecular opacity , and the mineralogy of the dust grains that grow in the expanding circumstellar envelope .', '1406.0676-2-92-1': 'Both factors combine to influence the mass-loss rates .', '1406.0676-2-92-2': 'In general, at the transition to the C-star regime, TP-AGB models with variable molecular opacities predict a sudden cooling of the track that makes the mass-loss rates increase (provided the adopted prescription is a sensitive function of [MATH]) with consequent reduction of the lifetimes.', '1406.0676-2-92-3': 'This point is fully discussed in [CITATION].', '1406.0676-2-93-0': 'The TP-AGB lifetime may also be sensitive to changes of the third dredge-up efficiency after the transition C-star phase.', '1406.0676-2-93-1': 'To test this possibility, we computed additional sets of TP-AGB tracks by varying the efficiency of the third dredge-up, while keeping the same [MATH] mass-loss formalism.', '1406.0676-2-93-2': 'Expressing the efficiency of the third dredge-up with the classical parameter [MATH], we adopt the original formalism for [MATH] proposed by [CITATION] ([MATH]), and test the two additional choices of doubling (2 [MATH]) and halving (0.5 [MATH]) the reference efficiency.', '1406.0676-2-93-3': 'We find that the predicted TP-AGB lifetimes barely change (see Fig. [REF], right panel), even when the total amount of dredged-up material varies by a factor of [MATH].', '1406.0676-2-93-4': 'The limited effect of the third dredge-up can be explained as a combination of two main factors.', '1406.0676-2-93-5': 'The first is due to the efficiency of the mass-loss prescription adopted here.', '1406.0676-2-93-6': 'The [MATH] model is quite efficient and the TP-AGB phase terminates quickly (for instance, at [MATH], the total number of thermal pulses is [MATH] in the relevant initial mass range).', '1406.0676-2-93-7': 'Therefore, there simply is not enough time for the third dredge-up to produce dramatic effects, no matter how it is varied.', '1406.0676-2-93-8': 'In addition, as the star reaches the super-wind phase, the mass-loss rates settle to typical values that are little affected by variations of other stellar parameters.', '1406.0676-2-94-0': 'The second factor is related to the sensitivity of the effective temperature as a function of the carbon excess in the atmosphere of carbon stars.', '1406.0676-2-94-1': 'In general, more carbon excess corresponds to lower effective temperature which causes higher rates of mass-loss.', '1406.0676-2-94-2': 'This response of the effective temperature to the increase of carbon can be described by the cooling rate (the derivative [MATH]) which is expected to progressively decrease at increasing C/O ratio .', '1406.0676-2-94-3': 'In other words, after the initial sizeable drop of [MATH] once C/O exceeds unity, as more third dredge-up events continue to take place, the atmospheric structure becomes less and less sensitive to further increase of carbon.', '1406.0676-2-94-4': 'As a consequence, the impact of the third dredge-up on the effective temperature becomes progressively weaker as more mixing events occur.', '1406.0676-2-95-0': 'In short, the duration of the TP-AGB phase is mainly controlled by mass-loss-at least for the mass/metallicity interval being considered here-while the effect of the third dredge-up is limited (see Fig. [REF], right panel).', '1406.0676-2-95-1': 'This result is important, as it strengthens the robustness of the analysis described next, whose primary aim is to obtain a quantitative estimation of the TP-AGB lifetimes as a function of the initial stellar mass, in the low-metallicity regime.', '1406.0676-2-96-0': 'However, we must also emphasize that although the TP-AGB lifetimes are found to be little influenced by the third dredge-up, the chemical composition of the ejecta is much affected by the properties of the mixing events.', '1406.0676-2-96-1': 'In particular, for the same amount of mass lost, the quantity of primary carbon, and hence of carbonaceous dust, that is injected in the interstellar medium does depend strongly on the efficiency of the third dredge-up.', '1406.0676-2-96-2': 'Therefore, although it is outside the scope of this paper, future calibration of the third dredge-up process is an essential step towards a comprehensive description of the TP-AGB phase that includes not only the spectro-photometric but also the chemical role of TP-AGB stars in the context of galaxy evolution.', '1406.0676-2-97-0': 'On a final note, we exclude the original [CITATION] from the subsequent discussion and analysis.', '1406.0676-2-97-1': 'As is shown in Figure [REF], [CITATION] mass-loss rate predicts a longer lifetime than the [MATH] model.', '1406.0676-2-97-2': 'There are two main reasons that the [CITATION] prescription is no longer satisfactory for the present work.', '1406.0676-2-97-3': 'First, the computations discussed here are based on a completely new release of stellar evolution models in which major modification and update of the input physics were introduced.', '1406.0676-2-97-4': 'This leads, for instance, to produce Hayashi lines that are on average somewhat cooler than in G10.', '1406.0676-2-97-5': 'Second, we found that if we vary the mass-loss efficiency ([MATH]) enough to account for the number of AGB stars in our predominantly old and metal-poor galaxies (increasing the efficiency parameter [MATH]), the effect on higher mass AGB stars (i.e. [MATH]) would be dramatic.', '1406.0676-2-97-6': 'In fact, that would lead to an extreme shortening of their lifetimes producing a deficit in contrast with the observations of AGB stars at typical metallities of the SMC and LMC ([MATH]).', '1406.0676-2-97-7': 'Finally, we underline that the modification to the original [CITATION] arises from replacing a simple assumption with a more physically-sound consideration of the dependence of the mass-loss rate on the mechanical magnetic flux .', '1406.0676-2-97-8': 'For these reasons, as discussed above, a new [MATH] scaling was in order, so we proceed with the [MATH] model in lieu of [MATH].', '1406.0676-2-98-0': '# Modeling the Data', '1406.0676-2-99-0': 'With optically derived SFHs of each galaxy and three sets of TP-AGB models, we now turn to our method of using star counts to robustly constrain TP-AGB lifetimes.', '1406.0676-2-99-1': 'We now discuss how we apply the derived SFHs and use TRILEGAL to create model LFs.', '1406.0676-2-100-0': '## Population Synthesis with TRILEGAL', '1406.0676-2-101-0': 'FollowingtaliasGirardi2010, we use TRILEGAL to synthesize the stellar populations for direct comparison to observations.', '1406.0676-2-101-1': 'TRILEGAL takes as input the PARSEC and COLIBRI stellar evolution libraries, a specified initial mass function (IMF), binary fraction, and the SFH.', '1406.0676-2-101-2': 'Importantly, TRILEGAL also simulates the [MATH]-[MATH] variations due to the thermal pulse cycles on the TP-AGB , and the reprocessing of radiation by their circumstellar dust-shells .', '1406.0676-2-101-3': 'For previous evolutionary phases, TRILEGAL provides simulations which are essentially identical to those performed by MATCH.', '1406.0676-2-102-0': 'The TRILEGAL input parameters are set to remain consistent with the parameters used in the SFH recovery (see Section [REF]).', '1406.0676-2-102-1': 'The stars produced by TRILEGAL are converted into absolute magnitudes in [MATH] filters using the set of bolometric corrections and extinction coefficients described in [CITATION], which are mostly based on ATLAS9 synthetic spectra, but with two important updates for cool giants: M giants now come from an extended database from Aringer et al. (in prep.)', '1406.0676-2-102-2': 'that covers the all relevant space of parameters ([MATH], [MATH], and [Fe/H]).', '1406.0676-2-102-3': 'For C-type stars, we adopt the [CITATION] library of C star models, interpolating inside the grids as a function of [MATH], [MATH], [Fe/H] and C/O ratio.', '1406.0676-2-102-4': "Radiation reprocessing by circumstellar dust shells in mass-losing stars are taken into account as in [CITATION], using the results of [CITATION]'s radiation transfer models for mixtures of 60% AlOx and 40% silicate, and of 85% amorphous Carbon and 15% silicon carbide (for M and C stars, respectively).", '1406.0676-2-102-5': 'Finally, the synthetic CMDs are corrected for distance and extinction, [MATH] using extinction coefficients from [CITATION], as listed in Table [REF].', '1406.0676-2-103-0': '## Accounting for Uncertainties in SFH', '1406.0676-2-104-0': 'To obtain a robust number ratio of TP-AGB stars to RGB stars as well as the range of probable LFs expected from a given TP-AGB model, one must account for the random uncertainties in SFH.', '1406.0676-2-104-1': 'We synthesize at least 50 stellar populations with SFHs that are randomly sampled within the uncertainties of the best fit SFH.', '1406.0676-2-104-2': 'Specifically, for each SFH sample, we take the value of SF in each time bin as a random draw of a Gaussian distribution whose mean is the best fit SFR in that time bin and whose [MATH] is the uncertainty associated with that time bin.', '1406.0676-2-104-3': 'If the SFR is zero in the time bin, we adopt only positive uncertainties.', '1406.0676-2-105-0': 'Figure [REF] shows an example of 50 SFH realizations based on the MATCH-derived SFH and hybrid Monte Carlo uncertainties .', '1406.0676-2-105-1': 'The effect of the randomly sampled SFH can be seen in the Figures [REF]-[REF] as a spread in LF at bright magnitudes.', '1406.0676-2-105-2': 'Accounting for uncertainties in the SFH shows a clearer picture of the model predictions on the LF by introducing a spread in the number of stars that are expected to be found in each magnitude bin on a LF.', '1406.0676-2-105-3': 'The effect on the mean [MATH] ratio is to consistently produce a standard deviation of [MATH], independent of the TP-AGB model.', '1406.0676-2-106-0': '## Creating LFs of the Galaxy Sample', '1406.0676-2-107-0': 'For each SFH of each galaxy and each TP-AGB model, we generate a model stellar population with TRILEGAL of sufficient size to both to completely sample the IMF and to have at least twice the number of RGB stars in the sample as there are in the data.', '1406.0676-2-107-1': 'We then correct for the discrepancy in total stellar mass by scaling the model LF by the number of stars in the RGB region of the data (defined in Section [REF]).', '1406.0676-2-107-2': 'In other words, we multiply the simulated LF by a factor [MATH], such that, [EQUATION]', '1406.0676-2-107-3': 'Example LFs of an un-scaled simulation are shown in Figure [REF], all simulated LFs in the figures following are scaled.', '1406.0676-2-108-0': 'To test the adequacy of the TP-AGB model, we compare the amplitude and the shape of the model LF to the observations.', '1406.0676-2-108-1': 'The number of stars in the model LF are compared to that in the data by calculating the [MATH] ratio as described in Section [REF].', '1406.0676-2-108-2': 'The [MATH] ratio is related to the average TP-AGB lifetime of the observed population, as it combines all the star counts into one data point.', '1406.0676-2-109-0': 'The [MATH] ratio is a useful first comparison to make between models, as well as comparisons to other studies (e.g., G10).', '1406.0676-2-109-1': 'A successful TP-AGB model must also match the shape of the observed LF.', '1406.0676-2-109-2': 'Therefore, we also compare the predicted LF with those of the observations.', '1406.0676-2-110-0': 'We calculate the Poisson-equivalent of the Gaussian [MATH] statistic in two regions of the LF to compare the model LF shape to that in the data.', '1406.0676-2-110-1': 'The first region is the "full LF" that is, from the 90% completeness magnitude and brighter (including the regions excluded from the [MATH] ratio calculation).', '1406.0676-2-110-2': 'The second region is only the TP-AGB region, defined as brighter than a small offset above the TRGB (0.1 mag in [MATH] and 0.2 mag in [MATH]; see Section [REF]).', '1406.0676-2-111-0': '# Analysis', '1406.0676-2-112-0': '## Ratio of TP-AGB to RGB stars', '1406.0676-2-113-0': 'Tables [REF] and [REF] list the mean [MATH] and standard deviation of their Poisson uncertainties calculated from 50 simulations for each galaxy for each TP-AGB model (columns 2, 4, and 6).', '1406.0676-2-113-1': 'Next to each mean [MATH] ratio are the fractional difference between the mean model ratio and that found in the data (columns 3, 5, and 7).', '1406.0676-2-113-2': 'We define the fractional difference to be [MATH]N_TP-AGB/N_RGB[MATH]N_TP-AGB/N_RGB[MATH].', '1406.0676-2-113-3': 'Therefore, a value of [MATH] would be perfect agreement between data and model while [MATH] would mean the model is overpredicting the number, and thus, the lifetimes of TP-AGB stars.', '1406.0676-2-114-0': 'For each individual galaxy, the [MATH] and [MATH] models consistently overpredict the [MATH] ratio.', '1406.0676-2-114-1': 'On average, the [MATH] mass-loss prescription overpredicts the number of optical TP-AGB stars by nearly a factor of 3 and the [MATH] mass-loss law overpredicts them by more than a factor of 2.', '1406.0676-2-114-2': "In the NIR, the overpredictions of TP-AGB stars are similar or higher, nearly a factor of 3 when not accounting for pre-dust mass-loss, and a factor of 2.5 when only including Reimers' mass-loss.", '1406.0676-2-114-3': 'The [MATH] model, however, is on average consistent or slightly lower within uncertainties to the observed [MATH] ratio in both filters.', '1406.0676-2-115-0': "The conclusion from comparing the [MATH] ratio from model to model, is that there are too many TP-AGB stars predicted by the mass-loss prescriptions that neglect pre-dust mass-loss, or assume only Reimers' relation.", '1406.0676-2-115-1': 'In contrast, pre-dust mass-loss in the [MATH] model is most consistent with the average lifetimes of observed TP-AGB stars.', '1406.0676-2-116-0': '## Luminosity Functions in the Optical and NIR', '1406.0676-2-117-0': 'Figures [REF], [REF], and [REF] show a set of panels with optical (left) and IR (right) LFs for each galaxy in the sample.', '1406.0676-2-117-1': 'Observations (corrected for completeness; red) are shown with Poisson uncertainties for each of the 50 model LFs per panel overplotted (grey).', '1406.0676-2-117-2': 'The regions around the TRGB which were excluded in calculating the [MATH] ratio are shaded (see Section [REF]).', '1406.0676-2-117-3': 'The fainter magnitudes between 90%-50% completeness are also shaded; they are never included in the analysis, though the RGB and RC are important constraints used to derive the SFH and best fit metallicity enrichment law.', '1406.0676-2-118-0': 'To propagate the completeness and photometric uncertainties from the observations to the model LFs, we make use of the uncertainties reported by MATCH in the derived SFHs (which include uncertainties from the artificial star tests).', '1406.0676-2-118-1': 'An example set of SFHs is shown in Figure [REF].', '1406.0676-2-119-0': 'Qualitatively, the model and data LF agree in the RGB region (between the shaded regions) and diverge at fainter magnitudes.', '1406.0676-2-119-1': 'The agreement around the RGB is by design, as discussed in Section [REF], because the model LFs are scaled to match the total number of observed RGB stars in this region.', '1406.0676-2-119-2': 'However, there are two observational sources that give rise to the disagreement between data and scaled models fainter than [MATH] completeness.', '1406.0676-2-119-3': 'The first observational source is decreasing completeness with increasing magnitude, and the second cause is due to photometric uncertainties (median uncertainties in each CMD are shown in each panel of Figure [REF]).', '1406.0676-2-120-0': 'In general, the model LFs follow the results in Section [REF] for the [MATH].', '1406.0676-2-120-1': 'The [MATH] and [MATH] LFs predict far more TP-AGB stars than are observed in each magnitude bin (the only exception is for bright TP-AGB stars in NGC2976 in the NIR).', '1406.0676-2-120-2': 'The [MATH] model however, shows excellent agreement in the TP-AGB region of LF compared to the observed LF in many cases.', '1406.0676-2-121-0': 'To robustly compare one TP-AGB model to another, we calculate the Poission-like [MATH] likelihood that the observational LF is randomly drawn from the model LF.', '1406.0676-2-121-1': 'Figure [REF] shows the results of this calculation for only the TP-AGB region compared to the same region in the data (top panels) and the LF brighter than the 90% completeness limit (bottom panels) for the optical (left panels) and the NIR (right panels).', '1406.0676-2-121-2': 'Each point of Figure [REF] represents the mean value of the [MATH] calculated individually for each of the 50 model LFs, with uncertainties corresponding to the standard deviation of the mean.', '1406.0676-2-122-0': 'The absolute placement of the [MATH] values for each galaxy are likely dominated by the goodness of the SFH recovery mentioned in Section [REF].', '1406.0676-2-122-1': 'For example, NGC 2976 has the largest effective [MATH], corresponding to the least-good SFH fit, while SCL-DE1 is the opposite.', '1406.0676-2-122-2': 'The same trend is seen in Figure [REF] but using TRILEGAL simulated LFs.', '1406.0676-2-122-3': 'Therefore, it is more meaningful to consider the relative [MATH] values in Figure [REF] to compare TP-AGB models.', '1406.0676-2-122-4': 'The [MATH] values follow the qualitative picture from the LF agreement described above.', '1406.0676-2-122-5': 'The [MATH] mass-loss produces the most consistent LF compared to that observed.', '1406.0676-2-123-0': 'With the success in the [MATH] mass-loss prescription we can now constrain the expected lifetimes of low metallicity, low mass TP-AGB stars.', '1406.0676-2-123-1': 'Figure [REF] shows the corresponding TP-AGB lifetimes for low and intermediate metallicities, for the [MATH] model.', '1406.0676-2-123-2': 'The left panel shows the entire lifetime of the TP-AGB, including the amount of time spent below the TRGB.', '1406.0676-2-123-3': 'The right panel shows the expected lifetimes of TP-AGB stars above the TRGB.', '1406.0676-2-123-4': 'Figure [REF] shows the distribution of TP-AGB masses from each of the best fitting LFs for each galaxy.', '1406.0676-2-123-5': 'Therefore, based on the number of observed TP-AGB stars in six galaxies that are low metallicity and have little recent SFH, the lifetimes of the typical TP-AGB star in our sample ([MATH]M_[MATH] or [MATH]) will be less than 1.2 Myr.', '1406.0676-2-123-6': 'For stars of mass [MATH]M_[MATH], we expect the TP-AGB lifetime to be less than half a Myr, as the star will expel much of its atmosphere during the pre-dust driven phase of the TP-AGB.', '1406.0676-2-124-0': '## The Initial and Final Mass Relationship', '1406.0676-2-125-0': 'As mentioned in G10, one of the clearest constraints on the evolution of low-mass low-metallicity TP-AGB stars is given by the (few) measured masses of white dwarfs (WDs) in globular clusters, and in particular those in M4, for which [CITATION] derived a mean mass of [MATH]M_[MATH].', '1406.0676-2-125-1': 'Assuming that M4 has a [Fe/H]=-1.07 and [[MATH]/Fe]=0.39 dex , it should be well represented by PARSEC tracks of [MATH].', '1406.0676-2-125-2': 'For this metallicity, the track which lifetime best fits the 12 Gyr age expected for globular clusters is the one with an initial mass of 0.85 [MATH], which takes 11.7 Gyr to evolve from the zero-age MS to the WD cooling ages of the observed M4 white dwarfs .', '1406.0676-2-125-3': 'This track finishes the TP-AGB with a remnant mass of 0.547 [MATH], which is just slightly larger than the mean value determined by [CITATION].', '1406.0676-2-125-4': 'The small difference in final mass can be easily explained by invoking a small additional amount of mass-loss on the RGB.', '1406.0676-2-125-5': 'We emphasize however that calibrating the RGB mass-loss to a precision of a few hundredths of solar masses, is certainly beyond the scope of this paper.', '1406.0676-2-126-0': 'lrrrrrrr', '1406.0676-2-127-0': 'Mean Optical [MATH] Ratios', '1406.0676-2-128-0': 'Target & [MATH] & Frac.', '1406.0676-2-128-1': 'Difference & [MATH] & Frac.', '1406.0676-2-128-2': 'Difference & [MATH] & Frac.', '1406.0676-2-128-3': 'Difference', '1406.0676-2-129-0': 'DDO71 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-130-0': 'DDO78 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-131-0': 'HS117 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-132-0': 'KKH37 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-133-0': 'NGC2976 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-134-0': 'SCL-DE1 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-135-0': 'Mean & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] Mean optical TP-AGB to RGB model ratios and their fractional differences compared to the data.', '1406.0676-2-135-1': 'For each galaxy for each TP-AGB model, 50 TRILEGAL simulations produced a model stellar population from the best fit SFH and its uncertainties (see Section [REF]).', '1406.0676-2-135-2': 'The only change from one set of simulations to the other is the TP-AGB model.', '1406.0676-2-135-3': 'The total fractional differences are calculated compared to the total [MATH] ratio in the data listed in Table [REF].', '1406.0676-2-136-0': 'lrrrrrrr', '1406.0676-2-137-0': 'Mean NIR [MATH] Ratios', '1406.0676-2-138-0': 'Target & [MATH] & Frac.', '1406.0676-2-138-1': 'Difference & [MATH] & Frac.', '1406.0676-2-138-2': 'Difference & [MATH] & Frac.', '1406.0676-2-138-3': 'Difference', '1406.0676-2-139-0': 'DDO71 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-140-0': 'DDO78 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-141-0': 'HS117 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-142-0': 'KKH37 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-143-0': 'NGC2976 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-144-0': 'SCL-DE1 & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1406.0676-2-145-0': 'Mean & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] Same as Table [REF] for WFC3/IR data.', '1406.0676-2-146-0': '# Conclusions', '1406.0676-2-147-0': 'We have extended the analysis from G10, and confirmed that pre-dust mass-loss plays an important role in the TP-AGB evolution of low-mass metal-poor stars.', '1406.0676-2-147-1': 'We have shown that neglecting this phase of mass-loss altogether will overpredict the numbers of TP-AGB stars by a factor of [MATH].', '1406.0676-2-147-2': "We also showed that assuming Reimers' scaling relation to describe the pre-dust mass-loss phase will also overpredict the number to TP-AGB stars but by a factor of [MATH].", '1406.0676-2-147-3': 'Following recent results of detailed models that describe the chromospheric winds of red giants, we implemented a revised mass-loss prescription with a stronger dependence on the effective temperature.', '1406.0676-2-147-4': 'Using the [MATH] ratio and comparing LFs, we showed this implementation reaches good to excellent agreement with the data.', '1406.0676-2-147-5': 'Moreover, the [MATH] mass-loss prescription further lowers the expected TP-AGB lifetime for low-mass, low-metallicity TP-AGB stars.', '1406.0676-2-148-0': 'An interesting outcome of the our analysis is the dependence on the initial chemical composition: TP-AGB lifetimes of low-mass stars are expected to become shorter at decreasing metallicity.', '1406.0676-2-148-1': 'We also show that, given our calibrated mass-loss relation, the efficiency of the third dredge-up has little effect on TP-AGB lifetimes in this regime.', '1406.0676-2-149-0': 'This paper represents the first step in a major calibration of the COLIBRI and PARSEC codes which will aid in the understanding of the physics involved in TP-AGB evolution.', '1406.0676-2-149-1': 'Upon the availability of PARSEC v2, which will include higher mass stellar evolution tracks, this study will be extended to include the 17 other galaxies in the AGB-SNAP sample.', '1406.0676-2-149-2': 'This addition will expand the metallicities available to test and increase the mass range of our sample to the complete sample of TP-AGB stars.', '1406.0676-2-150-0': 'These improvements to the TP-AGB models will be included in sets of isochrones and tools to synthesize stellar populations available on the CMD and TRILEGAL websites.'} | [['1406.0676-1-25-0', '1406.0676-2-25-0'], ['1406.0676-1-25-1', '1406.0676-2-25-1'], ['1406.0676-1-25-2', '1406.0676-2-25-2'], ['1406.0676-1-25-3', '1406.0676-2-25-3'], ['1406.0676-1-37-0', '1406.0676-2-37-0'], ['1406.0676-1-37-1', '1406.0676-2-37-1'], ['1406.0676-1-37-2', '1406.0676-2-37-2'], ['1406.0676-1-75-0', '1406.0676-2-75-0'], ['1406.0676-1-94-0', '1406.0676-2-94-0'], ['1406.0676-1-94-1', '1406.0676-2-94-1'], ['1406.0676-1-94-2', '1406.0676-2-94-2'], ['1406.0676-1-94-3', '1406.0676-2-94-3'], ['1406.0676-1-94-4', '1406.0676-2-94-4'], ['1406.0676-1-36-0', '1406.0676-2-36-0'], ['1406.0676-1-36-1', '1406.0676-2-36-1'], ['1406.0676-1-36-2', '1406.0676-2-36-2'], ['1406.0676-1-36-3', '1406.0676-2-36-3'], ['1406.0676-1-36-4', '1406.0676-2-36-4'], ['1406.0676-1-91-0', '1406.0676-2-91-0'], ['1406.0676-1-91-1', '1406.0676-2-91-1'], ['1406.0676-1-40-0', '1406.0676-2-40-0'], ['1406.0676-1-40-1', '1406.0676-2-40-1'], ['1406.0676-1-40-2', 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'1406.0676-1-44-1', '1406.0676-1-44-2', '1406.0676-1-45-0', '1406.0676-1-46-0', '1406.0676-1-47-0', '1406.0676-1-48-0', '1406.0676-1-49-0', '1406.0676-1-50-0', '1406.0676-1-51-0', '1406.0676-1-52-0', '1406.0676-1-53-0', '1406.0676-1-54-0', '1406.0676-1-55-0', '1406.0676-1-56-0', '1406.0676-1-57-0', '1406.0676-1-58-0', '1406.0676-1-59-0', '1406.0676-1-60-0', '1406.0676-1-61-0', '1406.0676-1-78-1', '1406.0676-1-80-0', '1406.0676-1-81-0', '1406.0676-1-126-0', '1406.0676-1-127-0', '1406.0676-1-128-0', '1406.0676-1-128-1', '1406.0676-1-128-2', '1406.0676-1-128-3', '1406.0676-1-129-0', '1406.0676-1-130-0', '1406.0676-1-131-0', '1406.0676-1-132-0', '1406.0676-1-133-0', '1406.0676-1-134-0', '1406.0676-1-136-0', '1406.0676-1-137-0', '1406.0676-1-138-0', '1406.0676-1-138-1', '1406.0676-1-138-2', '1406.0676-1-138-3', '1406.0676-1-139-0', '1406.0676-1-140-0', '1406.0676-1-141-0', '1406.0676-1-142-0', '1406.0676-1-143-0', '1406.0676-1-144-0', '1406.0676-1-145-0', '1406.0676-2-41-0', '1406.0676-2-42-0', '1406.0676-2-43-0', '1406.0676-2-44-0', '1406.0676-2-44-1', '1406.0676-2-44-2', '1406.0676-2-45-0', '1406.0676-2-46-0', '1406.0676-2-47-0', '1406.0676-2-48-0', '1406.0676-2-49-0', '1406.0676-2-50-0', '1406.0676-2-51-0', '1406.0676-2-52-0', '1406.0676-2-53-0', '1406.0676-2-54-0', '1406.0676-2-55-0', '1406.0676-2-56-0', '1406.0676-2-57-0', '1406.0676-2-58-0', '1406.0676-2-59-0', '1406.0676-2-60-0', '1406.0676-2-61-0', '1406.0676-2-78-1', '1406.0676-2-80-0', '1406.0676-2-81-0', '1406.0676-2-126-0', '1406.0676-2-127-0', '1406.0676-2-128-0', '1406.0676-2-128-1', '1406.0676-2-128-2', '1406.0676-2-128-3', '1406.0676-2-129-0', '1406.0676-2-130-0', '1406.0676-2-131-0', '1406.0676-2-132-0', '1406.0676-2-133-0', '1406.0676-2-134-0', '1406.0676-2-136-0', '1406.0676-2-137-0', '1406.0676-2-138-0', '1406.0676-2-138-1', '1406.0676-2-138-2', '1406.0676-2-138-3', '1406.0676-2-139-0', '1406.0676-2-140-0', '1406.0676-2-141-0', '1406.0676-2-142-0', '1406.0676-2-143-0', '1406.0676-2-144-0', '1406.0676-2-145-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1406.0676 | null | null | null | null | null |
1812.07170 | {'1812.07170-1-0-0': 'Bug fixing is generally a manually-intensive task.', '1812.07170-1-0-1': 'However, recent work has proposed the idea of automated program repair, which aims to repair (at least a subset of) bugs in different ways such as code mutation, etc.', '1812.07170-1-0-2': 'Following in the same line of work as automated bug repair, in this paper we aim to leverage past fixes to propose fixes of current/future bugs.', '1812.07170-1-0-3': 'Specifically, we propose Ratchet, a corrective patch generation system using neural machine translation.', '1812.07170-1-0-4': 'By learning corresponding pre-correction and post-correction code in past fixes with a neural sequence-to-sequence model, Ratchet is able to generate a fix code for a given bug-prone code query.', '1812.07170-1-0-5': 'We perform an empirical study with five open source projects, namely Ambari, Camel, Hadoop, Jetty and Wicket, to evaluate the effectiveness of Ratchet.', '1812.07170-1-0-6': 'Our findings show that Ratchet can generate syntactically valid statements 98.7% of the time, and achieve an F1-measure between 0.41-0.83 with respect to the actual fixes adopted in the code base.', '1812.07170-1-0-7': 'In addition, we perform a qualitative validation using 20 participants to see whether the generated statements can be helpful in correcting bugs.', '1812.07170-1-0-8': "Our survey showed that Ratchet's output was considered to be helpful in fixing the bugs on many occasions, even if fix was not 100% correct.", '1812.07170-1-1-0': 'patch generation, corrective patches, neural machine translation, change reuse.', '1812.07170-1-2-0': '# Introduction', '1812.07170-1-3-0': 'Most software bug fixing tasks are manual and tedious.', '1812.07170-1-3-1': 'Recently, a number of techniques related to automated program repair have been proposed to help automate and reduce the burden of some of these tasks [CITATION].', '1812.07170-1-3-2': 'These systems are also seeing practical use.', '1812.07170-1-3-3': 'For example, Facebook has announced that they started applying a system of automated program repair called SapFix in their large-scale products [CITATION].', '1812.07170-1-4-0': 'However, there are limitations in current approaches to automated program repair.', '1812.07170-1-4-1': 'First, there is a risk of overfitting to the training set (and breaking under tested functionality) in patch generation, especially generated tests tends to lead overfitting compared to human-generated, requirements-based test suites [CITATION].', '1812.07170-1-4-2': 'Second, correct patches may not exist in the search space, or correct patches cannot be generated because the search space is huge [CITATION].', '1812.07170-1-4-3': 'Several studies address this search space issue by making use of existing human-written patches [CITATION], but those generated patches need to be validated with test suites.', '1812.07170-1-4-4': 'Therefore, investigating techniques that assist in the generation of patches without the need for tests, etc. are needed.', '1812.07170-1-4-5': 'Instead of exploring fix ingredients in the search space (search-based), we study the possibility of learning fix ingredients from past fixes (learning-based).', '1812.07170-1-5-0': 'Recently, Neural Machine Translation (NMT) has been proposed and showed promising results in various areas including not only translation between natural languages (such as English and Japanese), but also other NLP tasks such as speech recognition [CITATION], natural language parsing [CITATION], and text summarization [CITATION].', '1812.07170-1-5-1': 'Similar techniques have been applied to code-related tasks [CITATION].', '1812.07170-1-5-2': 'The notable success of NMT in such a wide variety of tasks can be attributed to several traits: (1) It is an end-to-end machine learning framework that can be learned effectively from large data - if we have a large enough data source it is able to learn even complicated tasks such as translation in an effective way.', '1812.07170-1-5-3': '(2) Unlike previous models for translation such as phrase-based translation [CITATION] (which has also been used in code-related tasks such as language porting [CITATION]), NMT is able to take a holistic look at the entire input and make global decisions about which words or tokens to output.', '1812.07170-1-5-4': 'In particular, for bug fixing we posit this holistic view of the entire hunk of code we attempt to fix is important, and thus focus on approaches using NMT in this work.', '1812.07170-1-6-0': 'Hence, in this paper, we propose Ratchet, a NMT-based technique that generates bug fixes based on prior bug-and-fix examples.', '1812.07170-1-6-1': 'To evaluate the effectiveness of the technique, we perform an empirical study with five large software projects, namely Ambari, Camel, Hadoop, Jetty and Wicket.', '1812.07170-1-6-2': 'We use the previous Plastic Surgery approach, proposed by Barr et al. [CITATION], as a comparison baseline and examine the effectiveness of our NMT-based technique.', '1812.07170-1-6-3': 'In particular, we quantify the number of cases where our NMT-based technique is able to generate a valid fix and how accurate the generated fixes are.', '1812.07170-1-6-4': 'Our findings showed that Ratchet is able to generate a valid statements in 98.7% of the cases and achieves an F1 measure between 0.41 - 0.83 with respect to the actual fixes adopted in the code base.', '1812.07170-1-6-5': 'For all five projects, Ratchet was able to either outperform or perform as well as the Plastic Surgery approach.', '1812.07170-1-7-0': 'In addition to the quantitative validation, we also performed a survey with 20 participants to see whether the generated statements can help in correcting a bug (even if they were not 100% identical to the fix).', '1812.07170-1-7-1': 'Our findings show that survey participants find that the fixes generated by Ratchet are very helpful, even if they were not fully correct (although the correct fixes were most helpful).', '1812.07170-1-8-0': 'The rest of the paper is organized as follows.', '1812.07170-1-8-1': 'Section [REF] presents relevant terminology.', '1812.07170-1-8-2': 'Section [REF] provides background about NMT.', '1812.07170-1-8-3': 'Section [REF] details our approach.', '1812.07170-1-8-4': 'Section [REF] sets up our experiments, discussing their design and the data used.', '1812.07170-1-8-5': 'Section [REF] presents our results and Section [REF] discusses the generality and some challenges facing NMT-based solutions.', '1812.07170-1-8-6': 'Related work is presented and contrasted in Section [REF] and Section [REF] concludes the paper.', '1812.07170-1-9-0': '# Terminology', '1812.07170-1-10-0': 'We use the term, change hunk, similar to the previous study by Ray et al. [CITATION].', '1812.07170-1-10-1': 'A change hunk is a list of program statements deleted and added contiguously.', '1812.07170-1-10-2': 'In a single commit to a code repository, typically there are multiple change regions in multiple files.', '1812.07170-1-10-3': 'Even in a single file, there can be multiple change regions.', '1812.07170-1-10-4': 'Those changed regions can be identified with diff.', '1812.07170-1-10-5': 'Although the previous study by Ray et al. included unchanged statements in a change hunk [CITATION], we do not include them.', '1812.07170-1-10-6': 'We call deleted and added statements pre-correction and post-correction statements respectively.', '1812.07170-1-10-7': 'In Listing [REF], the red statement is a pre-correction statement and the green statement is a corresponding post-correction statement, and these associated two statements are considered to be a change hunk.', '1812.07170-1-11-0': 'In this study, we are interested in learning transforming patterns between corresponding pre-correction and post-correction statements.', '1812.07170-1-11-1': 'Thus, we ignore change hunks that only contain deleted or added statements.', '1812.07170-1-11-2': 'All change hunks studied in this paper are pairs of pre-correction and post-correction statements.', '1812.07170-1-12-0': '# Background', '1812.07170-1-13-0': 'Neural machine translation, also called neural sequence-to-sequence models [CITATION] is a method for converting one input sequence [MATH] into another output sequence [MATH] using neural networks.', '1812.07170-1-13-1': 'As the name suggests, the method was first conceived for and tested on machine translation; for converting one natural language (e.g. English) into another (e.g. French).', '1812.07170-1-13-2': 'However, because these methods can work on essentially any problem of converting one sequence into another, they have also been applied to a wide variety of other tasks such as speech recognition [CITATION], natural language parsing [CITATION], and text summarization [CITATION].', '1812.07170-1-13-3': 'They have also seen applications to software for generation of natural language comments from code [CITATION], generation of code from natural language [CITATION], generation of API sequences [CITATION], and suggesting fixes to learner code in programming MOOCs [CITATION].', '1812.07170-1-14-0': 'In this section we briefly overview neural networks, then explain neural machine translation in detail.', '1812.07170-1-15-0': '## Neural Networks', '1812.07170-1-16-0': 'Neural networks [CITATION], put simply, are a complicated function that is composed of simpler component parts that each have parameters that control their behavior.', '1812.07170-1-16-1': 'One common example of such a function is the simple multi-layer calculation below, which converts an input vector [MATH] into an output vector [MATH]: [EQUATION]', '1812.07170-1-16-2': 'Here, [MATH] and [MATH] are parameter matrices, and [MATH] and [MATH] are parameter vectors (called bias vectors).', '1812.07170-1-16-3': 'Importantly, the vector [MATH] is a hidden layer of the neural network, which results from multiplying [MATH], adding [MATH], then taking the hyperbolic tangent with respect to the input.', '1812.07170-1-16-4': 'This hidden layer plays an essential role in neural networks, as it allows the network to automatically discover features of the input that may be useful in predicting [MATH].', '1812.07170-1-17-0': 'Because neural networks have parameters ([MATH], [MATH], etc.) that specify their behavior, it is necessary to learn these parameters from training data.', '1812.07170-1-17-1': 'In general, we do so by calculating how well we do in predicting the correct answer [MATH] provided by the training data, and modify the parameters to increase our prediction accuracy.', '1812.07170-1-17-2': "Formally, we do so by calculating a loss function [MATH] which will (generally) be 0 if we predict perfectly, and higher if we're not doing a good job at prediction.", '1812.07170-1-17-3': 'We then take the derivative of this loss function with respect to the parameters, e.g. [MATH], and move the parameters in the direction to reduce the loss function, e.g. [EQUATION] where [MATH] is a learning rate that controls how big of a step we take after every update.', '1812.07170-1-18-0': 'The main difficulty here is that we must calculate derivatives [MATH].', '1812.07170-1-18-1': 'Even for a relatively simple function such as the one in ([REF]), calculating the derivative by hand can be cumbersome.', '1812.07170-1-18-2': 'Fortunately, this problem can be solved through a process of back-propagation (or auto-differentiation), which calculates the derivative of the whole complicated function by successively calculating derivatives of the smaller functions and multiplying them together using the chain rule [CITATION].', '1812.07170-1-18-3': 'Thus, it becomes possible to train arbitrarily complicated functions, as long as they are composed of simple component parts that can be differentiated, and a number of software libraries such as TensorFlow [CITATION] and DyNet [CITATION] make it possible to easily do so within applications.', '1812.07170-1-19-0': '## Neural Machine Translation', '1812.07170-1-20-0': 'Neural machine translation is an example of applying a complicated function learnable by neural nets and using it to solve a complicated problem: translation.', '1812.07170-1-20-1': 'To generate an output [MATH] (e.g. corrected hunk of code) given an input [MATH], these models incrementally generate each token in the output [MATH] one at a time.', '1812.07170-1-20-2': 'For example, if our output is "return this .', '1812.07170-1-20-3': 'index", the model would first predict and generate "return", then "this", then ".', '1812.07170-1-20-4': '", etc.', '1812.07170-1-20-5': 'This is done in a probabilistic way by calculating the probability of the first token of the output given the input [MATH], outputting the token in the vocabulary that maximizes this probability, then calculating the probability of the second token given the first token and the snippet [MATH] and similarly outputting the word with the highest probability, etc.', '1812.07170-1-20-6': 'When training the model, we already know a particular output [MATH] and want to calculate its probability given a particular snippet [MATH] so we can update the parameters based on the derivatives of this probability.', '1812.07170-1-20-7': 'To do so, we simply multiply these probabilities together using the chain rule as follows: [EQUATION]', '1812.07170-1-20-8': 'So how do neural MT models calculate this probability?', '1812.07170-1-20-9': 'We will explain a basic outline of a basic model called the encoder-decoder model [CITATION], and refer readers to references for details [CITATION].', '1812.07170-1-20-10': 'The encoder-decoder model, as shown in Figure [REF] works in two stages: first it encodes the input (in this case [MATH]) into a hidden vector of continuous numbers [MATH] using an encoding function [EQUATION]', '1812.07170-1-20-11': 'This function generally works in two steps: looking up a vector of numbers representing each token (often called "word embeddings" or "word vectors"), then incrementally adding information about these embeddings one token at a time using a particular variety of network called a recurrent neural network (RNN).', '1812.07170-1-20-12': 'To take the specific example shown in the figure, at the first time step, we would look up an embedding vector for the first token "return", [MATH] and then perform a calculation such as the one below to calculate the hidden vector for the first time step: [EQUATION] where [MATH] and [MATH] are a matrix and vector that are parameters of the model, and [MATH] is the hyperbolic tangent function used to "squish" the values to be between -1 and 1.', '1812.07170-1-21-0': 'In the next time step, we would do the same for the symbol "."', '1812.07170-1-21-1': ', using its embedding [MATH], and in the calculation from the second step onward we also use the result of the previous calculation (in this case [MATH]): [EQUATION]', '1812.07170-1-21-2': 'By using the hidden vector from the previous time step, the RNN is able to "remember" features of the previously occurring tokens within this vector, and by repeating this process until the end of the input sequence, it (theoretically) has the ability to remember the entire content of the input within this vector.', '1812.07170-1-22-0': 'Once we have encoded the entire source input, we can then use this encoded vector to predict the first token of the output.', '1812.07170-1-22-1': 'This is generally done by defining the first hidden vector for the output [MATH] to be equal to the final vector of the input [MATH], then multiplying it with another weight vector used for prediction to calculate a score [MATH] for each token in the output vocabulary: [EQUATION]', '1812.07170-1-22-2': 'We then predict the actual probability of the first token in the output statement, for example "return", by using the softmax function, which exponentiates all of the scores in the output vocabulary and then normalizes these scores so that they add to one: [EQUATION]', '1812.07170-1-22-3': 'We then calculate a new hidden vector given this input: [EQUATION]', '1812.07170-1-22-4': 'We continue this process recursively until we output a special "end of hunk" symbol "[MATH]/s[MATH]".', '1812.07170-1-23-0': 'Why neural MT models?', '1812.07170-1-23-1': ': As mentioned briefly in the intro, neural MT models are well-suited to the task of automatic patch generation for a number of reasons.', '1812.07170-1-23-2': 'First, they are an end-to-end probabilistic model that can be trained from parallel datasets of pre- and post-correction code without extra human intervention, making them easy to apply to new datasets or software projects.', '1812.07170-1-23-3': 'Second, they are powerful models that can learn correspondences on a variety of levels; from simple phenomena such as direct token-by-token matches, to soft paraphrases [CITATION], to weak correspondences between keywords and large documents for information retrieval [CITATION].', '1812.07170-1-23-4': 'Finally, they have demonstrated success in a number of code related tasks as iterated at the beginning of this section, which indicates that they should be useful as part of bug fixing algorithm as well.', '1812.07170-1-24-0': 'Attention: In addition, we use a neural MT model with this basic architecture, with the addition of a feature called attention, which, put simply, allows the model to "focus" on particular tokens in the input [MATH] when generating the output [MATH] [CITATION].', '1812.07170-1-24-1': 'Mathematically, this corresponds to calculating an "attention vector" [MATH], given the input hidden vectors [MATH] and the current output hidden vector [MATH].', '1812.07170-1-24-2': 'This vector consists of values between zero and one, one value for each word in the input, with values closer to one indicating that the model is choosing to focus more on that particular word.', '1812.07170-1-24-3': 'Finally, these values are used to calculate a "context vector" [EQUATION] which is used as additional information when calculating score [MATH].', '1812.07170-1-24-4': 'Attention is particularly useful when there are many token-to-token correspondences between the input and output, which we expect to be the case for our patch generation task, where the input and output code are likely to be very similar.', '1812.07170-1-24-5': 'This attention model can be further augmented to allow for exact copies of tokens [CITATION], or be used to incorporate a dictionary of common token-to-token correspondences (copies or replacements) [CITATION].', '1812.07170-1-24-6': 'In our model, we use the latter, which allows us to both capture the fact that tokens are frequently copied between pre- and post-correction code, and also the fact that some replacements will be particularly common (e.g. loadBalancerType to setLoadBalancerType).', '1812.07170-1-25-0': 'Implementation Details: As a specific implementation of the neural MT techniques listed above, we use the lamtram toolkit [CITATION].', '1812.07170-1-25-1': 'For reproducibility, we briefly list the parameters below, and interested readers can refer to the references for detail.', '1812.07170-1-25-2': 'As our model we use an encoder-decoder model with multi-layer perceptron attention [CITATION] and input feeding [CITATION], with encoders and decoders using a single layer of 512 LSTM cells [CITATION].', '1812.07170-1-25-3': 'We use the Adam optimizer [CITATION] with a learning rate of 0.001 and minibatch size of 2048 words, and decay the learning rate every time the development loss increases.', '1812.07170-1-25-4': 'To prevent overfitting, we use a dropout rate of 0.5 [CITATION].', '1812.07170-1-25-5': 'To generate our outputs, we perform beam search with a beam size of 10.', '1812.07170-1-26-0': '# Approach', '1812.07170-1-27-0': 'The idea of corrective patch generation using NMT considers code changes as translation from pre-correction code to post-correction code.', '1812.07170-1-27-1': 'Figure [REF] provides an overview of our system, Ratchet, which consists of two main parts: creating the training corpora, and generating patches using the trained model.', '1812.07170-1-27-2': 'In this paper, we target Java source code and focus on changes within Java methods.', '1812.07170-1-27-3': 'Particularly, the granularity of code we target is a statement similar to the previous study [CITATION].', '1812.07170-1-28-0': '## Extracting Change Hunks from Code Repositories', '1812.07170-1-29-0': 'In order to create our training corpora, we start by extracting pre- and post-correction statements using a sequence of steps.', '1812.07170-1-29-1': 'We detail each of these steps in the following text:', '1812.07170-1-30-0': 'Preparing Historage for method-level histories.', '1812.07170-1-30-1': 'Since the software repositories store the code modifications at the commit level, our first step is to transform these commits into method-level modifications.', '1812.07170-1-30-2': 'To do so, we convert the existing code repositories to historage repositories [CITATION].', '1812.07170-1-30-3': 'Historage creates a new repository that stores all methods in the logs of the original repository as individual Git objects.', '1812.07170-1-30-4': 'In essence, historage is a Git repository that allows us to operate any Git commands as usual.', '1812.07170-1-31-0': 'Collecting the modified methods.', '1812.07170-1-31-1': 'We use the command git log -diff-filter=M on the historage repositories to collect all modified methods in the entire history.', '1812.07170-1-31-2': 'The option -diff-filter=M will provide only modified (M) files, which are methods in historage repositories.', '1812.07170-1-31-3': 'Since we are interested in training our model on pre- and post-correction statements, we only consider methods that modify code, i.e., not methods that are newly created or completely deleted.', '1812.07170-1-32-0': 'Identifying change hunks.', '1812.07170-1-32-1': 'As stated in Section [REF], a change hunk is a pair of pre-correction and post-correction statements.', '1812.07170-1-32-2': 'We identify these change hunks from the outputs of the git diff.', '1812.07170-1-32-3': 'Since we assume pre-correction statements have been corrected to post-correction statements, we need to identify the corresponding line pairs appropriately.', '1812.07170-1-33-0': '## Preprocessing the Statement Corpora', '1812.07170-1-34-0': 'Before storing the statement pairs as pre-correction and post-correction statement corpora, we perform the following preprocessing steps.', '1812.07170-1-34-1': 'As seen in Figure [REF], the same processes will be applied to query statements except for the step (6) and (7), which are needed only for creating the corpora.', '1812.07170-1-35-0': '(1) Limit to single-statement changes and single-statement queries.', '1812.07170-1-35-1': 'In this study, we only consider single-statement (one-line) changes.', '1812.07170-1-35-2': 'We do so for the following three reasons.', '1812.07170-1-35-3': 'First, previous studies showed that most reusable code is found at the single-statement level [CITATION].', '1812.07170-1-35-4': 'Second, it is difficult to treat multiple statement changes (one-to-many, many-to-one, and many-to-many statement changes) for identifying pairs.', '1812.07170-1-35-5': 'Those multiple statement changes can have inappropriate corresponding statements.', '1812.07170-1-35-6': 'For example, if there exists one pre-correction statement and two post-correction statements in one change hunk, this change can be a single-statement change and one independent statement insertion.', '1812.07170-1-35-7': 'If we consider these statements one pair, the independently inserted statement can be noise in the training data.', '1812.07170-1-35-8': 'Third, it is difficult to manage past histories associated with multiple statements.', '1812.07170-1-35-9': 'Using the command git blame on historage, we identify commits on which deleted lines initially appeared.', '1812.07170-1-35-10': 'In general, multiple statements can have different past histories, which makes it difficult to treat those multiple statements as one unit.', '1812.07170-1-35-11': 'For all statement pairs, we collect past history information including the original commit, changed year and deleted year, to be used for our experiments.', '1812.07170-1-35-12': 'Although we apply this filtering, we found that single-statement changes are the majority in our change hunks (as we show later in Figure [REF] and Table [REF]).', '1812.07170-1-36-0': '(2) Tokenize statements.', '1812.07170-1-36-1': 'Since the NMT model requires separate tokens as input, we use the StreamTokenizer to tokenize the Java statements.', '1812.07170-1-37-0': '(3) Remove statement pairs or statement queries with less than three tokens.', '1812.07170-1-37-1': 'We remove statements that have very few tokens (i.e., less than 3) since they are less meaningful.', '1812.07170-1-37-2': 'Our observations indication that most such lines only contain opening or closing parenthesis.', '1812.07170-1-38-0': '(4) Replace the contents of method arguments with a special token.', '1812.07170-1-38-1': 'From our many trials, we realized that a wide variety of the contents of method arguments make it difficult to generate corresponding contents.', '1812.07170-1-38-2': 'This is because sometimes method argument contents include tokens that rarely appear.', '1812.07170-1-38-3': 'We replace method and array arguments with a special token, arg and val, respectively.', '1812.07170-1-39-0': '(5) Filter unparseable statement pairs and queries.', '1812.07170-1-39-1': 'There exist incomplete statements in our collected statements, e.g., when there is a long statement that is written across two lines, and only one line is changed.', '1812.07170-1-39-2': 'To remove these incomplete Java statements, we put each statement in a dummy method of a dummy class, and try parsing the class to get an AST using JavaParser.', '1812.07170-1-39-3': 'If we fail to parse classes with either pre- or post-correction statements, we filter out the failed statement pairs.', '1812.07170-1-40-0': '(6) Select post-correction statements from multiple candidates.', '1812.07170-1-40-1': 'This step is performed to address the nature of sequential order in documents.', '1812.07170-1-40-2': 'After collecting all pre- and post-correction statements from the entire history of a code repository, we can have statement pairs that have the same pre-correction statements but different post-correction statements.', '1812.07170-1-40-3': 'In order to allow the NMT models to effectively extract relationships or patterns, we chose only one post-correction statement for one pre-correction statement, and remove all other post-correction statements.', '1812.07170-1-40-4': 'The idea behind this selection is that it is better to learn from recently and frequently appearing statements.', '1812.07170-1-40-5': 'Given a pre-correction statement, we obtain post-correction statements that appeared in the most recent year.', '1812.07170-1-40-6': 'Then, from those newer statements, we select statements that most frequently appeared in the entire history.', '1812.07170-1-40-7': 'If we cannot break ties, we select the first statement in alphabetical order to make the process deterministic.', '1812.07170-1-41-0': '(7) Remove identical pre- and post-correction statements.', '1812.07170-1-41-1': 'After the above processes, there exist pairs of identical pre- and post-correction statements.', '1812.07170-1-41-2': 'For example, statement pairs from changes only within method arguments, and white space changes.', '1812.07170-1-41-3': 'We remove those statement pairs.', '1812.07170-1-42-0': '## Post-Processing', '1812.07170-1-43-0': 'Since we replace the contents of method arguments and replace it with a special token, the NMT model does not generate method arguments.', '1812.07170-1-43-1': 'However we expect that the method arguments of a query statement can be reused in the generated statement.', '1812.07170-1-43-2': 'Therefore we prepare the following heuristics for new method arguments.', '1812.07170-1-44-0': 'The lamtram toolkit provides scores associated with generated statements with the logarithm of a posteriori probability of output E given input F as [MATH].', '1812.07170-1-44-1': 'Those scores can be considered as confidences of the results.', '1812.07170-1-44-2': 'We empirically determine thresholds and ignore the generated statements with low scores.', '1812.07170-1-44-3': 'In addition, we can also ignore invalid generated statements that cannot be parsed.', '1812.07170-1-45-0': '# Experimental Setup', '1812.07170-1-46-0': 'In this section, we discuss our dataset and the design of our experiment.', '1812.07170-1-46-1': 'Particularly, we are interested in examining the viability of our approach in generating bug-fixing statements.', '1812.07170-1-46-2': 'To do so, we need to identify bug-fixing statement pairs.', '1812.07170-1-46-3': 'We discuss the tool used to identify the bug-inducing and bug-fixing commits that are used to determine our bug-fix statement pairs.', '1812.07170-1-46-4': 'Then, we provide descriptive statistics about the studied datasets.', '1812.07170-1-47-0': '## Subject Projects', '1812.07170-1-48-0': 'To perform our case study, we study five projects, namely Apache Ambari, Apache Camel, Apache Hadoop, Eclipse Jetty and Apache Wicket.', '1812.07170-1-48-1': 'We chose to study these five projects since they have a long development history and are large projects that contain many commits.', '1812.07170-1-48-2': 'Table [REF] shows the period considered, the number of commits, files and methods in our dataset.', '1812.07170-1-49-0': 'Figure [REF] shows the distribution of the number of pre- and post-correction statements in all change hunks.', '1812.07170-1-49-1': 'We find that most of changes are single statements in either insertion, deletion, or modification.', '1812.07170-1-49-2': 'Multi-statement changes are not frequent.', '1812.07170-1-49-3': 'Table [REF] shows the number of all change hunks and the number of change hunks that are derived from single-statement changes.', '1812.07170-1-49-4': 'We see from the table that approximately 62 - 68% of the changes are single-statement changes.', '1812.07170-1-49-5': 'This ratio of single-statement changes shows that although our NMT-based technique may not be applied to all changes, it is applicable to the majority of the changes.', '1812.07170-1-50-0': '## Experimental Design', '1812.07170-1-51-0': 'From the collected pre- and post-correction statements, we prepare the training data (Table [REF]) and testing data (Table [REF]).', '1812.07170-1-51-1': 'Considering the number of statements, we set the testing year for each project as shown in Table [REF].', '1812.07170-1-51-2': 'All statement pairs in each testing year are used as testing data, which means we chose statement pairs whose pre-correction statements are created in the testing year and changed to the corresponding post-correction statements in the same testing year.', '1812.07170-1-51-3': 'All years before the testing year are considered as training periods.', '1812.07170-1-51-4': 'In each training period, the numbers of statement pairs, whose pre-correction statements are changed to post-correction statements in the training period, are shown in Table [REF].', '1812.07170-1-52-0': 'This experimental design can be regarded as a simulation of generating corrected statements only by learning past histories when new statements are created and they will be modified soon (in the same year).', '1812.07170-1-52-1': 'If this works, we can prevent recurring or similar issues before being inserted into the code, or even when the code is being edited.', '1812.07170-1-53-0': '## Data Preparation', '1812.07170-1-54-0': 'Table [REF] details the impact of the various preprocessing steps on our approach.', '1812.07170-1-54-1': 'The before filtering row shows the number of all single-statement change pairs.', '1812.07170-1-54-2': 'The [MATH] tokens row shows the effect of removing statements that have less than 3 tokens.', '1812.07170-1-54-3': 'Then we remove the unparsable statements in both, pre-correction and post-correction statements.', '1812.07170-1-54-4': 'The final step removes identical statement pairs in the pre- and post-correction statements.', '1812.07170-1-54-5': 'The last row shows the final number of statements used in our study.', '1812.07170-1-55-0': 'In addition, we perform specific processing for the training and testing data, which we detail below:', '1812.07170-1-56-0': 'Replacing rare tokens in the training data.', '1812.07170-1-56-1': 'From the processed statement pairs, we prepare pre-correction statement corpus and post-correction statement corpus.', '1812.07170-1-56-2': 'For each corpus, tokens that appear only once are replaced with [MATH]unk[MATH], which is a common way to handle unknown tokens [CITATION].', '1812.07170-1-56-3': 'This script is available in the lamtram toolkit.', '1812.07170-1-57-0': 'Categorization of testing data.', '1812.07170-1-57-1': 'When testing our approach, we call the pre-correction statements in the testing data as queries.', '1812.07170-1-57-2': 'On the other hand, we call the post-correction statements as references.', '1812.07170-1-58-0': 'When we evaluate our approach, we separate the testing data with their characteristics.', '1812.07170-1-58-1': 'First, all statement pairs in the testing data are divided into bug-fix statement pairs and non-bug-fix statement pairs.', '1812.07170-1-58-2': 'This classification procedure is presented in Section [REF].', '1812.07170-1-58-3': 'Then both classes of statement pairs are categorized into three:', '1812.07170-1-59-0': '[NU:] No unknown.', '1812.07170-1-59-1': 'There are no unknown tokens in a statement pair.', '1812.07170-1-59-2': 'All tokens in a query statement appear in the pre-correction statement from the training data corpus, and all tokens in a reference statement appear in the post-correction statement of the training data corpus.', '1812.07170-1-60-0': '[UQ:] Unknown in query.', '1812.07170-1-60-1': 'One or more token(s) in the query statement do not appear in the pre-correction statement corpus.', '1812.07170-1-60-2': 'In other words, there are unknown tokens in the query.', '1812.07170-1-61-0': '[UR:] Unknown in reference.', '1812.07170-1-61-1': 'Although there is no unknown token in the query statement, there are one or more unknown token(s) in the reference, i.e., in the corresponding post-correction statement.', '1812.07170-1-62-0': 'We categorize the statements as shown above to know which data can be used in our experiments.', '1812.07170-1-62-1': 'This is particularly important since the trained NMT models have not seen unknown tokens during training, addressing queries in UQ or UR is very difficult.', '1812.07170-1-62-2': 'In fact, it is impossible for the NMT models to generate statements that are the exact same as the references for the UR category.', '1812.07170-1-63-0': 'Table [REF] shows the number of statement pairs for these categories of bug-fixing and non-bug fixing classes.', '1812.07170-1-63-1': "As can be seen from the Table [REF], the majority of the training data's statements fall in the UQ category (except for the Jetty project).", '1812.07170-1-63-2': 'On the other hand, the good news is that statements in the UR category are the least.', '1812.07170-1-63-3': 'We evaluate our approach using statements in the NU category.', '1812.07170-1-64-0': 'The dataset is available online.', '1812.07170-1-65-0': '## Identifying Bug-Fixing statements', '1812.07170-1-66-0': 'We collect bug-fixing statement pairs by identifying the pairs of bug-inducing and bug-fixing commits.', '1812.07170-1-66-1': 'To obtain these commits, we use Commit.guru [CITATION], a tool that analyzes and provides change level analytics.', '1812.07170-1-66-2': 'For full details about commit.guru, we point the reader to the paper by Rosen et al. [CITATION], however, here we describe the relevant details for our paper.', '1812.07170-1-66-3': 'Commit.guru takes as input a Git repository address, an original code repository in this study, and provides data for all commits of the project.', '1812.07170-1-66-4': 'It applies the SZZ algorithm [CITATION] to identify bug-inducing commits and their associated bug-fixing commits.', '1812.07170-1-66-5': 'In addition, Commit.guru provides a number of change level metrics related to the size of the change, the history of the files changed, the diffusion of the change and the experience of the developers making the modification.', '1812.07170-1-67-0': 'As mentioned earlier in step (1) of preprocessing (Section [REF]), we have meta information of statements including the original commits of post-correction statements and the original commits of pre-correction statements.', '1812.07170-1-67-1': 'We consider a pair of statements bug fixing if and only if a pre-correction statement is created in a bug-inducing commit and an post-correction statement is created in the associated bug-fixing commit.', '1812.07170-1-67-2': 'The other statement pairs are treated as non-bug-fix statements.', '1812.07170-1-67-3': 'We do not distinguish the types of the training data, that is, bug-fix or non-bug-fix.', '1812.07170-1-67-4': 'This is because we prefer to increase the training data available to the model and make the model learn from all varieties of changes.', '1812.07170-1-68-0': '# Evaluation', '1812.07170-1-69-0': 'We evaluate the performance of Ratchet with respect to two aspects: accuracy and usefulness of generated statements.', '1812.07170-1-69-1': 'In all of the results presented in this section, the NMT models are trained and tested with data from the same project (i.e., within-project evaluation).', '1812.07170-1-70-0': '## Can the models generate valid statements?', '1812.07170-1-71-0': 'Table [REF] shows the number (and percentage) of generated valid statements.', '1812.07170-1-71-1': 'We do not use thresholds here, that is, all generated statements including low scores are considered.', '1812.07170-1-71-2': 'As we see from the table, in most cases the models generated valid and complete Java statements.', '1812.07170-1-71-3': 'These high accuracy results are especially interesting since we did not explicitly teach the models the Java language specification.', '1812.07170-1-71-4': 'Simply, the models were able to achieve this high level of performance by themselves, using approx. 1,500 to 10,000 statement pairs.', '1812.07170-1-72-0': '.96In most cases of the five projects nearly 100 of the generated statements are valid Java statements.', '1812.07170-1-72-1': 'In total the models generated 230 valid statements for 233 queries (98.7).', '1812.07170-1-73-0': '## How accurate are the generated statements?', '1812.07170-1-74-0': 'In this section we evaluate the accuracy in a strict manner, that is, only generated statements that are identical to references are considered as correct.', '1812.07170-1-74-1': 'Our results are based on the NU (no unknown) category of statements, since as mentioned earlier, other categories are difficult (impossible for UR) to generate accurate statements that are identical to the reference statements.', '1812.07170-1-75-0': 'Before analyzing accuracy, we categorize outputs as the following three types:', '1812.07170-1-76-0': 'To measure the accuracy of generated results, we compute precision, recall, and [MATH], which are defined as: [MATH], [MATH], and [MATH], where [MATH] is the sum of [MATH] and [MATH].', '1812.07170-1-76-1': 'Higher precision indicates that the provided results are correct.', '1812.07170-1-76-2': 'Higher recall means that the results contain less NA but many correct.', '1812.07170-1-76-3': 'Providing a small number of results with high confidence can improve precision but lower recall.', '1812.07170-1-76-4': 'Since there is a tradeoff between precision and recall, [MATH], the harmonic mean of precision and recall, is also presented.', '1812.07170-1-77-0': 'We compare the accuracy of our NMT models with the previous state-of-the-art.', '1812.07170-1-77-1': 'Particularly, we compare to the Plastic Surgery approach [CITATION], i.e., patch recommendation with line-granular snippets that already exist in the training data.', '1812.07170-1-77-2': 'To do so, we examine whether a query statement exists in the pre-correction statement training data corpus, and if it does, we check whether the corresponding post-correction statement also exists (in the training data).', '1812.07170-1-77-3': 'If this happens, then we consider the statement to be covered by the Plastic Surgery approach.', '1812.07170-1-77-4': 'If there is no identical pre-correction statement, we mark the result as NA.', '1812.07170-1-78-0': 'When evaluating Ratchet and as stated in Section [REF], we use a threshold to ignore results with low confidence.', '1812.07170-1-78-1': 'Figure [REF] illustrates the transitions of [MATH] values with different thresholds (from -1.2 to -0.1).', '1812.07170-1-78-2': 'The solid lines are [MATH] values of Ratchet and the dotted lines are [MATH] values of the Plastic Surgery approach, which do not change with thresholds.', '1812.07170-1-78-3': 'We find that [MATH] values slightly change when we vary the thresholds.', '1812.07170-1-78-4': 'Lowering thresholds improves recall, however, it impacts the precision in the opposite direction.', '1812.07170-1-78-5': 'On the other hand, raising the threshold improves precision but makes recall worse.', '1812.07170-1-78-6': 'Based on our analysis of the threshold, we empirically set the threshold as 0.7 for the analyses that follow.', '1812.07170-1-79-0': 'Since Ratchet does not generate the contents of method arguments, we first evaluate the accuracy by ignoring those contents.', '1812.07170-1-79-1': 'To be fair, we also apply the same condition when evaluating the Plastic Surgery approach.', '1812.07170-1-79-2': 'Table [REF] shows the results of of our approach and compares it with the results with Plastic Surgery.', '1812.07170-1-79-3': 'We observe that the Plastic Surgery hypothesis holds in many cases, that is, changes (corrections) contain snippets that already exist in code repositories at the time of the changes, and these snippets can be efficiently found and exploited [CITATION].', '1812.07170-1-80-0': 'That said, Table [REF] shows that Ratchet improves the results in two projects and does not change in three projects.', '1812.07170-1-80-1': 'We observe that in camel, the results are greatly improved: 28 correct statements are generated compared with three patterns matching using the Plastic Surgery approach.', '1812.07170-1-80-2': 'Our results show that the NMT models work, as well as Plastic Surgery, if there are easily exploited statement-level patterns (i.e., reusable snippets), and works better than Plastic Surgery if there exist only finer-grained exploited patterns (i.e., fine-grained fixing patterns), which Plastic Surgery cannot use.', '1812.07170-1-81-0': 'Table [REF] presents examples of generated fixes that cannot be fixed by Plastic Surgery, but have a fix generated with our models.', '1812.07170-1-81-1': 'Sometimes the model learns the incrementation of value (query 1).', '1812.07170-1-81-2': 'Generics-related fixes are typical examples of successful generation with the NMT models (query 2 and 3).', '1812.07170-1-81-3': 'Sometimes it is preferred to remove this (query 4) if it makes the style consistent with the styling used in the specific project.', '1812.07170-1-81-4': "In fact, our models learned to remove the keyword 'this' because similar patterns were prevalent in the project's history.", '1812.07170-1-82-0': '.96NMT-based patch generation works better than Plastic Surgery, achieving [MATH] values between 0.41 to 0.83 for buggy queries.', '1812.07170-1-82-1': 'In total 157 correctly generated statements without method arguments, the contents of method arguments for 141 statements (89.8%) are correctly provided by reusing the contents of method arguments in queries.', '1812.07170-1-83-0': '## Can the generated statements help in correction?', '1812.07170-1-84-0': 'During the previous evaluation of accuracy, we considered the generated statements to be correct only if they are identical to the reference statements, otherwise they are considered to be incorrect or NA.', '1812.07170-1-84-1': 'To investigate whether the generated statements are useful, even if they are not identical to actual future corrections, we also perform a human evaluation with such (non-identical) corrections.', '1812.07170-1-85-0': 'We show survey participants the following three code snippets for one fix: i) an original problematic code snippet (before correction), ii) the actually fixed code snippet (after correction), and iii) a code snippet that is proposed as a fix by our NMT models.', '1812.07170-1-85-1': 'All code snippets contain one type of buggy or fixed statements with the surrounding statements.', '1812.07170-1-86-0': 'From the five projects, we collect ten corrections including five correctly and five incorrectly generated statements in the NU (no unknown) category, which are evaluated in Table [REF] and Table [REF].', '1812.07170-1-86-1': 'In addition, we collect five fixes that belong to the UQ (unknown in query) or the UR (unknown in reference) categories, which are known to be difficult for NMT models to generate.', '1812.07170-1-86-2': 'For simplicity, we call the above three groups correct fixes, incorrect fixes, and challenging fixes respectively.', '1812.07170-1-87-0': 'For each correction we prepare the following four statements, and ask the participants to evaluate using a five-level Likert scale scores from 1 (strongly disagree) to 5 (strongly agree) whether: (a) The proposed fix helps you to understand the required change, (b) The proposed fix can be a reference if you were to create your own fix, (c) The proposed fix is harmful or confusing, and (d) The proposed fix does not make sense and I will just ignore it.', '1812.07170-1-87-1': 'We asked not only positive impressions but also negative impressions in order to assess the usefulness and potential risks of incorrect generation.', '1812.07170-1-87-2': 'The survey material is available online.', '1812.07170-1-88-0': 'We recruited participants in Canada, US, and Japan, and 20 people participated in the survey including five undergraduate, 14 graduate students, and one professor.', '1812.07170-1-88-1': 'As Siegmund et al. reported that self estimation seems to be a reliable way to measure programming experience [CITATION], we asked the participants to estimate their experience in both, overall and Java programming experience.', '1812.07170-1-88-2': 'The participants can select any of 5 choices, varying between 1 (very inexperienced) to 5 (very experienced).', '1812.07170-1-88-3': 'Those who score 4 or 5 in both self estimation are considered to be high-experienced and others are considered to have low-experience.', '1812.07170-1-88-4': 'Five in six high experienced participants have more than five years of development experience, and the other have three-to-five years of experience.', '1812.07170-1-88-5': 'In 14 low-experience participants, the experience periods vary from less than one year, one-to-three years, three-to-five years, and more than five years.', '1812.07170-1-89-0': 'Figure [REF] shows the result of the correct fix group.', '1812.07170-1-89-1': 'The results shown in the figure show that the generated statements are useful.', '1812.07170-1-89-2': 'All high-experience and most of the low-experience participants agreed (scores 4 or 5 for questions (a) and (b)) that the correct fix statements helped them and that the statements and did not have negative effects (i.e., most scores are 1 or 2 for questions (c) and (d)).', '1812.07170-1-90-0': 'Figure [REF] shows the results of the incorrect fix group, which includes statements with incorrect method calls and/or incorrect generic types, for example.', '1812.07170-1-90-1': 'We assumed that these fixes are harmful or confusing because they tend to be partially the same as the references, but are slightly different from the actual fixes.', '1812.07170-1-90-2': 'However, as evidenced by the results shown in Figure [REF], the majority of highly and low experienced participants agree to that such imperfect statements may still be helpful (i.e., by providing positive answers to statement (a) and (b)).', '1812.07170-1-90-3': 'Although the highly experienced participants tend to consider such imperfect statements harmful or confusing (highly experienced agreed 40% and low experienced agreed 24%), both high and low experienced participants did not consider the proposed fix did not make sense.', '1812.07170-1-91-0': 'The following are some comments we received: "A potentially better fix than original," "I prefer having a \'this\' but this is personal preference.," "The word \'info\' seems more clear than \'trace.\'', '1812.07170-1-91-1': 'Good change," and "Changed to a wrong direction.', '1812.07170-1-91-2': 'One participant pointed out that the proposed fixes seem to provide several pieces of information, for example, the location of fix, the need of initialization of methods, and types for generics.', '1812.07170-1-91-3': 'S/he claimed that this information is useful if s/he knows the context of the code, even if an error exists.', '1812.07170-1-91-4': 'We find that even for the same fixes, some participants perceived them differently, from which we can infer that sometimes better fixes depend on preferences and/or the context.', '1812.07170-1-92-0': 'Figure [REF] shows the result of the challenging fix group.', '1812.07170-1-92-1': 'The fixes belonging to this group are difficult to generate because of unknown terms, which means that the queries and correct answers are mostly unseen by the models.', '1812.07170-1-92-2': 'Therefore we considered that those fixes did not make sense and did not provide any useful information.', '1812.07170-1-92-3': 'One case is changing BigInteger to toHexString, which even fails the compilation check.', '1812.07170-1-92-4': 'A participant left a comment "I think it would produce more confusion than help."', '1812.07170-1-92-5': 'As seen from the figure, the majority of both, highly and low experienced participants have negative impressions with regards to such statements.', '1812.07170-1-92-6': 'However, some still have positive opinions even for such failure cases.', '1812.07170-1-92-7': "Another case is given a query of 'FSDataOutputStream fos = null ;,' generating 'HdfsDataOutputStream copyError = null ;' while the correct answer is 'HdfsDataOutputStream fos = null ;.'", '1812.07170-1-92-8': "This happens because the term 'fos' does not appear in the target (i.e., post-correction statement) corpus.", '1812.07170-1-92-9': 'In the training corpora, there is no statement co-occurring FSDataOutputStream with fos or null.', '1812.07170-1-92-10': 'The model learned the replacement of FSDataOutputStream and HdfsDataOutputStream from the different context of statements.', '1812.07170-1-92-11': 'For this case, the participants evaluated more positively than negatively, although there were some comments which stated that the generated statements can be confusing.', '1812.07170-1-92-12': 'In sum, we find that even for the challenging fixes, they might be useful.', '1812.07170-1-93-0': '.96Even if generated fixes are not identical to actual fixes, they can be helpful because they can suggest the locations of required changes and possible replacements/insertions/deletions.', '1812.07170-1-93-1': 'Sometimes better fixes depend on personal preferences or the styles of projects.', '1812.07170-1-93-2': 'Although NMT models can learn fine-grained patterns of changes, the lack of information or novel queries are major challenges of fix generation.', '1812.07170-1-94-0': '# Discussion', '1812.07170-1-95-0': '## Generating non-bug-fixing statements', '1812.07170-1-96-0': 'For the accuracy evaluation in Section [REF], we only considered bug-fixing statements.', '1812.07170-1-96-1': 'Here we investigate the applicability of Ratchet in a more general context, i.e., for non-bug-fixing statements as well.', '1812.07170-1-96-2': 'In the same test year, we collected non-bug-fixing statements as shown in Table [REF].', '1812.07170-1-96-3': 'Again, we use a similar setup as we did for bug-fixing statement evaluations and compare the generated statements with Plastic Surgery outputs.', '1812.07170-1-97-0': 'Table [REF] shows the results for non-bug-fixing statements.', '1812.07170-1-97-1': 'As we can see from the Table [REF], the [MATH] values for non-bug-fixing queries ranges between 0.19 - 0.41.', '1812.07170-1-97-2': 'These [MATH] values are lower than the results obtained for the bug-fixing queries shown in Table [REF].', '1812.07170-1-97-3': 'That said, we still observe that in all five projects, Ratchet outperforms the Plastic Surgery approach.', '1812.07170-1-98-0': 'One possible explanation for the lower performance is the fact that there are relatively more UQ and UR statement pairs for non-bug-fixing datasets (as seen by comparing Table [REF] and Table [REF]), which indicates the unique nature of non-bug-fix changes.', '1812.07170-1-99-0': 'We investigate the impact of adding training data from a different project.', '1812.07170-1-99-1': 'Although the test years and the training periods are different among five projects, we simply combine training corpora for this analysis.', '1812.07170-1-99-2': 'To investigate the impact of inter-project data combination, we only add one project data for the own project data.', '1812.07170-1-100-0': 'Table [REF] shows the results for buggy queries.', '1812.07170-1-100-1': 'We see that some combinations get improvement in generating correct statements or removing incorrect statements, except for the project ambari.', '1812.07170-1-100-2': 'However, some combinations, hadoop+jetty, jetty+camel, and jetty+hadoop, get worse accuracies.', '1812.07170-1-100-3': 'This may be because of the different styles or preferences among projects.', '1812.07170-1-100-4': 'Especially only the project jetty belongs to Eclipse, other projects belong to Apache organization.', '1812.07170-1-101-0': 'Table [REF] presents the results for non-buggy queries.', '1812.07170-1-101-1': 'We find that similar combinations like camel+wicket, hadoop+camel, and jetty+ambari improve accuracies in both buggy and non-buggy results, which infers that the usefulness of additional data does not depend on the types of queries.', '1812.07170-1-101-2': 'Compared to the results for buggy queries, there are many cases that are improved; six cases in buggy results and 11 cases in non-buggy results, which may infer that non-buggy-fixes are more reusable across projects but buggy-fixes are more project-specific.', '1812.07170-1-102-0': 'These improvements are consistent with the finding in the previous study [CITATION], that is, other project data contribute slightly for reusing.', '1812.07170-1-102-1': 'Enabling accuracy improvement by adding other project data is promising.', '1812.07170-1-102-2': 'Even if a target project does not have enough data, we can add other project data.', '1812.07170-1-102-3': 'Since there seems to be preferences in styles, choosing appropriate project data should be important and this can be another further research challenge.', '1812.07170-1-103-0': '## When/why does NMT fail?', '1812.07170-1-104-0': 'We see that NMT can work better than Plastic Surgery for learning past changes and generating fixes.', '1812.07170-1-104-1': 'However, we also find limitations of our approach using NMT for code repository data.', '1812.07170-1-104-2': 'We examined some of the cases where our approach failed and discuss the challenges (and possible improvements) from two aspects, modeling and training.', '1812.07170-1-105-0': 'Modeling: Although NMT can learn the semantic and structural information by taking global context into consideration [CITATION], some limitations of NMT are known and studied [CITATION].', '1812.07170-1-106-0': 'Out-of-vocabulary problem or UNK problem.', '1812.07170-1-106-1': 'NMT usually uses top-N frequent words in the training data and regards other words as unseen ones, UNK.', '1812.07170-1-106-2': 'Therefore NMT often makes mistakes in translating low-frequency words [CITATION].', '1812.07170-1-106-3': 'In the context of our fix generation, we find similar issues in low-frequency or novel identifier names, as discussed in the survey result of challenging fix.', '1812.07170-1-106-4': 'Since those names can be identified from the context, integrating NMT with program analysis could be a promising direction.', '1812.07170-1-107-0': 'Coverage problem.', '1812.07170-1-107-1': 'NMT lacks a mechanism to guarantee that all words in a source statement are translated, and usually favors short translations.', '1812.07170-1-107-2': 'For example, in translation of long method chains with low-frequency tokens, we see insufficient outputs including incomplete statements and disappearing method calls.', '1812.07170-1-107-3': 'Moreover, this problem makes it difficult to address larger fix generation for more than one line.', '1812.07170-1-107-4': 'As there are several studies trying to address this problem [CITATION], we can consider applying these rapidly developed techniques.', '1812.07170-1-108-0': 'Training: In addition to techniques related to NMT, we think there is room for improvement in preparing training data.', '1812.07170-1-108-1': 'In this study, we design the experiment as batch learning, that is, whole training data is prepared from the past until the previous year of the test year.', '1812.07170-1-108-2': 'However, Barr et al. reported that more reusable pieces of code exist in the immediately previous version [CITATION].', '1812.07170-1-108-3': 'Previous studies have tried an online learning setting called training on errors [CITATION].', '1812.07170-1-108-4': 'Applying such online learning could be a promising challenge too.', '1812.07170-1-109-0': '## Threats to Validity', '1812.07170-1-110-0': 'Concerning external validity, this study only targets open source projects written in Java.', '1812.07170-1-110-1': 'Regarding programming languages, there is a threat of generalization, and it should be interesting to extend this study to other languages.', '1812.07170-1-111-0': 'With respect to construct validity, we collect fixes from histories, which can contain mistakes.', '1812.07170-1-111-1': 'For example, sometimes fixes can be reverted, but we do not consider such intention.', '1812.07170-1-111-2': 'In addition, the SZZ algorithm used for identifying bug-inducing and bug-fixing commits is known to produce errors [CITATION].', '1812.07170-1-111-3': 'Although we do not distinguish buggy and non-buggy changes for training, we classify test data as buggy or non-buggy.', '1812.07170-1-111-4': 'This could impact our discussion regarding the type of changes.', '1812.07170-1-111-5': 'However, as presented in Section [REF], Ratchet can work for generation of non-bug-fixing statements as well as bug-fixing statements.', '1812.07170-1-112-0': 'To mitigate threats to reliability, we made our dataset and survey material publicly available (see Section [REF] and Section [REF]).', '1812.07170-1-113-0': '# Related Work', '1812.07170-1-114-0': 'Our learning-based patch generation is related to mainly two research areas, namely probabilistic models of source code and change mining, which are for building models by learning and mining data for learning.', '1812.07170-1-115-0': '## Probabilistic Models of Source Code', '1812.07170-1-116-0': 'There are several studies on probabilistic machine learning models of source code for different applications using different techniques.', '1812.07170-1-116-1': 'Allamanis et al. conducted a large survey on this topic [CITATION].', '1812.07170-1-116-2': 'Table [REF] is originally presented in the survey of representative code models [CITATION].', '1812.07170-1-116-3': 'From the original table, non-refereed papers are excluded, some missing papers are added, and the column Data is newly prepared, which summarizes analyzed data in terms of programing languages, data sources, and historical information.', '1812.07170-1-117-0': 'As we see from the table, probabilistic machine learning models have been studied for various applications, such as code completion, code synthesis, coding conventions, and so on.', '1812.07170-1-117-1': 'From the point of view of models, newer techniques of neural networks (NN), especially neural sequence-to-sequence models (seq2seq), have not been extensively studied yet.', '1812.07170-1-117-2': 'So there are possibilities of extending and improving previous studies applying these models.', '1812.07170-1-118-0': 'From the data column, we see that several programing languages have been studied including Java, C, C, JavaScript, Python, among others.', '1812.07170-1-118-1': 'Although most of studies collected data from code repositories, some used other data sources, for example, programs in TopCorder.com [CITATION], Microsoft Excel help forums [CITATION], Android programming tutorial videos [CITATION], to build probabilistic models of source code.', '1812.07170-1-118-2': 'From source code repositories, collecting source code in selected snapshots is a common procedure.', '1812.07170-1-118-3': 'However, when considering software evolution, that is, software is updated continuously, learning over long periods is more practical.', '1812.07170-1-118-4': 'As discussed in Section [REF], online machine learning is one of challenges in this scenario.', '1812.07170-1-118-5': 'Previous studies demonstrated learning methods in long periods, called training on errors [CITATION].', '1812.07170-1-118-6': 'This can be a good hint for future research on online machine learning of patch generation.', '1812.07170-1-119-0': '## Change Mining', '1812.07170-1-120-0': 'Analyzing and exploiting historical change patterns is another similar topic to this work.', '1812.07170-1-120-1': 'Kim et al. proposed bug finding techniques based on textual code change histories [CITATION].', '1812.07170-1-120-2': 'From the analysis of open source repositories, they reported that a large amount of bugs appeared repeatedly.', '1812.07170-1-120-3': 'From the analysis of graph-based object usage models, Nguyen et al. also reported recurring bug-fix patterns and demonstrated fix recommendation based on those patterns [CITATION].', '1812.07170-1-120-4': 'To make use of similar code changes, Meng et al. proposes a tool called LASE for creating and applying context-aware edit scripts [CITATION].', '1812.07170-1-120-5': 'LASE analyzes AST-level changes and generates AST node edit operations.', '1812.07170-1-120-6': 'From a large-scale study of AST-level code changes in multiple Java projects, Nguyen et al. reported that repetitiveness is high for small size changes and similar bug-fix changes repeatedly occurred in cross projects [CITATION].', '1812.07170-1-120-7': 'Barr et al. studied the Plastic Surgery hypothesis, that is, same changes already exist in code histories and those changes can be efficiently found and exploited [CITATION].', '1812.07170-1-120-8': 'From line-granular snippet matching analyses, they reported that changes are repetitive and this repetitiveness is usefully exploitable.', '1812.07170-1-120-9': 'Yue et al. reported, from an empirical study of large-scale bug fixes, that 15-20% of bugs involved repeated fixes [CITATION].', '1812.07170-1-121-0': 'As these studies presented, using change patterns can be promising.', '1812.07170-1-121-1': 'However, from the study of the uniqueness of changes, instead of common changes, Ray et al. reported that unique changes are more common than non-unique changes [CITATION].', '1812.07170-1-121-2': 'This implies that simply applying past change patterns has limited capabilities in terms of reuse.', '1812.07170-1-121-3': 'As our results demonstrated, NMT-based learning approaches have the ability to address this issue by learning bug-fix correspondences on a variety of levels.', '1812.07170-1-122-0': '# Conclusion', '1812.07170-1-123-0': 'In this paper, we introduced Ratchet, an NMT-based technique to generate bug fixes from past fixes.', '1812.07170-1-123-1': 'Through an empirical validation on five open source projects, we find that Ratchet is effective in generating fixes.', '1812.07170-1-123-2': 'Moreover, we show that Ratchet can even be used to generate statements for non-bug-fixing statements.', '1812.07170-1-123-3': 'We compare Ratchet to the Plastic Surgery approach and show that Ratchet performs at least as well as the Plastic Surgery approach.', '1812.07170-1-124-0': 'We also investigate cases where Ratchet fails and find that Ratchet, or more generally NMT, suffers from the out-of-vocabulary problem since it depends on the presence of words in the past to train on.', '1812.07170-1-124-1': 'Also, NMT cannot guarantee that all words are covered/translated.', '1812.07170-1-124-2': 'These aforementioned issues are areas that we plan to tackle in future work.'} | {'1812.07170-2-0-0': 'Bug fixing is generally a manually-intensive task.', '1812.07170-2-0-1': 'However, recent work has proposed the idea of automated program repair, which aims to repair (at least a subset of) bugs in different ways such as code mutation, etc.', '1812.07170-2-0-2': 'Following in the same line of work as automated bug repair, in this paper we aim to leverage past fixes to propose fixes of current/future bugs.', '1812.07170-2-0-3': 'Specifically, we propose Ratchet, a corrective patch generation system using neural machine translation.', '1812.07170-2-0-4': 'By learning corresponding pre-correction and post-correction code in past fixes with a neural sequence-to-sequence model, Ratchet is able to generate a fix code for a given bug-prone code query.', '1812.07170-2-0-5': 'We perform an empirical study with five open source projects, namely Ambari, Camel, Hadoop, Jetty and Wicket, to evaluate the effectiveness of Ratchet.', '1812.07170-2-0-6': 'Our findings show that Ratchet can generate syntactically valid statements 98.7% of the time, and achieve an F1-measure between 0.29 - 0.83 with respect to the actual fixes adopted in the code base.', '1812.07170-2-0-7': 'In addition, we perform a qualitative validation using 20 participants to see whether the generated statements can be helpful in correcting bugs.', '1812.07170-2-0-8': "Our survey showed that Ratchet's output was considered to be helpful in fixing the bugs on many occasions, even if fix was not 100% correct.", '1812.07170-2-1-0': 'patch generation, corrective patches, neural machine translation, change reuse.', '1812.07170-2-2-0': '# Introduction', '1812.07170-2-3-0': 'Most software bug fixing tasks are manual and tedious.', '1812.07170-2-3-1': 'Recently, a number of techniques related to automated program repair have been proposed to help automate and reduce the burden of some of these tasks [CITATION].', '1812.07170-2-3-2': 'These systems are also seeing practical use.', '1812.07170-2-3-3': 'For example, Facebook has announced that they started applying a system of automated program repair called SapFix in their large-scale products [CITATION].', '1812.07170-2-4-0': 'However, there are limitations in current approaches to automated program repair.', '1812.07170-2-4-1': 'First, there is a risk of overfitting to the training set (and breaking under tested functionality) in patch generation, especially generated tests tends to lead overfitting compared to human-generated, requirements-based test suites [CITATION].', '1812.07170-2-4-2': 'Second, correct patches may not exist in the search space, or correct patches cannot be generated because the search space is huge [CITATION].', '1812.07170-2-4-3': 'Several studies address this search space issue by making use of existing human-written patches [CITATION], but those generated patches need to be validated with test suites.', '1812.07170-2-4-4': 'Therefore, investigating techniques that assist in the generation of patches without the need for tests, etc. are needed.', '1812.07170-2-4-5': 'Instead of exploring fix ingredients in the search space (search-based), we study the possibility of learning fix ingredients from past fixes (learning-based).', '1812.07170-2-5-0': 'Recently, Neural Machine Translation (NMT) has been proposed and showed promising results in various areas including not only translation between natural languages (such as English and Japanese), but also other NLP tasks such as speech recognition [CITATION], natural language parsing [CITATION], and text summarization [CITATION].', '1812.07170-2-5-1': 'Similar techniques have been applied to code-related tasks [CITATION].', '1812.07170-2-5-2': 'The notable success of NMT in such a wide variety of tasks can be attributed to several traits: (1) It is an end-to-end machine learning framework that can be learned effectively from large data - if we have a large enough data source it is able to learn even complicated tasks such as translation in an effective way; (2) Unlike previous models for translation such as phrase-based translation [CITATION] (which has also been used in code-related tasks such as language porting [CITATION] and pseudo-code generation [CITATION]), NMT is able to take a holistic look at the entire input and make global decisions about which words or tokens to output.', '1812.07170-2-5-3': 'In particular, for bug fixing we posit this holistic view of the entire hunk of code we attempt to fix is important, and thus focus on approaches using NMT in this work.', '1812.07170-2-6-0': 'Hence, in this paper, we propose Ratchet, a NMT-based technique that generates bug fixes based on prior bug-and-fix examples.', '1812.07170-2-6-1': 'To evaluate the effectiveness of the technique, we perform an empirical study with five large software projects, namely Ambari, Camel, Hadoop, Jetty and Wicket.', '1812.07170-2-6-2': 'We use the pattern-based patch suggestion inspired by the Plastic Surgery work [CITATION], as a comparison baseline and examine the effectiveness of our NMT-based technique.', '1812.07170-2-6-3': 'In particular, we quantify the number of cases where our NMT-based technique is able to generate a valid fix and how accurate the generated fixes are.', '1812.07170-2-6-4': 'Our findings showed that Ratchet is able to generate a valid statements in 98.7% of the cases and achieves an F1 measure between 0.29 - 0.83 with respect to the actual fixes adopted in the code base.', '1812.07170-2-6-5': 'For all five projects, Ratchet was able to either outperform or perform as well as the baseline.', '1812.07170-2-7-0': 'In addition to the quantitative validation, we also performed a survey with 20 participants to see whether the generated statements can help in correcting a bug (even if they were not 100% identical to the fix).', '1812.07170-2-7-1': 'Our findings through a participant survey show that the fixes generated by Ratchet are very helpful, even if they were not fully correct (although the correct fixes were most helpful).', '1812.07170-2-8-0': 'There are several recent studies on techniques to generate patches without test cases, which differ from our approach: inductive programming for program synthesis making used of historical change patterns [CITATION], additive program transformations using separation logic to identify and repair the lack of certain operations on heap issues [CITATION], and learning fix patterns of FindBugs violations using convolutional neural networks [CITATION].', '1812.07170-2-8-1': 'Similar to our approach, these proposals have learning aspects to generate patches without test cases.', '1812.07170-2-8-2': 'Major differences are specific targets (heap properties [CITATION] and static analysis tool violations [CITATION]) and/or specific patterns to be learned (specified constraints [CITATION] and manual fix specifications [CITATION]), while Ratchet learns any statement-level changes in a general NMT framework.', '1812.07170-2-8-3': 'Although limiting to specific targets and patterns could be effective for the targeted domains, our approach is able to target daily bug fixing activities.', '1812.07170-2-9-0': 'Our approach can be thought of as a method for "learning-based automated code changes" instead of one of automated program repair per se.', '1812.07170-2-9-1': 'Although the setting on automated program repair is expensive, especially for validation [CITATION], our NMT approach can work lightly for usual repetitive maintenance activities.', '1812.07170-2-9-2': 'As automated program repair research is recommended to focus on difficult and important bugs [CITATION], research on learning-based automated code changes could support repetitive and similar bug fixing tasks by learning common corrective maintenance activities.', '1812.07170-2-9-3': 'We expect that our approach can be integrated in daily maintenance activities.', '1812.07170-2-9-4': 'Ratchet can recommend generated patches to local code before committing to repositories and to submitted code for reviewing.', '1812.07170-2-9-5': 'While it could work in human-involved maintenance processes, we consider our approach is not necessarily an end-to-end bug fixing solution by assessing the correctness of generated patches.', '1812.07170-2-10-0': 'The rest of the paper is organized as follows.', '1812.07170-2-10-1': 'Section [REF] presents relevant terminology.', '1812.07170-2-10-2': 'Section [REF] provides background about NMT.', '1812.07170-2-10-3': 'Section [REF] details our approach.', '1812.07170-2-10-4': 'Section [REF] sets up our experiments, discussing their design and the data used.', '1812.07170-2-10-5': 'Section [REF] presents our results and Section [REF] discusses the generality and some challenges facing NMT-based solutions.', '1812.07170-2-10-6': 'Related work is presented and contrasted in Section [REF] and Section [REF] concludes the paper.', '1812.07170-2-11-0': '# Terminology', '1812.07170-2-12-0': 'We use the term, change hunk, similar to the previous study by Ray et al. [CITATION].', '1812.07170-2-12-1': 'A change hunk is a list of program statements deleted and added contiguously.', '1812.07170-2-12-2': 'In a single commit to a code repository, typically there are multiple change regions in multiple files.', '1812.07170-2-12-3': 'Even in a single file, there can be multiple change regions.', '1812.07170-2-12-4': 'Those changed regions can be identified with diff.', '1812.07170-2-12-5': 'Although the previous study by Ray et al. included unchanged statements in a change hunk [CITATION], we do not include them.', '1812.07170-2-12-6': 'We call deleted and added statements pre-correction and post-correction statements respectively.', '1812.07170-2-12-7': 'In Listing [REF], the red statement is a pre-correction statement and the green statement is a corresponding post-correction statement, and these associated two statements are considered to be a change hunk.', '1812.07170-2-13-0': 'In this study, we are interested in learning transforming patterns between corresponding pre-correction and post-correction statements.', '1812.07170-2-13-1': 'Thus, we ignore change hunks that only contain deleted or added statements.', '1812.07170-2-13-2': 'All change hunks studied in this paper are pairs of pre-correction and post-correction statements.', '1812.07170-2-14-0': '# Background', '1812.07170-2-15-0': 'Neural machine translation (NMT), also called neural sequence-to-sequence models (seq2seq) [CITATION] is a method for converting one input sequence [MATH] into another output sequence [MATH] using neural networks.', '1812.07170-2-15-1': 'As the name suggests, the method was first conceived for and tested on machine translation; for converting one natural language (e.g. English) into another (e.g. French).', '1812.07170-2-15-2': 'However, because these methods can work on essentially any problem of converting one sequence into another, they have also been applied to a wide variety of other tasks such as speech recognition [CITATION], natural language parsing [CITATION], and text summarization [CITATION].', '1812.07170-2-15-3': 'They have also seen applications to software for generation of natural language comments from code [CITATION], generation of code from natural language [CITATION], generation of API sequences [CITATION], and suggesting fixes to learner code in programming MOOCs [CITATION].', '1812.07170-2-16-0': 'In this section we briefly overview neural networks, then explain NMT in detail.', '1812.07170-2-17-0': '## Neural Networks', '1812.07170-2-18-0': 'Neural networks [CITATION], put simply, are a complicated function that is composed of simpler component parts that each have parameters that control their behavior.', '1812.07170-2-18-1': 'One common example of such a function is the simple multi-layer calculation below, which converts an input vector [MATH] into an output vector [MATH]: [EQUATION]', '1812.07170-2-18-2': 'Here, [MATH] and [MATH] are parameter matrices, and [MATH] and [MATH] are parameter vectors (called bias vectors).', '1812.07170-2-18-3': 'Importantly, the vector [MATH] is a hidden layer of the neural network, which results from multiplying [MATH], adding [MATH], then taking the hyperbolic tangent with respect to the input.', '1812.07170-2-18-4': 'This hidden layer plays an essential role in neural networks, as it allows the network to automatically discover features of the input that may be useful in predicting [MATH].', '1812.07170-2-19-0': 'Because neural networks have parameters ([MATH], [MATH], etc.) that specify their behavior, it is necessary to learn these parameters from training data.', '1812.07170-2-19-1': 'In general, we do so by calculating how well we do in predicting the correct answer [MATH] provided by the training data, and modify the parameters to increase our prediction accuracy.', '1812.07170-2-19-2': "Formally, we do so by calculating a loss function [MATH] which will (generally) be 0 if we predict perfectly, and higher if we're not doing a good job at prediction.", '1812.07170-2-19-3': 'We then take the derivative of this loss function with respect to the parameters, e.g. [MATH], and move the parameters in the direction to reduce the loss function, e.g. [EQUATION] where [MATH] is a learning rate that controls how big of a step we take after every update.', '1812.07170-2-20-0': 'The main difficulty here is that we must calculate derivatives [MATH].', '1812.07170-2-20-1': 'Even for a relatively simple function such as the one in ([REF]), calculating the derivative by hand can be cumbersome.', '1812.07170-2-20-2': 'Fortunately, this problem can be solved through a process of back-propagation (or auto-differentiation), which calculates the derivative of the whole complicated function by successively calculating derivatives of the smaller functions and multiplying them together using the chain rule [CITATION].', '1812.07170-2-20-3': 'Thus, it becomes possible to train arbitrarily complicated functions, as long as they are composed of simple component parts that can be differentiated, and a number of software libraries such as TensorFlow [CITATION] and DyNet [CITATION] make it possible to easily do so within applications.', '1812.07170-2-21-0': '## Neural Machine Translation', '1812.07170-2-22-0': 'NMT is an example of applying a complicated function learnable by neural nets and using it to solve a complicated problem: translation.', '1812.07170-2-22-1': 'To generate an output [MATH] (e.g. corrected hunk of code) given an input [MATH], these models incrementally generate each token in the output [MATH] one at a time.', '1812.07170-2-22-2': 'For example, if our output is "return this .', '1812.07170-2-22-3': 'index", the model would first predict and generate "return", then "this", then ".', '1812.07170-2-22-4': '", etc.', '1812.07170-2-22-5': 'This is done in a probabilistic way by calculating the probability of the first token of the output given the input [MATH], outputting the token in the vocabulary that maximizes this probability, then calculating the probability of the second token given the first token and the snippet [MATH] and similarly outputting the word with the highest probability, etc.', '1812.07170-2-22-6': 'When training the model, we already know a particular output [MATH] and want to calculate its probability given a particular snippet [MATH] so we can update the parameters based on the derivatives of this probability.', '1812.07170-2-22-7': 'To do so, we simply multiply these probabilities together using the chain rule as follows: [EQUATION]', '1812.07170-2-22-8': 'So how do NMT models calculate this probability?', '1812.07170-2-22-9': 'We will explain a basic outline of a basic model called the encoder-decoder model [CITATION], and refer readers to references for details [CITATION].', '1812.07170-2-22-10': 'The encoder-decoder model, as shown in Figure [REF] works in two stages: first it encodes the input (in this case [MATH]) into a hidden vector of continuous numbers [MATH] using an encoding function [EQUATION]', '1812.07170-2-22-11': 'This function generally works in two steps: looking up a vector of numbers representing each token (often called "word embeddings" or "word vectors"), then incrementally adding information about these embeddings one token at a time using a particular variety of network called a recurrent neural network (RNN).', '1812.07170-2-22-12': 'To take the specific example shown in the figure, at the first time step, we would look up an embedding vector for the first token "return", [MATH] and then perform a calculation such as the one below to calculate the hidden vector for the first time step: [EQUATION] where [MATH] and [MATH] are a matrix and vector that are parameters of the model, and [MATH] is the hyperbolic tangent function used to "squish" the values to be between -1 and 1.', '1812.07170-2-23-0': 'In the next time step, we would do the same for the symbol "."', '1812.07170-2-23-1': ', using its embedding [MATH], and in the calculation from the second step onward we also use the result of the previous calculation (in this case [MATH]): [EQUATION]', '1812.07170-2-23-2': 'By using the hidden vector from the previous time step, the RNN is able to "remember" features of the previously occurring tokens within this vector, and by repeating this process until the end of the input sequence, it (theoretically) has the ability to remember the entire content of the input within this vector.', '1812.07170-2-24-0': 'Once we have encoded the entire source input, we can then use this encoded vector to predict the first token of the output.', '1812.07170-2-24-1': 'This is generally done by defining the first hidden vector for the output [MATH] to be equal to the final vector of the input [MATH], then multiplying it with another weight vector used for prediction to calculate a score [MATH] for each token in the output vocabulary: [EQUATION]', '1812.07170-2-24-2': 'We then predict the actual probability of the first token in the output statement, for example "return", by using the softmax function, which exponentiates all of the scores in the output vocabulary and then normalizes these scores so that they add to one: [EQUATION]', '1812.07170-2-24-3': 'We then calculate a new hidden vector given this input: [EQUATION]', '1812.07170-2-24-4': 'We continue this process recursively until we output a special "end of hunk" symbol "[MATH]/s[MATH]".', '1812.07170-2-25-0': 'Why NMT models?', '1812.07170-2-25-1': ': As mentioned briefly in the intro, NMT models are well-suited to the task of automatic patch generation for a number of reasons.', '1812.07170-2-25-2': 'First, they are an end-to-end probabilistic model that can be trained from parallel datasets of pre- and post-correction code without extra human intervention, making them easy to apply to new datasets or software projects.', '1812.07170-2-25-3': 'Second, they are powerful models that can learn correspondences on a variety of levels; from simple phenomena such as direct token-by-token matches, to soft paraphrases [CITATION], to weak correspondences between keywords and large documents for information retrieval [CITATION].', '1812.07170-2-25-4': 'Finally, they have demonstrated success in a number of code related tasks as iterated at the beginning of this section, which indicates that they should be useful as part of bug fixing algorithm as well.', '1812.07170-2-26-0': 'Attention: In addition, we use a NMT model with this basic architecture, with the addition of a feature called attention, which, put simply, allows the model to "focus" on particular tokens in the input [MATH] when generating the output [MATH] [CITATION].', '1812.07170-2-26-1': 'Mathematically, this corresponds to calculating an "attention vector" [MATH], given the input hidden vectors [MATH] and the current output hidden vector [MATH].', '1812.07170-2-26-2': 'This vector consists of values between zero and one, one value for each word in the input, with values closer to one indicating that the model is choosing to focus more on that particular word.', '1812.07170-2-26-3': 'Finally, these values are used to calculate a "context vector" [EQUATION] which is used as additional information when calculating score [MATH].', '1812.07170-2-26-4': 'Attention is particularly useful when there are many token-to-token correspondences between the input and output, which we expect to be the case for our patch generation task, where the input and output code are likely to be very similar.', '1812.07170-2-26-5': 'This attention model can be further augmented to allow for exact copies of tokens [CITATION], or be used to incorporate a dictionary of common token-to-token correspondences (copies or replacements) [CITATION].', '1812.07170-2-26-6': 'In our model, we use the latter, which allows us to both capture the fact that tokens are frequently copied between pre- and post-correction code, and also the fact that some replacements will be particularly common (e.g. loadBalancerType to setLoadBalancerType).', '1812.07170-2-26-7': 'This dictionary is automatically inferred from our training data by running the fastalign toolkit, which can automatically learn such a dictionary from parallel data using probabilistic models [CITATION].', '1812.07170-2-27-0': 'Implementation details: As a specific implementation of the NMT techniques listed above, we use the lamtram toolkit [CITATION].', '1812.07170-2-27-1': 'For reproducibility, we briefly list the parameters below, and interested readers can refer to the references for detail.', '1812.07170-2-27-2': 'As our model we use an encoder-decoder model with multi-layer perceptron attention [CITATION] and input feeding [CITATION], with encoders and decoders using a single layer of 512 LSTM cells [CITATION].', '1812.07170-2-27-3': 'We use the Adam optimizer [CITATION] with a learning rate of 0.001 and minibatch size of 2048 words, and decay the learning rate every time the development loss increases.', '1812.07170-2-27-4': 'To prevent overfitting, we use a dropout rate of 0.5 [CITATION].', '1812.07170-2-27-5': 'To generate our outputs, we perform beam search with a beam size of 10.', '1812.07170-2-28-0': '# Approach', '1812.07170-2-29-0': 'The idea of corrective patch generation using NMT considers code changes as translation from pre-correction code to post-correction code.', '1812.07170-2-29-1': 'Figure [REF] provides an overview of our system, Ratchet, which consists of two main parts: creating the training corpora, and generating patches using the trained model.', '1812.07170-2-29-2': 'In this paper, we target Java source code and focus on changes within Java methods.', '1812.07170-2-29-3': 'Particularly, the granularity of code we target is a statement similar to the previous study [CITATION].', '1812.07170-2-29-4': 'Main focus in this study is preparing appropriate data for a NMT model to learn.', '1812.07170-2-29-5': 'To this aim, we build a system to collect fine-grained code change and try ignoring noisy data.', '1812.07170-2-30-0': '## Extracting Change Hunks from Code Repositories', '1812.07170-2-31-0': 'In order to create our training corpora, we start by extracting pre- and post-correction statements using a sequence of steps.', '1812.07170-2-31-1': 'We detail each of these steps in the following text:', '1812.07170-2-32-0': 'Preparing Historage for method-level histories.', '1812.07170-2-32-1': 'Since the software repositories store the code modifications at the commit level, our first step is to transform these commits into method-level modifications.', '1812.07170-2-32-2': 'To do so, we convert the existing code repositories to historage repositories [CITATION].', '1812.07170-2-32-3': 'Historage creates a new repository that stores all methods in the logs of the original repository as individual Git objects.', '1812.07170-2-32-4': 'In essence, historage is a Git repository that allows us to operate any Git commands as usual.', '1812.07170-2-33-0': 'Collecting the modified methods.', '1812.07170-2-33-1': 'We use the command git log -diff-filter=M on the historage repositories to collect all modified methods in the entire history.', '1812.07170-2-33-2': 'The option -diff-filter=M will provide only modified (M) files, which are methods in historage repositories.', '1812.07170-2-33-3': 'Since we are interested in training our model on pre- and post-correction statements, we only consider methods that modify code, i.e., not methods that are newly created or completely deleted.', '1812.07170-2-34-0': 'Identifying change hunks.', '1812.07170-2-34-1': 'As stated in Section [REF], a change hunk is a pair of pre-correction and post-correction statements.', '1812.07170-2-34-2': 'We identify these change hunks from the outputs of the git diff.', '1812.07170-2-34-3': 'Since we assume pre-correction statements have been corrected to post-correction statements, we need to identify the corresponding line pairs appropriately.', '1812.07170-2-35-0': '## Preprocessing the Statement Corpora', '1812.07170-2-36-0': 'Before storing the statement pairs as pre-correction and post-correction statement corpora, we perform the following preprocessing steps.', '1812.07170-2-36-1': 'As seen in Figure [REF], the same processes will be applied to query statements except for the step (6) and (7), which are needed only for creating the corpora.', '1812.07170-2-37-0': '(1) Limit to single-statement changes and single-statement queries.', '1812.07170-2-37-1': 'In this study, we only consider single-statement (one-line) changes.', '1812.07170-2-37-2': 'We do so for the following three reasons.', '1812.07170-2-37-3': 'First, previous studies showed that most reusable code is found at the single-statement level [CITATION].', '1812.07170-2-37-4': 'Second, it is difficult to treat multiple statement changes (one-to-many, many-to-one, and many-to-many statement changes) for identifying pairs.', '1812.07170-2-37-5': 'Those multiple statement changes can have inappropriate corresponding statements.', '1812.07170-2-37-6': 'For example, if there exists one pre-correction statement and two post-correction statements in one change hunk, this change can be a single-statement change and one independent statement insertion.', '1812.07170-2-37-7': 'If we consider these statements one pair, the independently inserted statement can be noise in the training data.', '1812.07170-2-37-8': 'Third, it is difficult to manage past histories associated with multiple statements.', '1812.07170-2-37-9': 'Using the command git blame on historage, we identify commits on which deleted lines initially appeared.', '1812.07170-2-37-10': 'In general, multiple statements can have different past histories, which makes it difficult to treat those multiple statements as one unit.', '1812.07170-2-37-11': 'For all statement pairs, we collect past history information including the original commit, changed year and deleted year, to be used for our experiments.', '1812.07170-2-37-12': 'Although we apply this filtering, we found that single-statement changes are the majority in our change hunks (as we show later in Figure [REF] and Table [REF]).', '1812.07170-2-38-0': '(2) Tokenize statements.', '1812.07170-2-38-1': 'Since the NMT model requires separate tokens as input, we use the StreamTokenizer to tokenize the Java statements.', '1812.07170-2-39-0': '(3) Remove statement pairs or statement queries with less than three tokens.', '1812.07170-2-39-1': 'We remove statements that have very few tokens (i.e., less than 3) since they are less meaningful.', '1812.07170-2-39-2': 'Our observations indication that most such lines only contain opening or closing parenthesis.', '1812.07170-2-40-0': '(4) Replace the contents of method arguments with a special token.', '1812.07170-2-40-1': 'From our many trials, we realized that a wide variety of the contents of method arguments make it difficult to generate corresponding contents.', '1812.07170-2-40-2': 'This is because sometimes method argument contents include tokens that rarely appear.', '1812.07170-2-40-3': 'We replace method and array arguments with a special token, arg and val, respectively.', '1812.07170-2-41-0': '(5) Filter unparseable statement pairs and queries.', '1812.07170-2-41-1': 'There exist incomplete statements in our collected statements, e.g., when there is a long statement that is written across two lines, and only one line is changed.', '1812.07170-2-41-2': 'To remove these incomplete Java statements, we put each statement in a dummy method of a dummy class, and try parsing the class to get an AST using JavaParser.', '1812.07170-2-41-3': 'If we fail to parse classes with either pre- or post-correction statements, we filter out the failed statement pairs', '1812.07170-2-42-0': '(6) Select post-correction statements from multiple candidates.', '1812.07170-2-42-1': 'This step is performed to address the nature of sequential order in documents.', '1812.07170-2-42-2': 'After collecting all pre- and post-correction statements from the entire history of a code repository, we can have statement pairs that have the same pre-correction statements but different post-correction statements.', '1812.07170-2-42-3': 'In order to allow the NMT models to effectively extract relationships or patterns, we chose only one post-correction statement for one pre-correction statement, and remove all other post-correction statements.', '1812.07170-2-42-4': 'The idea behind this selection is that it is better to learn from recently and frequently appearing statements.', '1812.07170-2-42-5': 'Given a pre-correction statement, we obtain post-correction statements that appeared in the most recent year.', '1812.07170-2-42-6': 'Then, from those newer statements, we select statements that most frequently appeared in the entire history.', '1812.07170-2-42-7': 'If we cannot break ties, we select the first statement in alphabetical order to make the process deterministic.', '1812.07170-2-43-0': '(7) Remove identical pre- and post-correction statements.', '1812.07170-2-43-1': 'After the above processes, there exist pairs of identical pre- and post-correction statements.', '1812.07170-2-43-2': 'For example, statement pairs from changes only within method arguments, and white space changes.', '1812.07170-2-43-3': 'We remove those statement pairs.', '1812.07170-2-44-0': '## Post-Processing', '1812.07170-2-45-0': 'Since we replace the contents of method arguments and replace it with a special token, the NMT model does not generate method arguments.', '1812.07170-2-45-1': 'However we expect that the method arguments of a query statement can be reused in the generated statement.', '1812.07170-2-45-2': 'Therefore we prepare the following heuristics for new method arguments.', '1812.07170-2-46-0': 'The lamtram toolkit provides scores associated with generated statements with the logarithm of a posteriori probability of output E given input F as [MATH].', '1812.07170-2-46-1': 'Those scores can be considered as confidences of the results.', '1812.07170-2-46-2': 'We empirically determine thresholds and ignore the generated statements with low scores.', '1812.07170-2-46-3': 'In addition, we can also ignore invalid generated statements that cannot be parsed.', '1812.07170-2-47-0': '# Experimental Setup', '1812.07170-2-48-0': 'In this section, we discuss our dataset and the design of our experiment.', '1812.07170-2-48-1': 'Particularly, we are interested in examining the viability of our approach in generating bug-fixing statements.', '1812.07170-2-48-2': 'To do so, we need to identify bug-fixing statement pairs.', '1812.07170-2-48-3': 'We discuss the tool used to identify the bug-inducing and bug-fixing commits that are used to determine our bug-fix statement pairs.', '1812.07170-2-48-4': 'Then, we provide descriptive statistics about the studied datasets.', '1812.07170-2-49-0': '## Subject Projects', '1812.07170-2-50-0': 'To perform our case study, we study five projects, namely Apache Ambari, Apache Camel, Apache Hadoop, Eclipse Jetty and Apache Wicket.', '1812.07170-2-50-1': 'We chose to study these five projects since they have long development histories and are large projects that contain many commits.', '1812.07170-2-50-2': 'Table [REF] shows the period considered, the number of commits, files and methods in our dataset.', '1812.07170-2-51-0': 'Figure [REF] shows the distribution of the number of pre- and post-correction statements in all change hunks (counted separately).', '1812.07170-2-51-1': 'We find that most of changes are single statements in either insertion, deletion, or modification.', '1812.07170-2-51-2': 'Multi-statement changes are not frequent.', '1812.07170-2-51-3': 'Table [REF] shows the number of all change hunks and the number of change hunks that are derived from single-statement changes.', '1812.07170-2-51-4': 'We see from the table that approximately 62 - 68% of the changes are single-statement changes.', '1812.07170-2-51-5': 'Since we investigated changes per methods using historage repositories [CITATION], we could divide large modifications in files [CITATION] to fine-grained changes, which results in high rations of single-statement changes.', '1812.07170-2-51-6': 'These ratios are encouraging for Ratchet, which is limited to single-statement changes.', '1812.07170-2-52-0': '## Experimental Design', '1812.07170-2-53-0': 'From the collected pre- and post-correction statements, we prepare the training data (Table [REF]) and testing data (Table [REF]).', '1812.07170-2-53-1': 'Considering the number of statements, we set the testing year for each project as shown in Table [REF].', '1812.07170-2-53-2': 'All statement pairs in each testing year are used as testing data, which means we chose statement pairs whose pre-correction statements are created in the testing year and changed to the corresponding post-correction statements in the same testing year.', '1812.07170-2-53-3': 'All years before the testing year are considered as training periods.', '1812.07170-2-53-4': 'In each training period, the numbers of statement pairs, whose pre-correction statements are changed to post-correction statements in the training period, are shown in Table [REF].', '1812.07170-2-54-0': 'This experimental design can be regarded as a simulation of generating corrected statements only by learning past histories when new statements are created and they will be modified soon (in the same year).', '1812.07170-2-54-1': 'If this works, we can prevent recurring or similar issues before being inserted into the code, or even when the code is being edited.', '1812.07170-2-54-2': 'For this purpose, we prepare the training and testing data by considering chronological order instead of random partitioning.', '1812.07170-2-54-3': 'For the risk of increasing unseen changes in the training data, we limit the testing year to one year.', '1812.07170-2-55-0': '## Data Preparation', '1812.07170-2-56-0': 'Table [REF] details the impact of the various preprocessing steps on our approach.', '1812.07170-2-56-1': 'The before filtering row shows the number of all single-statement change pairs.', '1812.07170-2-56-2': 'The [MATH] tokens row shows the effect of removing statements that have less than 3 tokens.', '1812.07170-2-56-3': 'Then we remove the unparsable statements in both, pre-correction and post-correction statements.', '1812.07170-2-56-4': 'The final step removes identical statement pairs in the pre- and post-correction statements.', '1812.07170-2-56-5': 'The last row shows the final number of statements used in our study.', '1812.07170-2-57-0': 'In addition, we perform specific processing for the training and testing data, which we detail below:', '1812.07170-2-58-0': 'Replacing rare tokens in the training data.', '1812.07170-2-58-1': 'From the processed statement pairs, we prepare pre-correction statement corpus and post-correction statement corpus.', '1812.07170-2-58-2': 'For each corpus, tokens that appear only once are replaced with [MATH]unk[MATH], which is a common way to handle unknown tokens [CITATION].', '1812.07170-2-58-3': 'This script is available in the lamtram toolkit.', '1812.07170-2-59-0': 'Categorization of testing data.', '1812.07170-2-59-1': 'When testing our approach, we call the pre-correction statements in the testing data as queries.', '1812.07170-2-59-2': 'On the other hand, we call the post-correction statements as references.', '1812.07170-2-60-0': 'When we evaluate our approach, we separate the testing data with their characteristics.', '1812.07170-2-60-1': 'First, all statement pairs in the testing data are divided into bug-fix statement pairs and non-bug-fix statement pairs.', '1812.07170-2-60-2': 'This classification procedure is presented in Section [REF].', '1812.07170-2-60-3': 'Then both classes of statement pairs are categorized into three:', '1812.07170-2-61-0': '[NU:] No unknown.', '1812.07170-2-61-1': 'There are no unknown tokens in a statement pair.', '1812.07170-2-61-2': 'All tokens in a query statement appear in the pre-correction statement from the training data corpus, and all tokens in a reference statement appear in the post-correction statement of the training data corpus.', '1812.07170-2-62-0': '[UQ:] Unknown in query.', '1812.07170-2-62-1': 'One or more token(s) in the query statement do not appear in the pre-correction statement corpus.', '1812.07170-2-62-2': 'In other words, there are unknown tokens in the query.', '1812.07170-2-63-0': '[UR:] Unknown in reference.', '1812.07170-2-63-1': 'Although there is no unknown token in the query statement, there are one or more unknown token(s) in the reference, i.e., in the corresponding post-correction statement.', '1812.07170-2-64-0': 'We categorize the statements as shown above to know which data can be used in our experiments.', '1812.07170-2-64-1': 'This is particularly important since the trained NMT models have not seen unknown tokens during training, addressing queries in UQ or UR is very difficult.', '1812.07170-2-64-2': 'In fact, it is impossible for our model to generate statements that are the exact same as the references for the UR category.', '1812.07170-2-65-0': 'Table [REF] shows the number of statement pairs for these categories of bug-fixing and non-bug fixing classes.', '1812.07170-2-65-1': "As can be seen from the Table [REF], the majority of the training data's statements fall in the UQ category (except for the Jetty project).", '1812.07170-2-65-2': 'On the other hand, the good news is that statements in the UR category are the least.', '1812.07170-2-65-3': 'We evaluate our approach using statements in the NU category.', '1812.07170-2-66-0': 'The dataset is available online.', '1812.07170-2-67-0': '## Identifying Bug-Fixing Statements', '1812.07170-2-68-0': 'We collect bug-fixing statement pairs by identifying the pairs of bug-inducing and bug-fixing commits.', '1812.07170-2-68-1': 'To obtain these commits, we use Commit.guru [CITATION], a tool that analyzes and provides change level analytics.', '1812.07170-2-68-2': 'For full details about commit.guru, we point the reader to the paper by Rosen et al. [CITATION], however, here we describe the relevant details for our paper.', '1812.07170-2-68-3': 'Commit.guru takes as input a Git repository address, an original code repository in this study, and provides data for all commits of the project.', '1812.07170-2-68-4': 'It applies the SZZ algorithm [CITATION] to identify bug-inducing commits and their associated bug-fixing commits.', '1812.07170-2-68-5': 'In addition, Commit.guru provides a number of change level metrics related to the size of the change, the history of the files changed, the diffusion of the change and the experience of the developers making the modification.', '1812.07170-2-69-0': 'As mentioned earlier in step (1) of preprocessing (Section [REF]), we have meta information of statements including the original commits of post-correction statements and the original commits of pre-correction statements.', '1812.07170-2-69-1': 'We consider a pair of statements bug fixing if and only if a pre-correction statement is created in a bug-inducing commit and an post-correction statement is created in the associated bug-fixing commit.', '1812.07170-2-69-2': 'The other statement pairs are treated as non-bug-fix statements.', '1812.07170-2-69-3': 'We do not distinguish the types of the training data, that is, bug-fix or non-bug-fix.', '1812.07170-2-69-4': 'This is because we prefer to increase the training data available to the model and make the model learn from all varieties of changes.', '1812.07170-2-70-0': '# Evaluation', '1812.07170-2-71-0': 'We evaluate the performance of Ratchet with respect to two aspects: accuracy and usefulness of generated statements.', '1812.07170-2-71-1': 'In all of the results presented in this section, the NMT models are trained and tested with data from the same project (i.e., within-project evaluation).', '1812.07170-2-72-0': '## Can the models generate valid statements?', '1812.07170-2-73-0': 'We consider complete and parsable statements as valid statements.', '1812.07170-2-73-1': 'We investigate whether the generated statements are valid using the same process of step (5) in the preprocessing described in Section [REF].', '1812.07170-2-73-2': 'Table [REF] shows the number (and percentage) of generated valid statements.', '1812.07170-2-73-3': 'We do not use thresholds here, that is, all generated statements including low scores are considered.', '1812.07170-2-73-4': 'As we see from the table, in most cases the models generated valid and complete Java statements.', '1812.07170-2-73-5': 'These high accuracy results are especially interesting since we did not explicitly teach the models the Java language specification.', '1812.07170-2-73-6': 'Simply, the models were able to achieve this high level of performance by themselves, using approx. 1,500 to 10,000 statement pairs.', '1812.07170-2-74-0': 'In most cases of the five projects nearly 100 of the generated statements are valid Java statements.', '1812.07170-2-74-1': 'In total the models generated 230 valid statements for 233 queries (98.7).', '1812.07170-2-75-0': '## How accurate are the generated statements?', '1812.07170-2-76-0': 'In this section we evaluate the accuracy in a strict manner, that is, only generated statements that are identical to references are considered as correct.', '1812.07170-2-76-1': 'Our results are based on the NU (no unknown) category of statements, since as mentioned earlier, other categories are difficult (impossible for UR) to generate accurate statements that are identical to the reference statements.', '1812.07170-2-77-0': 'Before analyzing accuracy, we categorize the outputs into four types:', '1812.07170-2-78-0': 'To measure the accuracy of generated results, we compute precision, recall, and [MATH], which are defined as: [MATH], [MATH], and [MATH], where [MATH] is the sum of [MATH], [MATH], and [MATH].', '1812.07170-2-78-1': 'Higher precision indicates that the provided results are correct.', '1812.07170-2-78-2': 'Higher recall means that the results contain less NA but many correct.', '1812.07170-2-78-3': 'Providing a small number of results with high confidence can improve precision but lower recall.', '1812.07170-2-78-4': 'Since there is a tradeoff between precision and recall, [MATH], the harmonic mean of precision and recall, is also presented.', '1812.07170-2-79-0': 'We compare the accuracy of our NMT models with the pattern-based patch suggestion approach, i.e., patch suggestion with line-granular snippets that already exist in the training data, which serves as our baseline [CITATION].', '1812.07170-2-79-1': 'To do so, we examine whether a query statement exists in the pre-correction statement training data corpus, and if it does, we check whether the corresponding post-correction statement also exists (in the training data).', '1812.07170-2-79-2': 'If this happens, then we consider the statement to be covered by the Plastic Surgery approach.', '1812.07170-2-79-3': 'If there is no identical pre-correction statement, we mark the result as NA.', '1812.07170-2-79-4': 'Note that this baseline is slightly different from the Plastic Surgery hypothesis since it searches post-correction statements from all packages, which requires the existence of pre-correction statements in the codebase.', '1812.07170-2-79-5': 'We apply the same argument replacing processing in Section [REF] to make it fair.', '1812.07170-2-80-0': 'When evaluating Ratchet and as stated in Section [REF], we use a threshold to ignore results with low confidence.', '1812.07170-2-80-1': 'Figure [REF] illustrates the transitions of [MATH] values with different thresholds (from -1.2 to -0.1).', '1812.07170-2-80-2': 'The solid lines are [MATH] values of Ratchet and the dotted lines are [MATH] values of the baseline, which do not change with thresholds.', '1812.07170-2-80-3': 'We find that [MATH] values slightly change when we vary the thresholds.', '1812.07170-2-80-4': 'Lowering thresholds improves recall, however, it impacts the precision in the opposite direction.', '1812.07170-2-80-5': 'On the other hand, raising the threshold improves precision but makes recall worse.', '1812.07170-2-80-6': 'Based on our analysis of the threshold, we empirically set the threshold as -0.7 for the analyses that follow.', '1812.07170-2-81-0': 'Table [REF] shows the results of our approach and compares it with the results of the baseline.', '1812.07170-2-81-1': 'We observe that, as reported in the Plastic Surgery hypothesis paper [CITATION], the baseline of the pattern-based patch recommendation works in many cases, that is, changes (corrections) contain snippets that already exist in code repositories at the time of the changes, and these snippets can be efficiently found and exploited.', '1812.07170-2-81-2': 'That said, Table [REF] shows that Ratchet improves the results in two projects and does not change in three projects.', '1812.07170-2-81-3': 'We observe that in camel, the results are greatly improved: 26 correct statements are generated compared with one correct recommendation from the pattern matching of the baseline.', '1812.07170-2-81-4': 'Our results show that the NMT models work, as well as the baseline, if there are easily exploited statement-level patterns (i.e., reusable snippets), and works better than the baseline if there exist only finer-grained exploited patterns (i.e., fine-grained fixing patterns), which the statement-based pattern matching cannot use.', '1812.07170-2-82-0': 'Table [REF] presents examples of generated fixes that cannot be fixed by the baseline, but have a fix generated with our models.', '1812.07170-2-82-1': 'Sometimes the model learns the incrementation of value (query 1).', '1812.07170-2-82-2': 'Generics-related fixes are typical examples of successful generation with the NMT models (query 2 and 3).', '1812.07170-2-82-3': 'Sometimes it is preferred to remove this (query 4) if it makes the style consistent with the styling used in the specific project.', '1812.07170-2-82-4': "In fact, our models learned to remove the keyword 'this' because similar patterns were prevalent in the project's history.", '1812.07170-2-83-0': 'NMT-based patch generation works better than pattern-based patch suggestion, achieving [MATH] values between 0.29 to 0.83 for buggy queries.', '1812.07170-2-83-1': 'In total 157 correctly generated statements without method arguments, the contents of method arguments for 141 statements (89.8%) are correctly provided by reusing the contents of method arguments in queries.', '1812.07170-2-84-0': '## Do humans detect similarity between generated statements and actual statements?', '1812.07170-2-85-0': 'During the previous evaluation of accuracy, we considered the generated statements to be correct only if they are identical to the reference statements, otherwise they are considered to be incorrect or NA.', '1812.07170-2-85-1': 'To investigate whether the generated statements are useful, even if they are not identical to actual future corrections, we also perform a human evaluation with such (non-identical) corrections.', '1812.07170-2-86-0': 'We show survey participants the following three code snippets for one fix: i) an original problematic code snippet (before correction), ii) the actually fixed code snippet (after correction), and iii) a code snippet that is proposed as a fix by our NMT models.', '1812.07170-2-86-1': 'All code snippets contain one type of buggy or fixed statements with the surrounding statements.', '1812.07170-2-87-0': 'From the five projects, we collect ten corrections including five correctly and five incorrectly generated statements in the NU (no unknown) category, which are evaluated in Table [REF].', '1812.07170-2-87-1': 'In addition, we collect five fixes that belong to the UQ (unknown in query) or the UR (unknown in reference) categories, which are known to be difficult for NMT models to generate.', '1812.07170-2-87-2': 'For simplicity, we call the above three groups correct fixes, incorrect fixes, and challenging fixes respectively.', '1812.07170-2-88-0': 'For each correction we prepare the following four statements, and ask the participants to evaluate using a five-level Likert scale scores from 1 (strongly disagree) to 5 (strongly agree) whether: (a) The proposed fix helps you to understand the required change, (b) The proposed fix can be a reference if you were to create your own fix, (c) The proposed fix is harmful or confusing, and (d) The proposed fix does not make sense and I will just ignore it.', '1812.07170-2-88-1': 'We asked not only positive impressions but also negative impressions in order to assess the usefulness and potential risks of incorrect generation.', '1812.07170-2-88-2': 'The survey material is available online.', '1812.07170-2-89-0': 'We recruited participants in Canada, US, and Japan, and 20 people participated in the survey including five undergraduate, 14 graduate students, and one professor.', '1812.07170-2-89-1': 'As Siegmund et al. reported that self estimation seems to be a reliable way to measure programming experience [CITATION], we asked the participants to estimate their experience in both, overall and Java programming experience.', '1812.07170-2-89-2': 'The participants can select any of 5 choices, varying between 1 (very inexperienced) to 5 (very experienced).', '1812.07170-2-89-3': 'Those who score 4 or 5 in both self estimation are considered to be high-experienced and others are considered to have low-experience.', '1812.07170-2-89-4': 'Five in six high experienced participants have more than five years of development experience, and the other have three-to-five years of experience.', '1812.07170-2-89-5': 'In 14 low-experience participants, the experience periods vary from less than one year, one-to-three years, three-to-five years, and more than five years.', '1812.07170-2-90-0': 'Figure [REF] shows the result of the correct fix group.', '1812.07170-2-90-1': 'The results shown in the figure show that the generated statements are useful.', '1812.07170-2-90-2': 'All high-experience and most of the low-experience participants agreed (scores 4 or 5 for questions (a) and (b)) that the correct fix statements helped them and that the statements and did not have negative effects (i.e., most scores are 1 or 2 for questions (c) and (d)).', '1812.07170-2-91-0': 'Figure [REF] shows the results of the incorrect fix group, which includes statements with incorrect method calls and/or incorrect generic types, for example.', '1812.07170-2-91-1': 'We assumed that these fixes are harmful or confusing because they tend to be partially the same as the references, but are slightly different from the actual fixes.', '1812.07170-2-91-2': 'However, as evidenced by the results shown in Figure [REF], the majority of highly and low experienced participants agree to that such imperfect statements may still be helpful (i.e., by providing positive answers to statement (a) and (b)).', '1812.07170-2-91-3': 'Although the highly experienced participants tend to consider such imperfect statements harmful or confusing (highly experienced agreed 40% and low experienced agreed 24%), both high and low experienced participants did not consider the proposed fix did not make sense.', '1812.07170-2-92-0': 'The following are some comments we received: "A potentially better fix than original," "I prefer having a \'this\' but this is personal preference.," "The word \'info\' seems more clear than \'trace.\'', '1812.07170-2-92-1': 'Good change," and "Changed to a wrong direction.', '1812.07170-2-92-2': 'One participant pointed out that the proposed fixes seem to provide several pieces of information, for example, the location of fix, the need of initialization of methods, and types for generics.', '1812.07170-2-92-3': 'S/he claimed that this information is useful if s/he knows the context of the code, even if an error exists.', '1812.07170-2-92-4': 'We find that even for the same fixes, some participants perceived them differently, from which we can infer that sometimes better fixes depend on preferences and/or the context.', '1812.07170-2-93-0': 'Figure [REF] shows the result of the challenging fix group.', '1812.07170-2-93-1': 'The fixes belonging to this group are difficult to generate because of unknown terms, which means that the queries and correct answers are mostly unseen by the models.', '1812.07170-2-93-2': 'Therefore we considered that those fixes did not make sense and did not provide any useful information.', '1812.07170-2-93-3': 'One case is changing BigInteger to toHexString, which even fails the compilation check.', '1812.07170-2-93-4': 'A participant left a comment "I think it would produce more confusion than help."', '1812.07170-2-93-5': 'As seen from the figure, the majority of both, highly and low experienced participants have negative impressions with regards to such statements.', '1812.07170-2-93-6': 'However, some still have positive opinions even for such failure cases.', '1812.07170-2-93-7': "Another case is given a query of 'FSDataOutputStream fos = null ;,' generating 'HdfsDataOutputStream copyError = null ;' while the correct answer is 'HdfsDataOutputStream fos = null ;.'", '1812.07170-2-93-8': "This happens because the term 'fos' does not appear in the target (i.e., post-correction statement) corpus.", '1812.07170-2-93-9': 'In the training corpora, there is no statement co-occurring FSDataOutputStream with fos or null.', '1812.07170-2-93-10': 'The model learned the replacement of FSDataOutputStream and HdfsDataOutputStream from the different context of statements.', '1812.07170-2-93-11': 'For this case, the participants evaluated more positively than negatively, although there were some comments which stated that the generated statements can be confusing.', '1812.07170-2-93-12': 'In sum, we find that even for the challenging fixes, they might be useful.', '1812.07170-2-94-0': 'Even if generated fixes are not identical to actual fixes, they can be helpful because they can suggest the locations of required changes and possible replacements/insertions/deletions.', '1812.07170-2-94-1': 'Sometimes better fixes depend on personal preferences or the styles of projects.', '1812.07170-2-94-2': 'Although NMT models can learn fine-grained patterns of changes, the lack of information or novel queries are major challenges of fix generation.', '1812.07170-2-95-0': '# Discussions', '1812.07170-2-96-0': '## Generating Non-Bug-Fixing Statements', '1812.07170-2-97-0': 'For the accuracy evaluation in Section [REF], we only considered bug-fixing statements.', '1812.07170-2-97-1': 'Here we investigate the applicability of Ratchet in a more general context, i.e., for non-bug-fixing statements as well.', '1812.07170-2-97-2': 'In the same test year, we collected non-bug-fixing statements as shown in Table [REF].', '1812.07170-2-97-3': 'Again, we use a similar setup as we did for bug-fixing statement evaluations and compare the generated statements with the baseline.', '1812.07170-2-98-0': 'Table [REF] shows the number of generated valid statements.', '1812.07170-2-98-1': 'Similar to the result for buggy queries in Table [REF], most generated statements are valid Java statements.', '1812.07170-2-98-2': 'Table [REF] shows the results for non-bug-fixing statements.', '1812.07170-2-98-3': 'The [MATH] values for non-bug-fixing queries ranges between 0.07 to 0.49.', '1812.07170-2-98-4': 'These [MATH] values are lower than the results obtained for the bug-fixing queries shown in Table [REF].', '1812.07170-2-98-5': 'That said, we still observe that in all five projects, Ratchet outperforms the baseline.', '1812.07170-2-98-6': 'One possible explanation for the lower performance is the fact that there are relatively more UQ and UR statement pairs for non-bug-fixing datasets (as seen by comparing Table [REF] and Table [REF]), which indicates the unique nature of non-bug-fix changes.', '1812.07170-2-99-0': 'We investigate the impact of adding training data from a different project.', '1812.07170-2-99-1': 'Although the test years and the training periods are different among five projects, we simply combine training corpora for this analysis.', '1812.07170-2-99-2': 'To investigate the impact of inter-project data combination, we only add one project data for the own project data.', '1812.07170-2-100-0': 'Table [REF] shows the results for buggy queries.', '1812.07170-2-100-1': 'We see that some combinations get improvement in generating correct statements or removing incorrect statements, except for the project ambari.', '1812.07170-2-100-2': 'However, some combinations, hadoop+jetty, jetty+camel, and jetty+hadoop, get worse accuracies.', '1812.07170-2-100-3': 'This may be because of the different styles or preferences among projects.', '1812.07170-2-100-4': 'Especially only the project jetty belongs to Eclipse, other projects belong to Apache organization.', '1812.07170-2-101-0': 'Table [REF] presents the results for non-buggy queries.', '1812.07170-2-101-1': 'We find that similar combinations like camel+wicket, hadoop+camel, and jetty+ambari improve accuracies in both buggy and non-buggy results, which infers that the usefulness of additional data does not depend on the types of queries.', '1812.07170-2-101-2': 'Compared to the results for buggy queries, there are many cases that are improved; six cases in buggy results and 11 cases in non-buggy results, which may infer that non-buggy-fixes are more reusable across projects but buggy-fixes are more project-specific.', '1812.07170-2-102-0': 'These improvements are consistent with the finding in the previous study [CITATION], that is, other project data contribute slightly for reusing.', '1812.07170-2-102-1': 'Enabling accuracy improvement by adding other project data is promising.', '1812.07170-2-102-2': 'Even if a target project does not have enough data, we can add other project data.', '1812.07170-2-102-3': 'Since there seems to be preferences in styles, choosing appropriate project data should be important and this can be another further research challenge.', '1812.07170-2-103-0': '## Applying to Bug Dataset', '1812.07170-2-104-0': 'We also investigate the applicability of Ratchet on a commonly used bug dataset, Defects4J [CITATION].', '1812.07170-2-104-1': 'The same steps shown in Section [REF], are applied (we only targeted Git repositories), and bug-fixing modifications within methods are identified.', '1812.07170-2-105-0': 'Table [REF] presents the characteristics of bugs in the dataset.', '1812.07170-2-105-1': 'We see that only 13 to 36 of bugs include single statement (one-line) changes for fixing, which are applicable for Ratchet.', '1812.07170-2-105-2': 'From the other bugs, 22 to 57 bugs require only addition of statements, which is not targeted by our current approach.', '1812.07170-2-105-3': 'From the bug-fixing commits, using the same steps detailed in Section [REF], we collected pre- and post-correction statements for testing data.', '1812.07170-2-105-4': 'Training data was prepared from the other commits.', '1812.07170-2-105-5': 'As seen in Table [REF], there are 48 statements in the NU category.', '1812.07170-2-105-6': 'Similar to other settings, we use the buggy statements as queries.', '1812.07170-2-106-0': 'Ratchet generated one complete fix and two partial fixes, which are shown in Table [REF].', '1812.07170-2-106-1': 'The bug of closure-46 requires one single statement fix and Ratchet could successfully generate the fix including an argument.', '1812.07170-2-106-2': 'The other two bugs require multiple and larger changes, and Ratchet generated argument incorrect fixes of single statements.', '1812.07170-2-106-3': 'Since the proposed approach does not account for the context of changes, it cannot handle appropriate argument changes.', '1812.07170-2-106-4': 'Compared to 26 successful fixes by ELIXIR, one of the recent program repair techniques [CITATION], the performance of Ratchet is low on the Defect4J dataset.', '1812.07170-2-107-0': 'This result highlights the limitations of Ratchet in fixing specific bugs, which require multiple modifications, requires insertions of statements, are not repetitive in code repositories, and so on.', '1812.07170-2-107-1': 'CODIT, a tree2tree model-based change generation technique, reported several correct modifications related to method arguments, for Defects4J [CITATION], but it did not generate fixes in Table [REF].', '1812.07170-2-107-2': 'Hence, we believe that our seq2seq approach can be a complement to tree2tree approaches.', '1812.07170-2-108-0': '## When/Why Does NMT Fail?', '1812.07170-2-109-0': 'We see that NMT can work better than the pattern-based patch suggestion for learning past changes and generating fixes.', '1812.07170-2-109-1': 'However, we also find limitations of our approach using NMT for code repository data.', '1812.07170-2-109-2': 'We examined some of the cases where our approach failed and discuss the challenges (and possible improvements) from two aspects, modeling and training.', '1812.07170-2-110-0': 'Modeling: Although NMT can learn the semantic and structural information by taking global context into consideration [CITATION], some limitations of NMT are known and studied [CITATION].', '1812.07170-2-111-0': 'Out-of-vocabulary problem or UNK problem.', '1812.07170-2-111-1': 'NMT usually uses the top-N most frequent words in the training data and regards other words as unseen ones, UNK.', '1812.07170-2-111-2': 'Therefore NMT often makes mistakes in translating low-frequency words.', '1812.07170-2-111-3': 'While we alleviated this problem somewhat by incorporating dictionaries into our method [CITATION], we still find similar issues in low-frequency or novel identifier names, as discussed in the survey result of challenging fix.', '1812.07170-2-111-4': 'Since those names can be identified from the context, integrating NMT with program analysis could be a promising direction.', '1812.07170-2-112-0': 'Coverage problem.', '1812.07170-2-112-1': 'NMT lacks a mechanism to guarantee that all words in a source statement are translated, and usually favors short translations.', '1812.07170-2-112-2': 'For example, in translation of long method chains with low-frequency tokens, we see insufficient outputs including incomplete statements and disappearing method calls.', '1812.07170-2-112-3': 'Moreover, this problem makes it difficult to address larger fix generation for more than one line.', '1812.07170-2-112-4': 'As there are several studies trying to address this problem [CITATION], we can consider applying these rapidly developed techniques.', '1812.07170-2-113-0': 'Training: In addition to techniques related to NMT, we think there is room for improvement in preparing training data.', '1812.07170-2-113-1': 'In this study, we design the experiment as batch learning, that is, whole training data is prepared from the past until the previous year of the test year.', '1812.07170-2-113-2': 'However, Barr et al. reported that more reusable pieces of code exist in the immediately previous version [CITATION].', '1812.07170-2-113-3': 'Previous studies have tried an online learning setting called training on errors [CITATION].', '1812.07170-2-113-4': 'Applying such online learning could be a promising challenge too.', '1812.07170-2-114-0': '## Limitations and Challenges', '1812.07170-2-115-0': 'Limitations of Ratchet are summarized as follows.', '1812.07170-2-116-0': 'It cannot generate patches that need additions or deletions of statements.', '1812.07170-2-116-1': 'It cannot generate patches that consist of multiple statements.', '1812.07170-2-116-2': 'It cannot generate patches that include the changes of method or array arguments.', '1812.07170-2-116-3': 'It cannot account for the context of patches outside statements.', '1812.07170-2-116-4': 'It cannot generate patches that include unknown tokens.', '1812.07170-2-116-5': 'It cannot generate assembled patches for single bugs.', '1812.07170-2-116-6': 'It is not evaluated in a cross-project setting.', '1812.07170-2-116-7': 'It does not include a bug-localization step.', '1812.07170-2-117-0': 'Considering the amount of hunks with single-statement pairs in Table [REF] (66.3% in average), the filtering results in Table [REF] (40.4% in average), the percentage of NU in Table [REF] (41.0% in average), around 11% of change hunks can be targeted in our approach.', '1812.07170-2-118-0': 'Regarding 1), 2), 3), and 4), extending the granularity of code to be learned can address these issues to some extent.', '1812.07170-2-118-1': 'As we discuss in Section [REF], some studies reported that learning the contents of methods or classes can work with seq2seq models [CITATION].', '1812.07170-2-118-2': 'More advanced tree-to-tree models may also applicable to address these limitations [CITATION].', '1812.07170-2-118-3': 'Regarding 5), increasing appropriate training data is one direction, as well as accounting for context information.', '1812.07170-2-118-4': 'Regarding 6), recent studies have examined multi-hunk program repair [CITATION].', '1812.07170-2-118-5': 'Our fine-grained code history analysis could be suitable for their approach of using revision histories.', '1812.07170-2-118-6': 'Although this paper does not conduct the evaluation of cross-project learning (limitation 7)), Ratchet can learn from data from multiple projects.', '1812.07170-2-118-7': 'Selecting and preprocessing data from multiple projects in an appropriate way can be a practical challenge.', '1812.07170-2-119-0': 'Regarding 8), there are studies of fine-grained defect prediction [CITATION].', '1812.07170-2-119-1': 'To localize problematic statements, we could try applying these techniques.', '1812.07170-2-119-2': 'We could also consider training another neural network to identify problematic statements to be query statements to Ratchet.', '1812.07170-2-120-0': '## Threats to Validity', '1812.07170-2-121-0': 'Concerning external validity, this study only targets five open source projects written in Java.', '1812.07170-2-121-1': 'As we do not have clear selection criteria, there can be a selection bias.', '1812.07170-2-121-2': 'Projects with different sizes, different management governance, etc. can lead to different results.', '1812.07170-2-121-3': 'Regarding programming languages, there is a threat of generalization, and it should be interesting to extend this study to other languages.', '1812.07170-2-122-0': 'With respect to construct validity, we collect fixes from histories, which can contain mistakes.', '1812.07170-2-122-1': 'For example, sometimes fixes can be reverted, but we do not consider such intention.', '1812.07170-2-122-2': 'In addition, the SZZ algorithm used for identifying bug-inducing and bug-fixing commits is known to produce errors [CITATION].', '1812.07170-2-122-3': 'Although we do not distinguish buggy and non-buggy changes for training, we classify test data as buggy or non-buggy.', '1812.07170-2-122-4': 'This could impact our discussion regarding the type of changes.', '1812.07170-2-122-5': 'However, as presented in Section [REF], Ratchet can work for generation of non-bug-fixing statements as well as bug-fixing statements.', '1812.07170-2-123-0': 'Another threat to construct validity exists in our preprocessing.', '1812.07170-2-123-1': 'We removed short statements (step (3)), the contents of method and array arguments (step (4)), unparsable statements (step (5)), and older and less frequent post-correction statements from multiple candidates (step (6)), to ignore noises.', '1812.07170-2-123-2': 'Although these steps were prepared in our trial-and-error experiments and evaluated empirically, different parameters and processes may improve the performance.', '1812.07170-2-123-3': 'Exploring a better configuration of preprocessing can be practical future work.', '1812.07170-2-124-0': 'To mitigate threats to reliability, we made our dataset and survey material publicly available (see Section [REF] and Section [REF]).', '1812.07170-2-125-0': '# Related Work', '1812.07170-2-126-0': 'In this section, we discuss similar NMT-based patch generation studies, and then discuss two research areas, namely probabilistic models of source code and change mining, which are typically used to build models by learning and mining data for learning.', '1812.07170-2-127-0': '## NMT-based Patch Generation', '1812.07170-2-128-0': 'Learning source code changes using NMT-based techniques, NMT-based automated code changes, is an emerging research topic.', '1812.07170-2-128-1': 'Tufano et al. conducted empirical studies to investigate the feasibility of learning bug-fixing patches using NMT techniques [CITATION].', '1812.07170-2-128-2': 'Similar to our approach, they built seq2seq models.', '1812.07170-2-128-3': 'By extending those studies, Tufano et al. studied the ability of a seq2seq NMT model to automate code changes for pull requests [CITATION].', '1812.07170-2-128-4': 'One of the differences of the earlier approaches and Ratchet is the granularity of code to be learned.', '1812.07170-2-128-5': 'While the above studies targeted methods within 50 or 100 tokens, Ratchet targets statements.', '1812.07170-2-128-6': 'Positive results at both granularity leveles show the capability of NMT models to learn different types of code changes.', '1812.07170-2-128-7': 'Although Tufano et al. prepared their training and testing data by random partitioning [CITATION], we prepared data considering chronological order, to emulate practical scenarios.', '1812.07170-2-128-8': 'Tufano et al. largely abstracted tokens for cross-project learning, while Ratchet kept identifiers and literals except for arguments to learn project-specific time-sensitive changes, which results in the successful number incrementation shown at the first result in Table [REF].', '1812.07170-2-129-0': 'SequenceR is another NMT-based system to learn source code changes based on a seq2seq model and copy mechanism [CITATION].', '1812.07170-2-129-1': 'Similar to Ratchet, one-line changes are targeted for fixing.', '1812.07170-2-129-2': 'While Ratchet expects only a buggy line as an input, SequenceR accepts surrounding method and class as well as an annotated buggy line as the context.', '1812.07170-2-129-3': 'SequenceR is an end-to-end approach including validation with test cases.', '1812.07170-2-129-4': 'CODIT learns source code change patterns with tree-to-tree NMT models considering AST-level changes [CITATION].', '1812.07170-2-129-5': 'Similar to Tufano et al. [CITATION], cross-project datasets without considering time were used for the evaluation of both SequenceR and CODIT.', '1812.07170-2-130-0': 'While making the vocabulary small is considered to be one of challenges in other studies [CITATION], we did not explicitly limit the number of tokens or identifiers to be learned.', '1812.07170-2-130-1': 'There seems to be a trade-off relationship between the vocabulary size and the context size.', '1812.07170-2-130-2': 'Since we targeted almost the smallest context (single-statement changes and changes within a single project), we did not need to make the vocabulary small.', '1812.07170-2-130-3': 'Addressing this trade-off to consider both larger vocabulary and wider context will be challenging future work.', '1812.07170-2-130-4': 'Another major difference between this study and the other studies is our human evaluation with the survey.', '1812.07170-2-130-5': 'We observed that sometimes the survey participants perceived positively even if generated statements were not identical to actual statements.', '1812.07170-2-130-6': 'Further user studies in practical scenario could be another future challenge.', '1812.07170-2-131-0': '## Probabilistic Models of Source Code', '1812.07170-2-132-0': 'There are several studies on probabilistic machine learning models of source code for different applications using different techniques.', '1812.07170-2-132-1': 'Allamanis et al. conducted a large survey on this topic [CITATION].', '1812.07170-2-132-2': 'Table [REF] is originally presented in the survey of representative code models [CITATION].', '1812.07170-2-132-3': 'From the original table, non-refereed papers are excluded, some missing papers are added, and the column Data is newly prepared, which summarizes analyzed data in terms of programing languages, data sources, and historical information.', '1812.07170-2-133-0': 'As we see from the table, probabilistic machine learning models have been studied for various applications, such as code completion, code synthesis, coding conventions, and so on.', '1812.07170-2-133-1': 'From the point of view of models, newer techniques of neural networks (NN), especially neural seq2seq models, have not been extensively studied yet.', '1812.07170-2-133-2': 'So there are possibilities of extending and improving previous studies applying these models.', '1812.07170-2-134-0': 'From the data column, we see that several programing languages have been studied including Java, C, C, JavaScript, Python, among others.', '1812.07170-2-134-1': 'Although most of studies collected data from code repositories, some used other data sources, for example, programs in TopCorder.com [CITATION], Microsoft Excel help forums [CITATION], Android programming tutorial videos [CITATION], to build probabilistic models of source code.', '1812.07170-2-134-2': 'From source code repositories, collecting source code in selected snapshots is a common procedure.', '1812.07170-2-134-3': 'However, when considering software evolution, that is, software is updated continuously, learning over long periods is more practical.', '1812.07170-2-134-4': 'As discussed in Section [REF], online machine learning is one of challenges in this scenario.', '1812.07170-2-134-5': 'Previous studies demonstrated learning methods in long periods, called training on errors [CITATION].', '1812.07170-2-134-6': 'This can be a good hint for future research on online machine learning of patch generation.', '1812.07170-2-135-0': '## Change Mining', '1812.07170-2-136-0': 'Analyzing and exploiting historical change patterns is another similar topic to this work.', '1812.07170-2-136-1': 'Kim et al. proposed bug finding techniques based on textual code change histories [CITATION].', '1812.07170-2-136-2': 'From the analysis of open source repositories, they reported that a large amount of bugs appeared repeatedly.', '1812.07170-2-136-3': 'From the analysis of graph-based object usage models, Nguyen et al. also reported recurring bug-fix patterns and demonstrated fix recommendation based on those patterns [CITATION].', '1812.07170-2-136-4': 'To make use of similar code changes, Meng et al. proposes a tool called LASE for creating and applying context-aware edit scripts [CITATION].', '1812.07170-2-136-5': 'LASE analyzes AST-level changes and generates AST node edit operations.', '1812.07170-2-136-6': 'From a large-scale study of AST-level code changes in multiple Java projects, Nguyen et al. reported that repetitiveness is high for small size changes and similar bug-fix changes repeatedly occurred in cross projects [CITATION].', '1812.07170-2-136-7': 'Barr et al. studied the Plastic Surgery hypothesis, that is, same changes already exist in code histories and those changes can be efficiently found and exploited [CITATION].', '1812.07170-2-136-8': 'From line-granular snippet matching analyses, they reported that changes are repetitive and this repetitiveness is usefully exploitable.', '1812.07170-2-136-9': 'Yue et al. reported, from an empirical study of large-scale bug fixes, that 15-20% of bugs involved repeated fixes [CITATION].', '1812.07170-2-137-0': 'As these studies presented, using change patterns can be promising.', '1812.07170-2-137-1': 'However, from the study of the uniqueness of changes, instead of common changes, Ray et al. reported that unique changes are more common than non-unique changes [CITATION].', '1812.07170-2-137-2': 'This implies that simply applying past change patterns has limited capabilities in terms of reuse.', '1812.07170-2-137-3': 'As our results demonstrated, NMT-based learning approaches have the ability to address this issue by learning bug-fix correspondences on a variety of levels.', '1812.07170-2-138-0': '# Conclusion', '1812.07170-2-139-0': 'In this paper, we introduced Ratchet, an NMT-based technique to generate bug fixes from past fixes.', '1812.07170-2-139-1': 'Through an empirical validation on five open source projects, we find that Ratchet is effective in generating fixes.', '1812.07170-2-139-2': 'Moreover, we show that Ratchet can even be used to generate statements for non-bug-fixing statements.', '1812.07170-2-139-3': 'We compare Ratchet to pattern-based patch suggestion as a baseline and show that Ratchet performs at least as well as the baseline.', '1812.07170-2-140-0': 'We also investigate cases where Ratchet fails and find that Ratchet, or more generally NMT, suffers from the out-of-vocabulary problem since it depends on the presence of words in the past to train on.', '1812.07170-2-140-1': 'Also, NMT cannot guarantee that all words are covered/translated.', '1812.07170-2-140-2': 'These aforementioned issues are areas that we plan to tackle in future work.', '1812.07170-2-141-0': '# Acknowledgments'} | [['1812.07170-1-108-0', '1812.07170-2-113-0'], ['1812.07170-1-108-1', '1812.07170-2-113-1'], ['1812.07170-1-108-2', '1812.07170-2-113-2'], ['1812.07170-1-108-3', '1812.07170-2-113-3'], ['1812.07170-1-108-4', 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'1812.07170-1-20-4', '1812.07170-1-21-0', '1812.07170-1-21-1', '1812.07170-1-23-0', '1812.07170-1-29-1', '1812.07170-1-32-0', '1812.07170-1-36-0', '1812.07170-1-55-0', '1812.07170-1-58-3', '1812.07170-1-59-0', '1812.07170-1-64-0', '1812.07170-1-75-0', '1812.07170-1-107-0', '1812.07170-2-1-0', '1812.07170-2-22-3', '1812.07170-2-22-4', '1812.07170-2-23-0', '1812.07170-2-23-1', '1812.07170-2-25-0', '1812.07170-2-31-1', '1812.07170-2-34-0', '1812.07170-2-38-0', '1812.07170-2-57-0', '1812.07170-2-60-3', '1812.07170-2-61-0', '1812.07170-2-66-0', '1812.07170-2-77-0', '1812.07170-2-112-0', '1812.07170-2-115-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1812.07170 | null | null | null | null | null |
hep-ph-9712349 | {'hep-ph-9712349-1-0-0': 'We study the time dependent CP asymmetries in [MATH] decays in the left-right model with spontaneous breakdown of CP.', 'hep-ph-9712349-1-0-1': 'Due to the new contributions to [MATH]-[MATH] mixing, the CP asymmetries can be substantially modified.', 'hep-ph-9712349-1-0-2': 'Most promising for detection of the new physics in the planned [MATH] factories is that the CP asymmetries in the decays [MATH] and [MATH] which are supposed to be equal in the standard model can differ significantly in this class of models independently of the results in the measurements of [MATH].', 'hep-ph-9712349-1-1-0': 'CP violation, currently observed only in the neutral kaon system, is one of the least tested aspects of Nature.', 'hep-ph-9712349-1-1-1': 'The standard model (SM) has specific predictions on the size as well as on the patterns of CP violation in [MATH] meson decays [CITATION] which, if disproved in the future [MATH] factories, would signal the existence of new physics [CITATION].', 'hep-ph-9712349-1-1-2': 'In [MATH] decays new physics can possibly contribute to the [MATH]-[MATH]) mixing as well as to the decay amplitudes.', 'hep-ph-9712349-1-1-3': 'The effect of the new physics in the mixing is universal, i.e., the time dependent rate asymmetries between [MATH] and [MATH] in all their decays to the common CP eigenstates receive the same contribution.', 'hep-ph-9712349-1-1-4': 'On the other hand, the effects of new physics in the decay amplitudes are non-universal and can show up in the comparison of the CP asymmetries in different decay modes [CITATION].', 'hep-ph-9712349-1-2-0': 'In this Letter we analyze the CP asymmetries in [MATH] decays in the [MATH] left-right symmetric model (LRSM) [CITATION] with spontaneous breakdown of CP [CITATION].', 'hep-ph-9712349-1-2-1': 'Indeed, in such a model with spontaneous parity violation it is natural to consider also CP as a spontaneously broken symmetry.', 'hep-ph-9712349-1-2-2': 'We show that with the present constraints on the parameters of the right-handed sector the new contribution to [MATH] meson mixing can be large and time dependent CP asymmetries can vary from -1 to 1 in both [MATH] and [MATH] systems.', 'hep-ph-9712349-1-2-3': 'In addition, due to the [MATH] enhancement of the flavor changing decay [MATH] the CP asymmetries in [MATH] and [MATH] which with high accuracy measure the same unitary triangle angle, [MATH], in the SM may differ from each other almost by unity in the LRSM even in the case in which the measurements in [MATH] correspond exactly to the SM predictions.', 'hep-ph-9712349-1-2-4': 'These two effects are complementary, while the former one is dominated by the new heavy particle exchange, the latter one is due to the left-right mixing.', 'hep-ph-9712349-1-3-0': 'The Higgs sector of the LRSM contains a bidoublet [MATH], [MATH], 0) and two triplets, [MATH](1, 0, 2) and [MATH](0, 1, 2).', 'hep-ph-9712349-1-3-1': 'In order to have parity as a spontaneously broken symmetry, a discrete left-right symmetry, [MATH] should be imposed.', 'hep-ph-9712349-1-3-2': 'After spontaneous symmetry breaking, the vacuum expectation values (vev) of the neutral components of [MATH] and [MATH] give masses to the quarks and left-handed gauge bosons.', 'hep-ph-9712349-1-3-3': "The phase [MATH] which is the relative phase between the vev's is the only source of CP-violation in our model.", 'hep-ph-9712349-1-3-4': 'The left- and right-handed Cabbibo-Kobayashi-Maskawa (CKM) matrices [MATH] and [MATH] respectively, are related as [MATH], due to the discrete left-right symmetry.', 'hep-ph-9712349-1-3-5': 'They contain all together six CP phases which are related to [MATH].', 'hep-ph-9712349-1-3-6': 'In the following it would be convenient to think about [MATH] as the SM CKM matrix and to shift all the phases but one to [MATH].', 'hep-ph-9712349-1-3-7': 'The charged current Lagrangian in the LRSM is given by [MATH] where [MATH] are the charged vector boson fields with the masses [MATH] respectively, and [MATH] denotes their mixing.', 'hep-ph-9712349-1-3-8': 'The appearance of [MATH] in the charged current Lagrangian is pure convention since it can be removed to [MATH].', 'hep-ph-9712349-1-3-9': 'The most stringent lower bound on [MATH] mass, [MATH]>[MATH][MATH] TeV, is derived from the [MATH]-[MATH] mass difference [CITATION].', 'hep-ph-9712349-1-3-10': 'The experimental upper bound on the mixing angle [MATH] depends on the phase [MATH].', 'hep-ph-9712349-1-3-11': 'For small phases it is [MATH]<[MATH][MATH] while for large phases [MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-1-3-12': 'All these results are subject of large hadronic uncertainties.', 'hep-ph-9712349-1-3-13': 'The best limit on [MATH], free of these uncertainties, arises from the muon decay data and is [MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-1-3-14': 'However, for our numerical evaluations we use the appropriate stringent bounds from Ref. [CITATION].', 'hep-ph-9712349-1-3-15': 'There are two neutral flavor changing Higgs bosons in the model whose masses are constrained as [MATH]>[MATH][MATH] TeV [CITATION].', 'hep-ph-9712349-1-4-0': 'CP violation in [MATH] decays takes place due to the interference between mixing and decay.', 'hep-ph-9712349-1-4-1': 'The corresponding CP asymmetry depends on the parameter [MATH] defined as [CITATION] [EQUATION] where [MATH] and [MATH] are the amplitudes of [MATH] and [MATH] decay to a common CP eigenstate, respectively, and we have used [MATH] to introduce the [MATH]-[MATH] mixing phase [MATH].', 'hep-ph-9712349-1-4-2': 'If [MATH] also [MATH] is a pure phase and the time dependent CP asymmetry takes a particularly simple form [EQUATION] where [MATH] is the mass difference between the two physical states.', 'hep-ph-9712349-1-4-3': 'From Eq. ([REF]) and Eq. ([REF]) it is clear that any new physics effect in the mixing will translate into [MATH] and will be universal to all decays while the effect in the decay, [MATH] will depend on the process.', 'hep-ph-9712349-1-4-4': 'In the SM the mixing is already one-loop effect and therefore new physics contribution to it may be sizeable.', 'hep-ph-9712349-1-4-5': 'Without rigourous arguments some of the recent reviews [CITATION] claim that the LRSM contributions to the [MATH] mixing are negligible.', 'hep-ph-9712349-1-4-6': 'We show the opposite by performing an explicit calculation.', 'hep-ph-9712349-1-5-0': 'Let us assume that the off-diagonal element [MATH] of the [MATH]-[MATH] mixing is changed by a factor of [MATH] as a result of the new contribution from the LRSM, [MATH].', 'hep-ph-9712349-1-5-1': 'Here [MATH] denotes the contribution from the left-handed sector which in our convention is equal to the SM result, and [MATH] denotes the dominant new contribution from the box diagrams with one [MATH] and one [MATH] and from the tree level flavor changing Higgs exchange.', 'hep-ph-9712349-1-5-2': '[MATH] including the LO QCD corrections has been calculated in Ref. [CITATION] using the vacuum insertion approximation.', 'hep-ph-9712349-1-5-3': 'With [MATH] GeV, [MATH] GeV, [MATH] GeV, [MATH] MeV, [MATH] MeV and the SM input as in Ref. [CITATION] the LO QCD improved result reads [CITATION] [EQUATION] where [MATH] and the function [MATH] is a complicated function of [MATH].', 'hep-ph-9712349-1-5-4': 'Numerically [MATH] and [MATH].', 'hep-ph-9712349-1-5-5': 'Note that this estimate holds for both [MATH] and [MATH] systems.', 'hep-ph-9712349-1-5-6': 'One can write [MATH] where [MATH].', 'hep-ph-9712349-1-5-7': 'Consequently the phase [MATH] in the mixing in the LRSM becomes [MATH] where [EQUATION]', 'hep-ph-9712349-1-5-8': 'The phase [MATH] in our model has been calculated in terms of the quark masses and phase [MATH] and reads [CITATION] [MATH] where [MATH] and [MATH] are the signs occuring in the Yukawa sector of the model.', 'hep-ph-9712349-1-5-9': 'While [MATH] [CITATION] there is an enhancement factor [MATH] in the expressions for [MATH] which thus can be as large as unity.', 'hep-ph-9712349-1-5-10': 'Therefore, taking into account the present constraints on the right-handed particle masses it follows from Eqs ([REF]), ([REF]) that in the LRSM with spontaneous CP violation the phases [MATH] can take any value from 0 to [MATH] and, consequently, the CP asymmetries in Eq. ([REF]) can vary between -1 and 1.', 'hep-ph-9712349-1-6-0': 'Unfortunately the CP asymmetries in [MATH] decays which are predicted to be very small in the SM and can easily show up the new physics cannot be studied in [MATH] factories running on the [MATH] peak.', 'hep-ph-9712349-1-6-1': '[MATH] decays, however, involve large CP asymmetries which are predicted with poor accuracy in the SM.', 'hep-ph-9712349-1-6-2': 'The "benchmark" modes [MATH] and [MATH] measure [MATH] and [MATH] respectively, where [MATH] and [MATH] are the angles of the SM unitary triangle.', 'hep-ph-9712349-1-6-3': 'The SM predictions for them are [MATH]<[MATH][MATH]<[MATH][MATH] and [MATH] [CITATION].', 'hep-ph-9712349-1-6-4': 'Unless the experimental measurement [MATH] clearly lays outside the allowed region the new physics cannot be traced off.', 'hep-ph-9712349-1-6-5': 'Moreover, since [MATH] gets modified as [MATH] then [MATH] cancels out in [MATH] [CITATION].', 'hep-ph-9712349-1-6-6': 'Therefore, finding new physics could rely only on the experimentally very challenging measurement of the third angle [MATH].', 'hep-ph-9712349-1-7-0': 'On the other hand, it is known that in the SM the CP asymmetries in the theoretically clean decays [MATH]) and [MATH]) measure with high accuracy the same angle [MATH].', 'hep-ph-9712349-1-7-1': 'The uncertainty in the SM is estimated to be [CITATION] [EQUATION] where [MATH].', 'hep-ph-9712349-1-7-2': 'Any deviation from this relation will be a clear indication of new physics.', 'hep-ph-9712349-1-7-3': 'The decay [MATH] is dominated by tree level [MATH] exchange and the new physics contribution to it cannot be sizeable.', 'hep-ph-9712349-1-7-4': 'However, the flavor changing decay [MATH] is one-loop effect in the SM and can, therefore, be modified by new physics.', 'hep-ph-9712349-1-8-0': 'The flavor changing decay [MATH] is induced by the QCD-, electroweak- and magnetic penguins.', 'hep-ph-9712349-1-8-1': 'The dominant contribution comes from the QCD penguins with top quark in the loop.', 'hep-ph-9712349-1-8-2': 'It is also known [CITATION] that the electroweak penguins decrease about 30% the decay rate and we shall add their contribution to the QCD improved effective Hamiltonian.', 'hep-ph-9712349-1-8-3': 'However, to the new [MATH] effect the electroweak magnetic contribution is suppressed compared with the gluonic magnetic penguins by a factor of [MATH] and we shall neglect it here.', 'hep-ph-9712349-1-8-4': 'The effective Hamiltonian due to the gluon exchange describing the decay [MATH] at the scale [MATH] is given as [EQUATION] where [MATH] and the [MATH] term describes the new dominant left-right contribution via the mixing angle [MATH].', 'hep-ph-9712349-1-8-5': 'Here [MATH] and analogously [MATH].', 'hep-ph-9712349-1-8-6': 'Note that the phases [MATH] are independent and can take any value in the range [MATH].', 'hep-ph-9712349-1-8-7': 'The functions [MATH] and [MATH] are Inami-Lim type functions [CITATION] of [MATH] and are given by [MATH].', 'hep-ph-9712349-1-8-8': 'The left-right analog of [MATH], [MATH], is numerically about factor of four larger than the latter one.', 'hep-ph-9712349-1-8-9': 'Together with the [MATH] enhancement in [MATH] this practically overcomes the left-right suppression by small [MATH].', 'hep-ph-9712349-1-9-0': 'To obtain reliable estimates for the CP asymmetries in [MATH] induced modes in the LRSM we have to calculate the LO QCD corrections to Eq. ([REF]).', 'hep-ph-9712349-1-9-1': 'Using the operator product expansion to integrate out the heavy fields and calculating the Wilson coefficients [MATH] in the leading logarithm approximation we run them with the renormalization group equations from the scale of [MATH] down to the scale [MATH] (since the contributions of [MATH] are negligible we start immediately from the [MATH] scale).', 'hep-ph-9712349-1-9-2': 'Because the new physics appears only in the gluonic magnetic operators we can safely take over some well-known results from the SM studies.', 'hep-ph-9712349-1-9-3': 'The effective Hamiltonian we start with is [EQUATION] where we have explicitly separated the electroweak penguin operators (the second term) which to a good approximation will not receive any new contribution in the LRSM from the twenty operators which do mix with the gluonic and photonic magnetic operators.', 'hep-ph-9712349-1-9-4': 'Due to the left-right symmetry the twenty operators split into two groups, [MATH]-[MATH] and [MATH]-[MATH] which can be obtained by [MATH] from each other.', 'hep-ph-9712349-1-9-5': 'For the QCD penguin operators [MATH]-[MATH] magnetic penguin operators [MATH] as well as for the electroweak penguin operators [MATH]-[MATH] we use the standard set of the operators from Ref. [CITATION].', 'hep-ph-9712349-1-9-6': 'The new left-right operators [MATH] are [CITATION] [MATH] and [MATH].', 'hep-ph-9712349-1-9-7': 'Keeping only the top and bottom quark masses to be non-vanishing, the matching conditions at [MATH] scale are given as [MATH] and the rest of the coefficients vanish.', 'hep-ph-9712349-1-9-8': 'Here the SM function [MATH] and its left-right analog [MATH] are given by [MATH].', 'hep-ph-9712349-1-10-0': 'The [MATH] anomalous dimension matrix decomposes into two identical [MATH] submatrices.', 'hep-ph-9712349-1-10-1': 'The SM [MATH] submatrix of the latter one can be found in Ref. [CITATION] and the rest of the entries have been calculated by Cho and Misiak in Ref. [CITATION].', 'hep-ph-9712349-1-10-2': "In the leading logarithm approximation the low energy Wilson coefficients for five flavors are given by [MATH] where the [MATH]'s in the exponent of [MATH] are the eigenvalues of the anomalous dimension matrix over [MATH] and the matrix [MATH] contains the corresponding eigenvectors.", 'hep-ph-9712349-1-10-3': 'We find [EQUATION] where [MATH] -0.9135, 0.0209, 0.0873, -0.0571) and [MATH] 0.4086, 0.1456, -0.4230, -0.8994).', 'hep-ph-9712349-1-10-4': 'We reproduced [MATH] and [MATH] exactly as in Ref. [CITATION] and [MATH]-[MATH] numerically within 1% as in Ref. [CITATION] and we shall not present them here.', 'hep-ph-9712349-1-11-0': 'Denoting [MATH] the decay amplitude of [MATH] can be written [EQUATION]', 'hep-ph-9712349-1-11-1': 'Here the hadronic matrix elements [MATH] and [MATH] can be approximated to be of the form [EQUATION] and similarly for [MATH] where the timelike gluon has produced [MATH].', 'hep-ph-9712349-1-11-2': 'Using factorization and the following parametrization for the hadronic matrix elements [CITATION] [MATH] where [MATH]=0.23 GeV, [MATH] and [MATH] one gets [CITATION] [MATH] and [MATH] where [MATH].', 'hep-ph-9712349-1-11-3': 'In the parametrization of [CITATION] [MATH] where [MATH]=0.38, [MATH]=5.8 GeV and [MATH]=5.4 GeV.', 'hep-ph-9712349-1-11-4': 'The element [MATH] decomposes to [MATH] where [MATH].', 'hep-ph-9712349-1-11-5': 'With factorization the new matrix element appearing is [MATH] where [MATH] is obtained using the heavy quark effective theory [CITATION].', 'hep-ph-9712349-1-11-6': 'As a result we get [MATH] where the second term in brackets is negligibly small.', 'hep-ph-9712349-1-11-7': 'The same result is valid also for [MATH].', 'hep-ph-9712349-1-12-0': 'It has been shown that the "physical" range of [MATH] in [MATH] is [MATH]<[MATH][MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-1-12-1': 'We use [MATH] and [MATH] GeV [CITATION] to estimate the possible effects of new physics.', 'hep-ph-9712349-1-12-2': 'Numerically we obtain for the LO QCD improved amplitude [MATH] [EQUATION]', 'hep-ph-9712349-1-12-3': 'It is important to notice that in [MATH] both phases [MATH] contribute to the CP asymmetry because only the hadronic matrix elements of the vector currents matter.', 'hep-ph-9712349-1-12-4': 'This should be compared with [MATH] in which CP asymmetry is given only by the right-projected operators [CITATION] and, consequently, the phase [MATH] does not contribute.', 'hep-ph-9712349-1-12-5': 'Also, the major source of uncertainty in the decay rate, the hadronic matrix element [MATH] cancels out in the CP asymmetry.', 'hep-ph-9712349-1-12-6': 'The maximum effect is obtained if [MATH].', 'hep-ph-9712349-1-12-7': 'We get [MATH] which implies [MATH].', 'hep-ph-9712349-1-12-8': 'This result should be compared with Eq. ([REF]) which implies that there could be a clear effect of the new physics.', 'hep-ph-9712349-1-12-9': 'The maximum allowed difference of the CP asymmetries in [MATH] and [MATH] in the LRSM could thus be as large as [MATH].', 'hep-ph-9712349-1-12-10': 'If the difference of the phases in these two processes will be measured within 10% and if no difference will be seen then a new upper bound, [MATH]<[MATH][MATH] can be put on the left-right mixing angle for large phases which is stronger than the present limit for small phases [MATH]<[MATH][MATH].', 'hep-ph-9712349-1-13-0': 'Finally, let us consider the constraints on the LRSM coming from the decay [MATH].', 'hep-ph-9712349-1-13-1': 'It is possible that due to the cancellation between the [MATH] and [MATH] contributions both the rate and the CP asymmetry in this process can, within errors, correspond to the SM predictions [CITATION].', 'hep-ph-9712349-1-13-2': 'If the SM predictions will be confirmed experimentally (the CP asymmetry in the SM is expected to be very small) this will constrain the phase [MATH] and the size of the [MATH] contribution to [MATH] but cannot probe the phase [MATH] which will still be a free parameter.', 'hep-ph-9712349-1-13-3': 'Assuming the above scenario we obtain in the LRSM for the decay [MATH] that [MATH] which means [MATH] and [MATH].', 'hep-ph-9712349-1-13-4': 'Therefore, large observable effects are possible independently of the results in [MATH].', 'hep-ph-9712349-1-14-0': 'In conclusion, we show that the LRSM with spontaneous violation of CP can dramatically affect the time dependent CP asymmetries in [MATH] decays.', 'hep-ph-9712349-1-14-1': 'Due to the new contribution to the [MATH]-[MATH] mixing the CP asymmetries can vary from -1 to 1 in both [MATH] and [MATH] decays.', 'hep-ph-9712349-1-14-2': 'Most importantly for discovering the new physics in the [MATH] factories, the CP asymmetries in [MATH] and [MATH] which are equal with high accuracy in the SM can differ from each other as much as unity in our model independently of the results in [MATH].', 'hep-ph-9712349-1-15-0': 'We thank Y. Grossman for pointing out the interesting CP violation effects in the decay amplitudes and A. Pich and L. Silvestrini for discussions on the QCD corrections.'} | {'hep-ph-9712349-2-0-0': 'We study the time dependent CP asymmetries in [MATH] decays in the left-right model with spontaneous breakdown of CP.', 'hep-ph-9712349-2-0-1': 'Due to the new contributions to [MATH]-[MATH] mixing, the CP asymmetries can be substantially modified.', 'hep-ph-9712349-2-0-2': 'Moreover, there can be significant new contributions to the [MATH]-meson decay amplitudes from the magnetic penguins.', 'hep-ph-9712349-2-0-3': 'Most promising for detection of the new physics in the planned [MATH] factories is that the CP asymmetries in the decays [MATH] and [MATH] which are supposed to be equal in the standard model can differ significantly in this class of models independently of the results in the measurements of [MATH].', 'hep-ph-9712349-2-1-0': 'CP violation, currently observed only in the neutral kaon system, is one of the least tested aspects of Nature.', 'hep-ph-9712349-2-1-1': 'The standard model (SM) has specific predictions on the size as well as on the patterns of CP violation in [MATH] meson decays [CITATION] which, if disproved in the future [MATH] factories, would signal the existence of new physics [CITATION].', 'hep-ph-9712349-2-1-2': 'In [MATH] decays new physics can possibly contribute to the [MATH]-[MATH]) mixing as well as to the decay amplitudes.', 'hep-ph-9712349-2-1-3': 'The effect of the new physics in the mixing is universal, i.e., the time dependent rate asymmetries between [MATH] and [MATH] in all their decays to the common CP eigenstates receive the same contribution.', 'hep-ph-9712349-2-1-4': 'On the other hand, the effects of new physics in the decay amplitudes are non-universal and can show up in the comparison of the CP asymmetries in different decay modes [CITATION].', 'hep-ph-9712349-2-2-0': 'In this Letter we analyze the CP asymmetries in [MATH] decays in the [MATH] left-right symmetric model (LRSM) [CITATION] with spontaneous breakdown of CP [CITATION].', 'hep-ph-9712349-2-2-1': 'Indeed, in such a model with spontaneous parity violation it is natural to consider also CP as a spontaneously broken symmetry.', 'hep-ph-9712349-2-2-2': 'We show that with the present constraints on the parameters of the right-handed sector the new contribution to [MATH] meson mixing can be large and time dependent CP asymmetries can vary from -1 to 1 in both [MATH] and [MATH] systems.', 'hep-ph-9712349-2-2-3': 'In addition, due to the new penguins contributing to the flavor changing decay [MATH] the CP asymmetries in [MATH] and [MATH] which with high accuracy measure the same unitary triangle angle, [MATH], in the SM may differ from each other almost by unity in the LRSM even in the case in which the measurements in [MATH] correspond exactly to the SM predictions.', 'hep-ph-9712349-2-2-4': 'These two effects are complementary, while the former one is dominated by the new heavy particle exchange, the latter one is due to the left-right mixing.', 'hep-ph-9712349-2-3-0': 'The Higgs sector of the LRSM contains a bidoublet [MATH], [MATH], 0) and two triplets, [MATH](1, 0, 2) and [MATH](0, 1, 2).', 'hep-ph-9712349-2-3-1': 'In order to have parity as a spontaneously broken symmetry, a discrete left-right symmetry, [MATH] should be imposed.', 'hep-ph-9712349-2-3-2': 'After spontaneous symmetry breaking, the vacuum expectation values (vev) of the neutral components of [MATH] and [MATH] give masses to the quarks and left-handed gauge bosons.', 'hep-ph-9712349-2-3-3': "The phase [MATH] which is the relative phase between the vev's is the only source of CP-violation in our model.", 'hep-ph-9712349-2-3-4': 'The left- and right-handed Cabbibo-Kobayashi-Maskawa (CKM) matrices [MATH] and [MATH] respectively, are related as [MATH], due to the discrete left-right symmetry.', 'hep-ph-9712349-2-3-5': 'They contain all together six CP phases which are related to [MATH].', 'hep-ph-9712349-2-3-6': 'In the following it would be convenient to think about [MATH] as the SM CKM matrix and to shift all the phases but one to [MATH].', 'hep-ph-9712349-2-3-7': 'The charged current Lagrangian in the LRSM is given by [MATH] where [MATH] are the charged vector boson fields with the masses [MATH] respectively, and [MATH] denotes their mixing.', 'hep-ph-9712349-2-3-8': 'The appearance of [MATH] in the charged current Lagrangian is pure convention since it can be removed to [MATH].', 'hep-ph-9712349-2-3-9': 'The most stringent lower bound on [MATH] mass, [MATH]>[MATH][MATH] TeV, is derived from the [MATH]-[MATH] mass difference [CITATION].', 'hep-ph-9712349-2-3-10': 'The experimental upper bound on the mixing angle [MATH] depends on the phase [MATH].', 'hep-ph-9712349-2-3-11': 'For small phases it is [MATH]<[MATH][MATH] while for large phases [MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-2-3-12': 'All these results are subject of large hadronic uncertainties.', 'hep-ph-9712349-2-3-13': 'The best limit on [MATH], free of these uncertainties, arises from the muon decay data and is [MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-2-3-14': 'However, for our numerical evaluations we use the appropriate stringent bounds from Ref. [CITATION].', 'hep-ph-9712349-2-3-15': 'There are two neutral flavor changing Higgs bosons in the model whose masses are constrained as [MATH]>[MATH][MATH] TeV [CITATION].', 'hep-ph-9712349-2-4-0': 'CP violation in [MATH] decays takes place due to the interference between mixing and decay.', 'hep-ph-9712349-2-4-1': 'The corresponding CP asymmetry depends on the parameter [MATH] defined as [CITATION] [EQUATION] where [MATH] and [MATH] are the amplitudes of [MATH] and [MATH] decay to a common CP eigenstate, respectively, and we have used [MATH] to introduce the [MATH]-[MATH] mixing phase [MATH].', 'hep-ph-9712349-2-4-2': 'If [MATH] also [MATH] is a pure phase and the time dependent CP asymmetry takes a particularly simple form [EQUATION] where [MATH] is the mass difference between the two physical states.', 'hep-ph-9712349-2-4-3': 'From Eq. ([REF]) and Eq. ([REF]) it is clear that any new physics effect in the mixing will translate into [MATH] and will be universal to all decays while the effect in the decay, [MATH] will depend on the process.', 'hep-ph-9712349-2-4-4': 'In the SM the mixing is already one-loop effect and therefore new physics contribution to it may be sizeable.', 'hep-ph-9712349-2-4-5': 'Without rigourous arguments some of the recent reviews [CITATION] claim that the LRSM contributions to the [MATH] mixing are negligible.', 'hep-ph-9712349-2-4-6': 'We show the opposite by performing an explicit calculation.', 'hep-ph-9712349-2-5-0': 'Let us assume that the off-diagonal element [MATH] of the [MATH]-[MATH] mixing is changed by a factor of [MATH] as a result of the new contribution from the LRSM, [MATH].', 'hep-ph-9712349-2-5-1': 'Here [MATH] denotes the contribution from the left-handed sector which in our convention is equal to the SM result, and [MATH] denotes the dominant new contribution from the box diagrams with one [MATH] and one [MATH] and from the tree level flavor changing Higgs exchange.', 'hep-ph-9712349-2-5-2': '[MATH] including the LO QCD corrections has been calculated in Ref. [CITATION] using the vacuum insertion approximation.', 'hep-ph-9712349-2-5-3': 'With [MATH] GeV, [MATH] GeV, [MATH] GeV, [MATH] MeV, [MATH] MeV and the SM input as in Ref. [CITATION] the LO QCD improved result reads [CITATION] [EQUATION] where [MATH] and the function [MATH] is a complicated function of [MATH].', 'hep-ph-9712349-2-5-4': 'Numerically [MATH] and [MATH].', 'hep-ph-9712349-2-5-5': 'Note that this estimate holds for both [MATH] and [MATH] systems.', 'hep-ph-9712349-2-5-6': 'One can write [MATH] where [MATH].', 'hep-ph-9712349-2-5-7': 'Consequently the phase [MATH] in the mixing in the LRSM becomes [MATH] where [EQUATION]', 'hep-ph-9712349-2-5-8': 'The phase [MATH] in our model has been calculated in terms of the quark masses and phase [MATH] and reads [CITATION] [MATH] where [MATH] and [MATH] are the signs occuring in the Yukawa sector of the model.', 'hep-ph-9712349-2-5-9': 'While [MATH] [CITATION] there is an enhancement factor [MATH] in the expressions for [MATH] which thus can be as large as unity.', 'hep-ph-9712349-2-5-10': 'Therefore, taking into account the present constraints on the right-handed particle masses it follows from Eqs ([REF]), ([REF]) that in the LRSM with spontaneous CP violation the phases [MATH] can take any value from 0 to [MATH] and, consequently, the CP asymmetries in Eq. ([REF]) can vary between -1 and 1.', 'hep-ph-9712349-2-6-0': 'Unfortunately the CP asymmetries in [MATH] decays which are predicted to be very small in the SM and can easily show up the new physics cannot be studied in [MATH] factories running on the [MATH] peak.', 'hep-ph-9712349-2-6-1': '[MATH] decays, however, involve large CP asymmetries which are predicted with poor accuracy in the SM.', 'hep-ph-9712349-2-6-2': 'The "benchmark" modes [MATH] and [MATH] measure [MATH] and [MATH] respectively, where [MATH] and [MATH] are the angles of the SM unitary triangle.', 'hep-ph-9712349-2-6-3': 'The SM predictions for them are [MATH]<[MATH][MATH]<[MATH][MATH] and [MATH] [CITATION].', 'hep-ph-9712349-2-6-4': 'Unless the experimental measurement [MATH] clearly lays outside the allowed region the new physics cannot be traced off.', 'hep-ph-9712349-2-6-5': 'Moreover, since [MATH] gets modified as [MATH] then [MATH] cancels out in [MATH] [CITATION].', 'hep-ph-9712349-2-6-6': 'Therefore, finding new physics could rely only on the experimentally very challenging measurement of the third angle [MATH].', 'hep-ph-9712349-2-7-0': 'On the other hand, it is known that in the SM the CP asymmetries in the theoretically clean decays [MATH]) and [MATH]) measure with high accuracy the same angle [MATH].', 'hep-ph-9712349-2-7-1': 'The uncertainty in the SM is estimated to be [CITATION] [EQUATION] where [MATH].', 'hep-ph-9712349-2-7-2': 'Any deviation from this relation (which should be further tested as proposed in Ref. [CITATION]) will be a clear indication of new physics.', 'hep-ph-9712349-2-7-3': 'The decay [MATH] is dominated by tree level [MATH] exchange and the new physics contribution to it cannot be sizeable.', 'hep-ph-9712349-2-7-4': 'However, the flavor changing decay [MATH] is one-loop effect in the SM and can, therefore, be modified by new physics.', 'hep-ph-9712349-2-8-0': 'The flavor changing decay [MATH] is induced by the QCD-, electroweak- and magnetic penguins.', 'hep-ph-9712349-2-8-1': 'The dominant contribution comes from the QCD penguins with top quark in the loop.', 'hep-ph-9712349-2-8-2': 'It is also known [CITATION] that the electroweak penguins decrease about 30% the decay rate and we shall add their contribution to the QCD improved effective Hamiltonian.', 'hep-ph-9712349-2-8-3': 'We start with the effective Hamiltonian due to the gluon exchange describing the decay [MATH] at the scale [MATH] [EQUATION] where [MATH] and the [MATH] term describes the new dominant left-right contribution via the mixing angle [MATH].', 'hep-ph-9712349-2-8-4': 'Here [MATH] and analogously [MATH].', 'hep-ph-9712349-2-8-5': 'Note that the phases [MATH] are independent and can take any value in the range [MATH].', 'hep-ph-9712349-2-8-6': 'The functions [MATH] and [MATH] are Inami-Lim type functions [CITATION] of [MATH] and are given by [MATH].', 'hep-ph-9712349-2-8-7': 'The left-right analog of [MATH], [MATH], is numerically about factor of four larger than the latter one.', 'hep-ph-9712349-2-8-8': 'Together with the [MATH] enhancement in [MATH] this practically overcomes the left-right suppression by small [MATH].', 'hep-ph-9712349-2-9-0': 'To obtain reliable estimates for the CP asymmetries in [MATH] induced modes in the LRSM we have to calculate the LO QCD corrections to Eq. ([REF]).', 'hep-ph-9712349-2-9-1': 'Using the operator product expansion to integrate out the heavy fields and calculating the Wilson coefficients [MATH] in the leading logarithm approximation we run them with the renormalization group equations from the scale of [MATH] down to the scale [MATH] (since the contributions of [MATH] are negligible we start immediately from the [MATH] scale).', 'hep-ph-9712349-2-9-2': 'Because the new physics appears only in the gluonic magnetic operators we can safely take over some well-known results from the SM studies.', 'hep-ph-9712349-2-9-3': 'The effective Hamiltonian we work with is [EQUATION] where we have explicitly separated the electroweak penguin operators (the second term) which to a good approximation will not receive any new contribution in the LRSM from the twenty operators which do mix with the gluonic and photonic magnetic operators.', 'hep-ph-9712349-2-9-4': 'Due to the left-right symmetry the twenty operators split into two groups, [MATH]-[MATH] and [MATH]-[MATH] which can be obtained by [MATH] from each other.', 'hep-ph-9712349-2-9-5': 'For the QCD penguin operators [MATH]-[MATH] magnetic penguin operators [MATH] as well as for the electroweak penguin operators [MATH]-[MATH] we use the standard set of the operators from Ref. [CITATION].', 'hep-ph-9712349-2-9-6': 'The new left-right operators [MATH] are [CITATION] [MATH] and [MATH].', 'hep-ph-9712349-2-9-7': 'Keeping only the top and bottom quark masses to be non-vanishing, the matching conditions at [MATH] scale are given as [MATH] and the rest of the coefficients vanish.', 'hep-ph-9712349-2-9-8': 'Here the SM function [MATH] and its left-right analog [MATH] are given by [MATH].', 'hep-ph-9712349-2-10-0': 'The [MATH] anomalous dimension matrix decomposes into two identical [MATH] submatrices.', 'hep-ph-9712349-2-10-1': 'The SM [MATH] submatrix of the latter one can be found in Ref. [CITATION] and the rest of the entries have been calculated by Cho and Misiak in Ref. [CITATION].', 'hep-ph-9712349-2-10-2': "In the leading logarithm approximation the low energy Wilson coefficients for five flavors are given by [MATH] where the [MATH]'s in the exponent of [MATH] are the eigenvalues of the anomalous dimension matrix over [MATH] and the matrix [MATH] contains the corresponding eigenvectors.", 'hep-ph-9712349-2-10-3': 'We find [EQUATION] where [MATH] -0.9135, 0.0209, 0.0873, -0.0571) and [MATH] 0.4086, 0.1456, -0.4230, -0.8994).', 'hep-ph-9712349-2-10-4': 'We reproduced [MATH] and [MATH] exactly as in Ref. [CITATION] and [MATH]-[MATH] numerically within 1% as in Ref. [CITATION] and we shall not present them here.', 'hep-ph-9712349-2-11-0': 'Denoting [MATH] the decay amplitude of [MATH] can be written [EQUATION]', 'hep-ph-9712349-2-11-1': 'Contributions from [MATH] are suppressed by a factor of [MATH] if compared with [MATH] and therefore negligible.', 'hep-ph-9712349-2-11-2': 'The hadronic matrix elements [MATH] and [MATH] can be approximated to be of the form [EQUATION] and similarly for [MATH] where the timelike gluon has produced [MATH].', 'hep-ph-9712349-2-11-3': 'Using factorization and the following parametrization for the hadronic matrix elements [CITATION] [MATH] where [MATH]=0.23 GeV, [MATH] and [MATH] one gets [CITATION] [MATH] and [MATH] where [MATH].', 'hep-ph-9712349-2-11-4': 'In the parametrization of [CITATION] [MATH] where [MATH]=0.38, [MATH]=5.8 GeV and [MATH]=5.4 GeV.', 'hep-ph-9712349-2-11-5': 'The element [MATH] decomposes to [MATH] where [MATH].', 'hep-ph-9712349-2-11-6': 'With factorization the new matrix element appearing is [MATH] where [MATH] is obtained using the heavy quark effective theory [CITATION].', 'hep-ph-9712349-2-11-7': 'As a result we get [MATH] where the second term in brackets is negligibly small.', 'hep-ph-9712349-2-11-8': 'The same result is valid also for [MATH].', 'hep-ph-9712349-2-12-0': 'It has been shown that the" physical" range of [MATH] in [MATH] is [MATH]<[MATH][MATH]<[MATH][MATH] [CITATION].', 'hep-ph-9712349-2-12-1': 'To be conservative we use [MATH] and [MATH] GeV [CITATION] to estimate the possible effects of new physics.', 'hep-ph-9712349-2-12-2': 'Numerically we obtain for the LO QCD improved amplitude [MATH] [EQUATION]', 'hep-ph-9712349-2-12-3': 'It is important to notice that in [MATH] both phases [MATH] contribute to the CP asymmetry because only the hadronic matrix elements of the vector currents matter.', 'hep-ph-9712349-2-12-4': 'This should be compared with [MATH] in which CP asymmetry is given only by the right-projected operators [CITATION] and, consequently, the phase [MATH] does not contribute.', 'hep-ph-9712349-2-12-5': 'Also, the major source of uncertainty in the decay rate, the hadronic matrix element [MATH] cancels out in the CP asymmetry.', 'hep-ph-9712349-2-12-6': 'The maximum effect is obtained if [MATH].', 'hep-ph-9712349-2-12-7': 'We get [MATH] which implies [MATH].', 'hep-ph-9712349-2-12-8': 'This result should be compared with Eq. ([REF]) which implies that there could be a clear effect of the new physics.', 'hep-ph-9712349-2-12-9': 'The maximum allowed difference of the CP asymmetries in [MATH] and [MATH] in the LRSM could thus be as large as [MATH].', 'hep-ph-9712349-2-12-10': 'If the difference of the phases in these two processes will be measured within 10% and if no difference will be seen then a new upper bound, [MATH]<[MATH][MATH] can be put on the left-right mixing angle for large phases which is stronger than the present limit for small phases [MATH]<[MATH][MATH].', 'hep-ph-9712349-2-13-0': 'Note that the new effect in Eq. ([REF]) is due to the LR contribution to the QCD magnetic penguins.', 'hep-ph-9712349-2-13-1': 'This new contribution can also provide an answer to the enhancement of [MATH] observed by CLEO [CITATION].', 'hep-ph-9712349-2-14-0': 'Finally, let us consider the constraints on the LRSM coming from the decay [MATH].', 'hep-ph-9712349-2-14-1': 'It is possible that due to the cancellation between the [MATH] and [MATH] contributions both the total rate [MATH] and the CP asymmetry in this process can, within errors, correspond to the SM predictions [CITATION].', 'hep-ph-9712349-2-14-2': 'If the SM predictions will be confirmed experimentally (the CP asymmetry in the SM is expected to be very small) this will constrain the phase [MATH] and the size of the [MATH] contribution to [MATH] but cannot probe the phase [MATH] which will still be a free parameter.', 'hep-ph-9712349-2-14-3': 'Assuming [MATH] we obtain in the most conservative case for the decay [MATH] in the LRSM that [MATH] which means [MATH] and [MATH].', 'hep-ph-9712349-2-14-4': 'Therefore, large observable effects are possible independently of the results in [MATH].', 'hep-ph-9712349-2-15-0': 'In conclusion, we show that the LRSM with spontaneous violation of CP can dramatically affect the time dependent CP asymmetries in [MATH] decays.', 'hep-ph-9712349-2-15-1': 'Due to the new contribution to the [MATH]-[MATH] mixing the CP asymmetries can vary from -1 to 1 in both [MATH] and [MATH] decays.', 'hep-ph-9712349-2-15-2': 'Moreover, the [MATH]-meson decay amplitudes can receive significant new contributions as well.', 'hep-ph-9712349-2-15-3': 'Most importantly for discovering the new physics in the [MATH] factories, the CP asymmetries in [MATH] and [MATH] which are equal with high accuracy in the SM can differ from each other as much as unity in our model independently of the results in [MATH].', 'hep-ph-9712349-2-15-4': 'Interestingly, the excess of [MATH] observed by CLEO can also be explained by the LRSM.', 'hep-ph-9712349-2-16-0': 'We thank Y. Grossman for pointing out the interesting CP violation effects in the decay amplitudes and A. Pich and L. Silvestrini for discussions on the QCD corrections.'} | [['hep-ph-9712349-1-2-0', 'hep-ph-9712349-2-2-0'], ['hep-ph-9712349-1-2-1', 'hep-ph-9712349-2-2-1'], ['hep-ph-9712349-1-2-2', 'hep-ph-9712349-2-2-2'], ['hep-ph-9712349-1-2-4', 'hep-ph-9712349-2-2-4'], ['hep-ph-9712349-1-8-0', 'hep-ph-9712349-2-8-0'], ['hep-ph-9712349-1-8-1', 'hep-ph-9712349-2-8-1'], ['hep-ph-9712349-1-8-2', 'hep-ph-9712349-2-8-2'], ['hep-ph-9712349-1-8-5', 'hep-ph-9712349-2-8-4'], ['hep-ph-9712349-1-8-6', 'hep-ph-9712349-2-8-5'], ['hep-ph-9712349-1-8-7', 'hep-ph-9712349-2-8-6'], ['hep-ph-9712349-1-8-8', 'hep-ph-9712349-2-8-7'], ['hep-ph-9712349-1-8-9', 'hep-ph-9712349-2-8-8'], ['hep-ph-9712349-1-9-0', 'hep-ph-9712349-2-9-0'], ['hep-ph-9712349-1-9-1', 'hep-ph-9712349-2-9-1'], ['hep-ph-9712349-1-9-2', 'hep-ph-9712349-2-9-2'], ['hep-ph-9712349-1-9-4', 'hep-ph-9712349-2-9-4'], ['hep-ph-9712349-1-9-5', 'hep-ph-9712349-2-9-5'], ['hep-ph-9712349-1-9-6', 'hep-ph-9712349-2-9-6'], ['hep-ph-9712349-1-9-7', 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'hep-ph-9712349-2-15-1'], ['hep-ph-9712349-1-14-2', 'hep-ph-9712349-2-15-3'], ['hep-ph-9712349-1-2-3', 'hep-ph-9712349-2-2-3'], ['hep-ph-9712349-1-8-4', 'hep-ph-9712349-2-8-3'], ['hep-ph-9712349-1-9-3', 'hep-ph-9712349-2-9-3'], ['hep-ph-9712349-1-11-1', 'hep-ph-9712349-2-11-2'], ['hep-ph-9712349-1-13-1', 'hep-ph-9712349-2-14-1'], ['hep-ph-9712349-1-13-3', 'hep-ph-9712349-2-14-3'], ['hep-ph-9712349-1-12-0', 'hep-ph-9712349-2-12-0'], ['hep-ph-9712349-1-12-1', 'hep-ph-9712349-2-12-1'], ['hep-ph-9712349-1-7-2', 'hep-ph-9712349-2-7-2']] | [['hep-ph-9712349-1-2-0', 'hep-ph-9712349-2-2-0'], ['hep-ph-9712349-1-2-1', 'hep-ph-9712349-2-2-1'], ['hep-ph-9712349-1-2-2', 'hep-ph-9712349-2-2-2'], ['hep-ph-9712349-1-2-4', 'hep-ph-9712349-2-2-4'], ['hep-ph-9712349-1-8-0', 'hep-ph-9712349-2-8-0'], ['hep-ph-9712349-1-8-1', 'hep-ph-9712349-2-8-1'], ['hep-ph-9712349-1-8-2', 'hep-ph-9712349-2-8-2'], ['hep-ph-9712349-1-8-5', 'hep-ph-9712349-2-8-4'], ['hep-ph-9712349-1-8-6', 'hep-ph-9712349-2-8-5'], ['hep-ph-9712349-1-8-7', 'hep-ph-9712349-2-8-6'], ['hep-ph-9712349-1-8-8', 'hep-ph-9712349-2-8-7'], ['hep-ph-9712349-1-8-9', 'hep-ph-9712349-2-8-8'], ['hep-ph-9712349-1-9-0', 'hep-ph-9712349-2-9-0'], ['hep-ph-9712349-1-9-1', 'hep-ph-9712349-2-9-1'], ['hep-ph-9712349-1-9-2', 'hep-ph-9712349-2-9-2'], ['hep-ph-9712349-1-9-4', 'hep-ph-9712349-2-9-4'], ['hep-ph-9712349-1-9-5', 'hep-ph-9712349-2-9-5'], ['hep-ph-9712349-1-9-6', 'hep-ph-9712349-2-9-6'], ['hep-ph-9712349-1-9-7', 'hep-ph-9712349-2-9-7'], ['hep-ph-9712349-1-9-8', 'hep-ph-9712349-2-9-8'], ['hep-ph-9712349-1-10-0', 'hep-ph-9712349-2-10-0'], ['hep-ph-9712349-1-10-1', 'hep-ph-9712349-2-10-1'], ['hep-ph-9712349-1-10-2', 'hep-ph-9712349-2-10-2'], ['hep-ph-9712349-1-10-4', 'hep-ph-9712349-2-10-4'], ['hep-ph-9712349-1-7-0', 'hep-ph-9712349-2-7-0'], ['hep-ph-9712349-1-7-1', 'hep-ph-9712349-2-7-1'], ['hep-ph-9712349-1-7-3', 'hep-ph-9712349-2-7-3'], ['hep-ph-9712349-1-7-4', 'hep-ph-9712349-2-7-4'], ['hep-ph-9712349-1-11-0', 'hep-ph-9712349-2-11-0'], ['hep-ph-9712349-1-11-2', 'hep-ph-9712349-2-11-3'], ['hep-ph-9712349-1-11-4', 'hep-ph-9712349-2-11-5'], ['hep-ph-9712349-1-11-5', 'hep-ph-9712349-2-11-6'], ['hep-ph-9712349-1-11-6', 'hep-ph-9712349-2-11-7'], ['hep-ph-9712349-1-11-7', 'hep-ph-9712349-2-11-8'], ['hep-ph-9712349-1-13-0', 'hep-ph-9712349-2-14-0'], ['hep-ph-9712349-1-13-2', 'hep-ph-9712349-2-14-2'], ['hep-ph-9712349-1-13-4', 'hep-ph-9712349-2-14-4'], ['hep-ph-9712349-1-6-0', 'hep-ph-9712349-2-6-0'], ['hep-ph-9712349-1-6-1', 'hep-ph-9712349-2-6-1'], ['hep-ph-9712349-1-6-2', 'hep-ph-9712349-2-6-2'], ['hep-ph-9712349-1-6-4', 'hep-ph-9712349-2-6-4'], ['hep-ph-9712349-1-6-5', 'hep-ph-9712349-2-6-5'], ['hep-ph-9712349-1-6-6', 'hep-ph-9712349-2-6-6'], ['hep-ph-9712349-1-12-2', 'hep-ph-9712349-2-12-2'], ['hep-ph-9712349-1-12-3', 'hep-ph-9712349-2-12-3'], ['hep-ph-9712349-1-12-4', 'hep-ph-9712349-2-12-4'], ['hep-ph-9712349-1-12-5', 'hep-ph-9712349-2-12-5'], ['hep-ph-9712349-1-12-6', 'hep-ph-9712349-2-12-6'], ['hep-ph-9712349-1-12-7', 'hep-ph-9712349-2-12-7'], ['hep-ph-9712349-1-12-8', 'hep-ph-9712349-2-12-8'], ['hep-ph-9712349-1-12-9', 'hep-ph-9712349-2-12-9'], ['hep-ph-9712349-1-12-10', 'hep-ph-9712349-2-12-10'], ['hep-ph-9712349-1-1-0', 'hep-ph-9712349-2-1-0'], ['hep-ph-9712349-1-1-1', 'hep-ph-9712349-2-1-1'], ['hep-ph-9712349-1-1-2', 'hep-ph-9712349-2-1-2'], ['hep-ph-9712349-1-1-3', 'hep-ph-9712349-2-1-3'], ['hep-ph-9712349-1-1-4', 'hep-ph-9712349-2-1-4'], ['hep-ph-9712349-1-4-0', 'hep-ph-9712349-2-4-0'], ['hep-ph-9712349-1-4-1', 'hep-ph-9712349-2-4-1'], ['hep-ph-9712349-1-4-2', 'hep-ph-9712349-2-4-2'], ['hep-ph-9712349-1-4-3', 'hep-ph-9712349-2-4-3'], ['hep-ph-9712349-1-4-4', 'hep-ph-9712349-2-4-4'], ['hep-ph-9712349-1-4-5', 'hep-ph-9712349-2-4-5'], ['hep-ph-9712349-1-4-6', 'hep-ph-9712349-2-4-6'], ['hep-ph-9712349-1-5-0', 'hep-ph-9712349-2-5-0'], ['hep-ph-9712349-1-5-1', 'hep-ph-9712349-2-5-1'], ['hep-ph-9712349-1-5-2', 'hep-ph-9712349-2-5-2'], ['hep-ph-9712349-1-5-3', 'hep-ph-9712349-2-5-3'], ['hep-ph-9712349-1-5-4', 'hep-ph-9712349-2-5-4'], ['hep-ph-9712349-1-5-5', 'hep-ph-9712349-2-5-5'], ['hep-ph-9712349-1-5-6', 'hep-ph-9712349-2-5-6'], ['hep-ph-9712349-1-5-7', 'hep-ph-9712349-2-5-7'], ['hep-ph-9712349-1-5-8', 'hep-ph-9712349-2-5-8'], ['hep-ph-9712349-1-5-9', 'hep-ph-9712349-2-5-9'], ['hep-ph-9712349-1-5-10', 'hep-ph-9712349-2-5-10'], ['hep-ph-9712349-1-0-0', 'hep-ph-9712349-2-0-0'], ['hep-ph-9712349-1-0-1', 'hep-ph-9712349-2-0-1'], ['hep-ph-9712349-1-0-2', 'hep-ph-9712349-2-0-3'], ['hep-ph-9712349-1-3-0', 'hep-ph-9712349-2-3-0'], ['hep-ph-9712349-1-3-1', 'hep-ph-9712349-2-3-1'], ['hep-ph-9712349-1-3-2', 'hep-ph-9712349-2-3-2'], ['hep-ph-9712349-1-3-3', 'hep-ph-9712349-2-3-3'], ['hep-ph-9712349-1-3-4', 'hep-ph-9712349-2-3-4'], ['hep-ph-9712349-1-3-5', 'hep-ph-9712349-2-3-5'], ['hep-ph-9712349-1-3-6', 'hep-ph-9712349-2-3-6'], ['hep-ph-9712349-1-3-7', 'hep-ph-9712349-2-3-7'], ['hep-ph-9712349-1-3-8', 'hep-ph-9712349-2-3-8'], ['hep-ph-9712349-1-3-9', 'hep-ph-9712349-2-3-9'], ['hep-ph-9712349-1-3-10', 'hep-ph-9712349-2-3-10'], ['hep-ph-9712349-1-3-11', 'hep-ph-9712349-2-3-11'], ['hep-ph-9712349-1-3-12', 'hep-ph-9712349-2-3-12'], ['hep-ph-9712349-1-3-13', 'hep-ph-9712349-2-3-13'], ['hep-ph-9712349-1-3-14', 'hep-ph-9712349-2-3-14'], ['hep-ph-9712349-1-3-15', 'hep-ph-9712349-2-3-15'], ['hep-ph-9712349-1-15-0', 'hep-ph-9712349-2-16-0'], ['hep-ph-9712349-1-14-0', 'hep-ph-9712349-2-15-0'], ['hep-ph-9712349-1-14-1', 'hep-ph-9712349-2-15-1'], ['hep-ph-9712349-1-14-2', 'hep-ph-9712349-2-15-3']] | [['hep-ph-9712349-1-2-3', 'hep-ph-9712349-2-2-3'], ['hep-ph-9712349-1-8-4', 'hep-ph-9712349-2-8-3'], ['hep-ph-9712349-1-9-3', 'hep-ph-9712349-2-9-3'], ['hep-ph-9712349-1-11-1', 'hep-ph-9712349-2-11-2'], ['hep-ph-9712349-1-13-1', 'hep-ph-9712349-2-14-1'], ['hep-ph-9712349-1-13-3', 'hep-ph-9712349-2-14-3'], ['hep-ph-9712349-1-12-0', 'hep-ph-9712349-2-12-0'], ['hep-ph-9712349-1-12-1', 'hep-ph-9712349-2-12-1']] | [] | [['hep-ph-9712349-1-7-2', 'hep-ph-9712349-2-7-2']] | [] | ['hep-ph-9712349-1-6-3', 'hep-ph-9712349-1-10-3', 'hep-ph-9712349-1-11-3', 'hep-ph-9712349-2-6-3', 'hep-ph-9712349-2-10-3', 'hep-ph-9712349-2-11-4'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/9712349 | null | null | null | null | null |
1208.5069 | {'1208.5069-1-0-0': 'Type Ia supernovae (SNe Ia) are thought to originate from the thermonuclear explosions of carbon-oxygen (C-O) white dwarfs (WDs).', '1208.5069-1-0-1': 'The gravitationally-confined detonation (GCD) model is a well-explored explosion model where unstable thermonuclear burning initiates in an accreting, Chandrasekhar-mass WD and forms an advancing flame.', '1208.5069-1-0-2': 'Rising, burning material breaks out of the WD surface, engulfs the WD, and leads to a gradient-induced detonation in the collision region, which subsequently unbinds the WD.', '1208.5069-1-0-3': 'We show that when the flame consumes enough fuel, the conditions for detonation are not met, and the WD does not detonate.', '1208.5069-1-0-4': 'We show through 2D and 3D simulations, such failed-detonation SNe expel a few 0.1 [MATH] of burned and partially-burned material, but do not unbind the WD.', '1208.5069-1-0-5': 'A fraction of the material falls back onto the WD, polluting the remnant WD with intermediate-mass and iron-group elements, that likely segregate to the core forming an iron-core C-O WD.', '1208.5069-1-0-6': 'The remaining material is asymmetrically ejected at velocities comparable to the escape velocity from the WD, and in response, the WD is kicked to velocities of a few hundred km s[MATH].', '1208.5069-1-0-7': 'Such kicks may unbind the binary and eject a runaway/hyper-velocity WD.', '1208.5069-1-0-8': 'Although the energy and ejected mass of the failed-detonation SN are a fraction of typical thermonuclear SNe, they are likely to appear as sub-luminous low-velocity SNe Ia.', '1208.5069-1-0-9': 'Such failed-GCDs might therefore explain or are related to the observed branch of peculiar SNe Ia, such as the family of low-velocity sub-luminous SNe (SN 2002cx/SN 2008ha-like SNe).', '1208.5069-1-1-0': '# Introduction', '1208.5069-1-2-0': 'Type Ia supernovae (SNe Ia) are among the most energetic explosions in the known universe, releasing [MATH] ergs of kinetic energy in their ejecta, and synthesizing [MATH] of radioactive [MATH]Ni.', '1208.5069-1-2-1': 'The discovery of the Phillips relation enabled the use of SNe Ia as standardizable cosmological candles, and has ushered in a new era of astronomy leading to the discovery of the acceleration of the universe , leading to the 2011 Nobel Prize in physics.', '1208.5069-1-3-0': 'In the following we introduce an unexplored variant of well-studied single-degenerate (SD) model of SNe Ia in which a Chandrasekhar-mass, carbon-oxygen (C-O) WD accretes mass from a non-degenerate main sequence or red giant companion.', '1208.5069-1-3-1': 'As the WD approaches the Chandrasekhar limit, carbon burning is initiated in its convective core.', '1208.5069-1-3-2': 'After a few hundred years of this "simmering" phase, unstable thermonuclear burning is expected to ignite at one or more off-centered points, generating a buoyantly-rising, subsonic deflagration flame bubble .', '1208.5069-1-3-3': 'We refer to this phase as the "deflagration phase".', '1208.5069-1-4-0': 'We adopt the assumption made by the gravitationally-confined detonation (GCD) scenario of SNe Ia in which no transition to detonation occurs as the rising flame approaches the low-density surface layers of the WD as is posited by the "deflagration-to-detonation transition" (DDT) model of SNe Ia .', '1208.5069-1-4-1': 'We assume an off-centered distribution of ignition points as in the GCD scenario.', '1208.5069-1-4-2': 'We further focus on the "pulsationally-assisted" variant of the GCD model described in [CITATION] which we review in section [REF].', '1208.5069-1-5-0': 'We demonstrate that under these assumptions, if enough mass is burned during the deflagration phase, the conditions to trigger a detonation via the GCD mechanism are not realized and the WD fails to detonate.', '1208.5069-1-5-1': 'Such "failed-detonation" (FGCD) models have numerous remarkable implications for the observable properties of the resulting explosion and its outcomes.', '1208.5069-1-5-2': 'These include the production of peculiar SNe Ia events with low expansion velocities, low luminosities and low ejecta-mass; all broadly consistent with the observed properties of a branch of peculiar SNe Ia similar to SN 2002cx and/or SN 2008ha.', '1208.5069-1-5-3': "Even more remarkably, we demonstrate that the WD survives the initial Ia event, receiving a large velocity kick from the asymmetric nature of the initial deflagration, and is significantly enriched with both intermediate-mass (IME's) and iron-group elements (IGE's), forming a peculiar heavy/iron core WD.", '1208.5069-1-6-0': '# Physics of the Failed-Detonation Scenario', '1208.5069-1-7-0': 'The pulsationally-assisted GCD model described in [CITATION] begins with an off-centered ignition region in a Chandrasekhar-mass WD which forms an advancing thermonuclear flame.', '1208.5069-1-7-1': 'The hot, buoyant ash quickly rises forming a large plume that advances and burns towards the surface of the star.', '1208.5069-1-7-2': 'Energy imparted to the WD during the deflagration causes the star to pulse.', '1208.5069-1-7-3': 'As the WD expands, hot ash erupts from the star, flows around the stellar surface, and mixes with cold fuel.', '1208.5069-1-7-4': 'The mixture collides at the antipodal point on the surface from breakout.', '1208.5069-1-7-5': 'Meanwhile, the WD reaches maximum expansion and contracts.', '1208.5069-1-7-6': "As the star contracts, it squeezes the fuel-ash mixture in the collision region to temperatures and densities that considerably shorten the minimum length scale on which the Zel'dovich gradient mechanism can trigger a detonation.", '1208.5069-1-7-7': 'A detonation is triggered in the mixture which subsequently consumes the WD.', '1208.5069-1-8-0': 'Roughly speaking, typical fuel-ash mixture properties associated with an induction gradient [MATH] km are densities [MATH] g/cm[MATH] and peak temperatures [MATH] K .', '1208.5069-1-9-0': 'The FGCD scenario proceeds similarly to the pulsational GCD model; however, the scenarios deviate in the amount of burning - and therefore the amount of energy released - during the deflagration phase.', '1208.5069-1-9-1': 'More burning occurs during the deflagration phase of the FGCD than in the GCD which means more mass is consumed by the flame and ejected from the star in the form of ash, and more energy is delivered to the WD.', '1208.5069-1-9-2': 'The WD is thus modified to a higher degree than in the GCD model and as a result, the WD does not attain high densities and temperatures in the collision region during its contraction.', '1208.5069-1-9-3': 'The fuel-ash mixture never reaches the critical conditions for detonation and therefore the star never detonates.', '1208.5069-1-10-0': 'The deflagration is strong enough to prevent the formation of the conditions for detonation, but too weak to completely unbind the star.', '1208.5069-1-10-1': 'The WD will thus remain intact; though, it will have a lower mass.', '1208.5069-1-10-2': 'Interestingly, since the ignition of the deflagration is off-centered, the remaining bound stellar material is kicked by the ejection of the ash and obtains a velocity of hundreds of km s[MATH].', '1208.5069-1-11-0': "This system produces an asymmetric outburst of a few tenths of a solar mass of deflagration products rich in IME's (such as Mg, Si, and S) and iron-group elements IGE's (such as Fe, Co, and Ni).", '1208.5069-1-11-1': 'Since the deflagration only processes a few tenths of a solar mass of material, the FGCD produces a small total mass of [MATH]Ni compared to normal SNe Ia which convert nearly half of the WD to [MATH]Ni.', '1208.5069-1-12-0': 'Some of this material attains escape velocity and some falls back onto the star.', '1208.5069-1-12-1': 'The velocity of the outflow will be slow (approximately a few thousand km s[MATH]) related to normal SNe Ia due to the comparatively small amount of energy released in the FGCD scenario.', '1208.5069-1-13-0': '# Simulations of the FGCD Model', '1208.5069-1-14-0': '## Simulation Setup', '1208.5069-1-15-0': 'We used the Adaptive Mesh Refinement (AMR) FLASH application framework to perform our simulations of FGCD models.', '1208.5069-1-15-1': 'FLASH has been previously used to simulate SNe Ia in both 2D cylindrical and 3D Cartesian geometry .', '1208.5069-1-15-2': 'Our simulations include an advection-diffusion-reaction (ADR) treatment of the thermonuclear flame , an equation of state that includes contributions from blackbody radiation, ions, and electrons of an arbitrary degree of degeneracy , and the multipole treatment of gravity included in the FLASH standard .', '1208.5069-1-16-0': 'ccccccccccccccc[t!]', '1208.5069-1-17-0': 'Simulations Properties 0pt', '1208.5069-1-18-0': 'sim & [MATH]a & n[MATH]b & r[MATH]c & z[MATH]d & E[MATH]e & C-O[MATH]f & IME[MATH] & IGE[MATH] & v[MATH]g & C-O[MATH]h & IME[MATH] & IGE[MATH] & E[MATH]i & v[MATH]j', '1208.5069-1-19-0': 'name & (km) & & (km) & (km) & (E[MATH]) & ([MATH]) & ([MATH]) & ([MATH]) & (km s[MATH]) & ([MATH]) & ([MATH]) & ([MATH]) & (ergs [MATH]) & (km s[MATH])', '1208.5069-1-20-0': '2D70 & 4 & 4 & 64.0 & 70.0 & 0.89 & 0.93 & 0.07 & 0.13 & 119 & 0.13 & 0.03 & 0.07 & 0.32 & 3,730', '1208.5069-1-21-0': '3D48 & 8 & 63 & 128.0 & 48.0 & 1.06 & 0.84 & 0.06 & 0.09 & 351 & 0.16 & 0.05 & 0.16 & 0.90 & 4,932', '1208.5069-1-22-0': '3D38 & 8 & 63 & 128.0 & 38.0 & 1.21 & 0.71 & 0.07 & 0.08 & 383 & 0.24 & 0.06 & 0.20 & 1.3 & 5,099', '1208.5069-1-23-0': '3D28 & 8 & 63 & 128.0 & 28.0 & 1.39 & 0.56 & 0.05 & 0.09 & 520 & 0.32 & 0.09 & 0.25 & 1.8 & 5,222', '1208.5069-1-24-0': '3D18 & 8 & 63 & 128.0 & 18.0 & 1.53 & 0.43 & 0.04 & 0.08 & 406 & 0.43 & 0.10 & 0.28 & 2.2 & 5,212', '1208.5069-1-25-0': 'aMaximum spatial resolution.', '1208.5069-1-25-1': 'bNumber of ignition points.', '1208.5069-1-25-2': 'cRadius of the spherical volume containing the ignition points.', '1208.5069-1-25-3': 'dZ coordinate of the origin of the spherical volume containing the ignition points.', '1208.5069-1-25-4': 'eEnergy released during the deflagration phase divided by the binding energy of the star ([MATH] ergs).', '1208.5069-1-25-5': 'f"B" refers to gravitationally bound material.', '1208.5069-1-25-6': 'gVelocity of the gravitationally bound material .', '1208.5069-1-25-7': 'h"E" refers to material that will escape the system.', '1208.5069-1-25-8': 'iAverage kinetic energy of the escaping material.', '1208.5069-1-25-9': 'jMass weighted velocity of escaping material.', '1208.5069-1-26-0': 'We performed four 3-dimensional (3D) exploratory simulations to test the feasibility of the FGCD and 1 full 2-dimensional (2D) model to adequately observe the FGCD at late times.', '1208.5069-1-27-0': 'Our 3D simulations included a reduced domain size and a moderate resolution (8 km) which reduced their computational expense.', '1208.5069-1-27-1': 'We initialized these simulations similarly to those in [CITATION] with a 1.365 [MATH] WD placed at the origin of the domain.', '1208.5069-1-27-2': 'We used the same distribution of sixty three, [MATH] km radius ignition "points" distributed in a [MATH] km radius spherical volume.', '1208.5069-1-27-3': 'We chose 48 km, 38, km, 28 km, and 18 km as offset distances along the z-axis of the spherical volume.', '1208.5069-1-27-4': 'Table [REF] lists the initial conditions for each simulation and the corresponding names we gave them.', '1208.5069-1-27-5': 'We ran the simulations from ignition, through peak stellar expansion, and at least until the WD reached a maximum central density upon contraction.', '1208.5069-1-28-0': 'Our 2D simulation was performed with a large domain in 2D cylindrical geometry at 4 km resolution and ran for 60 seconds.', '1208.5069-1-28-1': 'This simulation was initialized similarly to the 3D simulations except we placed only 4 bubbles in a 64 km spherical volume offset by 70 km along the z-axis (axis of symmetry).', '1208.5069-1-28-2': 'We chose these initial conditions to obtain an FGCD in 2D given what we learned from our 3D simulations.', '1208.5069-1-28-3': 'The larger domain size allowed us to follow the outer layers of the ejecta for the entirety of the simulation.', '1208.5069-1-29-0': '## Simulation Results', '1208.5069-1-30-0': 'We obtained an FGCD from each of our simulations.', '1208.5069-1-30-1': 'and were able to quantify features discussed in section [REF].', '1208.5069-1-30-2': 'Table [REF] contains a collection of these quantities from our simulations.', '1208.5069-1-31-0': 'The energy liberated during the deflagration is between 89% and 150% of the binding energy of the WD.', '1208.5069-1-31-1': 'However, in each simulation the deflagration fails to unbind the WD.', '1208.5069-1-31-2': 'The WD expands in response to the deflagration, reaches its maximum level of expansion, and then contracts.', '1208.5069-1-31-3': 'Figure [REF] shows the evolution of the central density, _c, of the WD and illustrates the oscillatory nature of the WD after the deflagration.', '1208.5069-1-31-4': 'The more energy released during the deflagration, the more the star expanded, the longer the pulsational period, and the smaller the _c at maximum contraction.', '1208.5069-1-32-0': 'The fact that the deflagration releases more than 100 WD binding energy in some models suggests that the WD should be unbound after the deflagration; however, the entire energy budget does not work to only unbind the star.', '1208.5069-1-32-1': 'For example, some of the energy is lost when the ejecta and the remnant WD are accelerated to high velocities.', '1208.5069-1-32-2': 'Thus, even though the deflagration liberates enough energy to completely unbind the star, the energy is partitioned in such a way that a portion of the original WD remains gravitationally bound.', '1208.5069-1-33-0': 'In each of our models, the WD gets a kick in the opposite direction from which the buoyant ash rises and breaks through the surface of the star.', '1208.5069-1-33-1': 'We measured this velocity to be on the order of hundreds of km s[MATH] and list these values for each simulation in table [REF].', '1208.5069-1-34-0': 'Some of the material from the FGCD escapes and achieves high velocities, while some of the material is bound to the remnant WD and will eventually accrete onto its surface.', '1208.5069-1-34-1': 'Table [REF] lists the composition of material that remains gravitationally bound and eventually will mix with the remnant WD.', '1208.5069-1-34-2': "In all cases between 0.1 [MATH] and 0.2 [MATH] of IME's and IGE's (some of which is radioactive [MATH]Ni) remained bound to the star.", '1208.5069-1-34-3': 'The evolution of the WD as the material falls back onto and heats the star is an interesting question and one which we will examine in future work.', '1208.5069-1-35-0': 'The composition of the material that escapes the system is also listed in table [REF] along with the kinetic energy and the mass-weighted velocity of the ejecta.', '1208.5069-1-35-1': "This material includes carbon and oxygen, IME's, and IGE's, and ranges from 0.2 to 0.8 [MATH].", '1208.5069-1-35-2': 'In general, the more material that is burned during the deflagration, the more material that escapes.', '1208.5069-1-36-0': 'Figure [REF] shows shows the density structure and composition with overlaid velocity contours of the 2D model at 60s.', '1208.5069-1-36-1': 'Note that the density profile of the FGCD model is asymmetric in velocity space between the hemisphere corresponding to the ejected deflagration and the opposite hemisphere of the system.', '1208.5069-1-36-2': 'The remnant of the WD can be seen as the tiny high-density feature slightly below the origin of the domain.', '1208.5069-1-37-0': 'The figure also shows the nature of the asymmetry in composition of the structure.', '1208.5069-1-37-1': 'The north side of the remnant contains the products of the deflagration that were sprayed from the surface of the WD.', '1208.5069-1-37-2': "Clumpy structures of IME's and IGE's exist at a range of velocities in the northern hemisphere of the domain but are less abundant in the southern hemisphere.", '1208.5069-1-37-3': 'The asymmetries suggest that this object would look much different depending on the viewing angle of the observer.', '1208.5069-1-38-0': 'All of our models produce a relatively small amount of radioactive [MATH]Ni.', '1208.5069-1-38-1': "Though we do not perform detailed nucleosynthetic post processing of the 3D models, we can set an upper limit on the [MATH]Ni yields with the amount of IGE's produced, which ranges from 0.2 [MATH] to 0.36 [MATH].", '1208.5069-1-38-2': 'Neutronization through electron capture reactions during the deflagration would shift production away from [MATH]Ni though and reduce its contribution to the IGE totals.', '1208.5069-1-38-3': "Whether even lower masses of IGE's (and thus [MATH]Ni) could be produced under appropriate conditions (e.g. comparable to that observed in SN 2008ha, a very faint SN with extremely low ejecta velocities), is yet to be explored.", '1208.5069-1-39-0': '# Discussion and Predictions', '1208.5069-1-40-0': '## Sub-luminous Low-Velocity SNe', '1208.5069-1-41-0': 'The most prominent features of the FGCD models are their low-mass, low-velocity ejecta and overall low kinetic and nuclear energy.', '1208.5069-1-41-1': 'These properties translate into the production of typically sub-luminous, low-velocity SNe Ia.', '1208.5069-1-41-2': 'It is therefore natural to examine whether SNe with such characteristics have already been discovered.', '1208.5069-1-41-3': 'In particular, one may explore peculiar SNe exhibiting either extremely low mass ejecta, such as SN 2008ha , SN 2005E-like SNe , or SN 2002bj-like SNe , or low ejecta velocity such as SN 2002cx-like SNe [CITATION].', '1208.5069-1-41-4': 'Some of the former low-mass-ejecta SNe, however, may present He signature in their spectra (type Ib SNe), inconsistent with our models.', '1208.5069-1-41-5': 'We therefore focus on observed low velocity, faint SNe Ia.', '1208.5069-1-42-0': 'Normal type Ia SNe differ in their ejecta velocities as measured in some standard method, but they generally fall between [MATH] km s[MATH] near peak luminosity, with similar dispersion at later times (as derived from the Si II line [CITATION]).', '1208.5069-1-42-1': 'The velocities of even the lowest velocity SNe in the [CITATION] sample much exceed the mass-averaged FGCD velocities listed in table [REF].', '1208.5069-1-42-2': 'We also note a trend of more energetic and likely more luminous (larger IGE yield) SNe to be accompanied by higher ejecta-velocities over almost an order of magnitude in kinetic energy.', '1208.5069-1-42-3': 'The only other type of SNe with such low expansion velocities are the branch of peculiar type Ia SNe, named for the prototype for this class of supernovae, 2002cx ; such SNe may also have an enregy-velocity correlation [CITATION] as observed in our simulations.', '1208.5069-1-43-0': 'SN 2002cx-like events are characterized by luminosities which lie too low in comparison to the Phillips relation for Branch-normal Ia events , low photospheric velocities , weak intermediate-mass element lines (including Si II, S II, Si III, and Ca II) , and late-time optical nebular spectra dominated by narrow Fe II lines .', '1208.5069-1-43-1': 'Since the discovery of SN 2002cx, a number of other 2002cx-like events have been discovered, including 2002es, 2005P, 2005hk , 2008ge and 2008ha.', '1208.5069-1-43-2': 'The latter event (SN 2008ha), in particular, is consistent with extremely low mass ejecta and energetics.', '1208.5069-1-43-3': 'We predict the FGCD models to produce similar properties to those characterizing SN 2002cx like SNe, given the low expansion velocities and the low estimated [MATH]Ni yield, and potentially even explaining SN 2008ha like events with low mass ejecta.', '1208.5069-1-44-0': "Though our initial set of simulations is limited, the robust features of FGCD's, including low velocity ejecta, the expected low luminosity (due to the small yield of [MATH]Ni) and their low mass ejecta (comparable to that SN 2008ha) make them tantalizing candidate progenitors for this branch of peculiar SNe.", '1208.5069-1-44-1': 'Note that the single degenerate origin of such SNe is also consistent with the overall typically young (but not necessarily young; [CITATION] 2010) environments found for SN-2002cx like SNe, compared to the expectations from, e.g. double degenerate SNe (older environments) or core-collapse SNe (only very young environments).', '1208.5069-1-45-0': '## Heavy metal/Iron-core WD', '1208.5069-1-46-0': 'In our FGCD scenario, a large amount of burnt material falls back to the remnant WD.', '1208.5069-1-46-1': "From table [REF], the WD may incorporate as much as [MATH] of IGE's and [MATH] of IME's of fallback material, together comprising as much as [MATH] of the remnant C-O WD.", '1208.5069-1-46-2': "Such heavy elements are likely to gravitationally settle, in time, to the WD core, making these heavy-metal polluted WD's become iron/heavy-core WDs.", '1208.5069-1-46-3': "The existence of iron-core WD's has been considered before, with even the potential observation of such WDs .", '1208.5069-1-46-4': "The FGCD scenario therefore provides a novel evolutionary scenario for the formation of these iron/heavy-core C-O WD's.", '1208.5069-1-46-5': 'A somewhat related scenario of failed SN was suggested for the formation of O-Ne-Mg WDs with iron cores .', '1208.5069-1-47-0': '## WD Natal Kicks', '1208.5069-1-48-0': "FGCD's produce a highly asymmetric ejection of material.", '1208.5069-1-48-1': 'This is not unique amongst various models for SNe explosions.', '1208.5069-1-48-2': 'However, in our FGCD case, the WD survives the explosion.', '1208.5069-1-48-3': 'Considering momentum conservation, this gives rise to a unique outcome, namely that the surviving WD is kicked at very high velocities, ranging hundreds of km s[MATH].', '1208.5069-1-48-4': 'The FGCD scenario suggests the existence of strong WD natal kicks, and provides an interesting prediction per the existence of hypervelocity WDs.', '1208.5069-1-48-5': 'Taken together, the potential existence of a a heavy core WD (discussed in section [REF]), and the high ejection velocity produce a highly peculiar object, which, if observed may provide a possibly unique smoking gun signature.', '1208.5069-1-48-6': 'One should note, however, that the population of halo WDs may also have relatively high velocities, and it might therefore be difficult to pinpoint the kinematic property as related to a natal kick (unless the WD is massive and young; an unlikely possibility for the old population of halo WDs).', '1208.5069-1-49-0': 'We note that velocities of hundreds of km s[MATH] are much larger than the orbital velocities of most binaries, and a kick velocity of such magnitude is likely to unbind the binary (ejecting the companion star at high velocity, but much lower than the WD kick velocity).', '1208.5069-1-49-1': "WD's accreting from evolved stars could have a large binary separation, and be disrupted following the WD kick.", '1208.5069-1-49-2': "However, WD's accreting from a main sequence companion might have highly compact configurations.", '1208.5069-1-49-3': 'In this latter case, the WD kick might not unbind the binary, and may only change its orbit, and provide an overall kick to the binary center of mass as a whole.', '1208.5069-1-49-4': 'An interesting aspect is then the result of a following accretion epoch involving the iron-core WD, and its possible evolutionary outcomes.', '1208.5069-1-49-5': 'Finally, in both cases of evolved and main sequence mass donors, the WD natal kick may eject the remnant WD towards the companion, in which case a collision may occur, with potentially interesting explosive results.', '1208.5069-1-50-0': '# Summary', '1208.5069-1-51-0': 'We have discovered a scenario in which the standard GCD model fails to detonate due to a large amount of burning during the deflagration phase.', '1208.5069-1-51-1': 'The failed GCD results in an asymmetric outburst of deflagration material consisting of intermediate elements and iron group elements along with a fraction of the original WD still gravitationally bound.', '1208.5069-1-51-2': 'The models produce a faint SNe Ia with a slowly evolving light curve due to the low [MATH]Ni yield and the low energetics.', '1208.5069-1-51-3': 'The remaining WD gets a kick on the order of hundreds of km s[MATH] and is contaminated with fall-back from the deflagration, producing an iron/heavy core WD.', '1208.5069-1-51-4': 'We presented our initial simulations to quantify the some of the bulk observable properties and hypothesize that the FGCD model is a possible explanation for 2002cx-like SN.', '1208.5069-1-51-5': 'Future studies will explore the detailed observational features of FGCD SNe and their direct comparison to observations.', '1208.5069-1-52-0': 'The authors thank the FLASH Code Group, the FLASH Astrophysics Group, and the Argonne Leadership Computing Facility at Argonne National Laboratory.', '1208.5069-1-52-1': 'HBP is supported by the CfA and BIKURA prize fellowships.', '1208.5069-1-52-2': 'AST - 0909132 for the "Petascale Computing of Thermonuclear Supernova Explosions".'} | {'1208.5069-2-0-0': 'Type Ia supernovae (SNe Ia) originate from the thermonuclear explosions of carbon-oxygen (C-O) white dwarfs (WDs).', '1208.5069-2-0-1': 'The single-degenerate scenario is a well-explored model of SNe Ia where unstable thermonuclear burning initiates in an accreting, Chandrasekhar-mass WD and forms an advancing flame.', '1208.5069-2-0-2': 'By several proposed physical processes the rising, burning material triggers a detonation, which subsequently consumes and unbinds the WD.', '1208.5069-2-0-3': 'However, if a detonation is not triggered and the deflagration is too weak to unbind the star, a completely different scenario unfolds.', '1208.5069-2-0-4': 'We explore the failure of the Gravitationally-Confined Detonation (GCD) mechanism of SNe Ia, and demonstrate through 2D and 3D simulations the properties of failed-detonation SNe.', '1208.5069-2-0-5': 'We show that failed-detonation SNe expel a few 0.1 [MATH] of burned and partially-burned material and that a fraction of the material falls back onto the WD, polluting the remnant WD with intermediate-mass and iron-group elements, that likely segregate to the core forming an WD whose core is iron rich.', '1208.5069-2-0-6': 'The remaining material is asymmetrically ejected at velocities comparable to the escape velocity from the WD, and in response, the WD is kicked to velocities of a few hundred km s[MATH].', '1208.5069-2-0-7': 'These kicks may unbind the binary and eject a runaway/hyper-velocity WD.', '1208.5069-2-0-8': 'Although the energy and ejected mass of the failed-detonation SN are a fraction of typical thermonuclear SNe, they are likely to appear as sub-luminous low-velocity SNe Ia.', '1208.5069-2-0-9': 'Such failed detonations might therefore explain or are related to the observed branch of peculiar SNe Ia, such as the family of low-velocity sub-luminous SNe (SN 2002cx/SN 2008ha-like SNe).', '1208.5069-2-1-0': '# Introduction', '1208.5069-2-2-0': 'Type Ia supernovae (SNe Ia) are among the most energetic explosions in the known universe, releasing [MATH] ergs of kinetic energy in their ejecta, and synthesizing [MATH] of radioactive [MATH]Ni.', '1208.5069-2-2-1': 'The discovery of the Phillips relation enabled the use of SNe Ia as standardizable cosmological candles, and has ushered in a new era of astronomy leading to the discovery of the acceleration of the universe , and to the 2011 Nobel Prize in physics.', '1208.5069-2-3-0': 'Models of normal SNe Ia, such as the single degenerate (SD) model, focus on exploding the WD in order to produce the explosion energies, luminosities, and typical velocities observed in normal SNe Ia.', '1208.5069-2-3-1': 'This is accomplished either by consuming enough of the WD with the initial subsonic buring phase - or deflagration phase - to unbind the WD as theorized by the Pure Deflagration (PD) model, or consuming the entire WD by a detonation triggered by the deflagration phase as posited by the deflagration-to-detonation transition (DDT) , the Pulsating Reverse Detonation (PRD) , and the GCD models of SNe Ia.', '1208.5069-2-4-0': 'In the following, we present a novel variant of SD model of SNe Ia in which the deflagration is too weak to unbind the star and a detonation is not triggered by any of the proposed mechansisms, resulting in the survival of a bound remnant of the original WD.', '1208.5069-2-4-1': 'We present for the first time predictions of these failed-detonation (FD) SNe from 2D and 3D simulations.', '1208.5069-2-4-2': 'We show that FD models have numerous remarkable implications for the observable properties of the resulting explosion and its outcomes.', '1208.5069-2-4-3': 'These include the production of a family of peculiar SNe Ia events with low expansion velocities, low luminosities and low ejecta-mass - whose properties are broadly consistent with the observed properties of a branch of peculiar SNe Ia similar to SN 2002cx and/or SN 2008ha.', '1208.5069-2-4-4': "Even more remarkably, the remnant WD receives a large velocity kick from the asymmetric nature of the deflagration, and is enriched with both intermediate-mass (IME's) and iron-group elements (IGE's), forming a peculiar WD with a heavey/iron-rich core.", '1208.5069-2-5-0': 'Previous work has suggested that though the PD model has shortcomings explaining normal SNe Ia, they may explain 2002cx-like SNe .', '1208.5069-2-5-1': 'The WD, however, is fully incinerated in these models, producing a Chandrasekhar-mass of ejecta.', '1208.5069-2-5-2': 'Such models might therefore not be able to explain the large diversity recently observed among SNe of this peculiar class of SNe.', '1208.5069-2-5-3': 'Additionally, a study by [CITATION] of initial conditions for the PD model produced situations that they termed "fizzles" which did not produce a healthy PD explosion and left the WD bound.', '1208.5069-2-5-4': 'They did not pursue these models beyond the scope of their study, nor did they relate these fizzles to peculiar SNe Ia.', '1208.5069-2-6-0': '# Avoiding the Transition to Detonation', '1208.5069-2-7-0': 'The FD scenario requires that no detonation is triggered as a result of the deflagration event.', '1208.5069-2-7-1': 'We briefly touch on the possibility of the PRD and GCD failing to trigger a detonation.', '1208.5069-2-7-2': 'We first make the general assumption that is made in the PD, PRD, and GCD scenarios, namely that the DDT mechanism is not active.', '1208.5069-2-7-3': 'We refer the reader to [CITATION] for a discussion of the justification of this assumption, the details of which are beyond the scope of this work.', '1208.5069-2-8-0': 'In the PRD model, ash ejected during the deflagration phase falls back onto the WD.', '1208.5069-2-8-1': 'An accretion shock formed by infalling ash surrounds and heats fuel remaining in the WD core, which in turn triggers a detonation.', '1208.5069-2-8-2': '[CITATION] reported that the PRD could not trigger a detonation if the energy released during the deflagration was near that of the binding energy of the WD.', '1208.5069-2-8-3': 'This situation is realized in our simulations.', '1208.5069-2-9-0': '[CITATION] (hereafter J12) detail how the GCD mechanism triggers a gradient-induced detonation when ash from the deflagration flows over the stellar surface, mixes with cold fuel, collides at the antipodal point from break out, and is squeezed to the necessary temperatures and densities by the contracting WD.', '1208.5069-2-9-1': 'Models presented here show that a detonation via the GCD mechanism is avoided alltogether.', '1208.5069-2-9-2': 'The primary difference between these models and those of J12 is the amount of energy released during the deflagration.', '1208.5069-2-9-3': 'More energy is released and delivered to the WD than in J12; thus, more mass is consumed by the flame and ejected from the WD than previously.', '1208.5069-2-9-4': 'The WD is modified to a higher degree and as a result the WD can not contract enough to squeeze the fuel-ash mixture to the critical conditions for detonation.', '1208.5069-2-9-5': 'Therefore the star never detonates.', '1208.5069-2-9-6': 'We note that [CITATION] also investigated the failure of the GCD mechanism; however, their simulations in which the WD was still bound and star did not detonate were stopped before the contraction phase.', '1208.5069-2-9-7': 'J12 discussed these models and showed that had [CITATION] run their simulations longer, they would have most likely triggered a GCD on contraction.', '1208.5069-2-10-0': 'The FD scenario thus occurs when a weak deflagration leaves a partially bound WD (in constrast to the PD model) and the conditions for detonation are never realized (and the DDT is avoided).', '1208.5069-2-11-0': '# Simulations of the FD Model', '1208.5069-2-12-0': '## Simulation Setup', '1208.5069-2-13-0': 'We used the Adaptive Mesh Refinement (AMR) FLASH application framework to perform our simulations of FD models.', '1208.5069-2-13-1': 'FLASH has been previously used to simulate SNe Ia in both 2D cylindrical and 3D Cartesian geometry ([CITATION], J12).', '1208.5069-2-13-2': 'Our simulations include an advection-diffusion-reaction (ADR) treatment of the thermonuclear flame , an equation of state that includes contributions from blackbody radiation, ions, and electrons of an arbitrary degree of degeneracy , and the multipole treatment of gravity .', '1208.5069-2-14-0': 'We performed four 3-dimensional (3D) exploratory simulations to test the feasibility of the FD and 1 full 2-dimensional (2D) model to adequately observe the FD at late times.', '1208.5069-2-15-0': 'Our 3D simulations included a reduced domain size and a moderate resolution (8 km) which reduced their computational expense.', '1208.5069-2-15-1': 'We initialized these simulations similarly to those in J12 with a 1.365 [MATH] WD placed at the origin of the domain.', '1208.5069-2-15-2': 'We used the same distribution of sixty three, [MATH] km radius ignition "points" distributed in a [MATH] km radius spherical volume.', '1208.5069-2-15-3': 'We chose 48 km, 38, km, 28 km, and 18 km as offset distances along the z-axis of the spherical volume.', '1208.5069-2-15-4': 'Note that these offsets are closer to the WD core than in J12 and lead to more buring during the deflagration phase.', '1208.5069-2-15-5': 'Table [REF] lists the initial conditions for each simulation and the corresponding names we gave them.', '1208.5069-2-15-6': 'We ran the simulations from ignition, through peak stellar expansion, and at least until the WD reached a maximum central density upon contraction.', '1208.5069-2-16-0': 'Our 2D simulation was performed with a large domain in 2D cylindrical geometry at 4 km resolution and ran for 60 seconds.', '1208.5069-2-16-1': 'This simulation was initialized similarly to the 3D simulations except we placed only 4 bubbles in a 64 km spherical volume offset by 70 km along the z-axis (axis of symmetry).', '1208.5069-2-16-2': 'We chose these initial conditions to obtain an FD in 2D given what we learned from our 3D simulations.', '1208.5069-2-16-3': 'The larger domain size allowed us to follow the outer layers of the ejecta for the entirety of the simulation.', '1208.5069-2-17-0': '## Simulation Results', '1208.5069-2-18-0': 'We obtained an FD from each of our simulations.', '1208.5069-2-18-1': 'Table [REF] contains a collection of their properties.', '1208.5069-2-19-0': 'The deflagration liberated between 89% and 167% of the binding energy of the WD.', '1208.5069-2-19-1': 'However, in each simulation the deflagration fails to unbind the WD.', '1208.5069-2-19-2': 'The WD expands in response to the deflagration, reaches its maximum level of expansion, and then contracts.', '1208.5069-2-19-3': 'Figure [REF] shows the evolution of the central density, _c, of the WD and illustrates the oscillatory nature of the WD after the deflagration.', '1208.5069-2-19-4': 'The more energy released during the deflagration, the more the star expanded, the longer the pulsational period, and the smaller _c at maximum contraction.', '1208.5069-2-20-0': 'The fact that the deflagration releases more than 100 WD binding energy in some models suggests that the WD should be unbound after the deflagration; however, the entire energy budget does not work to only unbind the star.', '1208.5069-2-20-1': 'For example, some of the energy is lost when the ejecta and the remnant WD are accelerated to high velocities.', '1208.5069-2-20-2': 'Even though the deflagration liberates enough energy to completely unbind the star, the energy is partitioned in such a way that a portion of the original WD remains gravitationally bound.', '1208.5069-2-21-0': 'In each of our models, the WD gets a kick in the opposite direction from which the buoyant ash rises and breaks through the surface of the star.', '1208.5069-2-21-1': 'We measured this velocity to be on the order of hundreds of km s[MATH] and list these values for each simulation in table [REF].', '1208.5069-2-21-2': 'We note that momentum conservation is better than 10% of the momentum of the WD kick in our simulations.', '1208.5069-2-21-3': 'Some of the material from the FD escapes and achieves high velocities, while some of the material is bound to the remnant WD and will eventually accrete onto its surface.', '1208.5069-2-21-4': 'Table [REF] lists the composition of material that remains gravitationally bound and eventually will mix with the remnant WD.', '1208.5069-2-21-5': "In all cases between 0.04 [MATH] and 0.2 [MATH] of IME's and IGE's (some of which is radioactive [MATH]Ni) remained bound to the star.", '1208.5069-2-21-6': 'The evolution of the WD as the material falls back onto and heats the star is an interesting question and one which we will examine in future work.', '1208.5069-2-22-0': 'The composition of the material that escapes the system is also listed in table [REF] along with the kinetic energy and the mass-weighted velocity of the ejecta.', '1208.5069-2-22-1': "This material includes carbon and oxygen, IME's, and IGE's, and ranges from 0.2 to 1.0 [MATH].", '1208.5069-2-22-2': 'In general, the more material that is burned during the deflagration, the more material that escapes.', '1208.5069-2-23-0': 'Figure [REF] shows shows the density structure and composition with overlaid velocity contours of the 2D model at 60s.', '1208.5069-2-23-1': 'Note that the density profile of the FD model is asymmetric in velocity space between the hemisphere corresponding to the ejected deflagration and the opposite hemisphere of the system.', '1208.5069-2-23-2': 'The remnant of the WD can be seen as the tiny high-density feature slightly below the origin of the domain.', '1208.5069-2-24-0': 'The figure also shows the nature of the asymmetry in composition of the structure.', '1208.5069-2-24-1': 'The north side of the remnant contains the products of the deflagration that were sprayed from the surface of the WD.', '1208.5069-2-24-2': "Clumpy structures of IME's and IGE's exist at a range of velocities in the northern hemisphere of the domain but are less abundant in the southern hemisphere.", '1208.5069-2-24-3': 'The asymmetries suggest that this object would look much different depending on the viewing angle of the observer.', '1208.5069-2-25-0': 'All of our models produce a relatively small amount of radioactive [MATH]Ni.', '1208.5069-2-25-1': "Though we do not perform detailed nucleosynthetic post processing of the 3D models, we can set an upper limit on the [MATH]Ni yields with the amount of IGE's produced, which ranges from 0.2 [MATH] to 0.34 [MATH].", '1208.5069-2-25-2': 'Neutronization through electron capture reactions during the deflagration would shift production away from [MATH]Ni though and reduce its contribution to the IGE totals.', '1208.5069-2-25-3': "Whether even lower masses of IGE's (and thus [MATH]Ni) could be produced under appropriate conditions (e.g. comparable to that observed in SN 2008ha, a very faint SN with extremely low ejecta velocities), is yet to be explored.", '1208.5069-2-26-0': 'In summary, we find that a remnant of the WD survives the FD SN event with a lower mass than the original.', '1208.5069-2-26-1': 'The remaining bound stellar material is kicked by the ejection of the ash and obtains a velocity of hundreds of km s[MATH].', '1208.5069-2-26-2': "An asymmetric outburst of deflagration products rich in IGE's (such as Fe, Co, and Ni) and containing some IME's (such as Mg, Si, and S) is produced.", '1208.5069-2-26-3': 'Some of this material attains escape velocity and some falls back onto the star.', '1208.5069-2-26-4': 'The velocity of the escapting outflow was slow (approximately a few thousand km s[MATH]) related to normal SNe Ia as a result of the comparatively small amount of energy released in the FD scenario.', '1208.5069-2-26-5': 'Due to the weak nature of the deflagration, the FD only converted 15% and 25% of the WD to [MATH]Ni, compared to normal SNe Ia which convert nearly half of the WD to [MATH]Ni.', '1208.5069-2-26-6': 'Finally, the collection of simulations we present produces a range of values for each of the discussed properties which suggests a population of events within this class of model.', '1208.5069-2-27-0': '# Discussion and Predictions', '1208.5069-2-28-0': '## Sub-luminous Low-Velocity SNe', '1208.5069-2-29-0': 'The most prominent features of the FD models are the low mass and low velocity of the ejecta, which translate into the production of typically sub-luminous, low-velocity SNe Ia.', '1208.5069-2-29-1': 'It is therefore natural to examine whether SNe with such characteristics have already been discovered.', '1208.5069-2-29-2': 'In particular, one may explore peculiar SNe exhibiting either extremely low mass ejecta, such as SN 2008ha , or low ejecta velocity such as SN 2002cx-like SNe .', '1208.5069-2-30-0': 'Normal SNe Ia differ in their ejecta velocities as measured in some standard method, but they generally fall between [MATH] km s[MATH] near peak luminosity, with similar dispersion at later times (as derived from the Si II line ).', '1208.5069-2-30-1': 'The velocities of even the lowest velocity SNe in the [CITATION] sample much exceed the mass-averaged FD velocities listed in table [REF].', '1208.5069-2-30-2': 'We also note a trend in our simulations of more energetic and likely more luminous (larger IGE yield) SNe to be accompanied by higher ejecta-velocities over almost an order of magnitude in kinetic energy.', '1208.5069-2-30-3': 'The only other type of SNe with such low expansion velocities are the branch of peculiar type Ia SNe, named for the prototype for this class of supernovae, 2002cx ; such SNe may also have an energy-velocity correlation as observed in our simulations.', '1208.5069-2-31-0': 'SN 2002cx-like events are characterized by luminosities which lie too low in comparison to the Phillips relation for Branch-normal Ia events , low photospheric velocities , weak intermediate-mass element lines , and late-time optical nebular spectra dominated by narrow Fe II lines .', '1208.5069-2-31-1': 'Since the discovery of SN 2002cx, a number of other 2002cx-like events have been discovered, including 2002es, 2005P, 2005hk , 2008ge and 2008ha .', '1208.5069-2-31-2': 'The latter event (SN 2008ha), in particular, is consistent with extremely low mass ejecta and energetics.', '1208.5069-2-31-3': 'We predict the FD models to produce similar properties to those characterizing SN 2002cx like SNe, given the low expansion velocities and the low estimated [MATH]Ni yield, and potentially even explaining SN 2008ha like events with low mass ejecta.', '1208.5069-2-32-0': "Though our initial set of simulations is limited, the robust features of FD's, including low velocity ejecta, the expected low luminosity (due to the small yield of [MATH]Ni) and their low mass ejecta (comparable to that SN 2008ha) make them tantalizing candidate progenitors for this branch of peculiar SNe.", '1208.5069-2-32-1': 'Note that the single degenerate origin of such SNe is also consistent with the overall typically young (but not necessarily young; [CITATION] 2010) environments found for SN-2002cx like SNe, compared to the expectations from, e.g. core-collapse SNe (only very young environments).', '1208.5069-2-33-0': "## WD's with Heavy/Iron-Rich Cores", '1208.5069-2-34-0': 'In our FD scenario, a large amount of burnt material falls back to the remnant WD.', '1208.5069-2-34-1': "From table [REF], the WD may incorporate as much as [MATH] of IGE's and [MATH] of IME's of fallback material, together comprising as much as [MATH] of the remnant C-O WD.", '1208.5069-2-34-2': "In time, these elements are likely to gravitationally settle to the WD core, creating WD's with iron-rich/heavy core.", '1208.5069-2-34-3': "The existence of iron-core WD's has been considered before, with even the potential observation of such WDs .", '1208.5069-2-34-4': "The FD scenario therefore provides a novel evolutionary scenario for the formation of these iron/heavy-core C-O WD's.", '1208.5069-2-34-5': 'A somewhat related scenario of failed SN was suggested for the formation of O-Ne-Mg WDs with iron cores .', '1208.5069-2-35-0': '## WD Natal Kicks', '1208.5069-2-36-0': "FD's produce a highly asymmetric ejection of material.", '1208.5069-2-36-1': 'This is not unique amongst various models for SNe explosions.', '1208.5069-2-36-2': 'However, in our FD case, the WD survives the explosion.', '1208.5069-2-36-3': 'Considering momentum conservation, this gives rise to a unique outcome, namely that the surviving WD is kicked at very high velocities, ranging hundreds of km s[MATH].', '1208.5069-2-36-4': 'The FD scenario suggests the existence of strong WD natal kicks, and provides an interesting prediction per the existence of hypervelocity WDs.', '1208.5069-2-36-5': 'Taken together, the potential existence of a a heavy core WD (discussed in section [REF]), and the high ejection velocity produce a highly peculiar object, which, if observed may provide a possibly unique smoking gun signature.', '1208.5069-2-36-6': 'One should note, however, that the population of halo WDs may also have relatively high velocities, and it might therefore be difficult to pinpoint the kinematic property as related to a natal kick (unless the WD is massive and young; an unlikely possibility for the old population of halo WDs).', '1208.5069-2-37-0': 'We note that velocities of hundreds of km s[MATH] could be larger than the orbital velocities of the SN binary progenitor, and a kick velocity of such magnitude can therefore unbind the binary.', '1208.5069-2-37-1': 'Various binary configurations have been explored for the single degenerate progenitor models, including progenitors with MS and RG companions .', '1208.5069-2-37-2': 'We conclude that the range of WD kick velocities could either unbind the binary (more likely for WD-MS binaries), or leave behind a bound WD binary (more likely for the WD-RG binaries).', '1208.5069-2-37-3': 'The latter case could lead to the formation of a very compact, but potentially eccentric WD-binary, which would be difficult to produce through other channels of binary evolution.', '1208.5069-2-38-0': '# Summary', '1208.5069-2-39-0': 'Our simulations demonstrate the properties of a scenario in which the deflagration is too weak to unbind the WD and the conditions to trigger a detonation are not met.', '1208.5069-2-39-1': "These failed-detonations result in an asymmetric outburst of deflagration material consisting of IME's and IGE's along with a fraction of the original WD still gravitationally bound.", '1208.5069-2-39-2': 'The models produce a family of faint SNe Ia with a slowly evolving light curves due to the low [MATH]Ni yield and the low energetics.', '1208.5069-2-39-3': 'The remaining WD gets a kick on the order of hundreds of km s[MATH] and is contaminated with fall-back from the deflagration, producing a WD with an iron-rich/heavy core.', '1208.5069-2-39-4': 'We presented our initial simulations to quantify the some of the bulk observable properties and demonstrate the conditions under which the GCD fails.', '1208.5069-2-39-5': 'We further hypothesize that the FD model is a possible explanation for 2002cx-like SN.', '1208.5069-2-39-6': 'Future studies will explore the detailed observational features of FD SNe and their direct comparison to observations.', '1208.5069-2-40-0': 'The authors thank the FLASH Code Group, the FLASH Astrophysics Group, and the Argonne Leadership Computing Facility at Argonne National Laboratory.', '1208.5069-2-40-1': 'HBP is supported by the CfA and BIKURA prize fellowships.', '1208.5069-2-40-2': 'AST - 0909132 for the "Petascale Computing of Thermonuclear Supernova Explosions".', '1208.5069-2-41-0': 'ccccccccccccccc', '1208.5069-2-42-0': 'Simulations Properties 0pt', '1208.5069-2-43-0': 'sim & [MATH]a & n[MATH]b & r[MATH]c & z[MATH]d & E[MATH]e & C-O[MATH]f & IME[MATH] & IGE[MATH] & v[MATH]g & C-O[MATH]h & IME[MATH] & IGE[MATH] & E[MATH]i & v[MATH]j', '1208.5069-2-44-0': 'name & (km) & & (km) & (km) & (E[MATH]) & ([MATH]) & ([MATH]) & ([MATH]) & (km s[MATH]) & ([MATH]) & ([MATH]) & ([MATH]) & (ergs [MATH]) & (km s[MATH])', '1208.5069-2-45-0': '2D70 & 4 & 4 & 64.0 & 70.0 & 0.89 & 0.93 & 0.07 & 0.13 & 119 & 0.13 & 0.03 & 0.07 & 0.32 & 3,730', '1208.5069-2-46-0': '3D48 & 8 & 63 & 128.0 & 48.0 & 1.06 & 0.84 & 0.06 & 0.09 & 351 & 0.16 & 0.05 & 0.16 & 0.90 & 4,932', '1208.5069-2-47-0': '3D38 & 8 & 63 & 128.0 & 38.0 & 1.30 & 0.68 & 0.05 & 0.05 & 411 & 0.27 & 0.09 & 0.24 & 1.6 & 5,139', '1208.5069-2-48-0': '3D28 & 8 & 63 & 128.0 & 28.0 & 1.51 & 0.49 & 0.03 & 0.04 & 549 & 0.41 & 0.13 & 0.30 & 2.3 & 5,229', '1208.5069-2-49-0': '3D18 & 8 & 63 & 128.0 & 18.0 & 1.67 & 0.26 & 0.02 & 0.02 & 483 & 0.58 & 0.17 & 0.34 & 2.9 & 5,193', '1208.5069-2-50-0': 'aMaximum spatial resolution.', '1208.5069-2-50-1': 'bNumber of ignition points.', '1208.5069-2-50-2': 'cRadius of the spherical volume containing the ignition points.', '1208.5069-2-50-3': 'dZ coordinate of the origin of the spherical volume containing the ignition points.', '1208.5069-2-50-4': 'eEnergy released during the deflagration phase divided by the binding energy of the star ([MATH] ergs).', '1208.5069-2-50-5': 'f"B" refers to gravitationally bound material.', '1208.5069-2-50-6': 'gVelocity of the gravitationally bound material .', '1208.5069-2-50-7': 'h"E" refers to material that will escape the system.', '1208.5069-2-50-8': 'iAverage kinetic energy of the escaping material.', 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[['1208.5069-1-49-0', '1208.5069-2-37-0']] | ['1208.5069-1-16-0', '1208.5069-1-17-0', '1208.5069-1-18-0', '1208.5069-1-19-0', '1208.5069-1-20-0', '1208.5069-1-21-0', '1208.5069-1-22-0', '1208.5069-1-23-0', '1208.5069-1-24-0', '1208.5069-1-25-0', '1208.5069-2-41-0', '1208.5069-2-42-0', '1208.5069-2-43-0', '1208.5069-2-44-0', '1208.5069-2-45-0', '1208.5069-2-46-0', '1208.5069-2-47-0', '1208.5069-2-48-0', '1208.5069-2-49-0', '1208.5069-2-50-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1208.5069 | null | null | null | null | null |
0811.0792 | {'0811.0792-1-0-0': 'We have performed a series of high-pressure electrical resistivity experiments on CeTe[MATH] single crystals.', '0811.0792-1-0-1': 'At ambient pressure, CeTe[MATH] is known to order magnetically near 3 K.', '0811.0792-1-0-2': 'We find that the magnetic ordering temperature increases rapidly under pressure to 13 K at 50 kbar.', '0811.0792-1-0-3': 'Above 50 kbar, the magnetic ordering temperature becomes nearly pressure independent.', '0811.0792-1-0-4': 'At 70 kbar and lower temperatures, an additional phase transition appears as evidenced by an abrupt drop in the electrical resistivity near 5 K.', '0811.0792-1-0-5': 'We also observe a monotonic decrease of the room-temperature resistivity with pressure and evidence for a possible structural transition at [MATH] 50 kbar.', '0811.0792-1-0-6': 'We have also performed ambient pressure electrical resistivity measurements with the magnetic field aligned both parallel and perpendicular to the crystal [MATH]-axis, and specific heat measurements with fields along the [MATH]-axis, indicating phase transitions at 1.3 K and 3 K. Together, the high-field specific heat and resistivity measurements suggest that these phase transitions have non-parallel magnetic easy axes.', '0811.0792-1-1-0': '# Introduction', '0811.0792-1-2-0': 'Charge- and spin-density waves (CDWs and SDWs) frequently occur in low-dimensional materials, and are driven by electron-phonon and electron-electron interactions.', '0811.0792-1-2-1': 'These phases are formed by nesting of the Fermi surface, which can be perfect in one-dimensional systems.', '0811.0792-1-2-2': '[CITATION] A perfect nesting CDW system refers to the situation where all the electrons near the Fermi surface are excited with the same [MATH] vector of a particular phonon mode.', '0811.0792-1-2-3': 'On the other hand, incomplete nesting takes place for higher dimensional materials, for which a density wave gap opens only over certain regions of the Fermi surface.', '0811.0792-1-2-4': 'Rare-earth tritellurides [MATH]Te[MATH] = La-Tm, except for Eu) constitute a class of quasi-two-dimensional materials that has recently attracted a considerable amount of attention because the electronic properties of these materials can be changed by substituting one rare-earth element for another, making them ideal candidates for investigating the properties of the CDW state.[', '0811.0792-1-2-5': '[CITATION] The rare-earth tritellurides crystallize in the NdTe[MATH] structure that belongs to the space group [MATH] (No. 63) and consists of alternating double layers of nominally square-planar Te sheets and corrugated double [MATH]Te layers and forms a weakly orthorhombic lattice.[', '0811.0792-1-2-6': '[CITATION] In this standard space group denomination, the [MATH]-axis is oriented perpendicular to the [MATH]-planes, and the average lattice parameters for all the lanthanide series are (a, b, c) [MATH] (4, 26, 4) .', '0811.0792-1-2-7': 'It is evident that these compounds are electronically anisotropic, with the Te planes quite decoupled from the [MATH]Te slabs.', '0811.0792-1-2-8': '[CITATION] For this family of materials, the lattice modulation is characterized by a single in-plane wave vector, which has approximately the same value for all the rare earths (2c*/7, with c*=2[MATH]/c).', '0811.0792-1-2-9': '[CITATION]', '0811.0792-1-3-0': 'It has been shown that the application of chemical pressure reduces the CDW ordering temperature.', '0811.0792-1-3-1': '[CITATION] The reduction of the single particle excitation frequency characteristic of the CDW state is accompanied by a decrease in the fraction of the Fermi surface that remains gapped, driving the samples towards a state of enhanced metallicity.', '0811.0792-1-3-2': 'This behavior was also observed in CeTe[MATH] with the application of external pressure, [CITATION] extending the study of the above mentioned phenomenon to even smaller lattice parameters than attainable through chemical pressure.', '0811.0792-1-4-0': 'A second CDW ordering temperature has recently been discovered for the compounds with smaller lattice parameters (Tm, Er, Ho, Dy).', '0811.0792-1-4-1': '[CITATION] In this case, the CDW is characterized by a wavevector transverse to the first one, and of larger value (a*/3).', '0811.0792-1-4-2': 'This phase occurs at lower temperatures, dropping below 50 K for DyTe[MATH], and it increases with the application of chemical pressure.', '0811.0792-1-4-3': 'The shift of our attention to CDW formation at low temperatures allows us to consider the effects caused on this state by other competing types of order.', '0811.0792-1-4-4': 'Cerium-based compounds frequently display an enhancement of their electronic effective mass at low temperatures caused by the strong hybridization of the localized 4f and conduction electron states and produce a variety of ground states, such as localized moment magnetic order[CITATION] and superconductivity,[CITATION] with many of these phases induced at high pressures.', '0811.0792-1-4-5': '[CITATION] The competing interaction of the CDW with some of these strongly correlated electron states by tuning chemical composition, pressure or magnetic field is of particular interest in these materials.', '0811.0792-1-4-6': 'In this report, we present high-pressure electrical transport measurements on CeTe[MATH], along with the results of a sub-Kelvin specific heat experiment at ambient pressure and high magnetic fields.', '0811.0792-1-4-7': 'We have found that two magnetic phases occur below 20 K, with non-parallel magnetic easy axes, as can be inferred from the additional transport measurements done in fields for different angles.', '0811.0792-1-4-8': 'A possible structural phase transition suggested by features in the electrical resistivity at room temperature and a pressure of 50 kbar may shed light on the various features present in T-P phase diagram.', '0811.0792-1-5-0': '# Experimental details', '0811.0792-1-6-0': 'Single crystals of CeTe[MATH] were grown by slow cooling of a binary melt as described elsewhere.', '0811.0792-1-6-1': '[CITATION] Electrical resistivity measurements under pressure were performed throughout the temperature range 1.1 K [MATH] T [MATH] 300 K, employing two different techniques.', '0811.0792-1-6-2': 'In the first technique, pressure was applied with a beryllium-copper, piston-cylinder clamped cell using a Teflon capsule filled with a 1:1 mixture of n-pentane:isoamyl alcohol as the pressure transmitting medium to ensure hydrostatic conditions during pressurization at room temperature.', '0811.0792-1-6-3': 'The pressure in the sample chamber was inferred from the inductively determined, pressure-dependent superconducting critical temperature of a lead manometer,[CITATION] and reached a maximum value of 23 kbar.', '0811.0792-1-6-4': 'In the second technique, pressure was applied in a beryllium-copper Bridgman anvil clamped cell using solid steatite as the quasi-hydrostatic pressure transmitting medium.', '0811.0792-1-6-5': 'The pressure was determined from the superconducting transition of a strip of lead foil placed adjacent to the sample and measured using a 4-lead resistive method.', '0811.0792-1-6-6': 'With this technique, a maximum pressure of 152 kbar was attained in a first attempt and 124 kbar in a second run.', '0811.0792-1-6-7': 'Pressure gradients were inferred from the width of the lead superconducting transition.', '0811.0792-1-6-8': 'These gradients were as large as 2% and 10% of the total pressure for the piston-cylinder and the Bridgman-anvil cell experiments, respectively.', '0811.0792-1-6-9': 'In both cases, the electrical resistance in the [MATH]-plane was measured using a 4-lead technique and a Linear Research Inc.', '0811.0792-1-6-10': 'LR-700 AC resistance bridge.', '0811.0792-1-6-11': 'Resistivity measurements at ambient pressure were obtained using a Quantum Design Physical Property Measurement System (PPMS) throughout the temperature range 1.9 K [MATH] T [MATH] 20 K and for magnetic fields ranging from 0 to 9 T, applied both parallel and perpendicular to the [MATH]-axis of the crystals.', '0811.0792-1-7-0': 'The specific heat C of two single crystals with total mass of 7.5 mg was measured as a function of temperature T from 0.65 to 200 K using a [MATH]He semiadiabatic calorimeter and a standard heat pulse technique for magnetic fields up to 5 T applied along the [MATH]-axis.', '0811.0792-1-8-0': '# Results and Discussion', '0811.0792-1-9-0': 'Electrical resistivity measurements as a function of temperature for different values of applied pressure are plotted in Fig. [REF].', '0811.0792-1-9-1': 'The upper panel shows data obtained from the hydrostatic cell experiment for pressures up to 23 kbar, while the lower panel displays data taken in the Bridgman cell experiments described in the previous section and for pressures up to 152 kbar.', '0811.0792-1-9-2': 'No CDW features were detected up to 300 K for any of the pressures, which is consistent with the high CDW temperature known for this compound.', '0811.0792-1-9-3': '[CITATION] At low pressures (hydrostatic cell, upper panel), the sample behaves as previously reported by Ru et al.,[CITATION] although no appreciable local minimum at 10 K has been seen for these initial values of pressure.', '0811.0792-1-9-4': 'In our measurements, the resistivity decreases monotonically throughout the entire temperature range, which is more evident for the highest pressures obtained in the Bridgman anvil cell (lower panel).', '0811.0792-1-9-5': 'Below 100 K, a broad hump denoted as T[MATH], which is clearly noticeable for the higher pressures, moves to lower temperatures to a value of 55 K at 50 kbar, after which it remains mostly unchanged for the higher pressures.', '0811.0792-1-9-6': 'This feature occurring at T[MATH] is suggestive of either the onset of Kondo coherence, the effect of crystalline electric fields, or the appearance of the charge density wave order.', '0811.0792-1-9-7': 'A lower temperature feature, labeled as T[MATH] was first reported by Iyeiri et al.[CITATION] and later in the above mentioned work by Ru et al..', '0811.0792-1-9-8': 'They attribute this feature to a transition to an antiferromagnetic state, given the negative Curie-Weiss temperatures obtained from magnetic susceptibility measurements.', '0811.0792-1-9-9': 'We found that this ordering temperature increases from 3 to [MATH] 13 K under pressure, as can be seen in the insets of the upper and lower panels of Fig. [REF].', '0811.0792-1-9-10': 'No appreciable change in T[MATH] is seen for pressures above [MATH] 50 kbar.', '0811.0792-1-9-11': 'For pressures below 23 kbar, power-law fits to the resistivity curves below T[MATH] yielded exponent values averaging 2.2 [MATH] 0.1.', '0811.0792-1-10-0': 'A novel feature, occurring at a temperature denoted T[MATH], was discovered above 60 kbar for the two crystals measured in the two Bridgman experiments (Fig. [REF], lower inset).', '0811.0792-1-10-1': 'The features are truncated at the base temperature of 1.1 K, where the resistivity has dropped by 65% of its value at the onset of the transition.', '0811.0792-1-10-2': 'In the first Bridgman cell experiment, the drop of the resistivity was detected at 2.7 K and 152 kbar and it was seen again at 2.4 K and 74.5 kbar.', '0811.0792-1-10-3': 'For the second Bridgman run, T[MATH] remained at a value of 5.5 [MATH] 0.1 K while increasing the pressure from 86 to 124 kbar.', '0811.0792-1-10-4': 'This suggests a possible new phase emerging at lower pressures and below the temperature range covered in this experiment.', '0811.0792-1-10-5': 'Fig. [REF] summarizes the different regions of the T-P phase diagram studied in the present work.', '0811.0792-1-11-0': 'Figure [REF] displays the electronic and magnetic contributions to the specific heat of CeTe[MATH] for magnetic fields up to 5 T applied along the [MATH]-axis of the crystals, obtained after subtracting the phonon contribution estimated from the high temperature C(T) data.', '0811.0792-1-11-1': 'The C/T versus T[MATH] fits yielded a Debye temperature of 161 K, comparable to previous values for LaTe[MATH], and an electronic specific heat coefficient [MATH] of 52 mJ/mol-K[MATH], substantially larger than is observed for LaTe[MATH],[CITATION] implying a moderately enhanced admixture of the localized 4[MATH] electron states of Ce with conduction electron states, as suggested in previously reported ARPES experiments.', '0811.0792-1-11-2': '[CITATION] A broad feature in C(T) characterizing the magnetic order that occurs at T[MATH] corresponds directly to the transition temperature T[MATH] obtained from electrical resistivity measurements at ambient pressure ([MATH] 3 K).', '0811.0792-1-11-3': 'This anomaly in the specific heat remains unchanged by the magnetic fields used in this experiment.', '0811.0792-1-11-4': 'The lower inset in Fig. [REF] shows the electronic and magnetic contributions to the entropy at zero magnetic field, which adds up to Rln2 (indicated by a horizontal dashed line) at temperatures right above T[MATH], consistent with what is expected for the Ce[MATH]-doublet ground state.', '0811.0792-1-12-0': 'At even lower temperatures, a sharper feature is observed in the C(T) data that exhibits a rather weak field dependence.', '0811.0792-1-12-1': 'This transition was not detected in the electrical resistivity experiments (down to 1.1 K).', '0811.0792-1-12-2': 'This suggests this new phase also has a magnetic origin.', '0811.0792-1-12-3': 'The transition temperature T[MATH] (defined after performing an equal-entropy analysis of the data) increases to a value of [MATH] 1.3 K at 2 T, and then decreases for the higher applied fields.', '0811.0792-1-12-4': 'The left panel of Fig. [REF] illustrates the evolution of this feature in the range of applied magnetic fields used during the specific heat experiment.', '0811.0792-1-13-0': 'The above evidence associated with the low-temperature transition below 1.5 K, and the apparent lack of field dependence for the 3 K ordering temperature revealed by the specific heat data led us to inquire further into the origin of these magnetic transitions.', '0811.0792-1-13-1': 'The work by Iyeiri and co-workers previously mentioned, foretells a strong dependence of the magnetic phases of CeTe[MATH] with the orientation of the applied magnetic field with respect to the crystalline axes.', '0811.0792-1-13-2': 'In order to test the angle dependence of T[MATH], electrical resistivity measurements were performed down to 2 K in magnetic fields [MATH] 9 T applied perpendicular and parallel to the [MATH]-axis of the crystals, and perpendicular to the direction of the current passing through the the ac-planes of the samples, utilizing a commercial Quantum Design PPMS sample rotator.', '0811.0792-1-13-3': 'The two right panels of Fig. [REF] show these results.', '0811.0792-1-13-4': 'With magnetic fields applied perpendicular to the planes (H [MATH] b, upper-right panel), the transition temperature T[MATH] does not depend on the applied magnetic field as it is seen in the specific heat experiment, although a rather strong magnetoresistance was found (([MATH] at 10 K).', '0811.0792-1-13-5': 'On the other hand, for fields applied parallel to the [MATH]-planes (lower-right panel), a negative magnetoresistance is observed, and the transition temperature T[MATH] moves towards zero as indicated by the arrows.', '0811.0792-1-13-6': 'The left panel in Fig. [REF] combines the field dependencies of T[MATH] for H [MATH] and H [MATH], with the field dependence of T[MATH] with H [MATH].', '0811.0792-1-14-0': 'Although not shown in Fig. [REF] for clarity, the local Kondo-like minimum around 10 K mentioned earlier[CITATION] has been seen in this set of measurements at ambient pressure.', '0811.0792-1-14-1': 'For H [MATH] b, this minimum appears at 9.8 K without applied magnetic field, and increases to 10.5, 12 and 13 K for 3, 6 and 9 T. For H [MATH], the minimum observed at zero field at 9.8 K remains unchanged for fields below 1.5 T, and then disappears for magnetic fields above 3 T.', '0811.0792-1-15-0': 'With the evidence presented above, we can attempt to draw some conclusions.', '0811.0792-1-15-1': 'T[MATH] seems to characterize the onset of the transition to a magnetic phase with the easy magnetic axis contained in the [MATH]-planes.', '0811.0792-1-15-2': 'The [MATH]-axis would then play the role of a hard axis for this magnetic phase, consistently with magnetic susceptibility measurements.', '0811.0792-1-15-3': '[CITATION] In the case of the transition at T[MATH], we did not measure the specific heat with the magnetic field applied parallel to the basal plane, but we can conclude that the T[MATH] magnetic easy axis is not parallel to the T[MATH] one, as can be found in other anisotropic magnetic [MATH]-electron systems reported elsewhere,[CITATION] in which ferromagnetic and antiferromagnetic phases occur at different ordering temperatures, with non-collinear ordering directions, due to the interaction of the localized [MATH] electrons with the conduction electrons (exchange) and with nearby ions (direct), and to the effect of crystalline electric fields.', '0811.0792-1-16-0': 'The T-P phase diagram, which contains the critical temperatures inferred from the specific heat at ambient pressure and the features in the electrical resistivity at high pressure, is presented in Fig. [REF].', '0811.0792-1-16-1': 'It is clearly seen that the critical temperature T[MATH] saturates to a rather constant value above 50 kbar, and that it matches the critical temperature determined from the specific heat data taken at ambient pressure.', '0811.0792-1-16-2': 'The temperature associated with the broad hump, denoted as T[MATH] (open squares), also shows a kink around the same pressure, and then attains a constant value of [MATH] 55 K.', '0811.0792-1-16-3': 'The inset of Fig. [REF] shows the room-temperature dependence of the electrical resistance with applied pressure (compression only).', '0811.0792-1-16-4': 'An abrupt drop of the resistance occurs around 45 kbar, suggesting the existence of a structural phase transition at this pressure.', '0811.0792-1-16-5': 'Despite the fact that this feature in the electrical resistance was observed at room temperature, the value of this pressure coincides with the the kinks in T[MATH] and T[MATH] vs. pressure at 50 kbar, and with the emergence of the critical temperature T[MATH] above 60 kbar.', '0811.0792-1-16-6': 'In Fig. [REF], we also grouped in a single low-temperature magnetic phase the critical temperatures T[MATH] obtained from measurements of the electrical resistivity under pressure and specific heat at ambient pressure.', '0811.0792-1-16-7': 'The fact that T[MATH] is visible in electrical resistivity only at high pressures obtained with the anisotropic Bridgman-anvil cell technique (in which a higher pressure is effectively applied along the [MATH]-axis of the crystals) suggests a possible connection between T[MATH] and the one detected in specific heat at T[MATH], although they might have different origins due to the possible change in crystal structure mentioned above.', '0811.0792-1-17-0': '# Conclusions', '0811.0792-1-18-0': 'In summary, we have presented high-pressure, transport and thermodynamic measurements of the rare-earth tritelluride CeTe[MATH] single crystals.', '0811.0792-1-18-1': 'These measurements yielded evidence for two magnetic phases, detected in electrical resistivity and specific heat order at low temperatures, with non-parallel magnetic [MATH] axes.', '0811.0792-1-18-2': 'We also observed the emergence of a new phase at very high pressures and low temperatures and a possible structural transition at 45 kbar, revealing that external pressure plays a key role in establishing the phase diagram of the highly anisotropic rare-earth tritellurides.'} | {'0811.0792-2-0-0': 'We have performed high-pressure, electrical resistivity and specific heat measurements on CeTe[MATH] single crystals.', '0811.0792-2-0-1': 'Two magnetic phases with non-parallel magnetic easy axes were detected in electrical resistivity and specific heat at low temperatures.', '0811.0792-2-0-2': 'We also observed the emergence of an additional phase at high pressures and low temperatures and a possible structural phase transition detected at room temperature and at 45 kbar which can possibly be related with the lowering of the charge density wave transition temperature known for this compound.', '0811.0792-2-1-0': '# Introduction', '0811.0792-2-2-0': 'Charge- and spin-density waves (CDWs and SDWs) frequently occur in low-dimensional materials, and are driven by electron-phonon and electron-electron interactions.', '0811.0792-2-2-1': 'These phases are formed by nesting of the Fermi surface, which can be perfect in one-dimensional systems.', '0811.0792-2-2-2': '[CITATION] A perfect nesting CDW system refers to the situation where all the electrons near the Fermi surface are excited with the same [MATH] vector of a particular phonon mode.', '0811.0792-2-2-3': 'On the other hand, incomplete nesting takes place for higher dimensional materials, for which a density wave gap opens only over certain regions of the Fermi surface.', '0811.0792-2-2-4': 'Rare-earth tritellurides [MATH]Te[MATH] = La-Tm, except for Eu) constitute a class of quasi-two-dimensional materials that has recently attracted a considerable amount of attention because the electronic properties of these materials can be changed by substituting one rare-earth element for another, making them ideal candidates for investigating the properties of the CDW state.[', '0811.0792-2-2-5': '[CITATION] The rare-earth tritellurides crystallize in the NdTe[MATH] structure that belongs to the space group [MATH] (No. 63); the structure consists of alternating double layers of nominally square-planar Te sheets and corrugated double [MATH]Te layers and forms a weakly orthorhombic lattice.[', '0811.0792-2-2-6': '[CITATION] In this standard space group denomination, the [MATH]-axis is oriented perpendicular to the [MATH]-planes, and the average lattice parameters for all the lanthanide series are (a, b, c) [MATH] (4, 26, 4) .', '0811.0792-2-2-7': 'It is evident that these compounds are electronically anisotropic, with the Te planes quite decoupled from the [MATH]Te slabs.', '0811.0792-2-2-8': '[CITATION] For this family of materials, the lattice modulation is characterized by a single in-plane wave vector, which has approximately the same value for all the rare earths (2c*/7, with c*=2[MATH]/c).', '0811.0792-2-2-9': '[CITATION]', '0811.0792-2-3-0': 'It has been shown that the application of chemical pressure reduces the CDW ordering temperature from values above 450 K for (La,Ce,Nd,Pr)Te[MATH] to 244 K for TmTe[MATH].', '0811.0792-2-3-1': '[CITATION] Moreover, a reduction with chemical pressure of the single particle excitation frequency characteristic of the CDW state is accompanied by a decrease in the fraction of the Fermi surface that remains gapped, driving the samples towards a state of enhanced metallicity.', '0811.0792-2-3-2': 'This behavior was also observed in CeTe[MATH] with the application of external pressure, [CITATION] extending the study of the above mentioned phenomenon to even smaller lattice parameters than attainable through chemical pressure.', '0811.0792-2-4-0': 'A second CDW ordering temperature has recently been discovered for the compounds with smaller lattice parameters (Tm, Er, Ho, Dy).', '0811.0792-2-4-1': '[CITATION] In this case, the CDW is characterized by a wavevector transverse to the first one, and of larger value (a*/3).', '0811.0792-2-4-2': 'This phase occurs at lower temperatures, dropping below 50 K for DyTe[MATH], and it increases with the application of chemical pressure.', '0811.0792-2-4-3': 'The shift of our attention to CDW formation at low temperatures allows us to consider the effects caused on this state by other competing types of order.', '0811.0792-2-4-4': 'Cerium-based compounds frequently display an enhancement of their electronic effective mass at low temperatures caused by the strong hybridization of the localized 4f and conduction electron states and produce a variety of ground states, such as localized moment magnetic order[CITATION] and superconductivity,[CITATION] with many of these phases induced at high pressures.[', '0811.0792-2-4-5': '[CITATION] The competing interaction of the CDW with some of these strongly correlated electron states by tuning chemical composition, pressure or magnetic field is of particular interest in these materials.[', '0811.0792-2-4-6': '[CITATION] In this report, we present high-pressure electrical transport measurements on CeTe[MATH], along with the results of a sub-kelvin specific heat experiment at ambient pressure and high magnetic fields.', '0811.0792-2-4-7': 'We have found that two magnetic phases occur below 20 K, with non-parallel magnetic easy axes, as can be inferred from the additional transport measurements made in fields for different angles.', '0811.0792-2-4-8': 'A possible structural phase transition suggested by features in the electrical resistivity at room temperature and at a pressure of 45 kbar, along with the low temperature features detected at high pressures, may indicate the reduction of the CDW transition temperature below 300 K for the range of pressures used in our experiments.', '0811.0792-2-5-0': '# Experimental details', '0811.0792-2-6-0': 'Single crystals of CeTe[MATH] were grown by slow cooling of a binary melt as described elsewhere.', '0811.0792-2-6-1': '[CITATION] Electrical resistivity measurements under pressure were performed throughout the temperature range 1.1 K [MATH] T [MATH] 300 K, employing two different techniques.', '0811.0792-2-6-2': 'In the first technique, pressure was applied with a beryllium-copper, piston-cylinder clamped cell using a Teflon capsule filled with a 1:1 mixture of n-pentane:isoamyl alcohol as the pressure transmitting medium to ensure hydrostatic conditions during pressurization at room temperature.', '0811.0792-2-6-3': 'The pressure in the sample chamber was inferred from the inductively determined, pressure-dependent superconducting critical temperature of a lead manometer,[CITATION] and reached a maximum value of 23 kbar.', '0811.0792-2-6-4': 'In the second technique, pressure was applied in a beryllium-copper Bridgman anvil clamped cell using solid steatite as the quasi-hydrostatic pressure transmitting medium.', '0811.0792-2-6-5': 'The pressure was determined from the superconducting transition of a strip of lead foil placed adjacent to the sample and measured using a 4-lead resistive method.', '0811.0792-2-6-6': 'With this technique, a maximum pressure of 152 kbar was attained in a first attempt and 124 kbar in a second run.', '0811.0792-2-6-7': 'Pressure gradients were inferred from the width of the lead superconducting transition.', '0811.0792-2-6-8': 'These gradients were as large as 2% and 10% of the total pressure for the piston-cylinder and the Bridgman-anvil cell experiments, respectively.', '0811.0792-2-6-9': 'In both cases, the electrical resistance in the [MATH]-plane was measured using a 4-lead technique and a Linear Research Inc.', '0811.0792-2-6-10': 'LR-700 AC resistance bridge.', '0811.0792-2-6-11': 'Resistivity measurements at ambient pressure were obtained using a Quantum Design Physical Property Measurement System (PPMS) throughout the temperature range 1.9 K [MATH] T [MATH] 20 K and for magnetic fields ranging from 0 to 9 T, applied both parallel and perpendicular to the [MATH]-axis of the crystals.', '0811.0792-2-7-0': 'The specific heat C of two single crystals with total mass of 7.5 mg was measured as a function of temperature T from 0.65 to 200 K using a [MATH]He semiadiabatic calorimeter and a standard heat pulse technique for magnetic fields up to 5 T applied along the [MATH]-axis.', '0811.0792-2-8-0': '# Results and Discussion', '0811.0792-2-9-0': 'Electrical resistivity measurements as a function of temperature for different values of applied pressure are plotted in Fig. [REF].', '0811.0792-2-9-1': 'The upper panel shows data obtained from the hydrostatic cell experiment for pressures up to 23 kbar, while the lower panel displays data taken in the Bridgman cell experiments described in the previous section and for pressures up to 152 kbar.', '0811.0792-2-9-2': 'At low pressures (hydrostatic cell, upper panel), the sample behaves as previously reported by Ru et al.,[CITATION] although no appreciable local minimum at 10 K has been seen for these initial values of pressure.', '0811.0792-2-9-3': 'In our measurements, the resistivity decreases monotonically throughout the entire temperature range, which is more evident for the highest pressures obtained in the Bridgman anvil cell (lower panel).', '0811.0792-2-9-4': 'Below 100 K, a broad hump denoted as T[MATH], which is clearly noticeable for the higher pressures, moves to lower temperatures to a value of 55 K at 50 kbar, after which it remains mostly unchanged for the higher pressures.', '0811.0792-2-9-5': 'This feature occurring at T[MATH] is suggestive of the appearance of the charge density wave order, which is supported by recent x-ray diffraction data obtained for CeTe[MATH] under pressure,[CITATION] where it is clearly seen that the onset of the CDW state occurs below room temperature at 30 kbar.', '0811.0792-2-9-6': 'Nevertheless, the effects of the crystalline electric field or the onset of Kondo coherence should not be ruled out, taking into account the strong hybridization of the localized 4f orbitals with the conduction band that usually takes place in cerium-based compounds.', '0811.0792-2-9-7': 'A lower temperature feature, labeled as T[MATH] was first reported by Iyeiri et al.[CITATION] and later in the above mentioned work by Ru et al..', '0811.0792-2-9-8': 'They attribute this feature to a transition to an antiferromagnetic state, given the negative Curie-Weiss temperatures obtained from magnetic susceptibility measurements.', '0811.0792-2-9-9': 'We found that this ordering temperature increases from 3 to 13 K under pressure, as can be seen in the insets of the upper and lower panels of Fig. [REF].', '0811.0792-2-9-10': 'No appreciable change in T[MATH] is observed for pressures above 50 kbar.', '0811.0792-2-9-11': 'For pressures below 23 kbar, power-law fits to the resistivity curves below T[MATH] yielded exponent values averaging 2.2 [MATH] 0.1.', '0811.0792-2-10-0': 'A feature, occurring at a temperature denoted T[MATH], was discovered above 70 kbar for the two crystals measured in the two Bridgman experiments (Fig. [REF], lower inset).', '0811.0792-2-10-1': 'The features are truncated at the base temperature of 1.1 K, where the resistivity has dropped by 65% of its value at the onset of the transition.', '0811.0792-2-10-2': 'In the first Bridgman cell experiment, the drop of the resistivity was detected at 2.7 K and 152 kbar and it was seen again at 2.4 K and 74.5 kbar.', '0811.0792-2-10-3': 'For the second Bridgman run, T[MATH] remained at a value of 5.5 [MATH] 0.1 K while increasing the pressure from 86 to 124 kbar.', '0811.0792-2-10-4': 'This suggests a possible new phase emerging at lower pressures and below the temperature range covered in this experiment.', '0811.0792-2-10-5': 'Fig. [REF] summarizes the different regions of the T-P phase diagram studied in the present work.', '0811.0792-2-11-0': 'Figure [REF] displays the electronic and magnetic contributions to the specific heat of CeTe[MATH] for magnetic fields up to 5 T applied along the [MATH]-axis of the crystals, obtained after subtracting the phonon contribution estimated from the high temperature C(T) data.', '0811.0792-2-11-1': 'The C/T versus T[MATH] fits yielded a Debye temperature of 161 K, comparable to previous values for LaTe[MATH], and an electronic specific heat coefficient [MATH] of 52 mJ/mol-K[MATH], substantially larger than is observed for LaTe[MATH],[CITATION] implying a moderately enhanced admixture of the localized 4f electron states of Ce with conduction electron states, as suggested in previously reported angle-resolved photoemission spectroscopy (ARPES) experiments.', '0811.0792-2-11-2': '[CITATION] A broad feature in C(T) characterizing the magnetic order that occurs at T[MATH] corresponds directly to the transition temperature T[MATH] obtained from electrical resistivity measurements at ambient pressure ([MATH] 3 K).', '0811.0792-2-11-3': 'This anomaly in the specific heat remains unchanged by the magnetic fields used in this experiment.', '0811.0792-2-11-4': 'The lower inset in Fig. [REF] shows the electronic and magnetic contributions to the entropy at zero magnetic field, which adds up to Rln2 (indicated by a horizontal dashed line) at temperatures right above T[MATH], consistent with what is expected for a Ce[MATH]-doublet ground state.', '0811.0792-2-12-0': 'At even lower temperatures, a sharper feature is observed in the C(T) data that exhibits a rather weak field dependence.', '0811.0792-2-12-1': 'This transition was not detected in the electrical resistivity experiments (down to 1.1 K).', '0811.0792-2-12-2': 'This suggests this new phase also has a magnetic origin.', '0811.0792-2-12-3': 'The transition temperature T[MATH] (defined after performing an equal-entropy analysis of the data) increases to a value of [MATH] 1.3 K at 2 T, and then decreases for the higher applied fields.', '0811.0792-2-12-4': 'The left panel of Fig. [REF] illustrates the evolution of this feature throughout the range of applied magnetic fields in which the specific heat measurements were made.', '0811.0792-2-13-0': 'The above evidence associated with the low-temperature transition below 1.5 K, and the apparent lack of field dependence for the 3 K ordering temperature revealed by the specific heat data led us to inquire further into the origin of these magnetic transitions.', '0811.0792-2-13-1': 'The work by Iyeiri and co-workers previously mentioned, foretells a strong dependence of the magnetic phases of CeTe[MATH] with the orientation of the applied magnetic field with respect to the crystalline axes.', '0811.0792-2-13-2': 'In order to test the angle dependence of T[MATH], electrical resistivity measurements were performed down to 2 K in magnetic fields [MATH] 9 T applied perpendicular and parallel to the [MATH]-axis of the crystals, and perpendicular to the direction of the current passing through the the ac-planes of the samples, utilizing a commercial Quantum Design PPMS sample rotator.', '0811.0792-2-13-3': 'The two right panels of Fig. [REF] show these results.', '0811.0792-2-13-4': 'With magnetic fields applied perpendicular to the planes (H [MATH] b, upper-right panel), the transition temperature T[MATH] does not shift with applied magnetic field, consistent with the specific heat measurements, although a rather strong magnetoresistance was found (([MATH] at 10 K).', '0811.0792-2-13-5': 'On the other hand, for fields applied parallel to the [MATH]-planes (lower-right panel), a negative magnetoresistance is observed, and the transition temperature T[MATH] moves towards zero as indicated by the arrows.', '0811.0792-2-13-6': 'The left panel in Fig. [REF] combines the field dependencies of T[MATH] for H [MATH] and H [MATH], with the field dependence of T[MATH] with H [MATH].', '0811.0792-2-14-0': 'Although not shown in Fig. [REF] for clarity, the local Kondo-like minimum around 10 K mentioned earlier[CITATION] has been seen in this set of measurements at ambient pressure.', '0811.0792-2-14-1': 'For H [MATH] b, this minimum appears at 9.8 K without applied magnetic field, and increases to 10.5, 12 and 13 K for 3, 6 and 9 T. For H [MATH], the minimum observed at zero field at 9.8 K remains unchanged for fields below 1.5 T, and then disappears for magnetic fields above 3 T.', '0811.0792-2-15-0': 'The specific heat and transport data presented in Figs. [REF] and [REF] suggest that T[MATH] characterizes the onset of the transition to a magnetic phase with the easy magnetic axis contained in the [MATH]-planes.', '0811.0792-2-15-1': 'The [MATH]-axis would then play the role of a hard axis for this magnetic phase, consistent with magnetic susceptibility measurements.', '0811.0792-2-15-2': '[CITATION] In the case of the transition at T[MATH], we did not measure the specific heat with the magnetic field applied parallel to the basal plane, but we can conclude that the T[MATH] magnetic easy axis is not parallel to the T[MATH] easy axis, as can be found in other anisotropic magnetic [MATH]-electron systems reported elsewhere,[CITATION] in which ferromagnetic and antiferromagnetic phases occur at different ordering temperatures, with non-collinear ordering directions, due to the interaction of the localized [MATH]-electrons with the conduction electrons (intra-atomic exchange) and with nearby ions (interatomic exchange), and to the effect of crystalline electric fields.', '0811.0792-2-16-0': 'The temperature vs. pressure phase diagram is presented in Fig. [REF].', '0811.0792-2-16-1': 'It is clearly seen that the curve T[MATH](P) saturates to a rather constant value above 50 kbar.', '0811.0792-2-16-2': 'T[MATH](P) also shows a kink around the same pressure, and then attains a constant value of [MATH] 55 K.', '0811.0792-2-16-3': 'This particular value of pressure separates the phase diagram in two regions: a low pressure region where the phase characterized by T[MATH](P) competes with the phases below T[MATH](P), and a high-pressure region where these three phases may coexist.', '0811.0792-2-16-4': 'The inset of Fig. [REF] shows the room-temperature dependence of the electrical resistance with applied pressure (compression only).', '0811.0792-2-16-5': 'An abrupt drop of the resistance occurs around 45 kbar, suggesting the existence of a structural phase transition at this pressure.', '0811.0792-2-16-6': 'Despite the fact that this feature in the electrical resistance was observed at room temperature, the value of this pressure coincides with the kinks in the T[MATH](P) and T[MATH](P) curves at 50 kbar, and with the emergence of the critical temperature T[MATH] above 60 kbar.', '0811.0792-2-16-7': 'As we previously mentioned, in a recent work by Sacchetti et al.,[CITATION] x-ray diffraction experiments performed under pressure showed that the satellite peaks associated with the CDW lattice distortion disappear at room temperature when an external pressure of 30 kbar is applied.', '0811.0792-2-16-8': 'This suggests that the onset of the CDW transition is driven to lower temperatures when high enough pressures are applied to the rare-earth tritellurides.', '0811.0792-2-16-9': 'In the same report, the authors showed that the a and c lattice parameters become equal at room temperature and for pressures above 30 kbar which is indicative that a structural phase transition might be taking place near that value of pressure.', '0811.0792-2-16-10': 'Unfortunately, no x-ray data under pressure and at low temperatures has yet been reported for CeTe[MATH], which would definitely clarify the origin of T[MATH] that we found in our Bridgman cell experiments.', '0811.0792-2-17-0': '# Conclusions', '0811.0792-2-18-0': 'In summary, we have presented high-pressure, transport and thermodynamic measurements on CeTe[MATH] single crystals.', '0811.0792-2-18-1': 'These measurements yielded evidence for two magnetic phases, detected in electrical resistivity and specific heat measurements at low temperatures, with non-parallel magnetic [MATH] axes.', '0811.0792-2-18-2': 'We also reported the emergence of a phase at high pressures and low temperatures and a possible structural transition detected at room temperature and at 45 kbar, which could be related to the reduction of the CDW transition temperature, illustrating that external pressure plays a key role in establishing the phase diagram of the highly anisotropic rare-earth tritellurides.'} | [['0811.0792-1-10-1', '0811.0792-2-10-1'], ['0811.0792-1-10-2', '0811.0792-2-10-2'], ['0811.0792-1-10-3', '0811.0792-2-10-3'], ['0811.0792-1-10-4', '0811.0792-2-10-4'], ['0811.0792-1-10-5', '0811.0792-2-10-5'], ['0811.0792-1-4-0', '0811.0792-2-4-0'], ['0811.0792-1-4-1', '0811.0792-2-4-1'], ['0811.0792-1-4-2', '0811.0792-2-4-2'], ['0811.0792-1-4-3', '0811.0792-2-4-3'], ['0811.0792-1-4-4', '0811.0792-2-4-4'], ['0811.0792-1-9-0', '0811.0792-2-9-0'], ['0811.0792-1-9-1', '0811.0792-2-9-1'], ['0811.0792-1-9-4', '0811.0792-2-9-3'], ['0811.0792-1-9-5', '0811.0792-2-9-4'], ['0811.0792-1-9-7', '0811.0792-2-9-7'], ['0811.0792-1-9-8', '0811.0792-2-9-8'], ['0811.0792-1-9-11', '0811.0792-2-9-11'], ['0811.0792-1-12-0', '0811.0792-2-12-0'], ['0811.0792-1-12-1', '0811.0792-2-12-1'], ['0811.0792-1-12-2', '0811.0792-2-12-2'], ['0811.0792-1-12-3', '0811.0792-2-12-3'], ['0811.0792-1-7-0', '0811.0792-2-7-0'], ['0811.0792-1-13-0', '0811.0792-2-13-0'], ['0811.0792-1-13-1', '0811.0792-2-13-1'], ['0811.0792-1-13-2', '0811.0792-2-13-2'], ['0811.0792-1-13-3', '0811.0792-2-13-3'], ['0811.0792-1-13-5', '0811.0792-2-13-5'], ['0811.0792-1-13-6', '0811.0792-2-13-6'], ['0811.0792-1-3-2', '0811.0792-2-3-2'], ['0811.0792-1-11-0', '0811.0792-2-11-0'], ['0811.0792-1-11-2', '0811.0792-2-11-2'], ['0811.0792-1-11-3', '0811.0792-2-11-3'], ['0811.0792-1-2-0', '0811.0792-2-2-0'], ['0811.0792-1-2-1', '0811.0792-2-2-1'], ['0811.0792-1-2-2', '0811.0792-2-2-2'], ['0811.0792-1-2-3', '0811.0792-2-2-3'], ['0811.0792-1-2-4', '0811.0792-2-2-4'], ['0811.0792-1-2-6', '0811.0792-2-2-6'], ['0811.0792-1-2-7', '0811.0792-2-2-7'], ['0811.0792-1-2-8', '0811.0792-2-2-8'], ['0811.0792-1-14-0', '0811.0792-2-14-0'], ['0811.0792-1-14-1', '0811.0792-2-14-1'], ['0811.0792-1-16-3', '0811.0792-2-16-4'], ['0811.0792-1-16-4', '0811.0792-2-16-5'], ['0811.0792-1-6-0', '0811.0792-2-6-0'], ['0811.0792-1-6-1', '0811.0792-2-6-1'], ['0811.0792-1-6-2', '0811.0792-2-6-2'], ['0811.0792-1-6-3', '0811.0792-2-6-3'], ['0811.0792-1-6-4', '0811.0792-2-6-4'], ['0811.0792-1-6-5', '0811.0792-2-6-5'], ['0811.0792-1-6-6', '0811.0792-2-6-6'], ['0811.0792-1-6-7', '0811.0792-2-6-7'], ['0811.0792-1-6-8', '0811.0792-2-6-8'], ['0811.0792-1-6-9', '0811.0792-2-6-9'], ['0811.0792-1-6-10', '0811.0792-2-6-10'], ['0811.0792-1-6-11', '0811.0792-2-6-11'], ['0811.0792-1-10-0', '0811.0792-2-10-0'], ['0811.0792-1-4-5', '0811.0792-2-4-5'], ['0811.0792-1-4-6', '0811.0792-2-4-6'], ['0811.0792-1-4-7', '0811.0792-2-4-7'], ['0811.0792-1-9-3', '0811.0792-2-9-2'], ['0811.0792-1-9-9', '0811.0792-2-9-9'], ['0811.0792-1-9-10', '0811.0792-2-9-10'], ['0811.0792-1-15-2', '0811.0792-2-15-1'], ['0811.0792-1-15-3', '0811.0792-2-15-2'], ['0811.0792-1-18-1', '0811.0792-2-18-1'], ['0811.0792-1-13-4', '0811.0792-2-13-4'], ['0811.0792-1-3-1', '0811.0792-2-3-1'], ['0811.0792-1-11-1', '0811.0792-2-11-1'], ['0811.0792-1-11-4', '0811.0792-2-11-4'], ['0811.0792-1-2-5', '0811.0792-2-2-5'], ['0811.0792-1-16-5', '0811.0792-2-16-6'], ['0811.0792-1-4-8', '0811.0792-2-4-8'], ['0811.0792-1-9-6', '0811.0792-2-9-5'], ['0811.0792-1-12-4', '0811.0792-2-12-4'], ['0811.0792-1-15-1', '0811.0792-2-15-0'], ['0811.0792-1-18-0', '0811.0792-2-18-0'], ['0811.0792-1-18-2', '0811.0792-2-18-2'], ['0811.0792-1-3-0', '0811.0792-2-3-0'], ['0811.0792-1-16-1', '0811.0792-2-16-1'], ['0811.0792-1-16-2', '0811.0792-2-16-2'], ['0811.0792-1-0-0', '0811.0792-2-0-0']] | [['0811.0792-1-10-1', '0811.0792-2-10-1'], ['0811.0792-1-10-2', '0811.0792-2-10-2'], ['0811.0792-1-10-3', '0811.0792-2-10-3'], ['0811.0792-1-10-4', '0811.0792-2-10-4'], ['0811.0792-1-10-5', '0811.0792-2-10-5'], ['0811.0792-1-4-0', '0811.0792-2-4-0'], ['0811.0792-1-4-1', '0811.0792-2-4-1'], ['0811.0792-1-4-2', '0811.0792-2-4-2'], ['0811.0792-1-4-3', '0811.0792-2-4-3'], ['0811.0792-1-9-0', '0811.0792-2-9-0'], ['0811.0792-1-9-1', '0811.0792-2-9-1'], ['0811.0792-1-9-4', '0811.0792-2-9-3'], ['0811.0792-1-9-5', '0811.0792-2-9-4'], ['0811.0792-1-9-7', '0811.0792-2-9-7'], ['0811.0792-1-9-8', '0811.0792-2-9-8'], ['0811.0792-1-9-11', '0811.0792-2-9-11'], ['0811.0792-1-12-0', '0811.0792-2-12-0'], ['0811.0792-1-12-1', '0811.0792-2-12-1'], ['0811.0792-1-12-2', '0811.0792-2-12-2'], ['0811.0792-1-12-3', '0811.0792-2-12-3'], ['0811.0792-1-7-0', '0811.0792-2-7-0'], ['0811.0792-1-13-0', '0811.0792-2-13-0'], ['0811.0792-1-13-1', '0811.0792-2-13-1'], ['0811.0792-1-13-2', '0811.0792-2-13-2'], ['0811.0792-1-13-3', '0811.0792-2-13-3'], ['0811.0792-1-13-5', '0811.0792-2-13-5'], ['0811.0792-1-13-6', '0811.0792-2-13-6'], ['0811.0792-1-3-2', '0811.0792-2-3-2'], ['0811.0792-1-11-0', '0811.0792-2-11-0'], ['0811.0792-1-11-2', '0811.0792-2-11-2'], ['0811.0792-1-11-3', '0811.0792-2-11-3'], ['0811.0792-1-2-0', '0811.0792-2-2-0'], ['0811.0792-1-2-1', '0811.0792-2-2-1'], ['0811.0792-1-2-2', '0811.0792-2-2-2'], ['0811.0792-1-2-3', '0811.0792-2-2-3'], ['0811.0792-1-2-4', '0811.0792-2-2-4'], ['0811.0792-1-2-6', '0811.0792-2-2-6'], ['0811.0792-1-2-7', '0811.0792-2-2-7'], ['0811.0792-1-2-8', '0811.0792-2-2-8'], ['0811.0792-1-14-0', '0811.0792-2-14-0'], ['0811.0792-1-14-1', '0811.0792-2-14-1'], ['0811.0792-1-16-3', '0811.0792-2-16-4'], ['0811.0792-1-16-4', '0811.0792-2-16-5'], ['0811.0792-1-6-0', '0811.0792-2-6-0'], ['0811.0792-1-6-1', '0811.0792-2-6-1'], ['0811.0792-1-6-2', '0811.0792-2-6-2'], ['0811.0792-1-6-3', '0811.0792-2-6-3'], ['0811.0792-1-6-4', '0811.0792-2-6-4'], ['0811.0792-1-6-5', '0811.0792-2-6-5'], ['0811.0792-1-6-6', '0811.0792-2-6-6'], ['0811.0792-1-6-7', '0811.0792-2-6-7'], ['0811.0792-1-6-8', '0811.0792-2-6-8'], ['0811.0792-1-6-9', '0811.0792-2-6-9'], ['0811.0792-1-6-10', '0811.0792-2-6-10'], ['0811.0792-1-6-11', '0811.0792-2-6-11']] | [['0811.0792-1-4-4', '0811.0792-2-4-4'], ['0811.0792-1-10-0', '0811.0792-2-10-0'], ['0811.0792-1-4-5', '0811.0792-2-4-5'], ['0811.0792-1-4-6', '0811.0792-2-4-6'], ['0811.0792-1-4-7', '0811.0792-2-4-7'], ['0811.0792-1-9-3', '0811.0792-2-9-2'], ['0811.0792-1-9-9', '0811.0792-2-9-9'], ['0811.0792-1-9-10', '0811.0792-2-9-10'], ['0811.0792-1-15-2', '0811.0792-2-15-1'], ['0811.0792-1-15-3', '0811.0792-2-15-2'], ['0811.0792-1-18-1', '0811.0792-2-18-1'], ['0811.0792-1-13-4', '0811.0792-2-13-4'], ['0811.0792-1-3-1', '0811.0792-2-3-1'], ['0811.0792-1-11-1', '0811.0792-2-11-1'], ['0811.0792-1-11-4', '0811.0792-2-11-4'], ['0811.0792-1-2-5', '0811.0792-2-2-5'], ['0811.0792-1-16-5', '0811.0792-2-16-6']] | [] | [['0811.0792-1-4-8', '0811.0792-2-4-8'], ['0811.0792-1-9-6', '0811.0792-2-9-5'], ['0811.0792-1-12-4', '0811.0792-2-12-4'], ['0811.0792-1-15-1', '0811.0792-2-15-0'], ['0811.0792-1-18-0', '0811.0792-2-18-0'], ['0811.0792-1-18-2', '0811.0792-2-18-2'], ['0811.0792-1-3-0', '0811.0792-2-3-0'], ['0811.0792-1-16-1', '0811.0792-2-16-1'], ['0811.0792-1-16-2', '0811.0792-2-16-2']] | [['0811.0792-1-0-0', '0811.0792-2-0-0']] | ['0811.0792-1-2-9', '0811.0792-2-2-9'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0811.0792 | null | null | null | null | null |
1201.3749 | {'1201.3749-1-0-0': 'Background: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection.', '1201.3749-1-0-1': 'Due to the very high state space dimension of these equations, numerical simulations are computationally expensive.', '1201.3749-1-0-2': 'This is a particular problem for analysis tasks requiring repeated simulations for different parameter values.', '1201.3749-1-0-3': 'Such tasks are computationally expensive to the point of infeasibility with the chemical master equation.', '1201.3749-1-1-0': 'Results: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation.', '1201.3749-1-1-1': 'These surrogate models can be used in various parametric analysis task such as parameter identificiation or sensitivity analysis.', '1201.3749-1-1-2': 'As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations.', '1201.3749-1-2-0': 'Conclusions: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.', '1201.3749-1-3-0': 'publ2', '1201.3749-1-4-0': '# Background', '1201.3749-1-5-0': 'The chemical master equation (CME) is the most basic mathematical description of stochastic biomolecular reaction networks [CITATION].', '1201.3749-1-5-1': 'The CME is a generally infinite-dimensional linear differential equation.', '1201.3749-1-5-2': "It characterizes the temporal development of the probabilities that the network is in any of its possible configurations, where the different configurations are characterized by the molecular copy numbers of the network's chemical species.", '1201.3749-1-6-0': 'Due to its infinite dimension, the CME is usually not directly solvable, not even with numerical methods.', '1201.3749-1-6-1': 'A recent breakthrough in the numerical treatment of the CME was the establishment of the finite state projection (FSP) method by Munsky and Khammash [CITATION].', '1201.3749-1-6-2': "They showed that it is possible to compute a good approximation to the real solution by projecting the CME to a suitable finite subdomain of the network's state space, and solving the resulting finite-dimensional linear differential equation on that domain.", '1201.3749-1-6-3': 'Nevertheless, the FSP approach still yields very high-dimensional models which are computationally expensive to simulate, even for small biochemical networks.', '1201.3749-1-6-4': 'The computational complexity is particularly problematic for common model analysis tasks in systems biology which require the repeated simulation of the model using different parameter values, for example parameter estimation or model sensitivity analysis.', '1201.3749-1-6-5': 'Thereby, while a single or a few evaluations of a CME model with the FSP approach may still be computationally feasible, the necessity of many repeated simulations will quickly render higher-level analysis tasks infeasible.', '1201.3749-1-7-0': 'Mathematical methods that approximate the behaviour of a high-dimensional original model through a low-dimensional reduced model are a common way to deal with complex models.', '1201.3749-1-7-1': 'Especially for linear differential equations, model order reduction is a well established field and several methods to compute reduced order models are available [CITATION].', '1201.3749-1-7-2': 'Note that the step of generating a reduced model is usually computationally more expensive than a single or even a few simulations of the original high-dimensional model.', '1201.3749-1-7-3': 'But the simulation of the resulting reduced models is frequently orders of magnitude faster than the solution of the original model.', '1201.3749-1-7-4': 'So, model reduction is worth the effort if many repeated simulations are to be expected.', '1201.3749-1-7-5': 'Unfortunately, for analysis tasks which require the repeated model simulation with different parameters, classical model reduction methods are not helpful.', '1201.3749-1-7-6': 'With these methods, the reduced model depends on specific parameter values in the original model, and the reduction needs to be redone for different parameter values.', '1201.3749-1-7-7': 'Thus, for the mentioned analysis tasks, the model reduction process would have to be repeated for each new parameter value, and no gain in computational efficiency would typically be possible.', '1201.3749-1-7-8': 'While classical model reduction techniques have been applied to the CME in the past [CITATION], they are not so suitable for parametric analysis tasks.', '1201.3749-1-8-0': 'Fortunately, model reduction methods where parameters from the original model are retained as adjustable parameters also in the reduced model are now being developed.', '1201.3749-1-8-1': 'These methods allow to compute a reduced model which uses the same parameters as the original model, and where the reduced model can directly be simulated with any choice of parameter values [CITATION].', '1201.3749-1-9-0': 'The purpose of this paper is to introduce the application of these parametric model reduction methods to finite-state approximations of the chemical master equation, and to show possible usage scenarios of such an approach.', '1201.3749-1-9-1': 'The structure is as follows.', '1201.3749-1-9-2': 'In the following section, we introduce some background and notation concerning the modelling of chemical reaction networks and parametric model order reduction.', '1201.3749-1-9-3': 'We also show how the parametric model order reduction methods can in fact be applied to the CME.', '1201.3749-1-9-4': 'Afterwards, we apply the reduction technique on two reaction network models and corresponding parametric analysis tasks.', '1201.3749-1-10-0': '# Theoretical basics and methods', '1201.3749-1-11-0': 'We start with some preparatory background on the chemical master equation (CME) and parametric model order reduction.', '1201.3749-1-11-1': 'This serves in particular to fix the notation used throughout the remainder of the article.', '1201.3749-1-11-2': 'Then the application of parametric model order reduction to the CME is introduced.', '1201.3749-1-12-0': '## The chemical master equation', '1201.3749-1-13-0': 'The structure of a biochemical reaction network is characterized completely by the list of involved species, denoted as [MATH], and the list of reactions, denoted as [EQUATION] where [MATH] is the number of reactions in the network, and the factors [MATH] and [MATH] are the stoichiometric coefficients of the reactant and product species, respectively [CITATION].', '1201.3749-1-13-1': 'The net change in the amount of species [MATH] occuring through reaction [MATH] is given by [EQUATION]', '1201.3749-1-13-2': 'Reversible reactions can always be written in the form [REF] by splitting the forward and reverse path into two separate irreversible reactions.', '1201.3749-1-14-0': 'For a stochastic network model, the variables of interest are the probabilities that the network is in any of the possible states which are characterized by the molecular copy numbers of the individual species [MATH].', '1201.3749-1-14-1': 'We denote the molecular copy number of [MATH] by [MATH].', '1201.3749-1-14-2': 'Then, the state variables of the stochastic model are given by the real numbers [EQUATION] for [MATH], [MATH].', '1201.3749-1-14-3': 'As a short-hand notation for [REF], we write [MATH], with [MATH].', '1201.3749-1-15-0': 'The transitions from one state to another are determined by chemical reactions according to [REF].', '1201.3749-1-15-1': 'The changes in the molecule numbers are described by the stoichiometric reaction vectors [EQUATION]', '1201.3749-1-15-2': 'To avoid needlessly complicated cases, we assume [MATH] for [MATH].', '1201.3749-1-16-0': 'The probabilities of the network being in any of the possible states [MATH] evolve over time, and their evolution is governed by the chemical master equation (CME) as derived by [CITATION].', '1201.3749-1-16-1': 'From a given molecular state [MATH], one can compute the propensity [MATH] that reaction [MATH] takes place according to the law of mass action as [EQUATION] where [MATH] is the vector of reaction rate constants, which are model parameters depending on the physical properties of the molecules involved in the reactions.', '1201.3749-1-16-2': 'The propensities are related to the probability that reaction [MATH] will occur in a short time interval of length [MATH] when the system is in state [MATH]: [EQUATION]', '1201.3749-1-16-3': 'Taking the possible transitions and the corresponding reaction propensities together yields the chemical master equation (CME), a linear differential equation where the variables are the probabilities that the system is in each of the possible molecular states [MATH]: [EQUATION] for [MATH].', '1201.3749-1-16-4': 'The CME [REF] is subject to an initial condition [MATH] for [MATH].', '1201.3749-1-17-0': 'Despite being linear, the CME is hard to solve numerically.', '1201.3749-1-17-1': 'This is due to the problem that the state space is for most systems infinite-dimensional, since all possible states [MATH] of the reaction network [REF] must be considered.', '1201.3749-1-17-2': 'Instead of directly solving the CME [REF], a number of alternative approaches to study the stochastic dynamics of biochemical reaction networks have been suggested.', '1201.3749-1-17-3': 'The most common approach is to generate a simulated realization of the stochastic process described by the reaction network [REF], using for example the Gillespie algorithm [CITATION].', '1201.3749-1-17-4': 'In this approach, the probabilities [MATH] for the possible system states are obtained from many simulated realizations.', '1201.3749-1-17-5': 'However, since this requires a large number of realizations, it is computationally expensive.', '1201.3749-1-18-0': 'As a more direct approach, Munsky and Khammash [CITATION] have proposed the finite state projection (FSP), where the CME is solved on a finite subset of the state space.', '1201.3749-1-18-1': 'Here, this subset is denoted by [MATH], and is defined as [EQUATION] where the [MATH] are the system states for which the probabilities are computed in the projected model.', '1201.3749-1-18-2': 'The underlying assumption is that the probabilities for other states will be very low on the time scale of interest-otherwise the FSP may not yield good approximations to the solution of the CME.', '1201.3749-1-18-3': 'In particular we assume the time interval of interest to be given by [MATH] for final time [MATH].', '1201.3749-1-18-4': 'The probabilities for the states [MATH] in [MATH] are written in the vector [MATH] approximating [MATH] at the finite number of states [MATH]: [EQUATION]', '1201.3749-1-18-5': 'The equation to be solved in the FSP is [EQUATION] where [MATH] is the matrix of state transition propensities, and [MATH] is a vector of initial probabilities for the states in [MATH].', '1201.3749-1-18-6': 'The elements of the matrix [MATH] are computed as [EQUATION]', '1201.3749-1-18-7': 'We will frequently omit the parameter dependence of the solution (and other parametric quantities).', '1201.3749-1-18-8': 'Hence the solution [MATH] (as abbreviation of [MATH]) of [REF] is an approximation to the solution [MATH] of the orginal CME on the domain [MATH].', '1201.3749-1-18-9': 'Munsky and Khammash [CITATION] have also derived an upper bound on the error between the solution [MATH] computed via the FSP, and the solution of the original CME [MATH] on [MATH].', '1201.3749-1-19-0': 'Here, we consider in addition an output vector [MATH] defined by [EQUATION] with [MATH].', '1201.3749-1-19-1': 'Examples for relevant outputs are the probability that the molecular copy numbers are in a certain domain [MATH], which is achieved by the output matrix [MATH] defined by [MATH] if [MATH], otherwise [MATH], or the expected molecular copy numbers, given by [EQUATION] i.e. [MATH].', '1201.3749-1-20-0': 'The basic motivation for the model reduction presented here is that we are interested in parametric analysis of the model, where the model [REF] has to be solved many times with different values for the parameters [MATH].', '1201.3749-1-20-1': 'Due to the typical high dimensions of the matrix [MATH], already a single simulation is computationally expensive, and analysis tasks requiring many repeated simulations are often computationally infeasible.', '1201.3749-1-20-2': 'Thus, the primary goal is to derive a reduced model which is rapidly solvable and provides an approximation [MATH] to the output [MATH], potentially without any consideration of the original state vector [MATH].', '1201.3749-1-21-0': '## Order reduction of parametric models', '1201.3749-1-22-0': 'Model order reduction of parametric problems is a very active research field in systems theory, engineering and applied mathematics.', '1201.3749-1-22-1': 'We refer to [CITATION] and references therein for more information on the topic.', '1201.3749-1-23-0': 'Here, we apply the reduction technique for parametric problems presented in [CITATION] adopted to our notation.', '1201.3749-1-23-1': 'It is based on two biorthogonal global projection matrices [MATH] with [MATH] and [MATH], where [MATH] is the dimension of the reduced model.', '1201.3749-1-23-2': 'The matrix [MATH] is assumed to span a space that approximates the system state variation for all parameters and times.', '1201.3749-1-23-3': 'The construction of such matrices will be detailed in the next subsection.', '1201.3749-1-24-0': 'The gain of computational efficiency in repeated simulations comes from a separation of the simulation task into a computationally expensive "offline" phase and a computationally cheap "online" phase.', '1201.3749-1-24-1': 'In the offline phase, suitable projection matrices [MATH] and [MATH] are computed without fixing specific parameter values.', '1201.3749-1-24-2': 'With the projection matrices, a reduced model with the same free parameters as the original model is computed.', '1201.3749-1-24-3': 'In the online phase, the reduced model is simulated with the actually chosen parameter values, which is typically several orders of magnitude faster than the simulation of the original model.', '1201.3749-1-24-4': 'For analysis tasks with repeated simulations, only the online phase has to be repeated for different choices of the parameter values, yielding an overall gain in computational efficiency.', '1201.3749-1-25-0': '### Decomposition in parametric and non-parametric part', '1201.3749-1-26-0': 'The reduction technique assumes a separable parameter dependence of the full system matrices and the initial condition.', '1201.3749-1-26-1': 'This means, we assume that there exist a suitable small constant [MATH], parameter independent components [MATH] and parameter dependent coefficient functions [MATH] for [MATH] such that [EQUATION] and similarly for the system matrix [MATH] and initial condition [MATH].', '1201.3749-1-26-2': 'We assume that [MATH] stems from some domain [MATH] of admissible parameters.', '1201.3749-1-26-3': 'In the next step, the reduced component matrices and initial conditions are determined by [EQUATION] for [MATH].', '1201.3749-1-26-4': 'The resulting quantities are [MATH]-dimensional vectors or matrices and independent of the high dimension [MATH].', '1201.3749-1-26-5': 'The basis computation and the computation of these reduced system components is performed once and parameter-independently in the offline-phase.', '1201.3749-1-26-6': 'Then, in the online-phase, for any new parameter [MATH] the reduced system matrices and the initial condition are assembled by [EQUATION] and similarly for [MATH] and [MATH].', '1201.3749-1-26-7': 'The low dimensional reduced system that remains to be solved is [EQUATION]', '1201.3749-1-26-8': 'Also, an approximate state can be reconstructed at any desired time by [MATH].', '1201.3749-1-26-9': 'Also the difference between the approximated output [MATH] and the output [MATH] of the original model can be bounded by so called error estimators.', '1201.3749-1-26-10': 'Suitable a-posteriori error bounds for the reduced systems as considered here are given in [CITATION].', '1201.3749-1-27-0': '### Basis generation', '1201.3749-1-28-0': 'Different methods for computation of projection bases [MATH] and [MATH] exist.', '1201.3749-1-28-1': 'In systems theory, methods like balanced truncation, Hankel-norm approximation or moment matching are applied, that approximate the input-output behaviour of a linear time-invariant system [CITATION].', '1201.3749-1-28-2': 'The resulting reduced models can be applied for varying imput signals.', '1201.3749-1-28-3': 'Extensions to parametric problems exist, e.g. [CITATION].', '1201.3749-1-28-4': 'As we do not have varying inputs in the problem studied here, we consider snapshot-based approaches to be more suitable.', '1201.3749-1-28-5': 'This means, the projection bases are constructed by solution snapshots, i.e. special solutions computed for selected parameter values.', '1201.3749-1-29-0': 'The generation of projection matrices [MATH] and [MATH] must be done in such a way, that they are globally well approximating the system states over the parameter and time domain.', '1201.3749-1-29-1': 'A possible way to achieve this is the POD-Greedy algorithm, which has been introduced in [CITATION] and is meanwhile a standard procedure in reduced basis methods [CITATION].', '1201.3749-1-29-2': 'Additionally, theoretical foundation can be given by convergence rate statements [CITATION].', '1201.3749-1-29-3': 'The algorithm makes use of a repeated proper orthogonal decomposition (POD) of trajectories [MATH], which can be defined for our purposes as [EQUATION]', '1201.3749-1-29-4': 'Intuitively, one dominant mode that minimizes the squared mean projection error of the given trajectory is computed.', '1201.3749-1-29-5': 'We further require a finite subset of parameters [MATH], that are used in the basis generation process.', '1201.3749-1-29-6': 'As error indicator [MATH] we use the projection error of the full system trajectory on the reduced space spanned by the orthonormal columns of [MATH], i.e. [EQUATION]', '1201.3749-1-29-7': 'The POD-Greedy procedure then starts with an arbitrary orthonormal initial basis [MATH] and performs an incremental basis extension.', '1201.3749-1-29-8': 'The algorithm repeatedly identifies the currently worst resolved parameter, orthogonalizes the corresponding full trajectory with the current reduced space, does a POD on the error trajectory, and inserts the dominant mode into the basis.', '1201.3749-1-30-0': 'function [MATH] POD-Greedy[MATH] while [MATH]', '1201.3749-1-31-0': 'end while', '1201.3749-1-32-0': 'Note that the algorithm is implemented such that the simulation of the full model, yielding the variable [MATH] in [REF], is only done once for each [MATH] in the training set [MATH].', '1201.3749-1-33-0': 'For concluding the basis generation, we set [MATH] satisfying the biorthogonality condition [MATH] as [MATH] has orthonormal columns by construction.', '1201.3749-1-33-1': 'In practice the time-integrals in [REF] are realized by a finite sampling of the time interval.', '1201.3749-1-33-2': 'Extensions of the above algorithm exist, e.g. allowing more than one mode per extension step, performing adaptive parameter and time-interval partitioning, or enabling training-set adaptation [CITATION].', '1201.3749-1-34-0': '## Reduced models of the parametrized chemical master equation', '1201.3749-1-35-0': 'In this section, we describe how to apply the reduction method for parametrized models presented in the previous section to FSP models for the chemical master equation.', '1201.3749-1-36-0': 'As discussed in the previous section, the first step in the proposed reduction method is a decomposition of the [MATH]-dimensional system matrix [MATH] as in [REF].', '1201.3749-1-36-1': 'Such a decomposition is possible for the case of mass action reaction propensities, as defined in [REF], or generalized mass action, as recently suggested for the chemical master equation [CITATION].', '1201.3749-1-36-2': 'In this case, the length of the parameter vector [MATH] is equal to the number of reactions [MATH], and we decompose [MATH] into [MATH] terms as [EQUATION]', '1201.3749-1-36-3': 'Hence, concerning the notation given before, we have [MATH] components [MATH] and coefficient functions [MATH].', '1201.3749-1-36-4': 'Each matrix [MATH] in this decomposition comes from just the transition propensities corresponding to reaction [MATH], and is defined by [EQUATION]', '1201.3749-1-36-5': 'More generally, such a decomposition is also possible if reaction rate propensities can be decomposed into the product of two terms, with the first term depending on parameters only, and the second term on molecule numbers only.', '1201.3749-1-36-6': 'This case is for example encountered when the temperature-dependance of the reaction rate constant is relevant, and the temperature [MATH] is a variable parameter in the Arrhenius equation [MATH].', '1201.3749-1-36-7': 'Since the output matrix [MATH] and the initial condition [MATH] are usually not depending on parameters in this framework, a decomposition of [MATH] and [MATH] is not considered.', '1201.3749-1-37-0': 'The situation is more difficult for reaction propensities involving for example rational terms with parameters in the denominator.', '1201.3749-1-37-1': 'The denominator parameters can not be included in the reduced order model by the decomposition outlined in [REF] and [REF].', '1201.3749-1-37-2': 'If variations in these parameters are however not relevant to the planned analysis, then they can be set to their nominal value, and the decomposition can directly be done as described above.', '1201.3749-1-37-3': 'Alternatively, approximation steps can be performed, such as Taylor series expansion or empirical interpolation [CITATION], that generate an approximating parameter-separable expansion.', '1201.3749-1-38-0': '# Results for exemplary applications in genetic switching and oscillations', '1201.3749-1-39-0': 'In this section, we present the study of two example networks with the proposed model reduction method.', '1201.3749-1-39-1': 'With these examples, the applicability of the reduced modeling approach especially for analysis tasks requiring repeated simulations with different parameter values is illustrated.', '1201.3749-1-39-2': 'The first network is a bistable genetic toggle switch, where cells may switch randomly between two states, based on the model in [CITATION].', '1201.3749-1-39-3': 'For this network, the problem of parameter estimation with a reduced model is studied.', '1201.3749-1-39-4': 'The second network is a second-order genetic oscillator, based on [CITATION], where we perform a sensitivity analysis over a wide parameter range.', '1201.3749-1-40-0': '## Parameter estimation in a genetic toggle switch model', '1201.3749-1-41-0': '### Network description', '1201.3749-1-42-0': 'The genetic toggle switch considered here is an ovarian follicle switch model from [CITATION].', '1201.3749-1-42-1': 'It is a system of two genes which activate each other.', '1201.3749-1-42-2': 'The switch is modelled as a reaction network with two species [MATH], [MATH], representing the gene products.', '1201.3749-1-42-3': 'The network reactions are specified in Table [REF].', '1201.3749-1-43-0': 'In [CITATION], this network was shown to describe a bistable switch with two probability peaks, one close to [MATH] and the other close to [MATH].', '1201.3749-1-44-0': 'In the study [CITATION], only the lower probability peak was of interest.', '1201.3749-1-44-1': 'Here, we are interested in the transition of the system from [MATH] to [MATH].', '1201.3749-1-44-2': 'Therefore, the system is truncated to a rectangle [MATH] such that [MATH], yielding a good approximation in the finite state projection to the infinite-dimensional chemical master equation.', '1201.3749-1-45-0': 'The next step is to apply the decomposition of the matrix [MATH] as described in the methods section.', '1201.3749-1-45-1': 'Note that [MATH] for the switch network contains rational terms with the parameters [MATH] and [MATH].', '1201.3749-1-45-2': 'Considering these two parameters as fixed quantities, the truncated CME for the follicle switch can be written as [EQUATION] where [MATH], [MATH] are of dimension [MATH].', '1201.3749-1-46-0': 'As initial condition we choose a probability distributed over some lower states [EQUATION]', '1201.3749-1-46-1': 'For the parametric model reduction, we consider only variations in the parameters [MATH] and [MATH].', '1201.3749-1-46-2': 'These influence both the steady state level of gene activity in the on-state as well as the switching kinetics and are thus of high biological significance in the model.', '1201.3749-1-46-3': 'Hence we set [MATH] as the parametric domain [MATH].', '1201.3749-1-46-4': 'As final time we choose [MATH] which corresponds to a time range of approximately 19 years, i.e. about three times the half-life time of the off-state estimated in [CITATION].', '1201.3749-1-47-0': 'Some state plots from the simulation of the full model are shown in Figure [REF].', '1201.3749-1-47-1': 'These snapshots clearly show the transition of the switch from the off-state with low values for [MATH] and [MATH] to the on state with high values.', '1201.3749-1-47-2': 'The parameter influence is mainly reflected in the speed of the transition: for the parameter vector [MATH] in the lower row, most of the probability is already arranged around the on-state at the end of the simulation time.', '1201.3749-1-47-3': 'In contrast, for the parameter vector [MATH] in the upper row, a significant portion of the probability is still located around the off-state at this time point.', '1201.3749-1-47-4': 'Also, the transition paths are different: in the first case, the values for [MATH] are lower than the values for [MATH] during the transition, while in the second case, this relation is reversed.', '1201.3749-1-48-0': 'As typical simulation time for a single trajectory of the full system, we obtain 98.2167 seconds on a IBM Lenovo 2.53 GHz Dual Core Laptop.', '1201.3749-1-49-0': '### Basis generation', '1201.3749-1-50-0': 'We generated a reduced basis with the POD-Greedy algorithm, where the training set was chosen as the vertices of a mesh with [MATH] logarithmically equidistant parameter values over the parameter domain [MATH].', '1201.3749-1-50-1': 'We set [MATH] as target accuracy.', '1201.3749-1-50-2': 'We use the projection error as error measure, hence precompute the 81 trajectories for construction of the reduced basis.', '1201.3749-1-50-3': 'As initial basis we set [MATH] and [MATH] using the parameter independent initial condition.', '1201.3749-1-51-0': 'The POD-Greedy algorithm produces a basis of 33 vectors and the overall computation of the reduced basis takes 7.9 hours, the dominating computation time being spent in the error evaluations and POD computations.', '1201.3749-1-51-1': 'Some of the resulting orthonormal basis vectors are illustrated in Figure [REF].', '1201.3749-1-51-2': 'The error decay curve and the selected parameters in the parameter domain are illustrated in Figure [REF].', '1201.3749-1-51-3': 'We nicely observe an exponential error decay, which indicates a parametric smoothness of the solution manifold.', '1201.3749-1-51-4': 'The selected parameters seem to be located at the boundary of the parameter domain, indicating that the model behaviour in between can well be interpolated from the model behaviours along the boundary of the parameter domain.', '1201.3749-1-52-0': 'The final reduced model of dimension 33 can then be simulated in 0.135 seconds.', '1201.3749-1-52-1': 'This is an overall computational speedup factor of more than 700.', '1201.3749-1-53-0': '### Parameter estimation', '1201.3749-1-54-0': 'We exemplify a possible application of the reduced order model in parameter estimation, where we assume that an output [MATH] as the expected values [MATH] is available from population-averaged measurements.', '1201.3749-1-54-1': 'The task is to estimate the parameter values [MATH] and [MATH] from such a measurement.', '1201.3749-1-55-0': 'The reference parameter is [MATH], and, for the purpose of this example, the measured output is produced by simulating the original model with the reference parameter values, [MATH].', '1201.3749-1-56-0': "We want to recover the values of the parameters [MATH] and [MATH] based on fitting the reduced parametric model's output [MATH] to the measured output [MATH].", '1201.3749-1-56-1': 'As is commonly done in parameter estimation, we formulate a least squares optimization problem as [EQUATION]', '1201.3749-1-56-2': 'In such an optimization problem, typically many forward simulations are required for adjusting [MATH] to the measurement.', '1201.3749-1-56-3': 'This is a particular beneficial scenario for reduced order models, as these simulations can be computed rapidly.', '1201.3749-1-57-0': 'In order to gain a deeper insight into the optimization problem [REF], we plot the values of the error functional [MATH] over the parameter domain (Figure [REF]).', '1201.3749-1-57-1': 'Using the reduced model, this computation is done in less than one minute.', '1201.3749-1-57-2': 'The peak at the reference parameter [MATH] is visible.', '1201.3749-1-57-3': 'We also realize that overall, the optimization problem is very difficult as we identify a narrow area of parameters, which seem to produce a similar output as the reference parameter [MATH].', '1201.3749-1-57-4': 'This is a first insight obtained from the parametric analysis: there seems to exist a functional dependence between the parameters [MATH] and [MATH] such that the model yields similar outputs [MATH].', '1201.3749-1-58-0': "Despite this ill-posed optimization problem, the optimization with a standard active set algorithm by MATLAB's command fmincon yields an acceptable estimation of the parameter [MATH] which is within [MATH] relative error to the reference value [MATH].", '1201.3749-1-58-1': 'Clearly, in an application of this approach to real measured data, the ill-posedness of the optimization problem will likely lead to much higher errors, so this analysis already suggests that in real applications the considered output may not be so well suited for parameter identification.', '1201.3749-1-59-0': 'The corresponding values of the optimization target are [MATH] and [MATH].', '1201.3749-1-59-1': 'This indicates that the estimated value in the reduced model indeed gives a better fit to the full model than using the reference value in the reduced model, probably due to slight approximation errors of the reduced model for these parameter values.', '1201.3749-1-59-2': 'The computation took 383 evaluations of the optimization target, which would be a significant computational effort when using a non-reduced model.', '1201.3749-1-59-3': 'Assuming a functional dependence of [MATH] and [MATH] we now consider the 1-dimensional optimization problem along the line [MATH].', '1201.3749-1-59-4': 'We would like to recover [MATH] from the optimization problem.', '1201.3749-1-59-5': 'Indeed, this restricted optimization problem is much better conditioned and we obtain [MATH], [MATH].', '1201.3749-1-59-6': 'This accounts to a relative error in the [MATH] value of 0.0011433%, hence excellent recovery.', '1201.3749-1-60-0': 'The right plot in Fig. [REF] illustrates another application of reduced parametric models: We incorporated the model in an interactive graphical user interface in RBmatlab, a matlab package for model order reduction, available for download at www.morepas.org.', '1201.3749-1-60-1': 'This allows interactive parameter variations and instantaneous simulation response.', '1201.3749-1-61-0': '## Sensitivity analysis in a stochastic oscillator', '1201.3749-1-62-0': '### Network description', '1201.3749-1-63-0': 'The second case study is built on a genetic oscillator model showing stochastic resonance, which was presented in [CITATION].', '1201.3749-1-63-1': 'The oscillator is based on a negative feedback loop between two genes with one gene having positive autoregulation.', '1201.3749-1-63-2': 'The oscillator is modelled as a reaction network with two species [MATH], [MATH], representing the gene products.', '1201.3749-1-63-3': 'The network reactions are specified in Table [REF].', '1201.3749-1-63-4': 'In the original model in [CITATION], the dynamics were described as stochastic differential equation for the amounts of [MATH] and [MATH], coming from a Langevin approximation to the stochastic dynamics [CITATION].', '1201.3749-1-63-5': 'For the framework used in this paper, the dynamics have to be described directly by the underlying CME.', '1201.3749-1-63-6': 'To achieve this, we introduce the parameter [MATH] which maps the dimensionless state variables from [CITATION] to actual molecule numbers as required for the CME.', '1201.3749-1-63-7': "Thus, [MATH] is also a measure for the network's noise level: the higher [MATH], the larger the molecule number that is considered, and the smaller the noise level will be.", '1201.3749-1-64-0': 'The network model in Table [REF] shows oscillations only in a stochastic description.', '1201.3749-1-64-1': 'The deterministic model has a unique asymptotically stable equilibrium point, but in a stochastic model, fluctuations may push the molecular numbers beyond a certain threshold, inducing a dynamical response along a slow manifold, which corresponds to one oscillatory period [CITATION].', '1201.3749-1-64-2': 'Depending on the noise level, such responses will be initiated more or less often, corresponding to a more or less regular oscillatory pattern.', '1201.3749-1-65-0': 'The system is truncated to the rectangle [MATH], which contains the relevant system states for the parameter ranges of interest.', '1201.3749-1-66-0': 'Similarly as in the switch example, the reaction propensity expressions contain rational terms in the parameters [MATH], [MATH], and [MATH].', '1201.3749-1-66-1': 'These three cannot be decomposed directly, so we do the decomposition described in the methods section for the other five parameters only.', '1201.3749-1-66-2': 'With this decomposition, the truncated CME for the genetic oscillator can be written as [EQUATION] where [MATH], [MATH] are of dimension [MATH].', '1201.3749-1-66-3': 'The initial condition for [REF] is chosen as a uniform distribution over the rectangle [MATH]: [EQUATION]', '1201.3749-1-66-4': 'The time scale of interest for the model in [REF] is for [MATH].', '1201.3749-1-66-5': 'At the end of the interval, the probability distribution seems to approach a steady state, as checked from individual simulations.', '1201.3749-1-67-0': 'Some state plots are given in Figure [REF].', '1201.3749-1-67-1': 'One observes a significant effect of the parameter [MATH] on the amplitude of the oscillations.', '1201.3749-1-67-2': 'The simulation time for the detailed model was in average 7.3 minutes on a Dell desktop computer with 3.2 GHz dual-core Intel 4 processor and 1 GB RAM, without including the computation time for the construction of the state transition matrix [MATH].', '1201.3749-1-68-0': '### Basis generation', '1201.3749-1-69-0': 'For the basis generation, the parameter [MATH] was assumed to vary within the interval [MATH].', '1201.3749-1-69-1': 'A reduced basis with the POD-Greedy algorithm was computed from a training set of 30 logarithmically equidistant parameters over the parameter domain (Figure [REF]).', '1201.3749-1-69-2': 'As in the switch example, the target accuracy was chosen as [MATH], and the initial basis was chosen from the initial condition [MATH].', '1201.3749-1-70-0': 'The POD-Greedy algorithm produces a basis of 109 vectors, with an overall computation time of 16.5 hours on the hardware as in the previous subsection.', '1201.3749-1-70-1': 'The first 20 basis vectors are shown in Figure [REF].', '1201.3749-1-70-2': 'It is apparent that several of the basis vectors are directly included in order to reproduce the different amplitudes of oscillations that will occur under variations of the parameter [MATH].', '1201.3749-1-70-3': 'The error decay curve is shown in Figure [REF], displaying an exponential error decay as also observed for the switch example.', '1201.3749-1-71-0': 'With the reduced basis [MATH], we can construct a reduced parametric model for the CME of the oscillator as [EQUATION] with [MATH] and [MATH].', '1201.3749-1-71-1': 'Note that since only [MATH] has been varied in the reduction process, the other parameters are no longer present as parameters in the reduced model, but just take their nominal values.', '1201.3749-1-71-2': 'While the same basis [MATH] could be used to construct another reduced model where all parameters are retained, it is unlikely that this other model will be a good approximation of the original one for varying values of the other parameters.', '1201.3749-1-72-0': '### Sensitivity analysis of the oscillation amplitude', '1201.3749-1-73-0': 'As an application of the reduced order parametric model obtained in the previous section, we study the variations of oscillatory amplitude over a parameter range.', '1201.3749-1-73-1': 'Specifically, we consider 200 equally spaced values for the parameter [MATH] in the interval [MATH] and compute the probability that the amount of [MATH] is larger than 100: [EQUATION] with [MATH] the final time of the simulation.', '1201.3749-1-73-2': 'The results are shown in Figure [REF] and show a clear decay of oscillatory amplitude for increasing values of [MATH].', '1201.3749-1-73-3': 'Due to the significant time savings from the reduced model, this sensitivity curve can be computed with a high resolution.', '1201.3749-1-74-0': 'To evaluate the quality of the reduced model, we also computed the high-amplitude probability [REF] using the original model [REF] at two points within the considered interval for the parameter [MATH].', '1201.3749-1-74-1': 'As shown in Figure [REF], the results from the original model are in perfect agreement with the predictions from the reduced model at these points.', '1201.3749-1-74-2': 'Since the points at which the original model was evaluated in this experiment were not part of the training set (shown as triangles on the parameter axis in Figure [REF]), this shows that it is in fact possible to extrapolate the reduced model to parameter values that were not used to construct the basis.', '1201.3749-1-75-0': '# Conclusions', '1201.3749-1-76-0': 'In this paper, we have introduced the application of parametric model reduction methods to finite-state approximations of the chemical master equation.', '1201.3749-1-76-1': 'We have also presented two case studies where these methods are applied to CME models of different networks in order to make parametric analysis tasks computationally efficient.', '1201.3749-1-76-2': 'By this, it has become clear that parametric model reduction methods are a very useful tool for the analysis of stochastic biochemical reaction network described by the CME.', '1201.3749-1-77-0': 'Especially analysis tasks where many repeated simulations of a network with different parameter values are required can profit significantly from parametric model reduction.', '1201.3749-1-77-1': 'This includes for example sensitivity analysis or parameter optimization tasks such as identification.', '1201.3749-1-77-2': "Moreover, the significant speedup of the simulation for the reduced model allows an interactive exploration of the network's dynamics within the parameter space within a suitable graphical user interface.", '1201.3749-1-78-0': 'This contribution is just a first step in the application of parametric model reduction methods to the CME.', '1201.3749-1-78-1': 'One particularly important aspect that we have not discussed here is the computation of error estimates for certifying that the simulation output of the reduced model is within some tolerance of the corresponding simulation output of the original model.', '1201.3749-1-78-2': 'To maintain computational efficiency, the error estimation should be done without actually simulating the original model.', '1201.3749-1-78-3': 'Error estimation methods have been developed for parametric model reduction of generic models [CITATION], but tighter estimates could likely be obtained by taking into account the special structure of the CME models.', '1201.3749-1-79-0': '# Authors contributions', '1201.3749-1-80-0': 'SW and BH conceived of the study, performed the study, and wrote the manuscript.', '1201.3749-1-80-1': 'Both authors read and approved the final manuscript.'} | {'1201.3749-2-0-0': 'Background: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection.', '1201.3749-2-0-1': 'Due to the very high state space dimension of these equations, numerical simulations are computationally expensive.', '1201.3749-2-0-2': 'This is a particular problem for analysis tasks requiring repeated simulations for different parameter values.', '1201.3749-2-0-3': 'Such tasks are computationally expensive to the point of infeasibility with the chemical master equation.', '1201.3749-2-1-0': 'Results: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation.', '1201.3749-2-1-1': 'These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis.', '1201.3749-2-1-2': 'As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations.', '1201.3749-2-2-0': 'Conclusions: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.', '1201.3749-2-3-0': 'publ2', '1201.3749-2-4-0': '# Background', '1201.3749-2-5-0': 'The chemical master equation (CME) is the most basic mathematical description of stochastic biomolecular reaction networks [CITATION].', '1201.3749-2-5-1': 'The CME is a generally infinite-dimensional linear differential equation.', '1201.3749-2-5-2': "It characterizes the temporal development of the probabilities that the network is in any of its possible configurations, where the different configurations are characterized by the molecular copy numbers of the network's chemical species.", '1201.3749-2-6-0': 'Due to its infinite dimension, the CME is usually not directly solvable, not even with numerical methods.', '1201.3749-2-6-1': 'A recent breakthrough in the numerical treatment of the CME was the establishment of the finite state projection (FSP) method by Munsky and Khammash [CITATION].', '1201.3749-2-6-2': "They showed that it is possible to compute a good approximation to the real solution by projecting the CME to a suitable finite subdomain of the network's state space, and solving the resulting finite-dimensional linear differential equation on that domain.", '1201.3749-2-6-3': 'Nevertheless, the FSP approach still yields very high-dimensional models which are computationally expensive to simulate, even for small biochemical networks.', '1201.3749-2-6-4': 'The efficient simulation of the CME is an area of active research, and recently other simulation methods have been developed that can also be used for larger networks [CITATION].', '1201.3749-2-7-0': 'Despite this progress, the direct simulation of the CME remains a computational bottleneck for common model analysis tasks in systems biology.', '1201.3749-2-7-1': 'It is especially problematic for tasks which require the repeated simulation of the model using different parameter values, for example identifiability analysis, parameter estimation, or model sensitivity analysis.', '1201.3749-2-7-2': 'Thereby, while a single or a few evaluations of a CME model with the FSP or other approaches may still be computationally feasible, the necessity of many repeated simulations will quickly render higher-level analysis tasks infeasible.', '1201.3749-2-8-0': 'Mathematical methods that approximate the behaviour of a high-dimensional original model through a low-dimensional reduced model are a common way to deal with complex models.', '1201.3749-2-8-1': 'Especially for linear differential equations, model order reduction is a well established field and several methods to compute reduced order models are available [CITATION].', '1201.3749-2-8-2': 'Note that the step of generating a reduced model is usually computationally more expensive than a single or even a few simulations of the original high-dimensional model.', '1201.3749-2-8-3': 'But the simulation of the resulting reduced models is frequently orders of magnitude faster than the solution of the original model.', '1201.3749-2-8-4': 'So, model reduction is worth the effort if many repeated simulations are to be expected.', '1201.3749-2-8-5': 'Unfortunately, for analysis tasks which require the repeated model simulation with different parameters, classical model reduction methods are not helpful.', '1201.3749-2-8-6': 'With these methods, the reduced model depends on specific parameter values in the original model, and the reduction needs to be redone for different parameter values.', '1201.3749-2-8-7': 'Thus, for the mentioned analysis tasks, the model reduction process would have to be repeated for each new parameter value, and no gain in computational efficiency would typically be possible.', '1201.3749-2-8-8': 'While classical model reduction techniques have been applied to the CME in the past [CITATION], they are not so suitable for parametric analysis tasks.', '1201.3749-2-9-0': 'Fortunately, model reduction methods where parameters from the original model are retained as adjustable parameters also in the reduced model are now being developed.', '1201.3749-2-9-1': 'These methods allow to compute a reduced model which uses the same parameters as the original model, and where the reduced model can directly be simulated with any choice of parameter values [CITATION].', '1201.3749-2-10-0': 'The purpose of this paper is to introduce the application of these parametric model reduction methods to finite-state approximations of the chemical master equation, and to show possible usage scenarios of such an approach.', '1201.3749-2-10-1': 'The structure is as follows.', '1201.3749-2-10-2': 'In the following section, we introduce some background and notation concerning the modelling of chemical reaction networks and parametric model order reduction.', '1201.3749-2-10-3': 'We also show how the parametric model order reduction methods can in fact be applied to the CME.', '1201.3749-2-10-4': 'Afterwards, we apply the reduction technique on two reaction network models and corresponding parametric analysis tasks.', '1201.3749-2-11-0': '# Theoretical basics and methods', '1201.3749-2-12-0': 'We start with some preparatory background on the chemical master equation (CME) and parametric model order reduction.', '1201.3749-2-12-1': 'This serves in particular to fix the notation used throughout the remainder of the article.', '1201.3749-2-12-2': 'Then the application of parametric model order reduction to the CME is introduced.', '1201.3749-2-13-0': '## The chemical master equation', '1201.3749-2-14-0': 'The structure of a biochemical reaction network is characterized completely by the list of involved species, denoted as [MATH], and the list of reactions, denoted as [EQUATION] where [MATH] is the number of reactions in the network, and the factors [MATH] and [MATH] are the stoichiometric coefficients of the reactant and product species, respectively [CITATION].', '1201.3749-2-14-1': 'The net change in the amount of species [MATH] occuring through reaction [MATH] is given by [EQUATION]', '1201.3749-2-14-2': 'Reversible reactions can always be written in the form [REF] by splitting the forward and reverse path into two separate irreversible reactions.', '1201.3749-2-15-0': 'For a stochastic network model, the variables of interest are the probabilities that the network is in any of the possible states which are characterized by the molecular copy numbers of the individual species [MATH].', '1201.3749-2-15-1': 'We denote the molecular copy number of [MATH] by [MATH].', '1201.3749-2-15-2': 'Then, the state variables of the stochastic model are given by the real numbers [EQUATION] for [MATH], [MATH].', '1201.3749-2-15-3': 'As a short-hand notation for [REF], we write [MATH], with [MATH].', '1201.3749-2-16-0': 'The transitions from one state to another are determined by chemical reactions according to [REF].', '1201.3749-2-16-1': 'The changes in the molecule numbers are described by the stoichiometric reaction vectors [EQUATION]', '1201.3749-2-16-2': 'To avoid needlessly complicated cases, we assume [MATH] for [MATH].', '1201.3749-2-17-0': 'The probabilities of the network being in any of the possible states [MATH] evolve over time, and their evolution is governed by the chemical master equation (CME) as derived by [CITATION].', '1201.3749-2-17-1': 'From a given molecular state [MATH], one can compute the propensity [MATH] that reaction [MATH] takes place according to the law of mass action as [EQUATION] where [MATH] is the vector of reaction rate constants, which are model parameters depending on the physical properties of the molecules involved in the reactions.', '1201.3749-2-17-2': 'The propensities are related to the probability that reaction [MATH] will occur in a short time interval of length [MATH] when the system is in state [MATH]: [EQUATION]', '1201.3749-2-17-3': 'Taking the possible transitions and the corresponding reaction propensities together yields the chemical master equation (CME), a linear differential equation where the variables are the probabilities that the system is in each of the possible molecular states [MATH]: [EQUATION] for [MATH].', '1201.3749-2-17-4': 'The CME [REF] is subject to an initial condition [MATH] for [MATH].', '1201.3749-2-18-0': 'Despite being linear, the CME is hard to solve numerically.', '1201.3749-2-18-1': 'This is due to the problem that the state space is for most systems infinite-dimensional, since all possible states [MATH] of the reaction network [REF] must in general be considered.', '1201.3749-2-18-2': 'Instead of directly solving the CME [REF], a number of alternative approaches to study the stochastic dynamics of biochemical reaction networks have been suggested.', '1201.3749-2-18-3': 'The most common approach is to generate a simulated realization of the stochastic process described by the reaction network [REF], using for example the Gillespie algorithm [CITATION].', '1201.3749-2-18-4': 'In this approach, the probabilities [MATH] for the possible system states are obtained from many simulated realizations.', '1201.3749-2-18-5': 'However, since this requires a large number of realizations, it is computationally expensive.', '1201.3749-2-19-0': 'As a more direct approach, Munsky and Khammash [CITATION] have proposed the finite state projection (FSP), where the CME is solved on a finite subset of the state space.', '1201.3749-2-19-1': 'Here, this subset is denoted by [MATH], and is defined as [EQUATION] where the [MATH] are the system states for which the probabilities are computed in the projected model.', '1201.3749-2-19-2': 'The underlying assumption is that the probabilities for other states will be very low on the time scale of interest-otherwise the FSP may not yield good approximations to the solution of the CME.', '1201.3749-2-19-3': 'In particular we assume the time interval of interest to be given by [MATH] for final time [MATH].', '1201.3749-2-19-4': 'The probabilities for the states [MATH] in [MATH] are written in the vector [MATH] approximating [MATH] at the finite number of states [MATH]: [EQUATION]', '1201.3749-2-19-5': 'The equation to be solved with the FSP approximation is [EQUATION] where [MATH] is the matrix of state transition propensities, and [MATH] is a vector of initial probabilities for the states in [MATH].', '1201.3749-2-19-6': 'The elements of the matrix [MATH] are computed as [EQUATION]', '1201.3749-2-19-7': 'We will frequently omit the parameter dependence of the solution (and other parametric quantities).', '1201.3749-2-19-8': 'Hence the solution [MATH], as abbreviation of [MATH], of [REF] is an approximation to the solution [MATH] of the orginal CME on the domain [MATH].', '1201.3749-2-19-9': 'Munsky and Khammash [CITATION] have also derived an upper bound on the error between the solution [MATH] computed via the FSP, and the solution of the original CME [MATH] on [MATH].', '1201.3749-2-20-0': 'Here, we consider in addition an output vector [MATH] defined by [EQUATION] with [MATH].', '1201.3749-2-20-1': 'Examples for relevant outputs are the probability that the molecular copy numbers are in a certain domain [MATH], which is achieved by the row vector output matrix [MATH] defined by [MATH] if [MATH], otherwise [MATH], with [MATH], or the expected molecular copy numbers, given by [EQUATION] i.e. [MATH] with [MATH].', '1201.3749-2-21-0': 'The basic motivation for the model reduction presented here is that we are interested in parametric analysis of the model, where the model [REF] has to be solved many times with different values for the parameters [MATH].', '1201.3749-2-21-1': 'Due to the typical high dimensions of the matrix [MATH], already a single simulation is computationally expensive, and analysis tasks requiring many repeated simulations are often computationally infeasible.', '1201.3749-2-21-2': 'Thus, the primary goal is to derive a reduced model which is rapidly solvable and provides an approximation [MATH] to the output [MATH], potentially without any consideration of the original state vector [MATH].', '1201.3749-2-22-0': '## Order reduction of parametric models', '1201.3749-2-23-0': 'Model order reduction of parametric problems is a very active research field in systems theory, engineering and applied mathematics.', '1201.3749-2-23-1': 'We refer to [CITATION] and references therein for more information on the topic.', '1201.3749-2-24-0': 'Here, we apply the reduction technique for parametric problems presented in [CITATION] adopted to our notation.', '1201.3749-2-24-1': 'It is based on two biorthogonal global projection matrices [MATH] with [MATH] and [MATH], where [MATH] is the dimension of the reduced model.', '1201.3749-2-24-2': 'The matrix [MATH] is assumed to span a space that approximates the system state variation for all parameters and times.', '1201.3749-2-24-3': 'The construction of such matrices will be detailed in the next subsection.', '1201.3749-2-25-0': 'The gain of computational efficiency in repeated simulations comes from a separation of the simulation task into a computationally expensive "offline" phase and a computationally cheap "online" phase.', '1201.3749-2-25-1': 'In the offline phase, suitable projection matrices [MATH] and [MATH] are computed without fixing specific parameter values.', '1201.3749-2-25-2': 'With the projection matrices, a reduced model with the same free parameters as the original model is computed.', '1201.3749-2-25-3': 'In the online phase, the reduced model is simulated with the actually chosen parameter values, which is typically several orders of magnitude faster than the simulation of the original model.', '1201.3749-2-25-4': 'For analysis tasks with repeated simulations, only the online phase has to be repeated for different choices of the parameter values, yielding an overall gain in computational efficiency.', '1201.3749-2-26-0': '### Decomposition in parametric and non-parametric part', '1201.3749-2-27-0': 'The reduction technique assumes a separable parameter dependence of the full system matrices and the initial condition.', '1201.3749-2-27-1': 'This means, we assume that there exist a suitable small constant [MATH], parameter independent components [MATH] and parameter dependent scalar coefficient functions [MATH] for [MATH] such that [EQUATION] and similarly for the system matrix [MATH] and initial condition [MATH].', '1201.3749-2-27-2': 'We assume that [MATH] stems from some domain [MATH] of admissible parameters.', '1201.3749-2-27-3': 'In the next step, the reduced component matrices and initial conditions are determined by [EQUATION] for [MATH].', '1201.3749-2-27-4': 'The resulting quantities [MATH], [MATH], and [MATH] are [MATH]-dimensional vectors or matrices and independent of the high dimension [MATH].', '1201.3749-2-27-5': 'The basis computation and the computation of these reduced system components is performed once and parameter-independently in the offline-phase.', '1201.3749-2-27-6': 'Then, in the online-phase, for any new parameter [MATH] the reduced system matrices and the initial condition are assembled by [EQUATION] and similarly for [MATH] and [MATH].', '1201.3749-2-27-7': 'The low dimensional reduced system that remains to be solved is [EQUATION]', '1201.3749-2-27-8': 'From the reduced state [MATH], an approximate state for the full system can be reconstructed at any desired time by [MATH].', '1201.3749-2-27-9': 'Also the difference between the approximated output [MATH] and the output [MATH] of the original model can be bounded by so called error estimators.', '1201.3749-2-27-10': 'A-posteriori error bounds for the reduced systems as considered here are given in [CITATION].', '1201.3749-2-28-0': '### Basis generation', '1201.3749-2-29-0': 'Different methods for the computation of the projection bases [MATH] and [MATH] exist.', '1201.3749-2-29-1': 'In systems theory, methods like balanced truncation, Hankel-norm approximation or moment matching are applied, that approximate the input-output behaviour of a linear time-invariant system [CITATION].', '1201.3749-2-29-2': 'The resulting reduced models can be applied for varying input signals.', '1201.3749-2-29-3': 'Extensions to parametric problems exist, e.g. [CITATION].', '1201.3749-2-29-4': 'As we do not have varying inputs in the problem studied here, we consider snapshot-based approaches to be more suitable.', '1201.3749-2-29-5': 'This means, the projection bases are constructed by solution snapshots, i.e. special solutions computed for selected parameter values.', '1201.3749-2-30-0': 'The generation of projection matrices [MATH] and [MATH] must be done in such a way, that they are globally well approximating the system states over the parameter and time domain.', '1201.3749-2-30-1': 'A possible way to achieve this is the POD-Greedy algorithm, which has been introduced in [CITATION] and is meanwhile a standard procedure in reduced basis methods [CITATION].', '1201.3749-2-30-2': 'The algorithm makes use of a repeated proper orthogonal decomposition (POD) of trajectories [MATH], which for our purposes can be defined as [EQUATION]', '1201.3749-2-30-3': 'Intuitively, [MATH] is a state space vector representing the single dominant mode that minimizes the squared mean projection error.', '1201.3749-2-30-4': 'Computationally, this minimization task is solved by a reformulation as a suitable eigenvalue problem.', '1201.3749-2-30-5': 'Consider the correlation matrix [MATH].', '1201.3749-2-30-6': 'Then, [MATH] is an eigenvector corresponding to the largest eigenvalue [MATH] of [MATH], i.e., [MATH].', '1201.3749-2-30-7': 'For additional theoretical and computational details on POD we refer to [CITATION].', '1201.3749-2-30-8': 'We further require a finite subset of parameters [MATH], that are used in the basis generation process.', '1201.3749-2-30-9': 'As error indicator [MATH] we use the projection error of the full system trajectory on the reduced space spanned by the orthonormal columns of [MATH], i.e. [EQUATION]', '1201.3749-2-30-10': 'The POD-Greedy procedure which is given in the pseudo-code below, starts with an arbitrary orthonormal initial basis [MATH] and performs an incremental basis extension.', '1201.3749-2-30-11': 'The algorithm repeatedly identifies the currently worst resolved parameter (a), orthogonalizes the corresponding full trajectory with the current reduced space (b), computes a POD of the error trajectory (c), and inserts the dominant mode into the basis (d).', '1201.3749-2-31-0': 'function [MATH] POD-Greedy[MATH] while [MATH]', '1201.3749-2-32-0': 'end while', '1201.3749-2-33-0': 'Note that the algorithm is implemented such that the simulation of the full model, yielding [MATH] in [REF], is only performed once for each [MATH] in the training set [MATH].', '1201.3749-2-34-0': 'For concluding the basis generation, we set [MATH].', '1201.3749-2-34-1': 'This satisfies the biorthogonality condition [MATH], as [MATH] has orthonormal columns by construction.', '1201.3749-2-34-2': 'In practice the time-integrals in [REF] are realized by a finite sampling of the time interval.', '1201.3749-2-35-0': 'A theoretical underpinning for the POD-Greedy algorithm has recently been provided by the analysis of convergence rates [CITATION].', '1201.3749-2-35-1': 'This is based on the approximation-theoretical notion of the Kolmogorov [MATH]-width [MATH] of a given set [MATH], which quantifies how well the set can be approximated by arbitrary [MATH]-dimensional linear subspaces of [MATH].', '1201.3749-2-35-2': 'The convergence statement for the case of exponential convergence then can be summarized as follows: If the set of solutions [MATH] is compact and has an exponentially decaying Kolmogorov [MATH]-width [MATH] for some [MATH] and all [MATH], then the error sequence [MATH] generated by the POD-Greedy procedure (cf. the definition in Step 2.', '1201.3749-2-35-3': 'in the pseudo code) also decays with an exponential rate, [MATH] with suitable constants [MATH] depending on [MATH].', '1201.3749-2-35-4': 'Thus, if the set of solutions can be approximated by linear subspaces with an exponentially decaying error term, then the POD-Greedy algorithm will in fact find an approximation with an exponentially decaying error term, though possibly with suboptimal parameters in the error bound.', '1201.3749-2-36-0': 'Extensions of the POD-Greedy algorithm exist, e.g. allowing more than one mode per extension step, performing adaptive parameter and time-interval partitioning, or enabling training-set adaptation [CITATION].', '1201.3749-2-37-0': '## Reduced models of the parametrized chemical master equation', '1201.3749-2-38-0': 'In this section, we describe how to apply the reduction method for parametrized models presented in the previous section to FSP models for the chemical master equation.', '1201.3749-2-39-0': 'As discussed in the previous section, the first step in the proposed reduction method is a decomposition of the [MATH]-dimensional system matrix [MATH] as in [REF].', '1201.3749-2-39-1': 'Such a decomposition is possible for the case of mass action reaction propensities, as defined in [REF], or generalized mass action, as recently suggested for the chemical master equation [CITATION].', '1201.3749-2-39-2': 'In this case, the length of the parameter vector [MATH] is equal to the number of reactions [MATH], and we decompose [MATH] into [MATH] terms as [EQUATION]', '1201.3749-2-39-3': 'Hence, concerning the notation given before, we have [MATH] components [MATH] and coefficient functions [MATH].', '1201.3749-2-39-4': 'Each matrix [MATH] in this decomposition comes from just the transition propensities corresponding to reaction [MATH], and is defined by [EQUATION]', '1201.3749-2-39-5': 'More generally, such a decomposition is also possible if reaction rate propensities can be decomposed into the product of two terms, with the first term depending on parameters only, and the second term on molecule numbers only.', '1201.3749-2-39-6': 'This case is for example encountered when the temperature-dependance of the reaction rate constant is relevant, and the temperature [MATH] is a variable parameter in the Arrhenius equation [MATH].', '1201.3749-2-39-7': 'Since the output matrix [MATH] and the initial condition [MATH] are usually not depending on parameters in this framework, a decomposition of [MATH] and [MATH] is not considered.', '1201.3749-2-40-0': 'The situation is more difficult for reaction propensities involving for example rational terms with parameters in the denominator.', '1201.3749-2-40-1': 'The denominator parameters can not be included in the reduced order model by the decomposition outlined in [REF] and [REF].', '1201.3749-2-40-2': 'If variations in these parameters are however not relevant to the planned analysis, then they can be set to their nominal value, and the decomposition can directly be done as described above.', '1201.3749-2-40-3': 'Alternatively, approximation steps can be performed, such as Taylor series expansion or empirical interpolation [CITATION], that generate an approximating parameter-separable expansion.', '1201.3749-2-41-0': '# Results for exemplary applications in genetic switching and oscillations', '1201.3749-2-42-0': 'In this section, we present the study of two example networks with the proposed model reduction method.', '1201.3749-2-42-1': 'With these examples, the applicability of the reduced modeling approach especially for analysis tasks requiring repeated simulations with different parameter values is illustrated.', '1201.3749-2-42-2': 'The first network is a bistable genetic toggle switch, where cells may switch randomly between two states, based on the model in [CITATION].', '1201.3749-2-42-3': 'For this network, the problem of parameter estimation with a reduced model is studied.', '1201.3749-2-42-4': 'The second network is a second-order genetic oscillator, based on [CITATION], where we perform a sensitivity analysis over a wide parameter range.', '1201.3749-2-43-0': '## Parameter estimation in a genetic toggle switch model', '1201.3749-2-44-0': '### Network description', '1201.3749-2-45-0': 'The genetic toggle switch considered here is an ovarian follicle switch model from [CITATION].', '1201.3749-2-45-1': 'It is a system of two genes which activate each other.', '1201.3749-2-45-2': 'The switch is modelled as a reaction network with two species [MATH], [MATH], representing the gene products.', '1201.3749-2-45-3': 'The network reactions are specified in Table [REF].', '1201.3749-2-46-0': 'In [CITATION], this network was shown to describe a bistable switch with two probability peaks, one close to [MATH] and the other close to [MATH].', '1201.3749-2-47-0': 'In the study [CITATION], only the lower probability peak was of interest.', '1201.3749-2-47-1': 'Here, we are interested in the transition of the system from [MATH] to [MATH].', '1201.3749-2-47-2': 'Therefore, the system is truncated to a rectangle [MATH] such that [MATH], yielding a good approximation in the finite state projection to the infinite-dimensional chemical master equation.', '1201.3749-2-48-0': 'The next step is to apply the decomposition of the matrix [MATH] as described in the methods section.', '1201.3749-2-48-1': 'Note that [MATH] for the switch network contains rational terms with the parameters [MATH] and [MATH].', '1201.3749-2-48-2': 'Considering these two parameters as fixed quantities, the truncated CME for the follicle switch can be written as [EQUATION] where [MATH], [MATH] are of dimension [MATH].', '1201.3749-2-49-0': 'As initial condition we choose a probability distributed over some lower states [EQUATION]', '1201.3749-2-49-1': 'For the parametric model reduction, we consider only variations in the parameters [MATH] and [MATH].', '1201.3749-2-49-2': 'These influence both the steady state level of gene activity in the on-state as well as the switching kinetics and are thus of high biological significance in the model.', '1201.3749-2-49-3': 'Hence we set [MATH] as the parametric domain [MATH].', '1201.3749-2-49-4': 'As final time we choose [MATH] which corresponds to a time range of approximately 19 years, i.e. about three times the half-life time of the off-state estimated in [CITATION].', '1201.3749-2-50-0': 'Some state plots from the simulation of the full model are shown in Figure [REF].', '1201.3749-2-50-1': 'These snapshots clearly show the transition of the switch from the off-state with low values for [MATH] and [MATH] to the on state with high values.', '1201.3749-2-50-2': 'The parameter influence is mainly reflected in the speed of the transition: for the parameter vector [MATH] in the lower row, most of the probability is already arranged around the on-state at the end of the simulation time.', '1201.3749-2-50-3': 'In contrast, for the parameter vector [MATH] in the upper row, a significant portion of the probability is still located around the off-state at this time point.', '1201.3749-2-50-4': 'Also, the transition paths are different: in the first case, the values for [MATH] are lower than the values for [MATH] during the transition, while in the second case, this relation is reversed.', '1201.3749-2-51-0': 'As typical simulation time for a single trajectory of the full system, we obtain 98.2 seconds on a IBM Lenovo 2.53 GHz Dual Core Laptop.', '1201.3749-2-52-0': '### Basis generation', '1201.3749-2-53-0': 'We generated a reduced basis with the POD-Greedy algorithm, where the training set was chosen as the vertices of a mesh with [MATH] logarithmically equidistant parameter values over the parameter domain [MATH].', '1201.3749-2-53-1': 'We set [MATH] as target accuracy.', '1201.3749-2-53-2': 'We use the projection error as error measure, hence precompute the 81 trajectories for construction of the reduced basis.', '1201.3749-2-53-3': 'As initial basis we set [MATH] and [MATH] using the parameter independent initial condition.', '1201.3749-2-54-0': 'The POD-Greedy algorithm produces a basis of 33 vectors and the overall computation of the reduced basis takes 7.9 hours, the dominating computation time being spent in the error evaluations and POD computations.', '1201.3749-2-54-1': 'Some of the resulting orthonormal basis vectors are illustrated in Figure [REF].', '1201.3749-2-54-2': 'The error decay curve and the selected parameters in the parameter domain are illustrated in Figure [REF].', '1201.3749-2-54-3': 'We nicely observe an exponential error decay, which indicates a parametric smoothness of the solution manifold, cf. the convergence rate statement given before for the POD-Greedy algorithm.', '1201.3749-2-54-4': 'The selected parameters seem to be located at the boundary of the parameter domain, indicating that the model behaviour in between can well be interpolated from the model behaviours along the boundary of the parameter domain.', '1201.3749-2-55-0': 'The final reduced model of dimension 33 can then be simulated in 0.135 seconds, corresponding to a computational speedup factor of more than 700.', '1201.3749-2-56-0': '### Parameter estimation', '1201.3749-2-57-0': 'We exemplify a possible application of the reduced order model in parameter estimation, where we assume that a distorted output [MATH] as the expected values [MATH] is available from population-averaged measurements.', '1201.3749-2-57-1': 'The task is to estimate the parameter values [MATH] and [MATH] from such a noisy measurement.', '1201.3749-2-58-0': 'The reference parameter is [MATH], and, for the purpose of this example, the measured output is produced by simulating the original model with the reference parameter values and adding 5% relative random white noise [MATH] sampled from a standard normal distribution, [MATH].', '1201.3749-2-58-1': 'An illustration of the reference output [MATH] and the noisy signal [MATH] is given in the left of Figure [REF].', '1201.3749-2-59-0': "We want to recover the values of the parameters [MATH] and [MATH] based on fitting the reduced parametric model's output [MATH] to the measured output [MATH].", '1201.3749-2-59-1': 'As is commonly done in parameter estimation, we formulate a least squares cost function as [EQUATION] and estimate the parameters by [EQUATION]', '1201.3749-2-59-2': 'In such an optimization problem, typically many forward simulations are required for adjusting [MATH] to the measurement.', '1201.3749-2-59-3': 'This is a particular beneficial scenario for reduced order models, as these simulations can be computed rapidly.', '1201.3749-2-60-0': 'In order to gain a deeper insight into the optimization problem [REF], we plot the values of the error functional [MATH] over the parameter domain (middle of Figure [REF]).', '1201.3749-2-60-1': 'Using the reduced model, the computation of the required [MATH] trajectories is realized in less than one minute.', '1201.3749-2-60-2': 'This would be a significant computational effort when using a non-reduced model.', '1201.3749-2-61-0': 'From the cost function plot, we observe a narrow area of parameters which seem to produce a similar output as the reference parameter [MATH].', '1201.3749-2-61-1': 'This shows that the two model parameters are not simultaneously identifiable from the considered output, and indicates that there may exist a functional dependence between the parameters [MATH] and [MATH] such that the model yields similar outputs [MATH].', '1201.3749-2-62-0': 'Assuming a functional dependence of [MATH] and [MATH] we now consider the 1-dimensional optimization problem along the line [MATH].', '1201.3749-2-62-1': 'We would like to recover [MATH] from the optimization problem.', '1201.3749-2-62-2': 'The corresponding value of the cost function is [MATH], indicating a significant contribution of the noise.', '1201.3749-2-62-3': "This restricted optimization problem is well conditioned and the optimization with a standard active set algorithm by MATLAB's command fmincon yields the estimated parameter [MATH] with [MATH], using 27 evaluations of the cost function.", '1201.3749-2-62-4': 'This accounts to a relative error in the [MATH] value of 0.204%, hence excellent recovery.', '1201.3749-2-62-5': 'We refrain from plotting the recovered output [MATH] as it is visually indiscriminable from the output in the left of Fig. [REF]).', '1201.3749-2-62-6': 'Interestingly, the optimization target value [MATH] implies [MATH], which may stem from a slight approximation error in the reduced model or from the effects of the measurement noise.', '1201.3749-2-63-0': 'The right plot in Fig. [REF] illustrates another application of reduced parametric models: We incorporated the model in an interactive graphical user interface in RBmatlab, a matlab package for model order reduction, available for download at www.morepas.org.', '1201.3749-2-63-1': 'This allows interactive parameter variations and instantaneous simulation response.', '1201.3749-2-64-0': '## Sensitivity analysis in a stochastic oscillator', '1201.3749-2-65-0': '### Network description', '1201.3749-2-66-0': 'The second case study is built on a genetic oscillator model showing stochastic resonance, which was presented in [CITATION].', '1201.3749-2-66-1': 'The oscillator is based on a negative feedback loop between two genes with one gene having positive autoregulation.', '1201.3749-2-66-2': 'The oscillator is modelled as a reaction network with two species [MATH], [MATH], representing the gene products.', '1201.3749-2-66-3': 'The network reactions are specified in Table [REF].', '1201.3749-2-66-4': 'In the original model in [CITATION], the dynamics were described as stochastic differential equation for the amounts of [MATH] and [MATH], coming from a Langevin approximation to the stochastic dynamics [CITATION].', '1201.3749-2-66-5': 'For the framework used in this paper, the dynamics have to be described directly by the underlying CME.', '1201.3749-2-66-6': 'To achieve this, we introduce the parameter [MATH] which maps the dimensionless state variables from [CITATION] to actual molecule numbers as required for the CME.', '1201.3749-2-66-7': "Thus, [MATH] is also a measure for the network's noise level: the higher [MATH], the larger the molecule number that is considered, and the smaller the noise level will be.", '1201.3749-2-67-0': 'The network model in Table [REF] shows oscillations only in a stochastic description.', '1201.3749-2-67-1': 'The deterministic model has a unique asymptotically stable equilibrium point, but in a stochastic model, fluctuations may push the molecular numbers beyond a certain threshold, inducing a dynamical response along a slow manifold, which corresponds to one oscillatory period [CITATION].', '1201.3749-2-67-2': 'Depending on the noise level, such responses will be initiated more or less often, corresponding to a more or less regular oscillatory pattern.', '1201.3749-2-68-0': 'The system is truncated to the rectangle [MATH], which contains the relevant system states for the parameter ranges of interest.', '1201.3749-2-69-0': 'Similarly as in the switch example, the reaction propensity expressions contain rational terms in the parameters [MATH], [MATH], and [MATH].', '1201.3749-2-69-1': 'These three cannot be decomposed directly, so we do the decomposition described in the methods section for the other five parameters only.', '1201.3749-2-69-2': 'With this decomposition, the truncated CME for the genetic oscillator can be written as [EQUATION] where [MATH], [MATH] are of dimension [MATH].', '1201.3749-2-69-3': 'The initial condition for [REF] is chosen as a uniform distribution over the rectangle [MATH]: [EQUATION]', '1201.3749-2-69-4': 'The time scale of interest for the model in [REF] is for [MATH].', '1201.3749-2-69-5': 'At the end of the interval, the probability distribution seems to approach a steady state.', '1201.3749-2-70-0': 'Some state plots are given in Figure [REF].', '1201.3749-2-70-1': 'One observes a significant effect of the parameter [MATH] on the amplitude of the oscillations.', '1201.3749-2-70-2': 'The simulation time for the detailed model was in average 7.3 minutes on a Dell desktop computer with 3.2 GHz dual-core Intel 4 processor and 1 GB RAM, without including the computation time for the construction of the state transition matrix [MATH].', '1201.3749-2-71-0': '### Basis generation', '1201.3749-2-72-0': 'For the basis generation, the parameter [MATH] was assumed to vary within the interval [MATH].', '1201.3749-2-72-1': 'A reduced basis with the POD-Greedy algorithm was computed from a training set of 30 logarithmically equidistant parameters over the parameter domain (Figure [REF]).', '1201.3749-2-72-2': 'As in the switch example, the target accuracy was chosen as [MATH], and the initial basis was chosen from the initial condition [MATH].', '1201.3749-2-73-0': 'The POD-Greedy algorithm produces a basis of 109 vectors, with an overall computation time of 16.5 hours on the hardware as in the previous subsection.', '1201.3749-2-73-1': 'The first 20 basis vectors are shown in Figure [REF].', '1201.3749-2-73-2': 'It is apparent that several of the basis vectors are directly included in order to reproduce the different amplitudes of oscillations that will occur under variations of the parameter [MATH].', '1201.3749-2-73-3': 'The error decay curve is shown in Figure [REF], displaying an exponential error decay as also observed for the switch example.', '1201.3749-2-74-0': 'With the reduced basis [MATH], we can construct a reduced parametric model for the CME of the oscillator as [EQUATION] with [MATH] and [MATH].', '1201.3749-2-74-1': 'Note that since only [MATH] has been varied in the reduction process, the other parameters are no longer present as parameters in the reduced model, but just take their nominal values.', '1201.3749-2-74-2': 'While the same basis [MATH] could be used to construct another reduced model where all parameters are retained, it is unlikely that this other model will be a good approximation of the original one for varying values of the other parameters.', '1201.3749-2-75-0': '### Sensitivity analysis of the oscillation amplitude', '1201.3749-2-76-0': 'As an application of the reduced order parametric model obtained in the previous section, we study the variations of oscillatory amplitude over a parameter range.', '1201.3749-2-76-1': 'Specifically, we consider 200 equally spaced values for the parameter [MATH] in the interval [MATH] and compute the probability that the amount of [MATH] is larger than 100: [EQUATION] with [MATH] the final time of the simulation.', '1201.3749-2-76-2': 'The results are shown in Figure [REF] and show a clear decay of oscillatory amplitude for increasing values of [MATH].', '1201.3749-2-76-3': 'Due to the significant time savings from the reduced model, this sensitivity curve can be computed with a high resolution.', '1201.3749-2-77-0': 'To evaluate the quality of the reduced model, we also computed the probability [REF] using the original model [REF] at two points within the considered interval for the parameter [MATH].', '1201.3749-2-77-1': 'As shown in Figure [REF], the results from the original model are in perfect agreement with the predictions from the reduced model at these points.', '1201.3749-2-77-2': 'Since the points at which the original model was evaluated in this experiment were not part of the training set (shown as triangles on the parameter axis in Figure [REF]), this shows that it is in fact possible to extrapolate the reduced model to parameter values that were not used to construct the basis.', '1201.3749-2-78-0': '# Conclusions', '1201.3749-2-79-0': 'In this paper, we have introduced the application of parametric model reduction methods to finite-state approximations of the chemical master equation.', '1201.3749-2-79-1': 'We have also presented two case studies where these methods are applied to CME models of different networks in order to make parametric analysis tasks computationally efficient.', '1201.3749-2-79-2': 'By this, it has become clear that parametric model reduction methods are a very useful tool for the analysis of stochastic biochemical reaction network described by the CME.', '1201.3749-2-80-0': 'Especially analysis tasks where many repeated simulations of a network with different parameter values are required can profit significantly from parametric model reduction.', '1201.3749-2-80-1': 'This includes for example sensitivity analysis or parameter optimization tasks such as identifiability analysis or estimation.', '1201.3749-2-80-2': "Moreover, the significant speedup of the simulation for the reduced model allows an interactive exploration of the network's dynamics within the parameter space within a suitable graphical user interface.", '1201.3749-2-81-0': 'This contribution is just a first step in the application of parametric model reduction methods to the CME.', '1201.3749-2-81-1': 'One particularly important aspect that we have not discussed here is the computation of error estimates for certifying that the simulation output of the reduced model is within some tolerance of the corresponding simulation output of the original model.', '1201.3749-2-81-2': 'To maintain computational efficiency, the error estimation should be done without actually simulating the original model.', '1201.3749-2-81-3': 'Error estimation methods have been developed for parametric model reduction of generic models [CITATION], but tighter estimates could likely be obtained by taking into account the special structure of the CME models.', '1201.3749-2-81-4': 'Recent work for example refined the previous generic error bounds for stable models [CITATION].', '1201.3749-2-82-0': '# Authors contributions', '1201.3749-2-83-0': 'SW and BH conceived of the study, performed the study, and wrote the manuscript.', '1201.3749-2-83-1': 'Both authors read and approved the final manuscript.'} | [['1201.3749-1-36-0', '1201.3749-2-39-0'], ['1201.3749-1-36-1', '1201.3749-2-39-1'], 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['1201.3749-1-19-0', '1201.3749-2-20-0'], ['1201.3749-1-8-0', '1201.3749-2-9-0'], ['1201.3749-1-8-1', '1201.3749-2-9-1'], ['1201.3749-1-39-0', '1201.3749-2-42-0'], ['1201.3749-1-39-1', '1201.3749-2-42-1'], ['1201.3749-1-39-2', '1201.3749-2-42-2'], ['1201.3749-1-39-3', '1201.3749-2-42-3'], ['1201.3749-1-39-4', '1201.3749-2-42-4'], ['1201.3749-1-64-0', '1201.3749-2-67-0'], ['1201.3749-1-64-1', '1201.3749-2-67-1'], ['1201.3749-1-64-2', '1201.3749-2-67-2'], ['1201.3749-1-16-0', '1201.3749-2-17-0'], ['1201.3749-1-16-1', '1201.3749-2-17-1'], ['1201.3749-1-16-2', '1201.3749-2-17-2'], ['1201.3749-1-16-3', '1201.3749-2-17-3'], ['1201.3749-1-16-4', '1201.3749-2-17-4'], ['1201.3749-1-6-0', '1201.3749-2-6-0'], ['1201.3749-1-6-1', '1201.3749-2-6-1'], ['1201.3749-1-6-2', '1201.3749-2-6-2'], ['1201.3749-1-6-3', '1201.3749-2-6-3'], ['1201.3749-1-33-1', '1201.3749-2-34-2']] | [['1201.3749-1-17-1', '1201.3749-2-18-1'], ['1201.3749-1-54-0', '1201.3749-2-57-0'], ['1201.3749-1-54-1', '1201.3749-2-57-1'], ['1201.3749-1-29-3', '1201.3749-2-30-2'], ['1201.3749-1-26-1', '1201.3749-2-27-1'], ['1201.3749-1-26-4', '1201.3749-2-27-4'], ['1201.3749-1-26-10', '1201.3749-2-27-10'], ['1201.3749-1-28-0', '1201.3749-2-29-0'], ['1201.3749-1-28-2', '1201.3749-2-29-2'], ['1201.3749-1-48-0', '1201.3749-2-51-0'], ['1201.3749-1-59-6', '1201.3749-2-62-4'], ['1201.3749-1-18-5', '1201.3749-2-19-5'], ['1201.3749-1-18-8', '1201.3749-2-19-8'], ['1201.3749-1-77-1', '1201.3749-2-80-1'], ['1201.3749-1-32-0', '1201.3749-2-33-0'], ['1201.3749-1-1-1', '1201.3749-2-1-1'], ['1201.3749-1-74-0', '1201.3749-2-77-0'], ['1201.3749-1-19-1', '1201.3749-2-20-1'], ['1201.3749-1-6-5', '1201.3749-2-7-2'], ['1201.3749-1-57-0', '1201.3749-2-60-0'], ['1201.3749-1-33-2', '1201.3749-2-36-0']] | [] | [['1201.3749-1-51-3', '1201.3749-2-54-3'], ['1201.3749-1-29-4', '1201.3749-2-30-3'], ['1201.3749-1-29-7', '1201.3749-2-30-10'], ['1201.3749-1-29-8', '1201.3749-2-30-11'], ['1201.3749-1-26-8', '1201.3749-2-27-8'], ['1201.3749-1-66-5', '1201.3749-2-69-5'], ['1201.3749-1-59-0', '1201.3749-2-62-6'], ['1201.3749-1-59-5', '1201.3749-2-62-3'], ['1201.3749-1-56-1', '1201.3749-2-59-1'], ['1201.3749-1-52-0', '1201.3749-2-55-0'], ['1201.3749-1-52-1', '1201.3749-2-55-0'], ['1201.3749-1-55-0', '1201.3749-2-58-0'], ['1201.3749-1-6-4', '1201.3749-2-7-1'], ['1201.3749-1-57-1', '1201.3749-2-60-1'], ['1201.3749-1-57-3', '1201.3749-2-61-0'], ['1201.3749-1-57-4', '1201.3749-2-61-1'], ['1201.3749-1-33-0', '1201.3749-2-34-0'], ['1201.3749-1-33-0', '1201.3749-2-34-1']] | [] | ['1201.3749-1-3-0', '1201.3749-1-30-0', '1201.3749-1-31-0', '1201.3749-2-3-0', '1201.3749-2-31-0', '1201.3749-2-32-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1201.3749 | null | null | null | null | null |
cond-mat-0603557 | {'cond-mat-0603557-1-0-0': 'Many proteins undergo a transition at a temperature [MATH].', 'cond-mat-0603557-1-0-1': 'Below [MATH] the fluctuations of the protein structure reduce dramatically [CITATION], a phenomenon possibly related to the hydration water surrounding the protein [CITATION].', 'cond-mat-0603557-1-0-2': 'This possibility raises the question of why the hydration water should change behavior at [MATH].', 'cond-mat-0603557-1-0-3': 'Here we use molecular dynamics (MD) simulation to argue that the change in hydration water is related to crossing a Widom line defined to be the locus of specific heat maxima emanating from a critical point [CITATION].', 'cond-mat-0603557-1-1-0': 'We carried out MD simulations of an orthorhombic form of hen egg-white lysozyme in the NPT ensemble using the software package GROMACS [CITATION].', 'cond-mat-0603557-1-1-1': 'We simulated a system consisting of a single protein in the native conformation solvated in [MATH] TIP5P water molecules at different temperatures, in a simulation box with periodic boundary conditions under atmospheric pressure [CITATION].', 'cond-mat-0603557-1-1-2': 'The simulation conditions correspond to the experimental hydration level of [MATH].', 'cond-mat-0603557-1-1-3': 'We first allowed the system to equilibrate at constant temperature and pressure using the Berendsen method, followed by a long production run during which we measured the dynamic and static properties of the system.', 'cond-mat-0603557-1-1-4': 'Equilibration time varied for different temperatures from a few ns for high temperatures to as much as [MATH] ns for low temperatures.', 'cond-mat-0603557-1-2-0': 'Figure 1(a) shows that protein fluctuations reduce dramatically below [MATH] K. Figure 1(b) shows that the diffusivity of hydration water changes upon cooling from non-Arrhenius ("fragile") to Arrhenius ("strong") behavior at [MATH] K, in agreement with recent experiments [CITATION].', 'cond-mat-0603557-1-2-1': 'Figure 1(c) shows that the entire system of water plus protein has a specific heat maximum at [MATH] K. Therefore, our results that [MATH] are consistent with the possibility that the changes in protein fluctuations [Fig.1(a)] and hydration water dynamics [Fig.1(b)] are both related to crossing the Widom line [Fig.1(c) and Fig. [REF](d)].', 'cond-mat-0603557-1-2-2': 'Crossing the Widom line corresponds to a continuous but rapid transition of the structural and thermodynamical properties of water from those resembling the properties of the high density liquid (HDL) at high temperatures to those resembling the properties of the low density liquid (LDL) at low temperatures [CITATION].', 'cond-mat-0603557-1-2-3': 'Hence it follows that our present results suggest that the protein glass transition is related to the hypothesized liquid-liquid critical point in supercooled water.', 'cond-mat-0603557-1-2-4': 'A consequence is the plausible picture that the fluctuations of the protein residues in the low density water (more ordered and more rigid) just below the Widom line should be smaller than the fluctuations in the high density water (less ordered and less rigid) just above the Widom line.', 'cond-mat-0603557-1-3-0': 'Our interpretation is supported by calculating the oxygen-oxygen pair correlation functions of the protein hydration water on crossing the Widom line from the higher density side to the lower density side.', 'cond-mat-0603557-1-3-1': 'We see [Fig. 1(e)] the features of HDL on the high-temperature side and LDL on the low-temperature side of the Widom line.', 'cond-mat-0603557-1-3-2': 'This interpretation is further supported by our finding that the boson peak (a vibrational mode) appears only on the more ordered (LDL) side of the Widom line [[REF](f)].', 'cond-mat-0603557-1-4-0': 'If the Widom line controls the change in water which in turn controls the protein glass transition, then an additional test is to change from one protein to a totally different object, DNA, also studied experimentally (Chen et al. unpublished).', 'cond-mat-0603557-1-4-1': 'Accordingly, we repeated the above calculations on DNA and found identical results, with [MATH]K.', 'cond-mat-0603557-1-5-0': 'P. Kumar[MATH], L. Xu[MATH], Z. Yan[MATH], M. G. Mazza[MATH], S. V. Buldyrev[MATH], S.-H. Chen[MATH], S. Sastry[MATH], and H. E. Stanley[MATH]Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA [MATH]Department of Physics, Yeshiva University, 500 West 185th Street, New York, NY 10033 USA', 'cond-mat-0603557-1-6-0': '[MATH]Nuclear Science and Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139 USA [MATH]Javaharlal Nehru Center for Advanced Scientific Research, Jakur Campus, Bangalore, 560061, India'} | {'cond-mat-0603557-2-0-0': 'Using molecular dynamics simulations, we investigate the relation between the dynamic transition of biomolecules (lysozyme and DNA) and the dynamic and thermodynamic properties of hydration water.', 'cond-mat-0603557-2-0-1': 'We find that the dynamic transition of the macromolecules, sometimes called a "protein glass transition", occurs at the temperature of dynamic crossover in the diffusivity of hydration water, and also coincides with the maximum of the isobaric specific heat [MATH].', 'cond-mat-0603557-2-0-2': 'We relate these findings to the hypothesis of a liquid-liquid critical point in water.', 'cond-mat-0603557-2-0-3': 'Our simulations suggest that the protein glass transition may be a result of crossing the Widom line, which is defined as the locus of correlation length maxima, emanating from the hypothesized second critical point of water.', 'cond-mat-0603557-2-1-0': 'It was hypothesized that the deeply supercooled region of the phase diagram of liquid water may contain a first order phase transition between a high density liquid (HDL) and a low density liquid (LDL) see Fig. [REF](a) [CITATION].', 'cond-mat-0603557-2-1-1': 'This line of phase transition has a negative slope in the P-T phase diagram and is terminated with a critical point, [MATH], which according to different estimates is located around [MATH] K and [MATH] MPa.', 'cond-mat-0603557-2-1-2': 'Upon crossing the first order phase transition line above the critical pressure the thermodynamic state functions change discontinuously.', 'cond-mat-0603557-2-1-3': 'Below the critical pressure they rapidly but continuously change upon cooling, see path [MATH] in Fig. [REF](a).', 'cond-mat-0603557-2-1-4': 'Indeed computer simulations of various models of water show that many response functions such as isobaric specific heat and isothermal compressibility have sharp maxima as functions of temperature if the system is cooled isobarically at [MATH].', 'cond-mat-0603557-2-1-5': 'The lines of these maxima of different response functions asymptotically approach one another as the critical point is approached, because all of them become expressible in terms of the correlation length which is diverging at the critical point.', 'cond-mat-0603557-2-1-6': 'This asymptotic line is sometimes called the Widom line, and it is often regarded as an extension of the coexistence line into the one-phase region.', 'cond-mat-0603557-2-1-7': 'A schematic diagram of Widom line in water is shown in Fig. [REF](a).', 'cond-mat-0603557-2-2-0': 'Experimental studies of supercritical water [CITATION] indeed show that various response functions have sharp maxima in the analogous region of the phase diagram above the liquid-vapor critical point [MATH], but no direct experimental indication of [MATH] had been available due to unavoidable crystallization of bulk water.', 'cond-mat-0603557-2-2-1': 'Recently it was found that water remains unfrozen in hydrophilic nanopores for [MATH] K [CITATION].', 'cond-mat-0603557-2-2-2': 'Moreover when cooled at constant pressure for [MATH] the dynamics changes from non-Arrhenius to Arrhenius at [MATH].', 'cond-mat-0603557-2-2-3': 'The line [MATH] is located in the range of temperatures between [MATH] K and has a negative slope in the P-T phase diagram.', 'cond-mat-0603557-2-2-4': 'Computer simulations suggest that this line may be associated with the Widom line, [MATH], near which the dynamic characteristics must rapidly change from those resembling the properties of HDL at high temperature to those of LDL at low temperature [CITATION].', 'cond-mat-0603557-2-3-0': 'Both experiments and computer simulation studies have shown that hydrated proteins undergo a "glass-like" transition near [MATH] K [CITATION], above which proteins exhibit diffusive motion, and below which the proteins are trapped in harmonic modes.', 'cond-mat-0603557-2-3-1': 'An important issue is to determine the effects of hydration water on this dynamical transition [CITATION].', 'cond-mat-0603557-2-3-2': 'Experiments and computer simulations suggested that when a protein is solvated, the protein glass transition is strongly coupled to the solvent, leading to the question of whether the protein glass transition is directly related to a dynamic transition in the surrounding solvent [CITATION].', 'cond-mat-0603557-2-3-3': 'Here we hypothesize that the observed glass transition in biomolecules is related to the Widom line of the liquid-liquid phase transition and test this hypothesis by computer simulations.', 'cond-mat-0603557-2-4-0': 'Using molecular dynamics (MD) simulations, we study the dynamic and thermodynamic behavior of lysozyme and DNA in hydration water [Fig. [REF]].', 'cond-mat-0603557-2-4-1': 'We carry out MD simulations of an orthorhombic form of hen egg-white lysozyme and also Dickerson dodecamer DNA [CITATION] at constant pressure [MATH] atm, several constant temperatures [MATH] and constant number of water molecules [MATH] (NPT ensemble) in a simulation box with periodic boundary conditions using the software package GROMACS [CITATION].', 'cond-mat-0603557-2-4-2': 'We first allow the system to equilibrate at constant temperature and pressure using the Berendsen method.', 'cond-mat-0603557-2-4-3': 'This initial equilibration is followed by a long production run during which we measure the dynamic and static properties of the system.', 'cond-mat-0603557-2-4-4': 'Equilibration times vary for different temperatures from a few ns for high temperatures to as much as [MATH] ns for low temperatures.', 'cond-mat-0603557-2-4-5': 'The MD for DNA was performed using the Amber force field [CITATION].', 'cond-mat-0603557-2-4-6': 'For lysozyme simulations, the system consists of a single protein in the native conformation solvated in [MATH] TIP5P water molecules [CITATION].', 'cond-mat-0603557-2-4-7': 'These simulation conditions correspond to the experimental hydration level of [MATH] (the ratio of water mass to protein mass is [MATH]).', 'cond-mat-0603557-2-4-8': 'The DNA system consists of a single DNA helix with 24 nucleotides solvated in [MATH] TIP5P water molecules, which corresponds to an experimental hydration level of [MATH].', 'cond-mat-0603557-2-4-9': 'For both the protein and the DNA, the hydration level is sufficient to have 3-4 molecular layers of hydration water on average.', 'cond-mat-0603557-2-4-10': 'Hence one can expect the results for hydration water in these cases to be similar to the results for bulk water, as the number of water molecules in the first hydration shell will be much smaller compared to the water molecules further away.', 'cond-mat-0603557-2-5-0': 'The simulation results for protein and protein hydration water are shown in Fig. [REF](a).', 'cond-mat-0603557-2-5-1': 'We calculate the root mean square (RMS) fluctuations [MATH] of protein from the equilibrated configurations, first for each atom over [MATH] ns, and then averaged over the total number of atoms in the protein.', 'cond-mat-0603557-2-5-2': 'We find that the protein fluctuations [Fig. [REF](a)] reduce dramatically below [MATH] K. Interestingly, upon cooling, the diffusivity of hydration water exhibits a dynamic crossover from non-Arrhenius to Arrhenius behavior at the same temperature [MATH] K [Fig. [REF](a)].', 'cond-mat-0603557-2-5-3': 'The coincidence of the crossover temperature indicates that the protein is strongly coupled with the surrounding solvent, in agreement with recent experiments [CITATION].', 'cond-mat-0603557-2-6-0': 'We also study the static properties - thermodynamics and structural properties - of the system.', 'cond-mat-0603557-2-6-1': 'The thermodynamic response function [MATH] of the entire system of water plus protein is shown in Fig. [REF](c) and displays a maximum at [MATH] K. For a comparison, we also show [MATH] for bulk TIP5P water at [MATH] MPa [Fig. [REF](b)].', 'cond-mat-0603557-2-6-2': 'Bulk TIP5P also shows a [MATH] maximum upon cooling at constant pressure.', 'cond-mat-0603557-2-6-3': 'Comparing Fig. [REF](c) and Fig. [REF](b), one can see that [MATH] for the protein and water combined system is dominated by the hydration water.', 'cond-mat-0603557-2-6-4': 'The fact that [MATH] is evidence of the strong correlation of the changes in protein fluctuations [Fig. [REF] (a)] with the hydration water thermodynamics [Fig. [REF] (c)].', 'cond-mat-0603557-2-6-5': 'Thus our results are consistent with the possibility that the protein glass transition is related to the Widom line (and hence to the hypothesized liquid-liquid critical point).', 'cond-mat-0603557-2-6-6': 'Crossing the Widom line corresponds to a continuous but rapid transition of the structural [Fig. [REF](a)] and thermodynamical properties [Fig. [REF](b)] of water from those resembling the properties of the HDL at high temperatures to those resembling the properties of the LDL at low temperatures [CITATION].', 'cond-mat-0603557-2-6-7': 'A consequence is the plausible picture that the fluctuations of the protein residues in low density water (more ordered and more rigid) just below the Widom line should be smaller than the fluctuations in high density water (less ordered and less rigid) just above the Widom line.', 'cond-mat-0603557-2-6-8': 'Our interpretation is further strengthened by the analysis of the structure of hydration water on the two sides of the Widom line.', 'cond-mat-0603557-2-6-9': 'Fig. [REF](a) shows the oxygen-oxygen radial distribution function [MATH] on two sides of the Widom line for lysozyme hydration water.', 'cond-mat-0603557-2-6-10': 'The first peak of [MATH] on the low temperature (T=[MATH] K) side is sharper and the first minimum is shallower compared to the [MATH] on the high temperature (T=[MATH] K) side of the Widom line, suggesting that water is more structured on the low temperature side.', 'cond-mat-0603557-2-6-11': 'To further test this finding, we calculate the structure factor of lysozyme hydration water [Fig. [REF](c)].', 'cond-mat-0603557-2-6-12': 'The first peak of the structure factor associated with the hydrogen bond is very sharp and pronounced, on the low temperature side of the Widom line, while on the high temperature side, it is diminished and moves to larger wave vectors, suggesting a LDL-like structure on the low temperature side and a HDL-like structure on the high temperature side of the Widom line.', 'cond-mat-0603557-2-7-0': 'Previous studies in computer simulation [CITATION] and experiment [CITATION] indicates that there is a glass transition of DNA around temperature [MATH] K. Hence to test the sensitivity of the dependence of dynamic crossover on different solutes, we performed a parallel study of DNA Dickerson dodecamer [CITATION].', 'cond-mat-0603557-2-7-1': 'We find that fluctuations [CITATION] of the DNA molecule [Fig. [REF](b)] change their behavior approximately at the same temperature as lysozyme, with [MATH] K.', 'cond-mat-0603557-2-7-2': 'The dynamic crossover in the hydration water upon cooling from non-Arrhenius to Arrhenius behavior takes place at [MATH] K [Fig. [REF](b)].', 'cond-mat-0603557-2-7-3': 'The specific heat maximum is located at [MATH] K [Fig. [REF](d)].', 'cond-mat-0603557-2-7-4': 'The specific heat [MATH] of bulk water, protein and DNA hydration water all have maximum at the transition temperature around [MATH] K [CITATION].', 'cond-mat-0603557-2-7-5': 'Fig. [REF](b) and Fig. [REF](d) show the oxygen-oxygen pair correlation function, [MATH], and the structure factor, [MATH], for the DNA hydration water.', 'cond-mat-0603557-2-7-6': 'It is clear the strong similarity with the lysozyme case, Fig. [REF](a) and Fig. [REF](c) respectively.', 'cond-mat-0603557-2-7-7': 'The difference in the height of the first peak of [MATH] for lysozyme and DNA hydration water is due to the different hydration levels of the two systems.', 'cond-mat-0603557-2-8-0': 'The quantitative agreement of these results with the corresponding results for lysozyme suggests that it is indeed the changes in the properties of hydration water that are responsible for the changes in dynamics of the protein and DNA biomolecules.', 'cond-mat-0603557-2-8-1': 'Our results are in qualitative agreement with a recent experiment performed on hydrated protein and DNA [CITATION] which found the crossover in side-chain fluctuations at [MATH] K (the differences in the numerical values of [MATH] and [MATH] obtained in the simulations and experiments can be ascribed to the limitations of the TIP5P water model).', 'cond-mat-0603557-2-9-0': 'We thank C. A. Angell, L. Cruz, P. G. Debenedetti, L. Liu, and B. Urbanc for helpful discussions of both experimental and computational work, and NSF for financial support.'} | [['cond-mat-0603557-2-0-2', 'cond-mat-0603557-3-0-2'], ['cond-mat-0603557-2-2-2', 'cond-mat-0603557-3-4-2'], ['cond-mat-0603557-2-2-3', 'cond-mat-0603557-3-4-3'], ['cond-mat-0603557-2-3-0', 'cond-mat-0603557-3-1-0'], ['cond-mat-0603557-2-3-1', 'cond-mat-0603557-3-1-1'], ['cond-mat-0603557-2-3-2', 'cond-mat-0603557-3-1-2'], ['cond-mat-0603557-2-7-1', 'cond-mat-0603557-3-9-1'], ['cond-mat-0603557-2-4-2', 'cond-mat-0603557-3-5-2'], ['cond-mat-0603557-2-4-4', 'cond-mat-0603557-3-5-4'], ['cond-mat-0603557-2-4-5', 'cond-mat-0603557-3-5-5'], ['cond-mat-0603557-2-4-6', 'cond-mat-0603557-3-5-6'], ['cond-mat-0603557-2-4-8', 'cond-mat-0603557-3-5-8'], ['cond-mat-0603557-2-6-5', 'cond-mat-0603557-3-7-3'], ['cond-mat-0603557-2-6-9', 'cond-mat-0603557-3-8-1'], ['cond-mat-0603557-2-0-0', 'cond-mat-0603557-3-0-0'], ['cond-mat-0603557-2-0-1', 'cond-mat-0603557-3-0-1'], 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['cond-mat-0603557-2-7-7', 'cond-mat-0603557-3-9-6'], ['cond-mat-0603557-2-4-0', 'cond-mat-0603557-3-5-1'], ['cond-mat-0603557-2-4-1', 'cond-mat-0603557-3-5-1'], ['cond-mat-0603557-2-4-7', 'cond-mat-0603557-3-5-7'], ['cond-mat-0603557-2-1-1', 'cond-mat-0603557-3-2-1'], ['cond-mat-0603557-2-1-3', 'cond-mat-0603557-3-2-3'], ['cond-mat-0603557-2-6-6', 'cond-mat-0603557-3-7-4'], ['cond-mat-0603557-2-6-8', 'cond-mat-0603557-3-8-0'], ['cond-mat-0603557-2-6-11', 'cond-mat-0603557-3-8-0'], ['cond-mat-0603557-2-6-11', 'cond-mat-0603557-3-8-3'], ['cond-mat-0603557-2-6-12', 'cond-mat-0603557-3-8-4']] | [['cond-mat-0603557-1-1-0', 'cond-mat-0603557-2-4-1'], ['cond-mat-0603557-1-1-1', 'cond-mat-0603557-2-4-6'], ['cond-mat-0603557-1-1-2', 'cond-mat-0603557-2-4-7'], ['cond-mat-0603557-1-1-3', 'cond-mat-0603557-2-4-3'], ['cond-mat-0603557-1-1-3', 'cond-mat-0603557-2-4-2'], ['cond-mat-0603557-1-1-4', 'cond-mat-0603557-2-4-4'], ['cond-mat-0603557-1-2-0', 'cond-mat-0603557-2-5-2'], ['cond-mat-0603557-1-2-2', 'cond-mat-0603557-2-6-6'], ['cond-mat-0603557-1-2-4', 'cond-mat-0603557-2-6-7']] | [] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/cond-mat/0603557 | {'cond-mat-0603557-3-0-0': 'Using molecular dynamics simulations, we investigate the relation between the dynamic transitions of biomolecules (lysozyme and DNA) and the dynamic and thermodynamic properties of hydration water.', 'cond-mat-0603557-3-0-1': 'We find that the dynamic transition of the macromolecules, sometimes called a "protein glass transition", occurs at the temperature of dynamic crossover in the diffusivity of hydration water, and also coincides with the maxima of the isobaric specific heat [MATH] and the temperature derivative of the orientational order parameter.', 'cond-mat-0603557-3-0-2': 'We relate these findings to the hypothesis of a liquid-liquid critical point in water.', 'cond-mat-0603557-3-0-3': 'Our simulations are consistent with the possibility that the protein glass transition results from crossing the Widom line, which is defined as the locus of correlation length maxima emanating from the hypothesized second critical point of water.', 'cond-mat-0603557-3-1-0': 'Both experiments and computer simulation studies have shown that hydrated proteins undergo a "glass-like" transition near [MATH] K [CITATION], above which proteins exhibit diffusive motion, and below which the proteins are trapped in harmonic modes.', 'cond-mat-0603557-3-1-1': 'An important issue is to determine the effects of hydration water on this dynamical transition [CITATION].', 'cond-mat-0603557-3-1-2': 'Experiments and computer simulations suggested that when a protein is solvated, the protein glass transition is strongly coupled to the solvent, leading to the question of whether the protein glass transition is directly related to a dynamic transition in the surrounding solvent [CITATION].', 'cond-mat-0603557-3-2-0': 'It has been hypothesized that the deeply supercooled region of the phase diagram of liquid water may contain a first order phase transition between a high density liquid (HDL) and a low density liquid (LDL) [CITATION].', 'cond-mat-0603557-3-2-1': 'This line of phase transition has a negative slope in the P-T phase diagram and terminates with a critical point, [MATH] [CITATION], which is located at [MATH] K and [MATH] MPa [CITATION].', 'cond-mat-0603557-3-2-2': 'Upon crossing the first order phase transition line above the critical pressure, the thermodynamic state functions change discontinuously.', 'cond-mat-0603557-3-2-3': 'Below the critical pressure they rapidly but continuously change upon cooling.', 'cond-mat-0603557-3-3-0': 'Computer simulations of the TIP5P and ST2 models show that response functions such as isobaric specific heat [MATH] and isothermal compressibility have maxima as functions of temperature if the system is cooled isobarically at [MATH] [CITATION].', 'cond-mat-0603557-3-3-1': 'The loci of these maxima asymptotically approach one another [CITATION] as the critical point is approached, because response functions become expressible in terms of the correlation length which diverges at the critical point.', 'cond-mat-0603557-3-3-2': 'The locus of the correlation length maxima is called the Widom line.', 'cond-mat-0603557-3-4-0': 'Experimental studies of supercritical water [CITATION] indeed show that various response functions have sharp maxima in the region of the phase diagram above the liquid-vapor critical point [MATH], but no direct experimental indication of a liquid-liquid critical point [MATH] had been available due to unavoidable crystallization of bulk water.', 'cond-mat-0603557-3-4-1': 'It was found that water remains unfrozen in hydrophilic nanopores for [MATH] K [CITATION].', 'cond-mat-0603557-3-4-2': 'Moreover when cooled at constant pressure for [MATH] the dynamics changes from non-Arrhenius to Arrhenius at [MATH].', 'cond-mat-0603557-3-4-3': 'The line [MATH] is located in the range of temperatures between [MATH] K and has a negative slope in the P-T phase diagram.', 'cond-mat-0603557-3-4-4': 'Computer simulations suggest that this line may be associated with [MATH], the Widom line, near which the local dynamic characteristics must rapidly change from those resembling the properties of HDL at high temperature to those of LDL at low temperature [CITATION].', 'cond-mat-0603557-3-5-0': 'Here we explore the hypothesis [CITATION] that the observed glass transition in biomolecules is related to the liquid-liquid phase transition using molecular dynamics (MD) simulations.', 'cond-mat-0603557-3-5-1': 'Specifically, we study the dynamic and thermodynamic behavior of lysozyme and DNA in hydration TIP5P water, by means of the software package GROMACS [CITATION] for (i) an orthorhombic form of hen egg-white lysozyme [CITATION] and (ii) a Dickerson dodecamer DNA [CITATION] at constant pressure [MATH] atm, several constant temperatures [MATH], and constant number of water molecules [MATH] (NPT ensemble) in a simulation box with periodic boundary conditions.', 'cond-mat-0603557-3-5-2': 'We first allow the system to equilibrate at constant temperature and pressure using the Berendsen method.', 'cond-mat-0603557-3-5-3': 'This initial equilibration is followed by a long production run during which we calculate the dynamic and static properties.', 'cond-mat-0603557-3-5-4': 'Equilibration times vary for different temperatures from a few ns for high temperatures to as much as [MATH] ns for low temperatures.', 'cond-mat-0603557-3-5-5': 'The MD for DNA was performed using the Amber force field [CITATION].', 'cond-mat-0603557-3-5-6': 'For lysozyme simulations, the system consists of a single protein in the native conformation solvated in [MATH] TIP5P water molecules [CITATION].', 'cond-mat-0603557-3-5-7': 'These simulation conditions correspond to a ratio of water mass to protein mass of [MATH].', 'cond-mat-0603557-3-5-8': 'The DNA system consists of a single DNA helix with 24 nucleotides solvated in [MATH] TIP5P water molecules, which corresponds to an experimental hydration level of [MATH].', 'cond-mat-0603557-3-6-0': 'The simulation results for the mean square fluctuations [MATH] of protein are shown in Fig. [REF](a).', 'cond-mat-0603557-3-6-1': 'We calculate the mean square fluctuations [MATH] of protein from the equilibrated configurations, first for each atom over [MATH] ns, and then averaged over the total number of atoms in the protein.', 'cond-mat-0603557-3-6-2': 'We find that [MATH] [Fig. [REF](a)] changes its functional form below [MATH] K. Moreover, upon cooling, the diffusivity of hydration water exhibits a dynamic crossover from non-Arrhenius [CITATION] to Arrhenius behavior at the crossover temperature [MATH] K [Fig. [REF](c)].', 'cond-mat-0603557-3-6-3': 'A similar temperature dependence of diffusivity of bulk TIP5P water was observed in Ref. [CITATION].', 'cond-mat-0603557-3-6-4': 'The coincidence of [MATH] with [MATH] within the error bars indicates that the behavior of the protein is strongly coupled with the behavior of the surrounding solvent, in agreement with recent experiments [CITATION].', 'cond-mat-0603557-3-6-5': 'Note that [MATH] is much higher than the glass transition temperature estimated for TIP5P as [MATH]K [CITATION].', 'cond-mat-0603557-3-6-6': 'Thus this crossover is not likely to be related to the glass transition in water.', 'cond-mat-0603557-3-6-7': 'Here we will explore the possibility that instead it is related to a change in the properties of protein hydration water.', 'cond-mat-0603557-3-7-0': 'We next calculate [MATH] by numerical differentiation of the total enthalpy of the system (protein and water) by fitting the simulation data for enthalpy with a fifth order polynomial, and then taking the derivative with respect to [MATH].', 'cond-mat-0603557-3-7-1': 'Figure [REF](a) displays a maximum of [MATH] at [MATH] K for the case of lysozyme-water system [CITATION].', 'cond-mat-0603557-3-7-2': 'The fact that [MATH] is evidence of the correlation between the changes in protein fluctuations [Fig. [REF] (a)] and the hydration water thermodynamics [Fig. [REF] (a)].', 'cond-mat-0603557-3-7-3': 'Thus our results are consistent with the possibility that the protein glass transition is related to the Widom line (and hence to the hypothesized liquid-liquid critical point).', 'cond-mat-0603557-3-7-4': 'Crossing the Widom line corresponds to a continuous but rapid transition of the properties of water from those resembling the properties of a local HDL structure for [MATH] to those resembling the properties of a local LDL structure for [MATH] [CITATION].', 'cond-mat-0603557-3-7-5': 'A consequence is the expectation that the fluctuations of the protein residues in predominantly LDL-like water (more ordered and more rigid) just below the Widom line should be smaller than the fluctuations in predominantly HDL-like water (less ordered and less rigid) just above the Widom line.', 'cond-mat-0603557-3-8-0': 'To test this interpretation, we analyze the structure of hydration water on the two sides of the Widom line.', 'cond-mat-0603557-3-8-1': 'Fig. [REF](a) shows the oxygen-oxygen radial distribution function [MATH] on two sides of the Widom line for lysozyme hydration water.', 'cond-mat-0603557-3-8-2': 'The first peak of [MATH] on the low temperature (T=[MATH] K) side is sharper and the first minimum is shallower compared to the [MATH] on the high temperature (T=[MATH] K, [MATH] K) side of the Widom line, suggesting that water is more structured on the low temperature side.', 'cond-mat-0603557-3-8-3': 'Further, we calculate the structure factor [MATH] of lysozyme hydration water [Fig. [REF](c)].', 'cond-mat-0603557-3-8-4': 'The first peak of the structure factor associated with the hydrogen bond is very sharp and pronounced, for [MATH], it is diminished and moves to larger wave vectors for [MATH], consistent with a LDL-like local structure for [MATH] and a HDL-like local structure for [MATH].', 'cond-mat-0603557-3-9-0': 'Previous simulations [CITATION] and experiments [CITATION] suggest a "glass-like" transition of DNA around temperature [MATH] K. Hence to test if the dynamic crossover depends on the solute, we performed a parallel study of the DNA Dickerson dodecamer [CITATION].', 'cond-mat-0603557-3-9-1': 'We find that fluctuations [CITATION] of the DNA molecule [Fig. [REF](b)] change their behavior approximately at the same temperature as lysozyme, with [MATH] K.', 'cond-mat-0603557-3-9-2': 'The dynamic crossover in the hydration water upon cooling from non-Arrhenius to Arrhenius behavior takes place at [MATH] K [Fig. [REF](d)].', 'cond-mat-0603557-3-9-3': 'For DNA hydration water system [MATH] has a maximum at [MATH] K, similar to the case of protein (see Fig. [REF](b)) [CITATION].', 'cond-mat-0603557-3-9-4': 'Fig. [REF](b) and Fig. [REF](d) show [MATH] and [MATH] for the DNA hydration water [CITATION].', 'cond-mat-0603557-3-9-5': 'Further to describe the quantitative changes in structure of hydration water, we calculate the local tetrahedral order parameter [MATH] [CITATION] for hydration water surrounding lysozyme and DNA.', 'cond-mat-0603557-3-9-6': 'Figs. [REF](e) and [REF](f) show that the rate of increase of [MATH] has a maximum at [MATH] K for lysozyme and DNA hydration water respectively; the same temperatures where we find a crossover in the behavior of mean square fluctuations and a change in the behavior of the dynamics of hydration water.', 'cond-mat-0603557-3-10-0': 'The quantitative agreement of the results for DNA and lysozyme suggests that it is indeed the changes in the properties of hydration water that are responsible for the changes in dynamics of the protein and DNA biomolecules.', 'cond-mat-0603557-3-10-1': 'Our results are in qualitative agreement with recent experiments on hydrated protein and DNA [CITATION] which found the crossover in side-chain fluctuations at [MATH] K .', 'cond-mat-0603557-3-11-0': 'Note Added in Proof: After this work was submitted, we learned of interesting parallel work on silicon, which also interprets structural change in [MATH] and [MATH] as crossing the Widom line [CITATION].', 'cond-mat-0603557-3-12-0': 'We thank C. A. Angell, L. Cruz, P. G. Debenedetti, L. Liu, P. J. Rossky, and B. Urbanc for helpful discussions of both experimental and computational work, and NSF for financial support.'} | null | null | null | null |
astro-ph-9907028 | {'astro-ph-9907028-1-0-0': 'We report the discovery of an isolated brown dwarf with similar properties to the binary object Gliese 229B and to the newly discovered field brown dwarf SDSS 1624+00.', 'astro-ph-9907028-1-0-1': 'Although exhibiting similar colors, its magnitude of [MATH] 20.5 is about 6 and 5 magnitudes fainter than Gliese 229B and SDSS 1624+00 respectively.', 'astro-ph-9907028-1-0-2': 'This is the third methane brown dwarf reported to date, the second isolated one in the field and by far the most distant at [MATH] 100 pc.', 'astro-ph-9907028-1-0-3': 'Its IR spectrum, although at low S/N given the faintness of the object, is remarkably similar to those of the 2 other methane brown dwarfs.', 'astro-ph-9907028-1-1-0': '# Introduction', 'astro-ph-9907028-1-2-0': 'Despite large observational efforts during recent years in both wide field and targeted searches for very cold brown dwarfs, the number of such objects known so far remains extremely small.', 'astro-ph-9907028-1-2-1': 'Since 1995, and until June 1999, the only genuine one identified was Gliese 229B (Nakajima [CITATION], Oppenheimer [CITATION]), the coolest substellar object known, with a temperature below 1000 K, a mass in the range 20-50 M[MATH] (Jupiter mass), and an age in the range 0.5-1 Gyr.', 'astro-ph-9907028-1-2-2': 'A second object of this class, SDSS 1624+00, has been discovered recently in the Sloan Digital Sky Survey (Strauss [CITATION]), having been identified from the survey database by its unusually red color.', 'astro-ph-9907028-1-2-3': 'Follow-up spectroscopy of this object in the visible with the Apache Point 3.5m telescope and in the IR with UKIRT identified it as a methane brown dwarf like Gliese 229B.', 'astro-ph-9907028-1-2-4': 'A couple of similar objects have since then been identified (Tzetanov, private communication).', 'astro-ph-9907028-1-3-0': 'In this paper we report our discovery of a new methane brown dwarf in the NTT Deep Field, a small area of the sky that was the target of very deep exposures in the visible and the near-infrared using the SUSI and SOFI instruments at the ESO New Technology Telescope (NTT) (Arnouts [CITATION], Saracco [CITATION]).', 'astro-ph-9907028-1-3-1': 'One object, NTTDF 1205-0744, stands out in these images for its very red (i-J) [MATH] 6 color index.', 'astro-ph-9907028-1-3-2': 'However, it is very blue at longer wavelengths, with (J-K) = -0.15.', 'astro-ph-9907028-1-3-3': 'Near-infrared spectroscopy with SOFI, and with ISAAC at the ESO Very Large Telescope (VLT), has confirmed the remarkable similarity of this object to Gliese 229B.', 'astro-ph-9907028-1-3-4': 'Due to the very faint apparent magnitude of NTTDF 1205-0744, the observations reported here have been possible only with the availability of two extremely powerful telescopes, the NTT and the VLT, and their near-IR spectrographs, SOFI and ISAAC.', 'astro-ph-9907028-1-3-5': 'Although the raw S/N is limited (1 to 2 per pixel, 5 to 10 after rebinning), our results secure the identification of NTTDF 1205-0744 as a new field methane brown dwarf.', 'astro-ph-9907028-1-4-0': '# Observations and data reduction', 'astro-ph-9907028-1-5-0': 'The NTT Deep Field covers an area of 2.3 [MATH] 2.3 arcminutes in the visible down to AB magnitude limits of 27.2, 27.0, 26.7 and 26.3 in B, V, r, and i, and 5 [MATH] 5 arcminutes in the IR down to magnitude limits of 24.6 and 22.8 in J and Ks.', 'astro-ph-9907028-1-6-0': 'The entire dataset of the NTT Deep Field Project, primarily targeted to the study of faint galaxy populations, as well as a detailed information on data acquisition and reduction, are publicly available at http://www.eso.org.', 'astro-ph-9907028-1-7-0': 'J and i band images of the field containing NTTDF 1205-0744 are shown in figure [REF].', 'astro-ph-9907028-1-8-0': 'After identification of NTTDF 1205-0744 from its unusual extremely red colour (i-J) in April 98, we carried out spectroscopy with SOFI at the NTT using Target of Opportunity Time on 30 June - 1 July 1998.', 'astro-ph-9907028-1-8-1': 'The spectrum, covering the range 0.95-1.65 microns (dispersion: 7 [MATH] per pixel), was obtained under non-photometric conditions using a 1" slit, and nodding along the slit between two positions, for a total effective on-target integration time of 84 minutes.', 'astro-ph-9907028-1-8-2': 'Spectrophotometric calibration and removal of telluric features was achieved using the observation of a B9 type star.', 'astro-ph-9907028-1-8-3': 'The spectrum was scaled to match the IR photometry in the J filter.', 'astro-ph-9907028-1-9-0': 'The spectrum shows clear H[MATH]O absorptions, leaving peaks in the spectrum at 1.05 and 1.27 [MATH]m (the latter peak at a S/N of 1-1.5 per pixel), and a marginally significant detection of a third peak at 1.57 microns.', 'astro-ph-9907028-1-10-0': 'We subsequently obtained spectroscopy of NTTDF 1205-0744 with ISAAC at the VLT in the H and K bands.', 'astro-ph-9907028-1-10-1': 'All the ISAAC observations were made with a 1 arcsec slit and nodding along the slit.', 'astro-ph-9907028-1-11-0': 'The K observations were carried out during the nights of 6 and 9 February 1999, for a total amount of time of 1 hour.', 'astro-ph-9907028-1-11-1': 'We used the Low Resolution grating in second order providing a dispersion per pixel of 7 [MATH].', 'astro-ph-9907028-1-11-2': 'Spectrophotometric calibration was achieved from the observation of a B6 type star observed on a different night.', 'astro-ph-9907028-1-11-3': 'The signal to noise per pixel is below 1 on the peak at 2.1 [MATH]m.', 'astro-ph-9907028-1-12-0': 'The observations in H were carried out during the night of 23 March 1999, again for a total integration time of 1 hour.', 'astro-ph-9907028-1-12-1': 'We used the same Low Resolution grating in third order, providing a dispersion per pixel of 4.7 [MATH].', 'astro-ph-9907028-1-12-2': 'Spectrophotometric calibration was done from the observation of a B8 type star.', 'astro-ph-9907028-1-12-3': 'The spectrum was arbitrarily scaled so as to correspond to an H magnitude of 20.3.', 'astro-ph-9907028-1-12-4': 'This scaling proved to properly match the SOFI spectrum.', 'astro-ph-9907028-1-12-5': 'The signal to noise ratio per pixel is [MATH] 2 on the peak at 1.57 [MATH]m.', 'astro-ph-9907028-1-13-0': 'The combined, flux calibrated, spectrum is presented on figure [REF], overplotted with the spectrum of Gliese 229B for reference (Geballe [CITATION]).', 'astro-ph-9907028-1-14-0': '# Discussion', 'astro-ph-9907028-1-15-0': 'The magnitudes, or magnitude lower limits of NTTDF 1205-0744 are given in table [REF].', 'astro-ph-9907028-1-16-0': 'Both the i-J and the J-K color indices match within less than 0.2 magnitude the color indices of both Gliese 229B and of SDSS 1624+00.', 'astro-ph-9907028-1-17-0': 'Our infrared spectrum shown in figure [REF] has relatively low s/n and some flux calibration uncertainties due to the fact that the observations were made at different times and with different instruments.', 'astro-ph-9907028-1-17-1': 'A detailed discussion of the smaller features is therefore not warranted.', 'astro-ph-9907028-1-17-2': 'For example, the feature in the Ks peak could be real but corresponds to a region of crowded OH sky lines and may just be noise.', 'astro-ph-9907028-1-17-3': 'The most important result here is its striking overall similarity with the spectra of both Gliese 229B and SDSS 1624+00, in particular, the clear presence of the strongest H[MATH]0 and CH[MATH] absorption features, which clearly identifiy it as a methane brown dwarf, and the relative flux distribution which implies a similar temperature.', 'astro-ph-9907028-1-18-0': 'Assuming not only that the colours but also the absolute magnitude is similar to Gliese 229B which is at 5.8 pc we obtain a distance of [MATH] 100 pc to NTTDF 1205-0744.', 'astro-ph-9907028-1-18-1': 'The assumption of a similar absolute magnitude may be justified on the basis of brown dwarf model predictions (Burrows [CITATION]).', 'astro-ph-9907028-1-18-2': 'Although both the colour and the magnitude change over a very large range at any particular brown dwarf age, theoretical isochrones practically overlap in color-magnitude diagrams for the range of colors of interest here.', 'astro-ph-9907028-1-18-3': 'Therefore, even if the mass and the age of NTTDF 1205-0744 may be very different from those of Gliese 229B and SDSS 1624+00, the similar (J-K) color is indicative of a similar absolute magnitude.', 'astro-ph-9907028-1-18-4': 'Thus, although both mass and age are very poorly constrained by our observations (the spectral features placing however the mass safely in the brown dwarf domain), the distance of NTTDF 1205-0744 is considered to be relatively secure.', 'astro-ph-9907028-1-19-0': 'We have SOFI and ISAAC images taken [MATH] 14 months apart.', 'astro-ph-9907028-1-19-1': 'We checked for possible proper motion, but nothing could be detected at the level of 0.3 arcsec (2 [MATH]).', 'astro-ph-9907028-1-20-0': 'NTTDF 1205-0744 stands out as the only object of its type within the 2.3 [MATH] 2.3 arcmin NTT deep field.', 'astro-ph-9907028-1-20-1': 'Although of dubious reliability based on a single object, the implied volume density is 0.5 per cubic parsec (assuming a recognition limit at J=22, see below).', 'astro-ph-9907028-1-20-2': 'This is considerably higher than the 0.01-0.03 per cubic parsec tentatively quoted by Strauss et al. ([CITATION]) based on 400 sq. deg.', 'astro-ph-9907028-1-20-3': 'of the Sloan survey implying that our technique is considerably more sensitive or that we were extremely lucky.', 'astro-ph-9907028-1-21-0': 'The probability of finding a very cold object in a random field with a given limiting magnitude can also be estimated using published brown dwarf models, the local density of low mass stars, and an extrapolation of the initial mass function towards lower masses.', 'astro-ph-9907028-1-21-1': 'We have carried out this exercise using the Burrows et al. ([CITATION]) models and the local volume density at 0.1 solar masses from Scalo ([CITATION]).', 'astro-ph-9907028-1-21-2': 'We have assumed a constant local formation rate of low mass stars over the last 10 Gyr.', 'astro-ph-9907028-1-21-3': 'The initial mass function (IMF) below 0.1 solar masses has been represented by a power-law of the form [MATH], and we have considered values for [MATH] ranging from -1.5 to +1.', 'astro-ph-9907028-1-21-4': 'We have then calculated the number of objects with a temperature lower than 1000 K that may be expected to appear in the field with an apparent J magnitude brighter than 22, under the assumption of the different values of [MATH].', 'astro-ph-9907028-1-21-5': 'Although objects much fainter than J=22 are still visible in the J image, the limit chosen is given by the need to be able to recognize the characteristic colors of possible brown dwarfs, namely the extremely red (i-J) and the blue (J-K).', 'astro-ph-9907028-1-21-6': 'The limiting J magnitude that we use is thus actually defined by the limiting K magnitude, combined with the (J-K) colors expected for the objects of interest.', 'astro-ph-9907028-1-21-7': 'The results are given in Table [REF].', 'astro-ph-9907028-1-22-0': 'These values, although also highly uncertain, tend to indicate that the probability of detecting NTTDF 1205-0744 was actually lower than expected for a volume density of even 0.01 per cubic parsec. This also probably explains why dedicated infrared surveys at much brighter limits (DENIS, 2MASS, see e.g. Reid ([CITATION]), Delfosse [CITATION]) have so far failed to identify such objects.', 'astro-ph-9907028-1-23-0': 'One of the most remarkable features of these objects is the huge I-J color index which should make them difficult to find using visible data alone.', 'astro-ph-9907028-1-23-1': 'Despite the spectacular success of the Sloan Digital Sky Survey which has led to the discovery of SDSS 1624+00, it is clear that the main prospective avenue for further discoveries of such methane brown dwarfs is to resort to relatively deep near IR observations, combined with deep visible imaging.', 'astro-ph-9907028-1-23-2': 'The high I-J (or any visible - J) color index, combined with an almost flat J-K color index, is a very clear indicator for these methane brown dwarfs.'} | {'astro-ph-9907028-2-0-0': 'We report the discovery of an isolated brown dwarf with similar properties to the binary object Gliese 229B and to the newly discovered field brown dwarfs from the SDSS and 2MASS surveys.', 'astro-ph-9907028-2-0-1': 'Although exhibiting similar colors, its magnitude of [MATH] 20.5 is about 6 magnitudes fainter than Gliese 229B .', 'astro-ph-9907028-2-0-2': 'This is the most distant of the several methane brown dwarfs reported to date, at a distance of [MATH] 90 pc.', 'astro-ph-9907028-2-0-3': 'Its IR spectrum, although at low S/N given the faintness of the object, is remarkably similar to those of the other methane brown dwarfs.', 'astro-ph-9907028-2-1-0': '# Introduction', 'astro-ph-9907028-2-2-0': 'Despite large observational efforts during recent years in both wide field and targeted searches for very cold brown dwarfs, the number of such objects known so far remains extremely small.', 'astro-ph-9907028-2-2-1': 'Since 1995, and until June 1999, the only genuine one identified was Gliese 229B (Nakajima [CITATION], Oppenheimer [CITATION]), the coolest substellar object known, with a temperature below 1000 K, a mass in the range 20-50 M[MATH] (Jupiter mass), and an age in the range 0.5-1 Gyr.', 'astro-ph-9907028-2-2-2': 'A second object of this class, SDSS 1624+00, has been discovered recently in the Sloan Digital Sky Survey (Strauss [CITATION]), after identification from the survey database by its unusual red color.', 'astro-ph-9907028-2-2-3': 'Follow-up spectroscopy of this object in the visible with the Apache Point 3.5m telescope and in the IR with UKIRT identified it as a methane brown dwarf like Gliese 229B.', 'astro-ph-9907028-2-2-4': 'A couple of similar objects have since then been identified (Tzetanov, private communication) from the SDSS survey.', 'astro-ph-9907028-2-2-5': 'At almost the same time, 4 other similar objects were identified from the Two Micron All-Sky Survey (2MASS) (Burgasser [CITATION]), and confirmed as methane brown dwarfs from visible spectroscopy at Palomar and IR spectroscopy at Keck.', 'astro-ph-9907028-2-3-0': 'In this paper we report our discovery of a new methane brown dwarf in the NTT Deep Field, a small area of the sky that was the target of very deep exposures in the visible and the near-infrared using the SUSI and SOFI instruments at the ESO New Technology Telescope (NTT) (Arnouts [CITATION], Saracco [CITATION]).', 'astro-ph-9907028-2-3-1': 'One object, NTTDF J1205-0744, stands out in these images for its very red (i-J) [MATH] 6 color index.', 'astro-ph-9907028-2-3-2': 'However, it is very blue at longer wavelengths, with (J-Ks) = -0.15.', 'astro-ph-9907028-2-3-3': 'Near-infrared spectroscopy with SOFI, and with ISAAC at the ESO Very Large Telescope (VLT), has confirmed the remarkable similarity of this object to Gliese 229B.', 'astro-ph-9907028-2-3-4': 'The powerful combinations NTT/SOFI and VLT/ISAAC made the observations reported here possible, in spite of the faint apparent magnitude of NTTDF J1205-0744.', 'astro-ph-9907028-2-3-5': 'Although the raw S/N is limited (1 to 2 per pixel, 5 to 10 after rebinning), our results secure the identification of NTTDF J1205-0744 as a new field methane brown dwarf.', 'astro-ph-9907028-2-4-0': '# Observations and data reduction', 'astro-ph-9907028-2-5-0': 'The NTT Deep Field covers an area of 2.3 [MATH] 2.3 arcminutes in the visible down to AB magnitude limits of 27.2, 27.0, 26.7 and 26.3 in B, V, r, and i, and 5 [MATH] 5 arcminutes in the IR down to magnitude limits of 24.6 and 22.8 in J and Ks.', 'astro-ph-9907028-2-6-0': 'The entire dataset of the NTT Deep Field Project, primarily targeted to the study of faint galaxy populations, as well as a detailed information on data acquisition and reduction, are publicly available at http://www.eso.org.', 'astro-ph-9907028-2-7-0': 'J and i band images of the field containing NTTDF J1205-0744 are shown in figure [REF].', 'astro-ph-9907028-2-8-0': 'After identification of NTTDF J1205-0744 from its unusual extremely red colour (i-J) in April 98, we carried out spectroscopy with SOFI at the NTT using Target of Opportunity Time on 30 June - 1 July 1998.', 'astro-ph-9907028-2-8-1': 'The spectrum, covering the range 0.95-1.65 microns (dispersion: 7 [MATH] per pixel), was obtained under non-photometric conditions using a 1[MATH] slit, and nodding along the slit between two positions, for a total effective on-target integration time of 84 minutes.', 'astro-ph-9907028-2-8-2': 'Spectrophotometric calibration and removal of telluric features was achieved using the observation of a B9 type star.', 'astro-ph-9907028-2-8-3': 'The spectrum was scaled to match the IR photometry in the J filter.', 'astro-ph-9907028-2-9-0': 'The spectrum shows clear H[MATH]O absorptions, leaving peaks in the spectrum at 1.05 and 1.27 [MATH] (the latter peak at a S/N of 1-1.5 per pixel), and a marginally significant detection of a third peak at 1.57 [MATH] .', 'astro-ph-9907028-2-10-0': 'We subsequently obtained spectroscopy of NTTDF J1205-0744 with ISAAC at the VLT in the H and K bands.', 'astro-ph-9907028-2-10-1': 'All the ISAAC observations were made with a 1[MATH] slit and nodding along the slit.', 'astro-ph-9907028-2-11-0': 'The K observations were carried out during the nights of 6 and 9 February 1999, for a total amount of time of 1 hour.', 'astro-ph-9907028-2-11-1': 'We used the Low Resolution grating in second order providing a dispersion per pixel of 7 [MATH].', 'astro-ph-9907028-2-11-2': 'Spectrophotometric calibration was achieved from the observation of a B6 type star observed on a different night.', 'astro-ph-9907028-2-11-3': 'The signal to noise per pixel is below 1 on the peak at 2.1 [MATH] .', 'astro-ph-9907028-2-12-0': 'The observations in H were carried out during the night of 23 March 1999, again for a total integration time of 1 hour.', 'astro-ph-9907028-2-12-1': 'We used the same Low Resolution grating in third order, providing a dispersion per pixel of 4.7 [MATH].', 'astro-ph-9907028-2-12-2': 'Spectrophotometric calibration was achieved from the observation of a B8 type star.', 'astro-ph-9907028-2-12-3': 'The spectrum was arbitrarily scaled so as to correspond to an H magnitude of 20.3.', 'astro-ph-9907028-2-12-4': 'This scaling proved to properly match the SOFI spectrum.', 'astro-ph-9907028-2-12-5': 'The signal to noise ratio per pixel is [MATH] 2 on the peak at 1.57 [MATH].', 'astro-ph-9907028-2-13-0': 'The combined, flux calibrated, spectrum is presented on figure [REF], overplotted with the spectrum of Gliese 229B for reference (Geballe [CITATION]).', 'astro-ph-9907028-2-14-0': '# Discussion', 'astro-ph-9907028-2-15-0': 'The magnitudes, or magnitude lower limits of NTTDF J1205-0744 are given in table [REF].', 'astro-ph-9907028-2-16-0': 'Both the i-J and the J-Ks color indices match within less than 0.2 magnitude the color indices of both Gliese 229B and of the SDSS and 2MASS brown dwarfs.', 'astro-ph-9907028-2-17-0': 'Our infrared spectrum shown in figure [REF] has relatively low s/n and some flux calibration uncertainties due to the fact that the observations were made at different times and with different instruments.', 'astro-ph-9907028-2-17-1': 'A detailed discussion of the smaller features is therefore not warranted.', 'astro-ph-9907028-2-17-2': 'For example, the feature in the Ks peak could be real but corresponds to a region of crowded OH sky lines and may just be noise.', 'astro-ph-9907028-2-17-3': 'The most important result here is its striking overall similarity with the spectra of Gliese 229B and of the recently discovered methane brown dwarfs, in particular, the clear presence of the strongest H[MATH]0 and CH[MATH] absorption features, which clearly identifiy it as a methane brown dwarf, and the relative flux distribution which implies a similar temperature.', 'astro-ph-9907028-2-18-0': 'Assuming not only that the colours but also the absolute magnitude is similar to Gliese 229B which is at 5.8 pc we obtain a distance of [MATH] 90 pc to NTTDF J1205-0744 ([MATH] = 6 magnitudes).', 'astro-ph-9907028-2-18-1': 'The assumption of a similar absolute magnitude may be justified on the basis of brown dwarf model predictions (Burrows [CITATION]).', 'astro-ph-9907028-2-18-2': 'Although both the colour and the magnitude change over a very large range at any particular brown dwarf age, theoretical isochrones practically overlap in color-magnitude diagrams for the range of colors of interest here.', 'astro-ph-9907028-2-18-3': 'Therefore, even if the mass and the age of NTTDF J1205-0744 may be very different from those of the other methane brown dwarfs, the similar (J-Ks) color is indicative of a similar absolute magnitude.', 'astro-ph-9907028-2-18-4': 'Thus, although both mass and age are very poorly constrained by our observations (the spectral features placing however the mass safely in the brown dwarf domain), the distance of NTTDF J1205-0744 is considered to be relatively secure.', 'astro-ph-9907028-2-19-0': 'We have SOFI and ISAAC images taken [MATH] 14 months apart.', 'astro-ph-9907028-2-19-1': 'We looked for possible proper motion, but nothing was detected at the level of 0.3 arcsec (2 [MATH]).', 'astro-ph-9907028-2-20-0': 'NTTDF J1205-0744 stands out as the only object of its type within the 2.3 [MATH] 2.3 arcminutes NTT deep field.', 'astro-ph-9907028-2-20-1': 'Although of dubious reliability based on a single object, the implied volume density is [MATH] 1 per cubic parsec (assuming a recognition limit at J=22, see below, corresponding to a distance of [MATH] 200 pc).', 'astro-ph-9907028-2-20-2': 'This is considerably higher than the 0.01-0.03 per cubic parsec tentatively quoted by Strauss et al. ([CITATION]) and than the 0.01 per cubic parsec derived from the discoveries of the 2MASS methane brown dwarfs (Burgasser [CITATION]).', 'astro-ph-9907028-2-20-3': 'This implies that either our technique is considerably more sensitive or, more likely, that we were extremely lucky.', 'astro-ph-9907028-2-21-0': 'The probability of finding a very cold object in a random field with a given limiting magnitude can also be estimated using published brown dwarf models, the local density of low mass stars, and an extrapolation of the initial mass function towards lower masses.', 'astro-ph-9907028-2-21-1': 'We have carried out this exercise using the Burrows et al. ([CITATION]) models and the local volume density at 0.1 solar masses from Scalo ([CITATION]).', 'astro-ph-9907028-2-21-2': 'We have assumed a constant local formation rate of low mass stars over the last 10 Gyr.', 'astro-ph-9907028-2-21-3': 'The initial mass function (IMF) below 0.1 solar masses has been represented by a power-law of the form [MATH], and we have considered values for [MATH] ranging from -1.5 to +1.', 'astro-ph-9907028-2-21-4': 'We have then calculated the number of objects with a temperature lower than 1000 K that may be expected to appear in the field with an apparent J magnitude brighter than 22, under the assumption of the different values of [MATH].', 'astro-ph-9907028-2-21-5': 'Although objects much fainter than J=22 are still visible in the J image, the limit chosen is given by the need to be able to recognize the characteristic colors of possible brown dwarfs, namely the extremely red (i-J) and the blue (J-Ks).', 'astro-ph-9907028-2-21-6': 'The limiting J magnitude that we use is thus actually defined by the limiting Ks magnitude, combined with the (J-Ks) colors expected for the objects of interest.', 'astro-ph-9907028-2-21-7': 'The results are given in Table [REF].', 'astro-ph-9907028-2-22-0': 'These values are much lower than the 1% probability one would expect for a volume density of 0.01 per cubic parsec, suggesting that a negative slope much steeper than -1.5 would be required for the IMF to fit with the observed density.', 'astro-ph-9907028-2-23-0': 'One of the most remarkable features of these objects is the huge I-J color index which make them difficult to find using visible data alone.', 'astro-ph-9907028-2-23-1': 'Despite the spectacular success of the Sloan Digital Sky Survey which has led to the discovery of SDSS 1624+00, the main avenue for unveiling in a systematic way this new population of methane brown dwarfs is to resort to combined visible (I) and IR (J and H or J and Ks) deep observations, as demonstrated by the 2MASS discoveries and by the present work.', 'astro-ph-9907028-2-23-2': 'It is interesting to note that the DENIS survey (Delfosse [CITATION]) did not detect so far such methane brown dwarfs, which might be explained by the relatively low detection limit in Ks (13.5).', 'astro-ph-9907028-2-23-3': 'With a volume density of 0.01 per cubic parsec, the chance of finding a methane brown dwarf brighter than this limit is [MATH] 1 over the whole sky.', 'astro-ph-9907028-2-24-0': 'The high I-J (or any visible - J) color index, combined with an almost flat J-H or J-Ks color index, is a very clear indicator for these methane brown dwarfs.'} | [['astro-ph-9907028-1-19-0', 'astro-ph-9907028-2-19-0'], ['astro-ph-9907028-1-8-2', 'astro-ph-9907028-2-8-2'], ['astro-ph-9907028-1-8-3', 'astro-ph-9907028-2-8-3'], ['astro-ph-9907028-1-3-0', 'astro-ph-9907028-2-3-0'], ['astro-ph-9907028-1-3-3', 'astro-ph-9907028-2-3-3'], ['astro-ph-9907028-1-18-1', 'astro-ph-9907028-2-18-1'], ['astro-ph-9907028-1-18-2', 'astro-ph-9907028-2-18-2'], ['astro-ph-9907028-1-2-0', 'astro-ph-9907028-2-2-0'], ['astro-ph-9907028-1-2-1', 'astro-ph-9907028-2-2-1'], ['astro-ph-9907028-1-2-3', 'astro-ph-9907028-2-2-3'], ['astro-ph-9907028-1-6-0', 'astro-ph-9907028-2-6-0'], ['astro-ph-9907028-1-17-0', 'astro-ph-9907028-2-17-0'], ['astro-ph-9907028-1-17-1', 'astro-ph-9907028-2-17-1'], ['astro-ph-9907028-1-17-2', 'astro-ph-9907028-2-17-2'], ['astro-ph-9907028-1-12-0', 'astro-ph-9907028-2-12-0'], ['astro-ph-9907028-1-12-1', 'astro-ph-9907028-2-12-1'], ['astro-ph-9907028-1-12-3', 'astro-ph-9907028-2-12-3'], ['astro-ph-9907028-1-12-4', 'astro-ph-9907028-2-12-4'], ['astro-ph-9907028-1-13-0', 'astro-ph-9907028-2-13-0'], ['astro-ph-9907028-1-5-0', 'astro-ph-9907028-2-5-0'], ['astro-ph-9907028-1-11-0', 'astro-ph-9907028-2-11-0'], ['astro-ph-9907028-1-11-1', 'astro-ph-9907028-2-11-1'], ['astro-ph-9907028-1-11-2', 'astro-ph-9907028-2-11-2'], ['astro-ph-9907028-1-21-0', 'astro-ph-9907028-2-21-0'], ['astro-ph-9907028-1-21-1', 'astro-ph-9907028-2-21-1'], ['astro-ph-9907028-1-21-2', 'astro-ph-9907028-2-21-2'], ['astro-ph-9907028-1-21-3', 'astro-ph-9907028-2-21-3'], ['astro-ph-9907028-1-21-4', 'astro-ph-9907028-2-21-4'], ['astro-ph-9907028-1-21-7', 'astro-ph-9907028-2-21-7'], ['astro-ph-9907028-1-19-1', 'astro-ph-9907028-2-19-1'], ['astro-ph-9907028-1-8-0', 'astro-ph-9907028-2-8-0'], ['astro-ph-9907028-1-8-1', 'astro-ph-9907028-2-8-1'], ['astro-ph-9907028-1-3-1', 'astro-ph-9907028-2-3-1'], ['astro-ph-9907028-1-3-2', 'astro-ph-9907028-2-3-2'], ['astro-ph-9907028-1-3-5', 'astro-ph-9907028-2-3-5'], ['astro-ph-9907028-1-16-0', 'astro-ph-9907028-2-16-0'], ['astro-ph-9907028-1-9-0', 'astro-ph-9907028-2-9-0'], ['astro-ph-9907028-1-7-0', 'astro-ph-9907028-2-7-0'], ['astro-ph-9907028-1-18-0', 'astro-ph-9907028-2-18-0'], ['astro-ph-9907028-1-18-4', 'astro-ph-9907028-2-18-4'], ['astro-ph-9907028-1-2-2', 'astro-ph-9907028-2-2-2'], ['astro-ph-9907028-1-10-0', 'astro-ph-9907028-2-10-0'], ['astro-ph-9907028-1-10-1', 'astro-ph-9907028-2-10-1'], ['astro-ph-9907028-1-17-3', 'astro-ph-9907028-2-17-3'], ['astro-ph-9907028-1-20-0', 'astro-ph-9907028-2-20-0'], ['astro-ph-9907028-1-12-2', 'astro-ph-9907028-2-12-2'], ['astro-ph-9907028-1-12-5', 'astro-ph-9907028-2-12-5'], ['astro-ph-9907028-1-0-0', 'astro-ph-9907028-2-0-0'], ['astro-ph-9907028-1-0-1', 'astro-ph-9907028-2-0-1'], ['astro-ph-9907028-1-0-3', 'astro-ph-9907028-2-0-3'], ['astro-ph-9907028-1-11-3', 'astro-ph-9907028-2-11-3'], ['astro-ph-9907028-1-15-0', 'astro-ph-9907028-2-15-0'], ['astro-ph-9907028-1-21-5', 'astro-ph-9907028-2-21-5'], ['astro-ph-9907028-1-21-6', 'astro-ph-9907028-2-21-6'], ['astro-ph-9907028-1-23-0', 'astro-ph-9907028-2-23-0'], ['astro-ph-9907028-1-23-2', 'astro-ph-9907028-2-24-0'], ['astro-ph-9907028-1-18-3', 'astro-ph-9907028-2-18-3'], ['astro-ph-9907028-1-2-4', 'astro-ph-9907028-2-2-4'], ['astro-ph-9907028-1-20-1', 'astro-ph-9907028-2-20-1'], ['astro-ph-9907028-1-20-2', 'astro-ph-9907028-2-20-2'], ['astro-ph-9907028-1-20-3', 'astro-ph-9907028-2-20-3'], ['astro-ph-9907028-1-0-2', 'astro-ph-9907028-2-0-2'], ['astro-ph-9907028-1-23-1', 'astro-ph-9907028-2-23-1']] | [['astro-ph-9907028-1-19-0', 'astro-ph-9907028-2-19-0'], ['astro-ph-9907028-1-8-2', 'astro-ph-9907028-2-8-2'], ['astro-ph-9907028-1-8-3', 'astro-ph-9907028-2-8-3'], ['astro-ph-9907028-1-3-0', 'astro-ph-9907028-2-3-0'], ['astro-ph-9907028-1-3-3', 'astro-ph-9907028-2-3-3'], ['astro-ph-9907028-1-18-1', 'astro-ph-9907028-2-18-1'], ['astro-ph-9907028-1-18-2', 'astro-ph-9907028-2-18-2'], ['astro-ph-9907028-1-2-0', 'astro-ph-9907028-2-2-0'], ['astro-ph-9907028-1-2-1', 'astro-ph-9907028-2-2-1'], ['astro-ph-9907028-1-2-3', 'astro-ph-9907028-2-2-3'], ['astro-ph-9907028-1-6-0', 'astro-ph-9907028-2-6-0'], ['astro-ph-9907028-1-17-0', 'astro-ph-9907028-2-17-0'], ['astro-ph-9907028-1-17-1', 'astro-ph-9907028-2-17-1'], ['astro-ph-9907028-1-17-2', 'astro-ph-9907028-2-17-2'], ['astro-ph-9907028-1-12-0', 'astro-ph-9907028-2-12-0'], ['astro-ph-9907028-1-12-1', 'astro-ph-9907028-2-12-1'], ['astro-ph-9907028-1-12-3', 'astro-ph-9907028-2-12-3'], ['astro-ph-9907028-1-12-4', 'astro-ph-9907028-2-12-4'], ['astro-ph-9907028-1-13-0', 'astro-ph-9907028-2-13-0'], ['astro-ph-9907028-1-5-0', 'astro-ph-9907028-2-5-0'], ['astro-ph-9907028-1-11-0', 'astro-ph-9907028-2-11-0'], ['astro-ph-9907028-1-11-1', 'astro-ph-9907028-2-11-1'], ['astro-ph-9907028-1-11-2', 'astro-ph-9907028-2-11-2'], ['astro-ph-9907028-1-21-0', 'astro-ph-9907028-2-21-0'], ['astro-ph-9907028-1-21-1', 'astro-ph-9907028-2-21-1'], ['astro-ph-9907028-1-21-2', 'astro-ph-9907028-2-21-2'], ['astro-ph-9907028-1-21-3', 'astro-ph-9907028-2-21-3'], ['astro-ph-9907028-1-21-4', 'astro-ph-9907028-2-21-4'], ['astro-ph-9907028-1-21-7', 'astro-ph-9907028-2-21-7']] | [['astro-ph-9907028-1-19-1', 'astro-ph-9907028-2-19-1'], ['astro-ph-9907028-1-8-0', 'astro-ph-9907028-2-8-0'], ['astro-ph-9907028-1-8-1', 'astro-ph-9907028-2-8-1'], ['astro-ph-9907028-1-3-1', 'astro-ph-9907028-2-3-1'], ['astro-ph-9907028-1-3-2', 'astro-ph-9907028-2-3-2'], ['astro-ph-9907028-1-3-5', 'astro-ph-9907028-2-3-5'], ['astro-ph-9907028-1-16-0', 'astro-ph-9907028-2-16-0'], ['astro-ph-9907028-1-9-0', 'astro-ph-9907028-2-9-0'], ['astro-ph-9907028-1-7-0', 'astro-ph-9907028-2-7-0'], ['astro-ph-9907028-1-18-0', 'astro-ph-9907028-2-18-0'], ['astro-ph-9907028-1-18-4', 'astro-ph-9907028-2-18-4'], ['astro-ph-9907028-1-2-2', 'astro-ph-9907028-2-2-2'], ['astro-ph-9907028-1-10-0', 'astro-ph-9907028-2-10-0'], ['astro-ph-9907028-1-10-1', 'astro-ph-9907028-2-10-1'], ['astro-ph-9907028-1-17-3', 'astro-ph-9907028-2-17-3'], ['astro-ph-9907028-1-20-0', 'astro-ph-9907028-2-20-0'], ['astro-ph-9907028-1-12-2', 'astro-ph-9907028-2-12-2'], ['astro-ph-9907028-1-12-5', 'astro-ph-9907028-2-12-5'], ['astro-ph-9907028-1-0-0', 'astro-ph-9907028-2-0-0'], ['astro-ph-9907028-1-0-1', 'astro-ph-9907028-2-0-1'], ['astro-ph-9907028-1-0-3', 'astro-ph-9907028-2-0-3'], ['astro-ph-9907028-1-11-3', 'astro-ph-9907028-2-11-3'], ['astro-ph-9907028-1-15-0', 'astro-ph-9907028-2-15-0'], ['astro-ph-9907028-1-21-5', 'astro-ph-9907028-2-21-5'], ['astro-ph-9907028-1-21-6', 'astro-ph-9907028-2-21-6'], ['astro-ph-9907028-1-23-0', 'astro-ph-9907028-2-23-0'], ['astro-ph-9907028-1-23-2', 'astro-ph-9907028-2-24-0']] | [] | [['astro-ph-9907028-1-18-3', 'astro-ph-9907028-2-18-3'], ['astro-ph-9907028-1-2-4', 'astro-ph-9907028-2-2-4'], ['astro-ph-9907028-1-20-1', 'astro-ph-9907028-2-20-1'], ['astro-ph-9907028-1-20-2', 'astro-ph-9907028-2-20-2'], ['astro-ph-9907028-1-20-3', 'astro-ph-9907028-2-20-3'], ['astro-ph-9907028-1-0-2', 'astro-ph-9907028-2-0-2'], ['astro-ph-9907028-1-23-1', 'astro-ph-9907028-2-23-1']] | [] | [] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/9907028 | null | null | null | null | null |
0801.3958 | {'0801.3958-1-0-0': '# Introduction', '0801.3958-1-1-0': "Zilber's original trichotomy conjecture proposed an explicit classification of all one-dimensional objects arising in model theory.", '0801.3958-1-1-1': 'At one point, classifying the simple groups of finite Morley rank was viewed as a subproblem whose affirmative answer would justify this conjecture.', '0801.3958-1-1-2': "Zilber's conjecture was eventually refuted by Hrushovski [CITATION], and the classification of simple groups of finite Morley rank remains open today.", '0801.3958-1-1-3': 'However, these conjectures hold in two significant cases.', '0801.3958-1-1-4': 'First, Hrushovski and Zilber prove the full trichotomy conjecture holds under very strong geometric assumptions [CITATION], and this suffices for various diophantine applications.', '0801.3958-1-1-5': 'Second, the Even Mixed Type Theorem [CITATION] shows that simple groups of finite Morley rank containing an infinite elementary abelian 2-subgroup are Chevellay groups over an algebraically closed field of characteristic two.', '0801.3958-1-2-0': 'In this paper, we clarify some middle ground between these two results by eliminating involutions from simple groups of finite Morley rank which are definable embedded in a linear group over an algebraically closed field of characteristic zero, and are not Zariski closed themselves.', '0801.3958-1-3-0': 'Theorem Let [MATH] be a definable infinite simple subgroup of [MATH] over a field [MATH] of finite Morley rank and characteristic zero, which is not Zariski closed.', '0801.3958-1-3-1': 'Then [MATH] has no involutions, and its Carter/Borel subgroups are self-normalizing.', '0801.3958-1-4-0': 'Poizat has shown that simple groups with such a definable linear embedding are Chevellay groups when the field has characteristic [MATH] [CITATION].', '0801.3958-1-4-1': 'Much geometric information is available about such groups but not nearly as much as [CITATION] assumes.', '0801.3958-1-4-2': 'For our purposes, the most important geometric fact is that every element of a counterexample is semisimple in the ambient linear group.', '0801.3958-1-5-0': 'We note that such groups are exactly those groups interpretable in bad fields, which have been recently shown to exist.', '0801.3958-1-5-1': 'However, the bad fields constructed with current methods do not interpret new simple groups.', '0801.3958-1-6-0': 'A major feature of our proof is the elimination of 2-tori of outer automorphism of simple groups via the Delehan-Nesin argument [CITATION] (see Lemma [REF]).', '0801.3958-1-6-1': 'Such an application is a hopeful sign for the current project to classify of simple groups of finite Morley rank with an involution.', '0801.3958-1-6-2': 'However, our proof is driven primarily by one unreasonably strong fact about these linear groups: all strongly real elements have a nontrivial power inside a unique conjugacy class of Borel subgroups.', '0801.3958-1-7-0': 'In the first section, we reduce the problem to characteristic zero using Poizat, and then to the hypotheses used later by our inductive argument.', '0801.3958-1-7-1': 'In the second section, we recall various facts about groups without unipotent torsion.', '0801.3958-1-7-2': 'In the third section, we prove our main result using the earlier reductions.', '0801.3958-1-8-0': '# Definably linear groups', '0801.3958-1-9-0': 'In this section, we analyze so-called definable linear groups.', '0801.3958-1-9-1': 'Poizat has already shown that simple linear groups over fields of characteristic [MATH] are isomorphic to algebraic groups.', '0801.3958-1-10-0': '[Poizat [CITATION]] Let [MATH] be a field of finite Morley rank and characteristic [MATH].', '0801.3958-1-10-1': 'Then any simple definable subgroup of [MATH] is isomorphic to an algebraic group over [MATH].', '0801.3958-1-11-0': 'So we consider only fields of characteristic zero.', '0801.3958-1-12-0': 'Let [MATH] be a definable connected subgroup of [MATH] over a field [MATH] of finite Morley rank and characteristic zero.', '0801.3958-1-13-0': 'We now show that all definable subgroups satisfy conjugacy of Borel subgroups with the following two facts.', '0801.3958-1-14-0': '[Poizat [CITATION]] If such a [MATH] is simple, but not Zariski closed, then all elements of [MATH] are semisimple.', '0801.3958-1-15-0': '[Mustafin [CITATION]] If [MATH] consists entirely of semisimple elements, then all Borel subgroups of [MATH] are conjugate, and this conjugacy class is generic.', '0801.3958-1-16-0': 'Let [MATH] be a Carter subgroup of [MATH], and let [MATH] be a maximal algebraic torus of [MATH] which contains [MATH].', '0801.3958-1-16-1': 'There are only finitely many distinct intersections [MATH] for [MATH], by [CITATION].', '0801.3958-1-16-2': 'As any distinct conjugates of [MATH] lie in distinct conjugates of [MATH], there are only finitely many distinct intersections [MATH] for [MATH] too, and so the union of such intersections is not generic in [MATH].', '0801.3958-1-16-3': 'By the genericity argument [CITATION], [MATH] is generic in [MATH].', '0801.3958-1-16-4': 'Now [MATH] has only one conjugacy class of Carter subgroups, by [CITATION].', '0801.3958-1-17-0': 'This one fact is all we require to verify Hypothesis [REF] below, with which we will eliminate involutions.', '0801.3958-1-18-0': 'If [MATH] is simple, but not Zariski closed, then [MATH] has no involutions and its Borel subgroups are self-normalizing.', '0801.3958-1-19-0': 'Such a group [MATH] has no unipotent torsion since [MATH] has characteristic zero.', '0801.3958-1-19-1': 'By Fact [REF], all elements of [MATH] are semisimple.', '0801.3958-1-19-2': 'So all connected solvable subgroups are nilpotent.', '0801.3958-1-19-3': 'All definable subgroups satisfy conjugacy of their Borel subgroups, by Fact [REF] too.', '0801.3958-1-19-4': 'Hence [MATH] satisfies Hypothesis [REF] below.', '0801.3958-1-19-5': 'So [MATH] has no involutions by Theorem [REF].', '0801.3958-1-19-6': 'As Borel subgroups are conjugate and equal to Carter subgroups, the Borel subgroups of [MATH] are self-normalizing by Lemma [REF].', '0801.3958-1-20-0': '# No unipotent torsion', '0801.3958-1-21-0': 'We recall several convenient results from [CITATION] concerning groups without unipotent torsion, i.e. which contains no definable connected nilpotent subgroup of bounded exponent.', '0801.3958-1-22-0': 'Let [MATH] be a connected group of finite Morley rank without unipotent torsion.', '0801.3958-1-22-1': 'Then every element has a nontrivial power which lies inside a Borel subgroup.', '0801.3958-1-23-0': 'This is a corollary of the following fact.', '0801.3958-1-24-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank of [MATH]-type.', '0801.3958-1-24-1': 'Then all [MATH]-elements of [MATH] are toral, i.e. lie inside [MATH]-tori.', '0801.3958-1-25-0': '[Proof of Lemma [REF]] Consider an element [MATH].', '0801.3958-1-25-1': 'If [MATH] is infinite, we choose [MATH].', '0801.3958-1-25-2': 'If [MATH] is finite, [MATH] lies insider some torus by Fact [REF].', '0801.3958-1-26-0': 'We observe that, if one assumes that Borel subgroups are conjugate, one may instead prove that all elements lie inside Borel subgroups.', '0801.3958-1-27-0': 'Let [MATH] be a connected degenerate type group of finite Morley rank without unipotent torsion.', '0801.3958-1-27-1': 'Then [MATH] has some self-normalizing Carter subgroup.', '0801.3958-1-28-0': 'This is a corollary of the following.', '0801.3958-1-29-0': 'Let [MATH] be a group of finite Morley rank, and [MATH] a maximal divisible abelian torsion subgroup of [MATH].', '0801.3958-1-29-1': 'The Weyl group of [MATH] is the group [MATH], which can be viewed as a group of automorphisms of [MATH].', '0801.3958-1-30-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank.', '0801.3958-1-30-1': 'Suppose the Weyl group is nontrivial and has odd order, with [MATH] the smallest prime divisor of its order.', '0801.3958-1-30-2': 'Then [MATH] contains a unipotent [MATH]-subgroup.', '0801.3958-1-31-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank, [MATH] a maximal [MATH]-torus in [MATH], and suppose that [MATH] is central in [MATH].', '0801.3958-1-31-1': 'If [MATH] has [MATH]-type, then any [MATH]-element [MATH] belongs to [MATH].', '0801.3958-1-32-0': '[Proof of Lemma [REF]] We may assume that [MATH] contains torsion.', '0801.3958-1-32-1': 'Let [MATH] be a maximal divisible abelian torsion subgroup of [MATH].', '0801.3958-1-32-2': 'By [CITATION], [MATH] has a Carter subgroup [MATH] containing [MATH].', '0801.3958-1-32-3': 'By Fact [REF], [MATH].', '0801.3958-1-32-4': 'So [MATH].', '0801.3958-1-32-5': 'By Fact [REF], [MATH] is torsion free.', '0801.3958-1-32-6': 'So [MATH], as desired.', '0801.3958-1-33-0': '# Bad-like groups', '0801.3958-1-34-0': 'Here we analyze groups satisfying the following hypotheses.', '0801.3958-1-35-0': 'Let [MATH] be a connected group of finite Morley rank without unipotent torsion, whose Borel subgroups are nilpotent and hereditarily conjugate, i.e. any subgroup [MATH] of [MATH] satisfies [MATH]-conjugacy of Borel subgroups of [MATH].', '0801.3958-1-36-0': 'These hypotheses clearly pass to subgroups.', '0801.3958-1-36-1': 'We observe that they also pass to sections by nilpotent kernels.', '0801.3958-1-37-0': 'Let [MATH] be a nilpotent normal subgroup of [MATH].', '0801.3958-1-37-1': 'Then [MATH] satisfies Hypothesis [REF] too, and images and inverse images preserve Borel subgroups.', '0801.3958-1-38-0': 'Quotients and extensions of solvable groups are solvable, so images and inverse images preserve maximal solvable subgroups.', '0801.3958-1-38-1': 'As images also preserve connectivity, they preserve Borel subgroups, and Borel subgroups of [MATH] are nilpotent.', '0801.3958-1-38-2': 'If [MATH] is connected, inverse images are connected too, and hence preserve Borel subgroups too.', '0801.3958-1-38-3': 'We may now assume that [MATH] is finite, and even cyclic of order [MATH].', '0801.3958-1-38-4': 'As [MATH] is connected, [MATH] is central in [MATH].', '0801.3958-1-38-5': 'So, by Fact [REF], [MATH] is contained in every maximal [MATH]-torus of [MATH].', '0801.3958-1-38-6': 'By conjugacy, [MATH] is contained in every Borel subgroup of [MATH] too.', '0801.3958-1-38-7': 'So all inverse images are connected, and hence preserve Borel subgroups.', '0801.3958-1-38-8': 'Our first part now follows easily by applying the second inside all definable connected subgroups of [MATH]', '0801.3958-1-39-0': 'A variation on the Delehan-Nesin argument [CITATION] shows that 2-tori have no interesting actions inside such a group.', '0801.3958-1-40-0': 'Let [MATH] be a normal subgroup of [MATH] which has no involutions.', '0801.3958-1-40-1': 'Then every 2-torus in [MATH] centralizes [MATH].', '0801.3958-1-41-0': 'Let [MATH] be a 2-torus of [MATH].', '0801.3958-1-41-1': 'We may take [MATH].', '0801.3958-1-41-2': 'We may assume [MATH] is centerless, by Lemma [REF].', '0801.3958-1-41-3': 'So there is a nontrivial strongly real element [MATH] inside [MATH].', '0801.3958-1-41-4': 'It follows, by Lemma [REF], that some nontrivial power [MATH] of [MATH] lies inside some Borel subgroup [MATH] of [MATH].', '0801.3958-1-41-5': 'As all Borel subgroups of [MATH] are conjugate, Frattini argument says that [MATH] is normalized by an [MATH]-conjugate [MATH] of [MATH].', '0801.3958-1-41-6': 'So [MATH] centralizes [MATH] by [CITATION], and [MATH] is centralized by [MATH].', '0801.3958-1-41-7': 'As [MATH] has no involutions, [CITATION] says [MATH] where [MATH].', '0801.3958-1-41-8': 'So there is a [MATH] such that [MATH].', '0801.3958-1-41-9': 'As [MATH] is centralized by [MATH], we have [MATH].', '0801.3958-1-41-10': 'Now [MATH] and [MATH].', '0801.3958-1-41-11': 'As [MATH] has no involutions, [MATH], a contradiction.', '0801.3958-1-42-0': 'We say a subgroup [MATH] of a group [MATH] is subnormal if there is a finite chain [EQUATION]', '0801.3958-1-42-1': 'A quasisimple subnormal subgroup of a group [MATH] is referred to as a component of [MATH].', '0801.3958-1-42-2': 'If [MATH] is connected, then its components are definable, connected, and normal in [MATH], by [CITATION].', '0801.3958-1-42-3': 'They also centralize one another by [CITATION].', '0801.3958-1-42-4': 'We let [MATH] denote the central product of the components in [MATH], which happen to be connected, and set [MATH].', '0801.3958-1-42-5': 'According to the following, [MATH] controls the structure of [MATH].', '0801.3958-1-43-0': '[[CITATION]] For any group [MATH] of finite Morley rank, we have [EQUATION]', '0801.3958-1-43-1': 'If all simple sections of [MATH] with nilpotent kernels have degenerate type, then [MATH] has a unique Sylow 2-subgroup, which is connected and central.', '0801.3958-1-44-0': 'Here we use the following.', '0801.3958-1-45-0': '[[CITATION]] A connected degenerate type group of finite Morley rank has no involutions.', '0801.3958-1-46-0': '[Proof of Lemma [REF]] We may assume that [MATH] itself is the section of interest by Lemma [REF].', '0801.3958-1-46-1': 'We may also assume that [MATH] by Lemma [REF].', '0801.3958-1-46-2': 'As [MATH] is [MATH] by hypothesis, Fact [REF] says that [MATH] is [MATH] too.', '0801.3958-1-46-3': 'But [MATH] by Fact [REF].', '0801.3958-1-46-4': 'So any 2-torsion in [MATH] is central by Lemma [REF].', '0801.3958-1-46-5': 'The Sylow 2-subgroup is connected by [CITATION].', '0801.3958-1-47-0': 'We now prove our main result.', '0801.3958-1-48-0': 'Suppose that [MATH] is simple.', '0801.3958-1-48-1': 'Then [MATH] has no involutions.', '0801.3958-1-49-0': 'We consider a counterexample [MATH] of minimal Morley rank.', '0801.3958-1-49-1': 'So, by Lemma [REF], all simple section with nilpotent kernels have degenerate type.', '0801.3958-1-49-2': 'By Lemma [REF], any proper connected subgroup [MATH] centralizes its Sylow 2-subgroup, which is a 2-torus.', '0801.3958-1-49-3': 'It follows that [MATH] the centralizers of distinct Sylowo2-subgroups are disjoint.', '0801.3958-1-50-0': 'Also, any Sylowo2-subgroup is central in a Borel subgroup.', '0801.3958-1-50-1': 'By conjugacy, all Borel subgroups contain central Sylowo2-subgroup.', '0801.3958-1-50-2': 'So [MATH] two Borel subgroups meet iff they both contain the same Sylowo2-subgroup of [MATH].', '0801.3958-1-51-0': 'Any two distinct involutions [MATH] and [MATH] together normalize a Sylowo2-subgroup [MATH] of [MATH], and [MATH].', '0801.3958-1-52-0': 'By Lemma [REF], there is a nontrivial power [MATH] of [MATH] inside a Borel subgroup [MATH] of [MATH].', '0801.3958-1-52-1': 'As [MATH] inverts [MATH], [MATH] also contains [MATH].', '0801.3958-1-52-2': 'By [MATH], [MATH] and [MATH] contain the same Sylowo2-subgroup [MATH] of [MATH].', '0801.3958-1-52-3': 'So [MATH] normalizes [MATH].', '0801.3958-1-52-4': 'Similarly, [MATH] normalizes [MATH] too.', '0801.3958-1-53-0': 'Every involution is contained in a Sylowo2-subgroup, by Fact [REF], which is unique by [MATH].', '0801.3958-1-53-1': 'Let [MATH] denote the Sylowo2-subgroup containing [MATH].', '0801.3958-1-53-2': 'Now [MATH] since [MATH] is the unique Sylowo2-subgroup of [MATH], by [MATH].', '0801.3958-1-54-0': 'Let [MATH] be a Sylowo2-subgroup of [MATH], and let [MATH] be an involution normalizing [MATH].', '0801.3958-1-54-1': 'Then either [MATH] or [MATH] inverts [MATH].', '0801.3958-1-54-2': 'In particular, [MATH] is an abelian Borel subgroup of [MATH].', '0801.3958-1-55-0': 'Suppose that [MATH].', '0801.3958-1-55-1': 'Then [MATH].', '0801.3958-1-55-2': 'So, by [CITATION], [MATH] is abelian and inverted by [MATH].', '0801.3958-1-55-3': 'Now [MATH] is a Borel subgroup of [MATH], as it contained a Borel subgroup of [MATH].', '0801.3958-1-56-0': 'As a consequence, if involutions [MATH] and [MATH] do not commute, then neither [MATH] nor [MATH] lie inside a Sylowo2-subgroup which they normalize together, and both invert its Borel subgroup.', '0801.3958-1-57-0': '[MATH] has rank one.', '0801.3958-1-57-1': 'So a Sylow 2-subgroup of [MATH] is isomorphic to that of [MATH].', '0801.3958-1-58-0': 'Suppose towards a contradiction that [MATH].', '0801.3958-1-58-1': 'Choose a pair of non-commuting involutions [MATH] and [MATH], and fix a Sylowo2-subgroup [MATH] normalized by them both.', '0801.3958-1-58-2': 'As [MATH] inverts [MATH] by Claim [REF], [MATH] normalizes [MATH].', '0801.3958-1-58-3': 'By generation [CITATION], [MATH].', '0801.3958-1-58-4': 'So [MATH] for some [MATH], which yields the contradiction [MATH].', '0801.3958-1-58-5': 'Thus [MATH].', '0801.3958-1-59-0': 'Let [MATH] be a Sylow 2-subgroup of [MATH].', '0801.3958-1-59-1': 'By Fact [REF], [MATH], and [MATH].', '0801.3958-1-59-2': 'There are involutions in [MATH] by Claims [REF] and [REF].', '0801.3958-1-59-3': 'So [MATH] has the desired structure.', '0801.3958-1-60-0': 'Consider the set [MATH] of all involutions in [MATH].', '0801.3958-1-60-1': 'We define the set [MATH] of lines on [MATH] to be the collection of cosets [MATH] where [MATH] is a Borel subgroup of [MATH] and [MATH] inverts [MATH].', '0801.3958-1-60-2': 'We now prove that [MATH] and [MATH] form a projective plane.', '0801.3958-1-61-0': 'PP2 Any two points lie on a unique line.', '0801.3958-1-62-0': 'Any two involutions [MATH] normalize some Sylowo2-subgroup [MATH], by Claim [REF].', '0801.3958-1-62-1': 'By Claim [REF], [MATH] is a Borel subgroup inverted by [MATH] and [MATH].', '0801.3958-1-62-2': 'So [MATH] and [MATH] lie on the line [MATH].', '0801.3958-1-62-3': 'Let [MATH] denote the power of [MATH] lies inside [MATH].', '0801.3958-1-62-4': 'By [MATH], [MATH] is the unique Sylowo2-subgroup of [MATH].', '0801.3958-1-62-5': 'So the line [MATH] is unique.', '0801.3958-1-63-0': 'PP3 Any two lines intersect at a unique point.', '0801.3958-1-64-0': 'For any two lines [MATH] and [MATH], we consider the involutions [MATH] and [MATH], which are unique by Claim [REF].', '0801.3958-1-64-1': 'Again, there is a Borel subgroup [MATH] inverted by both, by Claims [REF] and [REF].', '0801.3958-1-64-2': 'As [MATH] and [MATH] both centralize the involution [MATH], [MATH] inverts both [MATH] and [MATH], by Claims [REF] and [REF].', '0801.3958-1-64-3': 'Thus the lines [MATH] and [MATH] meet at [MATH].', '0801.3958-1-64-4': 'Such a point is unique by PP2.', '0801.3958-1-65-0': 'PP4 There are four points, no three of which are colinear.', '0801.3958-1-66-0': 'Consider a Borel subgroup [MATH] of [MATH], and the unique involution [MATH].', '0801.3958-1-66-1': 'Let [MATH], [MATH], and [MATH] be three involutions inverting [MATH], and let [MATH] be a third point on the line through [MATH] and [MATH].', '0801.3958-1-66-2': 'The original points [MATH] are obviously not colinear.', '0801.3958-1-66-3': 'The pair [MATH] is not colinear with either [MATH] or [MATH] by PP3.', '0801.3958-1-66-4': 'Also [MATH] is not colinear by PP2.', '0801.3958-1-66-5': 'So [MATH] are the four desired involutions.', '0801.3958-1-67-0': 'We prove one additional property of this projective plane.', '0801.3958-1-68-0': 'PP+ Any three involutions [MATH] are colinear iff their product [MATH] is an involution.', '0801.3958-1-69-0': 'The product of three colinear involutions is an involution by Claim [REF].', '0801.3958-1-69-1': 'Consider three involutions [MATH] whose product [MATH] is an involution [MATH].', '0801.3958-1-69-2': 'Let [MATH] be the Borel subgroup such that [MATH] and [MATH] invert [MATH].', '0801.3958-1-69-3': 'As [MATH] is an involution, [MATH].', '0801.3958-1-69-4': 'So [MATH] normalizes [MATH] by [MATH].', '0801.3958-1-69-5': 'But [MATH] as [MATH] has only one involution.', '0801.3958-1-69-6': 'So [MATH] inverts [MATH], and [MATH] lies on the line with [MATH] and [MATH].', '0801.3958-1-70-0': 'These four conditions contradict the following theorem.', '0801.3958-1-71-0': "Bachmann's Theorem[[CITATION]] Let [MATH] be a group whose set of involutions [MATH] posses the structure of a projective plane, and three involutions are colinear iff their product is an involution.", '0801.3958-1-71-1': 'Then [MATH] for some interpretable field [MATH] and some non-isotropic quadratic form [MATH].', '0801.3958-1-71-2': 'In particular, [MATH] is does not have finite Morley rank.', '0801.3958-1-72-0': 'Thus concluding the proof of Theorem [REF].'} | {'0801.3958-2-0-0': '# Introduction', '0801.3958-2-1-0': "Zilber's original trichotomy conjecture proposed an explicit classification of all one-dimensional objects arising in model theory.", '0801.3958-2-1-1': 'At one point, classifying the simple groups of finite Morley rank was viewed as a subproblem whose affirmative answer would justify this conjecture.', '0801.3958-2-1-2': "Zilber's conjecture was eventually refuted by Hrushovski [CITATION], and the classification of simple groups of finite Morley rank remains open today.", '0801.3958-2-1-3': 'However, these conjectures hold in two significant cases.', '0801.3958-2-1-4': 'First, Hrushovski and Zilber prove the full trichotomy conjecture holds under very strong geometric assumptions [CITATION], and this suffices for various diophantine applications.', '0801.3958-2-1-5': 'Second, the Even Mixed Type Theorem [CITATION] shows that simple groups of finite Morley rank containing an infinite elementary abelian 2-subgroup are Chevellay groups over an algebraically closed field of characteristic two.', '0801.3958-2-2-0': 'In this paper, we clarify some middle ground between these two results by eliminating involutions from simple groups which are definably embedded in a linear group over an algebraically closed field in a structure of finite Morley rank, and which are not Zariski closed themselves.', '0801.3958-2-2-1': 'One may simplify terminology by saying that [MATH] is a definably linear group over a field [MATH] of finite Morley rank, implicitly using some expansion of the field language, or just a definably linear group of finite Morley rank.', '0801.3958-2-3-0': 'Theorem Let [MATH] be a definable infinite simple subgroup of [MATH] over a field [MATH] of finite Morley rank and characteristic zero, which is not Zariski closed.', '0801.3958-2-3-1': 'Then [MATH] has no involutions, and its Borel subgroups are self-normalizing.', '0801.3958-2-4-0': 'Poizat has shown that simple groups with such a definable linear embedding are Chevellay groups when the field has characteristic [MATH] [CITATION].', '0801.3958-2-4-1': 'Much geometric information is available about such groups in characteristic zero but not as much as [CITATION] assumes.', '0801.3958-2-4-2': 'For our purposes, the most important geometric fact is that every element of a counterexample is semisimple in the ambient linear group.', '0801.3958-2-5-0': 'We note that such groups include those groups interpretable in bad fields, which have been recently shown to exist [CITATION].', '0801.3958-2-6-0': 'A major feature of our proof is the elimination of 2-tori of outer automorphisms of simple groups via the Delehan-Nesin argument [CITATION] (see Lemma [REF]).', '0801.3958-2-6-1': 'Such an application is a hopeful sign for the current "[MATH]" project of classifying simple groups of finite Morley rank which have an involution.', '0801.3958-2-6-2': 'On the other hand, our proof is driven primarily by one unreasonably strong fact about these linear groups: all strongly real elements, i.e. all inverted elements, have a nontrivial power inside a unique conjugacy class of Borel subgroups which are pairwise disjoint.', '0801.3958-2-7-0': 'In this vein, we leave the proof of the following consequence as an exercise to the reader (see also [CITATION]).', '0801.3958-2-8-0': 'Corollary A definably linear group of finite Morley rank is an [MATH]-groups.', '0801.3958-2-9-0': 'In the first section, we reduce the problem first to characteristic zero using a result of Poizat, and then to the hypotheses used later by our inductive argument.', '0801.3958-2-9-1': 'In the second section, we recall various facts about groups without unipotent torsion.', '0801.3958-2-9-2': 'In the third section, we prove our main result using the earlier reductions.', '0801.3958-2-10-0': 'The authors thank Oliver for helpful comments and corrections.', '0801.3958-2-11-0': '# Definably linear groups', '0801.3958-2-12-0': 'In this section, we strengthen our hypotheses by recalling some past work on definably linear groups of finite Morley rank.', '0801.3958-2-13-0': 'Poizat has already shown that simple linear groups over fields of characteristic [MATH] are isomorphic to algebraic groups.', '0801.3958-2-14-0': '[Poizat [CITATION]] Let [MATH] be a field of finite Morley rank and characteristic [MATH].', '0801.3958-2-14-1': 'Then any simple definable subgroup of [MATH] is isomorphic to an algebraic group over some field of characteristic [MATH], but not necessarily [MATH].', '0801.3958-2-15-0': 'As such, our theorem considers only fields of characteristic zero.', '0801.3958-2-16-0': 'Let [MATH] be a definable connected subgroup of [MATH] over a field [MATH] of finite Morley rank and characteristic zero.', '0801.3958-2-17-0': 'We now show that all definable subgroups satisfy conjugacy of Borel subgroups, i.e. maximal connected definable solvable subgroups, with the following two facts.', '0801.3958-2-18-0': '[Poizat [CITATION]] If such a group [MATH] is simple, but is not Zariski closed, then all elements of [MATH] are semisimple.', '0801.3958-2-19-0': '[Mustafin [CITATION]] If our group [MATH] consists entirely of semisimple elements, then all Borel subgroups of [MATH] are abelian and conjugate, and this conjugacy class is generic.', '0801.3958-2-20-0': 'Let [MATH] be a Borel subgroup of [MATH].', '0801.3958-2-20-1': 'The Zarisk closure [MATH] of [MATH] is a connected solvable group.', '0801.3958-2-20-2': 'So its derived subgroup [MATH] consists entirely of unipotent elements.', '0801.3958-2-20-3': 'As [MATH] consists entirely of semisimple elements, we have [MATH], and [MATH] is abelian.', '0801.3958-2-20-4': 'It follows that [MATH] is an algebraic torus.', '0801.3958-2-21-0': 'Let [MATH] be a maximal algebraic torus of [MATH] which contains [MATH].', '0801.3958-2-21-1': 'There are only finitely many distinct intersections [MATH] for [MATH], by [CITATION].', '0801.3958-2-21-2': 'As any distinct conjugates of [MATH] lie in distinct conjugates of [MATH], there are only finitely many distinct intersections [MATH] for [MATH] too, and so the union of such intersections is not generic in [MATH].', '0801.3958-2-21-3': 'By the genericity argument [CITATION], [MATH] is generic in [MATH].', '0801.3958-2-21-4': 'Now [MATH] has only one conjugacy class of Borel subgroups, by [CITATION].', '0801.3958-2-22-0': 'If [MATH] is simple, but not Zariski closed, then Borel subgroups of [MATH] are abelian and hereditarily conjugate.', '0801.3958-2-23-0': 'By Fact [REF], all elements of [MATH] are semisimple.', '0801.3958-2-23-1': 'As this property passes to subgroups, the result follows from Lemma [REF].', '0801.3958-2-24-0': 'It follows that our [MATH] satisfies Hypothesis [REF] below.', '0801.3958-2-24-1': 'So then Theorem [REF] will show that [MATH] has no involutions.', '0801.3958-2-25-0': 'We recall that a Carter subgroup of [MATH] is an almost self-normalizing definable nilpotent subgroup of [MATH] In our case, these are merely our Borel subgroups.', '0801.3958-2-25-1': 'As the Borel subgroups are conjugate, the Borel subgroups of [MATH] will then be self-normalizing too, by Lemma [REF].', '0801.3958-2-26-0': 'So these two results will complete the proof.', '0801.3958-2-27-0': '# No unipotent torsion', '0801.3958-2-28-0': 'We now recall several convenient results from [CITATION] concerning groups without unipotent torsion, i.e. which contains no definable connected nilpotent subgroup of bounded exponent.', '0801.3958-2-29-0': 'Let [MATH] be a connected group of finite Morley rank without unipotent torsion.', '0801.3958-2-29-1': 'Then every element has a nontrivial power which lies inside a Borel subgroup.', '0801.3958-2-30-0': 'This is a corollary of the following fact.', '0801.3958-2-31-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank of [MATH]-type, i.e. without unipotent [MATH]-torsion for [MATH].', '0801.3958-2-31-1': 'Then all [MATH]-elements of [MATH] are toral, i.e. lie inside [MATH]-tori.', '0801.3958-2-32-0': 'We recall that, in this setting, divisible torsion groups are referred to as tori and their elements may be referred to as toral.', '0801.3958-2-32-1': 'As usual, [MATH] denotes the connected component of [MATH], so a Sylowo2-subgroup is just a maximal 2-torus.', '0801.3958-2-33-0': '[Proof of Lemma [REF]] Consider an element [MATH].', '0801.3958-2-33-1': 'If [MATH] is infinite, we choose [MATH].', '0801.3958-2-33-2': 'If [MATH] is finite, [MATH] lies inside some torus by Fact [REF].', '0801.3958-2-34-0': 'We observe that, if one assumes that Borel subgroups are conjugate, one may instead prove that all elements lie inside Borel subgroups.', '0801.3958-2-34-1': 'For this, we require that the group has degenerate type, meaning a finite Sylow 2-subgroup or equivalently no involutions [CITATION].', '0801.3958-2-35-0': 'Let [MATH] be a connected degenerate type group of finite Morley rank without unipotent torsion.', '0801.3958-2-35-1': 'Then [MATH] has some self-normalizing Carter subgroup.', '0801.3958-2-36-0': 'This is a corollary of the following.', '0801.3958-2-37-0': 'Let [MATH] be a group of finite Morley rank, and [MATH] a maximal divisible abelian torsion subgroup of [MATH].', '0801.3958-2-37-1': 'The Weyl group of [MATH] is the finite group [MATH], which can be viewed as a group of automorphisms of [MATH].', '0801.3958-2-38-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank.', '0801.3958-2-38-1': 'Suppose the Weyl group is nontrivial and has odd order, with [MATH] the smallest prime divisor of its order.', '0801.3958-2-38-2': 'Then [MATH] contains a unipotent [MATH]-subgroup.', '0801.3958-2-39-0': '[[CITATION]] Let [MATH] be a connected group of finite Morley rank, [MATH] a maximal [MATH]-torus in [MATH], and suppose that [MATH] is central in [MATH].', '0801.3958-2-39-1': 'If [MATH] has [MATH]-type, then any [MATH]-element [MATH] belongs to [MATH].', '0801.3958-2-40-0': '[Proof of Lemma [REF]] We may assume that [MATH] contains torsion.', '0801.3958-2-40-1': 'Let [MATH] be a maximal divisible abelian torsion subgroup of [MATH].', '0801.3958-2-40-2': 'By [CITATION], [MATH] has a Carter subgroup [MATH] containing [MATH].', '0801.3958-2-40-3': 'By Fact [REF], [MATH].', '0801.3958-2-40-4': 'So [MATH].', '0801.3958-2-40-5': 'By Fact [REF], [MATH] is torsion free.', '0801.3958-2-40-6': 'So [MATH], as desired.', '0801.3958-2-41-0': '# Bad-like groups', '0801.3958-2-42-0': 'Here we analyze groups satisfying the following hypotheses.', '0801.3958-2-43-0': 'Let [MATH] be a connected group of finite Morley rank without unipotent torsion whose Borel subgroups are nilpotent and hereditarily conjugate, i.e. any subgroup [MATH] of [MATH] satisfies [MATH]-conjugacy of Borel subgroups of [MATH].', '0801.3958-2-44-0': 'These hypotheses clearly pass to subgroups.', '0801.3958-2-44-1': 'We observe that they also pass to sections by nilpotent kernels.', '0801.3958-2-45-0': 'Let [MATH] be a nilpotent normal subgroup of [MATH].', '0801.3958-2-45-1': 'Then [MATH] satisfies Hypothesis [REF] too, and images and inverse images preserve Borel subgroups.', '0801.3958-2-46-0': 'Quotients and extensions of solvable groups are solvable, so images and inverse images preserve maximal solvable subgroups.', '0801.3958-2-46-1': 'As images also preserve connectivity, they preserve Borel subgroups, and Borel subgroups of [MATH] are nilpotent.', '0801.3958-2-46-2': 'If [MATH] is connected, inverse images are connected too, and hence preserve Borel subgroups too.', '0801.3958-2-46-3': 'We may now assume that [MATH] is finite, and even cyclic of order [MATH].', '0801.3958-2-46-4': 'As [MATH] is connected, [MATH] is central in [MATH].', '0801.3958-2-46-5': 'So, by Fact [REF], [MATH] is contained in every maximal [MATH]-torus of [MATH].', '0801.3958-2-46-6': 'By conjugacy, [MATH] is contained in every Borel subgroup of [MATH] too.', '0801.3958-2-46-7': 'So all inverse images are connected, and hence preserve Borel subgroups.', '0801.3958-2-46-8': 'Our first part now follows easily by applying the second inside all definable connected subgroups of [MATH]', '0801.3958-2-47-0': 'A variation on the Delehan-Nesin argument [CITATION] shows that 2-tori have no interesting actions inside such a group.', '0801.3958-2-48-0': 'Let [MATH] be a definable normal subgroup of [MATH] which has no involutions.', '0801.3958-2-48-1': 'Then every 2-torus in [MATH] centralizes [MATH].', '0801.3958-2-49-0': 'For this, we note several facts about such a degenerate type group [MATH] normalized by an involution.', '0801.3958-2-50-0': "No element [MATH] is [MATH]-conjugate to it's inverse [MATH].", '0801.3958-2-51-0': 'Suppose that [MATH] for some [MATH].', '0801.3958-2-51-1': 'As [MATH] is not an involution, we have [MATH].', '0801.3958-2-51-2': 'But clearly [MATH] has order two modulo [MATH].', '0801.3958-2-51-3': 'So [MATH] contains an involution by [CITATION], a contradiction.', '0801.3958-2-52-0': 'No element of [MATH] is inverted by any involution in [MATH].', '0801.3958-2-53-0': 'Any involution of [MATH] is conjugate to [MATH] under [MATH].', '0801.3958-2-53-1': 'So otherwise there is an [MATH] such that [MATH].', '0801.3958-2-53-2': 'We may assume that [MATH] too because [CITATION] says [MATH] where [MATH].', '0801.3958-2-53-3': 'So [MATH], then [MATH], contradicting Fact [REF].', '0801.3958-2-54-0': '[Proof of Proposition [REF]] Consider a 2-torus [MATH] of [MATH].', '0801.3958-2-54-1': 'We may take [MATH].', '0801.3958-2-54-2': 'We may assume [MATH] is centerless, by Lemma [REF].', '0801.3958-2-54-3': 'So there is a nontrivial strongly real element [MATH] inside [MATH].', '0801.3958-2-54-4': 'It follows, by Lemma [REF], that some nontrivial power [MATH] of [MATH] lies inside some Borel subgroup [MATH] of [MATH].', '0801.3958-2-54-5': 'As all Borel subgroups of [MATH] are conjugate, some conjugate [MATH] of [MATH] lies inside the Borel subgroup of [MATH] containing [MATH].', '0801.3958-2-54-6': 'But now [MATH] centralizes [MATH] by [CITATION], contradicting Lemma [REF].', '0801.3958-2-55-0': 'We say a subgroup [MATH] of a group [MATH] is subnormal if there is a finite chain [EQUATION]', '0801.3958-2-55-1': 'A quasisimple subnormal subgroup of a group [MATH] is referred to as a component of [MATH].', '0801.3958-2-55-2': 'If [MATH] is connected, then its components are definable, connected, and normal in [MATH], by [CITATION].', '0801.3958-2-55-3': 'They also centralize one another by [CITATION].', '0801.3958-2-55-4': 'We let [MATH] denote the central product of the components in [MATH], which happen to be connected, and set [MATH].', '0801.3958-2-55-5': 'According to the following, [MATH] controls the structure of [MATH].', '0801.3958-2-56-0': '[[CITATION]] Any group [MATH] of finite Morley rank satisfies [EQUATION]', '0801.3958-2-56-1': 'If all simple sections of [MATH] with nilpotent kernels have degenerate type, then [MATH] has a unique Sylow 2-subgroup, which is connected and central.', '0801.3958-2-57-0': 'A connected degenerate type group of finite Morley rank has no involutions by [CITATION].', '0801.3958-2-57-1': 'So, as [MATH] is [MATH] by hypothesis, we find that [MATH] is [MATH] too.', '0801.3958-2-57-2': 'By Lemma [REF], the Sylowo2-subgroup [MATH] of [MATH] centralizes [MATH].', '0801.3958-2-57-3': 'But [MATH] by Fact [REF].', '0801.3958-2-57-4': 'So any 2-torsion in [MATH] is lies inside the solvable radical [MATH], which lies inside [MATH] because Borel subgroups are nilpotent.', '0801.3958-2-57-5': 'It follows that the Sylow 2-subgroup is connected by Fact [REF] and [CITATION], and central by [CITATION].', '0801.3958-2-58-0': 'We now prove our main result.', '0801.3958-2-59-0': 'Suppose that [MATH] is simple.', '0801.3958-2-59-1': 'Then [MATH] has no involutions.', '0801.3958-2-60-0': 'We consider a counterexample [MATH] of minimal Morley rank.', '0801.3958-2-60-1': 'So, by Lemma [REF], all proper simple section with nilpotent kernels have degenerate type.', '0801.3958-2-60-2': 'By Lemma [REF], any proper connected subgroup [MATH] centralizes its Sylow 2-subgroup, which is a 2-torus.', '0801.3958-2-60-3': 'As [MATH] is simple, it follows that [MATH] the centralizers of distinct Sylowo2-subgroups are disjoint.', '0801.3958-2-61-0': 'Also, any Sylowo2-subgroup is central in a Borel subgroup.', '0801.3958-2-61-1': 'By conjugacy, all Borel subgroups have a central Sylowo2-subgroup.', '0801.3958-2-61-2': 'So [MATH] two Borel subgroups meet iff they both contain the same Sylowo2-subgroup of [MATH].', '0801.3958-2-62-0': 'Any two distinct involutions [MATH] and [MATH] normalize a common Sylowo2-subgroup [MATH] of [MATH], and [MATH].', '0801.3958-2-63-0': 'By Lemma [REF], there is a nontrivial power [MATH] of [MATH] inside a Borel subgroup [MATH] of [MATH].', '0801.3958-2-63-1': 'As [MATH] inverts [MATH], [MATH] also contains [MATH].', '0801.3958-2-63-2': 'By [MATH], [MATH] and [MATH] contain the same Sylowo2-subgroup [MATH] of [MATH].', '0801.3958-2-63-3': 'So [MATH] normalizes [MATH].', '0801.3958-2-63-4': 'Similarly, [MATH] normalizes [MATH] too.', '0801.3958-2-64-0': 'Every involution is contained in a Sylowo2-subgroup, by Fact [REF], which is unique by [MATH].', '0801.3958-2-64-1': 'Let [MATH] denote the Sylowo2-subgroup containing [MATH].', '0801.3958-2-64-2': 'Now [MATH] since [MATH] is the unique Sylowo2-subgroup of [MATH], by [MATH].', '0801.3958-2-65-0': 'Let [MATH] be a Sylowo2-subgroup of [MATH], and let [MATH] be an involution normalizing [MATH].', '0801.3958-2-65-1': 'Then either [MATH] or [MATH] inverts [MATH].', '0801.3958-2-65-2': 'In particular, [MATH] is an abelian Borel subgroup of [MATH].', '0801.3958-2-66-0': 'Suppose that [MATH].', '0801.3958-2-66-1': 'Then [MATH].', '0801.3958-2-66-2': 'So, by [CITATION], [MATH] is abelian and inverted by [MATH].', '0801.3958-2-66-3': 'Now [MATH] is a Borel subgroup of [MATH], as it containes a Borel subgroup of [MATH].', '0801.3958-2-67-0': 'As a consequence, if involutions [MATH] and [MATH] do not commute, then neither [MATH] nor [MATH] lie inside a Sylowo2-subgroup which they both normalize, and both invert its Borel subgroup.', '0801.3958-2-68-0': '[MATH] has rank one.', '0801.3958-2-68-1': 'So a Sylow 2-subgroup of [MATH] is isomorphic to that of [MATH].', '0801.3958-2-69-0': 'Suppose towards a contradiction that [MATH].', '0801.3958-2-69-1': 'Choose a pair of non-commuting involutions [MATH] and [MATH], and fix a Sylowo2-subgroup [MATH] which is normalized by them both.', '0801.3958-2-69-2': 'As [MATH] inverts [MATH] by Claim [REF], [MATH] normalizes [MATH].', '0801.3958-2-69-3': 'By generation [CITATION], [MATH].', '0801.3958-2-69-4': 'So [MATH] for some [MATH], which yields the contradiction [MATH].', '0801.3958-2-69-5': 'Thus [MATH].', '0801.3958-2-70-0': 'Let [MATH] be a Sylow 2-subgroup of [MATH].', '0801.3958-2-70-1': 'By Fact [REF], [MATH], and [MATH].', '0801.3958-2-70-2': 'There are involutions in [MATH] by Claims [REF] and [REF].', '0801.3958-2-70-3': 'So [MATH] has the desired structure.', '0801.3958-2-71-0': 'Consider the set [MATH] of all involutions in [MATH].', '0801.3958-2-71-1': 'We define the set [MATH] of lines on [MATH] to be the collection of cosets [MATH] where [MATH] is a Borel subgroup of [MATH] and [MATH] inverts [MATH].', '0801.3958-2-71-2': 'We now prove that [MATH] and [MATH] form a projective plane.', '0801.3958-2-72-0': 'PP2 Any two points lie on a unique line.', '0801.3958-2-73-0': 'Any two involutions [MATH] normalize some Sylowo2-subgroup [MATH], by Claim [REF].', '0801.3958-2-73-1': 'By Claim [REF], [MATH] is a Borel subgroup inverted by [MATH] and [MATH].', '0801.3958-2-73-2': 'So [MATH] and [MATH] lie on the line [MATH].', '0801.3958-2-73-3': 'Let [MATH] denote the power of [MATH] lies inside some [MATH].', '0801.3958-2-73-4': 'By [MATH], [MATH] is the unique Sylowo2-subgroup of [MATH].', '0801.3958-2-73-5': 'So the line [MATH] is unique.', '0801.3958-2-74-0': 'PP3 Any two lines intersect at a unique point.', '0801.3958-2-75-0': 'For any two lines [MATH] and [MATH], we consider the involutions [MATH] and [MATH], which are unique by Claim [REF].', '0801.3958-2-75-1': 'Again, there is a Borel subgroup [MATH] inverted by both, by Claims [REF] and [REF].', '0801.3958-2-75-2': 'As [MATH] and [MATH] both centralize the involution [MATH], [MATH] inverts both [MATH] and [MATH], by Claims [REF] and [REF].', '0801.3958-2-75-3': 'Thus the lines [MATH] and [MATH] meet at [MATH].', '0801.3958-2-75-4': 'Such a point is unique by PP2.', '0801.3958-2-76-0': 'PP4 There are four points, no three of which are colinear.', '0801.3958-2-77-0': 'Consider a Borel subgroup [MATH] of [MATH], and the unique involution [MATH].', '0801.3958-2-77-1': 'Let [MATH], [MATH], and [MATH] be three involutions inverting [MATH], and let [MATH] be a third point on the line through [MATH] and [MATH].', '0801.3958-2-77-2': 'The original points [MATH] are obviously not colinear.', '0801.3958-2-77-3': 'The pair [MATH] is not colinear with either [MATH] or [MATH] by PP3.', '0801.3958-2-77-4': 'Also [MATH] is not colinear by PP2.', '0801.3958-2-77-5': 'So [MATH] are the four desired involutions.', '0801.3958-2-78-0': 'We prove one additional property of this projective plane.', '0801.3958-2-79-0': 'PP+ Any three involutions [MATH] are colinear iff their product [MATH] is an involution.', '0801.3958-2-80-0': 'The product of three colinear involutions is an involution by Claim [REF].', '0801.3958-2-80-1': 'Consider three involutions [MATH] whose product [MATH] is an involution [MATH].', '0801.3958-2-80-2': 'Let [MATH] be the Borel subgroup such that [MATH] and [MATH] invert [MATH].', '0801.3958-2-80-3': 'As [MATH] is an involution, [MATH].', '0801.3958-2-80-4': 'So [MATH] normalizes [MATH] by [MATH].', '0801.3958-2-80-5': 'But [MATH] as [MATH] has only one involution.', '0801.3958-2-80-6': 'So [MATH] inverts [MATH], and [MATH] lies on the line with [MATH] and [MATH].', '0801.3958-2-81-0': 'These four conditions contradict the following theorem.', '0801.3958-2-82-0': "Bachmann's Theorem[[CITATION]] Let [MATH] be a group whose set of involutions [MATH] posses the structure of a projective plane, and three involutions are colinear iff their product is an involution.", '0801.3958-2-82-1': 'Then [MATH] for some interpretable field [MATH] and some non-isotropic quadratic form [MATH].', '0801.3958-2-82-2': 'In particular, [MATH] does not have finite Morley rank.', '0801.3958-2-83-0': 'This concludes the proof of Theorem [REF].', '0801.3958-2-84-0': "We conclude by conjecturing a a stronger variation the above style of argument exploiting Bachmann's Theorem.", '0801.3958-2-85-0': 'There is no simple group of finite Morley rank in which every strongly real elements lies inside some [MATH] for [MATH] is a Sylowo2-subgroup.'} | [['0801.3958-1-66-0', '0801.3958-2-77-0'], ['0801.3958-1-66-1', '0801.3958-2-77-1'], ['0801.3958-1-66-2', '0801.3958-2-77-2'], ['0801.3958-1-66-3', '0801.3958-2-77-3'], ['0801.3958-1-66-4', '0801.3958-2-77-4'], ['0801.3958-1-66-5', '0801.3958-2-77-5'], ['0801.3958-1-4-0', '0801.3958-2-4-0'], ['0801.3958-1-4-2', '0801.3958-2-4-2'], ['0801.3958-1-37-0', '0801.3958-2-45-0'], ['0801.3958-1-37-1', '0801.3958-2-45-1'], ['0801.3958-1-40-1', '0801.3958-2-48-1'], ['0801.3958-1-68-0', '0801.3958-2-79-0'], ['0801.3958-1-60-0', '0801.3958-2-71-0'], ['0801.3958-1-60-1', '0801.3958-2-71-1'], ['0801.3958-1-60-2', 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'0801.3958-2-42-0', '0801.3958-2-50-0', '0801.3958-2-53-0', '0801.3958-2-53-1', '0801.3958-2-53-2', '0801.3958-2-53-3', '0801.3958-2-58-0', '0801.3958-2-59-0', '0801.3958-2-59-1', '0801.3958-2-62-0', '0801.3958-2-63-0', '0801.3958-2-63-1', '0801.3958-2-63-2', '0801.3958-2-63-3', '0801.3958-2-63-4', '0801.3958-2-66-0', '0801.3958-2-66-1', '0801.3958-2-69-5', '0801.3958-2-70-1', '0801.3958-2-72-0', '0801.3958-2-74-0', '0801.3958-2-78-0', '0801.3958-2-81-0', '0801.3958-2-83-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0801.3958 | null | null | null | null | null |
1303.4029 | {'1303.4029-1-0-0': '# Abstract We prove that an exact functor of Waldhausen quasicategories induces a stable equivalence of [MATH]-theory spectra if it induces an equivalence of cofibration homotopy categories.', '1303.4029-1-0-1': 'As a special case, if an exact functor reflects cofibrations and induces an equivalence of homotopy categories, then it induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-1-1-0': 'The main technical result to prove these versions of Waldhausen Approximation is that a functor of quasicategories [MATH] induces a weak homotopy equivalence of maximal Kan complexes if it is essentially surjective, [MATH] admits colimits of diagrams in its maximal Kan complex indexed by finite posets, [MATH] preserves them, and [MATH] reflects equivalences.', '1303.4029-1-2-0': 'We also prove that [MATH] is Waldhausen equivalent to [MATH] using the mid anodyne maps known as spine inclusions, and clarify how hypotheses and notions in Waldhausen structures are related in new ways in the context of quasicategories.', '1303.4029-1-2-1': 'In the last section, we give conditions for an exact functor to induce an equivalence between (cofibration) homotopy categories of higher [MATH]-iterates.', '1303.4029-1-3-0': '[2mm]', '1303.4029-1-4-0': "Key words: quasicategory, [MATH]-category, [MATH]-theory, Waldhausen's Approximation Theorem, Waldhausen quasicategory, Waldhausen [MATH]-category", '1303.4029-1-5-0': '2010 Mathematics Subject Classification.', '1303.4029-1-5-1': 'Primary: 19D10, 55N15, 55U10 ; Secondary: 18A99.', '1303.4029-1-6-0': '# Introduction', '1303.4029-1-7-0': 'When does a map [MATH] induce an equivalence of algebraic [MATH]-theory spectra?', '1303.4029-1-7-1': "Waldhausen's Approximation Theorem [CITATION] gives just such a criterion: if [MATH] is an exact functor between reasonable Waldhausen categories, [MATH] reflects weak equivalences, and any map [MATH] factors as [MATH], then [MATH] induces an equivalence of [MATH]-theory spectra (even [MATH] is a weak homotopy quivalence).", '1303.4029-1-8-0': 'The main purpose of the present paper is to prove the most general Approximation Theorem presently available in the context of quasicategories: if an exact functor of Waldhausen quasicategories induces an equivalence of cofibration homotopy categories, then it induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-1-8-1': 'Recent general results in this direction worked with localizations of homotopical categories, or assumed that every map is a cofibration (see the literature review later in this introduction).', '1303.4029-1-9-0': "A few years after Waldhausen's Approximation Theorem appeared, Thomason-Trobaugh published a proof in [CITATION] that a functor which induces an equivalence of derived homotopy categories induces an equivalence in [MATH]-theory, under appropriate hypotheses.", '1303.4029-1-9-1': 'In 1.9.9, Thomason remarks "Morally, it says that [MATH] essentially depends only on the derived category [MATH], and thus that Waldhausen [MATH]-theory gives essentially a [MATH]-theory of the derived category."', '1303.4029-1-10-0': "Since then, many mathematicians have obtained results in this direction in various contexts: Neeman [CITATION] and Dugger-Shipley [CITATION] for algebraic [MATH]-theory of rings, Schlichting's counterexample for Frobenius categories [CITATION], Toen-Vezzosi [CITATION] and Blumberg-Mandell [CITATION] for Dwyer-Kan simplicial localizations of Waldhausen categories, and Cisinski [CITATION] for right exact functors between reasonable Waldhausen categories.", '1303.4029-1-10-1': 'Sagave [CITATION] showed how to loosen Waldhausen\'s requirement of factorizing [MATH] mentioned above to a requirement of factorizing [MATH] when [MATH] is a "special object".', '1303.4029-1-10-2': "In Appendix A of [CITATION], Schlichting showed how to replace Waldhausen's cylinder functor and cylinder axiom in the Approximation Theorem by requiring factorization of any map into a cofibration followed by a weak equivalence.", '1303.4029-1-11-0': 'The main contribution of the present article is to prove several versions of Waldhausen Approximation in the context of the Waldhausen quasicategories of Barwick [CITATION] and Fiore-Luck [CITATION], see Definition [REF], Theorems [REF] and [REF], and Corollaries [REF] and [REF].', '1303.4029-1-11-1': 'A quasicategory, or [MATH]-category, is a simplicial set in which every inner horn has a filler.', '1303.4029-1-11-2': 'For instance Kan complexes and nerves of categories are quasicategories.', '1303.4029-1-11-3': 'A Waldhausen quasicategory is equipped with a distinguished zero object and a 1-full subquasicategory of cofibrations containing the equivalences, and is required to have pushouts along cofibrations.', '1303.4029-1-11-4': 'The subquasicategory of "weak equivalences" is always the 1-full subquasicategory on the equivalences of the underlying quasicategory, namely the 1-full subquasicategory on the maps that are invertible in its homotopy category.', '1303.4029-1-12-0': 'The main results of this paper begin with Theorem [REF]: if an exact functor [MATH] induces an equivalence of cofibration homotopy categories, then it induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-1-12-1': 'The next main result, Theorem [REF], says that if [MATH] reflects cofibrations and induces an equivalence of homotopy categories, then [MATH] induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-1-12-2': 'Proposition [REF] (without Waldhausen structures) is the main technical step, and states that a functor [MATH] between quasicategories induces a weak homotopy equivalence of maximal Kan complexes if [MATH] is essentially surjective, [MATH] admits colimits of diagrams in its maximal Kan complex indexed by finite posets, [MATH] preserves them, and [MATH] reflects equivalences.', '1303.4029-1-12-3': "The idea for the proof of Proposition [REF] is due to Waldhausen: use Lemma [REF] and Quillen's Theorem A. However, we also incorporate a quasicategorical implementation of an idea of Schlichting [CITATION], see the discussion preceding Proposition [REF].", '1303.4029-1-13-0': 'Factorization is not required for the main results (indeed, factorization in the quasicategorical context is equivalent to requiring all maps to be cofibrations).', '1303.4029-1-14-0': 'Related results for quasicategories are the following.', '1303.4029-1-14-1': 'Barwick proved in [CITATION] in the special case that all maps are cofibrations that an exact functor [MATH] between Waldhausen quasicategories is an equivalence if and only if it induces an equivalence of homotopy categories [MATH].', '1303.4029-1-14-2': 'Similarly, Blumberg-Gepner-Tabuada conclude in [CITATION] that a map of stable quasicategories is an equivalence if and only if it it induces an equivalence of homotopy categories (again all maps are considered cofibrations).', '1303.4029-1-14-3': 'In general, Blumberg-Gepner-Tabuada study [MATH]-theory in [CITATION] as an invariant of stable quasicategories.', '1303.4029-1-14-4': 'In the present paper, we do not require stability.', '1303.4029-1-15-0': 'Simplicial categories, on the other hand, are another model for [MATH]-categories.', '1303.4029-1-15-1': 'Toen-Vezzosi [CITATION] observed already in 2004 that the [MATH]-theory of a "good" category with fibrations and weak equivalences is an invariant of the underlying [MATH]-category, namely of its Dwyer-Kan hammock localization, despite the fact that the [MATH]-theory cannot be reconstructed from the triangulated homotopy category, as proved by Neeman [CITATION].', '1303.4029-1-15-2': 'Equivalence of localizations is in fact closely correlated to approximation: Blumberg-Mandell [CITATION] prove that equivalence of Dwyer-Kan localizations follows from Waldhausen\'s approximation axioms and Cisinski [CITATION] proves that this is actually an "if and only if" statement (see Theorems 2.9 and 3.25, Proposition 4.5, and Scholie 4.15, all in [CITATION]).', '1303.4029-1-15-3': 'Assuming Dwyer-Kan equivalence of Dwyer-Kan localizations of weak cofibration subcategories, and a few other hypotheses, a consequence of [CITATION] is a stable equivalence of [MATH]-theory spectra, see Remark [REF] for details.', '1303.4029-1-15-4': 'The present article remains entirely in the world of Waldhausen quasicategories.', '1303.4029-1-16-0': 'Outline of Paper.', '1303.4029-1-16-1': 'In Section [REF], I recall all the prerequisites from the theory of quasicategories as developed by Boardman-Vogt, Joyal, and Lurie, including homotopy, join, slice, and colimits.', '1303.4029-1-16-2': "In Section [REF], I recall the notion of Waldhausen quasicategory, discuss some of its consequences, and introduce the variants [MATH] and [MATH] of Waldhausen's constructions using spines [MATH] rather than simplices [MATH].", '1303.4029-1-16-3': "Section [REF] contains the proofs of various quasicategorical Approximation Theorems and the main technical result Proposition [REF], and discusses overquasicategories for the quasicategorical version of Quillen's Theorem A. Section [REF] also contains a comparison with other results in the literature.", '1303.4029-1-16-4': 'Section [REF] asks when an exact functor induces an equivalence of homotopy categories for higher [MATH] iterates, and discusses homotopy of natural transformations in terms of lifting conditions.'} | {'1303.4029-2-0-0': 'We prove a series of Approximation Theorems in the setting of Waldhausen quasicategories.', '1303.4029-2-0-1': "These theorems, inspired by Waldhausen's 1985 Approximation Theorem, give sufficient conditions for an exact functor of Waldhausen quasicategories to induce a level-wise weak homotopy equivalence of [MATH]-theory spectra.", '1303.4029-2-1-0': 'The Pre-Approximation Theorem, which holds in the general setting of quasicategories without Waldhausen structures, provides sufficient conditions for a functor [MATH] to restrict to an equivalence of the maximal [MATH]-groupoids in [MATH] and [MATH].', '1303.4029-2-1-1': 'These conditions are: [MATH] reflects equivalences, every codomain morphism [MATH] factors as [MATH], the functor [MATH] of homotopy categories is essentially surjective and full on isomorphisms, the domain quasicategory [MATH] admits colimits of diagrams of equivalences indexed by connected finite posets, and [MATH] preserves such colimits.', '1303.4029-2-2-0': 'Our Approximation Theorems follow from the Pre-Approximation Theorem.', '1303.4029-2-2-1': "The Approximation Theorem in the quasicategorical setting most analogous to Waldhausen's is: if an exact functor [MATH] satisfies Waldhausen's App 1 and App 2, and the domain [MATH] admits colimits of the aforementioned type and [MATH] preserves them, then [MATH] is a level-wise equivalence.", '1303.4029-2-2-2': 'As a corollary, if [MATH] is an exact functor with [MATH] an equivalence of ordinary categories, and every morphism in the domain [MATH] is a cofibration, then [MATH] is a level-wise equivalence.', '1303.4029-2-3-0': 'We then introduce a version of App 2 called Cofibration App 2 that only requires factorization of cofibrations [MATH] as [MATH] and prove an analogous Cofibration Approximation Theorem, and a corollary for certain functors that induce an equivalence of cofibration homotopy categories.', '1303.4029-2-4-0': 'We also prove that [MATH] is Waldhausen equivalent to [MATH] using the mid anodyne maps known as spine inclusions, and clarify how hypotheses and notions in Waldhausen structures are related in new ways in the context of quasicategories.', '1303.4029-2-5-0': '[2mm]', '1303.4029-2-6-0': "Key words: quasicategory, [MATH]-category, [MATH]-theory, Waldhausen's Approximation Theorem, Approximation, Waldhausen quasicategory, Waldhausen [MATH]-category", '1303.4029-2-7-0': '# Introduction', '1303.4029-2-8-0': 'When does a map [MATH] induce a stable equivalence of algebraic [MATH]-theory spectra?', '1303.4029-2-8-1': "Waldhausen's Approximation Theorem [CITATION] gives just such a criterion: if [MATH] is an exact functor between reasonable Waldhausen categories, [MATH] reflects weak equivalences, and every codomain morphism [MATH] factors as [MATH], then [MATH] induces a stable equivalence of [MATH]-theory spectra (even [MATH] is a weak homotopy equivalence).", '1303.4029-2-9-0': 'The main purpose of the present paper is to prove the most general Approximation Theorem presently available in the context of quasicategories: if an exact functor [MATH] of Waldhausen quasicategories satisfies', '1303.4029-2-10-0': '(App 1)', '1303.4029-2-11-0': '[MATH] reflects equivalences, (App 2)', '1303.4029-2-12-0': 'For every [MATH] and every morphism [MATH] in the codomain [MATH], there exists a cofibration [MATH] in [MATH], an equivalence [MATH] in [MATH], and a 2-simplex in [MATH] of the form [EQUATION] and The domain quasicategory [MATH] admits colimits of diagrams of equivalences indexed by connected finite posets, and [MATH] preserves such colimits,', '1303.4029-2-13-0': 'then [MATH] induces a level-wise weak homotopy equivalence of [MATH]-theory spectra.', '1303.4029-2-13-1': 'Recent general results in this direction worked with localizations of homotopical categories, or assumed that every map is a cofibration (see the literature review later in this introduction).', '1303.4029-2-14-0': "A few years after Waldhausen's Approximation Theorem appeared, Thomason-Trobaugh published a proof in [CITATION] that a functor which induces an equivalence of derived homotopy categories induces an equivalence in [MATH]-theory, under appropriate hypotheses.", '1303.4029-2-14-1': 'In 1.9.9, Thomason remarks "Morally, it says that [MATH] essentially depends only on the derived category [MATH], and thus that Waldhausen [MATH]-theory gives essentially a [MATH]-theory of the derived category."', '1303.4029-2-15-0': "Since then, many mathematicians have obtained results in this direction in various contexts: Neeman [CITATION] and Dugger-Shipley [CITATION] for algebraic [MATH]-theory of rings, Schlichting's counterexample for Frobenius categories [CITATION], Toen-Vezzosi [CITATION] and Blumberg-Mandell [CITATION] for Dwyer-Kan simplicial localizations of Waldhausen categories, and Cisinski [CITATION] for right exact functors between reasonable Waldhausen categories.", '1303.4029-2-15-1': 'Sagave [CITATION], in the context of classical Waldhausen categories, showed how to loosen Waldhausen\'s requirement of factorizing [MATH] mentioned above to a requirement of factorizing [MATH] when [MATH] is a "special object".', '1303.4029-2-15-2': "In Appendix A of [CITATION], Schlichting showed how to replace Waldhausen's cylinder functor and cylinder axiom in Waldhausen's classical Approximation Theorem by requiring factorization of any domain morphism into a cofibration followed by a weak equivalence (the full hypotheses of Schlichting's Approximation variant are: [MATH] reflects equivalences, every codomain morphism [MATH] factors as [MATH], every morphism of [MATH] factors as [MATH], and the weak equivalences in both the domain and codomain satisfy the 3-for-2 property).", '1303.4029-2-16-0': 'The main contribution of the present article is to prove several versions of Waldhausen Approximation in the context of the Waldhausen quasicategories of Barwick [CITATION] and Fiore-Pieper [CITATION], see Definition [REF], Theorem [REF], Corollary [REF], Theorem [REF], Corollary [REF], and Corollary [REF].', '1303.4029-2-16-1': 'All of these follow from the Pre-Approximation Theorem [REF], which does not require a Waldhausen structure.', '1303.4029-2-16-2': 'We recall the main notion and examples, and describe these results now.', '1303.4029-2-17-0': 'A quasicategory, or [MATH]-category, is a simplicial set in which every inner horn has a filler.', '1303.4029-2-17-1': 'For instance Kan complexes and nerves of categories are quasicategories.', '1303.4029-2-17-2': 'A Waldhausen quasicategory is a quasicategory with zero objects together with a selected 1-full subquasicategory of cofibrations containing the equivalences.', '1303.4029-2-17-3': 'A Waldhausen quasicategory is required to have pushouts along cofibrations, and any morphism with domain any zero object is required to be a cofibration.', '1303.4029-2-17-4': '"Weak equivalences" are not part of the definition, instead the "weak equivalences" in a Waldhausen quasicategory are by default the equivalences of the quasicategory, i.e. those morphisms that are invertible in the homotopy category of the quasicategory.', '1303.4029-2-18-0': 'We mention here several examples of Waldhausen quasicategories.', '1303.4029-2-18-1': 'The nerve of any classical Waldhausen category with weak equivalences the isomorphisms is a Waldhausen quasicategory.', '1303.4029-2-18-2': 'For instance, the nerve of the classical Waldhausen category of based finite sets, bijections, and injections is a Waldhausen quasicategory.', '1303.4029-2-18-3': "When a Waldhausen category comes from a model category, then Barwick's relative neve is a Waldhausen quasicategory with the same [MATH]-theory, see [CITATION] and see [CITATION] for a summary of the related results in [CITATION] by Barwick.", '1303.4029-2-18-4': 'Genuinely non-classical examples of Waldhausen quasicategories are the stable quasicategories of Lurie (in a stable quasicategory every morphism is considered a cofibration).', '1303.4029-2-18-5': 'Recall that a quasicategory is stable if it admits all finite limits and colimits, and pushout squares and pullback squares coincide.', '1303.4029-2-18-6': "Stable quasicategories are the context of Lurie's second book [CITATION].", '1303.4029-2-19-0': 'The main results of this paper begin with the Pre-Approximation Theorem [REF]: if [MATH] is a functor of quasicategories that reflects equivalences, every codomain morphism [MATH] factors as [MATH], the functor [MATH] of homotopy categories is essentially surjective and full on isomorphisms, the domain quasicategory [MATH] admits colimits of diagrams of equivalences indexed by connected finite posets, and [MATH] preserves such colimits, then [MATH] restricts to an equivalence of the maximal [MATH]-groupoids of [MATH] and [MATH].', '1303.4029-2-19-1': 'Notice that the Pre-Approximation Theorem does not require any Waldhausen structures.', '1303.4029-2-19-2': "The idea for the proof of Theorem [REF] is due to Waldhausen: use Proposition [REF] and (quasicategorical) Quillen's Theorem A. However, we also incorporate a quasicategorical implementation of an idea of Schlichting [CITATION], see the discussion preceding Theorem [REF].", '1303.4029-2-20-0': 'The first consequence of the Pre-Approximation Theorem is the Approximation Theorem [REF]: if [MATH] satisfies App 1 and App 2, and [MATH] admits all finite colimits and [MATH] preserves them, then [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-20-1': 'The requirement of finite colimits in [MATH] and their preservation by [MATH] could be replaced by the requirement of colimits in [MATH] of a more specific type and their preservation by [MATH], namely colimits of diagrams of equivalences indexed by connected finite posets.', '1303.4029-2-20-2': 'We prefer to require "finite colimits" because it is easier to state than the aforementioned more specific type of colimit.', '1303.4029-2-20-3': 'Corollary [REF], on the other hand, assumes [MATH] is an equivalence and [MATH] in order to conclude [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-21-0': 'Theorem [REF], Corollary [REF], and Corollary [REF] are versions of Approximation in which App 2 is replaced by the more general Cofibration App 2.', '1303.4029-2-21-1': 'In other words, instead of requiring every codomain morphism of the form [MATH] to factor as [MATH], which would imply every morphism [MATH] is a cofibration, we merely require every cofibration [MATH] to factor as [MATH], thus allowing non-cofibration morphisms [MATH] in [MATH].', '1303.4029-2-21-2': 'In Theorem [REF], Corollary [REF], and Corollary [REF], we require the domain cofibration subquasicategory [MATH] to admit colimits of diagrams in [MATH] indexed by connected finite posets, and [MATH] is required to preserve such colimits.', '1303.4029-2-21-3': 'In Theorem [REF] we also require [MATH] and [MATH] to be full.', '1303.4029-2-21-4': 'On the other hand, Corollary [REF] and Corollary [REF] require just for [MATH] the functor [MATH] to be full, because [MATH] is an equivalence of categories in those corollaries.', '1303.4029-2-21-5': 'In Corollary [REF] [MATH] is assumed to be an equivalence of categories, but the assumption that [MATH] reflects cofibrations implies that [MATH] is also an equivalence of categories.', '1303.4029-2-22-0': 'Factorization in the sense that every morphism in [MATH] or [MATH] factors as [MATH] is not required for the main results (indeed, factorization in the quasicategorical context is equivalent to requiring all maps to be cofibrations, a strong assumption we do not want to make).', '1303.4029-2-23-0': 'Related Approximation results in the quasicategorical literature are the following.', '1303.4029-2-23-1': 'Barwick proved the special case of our Approximation Theorem in which all maps are cofibrations.', '1303.4029-2-23-2': 'Namely, Barwick\'s Corollary 8.4 of [CITATION] implies: if [MATH] is an exact functor between Waldhausen quasicategories, both of which have all maps cofibrations and have all finite colimits, and [MATH] induces an equivalence [MATH] of homotopy categories, then [MATH] induces a stable equivalence of [MATH]-theory spectra (in fact he shows [MATH] induces an equivalence in any "additive theory").', '1303.4029-2-23-3': 'Blumberg-Gepner-Tabuada proved a stronger statement under the assumption of stability (all maps in a stable quasicategory are considered to be cofibrations).', '1303.4029-2-23-4': "Namely Blumberg-Gepner-Tabuada's Corollary 5.11 of [CITATION] states that a map of stable quasicategories is an equivalence if and only if it induces an equivalence of their homotopy categories.", '1303.4029-2-23-5': 'In general, Blumberg-Gepner-Tabuada study [MATH]-theory in [CITATION] as an invariant of stable quasicategories.', '1303.4029-2-23-6': 'In the present paper, we do not require stability, nor does Barwick.', '1303.4029-2-24-0': 'Simplicial categories, on the other hand, are another model for [MATH]-categories, and recent [MATH]-theory literature contains some related Approximation results concerning Dwyer-Kan hammock localizations of classical Waldhausen categories.', '1303.4029-2-24-1': 'Toen-Vezzosi [CITATION] observed already in 2004 that the [MATH]-theory of a "good" category with fibrations and weak equivalences is an invariant of the underlying [MATH]-category, namely of its Dwyer-Kan hammock localization, despite the fact that the [MATH]-theory cannot be reconstructed from the triangulated homotopy category, as proved by Neeman [CITATION].', '1303.4029-2-24-2': 'Equivalence of localizations is in fact closely correlated to Approximation.', '1303.4029-2-24-3': "Blumberg-Mandell [CITATION] proved that equivalence of Dwyer-Kan localizations follows from Waldhausen's Approximation axioms when [MATH] and [MATH] satisfy 3-for-2 and factorization, see Section [REF] of the present paper.", '1303.4029-2-24-4': 'Cisinski [CITATION] proved that this is actually an "if and only if" statement (see Theorems 2.9 and 3.25, Proposition 4.5, and Scholie 4.15, all in [CITATION]).', '1303.4029-2-25-0': 'Assuming Dwyer-Kan equivalence of Dwyer-Kan localizations of weak cofibration subcategories, and a few other hypotheses, a consequence of [CITATION] is a stable equivalence of [MATH]-theory spectra, see Remark [REF] for details.', '1303.4029-2-25-1': 'The present article remains entirely in the world of Waldhausen quasicategories.', '1303.4029-2-26-0': 'Outline of Paper.', '1303.4029-2-26-1': 'For readers not already familiar with quasicategories, I review in Appendix Section [REF] all the necessary results of Boardman-Vogt, Joyal, and Lurie, including homotopy, join, slice, and colimits.', '1303.4029-2-26-2': "There I also prove a criterion for a simplicial set to be weakly contractible, which is a simplicial version of a categorical result Schlichting extracted from Waldhausen's paper [CITATION].", '1303.4029-2-26-3': 'Section [REF] rapidly presents new examples that are covered by the Cofibration Approximation Theorem [REF] but not the Approximation Theorem [REF].', '1303.4029-2-26-4': 'Some terminology from the rest of the paper is used here, but the purpose is to quickly motive the rest of the work.', '1303.4029-2-26-5': "In Section [REF], I recall the notion of Waldhausen quasicategory, discuss some of its consequences, and introduce the variants [MATH] and [MATH] of Waldhausen's constructions using spines [MATH] rather than simplices [MATH].", '1303.4029-2-26-6': 'Section [REF] is the heart of the paper, it begins with the main technical result of the paper, the Pre-Approximation Theorem [REF], then proves various quasicategorical Approximation Theorems, and compares the axioms Pre-App 2, App 2, and Cofibration App 2.', '1303.4029-2-26-7': 'Section [REF] also contains a more detailed comparison with other results in the literature.', '1303.4029-2-27-0': '# Examples of Quasicategorical Cofibration Approximation Theorem not Covered by Approximation Theorem', '1303.4029-2-28-0': 'In this section we provide some motivational examples for the quasicategorical Cofibration Approximation Theorem [REF] and the rest of the paper.', '1303.4029-2-28-1': 'We freely use terminology that is introduced in the sequel.', '1303.4029-2-29-0': 'The quasicategorical Cofibration Approximation Theorem [REF] has genuinely new examples.', '1303.4029-2-29-1': 'We begin with the inclusion [MATH], and then specify it to a homotopy version of based finite sets, called [MATH].', '1303.4029-2-29-2': 'The comparison of [MATH] with the classical [MATH] also delivers examples of several results in this paper.', '1303.4029-2-30-0': 'A consequence of the Cofibration Approximation Theorem [REF] is the following.', '1303.4029-2-31-0': 'Corollary [REF].', '1303.4029-2-31-1': 'Let [MATH] be Waldhausen quasicategory.', '1303.4029-2-31-2': 'Then [MATH] induces a level-wise equivalence in [MATH]-theory.', '1303.4029-2-32-0': 'If [MATH] has a morphism that is not a cofibration, then the inclusion [MATH] does not satisfy App 2, and the Approximation Theorem [REF] does not apply, but the Cofibration Approximation Theorem [REF] does.', '1303.4029-2-33-0': 'Consider now the quasicategory [MATH] of based "homotopy finite, homotopy discrete spaces."', '1303.4029-2-33-1': 'This quasicategory is the simplicial nerve of the Kan enriched category of based Kan complexes of the form [EQUATION]', '1303.4029-2-33-2': 'A based simplicial map [MATH] between such based Kan complexes is a cofibration in the Waldhausen structure on [MATH] if [MATH] is injective.', '1303.4029-2-33-3': 'The equivalences in the quasicategory [MATH] are the based weak homotopy equivalences between such based Kan complexes.', '1303.4029-2-34-0': "To prove that the equivalences in the quasicategory [MATH] are the based weak homotopy equivalences between such based Kan complexes, we appeal to the foundational results on classical Quillen model structure on [MATH] in combination with Boardman and Vogt's results about homotopy in a quasicategory.", '1303.4029-2-34-1': 'First recall that a based map in [MATH] is a cofibration, fibration, or weak equivalence if and only if its underlying unbased map is.', '1303.4029-2-34-2': 'Thus, the fibrant objects in [MATH] are exactly the based Kan complexes, and every object of [MATH] is cofibrant.', '1303.4029-2-34-3': 'Consequently a based map between based Kan complexes is a weak equivalence if and only if it has a based homotopy inverse.', '1303.4029-2-34-4': 'On the other hand, a morphism in the quasicategory [MATH] is an equivalence if and only if it has a homotopy inverse in the quasicategory [MATH].', '1303.4029-2-34-5': 'But by [CITATION] (attributed to Boardman-Vogt [CITATION]), homotopy between morphisms in the quasicategory [MATH] is the same as based homotopy of based maps between based Kan complexes.', '1303.4029-2-34-6': 'Thus, the equivalences in the quasicategory [MATH] are the based weak homotopy equivalences between based Kan complexes of the form [REF].', '1303.4029-2-35-0': 'The quasicategory [MATH] is a Waldhausen quasicategory in the sense of Definition [REF], but not a Waldhausen category: ordinary pushouts in the category of such based Kan complexes [REF] do not exist.', '1303.4029-2-35-1': 'For instance, the following pushout in [MATH] of 3 such based Kan complexes [REF] is not a Kan complex.', '1303.4029-2-35-2': '[EQUATION]', '1303.4029-2-35-3': 'The two left simplicial sets [MATH] and [MATH] are finite and discrete, hence are homotopy finite and homotopy discrete Kan complexes.', '1303.4029-2-35-4': 'The upper right simplicial set [MATH] is the disjoint union of two nerves of contractible groupoids, so is a homotopy finite and homotopy discrete Kan complex.', '1303.4029-2-35-5': 'The top map is a trivial cofibration in the Quillen model structure on [MATH], i.e. it is a mono weak homotopy equivalence so the pushout is even a homotopy pushout in [MATH].', '1303.4029-2-35-6': 'The top map is of course also [MATH]-injective, so is a cofibration in the Waldhausen structure on the simplicial nerve [MATH].', '1303.4029-2-36-0': 'The bottom right corner pushout object we have indicated only with its non-degenerate 1-skeleton [MATH].', '1303.4029-2-36-1': 'It is not a Kan complex because the inner horn [MATH] does not have a filler, so it is not even a quasicategory.', '1303.4029-2-37-0': 'Since the pushout object in [REF] is not of the form [REF], the category of based Kan complexes of the form [REF] is not a Waldhausen category.', '1303.4029-2-38-0': 'However, the quasicategory [MATH] does admit colimits along its cofibrations.', '1303.4029-2-38-1': 'Namely, since [MATH] is the simplicial nerve of a Kan-enriched simplicial category, homotopy colimits in the simplicial category are the same as the colimits in the quasicategory [MATH] by [CITATION].', '1303.4029-2-38-2': 'We can form homotopy pushouts along [MATH]-injective maps of based Kan complexes of the form [REF] similar to how pushouts along injective based maps in [MATH] can be formed.', '1303.4029-2-39-0': 'Thus [MATH] is a genuine example of a Waldhausen quasicategory, and the inclusion [EQUATION] induces a level-wise equivalence in [MATH]-theory by the Cofibration Approximation Theorem, but not by the Approximation Theorem.', '1303.4029-2-40-0': 'We may now also consider the following inclusion functor and path component functor.', '1303.4029-2-40-1': '[EQUATION]', '1303.4029-2-40-2': 'The inclusion functor [MATH] does not satisfy App 2.', '1303.4029-2-40-3': 'Namely, to satisfy App 2, the non-injective constant [MATH] map would have to factor as a composite of injections.', '1303.4029-2-40-4': '[EQUATION]', '1303.4029-2-40-5': 'But that is impossible, so [MATH] does not satisfy App 2.', '1303.4029-2-40-6': 'But [MATH] does satisfy Cofibration App 2.', '1303.4029-2-40-7': 'Any given [MATH]-injective map in [MATH] [EQUATION] with codomain a based Kan complex [MATH] of the form [REF] factors as [EQUATION] where the based injective map sends a point to the path component it lands in, and the equivalence sends a path component in [MATH] to the point in that path component in the image of [MATH].', '1303.4029-2-40-8': 'So [MATH] satisfies Cofibration App 2.', '1303.4029-2-41-0': 'By similar considerations, one can show that [MATH] in [REF] does not satisfy App 2 but does satisfy Cofibration App 2.', '1303.4029-2-42-0': 'Both [MATH] and [MATH] in [REF] satisfy the hypotheses of the Cofibration Approximation Theorem [REF], Corollary [REF], and Corollary [REF], so they each induce a level-wise equivalence of Waldhausen [MATH]-theory spectra.', '1303.4029-2-43-0': 'We should not be surprised [MATH] and [MATH] have the same [MATH]-theory: [MATH] and [MATH] in [REF] are, after all, inverse Waldhausen equivalences of Waldhausen quasicategories in the sense of [CITATION].', '1303.4029-2-43-1': 'But it is still interesting to note that the Approximation Theorem [REF] does not apply, though Cofibration Approximation [REF] does.', '1303.4029-2-44-0': 'Further examples of non-equivalences that satisfy the hypotheses of the Cofibration Approximation Theorem [REF] and Corollary [REF] are the composites [EQUATION]', '1303.4029-2-44-1': 'These composites satisfy Cofibration App 2 because both second morphisms do and both first morphisms are essentially surjective, see Proposition [REF] [REF].', '1303.4029-2-45-0': '# Waldhausen Quasicategories', '1303.4029-2-46-0': "We first recall some background on Waldhausen quasicategories from the paper of Fiore-Pieper [CITATION], discuss consequences of the definition, and introduce a variant [MATH] of Waldhausen's [MATH].", '1303.4029-2-46-1': 'Recall that a subquasicategory [MATH] of a quasicategory [MATH] is 1-full if any simplex of [MATH] is in [MATH] if and only if all of its edges are in [MATH].', '1303.4029-2-47-0': '## Waldhausen Quasicategories and Exact Functors', '1303.4029-2-48-0': '[Waldhausen quasicategory, [CITATION]] A Waldhausen quasicategory consists of a quasicategory [MATH] with zero objects and a subquasicategory [MATH], the 1-simplices of which are called cofibrations and denoted [MATH], such that', '1303.4029-2-49-0': 'The subquasicategory [MATH] is 1-full in [MATH] and contains all equivalences in [MATH],', '1303.4029-2-50-0': 'For each object [MATH] of [MATH] and any zero object [MATH] of [MATH], every morphism [MATH] is a cofibration,', '1303.4029-2-51-0': 'The pushout of a cofibration along any morphism exists, and every pushout of a cofibration along any morphism is a cofibration.', '1303.4029-2-52-0': "Barwick's notion of Waldhausen [MATH]-category in [CITATION] is equivalent to Definition [REF].", '1303.4029-2-52-1': 'See Fiore-Pieper [CITATION] for a proof.', '1303.4029-2-53-0': 'Though Definition [REF] looks much like the classical definition, there are some important differences which have far-reaching consequences.', '1303.4029-2-53-1': 'Perhaps most prominently, a classical Waldhausen category comes equipped with a class of "weak equivalences" which contains the isomorphisms, whereas a Waldhausen quasicategory has its class of "weak equivalences" pre-selected as the equivalences of the underlying quasicategory.', '1303.4029-2-53-2': 'These are exactly the maps which become isomorphisms in the homotopy category.', '1303.4029-2-54-0': 'Consequently, extra hypotheses on the equivalences (typically needed in a discussion of Approximation) are rarely needed because they automatically hold.', '1303.4029-2-54-1': 'For instance, the 3-for-2 property for equivalences in a quasicategory [MATH] follows immediately from the 3-for-2 property for isomorphisms in the homotopy category [MATH] and the fact that 2-simplices in [MATH] give rise to commutative triangles in [MATH].', '1303.4029-2-54-2': 'Even more strongly, the equivalences in a quasicategory [MATH] satisfy the 6-for-2 property by a similar argument applied to [MATH] using the 6-for-2 property of the isomorphisms in [MATH] (the isomorphisms in any category satisfy the 6-for-2 property).', '1303.4029-2-55-0': 'Another consequence of choosing the equivalences as the "weak equivalences" is that any map homotopic to a cofibration is also a cofibration, see [CITATION].', '1303.4029-2-56-0': 'We also know that [MATH] is naturally a subcategory of [MATH] which contains the isomorphisms of [MATH].', '1303.4029-2-56-1': 'Namely from the 1-fullness of [MATH] in [MATH] in Definition [REF], it follows that for any cofibrations [MATH] there is a 2-simplex in [MATH] with boundary [MATH] if and only if there is a 2-simplex in [MATH] with boundary [MATH].', '1303.4029-2-57-0': 'The choice of the "weak equivalences" as the equivalences also changes the way that classical hypotheses are related to one another.', '1303.4029-2-57-1': 'For instance, the factorization axiom in a Waldhausen quasicategory is equivalent to the requirement that every map is a cofibration.', '1303.4029-2-57-2': 'A Waldhausen quasicategory [MATH] is said to admit factorization if for any morphism [MATH] of [MATH] there exists a 2-simplex [MATH] with boundary [EQUATION].', '1303.4029-2-57-3': 'As a proof that factorization is equivalent to all maps being cofibrations, if [MATH] admits factorization, and [MATH] is any morphism in [MATH], then [MATH] in the homotopy category for some equivalence [MATH] and some cofibration [MATH].', '1303.4029-2-57-4': 'But every equivalence is a cofibration, and [MATH] is naturally a subcategory of [MATH], so [MATH] is the homotopy class of a cofibration.', '1303.4029-2-57-5': 'But any map homotopic to a cofibration is a cofibration, so [MATH] is a cofibration.', '1303.4029-2-57-6': 'The converse is clear.', '1303.4029-2-58-0': 'Every "weak cofibration" in a Waldhausen quasicategory is actually a cofibration, so we make no distinction.', '1303.4029-2-58-1': 'More precisely, in [CITATION] and [CITATION], Blumberg-Mandell call a morphism [MATH] in a classical Waldhausen category a weak cofibration if there is a zig-zag of weak equivalences in the arrow category from [MATH] to a cofibration.', '1303.4029-2-58-2': 'If we have a commutative square in a Waldhausen quasicategory [MATH] (that is a map [MATH]) [EQUATION] then in the homotopy category [MATH] we have [EQUATION] so [MATH] is in [MATH] if and only if [MATH] is in [MATH] (recall that [MATH] contains the isomorphisms of [MATH]).', '1303.4029-2-58-3': 'But the homotopy class [MATH] is in [MATH] if and only if the morphism [MATH] is in [MATH], and similarly for [MATH] and [MATH].', '1303.4029-2-58-4': 'So in any vertical zig-zag of commutative squares like [REF] in a Waldhausen quasicategory, with all the vertical arrows weak equivalences, if any one of the horizontal arrows is a cofibration, then all of the horizontal arrows are cofibrations, and weak cofibrations are cofibrations.', '1303.4029-2-59-0': 'Similarly, a homotopy cocartesian square in the sense of [CITATION] and [CITATION], but in a Waldhausen quasicategory, is merely a pushout square in which one of the legs is a cofibration (the opposite morphism will then also be a cofibration).', '1303.4029-2-59-1': 'This follows from the fact that pushouts in a quasicategory are invariant under equivalence.', '1303.4029-2-60-0': 'After this discussion of the far-reaching consequences of the definition of Waldhausen quasicategory, we can now return to the review of the theory itself.', '1303.4029-2-61-0': '[Exact functor] Let [MATH] and [MATH] be Waldhausen quasicategories.', '1303.4029-2-61-1': 'A functor [MATH] is called exact if it sends zero objects of [MATH] to zero objects of [MATH], it sends cofibrations to cofibrations, and maps each pushout square along a cofibration to a pushout square along a cofibration.', '1303.4029-2-62-0': 'By 1-fullness of the subquasicategory of cofibrations, we have [MATH] for the entire cofibration sub quasicategory [MATH].', '1303.4029-2-62-1': 'Every map of quasicategories sends equivalences to equivalences, so we also have [MATH] by 1-fullness of the maximal sub Kan complex.', '1303.4029-2-62-2': 'By the comments preceding the definition, an exact functor between Waldhausen quasicategories preserves weak cofibrations and homotopy cocartesian squares.', '1303.4029-2-63-0': '## The [MATH] Construction', '1303.4029-2-64-0': 'We next recall the [MATH] construction.', '1303.4029-2-64-1': 'See also [CITATION] of Lurie, and [CITATION] where Blumberg-Gepner-Tabuada compare [MATH] with the [MATH] construction of [CITATION] and [CITATION] in the case of a simplicial model category which admits all finite homotopy colimits.', '1303.4029-2-64-2': 'See also Fiore-Pieper [CITATION].', '1303.4029-2-64-3': 'For a fibrational version of [MATH], see Barwick [CITATION].', '1303.4029-2-65-0': 'Let [MATH] be the category of arrows in [MATH].', '1303.4029-2-65-1': 'It is the partially ordered set with elements [MATH] such that [MATH], and with the order [MATH] whenever [MATH] and [MATH].', '1303.4029-2-65-2': 'The category [MATH] is an upper-triangular subgrid of an [MATH]-grid of squares.', '1303.4029-2-65-3': 'The upper triangular subgrid [MATH] contains the main diagonal of objects, but none below it.', '1303.4029-2-66-0': '[[MATH] Construction] Let [MATH] be a Waldhausen quasicategory.', '1303.4029-2-66-1': 'An [MATH]-complex is a map of simplicial sets [MATH] such that', '1303.4029-2-67-0': 'For each [MATH], [MATH] is a zero object of [MATH], possibly different for each [MATH],', '1303.4029-2-68-0': 'For each [MATH], the morphism [MATH] is a cofibration,', '1303.4029-2-69-0': 'For each [MATH], the diagram [EQUATION] is a pushout square in [MATH].', '1303.4029-2-70-0': 'Let [MATH] be the 0-full sub simplicial set of [MATH] on the [MATH]-complexes.', '1303.4029-2-70-1': 'The [MATH] construction of [MATH] is the simplicial quasicategory [MATH] defined by [EQUATION].', '1303.4029-2-70-2': 'The objects of [MATH] are sequences of cofibrations in [MATH] [EQUATION] with a choice of quotient [MATH] for each [MATH], and a choice of all composites with simplices that fill them.', '1303.4029-2-70-3': 'The face and degeneracy maps of the simplicial object [MATH] in the category [MATH] are induced from [MATH] via the category of arrows construction.', '1303.4029-2-70-4': 'In particular this means, if [MATH] and [MATH], then [MATH].', '1303.4029-2-71-0': 'Each quasicategory [MATH] is a Waldhausen quasicategory.', '1303.4029-2-71-1': 'A morphism [MATH] in [MATH] is a cofibration in [MATH] if each [MATH] is a cofibration and each pushout morphism [EQUATION] is a cofibration in [MATH] for each [MATH].', '1303.4029-2-71-2': 'If a morphism is level-wise homotopic to a cofibration in [MATH], then it is also a cofibration in [MATH].', '1303.4029-2-71-3': 'A natural transformation [MATH] is an equivalence in [MATH] if and only if each component [MATH] is an equivalence in [MATH] by [CITATION] or [CITATION].', '1303.4029-2-71-4': 'Because all the squares in an object are pushouts, this is equivalent to requiring every [MATH] to be an equivalence in [MATH] (the pushout of an equivalence in a quasicategory along any morphism is also an equivalence, see [CITATION]).', '1303.4029-2-71-5': 'See [CITATION] for the case of Waldhausen categories.', '1303.4029-2-72-0': 'The [MATH]-th [MATH]-theory space is [EQUATION] for [MATH], where [MATH] appears [MATH] times.', '1303.4029-2-73-0': 'The [MATH]-theory spaces form an [MATH]-spectrum [MATH] beyond the 0-th term.', '1303.4029-2-74-0': '## The [MATH] Construction, or the [MATH] Construction without Composites', '1303.4029-2-75-0': 'An important difference between categories and quasicategories motivates our variant of the classical definition: a sequence of [MATH] morphisms in a category, head to tail, already has uniquely determined composites of all head-to-tail subseqences and their composites, whereas in a quasicategory such composites for a head-to-tail sequence exist but are not chosen.', '1303.4029-2-75-1': 'For a category [MATH] for instance, if [MATH] denotes the spine of the [MATH]-simplex [MATH], namely the 1-dimensional subcomplex of [MATH] given by the union of the edges [MATH] for [MATH], then we have [EQUATION].', '1303.4029-2-75-2': 'In other words, indicating a sequence of [MATH] morphisms in a category [MATH] is tantamount to indicating all of the composites of subsequences (this is one reason why it is customary in category theory to only draw the [MATH] morphisms to indicate the [MATH] morphisms along with all of their composites).', '1303.4029-2-76-0': 'To see how such composites of subsequences exist in a quasicategory [MATH], notice that a map [MATH] is a sequence of [MATH] morphisms in [MATH], while a map [MATH] is a sequence of [MATH] morphisms in [MATH] together with a choice of composites for all possible subsequences and their composites, together with simplices which make them commute.', '1303.4029-2-76-1': 'The spine inclusion [MATH] is mid anodyne [CITATION] so that the following lift exists.', '1303.4029-2-76-2': '[EQUATION]', '1303.4029-2-76-3': 'Nevertheless, there is little theoretical difference in considering maps [MATH] versus maps [MATH] because for any quasicategory [MATH], the spine inclusion induces an equivalence [MATH] by [CITATION].', '1303.4029-2-76-4': 'Practically, it is easier to work with [MATH], and this is what we do now for [MATH], and for [MATH] in Section [REF].', '1303.4029-2-77-0': '[[MATH] Construction] Let [MATH] be the intersection of [MATH] with the 2-dimensional simplicial set [MATH].', '1303.4029-2-77-1': 'A restricted [MATH]-complex in a Waldhausen quasicategory [MATH] is a map of simplicial sets [MATH] such that', '1303.4029-2-78-0': 'For each [MATH], [MATH] is a zero object of [MATH], possibly different for each [MATH],', '1303.4029-2-79-0': 'For each [MATH], the morphism [MATH] is a cofibration,', '1303.4029-2-80-0': 'For each [MATH], the diagram [EQUATION] is a pushout square in [MATH].', '1303.4029-2-81-0': 'Then [MATH] is the full sub-simplicial set of [MATH] on the restricted [MATH]-complexes.', '1303.4029-2-81-1': 'The cofibrations in [MATH] are defined just as in [MATH].', '1303.4029-2-82-0': 'Note that the [MATH] construction is not a simplicial object.', '1303.4029-2-83-0': '[Equivalent descriptions of [MATH] and [MATH]] Recall that [MATH] is the upper triangular part (including main diagonal) of the diagram [MATH], and similarly [MATH] is the upper triangular part (including main diagonal) of the diagram [MATH].', '1303.4029-2-83-1': 'We can non-uniquely extend any element of [MATH] to a functor [MATH] which is a zero object on each [MATH] with [MATH].', '1303.4029-2-83-2': 'Similarly, we can non-uniquely extend any element of [MATH] to a functor [MATH] which is [MATH] on each [MATH] with [MATH].', '1303.4029-2-83-3': 'More precisely, let [MATH] be the sub quasicategory of [MATH] that is 0-full on the functors [MATH] which are zero objects on each [MATH] with [MATH] and satisfy [REF], [REF], and [REF] of Definition [REF].', '1303.4029-2-83-4': 'Similarly let [MATH] be the sub quasicategory of [MATH] that is 0-full on the functors [MATH] which are zero objects on each [MATH] with [MATH] and satisfy [REF], [REF], and [REF] of Definition [REF].', '1303.4029-2-84-0': 'Then the forgetful functors [EQUATION] are equivalences of quasicategories.', '1303.4029-2-85-0': '[Waldhausen equivalence of [MATH] with [MATH]] For any Waldhausen quasicategory [MATH], the forgetful functor [MATH] induced by the inclusion [MATH] is a Waldhausen equivalence, that is, it is an exact functor which admits an exact pseudo inverse (see [CITATION] for more on Waldhausen equivalences).', '1303.4029-2-86-0': 'Every mid anodyne map is a weak categorical equivalence [CITATION], and the Cartesian product of two weak categorical equivalences is a weak categorical equivalence [CITATION], so [MATH] is a weak categorical equivalence.', '1303.4029-2-86-1': 'Then the induced map [MATH] is an equivalence of quasicategories by [CITATION].', '1303.4029-2-86-2': 'Let us call this restriction equivalence [MATH].', '1303.4029-2-86-3': 'Then so far we have the following two commutative squares and information from Remark [REF].', '1303.4029-2-86-4': '[EQUATION]', '1303.4029-2-86-5': 'The middle map, which is the restriction of the fully faithful essentially surjective map [MATH] to [MATH], is also an equivalence: it is fully faithful because [MATH] is and its domain and codomain are 0-full, and it is essentially surjective because [MATH].', '1303.4029-2-87-0': 'Finally, by the 3-for-2 property of equivalences, the bottom map [MATH] is also an equivalence.', '1303.4029-2-87-1': 'This equivalence is exact and reflects cofibrations, so it is a Waldhausen equivalence by [CITATION].', '1303.4029-2-88-0': '## The [MATH] Construction', '1303.4029-2-89-0': 'We next define the quasicategory [MATH] of sequences of [MATH]-many cofibrations without selected quotients.', '1303.4029-2-89-1': 'Let [MATH] be a Waldhausen quasicategory.', '1303.4029-2-89-2': 'The quasicategory [MATH] is the 0-full subsimplicial set of [MATH] on the functors [MATH].', '1303.4029-2-89-3': 'For a fibrational version of an [MATH], see Barwick [CITATION].', '1303.4029-2-90-0': 'As in the classical work of Waldhausen [CITATION], the quasicategory [MATH] is a Waldhausen quasicategory, the cofibrations in [MATH] are level-wise cofibrations with the additional property that the map from each pushout to each lower corner is a cofibration in [MATH].', '1303.4029-2-91-0': 'The forgetful functor [MATH] is a Waldhausen equivalence of Waldhausen quasicategories.', '1303.4029-2-92-0': 'This forgetful functor factors as the composite [MATH], the first map of which is a Waldhausen equivalence by Proposition [REF].', '1303.4029-2-92-1': 'An inductive application of Proposition [REF], forgetting one row at a time, shows that the second map is also an equivalence.', '1303.4029-2-92-2': 'Each of these intermediate functors is exact, reflects cofibrations, and is an equivalence, so is a Waldhausen equivalence by [CITATION].', '1303.4029-2-92-3': 'For instance, to apply Proposition [REF], we consider row 1 and row 2 of a restricted [MATH]-complex as part of a colimiting cocone consisting of rows 1 and 2, a row of [MATH], and a row of [MATH], so that we have a row of pushout squares.', '1303.4029-2-93-0': 'Proposition [REF], and the earlier results it references, provides some of the details to [CITATION] of Blumberg-Gepner-Tabuada.', '1303.4029-2-94-0': 'The equivalence of quasicategories [MATH] in Proposition [REF] induces an equivalence of homotopy categories since [MATH] is a 2-functor [CITATION].', '1303.4029-2-95-0': '# Approximation Theorems for Waldhausen Quasicategories', '1303.4029-2-96-0': 'We now turn to the proof of various Approximation Theorems for Waldhausen quasicategories, namely the Pre-Approximation Theorem [REF], the Approximation Theorem [REF], its Corollary [REF], the Cofibration Approximation Theorem [REF], and its Corollaries [REF] and [REF].', '1303.4029-2-96-1': 'We also have a brief interlude in Section [REF] that compares three different approximation axioms: Pre-App 2, App 2, and Cofibration App 2.', '1303.4029-2-97-0': 'The central result in this succession of Approximation Theorems is the Pre-Approximation Theorem [REF].', '1303.4029-2-97-1': "Its proof uses the quasicategorical version of Quillen's Theorem A recalled in Section [REF], but does not require Waldhausen structures.", '1303.4029-2-97-2': 'The Pre-Approximation Theorem gives sufficient conditions for a functor to restrict to an equivalence of maximal [MATH]-groupoids.', '1303.4029-2-98-0': '## The Pre-Approximation Theorem', '1303.4029-2-99-0': 'We now turn to the main technical result in this paper, which has variants of Waldhausen Approximation among its consequences.', '1303.4029-2-99-1': "The idea goes back to Waldhausen in the classical context of Approximation, namely use a version of Proposition [REF] and Quillen's Theorem A. However, the implementation here is different.", '1303.4029-2-99-2': "A key ingredient comes from Schlichting's Appendix [CITATION], where he proves Approximation for Waldhausen categories, assuming factorizations instead of a cylinder functor.", '1303.4029-2-99-3': 'There, Schlichting takes a colimit of a (cofibrant replacement of a) [MATH]-shaped diagram, and applies the 3-for-2 property to an appropriate commutative triangle.', '1303.4029-2-99-4': 'We proceed in this way (without cofibrant replacement) for a similar triangle in diagram [REF].', '1303.4029-2-100-0': 'We do not assume factorization in the domain quasicategory, nor do we assume a cylinder functor, nor even a Waldhausen structure for the Pre-Approximation Theorem [REF].', '1303.4029-2-100-1': 'For its notation, recall that if [MATH] is a quasicategory, then [MATH] is its maximal Kan subcomplex, which is the 1-full subquasicategory on the equivalences in [MATH].', '1303.4029-2-100-2': 'Similarly, if [MATH] is a functor of quasicategories (i.e. map of the underlying simplicial sets), then [MATH] is the restriction of [MATH] to the maximal Kan subcomplexes.', '1303.4029-2-100-3': 'For the proof of Theorem [REF], recall also the adjunction between [EQUATION] and [MATH] in equation [REF].', '1303.4029-2-101-0': '[Pre-Approximation] Let [MATH] be a functor between quasicategories.', '1303.4029-2-101-1': 'No Waldhausen structures are assumed.', '1303.4029-2-101-2': 'Suppose:', '1303.4029-2-102-0': '(Pre-App 1)', '1303.4029-2-103-0': '[MATH] reflects equivalences.', '1303.4029-2-104-0': '(Pre-App 2)', '1303.4029-2-105-0': 'For every [MATH] and every morphism [MATH] in the codomain [MATH], there exists a morphism [MATH] in [MATH], an equivalence [MATH] in [MATH], and a 2-simplex in [MATH] of the following form.', '1303.4029-2-105-1': '[EQUATION]', '1303.4029-2-106-0': 'The functor of groupoids [MATH] is essentially surjective and full.', '1303.4029-2-106-1': 'In other words, [MATH] is essentially surjective and [MATH] is full on isomorphisms in the sense that if [MATH] is an isomorphism in [MATH], then there exists an isomorphism [MATH] in [MATH] such that [MATH].', '1303.4029-2-107-0': 'The quasicategory [MATH] admits colimits of diagrams in [MATH] indexed by connected finite posets, and [MATH] preserves such colimits.', '1303.4029-2-108-0': '(Note that [MATH] itself is not assumed to admit such colimits.)', '1303.4029-2-109-0': 'Then [MATH] is an equivalence of [MATH]-groupoids.', '1303.4029-2-110-0': 'If [MATH] is an equivalence of categories, then hypotheses [REF], [REF], [REF] are satisfied.', '1303.4029-2-110-1': 'Thus, if [MATH] is an equivalence of categories and [REF] holds, then [MATH] is an equivalence of [MATH]-groupoids.', '1303.4029-2-111-0': 'Suppose the four hypotheses [REF], [REF], [REF], [REF] hold.', '1303.4029-2-111-1': "We show that [MATH] is a weak homotopy equivalence using the quasicategorical version of Quillen's Theorem A recalled above in Theorem [REF].", '1303.4029-2-111-2': 'But since the domain and codomain of [MATH] are cofibrant-fibrant objects in the Kan model structure, this implies [MATH] is actually even a simplicial homotopy equivalence, and therefore an equivalence of quasicategories.', '1303.4029-2-112-0': 'Let [MATH] be a vertex in [MATH].', '1303.4029-2-112-1': 'Our goal is to show that [MATH] (see Definition [REF]) is weakly contractible using Proposition [REF].', '1303.4029-2-112-2': 'The over simplicial set [MATH] is non-empty and connected by hypothesis [REF].', '1303.4029-2-112-3': 'Let [MATH] be a connected finite poset and consider a map of simplicial sets [MATH].', '1303.4029-2-112-4': 'By the universal property of the pushout [MATH], this corresponds to two maps [MATH] and [MATH] such that [MATH] and the two corresponding triangles commute in the left diagram of [REF].', '1303.4029-2-112-5': '[EQUATION]', '1303.4029-2-112-6': 'We show that [MATH] extends to [MATH] by extending [MATH] and [MATH] to [MATH] and [MATH] on [MATH] in such a way that [MATH] as in the right diagram of [REF], and then we observe that [MATH] actually has codomain [MATH], and [MATH] actually has codomain [MATH], so that an extension [MATH] is defined by the universal property of the pullback.', '1303.4029-2-113-0': 'We use the notation [MATH] where [MATH] is the terminal simplicial set with sole vertex [MATH].', '1303.4029-2-113-1': 'This is of course isomorphic to [MATH], but writing [MATH] instead of 1 avoids ambiguities when we later use [MATH].', '1303.4029-2-113-2': 'We will write [MATH] as [MATH], the nerve of the category with a single nontrivial morphism [MATH].', '1303.4029-2-113-3': 'We shall also have occasion to use [MATH] and [MATH], and also [MATH].', '1303.4029-2-114-0': 'We extend [MATH] to [MATH] in two steps: first extend [MATH] to a universal cocone [MATH], and then compose [MATH] with a morphism [MATH] in [MATH] obtained from hypothesis [REF] to form [MATH].', '1303.4029-2-114-1': 'For the first step, since [MATH] takes values in [MATH], we can form a universal cocone [MATH] by hypothesis [REF].', '1303.4029-2-114-2': 'Its image [MATH] is a universal cocone in [MATH], also by [REF].', '1303.4029-2-115-0': 'For the second step of extending [MATH] to [MATH], notice we have another cocone in [MATH], and this one consists entirely of equivalences: the transpose [MATH] of [MATH] via adjunction [REF] is a (non-universal) cocone in [MATH] with [MATH].', '1303.4029-2-115-1': 'The transpose [MATH] is a cocone for [MATH] because [MATH] by Remark [REF], which is [MATH].', '1303.4029-2-115-2': 'Since [MATH] is an initial object of [MATH], there is a (homotopically unique) cocone morphism [MATH] such that [EQUATION] commutes and [EQUATION]', '1303.4029-2-115-3': 'Next we apply hypothesis [REF] to factor [MATH] as [EQUATION] in [MATH] using a morphism [MATH] in [MATH].', '1303.4029-2-116-0': 'The cocone [MATH] in composes with the morphism [MATH] to give a new cocone [MATH].', '1303.4029-2-116-1': 'Namely, the left vertical map below is mid anodyne by [CITATION] [EQUATION] because [MATH] is left anodyne.', '1303.4029-2-116-2': 'Here we write [MATH] to signify the nerve of the category [MATH], as mentioned earlier.', '1303.4029-2-116-3': 'The restriction of the diagonal lift to [MATH] is [MATH].', '1303.4029-2-116-4': 'This completes the extension of [MATH] to the cocone [MATH].', '1303.4029-2-117-0': 'We claim that [MATH] sends every morphism to an equivalence.', '1303.4029-2-117-1': 'Let [MATH] be an object of [MATH] and consider the unique morphisms [MATH], [MATH], [MATH] from [MATH] to [MATH], [MATH], [MATH] in [MATH], as in the following diagram of three 2-simplices.', '1303.4029-2-117-2': '[EQUATION]', '1303.4029-2-117-3': 'We construct a diagram in [MATH] of 2-simplices in [REF] by mapping parts of [REF] individually.', '1303.4029-2-117-4': 'The composite of [MATH] with the filler in [REF] maps the top 2-simplex of [REF] to the top 2-simplex of [REF].', '1303.4029-2-117-5': '[EQUATION]', '1303.4029-2-117-6': 'The lower left 2-simplex is the [MATH]-image of the lower left triangle of [REF] because of the equalities [REF].', '1303.4029-2-117-7': 'The lower right 2-simplex of [REF] is the 2-simplex [REF].', '1303.4029-2-117-8': 'All together, diagram [REF] is a [MATH]-horn in [MATH], so it extends to a 3-simplex, and in particular, we obtain a 2-simplex with boundary the outermost triangle.', '1303.4029-2-117-9': 'Thus, by the 3-for-2 property of equivalences, the top morphism [MATH] is an equivalence.', '1303.4029-2-117-10': 'By hypothesis [REF], we have that [MATH] is also an equivalence.', '1303.4029-2-117-11': 'We already know that [MATH] maps all morphisms of [MATH] to equivalences, as [MATH], so [MATH] sends every morphism to an equivalence as claimed.', '1303.4029-2-118-0': 'Next we construct the extension [MATH] from our already completed work.', '1303.4029-2-118-1': 'The inclusion [MATH] is left anodyne, so the left vertical map below is mid anodyne by [CITATION].', '1303.4029-2-118-2': '[EQUATION]', '1303.4029-2-118-3': 'The top horizontal morphism in [REF] is the following map, which is like [REF] with [MATH] replaced by [MATH].', '1303.4029-2-118-4': '[EQUATION]', '1303.4029-2-118-5': 'The second and third objects are equal by the compatibility of [MATH] with union by [CITATION].', '1303.4029-2-118-6': 'A diagonal lift in [REF] exists, and we denote its restriction to [MATH] by [EQUATION].', '1303.4029-2-118-7': 'The restriction is the analogue of the outermost triangle (i.e. the composite triangle) of [REF].', '1303.4029-2-118-8': 'In fact, [MATH] goes into [MATH] by the comments after [REF].', '1303.4029-2-118-9': 'The transpose of [MATH] is the desired map [EQUATION].', '1303.4029-2-118-10': "Hence we have constructed the extension [MATH] by the universal property of the pullback, so [MATH] is contractible by Proposition [REF], and [MATH] is a weak homotopy equivalence by the quasicategorical version of Quillen's Theorem A, recalled above in Theorem [REF].", '1303.4029-2-119-0': 'Suppose now that [MATH] is an equivalence of categories.', '1303.4029-2-119-1': 'We show that hypotheses [REF], [REF], [REF] are satisfied.', '1303.4029-2-119-2': 'For [REF], [MATH] reflects equivalences because [MATH] reflects isomorphisms.', '1303.4029-2-119-3': 'To construct the factorization in [REF] of given [MATH], we use essential surjectivity of [MATH] to find an equivalence [MATH], then we use the fully faithfulness of [MATH] to find a homotopy class [MATH] in [MATH] with [EQUATION].', '1303.4029-2-119-4': 'Finally, [MATH], so a 2-simplex as in [REF] exists.', '1303.4029-2-119-5': 'For [REF], [MATH] is essentially surjective because [MATH] is, and [MATH] is full because [MATH] is full and reflects equivalences ([MATH] is even fully faithful).', '1303.4029-2-120-0': 'Pre-App 2, stated as hypothesis [REF] of Theorem [REF], implies that [MATH] is essentially surjective when there is an initial object in the image of [MATH].', '1303.4029-2-120-1': 'This is the case for an exact functor between Waldhausen quasicategories.', '1303.4029-2-121-0': 'The following example illustrates that the colimit condition stated in hypothesis [REF] of Theorem [REF] is not an empty condition.', '1303.4029-2-121-1': 'Consider the connected finite poset [EQUATION] and the constant diagram [MATH] indexed by [MATH] in the quasicategory of Kan complexes.', '1303.4029-2-121-2': 'This diagram is a diagram of equivalences, in fact a diagram of equalities of the selected terminal Kan complex [MATH].', '1303.4029-2-121-3': 'Then the colimit of this constant diagram in this quasicategory is the homotopy colimit [MATH] in the Kan enriched category of Kan complexes [CITATION].', '1303.4029-2-121-4': 'By Bousfield-Kan [CITATION], [MATH] has the homotopy type of [MATH].', '1303.4029-2-121-5': 'Thus, the colimit condition [REF] on a diagram of equivalences in a quasicategory is genuinely different from the analogous condition on a diagram of isomorphisms in a category.', '1303.4029-2-121-6': 'In a category, a diagram of isomorphisms indexed by a connected finite poset always has a colimit: simply select any object in the diagram, and take as a colimiting cocone the paths to that object.', '1303.4029-2-122-0': '## The Approximation Theorem', '1303.4029-2-123-0': 'In this section we prove the Approximation Theorem [REF] using the Pre-Approximation Theorem [REF] and Lemma [REF].', '1303.4029-2-123-1': 'A comparison with other recent Approximation Theorems in the literature is in the subsequent Section [REF].', '1303.4029-2-124-0': 'A familiar tool in work on Approximation is to consider the coproduct [MATH] and make a factorization as in [REF], and I thank Georgios Raptis for suggesting the following Lemma.', '1303.4029-2-124-1': 'Recall that if [MATH] and [MATH] are objects in a Waldhausen quasicategory [MATH], then we denote any pushout of [MATH] by [MATH], and indicate any morphism induced by the universal property as [MATH].', '1303.4029-2-124-2': 'In particular, if [MATH] is an exact functor, then [MATH] is a [MATH] and we can speak of morphism sums out of [MATH], without making any identifications.', '1303.4029-2-125-0': '[Condition for a Homotopy Class to be in Image of [MATH] or [MATH]] Let [MATH] be an exact functor of Waldhausen quasicategories and suppose [MATH] reflects equivalences.', '1303.4029-2-125-1': 'Let [MATH] and [MATH] be objects in [MATH].', '1303.4029-2-125-2': 'Suppose [MATH] is a morphism in [MATH] and suppose the map [EQUATION] factors as [MATH] followed by an equivalence [MATH].', '1303.4029-2-125-3': '[EQUATION]', '1303.4029-2-125-4': 'Then [MATH] is in the image of [MATH].', '1303.4029-2-125-5': 'More precisely, there exists a homotopy class [MATH] in [MATH] such that [MATH].', '1303.4029-2-125-6': 'Moreover, if [MATH] is an equivalence, then [MATH] is in the image of [MATH].', '1303.4029-2-126-0': 'The two triangles [EQUATION] commute, so we factor the middle map according to [REF], and form the dotted composites below, to have 4 commutative triangles.', '1303.4029-2-126-1': '[EQUATION]', '1303.4029-2-126-2': 'By 3-for-2 in the bottom right triangle, [MATH] is an equivalence, so [MATH] is an equivalence, as [MATH] reflects equivalences.', '1303.4029-2-126-3': 'From the two right triangles we now have [MATH] is homotopic to [MATH], so we may take [MATH] to be any composite [MATH].', '1303.4029-2-126-4': 'If [MATH] is an equivalence, then [MATH] is also an equivalence, as [MATH] reflects equivalences, so [MATH] is in the image of [MATH].', '1303.4029-2-127-0': 'If [MATH] satisfies App 1 and App 2, then both [MATH] and [MATH] are full.', '1303.4029-2-128-0': 'This follows directly from Lemma [REF].', '1303.4029-2-129-0': '[Waldhausen Approximation for Quasicategories] Let [MATH] be an exact functor between Waldhausen quasicategories.', '1303.4029-2-129-1': 'Suppose:', '1303.4029-2-130-0': '(App 1)', '1303.4029-2-131-0': '[MATH] reflects equivalences.', '1303.4029-2-132-0': '(App 2)', '1303.4029-2-133-0': 'For every [MATH] and every morphism [MATH] in the codomain [MATH], there exists a cofibration [MATH] in [MATH], an equivalence [MATH] in [MATH], and a 2-simplex in [MATH] of the following form.', '1303.4029-2-133-1': '[EQUATION]', '1303.4029-2-134-0': 'The domain quasicategory [MATH] admits all finite colimits, and [MATH] preserves them (both are true for instance if all morphisms of the domain quasicategory [MATH] are cofibrations).', '1303.4029-2-135-0': 'Then [MATH] is an equivalence of [MATH]-groupoids, as is [MATH] for every [MATH].', '1303.4029-2-135-1': 'Consequently [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-136-0': 'We combine the Pre-Approximation Theorem [REF] with Lemma [REF].', '1303.4029-2-136-1': 'For [REF] and [REF] of the Pre-Approximation Theorem, Pre-App 1 and Pre-App 2 hold by assumption, as Pre-App 1 = App 1, and App 2 implies Pre-App 2.', '1303.4029-2-136-2': 'For [REF], the functor [MATH] is essentially surjective by Remark [REF], and is full by Lemma [REF].', '1303.4029-2-136-3': 'The special finite colimits required in [REF] of the Pre-Approximation Theorem exist in [MATH] and are preserved by [MATH] by hypothesis.', '1303.4029-2-137-0': 'The same argument applies to [MATH] to conclude [MATH] is an equivalence of [MATH]-groupoids, since [MATH] satisfies App 1 and App 2 when [MATH] does by [CITATION], see Proposition [REF] for the translation to quasicategories.', '1303.4029-2-137-1': 'Also, the quasicategory [MATH] admits finite colimits when [MATH] does, and [MATH] preserves finite colimits when [MATH] does.', '1303.4029-2-138-0': 'Since [MATH] is a weak homotopy equivalence of simplicial sets for all [MATH], the Realization Lemma for bisimplicial sets implies [MATH] is weak homotopy equivalence of spaces.', '1303.4029-2-139-0': 'The [MATH]-theory spaces recalled in Definition [REF] form an [MATH]-spectrum beyond the 0-th term.', '1303.4029-2-139-1': 'Since [MATH] is a weak homotopy equivalence of spaces, the map [MATH] is also a weak homotopy equivalence of spaces (recall [MATH] naturally), so 3-for-2 and the commutative diagram [EQUATION] imply that [MATH] is a weak homotopy equivalence.', '1303.4029-2-139-2': 'Continuing in this way, every [MATH] is a weak homotopy equivalence for [MATH].', '1303.4029-2-140-0': 'In the Approximation Theorem [REF], hypothesis [REF] could be weakened to the existence and preservation of the special colimits in hypothesis [REF] of the Pre-Approximation Theorem.', '1303.4029-2-140-1': 'In the Approximation Theorem we require finite colimits rather than merely the special ones because the statement is easier to formulate and check in practice.', '1303.4029-2-141-0': "Waldhausen's original condition App 2, namely that every morphism [MATH] factors as [MATH], implies in the quasicategorical context that every morphism in the codomain is a cofibration.", '1303.4029-2-141-1': 'To see this, suppose [MATH] is any morphism in [MATH].', '1303.4029-2-141-2': 'By essential surjectivity, there is an equivalence [MATH], and by App 2 every morphism [MATH] is a cofibration, so any composite [EQUATION] is a cofibration.', '1303.4029-2-141-3': 'Passing to the homotopy category and inverting the resulting isomorphism expresses [MATH], so [MATH] is a cofibration.', '1303.4029-2-141-4': "Since Waldhausen's App 2 implies every codomain morphism is a cofibration in the quasicategorical setting, it is natural to search for an Approximation Theorem with a weaker version of App 2: the factorization only of cofibrations [MATH] as [MATH].", '1303.4029-2-141-5': 'This is the topic of Sections [REF] and [REF].', '1303.4029-2-142-0': '[Approximation when [MATH] is an Equivalence of Homotopy Categories and [MATH]] Let [MATH] be an exact functor between Waldhausen quasicategories.', '1303.4029-2-142-1': 'Suppose [MATH] induces an equivalence of homotopy categories, and suppose every morphism of [MATH] is a cofibration.', '1303.4029-2-142-2': 'Then [MATH] is an equivalence of [MATH]-groupoids, as is [MATH] for every [MATH].', '1303.4029-2-142-3': 'Consequently [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-143-0': 'Since [MATH] induces an equivalence of homotopy categories, [MATH] reflects equivalences, so satisfies App 1.', '1303.4029-2-143-1': 'For App 2, suppose [MATH] is a morphism in [MATH].', '1303.4029-2-143-2': 'Then by the essential surjectivity of [MATH], we have an equivalence [MATH].', '1303.4029-2-143-3': 'The fullness of [MATH] provides a pre-image homotopy class for [MATH], and any representative completes the required triangle in App 2 (recall every morphism of [MATH] is a cofibration).', '1303.4029-2-143-4': 'Since every morphism of [MATH] is a cofibration, [MATH] admits pushouts and initial objects, so admits all finite colimits.', '1303.4029-2-143-5': 'The functor [MATH] preserves pushouts and initial objects, so also preserves all finite colimits.', '1303.4029-2-143-6': 'Theorem [REF] now applies.', '1303.4029-2-144-0': 'In the situation of Corollary [REF], every morphism [MATH] of the codomain is also a cofibration, since the equivalence of homotopy categories guarantees for each [MATH] in the codomain the existence of some cofibration [MATH] in the domain and some equivalences [MATH] and [MATH] in the codomain such that [MATH].', '1303.4029-2-145-0': 'Also notice, in Corollary [REF], the assumption that [MATH] is equivalent to the assumption that [MATH] admits factorization, see Section [REF].', '1303.4029-2-146-0': 'In Corollary [REF], if we drop the hypothesis [MATH] we can still conclude [MATH] is an equivalence of [MATH]-groupoids by the Pre-Approximation Theorem [REF], but we cannot conclude anything about [MATH].', '1303.4029-2-146-1': 'Properties Pre-App 1 and 2 for [MATH] do not persist to [MATH] in general.', '1303.4029-2-147-0': '## Comparison with Approximation Theorems in Classical Setting, Simplicially Enriched Setting, and Quasicategorical Setting', '1303.4029-2-148-0': 'There are several related results in the recent literature.', '1303.4029-2-148-1': 'First we recall the results in the classical categorical setting, some of which relate App 1 and App 2 to an equivalence of homotopy categories and simplicially enriched categories.', '1303.4029-2-148-2': 'For an ordinary 1-category [MATH] with weak equivalences, the homotopy category [MATH] in these statements means the category obtained by formally inverting the weak equivalences.', '1303.4029-2-148-3': 'Some of these statements are slight reformulations of the original statements.', '1303.4029-2-149-0': 'Theorem 1.6.7 of Waldhausen in [CITATION].', '1303.4029-2-149-1': 'Let [MATH] and [MATH] be Waldhausen categories.', '1303.4029-2-149-2': 'Suppose the weak equivalences in both [MATH] and [MATH] satisfy the 3-for-2 property.', '1303.4029-2-149-3': 'Suppose further that [MATH] has a cylinder functor and the weak equivalences in [MATH] satisfy the cylinder axiom.', '1303.4029-2-149-4': 'Let [MATH] be an exact functor.', '1303.4029-2-149-5': 'Suppose [MATH] satisfies App 1 and App 2.', '1303.4029-2-149-6': 'Then the induced maps [MATH] and [MATH] are weak homotopy equivalences.', '1303.4029-2-150-0': 'Theorem 10 of Schlichting in [CITATION].', '1303.4029-2-150-1': 'Let [MATH] and [MATH] be classical Waldhausen categories.', '1303.4029-2-150-2': 'Suppose the weak equivalences in [MATH] and [MATH] both have the 3-for-2 property.', '1303.4029-2-150-3': 'Suppose every morphism in [MATH] factors as a cofibration followed by a weak equivalence.', '1303.4029-2-150-4': 'Let [MATH] be an exact functor satisfying App 1 and App 2.', '1303.4029-2-150-5': 'Then the induced maps [MATH] and [MATH] are weak homotopy equivalences.', '1303.4029-2-151-0': 'Theorem 2.9 of Cisinski in [CITATION].', '1303.4029-2-151-1': 'Let [MATH] and [MATH] be Waldhausen categories that are "derivable" in the sense that their weak equivalences satisfy the 3-for-2 property and every one of their morphisms can be factored as a cofibration followed by a weak equivalence.', '1303.4029-2-151-2': 'Suppose also that [MATH] and [MATH] are each "strongly saturated" in the sense that any morphism is a weak equivalence if and only if its image in [MATH] respectively [MATH] is an isomorphism.', '1303.4029-2-151-3': 'Let [MATH] be a right exact functor.', '1303.4029-2-151-4': 'Then the following conditions are equivalent.', '1303.4029-2-152-0': 'For every finite ordered set [MATH], the induced functor [MATH] is an equivalence of categories.', '1303.4029-2-152-1': 'The induced functor [MATH] is an equivalence of categories.', '1303.4029-2-152-2': 'The functor [MATH] satisfies App 1 and App 2.', '1303.4029-2-153-0': 'Theorem 3.25 of Cisinski in [CITATION].', '1303.4029-2-153-1': 'Let [MATH] be a left exact functor between categories of fibrant objects.', '1303.4029-2-153-2': 'If the induced functor [MATH] is an equivalence of categories, then [MATH] induces an equivalence of simplicially enriched categories [EQUATION]', '1303.4029-2-153-3': 'Here [MATH] is the hammock localization of Dywer and Kan.', '1303.4029-2-154-0': 'Theorem 1.5 of Blumberg-Mandell in [CITATION].', '1303.4029-2-154-1': 'Let [MATH] be a classical Waldhausen category in which the weak equivalences have the 3-for-2 property and in which every morphism factors as a cofibration followed by a weak equivalence.', '1303.4029-2-154-2': 'Let [MATH] be a classical Waldhausen category in which the weak equivalences have the 3-for-2 property.', '1303.4029-2-154-3': 'Let [MATH] be an exact functor.', '1303.4029-2-154-4': 'If [MATH] satisfies App 1 and App 2, then [MATH] is an equivalence of categories, and [MATH] is an equivalence of categories for all objects [MATH] of [MATH].', '1303.4029-2-154-5': 'Here [MATH] and [MATH] mean the categories under [MATH] and [MATH].', '1303.4029-2-155-0': 'Theorem 1.4 of Blumberg-Mandell in [CITATION].', '1303.4029-2-155-1': 'Let [MATH] and [MATH] be classical Waldhausen categories, both of which have the 3-for-2 property for their weak equivalences.', '1303.4029-2-155-2': 'Suppose further that every morphism in both categories factors as a cofibration followed by a weak equivalence.', '1303.4029-2-155-3': 'Let [MATH] be a functor that preserves weak equivalences.', '1303.4029-2-155-4': 'Then [MATH] induces a Dwyer-Kan equivalence if and only if [MATH] induces an equivalence [MATH] and an equivalence [MATH] for all objects [MATH] of [MATH].', '1303.4029-2-156-0': 'Theorems 1.5 and 1.4 of Blumberg-Mandell together imply: if [MATH] is an exact functor between classical Waldhausen categories that both satisfy 3-for-2 and factorization, and [MATH] satisfies App 1 and App 2, then [MATH] induces a Dwyer-Kan equivalence of hammock localizations.', '1303.4029-2-157-0': 'Theorem 1.3 of Blumberg-Mandell in [CITATION].', '1303.4029-2-157-1': 'Let [MATH] and [MATH] be classical Waldhausen categories, both of which have the 3-for-2 property for their weak equivalences.', '1303.4029-2-157-2': 'Suppose further that every morphism in both categories factors as a cofibration followed by a weak equivalence.', '1303.4029-2-157-3': 'Let [MATH] be a a weakly exact functor that induces an equivalence on homotopy categories.', '1303.4029-2-157-4': 'If any one of the following additional hypotheses holds', '1303.4029-2-158-0': 'the weak equivalences of [MATH] and [MATH] are closed under retracts, a morphism [MATH] in [MATH] is a weak equivalence if and only the morphism [MATH] in [MATH] is a weak equivalence, or for any objects [MATH] in [MATH], the image of [MATH] in [MATH] coincides with the image of [MATH],', '1303.4029-2-159-0': 'then [MATH] induces an equivalence of [MATH]-theory spectra.', '1303.4029-2-160-0': 'Barwick has proved Corollary [REF], but not Theorem [REF].', '1303.4029-2-160-1': 'His Corollary 8.2.2 of [CITATION] implies: if [MATH] is an exact functor between Waldhausen quasicategories, both of which have all maps cofibrations, and [MATH] induces an equivalence [MATH] of homotopy categories, then [MATH] induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-2-160-2': 'Another version of Corollary [REF] was proved by Blumberg-Gepner-Tabuada for stable quasicategories: their Corollary 5.11 of [CITATION] states that a map of stable quasicategories is an equivalence if and only if it induces an equivalence of homotopy categories.', '1303.4029-2-160-3': 'See present Corollaries [REF] and [REF] for further quasicategorical settings in which an equivalence of cofibration homotopy categories or an equivalence of homotopy categories guarantees a levelwise equivalence of [MATH]-theory spectra.', '1303.4029-2-161-0': 'In light of Remark [REF], we can point out two predecessors to Corollary [REF] in the setting of classical Waldhausen categories.', '1303.4029-2-161-1': 'Blumberg-Mandell prove in [CITATION]: if [MATH] and [MATH] are both classical Waldhausen categories which admit factorization and whose classes of weak equivalences both have the 3-for-2 property, then any weakly exact functor [MATH] that reflects weak equivalences and induces an equivalence of homotopy categories induces a stable equivalence of [MATH]-theory spectra.', '1303.4029-2-161-2': 'Cisinski also made this same conclusion in [CITATION] for a right exact functor [MATH] for which each Waldhausen category [MATH] and [MATH] is a category of cofibrant objects, has a null object, and satisfies saturation conditions.', '1303.4029-2-162-0': '## The Cofibration Approximation Axiom 2 and its Comparison with App 2 and Pre-App 2', '1303.4029-2-163-0': "Since Waldhausen's condition App 2 (in the quasicategorical setting) implies every morphism of the codomain is a cofibration by Remark [REF], we would like a more general version the Approximation Theorem [REF] without every codomain morphism a cofibration.", '1303.4029-2-163-1': 'So, we replace the hypothesis App 2 by a weaker condition called Cofibration App 2 that requires only factorization of cofibrations [MATH] and prove the Cofibration Approximation Theorem [REF] in the next section.', '1303.4029-2-163-2': 'We now introduce Cofibration App 2, compare it with App 2 and Pre-App 2 in Remark [REF], consider invariance of all three properties under composition in Proposition [REF] [REF] and [REF], invariance of Cofibration App 2 under natural equivalence in Proposition [REF] [REF], the persistence of Cofibration App 2 to [MATH] and [MATH] in Proposition [REF], and the persistence of Pre-App 2 in Proposition [REF].', '1303.4029-2-164-0': 'An exact functor [MATH] between Waldhausen quasicategories has the Cofibration Approximation Property if its restriction [MATH] to the cofibration subquasicategories satisfies Pre-App 1 and Pre-App 2 of Theorem [REF].', '1303.4029-2-164-1': 'More specifically the following two conditions on [MATH] are required to hold.', '1303.4029-2-165-0': 'Cofibration App 1.', '1303.4029-2-165-1': 'The exact functor [MATH] reflects equivalences.', '1303.4029-2-166-0': 'Cofibration App 2.', '1303.4029-2-166-1': 'For every [MATH] and every cofibration [MATH] in the codomain [MATH], there exists a cofibration [MATH] in [MATH], an equivalence [MATH] in [MATH], and a 2-simplex in [MATH] of the following form.', '1303.4029-2-166-2': '[EQUATION] [Relationship between Pre-App, App, and Cofibration App] Clearly, App 2 implies both Pre-App 2 and Cofibration App 2.', '1303.4029-2-167-0': 'If every morphism of the domain is a cofibration, then conditions Pre-App 2 and App 2 coincide.', '1303.4029-2-167-1': 'If [MATH] satisfies either, then [MATH] also satisfies Cofibration App 2.', '1303.4029-2-167-2': 'If every morphism of the codomain is a cofibration, then conditions App 2 and Cofibration App 2 coincide.', '1303.4029-2-167-3': 'If [MATH] satisfies either, then [MATH] also satisfies Pre-App 2.', '1303.4029-2-167-4': 'If every morphism of both the domain and the codomain is a cofibration then conditions Pre-App 2, App 2, and Cofibration App 2 coincide.', '1303.4029-2-167-5': 'In any case, the equivalence-reflection conditions Pre-App 1, App 1, and Cofibration App 1 are all the same condition.', '1303.4029-2-168-0': 'Let [MATH], [MATH], and [MATH] be exact functors between Waldhausen quasicategories as below.', '1303.4029-2-168-1': '[EQUATION]', '1303.4029-2-168-2': 'Suppose [MATH] satisfies Cofibration App 2.', '1303.4029-2-169-0': 'If [MATH] is essentially surjective, then [MATH] also satisfies Cofibration App 2.', '1303.4029-2-170-0': 'If [MATH] is both full and essentially surjective, then [MATH] also satisfies Cofibration App 2.', '1303.4029-2-171-0': 'If an exact functor [MATH] is naturally equivalent to [MATH], then [MATH] also satisfies Cofibration App 2.', '1303.4029-2-172-0': 'Similar statements hold if "Cofibration App 2" is replaced throughout by "Pre-App 2" and in [REF] the category [MATH] is replaced by [MATH].', '1303.4029-2-173-0': 'Similar statements also hold if "Cofibration App 2" is replaced throughout by "App 2" and in [REF] the category [MATH] is replaced by [MATH].', '1303.4029-2-174-0': 'Suppose [MATH] is essentially surjective.', '1303.4029-2-174-1': 'Let [MATH] be a cofibration in [MATH].', '1303.4029-2-174-2': 'We factor [MATH] as [MATH].', '1303.4029-2-174-3': 'We select any equivalence [MATH], and obtain the desired factorization as [EQUATION].', '1303.4029-2-174-4': 'To find the filling 2-simplex, we work with homotopy classes in the homotopy category to arrive at a commutative triangle, and then revert back to representatives.', '1303.4029-2-174-5': 'A commutative triangle in the homotopy category always comes from a 2-simplex between any selected representatives.', '1303.4029-2-174-6': 'Suppose [MATH] is both full and essentially surjective.', '1303.4029-2-174-7': 'Let [MATH] be a cofibration in [MATH] and select any equivalence [MATH].', '1303.4029-2-174-8': 'Let [MATH] be a cofibration in [MATH] such that [MATH].', '1303.4029-2-174-9': 'We factor [MATH] as [MATH] and obtain the desired factorization as [EQUATION]', '1303.4029-2-174-10': 'Suppose [MATH] is exact and [MATH], and let [MATH] be a cofibration in [MATH].', '1303.4029-2-174-11': 'Then we precompose [MATH] with [MATH], factor the resulting morphism as [MATH], and then use the naturality square to factor [MATH] as [MATH] with the same [MATH].', '1303.4029-2-175-0': '[Cof Apps for [MATH] Cof Apps for [MATH] and [MATH]] If [MATH] is an exact functor that satisfies Cofibration App 1 and 2 in Definition [REF], then both [MATH] and [MATH] are exact functors that satisfy Cofibration App 1 and 2.', '1303.4029-2-176-0': 'It suffices to check that [MATH] satisfies Cofibration App 1 and 2 because the forgetful functor [MATH] is a Waldhausen equivalence by Proposition [REF].', '1303.4029-2-176-1': 'Namely, conjugation of [MATH] by this natural Waldhausen equivalence, shows that [MATH] satisfies Cofibration App 1 when [MATH] does.', '1303.4029-2-176-2': 'An application of all three parts of Proposition [REF] to the conjugation shows that [MATH] satisfies Cofibration App 2 when [MATH] does.', '1303.4029-2-177-0': 'For simplicity of notation we use [MATH] instead of [MATH].', '1303.4029-2-177-1': 'Recall that [MATH] is the union of the standard 1-simplices [MATH] for [MATH], and is called the spine of [MATH].', '1303.4029-2-177-2': 'The objects of [MATH] are functors [MATH], which we abbreviate as [MATH].', '1303.4029-2-177-3': 'An object of [MATH] is a sequence of cofibrations, head-to-tail, without composites and without quotients.', '1303.4029-2-178-0': 'If [MATH] satisfies Cofibration App 1, then it is clear that [MATH] also satisfies Cofibration App 1, because natural equivalences are precisely the natural transformations that are level-wise equivalences by [CITATION].', '1303.4029-2-179-0': 'Suppose [MATH] satisfies Cofibration App 2.', '1303.4029-2-179-1': 'We show that [MATH] satisfies Cofibration App 2.', '1303.4029-2-179-2': 'This proof is similar to Lemma 1.6.6 of Waldhausen [CITATION] concerning the analogous statement for App 2.', '1303.4029-2-179-3': 'We only need to justify the quasicategorical translation and check that the relevant maps are cofibrations.', '1303.4029-2-179-4': 'Let [MATH] be a cofibration in [MATH], and suppose for an inductive proof that [MATH] satisfies Cofibration App 2.', '1303.4029-2-179-5': 'Then our task is construct a factorization [MATH] which together with the already constructed [MATH] and [MATH] forms a desired factorization in [MATH].', '1303.4029-2-179-6': 'We first form the pushout [MATH] on the left below.', '1303.4029-2-179-7': 'The middle diagram is the [MATH]-image of the pushout square, and the last commutative square of [MATH].', '1303.4029-2-179-8': 'The universal property of the pushout [MATH] induces a contractible space of morphisms [MATH], we select one and call it [MATH].', '1303.4029-2-179-9': 'In the third diagram, the outer bottom square is the bottom square of the middle diagram, while the inner bottom square is a pushout, and therefore there is an induced selected morphism [MATH].', '1303.4029-2-179-10': '[EQUATION]', '1303.4029-2-179-11': 'But [MATH] is a cofibration: the two pushout squares in the right diagram compose to make a pushout square, [MATH] is a cofibration in [MATH], and [MATH] is also the induced map for the outermost square involving [MATH] and [MATH].', '1303.4029-2-179-12': 'We have now expressed [MATH] as a composite of two cofibrations, so [MATH] is also a cofibration.', '1303.4029-2-180-0': 'We can now apply Cofibration App 2 to [MATH] and factor it as [EQUATION] ([MATH] is defined this way).', '1303.4029-2-180-1': 'We define [MATH] and have [EQUATION].', '1303.4029-2-180-2': 'As remarked before, the previous sequence of equalities is actually performed in the homotopy category, and then we revert back to our selected representatives and a 2-simplex for [MATH] exists by elementary quasicategory theory.', '1303.4029-2-181-0': 'We define the cofibration [MATH] to be [MATH], which is a cofibration because both [MATH] and [MATH] are.', '1303.4029-2-182-0': 'Any map [MATH] induced by [MATH] is homotopic to [MATH] (by virtue of [MATH] and [MATH] being defined as composites with second map [MATH]), so [MATH] is a cofibration as [MATH] is.', '1303.4029-2-182-1': 'Thus [MATH] is a cofibration in [MATH].', '1303.4029-2-183-0': 'We would like an analogous statement to Proposition [REF] for Pre-Apps, but we will need to assume that every morphism of the domain is a cofibration.', '1303.4029-2-183-1': 'Why?', '1303.4029-2-183-2': 'Notice that the proof used in Proposition [REF] for Cofibration App 2 works fine for Pre-App 2 without further assumption, until the penultimate paragraph where [MATH] is proved to be a cofibration.', '1303.4029-2-183-3': 'Assuming only Pre-App 2 instead of Cofibration App 2, we can still define [MATH], however it is not necessarily a cofibration, as [MATH] is not necessarily a cofibration.', '1303.4029-2-183-4': 'Additionally assuming that every morphism of [MATH] is a cofibration of course guarantees that [MATH] is a cofibration as needed, in order for [MATH] to be an element of [MATH].', '1303.4029-2-184-0': '[Pre-Apps for [MATH] and [MATH] Pre-Apps for [MATH]] Let [MATH] be an exact functor and suppose that every morphism of [MATH] is a cofibration.', '1303.4029-2-184-1': 'If [MATH] satisfies Pre-App 1 and Pre-App 2 in Theorem [REF], then [MATH] is also an exact functor that satisfies Pre-App 1 and Pre-App 2.', '1303.4029-2-185-0': 'Since every morphism of [MATH] is a cofibration, every morphism of [MATH] and [MATH] is also a cofibration.', '1303.4029-2-185-1': 'Then the conditions Pre-App 2 and App 2 for [MATH] coincide, as do the conditions for [MATH] and [MATH] (separately), by Remark [REF] [REF].', '1303.4029-2-186-0': 'We reduce consideration of [MATH] to [MATH] via Proposition [REF] for Pre-App 2 or App 2.', '1303.4029-2-187-0': 'Waldhausen\'s argument of Lemma 1.6.6 in [CITATION] then applies, see the proof of Proposition [REF] and remove the word "cofibration" as appropriate, as stated in Remark [REF].', '1303.4029-2-188-0': '## The Cofibration Approximation Theorem and Other Variants', '1303.4029-2-189-0': 'We next prove a version of Approximation with Cofibration App 2 in place of App 2, and obtain various other statements as corollaries.', '1303.4029-2-189-1': 'We do not require a cylinder functor.', '1303.4029-2-190-0': '[Cofibration Approximation] Let [MATH] be an exact functor between Waldhausen quasicategories.', '1303.4029-2-190-1': 'Suppose:', '1303.4029-2-191-0': '(Cofibration App 1)', '1303.4029-2-192-0': '[MATH] reflects equivalences.', '1303.4029-2-193-0': '(Cofibration App 2)', '1303.4029-2-194-0': 'For every [MATH] and every cofibration [MATH] in the codomain [MATH], there exists a cofibration [MATH] in [MATH], an equivalence [MATH] in [MATH], and a 2-simplex in [MATH] of the following form.', '1303.4029-2-194-1': '[EQUATION]', '1303.4029-2-195-0': 'The functor of groupoids [MATH] is full, that is, if [MATH] is an isomorphism in [MATH], then there exists an isomorphism [MATH] in [MATH] such that [MATH].', '1303.4029-2-196-0': '[MATH] is also full for all [MATH].', '1303.4029-2-197-0': 'Then [MATH] is an equivalence of [MATH]-groupoids, as is [MATH] for every [MATH].', '1303.4029-2-197-1': 'Consequently [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-198-0': 'Hypothesis [REF] holds for instance if [MATH] is fully faithful and [MATH] is full.', '1303.4029-2-198-1': 'See Remark [REF].', '1303.4029-2-199-0': 'Notice that the restriction to cofibration subquasicategories [MATH] satisfies Pre-App 1 and Pre-App 2 there because [MATH] satisfies Cofibration App 1 and Cofibration App 2.', '1303.4029-2-199-1': 'We first prove that [MATH] is an equivalence of [MATH]-groupoids by applying the Pre-Approximation Theorem [REF] to the restriction [MATH] and using the fact that [MATH].', '1303.4029-2-199-2': 'Verification of the hypotheses of Pre-Approximation Theorem [REF] for [MATH] are as follows.', '1303.4029-2-200-0': 'Pre-App 1 for [MATH] is Cofibration App 1 for [MATH], so holds.', '1303.4029-2-200-1': 'Pre-App 2 for [MATH] is Cofibration App 2 for [MATH], so holds.', '1303.4029-2-200-2': 'The functor [MATH] is essentially surjective by Remark [REF].', '1303.4029-2-200-3': 'It is also full because [MATH] is full by hypothesis (recall also that every equivalence is a cofibration).', '1303.4029-2-201-0': 'The quasicategory [MATH] has an initial object and all pushouts, so admits all finite colimits.', '1303.4029-2-201-1': 'The functor [MATH] preserves zero objects, so also initial objects (every initial object is a zero object), and [MATH] preserves pushouts, so [MATH] preserves all finite colimits.', '1303.4029-2-202-0': 'Thus, by the Pre-Approximation Theorem [REF], the functor [MATH] is an equivalence of [MATH]-groupoids, but [MATH], so [MATH] is an equivalence of [MATH]-groupoids.', '1303.4029-2-203-0': 'We similarly prove [MATH] is an equivalence of [MATH]-groupoids using the Pre-Approximation Theorem [REF].', '1303.4029-2-203-1': 'By Proposition [REF], the exact functor [MATH] inherits Cofibration App 1 and 2 from [MATH], so assumptions [REF] and [REF] of Theorem [REF] hold for [MATH].', '1303.4029-2-203-2': 'The essential surjectivity of [MATH] in hypothesis [REF] and the special colimits in hypothesis [REF] are also inherited from [MATH].', '1303.4029-2-203-3': 'What remains to show is the functor [MATH] is full in [REF].', '1303.4029-2-203-4': 'But this follows from present hypothesis [REF] on [MATH] by Proposition [REF], so [MATH] is an equivalence of [MATH]-groupoids for all [MATH] by the Pre-Approximation Theorem.', '1303.4029-2-203-5': 'For [MATH] and [MATH], the functor [MATH] is basically [MATH], so also an equivalence of [MATH]-groupoids, while [MATH] is an equivalence between terminal quasicategories.', '1303.4029-2-204-0': 'From the equivalences of [MATH]-groupoids [MATH] and [MATH], the spectrum map [MATH] is a levelwise equivalence of spectra by the same argument as in the Approximation Theorem proof using diagram [REF].', '1303.4029-2-204-1': 'This completes the proof of the theorem.', '1303.4029-2-204-2': 'Next we prove the final claim about a sufficient condition for hypothesis [REF] to hold.', '1303.4029-2-205-0': 'It remains to prove that hypothesis [REF] holds if [MATH] is fully faithful and [MATH] is full.', '1303.4029-2-205-1': 'To illustrate the argument it suffices to prove the claim for [MATH].', '1303.4029-2-205-2': 'Suppose we have an equivalence [MATH], and consider its two commutative squares, separately.', '1303.4029-2-205-3': 'Then by the fullness of [MATH], we can find two commutative squares in [MATH] [EQUATION] whose images are homotopic in [MATH] to the two given squares of [MATH], separately.', '1303.4029-2-206-0': 'We would like to simultaneously alter the second square in [REF] and create a homotopy between it and its alteration.', '1303.4029-2-206-1': 'Consider the prism [MATH] which is the right square of [REF] in the 1-face and 2-face, and is the identity on [MATH] in the 0-face (its base).', '1303.4029-2-206-2': 'This is the trivial "left homotopy" from the right square of [REF] to itself.', '1303.4029-2-206-3': 'In this prism, replace the initial triangle at 0 by any 2-simplex of [MATH] that is a left homotopy from [MATH] to [MATH], remove the lower 2-simplex of the prism face that now has [MATH], and remove the (interior) 3-simplex to which this face belongs.', '1303.4029-2-206-4': 'This creates a [MATH] horn in [MATH], which we fill, to now have a new prism [MATH].', '1303.4029-2-206-5': 'This new prism is a homotopy between the right square of [REF] and the following square with [MATH] as the left vertical map, and a different lower 2-simplex.', '1303.4029-2-206-6': '[EQUATION]', '1303.4029-2-206-7': 'Next we glue [REF] to the left square of [REF] and obtain an equivalence in [MATH] whose [MATH]-image is homotopic to the equivalence [MATH] from the outset.', '1303.4029-2-206-8': 'This completes the conclusion of fullness of [MATH] in hypothesis [REF] from the indicated condition.', '1303.4029-2-206-9': 'The argument for fullness of [MATH] for [MATH] is similar: one successively alters and pastes squares as above, one after the other.', '1303.4029-2-207-0': '[Sufficient Conditions for Fullness of [MATH]] In the final sentence of Theorem [REF], a two-part sufficient condition was given for hypothesis [REF] to hold, the latter part of this sufficient condition was the fullness of [MATH].', '1303.4029-2-207-1': 'Suppose:', '1303.4029-2-208-0': '[MATH] is full, i.e., [MATH] has the right lifting property with respect to the inclusion [MATH], and [MATH] has the right lifting property with respect to [MATH].', '1303.4029-2-209-0': 'Then [MATH] is full, and [MATH] is full.', '1303.4029-2-210-0': 'Instead of requiring [MATH] to induce an equivalence of homotopy categories as in Corollary [REF], it is actually sufficient to require only an equivalence of the cofibration homotopy categories, provided [MATH] is full.', '1303.4029-2-211-0': '[Approximation, when [MATH] induces an equivalence of cofibration homotopy categories] Let [MATH] be an exact functor between Waldhausen quasicategories.', '1303.4029-2-211-1': 'Suppose:', '1303.4029-2-212-0': '[MATH] induces an equivalence of cofibration homotopy categories, i.e. the functor [EQUATION] is an equivalence of categories.', '1303.4029-2-213-0': '[MATH] is full.', '1303.4029-2-214-0': 'Then [MATH] is an equivalence of [MATH]-groupoids, as is [MATH] for every [MATH].', '1303.4029-2-214-1': 'Consequently [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-215-0': 'This proof is a direct application of the Cofibration Approximation Theorem [REF].', '1303.4029-2-215-1': 'Hypotheses [REF], [REF], and [REF] of the Cofibration Approximation Theorem all follow from the assumption that [MATH] is an equivalence of cofibration homotopy categories.', '1303.4029-2-216-0': 'For hypothesis [REF], we use the final claim of the Cofibration Approximation Theorem about a sufficient condition for [REF] to hold.', '1303.4029-2-216-1': 'The functor [MATH] is fully faithful because [MATH] is an equivalence of categories.', '1303.4029-2-216-2': 'The fullness of [MATH] is assumed.', '1303.4029-2-217-0': 'Let [MATH] be Waldhausen quasicategory.', '1303.4029-2-217-1': 'Then [MATH] induces a level-wise equivalence in [MATH]-theory.', '1303.4029-2-218-0': 'This follows directly from Corollary [REF].', '1303.4029-2-218-1': 'Hypothesis [REF] follows because every equivalence of [MATH] is a cofibration, and because a natural transformation is a natural equivalence if and only if each component is an equivalence by [CITATION].', '1303.4029-2-219-0': 'Let [MATH] be an exact functor between Waldhausen quasicategories such that [MATH] is an equivalence of categories.', '1303.4029-2-219-1': 'Then the functor [MATH] reflects cofibrations if and only if it induces an equivalence of cofibration homotopy categories, that is, [MATH] reflects cofibrations if and only if [EQUATION] is an equivalence of categories.', '1303.4029-2-220-0': 'Suppose [MATH] is an equivalence of categories.', '1303.4029-2-220-1': 'Since [MATH] is faithful and [MATH] is naturally a subcategory of [MATH] that maps to [MATH], the exact functor [MATH] induces a faithful functor of homotopy cofibration categories.', '1303.4029-2-220-2': 'Since [MATH] is essentially surjective, and equivalences are cofibrations, and essentially surjective functor [MATH] is also essentially surjective.', '1303.4029-2-221-0': 'Then [MATH] reflects cofibrations if and only if [MATH] is full (recall that any map homotopic to a cofibration is a cofibration).', '1303.4029-2-221-1': 'Consequently [MATH] reflects cofibrations if and only if [MATH] is an equivalence of categories.', '1303.4029-2-222-0': '[Approximation, when [MATH] reflects cofibrations and induces equivalence of homotopy categories] Let [MATH] be an exact functor between Waldhausen quasicategories.', '1303.4029-2-222-1': 'Suppose:', '1303.4029-2-223-0': '[MATH] induces an equivalence of homotopy categories [EQUATION] [MATH] is full.', '1303.4029-2-223-1': '[MATH] reflects cofibrations.', '1303.4029-2-224-0': 'Then [MATH] is an equivalence of [MATH]-groupoids, as is [MATH] for every [MATH].', '1303.4029-2-224-1': 'Consequently [MATH] is a level-wise equivalence of spectra.', '1303.4029-2-225-0': 'By Lemma [REF], the functor [MATH] is an equivalence of categories, so the result follows from Corollary [REF].', '1303.4029-2-226-0': 'In the classical context, a related result to Corollaries [REF] and [REF] is the following.', '1303.4029-2-226-1': 'Let [MATH] and [MATH] be classical Waldhausen categories both of which have the 3-for-2 property for weak equivalences, and both of which admit functorial mapping cylinders for weak cofibrations (FMCWC) in the sense of [CITATION].', '1303.4029-2-226-2': 'If [MATH] is an exact functor which induces an equivalence of homotopy categories and induces a Dwyer-Kan equivalence of DK-localizations of subcategories of weak cofibrations, then it follows from [CITATION] of Blumberg-Mandell that [MATH] induces a stable equivalence in [MATH]-theory.', '1303.4029-2-227-0': 'In Corollary [REF] we made no assumption that [MATH] satisfies App 1 and App 2, but instead required reflection of cofibrations and an equivalence of homotopy categories.', '1303.4029-2-227-1': 'In the classical setting, under certain hypotheses, these two sets of assumptions are equivalent.', '1303.4029-2-227-2': 'Namely, Blumberg-Mandell prove in [CITATION] that if [MATH] and [MATH] are both classical Waldhausen categories whose weak equivalences have the 3-for-2 property and [MATH] admits factorization, then any exact functor [MATH] that satisfies App 1 and App 2 induces an equivalence of homotopy categories.', '1303.4029-2-227-3': 'There are several related results in the present paper in the setting of quasicategories: App 2 implies [MATH] is essentially surjective (see Remark [REF]), App 1 and App 2 together imply [MATH] is full (see Lemma [REF]), and if the domain Waldhausen quasicategory admits factorization, then App 1 and App 2 imply [MATH] is an equivalence of [MATH]-groupoids (see the Approximation Theorem [REF]).', '1303.4029-2-228-0': 'A different variant of Approximation for an exact functor [MATH] betweeen classical Waldhausen categories was proved by Sagave in [CITATION].', '1303.4029-2-228-1': "Instead of requiring Waldhausen's App 2 for all maps [MATH] in the Waldhausen category [MATH], he requires App 2 only for maps with codomain [MATH] in a full subcategory of special objects.", '1303.4029-2-228-2': 'The category [MATH] is equipped with a functorial replacement of any object by a special one.', '1303.4029-2-228-3': 'If [MATH] and [MATH] are both classical Waldhausen categories whose weak equivalences have the 3-for-2 property and [MATH] admits factorization, then any exact functor [MATH] that satisfies special approximation and reflects weak equivalences induces an equivalence of algebraic [MATH]-theory spectra.', '1303.4029-2-228-4': 'An example of such a functor arises when [MATH] and [MATH] are full subcategories of pointed model categories, [MATH] and [MATH] both have full subcategories of special objects with functorial replacement, and [MATH] is an equivalence, see [CITATION].', '1303.4029-2-229-0': '# Appendix: Quasicategorical Recollections and a Criterion for a Simplicial Set to be Weakly Contractible', '1303.4029-2-230-0': 'Boardman and Vogt [CITATION] originally defined the concept of quasicategory under the name weak Kan complex.', '1303.4029-2-230-1': 'Joyal [CITATION] and Lurie [CITATION], [CITATION] have extensively developed the theory of quasicategories.', '1303.4029-2-230-2': 'We rapidly review quasicategories, their homotopy categories, the join and slice of quasicategories, and colimits in a quasicategory in Sections [REF]-[REF].', '1303.4029-2-230-3': "The main reference for Sections [REF]-[REF] is Joyal's Barcelona notes [CITATION] on quasicategories.", '1303.4029-2-230-4': 'Over quasicategories and the quasicategorical Quillen Theorem A are recalled in Section [REF].', '1303.4029-2-230-5': "In Section [REF] we quickly review barycentric subdivision of simplicial sets and Kan's functor [MATH] to prepare for a weak contractibility criterion in Section [REF].", '1303.4029-2-230-6': 'This criterion in Proposition [REF] is an important ingredient for the central result of the present paper, the Pre-Approximation Theorem [REF], so we prove every detail.', '1303.4029-2-230-7': "The argument is an adaptation of Schlichting's [CITATION], which he extracted from Waldhausen [CITATION].", '1303.4029-2-231-0': '## Quasicategories and Functors', '1303.4029-2-232-0': 'A quasicategory is a simplicial set [MATH] in which every inner horn admits a filler.', '1303.4029-2-232-1': 'That is, for any [MATH] and any map [MATH], there exists a map [MATH] such that the diagram [EQUATION] commutes.', '1303.4029-2-232-2': 'The objects and morphisms of a quasicategory are, respectively, its vertices and edges.', '1303.4029-2-232-3': 'A functor between quasicategories is merely a map of simplicial sets.', '1303.4029-2-232-4': 'The quasicategory of functors from a simplicial set [MATH] to a quasicategory [MATH] is the usual internal hom simplicial set [MATH].', '1303.4029-2-232-5': 'The 1-simplices of [MATH] are the natural transformations, that is, simplicial set maps [MATH].', '1303.4029-2-232-6': 'A functor [MATH] between quasicategories is fully faithful if [MATH] is a weak homotopy equivalence, while [MATH] is essentially surjective if [MATH] is essentially surjective.', '1303.4029-2-232-7': 'Here [MATH] is the mapping space recalled below, and [MATH] is the left adjoint to the nerve functor [MATH].', '1303.4029-2-232-8': 'A functor [MATH] between quasicategories is essentially surjective if and only if every object in its codomain is equivalent to one in its image.', '1303.4029-2-232-9': 'A functor between quasicategories is called an equivalence if it is an equivalence in the 2-category [MATH] which has simplicial sets as its objects and [MATH] as its hom categories.', '1303.4029-2-232-10': 'A functor between quasicategories is an equivalence if and only if it is fully faithful and essentially surjective [CITATION].', '1303.4029-2-233-0': 'A sub simplicial set [MATH] of a quasicategory [MATH] is called full or 0-full if any simplex of [MATH] is in [MATH] if and only if all of its vertices are in [MATH].', '1303.4029-2-233-1': 'Any 0-full sub simplicial set of a quasicategory is automatically a quasicategory [CITATION].', '1303.4029-2-233-2': 'A sub quasicategory [MATH] of a quasicategory [MATH] is called 1-full if any simplex of [MATH] is in [MATH] if and only if all of its edges are in [MATH].', '1303.4029-2-234-0': '## Homotopy and Equivalence in a Quasicategory', '1303.4029-2-235-0': 'Boardman and Vogt associated to any quasicategory [MATH] its homotopy category [MATH].', '1303.4029-2-235-1': 'A 1-simplex of a quasicategory [MATH] is called a morphism.', '1303.4029-2-235-2': 'Two parallel morphisms [MATH] are left homotopic if there exists a 2-simplex [MATH] with boundary [MATH].', '1303.4029-2-235-3': 'They are right homotopic if there exists a 2-simplex [MATH] with boundary [MATH].', '1303.4029-2-235-4': 'They are homotopic if they are in the same path component of the mapping space [MATH], which is the following pullback.', '1303.4029-2-235-5': '[EQUATION]', '1303.4029-2-235-6': 'All three notions of homotopy coincide and are an equivalence relation on the morphisms of the quasicategory [MATH].', '1303.4029-2-235-7': 'The 0-simplices of [MATH] together with the homotopy classes of morphisms form the homotopy category [MATH] [CITATION].', '1303.4029-2-235-8': 'This category is isomorphic to the fundamental category [MATH], so the categories [MATH] and [MATH] are identified without further mention when [MATH] is a quasicategory.', '1303.4029-2-235-9': 'Here [MATH] is the left adjoint to the nerve functor.', '1303.4029-2-236-0': 'A morphism in [MATH] is an equivalence if its homotopy class is an isomorphism in [MATH], which is the case if and only if the morphism has a "homotopy inverse" in [MATH].', '1303.4029-2-236-1': 'We denote the 1-full subquasicategory of [MATH] on the equivalences by [MATH].', '1303.4029-2-236-2': 'This is the maximal Kan subcomplex of [MATH], and [MATH] is right adjoint to the inclusion [MATH] [CITATION].', '1303.4029-2-236-3': 'If [MATH] and [MATH] are quasicategories, then a natural transformation [MATH] is an equivalence in the quasicategory [MATH] if and only if each component [MATH] is an equivalence in [MATH], see [CITATION].', '1303.4029-2-237-0': '[Nerve of a category] The nerve of any category [MATH] is a quasicategory.', '1303.4029-2-237-1': 'The homotopy class of a morphism [MATH] in [MATH] is simply [MATH].', '1303.4029-2-237-2': 'The mapping space [MATH] is [MATH] viewed as a discrete simplicial set.', '1303.4029-2-237-3': 'Consequently, the homotopy category of [MATH] is just [MATH], and a morphism in [MATH] is an equivalence if and only if it is an isomorphism in [MATH].', '1303.4029-2-237-4': 'Both of these consequences also follow from the fact that [MATH] (nerve is fully faithful).', '1303.4029-2-237-5': 'As expected, we now see as a special case of the above-mentioned [CITATION], the classical lemma that a natural transformation [MATH] has components isomorphisms if and only if it is invertible in [MATH].', '1303.4029-2-238-0': '## Join and Slice for Categories and for Simplicial Sets', '1303.4029-2-239-0': 'Recall that if [MATH] and [MATH] are ordinary categories, then the join [MATH] is the category with objects and morphisms [EQUATION]', '1303.4029-2-239-1': 'For, [MATH] and [MATH], any composite [MATH] is the unique map [MATH], namely [MATH].', '1303.4029-2-239-2': 'All other possible compositions are already defined.', '1303.4029-2-239-3': 'For an example of a familiar join, notice that the join of any [MATH]-element linearly ordered set with any [MATH]-element linearly ordered set is an [MATH]-element linearly ordered set, so [MATH].', '1303.4029-2-239-4': 'If 1 denotes the terminal category, then [MATH] is [MATH] with a terminal object adjoined, while [MATH] is [MATH] with an initial object adjoined.', '1303.4029-2-240-0': 'For any simplicial sets [MATH] and [MATH], the join simplicial set [MATH] of [CITATION] satisfies [EQUATION] by [CITATION].', '1303.4029-2-240-1': 'Compare equation [REF] for [MATH] and [MATH] with the object and morphism formulas for the join of categories in equations [REF] and [REF] above.', '1303.4029-2-240-2': 'The face and degeneracy maps can be understood from the case where [MATH] and [MATH] are nerves of categories.', '1303.4029-2-240-3': 'For this paper, we may take [REF] as a definition.', '1303.4029-2-240-4': 'The empty simplicial set is a unit for join, [MATH].', '1303.4029-2-241-0': 'Clearly, [MATH] and [MATH], and there are two functors [EQUATION]', '1303.4029-2-241-1': 'These each admit a right adjoint called slice [CITATION], we denote their respective right adjoints as [MATH] and [MATH] respectively for [MATH] and [MATH] in [MATH].', '1303.4029-2-241-2': 'In particular, there are bi-natural bijections [EQUATION]', '1303.4029-2-241-3': 'From these, we see an [MATH]-simplex [MATH] is a map [MATH] which extends [MATH] along [MATH], while an [MATH]-simplex [MATH] is a map [MATH] which extends [MATH] along [MATH].', '1303.4029-2-242-0': 'When [MATH] is a quasicategory, so are the slices [MATH] and [MATH], see [CITATION] for the case of [MATH].', '1303.4029-2-243-0': 'The join and slice of simplicial sets is compatible with the join and slice of categories.', '1303.4029-2-243-1': 'The nerve functor sends the join of two categories to the join of their nerves [CITATION].', '1303.4029-2-243-2': 'So for instance, we may prove [MATH] by the sequence of isomorphisms [EQUATION].', '1303.4029-2-243-3': 'The nerve preserves the slice operations [CITATION].', '1303.4029-2-243-4': '[EQUATION]', '1303.4029-2-243-5': 'The relationship between topological joins, categorical joins, and simplicial joins has been worked out by Fritsch-Golasinski [CITATION].', '1303.4029-2-244-0': '## Colimits in a Quasicategory', '1303.4029-2-245-0': 'Joyal defined the notion of colimit in a quasicategory [MATH] using join and slice as follows [CITATION], see also [CITATION] and [CITATION].', '1303.4029-2-245-1': 'An object [MATH] in a quasicategory [MATH] is initial if for every object [MATH] in [MATH] the map [MATH] is a weak homotopy equivalence.', '1303.4029-2-245-2': 'Any two initial objects of [MATH] are equivalent, in the sense that there is an equivalence from one to the other.', '1303.4029-2-245-3': 'Clearly, this equivalence is even homotopically unique.', '1303.4029-2-245-4': 'If [MATH] is a simplicial set and [MATH] is a map of simplicial sets, then a cocone with base [MATH] is a 0-simplex of [MATH].', '1303.4029-2-245-5': 'In other words, a cocone with base [MATH] is a map [MATH] which extends [MATH] along [MATH].', '1303.4029-2-245-6': 'A colimiting cocone for [MATH] is an initial object of [MATH], in other words a cocone [MATH] with base [MATH] which is initial.', '1303.4029-2-245-7': 'A colimit of [MATH] is the value of a colimiting cocone [MATH] at the unique vertex of the terminal simplicial set [MATH].', '1303.4029-2-246-0': 'If [MATH] is a category, then all of these concepts in [MATH] coincide with the usual 1-category notions in [MATH] because nerve commutes with join and slice, nerve is fully faithful, and [MATH] is [MATH] viewed as a discrete simplicial set.', '1303.4029-2-247-0': 'I thank David Gepner for sketching the following proposition to me.', '1303.4029-2-247-1': 'It will be used in Proposition [REF], the equivalence of [MATH] with [MATH].', '1303.4029-2-248-0': 'Let [MATH] be a simplicial set, [MATH] a quasicategory which admits [MATH]-shaped colimits, and [MATH] the subquasicategory of [MATH] that is 0-full on the colimiting cocones.', '1303.4029-2-248-1': 'Then the restriction map [EQUATION] is an equivalence of quasicategories.', '1303.4029-2-249-0': '## Overquasicategories and Quasicategorical Quillen Theorem A', '1303.4029-2-250-0': 'For the Pre-Approximation Theorem [REF] we need overquasicategories and the quasicategorical Quillen Theorem A.', '1303.4029-2-251-0': '[Overquasicategory [MATH]] Let [MATH] be an object of a quasicategory [MATH].', '1303.4029-2-251-1': 'An [MATH]-simplex over [MATH] is an [MATH]-simplex [MATH] such that [MATH], where [MATH] is [MATH].', '1303.4029-2-251-2': 'The simplices over [MATH] form the overquasicategory [MATH].', '1303.4029-2-251-3': 'The projection [EQUATION] is [MATH].', '1303.4029-2-252-0': "The overquasicategory [MATH] is isomorphic to Joyal's slice [MATH] recalled in Section [REF].", '1303.4029-2-252-1': 'An [MATH]-simplex of [MATH] is a map [MATH] which extends [MATH], but the domain is [MATH], so such a map [MATH] is the same as a map [MATH] with [MATH].', '1303.4029-2-252-2': 'Under this isomorphism [MATH], the restriction of [MATH] to [MATH] corresponds to the projection [REF].', '1303.4029-2-253-0': 'The simplicial set [MATH] is a quasicategory when [MATH] is, by [CITATION].', '1303.4029-2-253-1': 'Lurie writes [MATH] for [MATH] and [MATH], see [CITATION].', '1303.4029-2-253-2': 'The nerve preserves the slice operations, so if [MATH] is a category, then [MATH], see [CITATION].', '1303.4029-2-254-0': '[Over simplicial set [MATH]] Let [MATH] be a simplicial set, [MATH] an object of a quasicategory [MATH], and [MATH] a map of simplicial sets.', '1303.4029-2-254-1': 'An [MATH]-simplex [MATH]-over [MATH] is a pair [MATH] such that [MATH].', '1303.4029-2-254-2': 'The simplices [MATH]-over [MATH] form the over simplicial set [MATH], it is the pullback [EQUATION]', '1303.4029-2-254-3': 'In the notation of Lurie, [MATH] would be [MATH], see [CITATION].', '1303.4029-2-254-4': 'The simplicial set [MATH] is like a homotopy fiber.', '1303.4029-2-254-5': 'The left fiber [MATH] on page 337 of Waldhausen [CITATION], which is the pullback of [MATH] along [MATH], is like a strict fiber.', '1303.4029-2-255-0': '[Quasicategorical Quillen Theorem A, [CITATION]] Let [MATH] be a quasicategory, [MATH] a simplicial set, and [MATH] a map of simplicial sets.', '1303.4029-2-255-1': 'If the over simplicial set [MATH] is weakly contractible for every object [MATH] of [MATH], then [MATH] is a weak homotopy equivalence.', '1303.4029-2-256-0': 'For a proof of the analogous version of Theorem [REF] for [MATH], see Proposition 4.1.1.3.', '1303.4029-2-256-1': '(3) and Theorem 4.1.3.1 of Lurie on pages 223 and 236 of [CITATION].', '1303.4029-2-256-2': "For another variant of Quillen's Theorem A in the setting of quasicategories, see Heuts-Moerdijk [CITATION].", '1303.4029-2-257-0': "## Barycentric Subdivision and Kan's Functor Ex", '1303.4029-2-258-0': 'In Proposition [REF] we briefly need some properties of the adjunction [MATH] on [MATH], so we review it here.', '1303.4029-2-258-1': 'The subdivision of the standard [MATH]-simplex, denoted [MATH], is the nerve of the poset of non-degenerate simplices of [MATH].', '1303.4029-2-258-2': 'In other words, [MATH] is the nerve of the poset of non-empty subsets of [MATH], denoted [MATH].', '1303.4029-2-258-3': 'The subdivision of a simplicial set [MATH] is the colimit [EQUATION] where the colimit is over the category of simplices of [MATH].', '1303.4029-2-258-4': "The subdivision functor [MATH] is left adjoint to Kan's functor [MATH], as one can infer from the definition of its [MATH]-simplices.", '1303.4029-2-258-5': '[EQUATION]', '1303.4029-2-258-6': 'For a map [MATH] of simplicial sets, [MATH] is given by postcomposition with [MATH].', '1303.4029-2-259-0': 'The [MATH]-th last vertex map [MATH] is the nerve of the functor [EQUATION].', '1303.4029-2-259-1': 'The [MATH]-th last vertex map [MATH] is a simplicial homotopy equivalence, and the last vertex maps all together induce via precomposition a natural weak homotopy equivalence [EQUATION]', '1303.4029-2-259-2': 'If [MATH] is based, then [REF] is a basepoint-preserving map.', '1303.4029-2-259-3': 'See [CITATION] for a recent treatment of the foregoing topics.', '1303.4029-2-260-0': '## A Criterion for a Simplicial Set to be Weakly Contractible', '1303.4029-2-261-0': "The next proposition is a simplicial version of the categorical result [CITATION] by Schlichting, which he extracted from Waldhausen's paper [CITATION].", '1303.4029-2-261-1': "This proof is an adaptation of Schlichting's extraction to the situation of [MATH] a simplicial set (instead of a category), with a few more details.", '1303.4029-2-261-2': 'This proposition is a key ingredient in the proof of the Pre-Approximation Theorem [REF].', '1303.4029-2-262-0': 'Let [MATH] be a nonempty, connected simplicial set.', '1303.4029-2-262-1': 'If for every connected finite poset [MATH], every map [MATH] can be extended to a map [MATH], then [MATH] is weakly contractible.', '1303.4029-2-263-0': 'We show that each homotopy group of the realization [MATH] is trivial.', '1303.4029-2-264-0': 'For [MATH], we have [MATH] because [MATH] is a connected simplicial set by assumption, which implies that [MATH] is path-connected.', '1303.4029-2-265-0': 'Let [MATH], and let [MATH] be the fibrant replacement functor of Kan reviewed in Section [REF].', '1303.4029-2-265-1': 'Fix a vertex [MATH] of [MATH], and let [MATH] be any simplicial model of the [MATH]-sphere with only finitely many non-degenerate simplices.', '1303.4029-2-265-2': 'In the following, we make use of the isomorphisms [EQUATION] where isomorphism (1) arises from the realization of the weak homotopy equivalence [MATH], and isomorphism (2) is the identification of the simplicial homotopy groups of the fibrant simplicial set [MATH] with the standard homotopy groups of its realization.', '1303.4029-2-265-3': 'The inverse of isomorphism (2) is induced by geometric realization.', '1303.4029-2-266-0': 'Consider a homotopy class in [MATH] and let [MATH] represent it via the composite of isomorphisms (1) and (2) above.', '1303.4029-2-266-1': 'We would like to show that [MATH] is based homotopic to constant [MATH] to conclude the original homotopy class in [MATH] is zero via isomorphism (1).', '1303.4029-2-267-0': 'Since [MATH] commutes with directed colimits and [MATH] is the colimit of the directed diagram of weak homotopy equivalences [EQUATION] (see Kan [CITATION] or Goerss-Jardine [CITATION]), the basepoint preserving map [MATH] factors through some [MATH].', '1303.4029-2-267-1': 'More concretely, for each non-degenerate simplex [MATH] of [MATH], pick an index [MATH] such that [MATH] is in the image of [MATH] in [MATH], and then let [MATH].', '1303.4029-2-267-2': 'Realizing, we also have the factorization [EQUATION].', '1303.4029-2-268-0': 'We claim that it suffices to show [MATH] is unbased homotopic to a constant map.', '1303.4029-2-268-1': 'For if this is the case, then the based map [MATH] is also unbased homotopic to a constant map, which implies [MATH] is based homotopic to [MATH] by [CITATION], in other words implies [MATH] is zero in [MATH].', '1303.4029-2-269-0': 'So we show [MATH] is unbased homotopic to a constant map.', '1303.4029-2-269-1': 'The barycentric subdivision [MATH] is left adjoint to [MATH] (see Kan [CITATION] or Goerss-Jardine [CITATION] or Fritsch-Piccinini [CITATION]), so [MATH] corresponds to its transpose [MATH].', '1303.4029-2-269-2': 'Since [MATH], the domain finite simplicial set [MATH] is the nerve of a finite poset (see Thomason [CITATION]).', '1303.4029-2-269-3': 'This poset must be connected, as [MATH] is connected.', '1303.4029-2-269-4': 'By hypothesis, [MATH] now extends to [MATH].', '1303.4029-2-269-5': 'The geometric realization [MATH] is homeomorphic to an [MATH]-ball, so [MATH] is unbased homotopic to any constant in the image of this [MATH]-ball.', '1303.4029-2-270-0': 'To next see that [MATH] is unbased homotopic to a constant map, consider the following two commutative squares of continuous maps, which arise as the geometric realization of commutative squares of simplicial maps.', '1303.4029-2-270-1': '[EQUATION]', '1303.4029-2-270-2': 'The top square is from the definition of transpose [MATH] for the adjunction [MATH], while the bottom square comes from the naturality square for iteration of the weak homotopy equivalence [MATH] in [REF].', '1303.4029-2-270-3': 'The dotted map is any homotopy inverse to its adjacent map.', '1303.4029-2-271-0': 'Tracing the left, bottom, and right, we now have a homotopy [EQUATION].', '1303.4029-2-271-1': 'But since, [MATH] is unbased homotopic to a constant, so is the right-hand side, and so is [MATH].', '1303.4029-2-272-0': 'Thus [MATH] is based homotopic to constant [MATH] and [MATH] is trivial for all [MATH].', '1303.4029-2-273-0': '# Acknowledgements', '1303.4029-2-274-0': 'Scientific Acknowledgements.', '1303.4029-2-274-1': 'I thank Andre Joyal for his course on quasicategories at the Centre de Recerca Matematica (Barcelona) during the thematic programme on Homotopy Theory and Higher Categories in academic year 2007/08.', '1303.4029-2-274-2': 'The organizers of that glorious year have my gratitude: Carles Casacuberta, Andre Joyal, Joachim Kock, Amnon Neeman, and Frank Neumann.', '1303.4029-2-274-3': 'There I learned the quasicategory theory used in this article, while supported on grant SB2006-0085 of the Spanish Ministerio de Educacion y Ciencia at the Universitat Autonoma de Barcelona.', '1303.4029-2-275-0': 'I especially thank Andrew Blumberg for his excellent comments, and for answering my questions concerning [CITATION], [CITATION], and the general topic of Approximation.', '1303.4029-2-275-1': 'His critical suggestion to consider the cofibration subcategories more closely led me to discover Corollary [REF].', '1303.4029-2-275-2': 'I thank Georgios Raptis for occasional conversations, and especially for suggesting and discussing Lemma [REF].', '1303.4029-2-275-3': 'I also thank David Gepner for previous discussions on [MATH]-theory and quasicategories.', '1303.4029-2-275-4': 'I thank Denis-Charles Cisinski for helpful comments and suggestions in July 2017.', '1303.4029-2-276-0': 'I thank the Regensburg Sonderforschungsbereich 1085: Higher Invariants and the Regensburg Mathematics Department for a very stimulating working environment during during my September 2015 - July 2016 sabbatical visit, and during his short visit July 3 - July 16, 2017.', '1303.4029-2-277-0': 'Financial'} | [['1303.4029-1-15-3', '1303.4029-2-25-0'], ['1303.4029-1-15-4', '1303.4029-2-25-1'], ['1303.4029-1-9-0', '1303.4029-2-14-0'], ['1303.4029-1-9-1', '1303.4029-2-14-1'], ['1303.4029-1-2-0', '1303.4029-2-4-0'], ['1303.4029-1-4-0', '1303.4029-2-6-0'], ['1303.4029-1-12-1', '1303.4029-2-212-0'], ['1303.4029-1-1-0', '1303.4029-2-1-1'], ['1303.4029-1-1-0', '1303.4029-2-19-0'], ['1303.4029-1-7-0', '1303.4029-2-8-0'], ['1303.4029-1-7-1', '1303.4029-2-8-1'], ['1303.4029-1-8-0', '1303.4029-2-9-0'], ['1303.4029-1-8-1', '1303.4029-2-13-1'], ['1303.4029-1-10-0', '1303.4029-2-15-0'], ['1303.4029-1-10-1', '1303.4029-2-15-1'], ['1303.4029-1-10-2', '1303.4029-2-15-2'], ['1303.4029-1-11-0', '1303.4029-2-16-0'], ['1303.4029-1-11-1', '1303.4029-2-17-0'], ['1303.4029-1-11-1', '1303.4029-2-232-0'], ['1303.4029-1-11-2', '1303.4029-2-17-1'], ['1303.4029-1-11-3', '1303.4029-2-17-2'], ['1303.4029-1-11-3', '1303.4029-2-17-3'], ['1303.4029-1-13-0', '1303.4029-2-22-0'], ['1303.4029-1-16-1', '1303.4029-2-26-1'], ['1303.4029-1-16-2', '1303.4029-2-26-5']] | [['1303.4029-1-15-3', '1303.4029-2-25-0'], ['1303.4029-1-15-4', '1303.4029-2-25-1'], ['1303.4029-1-9-0', '1303.4029-2-14-0'], ['1303.4029-1-9-1', '1303.4029-2-14-1'], ['1303.4029-1-2-0', '1303.4029-2-4-0']] | [['1303.4029-1-4-0', '1303.4029-2-6-0']] | [] | [['1303.4029-1-12-1', '1303.4029-2-212-0'], ['1303.4029-1-1-0', '1303.4029-2-1-1'], ['1303.4029-1-1-0', '1303.4029-2-19-0']] | [['1303.4029-1-7-0', '1303.4029-2-8-0'], ['1303.4029-1-7-1', '1303.4029-2-8-1'], ['1303.4029-1-8-0', '1303.4029-2-9-0'], ['1303.4029-1-8-1', '1303.4029-2-13-1'], ['1303.4029-1-10-0', '1303.4029-2-15-0'], ['1303.4029-1-10-1', '1303.4029-2-15-1'], ['1303.4029-1-10-2', '1303.4029-2-15-2'], ['1303.4029-1-11-0', '1303.4029-2-16-0'], ['1303.4029-1-11-1', '1303.4029-2-17-0'], ['1303.4029-1-11-1', '1303.4029-2-232-0'], ['1303.4029-1-11-2', '1303.4029-2-17-1'], ['1303.4029-1-11-3', '1303.4029-2-17-2'], ['1303.4029-1-11-3', '1303.4029-2-17-3'], ['1303.4029-1-13-0', '1303.4029-2-22-0'], ['1303.4029-1-16-1', '1303.4029-2-26-1'], ['1303.4029-1-16-2', '1303.4029-2-26-5']] | ['1303.4029-1-3-0', '1303.4029-1-5-0', '1303.4029-1-5-1', '1303.4029-1-16-0', '1303.4029-2-5-0', '1303.4029-2-10-0', '1303.4029-2-11-0', '1303.4029-2-12-0', '1303.4029-2-26-0', '1303.4029-2-31-0', '1303.4029-2-35-2', '1303.4029-2-40-1', '1303.4029-2-40-4', '1303.4029-2-49-0', '1303.4029-2-50-0', '1303.4029-2-67-0', '1303.4029-2-68-0', '1303.4029-2-72-0', '1303.4029-2-76-2', '1303.4029-2-78-0', '1303.4029-2-79-0', '1303.4029-2-82-0', '1303.4029-2-84-0', '1303.4029-2-86-4', '1303.4029-2-101-2', '1303.4029-2-102-0', '1303.4029-2-103-0', '1303.4029-2-104-0', '1303.4029-2-105-1', '1303.4029-2-109-0', '1303.4029-2-112-5', '1303.4029-2-117-2', '1303.4029-2-117-5', '1303.4029-2-118-2', '1303.4029-2-118-4', '1303.4029-2-125-3', '1303.4029-2-126-1', '1303.4029-2-128-0', '1303.4029-2-129-1', '1303.4029-2-130-0', '1303.4029-2-131-0', '1303.4029-2-132-0', '1303.4029-2-133-1', '1303.4029-2-158-0', '1303.4029-2-159-0', '1303.4029-2-165-0', '1303.4029-2-165-1', '1303.4029-2-166-0', '1303.4029-2-168-1', '1303.4029-2-179-10', '1303.4029-2-183-1', '1303.4029-2-190-1', '1303.4029-2-191-0', '1303.4029-2-192-0', '1303.4029-2-193-0', '1303.4029-2-194-1', '1303.4029-2-196-0', '1303.4029-2-198-1', '1303.4029-2-206-6', '1303.4029-2-207-1', '1303.4029-2-209-0', '1303.4029-2-211-1', '1303.4029-2-213-0', '1303.4029-2-222-1', '1303.4029-2-223-1', '1303.4029-2-235-5', '1303.4029-2-243-4', '1303.4029-2-251-0', '1303.4029-2-251-1', '1303.4029-2-251-2', '1303.4029-2-251-3', '1303.4029-2-256-1', '1303.4029-2-258-5', '1303.4029-2-270-1', '1303.4029-2-274-0', '1303.4029-2-277-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1303.4029 | null | null | null | null | null |
0811.4281 | {'0811.4281-1-0-0': 'cluster expansion with applications]Abstract cluster expansion with application to statistical mechanical systems', '0811.4281-1-1-0': '# Abstract We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence.', '0811.4281-1-1-1': 'We apply the results to systems of classical and quantum particles with stable interactions.', '0811.4281-1-2-0': 'Keywords: cluster expansion, polymer model, stable interaction, quantum gas.', '0811.4281-1-3-0': '2000 Math.', '0811.4281-1-3-1': 'Subj.', '0811.4281-1-3-2': 'Class.: 82B05, 82B10, 82B20, 82B21, 82B26', '0811.4281-1-4-0': '# Introduction', '0811.4281-1-5-0': "The method of cluster expansions was introduced in the 1930's in statistical mechanics in order to study gases of classical interacting particles.", '0811.4281-1-5-1': 'Its main achievement, from the point of view of physics, may be the derivation of the van der Waals equation of state for weakly interacting systems.', '0811.4281-1-5-2': "The method was made rigorous by mathematical-physicists in the 1960's, see [CITATION] and references therein.", '0811.4281-1-6-0': 'The method split afterwards.', '0811.4281-1-6-1': 'One branch involves continuous systems, with applications to classical systems [CITATION], quantum systems [CITATION], or quantum field theory [CITATION].', '0811.4281-1-6-2': 'The other branch involves polymer systems, i.e. discrete systems with additional internal structure [CITATION].', '0811.4281-1-6-3': 'An important step forward was the article of Kotecky and Preiss with its simplified setting and its elegant condition for the convergence of the cluster expansion [CITATION].', '0811.4281-1-7-0': 'The methods for proving the convergence are diverse.', '0811.4281-1-7-1': 'Let us mention the study of the Kirkwood-Salsburg equation that involves correlation functions, see [CITATION] and references therein; the algebraic approach of Ruelle [CITATION]; combinatorial approaches using tree identities [CITATION]; inductions for discrete systems [CITATION].', '0811.4281-1-8-0': 'Important and useful surveys were written by Brydges [CITATION], Pfister [CITATION], Abdesselam and Rivasseau [CITATION].', '0811.4281-1-8-1': 'Recent articles have been devoted to combinatorial aspects [CITATION] and to improving estimates [CITATION].', '0811.4281-1-9-0': 'The method of cluster expansions applies when the objects do not interact much; this the case when they are far apart (low density), or when interactions are weak.', '0811.4281-1-9-1': 'An extension of the criterion of [CITATION] that takes into account these two aspects was proposed in [CITATION]; it applies to both discrete and continuous systems.', '0811.4281-1-10-0': 'All abstract (i.e. general) approaches involve restrictions that correspond to repulsive interactions.', '0811.4281-1-10-1': 'Yet the old results for classical and quantum systems only assume stable interactions, that may include an attractive part.', '0811.4281-1-10-2': 'The aim of the present article is to propose a general approach that applies to discrete and continuous systems with repulsive or stable interactions.', '0811.4281-1-10-3': 'Our proof is split into several independent steps and this helps clarify the situation.', '0811.4281-1-11-0': 'The setting and the results are presented in Section [REF].', '0811.4281-1-11-1': 'We consider applications to classical systems of particles in Section [REF], to polymer systems in Section [REF], and to the quantum gas in Section [REF].', '0811.4281-1-11-2': 'A fundamental tree estimate is derived in Section [REF], and the theorems of Section [REF] are proved in Section [REF].', '0811.4281-1-12-0': '# Cluster expansions', '0811.4281-1-13-0': 'We consider a set [MATH] whose elements may represent widely different objects - in the three applications considered in this article, an element [MATH] represents (i) the position of a classical particle, (ii) a polymer, i.e. a connected set of [MATH], and (iii) a closed Brownian bridge.', '0811.4281-1-13-1': 'For the general abstract theory, we assume the structure of a measure space, [MATH], with [MATH] a complex measure.', '0811.4281-1-13-2': 'We denote [MATH] the total variation (absolute value) of [MATH].', '0811.4281-1-13-3': 'Let [MATH] and [MATH] be complex measurable symmetric functions on [MATH], that are related by the equation [EQUATION]', '0811.4281-1-13-4': 'We allow the real part of [MATH] to take the value [MATH], in which case [MATH].', '0811.4281-1-13-5': 'In typical applications [MATH] represents the interactions between [MATH] and [MATH], and the value [MATH] corresponds to a hard-core repulsion.', '0811.4281-1-13-6': 'We define the "partition function" by [EQUATION] or, equivalently, [EQUATION]', '0811.4281-1-13-7': 'The term [MATH] of the sums is understood to be 1.', '0811.4281-1-14-0': 'The main goal of cluster expansions is to express the partition function as the exponential of a convergent series of "cluster terms".', '0811.4281-1-14-1': 'The main difficulty is to prove the convergence.', '0811.4281-1-14-2': 'We first assume that the potential [MATH] is stable.', '0811.4281-1-15-0': 'There exists a nonnegative function [MATH] on [MATH] such that, for all [MATH] and almost all [MATH], [EQUATION].', '0811.4281-1-16-0': 'In other words, we assume the lower bound [EQUATION]', '0811.4281-1-16-1': 'When the function [MATH] is constant, this is the usual definition of stability.', '0811.4281-1-16-2': '"Almost all" means that, for given [MATH], the set of points where the condition fails has measure zero with respect to the product measure [MATH].', '0811.4281-1-16-3': 'If [MATH] is countable, the condition must be satisfied for all [MATH] such that [MATH].', '0811.4281-1-17-0': 'The second condition deals with the strength of interactions.', '0811.4281-1-17-1': 'Physically, it is clear that hard core interactions have a similar effect than integrable interactions with longer range.', '0811.4281-1-17-2': 'A criterion that handles both situations is the one of [CITATION]; it extends the criterion of Kotecky and Preiss, initially proposed for discrete systems with hard-core interactions in [CITATION].', '0811.4281-1-17-3': 'We generalize it here to systems with stable interactions.', '0811.4281-1-18-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-1-19-0': 'In order to guess the correct form of [MATH], one should consider the left side of the equation above with [MATH].', '0811.4281-1-19-1': 'The integral may depend on [MATH]; a typical situation is that [MATH] is characterized by a length [MATH], which is a positive number, so that the left side is roughly proportional to [MATH].', '0811.4281-1-19-2': 'This suggests to try [MATH], and one can then optimize on the value of [MATH].', '0811.4281-1-20-0': 'We also consider an alternate criterion that involves [MATH] rather than [MATH].', '0811.4281-1-20-1': 'We use it in Section [REF] when studying a system of quantum particles that interact via an integrable stable potential.', '0811.4281-1-21-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-1-22-0': 'For positive [MATH] we can take [MATH]; and since [MATH], Assumption [REF] is always better than Assumption [REF].', '0811.4281-1-22-1': 'But Assumption [REF] has its advantages too; notice also that it involves [MATH] and not [MATH].', '0811.4281-1-22-2': 'We actually conjecture that, together with Assumption [REF], a sufficient condition is [EQUATION]', '0811.4281-1-22-3': 'That is, it should be possible to combine the best of both assumptions.', '0811.4281-1-22-4': 'But this remains to be proved.', '0811.4281-1-23-0': 'We denote by [MATH] the set of all graphs with [MATH] vertices (unoriented, no loops) and [MATH] the set of connected graphs with [MATH] vertices.', '0811.4281-1-23-1': 'We introduce the following combinatorial function on finite sequences [MATH] of elements of [MATH]: [EQUATION]', '0811.4281-1-23-2': 'The product is over edges of [MATH].', '0811.4281-1-24-0': '[Cluster expansions]', '0811.4281-1-25-0': 'Suppose that Assumptions [REF] and [REF], or [REF] and [REF], hold true.', '0811.4281-1-25-1': 'We also suppose that [MATH].', '0811.4281-1-25-2': 'Then we have [EQUATION]', '0811.4281-1-25-3': 'The term in the exponential converges absolutely.', '0811.4281-1-25-4': 'Furthermore, for almost all [MATH], we have the following estimate [EQUATION] (Under Assumption [REF], Eq. [REF] holds with [MATH] instead of [MATH].)', '0811.4281-1-26-0': 'Let us turn to correlation functions.', '0811.4281-1-26-1': 'We only consider one-point and two-point correlation functions since these are the most useful and expressions become more transparent.', '0811.4281-1-26-2': 'We refer to [CITATION] for more general functions.', '0811.4281-1-26-3': 'First, we define the unnormalized one-point correlation function by [EQUATION] (the term [MATH] is 1 by definition).', '0811.4281-1-26-4': 'And we define the unnormalized two-point correlation function by [EQUATION] (the term [MATH] is equal to [MATH]).', '0811.4281-1-26-5': 'Notice that [MATH] can be viewed as a regular partition function, given by Eq. [REF], but with the modified measure [MATH] instead of [MATH].', '0811.4281-1-26-6': 'The normalized correlation functions are [MATH] and [MATH].', '0811.4281-1-26-7': 'As is shown in Theorem [REF], they can be expressed using the "cluster functions" [EQUATION] and [EQUATION]', '0811.4281-1-26-8': 'Notice that [MATH] is bounded by the right side of [REF].', '0811.4281-1-27-0': '[Correlation functions]', '0811.4281-1-28-0': 'Under the same assumptions as in Theorem [REF], we have [EQUATION].', '0811.4281-1-29-0': 'In statistical mechanics, the relevant expression is the truncated two-point correlation function [EQUATION].', '0811.4281-1-29-1': 'When the cluster expansion converges, it is equal to [MATH] by the theorem above.', '0811.4281-1-29-2': 'This function usually provides an order parameter for phase transitions and it is useful to estimate its decay properties.', '0811.4281-1-30-0': '[Decay of correlations]', '0811.4281-1-31-0': 'If Assumptions [REF] and [REF] hold true, we have for almost all [MATH], [EQUATION] (with [MATH]).', '0811.4281-1-31-1': 'If Assumptions [REF] and [REF] hold true, we have the same bound but with [MATH] instead of [MATH], and [MATH] instead of [MATH].', '0811.4281-1-32-0': 'In many applications the functions [MATH] and [MATH] depend on the difference [MATH] (this assumes that [MATH] has additional structure, namely that of a group).', '0811.4281-1-32-1': 'The estimates for [MATH] are given by convolutions.', '0811.4281-1-33-0': 'The theorems of this section are proved in Section [REF].', '0811.4281-1-34-0': '# The classical gas', '0811.4281-1-35-0': 'We consider a gas of point particles that interact with a pair potential.', '0811.4281-1-35-1': 'We work in the grand-canonical ensemble where the parameters are the fugacity [MATH] and the inverse temperature [MATH] (both are real and positive numbers).', '0811.4281-1-35-2': 'The set [MATH] is an open bounded subset of [MATH] and [MATH] with [MATH] the Lebesgue measure.', '0811.4281-1-35-3': 'We actually write [MATH] so as to have more traditional notation.', '0811.4281-1-35-4': 'The interaction is given by a function [MATH] which we take to be piecewise continuous; [MATH].', '0811.4281-1-35-5': 'We suppose that [MATH] is stable, i.e. that there exists a constant [MATH] such that for any [MATH] and any [MATH]: [EQUATION]', '0811.4281-1-35-6': 'Our Assumption [REF] holds with [MATH].', '0811.4281-1-35-7': 'The system is translation invariant so all [MATH] are equivalent.', '0811.4281-1-35-8': 'The function of Assumptions [REF] and [REF] can then be taken to be a constant, [MATH].', '0811.4281-1-35-9': 'We seek a condition that does not depend on the size of the system.', '0811.4281-1-35-10': 'Then integrals over [MATH] are on [MATH] instead of [MATH].', '0811.4281-1-35-11': 'By translation invariance we can take [MATH].', '0811.4281-1-36-0': 'Assumptions [REF] and [REF] give the respective conditions [EQUATION]', '0811.4281-1-36-1': 'We obviously choose the constant [MATH] that maximizes the right side, which is [MATH].', '0811.4281-1-36-2': 'The first condition is the one in [CITATION] and the second is the one in [CITATION].', '0811.4281-1-36-3': 'The domains of parameters where these conditions hold correspond to low fugacities and high temperatures.', '0811.4281-1-37-0': 'The thermodynamic pressure is defined as the infinite volume limit of [EQUATION]', '0811.4281-1-37-1': 'Using Theorem [REF], we have [EQUATION]', '0811.4281-1-37-2': 'Consider now any sequence of increasing domains [MATH] such that [MATH].', '0811.4281-1-37-3': 'Thanks to the estimate [REF], and using translation invariance, we get [EQUATION] (The term with [MATH] is equal to [MATH].)', '0811.4281-1-37-4': 'This expression for the infinite volume pressure [MATH] should be viewed as a convergent series of analytic functions of [MATH].', '0811.4281-1-37-5': 'Then [MATH] is analytic in [MATH] and [MATH] by Vitali theorem and no phase transition takes place in the domain of parameters where the cluster expansion is convergent.', '0811.4281-1-38-0': 'The truncated two-point correlation function [MATH] is given by [MATH].', '0811.4281-1-38-1': 'We consider only the case of Assumption [REF] but a similar claim can be obtained with Assumption [REF].', '0811.4281-1-38-2': 'Let [MATH] be a function that satisfies the triangle inequality.', '0811.4281-1-38-3': 'The estimate of Theorem [REF] yields [EQUATION]', '0811.4281-1-39-0': 'Recall that [MATH], and let [EQUATION]', '0811.4281-1-39-1': 'Then we get [EQUATION]', '0811.4281-1-39-2': 'Then if the function [MATH] is such that [MATH], the truncated two-point correlation function decays faster than [MATH].', '0811.4281-1-40-0': '# Polymer systems', '0811.4281-1-41-0': 'Polymer systems are discrete, which is technically simpler, but they also have internal structure.', '0811.4281-1-41-1': 'The first application of cluster expansions to polymer systems is due to Gruber and Kunz [CITATION].', '0811.4281-1-41-2': 'Among the many articles devoted to this subject, let us mention [CITATION].', '0811.4281-1-41-3': 'The main goal of this section is to illustrate our setting; we therefore restrict ourselves to a specific model of polymers with both repulsive and attractive interactions.', '0811.4281-1-42-0': 'Our space [MATH] is the set of all finite connected subsets of [MATH].', '0811.4281-1-42-1': 'The measure [MATH] is the counting measure multiplied by the activity [MATH] (a function [MATH]).', '0811.4281-1-42-2': 'We choose [MATH] with [MATH].', '0811.4281-1-42-3': 'The interaction is hard core when polymers overlap and it is attractive when they touch: [EQUATION]', '0811.4281-1-42-4': 'Here, [MATH] is the number of "contacts" between [MATH] and [MATH], i.e. the number of bonds between sites of [MATH] and [MATH]; [MATH] is a parameter.', '0811.4281-1-42-5': 'The interaction is zero when the distance between polymers is greater than 1.', '0811.4281-1-43-0': 'The stability condition can be written [EQUATION]', '0811.4281-1-43-1': 'Only disjoint polymers need to be considered, the right side is infinite otherwise.', '0811.4281-1-43-2': 'The sum over [MATH] is always larger than [MATH] times the number of bonds connecting [MATH] with its exterior.', '0811.4281-1-43-3': 'Thus we can take [MATH].', '0811.4281-1-44-0': 'The function [MATH] in Assumption [REF] grows like [MATH], so it is natural to choose [MATH] for some constant [MATH].', '0811.4281-1-44-1': 'A sufficient condition is that [EQUATION]', '0811.4281-1-44-2': 'We bound [EQUATION]', '0811.4281-1-44-3': 'Summing over the sites of [MATH], and requiring that [MATH] contains the given site or comes at distance 1, we get [EQUATION]', '0811.4281-1-44-4': 'We used the fact that the activity is translation invariant.', '0811.4281-1-44-5': 'Then it is enough that [EQUATION]', '0811.4281-1-44-6': 'If [MATH] is a connected set, there exists a closed walk with nearest neighbor jumps whose support is [MATH], and whose length is at most [MATH].', '0811.4281-1-44-7': 'This can be seen by induction: knowing the walk for [MATH], it is easy to construct one for [MATH].', '0811.4281-1-44-8': 'The number of connected sets of cardinality [MATH] that contain the origin is therefore smaller than the number of walks of length [MATH] starting at the origin, which is equal to [MATH].', '0811.4281-1-44-9': 'Then it suffices that [EQUATION]', '0811.4281-1-44-10': 'This is equivalent to [EQUATION]', '0811.4281-1-44-11': 'Assumption [REF] holds for any [MATH].', '0811.4281-1-44-12': 'One can optimize the right side if one wishes; the resulting condition is not too elegant and our bounds are not optimal in any case.', '0811.4281-1-45-0': 'We have just established the existence of a low density phase provided the activity is small enough.', '0811.4281-1-45-1': 'The condition depends on the contact parameter [MATH].', '0811.4281-1-45-2': 'For large [MATH] one should expect interesting phases with many contacts between the polymers.', '0811.4281-1-46-0': '# Applications to quantum systems', '0811.4281-1-47-0': 'We follow a course that is similar to Ginibre [CITATION], using the Feynman-Kac formula so as to get a gas of winding Brownian loops.', '0811.4281-1-47-1': "Our results do not seem to improve Ginibre's results but our method is cleaner and better organized.", '0811.4281-1-47-2': 'Winding Brownian loops are kind of continuous polymers; they combine the difficulties of both cases above - the continuous nature and the internal structure.', '0811.4281-1-48-0': '## Feynman-Kac representation', '0811.4281-1-49-0': 'The state space for [MATH] fermions (resp. bosons) in a domain [MATH] is the Hilbert space [MATH] (resp. [MATH]) of square-integrable complex functions that are antisymmetric (resp. symmetric) with respect to their arguments.', '0811.4281-1-49-1': 'The Hamiltonian is [EQUATION] with [MATH] the Laplacian for the [MATH]-th variable and [MATH] a multiplication operator.', '0811.4281-1-49-2': 'As in the classical case, we consider the grand-canonical ensemble whose parameters are the fugacity [MATH] and the inverse temperature [MATH].', '0811.4281-1-49-3': 'The partition function is [EQUATION]', '0811.4281-1-49-4': 'We need to cast the partition function in the form [REF], which can be done using the Feynman-Kac representation.', '0811.4281-1-50-0': 'Namely, we have [EQUATION] [MATH] is the symmetric group of [MATH] elements; [MATH] is equal to the signature of the permutation [MATH] for fermions, [MATH] for bosons; [MATH] is the Wiener measure for the Brownian bridge from [MATH] to [MATH] in time [MATH] - the normalization is chosen so that [EQUATION] [MATH][MATH] is one if [MATH] for all [MATH], it is zero otherwise.', '0811.4281-1-50-1': 'An introduction to the Feynman-Kac formula in this context can be found in the survey of Ginibre [CITATION].', '0811.4281-1-51-0': 'The right side of Eq. [REF] is well defined for a large class of functions [MATH], that includes all piecewise continuous functions.', '0811.4281-1-51-1': 'Thus we take [REF] as the definition for [MATH].', '0811.4281-1-51-2': 'Under additional assumptions on [MATH], [REF] is equal to [REF] with Hamiltonian [REF] and with Dirichlet boundary conditions.', '0811.4281-1-52-0': 'We now rewrite the grand-canonical partition function in terms of winding loops.', '0811.4281-1-52-1': 'Let [MATH] be the set of continuous paths [MATH] that are closed.', '0811.4281-1-52-2': 'Its elements are denoted [MATH], with [MATH] the starting point, [MATH] the winding number, and [MATH] the path; we have [MATH].', '0811.4281-1-52-3': 'We consider the measure [MATH] given by [EQUATION]', '0811.4281-1-52-4': 'Here, [MATH] is a self-interaction term that is defined below in Eq. [REF]; [MATH] for fermions and 1 for bosons.', '0811.4281-1-52-5': 'Let [MATH]; the measure [MATH] above naturally extends to a measure on [MATH].', '0811.4281-1-52-6': 'The grand-canonical partition function can then be written as [EQUATION]', '0811.4281-1-52-7': 'Let [MATH] and [MATH].', '0811.4281-1-52-8': 'The self-interaction [MATH] and the 2-loop interaction [MATH] are given by [EQUATION]', '0811.4281-1-52-9': 'We always suppose that [MATH] is stable with constant [MATH], i.e. it satisfies Eq. [REF].', '0811.4281-1-52-10': 'For given loops [MATH], stability implies that [EQUATION]', '0811.4281-1-52-11': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-1-52-12': 'Notice that [MATH], again by the stability of [MATH].', '0811.4281-1-52-13': 'We now treat separately the case of integrable potentials and the case of potentials with hard core.', '0811.4281-1-53-0': '## Stable integrable potentials', '0811.4281-1-54-0': 'We use Assumption [REF] and we choose a function [MATH] with a constant [MATH] to be determined later.', '0811.4281-1-54-1': 'Explicitly, the assumption is that for any [MATH] [EQUATION]', '0811.4281-1-54-2': 'We have lifted the restriction that [MATH] because we want a condition that does not depend on [MATH].', '0811.4281-1-54-3': 'Eq. [REF] is easier to handle that its appearance suggests.', '0811.4281-1-54-4': 'Notice that the term in the second exponential is just [MATH].', '0811.4281-1-54-5': 'Using [MATH], and the definition [REF] of [MATH], it is enough that [EQUATION]', '0811.4281-1-55-0': 'We can immediately integrate over [MATH], yielding [MATH].', '0811.4281-1-55-1': 'The Wiener integral then gives [MATH] and we get the equivalent condition [EQUATION]', '0811.4281-1-55-2': 'For any [MATH], the inequality holds for [MATH] small enough.', '0811.4281-1-55-3': 'Notice that [MATH] in any case.', '0811.4281-1-55-4': 'One can get a more explicit condition for [MATH] by choosing [MATH] such that [MATH] = 1.', '0811.4281-1-55-5': 'This yields [EQUATION]', '0811.4281-1-55-6': 'Here, [MATH] is the Riemann zeta function.', '0811.4281-1-56-0': '## Stable potentials with hard core', '0811.4281-1-57-0': 'The presence of a hard core makes the situation more complicated; we only sketch the argument in this section without trying to get explicit bounds.', '0811.4281-1-57-1': 'Our aim is to show that, using Theorem [REF], the problem of convergence of the cluster expansion reduces to estimates on Wiener sausages.', '0811.4281-1-58-0': 'We consider an interaction [MATH] that consists of a hard core of radius [MATH], and of an integrable part.', '0811.4281-1-58-1': 'We also suppose that the integrable part is piecewise continuous.', '0811.4281-1-58-2': 'It is convenient to introduce two norms: [EQUATION] where the supremum is over countable sets of points in [MATH] such that any two points are at distance larger than [MATH].', '0811.4281-1-58-3': 'The latter norm represents a sort of integrability property.', '0811.4281-1-58-4': 'It is useful mainly for the following estimate; for any [MATH] with [MATH], [EQUATION] the latter being independent of [MATH].', '0811.4281-1-58-5': 'We also suppose that [MATH] is stable with constant [MATH] (see [REF]).', '0811.4281-1-58-6': 'Explicit conditions for the stability of such potentials are discussed in [CITATION].', '0811.4281-1-58-7': 'It is useful to restrict the space [MATH] to loops that satisfy [MATH].', '0811.4281-1-59-0': 'Given a loop [MATH], let [MATH] be the Wiener sausage generated by a ball of radius [MATH] when its center moves along the trajectory [MATH]: [EQUATION]', '0811.4281-1-60-0': 'We denote the volume of a Wiener sausage [MATH] by [MATH].', '0811.4281-1-60-1': 'One can check that [EQUATION] with [MATH] the volume of the unit ball (see Appendix 2 in [CITATION]).', '0811.4281-1-61-0': 'Recall that the stability condition holds with [MATH].', '0811.4281-1-61-1': 'By [REF] we have [MATH].', '0811.4281-1-61-2': 'We choose [MATH] in Assumption [REF].', '0811.4281-1-61-3': 'Then a sufficient condition is that for any [MATH], [EQUATION]', '0811.4281-1-61-4': 'We consider separately the cases where [MATH] belong or not to [MATH].', '0811.4281-1-61-5': 'First, [EQUATION] which we bound using [REF].', '0811.4281-1-61-6': 'Second, using [MATH] and [REF], [EQUATION]', '0811.4281-1-61-7': 'We certainly get [REF] if we have the two inequalities [EQUATION]', '0811.4281-1-61-8': 'One can estimate the integrals of Wiener sausages, see [CITATION], so that both conditions hold if [MATH] is small enough.', '0811.4281-1-62-0': '# The tree estimate', '0811.4281-1-63-0': 'In this section we obtain estimates of sums of connected graphs in terms of sums of trees.', '0811.4281-1-63-1': 'Our main result is Proposition [REF] below.', '0811.4281-1-63-2': 'Such estimates have often been considered in the past [CITATION].', '0811.4281-1-63-3': 'We introduce a minimal setting that clarifies its role in the cluster expansion.', '0811.4281-1-63-4': 'Namely, we fix the polymers so we only deal with the numbers that represent their interactions, [MATH] or [MATH], and the stability function [MATH].', '0811.4281-1-63-5': 'Assumption [REF] is vital here, but Assumptions [REF] and [REF] are not used in this section.', '0811.4281-1-64-0': 'Let [MATH] denote the set of trees with [MATH] vertices.', '0811.4281-1-64-1': 'Let [MATH] be an integer, [MATH] be real nonnegative numbers, and [MATH], [MATH], be complex numbers.', '0811.4281-1-64-2': 'We assume that the following bound holds for any subset [MATH]: [EQUATION]', '0811.4281-1-64-3': 'Let [MATH] be such that [MATH].', '0811.4281-1-64-4': 'We state two distinct tree estimates, the first one involving [MATH] and the second one involving [MATH].', '0811.4281-1-64-5': 'These bounds will allow to prove the convergence under either Assumption [REF] or Assumption [REF].', '0811.4281-1-65-0': 'If [REF] holds true, we have the two bounds', '0811.4281-1-66-0': 'We actually conjecture that the following estimate holds under the same hypotheses: [EQUATION]', '0811.4281-1-66-1': 'Proposition [REF] (a) follows from a tree identity due to Brydges, Battle, and Federbush [CITATION].', '0811.4281-1-66-2': 'Our claim is just a reformulation of Corollary 3.2 (a) in [CITATION].', '0811.4281-1-66-3': 'The claim (b) is based on the algebraic approach [CITATION].', '0811.4281-1-66-4': 'It seems to be new and we prove it in the rest of this section.', '0811.4281-1-67-0': 'Let [MATH] be the set of complex functions on the power set [MATH].', '0811.4281-1-67-1': 'We introduce the following multiplication operation for [MATH]: [EQUATION]', '0811.4281-1-67-2': 'Together with the addition, [MATH] is a commutative algebra with unit [MATH].', '0811.4281-1-67-3': 'It is possible to check that each [MATH] has a unique inverse, which we denote [MATH].', '0811.4281-1-67-4': 'We have [EQUATION]', '0811.4281-1-67-5': 'Let [MATH] be the subset of functions [MATH] such that [MATH] is an ideal of [MATH]).', '0811.4281-1-67-6': 'Notice that [MATH] for any [MATH] , when [MATH].', '0811.4281-1-67-7': 'We define the exponential mapping [MATH] by [EQUATION]', '0811.4281-1-67-8': 'Let [MATH] and [MATH] be the functions defined by [EQUATION]', '0811.4281-1-67-9': 'Here, [MATH] (resp. [MATH]) is the set of graphs (resp. connected graphs) on [MATH].', '0811.4281-1-67-10': 'We have the relation [EQUATION]', '0811.4281-1-67-11': 'We also introduce an operation that is reminiscent of differentiation: [EQUATION]', '0811.4281-1-67-12': 'One can check that [MATH].', '0811.4281-1-68-0': 'For disjoint [MATH], we define [EQUATION]', '0811.4281-1-68-1': 'Let [MATH].', '0811.4281-1-68-2': 'The assumption of Proposition [REF] implies that [EQUATION]', '0811.4281-1-68-3': 'Then there exists [MATH] such that [EQUATION]', '0811.4281-1-68-4': 'Such [MATH] is not unique in general but it does not matter.', '0811.4281-1-68-5': 'We consider a function [MATH] that assigns one of the indices [MATH] above to each nonempty subset [MATH].', '0811.4281-1-68-6': 'Notice that [MATH] for any subset [MATH].', '0811.4281-1-68-7': 'It is also useful to introduce the notation [MATH].', '0811.4281-1-69-0': 'The function [MATH] of Eq. [REF] is solution of the following equation.', '0811.4281-1-69-1': '[EQUATION].', '0811.4281-1-70-0': 'Since the equation gives [MATH] in terms of [MATH] with [MATH], it is well defined inductively and it has a unique solution.', '0811.4281-1-70-1': 'Notice that [MATH], and also that [MATH] for any index [MATH].', '0811.4281-1-71-0': 'Recall the definition [REF] of [MATH].', '0811.4281-1-71-1': 'For disjoint [MATH] we have [EQUATION]', '0811.4281-1-71-2': 'Then [EQUATION]', '0811.4281-1-71-3': 'The last sum is equal to [MATH].', '0811.4281-1-71-4': 'One recognizes the equation of Lemma [REF].', '0811.4281-1-72-0': 'We now estimate the function [MATH] using another function [MATH] that satisfies an equation that is similar to that of Lemma [REF].', '0811.4281-1-72-1': '[EQUATION]', '0811.4281-1-72-2': 'It also has a unique solution.', '0811.4281-1-72-3': 'Since [MATH], we can check inductively that [EQUATION] for any sets [MATH] (with [MATH]).', '0811.4281-1-72-4': 'Now the function [MATH] can be written explicitly.', '0811.4281-1-72-5': 'Let [MATH] be the set of forests on [MATH] rooted in [MATH].', '0811.4281-1-72-6': 'That is, a graph [MATH] is a forest such that each tree contains exactly one element of [MATH].', '0811.4281-1-73-0': 'The solution of Eq. [REF] is [EQUATION].', '0811.4281-1-74-0': 'Since the solution to Eq. [REF] is unique, it is enough to check that the Ansatz of the lemma satisfies the equation.', '0811.4281-1-74-1': 'First, let us observe that both sides are multiplied by [MATH].', '0811.4281-1-74-2': 'Thus it is enough to consider the case [MATH].', '0811.4281-1-75-0': 'The sum over graphs in [MATH] can be realized by first summing over the set [MATH] of indices (necessarily in [MATH]) that are connected to [MATH]; then over sets of trees in [MATH], and over connections to [MATH].', '0811.4281-1-75-1': 'Explicitly, [EQUATION]', '0811.4281-1-75-2': 'This equation is precisely [REF].', '0811.4281-1-76-0': '[Proof of Proposition [REF]] When [MATH] has a single element, the function [MATH] is equal to [EQUATION]', '0811.4281-1-76-1': 'This is the left side of Proposition [REF].', '0811.4281-1-76-2': 'We have [MATH], the set of trees with [MATH] vertices.', '0811.4281-1-76-3': 'Thus [MATH] is equal to the right side of Proposition [REF], and the proof follows from Eq. [REF].', '0811.4281-1-77-0': '# Proofs of the theorems', '0811.4281-1-78-0': 'In this section we prove the theorems of Section [REF].', '0811.4281-1-78-1': 'We consider only the case where Assumption [REF] holds true - the case with Assumption [REF] is entirely the same, one only needs to replace all [MATH] with [MATH] and all [MATH] with [MATH].', '0811.4281-1-78-2': 'The proofs are based on the following tree estimate, which is a direct consequence of Proposition [REF]: for almost all [MATH], [EQUATION] [Proof of Theorem [REF]] We start by proving the bound [REF].', '0811.4281-1-78-3': 'Let us introduce [EQUATION] (The term [MATH] is equal to [MATH] by definition.)', '0811.4281-1-78-4': 'We show by induction that [EQUATION] for any [MATH].', '0811.4281-1-78-5': 'Then [MATH] for almost all [MATH], and using [REF] we get [REF].', '0811.4281-1-79-0': 'The case [MATH] reduces to [MATH] and it is clear.', '0811.4281-1-79-1': 'The sum over trees with [MATH] vertices can be written as a sum over forests on [MATH], and a sum over edges between 1 and each tree of the forest.', '0811.4281-1-80-0': 'Here, [MATH] denote the set of trees with [MATH] as the set of vertices.', '0811.4281-1-80-1': 'If [MATH] the sum over [MATH] is one by definition.', '0811.4281-1-80-2': "The term after the sum over partitions depends on the cardinalities of the [MATH]'s, but not on the actual labeling.", '0811.4281-1-80-3': 'Also, each [MATH] gives the same contribution.', '0811.4281-1-81-0': 'We obtain an upper bound by releasing the constraint [MATH] to [MATH], [MATH].', '0811.4281-1-81-1': 'We then get [EQUATION]', '0811.4281-1-81-2': 'We have [MATH] by the induction hypothesis.', '0811.4281-1-81-3': 'Eq. [REF] follows from Assumption [REF].', '0811.4281-1-82-0': 'The rest of the proof is standard combinatorics.', '0811.4281-1-82-1': 'The partition function can be expanded so as to recognize the exponential of connected graphs.', '0811.4281-1-82-2': 'Namely, we start with [EQUATION]', '0811.4281-1-82-3': 'The graph [MATH] can be decomposed into [MATH] connected graphs whose sets of vertices form a partition of [MATH].', '0811.4281-1-82-4': 'Summing first over the number [MATH] of vertices for each set of the partition, we get [EQUATION]', '0811.4281-1-82-5': 'The triple sum is absolutely convergent thanks to the estimate [REF] that we have just established.', '0811.4281-1-82-6': 'One can then interchange the sums by the dominated convergence theorem.', '0811.4281-1-82-7': 'This removes the sum over [MATH], and this completes the proof of Theorem [REF].', '0811.4281-1-83-0': 'Next we prove Theorems [REF] and [REF] in reverse order, since we will use the convergence properties in the latter theorem to get the former.', '0811.4281-1-84-0': '[Proof of Theorem [REF]] From the definition [REF] and the tree estimate [REF], we have [EQUATION]', '0811.4281-1-84-1': 'The expression above involves a sum over trees of arbitrary size that connect 1 and 2.', '0811.4281-1-84-2': 'Any such tree decomposes into a line of edges that connect 1 and 2, and two trees rooted in 1 and 2.', '0811.4281-1-84-3': 'Taking into account the combinatorial factors, we obtain', '0811.4281-1-85-0': '[Proof of Theorem [REF]] It is actually similar to the end of the proof of Theorem [REF].', '0811.4281-1-85-1': '[MATH] can be expanded as a sum over graphs, that can be decomposed into a connected graph that contains 1, and other connected graphs.', '0811.4281-1-85-2': 'Taking into account the combinatorial factors, the contribution of connected graphs containing 1 yields [MATH], and the contribution of the others yields the expression [REF] for [MATH].', '0811.4281-1-85-3': 'Thus [MATH].', '0811.4281-1-85-4': 'One step involved interchanging unbounded sums, which is justified because everything is absolutely convergent, thanks to [REF] and Theorem [REF].', '0811.4281-1-86-0': 'In the graph expansion for [MATH], the terms where 1 and 2 belong to the same connected graph yield [MATH], and the terms where 1 and 2 belong to different connected graphs yield [MATH].', '0811.4281-1-86-1': 'The detailed argument is the same as above.', '0811.4281-1-86-2': 'We then obtained the desired expression.'} | {'0811.4281-2-0-0': 'cluster expansion with applications]Abstract cluster expansion with applications to statistical mechanical systems', '0811.4281-2-1-0': '# Abstract We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence.', '0811.4281-2-1-1': 'We apply the results to systems of classical and quantum particles with stable interactions.', '0811.4281-2-2-0': 'Keywords: cluster expansion, polymer model, stable interaction, quantum gas.', '0811.4281-2-3-0': '2000 Math.', '0811.4281-2-3-1': 'Subj.', '0811.4281-2-3-2': 'Class.: 82B05, 82B10, 82B20, 82B21, 82B26', '0811.4281-2-4-0': '# Introduction', '0811.4281-2-5-0': "The method of cluster expansions was introduced in the 1930's in statistical mechanics in order to study gases of classical interacting particles.", '0811.4281-2-5-1': 'Its main achievement, from the point of view of physics, may be the derivation of the van der Waals equation of state for weakly interacting systems.', '0811.4281-2-5-2': "The method was made rigorous by mathematical-physicists in the 1960's, see [CITATION] and references therein.", '0811.4281-2-6-0': 'The method split afterwards.', '0811.4281-2-6-1': 'One branch involves continuous systems, with applications to classical systems [CITATION], quantum systems [CITATION], or quantum field theory [CITATION].', '0811.4281-2-6-2': 'The other branch involves polymer systems, i.e. discrete systems with additional internal structure [CITATION].', '0811.4281-2-6-3': 'An important step forward was the article of Kotecky and Preiss with its simplified setting and its elegant condition for the convergence of the cluster expansion [CITATION].', '0811.4281-2-7-0': 'The methods for proving the convergence are diverse.', '0811.4281-2-7-1': 'Let us mention the study of Kirkwood-Salsburg equations that involves correlation functions, see [CITATION] and references therein; the algebraic approach of Ruelle [CITATION]; combinatorial approaches using tree identities [CITATION]; inductions for discrete systems [CITATION].', '0811.4281-2-8-0': 'Important and useful surveys were written by Brydges [CITATION], Pfister [CITATION], Abdesselam and Rivasseau [CITATION].', '0811.4281-2-8-1': 'Recent articles have been devoted to combinatorial aspects [CITATION] and to improving estimates [CITATION].', '0811.4281-2-9-0': 'The method of cluster expansions applies when the objects do not interact much; this the case when they are far apart (low density), or when interactions are weak.', '0811.4281-2-9-1': 'An extension of the criterion of [CITATION] that takes into account these two aspects was proposed in [CITATION] (see also [CITATION]); it applies to both discrete and continuous systems.', '0811.4281-2-10-0': 'All abstract (i.e. general) approaches involve restrictions that correspond to repulsive interactions.', '0811.4281-2-10-1': 'Yet the old results for classical and quantum systems only assume stable interactions, that may include an attractive part.', '0811.4281-2-10-2': 'The aim of the present article is to propose a general approach that applies to discrete and continuous systems with repulsive or stable interactions.', '0811.4281-2-10-3': 'Our proof is split into several independent steps and this helps clarify the situation.', '0811.4281-2-11-0': 'The setting and the results are presented in Section [REF].', '0811.4281-2-11-1': 'We consider applications to classical systems of particles in Section [REF], to polymer systems in Section [REF], and to the quantum gas in Section [REF].', '0811.4281-2-11-2': 'A fundamental tree estimate is derived in Section [REF], and the theorems of Section [REF] are proved in Section [REF].', '0811.4281-2-12-0': '# Cluster expansions', '0811.4281-2-13-0': 'We consider a set [MATH] whose elements may represent widely different objects - in the three applications considered in this article, an element [MATH] represents (i) the position of a classical particle, (ii) a polymer, i.e. a connected set of [MATH], and (iii) a closed Brownian bridge.', '0811.4281-2-13-1': 'For the general abstract theory, we assume the structure of a measure space, [MATH], with [MATH] a complex measure.', '0811.4281-2-13-2': 'We denote [MATH] the total variation (absolute value) of [MATH].', '0811.4281-2-13-3': 'Let [MATH] and [MATH] be complex measurable symmetric functions on [MATH], that are related by the equation [EQUATION]', '0811.4281-2-13-4': 'We allow the real part of [MATH] to take the value [MATH], in which case [MATH].', '0811.4281-2-13-5': 'In typical applications [MATH] represents the interactions between [MATH] and [MATH], and the value [MATH] corresponds to a hard-core repulsion.', '0811.4281-2-13-6': 'We define the "partition function" by [EQUATION] or, equivalently, [EQUATION]', '0811.4281-2-13-7': 'The term [MATH] of the sums is understood to be 1.', '0811.4281-2-14-0': 'The main goal of cluster expansions is to express the partition function as the exponential of a convergent series of "cluster terms".', '0811.4281-2-14-1': 'The main difficulty is to prove the convergence.', '0811.4281-2-14-2': 'We first assume that the potential [MATH] is stable.', '0811.4281-2-15-0': 'There exists a nonnegative function [MATH] on [MATH] such that, for all [MATH] and almost all [MATH], [EQUATION].', '0811.4281-2-16-0': 'In other words, we assume the lower bound [EQUATION]', '0811.4281-2-16-1': 'When the function [MATH] is constant, this is the usual definition of stability.', '0811.4281-2-16-2': '"Almost all" means that, for given [MATH], the set of points where the condition fails has measure zero with respect to the product measure [MATH].', '0811.4281-2-16-3': 'If [MATH] is countable, the condition must be satisfied for all [MATH] such that [MATH].', '0811.4281-2-17-0': 'The second condition deals with the strength of interactions.', '0811.4281-2-18-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-2-19-0': 'In order to guess the correct form of [MATH], one should consider the left side of the equation above with [MATH].', '0811.4281-2-19-1': 'The integral may depend on [MATH]; a typical situation is that [MATH] is characterized by a length [MATH], which is a positive number, so that the left side is roughly proportional to [MATH].', '0811.4281-2-19-2': 'This suggests to try [MATH], and one can then optimize on the value of [MATH].', '0811.4281-2-20-0': 'We also consider an alternate criterion that involves [MATH] rather than [MATH].', '0811.4281-2-20-1': 'We use it in Section [REF] when studying a system of quantum particles that interact via an integrable stable potential.', '0811.4281-2-21-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-2-22-0': 'For positive [MATH] we can take [MATH]; and since [MATH], Assumption [REF] is always better than Assumption [REF].', '0811.4281-2-22-1': 'But Assumption [REF] has its advantages too; notice also that it involves [MATH] and not [MATH].', '0811.4281-2-22-2': 'We actually conjecture that, together with Assumption [REF], a sufficient condition is [EQUATION]', '0811.4281-2-22-3': 'That is, it should be possible to combine the best of both assumptions.', '0811.4281-2-22-4': 'But this remains to be proved.', '0811.4281-2-23-0': 'We denote by [MATH] the set of all graphs with [MATH] vertices (unoriented, no loops) and [MATH] the set of connected graphs with [MATH] vertices.', '0811.4281-2-23-1': 'We introduce the following combinatorial function on finite sequences [MATH] of elements of [MATH]: [EQUATION]', '0811.4281-2-23-2': 'The product is over edges of [MATH].', '0811.4281-2-24-0': '[Cluster expansions]', '0811.4281-2-25-0': 'Suppose that Assumptions [REF] and [REF], or [REF] and [REF], hold true.', '0811.4281-2-25-1': 'We also suppose that [MATH].', '0811.4281-2-25-2': 'Then we have [EQUATION]', '0811.4281-2-25-3': 'The term in the exponential converges absolutely.', '0811.4281-2-25-4': 'Furthermore, for almost all [MATH], we have the following estimate [EQUATION] (Under Assumption [REF], Eq. [REF] holds with [MATH] instead of [MATH].)', '0811.4281-2-26-0': 'Let us turn to correlation functions.', '0811.4281-2-26-1': 'We only consider one-point and two-point correlation functions since these are the most useful and expressions become more transparent.', '0811.4281-2-26-2': 'We refer to [CITATION] for more general functions.', '0811.4281-2-26-3': 'First, we define the unnormalized one-point correlation function by [EQUATION] (the term [MATH] is 1 by definition).', '0811.4281-2-26-4': 'And we define the unnormalized two-point correlation function by [EQUATION] (the term [MATH] is equal to [MATH]).', '0811.4281-2-26-5': 'Notice that [MATH] can be viewed as a regular partition function, given by Eq. [REF], but with the modified measure [MATH] instead of [MATH].', '0811.4281-2-26-6': 'The normalized correlation functions are [MATH] and [MATH].', '0811.4281-2-26-7': 'As is shown in Theorem [REF], they can be expressed using the "cluster functions" [EQUATION] and [EQUATION]', '0811.4281-2-26-8': 'Notice that [MATH] by [REF].', '0811.4281-2-27-0': '[Correlation functions]', '0811.4281-2-28-0': 'Under the same assumptions as in Theorem [REF], we have [EQUATION].', '0811.4281-2-29-0': 'In statistical mechanics, the relevant expression is the truncated two-point correlation function [EQUATION].', '0811.4281-2-29-1': 'When the cluster expansion converges, it is equal to [MATH] by the theorem above.', '0811.4281-2-29-2': 'This function usually provides an order parameter for phase transitions and it is useful to estimate its decay properties.', '0811.4281-2-30-0': '[Decay of correlations]', '0811.4281-2-31-0': 'If Assumptions [REF] and [REF] hold true, we have for almost all [MATH], [EQUATION] (with [MATH]).', '0811.4281-2-31-1': 'If Assumptions [REF] and [REF] hold true, we have the same bound but with [MATH] instead of [MATH], and [MATH] instead of [MATH].', '0811.4281-2-32-0': 'In many applications the functions [MATH] and [MATH] depend on the difference [MATH] (this assumes that [MATH] has additional structure, namely that of a group).', '0811.4281-2-32-1': 'The estimates for [MATH] are given by convolutions.', '0811.4281-2-33-0': 'The theorems of this section are proved in Section [REF].', '0811.4281-2-34-0': '# The classical gas', '0811.4281-2-35-0': 'We consider a gas of point particles that interact with a pair potential.', '0811.4281-2-35-1': 'We work in the grand-canonical ensemble where the parameters are the fugacity [MATH] and the inverse temperature [MATH] (both are real and positive numbers).', '0811.4281-2-35-2': 'The set [MATH] is an open bounded subset of [MATH] and [MATH] with [MATH] the Lebesgue measure.', '0811.4281-2-35-3': 'We actually write [MATH] so as to have more traditional notation.', '0811.4281-2-35-4': 'The interaction is given by a function [MATH] which we take to be piecewise continuous; [MATH].', '0811.4281-2-35-5': 'We suppose that [MATH] is stable, i.e. that there exists a constant [MATH] such that for any [MATH] and any [MATH]: [EQUATION]', '0811.4281-2-35-6': 'Our Assumption [REF] holds with [MATH] (assuming [MATH]).', '0811.4281-2-35-7': 'The system is translation invariant so all [MATH] are equivalent.', '0811.4281-2-35-8': 'The function of Assumptions [REF] and [REF] can then be taken to be a constant, [MATH].', '0811.4281-2-35-9': 'We seek a condition that does not depend on the size of the system.', '0811.4281-2-35-10': 'Then integrals over [MATH] are on [MATH] instead of [MATH].', '0811.4281-2-35-11': 'By translation invariance we can take [MATH].', '0811.4281-2-36-0': 'Assumptions [REF] and [REF] give the respective conditions [EQUATION]', '0811.4281-2-36-1': 'We obviously choose the constant [MATH] that maximizes the right side, which is [MATH].', '0811.4281-2-36-2': 'The first condition is the one in [CITATION] and the second is the one in [CITATION].', '0811.4281-2-36-3': 'The domains of parameters where these conditions hold correspond to low fugacities and high temperatures.', '0811.4281-2-37-0': 'The thermodynamic pressure is defined as the infinite volume limit of [EQUATION]', '0811.4281-2-37-1': 'Using Theorem [REF], we have [EQUATION]', '0811.4281-2-37-2': 'Consider now any sequence of increasing domains [MATH] such that [MATH].', '0811.4281-2-37-3': 'Thanks to the estimate [REF], and using translation invariance, we get [EQUATION] (The term with [MATH] is equal to [MATH].)', '0811.4281-2-37-4': 'This expression for the infinite volume pressure [MATH] should be viewed as a convergent series of analytic functions of [MATH].', '0811.4281-2-37-5': 'Then [MATH] is analytic in [MATH] and [MATH] by Vitali theorem and no phase transition takes place in the domain of parameters where the cluster expansion is convergent.', '0811.4281-2-38-0': 'The truncated two-point correlation function [MATH] is given by [MATH].', '0811.4281-2-38-1': 'We consider only the case of Assumption [REF] but a similar claim can be obtained with Assumption [REF].', '0811.4281-2-38-2': 'Let [MATH] be a function that satisfies the triangle inequality.', '0811.4281-2-38-3': 'The estimate of Theorem [REF] yields [EQUATION]', '0811.4281-2-39-0': 'Recall that [MATH], and let [EQUATION]', '0811.4281-2-39-1': 'Then we get [EQUATION]', '0811.4281-2-39-2': 'Then if the function [MATH] is such that [MATH], the truncated two-point correlation function decays faster than [MATH].', '0811.4281-2-40-0': '# Polymer systems', '0811.4281-2-41-0': 'Polymer systems are discrete, which is technically simpler, but they also have internal structure.', '0811.4281-2-41-1': 'The first application of cluster expansions to polymer systems is due to Gruber and Kunz [CITATION].', '0811.4281-2-41-2': 'Among the many articles devoted to this subject, let us mention [CITATION].', '0811.4281-2-41-3': 'The main goal of this section is to illustrate our setting; we therefore restrict ourselves to a specific model of polymers with both repulsive and attractive interactions.', '0811.4281-2-42-0': 'Our space [MATH] is the set of all finite connected subsets of [MATH].', '0811.4281-2-42-1': 'The measure [MATH] is the counting measure multiplied by the activity [MATH] (a function [MATH]).', '0811.4281-2-42-2': 'We choose [MATH] with [MATH].', '0811.4281-2-42-3': 'The interaction is hard core when polymers overlap and it is attractive when they touch: [EQUATION]', '0811.4281-2-42-4': 'Here, [MATH] is the number of "contacts" between [MATH] and [MATH], i.e. the number of bonds between sites of [MATH] and [MATH]; [MATH] is a parameter.', '0811.4281-2-42-5': 'The interaction is zero when the distance between polymers is greater than 1.', '0811.4281-2-43-0': 'The stability condition can be written [EQUATION]', '0811.4281-2-43-1': 'Only disjoint polymers need to be considered, the right side is infinite otherwise.', '0811.4281-2-43-2': 'The sum over [MATH] is always larger than [MATH] times the number of bonds connecting [MATH] with its exterior.', '0811.4281-2-43-3': 'Thus we can take [MATH].', '0811.4281-2-44-0': 'The function [MATH] in Assumption [REF] grows like [MATH], so it is natural to choose [MATH] for some constant [MATH].', '0811.4281-2-44-1': 'A sufficient condition is that [EQUATION]', '0811.4281-2-44-2': 'We bound [EQUATION]', '0811.4281-2-44-3': 'Summing over the sites of [MATH], and requiring that [MATH] contains the given site or comes at distance 1, we get [EQUATION]', '0811.4281-2-44-4': 'We used the fact that the activity is translation invariant.', '0811.4281-2-44-5': 'Then it is enough that [EQUATION]', '0811.4281-2-44-6': 'If [MATH] is a connected set, there exists a closed walk with nearest neighbor jumps whose support is [MATH], and whose length is at most [MATH].', '0811.4281-2-44-7': 'This can be seen by induction: knowing the walk for [MATH], it is easy to construct one for [MATH].', '0811.4281-2-44-8': 'The number of connected sets of cardinality [MATH] that contain the origin is therefore smaller than the number of walks of length [MATH] starting at the origin, which is equal to [MATH].', '0811.4281-2-44-9': 'Then it suffices that [EQUATION]', '0811.4281-2-44-10': 'This is equivalent to [EQUATION]', '0811.4281-2-44-11': 'Assumption [REF] holds for any [MATH].', '0811.4281-2-44-12': 'One can optimize the right side if one wishes; the resulting condition is not too elegant and our bounds are not optimal in any case.', '0811.4281-2-45-0': 'We have just established the existence of a low density phase provided the activity is small enough.', '0811.4281-2-45-1': 'The condition depends on the contact parameter [MATH].', '0811.4281-2-45-2': 'For large [MATH] one should expect interesting phases with many contacts between the polymers.', '0811.4281-2-46-0': '# The quantum gas', '0811.4281-2-47-0': 'We follow a course that is similar to Ginibre [CITATION], using the Feynman-Kac formula so as to get a gas of winding Brownian loops.', '0811.4281-2-47-1': "Our results do not seem to improve Ginibre's results but our method is cleaner and better organized.", '0811.4281-2-47-2': 'Winding Brownian loops are kind of continuous polymers; they combine the difficulties of both cases above - the continuous nature and the internal structure.', '0811.4281-2-48-0': '## Feynman-Kac representation', '0811.4281-2-49-0': 'The state space for [MATH] fermions (resp. bosons) in a domain [MATH] is the Hilbert space [MATH] (resp. [MATH]) of square-integrable complex functions that are antisymmetric (resp. symmetric) with respect to their arguments.', '0811.4281-2-49-1': 'The Hamiltonian is [EQUATION] with [MATH] the Laplacian for the [MATH]-th variable and [MATH] a multiplication operator.', '0811.4281-2-49-2': 'As in the classical case, we consider the grand-canonical ensemble whose parameters are the fugacity [MATH] and the inverse temperature [MATH].', '0811.4281-2-49-3': 'The partition function is [EQUATION]', '0811.4281-2-49-4': 'We need to cast the partition function in the form [REF], which can be done using the Feynman-Kac representation.', '0811.4281-2-50-0': 'Namely, we have [EQUATION] [MATH] is the symmetric group of [MATH] elements; [MATH] is equal to the signature of the permutation [MATH] for fermions, [MATH] for bosons; [MATH] is the Wiener measure for the Brownian bridge from [MATH] to [MATH] in time [MATH] - the normalization is chosen so that [EQUATION] [MATH][MATH] is one if [MATH] for all [MATH], it is zero otherwise.', '0811.4281-2-50-1': 'An introduction to the Feynman-Kac formula in this context can be found in the survey of Ginibre [CITATION].', '0811.4281-2-51-0': 'The right side of Eq. [REF] is well defined for a large class of functions [MATH], that includes all piecewise continuous functions.', '0811.4281-2-51-1': 'Thus we take [REF] as the definition for [MATH].', '0811.4281-2-51-2': 'Under additional assumptions on [MATH], [REF] is equal to [REF] with Hamiltonian [REF] and with Dirichlet boundary conditions.', '0811.4281-2-52-0': 'We now rewrite the grand-canonical partition function in terms of winding loops.', '0811.4281-2-52-1': 'Let [MATH] be the set of continuous paths [MATH] that are closed.', '0811.4281-2-52-2': 'Its elements are denoted [MATH], with [MATH] the starting point, [MATH] the winding number, and [MATH] the path; we have [MATH].', '0811.4281-2-52-3': 'We consider the measure [MATH] given by [EQUATION]', '0811.4281-2-52-4': 'Here, [MATH] is a self-interaction term that is defined below in Eq. [REF]; [MATH] for fermions and 1 for bosons.', '0811.4281-2-52-5': 'Let [MATH]; the measure [MATH] above naturally extends to a measure on [MATH].', '0811.4281-2-52-6': 'The grand-canonical partition function can then be written as [EQUATION]', '0811.4281-2-52-7': 'Let [MATH] and [MATH].', '0811.4281-2-52-8': 'The self-interaction [MATH] and the 2-loop interaction [MATH] are given by [EQUATION]', '0811.4281-2-52-9': 'We now treat separately the case of integrable potentials and the case of potentials with hard core.', '0811.4281-2-53-0': '## Stable integrable potentials', '0811.4281-2-54-0': 'We suppose that [MATH] is stable with constant [MATH], i.e. it satisfies Eq. [REF].', '0811.4281-2-54-1': 'For given loops [MATH], stability implies that [EQUATION]', '0811.4281-2-54-2': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-2-54-3': 'Notice that [MATH], again by the stability of [MATH].', '0811.4281-2-55-0': 'We use Assumption [REF] and we choose a function [MATH] with a constant [MATH] to be determined later.', '0811.4281-2-55-1': 'Explicitly, the assumption is that for any [MATH] [EQUATION]', '0811.4281-2-55-2': 'We have lifted the restriction that [MATH] because we want a condition that does not depend on [MATH].', '0811.4281-2-55-3': 'Eq. [REF] is easier to handle than its appearance suggests.', '0811.4281-2-55-4': 'Notice that the term in the second exponential is just [MATH].', '0811.4281-2-55-5': 'Using [MATH], and the definition [REF] of [MATH], it is enough that [EQUATION]', '0811.4281-2-56-0': 'We can immediately integrate over [MATH], yielding [MATH].', '0811.4281-2-56-1': 'The Wiener integral then gives [MATH] and we get the equivalent condition [EQUATION]', '0811.4281-2-56-2': 'For any [MATH], the inequality holds for [MATH] small enough.', '0811.4281-2-56-3': 'Notice that [MATH] in any case.', '0811.4281-2-56-4': 'One can get a more explicit condition for [MATH] by choosing [MATH] such that [MATH] = 1.', '0811.4281-2-56-5': 'This yields [EQUATION]', '0811.4281-2-56-6': 'Here, [MATH] is the Riemann zeta function.', '0811.4281-2-57-0': '## Stable potentials with hard core', '0811.4281-2-58-0': 'The presence of a hard core makes the situation more complicated; we only sketch the argument in this section without trying to get explicit bounds.', '0811.4281-2-58-1': 'Our aim is to show that, using Theorem [REF], the problem of convergence of the cluster expansion reduces to estimates of Wiener sausages.', '0811.4281-2-59-0': 'We consider an interaction [MATH] that consists of a hard core of radius [MATH], and of an integrable part.', '0811.4281-2-59-1': 'We suppose that the stability condition takes the slightly stronger form [EQUATION] for any [MATH] such that [MATH].', '0811.4281-2-59-2': 'For potentials with a hard core this is equivalent to the property [REF], possibly with a different constant [MATH].', '0811.4281-2-59-3': 'Then one has [CITATION] [EQUATION]', '0811.4281-2-59-4': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-2-59-5': 'It is useful to restrict the space [MATH] to loops that satisfy [MATH].', '0811.4281-2-59-6': 'Let [MATH] be the potential without the hard core; then [EQUATION]', '0811.4281-2-59-7': 'Given a loop [MATH], let [MATH] be the Wiener sausage generated by a ball of radius [MATH] when its center moves along the trajectory [MATH]: [EQUATION]', '0811.4281-2-60-0': 'We denote the volume of a Wiener sausage [MATH] by [MATH].', '0811.4281-2-60-1': 'One can check that [EQUATION] with [MATH] the volume of the unit ball (see Appendix 2 in [CITATION]).', '0811.4281-2-61-0': 'We choose [MATH] in Assumption [REF].', '0811.4281-2-61-1': 'Then a sufficient condition is that for any [MATH], [EQUATION]', '0811.4281-2-61-2': 'We consider separately the cases where [MATH] belongs or not to [MATH].', '0811.4281-2-61-3': 'First, [EQUATION] which we bound using [REF].', '0811.4281-2-61-4': 'Second, using [MATH], [EQUATION]', '0811.4281-2-61-5': 'We certainly get [REF] if we have the two inequalities [EQUATION]', '0811.4281-2-61-6': 'One can estimate the integrals of Wiener sausages, see [CITATION], so that both conditions hold if [MATH] is small enough.', '0811.4281-2-62-0': 'Now that the cluster expansion is known to converge, it is possible to write the pressure as an absolutely convergent series of analytic functions in [MATH] and [MATH].', '0811.4281-2-62-1': 'It is also possible to study the decay of the "off-diagonal correlation function", a natural correlation function for quantum systems that signals the occurrence of Bose-Einstein condensation.', '0811.4281-2-63-0': '# The tree estimates', '0811.4281-2-64-0': 'In this section we obtain estimates of sums of connected graphs in terms of sums of trees.', '0811.4281-2-64-1': 'Our main result is Proposition [REF] below.', '0811.4281-2-64-2': 'Such estimates have often been considered in the past [CITATION].', '0811.4281-2-64-3': 'We introduce a minimal setting that clarifies its role in the cluster expansion.', '0811.4281-2-64-4': 'Namely, we fix the polymers so we only deal with the numbers that represent their interactions, [MATH] or [MATH], and the stability function [MATH].', '0811.4281-2-64-5': 'Assumption [REF] is vital here, but Assumptions [REF] and [REF] are not used in this section.', '0811.4281-2-65-0': 'Let [MATH] denote the set of trees with [MATH] vertices.', '0811.4281-2-65-1': 'Let [MATH] be an integer, [MATH] be real nonnegative numbers, and [MATH], [MATH], be complex numbers.', '0811.4281-2-65-2': 'We assume that the following bound holds for any subset [MATH]: [EQUATION]', '0811.4281-2-65-3': 'Let [MATH] be such that [MATH].', '0811.4281-2-65-4': 'We state two distinct tree estimates, the first one involving [MATH] and the second one involving [MATH].', '0811.4281-2-65-5': 'These bounds will allow to prove the convergence under either Assumption [REF] or Assumption [REF].', '0811.4281-2-66-0': 'If [REF] holds true, we have the two bounds', '0811.4281-2-67-0': 'We actually conjecture that the following estimate holds under the same hypotheses: [EQUATION]', '0811.4281-2-67-1': 'Proposition [REF] (a) follows from a tree identity due to Brydges, Battle, and Federbush [CITATION].', '0811.4281-2-67-2': 'Our claim is just a reformulation of Corollary 3.2 (a) of [CITATION].', '0811.4281-2-67-3': 'Proposition [REF] (b) seems to be new.', '0811.4281-2-67-4': "We prove it in the rest of this section using Ruelle's algebraic approach [CITATION].", '0811.4281-2-67-5': 'This method is usually combined with a Banach fixed point argument for correlation functions.', '0811.4281-2-67-6': 'However, we use it differently so as to get a tree estimate.', '0811.4281-2-68-0': 'Let [MATH] be the set of complex functions on the power set [MATH].', '0811.4281-2-68-1': 'We introduce the following multiplication operation for [MATH]: [EQUATION]', '0811.4281-2-68-2': 'Together with the addition, [MATH] is a commutative algebra with unit [MATH].', '0811.4281-2-68-3': 'It is possible to check that each [MATH] has a unique inverse, which we denote [MATH].', '0811.4281-2-68-4': 'We have [EQUATION]', '0811.4281-2-68-5': 'Let [MATH] be the subset of functions [MATH] such that [MATH] is an ideal of [MATH]).', '0811.4281-2-68-6': 'Notice that [MATH] for any [MATH] , when [MATH].', '0811.4281-2-68-7': 'We define the exponential mapping [MATH] by [EQUATION]', '0811.4281-2-68-8': 'Let [MATH] and [MATH] be the functions defined by [EQUATION]', '0811.4281-2-68-9': 'Here, [MATH] (resp. [MATH]) is the set of graphs (resp. connected graphs) on [MATH].', '0811.4281-2-68-10': 'We have the relation [EQUATION]', '0811.4281-2-68-11': 'We also introduce an operation that is reminiscent of differentiation: [EQUATION]', '0811.4281-2-68-12': 'One can check that [MATH].', '0811.4281-2-69-0': 'For disjoint [MATH], we define [EQUATION]', '0811.4281-2-69-1': 'Let [MATH].', '0811.4281-2-69-2': 'The assumption of Proposition [REF] implies that [EQUATION]', '0811.4281-2-69-3': 'Then there exists [MATH] such that [EQUATION]', '0811.4281-2-69-4': 'Such [MATH] is not unique in general but it does not matter.', '0811.4281-2-69-5': 'We consider a function [MATH] that assigns one of the indices [MATH] above to each nonempty subset [MATH].', '0811.4281-2-69-6': 'Notice that [MATH] for any subset [MATH].', '0811.4281-2-69-7': 'It is also useful to introduce the notation [MATH].', '0811.4281-2-70-0': 'The function [MATH] of Eq. [REF] is solution of the following equation.', '0811.4281-2-70-1': '[EQUATION].', '0811.4281-2-71-0': 'Since the equation gives [MATH] in terms of [MATH] with [MATH], it is well defined inductively and it has a unique solution.', '0811.4281-2-71-1': 'Notice that [MATH], and also that [MATH] for any index [MATH].', '0811.4281-2-72-0': 'Recall the definition [REF] of [MATH].', '0811.4281-2-72-1': 'For disjoint [MATH] we have [EQUATION]', '0811.4281-2-72-2': 'Then [EQUATION]', '0811.4281-2-72-3': 'The last sum is equal to [MATH].', '0811.4281-2-72-4': 'One recognizes the equation of Lemma [REF].', '0811.4281-2-73-0': 'We now estimate the function [MATH] using another function [MATH] that satisfies an equation that is similar to that of Lemma [REF].', '0811.4281-2-73-1': '[EQUATION]', '0811.4281-2-73-2': 'It also has a unique solution.', '0811.4281-2-73-3': 'Since [MATH], we can check inductively that [EQUATION] for any sets [MATH] (with [MATH]).', '0811.4281-2-73-4': 'Now the function [MATH] can be written explicitly [CITATION].', '0811.4281-2-73-5': 'Let [MATH] be the set of forests on [MATH] rooted in [MATH].', '0811.4281-2-73-6': 'That is, a graph [MATH] is a forest such that each tree contains exactly one element of [MATH].', '0811.4281-2-74-0': 'The solution of Eq. [REF] is [EQUATION].', '0811.4281-2-75-0': 'Since the solution to Eq. [REF] is unique, it is enough to check that the Ansatz of the lemma satisfies the equation.', '0811.4281-2-75-1': 'First, let us observe that both sides are multiplied by [MATH].', '0811.4281-2-75-2': 'Thus it is enough to consider the case [MATH].', '0811.4281-2-76-0': 'The sum over graphs in [MATH] can be realized by first summing over the set [MATH] of indices (necessarily in [MATH]) that are connected to [MATH]; then over sets of trees in [MATH], and over connections to [MATH].', '0811.4281-2-76-1': 'Explicitly, [EQUATION]', '0811.4281-2-76-2': 'This equation is precisely [REF].', '0811.4281-2-77-0': '[Proof of Proposition [REF]] When [MATH] has a single element, the function [MATH] is equal to [EQUATION]', '0811.4281-2-77-1': 'This is the left side of Proposition [REF].', '0811.4281-2-77-2': 'We have [MATH], the set of trees with [MATH] vertices.', '0811.4281-2-77-3': 'Thus [MATH] is equal to the right side of Proposition [REF], and the proof follows from Eq. [REF].', '0811.4281-2-78-0': '# Proofs of the theorems', '0811.4281-2-79-0': 'In this section we prove the theorems of Section [REF].', '0811.4281-2-79-1': 'We consider only the case where Assumption [REF] holds true - the case with Assumption [REF] is entirely the same, one only needs to replace all [MATH] with [MATH] and all [MATH] with [MATH].', '0811.4281-2-79-2': 'The proofs are based on the following tree estimate, which is a direct consequence of Proposition [REF]: for almost all [MATH], [EQUATION] [Proof of Theorem [REF]] We start by proving the bound [REF].', '0811.4281-2-79-3': 'Let us introduce [EQUATION] (The term [MATH] is equal to [MATH] by definition.)', '0811.4281-2-79-4': 'We show by induction that [EQUATION] for any [MATH].', '0811.4281-2-79-5': 'Then [MATH] for almost all [MATH], and using [REF] we get [REF].', '0811.4281-2-80-0': 'The case [MATH] reduces to [MATH] and it is clear.', '0811.4281-2-80-1': 'The sum over trees with [MATH] vertices can be written as a sum over forests on [MATH], and a sum over edges between 1 and each tree of the forest.', '0811.4281-2-81-0': 'Here, [MATH] denote the set of trees with [MATH] as the set of vertices.', '0811.4281-2-81-1': 'If [MATH] the sum over [MATH] is one by definition.', '0811.4281-2-81-2': "The term after the sum over partitions depends on the cardinalities of the [MATH]'s, but not on the actual labeling.", '0811.4281-2-81-3': 'Also, each [MATH] gives the same contribution.', '0811.4281-2-82-0': 'We obtain an upper bound by releasing the constraint [MATH] to [MATH], [MATH].', '0811.4281-2-82-1': 'We then get [EQUATION]', '0811.4281-2-82-2': 'We have [MATH] by the induction hypothesis.', '0811.4281-2-82-3': 'Eq. [REF] follows from Assumption [REF].', '0811.4281-2-83-0': 'The rest of the proof is standard combinatorics.', '0811.4281-2-83-1': 'The partition function can be expanded so as to recognize the exponential of connected graphs.', '0811.4281-2-83-2': 'Namely, we start with [EQUATION]', '0811.4281-2-83-3': 'The graph [MATH] can be decomposed into [MATH] connected graphs whose sets of vertices form a partition of [MATH].', '0811.4281-2-83-4': 'Summing first over the number [MATH] of vertices for each set of the partition, we get [EQUATION]', '0811.4281-2-83-5': 'The triple sum is absolutely convergent thanks to the estimate [REF] that we have just established.', '0811.4281-2-83-6': 'One can then interchange the sums by the dominated convergence theorem.', '0811.4281-2-83-7': 'This removes the sum over [MATH], and this completes the proof of Theorem [REF].', '0811.4281-2-84-0': 'Next we prove Theorems [REF] and [REF] in reverse order, since we will use the convergence properties in the latter theorem to get the former.', '0811.4281-2-85-0': '[Proof of Theorem [REF]] From the definition [REF] and the tree estimate [REF], we have [EQUATION]', '0811.4281-2-85-1': 'The expression above involves a sum over trees of arbitrary size that connect 1 and 2.', '0811.4281-2-85-2': 'Any such tree decomposes into a line of edges that connect 1 and 2, and two trees rooted in 1 and 2.', '0811.4281-2-85-3': 'Taking into account the combinatorial factors, we obtain', '0811.4281-2-86-0': '[Proof of Theorem [REF]] It is actually similar to the end of the proof of Theorem [REF].', '0811.4281-2-86-1': '[MATH] can be expanded as a sum over graphs, that can be decomposed into a connected graph that contains 1, and other connected graphs.', '0811.4281-2-86-2': 'Taking into account the combinatorial factors, the contribution of connected graphs containing 1 yields [MATH], and the contribution of the others yields the expression [REF] for [MATH].', '0811.4281-2-86-3': 'Thus [MATH].', '0811.4281-2-86-4': 'One step involved interchanging unbounded sums, which is justified because everything is absolutely convergent, thanks to [REF] and Theorem [REF].', '0811.4281-2-87-0': 'In the graph expansion for [MATH], the terms where 1 and 2 belong to the same connected graph yield [MATH], and the terms where 1 and 2 belong to different connected graphs yield [MATH].', '0811.4281-2-87-1': 'The detailed argument is the same as above.', '0811.4281-2-87-2': 'We then obtained the desired expression.'} | [['0811.4281-1-66-0', '0811.4281-2-67-0'], ['0811.4281-1-66-1', '0811.4281-2-67-1'], ['0811.4281-1-15-0', '0811.4281-2-15-0'], ['0811.4281-1-32-0', '0811.4281-2-32-0'], ['0811.4281-1-32-1', '0811.4281-2-32-1'], ['0811.4281-1-9-0', '0811.4281-2-9-0'], ['0811.4281-1-18-0', '0811.4281-2-18-0'], ['0811.4281-1-29-0', '0811.4281-2-29-0'], ['0811.4281-1-29-1', '0811.4281-2-29-1'], ['0811.4281-1-29-2', '0811.4281-2-29-2'], ['0811.4281-1-44-0', '0811.4281-2-44-0'], ['0811.4281-1-44-1', '0811.4281-2-44-1'], ['0811.4281-1-44-3', '0811.4281-2-44-3'], ['0811.4281-1-44-4', '0811.4281-2-44-4'], ['0811.4281-1-44-5', '0811.4281-2-44-5'], ['0811.4281-1-44-6', '0811.4281-2-44-6'], ['0811.4281-1-44-7', '0811.4281-2-44-7'], ['0811.4281-1-44-8', '0811.4281-2-44-8'], ['0811.4281-1-44-9', '0811.4281-2-44-9'], ['0811.4281-1-44-10', '0811.4281-2-44-10'], ['0811.4281-1-44-11', '0811.4281-2-44-11'], ['0811.4281-1-44-12', '0811.4281-2-44-12'], ['0811.4281-1-47-0', '0811.4281-2-47-0'], ['0811.4281-1-47-1', '0811.4281-2-47-1'], ['0811.4281-1-47-2', '0811.4281-2-47-2'], ['0811.4281-1-69-0', '0811.4281-2-70-0'], ['0811.4281-1-71-0', '0811.4281-2-72-0'], ['0811.4281-1-71-1', '0811.4281-2-72-1'], ['0811.4281-1-71-3', '0811.4281-2-72-3'], ['0811.4281-1-71-4', '0811.4281-2-72-4'], ['0811.4281-1-63-0', '0811.4281-2-64-0'], ['0811.4281-1-63-1', '0811.4281-2-64-1'], ['0811.4281-1-63-2', '0811.4281-2-64-2'], ['0811.4281-1-63-3', '0811.4281-2-64-3'], ['0811.4281-1-63-4', '0811.4281-2-64-4'], ['0811.4281-1-63-5', '0811.4281-2-64-5'], ['0811.4281-1-22-0', '0811.4281-2-22-0'], ['0811.4281-1-22-1', '0811.4281-2-22-1'], ['0811.4281-1-22-2', '0811.4281-2-22-2'], ['0811.4281-1-22-3', '0811.4281-2-22-3'], ['0811.4281-1-22-4', '0811.4281-2-22-4'], ['0811.4281-1-28-0', '0811.4281-2-28-0'], ['0811.4281-1-82-0', '0811.4281-2-83-0'], ['0811.4281-1-82-1', '0811.4281-2-83-1'], ['0811.4281-1-82-2', '0811.4281-2-83-2'], ['0811.4281-1-82-3', '0811.4281-2-83-3'], ['0811.4281-1-82-4', '0811.4281-2-83-4'], ['0811.4281-1-82-5', '0811.4281-2-83-5'], ['0811.4281-1-82-6', '0811.4281-2-83-6'], ['0811.4281-1-82-7', '0811.4281-2-83-7'], ['0811.4281-1-86-0', '0811.4281-2-87-0'], ['0811.4281-1-86-1', '0811.4281-2-87-1'], ['0811.4281-1-86-2', '0811.4281-2-87-2'], ['0811.4281-1-51-0', '0811.4281-2-51-0'], ['0811.4281-1-51-1', '0811.4281-2-51-1'], ['0811.4281-1-51-2', '0811.4281-2-51-2'], ['0811.4281-1-85-0', '0811.4281-2-86-0'], ['0811.4281-1-85-1', '0811.4281-2-86-1'], ['0811.4281-1-85-2', 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'0811.4281-4-7-1'], ['0811.4281-3-25-0', '0811.4281-4-25-0'], ['0811.4281-3-25-1', '0811.4281-4-25-1'], ['0811.4281-3-25-2', '0811.4281-4-25-2'], ['0811.4281-3-25-3', '0811.4281-4-25-3'], ['0811.4281-3-25-4', '0811.4281-4-25-4'], ['0811.4281-3-20-0', '0811.4281-4-20-0'], ['0811.4281-3-20-1', '0811.4281-4-20-1'], ['0811.4281-1-52-0', '0811.4281-2-52-0'], ['0811.4281-1-52-1', '0811.4281-2-52-1'], ['0811.4281-1-52-2', '0811.4281-2-52-2'], ['0811.4281-1-52-3', '0811.4281-2-52-3'], ['0811.4281-1-52-4', '0811.4281-2-52-4'], ['0811.4281-1-52-5', '0811.4281-2-52-5'], ['0811.4281-1-52-6', '0811.4281-2-52-6'], ['0811.4281-1-52-8', '0811.4281-2-52-8'], ['0811.4281-1-52-10', '0811.4281-2-54-1'], ['0811.4281-1-52-11', '0811.4281-2-54-2'], ['0811.4281-1-52-12', '0811.4281-2-54-3'], ['0811.4281-1-52-13', '0811.4281-2-52-9'], ['0811.4281-2-79-0', '0811.4281-3-84-0'], ['0811.4281-2-79-1', '0811.4281-3-84-1'], ['0811.4281-2-79-2', '0811.4281-3-84-2'], ['0811.4281-2-79-3', '0811.4281-3-84-3'], ['0811.4281-2-79-4', '0811.4281-3-84-4'], ['0811.4281-2-79-5', '0811.4281-3-84-5'], ['0811.4281-2-59-1', '0811.4281-3-59-2'], ['0811.4281-2-59-2', '0811.4281-3-59-3'], ['0811.4281-2-59-3', '0811.4281-3-59-4'], ['0811.4281-2-59-4', '0811.4281-3-59-5'], ['0811.4281-2-59-7', '0811.4281-3-60-0'], ['0811.4281-2-67-0', '0811.4281-3-68-0'], ['0811.4281-2-67-5', '0811.4281-3-68-2'], ['0811.4281-2-67-6', '0811.4281-3-68-3'], ['0811.4281-1-58-0', '0811.4281-2-59-0'], ['0811.4281-1-58-7', '0811.4281-2-59-5'], ['0811.4281-1-59-0', '0811.4281-2-59-7'], ['0811.4281-2-38-0', '0811.4281-3-38-0'], ['0811.4281-2-38-1', '0811.4281-3-38-1'], ['0811.4281-2-38-2', '0811.4281-3-38-2'], ['0811.4281-2-39-0', '0811.4281-3-38-4'], ['0811.4281-2-39-1', '0811.4281-3-38-5']] | [['0811.4281-1-66-2', '0811.4281-2-67-2'], ['0811.4281-1-9-1', '0811.4281-2-9-1'], ['0811.4281-1-57-1', '0811.4281-2-58-1'], ['0811.4281-1-61-4', '0811.4281-2-61-2'], ['0811.4281-1-7-1', '0811.4281-2-7-1'], ['0811.4281-1-0-0', '0811.4281-2-0-0'], ['0811.4281-1-54-3', '0811.4281-2-55-3'], ['0811.4281-2-56-0', '0811.4281-3-55-0'], ['0811.4281-2-71-1', '0811.4281-3-72-1'], ['0811.4281-2-52-4', '0811.4281-3-51-4'], ['0811.4281-2-52-5', '0811.4281-3-51-5'], ['0811.4281-2-9-1', '0811.4281-3-9-1'], ['0811.4281-2-36-0', '0811.4281-3-36-0'], ['0811.4281-2-77-0', '0811.4281-3-78-0'], ['0811.4281-2-77-3', '0811.4281-3-78-3'], ['0811.4281-3-86-0', '0811.4281-4-86-0'], ['0811.4281-3-42-1', '0811.4281-4-42-1'], ['0811.4281-1-52-9', '0811.4281-2-54-0'], ['0811.4281-2-38-3', '0811.4281-3-38-3']] | [] | [['0811.4281-1-66-4', '0811.4281-2-67-4'], ['0811.4281-1-61-6', '0811.4281-2-61-4'], ['0811.4281-1-26-8', '0811.4281-2-26-8'], ['0811.4281-1-72-4', '0811.4281-2-73-4'], ['0811.4281-1-35-6', '0811.4281-2-35-6'], ['0811.4281-2-35-6', '0811.4281-3-35-6'], ['0811.4281-2-64-2', '0811.4281-3-65-2'], ['0811.4281-2-8-1', '0811.4281-3-8-1'], ['0811.4281-2-65-3', '0811.4281-3-66-3'], ['0811.4281-2-52-9', '0811.4281-3-51-9'], ['0811.4281-2-85-2', '0811.4281-3-90-2'], ['0811.4281-2-62-1', '0811.4281-3-63-1'], ['0811.4281-2-36-2', '0811.4281-3-36-2'], ['0811.4281-2-77-1', '0811.4281-3-78-1'], ['0811.4281-2-59-0', '0811.4281-3-59-0'], ['0811.4281-2-67-1', '0811.4281-3-68-4'], ['0811.4281-2-67-2', '0811.4281-3-79-1'], ['0811.4281-2-67-4', '0811.4281-3-68-1']] | [] | ['0811.4281-1-2-0', '0811.4281-1-3-0', '0811.4281-1-3-1', '0811.4281-1-3-2', '0811.4281-1-24-0', '0811.4281-1-27-0', '0811.4281-1-30-0', '0811.4281-1-31-0', '0811.4281-1-33-0', '0811.4281-1-44-2', '0811.4281-1-52-7', '0811.4281-1-55-5', '0811.4281-1-65-0', '0811.4281-1-67-4', '0811.4281-1-68-1', '0811.4281-1-69-1', '0811.4281-1-71-2', '0811.4281-1-72-1', '0811.4281-1-73-0', '0811.4281-1-75-1', '0811.4281-1-85-3', '0811.4281-2-2-0', '0811.4281-2-3-0', '0811.4281-2-3-1', '0811.4281-2-3-2', '0811.4281-2-17-0', '0811.4281-2-24-0', '0811.4281-2-27-0', '0811.4281-2-30-0', '0811.4281-2-31-0', '0811.4281-2-33-0', '0811.4281-2-44-2', '0811.4281-2-52-7', '0811.4281-2-56-5', '0811.4281-2-66-0', '0811.4281-2-68-4', '0811.4281-2-69-1', '0811.4281-2-70-1', '0811.4281-2-72-2', '0811.4281-2-73-1', '0811.4281-2-74-0', '0811.4281-2-76-1', '0811.4281-2-86-3', '0811.4281-3-2-0', '0811.4281-3-3-0', '0811.4281-3-3-1', '0811.4281-3-3-2', '0811.4281-3-17-0', '0811.4281-3-20-2', '0811.4281-3-24-0', '0811.4281-3-27-0', '0811.4281-3-30-0', '0811.4281-3-31-0', '0811.4281-3-33-0', '0811.4281-3-51-7', '0811.4281-3-55-5', '0811.4281-3-67-0', '0811.4281-3-69-5', '0811.4281-3-70-1', '0811.4281-3-71-1', '0811.4281-3-73-2', '0811.4281-3-74-1', '0811.4281-3-75-0', '0811.4281-3-77-1', '0811.4281-3-81-3', '0811.4281-3-81-4', '0811.4281-3-91-3', '0811.4281-4-2-0', '0811.4281-4-3-0', '0811.4281-4-3-1', '0811.4281-4-3-2', '0811.4281-4-17-0', '0811.4281-4-20-2', '0811.4281-4-24-0', '0811.4281-4-27-0', '0811.4281-4-30-0', '0811.4281-4-31-0', '0811.4281-4-33-0', '0811.4281-4-51-7', '0811.4281-4-55-5', '0811.4281-4-67-0', '0811.4281-4-69-5', '0811.4281-4-70-1', '0811.4281-4-71-1', '0811.4281-4-73-2', '0811.4281-4-74-1', '0811.4281-4-75-0', '0811.4281-4-77-1', '0811.4281-4-81-3', '0811.4281-4-81-4', '0811.4281-4-91-3'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0811.4281 | {'0811.4281-3-0-0': 'cluster expansion with applications]Abstract cluster expansion with applications to statistical mechanical systems', '0811.4281-3-1-0': '# Abstract We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence.', '0811.4281-3-1-1': 'We apply the results to systems of classical and quantum particles with stable interactions.', '0811.4281-3-2-0': 'Keywords: cluster expansion, polymer model, stable interaction, quantum gas.', '0811.4281-3-3-0': '2000 Math.', '0811.4281-3-3-1': 'Subj.', '0811.4281-3-3-2': 'Class.: 82B05, 82B10, 82B20, 82B21, 82B26', '0811.4281-3-4-0': '# Introduction', '0811.4281-3-5-0': "The method of cluster expansions was introduced in the 1930's in statistical mechanics in order to study gases of classical interacting particles.", '0811.4281-3-5-1': 'Its main achievement, from the point of view of physics, may be the derivation of the van der Waals equation of state for weakly interacting systems.', '0811.4281-3-5-2': "The method was made rigorous by mathematical-physicists in the 1960's, see [CITATION] and references therein.", '0811.4281-3-6-0': 'The method split afterwards.', '0811.4281-3-6-1': 'One branch involves continuous systems, with applications to classical systems [CITATION], quantum systems [CITATION], or quantum field theory [CITATION].', '0811.4281-3-6-2': 'The other branch involves polymer systems, i.e. discrete systems with additional internal structure [CITATION].', '0811.4281-3-6-3': 'An important step forward was the article of Kotecky and Preiss with its simplified setting and its elegant condition for the convergence of the cluster expansion [CITATION].', '0811.4281-3-7-0': 'The methods for proving the convergence are diverse.', '0811.4281-3-7-1': 'Let us mention the study of Kirkwood-Salsburg equations that involves correlation functions, see [CITATION] and references therein; the algebraic approach of Ruelle [CITATION]; combinatorial approaches using tree identities [CITATION]; inductions for discrete systems [CITATION].', '0811.4281-3-8-0': 'Important and useful surveys were written by Brydges [CITATION], Pfister [CITATION], Abdesselam and Rivasseau [CITATION].', '0811.4281-3-8-1': 'Recent articles have been devoted to combinatorial aspects [CITATION] and to weakening the assumptions [CITATION].', '0811.4281-3-9-0': 'The method of cluster expansions applies when the objects do not interact much; this the case when they are far apart (low density), or when interactions are weak.', '0811.4281-3-9-1': 'An extension of the criterion of [CITATION] that takes into account these two aspects was proposed in [CITATION]; it applies to both discrete and continuous systems.', '0811.4281-3-10-0': 'All abstract (i.e. general) approaches involve restrictions that correspond to repulsive interactions.', '0811.4281-3-10-1': 'Yet the old results for classical and quantum systems only assume stable interactions, that may include an attractive part.', '0811.4281-3-10-2': 'The aim of the present article is to propose a general approach that applies to discrete and continuous systems with repulsive or stable interactions.', '0811.4281-3-10-3': 'Our proof is split into several independent steps and this helps clarify the situation.', '0811.4281-3-11-0': 'The setting and the results are presented in Section [REF].', '0811.4281-3-11-1': 'We consider applications to classical systems of particles in Section [REF], to polymer systems in Section [REF], and to the quantum gas in Section [REF].', '0811.4281-3-11-2': 'A fundamental tree estimate is derived in Section [REF], and the theorems of Section [REF] are proved in Section [REF].', '0811.4281-3-12-0': '# Cluster expansions', '0811.4281-3-13-0': 'We consider a set [MATH] whose elements may represent widely different objects - in the three applications considered in this article, an element [MATH] represents (i) the position of a classical particle, (ii) a polymer, i.e. a connected set of [MATH], and (iii) a closed Brownian bridge.', '0811.4281-3-13-1': 'For the general abstract theory, we assume the structure of a measure space, [MATH], with [MATH] a complex measure.', '0811.4281-3-13-2': 'We denote [MATH] the total variation (absolute value) of [MATH].', '0811.4281-3-13-3': 'Let [MATH] and [MATH] be complex measurable symmetric functions on [MATH], that are related by the equation [EQUATION]', '0811.4281-3-13-4': 'We allow the real part of [MATH] to take the value [MATH], in which case [MATH].', '0811.4281-3-13-5': 'In typical applications [MATH] represents the interactions between [MATH] and [MATH], and the value [MATH] corresponds to a hard-core repulsion.', '0811.4281-3-13-6': 'We define the "partition function" by [EQUATION] or, equivalently, [EQUATION]', '0811.4281-3-13-7': 'The term [MATH] of the sums is understood to be 1.', '0811.4281-3-14-0': 'The main goal of cluster expansions is to express the partition function as the exponential of a convergent series of "cluster terms".', '0811.4281-3-14-1': 'The main difficulty is to prove the convergence.', '0811.4281-3-14-2': 'We first assume that the potential [MATH] is stable.', '0811.4281-3-15-0': 'There exists a nonnegative function [MATH] on [MATH] such that, for all [MATH] and almost all [MATH], [EQUATION].', '0811.4281-3-16-0': 'In other words, we assume the lower bound [EQUATION]', '0811.4281-3-16-1': 'When the function [MATH] is constant, this is the usual definition of stability.', '0811.4281-3-16-2': '"Almost all" means that, for given [MATH], the set of points where the condition fails has measure zero with respect to the product measure [MATH].', '0811.4281-3-16-3': 'If [MATH] is countable, the condition must be satisfied for all [MATH] such that [MATH].', '0811.4281-3-17-0': 'The second condition deals with the strength of interactions.', '0811.4281-3-18-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-3-19-0': 'In order to guess the correct form of [MATH], one should consider the left side of the equation above with [MATH].', '0811.4281-3-19-1': 'The integral may depend on [MATH]; a typical situation is that [MATH] is characterized by a length [MATH], which is a positive number, so that the left side is roughly proportional to [MATH].', '0811.4281-3-19-2': 'This suggests to try [MATH], and one can then optimize on the value of [MATH].', '0811.4281-3-20-0': 'We also consider an alternate criterion that involves [MATH] rather than [MATH].', '0811.4281-3-20-1': 'It is inspired by the recent work of Procacci [CITATION].', '0811.4281-3-20-2': 'Let [EQUATION]', '0811.4281-3-21-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-3-22-0': 'For positive [MATH] we can take [MATH]; and since [MATH], Assumption [REF] is always better than Assumption [REF].', '0811.4281-3-22-1': 'We actually conjecture that, together with Assumption [REF], a sufficient condition is [EQUATION]', '0811.4281-3-22-2': 'That is, it should be possible to combine the best of both assumptions.', '0811.4281-3-22-3': 'In this respect Assumption [REF] is optimal in the case of positive potentials, and Assumption [REF] is optimal in the case of hard core plus negative potentials.', '0811.4281-3-23-0': 'We denote by [MATH] the set of all graphs with [MATH] vertices (unoriented, no loops) and [MATH] the set of connected graphs with [MATH] vertices.', '0811.4281-3-23-1': 'We introduce the following combinatorial function on finite sequences [MATH] of elements of [MATH]: [EQUATION]', '0811.4281-3-23-2': 'The product is over edges of [MATH].', '0811.4281-3-24-0': '[Cluster expansions]', '0811.4281-3-25-0': 'Suppose that Assumptions [REF] and [REF], or [REF] and [REF], hold true.', '0811.4281-3-25-1': 'We also suppose that [MATH].', '0811.4281-3-25-2': 'Then we have [EQUATION]', '0811.4281-3-25-3': 'The term in the exponential converges absolutely.', '0811.4281-3-25-4': 'Furthermore, for almost all [MATH], we have the following estimate [EQUATION] (Under Assumption [REF], Eq. [REF] holds with [MATH] instead of [MATH].)', '0811.4281-3-26-0': 'Let us turn to correlation functions.', '0811.4281-3-26-1': 'We only consider one-point and two-point correlation functions since these are the most useful and expressions become more transparent.', '0811.4281-3-26-2': 'We refer to [CITATION] for more general functions.', '0811.4281-3-26-3': 'First, we define the unnormalized one-point correlation function by [EQUATION] (the term [MATH] is 1 by definition).', '0811.4281-3-26-4': 'And we define the unnormalized two-point correlation function by [EQUATION] (the term [MATH] is equal to [MATH]).', '0811.4281-3-26-5': 'Notice that [MATH] can be viewed as a regular partition function, given by Eq. [REF], but with the modified measure [MATH] instead of [MATH].', '0811.4281-3-26-6': 'The normalized correlation functions are [MATH] and [MATH].', '0811.4281-3-26-7': 'As is shown in Theorem [REF], they can be expressed using the "cluster functions" [EQUATION] and [EQUATION]', '0811.4281-3-26-8': 'Notice that [MATH] by [REF].', '0811.4281-3-27-0': '[Correlation functions]', '0811.4281-3-28-0': 'Under the same assumptions as in Theorem [REF], we have [EQUATION].', '0811.4281-3-29-0': 'In statistical mechanics, the relevant expression is the truncated two-point correlation function [EQUATION].', '0811.4281-3-29-1': 'When the cluster expansion converges, it is equal to [MATH] by the theorem above.', '0811.4281-3-29-2': 'This function usually provides an order parameter for phase transitions and it is useful to estimate its decay properties.', '0811.4281-3-30-0': '[Decay of correlations]', '0811.4281-3-31-0': 'If Assumptions [REF] and [REF] hold true, we have for almost all [MATH], [EQUATION] (with [MATH] and [MATH]).', '0811.4281-3-31-1': 'If Assumptions [REF] and [REF] hold true, we have the same bound but with [MATH] instead of [MATH], and [MATH] instead of [MATH].', '0811.4281-3-32-0': 'In many applications the functions [MATH] and [MATH] depend on the difference [MATH] (this assumes that [MATH] has additional structure, namely that of a group).', '0811.4281-3-32-1': 'The estimates for [MATH] are given by convolutions.', '0811.4281-3-33-0': 'The theorems of this section are proved in Section [REF].', '0811.4281-3-34-0': '# Classical gas', '0811.4281-3-35-0': 'We consider a gas of point particles that interact with a pair potential.', '0811.4281-3-35-1': 'We work in the grand-canonical ensemble where the parameters are the fugacity [MATH] and the inverse temperature [MATH] (both are real and positive numbers).', '0811.4281-3-35-2': 'The set [MATH] is an open bounded subset of [MATH] and [MATH] with [MATH] the Lebesgue measure.', '0811.4281-3-35-3': 'We actually write [MATH] so as to have more traditional notation.', '0811.4281-3-35-4': 'The interaction is given by a function [MATH] which we take to be piecewise continuous; [MATH].', '0811.4281-3-35-5': 'We suppose that [MATH] is stable, i.e. that there exists a constant [MATH] such that for any [MATH] and any [MATH]: [EQUATION]', '0811.4281-3-35-6': 'Our Assumption [REF] holds with [MATH].', '0811.4281-3-35-7': 'The system is translation invariant so all [MATH] are equivalent.', '0811.4281-3-35-8': 'The function of Assumptions [REF] and [REF] can then be taken to be a constant, [MATH].', '0811.4281-3-35-9': 'We seek a condition that does not depend on the size of the system.', '0811.4281-3-35-10': 'Then integrals over [MATH] are on [MATH] instead of [MATH].', '0811.4281-3-35-11': 'By translation invariance we can take [MATH].', '0811.4281-3-36-0': 'Assumption [REF] gives the condition [EQUATION]', '0811.4281-3-36-1': 'We obviously choose the constant [MATH] that maximizes the right side, which is [MATH].', '0811.4281-3-36-2': 'This condition is the one in [CITATION].', '0811.4281-3-36-3': 'Let us now assume that [MATH] consists of a hard core of radius [MATH] and that it is otherwise integrable.', '0811.4281-3-36-4': 'Again with [MATH], Assumption [REF] gives the condition [EQUATION]', '0811.4281-3-36-5': 'Here, [MATH] is the volume of the ball in [MATH] dimensions.', '0811.4281-3-36-6': 'This condition is often better than [REF].', '0811.4281-3-36-7': 'Without hard core it is the one in [CITATION].', '0811.4281-3-36-8': 'The domains of parameters where these conditions hold correspond to low fugacities and high temperatures.', '0811.4281-3-37-0': 'The thermodynamic pressure is defined as the infinite volume limit of [EQUATION]', '0811.4281-3-37-1': 'Using Theorem [REF], we have [EQUATION]', '0811.4281-3-37-2': 'Consider now any sequence of increasing domains [MATH] such that [MATH].', '0811.4281-3-37-3': 'Thanks to the estimate [REF], and using translation invariance, we get [EQUATION] (The term with [MATH] is equal to [MATH].)', '0811.4281-3-37-4': 'This expression for the infinite volume pressure [MATH] should be viewed as a convergent series of analytic functions of [MATH].', '0811.4281-3-37-5': 'Then [MATH] is analytic in [MATH] and [MATH] by Vitali theorem and no phase transition takes place in the domain of parameters where the cluster expansion is convergent.', '0811.4281-3-38-0': 'The truncated two-point correlation function [MATH] is given by [MATH].', '0811.4281-3-38-1': 'We consider only the case of Assumption [REF] but a similar claim can be obtained with Assumption [REF].', '0811.4281-3-38-2': 'Let [MATH] be a function that satisfies the triangle inequality.', '0811.4281-3-38-3': 'The estimate of Theorem [REF] yields [EQUATION] (with [MATH]).', '0811.4281-3-38-4': 'Recall that [MATH], and let [EQUATION]', '0811.4281-3-38-5': 'Then we get [EQUATION]', '0811.4281-3-38-6': 'If the inequality [REF] is strict, one can usually find a function [MATH] that satisfies the triangle inequality and such that [MATH]; the truncated two-point correlation function then decays faster than [MATH].', '0811.4281-3-39-0': '# Polymer systems', '0811.4281-3-40-0': 'Polymer systems are discrete, which is technically simpler, but they also have internal structure.', '0811.4281-3-40-1': 'The first application of cluster expansions to polymer systems is due to Gruber and Kunz [CITATION].', '0811.4281-3-40-2': 'Among the many articles devoted to this subject, let us mention [CITATION].', '0811.4281-3-40-3': 'The main goal of this section is to illustrate our setting; we therefore restrict ourselves to a specific model of polymers with both repulsive and attractive interactions.', '0811.4281-3-41-0': 'Our space [MATH] is the set of all finite connected subsets of [MATH].', '0811.4281-3-41-1': 'The measure [MATH] is the counting measure multiplied by the activity [MATH] (a function [MATH]).', '0811.4281-3-41-2': 'We choose [MATH] with [MATH].', '0811.4281-3-41-3': 'The interaction is hard core when polymers overlap and it is attractive when they touch: [EQUATION]', '0811.4281-3-41-4': 'Here, [MATH] is the number of "contacts" between [MATH] and [MATH], i.e. the number of bonds between sites of [MATH] and [MATH]; [MATH] is a parameter.', '0811.4281-3-41-5': 'The interaction is zero when the distance between polymers is greater than 1.', '0811.4281-3-42-0': 'The stability condition can be written [EQUATION]', '0811.4281-3-42-1': 'Only disjoint polymers need to be considered, the right side is infinite otherwise.', '0811.4281-3-42-2': 'The sum over [MATH] is always larger than [MATH] times the number of bonds connecting [MATH] with its exterior.', '0811.4281-3-42-3': 'Thus we can take [MATH].', '0811.4281-3-43-0': 'The function [MATH] in Assumption [REF] grows like [MATH], so it is natural to choose [MATH] for some constant [MATH].', '0811.4281-3-43-1': 'A sufficient condition is that [EQUATION]', '0811.4281-3-43-2': 'We can bound [MATH] by [MATH].', '0811.4281-3-43-3': 'Summing over the sites of [MATH], and requiring that [MATH] contains the given site or comes at distance 1, we get [EQUATION]', '0811.4281-3-43-4': 'We used the fact that the activity is translation invariant.', '0811.4281-3-43-5': 'If [MATH] is a connected set, there exists a closed walk with nearest neighbor jumps whose support is [MATH], and whose length is at most [MATH].', '0811.4281-3-43-6': 'This can be seen by induction: knowing the walk for [MATH], it is easy to construct one for [MATH].', '0811.4281-3-43-7': 'The number of connected sets of cardinality [MATH] that contain the origin is therefore smaller than the number of walks of length [MATH] starting at the origin, which is equal to [MATH].', '0811.4281-3-43-8': 'Then it suffices that [EQUATION]', '0811.4281-3-43-9': 'This is equivalent to [EQUATION]', '0811.4281-3-43-10': 'Assumption [REF] holds for any [MATH].', '0811.4281-3-43-11': 'Using [MATH] and optimizing on [MATH], we find the sufficient condition [EQUATION] with [MATH].', '0811.4281-3-44-0': 'We have just established the existence of a low density phase provided the activity is small enough.', '0811.4281-3-44-1': 'The condition depends on the contact parameter [MATH].', '0811.4281-3-44-2': 'For large [MATH] one should expect interesting phases with many contacts between the polymers.', '0811.4281-3-45-0': '# Quantum gas', '0811.4281-3-46-0': 'We follow a course that is similar to Ginibre [CITATION], using the Feynman-Kac formula so as to get a gas of winding Brownian loops.', '0811.4281-3-46-1': 'We get comparable results, with a larger domain of convergence in the case of integrable potentials.', '0811.4281-3-46-2': 'Winding Brownian loops are kind of continuous polymers; they combine the difficulties of both cases above - the continuous nature and the internal structure.', '0811.4281-3-47-0': '## Feynman-Kac representation', '0811.4281-3-48-0': 'The state space for [MATH] fermions (resp. bosons) in a domain [MATH] is the Hilbert space [MATH] (resp. [MATH]) of square-integrable complex functions that are antisymmetric (resp. symmetric) with respect to their arguments.', '0811.4281-3-48-1': 'The Hamiltonian is [EQUATION] with [MATH] the Laplacian for the [MATH]-th variable and [MATH] a multiplication operator.', '0811.4281-3-48-2': 'As in the classical case, we consider the grand-canonical ensemble whose parameters are the fugacity [MATH] and the inverse temperature [MATH].', '0811.4281-3-48-3': 'The partition function is [EQUATION]', '0811.4281-3-48-4': 'We need to cast the partition function in the form [REF], which can be done using the Feynman-Kac representation.', '0811.4281-3-49-0': 'Namely, we have [EQUATION] [MATH] is the symmetric group of [MATH] elements; [MATH] is equal to the signature of the permutation [MATH] for fermions, [MATH] for bosons; [MATH] is the Wiener measure for the Brownian bridge from [MATH] to [MATH] in time [MATH] - the normalization is chosen so that [EQUATION] [MATH][MATH] is one if [MATH] for all [MATH], it is zero otherwise.', '0811.4281-3-49-1': 'An introduction to the Feynman-Kac formula in this context can be found in the survey of Ginibre [CITATION].', '0811.4281-3-50-0': 'The right side of Eq. [REF] is well defined for a large class of functions [MATH], that includes all piecewise continuous functions.', '0811.4281-3-50-1': 'Thus we take [REF] as the definition for [MATH].', '0811.4281-3-50-2': 'Under additional assumptions on [MATH], [REF] is equal to [REF] with Hamiltonian [REF] and with Dirichlet boundary conditions.', '0811.4281-3-51-0': 'We now rewrite the grand-canonical partition function in terms of winding loops.', '0811.4281-3-51-1': 'Let [MATH] be the set of continuous paths [MATH] that are closed.', '0811.4281-3-51-2': 'Its elements are denoted [MATH], with [MATH] the starting point, [MATH] the winding number, and [MATH] the path; we have [MATH].', '0811.4281-3-51-3': 'We consider the measure [MATH] given by [EQUATION]', '0811.4281-3-51-4': 'Here, [MATH] is a self-interaction term that is defined below in [REF]; [MATH] for fermions and [MATH] for bosons.', '0811.4281-3-51-5': 'Let [MATH]; the measure [MATH] naturally extends to a measure on [MATH].', '0811.4281-3-51-6': 'The grand-canonical partition function can then be written as [EQUATION]', '0811.4281-3-51-7': 'Let [MATH] and [MATH].', '0811.4281-3-51-8': 'The self-interaction [MATH] and the 2-loop interaction [MATH] are given by [EQUATION]', '0811.4281-3-51-9': 'We now treat separately the case of integrable potentials and the case of more general potentials.', '0811.4281-3-52-0': '## Stable integrable potentials', '0811.4281-3-53-0': 'We suppose that [MATH] is stable with constant [MATH], i.e. it satisfies Eq. [REF].', '0811.4281-3-53-1': 'For given loops [MATH], stability implies that [EQUATION]', '0811.4281-3-53-2': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-3-53-3': 'Notice that [MATH], again by the stability of [MATH].', '0811.4281-3-54-0': 'We use Assumption [REF] and we choose a function [MATH] with a constant [MATH] to be determined later.', '0811.4281-3-54-1': 'Explicitly, the assumption is that for any [MATH] [EQUATION]', '0811.4281-3-54-2': 'We have lifted the restriction that [MATH] because we want a condition that does not depend on [MATH].', '0811.4281-3-54-3': 'Eq. [REF] is easier to handle than its appearance suggests.', '0811.4281-3-54-4': 'Notice that the term in the second exponential is just [MATH].', '0811.4281-3-54-5': 'Using [MATH], and the definition [REF] of [MATH], it is enough that [EQUATION]', '0811.4281-3-55-0': 'We can immediately integrate over [MATH], which yields [MATH].', '0811.4281-3-55-1': 'The Wiener integral then gives [MATH] and we get the equivalent condition [EQUATION]', '0811.4281-3-55-2': 'For any [MATH], the inequality holds for [MATH] small enough.', '0811.4281-3-55-3': 'Notice that [MATH] in any case.', '0811.4281-3-55-4': 'One can get a more explicit condition for [MATH] by choosing [MATH] such that [MATH] = 1.', '0811.4281-3-55-5': 'This yields [EQUATION]', '0811.4281-3-55-6': 'Here, [MATH] is the Riemann zeta function.', '0811.4281-3-56-0': 'When [MATH] and when the potential is repulsive, one can rewrite [REF] in a more transparent way.', '0811.4281-3-56-1': 'Let [MATH] denote the Born approximation to the scattering length.', '0811.4281-3-56-2': 'The condition is then [EQUATION]', '0811.4281-3-56-3': 'The critical fugacity is expected to be greater than 1.', '0811.4281-3-56-4': 'The present result helps nonetheless to obtain a range of densities where the pressure is analytic.', '0811.4281-3-56-5': "In the bosonic case it compares well with physicists' expectations [CITATION].", '0811.4281-3-57-0': '## Stable potentials with hard core', '0811.4281-3-58-0': 'The presence of a hard core makes the situation more complicated; we only sketch the argument in this section without trying to get explicit bounds.', '0811.4281-3-58-1': 'Our aim is to show that, using Theorem [REF], the problem of convergence of the cluster expansion reduces to estimates of Wiener sausages.', '0811.4281-3-59-0': 'We consider an interaction [MATH].', '0811.4281-3-59-1': 'We assume that [MATH] is a repulsive potential of radius [MATH], with a hard core of radius [MATH], that [MATH] for [MATH], and that [MATH] is integrable otherwise.', '0811.4281-3-59-2': 'We suppose that the stability condition takes the slightly stronger form [EQUATION] for any [MATH] such that [MATH].', '0811.4281-3-59-3': 'For potentials with a hard core this is equivalent to the property [REF], possibly with a different constant [MATH].', '0811.4281-3-59-4': 'Then one has [CITATION] [EQUATION]', '0811.4281-3-59-5': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-3-59-6': 'Notice that the stability condition also holds with [MATH] given in [REF] (and with a better constant [MATH]).', '0811.4281-3-59-7': 'The advantage of [REF] is the factor [MATH] in front of [MATH].', '0811.4281-3-59-8': 'Then [MATH] involves a term that cancels the self-interactions of [MATH].', '0811.4281-3-60-0': 'Given a loop [MATH], let [MATH] be the Wiener sausage generated by a ball of radius [MATH] when its center moves along the trajectory [MATH]: [EQUATION]', '0811.4281-3-61-0': 'We denote the volume of a Wiener sausage [MATH] by [MATH].', '0811.4281-3-61-1': 'One can check that [EQUATION] with [MATH] the volume of the unit ball (see Appendix 2 in [CITATION]).', '0811.4281-3-62-0': 'We choose [MATH] in Assumption [REF].', '0811.4281-3-62-1': 'Then a sufficient condition is that for any [MATH], [EQUATION]', '0811.4281-3-62-2': 'We consider separately the cases where [MATH] belongs or not to [MATH].', '0811.4281-3-62-3': 'First, [EQUATION] which we bound using [REF].', '0811.4281-3-62-4': 'Second, using [MATH], [EQUATION]', '0811.4281-3-62-5': 'We certainly get [REF] if we have the two inequalities [EQUATION]', '0811.4281-3-62-6': 'One can estimate the integrals of Wiener sausages, see [CITATION], so that both conditions hold if [MATH] is small enough.', '0811.4281-3-63-0': 'Now that the cluster expansion is known to converge, it is possible to write the pressure as an absolutely convergent series of analytic functions in [MATH] and [MATH].', '0811.4281-3-63-1': 'It is also possible to study the decay of correlation functions.', '0811.4281-3-63-2': "In the case of potentials that consist of hard core plus integrable part, one can apply Assumption 2' instead.", '0811.4281-3-63-3': 'This may give better results, especially if the integrable part is mostly attractive.', '0811.4281-3-64-0': '# Tree estimates', '0811.4281-3-65-0': 'In this section we obtain estimates of sums of connected graphs in terms of sums of trees.', '0811.4281-3-65-1': 'Our main result is Proposition [REF] below.', '0811.4281-3-65-2': 'Such estimates seem to have been introduced by Penrose [CITATION] and they have often been considered in the past [CITATION].', '0811.4281-3-65-3': 'We introduce a minimal setting that clarifies its role in the cluster expansion.', '0811.4281-3-65-4': 'Namely, we fix the polymers so we only deal with the numbers that represent their interactions, [MATH] or [MATH], and the stability function [MATH].', '0811.4281-3-65-5': 'Assumption [REF] is vital here, but Assumptions [REF] and [REF] are not used in this section.', '0811.4281-3-66-0': 'Let [MATH] denote the set of trees with [MATH] vertices.', '0811.4281-3-66-1': 'Let [MATH] be an integer, [MATH] be real nonnegative numbers, and [MATH], [MATH], be complex numbers.', '0811.4281-3-66-2': 'We assume that the following bound holds for any subset [MATH]: [EQUATION]', '0811.4281-3-66-3': 'Let [MATH] be such that [MATH]; let [MATH] if [MATH], and [MATH] otherwise.', '0811.4281-3-66-4': 'We state two distinct tree estimates, the first one involving [MATH] and the second one involving [MATH].', '0811.4281-3-66-5': 'These bounds will allow to prove the convergence under either Assumption [REF] or Assumption [REF].', '0811.4281-3-67-0': 'If [REF] holds true, we have the two bounds', '0811.4281-3-68-0': 'We actually conjecture that the following estimate holds under the same hypotheses: [EQUATION]', '0811.4281-3-68-1': "We prove Proposition [REF] (a) below using Ruelle's algebraic approach [CITATION].", '0811.4281-3-68-2': 'This method is usually combined with a Banach fixed point argument for correlation functions.', '0811.4281-3-68-3': 'However, we use it differently so as to get a tree estimate.', '0811.4281-3-68-4': 'Proposition [REF] (b) follows from a tree identity due to Brydges, Battle, and Federbush [CITATION], combined with an argument due to Procacci [CITATION]; its proof can be found at the end of this section.', '0811.4281-3-69-0': 'Let [MATH] be the set of complex functions on the power set [MATH].', '0811.4281-3-69-1': 'We introduce the following multiplication operation for [MATH]: [EQUATION]', '0811.4281-3-69-2': 'We use the standard conventions for sums and products, namely that the empty sum is zero and the empty product is one.', '0811.4281-3-69-3': 'Together with the addition, [MATH] is a commutative algebra with unit [MATH].', '0811.4281-3-69-4': 'It is possible to check that each [MATH] has a unique inverse, which we denote [MATH].', '0811.4281-3-69-5': 'We have [EQUATION]', '0811.4281-3-69-6': 'Let [MATH] be the subset of functions [MATH] such that [MATH] is an ideal of [MATH]).', '0811.4281-3-69-7': 'Notice that [MATH] for any [MATH] , when [MATH].', '0811.4281-3-69-8': 'We define the exponential mapping [MATH] by [EQUATION]', '0811.4281-3-69-9': 'Let [MATH] and [MATH] be the functions defined by [EQUATION]', '0811.4281-3-69-10': 'Here, [MATH] (resp. [MATH]) is the set of graphs (resp. connected graphs) on [MATH].', '0811.4281-3-69-11': 'We have the relation [EQUATION]', '0811.4281-3-69-12': 'We also introduce an operation that is reminiscent of differentiation: [EQUATION]', '0811.4281-3-69-13': 'One can check that [MATH].', '0811.4281-3-70-0': 'For disjoint [MATH], we define [EQUATION]', '0811.4281-3-70-1': 'Let [MATH].', '0811.4281-3-70-2': 'The assumption of Proposition [REF] implies that [EQUATION]', '0811.4281-3-70-3': 'Then there exists [MATH] such that [EQUATION]', '0811.4281-3-70-4': 'Such [MATH] is not unique in general but it does not matter.', '0811.4281-3-70-5': 'We consider a function [MATH] that assigns one of the indices [MATH] above to each nonempty subset [MATH].', '0811.4281-3-70-6': 'Notice that [MATH] for any subset [MATH].', '0811.4281-3-70-7': 'It is also useful to introduce the notation [MATH].', '0811.4281-3-71-0': 'The function [MATH] of Eq. [REF] is solution of the following equation.', '0811.4281-3-71-1': '[EQUATION].', '0811.4281-3-72-0': 'Since the equation gives [MATH] in terms of [MATH] with [MATH], it is well defined inductively and it has a unique solution.', '0811.4281-3-72-1': 'Notice that [MATH], and that [MATH] for any index [MATH].', '0811.4281-3-73-0': 'Recall the definition [REF] of [MATH].', '0811.4281-3-73-1': 'For disjoint [MATH] we have [EQUATION]', '0811.4281-3-73-2': 'Then [EQUATION]', '0811.4281-3-73-3': 'The last sum is equal to [MATH].', '0811.4281-3-73-4': 'One recognizes the equation of Lemma [REF].', '0811.4281-3-74-0': 'We now estimate the function [MATH] using another function [MATH] that satisfies an equation that is similar to that of Lemma [REF].', '0811.4281-3-74-1': '[EQUATION]', '0811.4281-3-74-2': 'It also has a unique solution.', '0811.4281-3-74-3': 'Since [MATH], we can check inductively that [EQUATION] for any sets [MATH] (with [MATH]).', '0811.4281-3-74-4': 'Now the function [MATH] can be written explicitly [CITATION].', '0811.4281-3-74-5': 'Let [MATH] be the set of forests on [MATH] rooted in [MATH].', '0811.4281-3-74-6': 'That is, a graph [MATH] is a forest such that each tree contains exactly one element of [MATH].', '0811.4281-3-75-0': 'The solution of Eq. [REF] is [EQUATION].', '0811.4281-3-76-0': 'Since the solution to Eq. [REF] is unique, it is enough to check that the Ansatz of the lemma satisfies the equation.', '0811.4281-3-76-1': 'First, let us observe that both sides are multiplied by [MATH].', '0811.4281-3-76-2': 'Thus it is enough to consider the case [MATH].', '0811.4281-3-77-0': 'The sum over graphs in [MATH] can be realized by first summing over the set [MATH] of indices (necessarily in [MATH]) that are connected to [MATH]; then over sets of trees in [MATH], and over connections to [MATH].', '0811.4281-3-77-1': 'Explicitly, [EQUATION]', '0811.4281-3-77-2': 'This equation is precisely [REF].', '0811.4281-3-78-0': '[Proof of Proposition [REF] (a)] When [MATH] has a single element, the function [MATH] is equal to [EQUATION]', '0811.4281-3-78-1': 'This is the left side of Proposition [REF] (a).', '0811.4281-3-78-2': 'We have [MATH], the set of trees with [MATH] vertices.', '0811.4281-3-78-3': 'Thus [MATH] is equal to the right side of Proposition [REF] (a), and the proof follows from Eq. [REF].', '0811.4281-3-79-0': 'We now turn to the proof of Proposition [REF] (b).', '0811.4281-3-79-1': 'Notice that in absence of "hard cores", i.e. when [MATH], our claim is just a reformulation of Corollary 3.2 (a) of [CITATION].', '0811.4281-3-79-2': 'The present proof follows [CITATION].', '0811.4281-3-80-0': '[Proof of Proposition [REF] (b)] Let [MATH] be the set of [MATH] such that [MATH], i.e. such that [MATH].', '0811.4281-3-80-1': 'We regularize those numbers by setting [EQUATION]', '0811.4281-3-80-2': 'This allows to use the tree identity of [CITATION]; we will eventually take [MATH] to infinity.', '0811.4281-3-80-3': 'The tree identity can be written [EQUATION]', '0811.4281-3-80-4': 'The full definition of the measure [MATH] can be found in [CITATION]; here we only mention its relevant properties.', '0811.4281-3-80-5': '[MATH] depends on the tree [MATH] and it is a probability measure supported on a subset [MATH].', '0811.4281-3-80-6': 'If [MATH] satisfies the stability condition, then [EQUATION] for all [MATH] and all [MATH].', '0811.4281-3-81-0': 'If [MATH] for any [MATH], we immediately get the claim by using the stability condition above, since [MATH].', '0811.4281-3-81-1': 'The extension to possibly infinite numbers can be obtained using a trick due to Procacci [CITATION].', '0811.4281-3-81-2': 'Let [MATH] be a fixed tree and [MATH].', '0811.4281-3-81-3': 'We introduce [EQUATION]', '0811.4281-3-81-4': 'Then [EQUATION]', '0811.4281-3-81-5': 'If [MATH] is large enough (depending on [MATH]), [MATH] is stable and the first term of the right side is bounded below by [MATH].', '0811.4281-3-82-0': 'We can insert this estimate into Eq. [REF].', '0811.4281-3-82-1': 'Letting [MATH] and then [MATH], we get Proposition [REF] (b).', '0811.4281-3-83-0': '# Proofs of the theorems', '0811.4281-3-84-0': 'In this section we prove the theorems of Section [REF].', '0811.4281-3-84-1': 'We consider only the case where Assumption [REF] holds true - the case with Assumption [REF] is entirely the same, one only needs to replace all [MATH] with [MATH] and all [MATH] with [MATH].', '0811.4281-3-84-2': 'The proofs are based on the following tree estimate, which is a direct consequence of Proposition [REF]: for almost all [MATH], [EQUATION] [Proof of Theorem [REF]] We start by proving the bound [REF].', '0811.4281-3-84-3': 'Let us introduce [EQUATION] (The term [MATH] is equal to [MATH] by definition.)', '0811.4281-3-84-4': 'We show by induction that [EQUATION] for any [MATH].', '0811.4281-3-84-5': 'Then [MATH] for almost all [MATH], and using [REF] we get [REF].', '0811.4281-3-85-0': 'The case [MATH] reduces to [MATH] and it is clear.', '0811.4281-3-85-1': 'The sum over trees with [MATH] vertices can be written as a sum over forests on [MATH], and a sum over edges between 1 and each tree of the forest.', '0811.4281-3-86-0': 'Here, [MATH] denote the set of trees with [MATH] as the set of vertices.', '0811.4281-3-86-1': 'If [MATH] the sum over [MATH] is one by definition.', '0811.4281-3-86-2': "The term after the sum over partitions depends on the cardinalities of the [MATH]'s, but not on the actual labeling.", '0811.4281-3-86-3': 'Also, each [MATH] gives the same contribution.', '0811.4281-3-87-0': 'We obtain an upper bound by releasing the constraint [MATH] to [MATH], [MATH].', '0811.4281-3-87-1': 'We then get [EQUATION]', '0811.4281-3-87-2': 'We have [MATH] by the induction hypothesis.', '0811.4281-3-87-3': 'Eq. [REF] follows from Assumption [REF].', '0811.4281-3-88-0': 'The rest of the proof is standard combinatorics.', '0811.4281-3-88-1': 'The partition function can be expanded so as to recognize the exponential of connected graphs.', '0811.4281-3-88-2': 'Namely, we start with [EQUATION]', '0811.4281-3-88-3': 'The graph [MATH] can be decomposed into [MATH] connected graphs whose sets of vertices form a partition of [MATH].', '0811.4281-3-88-4': 'Summing first over the number [MATH] of vertices for each set of the partition, we get [EQUATION]', '0811.4281-3-88-5': 'The triple sum is absolutely convergent thanks to the estimate [REF] that we have just established.', '0811.4281-3-88-6': 'One can then interchange the sums by the dominated convergence theorem.', '0811.4281-3-88-7': 'This removes the sum over [MATH], and this completes the proof of Theorem [REF].', '0811.4281-3-89-0': 'Next we prove Theorems [REF] and [REF] in reverse order, since we will use the convergence properties in the latter theorem to get the former.', '0811.4281-3-90-0': '[Proof of Theorem [REF]] From the definition [REF] and the tree estimate [REF], we have [EQUATION]', '0811.4281-3-90-1': 'The expression above involves a sum over trees of arbitrary size that connect 1 and 2.', '0811.4281-3-90-2': 'Any such tree decomposes into a line of [MATH] edges that connect 1 and 2 ([MATH]), and [MATH] trees rooted in the vertices of the connecting line.', '0811.4281-3-90-3': 'Taking into account the combinatorial factors, we obtain', '0811.4281-3-91-0': '[Proof of Theorem [REF]] It is actually similar to the end of the proof of Theorem [REF].', '0811.4281-3-91-1': '[MATH] can be expanded as a sum over graphs, that can be decomposed into a connected graph that contains 1, and other connected graphs.', '0811.4281-3-91-2': 'Taking into account the combinatorial factors, the contribution of connected graphs containing 1 yields [MATH], and the contribution of the others yields the expression [REF] for [MATH].', '0811.4281-3-91-3': 'Thus [MATH].', '0811.4281-3-91-4': 'One step involved interchanging unbounded sums, which is justified because everything is absolutely convergent, thanks to [REF] and Theorem [REF].', '0811.4281-3-92-0': 'In the graph expansion for [MATH], the terms where 1 and 2 belong to the same connected graph yield [MATH], and the terms where 1 and 2 belong to different connected graphs yield [MATH].', '0811.4281-3-92-1': 'The detailed argument is the same as above.', '0811.4281-3-92-2': 'We then obtained the desired expression.'} | {'0811.4281-4-0-0': 'cluster expansion with applications]Abstract cluster expansion with applications to statistical mechanical systems', '0811.4281-4-1-0': '# Abstract We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence.', '0811.4281-4-1-1': 'We apply the results to systems of classical and quantum particles with stable interactions.', '0811.4281-4-2-0': 'Keywords: cluster expansion, polymer model, stable interaction, quantum gas.', '0811.4281-4-3-0': '2000 Math.', '0811.4281-4-3-1': 'Subj.', '0811.4281-4-3-2': 'Class.: 82B05, 82B10, 82B20, 82B21, 82B26', '0811.4281-4-4-0': '# Introduction', '0811.4281-4-5-0': "The method of cluster expansions was introduced in the 1930's in statistical mechanics in order to study gases of classical interacting particles.", '0811.4281-4-5-1': 'Its main achievement, from the point of view of physics, may be the derivation of the van der Waals equation of state for weakly interacting systems.', '0811.4281-4-5-2': "The method was made rigorous by mathematical-physicists in the 1960's, see [CITATION] and references therein.", '0811.4281-4-6-0': 'The method split afterwards.', '0811.4281-4-6-1': 'One branch involves continuous systems, with applications to classical systems [CITATION], quantum systems [CITATION], or quantum field theory [CITATION].', '0811.4281-4-6-2': 'The other branch involves polymer systems, i.e. discrete systems with additional internal structure [CITATION].', '0811.4281-4-6-3': 'An important step forward was the article of Kotecky and Preiss with its simplified setting and its elegant condition for the convergence of the cluster expansion [CITATION].', '0811.4281-4-7-0': 'The methods for proving the convergence are diverse.', '0811.4281-4-7-1': 'Let us mention the study of Kirkwood-Salsburg equations that involves correlation functions, see [CITATION] and references therein; the algebraic approach of Ruelle [CITATION]; combinatorial approaches using tree identities [CITATION]; inductions for discrete systems [CITATION].', '0811.4281-4-8-0': 'Important and useful surveys were written by Brydges [CITATION], Pfister [CITATION], Abdesselam and Rivasseau [CITATION].', '0811.4281-4-8-1': 'Recent articles have been devoted to combinatorial aspects [CITATION] and to weakening the assumptions [CITATION].', '0811.4281-4-9-0': 'The method of cluster expansions applies when the objects do not interact much; this the case when they are far apart (low density), or when interactions are weak.', '0811.4281-4-9-1': 'An extension of the criterion of [CITATION] that takes into account these two aspects was proposed in [CITATION]; it applies to both discrete and continuous systems.', '0811.4281-4-10-0': 'All abstract (i.e. general) approaches involve restrictions that correspond to repulsive interactions.', '0811.4281-4-10-1': 'Yet the old results for classical and quantum systems only assume stable interactions, that may include an attractive part.', '0811.4281-4-10-2': 'The aim of the present article is to propose a general approach that applies to discrete and continuous systems with repulsive or stable interactions.', '0811.4281-4-10-3': 'Our proof is split into several independent steps and this helps clarify the situation.', '0811.4281-4-11-0': 'The setting and the results are presented in Section [REF].', '0811.4281-4-11-1': 'We consider applications to classical systems of particles in Section [REF], to polymer systems in Section [REF], and to the quantum gas in Section [REF].', '0811.4281-4-11-2': 'A fundamental tree estimate is derived in Section [REF], and the theorems of Section [REF] are proved in Section [REF].', '0811.4281-4-12-0': '# Cluster expansions', '0811.4281-4-13-0': 'We consider a set [MATH] whose elements may represent widely different objects - in the three applications considered in this article, an element [MATH] represents (i) the position of a classical particle, (ii) a polymer, i.e. a connected set of [MATH], and (iii) a closed Brownian bridge.', '0811.4281-4-13-1': 'For the general abstract theory, we assume the structure of a measure space, [MATH], with [MATH] a complex measure.', '0811.4281-4-13-2': 'We denote [MATH] the total variation (absolute value) of [MATH].', '0811.4281-4-13-3': 'Let [MATH] and [MATH] be complex measurable symmetric functions on [MATH], that are related by the equation [EQUATION]', '0811.4281-4-13-4': 'We allow the real part of [MATH] to take the value [MATH], in which case [MATH].', '0811.4281-4-13-5': 'In typical applications [MATH] represents the interactions between [MATH] and [MATH], and the value [MATH] corresponds to a hard-core repulsion.', '0811.4281-4-13-6': 'We define the "partition function" by [EQUATION] or, equivalently, [EQUATION]', '0811.4281-4-13-7': 'The term [MATH] of the sums is understood to be 1.', '0811.4281-4-14-0': 'The main goal of cluster expansions is to express the partition function as the exponential of a convergent series of "cluster terms".', '0811.4281-4-14-1': 'The main difficulty is to prove the convergence.', '0811.4281-4-14-2': 'We first assume that the potential [MATH] is stable.', '0811.4281-4-15-0': 'There exists a nonnegative function [MATH] on [MATH] such that, for all [MATH] and almost all [MATH], [EQUATION].', '0811.4281-4-16-0': 'In other words, we assume the lower bound [EQUATION]', '0811.4281-4-16-1': 'When the function [MATH] is constant, this is the usual definition of stability.', '0811.4281-4-16-2': '"Almost all" means that, for given [MATH], the set of points where the condition fails has measure zero with respect to the product measure [MATH].', '0811.4281-4-16-3': 'If [MATH] is countable, the condition must be satisfied for all [MATH] such that [MATH].', '0811.4281-4-17-0': 'The second condition deals with the strength of interactions.', '0811.4281-4-18-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-4-19-0': 'In order to guess the correct form of [MATH], one should consider the left side of the equation above with [MATH].', '0811.4281-4-19-1': 'The integral may depend on [MATH]; a typical situation is that [MATH] is characterized by a length [MATH], which is a positive number, so that the left side is roughly proportional to [MATH].', '0811.4281-4-19-2': 'This suggests to try [MATH], and one can then optimize on the value of [MATH].', '0811.4281-4-20-0': 'We also consider an alternate criterion that involves [MATH] rather than [MATH].', '0811.4281-4-20-1': 'It is inspired by the recent work of Procacci [CITATION].', '0811.4281-4-20-2': 'Let [EQUATION]', '0811.4281-4-21-0': 'There exists a nonnegative function [MATH] on [MATH] such that for almost all [MATH], [EQUATION].', '0811.4281-4-22-0': 'For positive [MATH] we can take [MATH]; and since [MATH], Assumption [REF] is always better than Assumption [REF].', '0811.4281-4-22-1': 'We actually conjecture that, together with Assumption [REF], a sufficient condition is [EQUATION]', '0811.4281-4-22-2': 'That is, it should be possible to combine the best of both assumptions.', '0811.4281-4-22-3': 'In this respect Assumption [REF] is optimal in the case of positive potentials, and Assumption [REF] is optimal in the case of hard core plus negative potentials.', '0811.4281-4-23-0': 'We denote by [MATH] the set of all graphs with [MATH] vertices (unoriented, no loops) and [MATH] the set of connected graphs with [MATH] vertices.', '0811.4281-4-23-1': 'We introduce the following combinatorial function on finite sequences [MATH] of elements of [MATH]: [EQUATION]', '0811.4281-4-23-2': 'The product is over edges of [MATH].', '0811.4281-4-24-0': '[Cluster expansions]', '0811.4281-4-25-0': 'Suppose that Assumptions [REF] and [REF], or [REF] and [REF], hold true.', '0811.4281-4-25-1': 'We also suppose that [MATH].', '0811.4281-4-25-2': 'Then we have [EQUATION]', '0811.4281-4-25-3': 'The term in the exponential converges absolutely.', '0811.4281-4-25-4': 'Furthermore, for almost all [MATH], we have the following estimate [EQUATION] (Under Assumption [REF], Eq. [REF] holds with [MATH] instead of [MATH].)', '0811.4281-4-26-0': 'Let us turn to correlation functions.', '0811.4281-4-26-1': 'We only consider one-point and two-point correlation functions since these are the most useful and expressions become more transparent.', '0811.4281-4-26-2': 'We refer to [CITATION] for more general functions.', '0811.4281-4-26-3': 'First, we define the unnormalized one-point correlation function by [EQUATION] (the term [MATH] is 1 by definition).', '0811.4281-4-26-4': 'And we define the unnormalized two-point correlation function by [EQUATION] (the term [MATH] is equal to [MATH]).', '0811.4281-4-26-5': 'Notice that [MATH] can be viewed as a regular partition function, given by Eq. [REF], but with the modified measure [MATH] instead of [MATH].', '0811.4281-4-26-6': 'The normalized correlation functions are [MATH] and [MATH].', '0811.4281-4-26-7': 'As is shown in Theorem [REF], they can be expressed using the "cluster functions" [EQUATION] and [EQUATION]', '0811.4281-4-26-8': 'Notice that [MATH] by [REF].', '0811.4281-4-27-0': '[Correlation functions]', '0811.4281-4-28-0': 'Under the same assumptions as in Theorem [REF], we have [EQUATION].', '0811.4281-4-29-0': 'In statistical mechanics, the relevant expression is the truncated two-point correlation function [EQUATION].', '0811.4281-4-29-1': 'When the cluster expansion converges, it is equal to [MATH] by the theorem above.', '0811.4281-4-29-2': 'This function usually provides an order parameter for phase transitions and it is useful to estimate its decay properties.', '0811.4281-4-30-0': '[Decay of correlations]', '0811.4281-4-31-0': 'If Assumptions [REF] and [REF] hold true, we have for almost all [MATH], [EQUATION] (with [MATH] and [MATH]).', '0811.4281-4-31-1': 'If Assumptions [REF] and [REF] hold true, we have the same bound but with [MATH] instead of [MATH], and [MATH] instead of [MATH].', '0811.4281-4-32-0': 'In many applications the functions [MATH] and [MATH] depend on the difference [MATH] (this assumes that [MATH] has additional structure, namely that of a group).', '0811.4281-4-32-1': 'The estimates for [MATH] are given by convolutions.', '0811.4281-4-33-0': 'The theorems of this section are proved in Section [REF].', '0811.4281-4-34-0': '# Classical gas', '0811.4281-4-35-0': 'We consider a gas of point particles that interact with a pair potential.', '0811.4281-4-35-1': 'We work in the grand-canonical ensemble where the parameters are the fugacity [MATH] and the inverse temperature [MATH] (both are real and positive numbers).', '0811.4281-4-35-2': 'The set [MATH] is an open bounded subset of [MATH] and [MATH] with [MATH] the Lebesgue measure.', '0811.4281-4-35-3': 'We actually write [MATH] so as to have more traditional notation.', '0811.4281-4-35-4': 'The interaction is given by a function [MATH] which we take to be piecewise continuous; [MATH].', '0811.4281-4-35-5': 'We suppose that [MATH] is stable, i.e. that there exists a constant [MATH] such that for any [MATH] and any [MATH]: [EQUATION]', '0811.4281-4-35-6': 'Our Assumption [REF] holds with [MATH].', '0811.4281-4-35-7': 'The system is translation invariant so all [MATH] are equivalent.', '0811.4281-4-35-8': 'The function of Assumptions [REF] and [REF] can then be taken to be a constant, [MATH].', '0811.4281-4-35-9': 'We seek a condition that does not depend on the size of the system.', '0811.4281-4-35-10': 'Then integrals over [MATH] are on [MATH] instead of [MATH].', '0811.4281-4-35-11': 'By translation invariance we can take [MATH].', '0811.4281-4-36-0': 'Assumption [REF] gives the condition [EQUATION]', '0811.4281-4-36-1': 'We obviously choose the constant [MATH] that maximizes the right side, which is [MATH].', '0811.4281-4-36-2': 'This condition is the one in [CITATION].', '0811.4281-4-36-3': 'Let us now assume that [MATH] consists of a hard core of radius [MATH] and that it is otherwise integrable.', '0811.4281-4-36-4': 'Again with [MATH], Assumption [REF] gives the condition [EQUATION]', '0811.4281-4-36-5': 'Here, [MATH] is the volume of the ball in [MATH] dimensions.', '0811.4281-4-36-6': 'This condition is often better than [REF].', '0811.4281-4-36-7': 'Without hard core it is the one in [CITATION].', '0811.4281-4-36-8': 'The domains of parameters where these conditions hold correspond to low fugacities and high temperatures.', '0811.4281-4-37-0': 'The thermodynamic pressure is defined as the infinite volume limit of [EQUATION]', '0811.4281-4-37-1': 'Using Theorem [REF], we have [EQUATION]', '0811.4281-4-37-2': 'Consider now any sequence of increasing domains [MATH] such that [MATH].', '0811.4281-4-37-3': 'Thanks to the estimate [REF], and using translation invariance, we get [EQUATION] (The term with [MATH] is equal to [MATH].)', '0811.4281-4-37-4': 'This expression for the infinite volume pressure [MATH] should be viewed as a convergent series of analytic functions of [MATH].', '0811.4281-4-37-5': 'Then [MATH] is analytic in [MATH] and [MATH] by Vitali theorem and no phase transition takes place in the domain of parameters where the cluster expansion is convergent.', '0811.4281-4-38-0': 'The truncated two-point correlation function [MATH] is given by [MATH].', '0811.4281-4-38-1': 'We consider only the case of Assumption [REF] but a similar claim can be obtained with Assumption [REF].', '0811.4281-4-38-2': 'Let [MATH] be a function that satisfies the triangle inequality.', '0811.4281-4-38-3': 'The estimate of Theorem [REF] yields [EQUATION] (with [MATH]).', '0811.4281-4-38-4': 'Recall that [MATH], and let [EQUATION]', '0811.4281-4-38-5': 'Then we get [EQUATION]', '0811.4281-4-38-6': 'If the inequality [REF] is strict, one can usually find a function [MATH] that satisfies the triangle inequality and such that [MATH]; the truncated two-point correlation function then decays faster than [MATH].', '0811.4281-4-39-0': '# Polymer systems', '0811.4281-4-40-0': 'Polymer systems are discrete, which is technically simpler, but they also have internal structure.', '0811.4281-4-40-1': 'The first application of cluster expansions to polymer systems is due to Gruber and Kunz [CITATION].', '0811.4281-4-40-2': 'Among the many articles devoted to this subject, let us mention [CITATION].', '0811.4281-4-40-3': 'The main goal of this section is to illustrate our setting; we therefore restrict ourselves to a specific model of polymers with both repulsive and attractive interactions.', '0811.4281-4-41-0': 'Our space [MATH] is the set of all finite connected subsets of [MATH].', '0811.4281-4-41-1': 'The measure [MATH] is the counting measure multiplied by the activity [MATH] (a function [MATH]).', '0811.4281-4-41-2': 'We choose [MATH] with [MATH].', '0811.4281-4-41-3': 'The interaction is hard core when polymers overlap and it is attractive when they touch: [EQUATION]', '0811.4281-4-41-4': 'Here, [MATH] is the number of "contacts" between [MATH] and [MATH], i.e. the number of bonds between sites of [MATH] and [MATH]; [MATH] is a parameter.', '0811.4281-4-41-5': 'The interaction is zero when the distance between polymers is greater than 1.', '0811.4281-4-42-0': 'The stability condition can be written [EQUATION]', '0811.4281-4-42-1': 'Only disjoint polymers need to be considered, the left side is infinite otherwise.', '0811.4281-4-42-2': 'The sum over [MATH] is always larger than [MATH] times the number of bonds connecting [MATH] with its exterior.', '0811.4281-4-42-3': 'Thus we can take [MATH].', '0811.4281-4-43-0': 'The function [MATH] in Assumption [REF] grows like [MATH], so it is natural to choose [MATH] for some constant [MATH].', '0811.4281-4-43-1': 'A sufficient condition is that [EQUATION]', '0811.4281-4-43-2': 'We can bound [MATH] by [MATH].', '0811.4281-4-43-3': 'Summing over the sites of [MATH], and requiring that [MATH] contains the given site or comes at distance 1, we get [EQUATION]', '0811.4281-4-43-4': 'We used the fact that the activity is translation invariant.', '0811.4281-4-43-5': 'If [MATH] is a connected set, there exists a closed walk with nearest neighbor jumps whose support is [MATH], and whose length is at most [MATH].', '0811.4281-4-43-6': 'This can be seen by induction: knowing the walk for [MATH], it is easy to construct one for [MATH].', '0811.4281-4-43-7': 'The number of connected sets of cardinality [MATH] that contain the origin is therefore smaller than the number of walks of length [MATH] starting at the origin, which is equal to [MATH].', '0811.4281-4-43-8': 'Then it suffices that [EQUATION]', '0811.4281-4-43-9': 'This is equivalent to [EQUATION]', '0811.4281-4-43-10': 'Assumption [REF] holds for any [MATH].', '0811.4281-4-43-11': 'Using [MATH] and optimizing on [MATH], we find the sufficient condition [EQUATION] with [MATH].', '0811.4281-4-44-0': 'We have just established the existence of a low density phase provided the activity is small enough.', '0811.4281-4-44-1': 'The condition depends on the contact parameter [MATH].', '0811.4281-4-44-2': 'For large [MATH] one should expect interesting phases with many contacts between the polymers.', '0811.4281-4-45-0': '# Quantum gas', '0811.4281-4-46-0': 'We follow a course that is similar to Ginibre [CITATION], using the Feynman-Kac formula so as to get a gas of winding Brownian loops.', '0811.4281-4-46-1': 'We get comparable results, with a larger domain of convergence in the case of integrable potentials.', '0811.4281-4-46-2': 'Winding Brownian loops are kind of continuous polymers; they combine the difficulties of both cases above - the continuous nature and the internal structure.', '0811.4281-4-47-0': '## Feynman-Kac representation', '0811.4281-4-48-0': 'The state space for [MATH] fermions (resp. bosons) in a domain [MATH] is the Hilbert space [MATH] (resp. [MATH]) of square-integrable complex functions that are antisymmetric (resp. symmetric) with respect to their arguments.', '0811.4281-4-48-1': 'The Hamiltonian is [EQUATION] with [MATH] the Laplacian for the [MATH]-th variable and [MATH] a multiplication operator.', '0811.4281-4-48-2': 'As in the classical case, we consider the grand-canonical ensemble whose parameters are the fugacity [MATH] and the inverse temperature [MATH].', '0811.4281-4-48-3': 'The partition function is [EQUATION]', '0811.4281-4-48-4': 'We need to cast the partition function in the form [REF], which can be done using the Feynman-Kac representation.', '0811.4281-4-49-0': 'Namely, we have [EQUATION] [MATH] is the symmetric group of [MATH] elements; [MATH] is equal to the signature of the permutation [MATH] for fermions, [MATH] for bosons; [MATH] is the Wiener measure for the Brownian bridge from [MATH] to [MATH] in time [MATH] - the normalization is chosen so that [EQUATION] [MATH][MATH] is one if [MATH] for all [MATH], it is zero otherwise.', '0811.4281-4-49-1': 'An introduction to the Feynman-Kac formula in this context can be found in the survey of Ginibre [CITATION].', '0811.4281-4-50-0': 'The right side of Eq. [REF] is well defined for a large class of functions [MATH], that includes all piecewise continuous functions.', '0811.4281-4-50-1': 'Thus we take [REF] as the definition for [MATH].', '0811.4281-4-50-2': 'Under additional assumptions on [MATH], [REF] is equal to [REF] with Hamiltonian [REF] and with Dirichlet boundary conditions.', '0811.4281-4-51-0': 'We now rewrite the grand-canonical partition function in terms of winding loops.', '0811.4281-4-51-1': 'Let [MATH] be the set of continuous paths [MATH] that are closed.', '0811.4281-4-51-2': 'Its elements are denoted [MATH], with [MATH] the starting point, [MATH] the winding number, and [MATH] the path; we have [MATH].', '0811.4281-4-51-3': 'We consider the measure [MATH] given by [EQUATION]', '0811.4281-4-51-4': 'Here, [MATH] is a self-interaction term that is defined below in [REF]; [MATH] for fermions and [MATH] for bosons.', '0811.4281-4-51-5': 'Let [MATH]; the measure [MATH] naturally extends to a measure on [MATH].', '0811.4281-4-51-6': 'The grand-canonical partition function can then be written as [EQUATION]', '0811.4281-4-51-7': 'Let [MATH] and [MATH].', '0811.4281-4-51-8': 'The self-interaction [MATH] and the 2-loop interaction [MATH] are given by [EQUATION]', '0811.4281-4-51-9': 'We now treat separately the case of integrable potentials and the case of more general potentials.', '0811.4281-4-52-0': '## Stable integrable potentials', '0811.4281-4-53-0': 'We suppose that [MATH] is stable with constant [MATH], i.e. it satisfies Eq. [REF].', '0811.4281-4-53-1': 'For given loops [MATH], stability implies that [EQUATION]', '0811.4281-4-53-2': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-4-53-3': 'Notice that [MATH], again by the stability of [MATH].', '0811.4281-4-54-0': 'We use Assumption [REF] and we choose a function [MATH] with a constant [MATH] to be determined later.', '0811.4281-4-54-1': 'Explicitly, the assumption is that for any [MATH] [EQUATION]', '0811.4281-4-54-2': 'We have lifted the restriction that [MATH] because we want a condition that does not depend on [MATH].', '0811.4281-4-54-3': 'Eq. [REF] is easier to handle than its appearance suggests.', '0811.4281-4-54-4': 'Notice that the term in the second exponential is just [MATH].', '0811.4281-4-54-5': 'Using [MATH], and the definition [REF] of [MATH], it is enough that [EQUATION]', '0811.4281-4-55-0': 'We can immediately integrate over [MATH], which yields [MATH].', '0811.4281-4-55-1': 'The Wiener integral then gives [MATH] and we get the equivalent condition [EQUATION]', '0811.4281-4-55-2': 'For any [MATH], the inequality holds for [MATH] small enough.', '0811.4281-4-55-3': 'Notice that [MATH] in any case.', '0811.4281-4-55-4': 'One can get a more explicit condition for [MATH] by choosing [MATH] such that [MATH] = 1.', '0811.4281-4-55-5': 'This yields [EQUATION]', '0811.4281-4-55-6': 'Here, [MATH] is the Riemann zeta function.', '0811.4281-4-56-0': 'When [MATH] and when the potential is repulsive, one can rewrite [REF] in a more transparent way.', '0811.4281-4-56-1': 'Let [MATH] denote the Born approximation to the scattering length.', '0811.4281-4-56-2': 'The condition is then [EQUATION]', '0811.4281-4-56-3': 'The critical fugacity is expected to be greater than 1.', '0811.4281-4-56-4': 'The present result helps nonetheless to obtain a range of densities where the pressure is analytic.', '0811.4281-4-56-5': "In the bosonic case it compares well with physicists' expectations [CITATION].", '0811.4281-4-57-0': '## Stable potentials with hard core', '0811.4281-4-58-0': 'The presence of a hard core makes the situation more complicated; we only sketch the argument in this section without trying to get explicit bounds.', '0811.4281-4-58-1': 'Our aim is to show that, using Theorem [REF], the problem of convergence of the cluster expansion reduces to estimates of Wiener sausages.', '0811.4281-4-59-0': 'We consider an interaction [MATH].', '0811.4281-4-59-1': 'We assume that [MATH] is a repulsive potential of radius [MATH], with a hard core of radius [MATH], that [MATH] for [MATH], and that [MATH] is integrable otherwise.', '0811.4281-4-59-2': 'We suppose that the stability condition takes the slightly stronger form [EQUATION] for any [MATH] such that [MATH].', '0811.4281-4-59-3': 'For potentials with a hard core this is equivalent to the property [REF], possibly with a different constant [MATH].', '0811.4281-4-59-4': 'Then one has [CITATION] [EQUATION]', '0811.4281-4-59-5': 'Then Assumption [REF] holds with [EQUATION]', '0811.4281-4-59-6': 'Notice that the stability condition also holds with [MATH] given in [REF] (and with a better constant [MATH]).', '0811.4281-4-59-7': 'The advantage of [REF] is the factor [MATH] in front of [MATH].', '0811.4281-4-59-8': 'Then [MATH] involves a term that cancels the self-interactions of [MATH].', '0811.4281-4-60-0': 'Given a loop [MATH], let [MATH] be the Wiener sausage generated by a ball of radius [MATH] when its center moves along the trajectory [MATH]: [EQUATION]', '0811.4281-4-61-0': 'We denote the volume of a Wiener sausage [MATH] by [MATH].', '0811.4281-4-61-1': 'One can check that [EQUATION] with [MATH] the volume of the unit ball (see Appendix 2 in [CITATION]).', '0811.4281-4-62-0': 'We choose [MATH] in Assumption [REF].', '0811.4281-4-62-1': 'Then a sufficient condition is that for any [MATH], [EQUATION]', '0811.4281-4-62-2': 'We consider separately the cases where [MATH] belongs or not to [MATH].', '0811.4281-4-62-3': 'First, [EQUATION] which we bound using [REF].', '0811.4281-4-62-4': 'Second, using [MATH], [EQUATION]', '0811.4281-4-62-5': 'We certainly get [REF] if we have the two inequalities [EQUATION]', '0811.4281-4-62-6': 'One can estimate the integrals of Wiener sausages, see [CITATION], so that both conditions hold if [MATH] is small enough.', '0811.4281-4-63-0': 'Now that the cluster expansion is known to converge, it is possible to write the pressure as an absolutely convergent series of analytic functions in [MATH] and [MATH].', '0811.4281-4-63-1': 'It is also possible to study the decay of correlation functions.', '0811.4281-4-63-2': "In the case of potentials that consist of hard core plus integrable part, one can apply Assumption 2' instead.", '0811.4281-4-63-3': 'This may give better results, especially if the integrable part is mostly attractive.', '0811.4281-4-64-0': '# Tree estimates', '0811.4281-4-65-0': 'In this section we obtain estimates of sums of connected graphs in terms of sums of trees.', '0811.4281-4-65-1': 'Our main result is Proposition [REF] below.', '0811.4281-4-65-2': 'Such estimates seem to have been introduced by Penrose [CITATION] and they have often been considered in the past [CITATION].', '0811.4281-4-65-3': 'We introduce a minimal setting that clarifies its role in the cluster expansion.', '0811.4281-4-65-4': 'Namely, we fix the polymers so we only deal with the numbers that represent their interactions, [MATH] or [MATH], and the stability function [MATH].', '0811.4281-4-65-5': 'Assumption [REF] is vital here, but Assumptions [REF] and [REF] are not used in this section.', '0811.4281-4-66-0': 'Let [MATH] denote the set of trees with [MATH] vertices.', '0811.4281-4-66-1': 'Let [MATH] be an integer, [MATH] be real nonnegative numbers, and [MATH], [MATH], be complex numbers.', '0811.4281-4-66-2': 'We assume that the following bound holds for any subset [MATH]: [EQUATION]', '0811.4281-4-66-3': 'Let [MATH] be such that [MATH]; let [MATH] if [MATH], and [MATH] otherwise.', '0811.4281-4-66-4': 'We state two distinct tree estimates, the first one involving [MATH] and the second one involving [MATH].', '0811.4281-4-66-5': 'These bounds will allow to prove the convergence under either Assumption [REF] or Assumption [REF].', '0811.4281-4-67-0': 'If [REF] holds true, we have the two bounds', '0811.4281-4-68-0': 'We actually conjecture that the following estimate holds under the same hypotheses: [EQUATION]', '0811.4281-4-68-1': "We prove Proposition [REF] (a) below using Ruelle's algebraic approach [CITATION].", '0811.4281-4-68-2': 'This method is usually combined with a Banach fixed point argument for correlation functions.', '0811.4281-4-68-3': 'However, we use it differently so as to get a tree estimate.', '0811.4281-4-68-4': 'Proposition [REF] (b) follows from a tree identity due to Brydges, Battle, and Federbush [CITATION], combined with an argument due to Procacci [CITATION]; its proof can be found at the end of this section.', '0811.4281-4-69-0': 'Let [MATH] be the set of complex functions on the power set [MATH].', '0811.4281-4-69-1': 'We introduce the following multiplication operation for [MATH]: [EQUATION]', '0811.4281-4-69-2': 'We use the standard conventions for sums and products, namely that the empty sum is zero and the empty product is one.', '0811.4281-4-69-3': 'Together with the addition, [MATH] is a commutative algebra with unit [MATH].', '0811.4281-4-69-4': 'It is possible to check that each [MATH] has a unique inverse, which we denote [MATH].', '0811.4281-4-69-5': 'We have [EQUATION]', '0811.4281-4-69-6': 'Let [MATH] be the subset of functions [MATH] such that [MATH] is an ideal of [MATH]).', '0811.4281-4-69-7': 'Notice that [MATH] for any [MATH] , when [MATH].', '0811.4281-4-69-8': 'We define the exponential mapping [MATH] by [EQUATION]', '0811.4281-4-69-9': 'Let [MATH] and [MATH] be the functions defined by [EQUATION]', '0811.4281-4-69-10': 'Here, [MATH] (resp. [MATH]) is the set of graphs (resp. connected graphs) on [MATH].', '0811.4281-4-69-11': 'We have the relation [EQUATION]', '0811.4281-4-69-12': 'We also introduce an operation that is reminiscent of differentiation: [EQUATION]', '0811.4281-4-69-13': 'One can check that [MATH].', '0811.4281-4-70-0': 'For disjoint [MATH], we define [EQUATION]', '0811.4281-4-70-1': 'Let [MATH].', '0811.4281-4-70-2': 'The assumption of Proposition [REF] implies that [EQUATION]', '0811.4281-4-70-3': 'Then there exists [MATH] such that [EQUATION]', '0811.4281-4-70-4': 'Such [MATH] is not unique in general but it does not matter.', '0811.4281-4-70-5': 'We consider a function [MATH] that assigns one of the indices [MATH] above to each nonempty subset [MATH].', '0811.4281-4-70-6': 'Notice that [MATH] for any subset [MATH].', '0811.4281-4-70-7': 'It is also useful to introduce the notation [MATH].', '0811.4281-4-71-0': 'The function [MATH] of Eq. [REF] is solution of the following equation.', '0811.4281-4-71-1': '[EQUATION].', '0811.4281-4-72-0': 'Since the equation gives [MATH] in terms of [MATH] with [MATH], it is well defined inductively and it has a unique solution.', '0811.4281-4-72-1': 'Notice that [MATH], and that [MATH] for any index [MATH].', '0811.4281-4-73-0': 'Recall the definition [REF] of [MATH].', '0811.4281-4-73-1': 'For disjoint [MATH] we have [EQUATION]', '0811.4281-4-73-2': 'Then [EQUATION]', '0811.4281-4-73-3': 'The last sum is equal to [MATH].', '0811.4281-4-73-4': 'One recognizes the equation of Lemma [REF].', '0811.4281-4-74-0': 'We now estimate the function [MATH] using another function [MATH] that satisfies an equation that is similar to that of Lemma [REF].', '0811.4281-4-74-1': '[EQUATION]', '0811.4281-4-74-2': 'It also has a unique solution.', '0811.4281-4-74-3': 'Since [MATH], we can check inductively that [EQUATION] for any sets [MATH] (with [MATH]).', '0811.4281-4-74-4': 'Now the function [MATH] can be written explicitly [CITATION].', '0811.4281-4-74-5': 'Let [MATH] be the set of forests on [MATH] rooted in [MATH].', '0811.4281-4-74-6': 'That is, a graph [MATH] is a forest such that each tree contains exactly one element of [MATH].', '0811.4281-4-75-0': 'The solution of Eq. [REF] is [EQUATION].', '0811.4281-4-76-0': 'Since the solution to Eq. [REF] is unique, it is enough to check that the Ansatz of the lemma satisfies the equation.', '0811.4281-4-76-1': 'First, let us observe that both sides are multiplied by [MATH].', '0811.4281-4-76-2': 'Thus it is enough to consider the case [MATH].', '0811.4281-4-77-0': 'The sum over graphs in [MATH] can be realized by first summing over the set [MATH] of indices (necessarily in [MATH]) that are connected to [MATH]; then over sets of trees in [MATH], and over connections to [MATH].', '0811.4281-4-77-1': 'Explicitly, [EQUATION]', '0811.4281-4-77-2': 'This equation is precisely [REF].', '0811.4281-4-78-0': '[Proof of Proposition [REF] (a)] When [MATH] has a single element, the function [MATH] is equal to [EQUATION]', '0811.4281-4-78-1': 'This is the left side of Proposition [REF] (a).', '0811.4281-4-78-2': 'We have [MATH], the set of trees with [MATH] vertices.', '0811.4281-4-78-3': 'Thus [MATH] is equal to the right side of Proposition [REF] (a), and the proof follows from Eq. [REF].', '0811.4281-4-79-0': 'We now turn to the proof of Proposition [REF] (b).', '0811.4281-4-79-1': 'Notice that in absence of "hard cores", i.e. when [MATH], our claim is just a reformulation of Corollary 3.2 (a) of [CITATION].', '0811.4281-4-79-2': 'The present proof follows [CITATION].', '0811.4281-4-80-0': '[Proof of Proposition [REF] (b)] Let [MATH] be the set of [MATH] such that [MATH], i.e. such that [MATH].', '0811.4281-4-80-1': 'We regularize those numbers by setting [EQUATION]', '0811.4281-4-80-2': 'This allows to use the tree identity of [CITATION]; we will eventually take [MATH] to infinity.', '0811.4281-4-80-3': 'The tree identity can be written [EQUATION]', '0811.4281-4-80-4': 'The full definition of the measure [MATH] can be found in [CITATION]; here we only mention its relevant properties.', '0811.4281-4-80-5': '[MATH] depends on the tree [MATH] and it is a probability measure supported on a subset [MATH].', '0811.4281-4-80-6': 'If [MATH] satisfies the stability condition, then [EQUATION] for all [MATH] and all [MATH].', '0811.4281-4-81-0': 'If [MATH] for any [MATH], we immediately get the claim by using the stability condition above, since [MATH].', '0811.4281-4-81-1': 'The extension to possibly infinite numbers can be obtained using a trick due to Procacci [CITATION].', '0811.4281-4-81-2': 'Let [MATH] be a fixed tree and [MATH].', '0811.4281-4-81-3': 'We introduce [EQUATION]', '0811.4281-4-81-4': 'Then [EQUATION]', '0811.4281-4-81-5': 'If [MATH] is large enough (depending on [MATH]), [MATH] is stable and the first term of the right side is bounded below by [MATH].', '0811.4281-4-82-0': 'We can insert this estimate into Eq. [REF].', '0811.4281-4-82-1': 'Letting [MATH] and then [MATH], we get Proposition [REF] (b).', '0811.4281-4-83-0': '# Proofs of the theorems', '0811.4281-4-84-0': 'In this section we prove the theorems of Section [REF].', '0811.4281-4-84-1': 'We consider only the case where Assumption [REF] holds true - the case with Assumption [REF] is entirely the same, one only needs to replace all [MATH] with [MATH] and all [MATH] with [MATH].', '0811.4281-4-84-2': 'The proofs are based on the following tree estimate, which is a direct consequence of Proposition [REF]: for almost all [MATH], [EQUATION] [Proof of Theorem [REF]] We start by proving the bound [REF].', '0811.4281-4-84-3': 'Let us introduce [EQUATION] (The term [MATH] is equal to [MATH] by definition.)', '0811.4281-4-84-4': 'We show by induction that [EQUATION] for any [MATH].', '0811.4281-4-84-5': 'Then [MATH] for almost all [MATH], and using [REF] we get [REF].', '0811.4281-4-85-0': 'The case [MATH] reduces to [MATH] and it is clear.', '0811.4281-4-85-1': 'The sum over trees with [MATH] vertices can be written as a sum over forests on [MATH], and a sum over edges between 1 and each tree of the forest.', '0811.4281-4-86-0': 'Here, [MATH] denotes the set of trees with [MATH] as the set of vertices.', '0811.4281-4-86-1': 'If [MATH] the sum over [MATH] is one by definition.', '0811.4281-4-86-2': "The term after the sum over partitions depends on the cardinalities of the [MATH]'s, but not on the actual labeling.", '0811.4281-4-86-3': 'Also, each [MATH] gives the same contribution.', '0811.4281-4-87-0': 'We obtain an upper bound by releasing the constraint [MATH] to [MATH], [MATH].', '0811.4281-4-87-1': 'We then get [EQUATION]', '0811.4281-4-87-2': 'We have [MATH] by the induction hypothesis.', '0811.4281-4-87-3': 'Eq. [REF] follows from Assumption [REF].', '0811.4281-4-88-0': 'The rest of the proof is standard combinatorics.', '0811.4281-4-88-1': 'The partition function can be expanded so as to recognize the exponential of connected graphs.', '0811.4281-4-88-2': 'Namely, we start with [EQUATION]', '0811.4281-4-88-3': 'The graph [MATH] can be decomposed into [MATH] connected graphs whose sets of vertices form a partition of [MATH].', '0811.4281-4-88-4': 'Summing first over the number [MATH] of vertices for each set of the partition, we get [EQUATION]', '0811.4281-4-88-5': 'The triple sum is absolutely convergent thanks to the estimate [REF] that we have just established.', '0811.4281-4-88-6': 'One can then interchange the sums by the dominated convergence theorem.', '0811.4281-4-88-7': 'This removes the sum over [MATH], and this completes the proof of Theorem [REF].', '0811.4281-4-89-0': 'Next we prove Theorems [REF] and [REF] in reverse order, since we will use the convergence properties in the latter theorem to get the former.', '0811.4281-4-90-0': '[Proof of Theorem [REF]] From the definition [REF] and the tree estimate [REF], we have [EQUATION]', '0811.4281-4-90-1': 'The expression above involves a sum over trees of arbitrary size that connect 1 and 2.', '0811.4281-4-90-2': 'Any such tree decomposes into a line of [MATH] edges that connect 1 and 2 ([MATH]), and [MATH] trees rooted in the vertices of the connecting line.', '0811.4281-4-90-3': 'Taking into account the combinatorial factors, we obtain', '0811.4281-4-91-0': '[Proof of Theorem [REF]] It is actually similar to the end of the proof of Theorem [REF].', '0811.4281-4-91-1': '[MATH] can be expanded as a sum over graphs, that can be decomposed into a connected graph that contains 1, and other connected graphs.', '0811.4281-4-91-2': 'Taking into account the combinatorial factors, the contribution of connected graphs containing 1 yields [MATH], and the contribution of the others yields the expression [REF] for [MATH].', '0811.4281-4-91-3': 'Thus [MATH].', '0811.4281-4-91-4': 'One step involved interchanging unbounded sums, which is justified because everything is absolutely convergent, thanks to [REF] and Theorem [REF].', '0811.4281-4-92-0': 'In the graph expansion for [MATH], the terms where 1 and 2 belong to the same connected graph yield [MATH], and the terms where 1 and 2 belong to different connected graphs yield [MATH].', '0811.4281-4-92-1': 'The detailed argument is the same as above.', '0811.4281-4-92-2': 'We then obtained the desired expression.'} | null | null | null |
hep-ph-0603242 | {'hep-ph-0603242-1-0-0': 'We begin a general exploration of the phenomenology of TeV-scale extra-dimensional models with gravitational actions that contain higher curvature terms.', 'hep-ph-0603242-1-0-1': 'In particular, we examine how the classic collider signatures of the models of Arkani-Hamed, Dimopoulos and Dvali (missing energy and new dimension-8 contact interactions) and of Randall and Sundrum (TeV-scale graviton Kaluza-Klein resonances) are altered by these modifications to the usual Einstein-Hilbert action.', 'hep-ph-0603242-1-0-2': 'We find that not only are the detailed signatures for these gravitationally induced processes altered but new contributions are found to arise due to the existence of additional scalar Kaluza-Klein states in the spectrum.', 'hep-ph-0603242-1-1-0': '# Introduction and Background', 'hep-ph-0603242-1-2-0': 'The question as to why the Planck and electroweak scales differ by so many orders of magnitude remains mysterious.', 'hep-ph-0603242-1-2-1': 'In recent years, attempts have been made to address this hierarchy issue within the context of theories with extra spatial dimensions that lower the effective scale of gravity to the TeV region.', 'hep-ph-0603242-1-2-2': 'In both of the Arkani-Hamed, Dimopoulos and Dvali (ADD)[CITATION] and Randall-Sundrum (RS)[CITATION] models, new effects of gravitational origin are expected to occur near the TeV scale which should be observable at future colliders such as the LHC and ILC.', 'hep-ph-0603242-1-2-3': 'Though these two models are very different in detail they do have some common features the most important of which are ([MATH]) in their original versions they both assume that Standard Model matter is confined to a 4-dimensional brane; ([MATH]) they both assume that D-dimensional gravity is described by the Einstein-Hilbert (EH) action plus a possible cosmological constant and ([MATH]) the background metric is either strictly flat, i.e. , Minkowskian as in the ADD model with toroidally flat compactification, or has constant curvature and is conformally flat, i.e., [MATH] as in the RS model.', 'hep-ph-0603242-1-2-4': 'How would the predictions of these two models change if we surrendered the EH action and considered something more general?', 'hep-ph-0603242-1-2-5': 'This is the discussion we would like to begin in this paper.', 'hep-ph-0603242-1-3-0': 'General Relativity (GR) as described by the EH action is considered to be an effective theory below the fundamental Planck scale, [MATH].', 'hep-ph-0603242-1-3-1': 'Once energies approaching or beyond the scale [MATH] begin to be probed one might expect to observe deviations from the expectations arising from the EH action.', 'hep-ph-0603242-1-3-2': 'In the case of both the ADD and RS models, future colliders will probe near or above their (effective) fundamental scales so that non-EH aspects of the true gravitational theory, whatever its form, should become apparent.', 'hep-ph-0603242-1-3-3': 'Since the ultraviolet form of the true gravity theory is as of yet unknown one may hope to capture some of its aspects by considering how the presence of new higher curvature (and higher derivative) invariants in the actions of the ADD and RS models may lead to variations in the well-known predictions of these theories.', 'hep-ph-0603242-1-3-4': 'Many authors have considered the possibility of higher curvature invariants and how their existence would modify the predictions arising from the EH action within other contexts, e.g., the properties of black holes[CITATION], deviations in solar system tests of GR[CITATION] and in cosmology[CITATION] to possibly avoid the need for dark energy.', 'hep-ph-0603242-1-3-5': 'Some analyses along these lines for the potential modifications of the collider predictions of both the ADD and RS models have already been performed[CITATION].', 'hep-ph-0603242-1-3-6': 'In the present paper, we wish to both extend and generalize these results to get a feeling for the possible variation in the gravitational phenomena as predicted by these classic models all of which will be potentially observable at future colliders.', 'hep-ph-0603242-1-3-7': 'In particular we are interested in how the well known signatures of both the ADD and RS models are morphed if we keep the basic setups intact but modify the actions on which the corresponding equations of motion are derived.', 'hep-ph-0603242-1-3-8': 'A further generalization to such an analysis is possible if the original ADD/RS setups are simultaneously surrendered resulting in entirely new equations of motion; while this is an interesting possibility to consider it lies beyond the scope of the present paper.', 'hep-ph-0603242-1-4-0': 'Of course a completely general study of how these possible modifications to the effective gravity action may morph TeV collider signatures is an obviously immense task and here we aim only at a first round analysis in the discussion that follows.', 'hep-ph-0603242-1-4-1': 'The major signatures arising in both ADD and RS models originate from graviton exchange and the production of black holes; the ADD model also leads to missing energy signatures from graviton emission.', 'hep-ph-0603242-1-4-2': 'Fortunately, apart from issues associated with black holes, the relevant graviton properties (couplings, wavefunctions and propagators) necessary to extract experimental signatures for either model can be obtained from the expansion of the rather general action considered here to quadratic order in the curvature.', 'hep-ph-0603242-1-4-3': '(This would no longer be true if we wanted to consider, e.g., the triple graviton coupling as then an expansion to third order would be required.)', 'hep-ph-0603242-1-4-4': 'This simplifying observation forms the basis of the analysis that follows and allows us to determine the relevant graviton properties in both ADD- and RS-like models for a wide class of effective actions.', 'hep-ph-0603242-1-5-0': 'The general outline of our analysis in presented in Section 2 where our basic assumptions and notations are also given.', 'hep-ph-0603242-1-5-1': 'In Section 3 we apply our analysis to the ADD model; we then apply it to the RS case in Section 4.', 'hep-ph-0603242-1-5-2': 'Our summary and conclusions are given in Section 5.', 'hep-ph-0603242-1-6-0': '# Analysis', 'hep-ph-0603242-1-7-0': 'When going beyond the EH action there are many possibilities to consider especially when we are living in extra dimensions.', 'hep-ph-0603242-1-7-1': 'In the literature various forms have been assumed for the potential structures of higher curvature invariants that may appear in the gravity action.', 'hep-ph-0603242-1-7-2': 'The fairly general action that we will assume for the D-dimensional action in the present analysis takes the form (with [MATH] below): [EQUATION] where [MATH] is a continuous, differentiable and generally mathematically well-behaved function; in particular we will assume that [MATH] is non-singular when all of its arguments are zero.', 'hep-ph-0603242-1-7-3': 'Here [MATH] is the usual D-dimensional Ricci scalar while [MATH] and [MATH] are quadratic invariants constructed from the curvature tensor [MATH]: [MATH], with [MATH] being the Ricci tensor, while [MATH].', 'hep-ph-0603242-1-7-4': '[MATH] is the D-dimensional fundamental gravity scale which is [MATH] TeV in ADD and [MATH]M_pl[MATH] in RS.', 'hep-ph-0603242-1-7-5': 'In the low energy, small curvature limit we expect [MATH] (plus a possible cosmological constant) and so the overall dimensionful factor in the expression above allows us to make direct contact with the EH action in this limit.', 'hep-ph-0603242-1-7-6': 'This specific form for [MATH], though not completely general, covers a wide array of possibilities and has been considered (some only in [MATH]) in may different contexts for a multitude of purposes in the literature[CITATION].', 'hep-ph-0603242-1-7-7': 'Many of the higher curvature models previous considered by other authors form subcases of this more general action.', 'hep-ph-0603242-1-8-0': 'As is by now well-known[CITATION] the gravity theories described by an action of the form [MATH] can potentially have several serious problems since they, amongst other things, lead to equations of motion which are generally fourth order in the derivatives of the metric.', 'hep-ph-0603242-1-8-1': 'In particular, in addition to the usual massless D-dimensional tensor graviton which results in the familiar resulting 4-d graviton and graviscalar Kaluza-Klein (KK) tower excitations, there may also be present in the linearized D-dimensional theory additional massive scalar and tensor excitations.', 'hep-ph-0603242-1-8-2': 'These fields will in 4-d have KK towers without zero modes and which can be ghostlike and/or tachyonic.', 'hep-ph-0603242-1-8-3': '(We can think of these new D-dimensional fields as having bulk masses which influence their corresponding 4-d KK tower masses.)', 'hep-ph-0603242-1-8-4': 'Furthermore, the equations of motion naturally involving higher derivatives of the fields can lead to problems with unitarity.', 'hep-ph-0603242-1-8-5': 'The new massive tensor excitations are potentially the most serious issue to deal with as they are ghost fields that must be eliminated from the spectrum (though they may help in dealing with the theories renormalizability and bad high-energy behavior).', 'hep-ph-0603242-1-8-6': 'It has been noted[CITATION] that we one can remove these states from the spectrum (i.e., by giving their bulk masses an infinite value) if a tuning occurs such that the quantities [MATH] and [MATH] only appear in the special combination [MATH] in the action.', 'hep-ph-0603242-1-8-7': 'There has been some discussion in the literature, however, that these ghost states may not be as dangerous as one would imagine from lowest order perturbation theory[CITATION] so that we should perhaps keep an open mind about the possible forms for [MATH].', 'hep-ph-0603242-1-8-8': 'We will return to this point in what follows.', 'hep-ph-0603242-1-9-0': 'Given a general action of the kind above there are several issues that one normally wants to address in order to extract information that can be compared with experimental data.', 'hep-ph-0603242-1-9-1': 'From studies of both the ADD and RS models there are certain things we want to know, e.g., ([MATH]) the spectrum, wave functions, propagators and Standard Model (SM) matter couplings of the KK graviton (and other possible) excitations and ([MATH]) the relationship between [MATH], the volume of the compactified dimensions and the (reduced) 4-d Planck scale [MATH]M_pl[MATH].', 'hep-ph-0603242-1-9-2': 'To obtain this information, as well as to make contact with several other analyses[CITATION], it is sufficient to expand the general action [MATH] above around the background metric to quadratic order in the curvature to obtain an effective action for the graviton (and like) excitations.', 'hep-ph-0603242-1-9-3': 'At this level, one can extract the relevant 2-point functions as well as the differential equation for the KK wavefunctions which then yield the KK mass spectrum as well as the the desired graviton couplings to SM fields.', 'hep-ph-0603242-1-9-4': 'If, however, one wanted to probe, e.g., graviton 3- or 4-point functions then we would need to expand to at least cubic or quartic order in the curvature, respectively; these will not be of interest to us here but might be of interest in future experiment[CITATION] which would tell us more about the underlying theory of gravity.', 'hep-ph-0603242-1-10-0': 'Once we make this expansion, there are various equivalent ways of expressing the resulting effective action, [MATH], depending upon the basis of invariants we choose to employ; the most obvious form is simply [EQUATION] where [MATH] have been defined above.', 'hep-ph-0603242-1-10-1': '[MATH] is an effective cosmological constant and [MATH] are (in some cases dimensionful) constants all of which are functions of [MATH] and its derivatives evaluated employing the relevant background metric.', 'hep-ph-0603242-1-10-2': "To relate this back to the EH action in the limit of small curvature, one can think of the (necessarily positive) parameter [MATH] as a 'renormalization' of the fundamental mass scale [MATH]: [MATH].", 'hep-ph-0603242-1-11-0': 'A second version of [MATH] is given by [EQUATION] where [MATH] is the well-known Gauss-Bonnet(GB) invariant: [EQUATION]', 'hep-ph-0603242-1-11-1': 'The co-efficients [MATH] and [MATH] can be easily converted to [MATH] above by some straightforward algebra: [MATH] and [MATH].', 'hep-ph-0603242-1-11-2': 'In 4-d, the GB invariant is just a total derivative but this is no longer true for arbitrary values of [MATH].', 'hep-ph-0603242-1-11-3': 'The GB invariant is just (the quadratic) one of a general class called Lovelock invariants, constructed of various powers of the curvature tensor, which lead to special properties for the equations of motion[CITATION].', 'hep-ph-0603242-1-11-4': 'Generally the existence in the action of higher curvature terms, as discussed above, leads to higher order equations of motion that produce tachyonic and/or ghost excitations in the spectrum as well as potentially non-symmetric and/or non-conserved pieces of the corresponding Einstein equations.', 'hep-ph-0603242-1-11-5': 'Having an action consisting solely of Lovelock invariants avoids all of these potential difficulties as well as those associated with the massive tensor ghosts.', 'hep-ph-0603242-1-11-6': 'The D-dimensional scalar excitation discussed above is also absent in this case.', 'hep-ph-0603242-1-11-7': 'It is interesting to note that the GB term is the leading correction to the EH action in perturbative string theory[CITATION].', 'hep-ph-0603242-1-11-8': 'Higher order Lovelock invariant may also be present in the action (when [MATH]) but these cannot be described by the function [MATH] as employed here since they are constructed out of cubic or higher order combinations of the curvature tensor.', 'hep-ph-0603242-1-11-9': 'The effect of the presence of general Lovelock invariants in the action of the ADD model has been discussed within the black hole context in Ref. [CITATION].', 'hep-ph-0603242-1-12-0': 'A further possible form for the quadratic action commonly used in the literature is [EQUATION] where [MATH] is the square of the Weyl tensor which can be expressed as[CITATION]: [EQUATION] where [MATH] is the number of extra dimensions; the [MATH] are linearly related to the coefficients [MATH] above, e.g., [MATH].', 'hep-ph-0603242-1-12-1': 'This translation is simplified via the use of the identity[CITATION] [EQUATION]', 'hep-ph-0603242-1-12-2': 'Noting that the [MATH] have dimensions of mass[MATH] it is sometimes common in the literature to write [EQUATION] where [MATH] are two mass parameters which are naturally [MATH] in the theory.', 'hep-ph-0603242-1-12-3': 'One then finds that [MATH] are directly related to the bulk masses of the D-dimensional massive scalar and tensor excitations discussed above.', 'hep-ph-0603242-1-12-4': 'To avoid tachyons we apparently must demand that [MATH] but even in such a case this the massive tensor field remains a ghost since the kinetic term for this field would have the wrong sign.', 'hep-ph-0603242-1-13-0': 'Clearly all these forms for [MATH] are simply related.', 'hep-ph-0603242-1-13-1': 'In what follows we will make use of all of the above forms of [MATH] and treat them interchangeably.', 'hep-ph-0603242-1-14-0': "Our first goal will be to explicitly calculate [MATH] in one of these 'bases' from the more general [MATH] in terms of [MATH] and its derivatives.", 'hep-ph-0603242-1-14-1': 'To begin we perform a Taylor series expansion of [MATH] to quadratic order in all three arguments evaluating the result in the background metric, e.g., [EQUATION] where [MATH] is a constant corresponding to the evaluation of [MATH] itself in the fixed curvature background metric and [MATH]; [MATH] means that [MATH] is to be evaluated in terms of the background metric which we here assume to be a space of constant curvature, i.e., a maximally symmetric space as is the case in both the ADD and RS models.', 'hep-ph-0603242-1-14-2': 'Thus the quantities [MATH], [MATH], [MATH], [MATH] and [MATH] are just numbers which depend on the explicit form of the metric and possibly the number of extra dimensions.', 'hep-ph-0603242-1-14-3': 'In such a maximally symmetric space the Weyl tensor and corresponding invariant both vanish identically, i.e., [MATH] and one further finds that [EQUATION]', 'hep-ph-0603242-1-14-4': 'Note that in ADD [MATH] since the metric is Minkowskian whereas in the [MATH] RS bulk [MATH] (away from the two branes) where the parameter [MATH] originates from the usual RS metric [MATH].', 'hep-ph-0603242-1-14-5': 'Without making further assumptions we obtain [EQUATION] where [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0603242-1-14-6': 'For the case of [MATH] this reproduces the results give by, e.g., Navarro and Van Acoleyen in [CITATION].', 'hep-ph-0603242-1-14-7': 'Note that if we make the assumption that [MATH] is a function only of [MATH] and the combination [MATH] then [MATH] etc and, also noting that [MATH]=0, we obtain [MATH] so that the remaining expressions greatly simplify; we now obtain [EQUATION] where the parameter [MATH] is given by [EQUATION]', 'hep-ph-0603242-1-14-8': 'Note that having [MATH] implies that the Weyl term, [MATH], in the effective action is absent in second order which is equivalent to taking [MATH] thus eliminating the massive tensor ghost issue.', 'hep-ph-0603242-1-14-9': 'This field is now removed from the spectrum though the D-dimensional scalar remains in general.', 'hep-ph-0603242-1-15-0': '# Application I: ADD', 'hep-ph-0603242-1-16-0': 'In this section we will apply the above analysis to the general ADD framework where we now require (since the space is flat) [MATH] so that [MATH] automatically.', 'hep-ph-0603242-1-16-1': 'This significantly reduces the possible deviations from the classic ADD picture.', 'hep-ph-0603242-1-16-2': 'In this specific case the general second order expansion of [MATH] is rather simple and is given by [EQUATION]', 'hep-ph-0603242-1-16-3': 'Note that if we also demand that [MATH] be a function only of [MATH] and the combination [MATH] in order to avoid the massive tensor ghost issue this expression simplifies even further to [EQUATION]', 'hep-ph-0603242-1-16-4': 'There has been some discussion about other ways to circumvent this tensor ghost problem than by completely eliminating it from the perturbative spectrum.', 'hep-ph-0603242-1-16-5': 'As we are working only to lowest non-trivial order perhaps we should keep an open mind about the forms for [MATH].', 'hep-ph-0603242-1-16-6': 'Note that since [MATH] will essentially rescale the overall mass factor in the action we must demand that [MATH] to insure that the usual D-dimensional massless tensor gravitons not be ghost-like.', 'hep-ph-0603242-1-17-0': 'How are the predictions of ADD modified by these additional curvature terms?', 'hep-ph-0603242-1-17-1': 'The basic ADD picture leads to three essential predictions [CITATION]: ([MATH]) the emission of graviton KK states during the collision of SM particles producing signatures with apparent missing energy[CITATION]; ([MATH]) the exchange of graviton KK excitations between SM brane fields leading to dimension-8 contact interaction-like operators with distinctive spin-2 properties[CITATION]; ([MATH]) the production of black holes(BH) at colliders and in cosmic rays with geometric cross sections, [MATH], with [MATH] being the BH Schwarzschild radius, once collision energies greater than [MATH] are exceeded[CITATION].', 'hep-ph-0603242-1-18-0': 'The production and properties of D-dimensional, TeV-scale BH in higher curvature theories has been partially explored within the context of Lovelock extended gravity[CITATION] though not yet so in the fully general quadratic gravity case described by the function [MATH] considered here.', 'hep-ph-0603242-1-18-1': 'Such a study, which would be very interesting, is far beyond the scope of the present analysis.', 'hep-ph-0603242-1-18-2': 'However, it is interesting to make several observations: ([MATH]) Consider the vacuum solution; if we expand the general action [MATH] above to only quadratic order and if we also assume that all interesting solutions must satisfy [MATH] in the vacuum, then the only deviations from the conventional Schwarzschild form arise from the GB term in the action.', 'hep-ph-0603242-1-18-3': 'This can be seen immediately by examining the equations of motion resulting from the general quadratic action, e.g., in Ref. [CITATION].', 'hep-ph-0603242-1-18-4': 'This result does not remain valid when [MATH] is treated exactly.', 'hep-ph-0603242-1-18-5': '([MATH]) If [MATH] only and is treated exactly without expansion then the equations of motion allow for the conventional external BH result with [MATH] and will appear as an ordinary D-dimensional Schwarzschild solution.', 'hep-ph-0603242-1-18-6': 'This is not the most general solution as can be see by considering the simple case of [MATH] with [MATH] a dimensionless parameter.', 'hep-ph-0603242-1-18-7': 'Here there is also exists a solution with [MATH], which is a constant corresponding to deSitter or anti-deSitter space depending on the sign of [MATH].', 'hep-ph-0603242-1-18-8': '([MATH]) If [MATH], with [MATH] being a constant, then the general BH solution has neither [MATH] nor [MATH] equal to zero as is well-known from the exact solution[CITATION].', 'hep-ph-0603242-1-18-9': 'A more detailed study here would be worthwhile.', 'hep-ph-0603242-1-19-0': 'Let us now consider the situation of graviton exchange where it is well-known[CITATION] that ADD leads to new dimension-8 contact interactions.', 'hep-ph-0603242-1-19-1': 'To obtain the analogous quantities here we must expand the integrand of the action, i.e., the Lagrangian [MATH], to second order in the fluctuations, [MATH], around the flat background metric: [EQUATION]', 'hep-ph-0603242-1-19-2': 'Here we have expressed the original metric as [EQUATION] with [MATH] being the background metric, here identified in the ADD model as the Minkowski metric [MATH].', 'hep-ph-0603242-1-19-3': 'The propagator is then just the inverse of the operator [MATH]; once the propagator is known we sandwich it between two 4-d localized (at the origin of the extra-dimensional co-ordinates) and conserved SM stress-energy sources, [MATH], to find the relevant scattering amplitude remembering to KK decompose the various towers.', 'hep-ph-0603242-1-19-4': 'Fortunately, much of this work has been done for us by Accioly, Azeredo and Mukai[CITATION] from which, with some modifications, we obtain the expression for the D-dimensional graviton exchange amplitude (before performing the KK sums) [EQUATION] where [MATH] are the familiar (unaltered in terms of the radius [MATH]) flat space KK masses.', 'hep-ph-0603242-1-19-5': 'The [MATH] label the various KK levels; [MATH] is the 4-d trace of the SM source stress-energy tensor and [MATH] are just [EQUATION] as described above.', 'hep-ph-0603242-1-19-6': '[MATH] is the compactification radius which sets the KK mass scale; here we have assumed a common value for this quantity for all [MATH] extra dimensions so that the volume of the compactified space is just [MATH].', 'hep-ph-0603242-1-19-7': 'In the expression above the first term in the amplitude is the usual one encountered in the ADD model which results from the D-dimensional EH action and combines the contributions of the 4-d spin-2 graviton and spin-0 graviscalar KK towers.', 'hep-ph-0603242-1-19-8': 'The second and third terms correspond to the new D-dimensional massive tensor and scalar contributions, respectively.', 'hep-ph-0603242-1-19-9': 'The difference in the factors of [MATH] versus [MATH] in the first two terms arises from the existence of a 5-d bulk mass for the tensor ghost field.', 'hep-ph-0603242-1-19-10': 'It is interesting to note that this amplitude is very well behaved at large [MATH] (in fact vanishing) due to the detailed cancellations between the various terms.', 'hep-ph-0603242-1-20-0': 'Note that here [MATH] represent the bulk mass terms of the new fields which enter into the KK tower masses of the scalars (spin-0) and tensors (spin-2 and spin-0), respectively; here we see the effect of the tensor ghost KK tower exchange explicitly.', 'hep-ph-0603242-1-20-1': 'From this point of view it appears that the only way to remove this ghost tower is to take the bulk mass [MATH] implying that [MATH] is solely a function of [MATH] and the combination [MATH], which we will assume from now on in our ADD discussion.', 'hep-ph-0603242-1-20-2': 'Note that the existence of a GB term in the action will not yield a contribution to [MATH].', 'hep-ph-0603242-1-20-3': 'Since, as discussed above [MATH] is already required, tachyonic KK scalars are avoided when the denominator in the expression for [MATH] above is positive; when [MATH] is assumed to be a function only of [MATH] and [MATH], then this denominator simplifies to [MATH].', 'hep-ph-0603242-1-20-4': 'In the limit where [MATH] alone, and accounting for a sign factor in the definition of the above actions, our result for the squared scalar mass, [MATH], agrees with that obtained by Demir and Tanyildizi[CITATION].', 'hep-ph-0603242-1-20-5': 'As shown by these authors, the effect of the new scalar tower exchange is generally rather suppressed in comparison to the more familiar graviton exchange since the ratio [MATH] is small for most SM particle sources at TeV colliders.', 'hep-ph-0603242-1-20-6': 'For example, for the process [MATH] this ratio is of order [MATH].', 'hep-ph-0603242-1-20-7': 'The corresponding ratio of the of the KK summed scalar to graviton exchange amplitudes is somewhat further reduced by ([MATH]) the existence of the finite bulk scalar mass which implies that there are no light scalar KK exchanges with masses below [MATH] and ([MATH]) the [MATH]-dependent numerical factor in the denominator of the scalar amplitude.', 'hep-ph-0603242-1-20-8': 'Naturalness suggests that [MATH] TeV or larger unless the parameters of [MATH] are somehow fine-tuned.', 'hep-ph-0603242-1-20-9': 'For example, if [MATH], then [MATH] for all [MATH] if [MATH] is not too far from O(1).', 'hep-ph-0603242-1-20-10': 'Interestingly we see here that as [MATH] we recover the usual EH expectation as then [MATH].', 'hep-ph-0603242-1-20-11': 'Thus we find that for many practical purposes the structure of the usual ADD results for graviton exchange are not modified when the action is generalized to the form considered here.', 'hep-ph-0603242-1-20-12': 'However, with the existence of these additional scalars being a hallmark of the extended action, it behooves us to find a way to isolate their effects experimentally.', 'hep-ph-0603242-1-21-0': 'In expressions for graviton exchange only the combination [MATH] will now appear.', 'hep-ph-0603242-1-21-1': "In the amplitude this will lead to a modification of the pure 'graviton' exchange cross section expectations by a factor of [MATH], which is likely to be of [MATH], provided [MATH] is considered to be held fixed.", 'hep-ph-0603242-1-21-2': 'When the graviton tower interference term with the SM dominates, the effect in the gravitational part of the cross section will scale as [MATH].', 'hep-ph-0603242-1-21-3': 'Given the previous results of Demir and Tanyildizi[CITATION], this is not surprising.', 'hep-ph-0603242-1-22-0': 'We further note that since [MATH] is [MATH] TeV or larger it has no effect on laboratory measurements of the strength of the gravitational interaction in the micron range when [MATH].', 'hep-ph-0603242-1-23-0': 'Before closing this part of the discussion we would like to remind the reader that it was pointed out long ago[CITATION] that we can take any action of the form [MATH] and map it over to the EH action coupled to an minimally coupled real scalar field with a rather complicated potential [MATH], depending exponentially on the scalar field.', 'hep-ph-0603242-1-23-1': 'This can be done via a special conformal transformation [EQUATION]', 'hep-ph-0603242-1-23-2': 'Going from the original (Jordan) to the new (Einstein) frame one explicitly sees the existence of the new scalar degree of freedom.', 'hep-ph-0603242-1-23-3': 'The mass of this scalar field is exactly that of the field [MATH] above and can be gotten directly from the canonically normalized potential [MATH] in the usual manner, i.e., using [MATH].', 'hep-ph-0603242-1-23-4': 'This is a very powerful tool as it allows us to extend our previous flat space result for [MATH] to the much more general case where the metric is unspecified.', 'hep-ph-0603242-1-23-5': 'For example, if [MATH], we find that the value of [MATH] is the same as discussed above, i.e., [MATH], in a space with an arbitrary metric.', 'hep-ph-0603242-1-23-6': 'This will be an important result that we will employ when we discuss the case of the RS setup.', 'hep-ph-0603242-1-24-0': 'We now turn to the emission of gravitons in SM particle collisions.', 'hep-ph-0603242-1-24-1': 'Since the compactifying space is flat in the ADD case the normalizations of the graviton (and scalar) wavefunctions which control their couplings are unaltered by the existence of the quadratic curvature terms but the relationship between [MATH] and [MATH]M_pl[MATH] is modified.', 'hep-ph-0603242-1-24-2': 'This was briefly mentioned above where we saw that in the small curvature limit the parameter [MATH] essentially renormalizes the fundamental scale.', 'hep-ph-0603242-1-24-3': 'To see this in the present case it is sufficient to examine the tensor/spin-2 kinetic part of the 4-d effective Lagrangian to second order in [MATH] (which has not yet been KK-expanded) in the familiar transverse traceless gauge, i.e., [MATH], [MATH]; one obtains[CITATION] [EQUATION] where here [MATH] and [MATH] are defined above.', 'hep-ph-0603242-1-24-4': 'When we assume that [MATH] is only a function of [MATH] and the combination [MATH] then the second term in [MATH] vanishes and we recover the familiar result of the standard EH scenario apart from the overall factor of [MATH].', 'hep-ph-0603242-1-24-5': 'Hence, to recover the conventional 4-d EH action when inserting the usual (extra dimensionally) flat zero mode graviton wavefunction into [MATH] the ADD relationship must be modified, as hinted above, to [EQUATION] where [MATH]M_pl[MATH] is the 4-d reduced Planck scale and [MATH] is the volume of the compactified space.', 'hep-ph-0603242-1-24-6': 'Of course, [MATH] is just unity in the standard ADD model which employs the EH action.', 'hep-ph-0603242-1-24-7': 'Since the lightest of the KK scalars has a mass which is naturally on the order of a TeV and have rather weak couplings to SM fields these particles will not play much of an important role in missing energy processes.', 'hep-ph-0603242-1-24-8': 'If the cross section for graviton production, i.e., missing energy, is expressed in terms of the original [MATH] with other parameters held fixed, then the presence of [MATH] leads to a modification of the production cross section by a factor of [MATH].', 'hep-ph-0603242-1-24-9': 'However, as [MATH] is not likely to remain a direct observable (only the product [MATH] is) there may be no way to experimentally disentangle this effect.', 'hep-ph-0603242-1-24-10': 'Furthermore, for any given [MATH], since [MATH]M_pl[MATH] is numerically fixed and [MATH] is an input parameter the resulting derived value of [MATH] which sets the scale for the masses of the KK states is altered.', 'hep-ph-0603242-1-25-0': 'We thus conclude that if we assume that [MATH] is a function of only of [MATH] and the combination [MATH] then the classic predictions ([MATH]) and ([MATH]) of the ADD model will be essentially unaffected by going to the more general action considered here except for possible overall scalings by inverse powers of [MATH] when the parameter [MATH] is held fixed: graviton emission rates scale like [MATH] while graviton exchange cross sections scale as [MATH] or [MATH] depending on the presence of important SM contributions to the relevant process.', 'hep-ph-0603242-1-26-0': '# Application II: RS', 'hep-ph-0603242-1-27-0': 'The predictions of the classic RS model are the existence of TeV scale graviton resonances with fixed weak scale masses and couplings to the SM fields[CITATION], the existence of a weak scale radion excitation[CITATION], as well as the production of [MATH] BH.', 'hep-ph-0603242-1-27-1': 'In what follows we will be specifically interested in the nature of the KK gravitons so it is again sufficient to examine the quadratically expanded action.', 'hep-ph-0603242-1-27-2': 'The classic RS model is not generally consistent with the assumed form of either the original action [MATH] or its quadratically expanded form [MATH].', 'hep-ph-0603242-1-27-3': 'As is well-known, and as mentioned above, the equations of motion that follow from [MATH] and [MATH] will generally be fourth order in the derivatives of the metric.', 'hep-ph-0603242-1-27-4': 'In the usual 5-d RS model, one solves the Einstein equations of the form [EQUATION] where [MATH] is the Einstein tensor arising from the EH action involving no more than two derivatives of the metric.', 'hep-ph-0603242-1-27-5': 'The problem is that RS completely specifies [MATH]: a cosmological constant in the 5-d bulk plus two [MATH]-function sources at the orbifold locations of the TeV and Planck branes.', 'hep-ph-0603242-1-27-6': 'SM matter confined to the TeV brane is supposed to not be a large contributor to the stress-energy.', 'hep-ph-0603242-1-27-7': 'To obtain this result the standard RS metric takes the form discussed above: [MATH] with the linear exponential warp factor leading to the bulk [MATH] and the two field derivatives acting on the absolute value leading to the brane [MATH]-functions.', 'hep-ph-0603242-1-27-8': '(This is related to the comment above that [MATH] is not truly constant in RS and has brane [MATH]-function singularities.', 'hep-ph-0603242-1-27-9': 'Recall that these [MATH]-functions are the results of assuming infinitely thin branes.)', 'hep-ph-0603242-1-27-10': 'If an identical metric is assumed in our more general case we still can obtain [MATH] but the fourth order equations would lead to the more singular derivatives of [MATH]-functions at the brane locations.', 'hep-ph-0603242-1-27-11': 'This amongst other reasons is what led Kim, Kyae and Lee[CITATION] to consider only GB extensions of the EH action in RS since it alone only produces Einstein equations of second order in the derivatives.', 'hep-ph-0603242-1-27-12': 'Thus if we keep the classic picture an analysis of RS given our assumed effective action expanded around a background of constant curvature is not relevant.', 'hep-ph-0603242-1-27-13': '(A possible way of dealing with these derivatives of [MATH]-functions arising from orbifold singularities in higher dimensional effective field theories has been discussed in Ref. [CITATION].', 'hep-ph-0603242-1-27-14': 'Implementing our scheme employing such techniques is, however, beyond the scope of the present paper.)', 'hep-ph-0603242-1-28-0': "To avoid these issues for now we simplify our discussion of this problem (and to convince ourselves that an RS-like solution is possible in this framework) we consider a singularity-free, 'softened' version of RS where the orbifolded bulk space with branes is replaced by an interval, as has been suggested for other reasons[CITATION], with SM matter placed at one singular end point possessing an ignorable amount of stress-energy.", 'hep-ph-0603242-1-28-1': 'With a cosmological constant on the interval we can recover the background [MATH] bulk; in addition by removing the absolute value sign of the co-ordinate [MATH] in the metric above we expunge the [MATH]-functions as well as the possibility of any of their higher derivatives appearing in the equations of motion.', 'hep-ph-0603242-1-28-2': 'The boundary conditions at the end points for the graviton KK states can then be freely chosen to be the same as that of the original RS model.', 'hep-ph-0603242-1-28-3': 'This space is truly one of constant curvature and the general analysis we have presented above will now be applicable to this softened RS.', 'hep-ph-0603242-1-29-0': 'It is easy to verify that the form of the equations of motion[CITATION] in this case (recalling that we are only searching for solutions with constant [MATH] backgrounds) are given by: [EQUATION] and that if we take stress-energy tensor in the 5d bulk to be of the usual RS form [EQUATION] with [MATH], then indeed a space of constant curvature, i.e. [MATH], can be an allowed solution.', 'hep-ph-0603242-1-29-1': 'Taking the trace of the equations of motion above, evaluating it in the constant curvature bulk and relating the values of [MATH] to [MATH] as before (and recalling that here [MATH] using the softened metric) results in the constraint equation [EQUATION]', 'hep-ph-0603242-1-29-2': 'It is interesting to note that if we assume that [MATH]constant then this constraint equation automatically implies that [MATH]constant also but not necessarily with the further requirement that [MATH] are constants following from the equations of motion assuming that they are at most functions of [MATH].', 'hep-ph-0603242-1-29-3': 'When [MATH] is only a function of [MATH] and the combination [MATH], this constraint equation simplifies to [EQUATION] while in the specific RS background case this becomes, explicitly [EQUATION]', 'hep-ph-0603242-1-29-4': 'It is important to recall that [MATH] itself can be a complicated function of [MATH] so that this equation can be quite nontrivial.', 'hep-ph-0603242-1-29-5': 'For the EH action limit this yields the usual relation that [MATH]; here it in general provides an additional constraint on the allowed forms of the function [MATH] since we are requiring [MATH] to be both real and negative.', 'hep-ph-0603242-1-29-6': 'Given a specific function [MATH] for which a solution exists, this equation directly relates [MATH] and [MATH] though the solution may not be unique.', 'hep-ph-0603242-1-29-7': 'For example, if we assume the simple case of [EQUATION] as employed above, then there are two branches of solutions for [MATH]: [EQUATION] one of which (the negative root) goes over to the usual EH result as the parameter [MATH].', 'hep-ph-0603242-1-30-0': 'Allowing for the possibility of a RS-like solution with a softened metric it is interesting to think briefly about the previously analyzed effects of the GB term in the RS scenario.', 'hep-ph-0603242-1-30-1': 'This analysis was originally performed for the classic RS[CITATION] setup which employed the standard form of the RS metric; that result would now be modified by the changes in the model assumptions, i.e., moving to an interval and removing the [MATH]-function sources at the end points.', 'hep-ph-0603242-1-30-2': 'The previous analysis of BH in RS with the added GB term would not be significantly affected if this transition were made.', 'hep-ph-0603242-1-30-3': 'However, the properties and spectrum of the graviton KK states certainly would be influenced since the [MATH]-function terms are now absent.', 'hep-ph-0603242-1-30-4': 'The equation governing the masses and wavefunction of the graviton KK states for the present interval case can be obtained by expanding the equations of motion as before.', 'hep-ph-0603242-1-30-5': 'Since we are here only interested in the tensor modes associated with the usual gravitons, we can employ the expansion [EQUATION] where [MATH].', 'hep-ph-0603242-1-30-6': 'Applying the usual RS boundary conditions on the interval the most significant changes from the classic RS can be read off from Eqs. (15)-(28) in Ref. [CITATION] by setting the parameter [MATH] in appropriate places.', 'hep-ph-0603242-1-30-7': 'At the end of the day we find that the only apparent difference from the classic EH based RS model would be a shift in the relationship between the fundamental scale and [MATH]M_pl[MATH]-remarkably similar to what we saw for the ADD model above.', 'hep-ph-0603242-1-30-8': 'In the language employed in Ref. [CITATION] we would now obtain [EQUATION] where [MATH] is the coefficient of the GB term in the action.', 'hep-ph-0603242-1-30-9': 'Otherwise the masses as well as the couplings of all of the KK gravitons to localized SM matter would be identical to those of the original RS model expressed in terms of the derived parameter [MATH].', 'hep-ph-0603242-1-30-10': 'The explicit coupling and spectrum changes found in Ref[CITATION] for the graviton KK states in the presence of the GB term in the action were all found to due to the brane [MATH]-function singularities.', 'hep-ph-0603242-1-31-0': 'How would these graviton KK results obtained in the GB extended action generalize to the case of [MATH] above?', 'hep-ph-0603242-1-31-1': 'Here we choose to begin our analysis with [MATH], setting [MATH] from the beginning to avoid potential ghost fields, then taking [MATH] and making use the same curvature expansion as above.', 'hep-ph-0603242-1-31-2': 'In order to make a connection with the previous discussion, the existing RS literature and to directly compare with the GB case, however, we massage our notation slightly and rewrite [MATH] in the following form: [EQUATION] where the parameters [MATH] and [MATH] are dimensionless; the action employed in Ref. [CITATION] is now directly recovered by taking the [MATH] and [MATH] limits.', 'hep-ph-0603242-1-31-3': 'It is important at this point to recall that to obtain the linearized graviton equations of motion it is sufficient to employ [MATH] while the complete [MATH] needs to be examined in order to demonstrate the existence of the required [MATH] solution.', 'hep-ph-0603242-1-31-4': 'The equations of motion resulting from [MATH] are given by[CITATION] [EQUATION]', 'hep-ph-0603242-1-31-5': 'Here [MATH] is the covariant derivative operator and here [MATH].', 'hep-ph-0603242-1-31-6': 'First we look at the [MATH] component of this equation, remembering that for the moment we will only be interested in the tensor excitations corresponding to the KK gravitons.', 'hep-ph-0603242-1-31-7': 'In the usual gauge, [MATH] in this case is a constant to linear order so we arrive at a consistency condition [EQUATION]', 'hep-ph-0603242-1-31-8': 'Note that this reduces to the previously obtained purely quadratic GB extended RS result[CITATION] when [MATH].', 'hep-ph-0603242-1-31-9': 'In the more general case, this expression is not overly useful given the exact result in Eq. (27).', 'hep-ph-0603242-1-31-10': 'Turning now to the [MATH] terms which contain the 4-d graviton tensor excitation, we linearize employing the previously mentioned transverse, traceless gauge with constant [MATH].', 'hep-ph-0603242-1-31-11': 'This gives the standard equation of motion for the RS graviton found long ago[CITATION] though scaled by an overall factor.', 'hep-ph-0603242-1-31-12': 'Employing the standard KK decomposition [EQUATION] and recalling that [MATH], the [MATH] are seen to satisfy [EQUATION]', 'hep-ph-0603242-1-31-13': 'The overall factor [MATH] is given by [EQUATION] or, more explicitly in the RS case, [EQUATION] (Again recall that [MATH] itself can be a function of [MATH].)', 'hep-ph-0603242-1-31-14': 'This leads to a rescaling of the usual RS relationship [EQUATION] via the renormalization of the zero mode (i.e., massless graviton) wavefunction, thus generalizing Eq. (31).', 'hep-ph-0603242-1-31-15': 'Of course, [MATH] is required to avoid ghost states.', 'hep-ph-0603242-1-31-16': 'This result reduces to that previously obtained in the RS case with just the added GB term[CITATION] once boundary effects are neglected.', 'hep-ph-0603242-1-31-17': 'From this analysis we see immediately that the masses of the KK gravitons are identical to those obtained in the original RS model, provided we use the same value of the parameter [MATH], as we might have expected.', 'hep-ph-0603242-1-31-18': 'Here we are faced with the question of what are the independent parameters.', 'hep-ph-0603242-1-31-19': '[MATH] is clearly a derived parameter that is obtained by solving Eq. (27) for any given model.', 'hep-ph-0603242-1-31-20': 'In that sense, the KK graviton spectrum would just be rescaled in comparison to the usual expectations given the same input values of [MATH]etc.', 'hep-ph-0603242-1-31-21': 'As we have just seen, and as in the ADD case, the effect of a factor like [MATH] on the KK graviton couplings to 4-d SM matter again depends upon which model parameters are held fixed.', 'hep-ph-0603242-1-31-22': 'At the very least, up to an overall constant, these couplings are identical to those of the standard RS model.', 'hep-ph-0603242-1-32-0': 'So far we have only considered the 4-d graviton, spin-2 excitations.', 'hep-ph-0603242-1-32-1': 'It is important to remember that our softened RS model now has an additional massive scalar in the 5d spectrum with a large bulk mass, [MATH], so that no massless scalar zero mode will exist.', 'hep-ph-0603242-1-32-2': 'Since the bulk scalar mass is naturally of order [MATH] the KK spectrum of the corresponding tower will begin with a KK scalar state whose mass is qualitatively comparable to that of the first graviton excitation.', 'hep-ph-0603242-1-32-3': 'This bulk mass is explicitly calculable from expansion of the full action to quadratic order, [MATH], by going to the Einstein frame since we know that the GB term does not contribute.', 'hep-ph-0603242-1-32-4': 'In that case, using the results from the previous section we find that [EQUATION] or, in terms of the original parameters of the action, evaluated in the RS background: [EQUATION]', 'hep-ph-0603242-1-32-5': 'Note that [MATH] is required to avoid the scalar tachyons and graviton ghosts, consistent with our above analysis.', 'hep-ph-0603242-1-32-6': 'Note further that this reproduces the results of Eq. (19) in the flat space, [MATH], limit.', 'hep-ph-0603242-1-33-0': 'Given this bulk mass we can determine the mass(es) of the lightest KK scalar state(s), by following the standard RS manipulations[CITATION].', 'hep-ph-0603242-1-33-1': 'These masses are essentially given by the first roots of the equation [EQUATION] where [MATH] and [MATH] is the usual Bessel function.', 'hep-ph-0603242-1-33-2': 'The solution for the first KK state is provided by Fig.1; as stated above there are no massless modes.', 'hep-ph-0603242-1-33-3': 'The lightest scalar mass is then [MATH].', 'hep-ph-0603242-1-33-4': 'Here we observe that the mass of the first scalar KK scales almost linearly with the bulk mass when [MATH] gets large.', 'hep-ph-0603242-1-33-5': 'Note that for [MATH] and a typical value[CITATION] of [MATH], we then find [MATH] implying [MATH] from Fig.1; this is about 3 times larger than that for the usual lightest massive KK graviton, [MATH].', 'hep-ph-0603242-1-33-6': 'Here we see that unless [MATH] takes on large values the first scalar KK state is always rather heavy.', 'hep-ph-0603242-1-33-7': 'As is well-known, the [MATH] values for the more massive KK scalar states will be somewhat larger: approximately given by [MATH] where [MATH] labels the KK level.', 'hep-ph-0603242-1-33-8': 'Since these scalars will couple to the trace of the stress-energy tensor for the 4-d SM fields they will interact far more weakly than do the graviton KK states unless this is at least partially offset by ratios of 5-d wavefunction factors.', 'hep-ph-0603242-1-33-9': 'A quick estimate of such factors, however, indicates that, if anything, these wave function ratios lead to a further suppression of the scalar couplings relative to those of the KK gravitons by [MATH] as shown in Fig.2.', 'hep-ph-0603242-1-33-10': 'This overall picture of the scalar sector is qualitatively very similar to that of the existence of a very heavy tower of RS radions[CITATION] or a tower of KK Higgs bosons as in the case of Universal Warped Extra Dimensions[CITATION].', 'hep-ph-0603242-1-34-0': 'In the analysis as presented here we have ignored the possibility that the new scalar KK states may mix with the (usually eaten) RS graviscalars through cross-talk in the equations of motion, i.e., we have assumed that the 5-d tensor and scalar KK decompositions can be performed independently, and this is something which needs further exploration.', 'hep-ph-0603242-1-34-1': 'A fully detailed analysis of the such possibilities is, however, beyond the scope of the present paper.', 'hep-ph-0603242-1-35-0': '# Discussion and Conclusions', 'hep-ph-0603242-1-36-0': 'In this paper we have begun an examination of how generic higher curvature terms in the gravitational action can alter the predictions of both the ADD model and the RS model defined on a interval to avoid possible brane singularities.', 'hep-ph-0603242-1-36-1': 'We have assumed that the traditional assumptions of the two models, e.g., SM localized matter in a conformally flat bulk, remain valid; we have not considered more complex setups that may now be allowed by the modified equations of motion.', 'hep-ph-0603242-1-36-2': 'To be more concrete, we have further assumed that the EH action is generalized generalized to an action which is of the form [MATH] where [MATH] is a well-behaved function, [MATH] and [MATH].', 'hep-ph-0603242-1-36-3': 'In D-dimensions this action results in a propagating massless tensor field (identified with the usual graviton), a massive ghost tensor field, as well as a massive (possibly tachyonic) scalar.', 'hep-ph-0603242-1-36-4': 'The potentially dangerous ghost is removable from the spectrum, i.e., it becomes infinitely massive, if we demand that [MATH] only.', 'hep-ph-0603242-1-36-5': 'The remaining new scalar field has a bulk mass whose value is naturally expected to be of order the fundamental scale, [MATH], in either scenario.', 'hep-ph-0603242-1-36-6': 'The resulting ADD and RS models are altered in similar ways from their traditional standard forms:', 'hep-ph-0603242-1-37-0': '([MATH]) New scalar KK excitations appear in the spectrum of both models in a rather benign fashion coupling to the trace of the stress-energy tensor of the localized SM fields.', 'hep-ph-0603242-1-37-1': 'Since this trace is proportional to SM masses, the couplings of these scalars are relatively strongly suppressed in comparison to those for the KK gravitons at typical collider energies in both models.', 'hep-ph-0603242-1-37-2': 'In the ADD model, the KK scalar excitations begin at a mass [MATH] TeV.', 'hep-ph-0603242-1-37-3': 'Consequently their contributions to missing-energy signatures as well as to the usual dimension-8 contact interactions are further kinematically suppressed.', 'hep-ph-0603242-1-37-4': 'Thus at leading order these new scalars do not much influence ADD collider signatures.', 'hep-ph-0603242-1-37-5': 'In RS, the bulk scalar mass tends to be large so that the lightest scalar KK is several times more massive than is the lightest KK graviton.', 'hep-ph-0603242-1-37-6': 'Given their rather weak couplings such states will be difficult to observe at colliders.', 'hep-ph-0603242-1-38-0': '([MATH]) The basic model relationships involving the fundamental and 4-d Planck masses in both models get rescaled by functions of [MATH] and its derivatives evaluated in the corresponding background metric of the two models: in ADD we obtain [MATH]M_pl[MATH] while in RS we obtain [MATH]M_pl[MATH] where [MATH] is explicitly given in Eq. (39).', 'hep-ph-0603242-1-38-1': 'Assuming that [MATH] is a fixed fundamental parameter these modifications lead to changes in the graviton KK sectors of both models.', 'hep-ph-0603242-1-38-2': 'In the ADD case, since [MATH]M_pl[MATH] is known and [MATH] is an input parameter for any given [MATH] the volume of the compactified space and, hence, the value of the compactification radius which sets the graviton KK mass scale is altered.', 'hep-ph-0603242-1-38-3': 'Due to the presence of the [MATH] factor the emission rate for gravitons in the collisions of SM particles and for the graviton exchange amplitude are both modified by potentially O(1) effects.', 'hep-ph-0603242-1-38-4': 'Similarly in RS, [MATH] is a derived parameter which sets the scale for all the KK states.', 'hep-ph-0603242-1-38-5': 'The constraint Eq. (28) allows us to calculate [MATH] in terms of the input parameters [MATH] and the function [MATH] thus providing for us with [MATH].', 'hep-ph-0603242-1-38-6': 'In a manner similar to ADD, the presence of [MATH] rescales the coupling strengths of the of the KK graviton states to the SM fields thus modifying the widths and production cross sections at colliders by potentially O(1) factors.', 'hep-ph-0603242-1-39-0': 'As we have seen, the extension of the EH action to a more complicated structure can lead to significant modifications to both the ADD and RS model predictions in the simplest possible case.', 'hep-ph-0603242-1-39-1': 'The observation of such effects at future colliders could tell us valuable information about the underlying theory of gravity.', 'hep-ph-0603242-1-40-0': 'Note Added: After this paper was essentially completed, Ref. [CITATION] appeared which discusses generalised actions for the ADD model and thus has some common areas with the present work.', 'hep-ph-0603242-1-40-1': 'Where the two papers overlap there is general qualitative agreement though the points of view are somewhat different.'} | {'hep-ph-0603242-2-0-0': 'We begin a general exploration of the phenomenology of TeV-scale extra-dimensional models with gravitational actions that contain higher curvature terms.', 'hep-ph-0603242-2-0-1': 'In particular, we examine how the classic collider signatures of the models of Arkani-Hamed, Dimopoulos and Dvali (missing energy and new dimension-8 contact interactions) and of Randall and Sundrum (TeV-scale graviton Kaluza-Klein resonances) are altered by these modifications to the usual Einstein-Hilbert action.', 'hep-ph-0603242-2-0-2': 'We find that not only are the detailed signatures for these gravitationally induced processes altered but new contributions are found to arise due to the existence of additional scalar Kaluza-Klein states in the spectrum.', 'hep-ph-0603242-2-1-0': '# Introduction and Background', 'hep-ph-0603242-2-2-0': 'The question as to why the Planck and electroweak scales differ by so many orders of magnitude remains mysterious.', 'hep-ph-0603242-2-2-1': 'In recent years, attempts have been made to address this hierarchy issue within the context of theories with extra spatial dimensions that lower the effective scale of gravity to the TeV region.', 'hep-ph-0603242-2-2-2': 'In both the models of Arkani-Hamed, Dimopoulos and Dvali (ADD)[CITATION] and of Randall and Sundrum (RS)[CITATION], new effects of gravitational origin are expected to occur near the TeV scale which should be observable at future colliders such as the LHC and ILC.', 'hep-ph-0603242-2-2-3': 'Though these two models are very different in detail they do have some common features the most important of which are: ([MATH]) in their original versions they both assume that Standard Model matter is confined to a 4-dimensional brane; ([MATH]) they both assume that D-dimensional gravity is described by the Einstein-Hilbert (EH) action plus a possible cosmological constant and ([MATH]) the background spaces are maximally symmetric and are either strictly flat, i.e. , Minkowskian as in the ADD model with toroidally flat compactification, or being of constant curvature and is conformally flat, i.e., [MATH] as in the RS model.', 'hep-ph-0603242-2-2-4': 'How would the predictions of these two models be changed if we surrendered the assumption ([MATH]), i.e., that the EH action provided the full description of gravity and considered something more general?', 'hep-ph-0603242-2-2-5': 'This is the discussion we would like to begin in this paper which will follow a phenomenological bottom-up approach.', 'hep-ph-0603242-2-3-0': 'General Relativity (GR) as described by the EH action is considered to be an effective theory below the fundamental Planck scale, [MATH].', 'hep-ph-0603242-2-3-1': 'Thus, once energies approaching the scale [MATH] begin to be probed one might expect to observe deviations from the expectations arising from the EH action.', 'hep-ph-0603242-2-3-2': 'In the cases of both the ADD and RS models, future colliders will probe near or at their (effective) fundamental scales so that non-EH aspects of the true gravitational theory, whatever its form, should become apparent and be experimentally measured.', 'hep-ph-0603242-2-3-3': 'Since the ultraviolet form of the true gravity theory is as of yet unknown one may hope to capture some of its deeper aspects by considering how the presence of new higher curvature (and higher derivative) invariants in the actions of the ADD and RS models can lead to variations in the well-known predictions of these theories.', 'hep-ph-0603242-2-3-4': 'Many authors have considered the possibility of higher curvature invariants and how their existence would modify the predictions arising from the EH action within other contexts, e.g., the properties of black holes[CITATION], deviations in solar system tests of GR[CITATION] and in cosmology[CITATION] to possibly avoid the need for dark energy.', 'hep-ph-0603242-2-3-5': 'Some analyses along these lines for the potential modifications of the collider predictions of both the ADD and RS models have already been performed[CITATION].', 'hep-ph-0603242-2-3-6': 'In the present paper, we wish to both extend and generalize these results to get a feeling for the possible detailed variation of the various new gravitational phenomena as predicted by these classic models which will be potentially observable at future colliders.', 'hep-ph-0603242-2-3-7': 'In particular we are interested in how the well known signatures of both the ADD and RS models are morphed if we keep the basic setups intact but modify the actions on which the corresponding equations of motion are based.', 'hep-ph-0603242-2-3-8': 'A further generalization of such an analysis is possible if the original ADD/RS setups can now be simultaneously surrendered due to these modified actions resulting in entirely new setups with corresponding equations of motion; while this is an interesting possibility to consider it lies mostly beyond the scope of the present paper though it will be touched upon briefly in the discussion below.', 'hep-ph-0603242-2-4-0': 'Of course a completely general study of how these possible modifications to the effective gravity action may morph TeV collider signatures is an obviously immense task and here we aim only at a first round analysis in the discussion that follows.', 'hep-ph-0603242-2-4-1': 'The major signatures arising in both ADD and RS models originate from graviton exchange and the production of black holes; the ADD model also leads to missing energy signatures from graviton emission.', 'hep-ph-0603242-2-4-2': 'Fortunately, apart from issues associated with black holes, since we are dealing with maximally symmetric spaces of globally constant curvature, the relevant graviton properties (couplings, wavefunctions and propagators) necessary to extract experimental signatures for either model can be obtained from the expansion of the rather general action considered below to quadratic order in the curvature.', 'hep-ph-0603242-2-4-3': '(This would no longer be true if we wanted to consider, e.g., the triple graviton coupling as then an expansion to third order in the curvature would be required.)', 'hep-ph-0603242-2-4-4': 'This simplifying observation forms the basis of the analysis that follows and allows us to determine the relevant graviton properties in both ADD- and RS-like models for a wide class of effective actions.', 'hep-ph-0603242-2-5-0': 'The general outline of our analysis is presented in Section 2 where our basic assumptions and notations are also given.', 'hep-ph-0603242-2-5-1': 'In Section 3 we apply our analysis to the ADD model; we then apply it to the RS case in Section 4.', 'hep-ph-0603242-2-5-2': 'Our summary and conclusions are given in Section 5.', 'hep-ph-0603242-2-6-0': '# Analysis', 'hep-ph-0603242-2-7-0': 'When going beyond the EH action there are many possibilities to consider especially when we are living in extra dimensions.', 'hep-ph-0603242-2-7-1': 'In the literature various forms have been assumed for the potential structures of higher curvature and/or derivative invariants that may appear in the gravity action.', 'hep-ph-0603242-2-7-2': 'The fairly general structure that we will assume for the D-dimensional action in the present analysis takes the form (with [MATH] below): [EQUATION] where [MATH] is an arbitrary continuous, differentiable and generally mathematically well-behaved function; in particular we will assume that [MATH] is non-singular when all of its arguments are zero.', 'hep-ph-0603242-2-7-3': 'Here [MATH] is the usual D-dimensional Ricci scalar while [MATH] and [MATH] are quadratic invariants constructed from the curvature tensor [MATH]: [MATH], with [MATH] being the Ricci tensor, while [MATH].', 'hep-ph-0603242-2-7-4': '[MATH] is the D-dimensional fundamental gravity scale which is [MATH] TeV in ADD and [MATH]M_pl[MATH] in RS.', 'hep-ph-0603242-2-7-5': 'In the low energy, small curvature limit we expect [MATH] (plus a possible cosmological constant) and so the overall dimensionful factor in the expression above allows us to make direct contact with the EH action in this limit.', 'hep-ph-0603242-2-7-6': 'This specific form for [MATH], though not completely general, covers a wide array of possibilities and has been considered (sometimes only in [MATH]) in may different contexts for a multitude of purposes in the literature[CITATION].', 'hep-ph-0603242-2-7-7': 'Many of the higher curvature models previous considered by other authors form subcases of this more general action.', 'hep-ph-0603242-2-8-0': 'As is by now well-known[CITATION] the generalized gravity theories described by an action of the form [MATH] can potentially have several serious problems.', 'hep-ph-0603242-2-8-1': 'Since we will be dealing with ADD- and RS-type phenomenology we will be performing a perturbative analysis in the discussion that follows.', 'hep-ph-0603242-2-8-2': 'Employing such an analysis one finds that, amongst other things, this action leads to equations of motion which are generally fourth order in the derivatives of the metric.', 'hep-ph-0603242-2-8-3': 'In particular, in addition to the usual massless D-dimensional tensor graviton which results in, e.g., the familiar 4-d graviton and graviscalar Kaluza-Klein (KK) tower excitations, there may also be present in the linearized D-dimensional theory additional massive scalar and tensor excitations.', 'hep-ph-0603242-2-8-4': 'These fields will in 4-d have KK towers without massless modes and which can be ghostlike and/or tachyonic.', 'hep-ph-0603242-2-8-5': '(We can think of these new D-dimensional fields as having bulk masses which influence their corresponding 4-d KK tower masses.)', 'hep-ph-0603242-2-8-6': 'Furthermore, the equations of motion naturally involving higher derivatives of the fields can lead to problems with unitarity as well as non-canceling asymmetric pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-2-8-7': "The new massive tensor excitations are potentially the most serious issue to deal with as they are ghost fields that must be eliminated from the perturbative spectrum (though they may help in dealing with the theory's renormalizability and bad high-energy behavior).", 'hep-ph-0603242-2-8-8': 'It has been noted[CITATION] that we one can remove these states from the spectrum (i.e., by giving their bulk masses an infinite value) if a tuning occurs such that the quantities [MATH] and [MATH] only appear in the special combination [MATH] in the function [MATH].', 'hep-ph-0603242-2-8-9': 'How this tuning arises in the fundamental theory is unspecified.', 'hep-ph-0603242-2-8-10': 'There has been some discussion in the literature, however, that these ghost states may not be as dangerous as one would imagine from lowest order perturbation theory[CITATION] so that we should perhaps keep an open mind about the possible forms for [MATH].', 'hep-ph-0603242-2-8-11': 'We will return to this point in what follows.', 'hep-ph-0603242-2-9-0': 'Given a general action of the kind above there are several issues that one normally wants to address in order to extract information that can be compared with experimental data.', 'hep-ph-0603242-2-9-1': 'From studies of both the ADD and RS models there are certain things we want to know, e.g., ([MATH]) the spectrum, wave functions, propagators and Standard Model (SM) matter couplings of the KK graviton (and other possible) excitations and ([MATH]) the relationship between [MATH], the volume of the compactified dimensions and the (reduced) 4-d Planck scale [MATH]M_pl[MATH].', 'hep-ph-0603242-2-9-2': 'To obtain this information, as well as to make contact with several other analyses[CITATION], it is sufficient to expand the general action [MATH] above around the maximally symmetric background metric to quadratic order in the curvature to obtain an effective action for the graviton (and like) excitations.', 'hep-ph-0603242-2-9-3': 'At this level, one can extract the relevant 2-point functions as well as the differential equation for the KK wavefunctions which then yield the KK mass spectrum as well as the the desired graviton couplings to the localized SM fields.', 'hep-ph-0603242-2-9-4': 'If, however, one wanted to probe, e.g., graviton 3- or 4-point functions then we would need to expand to at least cubic or quartic order in the curvature, respectively; these will not be of interest to us here but might be of interest in future experiments[CITATION] which would tell us more about the underlying theory of gravity.', 'hep-ph-0603242-2-10-0': 'Once we make this expansion, there are various equivalent ways of expressing the resulting effective action, [MATH], depending upon the basis of invariants we choose to employ; the most obvious form is simply [EQUATION] where [MATH] have been defined above.', 'hep-ph-0603242-2-10-1': '[MATH] is an effective cosmological constant and [MATH] are (in some cases dimensionful) constants all of which are functions of [MATH] and its derivatives evaluated by employing the relevant background metric.', 'hep-ph-0603242-2-10-2': "To relate this back to the EH action in the limit of small curvature, one can think of the (necessarily positive) parameter [MATH] as a 'renormalization' of the fundamental mass scale [MATH]: [MATH].", 'hep-ph-0603242-2-11-0': 'A second and perhaps more useful version of [MATH] is given by [EQUATION] where [MATH] is the well-known Gauss-Bonnet(GB) invariant: [EQUATION]', 'hep-ph-0603242-2-11-1': 'The co-efficients [MATH] and [MATH] can be easily converted to [MATH] above by some straightforward algebra: [MATH] and [MATH].', 'hep-ph-0603242-2-11-2': 'In D[MATH]4, the GB invariant is either a topological invariant or just a total derivative not contributing to the equations of motion, but this is no longer true for arbitrary values of [MATH].', 'hep-ph-0603242-2-11-3': 'The GB invariant is just (the quadratic) member of a general class of Lovelock invariants, constructed of various powers of the curvature tensor, which lead to special properties for the equations of motion[CITATION].', 'hep-ph-0603242-2-11-4': 'Generally the existence in the action of higher curvature terms, as discussed above, leads to higher order equations of motion that produce tachyonic and/or ghost excitations in the spectrum as well as potentially non-symmetric and/or non-conserved pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-2-11-5': 'Having an action consisting solely of Lovelock invariants avoids all of these potential difficulties as well as those associated with the massive tensor ghosts.', 'hep-ph-0603242-2-11-6': 'The D-dimensional scalar excitation discussed above is also absent in this case.', 'hep-ph-0603242-2-11-7': 'It is interesting to note that the GB term is the leading correction to the EH action in perturbative string theory[CITATION].', 'hep-ph-0603242-2-11-8': 'Higher order Lovelock invariant may also be present in the action (when [MATH]) but these cannot be described by the function [MATH] as employed here since they are constructed out of cubic or higher order combinations of the curvature tensor.', 'hep-ph-0603242-2-11-9': 'The effect of the presence of general Lovelock invariants in the action of the ADD model has been discussed within the black hole context in Ref. [CITATION].', 'hep-ph-0603242-2-12-0': 'A further possibly very useful form for the quadratic action that is commonly used in the literature is [EQUATION] where [MATH] is the square of the Weyl tensor which can be expressed as[CITATION]: [EQUATION] where [MATH] is the number of extra dimensions; the [MATH] are linearly related to the coefficients [MATH] above, e.g., [MATH].', 'hep-ph-0603242-2-12-1': 'This translation is simplified via the use of the identity[CITATION] [EQUATION]', 'hep-ph-0603242-2-12-2': 'Noting that the [MATH] have dimensions of mass[MATH] it is sometimes common in the literature to write [EQUATION] where [MATH] are two mass parameters which are naturally [MATH] in the theory.', 'hep-ph-0603242-2-12-3': 'One then finds that [MATH] are directly related to the bulk masses of the D-dimensional massive scalar and tensor excitations discussed above.', 'hep-ph-0603242-2-12-4': 'To avoid tachyons we apparently must demand that [MATH] but even in such a case as we will see this the massive tensor field remains a ghost since the kinetic term for this field will have the wrong sign.', 'hep-ph-0603242-2-13-0': 'Clearly all these forms for [MATH] are simply related.', 'hep-ph-0603242-2-13-1': 'In what follows we will make use of all of the above forms of [MATH] and treat them interchangeably.', 'hep-ph-0603242-2-14-0': "Our first goal will be to explicitly calculate [MATH] in one of these 'bases' from the more general [MATH] in terms of [MATH] and its derivatives.", 'hep-ph-0603242-2-14-1': 'To begin we perform a Taylor series expansion of [MATH] to quadratic order in all three arguments evaluating the result in the background metric, e.g., [EQUATION] where [MATH] is a constant corresponding to the evaluation of [MATH] itself in the fixed curvature background metric and [MATH]; [MATH] means that [MATH] is to be evaluated in terms of the background metric which we here assume to be a space of constant curvature, i.e., a maximally symmetric space as is the case in both the ADD and RS models.', 'hep-ph-0603242-2-14-2': 'Thus the quantities [MATH], [MATH], [MATH], [MATH] and [MATH] are just numbers which depend on the explicit form of the metric and possibly the number of extra dimensions.', 'hep-ph-0603242-2-14-3': 'In such a maximally symmetric space the Weyl tensor and corresponding invariant both vanish identically, i.e., [MATH] and one further finds that [EQUATION]', 'hep-ph-0603242-2-14-4': 'Note that in ADD [MATH] since the metric is Minkowskian whereas in the [MATH] RS bulk [MATH] (away from the two branes) where the parameter [MATH] originates from the usual RS metric [MATH].', 'hep-ph-0603242-2-15-0': 'Without making any further assumptions we obtain [EQUATION] where we have defined [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0603242-2-15-1': 'For the case of [MATH] this reproduces the results give by, e.g., Navarro and Van Acoleyen in [CITATION].', 'hep-ph-0603242-2-15-2': 'Note that if we make the assumption that [MATH] is a function only of [MATH] and the combination [MATH] then [MATH] etc and, also noting that [MATH]=0, we obtain [MATH] so that the remaining expressions greatly simplify; we now obtain [EQUATION] where the parameter [MATH] is given by [EQUATION]', 'hep-ph-0603242-2-15-3': 'Note that having [MATH] implies that the Weyl term, [MATH], in the effective action is absent in second order which is equivalent to taking [MATH] thus eliminating the massive tensor ghost issue.', 'hep-ph-0603242-2-15-4': 'This field is now removed from the spectrum though the D-dimensional scalar remains in general.', 'hep-ph-0603242-2-16-0': '# Application I: ADD', 'hep-ph-0603242-2-17-0': 'In this section we will apply the above analysis to the general ADD framework where we now require (since the space is flat) [MATH] so that [MATH] automatically.', 'hep-ph-0603242-2-17-1': 'This significantly reduces the possible deviations from the classic ADD picture.', 'hep-ph-0603242-2-17-2': 'In this specific case the general second order expansion of [MATH] is rather simple and is given by [EQUATION]', 'hep-ph-0603242-2-17-3': 'Note that if we also demand that [MATH] be a function only of [MATH] and the combination [MATH] in order to avoid the issue of the massive tensor ghost this expression simplifies even further to [EQUATION]', 'hep-ph-0603242-2-17-4': 'Of course, there has been some discussion about other ways to circumvent this tensor ghost problem than by completely eliminating it from the perturbative spectrum.', 'hep-ph-0603242-2-17-5': 'Since we are working only to lowest non-trivial order perhaps we should keep an open mind about the forms for [MATH].', 'hep-ph-0603242-2-17-6': 'Note that since [MATH] will essentially rescale the overall mass factor in the action we must demand that [MATH] to insure that the usual D-dimensional massless tensor gravitons not be ghost-like.', 'hep-ph-0603242-2-18-0': 'How are the predictions of ADD modified by these additional curvature terms?', 'hep-ph-0603242-2-18-1': 'The basic ADD picture leads to three essential predictions [CITATION]: ([MATH]) the emission of graviton KK states during the collision of SM particles producing signatures with apparent missing energy[CITATION]; ([MATH]) the exchange of graviton KK excitations between SM brane fields leading to dimension-8 contact interaction-like operators with distinctive spin-2 properties[CITATION]; ([MATH]) the production of black holes(BH) at colliders and in cosmic rays with geometric cross sections, [MATH], with [MATH] being the BH Schwarzschild radius, once collision energies greater than [MATH] are exceeded[CITATION].', 'hep-ph-0603242-2-19-0': 'The production and properties of D-dimensional, TeV-scale BH in higher curvature theories has been partially explored within the context of Lovelock extended gravity[CITATION] though not yet so in the fully general quadratic gravity case described by the function [MATH] considered here.', 'hep-ph-0603242-2-19-1': 'Such a study, which would be very interesting, is far beyond the scope of the present analysis.', 'hep-ph-0603242-2-19-2': 'However, it is interesting to make several observations: ([MATH]) Consider the vacuum solution; if we expand the general action [MATH] above to only quadratic order and if we also assume that all interesting solutions must satisfy [MATH] in the vacuum, then the only deviations from the conventional Schwarzschild form arise from the GB term in the action.', 'hep-ph-0603242-2-19-3': 'This can be seen immediately by examining the equations of motion resulting from the general quadratic action, e.g., in Ref. [CITATION].', 'hep-ph-0603242-2-19-4': 'This result does not remain valid when [MATH] is treated exactly.', 'hep-ph-0603242-2-19-5': '([MATH]) If [MATH] only and is treated exactly without expansion then the equations of motion allow for the conventional external BH result with [MATH] and will appear as an ordinary D-dimensional Schwarzschild solution.', 'hep-ph-0603242-2-19-6': 'This is not, however, the most general solution as can be see by considering the simple case of [MATH] with [MATH] a dimensionless parameter.', 'hep-ph-0603242-2-19-7': 'Here there is also exists a solution with [MATH], which is a constant corresponding to deSitter or anti-deSitter space depending on the sign of [MATH].', 'hep-ph-0603242-2-19-8': '([MATH]) If [MATH], with [MATH] being a constant, then the general BH solution has neither [MATH] nor [MATH] equal to zero as is well-known from the exact solution[CITATION].', 'hep-ph-0603242-2-19-9': 'A more detailed study of these possibilities would be worthwhile.', 'hep-ph-0603242-2-20-0': 'Let us now consider the situation of graviton exchange where it is well-known[CITATION] that ADD leads to new dimension-8 contact interactions.', 'hep-ph-0603242-2-20-1': 'To obtain the analogous quantities here we must expand the integrand of the action, i.e., the Lagrangian [MATH], to second order in the fluctuations, [MATH], around the flat background metric: [EQUATION]', 'hep-ph-0603242-2-20-2': 'Here we have expressed the original metric as [EQUATION] with [MATH] being the background metric, which is identified in the ADD model as the Minkowski metric [MATH].', 'hep-ph-0603242-2-20-3': 'The propagator is then just the inverse of the operator [MATH]; once the propagator is known we sandwich it between two 4-d localized (at the origin of the extra-dimensional co-ordinates) and conserved SM stress-energy sources, [MATH], to find the relevant scattering amplitude; we must remember to later KK decompose the various towers.', 'hep-ph-0603242-2-20-4': "Fortunately, much of this work has been done for us by Accioly, Azeredo and Mukai[CITATION] from which, with some modifications, we obtain the expression for the D-dimensional 'graviton' exchange amplitude (before performing the KK sums) [EQUATION] where [MATH] are the familiar (unaltered in terms of the compactification radius [MATH]) flat space KK masses.", 'hep-ph-0603242-2-20-5': 'The [MATH] label the various KK levels; [MATH] is the 4-d trace of the SM source stress-energy tensor and [MATH] are just [EQUATION] as described above.', 'hep-ph-0603242-2-20-6': '[MATH] is the compactification radius which sets the KK mass scale; here we have assumed a common value for this quantity for all [MATH] extra dimensions so that the volume of the compactified space is just [MATH].', 'hep-ph-0603242-2-20-7': 'In the expression above the first term in the amplitude is the usual one encountered in the ADD model which results from the D-dimensional EH action and combines the contributions of the 4-d spin-2 graviton and spin-0 graviscalar KK towers.', 'hep-ph-0603242-2-20-8': 'The second and third terms correspond to the new D-dimensional massive ghost tensor and scalar contributions, respectively.', 'hep-ph-0603242-2-20-9': 'The difference in the factors of [MATH] versus [MATH] in the first two terms arises from the existence of a 5-d bulk mass for the tensor ghost field.', 'hep-ph-0603242-2-20-10': 'It is interesting to note that the full amplitude is very well behaved at large [MATH] (in fact, going as [MATH]) due to the detailed cancellations between the various terms.', 'hep-ph-0603242-2-21-0': 'Here [MATH] represent the bulk mass terms of the new fields which enter into the KK tower masses of the scalars (spin-0) and tensors (spin-2 and spin-0), respectively; here we see the effect of the tensor ghost KK tower exchange explicitly.', 'hep-ph-0603242-2-21-1': 'From this point of view it appears that the only way to remove this ghost tower is to take the bulk mass [MATH] implying that [MATH] is solely a function of [MATH] and the combination [MATH], which we will assume from now on in our ADD discussion.', 'hep-ph-0603242-2-21-2': 'Note that the existence of a GB term in the action will not yield a contribution to [MATH].', 'hep-ph-0603242-2-21-3': 'Since, as discussed above [MATH] is already required, tachyonic KK scalars are avoided when the denominator in the expression for [MATH] above is positive; when [MATH] is assumed to be a function only of [MATH] and [MATH], then this denominator simplifies to [MATH].', 'hep-ph-0603242-2-21-4': 'In the limit where [MATH] alone, and accounting for a sign factor in the definition of the above actions, our result for the squared scalar mass, [MATH], agrees with that obtained by Demir and Tanyildizi[CITATION].', 'hep-ph-0603242-2-21-5': 'As shown by these authors, the effect of the new scalar tower exchange is generally rather suppressed in comparison to the more familiar graviton exchange since the ratio [MATH] is small for most SM particle sources at TeV colliders.', 'hep-ph-0603242-2-21-6': 'For example, for the process [MATH] this ratio is of order [MATH].', 'hep-ph-0603242-2-21-7': 'The corresponding ratio of the KK summed scalar to graviton exchange amplitudes is somewhat further reduced by ([MATH]) the existence of the finite bulk scalar mass which implies that there are no light scalar KK exchanges with masses below [MATH] and ([MATH]) the [MATH]-dependent numerical factor in the denominator of the scalar amplitude.', 'hep-ph-0603242-2-21-8': 'Naturalness suggests that [MATH] TeV or larger unless the parameters of [MATH] are somehow fine-tuned.', 'hep-ph-0603242-2-21-9': 'For example, if [MATH], then [MATH] for all [MATH] if [MATH] is not too far from O(1).', 'hep-ph-0603242-2-21-10': 'Interestingly we see here that as [MATH] we recover the usual EH expectation as then [MATH].', 'hep-ph-0603242-2-21-11': 'Thus we find that for many practical purposes the structure of the usual ADD results for graviton exchange are not qualitatively modified when the action is generalized to the form considered here.', 'hep-ph-0603242-2-21-12': 'However, with the existence of these additional scalars being a hallmark of the extended action, it behooves us to find a way to isolate their effects experimentally.', 'hep-ph-0603242-2-22-0': 'In expressions for graviton exchange only the combination [MATH] will now appear.', 'hep-ph-0603242-2-22-1': "In the amplitude this will lead to a modification of the pure 'graviton' exchange cross section expectations by a factor of [MATH], which is likely to be of [MATH], provided [MATH] is considered to be held fixed.", 'hep-ph-0603242-2-22-2': 'When the graviton tower interference term with the SM dominates, the effect in the gravitational part of the cross section will scale as [MATH].', 'hep-ph-0603242-2-22-3': 'Given the previous results of Demir and Tanyildizi[CITATION], this is not surprising.', 'hep-ph-0603242-2-23-0': 'We further note that since [MATH] is [MATH] TeV or larger it has no effect on laboratory measurements of the strength of the gravitational interaction in the micron range when [MATH].', 'hep-ph-0603242-2-24-0': 'Before closing this part of the discussion we would like to remind the reader that it was pointed out long ago[CITATION] that we can take any action of the form [MATH] and map it over to the EH action coupled to an minimally coupled real scalar field with a rather complicated potential [MATH], depending exponentially on the scalar field.', 'hep-ph-0603242-2-24-1': 'This can be done via a special conformal transformation [EQUATION]', 'hep-ph-0603242-2-24-2': 'Going from the original (Jordan) to the new (Einstein) frame one explicitly sees the existence of the new scalar degree of freedom.', 'hep-ph-0603242-2-24-3': 'The mass of this scalar field is exactly that of the field [MATH] above and can be gotten directly from the canonically normalized potential [MATH] in the usual manner, i.e., using [MATH].', 'hep-ph-0603242-2-24-4': 'This is a very powerful tool as it allows us to extend our previous flat space result for [MATH] to the much more general case where the space has constant curvature.', 'hep-ph-0603242-2-24-5': 'For example, if [MATH], we find that the value of [MATH] is the same as discussed above, i.e., [MATH], in a space with constant curvature.', 'hep-ph-0603242-2-24-6': 'This will be an important result that we will employ when we discuss the case of the RS setup.', 'hep-ph-0603242-2-25-0': 'We now turn to the emission of gravitons in SM particle collisions.', 'hep-ph-0603242-2-25-1': 'Since the compactifying space is flat in the ADD case the normalizations of the graviton (and scalar) wavefunctions which control their couplings are unaltered by the existence of the quadratic curvature terms but the relationship between [MATH] and [MATH]M_pl[MATH] is modified.', 'hep-ph-0603242-2-25-2': 'This was briefly mentioned above where we saw that in the small curvature limit the parameter [MATH] essentially renormalizes the fundamental scale.', 'hep-ph-0603242-2-25-3': 'To see this in the present case it is sufficient to examine the tensor/spin-2 kinetic part of the 4-d effective Lagrangian to second order in [MATH] (which has not yet been KK-expanded) in the familiar transverse traceless gauge, i.e., [MATH], [MATH]; one obtains[CITATION] [EQUATION] where here [MATH] and [MATH] are defined above.', 'hep-ph-0603242-2-25-4': 'When we assume that [MATH] is only a function of [MATH] and the combination [MATH] then the second term in [MATH] vanishes and we recover the familiar result of the standard EH scenario apart from the overall factor of [MATH].', 'hep-ph-0603242-2-25-5': 'Hence, to recover the conventional 4-d EH action when inserting the usual (extra dimensionally) flat zero mode graviton wavefunction into [MATH] the ADD relationship must be modified, as hinted above, to [EQUATION] where [MATH]M_pl[MATH] is the 4-d reduced Planck scale and [MATH] is the volume of the compactified space.', 'hep-ph-0603242-2-25-6': 'Of course, [MATH] is just unity in the standard ADD model which employs the EH action.', 'hep-ph-0603242-2-25-7': 'Since the lightest of the KK scalars has a mass which is naturally on the order of a TeV and has rather weak couplings to SM fields these particles will not play much of an important role in missing energy processes.', 'hep-ph-0603242-2-25-8': 'If the cross section for graviton production, i.e., missing energy, is expressed in terms of the original [MATH] with other parameters held fixed, then the presence of [MATH] leads to a modification of the production cross section by a factor of [MATH].', 'hep-ph-0603242-2-25-9': 'However, as [MATH] is not likely to remain a direct observable (only the product [MATH] is) there may be no way to experimentally disentangle this effect.', 'hep-ph-0603242-2-25-10': 'Furthermore, for any given [MATH], since [MATH]M_pl[MATH] is numerically fixed and [MATH] is an input parameter the resulting derived value of [MATH] which sets the scale for the masses of the KK states is altered.', 'hep-ph-0603242-2-26-0': 'We thus conclude that if we assume that [MATH] is a function of only of [MATH] and the combination [MATH] then the classic predictions ([MATH]) and ([MATH]) of the ADD model will be qualitatively unaffected by going to the more general action considered here except for possible overall scalings by inverse powers of [MATH] when the parameter [MATH] is held fixed: graviton emission rates scale like [MATH] while graviton exchange cross sections scale as [MATH] or [MATH] depending on the presence of important SM contributions to the relevant process.', 'hep-ph-0603242-2-27-0': '# Application II: RS', 'hep-ph-0603242-2-28-0': 'The predictions of the classic RS model are the existence of TeV scale graviton resonances with fixed weak scale masses and couplings to the SM fields[CITATION], the existence of a weak scale radion excitation[CITATION], as well as the production of [MATH] BH.', 'hep-ph-0603242-2-28-1': 'In what follows we will be specifically interested in the nature of the KK gravitons so it is again sufficient to examine the quadratically expanded action.', 'hep-ph-0603242-2-28-2': 'The classic RS model is not generally strictly consistent with the assumed form of either the original action [MATH] or its quadratically expanded form [MATH].', 'hep-ph-0603242-2-28-3': 'As is well-known, and as mentioned above, the equations of motion that follow from [MATH] and [MATH] will generally be fourth order in the derivatives of the metric.', 'hep-ph-0603242-2-28-4': 'In the usual 5-d RS model, one solves the Einstein equations of the form [EQUATION] where [MATH] is the Einstein tensor arising from the EH action involving no more than two derivatives of the metric.', 'hep-ph-0603242-2-28-5': 'The problem is that RS completely specifies [MATH]: a cosmological constant in the 5-d bulk plus two [MATH]-function sources at the orbifold locations of the TeV and Planck branes.', 'hep-ph-0603242-2-28-6': 'SM matter confined to the TeV brane is supposed to not be a large contributor to the stress-energy.', 'hep-ph-0603242-2-28-7': 'To obtain this result the standard RS metric takes the form discussed above: [MATH] with the linear exponential warp factor leading to the bulk [MATH] and the two field derivatives acting on the absolute value leading to the brane [MATH]-functions.', 'hep-ph-0603242-2-28-8': '(This is related to the comment above that [MATH] is not truly constant in RS and has brane [MATH]-function singularities.', 'hep-ph-0603242-2-28-9': 'Recall that these [MATH]-functions are the results of assuming infinitely thin branes.)', 'hep-ph-0603242-2-28-10': 'If an identical metric is assumed in our more general case we still can obtain [MATH] but the fourth order equations would lead to the more singular derivatives of [MATH]-functions at the brane locations that are not canceled by any source terms.', 'hep-ph-0603242-2-28-11': 'This amongst other reasons is what led Kim, Kyae and Lee[CITATION] to consider only GB extensions of the EH action in RS since it is the only extension which uniquely produces Einstein equations of second order in the derivatives.', 'hep-ph-0603242-2-28-12': 'Thus if we keep the classic picture, an analysis of RS given our assumed effective action expanded around a background of constant curvature is not relevant.', 'hep-ph-0603242-2-28-13': '(A possible way of dealing with these derivatives of [MATH]-functions arising from orbifold singularities in higher dimensional effective field theories has been discussed in Ref. [CITATION].', 'hep-ph-0603242-2-28-14': 'Implementing our scheme employing such techniques is, however, beyond the scope of the present paper.)', 'hep-ph-0603242-2-29-0': "To avoid these issues for now we simplify our discussion of this problem (and to convince ourselves that an RS-like solution is possible in this framework) we consider a singularity-free, 'softened' version of RS where the orbifolded bulk space with branes is replaced by an interval, as has been suggested for other reasons[CITATION], with SM matter placed at one end point possessing an ignorable amount of stress-energy.", 'hep-ph-0603242-2-29-1': 'With a cosmological constant on the interval we can recover the background [MATH] bulk; in addition by removing the absolute value sign of the co-ordinate [MATH] in the metric above we expunge the [MATH]-functions as well as the possibility of any of their higher derivatives appearing in the equations of motion.', 'hep-ph-0603242-2-29-2': 'The boundary conditions at the end points for the graviton KK states can then be freely chosen to be the same as that of the original RS model.', 'hep-ph-0603242-2-29-3': "This space is truly one of constant curvature and the general analysis we have presented above will now be applicable to this 'softened' RS on an interval.", 'hep-ph-0603242-2-30-0': 'It is easy to verify that the form of the equations of motion[CITATION] in this case (recalling that we are only searching for solutions with maximally-symmetric, constant [MATH] backgrounds) are given by: [EQUATION] and that if we take stress-energy tensor in the 5d bulk to be of the usual RS form [EQUATION] with [MATH], then indeed a space of constant curvature, i.e. [MATH], can be an allowed solution.', 'hep-ph-0603242-2-30-1': 'Taking the trace of the equations of motion above, evaluating it in the constant curvature bulk and relating the values of [MATH] to [MATH] as before (recalling that here [MATH] using the softened metric) results in the constraint equation [EQUATION] where here [MATH].', 'hep-ph-0603242-2-30-2': 'It is interesting to note that if we assume that [MATH]constant then this constraint equation automatically implies that [MATH]constant as well; but this does not necessarily further require that all of the [MATH] are constants as we will see below.', 'hep-ph-0603242-2-30-3': 'When [MATH] is only a function of [MATH] and the combination [MATH], this constraint equation simplifies to [EQUATION] while in the specific RS background case this explicitly becomes [EQUATION]', 'hep-ph-0603242-2-30-4': 'It is important to recall that [MATH] itself can be a complicated function of [MATH] so that this equation can be quite nontrivial.', 'hep-ph-0603242-2-30-5': 'For the EH action limit this yields the usual relation that [MATH]; here it in general provides an additional constraint on the allowed forms of the function [MATH] since we are requiring [MATH] to be both real and negative.', 'hep-ph-0603242-2-30-6': 'Given a specific function [MATH] for which a solution exists, this equation directly relates [MATH] and [MATH] though the solution may not be unique.', 'hep-ph-0603242-2-30-7': 'For example, if we assume for purposes of demonstration the simple case of [EQUATION] as employed above, then there are two branches of solutions for [MATH]: [EQUATION] one of which (the negative root) goes over to the usual EH result as the parameter [MATH].', 'hep-ph-0603242-2-31-0': 'Allowing for the possibility of a RS-like solution with a softened metric it is interesting to think briefly about the previously analyzed effects of the GB term in the RS scenario.', 'hep-ph-0603242-2-31-1': 'This analysis was originally performed for the classic RS[CITATION] setup which employed the standard form of the RS metric; that result would now be modified by the changes in the model assumptions, i.e., moving to an interval and removing the [MATH]-function sources at the end points.', 'hep-ph-0603242-2-31-2': 'The previous analysis of BH in RS with the added GB term would not be significantly affected if this transition were made.', 'hep-ph-0603242-2-31-3': 'However, the properties and spectrum of the graviton KK states certainly would be influenced since the [MATH]-function terms are now absent.', 'hep-ph-0603242-2-31-4': 'The equation governing the masses and wavefunction of the graviton KK states for the present interval case can be obtained by expanding the equations of motion as before.', 'hep-ph-0603242-2-31-5': 'Since we are here only interested in the tensor modes associated with the usual gravitons, we can employ the expansion [EQUATION] where [MATH].', 'hep-ph-0603242-2-31-6': 'Applying the usual RS boundary conditions on the interval the most significant changes from the classic RS can be read off from Eqs. (15)-(28) in Ref. [CITATION] by setting the parameter [MATH] in appropriate places.', 'hep-ph-0603242-2-31-7': 'At the end of the day we find that the only apparent difference from the classic EH based RS model would be a shift in the relationship between the fundamental scale and [MATH]M_pl[MATH]-remarkably similar to what we saw for the ADD model above.', 'hep-ph-0603242-2-31-8': 'In the language employed in Ref. [CITATION] we would now obtain [EQUATION] where [MATH] is the coefficient of the GB term in the action.', 'hep-ph-0603242-2-31-9': 'Otherwise the masses as well as the couplings of all of the KK gravitons to localized SM matter would be identical to those of the original RS model expressed in terms of the derived parameter [MATH].', 'hep-ph-0603242-2-31-10': 'The explicit coupling and spectrum changes found in Ref[CITATION] for the graviton KK states in the presence of the GB term in the action were all found to due to the brane [MATH]-function singularities.', 'hep-ph-0603242-2-32-0': 'How would these graviton KK results obtained in the GB extended action generalize to the case of [MATH] above?', 'hep-ph-0603242-2-32-1': 'Here we choose to begin our analysis with [MATH], setting [MATH] from the beginning to avoid potential ghost fields, then taking [MATH] and using the same curvature expansion as above.', 'hep-ph-0603242-2-32-2': 'In order to make a connection with the previous discussion, the existing RS literature and to directly compare with the GB case, however, we massage our notation slightly and rewrite [MATH] in the following form: [EQUATION] where the parameters [MATH] and [MATH] are dimensionless; the action employed in Ref. [CITATION] is now directly recovered by taking the [MATH] and [MATH] limits.', 'hep-ph-0603242-2-32-3': 'It is important at this point to recall that to obtain the linearized graviton equations of motion it is sufficient to employ [MATH] while the complete [MATH] needs to be examined in order to demonstrate the existence of the required [MATH] solution.', 'hep-ph-0603242-2-32-4': 'The equations of motion resulting from [MATH] are given by[CITATION] [EQUATION]', 'hep-ph-0603242-2-32-5': 'Here [MATH] is the covariant derivative operator and here [MATH].', 'hep-ph-0603242-2-32-6': 'First we look at the [MATH] component of this equation, remembering that for the moment we will only be interested in the tensor excitations corresponding to the KK gravitons which are massless in 5-d.', 'hep-ph-0603242-2-32-7': 'In the usually chosen gauge, [MATH] is still a constant to linear order so we arrive at a consistency condition [EQUATION]', 'hep-ph-0603242-2-32-8': 'Note that this reduces to the previously obtained purely quadratic GB extended RS result[CITATION] when [MATH].', 'hep-ph-0603242-2-32-9': 'In the more general case, this expression is not overly useful given the exact result in Eq. (27).', 'hep-ph-0603242-2-33-0': 'Turning now to the [MATH] terms which contain the 4-d graviton tensor excitation, we linearize employing the previously mentioned transverse, traceless gauge with constant [MATH].', 'hep-ph-0603242-2-33-1': 'This gives the standard equation of motion for the RS graviton found long ago[CITATION] though scaled by an overall factor.', 'hep-ph-0603242-2-33-2': 'Employing the standard KK decomposition [EQUATION] and recalling that [MATH], the [MATH] are seen to satisfy [EQUATION]', 'hep-ph-0603242-2-33-3': 'The overall factor [MATH] is given by [EQUATION] or, more explicitly in the RS case, [EQUATION] (Again we recall that [MATH] itself can be a function of [MATH].)', 'hep-ph-0603242-2-33-4': 'This leads to a rescaling of the usual RS relationship [EQUATION] via the renormalization of the zero mode (i.e., massless graviton) wavefunction, thus generalizing Eq. (31).', 'hep-ph-0603242-2-33-5': 'Of course, [MATH] is required to avoid ghost states among the usual gravitons KKs.', 'hep-ph-0603242-2-33-6': 'This result reduces to that previously obtained in the RS case with just the added GB term[CITATION] once boundary effects are neglected.', 'hep-ph-0603242-2-34-0': 'From this analysis we see immediately that the masses of the KK gravitons are identical to those obtained in the original RS model, provided we use the same value of the parameter [MATH], as we might have expected.', 'hep-ph-0603242-2-34-1': 'Here we are faced with the question of just what are the independent parameters.', 'hep-ph-0603242-2-34-2': '[MATH] is clearly a derived parameter that is obtained by solving Eq. (27) for any given model.', 'hep-ph-0603242-2-34-3': 'In that sense, the KK graviton spectrum would just be rescaled in comparison to the usual expectations given the same input values of [MATH]etc.', 'hep-ph-0603242-2-34-4': 'As we have just seen, and as in the ADD case, the effect of a factor like [MATH] on the KK graviton couplings to 4-d SM matter depends upon which model parameters are assumed to be held fixed.', 'hep-ph-0603242-2-34-5': 'At the very least, up to an overall constant, these KK graviton couplings are identical to those of the standard RS model.', 'hep-ph-0603242-2-35-0': 'So far we have only considered the 4-d graviton, spin-2 excitations.', 'hep-ph-0603242-2-35-1': 'It is important to remember that our softened RS model now has an additional massive scalar in the 5-d spectrum with a large bulk mass, [MATH], and that no massless scalar zero mode will exist.', 'hep-ph-0603242-2-35-2': 'Since the bulk scalar mass is naturally of order [MATH] the KK spectrum of the corresponding tower will begin with a KK scalar state whose mass is qualitatively comparable to that of the first graviton excitation.', 'hep-ph-0603242-2-35-3': 'This bulk mass is explicitly calculable from the expansion of the full action to quadratic order, [MATH], by going to the Einstein frame since we know that the GB term does not contribute to this parameter.', 'hep-ph-0603242-2-35-4': 'In that case, using the results from the previous section we find that [EQUATION] or, in terms of the original parameters of the action, evaluated in the RS background: [EQUATION]', 'hep-ph-0603242-2-35-5': 'Note that [MATH] is required to avoid the scalar tachyons and graviton ghosts, consistent with our above analysis.', 'hep-ph-0603242-2-35-6': 'Note further that this reproduces the results of Eq. (19) in the flat space, [MATH], limit.', 'hep-ph-0603242-2-36-0': 'Given any [MATH] the scalar bulk mass is known and we can determine the mass(es) of the lightest KK scalar state(s) by following the standard RS manipulations[CITATION].', 'hep-ph-0603242-2-36-1': 'These masses are essentially given by the first roots of the equation [EQUATION] where [MATH] and [MATH] is the usual Bessel function.', 'hep-ph-0603242-2-36-2': 'The solution for the first KK state is provided by Fig.1; as stated above there are no massless modes.', 'hep-ph-0603242-2-36-3': 'The lightest scalar mass is then [MATH].', 'hep-ph-0603242-2-36-4': 'Here we observe that the mass of the first scalar KK scales almost linearly with the bulk mass when [MATH] gets large.', 'hep-ph-0603242-2-36-5': 'Note that for [MATH] and a typical value[CITATION] of [MATH], we then find [MATH] implying [MATH] from Fig.1; this is about 3 times larger than the root for the usual lightest massive KK graviton, [MATH].', 'hep-ph-0603242-2-36-6': 'Thus we see that unless [MATH] takes on large values the first scalar KK state is always rather heavy.', 'hep-ph-0603242-2-36-7': 'As is well-known, the [MATH] values for the more massive KK scalar states will be somewhat larger: approximately given by [MATH] where [MATH] labels the KK level.', 'hep-ph-0603242-2-36-8': 'Since these scalars will couple to the trace of the stress-energy tensor for the 4-d SM fields they will interact far more weakly than do the graviton KK states unless this is at least partially offset by ratios of 5-d wavefunction factors.', 'hep-ph-0603242-2-36-9': 'A quick estimate of such factors, however, indicates that, if anything, these wave function ratios lead to a further suppression of the scalar couplings relative to those of the KK gravitons by [MATH] as shown in Fig.2.', 'hep-ph-0603242-2-36-10': 'This overall picture of the scalar sector is qualitatively very similar to that of the existence of a very heavy tower of RS radions[CITATION] or a tower of KK Higgs bosons as in the case of Universal Warped Extra Dimensions[CITATION].', 'hep-ph-0603242-2-37-0': 'In the analysis as presented here we have ignored the possibility that the new scalar KK states may mix with the (usually eaten) RS graviscalars through cross-talk in the equations of motion, i.e., we have assumed that the 5-d tensor and scalar KK decompositions can be performed independently, and this is something which needs further exploration.', 'hep-ph-0603242-2-37-1': 'A fully detailed analysis of the such possibilities is, however, beyond the scope of the present paper.', 'hep-ph-0603242-2-38-0': 'It is perhaps interesting to ask whether the usual [MATH]=constant ([MATH]) solution considered here necessitates the metric and matter distribution of the conventional RS model employed above without the further assumption of a maximally symmetric space.', 'hep-ph-0603242-2-38-1': 'To analyze a simple and more easily tractable situation let us consider the more general warped metric [MATH] and assume that [MATH] only.', 'hep-ph-0603242-2-38-2': 'The first question to address is what is the most general form of the function [MATH]; to deal with this issue we note that the Ricci scalar arising from this metric is in general given by [MATH].', 'hep-ph-0603242-2-38-3': "In the 'soft' version of the RS model on the interval defined above one had [MATH] and [MATH] so that [MATH] as usual.", 'hep-ph-0603242-2-38-4': 'Here, this curvature condition provides a differential equation for the function [MATH]; by solving this equation we arrive at the result [MATH], where the [MATH] are integration constants and [MATH].', 'hep-ph-0603242-2-38-5': 'Choosing the [MATH] appropriately and rescaling [MATH], we can rewrite this in a more familiar form as [MATH] with [MATH] a dimensionless constant.', 'hep-ph-0603242-2-38-6': 'Note that when [MATH] and [MATH] we recover the usual RS result.', 'hep-ph-0603242-2-38-7': 'However, the choice of [MATH] can easily modify the warp factor from its conventional behavior.', 'hep-ph-0603242-2-38-8': 'What is the nature of the bulk matter distribution that yields this metric?', 'hep-ph-0603242-2-38-9': 'The solution to this can be obtained by considering Eqs.(24) and (26) with [MATH].', 'hep-ph-0603242-2-38-10': 'Eq.(26) immediately tells us that the trace of the 5-d stress-energy tensor [MATH], a constant, so that if we define [MATH] then we must have [MATH], with [MATH] an arbitrary function.', 'hep-ph-0603242-2-38-11': 'The [MATH] and [MATH] components of the equations of motion when combined then provides a first order differential equation for [MATH] that can be solved in a straightforward manner.', 'hep-ph-0603242-2-38-12': 'Setting [MATH] and defining [MATH] with [MATH], we obtain the general solution (assuming [MATH]) [EQUATION] with [MATH] an integration constant.', 'hep-ph-0603242-2-38-13': 'Though this is far from a uniform energy distribution (away from the TeV brane) it does lead to a space of constant curvature but not one which is maximally symmetric.', 'hep-ph-0603242-2-38-14': 'Thus we see that it is possible that the requirement that [MATH] does allow for the possibility of more complex solutions than that employed in the original RS model.', 'hep-ph-0603242-2-39-0': '# Discussion and Conclusions', 'hep-ph-0603242-2-40-0': 'In this paper we have begun an examination of how generic higher curvature terms in the gravitational action can alter the predictions of both the ADD model and the RS model defined on a interval to avoid possible brane singularities.', 'hep-ph-0603242-2-40-1': 'We have assumed that the traditional assumptions of the two models, e.g., SM localized matter in a conformally flat bulk, remain valid; we have not considered in detail more complex setups that may now be allowed by the modified equations of motion.', 'hep-ph-0603242-2-40-2': 'To be more concrete, we have further assumed that the EH action is now generalized to an action which is of the form [MATH] where [MATH] is a well-behaved function, [MATH] and [MATH].', 'hep-ph-0603242-2-40-3': 'In D-dimensions this action results in a propagating massless tensor field (identified with the usual graviton), a massive ghost tensor field, as well as a massive (possibly tachyonic) scalar.', 'hep-ph-0603242-2-40-4': 'The potentially dangerous ghost is removable from the perturbative spectrum, i.e., it becomes infinitely massive, if we demand that [MATH] only.', 'hep-ph-0603242-2-40-5': 'The remaining new scalar field has a bulk mass whose value is naturally expected to be of order the fundamental scale, [MATH], in either scenario.', 'hep-ph-0603242-2-40-6': 'The resulting ADD and RS models are altered in similar ways from their traditional standard forms:', 'hep-ph-0603242-2-41-0': '([MATH]) New scalar KK excitations appear in the spectrum of both models in a rather benign fashion coupling to the trace of the stress-energy tensor of the localized SM fields.', 'hep-ph-0603242-2-41-1': 'Since this trace is proportional to SM masses, the couplings of these scalars are relatively strongly suppressed in comparison to those for the KK gravitons at typical collider energies in both models.', 'hep-ph-0603242-2-41-2': 'In the ADD model, the KK scalar excitations begin at a mass [MATH] TeV.', 'hep-ph-0603242-2-41-3': 'Consequently their contributions to missing-energy signatures as well as to the usual dimension-8 contact interactions are further kinematically suppressed.', 'hep-ph-0603242-2-41-4': 'Thus at leading order these new scalars do not much influence ADD collider signatures.', 'hep-ph-0603242-2-41-5': 'In RS, the bulk scalar mass tends to be large so that the lightest scalar KK state is several times more massive than is the lightest KK graviton.', 'hep-ph-0603242-2-41-6': 'Given their rather weak couplings such states will be difficult to observe at colliders.', 'hep-ph-0603242-2-42-0': '([MATH]) The basic model relationships involving the fundamental and 4-d Planck masses in both models get rescaled by functions of [MATH] and its derivatives evaluated in the corresponding background metric of the two models: in ADD we obtain [MATH]M_pl[MATH] while in RS we obtain [MATH]M_pl[MATH] where [MATH] is explicitly given in Eq. (39).', 'hep-ph-0603242-2-42-1': 'Assuming that [MATH] is a fixed fundamental parameter these modifications lead to changes in the graviton KK sectors of both models.', 'hep-ph-0603242-2-42-2': 'In the ADD case, since [MATH]M_pl[MATH] is known and [MATH] is an input parameter for any given [MATH] the volume of the compactified space and, hence, the value of the compactification radius which sets the graviton KK mass scale is altered.', 'hep-ph-0603242-2-42-3': 'Due to the presence of the [MATH] factor the emission rate for gravitons in the collisions of SM particles and for the graviton exchange amplitude are both modified by potentially O(1) effects.', 'hep-ph-0603242-2-42-4': 'Similarly in RS, [MATH] is a derived parameter which sets the scale for all the KK states.', 'hep-ph-0603242-2-42-5': 'The constraint Eq. (28) allows us to calculate [MATH] in terms of the input parameter [MATH] and the function [MATH] thus providing for us with [MATH].', 'hep-ph-0603242-2-42-6': 'In a manner similar to ADD, the presence of [MATH] rescales the coupling strengths of the of the KK graviton states to the SM fields thus modifying the widths and production cross sections at colliders by potentially O(1) factors.', 'hep-ph-0603242-2-43-0': 'As we have seen, the extension of the EH action to a more complicated structure can lead to significant quantitative modifications to both the ADD and RS model predictions in the simplest possible case.', 'hep-ph-0603242-2-43-1': 'The observation of such effects at future colliders could tell us valuable information about the underlying theory of gravity.', 'hep-ph-0603242-2-44-0': 'Note Added: After this paper was essentially completed, Ref. [CITATION] appeared which discusses generalized actions for the ADD model and thus has some common areas with the present work.', 'hep-ph-0603242-2-44-1': 'Where the two papers overlap there is general qualitative agreement though the points of view are somewhat different.'} | [['hep-ph-0603242-1-36-0', 'hep-ph-0603242-2-40-0'], ['hep-ph-0603242-1-36-3', 'hep-ph-0603242-2-40-3'], ['hep-ph-0603242-1-36-5', 'hep-ph-0603242-2-40-5'], ['hep-ph-0603242-1-24-0', 'hep-ph-0603242-2-25-0'], ['hep-ph-0603242-1-24-1', 'hep-ph-0603242-2-25-1'], ['hep-ph-0603242-1-24-2', 'hep-ph-0603242-2-25-2'], ['hep-ph-0603242-1-24-3', 'hep-ph-0603242-2-25-3'], ['hep-ph-0603242-1-24-4', 'hep-ph-0603242-2-25-4'], ['hep-ph-0603242-1-24-5', 'hep-ph-0603242-2-25-5'], ['hep-ph-0603242-1-24-6', 'hep-ph-0603242-2-25-6'], ['hep-ph-0603242-1-24-8', 'hep-ph-0603242-2-25-8'], ['hep-ph-0603242-1-24-9', 'hep-ph-0603242-2-25-9'], ['hep-ph-0603242-1-24-10', 'hep-ph-0603242-2-25-10'], ['hep-ph-0603242-1-37-0', 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['hep-ph-0603242-1-27-12', 'hep-ph-0603242-2-28-12'], ['hep-ph-0603242-1-39-0', 'hep-ph-0603242-2-43-0'], ['hep-ph-0603242-1-28-0', 'hep-ph-0603242-2-29-0'], ['hep-ph-0603242-1-28-3', 'hep-ph-0603242-2-29-3'], ['hep-ph-0603242-1-8-1', 'hep-ph-0603242-2-8-3'], ['hep-ph-0603242-1-8-2', 'hep-ph-0603242-2-8-4'], ['hep-ph-0603242-1-8-5', 'hep-ph-0603242-2-8-7'], ['hep-ph-0603242-1-8-6', 'hep-ph-0603242-2-8-8'], ['hep-ph-0603242-1-16-3', 'hep-ph-0603242-2-17-3'], ['hep-ph-0603242-1-16-4', 'hep-ph-0603242-2-17-4'], ['hep-ph-0603242-1-16-5', 'hep-ph-0603242-2-17-5'], ['hep-ph-0603242-1-20-0', 'hep-ph-0603242-2-21-0'], ['hep-ph-0603242-1-20-7', 'hep-ph-0603242-2-21-7'], ['hep-ph-0603242-1-20-11', 'hep-ph-0603242-2-21-11'], ['hep-ph-0603242-1-29-0', 'hep-ph-0603242-2-30-0'], ['hep-ph-0603242-1-29-1', 'hep-ph-0603242-2-30-1'], ['hep-ph-0603242-1-29-3', 'hep-ph-0603242-2-30-3'], ['hep-ph-0603242-1-29-7', 'hep-ph-0603242-2-30-7'], ['hep-ph-0603242-1-7-1', 'hep-ph-0603242-2-7-1'], ['hep-ph-0603242-1-7-2', 'hep-ph-0603242-2-7-2'], ['hep-ph-0603242-1-7-6', 'hep-ph-0603242-2-7-6'], ['hep-ph-0603242-1-25-0', 'hep-ph-0603242-2-26-0'], ['hep-ph-0603242-1-2-2', 'hep-ph-0603242-2-2-2'], ['hep-ph-0603242-1-2-3', 'hep-ph-0603242-2-2-3'], ['hep-ph-0603242-1-3-1', 'hep-ph-0603242-2-3-1'], ['hep-ph-0603242-1-3-2', 'hep-ph-0603242-2-3-2'], ['hep-ph-0603242-1-3-3', 'hep-ph-0603242-2-3-3'], ['hep-ph-0603242-1-3-6', 'hep-ph-0603242-2-3-6'], ['hep-ph-0603242-1-3-7', 'hep-ph-0603242-2-3-7'], ['hep-ph-0603242-1-4-2', 'hep-ph-0603242-2-4-2'], ['hep-ph-0603242-1-4-3', 'hep-ph-0603242-2-4-3'], ['hep-ph-0603242-1-10-1', 'hep-ph-0603242-2-10-1'], ['hep-ph-0603242-1-9-2', 'hep-ph-0603242-2-9-2'], ['hep-ph-0603242-1-9-3', 'hep-ph-0603242-2-9-3'], ['hep-ph-0603242-1-9-4', 'hep-ph-0603242-2-9-4'], ['hep-ph-0603242-1-18-6', 'hep-ph-0603242-2-19-6'], ['hep-ph-0603242-1-11-0', 'hep-ph-0603242-2-11-0'], ['hep-ph-0603242-1-11-3', 'hep-ph-0603242-2-11-3'], ['hep-ph-0603242-1-11-4', 'hep-ph-0603242-2-11-4'], ['hep-ph-0603242-1-23-4', 'hep-ph-0603242-2-24-4'], ['hep-ph-0603242-1-23-5', 'hep-ph-0603242-2-24-5'], ['hep-ph-0603242-1-40-0', 'hep-ph-0603242-2-44-0'], ['hep-ph-0603242-1-19-2', 'hep-ph-0603242-2-20-2'], ['hep-ph-0603242-1-19-3', 'hep-ph-0603242-2-20-3'], ['hep-ph-0603242-1-19-4', 'hep-ph-0603242-2-20-4'], ['hep-ph-0603242-1-19-8', 'hep-ph-0603242-2-20-8'], ['hep-ph-0603242-1-19-10', 'hep-ph-0603242-2-20-10'], ['hep-ph-0603242-1-5-0', 'hep-ph-0603242-2-5-0'], ['hep-ph-0603242-2-36-2', 'hep-ph-0603242-3-37-2'], ['hep-ph-0603242-2-36-9', 'hep-ph-0603242-3-37-9'], ['hep-ph-0603242-2-34-2', 'hep-ph-0603242-3-34-2'], ['hep-ph-0603242-2-34-3', 'hep-ph-0603242-3-34-3'], ['hep-ph-0603242-1-14-5', 'hep-ph-0603242-2-15-0'], ['hep-ph-0603242-1-31-1', 'hep-ph-0603242-2-32-1'], ['hep-ph-0603242-1-31-6', 'hep-ph-0603242-2-32-6'], ['hep-ph-0603242-1-31-7', 'hep-ph-0603242-2-32-7'], ['hep-ph-0603242-1-31-13', 'hep-ph-0603242-2-33-3'], ['hep-ph-0603242-1-31-18', 'hep-ph-0603242-2-34-1'], ['hep-ph-0603242-1-31-21', 'hep-ph-0603242-2-34-4'], ['hep-ph-0603242-1-31-22', 'hep-ph-0603242-2-34-5']] | [] | [['hep-ph-0603242-1-33-0', 'hep-ph-0603242-2-36-0'], ['hep-ph-0603242-1-8-0', 'hep-ph-0603242-2-8-0'], ['hep-ph-0603242-1-8-0', 'hep-ph-0603242-2-8-2'], ['hep-ph-0603242-1-8-4', 'hep-ph-0603242-2-8-6'], ['hep-ph-0603242-1-29-2', 'hep-ph-0603242-2-30-2'], ['hep-ph-0603242-1-2-4', 'hep-ph-0603242-2-2-4'], ['hep-ph-0603242-1-2-5', 'hep-ph-0603242-2-2-5'], ['hep-ph-0603242-1-3-8', 'hep-ph-0603242-2-3-8'], ['hep-ph-0603242-1-18-9', 'hep-ph-0603242-2-19-9'], ['hep-ph-0603242-1-11-2', 'hep-ph-0603242-2-11-2'], ['hep-ph-0603242-1-31-15', 'hep-ph-0603242-2-33-5']] | [] | ['hep-ph-0603242-1-36-6', 'hep-ph-0603242-2-40-6', 'hep-ph-0603242-3-41-6', 'hep-ph-0603242-4-41-6', 'hep-ph-0603242-5-41-6'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/', '5': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/0603242 | {'hep-ph-0603242-3-0-0': 'We begin a general exploration of the phenomenology of TeV-scale extra-dimensional models with gravitational actions that contain higher curvature terms.', 'hep-ph-0603242-3-0-1': 'In particular, we examine how the classic collider signatures of the models of Arkani-Hamed, Dimopoulos and Dvali (missing energy and new dimension-8 contact interactions) and of Randall and Sundrum (TeV-scale graviton Kaluza-Klein resonances) are altered by these modifications to the usual Einstein-Hilbert action.', 'hep-ph-0603242-3-0-2': 'We find that not only are the detailed signatures for these gravitationally induced processes altered but new contributions are found to arise due to the existence of additional scalar Kaluza-Klein states in the spectrum.', 'hep-ph-0603242-3-1-0': '# Introduction and Background', 'hep-ph-0603242-3-2-0': 'The question as to why the Planck and electroweak scales differ by so many orders of magnitude remains mysterious.', 'hep-ph-0603242-3-2-1': 'In recent years, attempts have been made to address this hierarchy issue within the context of theories with extra spatial dimensions that lower the effective scale of gravity to the TeV region.', 'hep-ph-0603242-3-2-2': 'In both the models of Arkani-Hamed, Dimopoulos and Dvali (ADD)[CITATION] and of Randall and Sundrum (RS)[CITATION], new effects of gravitational origin are expected to occur near the TeV scale which should be observable at future colliders such as the LHC and ILC.', 'hep-ph-0603242-3-2-3': 'Though these two models are very different in detail they do have some common features the most important of which are: ([MATH]) in their original versions they both assume that Standard Model matter is confined to a 4-dimensional brane; ([MATH]) they both assume that D-dimensional gravity is described by the Einstein-Hilbert (EH) action plus a possible cosmological constant and ([MATH]) the background spaces are maximally symmetric and are either strictly flat, i.e. , Minkowskian as in the ADD model with toroidally flat compactification, or being of constant curvature and is conformally flat, i.e., [MATH] as in the RS model.', 'hep-ph-0603242-3-2-4': 'How would the predictions of these two models be changed if we surrendered the assumption ([MATH]), i.e., that the EH action provided the full description of gravity and considered something more general?', 'hep-ph-0603242-3-2-5': 'This is the discussion we would like to begin in this paper which will follow a phenomenological bottom-up approach.', 'hep-ph-0603242-3-3-0': 'General Relativity (GR) as described by the EH action is considered to be an effective theory below the fundamental Planck scale, [MATH].', 'hep-ph-0603242-3-3-1': 'Thus, once energies approaching the scale [MATH] begin to be probed one might expect to observe deviations from the expectations arising from the EH action.', 'hep-ph-0603242-3-3-2': 'In the cases of both the ADD and RS models, future colliders will probe near or at their (effective) fundamental scales so that non-EH aspects of the true gravitational theory, whatever its form, should become apparent and be experimentally measured.', 'hep-ph-0603242-3-3-3': 'Since the ultraviolet form of the true gravity theory is as of yet unknown one may hope to capture some of its deeper aspects by considering how the presence of new higher curvature (and higher derivative) invariants in the actions of the ADD and RS models can lead to variations in the well-known predictions of these theories.', 'hep-ph-0603242-3-3-4': 'Many authors have considered the possibility of higher curvature invariants and how their existence would modify the predictions arising from the EH action within other contexts, e.g., the properties of black holes[CITATION], deviations in solar system tests of GR[CITATION] and in cosmology[CITATION] to possibly avoid the need for dark energy.', 'hep-ph-0603242-3-3-5': 'Some analyses along these lines for the potential modifications of the collider predictions of both the ADD and RS models have already been performed[CITATION].', 'hep-ph-0603242-3-3-6': 'In the present paper, we wish to both extend and generalize these results to get a feeling for the possible detailed variation of the various new gravitational phenomena as predicted by these classic models which will be potentially observable at future colliders.', 'hep-ph-0603242-3-3-7': 'In particular we are interested in how the well known signatures of both the ADD and RS models are morphed if we keep the basic setups intact but modify the actions on which the corresponding equations of motion are based.', 'hep-ph-0603242-3-3-8': 'A further generalization of such an analysis is possible if the original ADD/RS setups can now be simultaneously surrendered due to these modified actions resulting in entirely new setups with corresponding equations of motion; while this is an interesting possibility to consider it lies mostly beyond the scope of the present paper though it will be touched upon briefly in the discussion below.', 'hep-ph-0603242-3-4-0': 'Of course a completely general study of how these possible modifications to the effective gravity action may morph TeV collider signatures is an obviously immense task and here we aim only at a first round analysis in the discussion that follows.', 'hep-ph-0603242-3-4-1': 'The major signatures arising in both ADD and RS models originate from graviton exchange and the production of black holes; the ADD model also leads to missing energy signatures from graviton emission.', 'hep-ph-0603242-3-4-2': 'Fortunately, apart from issues associated with black holes, since we are dealing with maximally symmetric spaces of globally constant curvature, the relevant graviton properties (couplings, wavefunctions and propagators) necessary to extract experimental signatures for either model can be obtained from the expansion of the rather general action considered below to quadratic order in the curvature.', 'hep-ph-0603242-3-4-3': '(This would no longer be true if we wanted to consider, e.g., the triple graviton coupling as then an expansion to third order in the curvature would be required.)', 'hep-ph-0603242-3-4-4': 'This simplifying observation forms the basis of the analysis that follows and allows us to determine the relevant graviton properties in both ADD- and RS-like models for a wide class of effective actions.', 'hep-ph-0603242-3-5-0': 'The general outline of our analysis is presented in Section 2 where our basic assumptions and notations are also given.', 'hep-ph-0603242-3-5-1': 'In Section 3 we apply our analysis to the ADD model; we then apply it to the RS case in Section 4.', 'hep-ph-0603242-3-5-2': 'Our summary and conclusions are given in Section 5.', 'hep-ph-0603242-3-6-0': '# Analysis', 'hep-ph-0603242-3-7-0': 'When going beyond the EH action there are many possibilities to consider especially when we are living in extra dimensions.', 'hep-ph-0603242-3-7-1': 'In the literature various forms have been assumed for the potential structures of higher curvature and/or derivative invariants that may appear in the gravity action.', 'hep-ph-0603242-3-7-2': 'The fairly general structure that we will assume for the D-dimensional action in the present analysis takes the form (with [MATH] below): [EQUATION] where [MATH] is an arbitrary continuous, differentiable and generally mathematically well-behaved function; in particular we will assume that [MATH] is non-singular when all of its arguments are zero.', 'hep-ph-0603242-3-7-3': 'Here [MATH] is the usual D-dimensional Ricci scalar while [MATH] and [MATH] are quadratic invariants constructed from the curvature tensor [MATH]: [MATH], with [MATH] being the Ricci tensor, while [MATH].', 'hep-ph-0603242-3-7-4': '[MATH] is the D-dimensional fundamental gravity scale which is [MATH] TeV in ADD and [MATH]M_pl[MATH] in RS.', 'hep-ph-0603242-3-7-5': 'In the low energy, small curvature limit we expect [MATH] (plus a possible cosmological constant) and so the overall dimensionful factor in the expression above allows us to make direct contact with the EH action in this limit.', 'hep-ph-0603242-3-7-6': 'This specific form for [MATH], though not completely general, covers a wide array of possibilities and has been considered (sometimes only in [MATH]) in may different contexts for a multitude of purposes in the literature[CITATION].', 'hep-ph-0603242-3-7-7': 'Many of the higher curvature models previous considered by other authors form subcases of this more general action.', 'hep-ph-0603242-3-8-0': 'As is by now well-known[CITATION] the generalized gravity theories described by an action of the form [MATH] can potentially have several serious problems.', 'hep-ph-0603242-3-8-1': 'Since we will be dealing with ADD- and RS-type phenomenology we will be performing a perturbative analysis in the discussion that follows.', 'hep-ph-0603242-3-8-2': 'Employing such an analysis one finds that, amongst other things, this action leads to equations of motion which are generally fourth order in the derivatives of the metric.', 'hep-ph-0603242-3-8-3': 'In particular, in addition to the usual massless D-dimensional tensor graviton which results in, e.g., the familiar 4-d graviton and graviscalar Kaluza-Klein (KK) tower excitations, there may also be present in the linearized D-dimensional theory additional massive scalar and tensor excitations.', 'hep-ph-0603242-3-8-4': 'These fields will in 4-d have KK towers without massless modes and which can be ghostlike and/or tachyonic.', 'hep-ph-0603242-3-8-5': '(We can think of these new D-dimensional fields as having bulk masses which influence their corresponding 4-d KK tower masses.)', 'hep-ph-0603242-3-8-6': 'Furthermore, the equations of motion naturally involving higher derivatives of the fields can lead to problems with unitarity as well as non-canceling asymmetric pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-3-8-7': "The new massive tensor excitations are potentially the most serious issue to deal with as they are ghost fields that must be eliminated from the perturbative spectrum (though they may help in dealing with the theory's renormalizability and bad high-energy behavior).", 'hep-ph-0603242-3-8-8': 'It has been noted[CITATION] that we one can remove these states from the spectrum (i.e., by giving their bulk masses an infinite value) if a tuning occurs such that the quantities [MATH] and [MATH] only appear in the special combination [MATH] in the function [MATH].', 'hep-ph-0603242-3-8-9': 'How this tuning arises in the fundamental theory is unspecified.', 'hep-ph-0603242-3-8-10': 'There has been some discussion in the literature, however, that these ghost states may not be as dangerous as one would imagine from lowest order perturbation theory[CITATION] so that we should perhaps keep an open mind about the possible forms for [MATH].', 'hep-ph-0603242-3-8-11': 'We will return to this point in what follows.', 'hep-ph-0603242-3-9-0': 'Given a general action of the kind above there are several issues that one normally wants to address in order to extract information that can be compared with experimental data.', 'hep-ph-0603242-3-9-1': 'From studies of both the ADD and RS models there are certain things we want to know, e.g., ([MATH]) the spectrum, wave functions, propagators and Standard Model (SM) matter couplings of the KK graviton (and other possible) excitations and ([MATH]) the relationship between [MATH], the volume of the compactified dimensions and the (reduced) 4-d Planck scale [MATH]M_pl[MATH].', 'hep-ph-0603242-3-9-2': 'To obtain this information, as well as to make contact with several other analyses[CITATION], it is sufficient to expand the general action [MATH] above around the maximally symmetric background metric to quadratic order in the curvature to obtain an effective action for the graviton (and like) excitations.', 'hep-ph-0603242-3-9-3': 'At this level, one can extract the relevant 2-point functions as well as the differential equation for the KK wavefunctions which then yield the KK mass spectrum as well as the the desired graviton couplings to the localized SM fields.', 'hep-ph-0603242-3-9-4': 'If, however, one wanted to probe, e.g., graviton 3- or 4-point functions then we would need to expand to at least cubic or quartic order in the curvature, respectively; these will not be of interest to us here but might be of interest in future experiments[CITATION] which would tell us more about the underlying theory of gravity.', 'hep-ph-0603242-3-10-0': 'Once we make this expansion, there are various equivalent ways of expressing the resulting effective action, [MATH], depending upon the basis of invariants we choose to employ; the most obvious form is simply [EQUATION] where [MATH] have been defined above.', 'hep-ph-0603242-3-10-1': '[MATH] is an effective cosmological constant and [MATH] are (in some cases dimensionful) constants all of which are functions of [MATH] and its derivatives evaluated by employing the relevant background metric.', 'hep-ph-0603242-3-10-2': "To relate this back to the EH action in the limit of small curvature, one can think of the (necessarily positive) parameter [MATH] as a 'renormalization' of the fundamental mass scale [MATH]: [MATH].", 'hep-ph-0603242-3-11-0': 'A second and perhaps more useful version of [MATH] is given by [EQUATION] where [MATH] is the well-known Gauss-Bonnet(GB) invariant: [EQUATION]', 'hep-ph-0603242-3-11-1': 'The co-efficients [MATH] and [MATH] can be easily converted to [MATH] above by some straightforward algebra: [MATH] and [MATH].', 'hep-ph-0603242-3-11-2': 'In D[MATH]4, the GB invariant is either a topological invariant or just a total derivative not contributing to the equations of motion, but this is no longer true for arbitrary values of [MATH].', 'hep-ph-0603242-3-11-3': 'The GB invariant is just (the quadratic) member of a general class of Lovelock invariants, constructed of various powers of the curvature tensor, which lead to special properties for the equations of motion[CITATION].', 'hep-ph-0603242-3-11-4': 'Generally the existence in the action of higher curvature terms, as discussed above, leads to higher order equations of motion that produce tachyonic and/or ghost excitations in the spectrum as well as potentially non-symmetric and/or non-conserved pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-3-11-5': 'Having an action consisting solely of Lovelock invariants avoids all of these potential difficulties as well as those associated with the massive tensor ghosts.', 'hep-ph-0603242-3-11-6': 'The D-dimensional scalar excitation discussed above is also absent in this case.', 'hep-ph-0603242-3-11-7': 'It is interesting to note that the GB term is the leading correction to the EH action in perturbative string theory[CITATION].', 'hep-ph-0603242-3-11-8': 'Higher order Lovelock invariant may also be present in the action (when [MATH]) but these cannot be described by the function [MATH] as employed here since they are constructed out of cubic or higher order combinations of the curvature tensor.', 'hep-ph-0603242-3-11-9': 'The effect of the presence of general Lovelock invariants in the action of the ADD model has been discussed within the black hole context in Ref. [CITATION].', 'hep-ph-0603242-3-12-0': 'A further possibly very useful form for the quadratic action that is commonly used in the literature is [EQUATION] where [MATH] is the square of the Weyl tensor which can be expressed as[CITATION]: [EQUATION] where [MATH] is the number of extra dimensions; the [MATH] are linearly related to the coefficients [MATH] above, e.g., [MATH].', 'hep-ph-0603242-3-12-1': 'This translation is simplified via the use of the identity[CITATION] [EQUATION]', 'hep-ph-0603242-3-12-2': 'Noting that the [MATH] have dimensions of mass[MATH] it is sometimes common in the literature to write [EQUATION] where [MATH] are two mass parameters which are naturally [MATH] in the theory.', 'hep-ph-0603242-3-12-3': 'One then finds that [MATH] are directly related to the bulk masses of the D-dimensional massive scalar and tensor excitations discussed above.', 'hep-ph-0603242-3-12-4': 'To avoid tachyons we apparently must demand that [MATH] but even in such a case as we will see this the massive tensor field remains a ghost since the kinetic term for this field will have the wrong sign.', 'hep-ph-0603242-3-13-0': 'Clearly all these forms for [MATH] are simply related.', 'hep-ph-0603242-3-13-1': 'In what follows we will make use of all of the above forms of [MATH] and treat them interchangeably.', 'hep-ph-0603242-3-14-0': "Our first goal will be to explicitly calculate [MATH] in one of these 'bases' from the more general [MATH] in terms of [MATH] and its derivatives.", 'hep-ph-0603242-3-14-1': 'To begin we perform a Taylor series expansion of [MATH] to quadratic order in all three arguments evaluating the result in the background metric, e.g., [EQUATION] where [MATH] is a constant corresponding to the evaluation of [MATH] itself in the fixed curvature background metric and [MATH]; [MATH] means that [MATH] is to be evaluated in terms of the background metric which we here assume to be a space of constant curvature, i.e., a maximally symmetric space as is the case in both the ADD and RS models.', 'hep-ph-0603242-3-14-2': 'Thus the quantities [MATH], [MATH], [MATH], [MATH] and [MATH] are just numbers which depend on the explicit form of the metric and possibly the number of extra dimensions.', 'hep-ph-0603242-3-14-3': 'In such a maximally symmetric space the Weyl tensor and corresponding invariant both vanish identically, i.e., [MATH] and one further finds that [EQUATION]', 'hep-ph-0603242-3-14-4': 'Note that in ADD [MATH] since the metric is Minkowskian whereas in the [MATH] RS bulk [MATH] (away from the two branes) where the parameter [MATH] originates from the usual RS metric [MATH].', 'hep-ph-0603242-3-15-0': 'Without making any further assumptions we obtain [EQUATION] where we have defined [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0603242-3-15-1': 'For the case of [MATH] this reproduces the results give by, e.g., Navarro and Van Acoleyen in [CITATION].', 'hep-ph-0603242-3-15-2': 'Note that if we make the assumption that [MATH] is a function only of [MATH] and the combination [MATH] then [MATH] etc and, also noting that [MATH]=0, we obtain [MATH] so that the remaining expressions greatly simplify; we now obtain [EQUATION] where the parameter [MATH] is given by [EQUATION]', 'hep-ph-0603242-3-15-3': 'Note that having [MATH] implies that the Weyl term, [MATH], in the effective action is absent in second order which is equivalent to taking [MATH] thus eliminating the massive tensor ghost issue.', 'hep-ph-0603242-3-15-4': 'This field is now removed from the spectrum though the D-dimensional scalar remains in general.', 'hep-ph-0603242-3-16-0': '# Application I: ADD', 'hep-ph-0603242-3-17-0': 'In this section we will apply the above analysis to the general ADD framework where we now require (since the space is flat) [MATH] so that [MATH] automatically.', 'hep-ph-0603242-3-17-1': 'This significantly reduces the possible deviations from the classic ADD picture.', 'hep-ph-0603242-3-17-2': 'In this specific case the general second order expansion of [MATH] is rather simple and is given by [EQUATION]', 'hep-ph-0603242-3-17-3': 'Note that if we also demand that [MATH] be a function only of [MATH] and the combination [MATH] in order to avoid the issue of the massive tensor ghost this expression simplifies even further to [EQUATION]', 'hep-ph-0603242-3-17-4': 'Of course, there has been some discussion about other ways to circumvent this tensor ghost problem than by completely eliminating it from the perturbative spectrum.', 'hep-ph-0603242-3-17-5': 'Since we are working only to lowest non-trivial order perhaps we should keep an open mind about the forms for [MATH].', 'hep-ph-0603242-3-17-6': 'Note that since [MATH] will essentially rescale the overall mass factor in the action we must demand that [MATH] to insure that the usual D-dimensional massless tensor gravitons not be ghost-like.', 'hep-ph-0603242-3-18-0': 'How are the predictions of ADD modified by these additional curvature terms?', 'hep-ph-0603242-3-18-1': 'The basic ADD picture leads to three essential predictions [CITATION]: ([MATH]) the emission of graviton KK states during the collision of SM particles producing signatures with apparent missing energy[CITATION]; ([MATH]) the exchange of graviton KK excitations between SM brane fields leading to dimension-8 contact interaction-like operators with distinctive spin-2 properties[CITATION]; ([MATH]) the production of black holes(BH) at colliders and in cosmic rays with geometric cross sections, [MATH], with [MATH] being the BH Schwarzschild radius, once collision energies greater than [MATH] are exceeded[CITATION].', 'hep-ph-0603242-3-19-0': 'The production and properties of D-dimensional, TeV-scale BH in higher curvature theories has been partially explored within the context of Lovelock extended gravity[CITATION] though not yet so in the fully general quadratic gravity case described by the function [MATH] considered here.', 'hep-ph-0603242-3-19-1': 'Such a study, which would be very interesting, is far beyond the scope of the present analysis.', 'hep-ph-0603242-3-19-2': 'However, it is interesting to make several observations: ([MATH]) Consider the vacuum solution; if we expand the general action [MATH] above to only quadratic order and if we also assume that all interesting solutions must satisfy [MATH] in the vacuum, then the only deviations from the conventional Schwarzschild form arise from the GB term in the action.', 'hep-ph-0603242-3-19-3': 'This can be seen immediately by examining the equations of motion resulting from the general quadratic action, e.g., in Ref. [CITATION].', 'hep-ph-0603242-3-19-4': 'This result does not remain valid when [MATH] is treated exactly.', 'hep-ph-0603242-3-19-5': '([MATH]) If [MATH] only and is treated exactly without expansion then the equations of motion allow for the conventional external BH result with [MATH] and will appear as an ordinary D-dimensional Schwarzschild solution.', 'hep-ph-0603242-3-19-6': 'This is not, however, the most general solution as can be see by considering the simple case of [MATH] with [MATH] a dimensionless parameter.', 'hep-ph-0603242-3-19-7': 'Here there is also exists a solution with [MATH], which is a constant corresponding to deSitter or anti-deSitter space depending on the sign of [MATH].', 'hep-ph-0603242-3-19-8': '([MATH]) If [MATH], with [MATH] being a constant, then the general BH solution has neither [MATH] nor [MATH] equal to zero as is well-known from the exact solution[CITATION].', 'hep-ph-0603242-3-19-9': 'A more detailed study of these possibilities would be worthwhile.', 'hep-ph-0603242-3-20-0': 'Let us now consider the situation of graviton exchange where it is well-known[CITATION] that ADD leads to new dimension-8 contact interactions.', 'hep-ph-0603242-3-20-1': 'To obtain the analogous quantities here we must expand the integrand of the action, i.e., the Lagrangian [MATH], to second order in the fluctuations, [MATH], around the flat background metric: [EQUATION]', 'hep-ph-0603242-3-20-2': 'Here we have expressed the original metric as [EQUATION] with [MATH] being the background metric, which is identified in the ADD model as the Minkowski metric [MATH].', 'hep-ph-0603242-3-20-3': 'The propagator is then just the inverse of the operator [MATH]; once the propagator is known we sandwich it between two 4-d localized (at the origin of the extra-dimensional co-ordinates) and conserved SM stress-energy sources, [MATH], to find the relevant scattering amplitude; we must remember to later KK decompose the various towers.', 'hep-ph-0603242-3-20-4': "Fortunately, much of this work has been done for us by Accioly, Azeredo and Mukai[CITATION] from which, with some modifications, we obtain the expression for the D-dimensional 'graviton' exchange amplitude (before performing the KK sums) [EQUATION] where [MATH] are the familiar (unaltered in terms of the compactification radius [MATH]) flat space KK masses.", 'hep-ph-0603242-3-20-5': 'The [MATH] label the various KK levels; [MATH] is the 4-d trace of the SM source stress-energy tensor and [MATH] are just [EQUATION] as described above.', 'hep-ph-0603242-3-20-6': '[MATH] is the compactification radius which sets the KK mass scale; here we have assumed a common value for this quantity for all [MATH] extra dimensions so that the volume of the compactified space is just [MATH].', 'hep-ph-0603242-3-20-7': 'In the expression above the first term in the amplitude is the usual one encountered in the ADD model which results from the D-dimensional EH action and combines the contributions of the 4-d spin-2 graviton and spin-0 graviscalar KK towers.', 'hep-ph-0603242-3-20-8': 'The second and third terms correspond to the new D-dimensional massive ghost tensor and scalar contributions, respectively.', 'hep-ph-0603242-3-20-9': 'The difference in the factors of [MATH] versus [MATH] in the first two terms arises from the existence of a 5-d bulk mass for the tensor ghost field.', 'hep-ph-0603242-3-20-10': 'It is interesting to note that the full amplitude is very well behaved at large [MATH] (in fact, going as [MATH]) due to the detailed cancellations between the various terms.', 'hep-ph-0603242-3-21-0': 'Here [MATH] represent the bulk mass terms of the new fields which enter into the KK tower masses of the scalars (spin-0) and tensors (spin-2 and spin-0), respectively; here we see the effect of the tensor ghost KK tower exchange explicitly.', 'hep-ph-0603242-3-21-1': 'From this point of view it appears that the only way to remove this ghost tower is to take the bulk mass [MATH] implying that [MATH] is solely a function of [MATH] and the combination [MATH], which we will assume from now on in our ADD discussion.', 'hep-ph-0603242-3-21-2': 'Note that the existence of a GB term in the action will not yield a contribution to [MATH].', 'hep-ph-0603242-3-21-3': 'Since, as discussed above [MATH] is already required, tachyonic KK scalars are avoided when the denominator in the expression for [MATH] above is positive; when [MATH] is assumed to be a function only of [MATH] and [MATH], then this denominator simplifies to [MATH].', 'hep-ph-0603242-3-21-4': 'In the limit where [MATH] alone, and accounting for a sign factor in the definition of the above actions, our result for the squared scalar mass, [MATH], agrees with that obtained by Demir and Tanyildizi[CITATION].', 'hep-ph-0603242-3-21-5': 'As shown by these authors, the effect of the new scalar tower exchange is generally rather suppressed in comparison to the more familiar graviton exchange since the ratio [MATH] is small for most SM particle sources at TeV colliders.', 'hep-ph-0603242-3-21-6': 'For example, for the process [MATH] this ratio is of order [MATH].', 'hep-ph-0603242-3-21-7': 'The corresponding ratio of the KK summed scalar to graviton exchange amplitudes is somewhat further reduced by ([MATH]) the existence of the finite bulk scalar mass which implies that there are no light scalar KK exchanges with masses below [MATH] and ([MATH]) the [MATH]-dependent numerical factor in the denominator of the scalar amplitude.', 'hep-ph-0603242-3-21-8': 'Naturalness suggests that [MATH] TeV or larger unless the parameters of [MATH] are somehow fine-tuned.', 'hep-ph-0603242-3-21-9': 'For example, if [MATH], then [MATH] for all [MATH] if [MATH] is not too far from O(1).', 'hep-ph-0603242-3-21-10': 'Interestingly we see here that as [MATH] we recover the usual EH expectation as then [MATH].', 'hep-ph-0603242-3-21-11': 'Thus we find that for many practical purposes the structure of the usual ADD results for graviton exchange are not qualitatively modified when the action is generalized to the form considered here.', 'hep-ph-0603242-3-21-12': 'However, with the existence of these additional scalars being a hallmark of the extended action, it behooves us to find a way to isolate their effects experimentally.', 'hep-ph-0603242-3-22-0': 'In expressions for graviton exchange only the combination [MATH] will now appear.', 'hep-ph-0603242-3-22-1': "In the amplitude this will lead to a modification of the pure 'graviton' exchange cross section expectations by a factor of [MATH], which is likely to be of [MATH], provided [MATH] is considered to be held fixed.", 'hep-ph-0603242-3-22-2': 'When the graviton tower interference term with the SM dominates, the effect in the gravitational part of the cross section will scale as [MATH].', 'hep-ph-0603242-3-22-3': 'Given the previous results of Demir and Tanyildizi[CITATION], this is not surprising.', 'hep-ph-0603242-3-23-0': 'We further note that since [MATH] is [MATH] TeV or larger it has no effect on laboratory measurements of the strength of the gravitational interaction in the micron range when [MATH].', 'hep-ph-0603242-3-24-0': 'Before closing this part of the discussion we would like to remind the reader that it was pointed out long ago[CITATION] that we can take any action of the form [MATH] and map it over to the EH action coupled to an minimally coupled real scalar field with a rather complicated potential [MATH], depending exponentially on the scalar field.', 'hep-ph-0603242-3-24-1': 'This can be done via a special conformal transformation [EQUATION]', 'hep-ph-0603242-3-24-2': 'Going from the original (Jordan) to the new (Einstein) frame one explicitly sees the existence of the new scalar degree of freedom.', 'hep-ph-0603242-3-24-3': 'The mass of this scalar field is exactly that of the field [MATH] above and can be gotten directly from the canonically normalized potential [MATH] in the usual manner, i.e., using [MATH].', 'hep-ph-0603242-3-24-4': 'This is a very powerful tool as it allows us to extend our previous flat space result for [MATH] to the much more general case where the space has constant curvature.', 'hep-ph-0603242-3-24-5': 'For example, if [MATH], we find that the value of [MATH] is the same as discussed above, i.e., [MATH], in a space with constant curvature.', 'hep-ph-0603242-3-24-6': 'This will be an important result that we will employ when we discuss the case of the RS setup.', 'hep-ph-0603242-3-25-0': 'We now turn to the emission of gravitons in SM particle collisions.', 'hep-ph-0603242-3-25-1': 'Since the compactifying space is flat in the ADD case the normalizations of the graviton (and scalar) wavefunctions which control their couplings are unaltered by the existence of the quadratic curvature terms but the relationship between [MATH] and [MATH]M_pl[MATH] is modified.', 'hep-ph-0603242-3-25-2': 'This was briefly mentioned above where we saw that in the small curvature limit the parameter [MATH] essentially renormalizes the fundamental scale.', 'hep-ph-0603242-3-25-3': 'To see this in the present case it is sufficient to examine the tensor/spin-2 kinetic part of the 4-d effective Lagrangian to second order in [MATH] (which has not yet been KK-expanded) in the familiar transverse traceless gauge, i.e., [MATH], [MATH]; one obtains[CITATION] [EQUATION] where here [MATH] and [MATH] are defined above.', 'hep-ph-0603242-3-25-4': 'When we assume that [MATH] is only a function of [MATH] and the combination [MATH] then the second term in [MATH] vanishes and we recover the familiar result of the standard EH scenario apart from the overall factor of [MATH].', 'hep-ph-0603242-3-25-5': 'Hence, to recover the conventional 4-d EH action when inserting the usual (extra dimensionally) flat zero mode graviton wavefunction into [MATH] the ADD relationship must be modified, as hinted above, to [EQUATION] where [MATH]M_pl[MATH] is the 4-d reduced Planck scale and [MATH] is the volume of the compactified space.', 'hep-ph-0603242-3-25-6': 'Of course, [MATH] is just unity in the standard ADD model which employs the EH action.', 'hep-ph-0603242-3-25-7': 'Since the lightest of the KK scalars has a mass which is naturally on the order of a TeV and has rather weak couplings to SM fields these particles will not play much of an important role in missing energy processes.', 'hep-ph-0603242-3-25-8': 'If the cross section for graviton production, i.e., missing energy, is expressed in terms of the original [MATH] with other parameters held fixed, then the presence of [MATH] leads to a modification of the production cross section by a factor of [MATH].', 'hep-ph-0603242-3-25-9': 'However, as [MATH] is not likely to remain a direct observable (only the product [MATH] is) there may be no way to experimentally disentangle this effect.', 'hep-ph-0603242-3-25-10': 'Furthermore, for any given [MATH], since [MATH]M_pl[MATH] is numerically fixed and [MATH] is an input parameter the resulting derived value of [MATH] which sets the scale for the masses of the KK states is altered.', 'hep-ph-0603242-3-26-0': 'We thus conclude that if we assume that [MATH] is a function of only of [MATH] and the combination [MATH] then the classic predictions ([MATH]) and ([MATH]) of the ADD model will be qualitatively unaffected by going to the more general action considered here except for possible overall scalings by inverse powers of [MATH] when the parameter [MATH] is held fixed: graviton emission rates scale like [MATH] while graviton exchange cross sections scale as [MATH] or [MATH] depending on the presence of important SM contributions to the relevant process.', 'hep-ph-0603242-3-26-1': 'Furthermore, given Eq. (22) and fixed values of [MATH] and [MATH], the KK masses, being proportional to [MATH], will scale as [MATH]; such a mass shift can be quite sizable for reasonable values of [MATH].', 'hep-ph-0603242-3-27-0': '# Application II: RS', 'hep-ph-0603242-3-28-0': 'The predictions of the classic RS model are the existence of TeV scale graviton resonances with fixed weak scale masses and couplings to the SM fields[CITATION], the existence of a weak scale radion excitation[CITATION], as well as the production of [MATH] BH.', 'hep-ph-0603242-3-28-1': 'In what follows we will be specifically interested in the nature of the KK gravitons so it is again sufficient to examine the quadratically expanded action.', 'hep-ph-0603242-3-28-2': 'The classic RS model is not generally strictly consistent with the assumed form of either the original action [MATH] or its quadratically expanded form [MATH].', 'hep-ph-0603242-3-28-3': 'As is well-known, and as mentioned above, the equations of motion that follow from [MATH] and [MATH] will generally be fourth order in the derivatives of the metric.', 'hep-ph-0603242-3-28-4': 'In the usual 5-d RS model, one solves the Einstein equations of the form [EQUATION] where [MATH] is the Einstein tensor arising from the EH action involving no more than two derivatives of the metric.', 'hep-ph-0603242-3-28-5': 'The problem is that RS completely specifies [MATH]: a cosmological constant in the 5-d bulk plus two [MATH]-function sources at the orbifold locations of the TeV and Planck branes.', 'hep-ph-0603242-3-28-6': 'SM matter confined to the TeV brane is supposed to not be a large contributor to the stress-energy.', 'hep-ph-0603242-3-28-7': 'To obtain this result the standard RS metric takes the form discussed above: [MATH] with the linear exponential warp factor leading to the bulk [MATH] and the two field derivatives acting on the absolute value leading to the brane [MATH]-functions.', 'hep-ph-0603242-3-28-8': '(This is related to the comment above that [MATH] is not truly constant in RS and has brane [MATH]-function singularities.', 'hep-ph-0603242-3-28-9': 'Recall that these [MATH]-functions are the results of assuming infinitely thin branes.)', 'hep-ph-0603242-3-28-10': 'If an identical metric is assumed in our more general case we still can obtain [MATH] but the fourth order equations would lead to the more singular derivatives of [MATH]-functions at the brane locations that are not canceled by any source terms.', 'hep-ph-0603242-3-28-11': 'This amongst other reasons is what led Kim, Kyae and Lee[CITATION] to consider only GB extensions of the EH action in RS since it is the only extension which uniquely produces Einstein equations of second order in the derivatives.', 'hep-ph-0603242-3-28-12': 'Thus if we keep the classic picture, an analysis of RS given our assumed effective action expanded around a background of constant curvature is not relevant.', 'hep-ph-0603242-3-28-13': '(A possible way of dealing with these derivatives of [MATH]-functions arising from orbifold singularities in higher dimensional effective field theories has been discussed in Ref. [CITATION].', 'hep-ph-0603242-3-28-14': 'Implementing our scheme employing such techniques is, however, beyond the scope of the present paper.)', 'hep-ph-0603242-3-29-0': "To avoid these issues for now we simplify our discussion of this problem (and to convince ourselves that an RS-like solution is possible in this framework) we consider a singularity-free, 'softened' version of RS where the orbifolded bulk space with branes is replaced by an interval, as has been suggested for other reasons[CITATION], with SM matter placed at one end point possessing an ignorable amount of stress-energy.", 'hep-ph-0603242-3-29-1': 'With a cosmological constant on the interval we can recover the background [MATH] bulk; in addition by removing the absolute value sign of the co-ordinate [MATH] in the metric above we expunge the [MATH]-functions as well as the possibility of any of their higher derivatives appearing in the equations of motion.', 'hep-ph-0603242-3-29-2': 'The boundary conditions at the end points for the graviton KK states can then be freely chosen to be the same as that of the original RS model.', 'hep-ph-0603242-3-29-3': "This space is truly one of constant curvature and the general analysis we have presented above will now be applicable to this 'softened' RS on an interval.", 'hep-ph-0603242-3-30-0': 'It is easy to verify that the form of the equations of motion[CITATION] in this case (recalling that we are only searching for solutions with maximally-symmetric, constant [MATH] backgrounds) are given by: [EQUATION] and that if we take stress-energy tensor in the 5d bulk to be of the usual RS form [EQUATION] with [MATH], then indeed a space of constant curvature, i.e. [MATH], can be an allowed solution.', 'hep-ph-0603242-3-30-1': 'Taking the trace of the equations of motion above, evaluating it in the constant curvature bulk and relating the values of [MATH] to [MATH] as before (recalling that here [MATH] using the softened metric) results in the constraint equation [EQUATION] where here [MATH].', 'hep-ph-0603242-3-30-2': 'It is interesting to note that if we assume that [MATH]constant then this constraint equation automatically implies that [MATH]constant as well; but this does not necessarily further require that all of the [MATH] are constants as we will see below.', 'hep-ph-0603242-3-30-3': 'When [MATH] is only a function of [MATH] and the combination [MATH], this constraint equation simplifies to [EQUATION] while in the specific RS background case this explicitly becomes [EQUATION]', 'hep-ph-0603242-3-30-4': 'It is important to recall that [MATH] itself can be a complicated function of [MATH] so that this equation can be quite nontrivial.', 'hep-ph-0603242-3-30-5': 'For the EH action limit this yields the usual relation that [MATH]; here it in general provides an additional constraint on the allowed forms of the function [MATH] since we are requiring [MATH] to be both real and negative.', 'hep-ph-0603242-3-30-6': 'Given a specific function [MATH] for which a solution exists, this equation directly relates [MATH] and [MATH] though the solution may not be unique.', 'hep-ph-0603242-3-30-7': 'For example, if we assume for purposes of demonstration the simple case of [EQUATION] as employed above, then there are two branches of solutions for [MATH]: [EQUATION] one of which (the negative root) goes over to the usual EH result as the parameter [MATH].', 'hep-ph-0603242-3-31-0': 'Allowing for the possibility of a RS-like solution with a softened metric it is interesting to think briefly about the previously analyzed effects of the GB term in the RS scenario.', 'hep-ph-0603242-3-31-1': 'This analysis was originally performed for the classic RS[CITATION] setup which employed the standard form of the RS metric; that result would now be modified by the changes in the model assumptions, i.e., moving to an interval and removing the [MATH]-function sources at the end points.', 'hep-ph-0603242-3-31-2': 'The previous analysis of BH in RS with the added GB term would not be significantly affected if this transition were made.', 'hep-ph-0603242-3-31-3': 'However, the properties and spectrum of the graviton KK states certainly would be influenced since the [MATH]-function terms are now absent.', 'hep-ph-0603242-3-31-4': 'The equation governing the masses and wavefunction of the graviton KK states for the present interval case can be obtained by expanding the equations of motion as before.', 'hep-ph-0603242-3-31-5': 'Since we are here only interested in the tensor modes associated with the usual gravitons, we can employ the expansion [EQUATION] where [MATH].', 'hep-ph-0603242-3-31-6': 'Applying the usual RS boundary conditions on the interval the most significant changes from the classic RS can be read off from Eqs. (15)-(28) in Ref. [CITATION] by setting the parameter [MATH] in appropriate places.', 'hep-ph-0603242-3-31-7': 'At the end of the day we find that the only apparent difference from the classic EH based RS model would be a shift in the relationship between the fundamental scale and [MATH]M_pl[MATH]-remarkably similar to what we saw for the ADD model above.', 'hep-ph-0603242-3-31-8': 'In the language employed in Ref. [CITATION] we would now obtain [EQUATION] where [MATH] is the coefficient of the GB term in the action.', 'hep-ph-0603242-3-31-9': 'Otherwise the masses as well as the couplings of all of the KK gravitons to localized SM matter would be identical to those of the original RS model expressed in terms of the derived parameter [MATH].', 'hep-ph-0603242-3-31-10': 'The explicit coupling and spectrum changes found in Ref[CITATION] for the graviton KK states in the presence of the GB term in the action were all found to due to the brane [MATH]-function singularities.', 'hep-ph-0603242-3-32-0': 'How would these graviton KK results obtained in the GB extended action generalize to the case of [MATH] above?', 'hep-ph-0603242-3-32-1': 'Here we choose to begin our analysis with [MATH], setting [MATH] from the beginning to avoid potential ghost fields, then taking [MATH] and using the same curvature expansion as above.', 'hep-ph-0603242-3-32-2': 'In order to make a connection with the previous discussion, the existing RS literature and to directly compare with the GB case, however, we massage our notation slightly and rewrite [MATH] in the following form: [EQUATION] where the parameters [MATH] and [MATH] are dimensionless; the action employed in Ref. [CITATION] is now directly recovered by taking the [MATH] and [MATH] limits.', 'hep-ph-0603242-3-32-3': 'It is important at this point to recall that to obtain the linearized graviton equations of motion it is sufficient to employ [MATH] while the complete [MATH] needs to be examined in order to demonstrate the existence of the required [MATH] solution.', 'hep-ph-0603242-3-32-4': 'The equations of motion resulting from [MATH] are given by[CITATION] [EQUATION]', 'hep-ph-0603242-3-32-5': 'Here [MATH] is the covariant derivative operator and here [MATH].', 'hep-ph-0603242-3-32-6': 'First we look at the [MATH] component of this equation, remembering that for the moment we will only be interested in the tensor excitations corresponding to the KK gravitons which are massless in 5-d.', 'hep-ph-0603242-3-32-7': 'In the usually chosen gauge, [MATH] is still a constant to linear order so we arrive at a consistency condition [EQUATION]', 'hep-ph-0603242-3-32-8': 'Note that this reduces to the previously obtained purely quadratic GB extended RS result[CITATION] when [MATH].', 'hep-ph-0603242-3-32-9': 'In the more general case, this expression is not overly useful given the exact result in Eq. (27).', 'hep-ph-0603242-3-33-0': 'Turning now to the [MATH] terms which contain the 4-d graviton tensor excitation, we linearize employing the previously mentioned transverse, traceless gauge with constant [MATH].', 'hep-ph-0603242-3-33-1': 'This gives the standard equation of motion for the RS graviton found long ago[CITATION] though scaled by an overall factor.', 'hep-ph-0603242-3-33-2': 'Employing the standard KK decomposition [EQUATION] and recalling that [MATH], the [MATH] are seen to satisfy [EQUATION]', 'hep-ph-0603242-3-33-3': 'The overall factor [MATH] is given by [EQUATION] or, more explicitly in the RS case, [EQUATION] (Again we recall that [MATH] itself can be a function of [MATH].)', 'hep-ph-0603242-3-33-4': 'This leads to a rescaling of the usual RS relationship [EQUATION] via the renormalization of the zero mode (i.e., massless graviton) wavefunction, thus generalizing Eq. (31).', 'hep-ph-0603242-3-33-5': 'Of course, [MATH] is required to avoid ghost states among the usual gravitons KKs.', 'hep-ph-0603242-3-33-6': 'This result reduces to that previously obtained in the RS case with just the added GB term[CITATION] once boundary effects are neglected.', 'hep-ph-0603242-3-34-0': 'From this analysis we see immediately that the masses of the KK gravitons are identical to those obtained in the original RS model, provided we use the same value of the parameter [MATH], as we might have expected.', 'hep-ph-0603242-3-34-1': 'Here we are faced with the question of just what are the independent parameters.', 'hep-ph-0603242-3-34-2': '[MATH] is clearly a derived parameter that is obtained by simultaneously solving Eqs. (27) and (40) for any given model.', 'hep-ph-0603242-3-34-3': 'In that sense, the KK graviton spectrum would just be rescaled in comparison to the usual expectations given the same input value of [MATH].', 'hep-ph-0603242-3-34-4': 'As we have just seen, and as in the ADD case, the effect of a factor like [MATH] on the KK graviton couplings to 4-d SM matter depends upon which model parameters are assumed to be held fixed.', 'hep-ph-0603242-3-34-5': 'At the very least, up to an overall constant, these KK graviton couplings are identical to those of the standard RS model.', 'hep-ph-0603242-3-35-0': 'As an example of a simple model where the shifts in the KK spectrum can be calculated analytically consider substituting for the integrand of the conventional RS action, [MATH], the simple higher curvature action [MATH] as was considered above.', 'hep-ph-0603242-3-35-1': 'Let [MATH] be the values obtained for the parameter [MATH] in the usual RS model, i.e., [MATH].', 'hep-ph-0603242-3-35-2': 'Keeping the warp factor fixed we can use the equations above to calculate the value of, e.g., the mass of the first graviton KK state in both the standard RS model, [MATH], and in the current model with an augmented action, [MATH].', 'hep-ph-0603242-3-35-3': 'Using Eqs. (27) and (40) this ratio can be calculated analytically in the present case as a function of [MATH] and [MATH]M_pl[MATH]; we obtain [EQUATION]', 'hep-ph-0603242-3-35-4': 'The result of this calculation is shown in Fig.1 for a wide range of model parameters.', 'hep-ph-0603242-3-35-5': 'In this example we see that the size of the possible shift in the mass spectrum can be quite large assuming a fixed value of [MATH].', 'hep-ph-0603242-3-36-0': 'So far we have only considered the 4-d graviton, spin-2 excitations.', 'hep-ph-0603242-3-36-1': 'It is important to remember that our softened RS model now has an additional massive scalar in the 5-d spectrum with a large bulk mass, [MATH], and that no massless scalar zero mode will exist.', 'hep-ph-0603242-3-36-2': 'Since the bulk scalar mass is naturally of order [MATH] the KK spectrum of the corresponding tower will begin with a KK scalar state whose mass is qualitatively comparable to that of the first graviton excitation.', 'hep-ph-0603242-3-36-3': 'This bulk mass is explicitly calculable from the expansion of the full action to quadratic order, [MATH], by going to the Einstein frame since we know that the GB term does not contribute to this parameter.', 'hep-ph-0603242-3-36-4': 'In that case, using the results from the previous section we find that [EQUATION] or, in terms of the original parameters of the action, evaluated in the RS background: [EQUATION]', 'hep-ph-0603242-3-36-5': 'Note that [MATH] is required to avoid the scalar tachyons and graviton ghosts, consistent with our above analysis.', 'hep-ph-0603242-3-36-6': 'Note further that this reproduces the results of Eq. (19) in the flat space, [MATH], limit.', 'hep-ph-0603242-3-37-0': 'Given any [MATH] the scalar bulk mass is known and we can determine the mass(es) of the lightest KK scalar state(s) by following the standard RS manipulations[CITATION].', 'hep-ph-0603242-3-37-1': 'These masses are essentially given by the first roots of the equation [EQUATION] where [MATH] and [MATH] is the usual Bessel function.', 'hep-ph-0603242-3-37-2': 'The solution for the first KK state is provided by Fig.2; as stated above there are no massless modes.', 'hep-ph-0603242-3-37-3': 'The lightest scalar mass is then [MATH].', 'hep-ph-0603242-3-37-4': 'Here we observe that the mass of the first scalar KK scales almost linearly with the bulk mass when [MATH] gets large.', 'hep-ph-0603242-3-37-5': 'Note that for [MATH] and a typical value[CITATION] of [MATH], we then find [MATH] implying [MATH] from Fig.1; this is about 3 times larger than the root for the usual lightest massive KK graviton, [MATH].', 'hep-ph-0603242-3-37-6': 'Thus we see that unless [MATH] takes on large values the first scalar KK state is always rather heavy.', 'hep-ph-0603242-3-37-7': 'As is well-known, the [MATH] values for the more massive KK scalar states will be somewhat larger: approximately given by [MATH] where [MATH] labels the KK level.', 'hep-ph-0603242-3-37-8': 'Since these scalars will couple to the trace of the stress-energy tensor for the 4-d SM fields they will interact far more weakly than do the graviton KK states unless this is at least partially offset by ratios of 5-d wavefunction factors.', 'hep-ph-0603242-3-37-9': 'A quick estimate of such factors, however, indicates that, if anything, these wave function ratios lead to a further suppression of the scalar couplings relative to those of the KK gravitons by [MATH] as shown in Fig.3.', 'hep-ph-0603242-3-37-10': 'This overall picture of the scalar sector is qualitatively very similar to that of the existence of a very heavy tower of RS radions[CITATION] or a tower of KK Higgs bosons as in the case of Universal Warped Extra Dimensions[CITATION].', 'hep-ph-0603242-3-38-0': 'In the analysis as presented here we have ignored the possibility that the new scalar KK states may mix with the (usually eaten) RS graviscalars through cross-talk in the equations of motion, i.e., we have assumed that the 5-d tensor and scalar KK decompositions can be performed independently, and this is something which needs further exploration.', 'hep-ph-0603242-3-38-1': 'A fully detailed analysis of the such possibilities is, however, beyond the scope of the present paper.', 'hep-ph-0603242-3-39-0': 'It is perhaps interesting to ask whether the usual [MATH]=constant ([MATH]) solution considered here necessitates the metric and matter distribution of the conventional RS model employed above without the further assumption of a maximally symmetric space.', 'hep-ph-0603242-3-39-1': 'To analyze a simple and more easily tractable situation let us consider the more general warped metric [MATH] and assume that [MATH] only.', 'hep-ph-0603242-3-39-2': 'The first question to address is what is the most general form of the function [MATH]; to deal with this issue we note that the Ricci scalar arising from this metric is in general given by [MATH].', 'hep-ph-0603242-3-39-3': "In the 'soft' version of the RS model on the interval defined above one had [MATH] and [MATH] so that [MATH] as usual.", 'hep-ph-0603242-3-39-4': 'Here, this curvature condition provides a differential equation for the function [MATH]; by solving this equation we arrive at the result [MATH], where the [MATH] are integration constants and [MATH].', 'hep-ph-0603242-3-39-5': 'Choosing the [MATH] appropriately and rescaling [MATH], we can rewrite this in a more familiar form as [MATH] with [MATH] a dimensionless constant.', 'hep-ph-0603242-3-39-6': 'Note that when [MATH] and [MATH] we recover the usual RS result.', 'hep-ph-0603242-3-39-7': 'However, the choice of [MATH] can easily modify the warp factor from its conventional behavior.', 'hep-ph-0603242-3-39-8': 'What is the nature of the bulk matter distribution that yields this metric?', 'hep-ph-0603242-3-39-9': 'The solution to this can be obtained by considering Eqs.(24) and (26) with [MATH].', 'hep-ph-0603242-3-39-10': 'Eq.(26) immediately tells us that the trace of the 5-d stress-energy tensor [MATH], a constant, so that if we define [MATH] then we must have [MATH], with [MATH] an arbitrary function.', 'hep-ph-0603242-3-39-11': 'The [MATH] and [MATH] components of the equations of motion when combined then provides a first order differential equation for [MATH] that can be solved in a straightforward manner.', 'hep-ph-0603242-3-39-12': 'Setting [MATH] and defining [MATH] with [MATH], we obtain the general solution (assuming [MATH]) [EQUATION] with [MATH] an integration constant.', 'hep-ph-0603242-3-39-13': 'Though this is far from a uniform energy distribution (away from the TeV brane) it does lead to a space of constant curvature but not one which is maximally symmetric.', 'hep-ph-0603242-3-39-14': 'Thus we see that it is possible that the requirement that [MATH] does allow for the possibility of more complex solutions than that employed in the original RS model.', 'hep-ph-0603242-3-40-0': '# Discussion and Conclusions', 'hep-ph-0603242-3-41-0': 'In this paper we have begun an examination of how generic higher curvature terms in the gravitational action can alter the predictions of both the ADD model and the RS model defined on a interval to avoid possible brane singularities.', 'hep-ph-0603242-3-41-1': 'We have assumed that the traditional assumptions of the two models, e.g., SM localized matter in a conformally flat bulk, remain valid; we have not considered in detail more complex setups that may now be allowed by the modified equations of motion.', 'hep-ph-0603242-3-41-2': 'To be more concrete, we have further assumed that the EH action is now generalized to an action which is of the form [MATH] where [MATH] is a well-behaved function, [MATH] and [MATH].', 'hep-ph-0603242-3-41-3': 'In D-dimensions this action results in a propagating massless tensor field (identified with the usual graviton), a massive ghost tensor field, as well as a massive (possibly tachyonic) scalar.', 'hep-ph-0603242-3-41-4': 'The potentially dangerous ghost is removable from the perturbative spectrum, i.e., it becomes infinitely massive, if we demand that [MATH] only.', 'hep-ph-0603242-3-41-5': 'The remaining new scalar field has a bulk mass whose value is naturally expected to be of order the fundamental scale, [MATH], in either scenario.', 'hep-ph-0603242-3-41-6': 'The resulting ADD and RS models are altered in similar ways from their traditional standard forms:', 'hep-ph-0603242-3-42-0': '([MATH]) New scalar KK excitations appear in the spectrum of both models in a rather benign fashion coupling to the trace of the stress-energy tensor of the localized SM fields.', 'hep-ph-0603242-3-42-1': 'Since this trace is proportional to SM masses, the couplings of these scalars are relatively strongly suppressed in comparison to those for the KK gravitons at typical collider energies in both models.', 'hep-ph-0603242-3-42-2': 'In the ADD model, the KK scalar excitations begin at a mass [MATH] TeV.', 'hep-ph-0603242-3-42-3': 'Consequently their contributions to missing-energy signatures as well as to the usual dimension-8 contact interactions are further kinematically suppressed.', 'hep-ph-0603242-3-42-4': 'Thus at leading order these new scalars do not much influence ADD collider signatures.', 'hep-ph-0603242-3-42-5': 'In RS, the bulk scalar mass tends to be large so that the lightest scalar KK state is several times more massive than is the lightest KK graviton.', 'hep-ph-0603242-3-42-6': 'Given their rather weak couplings such states will be difficult to observe at colliders.', 'hep-ph-0603242-3-43-0': '([MATH]) The basic model relationships involving the fundamental and 4-d Planck masses in both models get rescaled by functions of [MATH] and its derivatives evaluated in the corresponding background metric of the two models: in ADD we obtain [MATH]M_pl[MATH] while in RS we obtain [MATH]M_pl[MATH] where [MATH] is explicitly given in Eq. (39).', 'hep-ph-0603242-3-43-1': 'Assuming that [MATH] is a fixed fundamental parameter these modifications lead to changes in the graviton KK sectors of both models.', 'hep-ph-0603242-3-43-2': 'In the ADD case, since [MATH]M_pl[MATH] is known and [MATH] is an input parameter for any given [MATH] the volume of the compactified space and, hence, the value of the compactification radius which sets the graviton KK mass scale is altered.', 'hep-ph-0603242-3-43-3': 'Due to the presence of the [MATH] factor the emission rate for gravitons in the collisions of SM particles and for the graviton exchange amplitude are both modified by potentially O(1) effects.', 'hep-ph-0603242-3-43-4': 'Similarly in RS, [MATH] is a derived parameter which sets the scale for all the KK states.', 'hep-ph-0603242-3-43-5': 'The constraint Eq. (28) allows us to calculate [MATH] in terms of the input parameter [MATH] and the function [MATH] thus providing for us with [MATH].', 'hep-ph-0603242-3-43-6': 'In a manner similar to ADD, the presence of [MATH] rescales the coupling strengths of the of the KK graviton states to the SM fields thus modifying the widths and production cross sections at colliders by potentially O(1) factors.', 'hep-ph-0603242-3-44-0': 'As we have seen, the extension of the EH action to a more complicated structure can lead to significant quantitative modifications to both the ADD and RS model predictions in the simplest possible case.', 'hep-ph-0603242-3-44-1': 'The observation of such effects at future colliders could tell us valuable information about the underlying theory of gravity.', 'hep-ph-0603242-3-45-0': 'Note Added: After this paper was essentially completed, Ref. [CITATION] appeared which discusses generalized actions for the ADD model and thus has some common areas with the present work.', 'hep-ph-0603242-3-45-1': 'Where the two papers overlap there is general qualitative agreement though the points of view are somewhat different.'} | {'hep-ph-0603242-4-0-0': 'We begin a general exploration of the phenomenology of TeV-scale extra-dimensional models with gravitational actions that contain higher curvature terms.', 'hep-ph-0603242-4-0-1': 'In particular, we examine how the classic collider signatures of the models of Arkani-Hamed, Dimopoulos and Dvali (missing energy and new dimension-8 contact interactions) and of Randall and Sundrum (TeV-scale graviton Kaluza-Klein resonances) are altered by these modifications to the usual Einstein-Hilbert action.', 'hep-ph-0603242-4-0-2': 'We find that not only are the detailed signatures for these gravitationally induced processes altered but new contributions are found to arise due to the existence of additional scalar Kaluza-Klein states in the spectrum.', 'hep-ph-0603242-4-1-0': '# Introduction and Background', 'hep-ph-0603242-4-2-0': 'The question as to why the Planck and electroweak scales differ by so many orders of magnitude remains mysterious.', 'hep-ph-0603242-4-2-1': 'In recent years, attempts have been made to address this hierarchy issue within the context of theories with extra spatial dimensions that lower the effective scale of gravity to the TeV region.', 'hep-ph-0603242-4-2-2': 'In both the models of Arkani-Hamed, Dimopoulos and Dvali (ADD)[CITATION] and of Randall and Sundrum (RS)[CITATION], new effects of gravitational origin are expected to occur near the TeV scale which should be observable at future colliders such as the LHC and ILC.', 'hep-ph-0603242-4-2-3': 'Though these two models are very different in detail they do have some common features the most important of which are: ([MATH]) in their original versions they both assume that Standard Model matter is confined to a 4-dimensional brane; ([MATH]) they both assume that D-dimensional gravity is described by the Einstein-Hilbert (EH) action plus a possible cosmological constant and ([MATH]) the background spaces are maximally symmetric and are either strictly flat, i.e. , Minkowskian as in the ADD model with toroidally flat compactification, or being of constant curvature and is conformally flat, i.e., [MATH] as in the RS model.', 'hep-ph-0603242-4-2-4': 'How would the predictions of these two models be changed if we surrendered the assumption ([MATH]), i.e., that the EH action provided the full description of gravity and considered something more general?', 'hep-ph-0603242-4-2-5': 'This is the discussion we would like to begin in this paper which will follow a phenomenological bottom-up approach.', 'hep-ph-0603242-4-3-0': 'General Relativity (GR) as described by the EH action is considered to be an effective theory below the fundamental Planck scale, [MATH].', 'hep-ph-0603242-4-3-1': 'Thus, once energies approaching the scale [MATH] begin to be probed one might expect to observe deviations from the expectations arising from the EH action.', 'hep-ph-0603242-4-3-2': 'In the cases of both the ADD and RS models, future colliders will probe near or at their (effective) fundamental scales so that non-EH aspects of the true gravitational theory, whatever its form, should become apparent and be experimentally measured.', 'hep-ph-0603242-4-3-3': 'Since the ultraviolet form of the true gravity theory is as of yet unknown one may hope to capture some of its deeper aspects by considering how the presence of new higher curvature (and higher derivative) invariants in the actions of the ADD and RS models can lead to variations in the well-known predictions of these theories.', 'hep-ph-0603242-4-3-4': 'Many authors have considered the possibility of higher curvature invariants and how their existence would modify the predictions arising from the EH action within other contexts, e.g., the properties of black holes[CITATION], deviations in solar system tests of GR[CITATION] and in cosmology[CITATION] to possibly avoid the need for dark energy.', 'hep-ph-0603242-4-3-5': 'Some analyses along these lines for the potential modifications of the collider predictions of both the ADD and RS models have already been performed[CITATION].', 'hep-ph-0603242-4-3-6': 'In the present paper, we wish to both extend and generalize these results to get a feeling for the possible detailed variation of the various new gravitational phenomena as predicted by these classic models which will be potentially observable at future colliders.', 'hep-ph-0603242-4-3-7': 'In particular we are interested in how the well known signatures of both the ADD and RS models are morphed if we keep the basic setups intact but modify the actions on which the corresponding equations of motion are based.', 'hep-ph-0603242-4-3-8': 'A further generalization of such an analysis is possible if the original ADD/RS setups can now be simultaneously surrendered due to these modified actions resulting in entirely new setups with corresponding equations of motion; while this is an interesting possibility to consider it lies mostly beyond the scope of the present paper though it will be touched upon briefly in the discussion below.', 'hep-ph-0603242-4-4-0': 'Of course a completely general study of how these possible modifications to the effective gravity action may morph TeV collider signatures is an obviously immense task and here we aim only at a first round analysis in the discussion that follows.', 'hep-ph-0603242-4-4-1': 'The major signatures arising in both ADD and RS models originate from graviton exchange and the production of black holes; the ADD model also leads to missing energy signatures from graviton emission.', 'hep-ph-0603242-4-4-2': 'Fortunately, apart from issues associated with black holes, since we are dealing with maximally symmetric spaces of globally constant curvature, the relevant graviton properties (couplings, wavefunctions and propagators) necessary to extract experimental signatures for either model can be obtained from the expansion of the rather general action considered below to quadratic order in the curvature.', 'hep-ph-0603242-4-4-3': '(This would no longer be true if we wanted to consider, e.g., the triple graviton coupling as then an expansion to third order in the curvature would be required.)', 'hep-ph-0603242-4-4-4': 'This simplifying observation forms the basis of the analysis that follows and allows us to determine the relevant graviton properties in both ADD- and RS-like models for a wide class of effective actions.', 'hep-ph-0603242-4-5-0': 'The general outline of our analysis is presented in Section 2 where our basic assumptions and notations are also given.', 'hep-ph-0603242-4-5-1': 'In Section 3 we apply our analysis to the ADD model; we then apply it to the RS case in Section 4.', 'hep-ph-0603242-4-5-2': 'Our summary and conclusions are given in Section 5.', 'hep-ph-0603242-4-6-0': '# Analysis', 'hep-ph-0603242-4-7-0': 'When going beyond the EH action there are many possibilities to consider especially when we are living in extra dimensions.', 'hep-ph-0603242-4-7-1': 'In the literature various forms have been assumed for the potential structures of higher curvature and/or derivative invariants that may appear in the gravity action.', 'hep-ph-0603242-4-7-2': 'The fairly general structure that we will assume for the D-dimensional action in the present analysis takes the form (with [MATH] below): [EQUATION] where [MATH] is an arbitrary continuous, differentiable and generally mathematically well-behaved function; in particular we will assume that [MATH] is non-singular when all of its arguments are zero.', 'hep-ph-0603242-4-7-3': 'Here [MATH] is the usual D-dimensional Ricci scalar while [MATH] and [MATH] are quadratic invariants constructed from the curvature tensor [MATH]: [MATH], with [MATH] being the Ricci tensor, while [MATH].', 'hep-ph-0603242-4-7-4': '[MATH] is the D-dimensional fundamental gravity scale which is [MATH] TeV in ADD and [MATH]M_pl[MATH] in RS.', 'hep-ph-0603242-4-7-5': 'In the low energy, small curvature limit we expect [MATH] (plus a possible cosmological constant) and so the overall dimensionful factor in the expression above allows us to make direct contact with the EH action in this limit.', 'hep-ph-0603242-4-7-6': 'This specific form for [MATH], though not completely general, covers a wide array of possibilities and has been considered (sometimes only in [MATH]) in may different contexts for a multitude of purposes in the literature[CITATION].', 'hep-ph-0603242-4-7-7': 'Many of the higher curvature models previous considered by other authors form subcases of this more general action.', 'hep-ph-0603242-4-7-8': 'Generally we may think of [MATH] as summing a number of corrections to the EH action which arise higher dimension invariants which are suppressed by appropriate powers of [MATH].', 'hep-ph-0603242-4-7-9': 'Thus we will only consider energy regimes where the collision energies are below the effective fundamental scale in what follows as is usually done when considering ADD and RS phenomenology.', 'hep-ph-0603242-4-8-0': 'As is by now well-known[CITATION] the generalized gravity theories described by an action of the form [MATH] can potentially have several serious problems.', 'hep-ph-0603242-4-8-1': 'Since we will be dealing with ADD- and RS-type phenomenology we will be performing a perturbative analysis in the discussion that follows.', 'hep-ph-0603242-4-8-2': 'Employing such an analysis one finds that, amongst other things, this action leads to equations of motion which are generally fourth order in the derivatives of the metric.', 'hep-ph-0603242-4-8-3': 'In particular, in addition to the usual massless D-dimensional tensor graviton which results in, e.g., the familiar 4-d graviton and graviscalar Kaluza-Klein (KK) tower excitations, there may also be present in the linearized D-dimensional theory additional massive scalar and tensor excitations.', 'hep-ph-0603242-4-8-4': 'These fields will in 4-d have KK towers without massless modes and which can be ghostlike and/or tachyonic.', 'hep-ph-0603242-4-8-5': '(We can think of these new D-dimensional fields as having bulk masses which influence their corresponding 4-d KK tower masses.)', 'hep-ph-0603242-4-8-6': 'Furthermore, the equations of motion naturally involving higher derivatives of the fields can lead to problems with unitarity as well as non-canceling asymmetric pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-4-8-7': "The new massive tensor excitations are potentially the most serious issue to deal with as they are ghost fields that must be eliminated from the perturbative spectrum (though they may help in dealing with the theory's renormalizability and bad high-energy behavior).", 'hep-ph-0603242-4-8-8': 'It has been noted[CITATION] that we one can remove these states from the spectrum (i.e., by giving their bulk masses an infinite value) if a tuning occurs such that the quantities [MATH] and [MATH] only appear in the special combination [MATH] in the function [MATH].', 'hep-ph-0603242-4-8-9': 'How this tuning arises in the fundamental theory is unspecified.', 'hep-ph-0603242-4-8-10': 'There has been some discussion in the literature, however, that these ghost states may not be as dangerous as one would imagine from lowest order perturbation theory[CITATION] so that we should perhaps keep an open mind about the possible forms for [MATH].', 'hep-ph-0603242-4-8-11': 'We will return to this point in what follows.', 'hep-ph-0603242-4-9-0': 'Given a general action of the kind above there are several issues that one normally wants to address in order to extract information that can be compared with experimental data.', 'hep-ph-0603242-4-9-1': 'From studies of both the ADD and RS models there are certain things we want to know, e.g., ([MATH]) the spectrum, wave functions, propagators and Standard Model (SM) matter couplings of the KK graviton (and other possible) excitations and ([MATH]) the relationship between [MATH], the volume of the compactified dimensions and the (reduced) 4-d Planck scale [MATH]M_pl[MATH].', 'hep-ph-0603242-4-9-2': 'To obtain this information, as well as to make contact with several other analyses[CITATION], it is sufficient to expand the general action [MATH] above around the maximally symmetric background metric to quadratic order in the curvature to obtain an effective action for the graviton (and like) excitations.', 'hep-ph-0603242-4-9-3': 'At this level, one can extract the relevant 2-point functions as well as the differential equation for the KK wavefunctions which then yield the KK mass spectrum as well as the the desired graviton couplings to the localized SM fields.', 'hep-ph-0603242-4-9-4': 'If, however, one wanted to probe, e.g., graviton 3- or 4-point functions then we would need to expand to at least cubic or quartic order in the curvature, respectively; these will not be of interest to us here but might be of interest in future experiments[CITATION] which would tell us more about the underlying theory of gravity.', 'hep-ph-0603242-4-9-5': 'Interestingly, if we allow for cubic or higher invariants in the original action in a maximally symmetric background, they will make no contributions to the quantities of interest to us here.', 'hep-ph-0603242-4-9-6': 'Thus our analysis allows for the most general possible effective bulk local action for gravity.', 'hep-ph-0603242-4-10-0': 'Once we make this expansion, there are various equivalent ways of expressing the resulting effective action, [MATH], depending upon the basis of invariants we choose to employ; the most obvious form is simply [EQUATION] where [MATH] have been defined above.', 'hep-ph-0603242-4-10-1': '[MATH] is an effective cosmological constant and [MATH] are (in some cases dimensionful) constants all of which are functions of [MATH] and its derivatives evaluated by employing the relevant background metric.', 'hep-ph-0603242-4-10-2': "To relate this back to the EH action in the limit of small curvature, one can think of the (necessarily positive) parameter [MATH] as a 'renormalization' of the fundamental mass scale [MATH]: [MATH].", 'hep-ph-0603242-4-11-0': 'A second and perhaps more useful version of [MATH] is given by [EQUATION] where [MATH] is the well-known Gauss-Bonnet(GB) invariant: [EQUATION]', 'hep-ph-0603242-4-11-1': 'The co-efficients [MATH] and [MATH] can be easily converted to [MATH] above by some straightforward algebra: [MATH] and [MATH].', 'hep-ph-0603242-4-11-2': 'In D[MATH]4, the GB invariant is either a topological invariant or just a total derivative not contributing to the equations of motion, but this is no longer true for arbitrary values of [MATH].', 'hep-ph-0603242-4-11-3': 'The GB invariant is just (the quadratic) member of a general class of Lovelock invariants, constructed of various powers of the curvature tensor, which lead to special properties for the equations of motion[CITATION].', 'hep-ph-0603242-4-11-4': 'Generally the existence in the action of higher curvature terms, as discussed above, leads to higher order equations of motion that produce tachyonic and/or ghost excitations in the spectrum as well as potentially non-symmetric and/or non-conserved pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-4-11-5': 'Having an action consisting solely of Lovelock invariants avoids all of these potential difficulties as well as those associated with the massive tensor ghosts.', 'hep-ph-0603242-4-11-6': 'The D-dimensional scalar excitation discussed above is also absent in this case.', 'hep-ph-0603242-4-11-7': 'It is interesting to note that the GB term is the leading correction to the EH action in perturbative string theory[CITATION].', 'hep-ph-0603242-4-11-8': 'Higher order Lovelock invariant may also be present in the action (when [MATH]) but these cannot be described by the function [MATH] as employed here since they are constructed out of cubic or higher order combinations of the curvature tensor.', 'hep-ph-0603242-4-11-9': 'The effect of the presence of general Lovelock invariants in the action of the ADD model has been discussed within the black hole context in Ref. [CITATION].', 'hep-ph-0603242-4-12-0': 'A further possibly very useful form for the quadratic action that is commonly used in the literature is [EQUATION] where [MATH] is the square of the Weyl tensor which can be expressed as[CITATION]: [EQUATION] where [MATH] is the number of extra dimensions; the [MATH] are linearly related to the coefficients [MATH] above, e.g., [MATH].', 'hep-ph-0603242-4-12-1': 'This translation is simplified via the use of the identity[CITATION] [EQUATION]', 'hep-ph-0603242-4-12-2': 'Noting that the [MATH] have dimensions of mass[MATH] it is sometimes common in the literature to write [EQUATION] where [MATH] are two mass parameters which are naturally [MATH] in the theory.', 'hep-ph-0603242-4-12-3': 'One then finds that [MATH] are directly related to the bulk masses of the D-dimensional massive scalar and tensor excitations discussed above.', 'hep-ph-0603242-4-12-4': 'To avoid tachyons we apparently must demand that [MATH] but even in such a case as we will see this the massive tensor field remains a ghost since the kinetic term for this field will have the wrong sign.', 'hep-ph-0603242-4-13-0': 'Clearly all these forms for [MATH] are simply related.', 'hep-ph-0603242-4-13-1': 'In what follows we will make use of all of the above forms of [MATH] and treat them interchangeably.', 'hep-ph-0603242-4-14-0': "Our first goal will be to explicitly calculate [MATH] in one of these 'bases' from the more general [MATH] in terms of [MATH] and its derivatives.", 'hep-ph-0603242-4-14-1': 'To begin we perform a Taylor series expansion of [MATH] to quadratic order in all three arguments evaluating the result in the background metric, e.g., [EQUATION] where [MATH] is a constant corresponding to the evaluation of [MATH] itself in the fixed curvature background metric and [MATH]; [MATH] means that [MATH] is to be evaluated in terms of the background metric which we here assume to be a space of constant curvature, i.e., a maximally symmetric space as is the case in both the ADD and RS models.', 'hep-ph-0603242-4-14-2': 'Thus the quantities [MATH], [MATH], [MATH], [MATH] and [MATH] are just numbers which depend on the explicit form of the metric and possibly the number of extra dimensions.', 'hep-ph-0603242-4-14-3': 'In such a maximally symmetric space the Weyl tensor and corresponding invariant both vanish identically, i.e., [MATH] and one further finds that [EQUATION]', 'hep-ph-0603242-4-14-4': 'Note that in ADD [MATH] since the metric is Minkowskian whereas in the [MATH] RS bulk [MATH] (away from the two branes) where the parameter [MATH] originates from the usual RS metric [MATH].', 'hep-ph-0603242-4-15-0': 'Without making any further assumptions we obtain [EQUATION] where we have defined [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0603242-4-15-1': 'For the case of [MATH] this reproduces the results give by, e.g., Navarro and Van Acoleyen in [CITATION].', 'hep-ph-0603242-4-15-2': 'Note that if we make the assumption that [MATH] is a function only of [MATH] and the combination [MATH] then [MATH] etc and, also noting that [MATH]=0, we obtain [MATH] so that the remaining expressions greatly simplify; we now obtain [EQUATION] where the parameter [MATH] is given by [EQUATION]', 'hep-ph-0603242-4-15-3': 'Note that having [MATH] implies that the Weyl term, [MATH], in the effective action is absent in second order which is equivalent to taking [MATH] thus eliminating the massive tensor ghost issue.', 'hep-ph-0603242-4-15-4': 'This field is now removed from the spectrum though the D-dimensional scalar remains in general.', 'hep-ph-0603242-4-16-0': '# Application I: ADD', 'hep-ph-0603242-4-17-0': 'In this section we will apply the above analysis to the general ADD framework where we now require (since the space is flat) [MATH] so that [MATH] automatically.', 'hep-ph-0603242-4-17-1': 'This significantly reduces the possible deviations from the classic ADD picture.', 'hep-ph-0603242-4-17-2': 'In this specific case the general second order expansion of [MATH] is rather simple and is given by [EQUATION]', 'hep-ph-0603242-4-17-3': 'Note that if we also demand that [MATH] be a function only of [MATH] and the combination [MATH] in order to avoid the issue of the massive tensor ghost this expression simplifies even further to [EQUATION]', 'hep-ph-0603242-4-17-4': 'Of course, there has been some discussion about other ways to circumvent this tensor ghost problem than by completely eliminating it from the perturbative spectrum.', 'hep-ph-0603242-4-17-5': 'Since we are working only to lowest non-trivial order perhaps we should keep an open mind about the forms for [MATH].', 'hep-ph-0603242-4-17-6': 'Note that since [MATH] will essentially rescale the overall mass factor in the action we must demand that [MATH] to insure that the usual D-dimensional massless tensor gravitons not be ghost-like.', 'hep-ph-0603242-4-18-0': 'How are the predictions of ADD modified by these additional curvature terms?', 'hep-ph-0603242-4-18-1': 'The basic ADD picture leads to three essential predictions [CITATION]: ([MATH]) the emission of graviton KK states during the collision of SM particles producing signatures with apparent missing energy[CITATION]; ([MATH]) the exchange of graviton KK excitations between SM brane fields leading to dimension-8 contact interaction-like operators with distinctive spin-2 properties[CITATION]; ([MATH]) the production of black holes(BH) at colliders and in cosmic rays with geometric cross sections, [MATH], with [MATH] being the BH Schwarzschild radius, once collision energies greater than [MATH] are exceeded[CITATION].', 'hep-ph-0603242-4-19-0': 'The production and properties of D-dimensional, TeV-scale BH in higher curvature theories has been partially explored within the context of Lovelock extended gravity[CITATION] though not yet so in the fully general quadratic gravity case described by the function [MATH] considered here.', 'hep-ph-0603242-4-19-1': 'Such a study, which would be very interesting, is far beyond the scope of the present analysis.', 'hep-ph-0603242-4-19-2': 'However, it is interesting to make several observations: ([MATH]) Consider the vacuum solution; if we expand the general action [MATH] above to only quadratic order and if we also assume that all interesting solutions must satisfy [MATH] in the vacuum, then the only deviations from the conventional Schwarzschild form arise from the GB term in the action.', 'hep-ph-0603242-4-19-3': 'This can be seen immediately by examining the equations of motion resulting from the general quadratic action, e.g., in Ref. [CITATION].', 'hep-ph-0603242-4-19-4': 'This result does not remain valid when [MATH] is treated exactly.', 'hep-ph-0603242-4-19-5': '([MATH]) If [MATH] only and is treated exactly without expansion then the equations of motion allow for the conventional external BH result with [MATH] and will appear as an ordinary D-dimensional Schwarzschild solution.', 'hep-ph-0603242-4-19-6': 'This is not, however, the most general solution as can be see by considering the simple case of [MATH] with [MATH] a dimensionless parameter.', 'hep-ph-0603242-4-19-7': 'Here there is also exists a solution with [MATH], which is a constant corresponding to deSitter or anti-deSitter space depending on the sign of [MATH].', 'hep-ph-0603242-4-19-8': '([MATH]) If [MATH], with [MATH] being a constant, then the general BH solution has neither [MATH] nor [MATH] equal to zero as is well-known from the exact solution[CITATION].', 'hep-ph-0603242-4-19-9': 'A more detailed study of these possibilities would be worthwhile.', 'hep-ph-0603242-4-20-0': 'Let us now consider the situation of graviton exchange where it is well-known[CITATION] that ADD leads to new dimension-8 contact interactions.', 'hep-ph-0603242-4-20-1': 'To obtain the analogous quantities here we must expand the integrand of the action, i.e., the Lagrangian [MATH], to second order in the fluctuations, [MATH], around the flat background metric: [EQUATION]', 'hep-ph-0603242-4-20-2': 'Here we have expressed the original metric as [EQUATION] with [MATH] being the background metric, which is identified in the ADD model as the Minkowski metric [MATH].', 'hep-ph-0603242-4-20-3': 'The propagator is then just the inverse of the operator [MATH]; once the propagator is known we sandwich it between two 4-d localized (at the origin of the extra-dimensional co-ordinates) and conserved SM stress-energy sources, [MATH], to find the relevant scattering amplitude; we must remember to later KK decompose the various towers.', 'hep-ph-0603242-4-20-4': "Fortunately, much of this work has been done for us by Accioly, Azeredo and Mukai[CITATION] from which, with some modifications, we obtain the expression for the D-dimensional 'graviton' exchange amplitude (before performing the KK sums) [EQUATION] where [MATH] are the familiar (unaltered in terms of the compactification radius [MATH]) flat space KK masses.", 'hep-ph-0603242-4-20-5': 'The [MATH] label the various KK levels; [MATH] is the 4-d trace of the SM source stress-energy tensor and [MATH] are just [EQUATION] as described above.', 'hep-ph-0603242-4-20-6': '[MATH] is the compactification radius which sets the KK mass scale; here we have assumed a common value for this quantity for all [MATH] extra dimensions so that the volume of the compactified space is just [MATH].', 'hep-ph-0603242-4-20-7': 'In the expression above the first term in the amplitude is the usual one encountered in the ADD model which results from the D-dimensional EH action and combines the contributions of the 4-d spin-2 graviton and spin-0 graviscalar KK towers.', 'hep-ph-0603242-4-20-8': 'The second and third terms correspond to the new D-dimensional massive ghost tensor and scalar contributions, respectively.', 'hep-ph-0603242-4-20-9': 'The difference in the factors of [MATH] versus [MATH] in the first two terms arises from the existence of a 5-d bulk mass for the tensor ghost field.', 'hep-ph-0603242-4-20-10': 'It is interesting to note that the full amplitude is very well behaved at large [MATH] (in fact, going as [MATH]) due to the detailed cancellations between the various terms.', 'hep-ph-0603242-4-21-0': 'Here [MATH] represent the bulk mass terms of the new fields which enter into the KK tower masses of the scalars (spin-0) and tensors (spin-2 and spin-0), respectively; here we see the effect of the tensor ghost KK tower exchange explicitly.', 'hep-ph-0603242-4-21-1': 'From this point of view it appears that the only way to remove this ghost tower is to take the bulk mass [MATH] implying that [MATH] is solely a function of [MATH] and the combination [MATH], which we will assume from now on in our ADD discussion.', 'hep-ph-0603242-4-21-2': 'Note that the existence of a GB term in the action will not yield a contribution to [MATH].', 'hep-ph-0603242-4-21-3': 'Since, as discussed above [MATH] is already required, tachyonic KK scalars are avoided when the denominator in the expression for [MATH] above is positive; when [MATH] is assumed to be a function only of [MATH] and [MATH], then this denominator simplifies to [MATH].', 'hep-ph-0603242-4-21-4': 'In the limit where [MATH] alone, and accounting for a sign factor in the definition of the above actions, our result for the squared scalar mass, [MATH], agrees with that obtained by Demir and Tanyildizi[CITATION].', 'hep-ph-0603242-4-21-5': 'As shown by these authors, the effect of the new scalar tower exchange is generally rather suppressed in comparison to the more familiar graviton exchange since the ratio [MATH] is small for most SM particle sources at TeV colliders.', 'hep-ph-0603242-4-21-6': 'For example, for the process [MATH] this ratio is of order [MATH].', 'hep-ph-0603242-4-21-7': 'The corresponding ratio of the KK summed scalar to graviton exchange amplitudes is somewhat further reduced by ([MATH]) the existence of the finite bulk scalar mass which implies that there are no light scalar KK exchanges with masses below [MATH] and ([MATH]) the [MATH]-dependent numerical factor in the denominator of the scalar amplitude.', 'hep-ph-0603242-4-21-8': 'Naturalness suggests that [MATH] TeV or larger unless the parameters of [MATH] are somehow fine-tuned.', 'hep-ph-0603242-4-21-9': 'For example, if [MATH], then [MATH] for all [MATH] if [MATH] is not too far from O(1).', 'hep-ph-0603242-4-21-10': 'Interestingly we see here that as [MATH] we recover the usual EH expectation as then [MATH].', 'hep-ph-0603242-4-21-11': 'Thus we find that for many practical purposes the structure of the usual ADD results for graviton exchange are not qualitatively modified when the action is generalized to the form considered here.', 'hep-ph-0603242-4-21-12': 'However, with the existence of these additional scalars being a hallmark of the extended action, it behooves us to find a way to isolate their effects experimentally.', 'hep-ph-0603242-4-22-0': 'In expressions for graviton exchange only the combination [MATH] will now appear.', 'hep-ph-0603242-4-22-1': "In the amplitude this will lead to a modification of the pure 'graviton' exchange cross section expectations by a factor of [MATH], which is likely to be of [MATH], provided [MATH] is considered to be held fixed.", 'hep-ph-0603242-4-22-2': 'When the graviton tower interference term with the SM dominates, the effect in the gravitational part of the cross section will scale as [MATH].', 'hep-ph-0603242-4-22-3': 'Given the previous results of Demir and Tanyildizi[CITATION], this is not surprising.', 'hep-ph-0603242-4-23-0': 'We further note that since [MATH] is [MATH] TeV or larger it has no effect on laboratory measurements of the strength of the gravitational interaction in the micron range when [MATH].', 'hep-ph-0603242-4-24-0': 'Before closing this part of the discussion we would like to remind the reader that it was pointed out long ago[CITATION] that we can take any action of the form [MATH] and map it over to the EH action coupled to an minimally coupled real scalar field with a rather complicated potential [MATH], depending exponentially on the scalar field.', 'hep-ph-0603242-4-24-1': 'This can be done via a special conformal transformation [EQUATION]', 'hep-ph-0603242-4-24-2': 'Going from the original (Jordan) to the new (Einstein) frame one explicitly sees the existence of the new scalar degree of freedom.', 'hep-ph-0603242-4-24-3': 'The mass of this scalar field is exactly that of the field [MATH] above and can be gotten directly from the canonically normalized potential [MATH] in the usual manner, i.e., using [MATH].', 'hep-ph-0603242-4-24-4': 'This is a very powerful tool as it allows us to extend our previous flat space result for [MATH] to the much more general case where the space has constant curvature.', 'hep-ph-0603242-4-24-5': 'For example, if [MATH], we find that the value of [MATH] is the same as discussed above, i.e., [MATH], in a space with constant curvature.', 'hep-ph-0603242-4-24-6': 'This will be an important result that we will employ when we discuss the case of the RS setup.', 'hep-ph-0603242-4-25-0': 'We now turn to the emission of gravitons in SM particle collisions.', 'hep-ph-0603242-4-25-1': 'Since the compactifying space is flat in the ADD case the normalizations of the graviton (and scalar) wavefunctions which control their couplings are unaltered by the existence of the quadratic curvature terms but the relationship between [MATH] and [MATH]M_pl[MATH] is modified.', 'hep-ph-0603242-4-25-2': 'This was briefly mentioned above where we saw that in the small curvature limit the parameter [MATH] essentially renormalizes the fundamental scale.', 'hep-ph-0603242-4-25-3': 'To see this in the present case it is sufficient to examine the tensor/spin-2 kinetic part of the 4-d effective Lagrangian to second order in [MATH] (which has not yet been KK-expanded) in the familiar transverse traceless gauge, i.e., [MATH], [MATH]; one obtains[CITATION] [EQUATION] where here [MATH] and [MATH] are defined above.', 'hep-ph-0603242-4-25-4': 'When we assume that [MATH] is only a function of [MATH] and the combination [MATH] then the second term in [MATH] vanishes and we recover the familiar result of the standard EH scenario apart from the overall factor of [MATH].', 'hep-ph-0603242-4-25-5': 'Hence, to recover the conventional 4-d EH action when inserting the usual (extra dimensionally) flat zero mode graviton wavefunction into [MATH] the ADD relationship must be modified, as hinted above, to [EQUATION] where [MATH]M_pl[MATH] is the 4-d reduced Planck scale and [MATH] is the volume of the compactified space.', 'hep-ph-0603242-4-25-6': 'Of course, [MATH] is just unity in the standard ADD model which employs the EH action.', 'hep-ph-0603242-4-25-7': 'Since the lightest of the KK scalars has a mass which is naturally on the order of a TeV and has rather weak couplings to SM fields these particles will not play much of an important role in missing energy processes.', 'hep-ph-0603242-4-25-8': 'If the cross section for graviton production, i.e., missing energy, is expressed in terms of the original [MATH] with other parameters held fixed, then the presence of [MATH] leads to a modification of the production cross section by a factor of [MATH].', 'hep-ph-0603242-4-25-9': 'However, as [MATH] is not likely to remain a direct observable (only the product [MATH] is) there may be no way to experimentally disentangle this effect.', 'hep-ph-0603242-4-25-10': 'Furthermore, for any given [MATH], since [MATH]M_pl[MATH] is numerically fixed and [MATH] is an input parameter the resulting derived value of [MATH] which sets the scale for the masses of the KK states is altered.', 'hep-ph-0603242-4-26-0': 'We thus conclude that if we assume that [MATH] is a function of only of [MATH] and the combination [MATH] then the classic predictions ([MATH]) and ([MATH]) of the ADD model will be qualitatively unaffected by going to the more general action considered here except for possible overall scalings by inverse powers of [MATH] when the parameter [MATH] is held fixed: graviton emission rates scale like [MATH] while graviton exchange cross sections scale as [MATH] or [MATH] depending on the presence of important SM contributions to the relevant process.', 'hep-ph-0603242-4-26-1': 'Furthermore, given Eq. (22) and fixed values of [MATH] and [MATH], the KK masses, being proportional to [MATH], will scale as [MATH]; such a mass shift can be quite sizable for reasonable values of [MATH].', 'hep-ph-0603242-4-27-0': '# Application II: RS', 'hep-ph-0603242-4-28-0': 'The predictions of the classic RS model are the existence of TeV scale graviton resonances with fixed weak scale masses and couplings to the SM fields[CITATION], the existence of a weak scale radion excitation[CITATION], as well as the production of [MATH] BH.', 'hep-ph-0603242-4-28-1': 'In what follows we will be specifically interested in the nature of the KK gravitons so it is again sufficient to examine the quadratically expanded action.', 'hep-ph-0603242-4-28-2': 'The classic RS model is not generally strictly consistent with the assumed form of either the original action [MATH] or its quadratically expanded form [MATH].', 'hep-ph-0603242-4-28-3': 'As is well-known, and as mentioned above, the equations of motion that follow from [MATH] and [MATH] will generally be fourth order in the derivatives of the metric.', 'hep-ph-0603242-4-28-4': 'In the usual 5-d RS model, one solves the Einstein equations of the form [EQUATION] where [MATH] is the Einstein tensor arising from the EH action involving no more than two derivatives of the metric.', 'hep-ph-0603242-4-28-5': 'The problem is that RS completely specifies [MATH]: a cosmological constant in the 5-d bulk plus two [MATH]-function sources at the orbifold locations of the TeV and Planck branes.', 'hep-ph-0603242-4-28-6': 'SM matter confined to the TeV brane is supposed to not be a large contributor to the stress-energy.', 'hep-ph-0603242-4-28-7': 'To obtain this result the standard RS metric takes the form discussed above: [MATH] with the linear exponential warp factor leading to the bulk [MATH] and the two field derivatives acting on the absolute value leading to the brane [MATH]-functions.', 'hep-ph-0603242-4-28-8': '(This is related to the comment above that [MATH] is not truly constant in RS and has brane [MATH]-function singularities.', 'hep-ph-0603242-4-28-9': 'Recall that these [MATH]-functions are the results of assuming infinitely thin branes.)', 'hep-ph-0603242-4-28-10': 'If an identical metric is assumed in our more general case we still can obtain [MATH] but the fourth order equations would lead to the more singular derivatives of [MATH]-functions at the brane locations that are not canceled by any source terms.', 'hep-ph-0603242-4-28-11': 'This amongst other reasons is what led Kim, Kyae and Lee[CITATION] to consider only GB extensions of the EH action in RS since it is the only extension which uniquely produces Einstein equations of second order in the derivatives.', 'hep-ph-0603242-4-28-12': 'Thus if we keep the classic picture, an analysis of RS given our assumed effective action expanded around a background of constant curvature is not relevant.', 'hep-ph-0603242-4-28-13': '(A possible way of dealing with these derivatives of [MATH]-functions arising from orbifold singularities in higher dimensional effective field theories has been discussed in Ref. [CITATION].', 'hep-ph-0603242-4-28-14': 'Implementing our scheme employing such techniques is, however, beyond the scope of the present paper.)', 'hep-ph-0603242-4-29-0': "To avoid these issues for now we simplify our discussion of this problem (and to convince ourselves that an RS-like solution is possible in this framework) we consider a singularity-free, 'softened' version of RS where the orbifolded bulk space with branes is replaced by an interval, as has been suggested for other reasons[CITATION], with SM matter placed at one end point possessing an ignorable amount of stress-energy.", 'hep-ph-0603242-4-29-1': 'With a cosmological constant on the interval we can recover the background [MATH] bulk; in addition by removing the absolute value sign of the co-ordinate [MATH] in the metric above we expunge the [MATH]-functions as well as the possibility of any of their higher derivatives appearing in the equations of motion.', 'hep-ph-0603242-4-29-2': 'The boundary conditions at the end points for the graviton KK states can then be freely chosen to be the same as that of the original RS model.', 'hep-ph-0603242-4-29-3': "This space is truly one of constant curvature and the general analysis we have presented above will now be applicable to this 'softened' RS on an interval.", 'hep-ph-0603242-4-30-0': 'It is easy to verify that the form of the equations of motion[CITATION] in this case (recalling that we are only searching for solutions with maximally-symmetric, constant [MATH] backgrounds) are given by: [EQUATION] and that if we take stress-energy tensor in the 5d bulk to be of the usual RS form [EQUATION] with [MATH], then indeed a space of constant curvature, i.e. [MATH], can be an allowed solution.', 'hep-ph-0603242-4-30-1': 'Taking the trace of the equations of motion above, evaluating it in the constant curvature bulk and relating the values of [MATH] to [MATH] as before (recalling that here [MATH] using the softened metric) results in the constraint equation [EQUATION] where here [MATH].', 'hep-ph-0603242-4-30-2': 'It is interesting to note that if we assume that [MATH]constant then this constraint equation automatically implies that [MATH]constant as well; but this does not necessarily further require that all of the [MATH] are constants as we will see below.', 'hep-ph-0603242-4-30-3': 'When [MATH] is only a function of [MATH] and the combination [MATH], this constraint equation simplifies to [EQUATION] while in the specific RS background case this explicitly becomes [EQUATION]', 'hep-ph-0603242-4-30-4': 'It is important to recall that [MATH] itself can be a complicated function of [MATH] so that this equation can be quite nontrivial.', 'hep-ph-0603242-4-30-5': 'For the EH action limit this yields the usual relation that [MATH]; here it in general provides an additional constraint on the allowed forms of the function [MATH] since we are requiring [MATH] to be both real and negative.', 'hep-ph-0603242-4-30-6': 'Given a specific function [MATH] for which a solution exists, this equation directly relates [MATH] and [MATH] though the solution may not be unique.', 'hep-ph-0603242-4-30-7': 'For example, if we assume for purposes of demonstration the simple case of [EQUATION] as employed above, then there are two branches of solutions for [MATH]: [EQUATION] one of which (the negative root) goes over to the usual EH result as the parameter [MATH].', 'hep-ph-0603242-4-31-0': 'Allowing for the possibility of a RS-like solution with a softened metric it is interesting to think briefly about the previously analyzed effects of the GB term in the RS scenario.', 'hep-ph-0603242-4-31-1': 'This analysis was originally performed for the classic RS[CITATION] setup which employed the standard form of the RS metric; that result would now be modified by the changes in the model assumptions, i.e., moving to an interval and removing the [MATH]-function sources at the end points.', 'hep-ph-0603242-4-31-2': 'The previous analysis of BH in RS with the added GB term would not be significantly affected if this transition were made.', 'hep-ph-0603242-4-31-3': 'However, the properties and spectrum of the graviton KK states certainly would be influenced since the [MATH]-function terms are now absent.', 'hep-ph-0603242-4-31-4': 'The equation governing the masses and wavefunction of the graviton KK states for the present interval case can be obtained by expanding the equations of motion as before.', 'hep-ph-0603242-4-31-5': 'Since we are here only interested in the tensor modes associated with the usual gravitons, we can employ the expansion [EQUATION] where [MATH].', 'hep-ph-0603242-4-31-6': 'Applying the usual RS boundary conditions on the interval the most significant changes from the classic RS can be read off from Eqs. (15)-(28) in Ref. [CITATION] by setting the parameter [MATH] in appropriate places.', 'hep-ph-0603242-4-31-7': 'At the end of the day we find that the only apparent difference from the classic EH based RS model would be a shift in the relationship between the fundamental scale and [MATH]M_pl[MATH]-remarkably similar to what we saw for the ADD model above.', 'hep-ph-0603242-4-31-8': 'In the language employed in Ref. [CITATION] we would now obtain [EQUATION] where [MATH] is the coefficient of the GB term in the action.', 'hep-ph-0603242-4-31-9': 'Otherwise the masses as well as the couplings of all of the KK gravitons to localized SM matter would be identical to those of the original RS model expressed in terms of the derived parameter [MATH].', 'hep-ph-0603242-4-31-10': 'The explicit coupling and spectrum changes found in Ref[CITATION] for the graviton KK states in the presence of the GB term in the action were all found to due to the brane [MATH]-function singularities.', 'hep-ph-0603242-4-32-0': 'How would these graviton KK results obtained in the GB extended action generalize to the case of [MATH] above?', 'hep-ph-0603242-4-32-1': 'Here we choose to begin our analysis with [MATH], setting [MATH] from the beginning to avoid potential ghost fields, then taking [MATH] and using the same curvature expansion as above.', 'hep-ph-0603242-4-32-2': 'In order to make a connection with the previous discussion, the existing RS literature and to directly compare with the GB case, however, we massage our notation slightly and rewrite [MATH] in the following form: [EQUATION] where the parameters [MATH] and [MATH] are dimensionless; the action employed in Ref. [CITATION] is now directly recovered by taking the [MATH] and [MATH] limits.', 'hep-ph-0603242-4-32-3': 'It is important at this point to recall that to obtain the linearized graviton equations of motion it is sufficient to employ [MATH] while the complete [MATH] needs to be examined in order to demonstrate the existence of the required [MATH] solution.', 'hep-ph-0603242-4-32-4': 'The equations of motion resulting from [MATH] are given by[CITATION] [EQUATION]', 'hep-ph-0603242-4-32-5': 'Here [MATH] is the covariant derivative operator and here [MATH].', 'hep-ph-0603242-4-32-6': 'First we look at the [MATH] component of this equation, remembering that for the moment we will only be interested in the tensor excitations corresponding to the KK gravitons which are massless in 5-d.', 'hep-ph-0603242-4-32-7': 'In the usually chosen gauge, [MATH] is still a constant to linear order so we arrive at a consistency condition [EQUATION]', 'hep-ph-0603242-4-32-8': 'Note that this reduces to the previously obtained purely quadratic GB extended RS result[CITATION] when [MATH].', 'hep-ph-0603242-4-32-9': 'In the more general case, this expression is not overly useful given the exact result in Eq. (27).', 'hep-ph-0603242-4-33-0': 'Turning now to the [MATH] terms which contain the 4-d graviton tensor excitation, we linearize employing the previously mentioned transverse, traceless gauge with constant [MATH].', 'hep-ph-0603242-4-33-1': 'This gives the standard equation of motion for the RS graviton found long ago[CITATION] though scaled by an overall factor.', 'hep-ph-0603242-4-33-2': 'Employing the standard KK decomposition [EQUATION] and recalling that [MATH], the [MATH] are seen to satisfy [EQUATION]', 'hep-ph-0603242-4-33-3': 'The overall factor [MATH] is given by [EQUATION] or, more explicitly in the RS case, [EQUATION] (Again we recall that [MATH] itself can be a function of [MATH].)', 'hep-ph-0603242-4-33-4': 'This leads to a rescaling of the usual RS relationship [EQUATION] via the renormalization of the zero mode (i.e., massless graviton) wavefunction, thus generalizing Eq. (31).', 'hep-ph-0603242-4-33-5': 'Of course, [MATH] is required to avoid ghost states among the usual gravitons KKs.', 'hep-ph-0603242-4-33-6': 'This result reduces to that previously obtained in the RS case with just the added GB term[CITATION] once boundary effects are neglected.', 'hep-ph-0603242-4-34-0': 'From this analysis we see immediately that the masses of the KK gravitons are identical to those obtained in the original RS model, provided we use the same value of the parameter [MATH], as we might have expected.', 'hep-ph-0603242-4-34-1': 'Here we are faced with the question of just what are the independent parameters.', 'hep-ph-0603242-4-34-2': '[MATH] is clearly a derived parameter that is obtained by simultaneously solving Eqs. (27) and (40) for any given model.', 'hep-ph-0603242-4-34-3': 'In that sense, the KK graviton spectrum would just be rescaled in comparison to the usual expectations given the same input value of [MATH].', 'hep-ph-0603242-4-34-4': 'As we have just seen, and as in the ADD case, the effect of a factor like [MATH] on the KK graviton couplings to 4-d SM matter depends upon which model parameters are assumed to be held fixed.', 'hep-ph-0603242-4-34-5': 'At the very least, up to an overall constant, these KK graviton couplings are identical to those of the standard RS model.', 'hep-ph-0603242-4-35-0': 'As an example of a simple model where the shifts in the KK spectrum can be calculated analytically consider substituting for the integrand of the conventional RS action, [MATH], the simple higher curvature action [MATH] as was considered above.', 'hep-ph-0603242-4-35-1': 'Let [MATH] be the values obtained for the parameter [MATH] in the usual RS model, i.e., [MATH].', 'hep-ph-0603242-4-35-2': 'Keeping the warp factor fixed we can use the equations above to calculate the value of, e.g., the mass of the first graviton KK state in both the standard RS model, [MATH], and in the current model with an augmented action, [MATH].', 'hep-ph-0603242-4-35-3': 'Using Eqs. (27) and (40) this ratio can be calculated analytically in the present case as a function of [MATH] and [MATH]M_pl[MATH]; we obtain [EQUATION]', 'hep-ph-0603242-4-35-4': 'The result of this calculation is shown in Fig.1 for a wide range of model parameters.', 'hep-ph-0603242-4-35-5': 'In this example we see that the size of the possible shift in the mass spectrum can be quite large assuming a fixed value of [MATH].', 'hep-ph-0603242-4-36-0': 'So far we have only considered the 4-d graviton, spin-2 excitations.', 'hep-ph-0603242-4-36-1': 'It is important to remember that our softened RS model now has an additional massive scalar in the 5-d spectrum with a large bulk mass, [MATH], and that no massless scalar zero mode will exist.', 'hep-ph-0603242-4-36-2': 'Since the bulk scalar mass is naturally of order [MATH] the KK spectrum of the corresponding tower will begin with a KK scalar state whose mass is qualitatively comparable to that of the first graviton excitation.', 'hep-ph-0603242-4-36-3': 'This bulk mass is explicitly calculable from the expansion of the full action to quadratic order, [MATH], by going to the Einstein frame since we know that the GB term does not contribute to this parameter.', 'hep-ph-0603242-4-36-4': 'In that case, using the results from the previous section we find that [EQUATION] or, in terms of the original parameters of the action, evaluated in the RS background: [EQUATION]', 'hep-ph-0603242-4-36-5': 'Note that [MATH] is required to avoid the scalar tachyons and graviton ghosts, consistent with our above analysis.', 'hep-ph-0603242-4-36-6': 'Note further that this reproduces the results of Eq. (19) in the flat space, [MATH], limit.', 'hep-ph-0603242-4-37-0': 'Given any [MATH] the scalar bulk mass is known and we can determine the mass(es) of the lightest KK scalar state(s) by following the standard RS manipulations[CITATION].', 'hep-ph-0603242-4-37-1': 'These masses are essentially given by the first roots of the equation [EQUATION] where [MATH] and [MATH] is the usual Bessel function.', 'hep-ph-0603242-4-37-2': 'The solution for the first KK state is provided by Fig.2; as stated above there are no massless modes.', 'hep-ph-0603242-4-37-3': 'The lightest scalar mass is then [MATH].', 'hep-ph-0603242-4-37-4': 'Here we observe that the mass of the first scalar KK scales almost linearly with the bulk mass when [MATH] gets large.', 'hep-ph-0603242-4-37-5': 'Note that for [MATH] and a typical value[CITATION] of [MATH], we then find [MATH] implying [MATH] from Fig.1; this is about 3 times larger than the root for the usual lightest massive KK graviton, [MATH].', 'hep-ph-0603242-4-37-6': 'Thus we see that unless [MATH] takes on large values the first scalar KK state is always rather heavy.', 'hep-ph-0603242-4-37-7': 'As is well-known, the [MATH] values for the more massive KK scalar states will be somewhat larger: approximately given by [MATH] where [MATH] labels the KK level.', 'hep-ph-0603242-4-37-8': 'Since these scalars will couple to the trace of the stress-energy tensor for the 4-d SM fields they will interact far more weakly than do the graviton KK states unless this is at least partially offset by ratios of 5-d wavefunction factors.', 'hep-ph-0603242-4-37-9': 'A quick estimate of such factors, however, indicates that, if anything, these wave function ratios lead to a further suppression of the scalar couplings relative to those of the KK gravitons by [MATH] as shown in Fig.3.', 'hep-ph-0603242-4-37-10': 'This overall picture of the scalar sector is qualitatively very similar to that of the existence of a very heavy tower of RS radions[CITATION] or a tower of KK Higgs bosons as in the case of Universal Warped Extra Dimensions[CITATION].', 'hep-ph-0603242-4-38-0': 'In the analysis as presented here we have ignored the possibility that the new scalar KK states may mix with the (usually eaten) RS graviscalars through cross-talk in the equations of motion, i.e., we have assumed that the 5-d tensor and scalar KK decompositions can be performed independently, and this is something which needs further exploration.', 'hep-ph-0603242-4-38-1': 'A fully detailed analysis of the such possibilities is, however, beyond the scope of the present paper.', 'hep-ph-0603242-4-39-0': 'It is perhaps interesting to ask whether the usual [MATH]=constant ([MATH]) solution considered here necessitates the metric and matter distribution of the conventional RS model employed above without the further assumption of a maximally symmetric space.', 'hep-ph-0603242-4-39-1': 'To analyze a simple and more easily tractable situation let us consider the more general warped metric [MATH] and assume that [MATH] only.', 'hep-ph-0603242-4-39-2': 'The first question to address is what is the most general form of the function [MATH]; to deal with this issue we note that the Ricci scalar arising from this metric is in general given by [MATH].', 'hep-ph-0603242-4-39-3': "In the 'soft' version of the RS model on the interval defined above one had [MATH] and [MATH] so that [MATH] as usual.", 'hep-ph-0603242-4-39-4': 'Here, this curvature condition provides a differential equation for the function [MATH]; by solving this equation we arrive at the result [MATH], where the [MATH] are integration constants and [MATH].', 'hep-ph-0603242-4-39-5': 'Choosing the [MATH] appropriately and rescaling [MATH], we can rewrite this in a more familiar form as [MATH] with [MATH] a dimensionless constant.', 'hep-ph-0603242-4-39-6': 'Note that when [MATH] and [MATH] we recover the usual RS result.', 'hep-ph-0603242-4-39-7': 'However, the choice of [MATH] can easily modify the warp factor from its conventional behavior.', 'hep-ph-0603242-4-39-8': 'What is the nature of the bulk matter distribution that yields this metric?', 'hep-ph-0603242-4-39-9': 'The solution to this can be obtained by considering Eqs.(24) and (26) with [MATH].', 'hep-ph-0603242-4-39-10': 'Eq.(26) immediately tells us that the trace of the 5-d stress-energy tensor [MATH], a constant, so that if we define [MATH] then we must have [MATH], with [MATH] an arbitrary function.', 'hep-ph-0603242-4-39-11': 'The [MATH] and [MATH] components of the equations of motion when combined then provides a first order differential equation for [MATH] that can be solved in a straightforward manner.', 'hep-ph-0603242-4-39-12': 'Setting [MATH] and defining [MATH] with [MATH], we obtain the general solution (assuming [MATH]) [EQUATION] with [MATH] an integration constant.', 'hep-ph-0603242-4-39-13': 'Though this is far from a uniform energy distribution (away from the TeV brane) it does lead to a space of constant curvature but not one which is maximally symmetric.', 'hep-ph-0603242-4-39-14': 'Thus we see that it is possible that the requirement that [MATH] does allow for the possibility of more complex solutions than that employed in the original RS model.', 'hep-ph-0603242-4-40-0': '# Discussion and Conclusions', 'hep-ph-0603242-4-41-0': 'In this paper we have begun an examination of how generic higher curvature terms in the gravitational action can alter the predictions of both the ADD model and the RS model defined on a interval to avoid possible brane singularities.', 'hep-ph-0603242-4-41-1': 'We have assumed that the traditional assumptions of the two models, e.g., SM localized matter in a conformally flat bulk, remain valid; we have not considered in detail more complex setups that may now be allowed by the modified equations of motion.', 'hep-ph-0603242-4-41-2': 'To be more concrete, we have further assumed that the EH action is now generalized to an action which is of the form [MATH] where [MATH] is a well-behaved function, [MATH] and [MATH].', 'hep-ph-0603242-4-41-3': 'In D-dimensions this action results in a propagating massless tensor field (identified with the usual graviton), a massive ghost tensor field, as well as a massive (possibly tachyonic) scalar.', 'hep-ph-0603242-4-41-4': 'The potentially dangerous ghost is removable from the perturbative spectrum, i.e., it becomes infinitely massive, if we demand that [MATH] only.', 'hep-ph-0603242-4-41-5': 'The remaining new scalar field has a bulk mass whose value is naturally expected to be of order the fundamental scale, [MATH], in either scenario.', 'hep-ph-0603242-4-41-6': 'The resulting ADD and RS models are altered in similar ways from their traditional standard forms:', 'hep-ph-0603242-4-42-0': '([MATH]) New scalar KK excitations appear in the spectrum of both models in a rather benign fashion coupling to the trace of the stress-energy tensor of the localized SM fields.', 'hep-ph-0603242-4-42-1': 'Since this trace is proportional to SM masses, the couplings of these scalars are relatively strongly suppressed in comparison to those for the KK gravitons at typical collider energies in both models.', 'hep-ph-0603242-4-42-2': 'In the ADD model, the KK scalar excitations begin at a mass [MATH] TeV.', 'hep-ph-0603242-4-42-3': 'Consequently their contributions to missing-energy signatures as well as to the usual dimension-8 contact interactions are further kinematically suppressed.', 'hep-ph-0603242-4-42-4': 'Thus at leading order these new scalars do not much influence ADD collider signatures.', 'hep-ph-0603242-4-42-5': 'In RS, the bulk scalar mass tends to be large so that the lightest scalar KK state is several times more massive than is the lightest KK graviton.', 'hep-ph-0603242-4-42-6': 'Given their rather weak couplings such states will be difficult to observe at colliders.', 'hep-ph-0603242-4-43-0': '([MATH]) The basic model relationships involving the fundamental and 4-d Planck masses in both models get rescaled by functions of [MATH] and its derivatives evaluated in the corresponding background metric of the two models: in ADD we obtain [MATH]M_pl[MATH] while in RS we obtain [MATH]M_pl[MATH] where [MATH] is explicitly given in Eq. (39).', 'hep-ph-0603242-4-43-1': 'Assuming that [MATH] is a fixed fundamental parameter these modifications lead to changes in the graviton KK sectors of both models.', 'hep-ph-0603242-4-43-2': 'In the ADD case, since [MATH]M_pl[MATH] is known and [MATH] is an input parameter for any given [MATH] the volume of the compactified space and, hence, the value of the compactification radius which sets the graviton KK mass scale is altered.', 'hep-ph-0603242-4-43-3': 'Due to the presence of the [MATH] factor the emission rate for gravitons in the collisions of SM particles and for the graviton exchange amplitude are both modified by potentially O(1) effects.', 'hep-ph-0603242-4-43-4': 'Similarly in RS, [MATH] is a derived parameter which sets the scale for all the KK states.', 'hep-ph-0603242-4-43-5': 'The constraint Eq. (28) allows us to calculate [MATH] in terms of the input parameter [MATH] and the function [MATH] thus providing for us with [MATH].', 'hep-ph-0603242-4-43-6': 'In a manner similar to ADD, the presence of [MATH] rescales the coupling strengths of the of the KK graviton states to the SM fields thus modifying the widths and production cross sections at colliders by potentially O(1) factors.', 'hep-ph-0603242-4-44-0': 'As we have seen, the extension of the EH action to a more complicated structure can lead to significant quantitative modifications to both the ADD and RS model predictions in the simplest possible case.', 'hep-ph-0603242-4-44-1': 'The observation of such effects at future colliders could tell us valuable information about the underlying theory of gravity.', 'hep-ph-0603242-4-45-0': 'Note Added: After this paper was essentially completed, Ref. [CITATION] appeared which discusses generalized actions for the ADD model and thus has some common areas with the present work.', 'hep-ph-0603242-4-45-1': 'Where the two papers overlap there is general qualitative agreement though the points of view are somewhat different.'} | {'hep-ph-0603242-5-0-0': 'We begin a general exploration of the phenomenology of TeV-scale extra-dimensional models with gravitational actions that contain higher curvature terms.', 'hep-ph-0603242-5-0-1': 'In particular, we examine how the classic collider signatures of the models of Arkani-Hamed, Dimopoulos and Dvali (missing energy and new dimension-8 contact interactions) and of Randall and Sundrum (TeV-scale graviton Kaluza-Klein resonances) are altered by these modifications to the usual Einstein-Hilbert action.', 'hep-ph-0603242-5-0-2': 'We find that not only are the detailed signatures for these gravitationally induced processes altered but new contributions are found to arise due to the existence of additional scalar Kaluza-Klein states in the spectrum.', 'hep-ph-0603242-5-1-0': '# Introduction and Background', 'hep-ph-0603242-5-2-0': 'The question as to why the Planck and electroweak scales differ by so many orders of magnitude remains mysterious.', 'hep-ph-0603242-5-2-1': 'In recent years, attempts have been made to address this hierarchy issue within the context of theories with extra spatial dimensions that lower the effective scale of gravity to the TeV region.', 'hep-ph-0603242-5-2-2': 'In both the models of Arkani-Hamed, Dimopoulos and Dvali (ADD)[CITATION] and of Randall and Sundrum (RS)[CITATION], new effects of gravitational origin are expected to occur near the TeV scale which should be observable at future colliders such as the LHC and ILC.', 'hep-ph-0603242-5-2-3': 'Though these two models are very different in detail they do have some common features the most important of which are: ([MATH]) in their original versions they both assume that Standard Model matter is confined to a 4-dimensional brane; ([MATH]) they both assume that D-dimensional gravity is described by the Einstein-Hilbert (EH) action plus a possible cosmological constant and ([MATH]) the background spaces are maximally symmetric and are either strictly flat, i.e. , Minkowskian as in the ADD model with toroidally flat compactification, or being of constant curvature and is conformally flat, i.e., [MATH] as in the RS model.', 'hep-ph-0603242-5-2-4': 'How would the predictions of these two models be changed if we surrendered the assumption ([MATH]), i.e., that the EH action provided the full description of gravity and considered something more general?', 'hep-ph-0603242-5-2-5': 'This is the discussion we would like to begin in this paper which will follow a phenomenological bottom-up approach.', 'hep-ph-0603242-5-3-0': 'General Relativity (GR) as described by the EH action is considered to be an effective theory below the fundamental Planck scale, [MATH].', 'hep-ph-0603242-5-3-1': 'Thus, once energies approaching the scale [MATH] begin to be probed one might expect to observe deviations from the expectations arising from the EH action.', 'hep-ph-0603242-5-3-2': 'In the cases of both the ADD and RS models, future colliders will probe near or at their (effective) fundamental scales so that non-EH aspects of the true gravitational theory, whatever its form, should become apparent and be experimentally measured.', 'hep-ph-0603242-5-3-3': 'Since the ultraviolet form of the true gravity theory is as of yet unknown one may hope to capture some of its deeper aspects by considering how the presence of new higher curvature (and higher derivative) invariants in the actions of the ADD and RS models can lead to variations in the well-known predictions of these theories.', 'hep-ph-0603242-5-3-4': 'Many authors have considered the possibility of higher curvature invariants and how their existence would modify the predictions arising from the EH action within other contexts, e.g., the properties of black holes[CITATION], deviations in solar system tests of GR[CITATION] and in cosmology[CITATION] to possibly avoid the need for dark energy.', 'hep-ph-0603242-5-3-5': 'Some analyses along these lines for the potential modifications of the collider predictions of both the ADD and RS models have already been performed[CITATION].', 'hep-ph-0603242-5-3-6': 'In the present paper, we wish to both extend and generalize these results to get a feeling for the possible detailed variation of the various new gravitational phenomena as predicted by these classic models which will be potentially observable at future colliders.', 'hep-ph-0603242-5-3-7': 'In particular we are interested in how the well known signatures of both the ADD and RS models are morphed if we keep the basic setups intact but modify the actions on which the corresponding equations of motion are based.', 'hep-ph-0603242-5-3-8': 'A further generalization of such an analysis is possible if the original ADD/RS setups can now be simultaneously surrendered due to these modified actions resulting in entirely new setups with corresponding equations of motion; while this is an interesting possibility to consider it lies mostly beyond the scope of the present paper though it will be touched upon briefly in the discussion below.', 'hep-ph-0603242-5-4-0': 'Of course a completely general study of how these possible modifications to the effective gravity action may morph TeV collider signatures is an obviously immense task and here we aim only at a first round analysis in the discussion that follows.', 'hep-ph-0603242-5-4-1': 'The major signatures arising in both ADD and RS models originate from graviton exchange and the production of black holes; the ADD model also leads to missing energy signatures from graviton emission.', 'hep-ph-0603242-5-4-2': 'Fortunately, apart from issues associated with black holes, since we are dealing with maximally symmetric spaces of globally constant curvature, the relevant graviton properties (couplings, wavefunctions and propagators) necessary to extract experimental signatures for either model can be obtained from the expansion of the rather general action considered below to quadratic order in the curvature.', 'hep-ph-0603242-5-4-3': '(This would no longer be true if we wanted to consider, e.g., the triple graviton coupling as then an expansion to third order in the curvature would be required.)', 'hep-ph-0603242-5-4-4': 'This simplifying observation forms the basis of the analysis that follows and allows us to determine the relevant graviton properties in both ADD- and RS-like models for a wide class of effective actions.', 'hep-ph-0603242-5-5-0': 'The general outline of our analysis is presented in Section 2 where our basic assumptions and notations are also given.', 'hep-ph-0603242-5-5-1': 'In Section 3 we apply our analysis to the ADD model; we then apply it to the RS case in Section 4.', 'hep-ph-0603242-5-5-2': 'Our summary and conclusions are given in Section 5.', 'hep-ph-0603242-5-6-0': '# Analysis', 'hep-ph-0603242-5-7-0': 'When going beyond the EH action there are many possibilities to consider especially when we are living in extra dimensions.', 'hep-ph-0603242-5-7-1': 'In the literature various forms have been assumed for the potential structures of higher curvature and/or derivative invariants that may appear in the gravity action.', 'hep-ph-0603242-5-7-2': 'The fairly general structure that we will assume for the D-dimensional action in the present analysis takes the form (with [MATH] below): [EQUATION] where [MATH] is an arbitrary continuous, differentiable and generally mathematically well-behaved function; in particular we will assume that [MATH] is non-singular when all of its arguments are zero.', 'hep-ph-0603242-5-7-3': 'Here [MATH] is the usual D-dimensional Ricci scalar while [MATH] and [MATH] are quadratic invariants constructed from the curvature tensor [MATH]: [MATH], with [MATH] being the Ricci tensor, while [MATH].', 'hep-ph-0603242-5-7-4': '[MATH] is the D-dimensional fundamental gravity scale which is [MATH] TeV in ADD and [MATH]M_pl[MATH] in RS.', 'hep-ph-0603242-5-7-5': 'In the low energy, small curvature limit we expect [MATH] (plus a possible cosmological constant) and so the overall dimensionful factor in the expression above allows us to make direct contact with the EH action in this limit.', 'hep-ph-0603242-5-7-6': 'This specific form for [MATH], though not completely general, covers a wide array of possibilities and has been considered (sometimes only in [MATH]) in may different contexts for a multitude of purposes in the literature[CITATION].', 'hep-ph-0603242-5-7-7': 'Many of the higher curvature models previous considered by other authors form subcases of this more general action.', 'hep-ph-0603242-5-7-8': 'Generally we may think of [MATH] as summing a number of corrections to the EH action which arise higher dimension invariants which are suppressed by appropriate powers of [MATH].', 'hep-ph-0603242-5-7-9': 'Thus we will only consider energy regimes where the collision energies are below the effective fundamental scale in what follows as is usually done when considering ADD and RS phenomenology.', 'hep-ph-0603242-5-8-0': 'As is by now well-known[CITATION] the generalized gravity theories described by an action of the form [MATH] can potentially have several serious problems.', 'hep-ph-0603242-5-8-1': 'Since we will be dealing with ADD- and RS-type phenomenology we will be performing a perturbative analysis in the discussion that follows.', 'hep-ph-0603242-5-8-2': 'Employing such an analysis one finds that, amongst other things, this action leads to equations of motion which are generally fourth order in the derivatives of the metric.', 'hep-ph-0603242-5-8-3': 'In particular, in addition to the usual massless D-dimensional tensor graviton which results in, e.g., the familiar 4-d graviton and graviscalar Kaluza-Klein (KK) tower excitations, there may also be present in the linearized D-dimensional theory additional massive scalar and tensor excitations.', 'hep-ph-0603242-5-8-4': 'These fields will in 4-d have KK towers without massless modes and which can be ghostlike and/or tachyonic.', 'hep-ph-0603242-5-8-5': '(We can think of these new D-dimensional fields as having bulk masses which influence their corresponding 4-d KK tower masses.)', 'hep-ph-0603242-5-8-6': 'Furthermore, the equations of motion naturally involving higher derivatives of the fields can lead to problems with unitarity as well as non-canceling asymmetric pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-5-8-7': "The new massive tensor excitations are potentially the most serious issue to deal with as they are ghost fields that must be eliminated from the perturbative spectrum (though they may help in dealing with the theory's renormalizability and bad high-energy behavior).", 'hep-ph-0603242-5-8-8': 'It has been noted[CITATION] that we one can remove these states from the spectrum (i.e., by giving their bulk masses an infinite value) if a tuning occurs such that the quantities [MATH] and [MATH] only appear in the special combination [MATH] in the function [MATH].', 'hep-ph-0603242-5-8-9': 'How this tuning arises in the fundamental theory is unspecified.', 'hep-ph-0603242-5-8-10': 'There has been some discussion in the literature, however, that these ghost states may not be as dangerous as one would imagine from lowest order perturbation theory[CITATION] so that we should perhaps keep an open mind about the possible forms for [MATH].', 'hep-ph-0603242-5-8-11': 'We will return to this point in what follows.', 'hep-ph-0603242-5-9-0': 'Given a general action of the kind above there are several issues that one normally wants to address in order to extract information that can be compared with experimental data.', 'hep-ph-0603242-5-9-1': 'From studies of both the ADD and RS models there are certain things we want to know, e.g., ([MATH]) the spectrum, wave functions, propagators and Standard Model (SM) matter couplings of the KK graviton (and other possible) excitations and ([MATH]) the relationship between [MATH], the volume of the compactified dimensions and the (reduced) 4-d Planck scale [MATH]M_pl[MATH].', 'hep-ph-0603242-5-9-2': 'To obtain this information, as well as to make contact with several other analyses[CITATION], it is sufficient to expand the general action [MATH] above around the maximally symmetric background metric to quadratic order in the curvature to obtain an effective action for the graviton (and like) excitations.', 'hep-ph-0603242-5-9-3': 'At this level, one can extract the relevant 2-point functions as well as the differential equation for the KK wavefunctions which then yield the KK mass spectrum as well as the the desired graviton couplings to the localized SM fields.', 'hep-ph-0603242-5-9-4': 'If, however, one wanted to probe, e.g., graviton 3- or 4-point functions then we would need to expand to at least cubic or quartic order in the curvature, respectively; these will not be of interest to us here but might be of interest in future experiments[CITATION] which would tell us more about the underlying theory of gravity.', 'hep-ph-0603242-5-9-5': 'Interestingly, if we allow for cubic or higher invariants in the original action in a maximally symmetric background, they will make no contributions to the quantities of interest to us here.', 'hep-ph-0603242-5-9-6': 'Thus our analysis allows for the most general possible effective bulk local action for gravity.', 'hep-ph-0603242-5-10-0': 'Once we make this expansion, there are various equivalent ways of expressing the resulting effective action, [MATH], depending upon the basis of invariants we choose to employ; the most obvious form is simply [EQUATION] where [MATH] have been defined above.', 'hep-ph-0603242-5-10-1': '[MATH] is an effective cosmological constant and [MATH] are (in some cases dimensionful) constants all of which are functions of [MATH] and its derivatives evaluated by employing the relevant background metric.', 'hep-ph-0603242-5-10-2': "To relate this back to the EH action in the limit of small curvature, one can think of the (necessarily positive) parameter [MATH] as a 'renormalization' of the fundamental mass scale [MATH]: [MATH].", 'hep-ph-0603242-5-11-0': 'A second and perhaps more useful version of [MATH] is given by [EQUATION] where [MATH] is the well-known Gauss-Bonnet(GB) invariant: [EQUATION]', 'hep-ph-0603242-5-11-1': 'The co-efficients [MATH] and [MATH] can be easily converted to [MATH] above by some straightforward algebra: [MATH] and [MATH].', 'hep-ph-0603242-5-11-2': 'In D[MATH]4, the GB invariant is either a topological invariant or just a total derivative not contributing to the equations of motion, but this is no longer true for arbitrary values of [MATH].', 'hep-ph-0603242-5-11-3': 'The GB invariant is just (the quadratic) member of a general class of Lovelock invariants, constructed of various powers of the curvature tensor, which lead to special properties for the equations of motion[CITATION].', 'hep-ph-0603242-5-11-4': 'Generally the existence in the action of higher curvature terms, as discussed above, leads to higher order equations of motion that produce tachyonic and/or ghost excitations in the spectrum as well as potentially non-symmetric and/or non-conserved pieces of the corresponding Einstein tensor.', 'hep-ph-0603242-5-11-5': 'Having an action consisting solely of Lovelock invariants avoids all of these potential difficulties as well as those associated with the massive tensor ghosts.', 'hep-ph-0603242-5-11-6': 'The D-dimensional scalar excitation discussed above is also absent in this case.', 'hep-ph-0603242-5-11-7': 'It is interesting to note that the GB term is the leading correction to the EH action in perturbative string theory[CITATION].', 'hep-ph-0603242-5-11-8': 'Higher order Lovelock invariant may also be present in the action (when [MATH]) but these cannot be described by the function [MATH] as employed here since they are constructed out of cubic or higher order combinations of the curvature tensor.', 'hep-ph-0603242-5-11-9': 'The effect of the presence of general Lovelock invariants in the action of the ADD model has been discussed within the black hole context in Ref. [CITATION].', 'hep-ph-0603242-5-12-0': 'A further possibly very useful form for the quadratic action that is commonly used in the literature is [EQUATION] where [MATH] is the square of the Weyl tensor which can be expressed as[CITATION]: [EQUATION] where [MATH] is the number of extra dimensions; the [MATH] are linearly related to the coefficients [MATH] above, e.g., [MATH].', 'hep-ph-0603242-5-12-1': 'This translation is simplified via the use of the identity[CITATION] [EQUATION]', 'hep-ph-0603242-5-12-2': 'Noting that the [MATH] have dimensions of mass[MATH] it is sometimes common in the literature to write [EQUATION] where [MATH] are two mass parameters which are naturally [MATH] in the theory.', 'hep-ph-0603242-5-12-3': 'One then finds that [MATH] are directly related to the bulk masses of the D-dimensional massive scalar and tensor excitations discussed above.', 'hep-ph-0603242-5-12-4': 'To avoid tachyons we apparently must demand that [MATH] but even in such a case as we will see this the massive tensor field remains a ghost since the kinetic term for this field will have the wrong sign.', 'hep-ph-0603242-5-13-0': 'Clearly all these forms for [MATH] are simply related.', 'hep-ph-0603242-5-13-1': 'In what follows we will make use of all of the above forms of [MATH] and treat them interchangeably.', 'hep-ph-0603242-5-14-0': "Our first goal will be to explicitly calculate [MATH] in one of these 'bases' from the more general [MATH] in terms of [MATH] and its derivatives.", 'hep-ph-0603242-5-14-1': 'To begin we perform a Taylor series expansion of [MATH] to quadratic order in all three arguments evaluating the result in the background metric, e.g., [EQUATION] where [MATH] is a constant corresponding to the evaluation of [MATH] itself in the fixed curvature background metric and [MATH]; [MATH] means that [MATH] is to be evaluated in terms of the background metric which we here assume to be a space of constant curvature, i.e., a maximally symmetric space as is the case in both the ADD and RS models.', 'hep-ph-0603242-5-14-2': 'Thus the quantities [MATH], [MATH], [MATH], [MATH] and [MATH] are just numbers which depend on the explicit form of the metric and possibly the number of extra dimensions.', 'hep-ph-0603242-5-14-3': 'In such a maximally symmetric space the Weyl tensor and corresponding invariant both vanish identically, i.e., [MATH] and one further finds that [EQUATION]', 'hep-ph-0603242-5-14-4': 'Note that in ADD [MATH] since the metric is Minkowskian whereas in the [MATH] RS bulk [MATH] (away from the two branes) where the parameter [MATH] originates from the usual RS metric [MATH].', 'hep-ph-0603242-5-15-0': 'Without making any further assumptions we obtain [EQUATION] where we have defined [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', 'hep-ph-0603242-5-15-1': 'For the case of [MATH] this reproduces the results give by, e.g., Navarro and Van Acoleyen in [CITATION].', 'hep-ph-0603242-5-15-2': 'Note that if we make the assumption that [MATH] is a function only of [MATH] and the combination [MATH] then [MATH] etc and, also noting that [MATH]=0, we obtain [MATH] so that the remaining expressions greatly simplify; we now obtain [EQUATION] where the parameter [MATH] is given by [EQUATION]', 'hep-ph-0603242-5-15-3': 'Note that having [MATH] implies that the Weyl term, [MATH], in the effective action is absent in second order which is equivalent to taking [MATH] thus eliminating the massive tensor ghost issue.', 'hep-ph-0603242-5-15-4': 'This field is now removed from the spectrum though the D-dimensional scalar remains in general.', 'hep-ph-0603242-5-16-0': '# Application I: ADD', 'hep-ph-0603242-5-17-0': 'In this section we will apply the above analysis to the general ADD framework where we now require (since the space is flat) [MATH] so that [MATH] automatically.', 'hep-ph-0603242-5-17-1': 'This significantly reduces the possible deviations from the classic ADD picture.', 'hep-ph-0603242-5-17-2': 'In this specific case the general second order expansion of [MATH] is rather simple and is given by [EQUATION]', 'hep-ph-0603242-5-17-3': 'Note that if we also demand that [MATH] be a function only of [MATH] and the combination [MATH] in order to avoid the issue of the massive tensor ghost this expression simplifies even further to [EQUATION]', 'hep-ph-0603242-5-17-4': 'Of course, there has been some discussion about other ways to circumvent this tensor ghost problem than by completely eliminating it from the perturbative spectrum.', 'hep-ph-0603242-5-17-5': 'Since we are working only to lowest non-trivial order perhaps we should keep an open mind about the forms for [MATH].', 'hep-ph-0603242-5-17-6': 'Note that since [MATH] will essentially rescale the overall mass factor in the action we must demand that [MATH] to insure that the usual D-dimensional massless tensor gravitons not be ghost-like.', 'hep-ph-0603242-5-18-0': 'How are the predictions of ADD modified by these additional curvature terms?', 'hep-ph-0603242-5-18-1': 'The basic ADD picture leads to three essential predictions [CITATION]: ([MATH]) the emission of graviton KK states during the collision of SM particles producing signatures with apparent missing energy[CITATION]; ([MATH]) the exchange of graviton KK excitations between SM brane fields leading to dimension-8 contact interaction-like operators with distinctive spin-2 properties[CITATION]; ([MATH]) the production of black holes(BH) at colliders and in cosmic rays with geometric cross sections, [MATH], with [MATH] being the BH Schwarzschild radius, once collision energies greater than [MATH] are exceeded[CITATION].', 'hep-ph-0603242-5-19-0': 'The production and properties of D-dimensional, TeV-scale BH in higher curvature theories has been partially explored within the context of Lovelock extended gravity[CITATION] though not yet so in the fully general quadratic gravity case described by the function [MATH] considered here.', 'hep-ph-0603242-5-19-1': 'Such a study, which would be very interesting, is far beyond the scope of the present analysis.', 'hep-ph-0603242-5-19-2': 'However, it is interesting to make several observations: ([MATH]) Consider the vacuum solution; if we expand the general action [MATH] above to only quadratic order and if we also assume that all interesting solutions must satisfy [MATH] in the vacuum, then the only deviations from the conventional Schwarzschild form arise from the GB term in the action.', 'hep-ph-0603242-5-19-3': 'This can be seen immediately by examining the equations of motion resulting from the general quadratic action, e.g., in Ref. [CITATION].', 'hep-ph-0603242-5-19-4': 'This result does not remain valid when [MATH] is treated exactly.', 'hep-ph-0603242-5-19-5': '([MATH]) If [MATH] only and is treated exactly without expansion then the equations of motion allow for the conventional external BH result with [MATH] and will appear as an ordinary D-dimensional Schwarzschild solution.', 'hep-ph-0603242-5-19-6': 'This is not, however, the most general solution as can be see by considering the simple case of [MATH] with [MATH] a dimensionless parameter.', 'hep-ph-0603242-5-19-7': 'Here there is also exists a solution with [MATH], which is a constant corresponding to deSitter or anti-deSitter space depending on the sign of [MATH].', 'hep-ph-0603242-5-19-8': '([MATH]) If [MATH], with [MATH] being a constant, then the general BH solution has neither [MATH] nor [MATH] equal to zero as is well-known from the exact solution[CITATION].', 'hep-ph-0603242-5-19-9': 'A more detailed study of these possibilities would be worthwhile.', 'hep-ph-0603242-5-20-0': 'Let us now consider the situation of graviton exchange where it is well-known[CITATION] that ADD leads to new dimension-8 contact interactions.', 'hep-ph-0603242-5-20-1': 'To obtain the analogous quantities here we must expand the integrand of the action, i.e., the Lagrangian [MATH], to second order in the fluctuations, [MATH], around the flat background metric: [EQUATION]', 'hep-ph-0603242-5-20-2': 'Here we have expressed the original metric as [EQUATION] with [MATH] being the background metric, which is identified in the ADD model as the Minkowski metric [MATH].', 'hep-ph-0603242-5-20-3': 'The propagator is then just the inverse of the operator [MATH]; once the propagator is known we sandwich it between two 4-d localized (at the origin of the extra-dimensional co-ordinates) and conserved SM stress-energy sources, [MATH], to find the relevant scattering amplitude; we must remember to later KK decompose the various towers.', 'hep-ph-0603242-5-20-4': "Fortunately, much of this work has been done for us by Accioly, Azeredo and Mukai[CITATION] from which, with some modifications, we obtain the expression for the D-dimensional 'graviton' exchange amplitude (before performing the KK sums) [EQUATION] where [MATH] are the familiar (unaltered in terms of the compactification radius [MATH]) flat space KK masses.", 'hep-ph-0603242-5-20-5': 'The [MATH] label the various KK levels; [MATH] is the 4-d trace of the SM source stress-energy tensor and [MATH] are just [EQUATION] as described above.', 'hep-ph-0603242-5-20-6': '[MATH] is the compactification radius which sets the KK mass scale; here we have assumed a common value for this quantity for all [MATH] extra dimensions so that the volume of the compactified space is just [MATH].', 'hep-ph-0603242-5-20-7': 'In the expression above the first term in the amplitude is the usual one encountered in the ADD model which results from the D-dimensional EH action and combines the contributions of the 4-d spin-2 graviton and spin-0 graviscalar KK towers.', 'hep-ph-0603242-5-20-8': 'The second and third terms correspond to the new D-dimensional massive ghost tensor and scalar contributions, respectively.', 'hep-ph-0603242-5-20-9': 'The difference in the factors of [MATH] versus [MATH] in the first two terms arises from the existence of a 5-d bulk mass for the tensor ghost field.', 'hep-ph-0603242-5-20-10': 'It is interesting to note that the full amplitude is very well behaved at large [MATH] (in fact, going as [MATH]) due to the detailed cancellations between the various terms.', 'hep-ph-0603242-5-21-0': 'Here [MATH] represent the bulk mass terms of the new fields which enter into the KK tower masses of the scalars (spin-0) and tensors (spin-2 and spin-0), respectively; here we see the effect of the tensor ghost KK tower exchange explicitly.', 'hep-ph-0603242-5-21-1': 'From this point of view it appears that the only way to remove this ghost tower is to take the bulk mass [MATH] implying that [MATH] is solely a function of [MATH] and the combination [MATH], which we will assume from now on in our ADD discussion.', 'hep-ph-0603242-5-21-2': 'Note that the existence of a GB term in the action will not yield a contribution to [MATH].', 'hep-ph-0603242-5-21-3': 'Since, as discussed above [MATH] is already required, tachyonic KK scalars are avoided when the denominator in the expression for [MATH] above is positive; when [MATH] is assumed to be a function only of [MATH] and [MATH], then this denominator simplifies to [MATH].', 'hep-ph-0603242-5-21-4': 'In the limit where [MATH] alone, and accounting for a sign factor in the definition of the above actions, our result for the squared scalar mass, [MATH], agrees with that obtained by Demir and Tanyildizi[CITATION].', 'hep-ph-0603242-5-21-5': 'As shown by these authors, the effect of the new scalar tower exchange is generally rather suppressed in comparison to the more familiar graviton exchange since the ratio [MATH] is small for most SM particle sources at TeV colliders.', 'hep-ph-0603242-5-21-6': 'For example, for the process [MATH] this ratio is of order [MATH].', 'hep-ph-0603242-5-21-7': 'The corresponding ratio of the KK summed scalar to graviton exchange amplitudes is somewhat further reduced by ([MATH]) the existence of the finite bulk scalar mass which implies that there are no light scalar KK exchanges with masses below [MATH] and ([MATH]) the [MATH]-dependent numerical factor in the denominator of the scalar amplitude.', 'hep-ph-0603242-5-21-8': 'Naturalness suggests that [MATH] TeV or larger unless the parameters of [MATH] are somehow fine-tuned.', 'hep-ph-0603242-5-21-9': 'For example, if [MATH], then [MATH] for all [MATH] if [MATH] is not too far from O(1).', 'hep-ph-0603242-5-21-10': 'Interestingly we see here that as [MATH] we recover the usual EH expectation as then [MATH].', 'hep-ph-0603242-5-21-11': 'Thus we find that for many practical purposes the structure of the usual ADD results for graviton exchange are not qualitatively modified when the action is generalized to the form considered here.', 'hep-ph-0603242-5-21-12': 'However, with the existence of these additional scalars being a hallmark of the extended action, it behooves us to find a way to isolate their effects experimentally.', 'hep-ph-0603242-5-22-0': 'In expressions for graviton exchange only the combination [MATH] will now appear.', 'hep-ph-0603242-5-22-1': "In the amplitude this will lead to a modification of the pure 'graviton' exchange cross section expectations by a factor of [MATH], which is likely to be of [MATH], provided [MATH] is considered to be held fixed.", 'hep-ph-0603242-5-22-2': 'When the graviton tower interference term with the SM dominates, the effect in the gravitational part of the cross section will scale as [MATH].', 'hep-ph-0603242-5-22-3': 'Given the previous results of Demir and Tanyildizi[CITATION], this is not surprising.', 'hep-ph-0603242-5-23-0': 'We further note that since [MATH] is [MATH] TeV or larger it has no effect on laboratory measurements of the strength of the gravitational interaction in the micron range when [MATH].', 'hep-ph-0603242-5-24-0': 'Before closing this part of the discussion we would like to remind the reader that it was pointed out long ago[CITATION] that we can take any action of the form [MATH] and map it over to the EH action coupled to an minimally coupled real scalar field with a rather complicated potential [MATH], depending exponentially on the scalar field.', 'hep-ph-0603242-5-24-1': 'This can be done via a special conformal transformation [EQUATION]', 'hep-ph-0603242-5-24-2': 'Going from the original (Jordan) to the new (Einstein) frame one explicitly sees the existence of the new scalar degree of freedom.', 'hep-ph-0603242-5-24-3': 'The mass of this scalar field is exactly that of the field [MATH] above and can be gotten directly from the canonically normalized potential [MATH] in the usual manner, i.e., using [MATH].', 'hep-ph-0603242-5-24-4': 'This is a very powerful tool as it allows us to extend our previous flat space result for [MATH] to the much more general case where the space has constant curvature.', 'hep-ph-0603242-5-24-5': 'For example, if [MATH], we find that the value of [MATH] is the same as discussed above, i.e., [MATH], in a space with constant curvature.', 'hep-ph-0603242-5-24-6': 'This will be an important result that we will employ when we discuss the case of the RS setup.', 'hep-ph-0603242-5-25-0': 'We now turn to the emission of gravitons in SM particle collisions.', 'hep-ph-0603242-5-25-1': 'Since the compactifying space is flat in the ADD case the normalizations of the graviton (and scalar) wavefunctions which control their couplings are unaltered by the existence of the quadratic curvature terms but the relationship between [MATH] and [MATH]M_pl[MATH] is modified.', 'hep-ph-0603242-5-25-2': 'This was briefly mentioned above where we saw that in the small curvature limit the parameter [MATH] essentially renormalizes the fundamental scale.', 'hep-ph-0603242-5-25-3': 'To see this in the present case it is sufficient to examine the tensor/spin-2 kinetic part of the 4-d effective Lagrangian to second order in [MATH] (which has not yet been KK-expanded) in the familiar transverse traceless gauge, i.e., [MATH], [MATH]; one obtains[CITATION] [EQUATION] where here [MATH] and [MATH] are defined above.', 'hep-ph-0603242-5-25-4': 'When we assume that [MATH] is only a function of [MATH] and the combination [MATH] then the second term in [MATH] vanishes and we recover the familiar result of the standard EH scenario apart from the overall factor of [MATH].', 'hep-ph-0603242-5-25-5': 'Hence, to recover the conventional 4-d EH action when inserting the usual (extra dimensionally) flat zero mode graviton wavefunction into [MATH] the ADD relationship must be modified, as hinted above, to [EQUATION] where [MATH]M_pl[MATH] is the 4-d reduced Planck scale and [MATH] is the volume of the compactified space.', 'hep-ph-0603242-5-25-6': 'Of course, [MATH] is just unity in the standard ADD model which employs the EH action.', 'hep-ph-0603242-5-25-7': 'Since the lightest of the KK scalars has a mass which is naturally on the order of a TeV and has rather weak couplings to SM fields these particles will not play much of an important role in missing energy processes.', 'hep-ph-0603242-5-25-8': 'If the cross section for graviton production, i.e., missing energy, is expressed in terms of the original [MATH] with other parameters held fixed, then the presence of [MATH] leads to a modification of the production cross section by a factor of [MATH].', 'hep-ph-0603242-5-25-9': 'However, as [MATH] is not likely to remain a direct observable (only the product [MATH] is) there may be no way to experimentally disentangle this effect.', 'hep-ph-0603242-5-25-10': 'Furthermore, for any given [MATH], since [MATH]M_pl[MATH] is numerically fixed and [MATH] is an input parameter the resulting derived value of [MATH] which sets the scale for the masses of the KK states is altered.', 'hep-ph-0603242-5-26-0': 'We thus conclude that if we assume that [MATH] is a function of only of [MATH] and the combination [MATH] then the classic predictions ([MATH]) and ([MATH]) of the ADD model will be qualitatively unaffected by going to the more general action considered here except for possible overall scalings by inverse powers of [MATH] when the parameter [MATH] is held fixed: graviton emission rates scale like [MATH] while graviton exchange cross sections scale as [MATH] or [MATH] depending on the presence of important SM contributions to the relevant process.', 'hep-ph-0603242-5-26-1': 'Furthermore, given Eq. (22) and fixed values of [MATH] and [MATH], the KK masses, being proportional to [MATH], will scale as [MATH]; such a mass shift can be quite sizable for reasonable values of [MATH].', 'hep-ph-0603242-5-27-0': '# Application II: RS', 'hep-ph-0603242-5-28-0': 'The predictions of the classic RS model are the existence of TeV scale graviton resonances with fixed weak scale masses and couplings to the SM fields[CITATION], the existence of a weak scale radion excitation[CITATION], as well as the production of [MATH] BH.', 'hep-ph-0603242-5-28-1': 'In what follows we will be specifically interested in the nature of the KK gravitons so it is again sufficient to examine the quadratically expanded action.', 'hep-ph-0603242-5-28-2': 'The classic RS model is not generally strictly consistent with the assumed form of either the original action [MATH] or its quadratically expanded form [MATH].', 'hep-ph-0603242-5-28-3': 'As is well-known, and as mentioned above, the equations of motion that follow from [MATH] and [MATH] will generally be fourth order in the derivatives of the metric.', 'hep-ph-0603242-5-28-4': 'In the usual 5-d RS model, one solves the Einstein equations of the form [EQUATION] where [MATH] is the Einstein tensor arising from the EH action involving no more than two derivatives of the metric.', 'hep-ph-0603242-5-28-5': 'The problem is that RS completely specifies [MATH]: a cosmological constant in the 5-d bulk plus two [MATH]-function sources at the orbifold locations of the TeV and Planck branes.', 'hep-ph-0603242-5-28-6': 'SM matter confined to the TeV brane is supposed to not be a large contributor to the stress-energy.', 'hep-ph-0603242-5-28-7': 'To obtain this result the standard RS metric takes the form discussed above: [MATH] with the linear exponential warp factor leading to the bulk [MATH] and the two field derivatives acting on the absolute value leading to the brane [MATH]-functions.', 'hep-ph-0603242-5-28-8': '(This is related to the comment above that [MATH] is not truly constant in RS and has brane [MATH]-function singularities.', 'hep-ph-0603242-5-28-9': 'Recall that these [MATH]-functions are the results of assuming infinitely thin branes.)', 'hep-ph-0603242-5-28-10': 'If an identical metric is assumed in our more general case we still can obtain [MATH] but the fourth order equations would lead to the more singular derivatives of [MATH]-functions at the brane locations that are not canceled by any source terms.', 'hep-ph-0603242-5-28-11': 'This amongst other reasons is what led Kim, Kyae and Lee[CITATION] to consider only GB extensions of the EH action in RS since it is the only extension which uniquely produces Einstein equations of second order in the derivatives.', 'hep-ph-0603242-5-28-12': 'Thus if we keep the classic picture, an analysis of RS given our assumed effective action expanded around a background of constant curvature is not relevant.', 'hep-ph-0603242-5-28-13': '(A possible way of dealing with these derivatives of [MATH]-functions arising from orbifold singularities in higher dimensional effective field theories has been discussed in Ref. [CITATION].', 'hep-ph-0603242-5-28-14': 'Implementing our scheme employing such techniques is, however, beyond the scope of the present paper.)', 'hep-ph-0603242-5-29-0': "To avoid these issues for now we simplify our discussion of this problem (and to convince ourselves that an RS-like solution is possible in this framework) we consider a singularity-free, 'softened' version of RS where the orbifolded bulk space with branes is replaced by an interval, as has been suggested for other reasons[CITATION], with SM matter placed at one end point possessing an ignorable amount of stress-energy.", 'hep-ph-0603242-5-29-1': 'With a cosmological constant on the interval we can recover the background [MATH] bulk; in addition by removing the absolute value sign of the co-ordinate [MATH] in the metric above we expunge the [MATH]-functions as well as the possibility of any of their higher derivatives appearing in the equations of motion.', 'hep-ph-0603242-5-29-2': 'The boundary conditions at the end points for the graviton KK states can then be freely chosen to be the same as that of the original RS model.', 'hep-ph-0603242-5-29-3': "This space is truly one of constant curvature and the general analysis we have presented above will now be applicable to this 'softened' RS on an interval.", 'hep-ph-0603242-5-30-0': 'It is easy to verify that the form of the equations of motion[CITATION] in this case (recalling that we are only searching for solutions with maximally-symmetric, constant [MATH] backgrounds) are given by: [EQUATION] and that if we take stress-energy tensor in the 5d bulk to be of the usual RS form [EQUATION] with [MATH], then indeed a space of constant curvature, i.e. [MATH], can be an allowed solution.', 'hep-ph-0603242-5-30-1': 'Taking the trace of the equations of motion above, evaluating it in the constant curvature bulk and relating the values of [MATH] to [MATH] as before (recalling that here [MATH] using the softened metric) results in the constraint equation [EQUATION] where here [MATH].', 'hep-ph-0603242-5-30-2': 'It is interesting to note that if we assume that [MATH]constant then this constraint equation automatically implies that [MATH]constant as well; but this does not necessarily further require that all of the [MATH] are constants as we will see below.', 'hep-ph-0603242-5-30-3': 'When [MATH] is only a function of [MATH] and the combination [MATH], this constraint equation simplifies to [EQUATION] while in the specific RS background case this explicitly becomes [EQUATION]', 'hep-ph-0603242-5-30-4': 'It is important to recall that [MATH] itself can be a complicated function of [MATH] so that this equation can be quite nontrivial.', 'hep-ph-0603242-5-30-5': 'For the EH action limit this yields the usual relation that [MATH]; here it in general provides an additional constraint on the allowed forms of the function [MATH] since we are requiring [MATH] to be both real and negative.', 'hep-ph-0603242-5-30-6': 'Given a specific function [MATH] for which a solution exists, this equation directly relates [MATH] and [MATH] though the solution may not be unique.', 'hep-ph-0603242-5-30-7': 'For example, if we assume for purposes of demonstration the simple case of [EQUATION] as employed above, then there are two branches of solutions for [MATH]: [EQUATION] one of which (the negative root) goes over to the usual EH result as the parameter [MATH].', 'hep-ph-0603242-5-31-0': 'Allowing for the possibility of a RS-like solution with a softened metric it is interesting to think briefly about the previously analyzed effects of the GB term in the RS scenario.', 'hep-ph-0603242-5-31-1': 'This analysis was originally performed for the classic RS[CITATION] setup which employed the standard form of the RS metric; that result would now be modified by the changes in the model assumptions, i.e., moving to an interval and removing the [MATH]-function sources at the end points.', 'hep-ph-0603242-5-31-2': 'The previous analysis of BH in RS with the added GB term would not be significantly affected if this transition were made.', 'hep-ph-0603242-5-31-3': 'However, the properties and spectrum of the graviton KK states certainly would be influenced since the [MATH]-function terms are now absent.', 'hep-ph-0603242-5-31-4': 'The equation governing the masses and wavefunction of the graviton KK states for the present interval case can be obtained by expanding the equations of motion as before.', 'hep-ph-0603242-5-31-5': 'Since we are here only interested in the tensor modes associated with the usual gravitons, we can employ the expansion [EQUATION] where [MATH].', 'hep-ph-0603242-5-31-6': 'Applying the usual RS boundary conditions on the interval the most significant changes from the classic RS can be read off from Eqs. (15)-(28) in Ref. [CITATION] by setting the parameter [MATH] in appropriate places.', 'hep-ph-0603242-5-31-7': 'At the end of the day we find that the only apparent difference from the classic EH based RS model would be a shift in the relationship between the fundamental scale and [MATH]M_pl[MATH]-remarkably similar to what we saw for the ADD model above.', 'hep-ph-0603242-5-31-8': 'In the language employed in Ref. [CITATION] we would now obtain [EQUATION] where [MATH] is the coefficient of the GB term in the action.', 'hep-ph-0603242-5-31-9': 'Otherwise the masses as well as the couplings of all of the KK gravitons to localized SM matter would be identical to those of the original RS model expressed in terms of the derived parameter [MATH].', 'hep-ph-0603242-5-31-10': 'The explicit coupling and spectrum changes found in Ref[CITATION] for the graviton KK states in the presence of the GB term in the action were all found to due to the brane [MATH]-function singularities.', 'hep-ph-0603242-5-32-0': 'How would these graviton KK results obtained in the GB extended action generalize to the case of [MATH] above?', 'hep-ph-0603242-5-32-1': 'Here we choose to begin our analysis with [MATH], setting [MATH] from the beginning to avoid potential ghost fields, then taking [MATH] and using the same curvature expansion as above.', 'hep-ph-0603242-5-32-2': 'In order to make a connection with the previous discussion, the existing RS literature and to directly compare with the GB case, however, we massage our notation slightly and rewrite [MATH] in the following form: [EQUATION] where the parameters [MATH] and [MATH] are dimensionless; the action employed in Ref. [CITATION] is now directly recovered by taking the [MATH] and [MATH] limits.', 'hep-ph-0603242-5-32-3': 'It is important at this point to recall that to obtain the linearized graviton equations of motion it is sufficient to employ [MATH] while the complete [MATH] needs to be examined in order to demonstrate the existence of the required [MATH] solution.', 'hep-ph-0603242-5-32-4': 'The equations of motion resulting from [MATH] are given by[CITATION] [EQUATION]', 'hep-ph-0603242-5-32-5': 'Here [MATH] is the covariant derivative operator and here [MATH].', 'hep-ph-0603242-5-32-6': 'First we look at the [MATH] component of this equation, remembering that for the moment we will only be interested in the tensor excitations corresponding to the KK gravitons which are massless in 5-d.', 'hep-ph-0603242-5-32-7': 'In the usually chosen gauge, [MATH] is still a constant to linear order so we arrive at a consistency condition [EQUATION]', 'hep-ph-0603242-5-32-8': 'Note that this reduces to the previously obtained purely quadratic GB extended RS result[CITATION] when [MATH].', 'hep-ph-0603242-5-32-9': 'In the more general case, this expression is not overly useful given the exact result in Eq. (27).', 'hep-ph-0603242-5-33-0': 'Turning now to the [MATH] terms which contain the 4-d graviton tensor excitation, we linearize employing the previously mentioned transverse, traceless gauge with constant [MATH].', 'hep-ph-0603242-5-33-1': 'This gives the standard equation of motion for the RS graviton found long ago[CITATION] though scaled by an overall factor.', 'hep-ph-0603242-5-33-2': 'Employing the standard KK decomposition [EQUATION] and recalling that [MATH], the [MATH] are seen to satisfy [EQUATION]', 'hep-ph-0603242-5-33-3': 'The overall factor [MATH] is given by [EQUATION] or, more explicitly in the RS case, [EQUATION] (Again we recall that [MATH] itself can be a function of [MATH].)', 'hep-ph-0603242-5-33-4': 'This leads to a rescaling of the usual RS relationship [EQUATION] via the renormalization of the zero mode (i.e., massless graviton) wavefunction, thus generalizing Eq. (31).', 'hep-ph-0603242-5-33-5': 'Of course, [MATH] is required to avoid ghost states among the usual gravitons KKs.', 'hep-ph-0603242-5-33-6': 'This result reduces to that previously obtained in the RS case with just the added GB term[CITATION] once boundary effects are neglected.', 'hep-ph-0603242-5-34-0': 'From this analysis we see immediately that the masses of the KK gravitons are identical to those obtained in the original RS model, provided we use the same value of the parameter [MATH], as we might have expected.', 'hep-ph-0603242-5-34-1': 'Here we are faced with the question of just what are the independent parameters.', 'hep-ph-0603242-5-34-2': '[MATH] is clearly a derived parameter that is obtained by simultaneously solving Eqs. (27) and (40) for any given model.', 'hep-ph-0603242-5-34-3': 'In that sense, the KK graviton spectrum would just be rescaled in comparison to the usual expectations given the same input value of [MATH].', 'hep-ph-0603242-5-34-4': 'As we have just seen, and as in the ADD case, the effect of a factor like [MATH] on the KK graviton couplings to 4-d SM matter depends upon which model parameters are assumed to be held fixed.', 'hep-ph-0603242-5-34-5': 'At the very least, up to an overall constant, these KK graviton couplings are identical to those of the standard RS model.', 'hep-ph-0603242-5-35-0': 'As an example of a simple model where the shifts in the KK spectrum can be calculated analytically consider substituting for the integrand of the conventional RS action, [MATH], the simple higher curvature action [MATH] as was considered above.', 'hep-ph-0603242-5-35-1': 'Let [MATH] be the values obtained for the parameter [MATH] in the usual RS model, i.e., [MATH].', 'hep-ph-0603242-5-35-2': 'Keeping the warp factor fixed we can use the equations above to calculate the value of, e.g., the mass of the first graviton KK state in both the standard RS model, [MATH], and in the current model with an augmented action, [MATH].', 'hep-ph-0603242-5-35-3': 'Using Eqs. (27) and (40) this ratio can be calculated analytically in the present case as a function of [MATH] and [MATH]M_pl[MATH]; we obtain [EQUATION]', 'hep-ph-0603242-5-35-4': 'The result of this calculation is shown in Fig.1 for a wide range of model parameters.', 'hep-ph-0603242-5-35-5': 'In this example we see that the size of the possible shift in the mass spectrum can be quite large assuming a fixed value of [MATH].', 'hep-ph-0603242-5-36-0': 'So far we have only considered the 4-d graviton, spin-2 excitations.', 'hep-ph-0603242-5-36-1': 'It is important to remember that our softened RS model now has an additional massive scalar in the 5-d spectrum with a large bulk mass, [MATH], and that no massless scalar zero mode will exist.', 'hep-ph-0603242-5-36-2': 'Since the bulk scalar mass is naturally of order [MATH] the KK spectrum of the corresponding tower will begin with a KK scalar state whose mass is qualitatively comparable to that of the first graviton excitation.', 'hep-ph-0603242-5-36-3': 'This bulk mass is explicitly calculable from the expansion of the full action to quadratic order, [MATH], by going to the Einstein frame since we know that the GB term does not contribute to this parameter.', 'hep-ph-0603242-5-36-4': 'In that case, using the results from the previous section we find that [EQUATION] or, in terms of the original parameters of the action, evaluated in the RS background: [EQUATION]', 'hep-ph-0603242-5-36-5': 'Note that [MATH] is required to avoid the scalar tachyons and graviton ghosts, consistent with our above analysis.', 'hep-ph-0603242-5-36-6': 'Note further that this reproduces the results of Eq. (19) in the flat space, [MATH], limit.', 'hep-ph-0603242-5-37-0': 'Given any [MATH] the scalar bulk mass is known and we can determine the mass(es) of the lightest KK scalar state(s) by following the standard RS manipulations[CITATION].', 'hep-ph-0603242-5-37-1': 'These masses are essentially given by the first roots of the equation [EQUATION] where [MATH] and [MATH] is the usual Bessel function.', 'hep-ph-0603242-5-37-2': 'The solution for the first KK state is provided by Fig.2; as stated above there are no massless modes.', 'hep-ph-0603242-5-37-3': 'The lightest scalar mass is then [MATH].', 'hep-ph-0603242-5-37-4': 'Here we observe that the mass of the first scalar KK scales almost linearly with the bulk mass when [MATH] gets large.', 'hep-ph-0603242-5-37-5': 'Note that for [MATH] and a typical value[CITATION] of [MATH], we then find [MATH] implying [MATH] from Fig.1; this is about 3 times larger than the root for the usual lightest massive KK graviton, [MATH].', 'hep-ph-0603242-5-37-6': 'Thus we see that unless [MATH] takes on large values the first scalar KK state is always rather heavy.', 'hep-ph-0603242-5-37-7': 'As is well-known, the [MATH] values for the more massive KK scalar states will be somewhat larger: approximately given by [MATH] where [MATH] labels the KK level.', 'hep-ph-0603242-5-37-8': 'Since these scalars will couple to the trace of the stress-energy tensor for the 4-d SM fields they will interact far more weakly than do the graviton KK states unless this is at least partially offset by ratios of 5-d wavefunction factors.', 'hep-ph-0603242-5-37-9': 'A quick estimate of such factors, however, indicates that, if anything, these wave function ratios lead to a further suppression of the scalar couplings relative to those of the KK gravitons by [MATH] as shown in Fig.3.', 'hep-ph-0603242-5-37-10': 'This overall picture of the scalar sector is qualitatively very similar to that of the existence of a very heavy tower of RS radions[CITATION] or a tower of KK Higgs bosons as in the case of Universal Warped Extra Dimensions[CITATION].', 'hep-ph-0603242-5-38-0': 'In the analysis as presented here we have ignored the possibility that the new scalar KK states may mix with the (usually eaten) RS graviscalars through cross-talk in the equations of motion, i.e., we have assumed that the 5-d tensor and scalar KK decompositions can be performed independently, and this is something which needs further exploration.', 'hep-ph-0603242-5-38-1': 'A fully detailed analysis of the such possibilities is, however, beyond the scope of the present paper.', 'hep-ph-0603242-5-39-0': 'It is perhaps interesting to ask whether the usual [MATH]=constant ([MATH]) solution considered here necessitates the metric and matter distribution of the conventional RS model employed above without the further assumption of a maximally symmetric space.', 'hep-ph-0603242-5-39-1': 'To analyze a simple and more easily tractable situation let us consider the more general warped metric [MATH] and assume that [MATH] only.', 'hep-ph-0603242-5-39-2': 'The first question to address is what is the most general form of the function [MATH]; to deal with this issue we note that the Ricci scalar arising from this metric is in general given by [MATH].', 'hep-ph-0603242-5-39-3': "In the 'soft' version of the RS model on the interval defined above one had [MATH] and [MATH] so that [MATH] as usual.", 'hep-ph-0603242-5-39-4': 'Here, this curvature condition provides a differential equation for the function [MATH]; by solving this equation we arrive at the result [MATH], where the [MATH] are integration constants and [MATH].', 'hep-ph-0603242-5-39-5': 'Choosing the [MATH] appropriately and rescaling [MATH], we can rewrite this in a more familiar form as [MATH] with [MATH] a dimensionless constant.', 'hep-ph-0603242-5-39-6': 'Note that when [MATH] and [MATH] we recover the usual RS result.', 'hep-ph-0603242-5-39-7': 'However, the choice of [MATH] can easily modify the warp factor from its conventional behavior.', 'hep-ph-0603242-5-39-8': 'What is the nature of the bulk matter distribution that yields this metric?', 'hep-ph-0603242-5-39-9': 'The solution to this can be obtained by considering Eqs.(24) and (26) with [MATH].', 'hep-ph-0603242-5-39-10': 'Eq.(26) immediately tells us that the trace of the 5-d stress-energy tensor [MATH], a constant, so that if we define [MATH] then we must have [MATH], with [MATH] an arbitrary function.', 'hep-ph-0603242-5-39-11': 'The [MATH] and [MATH] components of the equations of motion when combined then provides a first order differential equation for [MATH] that can be solved in a straightforward manner.', 'hep-ph-0603242-5-39-12': 'Setting [MATH] and defining [MATH] with [MATH], we obtain the general solution (assuming [MATH]) [EQUATION] with [MATH] an integration constant.', 'hep-ph-0603242-5-39-13': 'Though this is far from a uniform energy distribution (away from the TeV brane) it does lead to a space of constant curvature but not one which is maximally symmetric.', 'hep-ph-0603242-5-39-14': 'Thus we see that it is possible that the requirement that [MATH] does allow for the possibility of more complex solutions than that employed in the original RS model.', 'hep-ph-0603242-5-40-0': '# Discussion and Conclusions', 'hep-ph-0603242-5-41-0': 'In this paper we have begun an examination of how generic higher curvature terms in the gravitational action can alter the predictions of both the ADD model and the RS model defined on a interval to avoid possible brane singularities.', 'hep-ph-0603242-5-41-1': 'We have assumed that the traditional assumptions of the two models, e.g., SM localized matter in a conformally flat bulk, remain valid; we have not considered in detail more complex setups that may now be allowed by the modified equations of motion.', 'hep-ph-0603242-5-41-2': 'To be more concrete, we have further assumed that the EH action is now generalized to an action which is of the form [MATH] where [MATH] is a well-behaved function, [MATH] and [MATH].', 'hep-ph-0603242-5-41-3': 'In D-dimensions this action results in a propagating massless tensor field (identified with the usual graviton), a massive ghost tensor field, as well as a massive (possibly tachyonic) scalar.', 'hep-ph-0603242-5-41-4': 'The potentially dangerous ghost is removable from the perturbative spectrum, i.e., it becomes infinitely massive, if we demand that [MATH] only.', 'hep-ph-0603242-5-41-5': 'The remaining new scalar field has a bulk mass whose value is naturally expected to be of order the fundamental scale, [MATH], in either scenario.', 'hep-ph-0603242-5-41-6': 'The resulting ADD and RS models are altered in similar ways from their traditional standard forms:', 'hep-ph-0603242-5-42-0': '([MATH]) New scalar KK excitations appear in the spectrum of both models in a rather benign fashion coupling to the trace of the stress-energy tensor of the localized SM fields.', 'hep-ph-0603242-5-42-1': 'Since this trace is proportional to SM masses, the couplings of these scalars are relatively strongly suppressed in comparison to those for the KK gravitons at typical collider energies in both models.', 'hep-ph-0603242-5-42-2': 'In the ADD model, the KK scalar excitations begin at a mass [MATH] TeV.', 'hep-ph-0603242-5-42-3': 'Consequently their contributions to missing-energy signatures as well as to the usual dimension-8 contact interactions are further kinematically suppressed.', 'hep-ph-0603242-5-42-4': 'Thus at leading order these new scalars do not much influence ADD collider signatures.', 'hep-ph-0603242-5-42-5': 'In RS, the bulk scalar mass tends to be large so that the lightest scalar KK state is several times more massive than is the lightest KK graviton.', 'hep-ph-0603242-5-42-6': 'Given their rather weak couplings such states will be difficult to observe at colliders.', 'hep-ph-0603242-5-43-0': '([MATH]) The basic model relationships involving the fundamental and 4-d Planck masses in both models get rescaled by functions of [MATH] and its derivatives evaluated in the corresponding background metric of the two models: in ADD we obtain [MATH]M_pl[MATH] while in RS we obtain [MATH]M_pl[MATH] where [MATH] is explicitly given in Eq. (39).', 'hep-ph-0603242-5-43-1': 'Assuming that [MATH] is a fixed fundamental parameter these modifications lead to changes in the graviton KK sectors of both models.', 'hep-ph-0603242-5-43-2': 'In the ADD case, since [MATH]M_pl[MATH] is known and [MATH] is an input parameter for any given [MATH] the volume of the compactified space and, hence, the value of the compactification radius which sets the graviton KK mass scale is altered.', 'hep-ph-0603242-5-43-3': 'Due to the presence of the [MATH] factor the emission rate for gravitons in the collisions of SM particles and for the graviton exchange amplitude are both modified by potentially O(1) effects.', 'hep-ph-0603242-5-43-4': 'Similarly in RS, [MATH] is a derived parameter which sets the scale for all the KK states.', 'hep-ph-0603242-5-43-5': 'The constraint Eq. (28) allows us to calculate [MATH] in terms of the input parameter [MATH] and the function [MATH] thus providing for us with [MATH].', 'hep-ph-0603242-5-43-6': 'In a manner similar to ADD, the presence of [MATH] rescales the coupling strengths of the of the KK graviton states to the SM fields thus modifying the widths and production cross sections at colliders by potentially O(1) factors.', 'hep-ph-0603242-5-44-0': 'As we have seen, the extension of the EH action to a more complicated structure can lead to significant quantitative modifications to both the ADD and RS model predictions in the simplest possible case.', 'hep-ph-0603242-5-44-1': 'The observation of such effects at future colliders could tell us valuable information about the underlying theory of gravity.', 'hep-ph-0603242-5-45-0': 'Note Added: After this paper was essentially completed, Ref. [CITATION] appeared which discusses generalized actions for the ADD model and thus has some common areas with the present work.', 'hep-ph-0603242-5-45-1': 'Where the two papers overlap there is general qualitative agreement though the points of view are somewhat different.'} | null | null |
1511.04162 | {'1511.04162-1-0-0': 'This paper studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treatment variable is mismeasured and endogenous.', '1511.04162-1-0-1': 'I characterize the sharp identified set for the local average treatment effect under the following two assumptions: (1) the exclusion restriction of an instrument and (2) deterministic monotonicity of the true treatment variable in the instrument.', '1511.04162-1-0-2': 'The identification strategy allows for general measurement error.', '1511.04162-1-0-3': 'Notably, (i) the measurement error is nonclassical, (ii) it can be endogenous, and (iii) no assumptions are imposed on the marginal distribution of the measurement error, so that I do not need to assume the accuracy of the measurement.', '1511.04162-1-0-4': 'Based on the partial identification result, I provide a consistent confidence interval for the local average treatment effect with uniformly valid size control.', '1511.04162-1-0-5': 'I also show that the identification strategy can incorporate repeated measurements to narrow the identified set, even if the repeated measurements themselves are endogenous.', '1511.04162-1-0-6': 'Using the NLS-72 dataset, I demonstrate that my new methodology can produce nontrivial bounds for the return to college attendance when attendance is mismeasured and endogenous.', '1511.04162-1-1-0': '[] Keywords: Misclassification; Local average treatment effect; Endogenous measurement error; Instrumental variable; Partial identification [] JEL Classification Codes: C21, C26', '1511.04162-1-2-0': '# Introduction', '1511.04162-1-3-0': 'Treatment effect analyses often entail a measurement error problem as well as an endogeneity problem.', '1511.04162-1-3-1': 'For example, [CITATION] document a substantial measurement error in educational attainments in the 1990 U.S. Census.', '1511.04162-1-3-2': 'Educational attainments are treatment variables in a return to schooling analysis, and they are endogenous because unobserved individual ability affects both schooling decisions and wages .', '1511.04162-1-3-3': 'The econometric literature, however, has offered only a few solutions for addressing the two problems at the same time.', '1511.04162-1-3-4': 'Although an instrumental variable is a standard technique for correcting both endogeneity and measurement error , no paper has investigated the identifying power of an instrumental variable for the heterogeneous treatment effect when the treatment variable is both mismeasured and endogenous.', '1511.04162-1-4-0': 'I consider a measurement error in the treatment variable in the framework of [CITATION] and [CITATION], and focus on identification/inference problems for the local average treatment effect (LATE).', '1511.04162-1-4-1': 'The LATE is the average treatment effect for the subpopulation (the compliers) whose true treatment status is strictly affected by an instrument.', '1511.04162-1-4-2': 'Focusing on LATE is meaningful for a few reasons.', '1511.04162-1-4-3': 'First, LATE has been a widely used parameter to investigate the heterogeneous treatment effect with endogeneity.', '1511.04162-1-4-4': 'My analysis on LATE of a mismeasured treatment variable offers a tool for a robustness check to those who have already investigated LATE.', '1511.04162-1-4-5': 'Second, LATE can be used to extrapolate to the average treatment effect or other parameters of interest.', '1511.04162-1-4-6': '[CITATION] emphasize the utility of reporting LATE even if the parameter of interest is obtained based on LATE, because the extrapolation often requires additional assumptions and the result of the extrapolation can be less credible than LATE.', '1511.04162-1-5-0': 'I take a worst case scenario approach with respect to the measurement error and allow for arbitrary measurement error.', '1511.04162-1-5-1': 'The only assumption concerning the measurement error is its independence of the instrumental variable.', '1511.04162-1-5-2': 'The following types of measurement error are considered in my analysis.', '1511.04162-1-5-3': 'First, the measurement error is nonclassical; that is, it can be dependent on the true treatment variable.', '1511.04162-1-5-4': 'The measurement error for a binary variable is always nonclassical.', '1511.04162-1-5-5': 'It is because the measurement error cannot be negative (positive) when the true variable takes the lowest (highest) value.', '1511.04162-1-5-6': 'Second, I allow the measurement error to be endogenous; that is, the measured treatment variable is allowed to be dependent on the outcome variable conditional on the true treatment variable.', '1511.04162-1-5-7': 'It is also called a differential measurement error.', '1511.04162-1-5-8': 'The validation study by [CITATION] finds that the measurement error is likely to be correlated with individual observed and unobserved heterogeneity.', '1511.04162-1-5-9': 'The unobserved heterogeneity causes the endogeneity of the measurement error; it affects the measurement and the outcome at the same time.', '1511.04162-1-5-10': 'For example, the measurement error for educational attainment depends on the familiarity with the educational system in the U.S., and immigrants may have a higher rate of measurement error.', '1511.04162-1-5-11': 'At the same time, the familiarity with the U.S. educational system can be related to the English language skills, which can affect the labor market outcomes.', '1511.04162-1-5-12': '[CITATION] also argue that measurement error is likely to be differential in some empirical applications.', '1511.04162-1-5-13': 'Third, there is no assumption concerning the marginal distribution of the measurement error.', '1511.04162-1-5-14': 'It is not necessary to assume anything about the accuracy of the measurement.', '1511.04162-1-6-0': 'Even if I allow for an arbitrary measurement error, this paper demonstrates that an instrumental variable can still partially identify LATE when (a) the instrument satisfies the exclusion restriction such that the instrument affects the outcome and the measured treatment only through the true treatment, and (b) the instrument weakly increases the true treatment.', '1511.04162-1-6-1': 'These assumptions are standard in the LATE framework ([CITATION] and [CITATION]).', '1511.04162-1-6-2': 'I show that the point identification for LATE is impossible unless LATE is zero, and I characterize the sharp identified set for LATE.', '1511.04162-1-6-3': 'Based on the sharp identified set, (i) the sign of LATE is identified, (ii) there are finite upper and lower bounds on LATE even for the unbounded outcome variable, and (iii) the Wald estimand is an upper bound on LATE in absolute value but sharp upper bound is in general smaller than the Wald estimand.', '1511.04162-1-6-4': 'I obtain an upper bound on LATE in absolute value by deriving a new implication of the exclusion restriction.', '1511.04162-1-7-0': 'Inference for LATE in my framework does not fall directly into the existing moment inequality models particularly when the outcome variable is continuous.', '1511.04162-1-7-1': 'First, the upper bound for LATE in absolute value is not differentiable with respect to the data distribution.', '1511.04162-1-7-2': 'This non-differentiability problem precludes any estimator for the upper bound from having a uniformly valid asymptotic distribution, as is formulated in [CITATION] and [CITATION].', '1511.04162-1-7-3': 'Second, the upper bound cannot be characterized as the infimum over differentiable functionals indexed by a compact subset in a finite dimensional space, unless the outcome variable has a finite support.', '1511.04162-1-7-4': 'This prohibits from applying the existing methodologies in conditional moment inequalities, e.g., [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1511.04162-1-8-0': 'I construct a confidence interval for LATE which can be applied to both discrete and continuous outcome variables.', '1511.04162-1-8-1': 'To circumvent the aforementioned problems, I approximate the sharp identified set by discretizing the support of the outcome variable where the discretization becomes finer as the sample size increases.', '1511.04162-1-8-2': 'The approximation for the sharp identified set resembles many moment inequalities in [CITATION] and [CITATION], who consider a finite but divergent number of moment inequalities.', '1511.04162-1-8-3': 'I adapt a bootstrap method in [CITATION] into my framework to construct a confidence interval with uniformly valid asymptotic size control.', '1511.04162-1-8-4': 'Moreover, I demonstrate that the confidence interval is consistent against the local alternatives in which a parameter value approaches to the sharp identified set at a certain rate.', '1511.04162-1-9-0': 'As empirical illustrations, I apply the new methodology for evaluating the effect on wages of attending a college when the college attendance can be mismeasured.', '1511.04162-1-9-1': 'I use the National Longitudinal Survey of the High School Class of 1972 (NLS-72), as in [CITATION].', '1511.04162-1-9-2': 'Using the proximity to college as an instrumental variable , the confidence interval developed in the present paper offers nontrivial bounds on LATE, even if I allow for measurement error in college attendance.', '1511.04162-1-9-3': 'Moreover, the empirical results confirm the theoretical result that the Wald estimator is an upper bound on LATE but is not the sharp upper bound.', '1511.04162-1-10-0': 'As an extension, I demonstrate that my identification strategy offers a new use of repeated measurements as additional sources for identification.', '1511.04162-1-10-1': 'The existing practice of the repeated measurements exploits them as instrumental variables, as in [CITATION] and [CITATION].', '1511.04162-1-10-2': 'However, when the true treatment variable is endogenous, the repeated measurements are likely to be endogenous and are not good candidates for an instrumental variable.', '1511.04162-1-10-3': 'My identification strategy shows that those variables are useful for bounding LATE in the presence of measurement error, even if the repeated measurement are not valid instrumental variables.', '1511.04162-1-10-4': 'I give a necessary and sufficient condition under which the repeated measurement strictly narrows the identified set.', '1511.04162-1-11-0': 'The remainder of the present paper is organized as follows.', '1511.04162-1-11-1': 'Subsection [REF] explains several examples motivating mismeasured endogenous treatment variables and Subsection [REF] reviews the related econometric literature.', '1511.04162-1-11-2': 'Section [REF] introduces the LATE framework with mismeasured treatment variables.', '1511.04162-1-11-3': 'Section [REF] constructs the identified set for LATE.', '1511.04162-1-11-4': 'Section [REF] proposes an inference procedure for LATE.', '1511.04162-1-11-5': 'Section [REF] conducts the Monte Carlo simulations.', '1511.04162-1-11-6': 'Section [REF] implements the inference procedure in NLS-72 to estimate the return to schooling.', '1511.04162-1-11-7': 'Section [REF] discusses how repeated measurements narrow the identified set, even if the repeated measurements themselves are not instrumental variables.', '1511.04162-1-11-8': 'Section [REF] concludes.', '1511.04162-1-11-9': 'The Appendix collects proofs and remarks.', '1511.04162-1-12-0': '## Examples for mismeasured endogenous treatment variables', '1511.04162-1-13-0': 'I introduce several examples in which binary treatment variables can be both endogenous and mismeasured at the same time.', '1511.04162-1-13-1': 'The first example is the return to schooling, in which the outcome variable is wages and the treatment variable is educational attainment, for example, whether a person has completed college or not.', '1511.04162-1-13-2': 'It is well-known that unobserved individual ability affects both the schooling decision and wage determination, which leads to the endogeneity of educational attainment in the wage equation (see, for example, [CITATION]).', '1511.04162-1-13-3': "Moreover, survey datasets record educational attainments based on the interviewee's answers and these self-reported educational attainments are subject to measurement error.", '1511.04162-1-13-4': 'Empirical papers by [CITATION], [CITATION], [CITATION],alpC2001, [CITATION] have pointed out the mismeasurement.', '1511.04162-1-13-5': 'For example, [CITATION] estimate that the 1990 Decennial Census has 17.7 false positive rate of reporting a doctoral degree.', '1511.04162-1-14-0': 'The second example is labor supply response to welfare program participation, in which the outcome variable is employment status and the treatment variable is welfare program participation.', '1511.04162-1-14-1': 'Self-reported welfare program participation in survey datasets can be mismeasured .', '1511.04162-1-14-2': "The psychological cost for welfare program participation, welfare stigma, affects job search behavior and welfare program participation simultaneously; that is, welfare stigma may discourage individuals from participating in a welfare program, and, at the same time, affect an individual's effort in the labor market (see [CITATION] and [CITATION] for a discussion on the welfare stigma).", '1511.04162-1-14-3': 'Moreover, the welfare stigma gives welfare recipients some incentive not to reveal their participation status to the survey, which causes differential measurement error in that the unobserved individual heterogeneity affects both the measurement error and the outcome.', '1511.04162-1-15-0': 'The third example is the effect of a job training program on wages .', '1511.04162-1-15-1': 'As it is similar to the return to schooling, the unobserved individual ability plays a key role in this example.', '1511.04162-1-15-2': 'Self-reported completion of job training program is also subject to measurement error .', '1511.04162-1-15-3': '[CITATION] develop a methodology for evaluating a homogeneous treatment effect with mismeasured endogenous treatment variable, and apply their methodology to evaluate the effect of a job training program on wages.', '1511.04162-1-16-0': 'The last example is the effect of maternal drug use on infant birth weight.', '1511.04162-1-16-1': '[CITATION] estimate that a mother tends to underreport her drug use, but, at the same time, she tends to report it correctly if she is a heavy user.', '1511.04162-1-16-2': 'When the degree of drug addiction is not observed, it becomes an individual unobserved heterogeneity variable which affects infant birth weight and the measurement in addition to the drug use.', '1511.04162-1-17-0': '## Literature review', '1511.04162-1-18-0': 'This paper is related to a few strands of the econometric literature.', '1511.04162-1-18-1': 'First, [CITATION], [CITATION] and [CITATION] use an instrumental variable to correct for measurement error in a binary treatment in the heterogeneous treatment effect framework and they achieve nonparametric point identification of the average treatment effect.', '1511.04162-1-18-2': 'This result assumes the true treatment variable is exogenous, whereas I allow it to be endogenous.', '1511.04162-1-19-0': 'A few papers have applied an instrumental variable to a mismeasured binary regressor in the homogenous treatment effect framework.', '1511.04162-1-19-1': 'They include [CITATION], [CITATION], [CITATION], [CITATION] and [CITATION].', '1511.04162-1-19-2': '[CITATION] is the most closely related to the present paper among them, since they consider an endogenous mismeasured regressor.', '1511.04162-1-19-3': 'In contrast, I allow for heterogeneous treatment effects.', '1511.04162-1-19-4': 'Therefore, I contribute to the heterogeneous treatment effect literature by investigating the consequences of the measurement errors in the treatment variable.', '1511.04162-1-20-0': '[CITATION], [CITATION], [CITATION], and [CITATION] apply a partial identification strategy for the average treatment effect to the mismeasured binary regressor problem by utilizing the knowledge of the marginal distribution for the true treatment.', '1511.04162-1-20-1': 'Those papers use auxiliary datasets to obtain the marginal distribution for the true treatment.', '1511.04162-1-20-2': '[CITATION] is the most closely related to the present paper, in that they allow for both treatment endogeneity and differential measurement error.', '1511.04162-1-20-3': 'My instrumental variable approach can be an an alternative strategy to deal with mismeasured endogenous treatment.', '1511.04162-1-20-4': 'It is worthwhile because, as mentioned in [CITATION], the availability of an auxiliary dataset is limited in empirical research.', '1511.04162-1-20-5': 'Furthermore, it is not always the case that the results from auxiliary datasets is transported into the primary dataset ,', '1511.04162-1-21-0': 'Some papers investigate mismeasured endogenous continuous variables, instead of binary variables.', '1511.04162-1-21-1': '[CITATION] consider nonlinear models with mismeasured continuous explanatory variables.', '1511.04162-1-21-2': 'The continuity of the treatment variable is crucial for their analysis, because they assume classical measurement error.', '1511.04162-1-21-3': 'The treatment variable in the present paper is binary and therefore the measurement error is nonclassical.', '1511.04162-1-21-4': '[CITATION] consider mismeasured endogenous continuous variables in single index models.', '1511.04162-1-21-5': 'However, their approach depends on taking derivatives of the conditional expectations with respect to the continuous variable.', '1511.04162-1-21-6': 'It is not clear if it can be extended to binary variables.', '1511.04162-1-21-7': '[CITATION] considers the semi-parametric model when endogenous continuous variables are subject to nonclassical measurement error.', '1511.04162-1-21-8': 'He assumes conditional independence between the instrumental variable and the outcome variable given the true treatment variable.', '1511.04162-1-21-9': 'In the LATE framework, this conditional independence assumption implies that LATE does not exhibit essential heterogeneity and that LATE is equal to the mean difference between the control and treatment groups.', '1511.04162-1-21-10': 'Instead I do not assume any structure on the outcome equation.', '1511.04162-1-22-0': '[CITATION] investigates the consequences of measurement error in the instrumental variable instead of the treatment variable.', '1511.04162-1-22-1': 'He assumes that the treatment variable is perfectly observed, whereas I allow for it to be measured with error.', '1511.04162-1-22-2': 'Since I assume that the instrumental variable is perfectly observed, my analysis is not overlapped with [CITATION].', '1511.04162-1-23-0': '[CITATION], [CITATION], and [CITATION] have similar identification strategy to the present paper in the context of sample selection models.', '1511.04162-1-23-1': 'These papers also use the exclusion restriction of the instrumental variable for their partial identification results.', '1511.04162-1-23-2': 'Particularly, [CITATION] derives the "integrated envelope" from the exclusion restriction, which is similar to the total variation distance in the present paper because both of them are characterized as a supremum over the set of the partitions.', '1511.04162-1-23-3': 'First and the most importantly, the present paper considers mismeasurement of the treatment variable, whereas the sample selection model considers truncation of the outcome variable.', '1511.04162-1-23-4': 'It is not straightforward to apply their methodologies in sample selection models into mismeasured treatment problem.', '1511.04162-1-23-5': 'Second, the present paper offers an inference method with uniform size control, but [CITATION] derives only point-wise size control.', '1511.04162-1-23-6': 'Last, [CITATION] and [CITATION] use their result for specification test, but I cannot use it for specification test because the sharp identified set of the present paper is always non-empty.', '1511.04162-1-24-0': '# LATE model with misclassification', '1511.04162-1-25-0': 'This section introduces measurement error in the treatment variable into the LATE framework ([CITATION], and [CITATION]).', '1511.04162-1-25-1': 'The objective is to evaluate the causal effect of a binary treatment variable [MATH] on an outcome variable [MATH], where [MATH] represents the control group and [MATH] represents the treatment group.', '1511.04162-1-25-2': 'To control for endogeneity of [MATH], the LATE framework requires a binary instrumental variable [MATH] which shifts [MATH] exogenously without any direct effect on [MATH].', '1511.04162-1-25-3': 'The treatment variable [MATH] of interest is not directly observed, and instead there is a binary measurement [MATH] for [MATH].', '1511.04162-1-25-4': 'I put the [MATH] symbol on [MATH] to emphasize that the true treatment variable [MATH] is unobserved.', '1511.04162-1-25-5': '[MATH] can be discrete, continuous or mixed; [MATH] is only required to have some dominating finite measure [MATH] on the real line.', '1511.04162-1-25-6': '[MATH] can be the Lebesgue measure or the counting measure.', '1511.04162-1-26-0': 'To describe the data generating process, I consider the counterfactual variables.', '1511.04162-1-26-1': 'Let [MATH] denote the counterfactual true treatment variable when [MATH].', '1511.04162-1-26-2': 'Let [MATH] denote the counterfactual outcome when [MATH].', '1511.04162-1-26-3': 'Let [MATH] denote the potential measured treatment variable when [MATH].', '1511.04162-1-26-4': 'The individual treatment effect is [MATH].', '1511.04162-1-26-5': 'It is not directly observed; [MATH] and [MATH] are not observed at the same time.', '1511.04162-1-26-6': 'Only [MATH] is observable.', '1511.04162-1-26-7': 'Using the notation, the observed variables [MATH] are generated by the following three equations: [EQUATION]', '1511.04162-1-26-8': 'Figure [REF] describes the above three equations graphically.', '1511.04162-1-26-9': '([REF]) is the measurement equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-1-26-10': '[MATH] is the measurement error; [MATH] (or [MATH]) represents a false positive and [MATH] (or [MATH]) represents a false negative.', '1511.04162-1-26-11': 'The next two equations ([REF]) and ([REF]) are standard in the LATE framework.', '1511.04162-1-26-12': '([REF]) is the outcome equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-1-26-13': '([REF]) is the treatment assignment equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-1-26-14': 'Correlation between [MATH] and [MATH] causes an endogeneity problem.', '1511.04162-1-27-0': 'In a return to schooling analysis, [MATH] is wages, [MATH] is the true indicator for college completion, [MATH] is the proximity to college, and [MATH] is the measurement of [MATH].', '1511.04162-1-27-1': 'The treatment effect [MATH] in the return to schooling is the effect of college attendance [MATH] on wages [MATH].', '1511.04162-1-27-2': 'The college attendance is not correctly measured in a dataset, such that only the proxy [MATH] is observed.', '1511.04162-1-28-0': 'The only assumption for my identification analysis is as follows.', '1511.04162-1-29-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-1-29-1': '(ii) [MATH] almost surely.', '1511.04162-1-30-0': 'Part (i) is the exclusion restriction and I consider stochastic independence instead of mean independence.', '1511.04162-1-30-1': 'Although it is stronger than the minimal conditions for the identification for LATE without measurement error, a large part of the existing applied papers assume stochastic independence .', '1511.04162-1-30-2': '[MATH] is also independent of [MATH] conditional on [MATH], which is the only assumption on the measurement error for the identified set in Section [REF].', '1511.04162-1-31-0': 'Part (ii) is the monotonicity condition for the instrument, in which the instrument [MATH] increases the value of [MATH] for all the individuals.', '1511.04162-1-31-1': '[CITATION] relaxes the monotonicity condition, and the following analysis of the present paper only requires the complier-defiers-for-marginals condition in [CITATION] instead of the monotonicity condition.', '1511.04162-1-31-2': 'Moreover, Part (ii) implies that the sign of the first stage regression, which is the effect of the instrumental variable on the true treatment variable, is known.', '1511.04162-1-31-3': 'It is a reasonable assumption because most empirical applications of the LATE framework assume the sign is known.', '1511.04162-1-31-4': 'For example, [CITATION] claims that the proximity-to-college instrument weakly increases the likelihood of going to a college.', '1511.04162-1-31-5': 'Last, I do not assume a relevance condition for the instrumental variable, such as [MATH].', '1511.04162-1-31-6': 'The relevance condition is a testable assumption when [MATH], but it is not testable in my analysis.', '1511.04162-1-31-7': 'I will discuss the relevance condition in my framework after Theorem [REF].', '1511.04162-1-32-0': 'As I emphasized in the introduction, the framework here does not assume anything on measurement error except for the independence from [MATH].', '1511.04162-1-32-1': 'I do not impose any restriction on the marginal distribution of the measurement error or on the relationship between the measurement error and [MATH].', '1511.04162-1-32-2': 'Particularly, the measurement error can be differential, that is, [MATH] can depend on [MATH].', '1511.04162-1-33-0': 'In this paper, I focus on the local average treatment effect (LATE), which is defined by [EQUATION].', '1511.04162-1-33-1': 'LATE is the average of the treatment effect [MATH] over the subpopulation (the compliers) whose treatment status depend on the instrument.', '1511.04162-1-33-2': '[CITATION] show that LATE equals [EQUATION] where I define [MATH] for a random variable [MATH].', '1511.04162-1-33-3': 'The present paper introduces measurement error in the treatment variable, and therefore the fraction [MATH] is not equal to the Wald estimand [EQUATION].', '1511.04162-1-33-4': 'Since [MATH] is not point identified, I cannot point identify LATE.', '1511.04162-1-33-5': 'The failure for the point identification comes purely from the measurement error, because LATE would be point identified under [MATH].', '1511.04162-1-34-0': '# Sharp identified set for LATE', '1511.04162-1-35-0': 'This section considers the partial identification problem for LATE.', '1511.04162-1-35-1': 'Before defining the sharp identified set, I express LATE as a function of the underlying distribution [MATH] of [MATH].', '1511.04162-1-35-2': 'I use the [MATH] symbol on [MATH] to clarify that [MATH] is the distribution of the unobserved variables.', '1511.04162-1-35-3': 'In the following arguments, I denote the expectation operator [MATH] by [MATH] when I need to clarify the underlying distribution.', '1511.04162-1-35-4': 'The local average treatment effect is a function of the unobserved distribution [MATH]: [EQUATION].', '1511.04162-1-36-0': 'The sharp identified set is the set of parameter values for LATE which is consistent with the distribution of the observed variables.', '1511.04162-1-36-1': 'I use [MATH] for the distribution of the observed variables [MATH] The equations ([REF]), ([REF]), and ([REF]) induce the distribution of the observables [MATH] from the unobserved distribution [MATH], and I denote by [MATH] the induced distribution.', '1511.04162-1-36-2': "When the distribution of [MATH] is [MATH], the set of [MATH] which induces [MATH] is [EQUATION] where [MATH] is the set of [MATH]'s satisfying Assumptions [REF].", '1511.04162-1-36-3': 'For every distribution [MATH] of [MATH], the sharp identified set for LATE is defined as [EQUATION].', '1511.04162-1-37-0': 'The proof of Theorem 1 in [CITATION] provides a relationship between [MATH] and LATE: [EQUATION]', '1511.04162-1-37-1': 'This equation gives the two pieces of information of [MATH].', '1511.04162-1-37-2': 'First, the sign of [MATH] is the same as [MATH].', '1511.04162-1-37-3': 'Second, the absolute value of [MATH] is at least the absolute value of [MATH].', '1511.04162-1-37-4': 'The following lemma summaries there two pieces.', '1511.04162-1-37-5': '[EQUATION].', '1511.04162-1-38-0': 'I derive a new implication from the exclusion restriction for the instrumental variable in order to obtain an upper bound on [MATH] in absolute value.', '1511.04162-1-38-1': 'To explain the new implication, I introduce the total variation distance.', '1511.04162-1-38-2': 'The total variation distance [EQUATION] is the distance between the distribution [MATH] and [MATH].', '1511.04162-1-38-3': 'In Figure [REF], the total variation distance is the half of the area for the shaded region.', '1511.04162-1-39-0': 'I use the total variation distance to evaluate the distributional effect of a binary variable, particularly the distributional effect of [MATH] on [MATH].', '1511.04162-1-39-1': 'The distributional effect of [MATH] on [MATH] reflects the dependency of [MATH] on [MATH], and I interpret the total variation distance [MATH] as the magnitude of the distributional effect.', '1511.04162-1-39-2': 'Even when the variable [MATH] is discrete, I use the density [MATH] for [MATH] to represent the probability function for [MATH].', '1511.04162-1-40-0': 'The new implication is based on the exclusion restriction imposes that the instrumental variable has direct effect on the true treatment variable [MATH] and has indirect effect on the outcome variable [MATH] and on the measured treatment variable [MATH].', '1511.04162-1-40-1': 'The new implication formalizes the idea that the magnitude of the direct effect of [MATH] on [MATH] is no smaller than the magnitude of the indirect effect of [MATH] on [MATH].', '1511.04162-1-41-0': 'Under Assumption [REF], then [EQUATION] and therefore [EQUATION].', '1511.04162-1-42-0': 'The new implication in Lemma [REF] gives a lower bound on [MATH] and therefore yields an upper bound on LATE in absolute value, combined with Eq. ([REF]).', '1511.04162-1-42-1': 'Therefore, I use these relationships to derive an upper bound on LATE in absolute value, that is, [EQUATION] as long as [MATH].', '1511.04162-1-43-0': 'The next theorem shows that the above observations characterize the sharp identified set for LATE.', '1511.04162-1-44-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-1-44-1': "(i) The sharp identified set [MATH] for LATE is included in [MATH], where [MATH] is the set of [MATH]'s which satisfies the following three inequalities.", '1511.04162-1-44-2': '[EQUATION] (ii) If [MATH] is unbounded, then [MATH] is equal to [MATH].', '1511.04162-1-45-0': 'Consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-1-45-1': 'If [MATH], [EQUATION].', '1511.04162-1-45-2': 'If [MATH], then [MATH].', '1511.04162-1-46-0': 'The total variation distance [MATH] measures the strength for the instrumental variable in my analysis, that is, [MATH] is the relevance condition in my identification analysis.', '1511.04162-1-46-1': '[MATH] means that the instrumental variable [MATH] does not affect [MATH] and [MATH], in which case [MATH] has no identifying power for the local average treatment effect.', '1511.04162-1-46-2': 'When [MATH], the interval in the above theorem is always nonempty and bounded, which implies that [MATH] has some identifying power for the local average treatment effect.', '1511.04162-1-47-0': 'The Wald estimand [MATH] can be outside the identified set.', '1511.04162-1-47-1': 'The inequality [EQUATION] holds and a strict inequality holds unless the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-1-47-2': 'It might seem counter-intuitive that the Wald estimand equals to LATE without measurement error but that it is not necessarily in the identified set [MATH] when measurement error is allowed.', '1511.04162-1-47-3': 'Recall that my framework includes no measurement error as a special case.', '1511.04162-1-47-4': 'As in [CITATION] and [CITATION], the the LATE framework has the testable implications: [EQUATION].', '1511.04162-1-47-5': 'When the data distribution does not satisfy the testable implications and there is no measurement error on the treatment variable, the identified set for LATE becomes empty and, therefore, the Wald estimand is no longer equal to LATE anymore.', '1511.04162-1-47-6': 'My framework has no testable implications, because the identified set is always non-empty.', '1511.04162-1-47-7': 'The recent papers by [CITATION], [CITATION] and [CITATION] propose the testing procedures for the testable implications.', '1511.04162-1-48-0': '# Inference', '1511.04162-1-49-0': 'Having derived what can be identified about LATE under Assumption [REF], this section considers statistical inference about LATE.', '1511.04162-1-49-1': 'I construct a confidence interval for LATE based on [MATH] in Theorem [REF].', '1511.04162-1-49-2': 'There are two difficulties with directly using Theorem [REF] for statistical inference.', '1511.04162-1-49-3': 'First, as is often the case in the partially identified models, the length of the interval is unknown ex ante, which causes uncertainty of how many moment inequalities are binding for a given value of the parameter.', '1511.04162-1-49-4': 'Second, the identified set depends on the total variation distance which involves absolute values of the data distribution.', '1511.04162-1-49-5': 'I cannot apply the delta method to derive the asymptotic distribution for the total variation distance, because of the failure of differentiability ([CITATION], [CITATION]).', '1511.04162-1-49-6': 'This non-differentiability problem remains even if the support of [MATH] is finite.', '1511.04162-1-50-0': 'I will take three steps to construct a confidence interval.', '1511.04162-1-50-1': 'In the first step, I use the supremum representation of the total variation distance to characterize the identified set [MATH] by the moment inequalities with differentiable moment functions.', '1511.04162-1-50-2': 'When the outcome variable [MATH] has a finite support, I can apply methodologies developed for a finite number of moment inequalities to construct a confidence interval for LATE.', '1511.04162-1-50-3': 'When the outcome variable [MATH] has an infinite support, however, none of the existing methods can be directly applied because the moment inequalities are not continuously indexed by a compact subset of the finite dimensional space.', '1511.04162-1-50-4': 'In the second step, therefore, I discretize the support of [MATH] to make the number of the moment inequalities to be finite in the finite sample.', '1511.04162-1-50-5': 'I let the discretization finer as the sample size goes to the infinity, such that eventually the approximation error from the discretization vanishes.', '1511.04162-1-50-6': 'The number of the moment inequalities become finite but growing, particularly diverging to the infinity when [MATH] has an infinite support.', '1511.04162-1-50-7': 'This structure resembles many moment inequalities in [CITATION].', '1511.04162-1-50-8': 'The third step is to adapt a bootstrapped critical value construction in [CITATION] to my framework.', '1511.04162-1-51-0': '## Supremum representation of the total variation distance', '1511.04162-1-52-0': 'In order to avoid the non-differentiability problem, I characterize the identified set by the moment inequalities which are differentiable with respect to the data distribution.', '1511.04162-1-53-0': 'Let [MATH] be an arbitrary data distribution of [MATH].', '1511.04162-1-54-0': 'Let [MATH] be the support for the random variable [MATH] and [MATH] be the support for [MATH].', '1511.04162-1-54-1': 'Denote by [MATH] the set of measurable functions on [MATH] taking a value in [MATH].', '1511.04162-1-54-2': 'Then [EQUATION]', '1511.04162-1-54-3': "The identified set [MATH] for LATE is the set of [MATH]'s which satisfy the following conditions [EQUATION]", '1511.04162-1-54-4': 'The number of elements [MATH] can be large; [MATH] is an infinite set when [MATH] is continuous, and [MATH] has the same elements of the power set of [MATH] when [MATH] takes only finite values.', '1511.04162-1-55-0': '## Discretizing the outcome variable', '1511.04162-1-56-0': 'To make the inference problem statistically and computationally feasible, I discretize the support for [MATH] and make the number of the moment inequalities finite.', '1511.04162-1-56-1': 'Consider a partition [MATH] over [MATH], in which [MATH] depends on [MATH], and [MATH] can grow with sample size.', '1511.04162-1-56-2': 'Let [MATH] be a generic function of [MATH] into [MATH] that is constant over [MATH] for every [MATH] and every [MATH].', '1511.04162-1-56-3': 'Let [MATH] be the set of all such functions.', '1511.04162-1-56-4': 'Note that [MATH] is a subset of [MATH].', '1511.04162-1-56-5': "Using these [MATH]'s, I consider the following set [MATH] characterized by the moment inequalities [EQUATION] where [MATH] is the number of the moment inequalities, and [EQUATION] for every [MATH].", '1511.04162-1-56-6': 'That the set [MATH] is an outer identified set.', '1511.04162-1-56-7': 'That is, it is a superset of the identified set [MATH].', '1511.04162-1-56-8': 'The next subsections consider consistency with respect to the identified set [MATH] by letting [MATH] converge to [MATH].', '1511.04162-1-56-9': 'This point is different than the usual use of the outer identified set and is similar to sieve estimation.', '1511.04162-1-56-10': 'To clarify the convergence of [MATH] to [MATH], I call [MATH] the approximated identified set.', '1511.04162-1-57-0': '## Confidence interval for LATE', '1511.04162-1-58-0': 'The approximated identified set [MATH] consists of a finite number of moments inequalities, but the number of moment inequalities depends on the sample size.', '1511.04162-1-58-1': 'As Section [REF] requires, the number of moment inequalities [MATH] needs to diverge in order to obtain the consistency for the confidence interval when [MATH] is continuous.', '1511.04162-1-58-2': 'The approximated identified set is defined to converge to the sharp identified set, so that the confidence interval in the present paper exhausts all the information in the large sample.', '1511.04162-1-58-3': 'As in Subsection [REF], the confidence interval has asymptotic power [MATH] against all the fixed alternatives outside the sharp identified set.', '1511.04162-1-59-0': 'This divergent number of the moment inequalities in the approximated identified set resembles the identified set in [CITATION], who considers testing a growing number of moment inequalities in which each moment inequalities are based on different random variables.', '1511.04162-1-59-1': 'I modify their methodology into the two sample framework where one sample is the group with [MATH] and the other sample is the group with [MATH].', '1511.04162-1-59-2': 'For simplicity, I assume that [MATH] is deterministic, which makes the notation in the following analysis simpler.', '1511.04162-1-59-3': 'The assumption of deterministic [MATH] yields two independent samples: [MATH] are the observations with [MATH] and [MATH] are the observations with [MATH].', '1511.04162-1-59-4': '[MATH] is the sample size for the observations with [MATH] and [MATH] is for [MATH].', '1511.04162-1-59-5': 'The total sample size is [MATH].', '1511.04162-1-59-6': 'I assume [MATH].', '1511.04162-1-60-0': 'In order to discuss a test statistic and a critical value, I introduce estimators for the moment functions and the estimated standard deviations for the moment functions.', '1511.04162-1-60-1': 'For an estimator for the moment function, [EQUATION] estimates the [MATH]th moment function [MATH], where [EQUATION].', '1511.04162-1-60-2': 'Denote by [MATH] the standard deviation of [MATH].', '1511.04162-1-60-3': 'The standard deviation [MATH] for [MATH] is [MATH].', '1511.04162-1-60-4': 'Denote by [MATH] the estimated standard deviation of [MATH], that is, [EQUATION] [MATH] estimates the standard deviation [MATH], that is, [EQUATION] [t] [1] For each [MATH], generate independent random variables [MATH] from [MATH].', '1511.04162-1-60-5': 'Construct the bootstrap test statistics for the moment inequality selection by [EQUATION] where [MATH] and [MATH].', '1511.04162-1-60-6': 'Construct the bootstrap critical value [MATH] for the moment inequality selection as the conditional [MATH]-quantile of [MATH] given [MATH].', '1511.04162-1-60-7': 'Select the moment inequalities and save [EQUATION]', '1511.04162-1-60-8': 'Construct the bootstrap test statistics by [EQUATION] where [MATH] if [MATH] is empty.', '1511.04162-1-60-9': 'Construct the bootstrap critical value [MATH] as the conditional [MATH]-quantile of [MATH] given [MATH].', '1511.04162-1-61-0': 'Two-step multiplier bootstrap', '1511.04162-1-62-0': 'The test statistics for [MATH] is [EQUATION] where [MATH] is a small positive number which prohibits the fraction from becoming too large when the estimated standard deviation is near zero.', '1511.04162-1-62-1': 'The truncation via [MATH] controls the effect of the approximation error from the approximated identified set on the power against local alternatives, as in Subsection [REF].', '1511.04162-1-62-2': 'The size is [MATH] and the pretest size for the moment inequality selection is [MATH].', '1511.04162-1-62-3': 'The critical value [MATH] for [MATH] is based on the two-step multiplier bootstrap , described in Algorithm [REF].', '1511.04162-1-62-4': 'The [MATH]-confidence interval for LATE is [EQUATION].', '1511.04162-1-63-0': 'Under the following three assumptions, I show that this confidence uniformly valid asymptotic size control for the confidence interval, by adapting Theorem 4.4 in [CITATION] into the two independent samples.', '1511.04162-1-64-0': '[MATH] is bounded.', '1511.04162-1-65-0': 'There are constants [MATH] and [MATH] such that [MATH] with [MATH].', '1511.04162-1-66-0': '(i) There is a constant [MATH] such that [MATH].', '1511.04162-1-66-1': '(ii) [MATH].', '1511.04162-1-67-0': 'Under Assumptions [REF] and [REF], [EQUATION] where [MATH] is the set of [MATH] such that [MATH] satisfies Assumption [REF] and [MATH].', '1511.04162-1-68-0': '## Power against fixed and local alternatives', '1511.04162-1-69-0': 'This section discusses the power properties of the confidence interval.', '1511.04162-1-69-1': 'First, I assume that the density function [MATH] satisfies the Holder continuity.', '1511.04162-1-69-2': 'This assumption justifies the approximation of the total variation distance via step functions, which is similar to the sieve estimation.', '1511.04162-1-70-0': 'The density function [MATH] is Holder continuous in [MATH] with the Holder constant [MATH] and exponent [MATH].', '1511.04162-1-71-0': 'I restrict the number of the moment inequalities and, in turn, restrict the magnitude of the critical value.', '1511.04162-1-71-1': 'Note that the number of the moment inequalities is the tuning parameter in this framework.', '1511.04162-1-71-2': 'The tradeoff is as follows: the approximation error is large if [MATH] slow, and the sampling error is large if [MATH] fast.', '1511.04162-1-72-0': '[MATH].', '1511.04162-1-73-0': 'The last condition is that the grids in [MATH] becomes finer as the sample size goes to infinity.', '1511.04162-1-74-0': 'There is a positive constant [MATH] such that [MATH] is a subset of some open ball with radius [MATH] in [MATH].', '1511.04162-1-75-0': 'I obtain the following power property against local alternatives, based on Corollary 5.1 in [CITATION].', '1511.04162-1-76-0': 'Fix [MATH] and [MATH] with [MATH].', '1511.04162-1-76-1': "Denote by [MATH] the set of local alternatives [MATH]'s satisfying Assumptions [REF], [REF] and at least one of the following inequalities: [EQUATION] where [MATH].", '1511.04162-1-76-2': 'Under Assumptions [REF] and [REF], [REF] and [REF], [EQUATION].', '1511.04162-1-77-0': 'The violation of the moment inequalities includes local alternatives in the sense that [MATH] and [MATH] go to zero in the large sample.', '1511.04162-1-78-0': '# Monte Carlo simulations', '1511.04162-1-79-0': 'This section illustrates the theoretical properties for the confidence interval in Section [REF], using simulated datasets.', '1511.04162-1-79-1': 'Consider four independent random variables [MATH] from [MATH].', '1511.04162-1-79-2': 'Using the [MATH] cumulative distribution function [MATH], I generate [MATH] in the following way: [EQUATION]', '1511.04162-1-79-3': 'I have the three parameters in the model: [MATH] represents the strength of the instrumental variable, [MATH] represents the magnitude of treatment effect, and [MATH] represents the degree of the measurement error.', '1511.04162-1-79-4': 'This is the heterogeneous treatment effect model, because [MATH] is nonlinear.', '1511.04162-1-79-5': 'I select several values for [MATH] as in Table [REF].', '1511.04162-1-79-6': 'The treatment effect is small ([MATH]) in Designs 1-4 and large ([MATH]) in Designs 5-8.', '1511.04162-1-79-7': 'The measurement error is small ([MATH]) in Designs 3,4,7,8 and large ([MATH]) in Designs 1,2,5,6.', '1511.04162-1-79-8': 'The instrumental variable is strong ([MATH]) in Designs 2,4,6,8 and weak ([MATH]) in Design 1,3,5,7.', '1511.04162-1-80-0': 'In Table [REF], I compute the three population objects: LATE, the Wald estimand, and the sharp identified set for LATE.', '1511.04162-1-80-1': 'As expected, LATE is included in the sharp identified set in all the designs.', '1511.04162-1-80-2': 'The comparison between the Wald estimand and the sharp identified set hints that the Wald estimand is relatively large compared to the upper bound of the identified set [MATH] when the measurement error has a large degree in Design 1,2,5,6.', '1511.04162-1-80-3': 'For those designs, the Wald estimand is too large to be interpreted as an upper bound on LATE, because the upper bound of the identified set [MATH] is much smaller.', '1511.04162-1-81-0': 'I choose the sample size [MATH] for the Monte Carlo simulations.', '1511.04162-1-81-1': 'Note that the numbers covers the sample size (2,909) in NLS-72.', '1511.04162-1-81-2': 'I simulate 2,000 datasets of three sample sizes.', '1511.04162-1-81-3': 'For each dataset, I construct the different confidence set with confidence size [MATH], as in Section [REF].', '1511.04162-1-81-4': 'I use the partition of equally spaced grids over [MATH] with the number the partitions [MATH].', '1511.04162-1-81-5': 'In all the confidence intervals, I use 5,000 bootstraps repetitions.', '1511.04162-1-81-6': 'Figures [REF]-[REF] describe the coverage probabilities of the confidence intervals for each parameter value.', '1511.04162-1-82-0': 'For each design, two figures are displayed.', '1511.04162-1-82-1': 'First, the left figures demonstrate the coverage probabilities according to [MATH] given [MATH].', '1511.04162-1-82-2': 'The left figures of all the designs support the consistency results in the previous section; as the sample size increases, the coverage probabilities of the confidence intervals accumulate over [MATH].', '1511.04162-1-82-3': 'Second, the right figures demonstrate the coverage probabilities according to [MATH] given [MATH].', '1511.04162-1-82-4': 'When the Wald estimand is close to the upper bound of [MATH](Design 3,4,7,8), it seems advantageous to use [MATH].', '1511.04162-1-82-5': 'It is presumably because [MATH] uses more inequalities for inference compared to [MATH] but these inequalities are not informative for LATE.', '1511.04162-1-82-6': 'When the Wald estimand is significantly larger than the upper bound of [MATH] (Design 1,2,5,6), it seems advantageous to use [MATH], particularly for the coverage probabilities near the upper bound of [MATH].', '1511.04162-1-82-7': 'In these designs, the coverage probabilities are not sensitive the choice of [MATH] or [MATH].', '1511.04162-1-83-0': '# Empirical illustrations', '1511.04162-1-84-0': 'To illustrate the theoretical results on identification and inference, this section uses the National Longitudinal Survey of the High School Class of 1972 (NLS-72) to investigate the effect on wages of attending a college when the college attendance can be mismeasured.', '1511.04162-1-84-1': '[CITATION] and [CITATION] use the same dataset to investigate the educational effect on wages in the presence of the endogeneity and the measurement error in the educational attainments.', '1511.04162-1-84-2': 'However, they do not consider the two problems and their results are dependent on the constant return to schooling.', '1511.04162-1-84-3': 'For an instrument, I follow the strategy in [CITATION] and [CITATION] closely and use the proximity to college as an instrumental variable for the college attendance.', '1511.04162-1-85-0': 'NLS-72 was conducted by the National Center for Education Statistics with the U.S. Department of Education, and it contains 22,652 seniors (as of 1972) from 1,200 schools across the U.S.', '1511.04162-1-85-1': 'The sampled individuals were asked to participate in multiple surveys from 1972 through 1986.', '1511.04162-1-85-2': 'The survey collects labor market experiences, schooling information and demographic characteristics.', '1511.04162-1-85-3': 'I drop the individuals with college degree or more, to focus on the comparison between high school graduates and the individuals with some college education.', '1511.04162-1-85-4': 'I also drop those who have missing values for wages in 1986 or educational attainments.', '1511.04162-1-85-5': 'The resulting size is 2,909.', '1511.04162-1-86-0': 'I consider the effect of the college attendance [MATH] on [MATH] (the log of wages in 1986).', '1511.04162-1-86-1': 'The treatment group with [MATH] is the individuals who have attended a college without a degree, and the control group is the individuals who have never been to a college.', '1511.04162-1-86-2': 'Some summary statistics are on Tables [REF] and [REF].', '1511.04162-1-86-3': 'I allows for the possibility that [MATH] is mismeasured, that is, the college attendance [MATH] in the dataset can be different from the truth [MATH].', '1511.04162-1-86-4': 'I define the instrumental variable [MATH] as an indicator for whether an individual grew up near 4 year college.', '1511.04162-1-86-5': 'I use 10 miles as a threshold for the proximity-to-college to similar to the strategy in [CITATION].', '1511.04162-1-87-0': 'I present inference results in Table [REF].', '1511.04162-1-87-1': 'The first row is the Wald estimate and the [MATH] confidence interval for the Wald estimand.', '1511.04162-1-87-2': 'The second row is the [MATH] confidence interval for LATE based on the identified set [MATH] in Theorem [REF].', '1511.04162-1-87-3': 'For the calculation of this confidence interval, I use the partition of equally spaced grids over [MATH] with the number of the partitions equal to [MATH].', '1511.04162-1-87-4': 'In all the confidence intervals, I use 5000 bootstrap repetitions.', '1511.04162-1-87-5': 'The results are consistent with my identification analysis in the following two points.', '1511.04162-1-87-6': 'First, the Wald estimate is too large for the effect of attending a college.', '1511.04162-1-87-7': 'For example, [CITATION] documents the existing estimates for the return to schooling and most of them fall in the range of 5-15% as the percentage increases for one additional year of education.', '1511.04162-1-87-8': 'According to my analysis, the large value of the Wald estimate can result from the mismeasurement of the college attendance.', '1511.04162-1-87-9': 'Second, when I compare the upper bounds of the confidence intervals for the Wald estimand and LATE, the upper bound ([MATH]) based on [MATH] is strictly lower than that ([MATH]) of the Wald estimand.', '1511.04162-1-87-10': 'This implies that the Wald estimator is an upper bound for LATE but it does not offer the sharp upper bound for LATE.', '1511.04162-1-87-11': 'These two findings are still valid when I consider six subgroups (Table [REF]).', '1511.04162-1-88-0': '# Identifying power of repeated measurements', '1511.04162-1-89-0': 'This section explores the identifying power of repeated measurements.', '1511.04162-1-89-1': 'Repeated measurements is a popular approach in the literature on measurement error, but they cannot be instrumental variables in this framework.', '1511.04162-1-89-2': 'This is because the true treatment variable [MATH] is endogenous and it is natural to suspect that a measurement of [MATH] is also endogenous.', '1511.04162-1-89-3': 'The more accurate the measurement is, the more likely it is to be endogenous.', '1511.04162-1-89-4': 'Nevertheless, the identification strategy of the present paper incorporates repeated measurements as an additional information to narrow the identified set for LATE, when they are coupled with the instrumental variable [MATH].', '1511.04162-1-89-5': 'Unlike the other paper on repeated measurements, I do not need to assume the independence of measurement errors among multiple measurements.', '1511.04162-1-89-6': 'The strategy of the present paper also benefits from having more than two measurements unlike [CITATION] who achieve the point identification with two measurements.', '1511.04162-1-90-0': 'Consider a second measurement [MATH] for [MATH].', '1511.04162-1-90-1': 'I do not require that [MATH] is binary, so [MATH] can be discrete or continuous.', '1511.04162-1-90-2': 'Like [MATH], I model [MATH] using the counterfactual outcome notations.', '1511.04162-1-90-3': '[MATH] is a counterfactual second measurement when the true variable [MATH] is [MATH], and [MATH] is a counterfactual second measurement when the true variable [MATH] is [MATH].', '1511.04162-1-90-4': 'Then the data generation of [MATH] is [EQUATION].', '1511.04162-1-90-5': 'I assume that the instrumental variable [MATH] is independent of [MATH] conditional on [MATH].', '1511.04162-1-91-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-1-91-1': '(ii) [MATH] almost surely.', '1511.04162-1-92-0': 'Note that I do not assume the independence between [MATH] and [MATH], where the independence between the measurement errors is a key assumption when the repeated measurement is an instrumental variable.', '1511.04162-1-93-0': 'Under this assumption, I refine the identified set for LATE as follows.', '1511.04162-1-94-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-1-94-1': "(i) The sharp identified set [MATH] for LATE is included in [MATH], where [MATH] is the set of [MATH]'s which satisfies the following three inequalities.", '1511.04162-1-94-2': '[EQUATION] (ii) If [MATH] is unbounded, then [MATH] is equal to [MATH].', '1511.04162-1-95-0': 'The total variation distance [MATH] in Theorem [REF] is weakly larger than that in Theorem [REF], which implies that the identified set in Theorem [REF] is weakly smaller than the identified set in Theorem [REF]: [EQUATION] and the strict inequality holds unless the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-1-95-1': 'Therefore, it is possible to test whether the repeated measurement [MATH] has additional information, by testing whether the sign of [MATH] is constant in [MATH].', '1511.04162-1-96-0': 'Conclusion This paper studies the identifying power of instrumental variable in the heterogeneous treatment effect framework when a binary treatment variable is mismeasured and endogenous.', '1511.04162-1-96-1': 'The assumptions in this framework are the monotonicity of the instrumental variable [MATH] on the true treatment variable [MATH] and the exogeneity of [MATH].', '1511.04162-1-96-2': 'I use the total variation distance to characterize the identified set for LATE parameter [MATH].', '1511.04162-1-96-3': 'I also provide an inference procedure for LATE.', '1511.04162-1-96-4': 'Unlike the existing literature on measurement error, the identification strategy does not reply on a specific assumption on the measurement error; the only assumption on the measurement error is its independence of the instrumental variable.', '1511.04162-1-96-5': 'I apply the new methodology to study the return to schooling in the proximity-to-college instrumental variable regression using the NLS-72 dataset.', '1511.04162-1-97-0': 'There are several directions for future research.', '1511.04162-1-97-1': 'First, the choice of the partition [MATH] in Section [REF], particularly the choice of [MATH], is an interesting direction.', '1511.04162-1-97-2': 'To the best of my knowledge, the literature on many moment inequalities has not investigated how econometricians choose the numbers of the many moment inequalities.', '1511.04162-1-97-3': 'Second, it is worthwhile to investigating the other parameter for the treatment effect.', '1511.04162-1-97-4': 'This paper has focused on the local average treatment effect (LATE) for the reasons mentioned in the introduction, but the literature on heterogeneous treatment effect has emphasized the choice of treatment effect parameter as an answer to relevant policy questions.', '1511.04162-1-98-0': '# Proofs of Lemmas [REF], [REF], and [REF]', '1511.04162-1-99-0': '[Proof of Lemma [REF]] Eq. ([REF]) implies [EQUATION] and [EQUATION] because [MATH].', '1511.04162-1-100-0': '[Proof of Lemma [REF]] I obtain [EQUATION] by applying the same logic as Theorem 1 in [CITATION]: [EQUATION]', '1511.04162-1-100-1': 'This implies [EQUATION] where the last inequality follows because the total variation distance is at most one.', '1511.04162-1-100-2': 'Moreover, since [MATH] almost surely, [EQUATION] [Proof of Lemma [REF]] The lemma follows from [EQUATION] where the maximization is achieved if [MATH] if [MATH] and [MATH] if [MATH].', '1511.04162-1-101-0': '# Proofs of Theorems [REF] and [REF]', '1511.04162-1-102-0': 'Theorem [REF] is a special case of Theorem [REF] with [MATH] being constant, and therefore I demonstrate the proof only for Theorem [REF].', '1511.04162-1-102-1': 'Lemma [REF] is modified into the following lemma in the framework of Theorem [REF].', '1511.04162-1-103-0': 'Under Assumption [REF], then [EQUATION].', '1511.04162-1-104-0': 'The proof is the same as Lemma [REF] and this lemma follows from [EQUATION].', '1511.04162-1-105-0': 'From Lemmas [REF] and [REF], all the three inequalities in Theorem [REF] are satisfied when [MATH] is the true value for LATE, which is the first part of Theorem [REF].', '1511.04162-1-105-1': 'To prove Theorem [REF], I am going to show the sharpness of the three inequalities, that is, that any point satisfying the three inequalities is generated by some data generating process [MATH] which is consistent with the data distribution [MATH].', '1511.04162-1-105-2': 'I will consider two cases based on the value of [MATH].', '1511.04162-1-106-0': '## Case 1: Zero total variation distance', '1511.04162-1-107-0': 'Consider [MATH].', '1511.04162-1-107-1': 'In this case, [MATH] almost everywhere over [MATH] and particularly [MATH].', '1511.04162-1-107-2': 'Note that all the three inequalities in Theorem [REF] have no restriction on [MATH] in this case.', '1511.04162-1-107-3': 'For every [MATH], consider the following two data generating processes.', '1511.04162-1-107-4': 'First, [MATH] is defined by [EQUATION] where [MATH] and [MATH] are conditionally independent of [MATH].', '1511.04162-1-107-5': 'Second, [MATH] is defined by [EQUATION] where [MATH] and [MATH] are conditionally independent of [MATH].', '1511.04162-1-108-0': 'Consider the assumptions in Theorem [REF].', '1511.04162-1-108-1': 'If [MATH], then, for every [MATH] and every [MATH],', '1511.04162-1-109-0': 'the mixture distribution [MATH] satisfies Assumption [REF]; the mixture distribution [MATH] generates the data distribution [MATH]; under the mixture distribution [MATH], LATE is equal to [MATH].', '1511.04162-1-110-0': '(1) Both [MATH] and [MATH] satisfy the independence between [MATH] and [MATH] for each [MATH].', '1511.04162-1-110-1': 'Furthermore, [MATH] and [MATH] have the same marginal distribution for [MATH]: [MATH].', '1511.04162-1-110-2': 'Therefore, the mixture of [MATH] and [MATH] also satisfies the independence.', '1511.04162-1-110-3': 'Since the mixture of [MATH] and [MATH] satisfies [MATH] almost surely, the first part of this lemma is established.', '1511.04162-1-111-0': '(2) The second part follows from the fact that both [MATH] and [MATH] generate the data distribution [MATH].', '1511.04162-1-111-1': 'Since the proof is essentially the same for [MATH] and [MATH], I demonstrate it only for [MATH].', '1511.04162-1-111-2': 'Denote by [MATH] the density function of [MATH].', '1511.04162-1-111-3': 'Then [EQUATION] where the last equality uses [MATH] implies [MATH].', '1511.04162-1-112-0': '(3) LATE under [MATH] is [EQUATION] and LATE under [MATH] is [EQUATION]', '1511.04162-1-112-1': 'They imply that LATE under the mixture distribution is equal to [MATH].', '1511.04162-1-113-0': 'Now I will prove Theorem [REF] for Case 1.', '1511.04162-1-113-1': 'Let [MATH] be any real number.', '1511.04162-1-113-2': 'Since [MATH] is unbounded, there are [MATH] and [MATH] with [MATH] such that [MATH] and [MATH].', '1511.04162-1-113-3': 'Since [EQUATION] there is [MATH] such that [EQUATION].', '1511.04162-1-113-4': 'Using Lemma [REF], the right hand side of the above equation is LATE under the mixture distribution [MATH].', '1511.04162-1-113-5': 'This proves that [MATH] is LATE under some data generating process which is consistent with the observed distribution [MATH].', '1511.04162-1-114-0': '## Case 2: Positive total variation distance', '1511.04162-1-115-0': 'Consider [MATH].', '1511.04162-1-115-1': 'This means that the instrumental variable [MATH] has non-zero indirect effect on [MATH].', '1511.04162-1-115-2': 'Consider the following two data generating processes.', '1511.04162-1-115-3': 'First, [MATH] is defined by [EQUATION] where [MATH] and [MATH] are conditionally independent of [MATH].', '1511.04162-1-115-4': 'Second, [MATH] defined as follows.', '1511.04162-1-115-5': 'Define [EQUATION] and define [MATH] as [EQUATION] where [MATH] and [MATH] are conditionally independent of [MATH].', '1511.04162-1-116-0': 'Consider the assumptions in Theorem [REF].', '1511.04162-1-116-1': 'If [MATH], then', '1511.04162-1-117-0': '[MATH] generates the data distribution [MATH] and LATE under [MATH] is equal to [MATH]; and [MATH] generates the data distribution [MATH] and LATE under [MATH] is equal to [EQUATION] (1) Denote by [MATH] the density function of [MATH].', '1511.04162-1-117-1': '[MATH] generates the data distribution [MATH]: [EQUATION] where the first equality uses [MATH].', '1511.04162-1-117-2': 'Under [MATH], LATE is equal to [MATH]: [EQUATION] (2) Denote by [MATH] the density function of [MATH].', '1511.04162-1-117-3': 'When [MATH], [MATH] is positive on : [EQUATION] [MATH] generates the data distribution [MATH]: [EQUATION]', '1511.04162-1-117-4': 'Under [MATH], LATE is equal to [MATH]: [EQUATION]', '1511.04162-1-117-5': 'Consider the assumptions in Theorem [REF].', '1511.04162-1-117-6': 'If [MATH], then, for every [MATH],', '1511.04162-1-118-0': 'the mixture distribution [MATH] satisfies Assumption [REF]; the mixture distribution [MATH] generates the data distribution [MATH]; under the mixture distribution [MATH], LATE is equal to [EQUATION].', '1511.04162-1-119-0': 'The proof for this lemma is the same as the proof in Lemma [REF].', '1511.04162-1-120-0': 'Now I will prove Theorem [REF] for Case 2.', '1511.04162-1-120-1': 'Let [MATH] be any real number satisfying all the three inequalities in Theorem [REF].', '1511.04162-1-120-2': 'Then there is [MATH] such that [EQUATION].', '1511.04162-1-120-3': 'Using Lemma [REF], the right hand side of the above equation is LATE under the mixture distribution [MATH].', '1511.04162-1-120-4': 'This proves that [MATH] is LATE under some data generating process which is consistent with the observed distribution [MATH].', '1511.04162-1-121-0': '# Proofs of Theorems [REF] and [REF]', '1511.04162-1-122-0': 'These results are obtained by applying the results in [CITATION] for two independent samples.', '1511.04162-1-122-1': 'I use the same notation as them and modify their proofs to the two samples.', '1511.04162-1-122-2': 'First, I introduce the following lemma.', '1511.04162-1-123-0': '(1) If [MATH] are [MATH]-dimensional random variables and [MATH], then [EQUATION] (2) If [MATH] are [MATH]-dimensional random variables and if [MATH] and [MATH] are independent, then [EQUATION] (3) If [MATH] are [MATH]-dimensional random variables and [MATH], then [EQUATION].', '1511.04162-1-124-0': 'The first statement is as follows.', '1511.04162-1-124-1': 'If [MATH], then [EQUATION].', '1511.04162-1-124-2': 'If [MATH], then [EQUATION].', '1511.04162-1-125-0': 'The second statement is as follows.', '1511.04162-1-125-1': '[EQUATION]', '1511.04162-1-125-2': 'The third statement is as follows.', '1511.04162-1-125-3': 'If [MATH], then [EQUATION]', '1511.04162-1-125-4': 'If [MATH], then [EQUATION]', '1511.04162-1-125-5': 'If [MATH], then [EQUATION]', '1511.04162-1-125-6': 'If [MATH], then [EQUATION]', '1511.04162-1-125-7': 'Now I modify [CITATION] for two independent sample.', '1511.04162-1-125-8': 'In order to simplify the notations, this section focuses on a fixed value of [MATH] and omits [MATH] in the following discussion.', '1511.04162-1-125-9': 'All the following results are uniformly valid in [MATH].', '1511.04162-1-125-10': 'Denote [MATH], [MATH] and [MATH].', '1511.04162-1-125-11': 'For every [MATH], define [EQUATION]', '1511.04162-1-125-12': 'Denote by [MATH] the conditional [MATH]-quantile of [MATH] given [MATH].', '1511.04162-1-126-0': 'Denote by [MATH] the variance-covariance matrix of [MATH].', '1511.04162-1-126-1': '[MATH] is the variance-covariance matrix of [MATH].', '1511.04162-1-126-2': 'Denote by [EQUATION] where [MATH] is the element-wise maximum.', '1511.04162-1-126-3': 'Let [MATH] and [MATH] be [MATH]-dimensional independent normal random variable with [EQUATION].', '1511.04162-1-126-4': 'Denote by [MATH] the [MATH] quantile of [MATH].', '1511.04162-1-126-5': 'Define [EQUATION]', '1511.04162-1-126-6': 'Note that [MATH] is a [MATH]-dimensional vector in the definitions of [MATH], and therefore I need to use the central limit and bootstrap theorems for hyper-rectangles, which is slightly different from [CITATION].', '1511.04162-1-127-0': 'Note the following statements are taken from [CITATION] and [CITATION].', '1511.04162-1-128-0': '(1) There are positive numbers [MATH] and [MATH] such that [EQUATION] (2) The following inequalities hold: [EQUATION]', '1511.04162-1-128-1': 'Note that [EQUATION].', '1511.04162-1-128-2': 'By Assumptions [REF], [REF], and [REF], [EQUATION] where [EQUATION]', '1511.04162-1-128-3': 'This is the key assumption in [CITATION] and therefore we can borrow their results.', '1511.04162-1-128-4': 'The first statement is Theorem 2.1 in [CITATION].', '1511.04162-1-128-5': 'The second is Corollary 4.3 in [CITATION].', '1511.04162-1-128-6': 'The third and fourth are Step 3 of Theorem 4.3.', '1511.04162-1-128-7': 'The last two statements are Lemma A.4 in [CITATION].', '1511.04162-1-129-0': 'Lemma [REF] yields the following lemma in the two sample setting.', '1511.04162-1-129-1': '[EQUATION] where [MATH].', '1511.04162-1-130-0': 'Using the inequality in Lemma [REF] (2), [MATH] and [MATH], which implies [MATH] and [MATH].', '1511.04162-1-131-0': 'Using the inequality [MATH] for [MATH], [EQUATION]', '1511.04162-1-131-1': 'Applying the triangle inequality and [MATH] [EQUATION]', '1511.04162-1-131-2': 'Applying the triangle inequality [EQUATION]', '1511.04162-1-131-3': 'If [MATH] satisfies at least one of the inequalities in Theorem [REF], then [EQUATION].', '1511.04162-1-132-0': 'When one of the first two inequalities in Theorem [REF] holds, this lemma is immediate.', '1511.04162-1-132-1': 'I will show the case when the last inequality holds.', '1511.04162-1-132-2': 'By Assumptions [REF] and [REF], [EQUATION].', '1511.04162-1-132-3': 'Define [MATH] and then [EQUATION] for every [MATH] and [MATH].', '1511.04162-1-132-4': 'Define [EQUATION]', '1511.04162-1-132-5': 'If [MATH] for some [MATH] with [MATH], then [MATH] is constant on [MATH], and therefore [MATH] is constant on [MATH].', '1511.04162-1-132-6': 'Then, on every [MATH], either [MATH] or [MATH].', '1511.04162-1-132-7': 'It implies [EQUATION] and [EQUATION]', '1511.04162-1-132-8': 'Then [EQUATION]', '1511.04162-1-132-9': 'By the last inequality ([REF]) in Theorem [REF], [EQUATION].', '1511.04162-1-133-0': 'Define [EQUATION]', '1511.04162-1-133-1': 'Since [EQUATION]', '1511.04162-1-133-2': 'Lemma [REF] implies [EQUATION]', '1511.04162-1-133-3': 'By Lemma [REF], I have [EQUATION].', '1511.04162-1-133-4': 'Since Lemma [REF] implies [EQUATION]', '1511.04162-1-133-5': 'Markov inequality implies [EQUATION]', '1511.04162-1-133-6': 'Lemma [REF] implies [EQUATION]', '1511.04162-1-133-7': 'Define [EQUATION]', '1511.04162-1-133-8': 'By Lemma [REF], I have [EQUATION] and therefore [EQUATION].', '1511.04162-1-133-9': 'If [MATH] and [MATH], then [EQUATION] and, at [MATH], [EQUATION].', '1511.04162-1-133-10': 'Therefore [EQUATION] if [MATH] and [MATH].', '1511.04162-1-133-11': 'By Lemmas [REF] and [REF], this lemma is established.', '1511.04162-1-134-0': 'Define [MATH] and I obtain the following lemma.', '1511.04162-1-134-1': '[EQUATION]', '1511.04162-1-134-2': 'Define [MATH] and consider [MATH].', '1511.04162-1-134-3': 'Note that [MATH] implies [EQUATION].', '1511.04162-1-134-4': 'If [MATH] and [MATH], then [MATH] implies that [EQUATION].', '1511.04162-1-134-5': 'Therefore [EQUATION]', '1511.04162-1-134-6': 'By Lemma [REF], [EQUATION] and then [EQUATION]', '1511.04162-1-134-7': 'By Lemmas [REF] and [REF], this lemma is established.', '1511.04162-1-135-0': '## Proof of Theorem [REF]', '1511.04162-1-136-0': 'First, I show [EQUATION] for every [MATH].', '1511.04162-1-136-1': 'Since [MATH], [EQUATION]', '1511.04162-1-136-2': 'Using Lemma [REF], [EQUATION]', '1511.04162-1-136-3': 'Next, I show [EQUATION]', '1511.04162-1-136-4': 'Since [MATH] implies [MATH], it follows that [EQUATION]', '1511.04162-1-136-5': 'Last, I show that the statement of this theorem [MATH].', '1511.04162-1-136-6': 'If the following three statements are true [EQUATION] then I have [EQUATION]', '1511.04162-1-136-7': 'By Lemmas [REF] and Eq. ([REF]) and ([REF]), I have [EQUATION]', '1511.04162-1-136-8': 'Except for [MATH], all the terms on the right-hand side converges to [MATH] uniformly over [MATH].', '1511.04162-1-137-0': '## Proof of Theorem [REF]', '1511.04162-1-138-0': 'Denote by [MATH].', '1511.04162-1-138-1': 'If the following four statements are true [EQUATION] then [MATH], because [EQUATION] ([REF]) holds at least with probability [MATH] from Lemma [REF].', '1511.04162-1-138-2': '([REF]) hold at least with probability approaching to one because, using the Markov inequality, [EQUATION] ([REF]) hold by Lemma [REF] and ([REF]) holds by Lemma [REF].', '1511.04162-1-138-3': 'Therefore [EQUATION].', '1511.04162-1-138-4': 'Since the right-hand side of the above equation does not depend on [MATH], the uniform convergence in Theorem [REF] follows.'} | {'1511.04162-2-0-0': 'This paper studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous.', '1511.04162-2-0-1': 'I characterize the sharp identified set for the local average treatment effect under the exclusion restriction of an instrument and the deterministic monotonicity of the true treatment in the instrument.', '1511.04162-2-0-2': 'Even allowing for general measurement error (e.g., the measurement error is nonclassical and endogenous), it is still possible to obtain finite bounds on the local average treatment effect.', '1511.04162-2-0-3': 'Notably, the Wald estimand is an upper bound on the local average treatment effect, but it is not the sharp bound in general.', '1511.04162-2-0-4': 'I also provide a confidence interval for the local average treatment effect with uniformly asymptotically valid size control.', '1511.04162-2-0-5': 'Furthermore, I demonstrate that the identification strategy of this paper offers a new use of repeated measurements for tightening the identified set.', '1511.04162-2-0-6': 'Keywords: Misclassification; Local average treatment effect; Endogenous measurement error; Instrumental variable; Partial identification', '1511.04162-2-1-0': '# Introduction', '1511.04162-2-2-0': 'Treatment effect analyses often entail a measurement error problem as well as an endogeneity problem.', '1511.04162-2-2-1': 'For example, [CITATION] document a substantial measurement error in educational attainments in the 1990 U.S. Census.', '1511.04162-2-2-2': 'At the same time, educational attainments are endogenous treatment variables in a return to schooling analysis, because unobserved individual ability affects both schooling decisions and wages .', '1511.04162-2-2-3': 'The econometric literature, however, has offered only a few solutions for addressing the two problems at the same time.', '1511.04162-2-2-4': 'Although an instrumental variable is a standard technique for correcting endogeneity and measurement error , no paper has investigated the identifying power of an instrumental variable for the heterogeneous treatment effect when the treatment is both mismeasured and endogenous.', '1511.04162-2-3-0': 'I consider a mismeasured treatment in the framework of [CITATION] and [CITATION], and focus on the local average treatment effect as a parameter of interest.', '1511.04162-2-3-1': 'My analysis studies the identifying power of an instrumental variable under the following two assumptions: (i) the instrument affects the outcome and the measured treatment only through the true treatment (the exclusion restriction of an instrument), and (ii) the instrument weakly increases the true treatment (the deterministic monotonicity of the true treatment in the instrument).', '1511.04162-2-3-2': 'These assumptions are an extension of [CITATION] and [CITATION] into the framework with mismeasured treatment.', '1511.04162-2-3-3': 'The local average treatment effect is the average treatment effect for the compliers, that is, the subpopulation whose true treatment status is strictly affected by an instrument.', '1511.04162-2-3-4': 'Focusing on the local average treatment effect is meaningful for a few reasons.', '1511.04162-2-3-5': 'First, the local average treatment effect has been a widely used parameter to investigate the heterogeneous treatment effect with endogeneity.', '1511.04162-2-3-6': 'My analysis offers a tool for a robustness check to those who have already investigated the local average treatment effect.', '1511.04162-2-3-7': 'Second, the local average treatment effect can be used to extrapolate to the average treatment effect or other parameters of interest.', '1511.04162-2-3-8': '[CITATION] emphasize the utility of reporting the local average treatment effect in addition to the other parameters of interest, because the extrapolation often requires additional assumptions and can be less credible than the local average treatment effect.', '1511.04162-2-4-0': 'The mismeasured treatment prevents the local average treatment effect from being point-identified.', '1511.04162-2-4-1': 'As in [CITATION] and [CITATION], the local average treatment effect is the ratio of the intention-to-treat effect over the size of compliers.', '1511.04162-2-4-2': 'Since the measured treatment is not the true treatment, however, the size of compliers is not identified and therefore the local average treatment effect is not identified.', '1511.04162-2-4-3': 'The under-identification for the local average treatment effect is a consequence of the under-identification for the size of compliers; if I assumed no measurement error, I could compute the size of compliers based on the measured treatment and therefore the local average treatment effect would be the Wald estimand.', '1511.04162-2-5-0': 'I take a worst case scenario approach with respect to the measurement error and allow for a general form of measurement error.', '1511.04162-2-5-1': 'The only assumption concerning the measurement error is its independence of the instrumental variable.', '1511.04162-2-5-2': '(Section [REF] dispenses with this assumption and shows that it is still possible to bound the local average treatment effect.)', '1511.04162-2-5-3': 'I allow for the following types of measurement error.', '1511.04162-2-5-4': 'First, the measurement error is nonclassical; that is, it can be dependent on the true treatment.', '1511.04162-2-5-5': 'The measurement error for a binary variable is always nonclassical.', '1511.04162-2-5-6': 'It is because the measurement error cannot be negative (positive) when the true variable takes the low (high) value.', '1511.04162-2-5-7': 'Second, I allow the measurement error to be endogenous (or differential); that is, the measured treatment can be dependent on the outcome conditional on the true treatment.', '1511.04162-2-5-8': 'For example, as [CITATION] argue, the measurement error for educational attainment depends on the familiarity with the educational system in the U.S., and immigrants may have a higher rate of measurement error.', '1511.04162-2-5-9': 'At the same time, the familiarity with the U.S. educational system can be related to the English language skills, which can affect the labor market outcomes.', '1511.04162-2-5-10': '[CITATION] also argue that measurement error is likely to be endogenous in some empirical applications.', '1511.04162-2-5-11': 'Third, there is no assumption concerning the marginal distribution of the measurement error.', '1511.04162-2-5-12': 'It is not necessary to assume anything about the accuracy of the measurement.', '1511.04162-2-6-0': 'In the presence of measurement error, I derive the identified set for the local average treatment effect (Theorem [REF]).', '1511.04162-2-7-0': 'Figure [REF] describes the relationship among the identified set for the local average treatment effect, the intention-to-treat effect, and the Wald estimand.', '1511.04162-2-7-1': 'First, the intention-to-treat effect has the same sign as the local average treatment effect.', '1511.04162-2-7-2': 'This is why Figure [REF] has three subfigures according to the sign of the intention-to-treat effect: (a) positive, (b) zero, and (c) negative.', '1511.04162-2-7-3': 'Second, the intention-to-treat effect is the sharp lower bound on the local average treatment effect in absolute value.', '1511.04162-2-7-4': 'Third, the Wald estimand is an upper bound on the local average treatment effect in absolute value.', '1511.04162-2-7-5': 'The Wald estimand is the probability limit of the instrumental variable estimator in my framework, which ignores the measurement error but controls only for the endogeneity.', '1511.04162-2-7-6': 'This point implies that an upper bound on the local average treatment effect is obtained by ignoring the measurement error.', '1511.04162-2-7-7': '[CITATION] obtain a similar result in the homogeneous treatment effect model.', '1511.04162-2-7-8': 'Last, but most importantly, the sharp upper bound in absolute value can be smaller than the Wald estimand.', '1511.04162-2-7-9': 'It warns empirical researchers about ignoring the measurement error.', '1511.04162-2-7-10': 'Even for analyzing only an upper bound on the local average treatment effect, it is recommended to take the measurement error into account, which can yield a smaller upper bound than the Wald estimand.', '1511.04162-2-7-11': 'Section [REF] investigates when the Wald estimand coincide with the sharp upper bound.', '1511.04162-2-8-0': 'I construct a confidence interval for the local average treatment effect.', '1511.04162-2-8-1': 'To construct the confidence interval, first, I approximate the identified set by discretizing the support of the outcome where the discretization becomes finer as the sample size increases.', '1511.04162-2-8-2': 'The approximation for the identified set resembles many moment inequalities in [CITATION] and [CITATION], who consider a finite but divergent number of moment inequalities.', '1511.04162-2-8-3': 'I apply a bootstrap method in [CITATION] to construct a confidence interval with uniformly asymptotically valid asymptotic size control.', '1511.04162-2-8-4': 'The confidence interval also rejects parameter value which does not belong to the sharp identified set.', '1511.04162-2-8-5': 'Monte Carlo simulations support for those asymptotic results.', '1511.04162-2-9-0': 'As an extension, I dispense with the independence between the instrument and the measurement error.', '1511.04162-2-9-1': 'In this case, there is no assumption on the measurement error, and therefore the measured treatment has no information on the local average treatment effect.', '1511.04162-2-9-2': 'Even without any information from the measured treatment, however, I can still apply the same identification strategy and obtain finite (but less tight) bounds on the local average treatment effect.', '1511.04162-2-10-0': 'Moreover, I offer a new use of repeated measurements as additional sources for identification.', '1511.04162-2-10-1': 'The existing practice of repeated measurements uses one of them as an instrumental variable, as in [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION] However, when the true treatment is endogenous, the repeated measurements are likely to be endogenous and are not good candidates for an instrumental variable.', '1511.04162-2-10-2': 'My identification strategy demonstrates that those variables are useful for bounding the local average treatment effect in the presence of measurement error, even if any of the repeated measurement are not valid instrumental variables.', '1511.04162-2-11-0': 'The remainder of this paper is organized as follows.', '1511.04162-2-11-1': 'Section [REF] explains several empirical examples motivating mismeasured endogenous treatments and Section [REF] reviews the related econometric literature.', '1511.04162-2-11-2': 'Section [REF] introduces mismeasured treatments in the framework of [CITATION] and [CITATION].', '1511.04162-2-11-3': 'Section [REF] constructs the identified set for the local average treatment effect.', '1511.04162-2-11-4': 'I also discuss two extensions.', '1511.04162-2-11-5': 'One extension describes how repeated measurements tighten the identified set even if I cannot use any of the repeated measurements as an instrumental variable, and the other dispenses with independence between the instrument and the measurement error.', '1511.04162-2-11-6': 'Section [REF] proposes an inference procedure for the local average treatment effect.', '1511.04162-2-11-7': 'Section [REF] conducts the Monte Carlo simulations.', '1511.04162-2-11-8': 'Section [REF] concludes.', '1511.04162-2-11-9': 'Appendix collects proofs and remarks.', '1511.04162-2-12-0': '## Examples for mismeasured endogenous treatments', '1511.04162-2-13-0': 'I introduce several empirical examples in which binary treatments can be both endogenous and mismeasured at the same time.', '1511.04162-2-13-1': 'The first example is the return to schooling, in which the outcome is wages and the treatment is educational attainment, for example, whether a person has completed college or not.', '1511.04162-2-13-2': 'Unobserved individual ability affects both the schooling decision and wage determination, which leads to the endogeneity of educational attainment in the wage equation (see, for example, [CITATION]).', '1511.04162-2-13-3': "Moreover, survey datasets record educational attainments based on the interviewee's answers, and these self-reported educational attainments are subject to measurement error.", '1511.04162-2-13-4': '[CITATION], [CITATION], [CITATION],alpcard:2001, [CITATION] have pointed out the mismeasurement of educational attainments.', '1511.04162-2-13-5': 'For example, [CITATION] estimate that the 1990 Decennial Census has 17.7 false positive rate of reporting a doctoral degree.', '1511.04162-2-14-0': 'The second example is labor supply response to welfare program participation, in which the outcome is employment status and the treatment is welfare program participation.', '1511.04162-2-14-1': 'Self-reported welfare program participation in survey datasets can be mismeasured .', '1511.04162-2-14-2': "The psychological cost for welfare program participation, welfare stigma, affects job search behavior and welfare program participation simultaneously; that is, welfare stigma may discourage individuals from participating in a welfare program, and, at the same time, affect an individual's effort in the labor market (see [CITATION] and [CITATION] for a discussion on the welfare stigma).", '1511.04162-2-14-3': 'Moreover, the welfare stigma gives welfare recipients some incentive not to reveal their participation status to the survey, which causes endogenous measurement error in that the unobserved individual heterogeneity affects both the measurement error and the outcome.', '1511.04162-2-15-0': 'The third example is the effect of a job training program on wages.', '1511.04162-2-15-1': 'As it is similar to the return to schooling, unobserved individual ability plays a key role in this example.', '1511.04162-2-15-2': 'Self-reported completion of job training program is also subject to measurement error .', '1511.04162-2-15-3': '[CITATION] develop a methodology for evaluating a homogeneous treatment effect with mismeasured endogenous treatment, and apply their methodology to evaluate the effect of a job training program on wages.', '1511.04162-2-16-0': 'The last example is the effect of maternal drug use on infant birth weight.', '1511.04162-2-16-1': '[CITATION] estimate that a mother tends to underreport her drug use, but, at the same time, she tends to report it correctly if she is a heavy user.', '1511.04162-2-16-2': 'When the degree of drug addiction is not observed, it becomes an individual unobserved heterogeneity which affects infant birth weight and the measurement in addition to the drug use.', '1511.04162-2-17-0': '## Literature review', '1511.04162-2-18-0': 'Here I summarize the related econometric literature.', '1511.04162-2-18-1': 'First, [CITATION], [CITATION], and [CITATION] use an instrumental variable to correct for measurement error in a binary (or discrete) treatment in the heterogeneous treatment effect framework and they achieve nonparametric point identification of the average treatment effect.', '1511.04162-2-18-2': 'They assume that the true treatment is exogenous, whereas I allow it to be endogenous.', '1511.04162-2-19-0': 'Finite mixture models are related to my analysis.', '1511.04162-2-19-1': 'I consider the unobserved binary treatment, whereas finite mixture models deal with unobserved type.', '1511.04162-2-19-2': '[CITATION] and [CITATION] are the most closely related.', '1511.04162-2-19-3': 'They investigate the identification problem in finite mixture models, by using the exclusion restriction in which an instrumental variable only affects the mixing distribution of a type without affecting the component distribution (that is, the conditional distribution given the type).', '1511.04162-2-19-4': 'If I applied their approach directly to my framework, their exclusion restriction would imply conditional independence between the instrumental variable and the outcome given the true treatment.', '1511.04162-2-19-5': 'This conditional independence implies that the local average treatment effect does not exhibit essential heterogeneity and that the local average treatment effect is the mean difference between the control and treatment groups.', '1511.04162-2-19-6': 'Instead of applying the approaches in [CITATION] and [CITATION], I use a different exclusion restriction in which the instrumental variable does not affect the outcome or the measured treatment directly.', '1511.04162-2-20-0': 'A few papers have applied an instrumental variable to a mismeasured binary regressor in the homogenous treatment effect framework.', '1511.04162-2-20-1': 'They include [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1511.04162-2-20-2': '[CITATION] and [CITATION] are the most closely related among them, since they allow for endogeneity.', '1511.04162-2-20-3': 'Here I allow for heterogeneous treatment effects, and I contribute to the heterogeneous treatment effect literature by investigating the consequences of the measurement errors in the treatment.', '1511.04162-2-21-0': '[CITATION], [CITATION], [CITATION], and [CITATION] apply a partial identification strategy for the average treatment effect to the mismeasured binary regressor problem by utilizing the knowledge of the marginal distribution for the true treatment.', '1511.04162-2-21-1': 'Those papers use auxiliary datasets to obtain the marginal distribution for the true treatment.', '1511.04162-2-21-2': '[CITATION] is the most closely related, in that they allow for both treatment endogeneity and endogenous measurement error.', '1511.04162-2-21-3': 'My instrumental variable approach can be an an alternative strategy to deal with mismeasured endogenous treatment.', '1511.04162-2-21-4': 'It is worthwhile because, as mentioned in [CITATION], the availability of an auxiliary dataset is limited in empirical research.', '1511.04162-2-21-5': 'Furthermore, it is not always the case that the results from auxiliary datasets is transported into the primary dataset ,', '1511.04162-2-22-0': 'Some papers investigate mismeasured endogenous continuous variables, instead of binary variables.', '1511.04162-2-22-1': '[CITATION] consider nonlinear models with mismeasured continuous explanatory variables.', '1511.04162-2-22-2': 'The continuity of the treatment is crucial for their analysis, because they assume classical measurement error.', '1511.04162-2-22-3': 'The treatment in my analysis is binary and therefore the measurement error is nonclassical.', '1511.04162-2-22-4': '[CITATION] consider mismeasured endogenous continuous variables in single index models.', '1511.04162-2-22-5': 'However, their approach depends on taking derivatives of the conditional expectations with respect to the continuous variable.', '1511.04162-2-22-6': 'It is not clear if it can be extended to binary variables.', '1511.04162-2-22-7': '[CITATION] considers the semi-parametric model when endogenous continuous variables are subject to nonclassical measurement error.', '1511.04162-2-22-8': 'He assumes conditional independence between the instrumental variable and the outcome given the true treatment, which would impose some structure on the outcome equation when a treatment is binary (see Footnote [REF]).', '1511.04162-2-22-9': 'Instead I propose an identification strategy without assuming any structure on the outcome equation.', '1511.04162-2-23-0': '[CITATION] investigates the consequences of measurement error in the instrumental variable instead of the treatment.', '1511.04162-2-23-1': 'He assumes that the treatment is perfectly observed, whereas I allow for it to be measured with error.', '1511.04162-2-23-2': 'Since I assume that the instrumental variable is perfectly observed, my analysis is not overlapped with [CITATION].', '1511.04162-2-24-0': '[CITATION], [CITATION], and [CITATION] have similar identification strategy in the context of sample selection models.', '1511.04162-2-24-1': 'These papers also use the exclusion restriction of the instrumental variable for their partial identification results.', '1511.04162-2-24-2': 'Particularly, [CITATION] derives the integrated envelope from the exclusion restriction, which is similar to the total variation distance in my analysis because both of them are characterized as a supremum over the set of the partitions.', '1511.04162-2-24-3': 'First and the most importantly, I consider mismeasurement of the treatment, whereas the sample selection model considers truncation of the outcome.', '1511.04162-2-24-4': 'It is not straightforward to apply their methodologies in sample selection models into mismeasured treatment problem.', '1511.04162-2-24-5': 'Second, I offer an inference method with uniform size control, but [CITATION] derives only point-wise size control.', '1511.04162-2-24-6': 'Last, [CITATION] and [CITATION] use their result for specification test, but I cannot use it for specification test because the sharp identified set of my analysis is always non-empty.', '1511.04162-2-25-0': '# Local average treatment effect framework with misclassification', '1511.04162-2-26-0': 'My analysis considers a mismeasured treatment in the framework of [CITATION] and [CITATION].', '1511.04162-2-26-1': 'The objective is to evaluate the causal effect of a binary treatment [MATH] on an outcome [MATH], where [MATH] represents the control group and [MATH] represents the treatment group.', '1511.04162-2-26-2': 'To deal with endogeneity of [MATH], a binary instrumental variable [MATH] needs to shift [MATH] exogenously without any direct effect on [MATH].', '1511.04162-2-26-3': 'The treatment [MATH] of interest is not directly observed, and instead there is a binary measurement [MATH] for [MATH].', '1511.04162-2-26-4': 'I put the [MATH] symbol on [MATH] to emphasize that the true treatment [MATH] is unobserved.', '1511.04162-2-26-5': 'I allow [MATH] to be discrete, continuous or mixed; [MATH] is only required to have some known dominating finite measure [MATH] on the real line.', '1511.04162-2-26-6': 'For example, [MATH] can be the Lebesgue measure or the counting measure.', '1511.04162-2-26-7': 'Let [MATH] be the support for the random variable [MATH] and [MATH] be the support for [MATH].', '1511.04162-2-27-0': 'To describe the data generating process, I consider the counterfactual variables.', '1511.04162-2-27-1': '[MATH] is the counterfactual true treatment when [MATH].', '1511.04162-2-27-2': '[MATH] is the counterfactual outcome when [MATH].', '1511.04162-2-27-3': '[MATH] is the counterfactual measured treatment when [MATH].', '1511.04162-2-27-4': 'The individual treatment effect is [MATH].', '1511.04162-2-27-5': 'It is not directly observed; [MATH] and [MATH] are not observed at the same time.', '1511.04162-2-27-6': 'Only [MATH] is observable.', '1511.04162-2-27-7': 'Using the notation, the observed variables [MATH] are generated by the three equations: [EQUATION]', '1511.04162-2-27-8': 'Figure [REF] graphically describes the relationship among the instrument [MATH], the (unobserved) true treatment [MATH], the measured treatment [MATH], and the outcome [MATH].', '1511.04162-2-28-0': '([REF]) is the measurement equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-2-28-1': '[MATH] is the measurement error; [MATH] (or [MATH]) represents a false positive and [MATH] (or [MATH]) represents a false negative.', '1511.04162-2-28-2': 'Equations ([REF]) and ([REF]) are the same as [CITATION] and [CITATION].', '1511.04162-2-28-3': '([REF]) is the outcome equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-2-28-4': '([REF]) is the treatment assignment equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-2-28-5': 'Correlation between [MATH] and [MATH] causes an endogeneity problem.', '1511.04162-2-29-0': 'In a return to schooling analysis, [MATH] is wages, [MATH] is the true indicator for college completion, [MATH] is the proximity to college, and [MATH] is the self-reported college completion.', '1511.04162-2-29-1': 'The treatment effect [MATH] in the return to schooling is the effect of college completion [MATH] on wages [MATH].', '1511.04162-2-29-2': 'The college completion is not correctly measured in a survey dataset, such that only the self report [MATH] is observed.', '1511.04162-2-30-0': 'This section and Section [REF] impose only the following assumption.', '1511.04162-2-31-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-2-31-1': '(ii) [MATH] almost surely.', '1511.04162-2-31-2': '(iii) [MATH].', '1511.04162-2-32-0': 'Assumption [REF] (i) is the exclusion restriction and I consider stochastic independence instead of mean independence.', '1511.04162-2-32-1': 'Although it is stronger than the minimal conditions for the identification for the local average treatment effect without measurement error, a large part of the existing applied papers assume stochastic independence .', '1511.04162-2-32-2': '[MATH] is also independent of [MATH] conditional on [MATH], which is the only assumption on the measurement error for the identified set in Section [REF].', '1511.04162-2-32-3': '(Section [REF] even dispenses with this assumption.)', '1511.04162-2-32-4': 'Assumption [REF] (ii) is the monotonicity condition for the instrument, in which the instrument [MATH] increases the value of [MATH] for all the individuals.', '1511.04162-2-32-5': '[CITATION] relaxes the monotonicity condition, and it can be shown that the identification results in my analysis only require the complier-defiers-for-marginals condition in [CITATION].', '1511.04162-2-32-6': 'Note that Assumption [REF] does not include a relevance condition for the instrumental variable.', '1511.04162-2-32-7': 'The standard relevance condition [MATH] does not affect the identification results in my analysis.', '1511.04162-2-32-8': 'I will discuss the relevance condition in my framework after Theorem [REF].', '1511.04162-2-32-9': 'Assumption [REF] (iii) excludes that [MATH] is constant.', '1511.04162-2-33-0': 'As I emphasized in the introduction, the framework here does not assume anything on measurement error [MATH] except for its independence from [MATH].', '1511.04162-2-33-1': 'Assumption [REF] does not impose any restriction on the marginal distribution of the measurement error [MATH] or on the relationship between the measurement error [MATH] and [MATH].', '1511.04162-2-33-2': 'Particularly, the measurement error can be endogenous, that is, [MATH] and [MATH] can be correlated.', '1511.04162-2-34-0': 'I focus on the local average treatment effect, which is defined by [EQUATION].', '1511.04162-2-34-1': 'The local average treatment effect is the average of the treatment effect [MATH] over the subpopulation (the compliers) whose treatment status is strictly affected by the instrument.', '1511.04162-2-34-2': '[CITATION] show that the local average treatment effect equals [EQUATION] where I define [MATH] for a random variable [MATH].', '1511.04162-2-34-3': 'The treatment is measured with error, and therefore the above fraction [MATH] is not the Wald estimand [EQUATION].', '1511.04162-2-34-4': 'Since [MATH] is not identified, I cannot identify the local average treatment effect.', '1511.04162-2-34-5': 'The failure for the point identification comes purely from the measurement error, because the local average treatment effect would be point-identified under [MATH].', '1511.04162-2-35-0': '# Identified set for the local average treatment effect', '1511.04162-2-36-0': 'This section show how the instrumental variable partially identifies the local average treatment effect in the framework of Section [REF].', '1511.04162-2-36-1': 'Before defining the identified set, I express the local average treatment effect as a function of the underlying distribution [MATH] of [MATH].', '1511.04162-2-36-2': 'I use the [MATH] symbol on [MATH] to clarify that [MATH] is the distribution of the unobserved variables.', '1511.04162-2-36-3': 'I denote the expectation operator [MATH] by [MATH] when I need to clarify the underlying distribution.', '1511.04162-2-36-4': 'The local average treatment effect is a function of the unobserved distribution [MATH]: [EQUATION].', '1511.04162-2-36-5': 'I denote by [MATH] the parameter space for the local average treatment effect [MATH], that is, a subset of the set of [MATH] where [MATH] and [MATH] are density functions dominated by the known measure [MATH].', '1511.04162-2-36-6': 'For example, [MATH] when [MATH] is binary.', '1511.04162-2-37-0': 'The identified set is the set of parameter values for the local average treatment effect which is consistent with the distribution of the observed variables.', '1511.04162-2-37-1': 'I use [MATH] for the distribution of the observed variables [MATH] The equations ([REF]), ([REF]), and ([REF]) induce the distribution of the observables [MATH] from the unobserved distribution [MATH], and I denote by [MATH] the induced distribution.', '1511.04162-2-37-2': "When the distribution of [MATH] is [MATH], the set of [MATH] which induces [MATH] is [MATH], where [MATH] is the set of [MATH]'s satisfying Assumptions [REF].", '1511.04162-2-37-3': 'For every distribution [MATH] of [MATH], the (sharp) identified set for the local average treatment effect is defined as [MATH].', '1511.04162-2-38-0': '[CITATION] provides a relationship between [MATH] and the local average treatment effect: [EQUATION]', '1511.04162-2-38-1': 'This equation gives the two pieces of information of [MATH].', '1511.04162-2-38-2': 'First, the sign of [MATH] is the same as [MATH].', '1511.04162-2-38-3': 'Second, the absolute value of [MATH] is at least the absolute value of [MATH].', '1511.04162-2-38-4': 'The following lemma summaries these two pieces.', '1511.04162-2-39-0': 'Under Assumption [REF], [EQUATION].', '1511.04162-2-40-0': 'I derive a new implication from the exclusion restriction for the instrumental variable in order to obtain an upper bound on [MATH] in absolute value.', '1511.04162-2-40-1': 'To explain the new implication, I introduce the total variation distance, which is the [MATH] distance between the distribution [MATH] and [MATH]: For any random variable [MATH], define [EQUATION] where [MATH] is a dominating measure for the distribution of [MATH].', '1511.04162-2-41-0': 'Under Assumption [REF], [EQUATION].', '1511.04162-2-42-0': 'The first term, [MATH], in Lemma [REF] reflects the dependency of [MATH] on [MATH], and it can be interpreted as the magnitude of the distributional effect of [MATH] on [MATH].', '1511.04162-2-42-1': 'The second and third terms, [MATH] and [MATH], are the effect of the instrument [MATH] on the true treatment [MATH].', '1511.04162-2-42-2': 'Based on Lemma [REF], the magnitude of the effect of [MATH] on [MATH] is no smaller than the magnitude of the effect of [MATH] on [MATH].', '1511.04162-2-43-0': 'The new implication in Lemma [REF] gives a lower bound on [MATH] and therefore yields an upper bound on the local average treatment effect in absolute value, combined with equation ([REF]).', '1511.04162-2-43-1': 'Therefore, I use these relationships to derive an upper bound on the local average treatment effect in absolute value, that is, [EQUATION] as long as [MATH].', '1511.04162-2-44-0': 'Theorem [REF] shows that the above observations characterize the sharp identified set for the local average treatment effect.', '1511.04162-2-45-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-2-45-1': 'The identified set [MATH] for the local average treatment effect is the set of [MATH] satisfying the following three inequalities.', '1511.04162-2-45-2': '[EQUATION]', '1511.04162-2-45-3': 'As a corollary, [MATH] if [MATH]; otherwise, [EQUATION].', '1511.04162-2-46-0': 'The total variation distance [MATH] measures the strength for the instrumental variable, that is, [MATH] is the relevance condition in my identification analysis.', '1511.04162-2-46-1': 'When [MATH], the interval in the above theorem is always nonempty and bounded, which implies that [MATH] has some identifying power for the local average treatment effect.', '1511.04162-2-46-2': 'By contrast, [MATH] means that the instrumental variable [MATH] does not affect [MATH] and [MATH], in which case [MATH] has no identifying power for the local average treatment effect.', '1511.04162-2-46-3': 'In this case, [MATH] almost everywhere over [MATH] and particularly [MATH].', '1511.04162-2-46-4': 'Note that all the three inequalities in Theorem [REF] have no restriction on [MATH] in this case.', '1511.04162-2-47-0': 'Note that the sharp identified set is always non-empty.', '1511.04162-2-47-1': 'There is no testable implications on the distribution of the observed variables, and therefore it is impossible to conduct a specification test for Assumption [REF].', '1511.04162-2-48-0': '## Wald estimand and the identified set', '1511.04162-2-49-0': 'The Wald estimand [MATH] can be outside the identified set.', '1511.04162-2-49-1': 'One necessary and sufficient condition for the Wald estimand to be included in the identified set is given as follows.', '1511.04162-2-50-0': 'The Wald estimand is in the identified set if and only if [EQUATION]', '1511.04162-2-50-1': 'This condition in ([REF]) is the testable implications from the the local average treatment effect framework without measurement error ([CITATION] and [CITATION]).', '1511.04162-2-50-2': 'The recent papers by [CITATION], [CITATION], and [CITATION] propose the testing procedures for ([REF]).', '1511.04162-2-50-3': 'Based on the results in Theorem [REF], their testing procedures are re-interpreted as a test for the null hypothesis that the Wald estimand is inside the sharp upper bound on the local average treatment effect.', '1511.04162-2-51-0': '## Identifying power of repeated measurements', '1511.04162-2-52-0': 'The identification strategy in the above analysis offers a new use of repeated measurements as additional sources for identification.', '1511.04162-2-52-1': 'Repeated measurements is a popular approach in the literature on measurement error, but they cannot be instrumental variables in this framework.', '1511.04162-2-52-2': 'This is because the true treatment [MATH] is endogenous and it is natural to suspect that a measurement of [MATH] is also endogenous.', '1511.04162-2-52-3': 'The more accurate the measurement is, the more likely it is to be endogenous.', '1511.04162-2-52-4': 'Nevertheless, the identification strategy incorporates repeated measurements as an additional information to tighten the identified set for the local average treatment effect, when they are coupled with the instrumental variable [MATH].', '1511.04162-2-52-5': 'Unlike the other paper on repeated measurements, I do not need to assume the independence of measurement errors among multiple measurements.', '1511.04162-2-52-6': 'The strategy also benefits from having more than two measurements unlike [CITATION] who achieve the point identification with two measurements.', '1511.04162-2-53-0': 'Consider a repeated measurement [MATH] for [MATH].', '1511.04162-2-53-1': 'I do not require that [MATH] is binary, so [MATH] can be discrete or continuous.', '1511.04162-2-53-2': 'Like [MATH], I model [MATH] using the counterfactual outcome notations.', '1511.04162-2-53-3': '[MATH] is a counterfactual second measurement when the true treatment [MATH] is [MATH], and [MATH] is a counterfactual second measurement when the true treatment [MATH] is [MATH].', '1511.04162-2-53-4': 'Then the data generation of [MATH] is [EQUATION].', '1511.04162-2-53-5': 'I strengthen Assumption [REF] by assuming that the instrumental variable [MATH] is independent of [MATH] conditional on [MATH].', '1511.04162-2-54-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-2-54-1': '(ii) [MATH] almost surely.', '1511.04162-2-54-2': '(iii) [MATH].', '1511.04162-2-55-0': 'Note that I do not assume the independence between [MATH] and [MATH], where the independence between the measurement errors is a key assumption when the repeated measurement is an instrumental variable.', '1511.04162-2-55-1': 'Assumption [REF] tightens the identified set for the local average treatment effect as follows.', '1511.04162-2-56-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-2-56-1': 'The identified set [MATH] for the local average treatment effect is the set of [MATH] satisfying the following three inequalities: [EQUATION]', '1511.04162-2-56-2': 'The identified set in Theorem [REF] is weakly smaller than the identified set in Theorem [REF].', '1511.04162-2-56-3': 'The total variation distance [MATH] in Theorem [REF] is weakly larger than that in Theorem [REF], because, using the triangle inequality, [EQUATION] and the strict inequality holds unless the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-2-56-4': 'Therefore, it is possible to test whether the repeated measurement [MATH] has additional information, by testing whether the sign of [MATH] is constant in [MATH].', '1511.04162-2-57-0': '## Dependence between measurement error and instrumental variable', '1511.04162-2-58-0': 'It is still possible to apply the same identification strategy and obtain finite (but less tight) bounds on the local average treatment effect, even without assuming the independence between the instrumental variable and the measurement error.', '1511.04162-2-58-1': '(Assumption [REF] (i) implies that [MATH] is independent of [MATH] for each [MATH].)', '1511.04162-2-58-2': 'Instead Assumption [REF] is weakened to allow for the measurement error [MATH] to be correlated with the instrumental variable [MATH].', '1511.04162-2-59-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-2-59-1': '(ii) [MATH] almost surely.', '1511.04162-2-59-2': '(iii) [MATH].', '1511.04162-2-60-0': 'Theorem [REF] shows that the above observations characterize the identified set for the local average treatment effect under Assumption [REF].', '1511.04162-2-61-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-2-61-1': 'The identified set [MATH] for the local average treatment effect is the set of [MATH] satisfying the following three inequalities: [EQUATION]', '1511.04162-2-61-2': 'The difference from Theorem [REF] is that Theorem [REF] does not depend on the measured treatment [MATH].', '1511.04162-2-61-3': 'Although it is observed in the dataset, [MATH] does not have any information on the local average treatment effect because Assumption [REF] does not have restriction on [MATH].', '1511.04162-2-61-4': 'When [MATH], there are nontrivial upper and lower bounds on the local average treatment effect even without any information from the measured treatment [MATH].', '1511.04162-2-62-0': '# Inference', '1511.04162-2-63-0': 'Based on the partial identification result in Theorem [REF], this section constructs a confidence interval for the local average treatment effect based on an i.i.d. sample [MATH] of [MATH].', '1511.04162-2-63-1': 'The confidence interval controls the asymptotic size uniformly over a class of data generating processes, and rejects all the fixed alternatives.', '1511.04162-2-63-2': 'In the following, first, I characterize [MATH] as a projection of the limit of unconditional moment inequality models [MATH].', '1511.04162-2-63-3': 'Then I apply the two-step multiplier bootstrap to moment inequality models in which the number of moment inequalities is finite but growing as the sample size increases.', '1511.04162-2-64-0': 'I rewrite the identified set [MATH] as a projection of [MATH] where [MATH] is characterized by moment inequalities which are differentiable with respect to the data distribution.', '1511.04162-2-64-1': 'It is necessary because the total variation distance involves absolute values of the data distribution and it does not have the uniformly valid asymptotic distribution for the total variation distance due to the lack of differentiability ([CITATION] and [CITATION]).', '1511.04162-2-64-2': 'It is worth mentioning that this non-differentiability problem remains even if the support of [MATH] is finite.', '1511.04162-2-65-0': '[MATH] is defined as follows.', '1511.04162-2-65-1': 'Define [MATH] and let [MATH] be a generic value for [MATH].', '1511.04162-2-65-2': 'For [MATH] and [MATH], define the functions [MATH] by [EQUATION] where [MATH] the set of measurable functions on [MATH] taking a value in [MATH] and [MATH].', '1511.04162-2-65-3': 'Note that [MATH] is different from the usual sign function in that [MATH].', '1511.04162-2-65-4': 'Define [EQUATION].', '1511.04162-2-65-5': 'Then [MATH] is a projection of [MATH] as in the next lemma.', '1511.04162-2-66-0': 'Let [MATH] be an arbitrary data distribution of [MATH].', '1511.04162-2-66-1': 'Under Assumption [REF], [EQUATION].', '1511.04162-2-67-0': 'The number of the moment inequalities in [MATH] can be finite or infinite, which determines whether some of the existing methods can be applied directly to the inference on the local average treatment effect.', '1511.04162-2-67-1': 'First, when the outcome [MATH] has a finite support and therefore [MATH] is finite, the sharp identified set is characterized by a finite number of inequalities.', '1511.04162-2-67-2': 'Therefore I can apply the methodologies in unconditional moment inequalities.', '1511.04162-2-67-3': 'Second, to the best of my knowledge, inference for the local average treatment effect in my framework does not fall directly into the existing moment inequality models when the outcome is continuous.', '1511.04162-2-67-4': 'When the outcome is continuous, the sharp identified set is characterized by an uncountably infinite number of inequalities.', '1511.04162-2-67-5': 'In the current literature on the partially identified parameters, an infinite number of moment inequalities are mainly considered in the context of conditional moment inequalities.', '1511.04162-2-67-6': 'The identified set in this paper is not characterized by conditional moment inequalities.', '1511.04162-2-67-7': '[CITATION] also considers an infinite number of unconditional moment inequalities in which the moment functions are continuously indexed by a compact subset in a finite dimensional space.', '1511.04162-2-67-8': 'It is not straightforward to verify the continuity condition (Condition C.1 in their paper) for the set [MATH], in which the moment functions need to be continuously indexed by a compact subset of the finite dimensional space.', '1511.04162-2-67-9': '[CITATION] considers an infinite number of unconditional moment inequalities in which the moment functions satisfies manageability condition.', '1511.04162-2-67-10': 'I cannot apply their approach here because [MATH] takes discrete values in [MATH] which makes the packing numbers dependent on the sample size.', '1511.04162-2-68-0': 'To construct a confidence interval applicable to both discrete and continuous outcomes, I discretize the outcome [MATH] so that the number of the inequalities becomes finite, and, at the same time, take the discretization finer as the sample size increases.', '1511.04162-2-68-1': 'Although it is unnecessary for discrete [MATH], the discretization may be computationally attractive even for discrete [MATH] since it reduces the number of the inequalities.', '1511.04162-2-68-2': 'Consider a partition [MATH] over [MATH], in which the intervals [MATH] and the grid size [MATH] depend on the sample size [MATH].', '1511.04162-2-68-3': 'Let [MATH] be a generic function of [MATH] into [MATH] that is constant over [MATH] for every [MATH] and every [MATH].', '1511.04162-2-68-4': 'Let [MATH] be the set of all such functions.', '1511.04162-2-68-5': 'Note that [MATH] is a subset of [MATH].', '1511.04162-2-68-6': "Using these [MATH]'s, I consider [MATH] characterized by [MATH] unconditional moment inequalities [EQUATION] where, for simplicity, I denote [MATH] for [MATH].", '1511.04162-2-69-0': 'The set [MATH] consists of a finite number of unconditional moments inequalities, but the number of the inequalities become finite but growing, particularly diverging to the infinity when [MATH] has an infinite support.', '1511.04162-2-69-1': '[CITATION] considers testing a finite but growing number of moment inequalities.', '1511.04162-2-69-2': 'I construct a test statistic [MATH] and a critical value [MATH] for [MATH] via their two-step multiplier bootstrap for the size [MATH], which I describe in Section [REF], and then project the confidence region into [MATH], in order to obtain a confidence interval [MATH] for the local average treatment effect.', '1511.04162-2-69-3': 'Theorem [REF] shows asymptotic properties of the confidence interval [MATH].', '1511.04162-2-70-0': 'Given positive constants [MATH] and [MATH], consider the class [MATH] of data generating processes such that [MATH] and [MATH] for each [MATH].', '1511.04162-2-70-1': 'If (a) [MATH] is bounded, (b) [MATH] and [MATH] for some [MATH] and [MATH], (c) [MATH] for every [MATH], and (d) there is a positive constant [MATH] such that [MATH] is a subset of some open ball with radius [MATH] in [MATH], then (i) the confidence interval controls the asymptotic size uniformly: [EQUATION] and (ii) the confidence interval excludes all the fixed alternatives: [EQUATION] for every [MATH] with [MATH] such that the density function [MATH] is Holder continuous in [MATH] with the Holder constant [MATH] and exponent [MATH].', '1511.04162-2-71-0': 'The first result (i) is the uniform asymptotic size control and the second result (ii) is the consistency against all the fixed alternatives.', '1511.04162-2-71-1': 'The first assumption (a) requires researchers to know ex ante upper and lower bounds on the parameter.', '1511.04162-2-71-2': 'The second assumption (b) restricts the number of the inequalities and, in turn, restrict the magnitude of the critical value.', '1511.04162-2-71-3': 'Note that the number of the inequalities is the tuning parameter in this framework.', '1511.04162-2-71-4': 'The tradeoff is as follows: the approximation error is large if the growth of [MATH] is slow, and the sampling error is large if the growth of [MATH] is fast.', '1511.04162-2-71-5': 'The third assumption (c) guarantees that the test statistics is well-defined.', '1511.04162-2-71-6': 'The four assumption (d), together with the Holder continuity, guarantees that the approximation error from the discretization vanishes as the sample size [MATH] increases.', '1511.04162-2-72-0': '## Two-step multiplier bootstrap', '1511.04162-2-73-0': 'I construct a confidence interval [MATH] via the two-step multiplier bootstrap in [CITATION].', '1511.04162-2-73-1': 'The test statistics for the true parameter values being [MATH] is defined by [EQUATION] where [MATH] estimates [MATH], and [MATH] estimates the variance [MATH] of [MATH]: [EQUATION]', '1511.04162-2-73-2': 'To conduct a multiplier bootstrap, generate [MATH] independent standard normal random variables [MATH].', '1511.04162-2-73-3': 'The centered bootstrap moments are [EQUATION].', '1511.04162-2-73-4': 'For every [MATH], the bootstrapped test statistics is defined by [EQUATION] where [MATH] is [MATH] if [MATH] is empty.', '1511.04162-2-73-5': 'The first step of the two-step multiplier bootstrap collects the inequalities which are not too negatively large: [EQUATION] where [MATH] is the pretest size for the inequality selection and [MATH] is the conditional [MATH]-quantile of [MATH] given [MATH]: [EQUATION].', '1511.04162-2-73-6': 'As the second step, [MATH] is defined as the conditional [MATH]-quantile of [MATH] given [MATH]: [EQUATION].', '1511.04162-2-73-7': 'The [MATH]-confidence interval [MATH] for the local average treatment effect is constructed as [EQUATION]', '1511.04162-2-74-0': '# Monte Carlo simulations', '1511.04162-2-75-0': 'Using simulated datasets, this section illustrates the theoretical properties for the confidence interval in the previous section.', '1511.04162-2-75-1': 'The data generating process for [MATH] is as follows: [EQUATION] where [MATH] are independent random variables from [MATH] and [MATH] is the standard normal cumulative distribution function.', '1511.04162-2-76-0': 'There are three parameters in the model: [MATH] represents the strength of the instrumental variable, [MATH] represents the magnitude of treatment effect, and [MATH] represents the degree of the measurement error.', '1511.04162-2-76-1': 'This is the heterogeneous treatment effect model because [MATH] is nonlinear.', '1511.04162-2-76-2': 'Several values for [MATH] are considered as in Table [REF].', '1511.04162-2-76-3': 'The treatment effect is small ([MATH]) in Designs 1-4 and large ([MATH]) in Designs 5-8.', '1511.04162-2-76-4': 'The measurement error is small ([MATH]) in Designs 3,4,7,8 and large ([MATH]) in Designs 1,2,5,6.', '1511.04162-2-76-5': 'The instrumental variable is strong ([MATH]) in Designs 2,4,6,8 and weak ([MATH]) in Design 1,3,5,7.', '1511.04162-2-77-0': 'Table [REF] also computes the three population objects: the local average treatment effect, the Wald estimand, and the identified set for the local average treatment effect.', '1511.04162-2-77-1': 'The comparison between the Wald estimand and the identified set hints that the Wald estimand is relatively large compared to the upper bound of the identified set [MATH] when the measurement error has a large degree in Design 1,2,5,6.', '1511.04162-2-77-2': 'For those designs, the Wald estimand is too large to be used as an upper bound on the local average treatment effect, because the upper bound of the identified set [MATH] is much smaller.', '1511.04162-2-78-0': 'The sample sizes are [MATH] for the Monte Carlo simulations.', '1511.04162-2-78-1': 'The number of the simulations are 2,000 for three sample sizes.', '1511.04162-2-78-2': 'For each dataset, I construct the confidence set with confidence size [MATH], as in Section [REF].', '1511.04162-2-78-3': 'The partition of grids is equally spaced over [MATH] with the number the partitions [MATH].', '1511.04162-2-78-4': 'The results for [MATH] are omitted for graphical simplicity.', '1511.04162-2-78-5': 'All the confidence intervals use 5,000 bootstrap repetitions.', '1511.04162-2-78-6': 'Figures [REF]-[REF] describe the coverage probabilities of the confidence intervals for each parameter value.', '1511.04162-2-79-0': 'For each design, two figures are displayed.', '1511.04162-2-79-1': 'First, the left figures demonstrate the coverage probabilities according to [MATH] given [MATH].', '1511.04162-2-79-2': 'These figures for all the designs support the consistency results in the previous section; as the sample size increases, the coverage probabilities of the confidence intervals are high only in a neighborhood of [MATH].', '1511.04162-2-79-3': 'Second, the right figures demonstrate the coverage probabilities according to [MATH] given [MATH].', '1511.04162-2-79-4': 'When the Wald estimand is close to the upper bound of [MATH] in Design 3,4,7,8, it seems advantageous to use [MATH].', '1511.04162-2-79-5': 'It is presumably because [MATH] uses more inequalities for inference compared to [MATH] but these inequalities are not informative for those designs.', '1511.04162-2-79-6': 'When the Wald estimand is significantly larger than the upper bound of [MATH] in Design 1,2,5,6, it seems advantageous to use [MATH], particularly for the coverage probabilities near the upper bound of [MATH].', '1511.04162-2-79-7': 'In these designs, the coverage probabilities are not sensitive the choice of [MATH] or [MATH].', '1511.04162-2-80-0': '# Conclusion', '1511.04162-2-81-0': 'This paper studies the identifying power of instrumental variable in the heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous.', '1511.04162-2-81-1': 'The assumptions in this framework are the monotonicity of the instrumental variable [MATH] on the true treatment [MATH] and the exogeneity of [MATH].', '1511.04162-2-81-2': 'I use the total variation distance to characterize the identified set for the local average treatment effect [MATH].', '1511.04162-2-81-3': 'I also provide an inference procedure for the local average treatment effect.', '1511.04162-2-81-4': 'Unlike the existing literature on measurement error, the identification strategy does not reply on a specific structure of the measurement error; the only assumption on the measurement error is its independence of the instrumental variable.', '1511.04162-2-81-5': '(Section [REF] shows that the same identification strategy works even without this assumption.)', '1511.04162-2-82-0': 'There are several directions for future research.', '1511.04162-2-82-1': 'First, the choice of the partition [MATH] in Section [REF], particularly the choice of [MATH], is an interesting direction.', '1511.04162-2-82-2': 'To the best of my knowledge, the literature on many moment inequalities has not investigated how econometricians choose the numbers of the many moment inequalities, e.g., [CITATION].', '1511.04162-2-82-3': 'Second, it is worthwhile to investigating the other parameter for the treatment effect.', '1511.04162-2-82-4': 'This paper has focused on the local average treatment effect, but the literature on heterogeneous treatment effect has emphasized the importance of choosing a adequate treatment effect parameter in order to answer relevant policy questions.', '1511.04162-2-82-5': 'Third, it is also interesting to investigate various assumptions on the measurement errors.', '1511.04162-2-82-6': 'In some empirical settings, for example, it may be reasonable to assume that the measurement error is one-directional (e.g., misclassification happens only when [MATH]).', '1511.04162-2-83-0': '# Appendix', '1511.04162-2-84-0': '## Proof of Lemma [REF]', '1511.04162-2-85-0': 'By Equation ([REF]), [MATH], and [MATH].', '1511.04162-2-86-0': '## Proof of Lemma [REF]', '1511.04162-2-87-0': 'I obtain [MATH] by the same logic as Theorem 1 in [CITATION]: [EQUATION]', '1511.04162-2-87-1': 'By the triangle inequality, [EQUATION]', '1511.04162-2-87-2': 'Moreover, since [MATH] almost surely, [EQUATION]', '1511.04162-2-88-0': '## Proof of Lemma [REF]', '1511.04162-2-89-0': 'Based on the triangle inequality, [EQUATION]', '1511.04162-2-89-1': 'The equality holds if and only if the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-2-89-2': 'Since [MATH] if and only if [MATH] is positive, the condition in ([REF]) is a necessary and sufficient condition for [MATH], which is equivalent for the Wald estimand to belong to the identified set.', '1511.04162-2-90-0': '## Proof of Theorems [REF] and [REF]', '1511.04162-2-91-0': 'Theorem [REF] is a special case of Theorem [REF] with [MATH], and therefore I demonstrate the proof only for Theorem [REF].', '1511.04162-2-91-1': 'As in Theorem [REF], I will consider two cases based on the value of [MATH].', '1511.04162-2-92-0': '### Case 1: Zero total variation distance', '1511.04162-2-93-0': 'Consider [MATH].', '1511.04162-2-93-1': 'In other words, [MATH] for every [MATH].', '1511.04162-2-93-2': 'Let [MATH] and [MATH] be any pair of density functions for [MATH] dominated by [MATH].', '1511.04162-2-93-3': 'Define the data generating process [MATH]: [EQUATION]', '1511.04162-2-93-4': 'Theorem [REF] follows from the next lemma.', '1511.04162-2-94-0': 'If [MATH], then (i) [MATH] satisfies Assumption [REF]; (ii) [MATH] generates the data distribution [MATH]; (iii) under [MATH], the local average treatment effect is [MATH].', '1511.04162-2-95-0': '(i) [MATH] satisfies the independence between [MATH] and [MATH] for each [MATH].', '1511.04162-2-95-1': 'Furthermore, [MATH] satisfies [MATH] almost surely.', '1511.04162-2-95-2': '(ii) Denote by [MATH] the density function of [MATH].', '1511.04162-2-95-3': 'Then [EQUATION] where the last equality uses [MATH].', '1511.04162-2-95-4': '(iii) The local average treatment effect under [MATH] is [EQUATION]', '1511.04162-2-96-0': '### Case 2: Positive total variation distance', '1511.04162-2-97-0': 'Consider [MATH].', '1511.04162-2-97-1': 'Lemma [REF] is modified into the framework of Theorem [REF].', '1511.04162-2-98-0': 'Under Assumption [REF], [MATH].', '1511.04162-2-99-0': 'The proof is the same as Lemma [REF] and this lemma follows from [MATH].', '1511.04162-2-100-0': 'From Lemmas [REF] and [REF] and Equation [REF], all the three inequalities in Theorem [REF] are satisfied for the true value of the local average treatment effect.', '1511.04162-2-100-1': 'To complete Theorem [REF], it suffices to show that, for any data generating process [MATH], any point [MATH] satisfying the three inequalities in Theorem [REF] is the local average treatment effect under some data generating process [MATH] whose data distribution is equal to [MATH].', '1511.04162-2-101-0': 'Define the two data generating processes: [MATH] and [MATH].', '1511.04162-2-101-1': 'First, [MATH] is defined by [EQUATION]', '1511.04162-2-101-2': 'Second, [MATH] is defined as follows.', '1511.04162-2-101-3': 'Using [MATH], define [EQUATION] and define [MATH] as [EQUATION]', '1511.04162-2-101-4': 'If [MATH], then (i) [MATH] generates the data distribution [MATH] and the local average treatment effect under [MATH] is [MATH]; and (ii) [MATH] generates the data distribution [MATH] and the local average treatment effect under [MATH] is [MATH].', '1511.04162-2-102-0': '(i) Denote by [MATH] the density function of [MATH].', '1511.04162-2-102-1': 'The data generating process [MATH] generates the data distribution [MATH]: [EQUATION] where the first equality uses [MATH].', '1511.04162-2-102-2': 'Under [MATH], the local average treatment effect is [MATH]: [EQUATION] (ii) Note that [EQUATION]', '1511.04162-2-102-3': 'Denote by [MATH] the density function of [MATH].', '1511.04162-2-102-4': 'Notice that [MATH] is well-defined because [EQUATION] [MATH] generates the data distribution [MATH]: [EQUATION] and similarly [MATH].', '1511.04162-2-102-5': 'Under [MATH], the local average treatment effect is [EQUATION]', '1511.04162-2-102-6': 'Theorem [REF] follows from the next lemma.', '1511.04162-2-103-0': 'If [MATH], then, for every [MATH], (i) the mixture distribution [MATH] satisfies Assumption [REF]; (ii) [MATH] generates the data distribution [MATH]; (iii) under [MATH], the local average treatment effect is [EQUATION] (i) Under both [MATH] and [MATH], [MATH] is independent of [MATH] for each [MATH].', '1511.04162-2-103-1': 'Furthermore, [MATH] and [MATH] have the same marginal distribution for [MATH]: [MATH].', '1511.04162-2-103-2': 'Therefore, the mixture of [MATH] and [MATH] also satisfies the independence.', '1511.04162-2-103-3': 'Since both[MATH] and [MATH] satisfy [MATH] almost surely, so does the mixture.', '1511.04162-2-103-4': '(ii) By Lemma [REF], both [MATH] and [MATH] generate the data distribution [MATH] and so does the mixture.', '1511.04162-2-103-5': '(iii) It follows from the last statement in Lemma [REF].', '1511.04162-2-104-0': '## Proof of Theorem [REF]', '1511.04162-2-105-0': 'The proof of Theorems [REF] is similar to Theorems [REF] and [REF].', '1511.04162-2-105-1': 'Only the difference is to change the definition of [MATH] as follows.', '1511.04162-2-105-2': 'Define [MATH] and define [MATH] as [EQUATION]', '1511.04162-2-106-0': '## Proof of Lemma [REF]', '1511.04162-2-107-0': 'First, note that [EQUATION]', '1511.04162-2-107-1': 'This is verified as follows.', '1511.04162-2-107-2': 'For every [MATH], I have [EQUATION]', '1511.04162-2-107-3': 'Moreover, the above inequality becomes an equality if [MATH] if [MATH] and [MATH] if [MATH].', '1511.04162-2-108-0': 'Second, note that [EQUATION] for any random variable [MATH], where [MATH].', '1511.04162-2-108-1': 'This follows from [EQUATION]', '1511.04162-2-108-2': 'By Theorem [REF] and Equation ([REF]), [MATH] is characterized by [EQUATION]', '1511.04162-2-108-3': 'Since the second condition implies [MATH], the above three conditions becomes [EQUATION]', '1511.04162-2-108-4': 'By Equation ([REF]), [MATH] is characterized by [EQUATION]', '1511.04162-2-108-5': 'This verifies Lemma [REF].', '1511.04162-2-109-0': '## Proof of Theorem [REF]', '1511.04162-2-110-0': 'The following theorem is taken from Corollary 5.1 and Theorem 6.1 in [CITATION].', '1511.04162-2-111-0': 'Given [MATH] with [MATH] and [MATH], denote by [MATH] the set of [MATH] for which [EQUATION]', '1511.04162-2-111-1': 'Under the assumptions in Theorem [REF], [EQUATION]', '1511.04162-2-111-2': 'Theorem [REF] (i) is the same as Theorem [REF] (i).', '1511.04162-2-111-3': 'Denote by [MATH] a constant for which [MATH].', '1511.04162-2-111-4': 'Let [MATH] be any element of [MATH] with [MATH].', '1511.04162-2-111-5': 'By Theorem [REF] (ii), it suffices to show that ([REF]) holds for sufficiently large [MATH].', '1511.04162-2-111-6': 'If [MATH] or [MATH], then ([REF]) holds for sufficiently large [MATH].', '1511.04162-2-111-7': 'In the rest of the proof, I focus on the case where [MATH].', '1511.04162-2-111-8': 'By the growth rate of [MATH], [EQUATION] for sufficiently large [MATH].', '1511.04162-2-111-9': 'It suffices to show that ([REF]) implies ([REF]).', '1511.04162-2-111-10': 'Define [EQUATION]', '1511.04162-2-111-11': 'By the definition of [MATH], [EQUATION]', '1511.04162-2-111-12': 'By the Holder continuity of [MATH], [EQUATION].', '1511.04162-2-111-13': 'For [MATH] with [MATH], the above inequality implies that the sign of [MATH] is constant.', '1511.04162-2-111-14': 'For those [MATH], therefore, [MATH] is constant, and [MATH].', '1511.04162-2-111-15': 'Then, on every [MATH], either [MATH] or [MATH].', '1511.04162-2-111-16': 'This implies [EQUATION]', '1511.04162-2-111-17': 'If ([REF]) holds, then [EQUATION] and therefore ([REF]) holds.'} | [['1511.04162-1-19-0', '1511.04162-2-20-0'], ['1511.04162-1-38-0', '1511.04162-2-40-0'], ['1511.04162-1-95-1', '1511.04162-2-56-4'], ['1511.04162-1-5-1', '1511.04162-2-5-1'], ['1511.04162-1-5-4', '1511.04162-2-5-5'], ['1511.04162-1-5-11', '1511.04162-2-5-9'], ['1511.04162-1-5-13', '1511.04162-2-5-11'], ['1511.04162-1-5-14', '1511.04162-2-5-12'], ['1511.04162-1-23-1', '1511.04162-2-24-1'], ['1511.04162-1-23-4', '1511.04162-2-24-4'], ['1511.04162-1-89-1', '1511.04162-2-52-1'], ['1511.04162-1-89-3', '1511.04162-2-52-3'], ['1511.04162-1-89-5', '1511.04162-2-52-5'], 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1511.04162 | {'1511.04162-3-0-0': 'This paper studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous.', '1511.04162-3-0-1': 'Using a binary instrumental variable, I characterize the sharp identified set for the local average treatment effect under the exclusion restriction of an instrument and the deterministic monotonicity of the true treatment in the instrument.', '1511.04162-3-0-2': 'Even allowing for general measurement error (e.g., the measurement error is endogenous), it is still possible to obtain finite bounds on the local average treatment effect.', '1511.04162-3-0-3': 'Notably, the Wald estimand is an upper bound on the local average treatment effect, but it is not the sharp bound in general.', '1511.04162-3-0-4': 'I also provide a confidence interval for the local average treatment effect with uniformly asymptotically valid size control.', '1511.04162-3-0-5': 'Furthermore, I demonstrate that the identification strategy of this paper offers a new use of repeated measurements for tightening the identified set.', '1511.04162-3-0-6': 'Keywords: Local average treatment effect; Instrumental variable; Nonclassical measurement error; Endogenous measurement error; Partial identification', '1511.04162-3-1-0': '# Introduction', '1511.04162-3-2-0': 'Treatment effect analyses often entail a measurement error problem as well as an endogeneity problem.', '1511.04162-3-2-1': 'For example, [CITATION] document a substantial measurement error in educational attainments in the 1990 U.S. Census.', '1511.04162-3-2-2': 'At the same time, educational attainments are endogenous treatment variables in a return to schooling analysis, because unobserved individual ability affects both schooling decisions and wages .', '1511.04162-3-2-3': 'The econometric literature, however, has offered only a few solutions for addressing the two problems at the same time.', '1511.04162-3-2-4': 'An instrumental variable is a standard technique for correcting endogeneity and measurement error , but, to the best of my knowledge, no existing research has explicitly investigated the identifying power of an instrumental variable for the heterogeneous treatment effect when the treatment is both mismeasured and endogenous.', '1511.04162-3-3-0': 'I consider a mismeasured treatment in the framework of [CITATION] and [CITATION], and focus on the local average treatment effect as a parameter of interest.', '1511.04162-3-3-1': 'My analysis studies the identifying power of a binary instrumental variable under the following two assumptions: (i) the instrument affects the outcome and the measured treatment only through the true treatment (the exclusion restriction of an instrument), and (ii) the instrument weakly increases the true treatment (the deterministic monotonicity of the true treatment in the instrument).', '1511.04162-3-3-2': 'These assumptions are an extension of [CITATION] and [CITATION] into the framework with mismeasured treatment.', '1511.04162-3-3-3': 'The local average treatment effect is the average treatment effect for the compliers, that is, the subpopulation whose true treatment status is strictly affected by an instrument.', '1511.04162-3-3-4': 'Focusing on the local average treatment effect is meaningful for a few reasons.', '1511.04162-3-3-5': 'First, the local average treatment effect has been a widely used parameter to investigate the heterogeneous treatment effect with endogeneity.', '1511.04162-3-3-6': 'My analysis offers a tool for a robustness check to those who have already investigated the local average treatment effect.', '1511.04162-3-3-7': 'Second, the local average treatment effect can be used to extrapolate to the average treatment effect or other parameters of interest.', '1511.04162-3-3-8': '[CITATION] emphasize the utility of reporting the local average treatment effect in addition to the other parameters of interest, because the extrapolation often requires additional assumptions and can be less credible than the local average treatment effect.', '1511.04162-3-4-0': 'The mismeasured treatment prevents the local average treatment effect from being point-identified.', '1511.04162-3-4-1': 'As in [CITATION] and [CITATION], the local average treatment effect is the ratio of the intent-to-treat effect over the size of compliers.', '1511.04162-3-4-2': 'Since the measured treatment is not the true treatment, however, the size of compliers is not identified and therefore the local average treatment effect is not identified.', '1511.04162-3-4-3': 'The under-identification for the local average treatment effect is a consequence of the under-identification for the size of compliers; if I assumed no measurement error, I could compute the size of compliers based on the measured treatment and therefore the local average treatment effect would be the Wald estimand.', '1511.04162-3-5-0': 'I take a worst case scenario approach with respect to the measurement error and allow for a general form of measurement error.', '1511.04162-3-5-1': 'The only assumption concerning the measurement error is its independence of the instrumental variable.', '1511.04162-3-5-2': '(Section [REF] dispenses with this assumption and shows that it is still possible to bound the local average treatment effect.)', '1511.04162-3-5-3': 'I consider the following types of measurement error.', '1511.04162-3-5-4': 'First, the measurement error is nonclassical; that is, it can be dependent on the true treatment.', '1511.04162-3-5-5': 'The measurement error for a binary variable is always nonclassical.', '1511.04162-3-5-6': 'It is because the measurement error cannot be negative (positive) when the true variable takes the low (high) value.', '1511.04162-3-5-7': 'Second, I allow the measurement error to be endogenous (or differential); that is, the measured treatment can be dependent on the outcome conditional on the true treatment.', '1511.04162-3-5-8': 'For example, as [CITATION] argue, the measurement error for educational attainment depends on the familiarity with the educational system in the U.S., and immigrants may have a higher rate of measurement error.', '1511.04162-3-5-9': 'At the same time, the familiarity with the U.S. educational system can be related to the English language skills, which can affect the labor market outcomes.', '1511.04162-3-5-10': '[CITATION] also argue that measurement error is likely to be endogenous in some empirical applications.', '1511.04162-3-5-11': '(In Appendix D, I explore for the identifying power of the exogeneity assumption on the measurement error.', '1511.04162-3-5-12': 'The additional assumption yields a tighter sharp identified set, but I still cannot point identify the local average treatment effect in general.)', '1511.04162-3-5-13': 'Third, there is no assumption concerning the marginal distribution of the measurement error.', '1511.04162-3-5-14': 'It is not necessary to assume anything about the accuracy of the measurement.', '1511.04162-3-6-0': 'In the presence of measurement error, I derive the identified set for the local average treatment effect (Theorem [REF]).', '1511.04162-3-7-0': 'Figure [REF] describes the relationship among the identified set for the local average treatment effect, the intent-to-treat effect, and the Wald estimand.', '1511.04162-3-7-1': 'First, the intent-to-treat effect has the same sign as the local average treatment effect.', '1511.04162-3-7-2': 'This is why Figure [REF] has three subfigures according to the sign of the intent-to-treat effect: (a) positive, (b) zero, and (c) negative.', '1511.04162-3-7-3': 'Second, the intent-to-treat effect is the sharp lower bound on the local average treatment effect in absolute value.', '1511.04162-3-7-4': 'Third, the Wald estimand is an upper bound on the local average treatment effect in absolute value.', '1511.04162-3-7-5': 'The Wald estimand is the probability limit of the instrumental variable estimator in my framework, which ignores the measurement error but controls only for the endogeneity.', '1511.04162-3-7-6': 'This point implies that an upper bound on the local average treatment effect is obtained by ignoring the measurement error.', '1511.04162-3-7-7': '[CITATION] obtain a similar result in the homogeneous treatment effect model.', '1511.04162-3-7-8': 'Last, but most importantly, the sharp upper bound in absolute value can be smaller than the Wald estimand.', '1511.04162-3-7-9': 'It is a potential cost of ignoring the measurement error and using the Wald estimand.', '1511.04162-3-7-10': 'Even for analyzing only an upper bound on the local average treatment effect, it is recommended to take the measurement error into account, which can yield a smaller upper bound than the Wald estimand.', '1511.04162-3-7-11': 'Section [REF] investigates when the Wald estimand coincide with the sharp upper bound.', '1511.04162-3-8-0': 'I extend the identification analysis to incorporate covariates other than the treatment variable.', '1511.04162-3-8-1': 'In this setting, the instrumental variable satisfies the exclusion restriction after conditioning covariates.', '1511.04162-3-8-2': 'Based on the insights from [CITATION] and [CITATION], I show that the identification strategy of this paper works in the presence of covariates.', '1511.04162-3-9-0': 'I construct a confidence interval for the local average treatment effect.', '1511.04162-3-9-1': 'To construct the confidence interval, first, I approximate the identified set by discretizing the support of the outcome where the discretization becomes finer as the sample size increases.', '1511.04162-3-9-2': 'The approximation for the identified set resembles many moment inequalities in [CITATION] and [CITATION], who consider a finite but divergent number of moment inequalities.', '1511.04162-3-9-3': 'I apply a bootstrap method in [CITATION] to construct a confidence interval with uniformly asymptotically valid asymptotic size control.', '1511.04162-3-9-4': 'The confidence interval also rejects parameter values which do not belong to the sharp identified set.', '1511.04162-3-9-5': 'An empirical excise and a Monte Carlo simulation demonstrate a finite sample property of the proposed inference method.', '1511.04162-3-9-6': 'The empirical exercise is based on [CITATION], who studies the effects of 401(k) participation on financial savings, and considers a misclassification of the 401(k) participation.', '1511.04162-3-10-0': 'As an extension, I consider the dependence between the instrument and the measurement error.', '1511.04162-3-10-1': 'In this case, there is no assumption on the measurement error, and therefore the measured treatment has no information on the local average treatment effect.', '1511.04162-3-10-2': 'Even without using the measured treatment, however, I can still apply the same identification strategy and obtain finite (but less tight) bounds on the local average treatment effect.', '1511.04162-3-11-0': 'Moreover, I offer a new use of repeated measurements as additional sources for identification.', '1511.04162-3-11-1': 'The existing practice of repeated measurements uses one of them as an instrumental variable, as in [CITATION], [CITATION], [CITATION], and [CITATION].', '1511.04162-3-11-2': 'However, when the true treatment is endogenous, the repeated measurements are likely to be endogenous and are not good candidates for an instrumental variable.', '1511.04162-3-11-3': 'My identification strategy demonstrates that those variables are useful for bounding the local average treatment effect in the presence of measurement error, even if none of the repeated measurement are valid instrumental variables.', '1511.04162-3-12-0': 'The remainder of this paper is organized as follows.', '1511.04162-3-12-1': 'Section [REF] explains several empirical examples motivating mismeasured endogenous treatments and Section [REF] reviews the related econometric literature.', '1511.04162-3-12-2': 'Section [REF] introduces mismeasured treatments in the framework of [CITATION] and [CITATION].', '1511.04162-3-12-3': 'Section [REF] constructs the identified set for the local average treatment effect.', '1511.04162-3-12-4': 'I also discuss two extensions.', '1511.04162-3-12-5': 'One extension describes how repeated measurements tighten the identified set even if I cannot use any of the repeated measurements as an instrumental variable, and the other dispenses with independence between the instrument and the measurement error.', '1511.04162-3-12-6': 'Section [REF] proposes an inference procedure for the local average treatment effect.', '1511.04162-3-12-7': 'Section [REF] conducts an empirical illustrations, and Section [REF] conducts Monte Carlo simulations.', '1511.04162-3-12-8': 'Section [REF] concludes.', '1511.04162-3-12-9': 'Appendix collects proofs and remarks.', '1511.04162-3-13-0': '## Examples for mismeasured endogenous treatments', '1511.04162-3-14-0': 'I introduce several empirical examples in which binary treatments can be both endogenous and mismeasured at the same time.', '1511.04162-3-14-1': 'The first example is the return to schooling, in which the outcome is wages and the treatment is educational attainment, for example, whether a person has completed college or not.', '1511.04162-3-14-2': 'Unobserved individual ability affects both the schooling decision and wage determination, which leads to the endogeneity of educational attainment in the wage equation (see, for example, [CITATION]).', '1511.04162-3-14-3': "Moreover, survey datasets record educational attainments based on the interviewee's answers, and these self-reported educational attainments are subject to measurement error.", '1511.04162-3-14-4': '[CITATION], [CITATION], [CITATION],alpcard:2001, [CITATION] have pointed out the mismeasurement of educational attainments.', '1511.04162-3-14-5': 'For example, [CITATION] estimate that the 1990 Decennial Census has 17.7 false positive rate of reporting a doctoral degree.', '1511.04162-3-15-0': 'The second example is labor supply response to welfare program participation, in which the outcome is employment status and the treatment is welfare program participation.', '1511.04162-3-15-1': 'Self-reported welfare program participation in survey datasets can be mismeasured .', '1511.04162-3-15-2': "The psychological cost for welfare program participation, welfare stigma, affects job search behavior and welfare program participation simultaneously; that is, welfare stigma may discourage individuals from participating in a welfare program, and, at the same time, affect an individual's effort in the labor market (see [CITATION] and [CITATION] for a discussion on the welfare stigma).", '1511.04162-3-15-3': 'Moreover, the welfare stigma gives welfare recipients some incentive not to reveal their participation status to the survey, which causes endogenous measurement error in that the unobserved individual heterogeneity affects both the measurement error and the outcome.', '1511.04162-3-16-0': 'The third example is the effect of a job training program on wages.', '1511.04162-3-16-1': 'As it is similar to the return to schooling, unobserved individual ability plays a key role in this example.', '1511.04162-3-16-2': 'Self-reported completion of job training program is also subject to measurement error .', '1511.04162-3-16-3': '[CITATION] develop a methodology for evaluating a homogeneous treatment effect with mismeasured endogenous treatment, and apply their methodology to evaluate the effect of a job training program on wages.', '1511.04162-3-17-0': 'The last example is the effect of maternal drug use on infant birth weight.', '1511.04162-3-17-1': '[CITATION] estimate that a mother tends to underreport her drug use, but, at the same time, she tends to report it correctly if she is a heavy user.', '1511.04162-3-17-2': 'When the degree of drug addiction is not observed, it becomes an individual unobserved heterogeneity which affects infant birth weight and the measurement in addition to the drug use.', '1511.04162-3-18-0': '## Literature review', '1511.04162-3-19-0': 'Here I summarize the related econometric literature.', '1511.04162-3-19-1': '[CITATION], [CITATION], and [CITATION] use an instrumental variable to correct for measurement error in a binary (or discrete) treatment in the homogeneous treatment effect framework and they achieve nonparametric point identification of the average treatment effect.', '1511.04162-3-19-2': 'They assume that the true treatment is exogenous, whereas I allow it to be endogenous.', '1511.04162-3-20-0': 'Finite mixture models are related to my analysis.', '1511.04162-3-20-1': 'I consider the unobserved binary treatment, whereas finite mixture models deal with unobserved type.', '1511.04162-3-20-2': '[CITATION] and [CITATION] are the most closely related.', '1511.04162-3-20-3': 'They investigate the identification problem in finite mixture models, by using the exclusion restriction in which an instrumental variable only affects the mixing distribution of a type without affecting the component distribution (that is, the conditional distribution given the type).', '1511.04162-3-20-4': 'If I applied their approach directly to my framework, their exclusion restriction would imply conditional independence between the instrumental variable and the outcome given the true treatment.', '1511.04162-3-20-5': 'This conditional independence implies that the local average treatment effect does not exhibit essential heterogeneity and that the local average treatment effect is the mean difference between the control and treatment groups.', '1511.04162-3-20-6': 'Instead of applying the approaches in [CITATION] and [CITATION], I use a different exclusion restriction in which the instrumental variable does not affect the outcome or the measured treatment directly.', '1511.04162-3-21-0': 'A few papers have applied an instrumental variable to a mismeasured binary regressor in the homogenous treatment effect framework.', '1511.04162-3-21-1': 'They include [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1511.04162-3-21-2': '[CITATION] and [CITATION] are the most closely related among them, since they allow for endogeneity.', '1511.04162-3-21-3': 'Here I allow for heterogeneous treatment effects, and I contribute to the heterogeneous treatment effect literature by investigating the consequences of the measurement errors in the treatment.', '1511.04162-3-22-0': '[CITATION], [CITATION], [CITATION], and [CITATION] apply a partial identification strategy for the average treatment effect to the mismeasured binary regressor problem by utilizing the knowledge of the marginal distribution for the true treatment.', '1511.04162-3-22-1': 'Those papers use auxiliary datasets to obtain the marginal distribution for the true treatment.', '1511.04162-3-22-2': '[CITATION] is the most closely related, in that they allow for both treatment endogeneity and endogenous measurement error.', '1511.04162-3-22-3': 'My instrumental variable approach can be an an alternative strategy to deal with mismeasured endogenous treatment.', '1511.04162-3-22-4': 'It is worthwhile because, as mentioned in [CITATION], the availability of an auxiliary dataset is limited in empirical research.', '1511.04162-3-22-5': 'Furthermore, it is not always the case that the results from auxiliary datasets is transported into the primary dataset ,', '1511.04162-3-23-0': 'Some papers investigate mismeasured endogenous continuous variables, instead of binary variables.', '1511.04162-3-23-1': '[CITATION] consider nonlinear models with mismeasured continuous explanatory variables.', '1511.04162-3-23-2': 'The continuity of the treatment is crucial for their analysis, because they assume classical measurement error.', '1511.04162-3-23-3': 'The treatment in my analysis is binary and therefore the measurement error is nonclassical.', '1511.04162-3-23-4': '[CITATION] consider mismeasured endogenous continuous variables in single index models.', '1511.04162-3-23-5': 'However, their approach depends on taking derivatives of the conditional expectations with respect to the continuous variable.', '1511.04162-3-23-6': 'It is not clear if it can be extended to binary variables.', '1511.04162-3-23-7': '[CITATION] considers the semi-parametric model when endogenous continuous variables are subject to nonclassical measurement error.', '1511.04162-3-23-8': 'He assumes conditional independence between the instrumental variable and the outcome given the true treatment, which would impose some structure on the outcome equation when a treatment is binary (see Footnote [REF]).', '1511.04162-3-23-9': 'Instead I propose an identification strategy without assuming any structure on the outcome equation.', '1511.04162-3-24-0': '[CITATION] investigates the consequences of measurement error in the instrumental variable instead of the treatment.', '1511.04162-3-24-1': 'He assumes that the treatment is perfectly observed, whereas I allow for it to be measured with error.', '1511.04162-3-24-2': 'Since I assume that the instrumental variable is perfectly observed, my analysis is not overlapped with [CITATION].', '1511.04162-3-25-0': '[CITATION], [CITATION], and [CITATION] have similar identification strategy in the context of sample selection models.', '1511.04162-3-25-1': 'These papers also use the exclusion restriction of the instrumental variable for their partial identification results.', '1511.04162-3-25-2': 'Particularly, [CITATION] derives the integrated envelope from the exclusion restriction, which is similar to the total variation distance in my analysis because both of them are characterized as a supremum over the set of the partitions.', '1511.04162-3-25-3': 'First and the most importantly, I consider mismeasurement of the treatment, whereas the sample selection model considers truncation of the outcome.', '1511.04162-3-25-4': 'It is not straightforward to apply their methodologies in sample selection models into mismeasured treatment problem.', '1511.04162-3-25-5': 'Second, I offer an inference method with uniform size control, but [CITATION] derives only point-wise size control.', '1511.04162-3-25-6': 'Last, [CITATION] and [CITATION] use their result for specification test, but I cannot use it to carry out a specification test because the sharp identified set of my analysis is always non-empty.', '1511.04162-3-26-0': 'Finally, [CITATION] and [CITATION] have recently discussed identification issues of the local average treatment effect in the presence of a measurement error in the treatment variable.', '1511.04162-3-26-1': 'They are built on results in the previous draft of this paper to derive novel and important results when there are additional variables in a dataset: multiple measurements of the true treatment variable or multiple instrumental variables .', '1511.04162-3-26-2': 'In contrast, the results of this paper are valid without these additional variables and only requires the assumptions in [CITATION] and [CITATION].', '1511.04162-3-27-0': '# Local average treatment effect framework with misclassification', '1511.04162-3-28-0': 'My analysis considers a mismeasured treatment in the framework of [CITATION] and [CITATION].', '1511.04162-3-28-1': 'The objective is to evaluate the causal effect of a binary treatment [MATH] on an outcome [MATH], where [MATH] represents the control group and [MATH] represents the treatment group.', '1511.04162-3-28-2': 'To deal with endogeneity of [MATH], I use a binary instrumental variable [MATH] which shifts [MATH] exogenously without any direct effect on [MATH].', '1511.04162-3-28-3': 'The treatment [MATH] of interest is not directly observed, and instead there is a binary measurement [MATH] for [MATH].', '1511.04162-3-28-4': 'I put the [MATH] symbol on [MATH] to emphasize that the true treatment [MATH] is unobserved.', '1511.04162-3-28-5': 'I allow [MATH] to be discrete, continuous or mixed; [MATH] is only required to have some known dominating finite measure [MATH] on the real line.', '1511.04162-3-28-6': 'For example, [MATH] can be the Lebesgue measure or the counting measure.', '1511.04162-3-28-7': 'Let [MATH] be the support for the random variable [MATH] and [MATH] be the support for [MATH].', '1511.04162-3-29-0': 'To describe the data generating process, I consider the counterfactual variables.', '1511.04162-3-29-1': '[MATH] is the counterfactual true treatment when [MATH].', '1511.04162-3-29-2': '[MATH] is the counterfactual outcome when [MATH].', '1511.04162-3-29-3': '[MATH] is the counterfactual measured treatment when [MATH].', '1511.04162-3-29-4': 'The individual treatment effect is [MATH].', '1511.04162-3-29-5': 'It is not directly observed; [MATH] and [MATH] are not observed at the same time.', '1511.04162-3-29-6': 'Only [MATH] is observable.', '1511.04162-3-29-7': 'Using the notation, the observed variables [MATH] are generated by the three equations: [EQUATION]', '1511.04162-3-29-8': 'Figure [REF] graphically describes the relationship among the instrument [MATH], the (unobserved) true treatment [MATH], the measured treatment [MATH], and the outcome [MATH].', '1511.04162-3-30-0': '([REF]) is the measurement equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-3-30-1': '[MATH] is the measurement error; [MATH] (or [MATH]) represents a false positive and [MATH] (or [MATH]) represents a false negative.', '1511.04162-3-30-2': 'Equations ([REF]) and ([REF]) are the same as [CITATION] and [CITATION].', '1511.04162-3-30-3': '([REF]) is the outcome equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-3-30-4': '([REF]) is the treatment assignment equation, which is the arrow from [MATH] to [MATH] in Figure [REF].', '1511.04162-3-30-5': 'A potentially non-zero correlation between [MATH] and [MATH] causes an endogeneity problem.', '1511.04162-3-31-0': 'In a return to schooling analysis, [MATH] is wages, [MATH] is the true indicator for college completion, [MATH] is the proximity to college, and [MATH] is the self-reported college completion.', '1511.04162-3-31-1': 'The treatment effect [MATH] in the return to schooling is the effect of college completion [MATH] on wages [MATH].', '1511.04162-3-31-2': 'The college completion is not correctly measured in a survey dataset, such that only the self report [MATH] is observed.', '1511.04162-3-32-0': 'This section and Section [REF] impose only the following assumption.', '1511.04162-3-33-0': '(i) For each [MATH], [MATH] is independent of [MATH].', '1511.04162-3-33-1': '(ii) [MATH] almost surely.', '1511.04162-3-33-2': '(iii) [MATH].', '1511.04162-3-34-0': 'Assumption [REF] (i) is the exclusion restriction and I consider stochastic independence instead of mean independence.', '1511.04162-3-34-1': 'Although it is stronger than the minimal conditions for the identification for the local average treatment effect without measurement error, a large part of the existing applied papers assume stochastic independence .', '1511.04162-3-34-2': '[MATH] is also independent of [MATH] conditional on [MATH], which is the only assumption on the measurement error for the identified set in Section [REF].', '1511.04162-3-34-3': '(Section [REF] even dispenses with this assumption.)', '1511.04162-3-34-4': 'Assumption [REF] (ii) is the monotonicity condition for the instrument, in which the instrument [MATH] increases the value of [MATH] for all the individuals.', '1511.04162-3-34-5': '[CITATION] relaxes the monotonicity condition, and it can be shown in Appendix E that the identification results in my analysis still holds with a slight modification under the complier-defiers-for-marginals condition in [CITATION].', '1511.04162-3-34-6': 'Note that Assumption [REF] does not include a relevance condition for the instrumental variable.', '1511.04162-3-34-7': 'The standard relevance condition [MATH] does not affect the identification results in my analysis.', '1511.04162-3-34-8': 'I will discuss the relevance condition in my framework after Theorem [REF].', '1511.04162-3-34-9': 'Assumption [REF] (iii) excludes that [MATH] is constant.', '1511.04162-3-35-0': 'As I emphasized in the introduction, the framework here does not assume anything on measurement error [MATH] except for its independence from [MATH].', '1511.04162-3-35-1': 'Assumption [REF] does not impose any restriction on the marginal distribution of the measurement error [MATH] or on the relationship between the measurement error [MATH] and [MATH].', '1511.04162-3-35-2': 'Particularly, the measurement error can be endogenous, that is, [MATH] and [MATH] can be correlated.', '1511.04162-3-36-0': 'I focus on the local average treatment effect, which is defined by [EQUATION].', '1511.04162-3-36-1': 'The local average treatment effect is the average of the treatment effect [MATH] over the subpopulation (the compliers) whose treatment status is strictly affected by the instrument.', '1511.04162-3-36-2': '[CITATION] show that the local average treatment effect equals [EQUATION] where I define [MATH] for a random variable [MATH].', '1511.04162-3-36-3': 'Note that [MATH] is the intent-to-treat effect, that is, the regression of [MATH] on [MATH].', '1511.04162-3-36-4': 'The treatment is measured with error, and therefore the above fraction [MATH] is not the Wald estimand [EQUATION].', '1511.04162-3-36-5': 'Since [MATH] is not identified, I cannot identify the local average treatment effect.', '1511.04162-3-36-6': 'The failure of point identification comes purely from the measurement error, because the local average treatment effect would be point-identified under [MATH].', '1511.04162-3-36-7': 'In fact, my proposed methodology in this paper is essentially a bounding strategy of [MATH] and I use the bound to construct the sharp identified set for the local average treatment effect.', '1511.04162-3-37-0': '# Identified set for the local average treatment effect', '1511.04162-3-38-0': 'This section show how the instrumental variable partially identifies the local average treatment effect in the framework of Section [REF].', '1511.04162-3-38-1': 'Before defining the identified set, I express the local average treatment effect as a function of the underlying distribution [MATH] of [MATH].', '1511.04162-3-38-2': 'I use the [MATH] symbol on [MATH] to clarify that [MATH] is the distribution of the unobserved variables.', '1511.04162-3-38-3': 'I denote the expectation operator [MATH] by [MATH] when I need to clarify the underlying distribution.', '1511.04162-3-38-4': 'The local average treatment effect is a function of the unobserved distribution [MATH]: [EQUATION].', '1511.04162-3-38-5': 'I denote by [MATH] the parameter space for the local average treatment effect [MATH], that is, the set of [MATH] where [MATH] and [MATH] are density functions dominated by the known probability measure [MATH].', '1511.04162-3-38-6': 'For example, [MATH] when [MATH] is binary.', '1511.04162-3-39-0': 'The identified set is the set of parameter values for the local average treatment effect which is consistent with the distribution of the observed variables.', '1511.04162-3-39-1': 'I use [MATH] for the distribution of the observed variables [MATH] The equations ([REF]), ([REF]), and ([REF]) induce the distribution of the observables [MATH] from the unobserved distribution [MATH], and I denote by [MATH] the induced distribution.', '1511.04162-3-39-2': "When the distribution of [MATH] is [MATH], the set of [MATH] which induces [MATH] is [MATH], where [MATH] is the set of [MATH]'s satisfying Assumptions [REF].", '1511.04162-3-39-3': 'For every distribution [MATH] of [MATH], the (sharp) identified set for the local average treatment effect is defined as [MATH].', '1511.04162-3-40-0': '[CITATION] provides a relationship between [MATH] and the local average treatment effect: [EQUATION]', '1511.04162-3-40-1': 'This equation gives the two pieces of information of [MATH].', '1511.04162-3-40-2': 'First, the sign of [MATH] is the same as [MATH].', '1511.04162-3-40-3': 'Second, the absolute value of [MATH] is at least the absolute value of [MATH].', '1511.04162-3-40-4': 'The following lemma summarizes these two pieces.', '1511.04162-3-41-0': 'Under Assumption [REF], [EQUATION].', '1511.04162-3-42-0': 'I derive a new implication from the exclusion restriction for the instrumental variable in order to obtain an upper bound on [MATH] in absolute value.', '1511.04162-3-42-1': 'To explain the new implication, I introduce the total variation distance, which is the [MATH] distance between the distribution [MATH] and [MATH]: For any random variable [MATH], define [EQUATION] where [MATH] is a dominating measure for the distribution of [MATH].', '1511.04162-3-43-0': 'Under Assumption [REF], [EQUATION].', '1511.04162-3-44-0': 'The first term, [MATH], in Lemma [REF] reflects the dependency of [MATH] on [MATH], and it can be interpreted as the magnitude of the distributional effect of [MATH] on [MATH].', '1511.04162-3-44-1': 'The second and third terms, [MATH] and [MATH], are the effect of the instrument [MATH] on the true treatment [MATH].', '1511.04162-3-44-2': 'Based on Lemma [REF], the magnitude of the effect of [MATH] on [MATH] is no smaller than the magnitude of the effect of [MATH] on [MATH].', '1511.04162-3-45-0': 'The new implication in Lemma [REF] gives a lower bound on [MATH] and therefore yields an upper bound on the local average treatment effect in absolute value, combined with equation ([REF]).', '1511.04162-3-45-1': 'Therefore, I use these relationships to derive an upper bound on the local average treatment effect in absolute value, that is, [EQUATION] as long as [MATH].', '1511.04162-3-46-0': 'Theorem [REF] shows that the above observations characterize the sharp identified set for the local average treatment effect.', '1511.04162-3-47-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-47-1': 'The identified set [MATH] for the local average treatment effect is characterized as follows: [MATH] if [MATH]; otherwise, [EQUATION].', '1511.04162-3-48-0': 'The total variation distance [MATH] plays two roles in determining the sharp identified set in this theorem.', '1511.04162-3-48-1': 'First, [MATH] measures the strength of the instrumental variable, that is, [MATH] is the relevance condition in my identification analysis.', '1511.04162-3-48-2': 'When [MATH], the interval in the above theorem is always nonempty and bounded, which implies that [MATH] has some identifying power for the local average treatment effect.', '1511.04162-3-48-3': 'By contrast, [MATH] means that the instrumental variable [MATH] does not affect [MATH] and [MATH], in which case [MATH] has no identifying power for the local average treatment effect.', '1511.04162-3-48-4': 'In this case, [MATH] almost everywhere over [MATH] and particularly [MATH].', '1511.04162-3-48-5': 'Note that all the three inequalities in Theorem [REF] have no restriction on [MATH] in this case.', '1511.04162-3-48-6': 'Second, [MATH] determines the length of the sharp identified set.', '1511.04162-3-48-7': 'The length is [MATH], which is a decreasing function in [MATH].', '1511.04162-3-49-0': 'In general, the lower and upper bounds of the sharp identified set are not equal to the local average treatment effect.', '1511.04162-3-49-1': 'The lower bound is weakly smaller (in the absolute value) than the local average treatment effect, because the size of the compliers is weakly smaller than one.', '1511.04162-3-49-2': 'The upper bound is weakly larger (in the absolute value) than the local average treatment effect, because [MATH] is weakly smaller than the size of the compliers due to the mis-measurement of the treatment variable.', '1511.04162-3-50-0': 'The standard relevance condition [MATH] is not required in Theorem [REF].', '1511.04162-3-50-1': '[MATH] is a necessary condition to define the Wald estimand, but the sharp identified set does not depend directly on the Wald estimand.', '1511.04162-3-50-2': 'In fact, [MATH] in Theorem [REF] is weaker than [MATH].', '1511.04162-3-51-0': 'Note that the sharp identified set is always non-empty.', '1511.04162-3-51-1': 'There is no testable implications on the distribution of the observed variables, and therefore it is impossible to conduct a specification test for Assumption [REF].', '1511.04162-3-52-0': '## Wald estimand and the identified set', '1511.04162-3-53-0': 'The Wald estimand [MATH] can be outside the identified set.', '1511.04162-3-53-1': 'One necessary and sufficient condition for the Wald estimand to be included in the identified set is given as follows.', '1511.04162-3-54-0': 'The Wald estimand is in the identified set if and only if [EQUATION]', '1511.04162-3-54-1': 'This condition in ([REF]) are the testable implications from the the local average treatment effect framework without measurement error ([CITATION] and [CITATION]).', '1511.04162-3-54-2': 'The recent papers by [CITATION], [CITATION], and [CITATION] propose the testing procedures for ([REF]).', '1511.04162-3-54-3': 'Based on the results in Theorem [REF], their testing procedures are re-interpreted as a test for the null hypothesis that the Wald estimand is inside the sharp upper bound on the local average treatment effect.', '1511.04162-3-55-0': '## Conditional exogeneity of the instrumental variable', '1511.04162-3-56-0': 'As in [CITATION] and [CITATION], this section considers the conditional exogeneity of the instrumental variable [MATH] in which [MATH] is exogenous given a set of covariates [MATH], which weaker than the unconditional exogeneity in Assumption [REF].', '1511.04162-3-57-0': 'There is some variable [MATH] taking values in a set [MATH] satisfying the following properties.', '1511.04162-3-57-1': '(i) For each [MATH], [MATH] is conditionally independent of [MATH] given [MATH].', '1511.04162-3-57-2': '(ii) [MATH] almost surely.', '1511.04162-3-57-3': '(iii) [MATH].', '1511.04162-3-58-0': 'I define the [MATH]-conditional total variation distance by [EQUATION].', '1511.04162-3-58-1': 'Note that [MATH] is a random variable as a function of [MATH].', '1511.04162-3-58-2': 'Under the conditional exogeneity of [MATH], Theorem [REF] becomes as follows.', '1511.04162-3-59-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-59-1': 'The identified set [MATH] for the local average treatment effect is characterized as follows: [MATH] if [MATH]; otherwise, [EQUATION]', '1511.04162-3-60-0': '## Identifying power of repeated measurements', '1511.04162-3-61-0': 'The identification strategy in the above analysis offers a new use of repeated measurements as additional sources for identification.', '1511.04162-3-61-1': 'Repeated measurements is a popular approach in the literature on measurement error, but they cannot be instrumental variables in this framework.', '1511.04162-3-61-2': 'This is because the true treatment [MATH] is endogenous and it is natural to suspect that a measurement of [MATH] is also endogenous.', '1511.04162-3-61-3': 'The more accurate the measurement is, the more likely it is to be endogenous.', '1511.04162-3-61-4': 'Nevertheless, the identification strategy incorporates repeated measurements as an additional information to tighten the identified set for the local average treatment effect, when they are coupled with the instrumental variable [MATH].', '1511.04162-3-61-5': 'Unlike the other paper on repeated measurements, I do not need to assume the independence of measurement errors among multiple measurements.', '1511.04162-3-61-6': 'The strategy also benefits from having more than two measurements unlike [CITATION] who achieve point identification with two measurements.', '1511.04162-3-62-0': 'Consider a repeated measurement [MATH] for [MATH].', '1511.04162-3-62-1': 'I do not require that [MATH] is binary, so [MATH] can be discrete or continuous.', '1511.04162-3-62-2': 'Like [MATH], I model [MATH] using the counterfactual outcome notations.', '1511.04162-3-62-3': '[MATH] is a counterfactual second measurement when the true treatment [MATH] is [MATH], and [MATH] is a counterfactual second measurement when the true treatment [MATH] is [MATH].', '1511.04162-3-62-4': 'Then the data generation of [MATH] is [EQUATION].', '1511.04162-3-62-5': 'I strengthen Assumption [REF] by assuming that the instrumental variable [MATH] is independent of [MATH] conditional on [MATH].', '1511.04162-3-63-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-3-63-1': '(ii) [MATH] almost surely.', '1511.04162-3-63-2': '(iii) [MATH].', '1511.04162-3-64-0': 'Note that I do not assume the independence between [MATH] and [MATH], where the independence between the measurement errors is a key assumption when the repeated measurement is an instrumental variable.', '1511.04162-3-64-1': 'Assumption [REF] tightens the identified set for the local average treatment effect as follows.', '1511.04162-3-65-0': 'The requirement on [MATH] does not restrict [MATH] to have the same support as [MATH].', '1511.04162-3-65-1': 'In fact, [MATH] can be any variable which depends on [MATH].', '1511.04162-3-65-2': 'For example, [MATH] can be another outcome variable than [MATH].', '1511.04162-3-66-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-66-1': 'The identified set [MATH] for the local average treatment effect is characterized as follows: [MATH] if [MATH]; otherwise, [EQUATION].', '1511.04162-3-67-0': 'The identified set in Theorem [REF] is weakly smaller than the identified set in Theorem [REF].', '1511.04162-3-67-1': 'The total variation distance [MATH] in Theorem [REF] is weakly larger than that in Theorem [REF], because, using the triangle inequality, [EQUATION] and the strict inequality holds unless the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-3-67-2': 'Therefore, it is possible to test whether the repeated measurement [MATH] has additional information, by testing whether the sign of [MATH] is constant in [MATH].', '1511.04162-3-68-0': '## Dependence between measurement error and instrumental variable', '1511.04162-3-69-0': 'It is still possible to apply the same identification strategy and obtain finite (but less tight) bounds on the local average treatment effect, even without the independence between the instrumental variable and the measurement error.', '1511.04162-3-69-1': '(Assumption [REF] (i) implies that [MATH] is independent of [MATH] for each [MATH].)', '1511.04162-3-69-2': 'Instead Assumption [REF] is weakened to allow for the measurement error [MATH] to be correlated with the instrumental variable [MATH].', '1511.04162-3-70-0': '(i) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-3-70-1': '(ii) [MATH] almost surely.', '1511.04162-3-70-2': '(iii) [MATH].', '1511.04162-3-71-0': 'Theorem [REF] shows that the above observations characterize the identified set for the local average treatment effect under Assumption [REF].', '1511.04162-3-72-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-72-1': 'The identified set [MATH] for the local average treatment effect is characterized as follows: [MATH] if [MATH]; otherwise, [EQUATION].', '1511.04162-3-73-0': 'The difference from Theorem [REF] is that Theorem [REF] does not depend on the measured treatment [MATH].', '1511.04162-3-73-1': 'Although it is observed in the dataset, [MATH] does not have any information on the local average treatment effect because Assumption [REF] does not restrict [MATH].', '1511.04162-3-73-2': 'When [MATH], there are nontrivial upper and lower bounds on the local average treatment effect even without using the measured treatment [MATH].', '1511.04162-3-74-0': '# Inference', '1511.04162-3-75-0': 'Based on the sharp identified set in the presence of covariates (Theorem [REF]), this section constructs a confidence interval for the local average treatment effect based on an i.i.d. sample [MATH] of [MATH].', '1511.04162-3-75-1': 'The confidence interval described below controls the asymptotic size uniformly over a class of data generating processes, and rejects all the fixed alternatives.', '1511.04162-3-76-0': 'The identified set in [REF] is characterized by moment inequalities as follows.', '1511.04162-3-77-0': 'Let [MATH] be an arbitrary data distribution of [MATH].', '1511.04162-3-77-1': 'Under Assumption [REF], [MATH] is the set of [MATH] in which [EQUATION] where [MATH], [MATH] is the set of measurable functions on [MATH] taking a value in [MATH] and [MATH].', '1511.04162-3-78-0': 'I construct a [MATH]-confidence interval for the local average treatment effect [MATH] with treating [MATH] as a nuisance parameter for given [MATH].', '1511.04162-3-78-1': 'I assume that a [MATH]-confidence interval [MATH] for [MATH] is available for researchers for given [MATH].', '1511.04162-3-78-2': 'Given [MATH], I construct the [MATH]-confidence interval [MATH] for the local average treatment effect as [EQUATION] where [MATH] and [MATH] are defined below using the bootstrap-based testing .', '1511.04162-3-79-0': 'The number of the moment inequalities in Lemma [REF] can be finite or infinite, which determines whether some of the existing methods can be applied directly to the inference on the local average treatment effect.', '1511.04162-3-79-1': 'When [MATH] has finite supports and therefore [MATH] is finite, the sharp identified set is characterized by a finite number of inequalities, and therefore I can apply inference methods based on unconditional moment inequalities.', '1511.04162-3-79-2': 'To the best of my knowledge, however, inference for the local average treatment effect in my framework does not fall directly into the existing moment inequality models when either [MATH] or [MATH] is continuous.', '1511.04162-3-79-3': 'When either [MATH] or [MATH] is continuous, the sharp identified set is characterized by an uncountably infinite number of inequalities.', '1511.04162-3-79-4': 'In the current literature on the partially identified parameters, an infinite number of moment inequalities are mainly considered in the context of conditional moment inequalities.', '1511.04162-3-79-5': 'The identified set in this paper is not characterized by conditional moment inequalities.', '1511.04162-3-80-0': 'I considers a sequence of finite sets [MATH] which converges to [MATH] as a sample size increases.', '1511.04162-3-80-1': '(The convergence is formally defined in Assumption [REF], and an example for [MATH] appears after Assumption [REF].)', '1511.04162-3-80-2': 'Note that, when [MATH] is finite, [MATH] can be equal to [MATH].', '1511.04162-3-80-3': 'If [MATH] is replaced with [MATH] in Lemma [REF], the number of the moment inequalities becomes finite.', '1511.04162-3-80-4': 'At the same time, as [MATH] approaches to [MATH], the approximation error from using [MATH] converges to zero, and the number of the inequalities can be increasing, particularly diverging to the infinity when [MATH] includes infinite elements.', '1511.04162-3-80-5': 'The approximated identified set is characterized by a finite number of the following moment inequalities: [EQUATION]', '1511.04162-3-80-6': 'Denote by [MATH] the resulting number of moment inequalities, that is, the number of elements in [MATH] plus [MATH].', '1511.04162-3-80-7': 'Note that, when [MATH], the moment inequalities in ([REF]) is equivalent to using the Wald estimand as the upper bound for [MATH].', '1511.04162-3-81-0': 'For the size [MATH], I construct a test statistic [MATH] and a critical value [MATH] via the multiplier bootstrap for many moment inequality models (described in Section [REF]).', '1511.04162-3-81-1': '[CITATION] studies the testing problem for moment inequality models in which the number of the moment inequalities is finite but growing.', '1511.04162-3-81-2': 'Since the number of the moment inequalities in ([REF])-([REF]) is finite but growing, their results are applicable to construct a confidence interval based on ([REF])-([REF]).', '1511.04162-3-82-0': 'Given positive constants [MATH] and [MATH], the class of data generating processes, denoted by [MATH], and the parameter spaces [MATH] satisfy', '1511.04162-3-83-0': 'The first assumption (i) is a regularity condition.', '1511.04162-3-83-1': 'The second assumption (ii) requires researchers to know ex ante upper and lower bounds on the parameter.', '1511.04162-3-83-2': 'The third assumption (iii) guarantees that the test statistic is well-defined.', '1511.04162-3-83-3': 'The fourth assumption (iv) is that the confidence interval for [MATH] controls the size uniformly over [MATH].', '1511.04162-3-83-4': 'The last assumption (v) is that the propensity score [MATH] is bounded away from zero and one.', '1511.04162-3-84-0': 'In this paper I assume that [MATH] satisfies the following conditions.', '1511.04162-3-85-0': '(i) [MATH].', '1511.04162-3-85-1': '(ii) The convergence [EQUATION] holds uniformly over [MATH] and [MATH].', '1511.04162-3-85-2': '(iii) The number of elements in [MATH] satisfies [EQUATION] for some [MATH] and [MATH].', '1511.04162-3-86-0': 'An example of [MATH] is obtained by discretizing [MATH].', '1511.04162-3-86-1': 'Consider a partition [MATH] over [MATH], in which the intervals [MATH] and the grid size [MATH] depend on the sample size [MATH].', '1511.04162-3-86-2': 'Let [MATH] be a generic function of [MATH] into [MATH] that is constant over [MATH] for every [MATH].', '1511.04162-3-86-3': 'Let [MATH] be the set of all such functions.', '1511.04162-3-86-4': 'Lemma [REF] shows that this construction of [MATH] satisfies Eq. ([REF]) under conditions on [MATH] and [MATH].', '1511.04162-3-86-5': 'The conditions in Lemma [REF] guarantee that the approximation error from the discretization vanishes as the sample size [MATH] increases.', '1511.04162-3-87-0': 'It is worthwhile to mention that, when [MATH], the implied upper bound in ([REF]) is equal to the Wald estimand.', '1511.04162-3-87-1': 'It can be smaller than the Wald estimand as long as [MATH],', '1511.04162-3-88-0': 'Assumption [REF] holds if', '1511.04162-3-89-0': 'Theorem [REF] shows asymptotic properties of the confidence interval [MATH].', '1511.04162-3-89-1': 'The first result (i) is the uniform asymptotic size control and the second result (ii) is the consistency against all the fixed alternatives.', '1511.04162-3-90-0': 'Suppose that Assumptions [REF] and [REF] hold.', '1511.04162-3-90-1': '(i) The confidence interval controls the asymptotic size uniformly: [EQUATION] (ii) If Eq. ([REF]) holds, the confidence interval excludes all the fixed alternatives: [EQUATION]', '1511.04162-3-91-0': '# Empirical illustrations', '1511.04162-3-92-0': 'This section studies the effects of 401(k) participation on financial savings using the inference method in Section [REF].', '1511.04162-3-92-1': 'I introduce a measurement error problem to the analysis of [CITATION], which investigates the local average treatment effect using the eligibility for 401(k) program.', '1511.04162-3-92-2': 'The robustness to misclassification is empirically relevant, because the retirement pension plan type is subject to a measurement error in survey datasets.', '1511.04162-3-92-3': 'Using the Health and Retirement Study, for example, [CITATION] estimate that around one fourth of the survey respondents misclassified their pension plan type.', '1511.04162-3-93-0': 'The dataset in my analysis is from the Survey of Income and Program Participation (SIPP) of 1991.', '1511.04162-3-93-1': 'It has been used in various analyses, e.g., [CITATION] and [CITATION].', '1511.04162-3-93-2': 'I follow the data construction in [CITATION].', '1511.04162-3-93-3': 'The sample consists of households in which at least one person is employed, which has no income from self-employment, and whose annual family income is between 10,000 and 200,000.', '1511.04162-3-93-4': 'The resulting sample size is 9,275.', '1511.04162-3-94-0': 'The outcome variable [MATH] is the net financial assets, the measured treatment variable [MATH] is the self-reported participation in 401(k), [MATH] is the eligibility for 401(k) and [MATH] is the participation in an individual retirement account (IRA).', '1511.04162-3-94-1': 'The control variables [MATH] includes constant, family income, age and its square, marital status, and family size.', '1511.04162-3-94-2': 'I compute the summary statistics for these variables in Table [REF].', '1511.04162-3-94-3': 'The 401(k) participation can be endogenous, because participants in 401(k) might be more informed or plan more about retirement savings than non-participants.', '1511.04162-3-94-4': 'To control for the endoeneity problem, this paper uses 401(k) eligibility as an instrumental variable.', '1511.04162-3-95-0': 'I use the linear probability model for the regression of the instrumental variable [MATH] on the control variables [MATH], that is, [MATH].', '1511.04162-3-95-1': 'For a comparison purpose, I compute the Wald estimator, [MATH], with a 95 bootstrapped confidence interval [MATH].', '1511.04162-3-95-2': 'The intent-to-treat effect [MATH] is estimated as [MATH] with a 95 bootstrapped confidence interval [MATH]', '1511.04162-3-96-0': 'Table [REF]-[REF] shows that the confidence intervals for the local average treatment effect, under different assumptions (Theorem [REF], [REF], [REF], respectively).', '1511.04162-3-96-1': 'The confidence intervals in these tables are robust to a misclassification of the treatment variable.', '1511.04162-3-96-2': 'They are wider than the 95% confidence interval [MATH] for the Wald estimator, but are in general comparable to the confidence interval for the Wald estimator.', '1511.04162-3-96-3': 'The confidence intervals in this exercise do not shrink as [MATH] increases from [MATH] to [MATH].', '1511.04162-3-96-4': '(Note that, when [MATH], the moment inequalities in ([REF]) is equivalent to using the Wald estimand as the upper bound for [MATH].)', '1511.04162-3-96-5': 'This is possibly because the data generation process does not violate the conditions in ([REF]) to a large extent, and therefore the Wald estimand is close (even if not equal) to the sharp upper bound for the local average treatment effect.', '1511.04162-3-97-0': 'Table [REF] summarizes the confidence intervals with the IRA participation [MATH] as an additional measurement, as discussed in Theorem [REF].', '1511.04162-3-97-1': 'It shows similar values to Table [REF] and it can be interpreted as a result that the IRA participation [MATH] has only little identifying power on the local average treatment effect in this empirical exercise.', '1511.04162-3-98-0': 'Table [REF] summarizes the confidence intervals without using the measured treatment [MATH], as in Theorem [REF].', '1511.04162-3-98-1': 'The lower bound of the confidence intervals does not change from those in Table [REF], because the lower bound of the identified set does not change without the information from the measured treatment [MATH].', '1511.04162-3-98-2': 'The upper bound is 3-4 times larger than those in Table [REF], which is the cost of not using [MATH].', '1511.04162-3-99-0': '# Numerical example and Monte Carlo simulations', '1511.04162-3-100-0': 'This section considers a numerical example to illustrates the theoretical properties in the previous section.', '1511.04162-3-100-1': 'I consider the following data generating process: [EQUATION] where [MATH] is the standard normal cdf and, conditional on [MATH], [MATH] is drawn from the Gaussian copula with the correlation matrix [EQUATION].', '1511.04162-3-100-2': 'I set [MATH], which captures the degree of the misclassification.', '1511.04162-3-100-3': 'In this design, the treatment variable is endogenous since [MATH] and [MATH] are correlated.', '1511.04162-3-100-4': 'In addition, the misclassification is endogenous in that [MATH] and [MATH] are correlated.', '1511.04162-3-101-0': 'Table [REF] lists the three population objects: the local average treatment effect, the Wald estimand, and the identified set for the local average treatment effect.', '1511.04162-3-101-1': 'Note that, unless [MATH], the distribution for [MATH] violates the conditions in ([REF]).', '1511.04162-3-101-2': 'When there is no measurement error, the sharp upper bound is equal to the Wald estimand, which is the case for [MATH].', '1511.04162-3-101-3': 'When there is a measurement error, the sharp upper bound for the local average treatment effect can be smaller than the Wald estimand.', '1511.04162-3-102-0': 'In order to focus on the finite sample properties of the test [MATH], I only evaluate coverage probabilities given [MATH] for various value of [MATH].', '1511.04162-3-102-1': 'The partition of grids is equally spaced over [MATH] with the number the partitions [MATH].', '1511.04162-3-102-2': 'Coverage probabilities are calculated as how often the [MATH] confidence interval includes a given parameter value out of [MATH] simulations.', '1511.04162-3-102-3': 'The sample size is [MATH] for Monte Carlo simulations.', '1511.04162-3-102-4': 'I use [MATH] bootstrap repetitions to construct critical values.', '1511.04162-3-102-5': 'I set [MATH] for the moment selection.', '1511.04162-3-103-0': 'Figures [REF]-[REF] describe the coverage probabilities of the confidence intervals for each parameter value.', '1511.04162-3-103-1': 'When the degree of measurement error is zero ([MATH]), the power for the confidence interval with [MATH] has a slightly better performance than those with [MATH].', '1511.04162-3-103-2': 'It can be because the number of moment inequalities are larger for [MATH] and then the critical value is bigger.', '1511.04162-3-103-3': 'As the degree of measurement error becomes larger, the power for the confidence intervals with [MATH] becomes better than that with [MATH].', '1511.04162-3-103-4': 'It is a result of the fact that the sharp upper bound for the local average treatment effect is smaller than the Wald estimand.', '1511.04162-3-104-0': 'Next, I investigate the identifying power of an additional measurement.', '1511.04162-3-104-1': '[EQUATION] where [MATH] is drawn from the Gaussian copula with the correlation matrix [EQUATION].', '1511.04162-3-104-2': 'Table [REF] lists the three population objects: the local average treatment effect, the Wald estimand, and the identified set for the local average treatment effect.', '1511.04162-3-104-3': 'Figures [REF]-[REF] describe the coverage probabilities of the confidence intervals for each parameter value.', '1511.04162-3-104-4': "The comparison among different [MATH]'s are similar to the previous figures.", '1511.04162-3-105-0': 'Last, I consider the dependence between measurement error and instrumental variable, as in Section [REF].', '1511.04162-3-105-1': 'Table [REF] lists the three population objects and Figures [REF]-[REF] describe the coverage probabilities of the confidence intervals.', '1511.04162-3-105-2': 'Since they do not use any information from the measured treatment [MATH], the identified sets and the confidence intervals show that the upper bounds on the local average treatment effect is larger than those under the independence between measurement error and instrumental variable.', '1511.04162-3-105-3': 'The difference becomes smaller when the degree of the measurement error is larger.', '1511.04162-3-105-4': 'It can be considered as the result that, when the misclassification happens too often, the measured treatment [MATH] has only little information about the true treatment and therefore there is a small difference between the identified sets.', '1511.04162-3-106-0': '# Conclusion', '1511.04162-3-107-0': 'This paper studies the identifying power of an instrumental variable in the heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous.', '1511.04162-3-107-1': 'The assumptions in this framework are the monotonicity of the instrumental variable [MATH] on the true treatment [MATH] and the exogeneity of [MATH].', '1511.04162-3-107-2': 'I use the total variation distance to characterize the identified set for the local average treatment effect [MATH].', '1511.04162-3-107-3': 'I also provide an inference procedure for the local average treatment effect.', '1511.04162-3-107-4': 'Unlike the existing literature on measurement error, the identification strategy does not reply on a specific structure of the measurement error; the only assumption on the measurement error is its independence of the instrumental variable.', '1511.04162-3-108-0': 'There are several directions for future research.', '1511.04162-3-108-1': 'First, the choice of the partition [MATH] in Section [REF], particularly the choice of [MATH], is an interesting direction.', '1511.04162-3-108-2': 'To the best of my knowledge, the literature on many moment inequalities has not investigated how econometricians choose the numbers of the many moment inequalities, e.g., [CITATION].', '1511.04162-3-108-3': 'Second, it may be worthwhile to investigate the other parameter for the treatment effect.', '1511.04162-3-108-4': 'This paper has focused on the local average treatment effect, but the literature on heterogeneous treatment effect has emphasized the importance of choosing a adequate treatment effect parameter in order to answer relevant policy questions.', '1511.04162-3-108-5': 'Third, it is also interesting to investigate various assumptions on the measurement errors.', '1511.04162-3-108-6': 'In some empirical settings, for example, it may be reasonable to assume that the measurement error is one-directional (e.g., misclassification happens only when [MATH]).', '1511.04162-3-108-7': 'Fourth, it is not trivial how the analysis of this paper can be extended to an instrumental variable taking more than two values.', '1511.04162-3-108-8': 'For a general instrumental variable, it is always possible to focus on two values of the instrumental variable and apply the analysis of this paper to the subpopulation with the instrumental variable taking these two values.', '1511.04162-3-108-9': 'However, different pairs of the values can have different compliers, so that the parameter of interest is not common across different pairs, as in [CITATION].', '1511.04162-3-109-0': '# Appendix A: Multiplier bootstrap', '1511.04162-3-110-0': 'This section describes the multiplier bootstrap in [CITATION].', '1511.04162-3-110-1': 'For the sake of simplicity, I do not talk about their two- and three-step versions of the multiplier bootstrap.', '1511.04162-3-111-0': 'Define the moment functions based on ([REF])-([REF]).', '1511.04162-3-111-1': 'Define [MATH] and [EQUATION]', '1511.04162-3-111-2': 'Then the approximated identified set is characterized by [EQUATION].', '1511.04162-3-112-0': 'I construct a confidence interval [MATH] via the multiplier bootstrap in [CITATION].', '1511.04162-3-112-1': 'The test statistic for the true parameter values being [MATH] is defined by [EQUATION] where [MATH] estimates [MATH], and [MATH] estimates the variance [MATH] of [MATH]: [EQUATION]', '1511.04162-3-112-2': 'To conduct a multiplier bootstrap, generate [MATH] independent standard normal random variables [MATH].', '1511.04162-3-112-3': 'The centered bootstrap moments are [EQUATION].', '1511.04162-3-112-4': 'The bootstrapped test statistic is defined by [EQUATION].', '1511.04162-3-112-5': 'The critical value [MATH] is defined as the conditional [MATH]-quantile of [MATH] given [MATH]: [EQUATION]', '1511.04162-3-113-0': '# Appendix B: Proofs', '1511.04162-3-114-0': '## Proof of Lemma [REF]', '1511.04162-3-115-0': 'By Equation ([REF]), [MATH], and [MATH].', '1511.04162-3-116-0': '## Proof of Lemma [REF]', '1511.04162-3-117-0': 'I obtain [MATH] by the same logic as Theorem 1 in [CITATION]: [EQUATION]', '1511.04162-3-117-1': 'By the triangle inequality, [EQUATION]', '1511.04162-3-117-2': 'Moreover, since [MATH] almost surely, [EQUATION]', '1511.04162-3-118-0': '## Proof of Lemma [REF]', '1511.04162-3-119-0': 'Based on the triangle inequality, [EQUATION]', '1511.04162-3-119-1': 'The equality holds if and only if the sign of [MATH] is constant in [MATH] for every [MATH].', '1511.04162-3-119-2': 'Since [MATH] if and only if [MATH] is positive, the condition in ([REF]) is a necessary and sufficient condition for [MATH], which is equivalent for the Wald estimand to belong to the identified set.', '1511.04162-3-120-0': '## Proof of Theorems [REF], [REF] and [REF]', '1511.04162-3-121-0': 'Theorems [REF], [REF] and [REF] are special cases of the following theorem.', '1511.04162-3-122-0': 'There is some variable [MATH] taking values in a set [MATH] satisfying the following properties.', '1511.04162-3-123-0': 'Consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-123-1': 'The identified set [MATH] for the local average treatment effect is the set of [MATH] satisfying the following three inequalities.', '1511.04162-3-123-2': '[EQUATION]', '1511.04162-3-123-3': 'To show Theorem [REF], I consider the two cases separately: [MATH] or [MATH].', '1511.04162-3-124-0': '### Case 1: [MATH].', '1511.04162-3-125-0': 'In this case, [MATH] a.s. for every [MATH].', '1511.04162-3-125-1': 'Let [MATH] and [MATH] be any pair of density functions for [MATH] dominated by [MATH].', '1511.04162-3-125-2': 'Define the data generating process [MATH]: [EQUATION]', '1511.04162-3-125-3': 'Theorem [REF] follows from the following three observations:', '1511.04162-3-126-0': '(i) [MATH] satisfies the independence between [MATH] and [MATH] given [MATH] for each [MATH].', '1511.04162-3-126-1': 'Furthermore, [MATH] satisfies [MATH] almost surely.', '1511.04162-3-127-0': '(ii) Denote by [MATH] the density function of [MATH].', '1511.04162-3-127-1': 'Then [EQUATION] where the last equality uses [MATH].', '1511.04162-3-128-0': '(iii) The local average treatment effect under [MATH] is [EQUATION]', '1511.04162-3-129-0': '### Case 2: [MATH].', '1511.04162-3-130-0': 'Lemma [REF] is modified into the framework of Theorem [REF].', '1511.04162-3-131-0': 'Under Assumption [REF], [MATH].', '1511.04162-3-132-0': 'The proof is the same as Lemma [REF] and this lemma follows from [MATH].', '1511.04162-3-133-0': 'From Lemmas [REF] and [REF] and Equation [REF], all the three inequalities in Theorem [REF] are satisfied for the true value of the local average treatment effect.', '1511.04162-3-133-1': 'To complete Theorem [REF], it suffices to show that, for any data generating process [MATH], any point [MATH] satisfying the three inequalities in Theorem [REF] is the local average treatment effect under some data generating process whose data distribution is equal to [MATH].', '1511.04162-3-134-0': 'Define the two data generating processes: [MATH] and [MATH].', '1511.04162-3-134-1': 'First, [MATH] is defined by [EQUATION]', '1511.04162-3-134-2': 'Second, [MATH] is defined as follows.', '1511.04162-3-134-3': 'Using [MATH], define [EQUATION] and define [MATH] as [EQUATION]', '1511.04162-3-134-4': 'If [MATH], then (i) [MATH] generates the data distribution [MATH] and the local average treatment effect under [MATH] is [MATH]; and (ii) [MATH] generates the data distribution [MATH] and the local average treatment effect under [MATH] is [MATH].', '1511.04162-3-135-0': '(i) Denote by [MATH] the density function of [MATH].', '1511.04162-3-135-1': 'The data generating process [MATH] generates the data distribution [MATH]: [EQUATION] where the first equality uses [MATH].', '1511.04162-3-135-2': 'Under [MATH], the local average treatment effect is [MATH]: [EQUATION] (ii) Note that [EQUATION]', '1511.04162-3-135-3': 'Denote by [MATH] the density function of [MATH].', '1511.04162-3-135-4': 'Notice that [MATH] is well-defined because [EQUATION] [MATH] generates the data distribution [MATH]: [EQUATION] and similarly [MATH].', '1511.04162-3-135-5': "Under [MATH], the local average treatment effect is [EQUATION] where the fifth equality comes from Bayes' theorem.", '1511.04162-3-136-0': 'Theorem [REF] follows from the next lemma.', '1511.04162-3-137-0': 'If [MATH], then, for every [MATH], (i) the mixture distribution [MATH] satisfies Assumption [REF]; (ii) [MATH] generates the data distribution [MATH]; (iii) under [MATH], the local average treatment effect is [EQUATION] (i) Under both [MATH] and [MATH], [MATH] is independent of [MATH] given [MATH] for each [MATH].', '1511.04162-3-137-1': 'Furthermore, [MATH] and [MATH] have the same marginal distribution for [MATH]: [MATH].', '1511.04162-3-137-2': 'Therefore, the mixture of [MATH] and [MATH] also satisfies the independence.', '1511.04162-3-137-3': 'Since both[MATH] and [MATH] satisfy [MATH] almost surely, so does the mixture.', '1511.04162-3-137-4': '(ii) By Lemma [REF], both [MATH] and [MATH] generate the data distribution [MATH] and so does the mixture.', '1511.04162-3-137-5': '(iii) It follows from the last statement in Lemma [REF].', '1511.04162-3-138-0': '## Proof of Theorem [REF]', '1511.04162-3-139-0': 'The proof of Theorems [REF] is similar to Theorems [REF].', '1511.04162-3-139-1': 'Only the difference is to change the definition of [MATH] as follows.', '1511.04162-3-139-2': 'Define [MATH] and define [MATH] as [EQUATION]', '1511.04162-3-140-0': '## Proof of Lemma [REF]', '1511.04162-3-141-0': 'First, note that [EQUATION]', '1511.04162-3-141-1': 'This is verified as follows.', '1511.04162-3-141-2': 'For every [MATH], I have [EQUATION]', '1511.04162-3-141-3': 'Moreover, the above inequality becomes an equality if [MATH] if [MATH] and [MATH] if [MATH].', '1511.04162-3-142-0': '[CITATION] and [CITATION] show that [EQUATION] for any random variable [MATH].', '1511.04162-3-142-1': 'The proof is as follows.', '1511.04162-3-142-2': '[EQUATION]', '1511.04162-3-142-3': 'By Theorem [REF] and Equation ([REF]), [MATH] is characterized by [EQUATION]', '1511.04162-3-142-4': 'Since the second condition implies [MATH], the above three conditions becomes [EQUATION]', '1511.04162-3-142-5': 'By Equation ([REF]), [MATH] is characterized as in Lemma [REF].', '1511.04162-3-143-0': '## Proof of Lemma [REF]', '1511.04162-3-144-0': 'Condition (i) implies Assumption [REF] (i).', '1511.04162-3-144-1': 'Condition (ii) implies Assumption [REF] (iii).', '1511.04162-3-144-2': 'The rest of the proof is going to show Assumption [REF] (ii).', '1511.04162-3-144-3': 'Define [EQUATION].', '1511.04162-3-144-4': 'Then [MATH].', '1511.04162-3-144-5': 'Define [EQUATION].', '1511.04162-3-144-6': 'By the Holder continuity of [MATH], [EQUATION].', '1511.04162-3-144-7': 'Define [MATH].', '1511.04162-3-144-8': 'For [MATH] with [MATH], the above inequality implies that the sign of [MATH] is constant on [MATH].', '1511.04162-3-144-9': 'For those [MATH], [MATH] on [MATH].', '1511.04162-3-144-10': 'Then, on every [MATH], either [MATH] or [MATH].', '1511.04162-3-144-11': 'Therefore [EQUATION].', '1511.04162-3-144-12': 'Since [EQUATION] it follows that [EQUATION].', '1511.04162-3-144-13': 'Since [MATH] converges to zero uniformly over [MATH], Assumption [REF] (ii) holds.', '1511.04162-3-145-0': '## Proof of Theorem [REF]', '1511.04162-3-146-0': 'The following theorem is taken from Corollary 5.1 and Theorem 6.1 in [CITATION].', '1511.04162-3-147-0': 'Given [MATH] with [MATH] and [MATH], denote by [MATH] the set of [MATH] for which [MATH] and [EQUATION]', '1511.04162-3-147-1': 'Under the assumptions in Theorem [REF], [EQUATION]', '1511.04162-3-147-2': 'Theorem [REF] (i) follows from [EQUATION] where the last inequality comes from Theorem [REF] (i) and Assumption [REF] (iv).', '1511.04162-3-148-0': 'Theorem [REF] (ii) is shown as follows.', '1511.04162-3-148-1': 'Denote by [MATH] a constant for which [MATH].', '1511.04162-3-148-2': 'Let [MATH] be any element of [MATH] with [MATH].', '1511.04162-3-148-3': 'It suffices to show that ([REF]) holds for sufficiently large [MATH].', '1511.04162-3-148-4': 'If either ([REF]) or ([REF]) is violated, then ([REF]) holds for sufficiently large [MATH].', '1511.04162-3-148-5': 'In the rest of the proof, I focus on the case where ([REF]) is violated.', '1511.04162-3-148-6': 'That is, [EQUATION].', '1511.04162-3-148-7': 'Since [MATH] converges to [MATH] in the sense of ([REF]) and [MATH] is bounded, it follows that, for sufficiently large [MATH], there is [MATH] such that [EQUATION].', '1511.04162-3-148-8': 'Denoted by [MATH] the value of the left-hand side in the above inequality.', '1511.04162-3-148-9': 'For sufficiently large [MATH], there is [MATH] such that [MATH].', '1511.04162-3-148-10': 'For such [MATH], [MATH].', '1511.04162-3-148-11': 'Therefore ([REF]) holds for sufficiently large [MATH].', '1511.04162-3-149-0': '# Appendix C: Additional assumptions on the measurement error', '1511.04162-3-150-0': 'This section considers widely-used assumptions on the measurement structure: (i) a positive correlation between the measurement and the truth and (ii) the measurement error is independent of the other error in the simultaneous equation system in ([REF])-([REF]).', '1511.04162-3-151-0': '(i) [MATH].', '1511.04162-3-151-1': '(ii) [MATH] is independent of [MATH] for each [MATH].', '1511.04162-3-152-0': 'Assumption [REF] (i) has been used in the literature on measurement error, e.g., [CITATION]; [CITATION]; and [CITATION].', '1511.04162-3-152-1': 'Assumption [REF] (ii) is that the measured treatment [MATH] is positively correlated with the true variable [MATH] as in [CITATION].', '1511.04162-3-153-0': 'Unlike Assumption [REF] itself, the combination of Assumptions [REF] and [REF] yields on restrictions on the distribution for the observed variables.', '1511.04162-3-154-0': 'Suppose that Assumptions [REF] and [REF] hold, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-154-1': 'Then [EQUATION].', '1511.04162-3-155-0': 'Assumption [REF] (i) implies [EQUATION]', '1511.04162-3-155-1': 'As in [CITATION] and [CITATION], Assumptions [REF] implies the following inequalities [EQUATION]', '1511.04162-3-155-2': 'Therefore [EQUATION].', '1511.04162-3-156-0': 'The sharp identified set is characterized as follows.', '1511.04162-3-157-0': 'Suppose that Assumptions [REF] and [REF] hold, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-157-1': 'Denote [MATH] and denote [MATH].', '1511.04162-3-157-2': 'Then [MATH] if [MATH]; otherwise [EQUATION] where [EQUATION].', '1511.04162-3-158-0': 'Define [EQUATION] and define [MATH] and [MATH].', '1511.04162-3-158-1': 'Assumption [REF] (i) implies [EQUATION] so that [EQUATION].', '1511.04162-3-158-2': 'Assumption [REF] (ii) implies that the matrix [MATH] is invertible.', '1511.04162-3-158-3': 'Thus [EQUATION] and [EQUATION] because [EQUATION]', '1511.04162-3-158-4': 'Define [MATH] and [MATH].', '1511.04162-3-158-5': 'In the rest of the proof, I am going to show that the sharp identified set for [MATH] is [MATH] where [EQUATION].', '1511.04162-3-159-0': 'First, I am going to show that the identified set for [MATH] is a subset of [MATH].', '1511.04162-3-159-1': 'As in [CITATION] and [CITATION], Assumptions [REF] implies the following inequalities [EQUATION]', '1511.04162-3-159-2': 'In the notation of this proof, [EQUATION]', '1511.04162-3-159-3': 'By some algebraic operations, [EQUATION]', '1511.04162-3-159-4': 'Since [MATH], [EQUATION]', '1511.04162-3-159-5': 'By Lemma [REF], [MATH] and then [EQUATION].', '1511.04162-3-160-0': 'Then, I am going to show that [MATH] is included in the identified set for [MATH].', '1511.04162-3-160-1': 'Let [MATH] be any element of [MATH].', '1511.04162-3-160-2': 'Define [MATH] and [MATH].', '1511.04162-3-160-3': 'Then define [EQUATION]', '1511.04162-3-160-4': 'By construction, [EQUATION]', '1511.04162-3-160-5': 'This is a sufficient condition for [MATH] to be consistent with Assumptions [REF], which is shown in [CITATION] and [CITATION].', '1511.04162-3-160-6': 'Thus [MATH] belongs to the identified set for [MATH].', '1511.04162-3-161-0': '# Appendix D: Compliers-defiers-for-marginals condition', '1511.04162-3-162-0': 'This section demonstrates that a variant of Theorem [REF] still holds under a weaker condition than the deterministic monotonicity condition in Assumption [REF] (ii).', '1511.04162-3-162-1': 'I consider the following assumption.', '1511.04162-3-163-0': '(i) For each [MATH], [MATH] is independent of [MATH].', '1511.04162-3-163-1': '(ii) There is a subset [MATH] of [MATH] such that [EQUATION] (iii) [MATH].', '1511.04162-3-164-0': 'Assumption [REF] (ii) imposes the compliers-defiers-for-marginals condition on the joint distribution of [MATH].', '1511.04162-3-164-1': 'Under this assumption, Theorem 2.1 of [CITATION] shows that [EQUATION] where [MATH].', '1511.04162-3-165-0': 'Suppose that Assumption [REF] holds, and consider an arbitrary data distribution [MATH] of [MATH].', '1511.04162-3-165-1': 'The identified set [MATH] for [MATH] is characterized in the same way as Theorem [REF]: [MATH] if [MATH]; otherwise, [EQUATION].', '1511.04162-3-166-0': 'Since Theorem [REF] gives the sharp identified set under a stronger assumption of this theorem, the identified set [MATH] in this theorem should be equal to or larger than the set in Theorem [REF].', '1511.04162-3-166-1': 'As a result, it suffices to show that [MATH] is a subset of the set in Theorem [REF].', '1511.04162-3-166-2': 'By the Assumption [REF] (ii), [EQUATION]', '1511.04162-3-166-3': 'Based on the definition of the total variation distance, [EQUATION] and therefore [EQUATION].', '1511.04162-3-166-4': 'Since [MATH], [EQUATION].', '1511.04162-3-166-5': 'This concludes that [MATH] is included in [EQUATION]'} | null | null | null | null |
nucl-th-0505049 | {'nucl-th-0505049-1-0-0': 'A statistical model for decay and formation of heavy hadronic resonances is formulated.', 'nucl-th-0505049-1-0-1': 'The resonance properties become increasingly uncertain with increasing resonance mass.', 'nucl-th-0505049-1-0-2': 'Drawing on analogy with the situation in low-energy nuclear physics, we employ the Weisskopf approach to the resonance processes.', 'nucl-th-0505049-1-0-3': 'In the large-mass limit, the density of resonance states in mass is governed by a universal Hagedorn-like temperature [MATH].', 'nucl-th-0505049-1-0-4': 'As resonances decay, progressively more and more numerous lighter states get populated.', 'nucl-th-0505049-1-0-5': 'For [MATH], the model describes data for the hadron yield ratios at the RHIC and SPS energies under the extreme assumption of a single heavy resonance giving rise to measured yields.', 'nucl-th-0505049-1-1-0': 'Studies of ultrarelativistic nuclear reactions aim at learning on properties of highly-excited strongly-interacting matter and, in particular, on the predicted transition to quark-gluon plasma (QGP) [CITATION].', 'nucl-th-0505049-1-1-1': 'Yields from those reactions can be described in terms of grand-canonical thermodynamic models [CITATION], at temperatures close to those anticipated for the transition [CITATION].', 'nucl-th-0505049-1-1-2': 'Similar temperatures are utilized within the microcanonical model for electron-positron annihilation into hadrons [CITATION].', 'nucl-th-0505049-1-1-3': 'Large fractions of final light-particles in those thermodynamic equilibrium models result from secondary decays of heavy resonances whose features are, generally, less and less known the heavier the resonances.', 'nucl-th-0505049-1-1-4': 'This has forced the use of cut-offs on the primary resonance mass in the equilibrium models [CITATION]; similar cut-offs have been employed in the transport models [CITATION].', 'nucl-th-0505049-1-1-5': 'In this paper, we explore the possibility of an apparent equilibrium in the reactions stemming from sequential binary decays of heavy resonances, with the characteristic temperature describing the resonance mass spectrum, rather than being proposed ad hoc.', 'nucl-th-0505049-1-2-0': 'The situation of deteriorating knowledge of resonance properties, with increasing resonance energy and resonance density in energy, reminisces that of the resonances in low-energy nuclear physics.', 'nucl-th-0505049-1-2-1': 'There, statistical descriptions, in terms of the Weisskopf compound-nucleus model and the Hauser-Feschbach theory, have been highly successful [CITATION].', 'nucl-th-0505049-1-2-2': 'Underlying the statistical descriptions is the density of resonance states in energy [MATH], where [MATH] represents discrete quantum numbers of the resonances.', 'nucl-th-0505049-1-2-3': 'For the hadronic states, a universal temperature in terms of the density of states, [MATH], has been originally considered by Hagedorn [CITATION], in part due to an evidence for a rapid, nearly exponential, increase in the number of resonances with energy; see also [CITATION].', 'nucl-th-0505049-1-2-4': 'To reach a conclusion on the basis of spectrum, however, the counting of resonances should be carried out at constant values of the quantum numbers [MATH], particularly in the region of opening thresholds for different [MATH].', 'nucl-th-0505049-1-2-5': 'This has not been considered in Refs. [CITATION].', 'nucl-th-0505049-1-2-6': 'Nonetheless, in the limit of large [MATH] at fixed [MATH], the universal temperature [MATH], independent of [MATH], but possibly dependent on [MATH], may be expected for the resonances due to the lack of any scales that could govern the [MATH]-dependence on [MATH].', 'nucl-th-0505049-1-2-7': 'While our model for [MATH] will be fairly schematic, similar to the models employed in the literature [CITATION], there are generally important questions regarding strong interactions that can be suitably asked in terms of [MATH], concerning e.g., besides the [MATH] dependence, the emergence of a surface tension in the thermodynamic limit.', 'nucl-th-0505049-1-2-8': 'In the context of the phase transition, the temperature [MATH] may be considered as the temperature for a metastable equilibrium of quark-gluon drops with vacuum, and, as such, slightly lower temperature than the critical [MATH].', 'nucl-th-0505049-1-3-0': 'For the resonances described by the continuum density of states [MATH], we consider the processes of binary breakup and inverse fusion, constrained by the detailed balance, the geometry and the conservation of baryon number [MATH], of strangeness [MATH], isospin [MATH] and of isospin projection [MATH].', 'nucl-th-0505049-1-3-1': 'From the two versions of our model, with and without a strict conservation of angular momentum [MATH], we discuss the simpler Weisskopf version, in this first model presentation.', 'nucl-th-0505049-1-3-2': 'The lighter particles (comprising of 55 baryonic and 34 mesonic states) are treated explicitly as discrete states in the model.', 'nucl-th-0505049-1-3-3': 'Our model allows to explore various aspects of the system evolution in ultrarelativistic collisions, including formation of hadrons with extreme strangeness and isospin, as well as chemical and kinetic freeze-out.', 'nucl-th-0505049-1-4-0': 'We first discuss details of the density of states.', 'nucl-th-0505049-1-4-1': 'As a threshold mass, separating the discrete states from those in the statistical continuum, we take [MATH].', 'nucl-th-0505049-1-4-2': 'We assume that at low excitation energies the density of states is similar at different [MATH] for the same excitation energy above the spectrum bottom [MATH].', 'nucl-th-0505049-1-4-3': 'Following quark considerations, we adopt [EQUATION] where [MATH] and [MATH].', 'nucl-th-0505049-1-4-4': 'The coefficients had been adjusted with the constraint that the mass [MATH] from Eq. ([REF]) exceeds the lowest known masses for different [MATH]; regarding the practicality of sequential decays, we prefer to overestimate rather than to underestimate the position of spectrum bottom for the continuum, and its rise with [MATH], to preclude the emergence of any unphysical stable states towards the continuum bottom.', 'nucl-th-0505049-1-4-5': 'At high masses [MATH] the influence of the ground state mass on [MATH] should decrease.', 'nucl-th-0505049-1-4-6': 'We eventually arrive at the following density of states employed in our calculations: [EQUATION]', 'nucl-th-0505049-1-4-7': 'The role of the factor [MATH] is to suppress the effect of [MATH] at high [MATH] and we use [EQUATION] with [MATH].', 'nucl-th-0505049-1-4-8': 'The prefactor in Eq. ([REF]), with [MATH], acts to enhance asymmetric (rather than symmetric) breakups for continuum hadrons, leaving the issue of surface tension in the thermodynamic limit open.', 'nucl-th-0505049-1-4-9': 'The normalizing factor [MATH] and the power [MATH] are adjusted, for different assumed values of Hagedorn temperature [MATH], by comparing the low-[MATH] cumulant spectra from measurements and the continuum representation [CITATION]: [EQUATION]', 'nucl-th-0505049-1-4-10': 'For [MATH] MeV, as an example, we obtain the prefactor power of [MATH].', 'nucl-th-0505049-1-4-11': 'The normalization factor [MATH] drops out from probabilities for the most common processes involving continuum hadrons, with one continuum and one discrete hadron either in the initial or final state.', 'nucl-th-0505049-1-5-0': 'We now turn to cross sections and decay rates.', 'nucl-th-0505049-1-5-1': 'The cross section for the formation of a resonance [MATH] in the interaction of hadrons 1 and 2, can be, on one hand, represented as [CITATION]: [EQUATION]', 'nucl-th-0505049-1-5-2': "Here, [MATH] is the relative velocity, [MATH]'s are the single-particle energies, and [MATH] is the matrix element squared for the fusion, which is averaged over initial and final spin directions.", 'nucl-th-0505049-1-5-3': 'The factor of [MATH], associated with the last averaging, has been absorbed into [MATH].', 'nucl-th-0505049-1-5-4': 'The c.m. momentum in Eq. ([REF]) is [EQUATION]', 'nucl-th-0505049-1-5-5': 'For the final state in continuum, on the basis of geometric considerations, the cross section for fusion, on the other hand, is [EQUATION] where [MATH] represents the isospin Clebsch-Gordan coefficient.', 'nucl-th-0505049-1-5-6': 'The cross section radius is taken as that of the fused entity, [MATH], with [MATH] as a characteristic radius for a [MATH] hadron.', 'nucl-th-0505049-1-5-7': 'Equations ([REF]) and ([REF]) allow to extract the square of the transition matrix element needed for computation of the partial resonance-width.', 'nucl-th-0505049-1-6-0': 'The partial width for decay into [MATH] and [MATH] can be generally represented as [EQUATION]', 'nucl-th-0505049-1-6-1': 'As before, the factors of [MATH] are absorbed into [MATH].', 'nucl-th-0505049-1-6-2': 'A resonance in the continuum can undergo three types of binary decay: where (i) both daughters are particles with well established properties within the discrete spectrum below [MATH], (ii) one of the daughters belongs to the discrete spectrum and the other to the continuum and, finally, where (iii) both daughter resonances belong to the continuum.', 'nucl-th-0505049-1-6-3': 'In the case (i), the state densities are [MATH].', 'nucl-th-0505049-1-6-4': 'With the detailed balance relation, [MATH], we then get [EQUATION]', 'nucl-th-0505049-1-6-5': 'In analyzing the case (ii), let the particle characterized by [MATH] belong to the discrete spectrum and that characterized by [MATH] to the continuum spectrum.', 'nucl-th-0505049-1-6-6': 'From ([REF]), we then find [EQUATION] where [MATH].', 'nucl-th-0505049-1-6-7': 'For [MATH] small compared to [MATH], it is of advantage to represent the subintegral density of states as an exponential of the density logarithm and to expand the logarithm in [MATH].', 'nucl-th-0505049-1-6-8': 'The integration over relative momentum can be thereafter carried out explicitly yielding [EQUATION]', 'nucl-th-0505049-1-6-9': 'Here, [MATH] and the temperature is defined as [EQUATION]', 'nucl-th-0505049-1-6-10': 'For [MATH] small compared to [MATH], the obvious further possibility is the expansion of [MATH] in [MATH], with an emergence of the chemical potentials conjugate to [MATH].', 'nucl-th-0505049-1-7-0': 'In the case (iii), of both daughters in the continuum with large masses [MATH], the nonrelativistic limit in Eq. ([REF]) is justified.', 'nucl-th-0505049-1-7-1': 'On employing Eqs. ([REF]) and ([REF]), the result is [EQUATION] where [EQUATION]', 'nucl-th-0505049-1-7-2': 'To obtain the last expression in Eq. ([REF]), we have expanded the logarithm of subintegral density, with the temperature representing [MATH].', 'nucl-th-0505049-1-8-0': 'The total decay width of an [MATH] resonance is finally [EQUATION]', 'nucl-th-0505049-1-8-1': 'A moving resonance will live an average time of [MATH], where [MATH] is the resonance Lorentz factor.', 'nucl-th-0505049-1-9-0': 'The formulas above provide the basis for our Monte-Carlo simulations of the resonance decay sequences in heavy ion collisions.', 'nucl-th-0505049-1-9-1': 'A resonance follows an exponential decay law corresponding to [MATH].', 'nucl-th-0505049-1-9-2': 'The product properties are selected according to the decay branching ratios.', 'nucl-th-0505049-1-9-3': 'Since the parent angular momentum is not tracked in the Weisskopf approach, the angular distribution of products is taken as isotropic.', 'nucl-th-0505049-1-9-4': 'During the evolution two resonances can fuse with each other, according to the cross section of Eq. ([REF]), if the final state is in continuum.', 'nucl-th-0505049-1-9-5': 'If, on the other hand, the state is discrete, the cross section acquires the standard form [CITATION], from Eqs. ([REF]) and ([REF]), [EQUATION] where the width for the spectral function is considered, [EQUATION]', 'nucl-th-0505049-1-9-6': 'Besides the decay and fusion processes, related by detailed balance, a provisional constant cross section [MATH] mb has been assumed for all collisions.', 'nucl-th-0505049-1-10-0': 'In our model, we simulate, in particular, the features of the final state of central Au+Au collisions at [MATH] GeV.', 'nucl-th-0505049-1-10-1': 'Following the presumption that the decay sequences will tend to erase fine details of the initial state, we push the characteristics of the initial state to an extreme, allowing for a single heavy resonance to populate a given rapidity region.', 'nucl-th-0505049-1-10-2': 'In the end, when examining the transverse momentum spectra, we find that the resonance decay and reformation alone generates insufficient transverse collective energy, indicating that the early resonances need to be affected by a collective motion generated prior to the resonance stage.', 'nucl-th-0505049-1-10-3': 'This finding is consistent with those in other works [CITATION].', 'nucl-th-0505049-1-11-0': 'Within the single resonance scenario, the local final state reflects the initial quantum numbers of a resonance characterized by [MATH], where [MATH].', 'nucl-th-0505049-1-11-1': 'After the value of the Hagedorn temperature [MATH] is set, the relative yields of particles in the final state are, in practice, sensitive only to the relative values of the quantum numbers of the initial resonance.', 'nucl-th-0505049-1-11-2': 'We normally impose strangeness neutrality, so that the starting value is [MATH].', 'nucl-th-0505049-1-11-3': 'The magnitude of [MATH], for a given [MATH], can be adjusted by using the final-state antiproton-to-proton or antiproton-to-pion ratios.', 'nucl-th-0505049-1-11-4': 'The starting isospin, for a given [MATH], can be adjusted by using the isospin of original nuclei.', 'nucl-th-0505049-1-11-5': 'However, we find the initial isospin has only a marginal impact on the isospin of individual final particles.', 'nucl-th-0505049-1-11-6': 'This may be attributed to the cumulative effect of isospin fluctuations when many particles, compared to [MATH], get produced.', 'nucl-th-0505049-1-11-7': 'For specific initial [MATH] values, we repeat numerous times the Monte-Carlo simulations of the decay chain and recombination, and the results presented here are an average of about [MATH] generated event sequences.', 'nucl-th-0505049-1-12-0': 'Figure [REF] illustrates the average features of an exemplary local system that starts out as a resonance characterized by [MATH] GeV and [MATH].', 'nucl-th-0505049-1-12-1': 'The left panel shows the ratio of the average maximal resonance mass [MATH] to the initial mass [MATH] as a function of time, as well as the mass asymmetry, the mass difference between the heaviest, [MATH], resonance and the next heaviest, [MATH], divided by the sum of their masses, [MATH].', 'nucl-th-0505049-1-12-2': 'Persistence of the large asymmetry with time indicates that the heavy resonance decays primarily through light-hadron emission.', 'nucl-th-0505049-1-13-0': 'In addition, Fig. [REF] shows as a function of time the abundances of particles (left panel) and antiparticles (right panel).', 'nucl-th-0505049-1-13-1': 'The antibaryon abundances freeze out noticeably earlier than the baryon abundances, in spite of a low initial baryon number relative to the initial mass.', 'nucl-th-0505049-1-13-2': 'The higher the strangeness, the later the abundance saturates.', 'nucl-th-0505049-1-13-3': 'This is likely due to the fact that the effects of strangeness fluctuation need to accumulate with time; the situation would change if we assumed strangeness fluctuations right for the initial resonance conditions.', 'nucl-th-0505049-1-14-0': 'The calculations have been repeated while suppressing the back resonance fusion reactions.', 'nucl-th-0505049-1-14-1': 'As expected, in this case the maximal mass and asymmetry decrease faster with time, see the left panel of Fig. [REF].', 'nucl-th-0505049-1-14-2': 'The abundances (not shown) grow faster and saturate earlier for the modified evolution.', 'nucl-th-0505049-1-15-0': 'Table [REF] compares the yield ratios from the resonance-decay model, within the region of optimal model-parameter values, with the available central Au+Au collision data from RHIC at [MATH] GeV [CITATION].', 'nucl-th-0505049-1-15-1': 'The temperature of the resonance mass spectrum, [MATH], and the initial baryon-number-to-mass ratio, [MATH], can be adjusted by simultaneously considering the ratios of [MATH] and [MATH].', 'nucl-th-0505049-1-15-2': 'An optimal agreement between the model and the central RHIC data is obtained for [MATH] MeV and [MATH] GeV[MATH].', 'nucl-th-0505049-1-15-3': 'The model results tend to be only weakly sensitive to [MATH]: with an increase in [MATH] a slight increase in [MATH] is favored, that can be traced to the factor [MATH] in the density of states.', 'nucl-th-0505049-1-16-0': 'The overall agreement between data and the decay-model calculations in Table [REF] is quite remarkable, given that only two parameters, [MATH] and [MATH], are adjusted.', 'nucl-th-0505049-1-16-1': 'One should note that the optimal Hagedorn temperature of [MATH] MeV is close to the critical temperature of [MATH] MeV for a transition to the color-deconfined QGP phase obtained in the lattice QCD calculations at zero net baryon density [CITATION], and also similar to the chemical freeze-out temperature [MATH] MeV extracted from the analysis of RHIC data within a grand-canonical model [CITATION].', 'nucl-th-0505049-1-17-0': 'At the general level, the calculations are quite good in reproducing yield ratios involving strange particles.', 'nucl-th-0505049-1-17-1': 'A more detailed examination reveals some potential deficit of multistrange baryons and antibaryons.', 'nucl-th-0505049-1-17-2': 'Possible reasons for the deficiency, to be investigated in the future, include: the possible role played by the non-resonant strangeness-exchange processes [CITATION] and by the multiparticle processes [CITATION] and, further, the possible sensitivity of strangeness production to an early system dynamics [CITATION] and, specifically, to strangeness fluctuations for early resonances and/or to details in flavor-dependence of the state density.', 'nucl-th-0505049-1-18-0': 'Besides the yield ratios in the model standard evolution, the corresponding ratios from evolution with suppressed fusion processes are shown in parenthesis in Table [REF] for [MATH] MeV.', 'nucl-th-0505049-1-18-1': 'Though the suppression of fusion alters particle abundances early on in the system development, the final yields turn out to be rather similar, quite uniformly across the particle species.', 'nucl-th-0505049-1-18-2': 'Only a careful examination reveals that the fusion suppression enhances slightly the production of pions and other light mesons and reduces slightly the production of heavier strange particles.', 'nucl-th-0505049-1-19-0': 'Within a moderate range, there is no strong preference for a particular Hagedorn temperature.', 'nucl-th-0505049-1-19-1': 'As Table [REF] shows, similar yield ratios are obtained for [MATH] MeV as for [MATH] MeV, if, in the context of the RHIC data, a slightly reduced initial mass of [MATH] GeV is assumed.', 'nucl-th-0505049-1-20-0': 'In Fig. [REF], we show the particle yield ratios that turn out to be particularly sensitive to the Hagedorn temperature under a given constraint.', 'nucl-th-0505049-1-20-1': "In one case, while varying the temperature, we adjust the initial fireball's ratio [MATH] to reproduce the ratio [MATH] for the RHIC data (lower set of lines).", 'nucl-th-0505049-1-20-2': 'In that case, the baryon-to-antibaryon ratios, [MATH], remain rather stable with [MATH] variation; the strongest variations are observed for the ratio of strange baryons to the negatively charged hadrons or to the negative mesons.', 'nucl-th-0505049-1-20-3': 'On the other hand, if we adjust the ratio [MATH] to reproduce the ratio [MATH] (upper set of lines), strong variations are observed for the [MATH] ratios.', 'nucl-th-0505049-1-20-4': 'No matter what fitting strategy is followed, a reasonable agreement with the data is obtained within the Hagedorn temperature range of [MATH].', 'nucl-th-0505049-1-21-0': 'We next confront our resonance decay model with the SPS abundance data from the central Pb+Pb collisions at the laboratory energy of [MATH] GeV.', 'nucl-th-0505049-1-21-1': 'As illustrated in Fig. [REF], an optimal agreement with the data is obtained for [MATH] GeV when assuming (at [MATH] MeV) a starting baryon number of [MATH].', 'nucl-th-0505049-1-21-2': 'While the general agreement is rather good, we note that the calculated [MATH] ratio is about [MATH] larger than the data.', 'nucl-th-0505049-1-21-3': 'It is likely that the assumption of a larger number of lighter initial resonances would improve the agreement; in the thermal model the discrepancy is tauted as strangeness undersaturation [CITATION].', 'nucl-th-0505049-1-22-0': 'Kinematic spectra of particles from central heavy ion collisions exhibit the effects of collective expansion.', 'nucl-th-0505049-1-22-1': 'The sequences of decays and fusion generate some collective motion within our model, but not enough to explain the transverse RHIC spectra.', 'nucl-th-0505049-1-22-2': 'It is thus necessary to assume the presence of some collective motion early on, leading to resonances that exhibit space-momentum correlations.', 'nucl-th-0505049-1-22-3': 'It might be that the dynamics, beyond statistics, needs to be involved in the predominant decays, involving the interior degrees of freedom of resonances.', 'nucl-th-0505049-1-22-4': 'The first resonances might also emerge at finite transverse velocities.', 'nucl-th-0505049-1-22-5': 'In either case, degrees of freedom beyond resonances would be involved.', 'nucl-th-0505049-1-23-0': 'The transverse mass spectra displayed in Fig. [REF] are obtained by folding a common collective velocity field with the spectra from our decay model.', 'nucl-th-0505049-1-23-1': 'Specifically, we assume a uniform transverse velocity distribution, [MATH].', 'nucl-th-0505049-1-23-2': 'The spectra for all the hadrons can be best described, at [MATH] MeV, with a uniform velocity field of [MATH], corresponding to an average flow velocity of [MATH].', 'nucl-th-0505049-1-23-3': 'Notably, much less early flow, as characterized by [MATH], is required to explain the particle spectra at SPS energy.', 'nucl-th-0505049-1-24-0': 'In summary, we have formulated a statistical model of hadron resonance formation and decay.', 'nucl-th-0505049-1-24-1': 'Within the model, the density of hadronic states in mass is described in terms of a universal Hagedorn-type temperature.', 'nucl-th-0505049-1-24-2': 'We have demonstrated that both the RHIC and SPS abundance data can be suitably described in terms of resonance decays at the spectral temperature of [MATH] MeV, even when pursuing the extreme assumption of a single heavy resonance populating the investigated rapidity region.', 'nucl-th-0505049-1-24-3': 'To explain the data for particle spectra, we needed to invoke additional collective motion beyond that generated in the hadronic interactions.', 'nucl-th-0505049-1-25-0': 'This work was supported by the U.S. National Science Foundation under Grant PHY-0245009 and by the U.S. Department of Energy under Grant DE-FG02-03ER41259.'} | {'nucl-th-0505049-2-0-0': 'A statistical model for decay and formation of heavy hadronic resonances is formulated.', 'nucl-th-0505049-2-0-1': 'The resonance properties become increasingly uncertain with increasing resonance mass.', 'nucl-th-0505049-2-0-2': 'Drawing on analogy with the situation in low-energy nuclear physics, we employ the Weisskopf approach to the resonance processes.', 'nucl-th-0505049-2-0-3': 'In the large-mass limit, the density of resonance states in mass is governed by a universal Hagedorn-like temperature [MATH].', 'nucl-th-0505049-2-0-4': 'As resonances decay, progressively more and more numerous lighter states get populated.', 'nucl-th-0505049-2-0-5': 'For [MATH], the model describes data for the hadron yield ratios at the RHIC and SPS energies under the extreme assumption of a single heavy resonance giving rise to measured yields.', 'nucl-th-0505049-2-1-0': 'Studies of ultrarelativistic nuclear reactions aim at learning on properties of highly-excited strongly-interacting matter and, in particular, on the predicted transition to quark-gluon plasma (QGP) [CITATION].', 'nucl-th-0505049-2-1-1': 'Yields from those reactions can be described in terms of grand-canonical thermodynamic models [CITATION], at temperatures close to those anticipated for the transition [CITATION].', 'nucl-th-0505049-2-1-2': 'Similar temperatures are utilized in the microcanonical model for electron-positron annihilation into hadrons [CITATION].', 'nucl-th-0505049-2-1-3': 'Large fractions of final light-particles within those thermodynamic equilibrium models result from secondary decays of heavy resonances whose features are, generally, less and less known the heavier the resonances.', 'nucl-th-0505049-2-1-4': 'The lack of knowledge has forced the use of cut-offs on the primary resonance mass in equilibrium models [CITATION]; analogous cut-offs have been employed in transport models [CITATION].', 'nucl-th-0505049-2-1-5': 'In this paper, we explore the possibility of an apparent equilibrium in the reactions stemming from sequential binary decays of heavy resonances, with the characteristic temperature describing the resonance mass spectrum, rather than being proposed ad hoc.', 'nucl-th-0505049-2-2-0': 'The situation of deteriorating knowledge of resonance properties, with increasing resonance energy and resonance density in energy, reminisces that of resonances in low-energy nuclear physics.', 'nucl-th-0505049-2-2-1': 'There, statistical descriptions, in terms of the Weisskopf compound-nucleus model and the Hauser-Feschbach theory, have been highly successful [CITATION].', 'nucl-th-0505049-2-2-2': 'Underlying the statistical descriptions is the density of resonance states in energy [MATH], where [MATH] represents discrete quantum numbers of the resonances.', 'nucl-th-0505049-2-2-3': 'For the hadronic states, a universal temperature in terms of the density of states, [MATH], has been originally considered by Hagedorn [CITATION], in part due to an evidence for a rapid, nearly exponential, increase in the number of resonances with energy; see also [CITATION].', 'nucl-th-0505049-2-2-4': 'To reach a conclusion on the basis of spectrum, however, the counting of resonances should be carried out at constant values of the quantum numbers [MATH], particularly in the region of opening thresholds for different [MATH].', 'nucl-th-0505049-2-2-5': 'This has not been considered in Refs. [CITATION].', 'nucl-th-0505049-2-2-6': 'Nonetheless, in the limit of large [MATH] at fixed [MATH], the universal temperature [MATH], independent of [MATH], but possibly dependent on [MATH], may be expected for the resonances due to the lack of any scales that could govern the [MATH]-dependence on [MATH].', 'nucl-th-0505049-2-2-7': 'While our model for [MATH] will be fairly schematic, similar to the models employed in the literature [CITATION], there are generally important questions regarding strong interactions that can be suitably asked in terms of [MATH], concerning e.g., besides the [MATH] dependence, the emergence of a surface tension in the thermodynamic limit.', 'nucl-th-0505049-2-2-8': 'In the context of the phase transition, the temperature [MATH] may be considered as the temperature for a metastable equilibrium of quark-gluon drops with vacuum, and, as such, slightly lower temperature than the critical [MATH].', 'nucl-th-0505049-2-3-0': 'For resonances described by the continuum density of states [MATH], we consider the processes of binary breakup and inverse fusion, constrained by detailed balance, geometry and by the conservation of baryon number [MATH], strangeness [MATH], isospin [MATH] and of isospin projection [MATH].', 'nucl-th-0505049-2-3-1': 'From the two versions of our model, with and without a strict conservation of angular momentum [MATH], we discuss the simpler Weisskopf version, in this first model presentation.', 'nucl-th-0505049-2-3-2': 'The lighter particles (comprising of 55 baryonic and 34 mesonic states) are treated explicitly as discrete states in the model.', 'nucl-th-0505049-2-3-3': 'Our model allows to explore various aspects of the system evolution in ultrarelativistic collisions, including formation of hadrons with extreme strangeness and isospin, as well as chemical and kinetic freeze-out.', 'nucl-th-0505049-2-4-0': 'We first discuss details of the density of states.', 'nucl-th-0505049-2-4-1': 'As a threshold mass, separating the discrete states from those in the statistical continuum, we take [MATH].', 'nucl-th-0505049-2-4-2': 'We assume that at low excitation energies the density of states is similar at different [MATH] for the same excitation energy above the spectrum bottom [MATH].', 'nucl-th-0505049-2-4-3': 'Following quark considerations, we adopt [EQUATION] where [MATH] and [MATH].', 'nucl-th-0505049-2-4-4': 'The coefficient magnitudes have been adjusted requiring that the mass [MATH] from Eq. ([REF]) exceeds the lowest known masses for different [MATH]; regarding the practicality of sequential decays, we prefer to overestimate rather than to underestimate the reference position of spectrum bottom for the continuum, and to overestimate its rise with [MATH], to preclude the emergence of any unphysical stable states towards the continuum bottom.', 'nucl-th-0505049-2-4-5': 'At high masses [MATH] the influence of the ground state mass on [MATH] should decrease.', 'nucl-th-0505049-2-4-6': 'We eventually arrive at the following density of states employed in our calculations: [EQUATION]', 'nucl-th-0505049-2-4-7': 'The role of the factor [MATH] is to suppress the effect of [MATH] at high [MATH] and we use [EQUATION] with [MATH].', 'nucl-th-0505049-2-4-8': 'The prefactor of the exponential in Eq. ([REF]), with [MATH], acts to enhance asymmetric (rather than symmetric) breakups for continuum hadrons, leaving the issue of surface tension in the thermodynamic limit open.', 'nucl-th-0505049-2-4-9': 'The normalizing factor [MATH] and the power [MATH] are adjusted, for different assumed values of Hagedorn temperature [MATH], by comparing the low-[MATH] cumulant spectra from measurements and from the continuum representation [CITATION]: [EQUATION]', 'nucl-th-0505049-2-4-10': 'For [MATH] MeV, as an example, we obtain the prefactor power of [MATH].', 'nucl-th-0505049-2-4-11': 'The normalization factor [MATH] drops out from probabilities for the most common processes involving continuum hadrons, with one continuum and one discrete hadron either in the initial or final state.', 'nucl-th-0505049-2-5-0': 'We now turn to cross sections and decay rates.', 'nucl-th-0505049-2-5-1': 'The cross section for the formation of a resonance [MATH] in the interaction of hadrons 1 and 2, can be, on one hand, represented as [CITATION]: [EQUATION]', 'nucl-th-0505049-2-5-2': "Here, [MATH] is the relative velocity, [MATH]'s are the single-particle energies, and [MATH] is the matrix element squared for the fusion, which is averaged over initial and final spin directions.", 'nucl-th-0505049-2-5-3': 'The factor of [MATH], associated with the last averaging, has been absorbed into [MATH].', 'nucl-th-0505049-2-5-4': 'The c.m. momentum in Eq. ([REF]) is [EQUATION]', 'nucl-th-0505049-2-5-5': 'For the final state in continuum, following geometric considerations, the cross section for fusion, on the other hand, is [EQUATION] where [MATH] represents the isospin Clebsch-Gordan coefficient.', 'nucl-th-0505049-2-5-6': 'The cross section radius is taken as that of the fused entity, [MATH], with [MATH] as a characteristic radius for a [MATH] hadron.', 'nucl-th-0505049-2-5-7': 'Equations ([REF]) and ([REF]) allow to extract the square of the transition matrix element needed for computation of the partial resonance-width.', 'nucl-th-0505049-2-6-0': 'The partial width for decay into [MATH] and [MATH] can be generally represented as [EQUATION]', 'nucl-th-0505049-2-6-1': 'As before, the factors of [MATH] are absorbed into [MATH].', 'nucl-th-0505049-2-6-2': 'A resonance in the continuum can undergo three types of binary decay: where (i) both daughters are particles with well established properties within the discrete spectrum below [MATH], (ii) one of the daughters belongs to the discrete spectrum and the other to the continuum and, finally, where (iii) both daughter resonances belong to the continuum.', 'nucl-th-0505049-2-6-3': 'In the case (i), the state densities are [MATH].', 'nucl-th-0505049-2-6-4': 'With the detailed balance relation, [MATH], we then get [EQUATION]', 'nucl-th-0505049-2-6-5': 'In analyzing the case (ii), let the particle characterized by [MATH] belong to the discrete spectrum and that characterized by [MATH] to the continuum spectrum.', 'nucl-th-0505049-2-6-6': 'From ([REF]), we then find [EQUATION] where [MATH].', 'nucl-th-0505049-2-6-7': 'For [MATH] small compared to [MATH], it is of advantage to represent the subintegral density of states as an exponential of the density logarithm and to expand the logarithm in [MATH].', 'nucl-th-0505049-2-6-8': 'The integration over relative momentum can be thereafter carried out explicitly, yielding [EQUATION]', 'nucl-th-0505049-2-6-9': 'Here, [MATH] and the temperature is defined as [EQUATION]', 'nucl-th-0505049-2-6-10': 'For [MATH] small compared to [MATH], the obvious further possibility is the expansion of [MATH] in [MATH], with an emergence of the chemical potentials conjugate to [MATH].', 'nucl-th-0505049-2-7-0': 'In the case (iii), of both daughters in the continuum with large masses [MATH], the nonrelativistic limit in Eq. ([REF]) is justified.', 'nucl-th-0505049-2-7-1': 'On employing Eqs. ([REF]) and ([REF]), the result for the partial width is [EQUATION] where [EQUATION]', 'nucl-th-0505049-2-7-2': 'To obtain the last expression in Eq. ([REF]), we have expanded the logarithm of subintegral density, with the temperature representing [MATH].', 'nucl-th-0505049-2-8-0': 'The total decay width of an [MATH] resonance is finally [EQUATION]', 'nucl-th-0505049-2-8-1': 'A moving resonance will live an average time of [MATH], where [MATH] is the resonance Lorentz factor.', 'nucl-th-0505049-2-9-0': 'The formulas above provide the basis for our Monte-Carlo simulations of the resonance decay sequences in heavy ion collisions.', 'nucl-th-0505049-2-9-1': 'A resonance follows an exponential decay law corresponding to [MATH].', 'nucl-th-0505049-2-9-2': 'The product properties are selected according to the decay branching ratios.', 'nucl-th-0505049-2-9-3': 'Since the parent angular momentum is not tracked in the Weisskopf approach, the angular distribution of products is taken as isotropic.', 'nucl-th-0505049-2-9-4': 'During the evolution two resonances can fuse with each other, according to the cross section of Eq. ([REF]), if the final state is in continuum.', 'nucl-th-0505049-2-9-5': 'If, on the other hand, the state is discrete, the cross section acquires the standard form [CITATION], from Eqs. ([REF]) and ([REF]), [EQUATION] where the width for the spectral function is considered: [EQUATION]', 'nucl-th-0505049-2-9-6': 'Besides the decay and fusion processes, related by detailed balance, a provisional constant cross section [MATH] mb has been assumed for all collisions.', 'nucl-th-0505049-2-10-0': 'In our model, we simulate, in particular, the features of the final state of central Au+Au collisions at [MATH] GeV.', 'nucl-th-0505049-2-10-1': 'Following the presumption that the decay sequences will tend to erase fine details of the initial state, we push the characteristics of the initial state to an extreme, allowing for a single heavy resonance to populate a given rapidity region.', 'nucl-th-0505049-2-10-2': 'In the end, when examining the transverse momentum spectra, we find that the resonance decay and reformation alone generates insufficient transverse collective energy, indicating that the early resonances need to be affected by a collective motion generated prior to the resonance stage.', 'nucl-th-0505049-2-10-3': 'This finding is consistent with those in other works [CITATION].', 'nucl-th-0505049-2-11-0': 'Within the single resonance scenario, the local final state reflects the initial quantum numbers of a resonance characterized by [MATH], where [MATH].', 'nucl-th-0505049-2-11-1': 'After the value of the Hagedorn temperature [MATH] is set, the relative yields of particles in the final state are, in practice, sensitive only to the relative values of the quantum numbers of the initial resonance.', 'nucl-th-0505049-2-11-2': 'We normally impose strangeness neutrality, so that the starting value is [MATH].', 'nucl-th-0505049-2-11-3': 'The magnitude of [MATH], for a given [MATH], can be adjusted by using the final-state antiproton-to-proton or antiproton-to-pion ratios.', 'nucl-th-0505049-2-11-4': 'The starting isospin, for a given [MATH], can be adjusted by using the isospin of original nuclei.', 'nucl-th-0505049-2-11-5': 'However, we find that the initial isospin has only a marginal impact on the isospin of individual final particles.', 'nucl-th-0505049-2-11-6': 'This may be attributed to the cumulative effect of isospin fluctuations when many particles, compared to [MATH], are produced.', 'nucl-th-0505049-2-11-7': 'For specific initial [MATH] values, we repeat numerous times the Monte-Carlo simulations of the decay chain and recombination, and the results presented here are an average of about [MATH] generated event sequences.', 'nucl-th-0505049-2-12-0': 'Figure [REF] illustrates the average features of an exemplary local system that starts out as a resonance characterized by [MATH] GeV and [MATH].', 'nucl-th-0505049-2-12-1': 'The left panel shows the ratio of the average maximal resonance mass [MATH] to the initial mass [MATH], as a function of time, as well as the mass asymmetry, the mass difference between the heaviest resonance and the next heaviest, divided by the sum of their masses, [MATH].', 'nucl-th-0505049-2-12-2': 'Persistence of the large asymmetry with time indicates that the heavy resonance decays primarily through light-hadron emission.', 'nucl-th-0505049-2-13-0': 'In addition, Fig. [REF] shows, as a function of time, the abundances of particles (left panel) and antiparticles (right panel).', 'nucl-th-0505049-2-13-1': 'The antibaryon abundances freeze out noticeably earlier than the baryon abundances, in spite of a low initial baryon number relative to the initial mass.', 'nucl-th-0505049-2-13-2': 'The higher the strangeness, the later the abundance saturates.', 'nucl-th-0505049-2-13-3': 'This is likely due to the fact that the effects of strangeness fluctuation need to accumulate with time; the situation would change if we assumed strangeness fluctuations right for the initial resonance conditions.', 'nucl-th-0505049-2-14-0': 'The calculations have been repeated while suppressing the back resonance fusion reactions.', 'nucl-th-0505049-2-14-1': 'As expected, in this case the maximal mass and asymmetry decrease faster with time, see the left panel of Fig. [REF].', 'nucl-th-0505049-2-14-2': 'The abundances (not shown) grow faster and saturate earlier for the modified evolution.', 'nucl-th-0505049-2-15-0': 'Table [REF] compares the yield ratios from the resonance-decay model, within the region of optimal model-parameter values, with the available central Au+Au collision data from RHIC at [MATH] GeV [CITATION].', 'nucl-th-0505049-2-15-1': 'The temperature of the resonance mass spectrum, [MATH], and the initial baryon-number-to-mass ratio, [MATH], can be adjusted by simultaneously considering the ratios of [MATH] and [MATH].', 'nucl-th-0505049-2-15-2': 'An optimal agreement between the model and the central RHIC data is obtained for [MATH] MeV and [MATH] GeV[MATH].', 'nucl-th-0505049-2-15-3': 'The model results tend to be only weakly sensitive to [MATH]: with an increase in [MATH] a slight increase in [MATH] is favored, that can be traced to the factor [MATH] in the density of states.', 'nucl-th-0505049-2-16-0': 'The overall agreement between data and the decay-model calculations in Table [REF] is quite remarkable, given that only two parameters, [MATH] and [MATH], are adjusted.', 'nucl-th-0505049-2-16-1': 'One should note that the optimal Hagedorn temperature of [MATH] MeV is close to the critical temperature of [MATH] MeV for a transition to the color-deconfined QGP phase obtained in the lattice QCD calculations at zero net baryon density [CITATION], and it is also similar to the chemical freeze-out temperature [MATH] MeV extracted from the analysis of RHIC data within a grand-canonical model [CITATION].', 'nucl-th-0505049-2-17-0': 'At the general level, the calculations are quite good in reproducing yield ratios involving strange particles.', 'nucl-th-0505049-2-17-1': 'A more detailed examination, however, reveals some potential deficit of multistrange baryons and antibaryons.', 'nucl-th-0505049-2-17-2': 'Possible reasons for the deficiency, to be investigated in the future, include: the possible role played by the non-resonant strangeness-exchange processes [CITATION] and by the multiparticle processes [CITATION] and, further, the possible sensitivity of strangeness production to an early system dynamics [CITATION] and, specifically, to strangeness fluctuations for early resonances and/or to details in flavor-dependence of state density.', 'nucl-th-0505049-2-18-0': 'Besides the yield ratios in the standard model evolution, the corresponding ratios from the evolution with suppressed fusion processes are given in parenthesis in Table [REF] for [MATH] MeV.', 'nucl-th-0505049-2-18-1': 'Though the suppression of fusion alters particle abundances early on in the system development, the final yields turn out to be rather similar, quite uniformly across the particle species.', 'nucl-th-0505049-2-18-2': 'Only a careful examination reveals that the fusion suppression enhances slightly the production of pions and other light mesons and reduces slightly the production of heavier strange particles.', 'nucl-th-0505049-2-19-0': 'Within a moderate range, there is no strong preference for a particular Hagedorn temperature.', 'nucl-th-0505049-2-19-1': 'As Table [REF] shows, similar yield ratios are obtained for [MATH] MeV as for [MATH] MeV, if, in the context of the RHIC data, a slightly reduced initial mass of [MATH] GeV is assumed.', 'nucl-th-0505049-2-20-0': 'In Fig. [REF], we show the particle yield ratios that turn out to be particularly sensitive to the Hagedorn temperature under a given constraint.', 'nucl-th-0505049-2-20-1': "In one case, while varying the temperature, we adjust the initial fireball's ratio [MATH] to reproduce the ratio [MATH] for the RHIC data (lower set of lines).", 'nucl-th-0505049-2-20-2': 'In that case, the baryon-to-antibaryon ratios, [MATH], remain rather stable with [MATH] variation; the strongest variations are observed for the ratio of strange baryons to the negatively charged hadrons or to the negative mesons.', 'nucl-th-0505049-2-20-3': 'On the other hand, if we adjust the ratio [MATH] to reproduce the ratio [MATH] (upper set of lines), strong variations are observed for the [MATH] ratios.', 'nucl-th-0505049-2-20-4': 'No matter what fitting strategy is followed, a reasonable agreement with the data is obtained within the Hagedorn temperature range of [MATH].', 'nucl-th-0505049-2-21-0': 'We next confront our resonance decay model with the SPS abundance data from the central Pb+Pb collisions at the laboratory energy of [MATH] GeV.', 'nucl-th-0505049-2-21-1': 'As illustrated in Fig. [REF], an optimal agreement with the data is obtained for [MATH] GeV when assuming (at [MATH] MeV) a starting baryon number of [MATH].', 'nucl-th-0505049-2-21-2': 'While the general agreement is rather good, we note that the calculated [MATH] ratio is about [MATH] larger than the data.', 'nucl-th-0505049-2-21-3': 'It is likely that the assumption of a larger number of lighter initial resonances would improve the agreement; in the thermal model the discrepancy is tauted as strangeness undersaturation [CITATION].', 'nucl-th-0505049-2-22-0': 'Measured kinematic spectra of particles from central heavy ion collisions exhibit the effects of collective expansion.', 'nucl-th-0505049-2-22-1': 'As may be expected, with suppressed fusion processes, our model produces kinematic spectra characterized by slope temperatures close to [MATH] (see also Ref. [CITATION]).', 'nucl-th-0505049-2-22-2': 'In the standard version of the model, the sequences of decay and fusion generate some collective motion, but not enough to explain the transverse RHIC spectra.', 'nucl-th-0505049-2-22-3': 'For pions e.g. the slope temperature raises by 4 compared to the version without fusion.', 'nucl-th-0505049-2-22-4': 'Moderate increases in the elastic cross section, such as to an overall 10 mb, raise kinematic temperatures further, 6 for pions, but not well enough to approach data.', 'nucl-th-0505049-2-22-5': 'As a consequence, it is necessary to assume the presence of some collective motion early on in system evolution, leading to resonances that exhibit space-momentum correlations.', 'nucl-th-0505049-2-22-6': 'It might be that the dynamics, beyond statistics, needs to be involved in the predominant decays, involving the interior degrees of freedom of resonances.', 'nucl-th-0505049-2-22-7': 'The first resonances might also emerge at finite transverse velocities.', 'nucl-th-0505049-2-22-8': 'In either case, degrees of freedom beyond resonances would be involved.', 'nucl-th-0505049-2-23-0': 'The transverse mass spectra displayed in Fig. [REF] are obtained by folding a common collective velocity field with the spectra from our decay model.', 'nucl-th-0505049-2-23-1': 'Specifically, we assume a uniform transverse velocity distribution, [MATH].', 'nucl-th-0505049-2-23-2': 'The spectra for all the hadrons can be best described, at [MATH] MeV, with a uniform velocity field of [MATH], corresponding to an average flow velocity of [MATH].', 'nucl-th-0505049-2-23-3': 'Notably, much less early flow, as characterized by [MATH], is required to explain the particle spectra at SPS energy, see Fig. [REF].', 'nucl-th-0505049-2-24-0': 'In summary, we have formulated a statistical model of hadron resonance formation and decay.', 'nucl-th-0505049-2-24-1': 'Within the model, the density of hadronic states in mass is described in terms of a universal Hagedorn-type temperature.', 'nucl-th-0505049-2-24-2': 'We have demonstrated that both the RHIC and SPS abundance data can be suitably described in terms of resonance decays at the spectral temperature of [MATH] MeV, even when pursuing the extreme assumption of a single heavy resonance populating the investigated rapidity region.', 'nucl-th-0505049-2-24-3': 'To explain the data for particle spectra, we needed to invoke additional collective motion beyond that generated in the hadronic interactions.', 'nucl-th-0505049-2-25-0': 'This work was supported by the U.S. National Science Foundation under Grant PHY-0245009 and by the U.S. Department of Energy under Grant DE-FG02-03ER41259.'} | [['nucl-th-0505049-1-9-0', 'nucl-th-0505049-2-9-0'], ['nucl-th-0505049-1-9-1', 'nucl-th-0505049-2-9-1'], ['nucl-th-0505049-1-9-2', 'nucl-th-0505049-2-9-2'], ['nucl-th-0505049-1-9-3', 'nucl-th-0505049-2-9-3'], ['nucl-th-0505049-1-9-4', 'nucl-th-0505049-2-9-4'], ['nucl-th-0505049-1-9-6', 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1707.02387 | {'1707.02387-1-0-0': '* Generating Realtime Motion Plans from Complex Natural Language Commands Using Dynamic Grounding Graphs', '1707.02387-1-1-0': 'We present an algorithm for combining natural language processing (NLP) and realtime robot motion planning to automatically generate safe robot movements.', '1707.02387-1-1-1': 'We present a novel method to map the complex natural language commands into appropriate cost function and constraint parameters for optimization-based motion planning.', '1707.02387-1-1-2': 'Given NLP commands, we generate a factor graph named Dynamic Grounding Graph (DGG).', '1707.02387-1-1-3': 'The coefficients of this factor graph are learned based on conditional random fields and used to dynamically generate the constraints for motion planning.', '1707.02387-1-1-4': 'We directly map the cost function to the parameters of the motion planner to generate collision-free and smooth paths in complex scenes with moving obstacles.', '1707.02387-1-1-5': 'We highlight the performance of our approach in a simulated environment as well as a human interacting with a 7-DOF Fetch robot with complex NLP commands.', '1707.02387-1-2-0': '# Introduction', '1707.02387-1-3-0': "In human-robot interaction (HRI), natural language processing (NLP) has been used as one of the interfaces to communicate human's intent to a robot [CITATION].", '1707.02387-1-3-1': 'There is substantial work in this area that has focused on simple commands or tasks for robot manipulation, such as pick and place objects.', '1707.02387-1-3-2': 'As robots are increasingly used in complex scenarios and applications, it is important to develop a new generation of motion planning and robot movement techniques that can respond appropriately to diverse and free-form NLP commands for HRI.', '1707.02387-1-3-3': 'This includes automatic parsing of rich and complex natural language commands and generate appropriate robot actions.', '1707.02387-1-3-4': 'For example, humans frequently issue commands that include sentences with orientation-based constraints such as "put a bottle on the table and keep it upright", or "move the knife but don\'t point it towards people", or sentences with velocity-based constraints such as "move slowly when you are human gets close".', '1707.02387-1-3-5': 'In order to generate robot actions and movements in response to such complex natural language commands, we need to deal with two kinds of challenges:', '1707.02387-1-4-0': 'Motion planning has been extensively studied in robotics and related areas for more than four decades.', '1707.02387-1-4-1': 'There is a large body of work to compute collision free paths between two configurations or perform clearly specified tasks, that take into account various constraints corresponding to robot kinematics and dynamics, uncertainty or the environment.', '1707.02387-1-4-2': 'However, current motion planning algorithms used with NLP need to deal with additional constraints that arise due to the use of complex commands corresponding to negation, orientation, velocity, etc.', '1707.02387-1-5-0': 'Main Results: We present a novel approach, called Dynamic Grounding Graphs (DGG), to map natural language commands to appropriate optimization-based motion planning formulation in a dynamic environment.', '1707.02387-1-5-1': 'Our planner uses constrained optimization to compute a high-DOF collision-free trajectory for the robot and we construct parametric constraints based on interpretation of natural language semantics.', '1707.02387-1-5-2': 'The novel contributions of our work include:', '1707.02387-1-6-0': 'We highlight the performance of our algorithms in a simulated environment as well as on a 7-DOF Fetch robot operating next to a human in a safe manner.', '1707.02387-1-6-1': 'Our approach can handle a rich set of natural language commands and generate appropriate paths in realtime.', '1707.02387-1-6-2': 'These include complex commands such as picking (e.g., "pick up a red object near you"), correcting the motion (e.g., "don\'t pick up that one" or negation "don\'t put it on the book").', '1707.02387-1-7-0': '# Related Work', '1707.02387-1-8-0': 'Most algorithm used to map the natural language instruction to robot actions tends to separate the problem into two parts, the parsing part and motion planning computation.', '1707.02387-1-8-1': 'In this section, we give a brief overview of prior work in these areas.', '1707.02387-1-9-0': '## Natural Language Processing', '1707.02387-1-10-0': 'Kollar et al. [CITATION] present a probabilistic graphical learning model called Generalized Grounding Graphs or G[MATH], by which the robot interprets and grounds natural language commands to the physical world.', '1707.02387-1-10-1': 'The graph is defined based in a syntactic parse structure of the command, enabling the system to associate parts of the command with objects, events, and locations in the external world to which they refer.', '1707.02387-1-10-2': 'Howard et al. [CITATION] reduced the search space by modifying the G[MATH] graph structure by adding all possible grounding nodes (meanings of word phrases) and optimizing the correspondence variables (indicating the word phrase and the grounding matches correctly).', '1707.02387-1-11-0': 'Other related work in language grounding of task instructions includes Branavan et al. [CITATION], who applied reinforcement learning to learn the mapping from natural language instructions to sequences of executable actions.', '1707.02387-1-11-1': 'Reinforcement learning based method requires little annotated data, but the search space can be very large when applied to motion planning for the robot arm.', '1707.02387-1-11-2': 'Matuszek et al. [CITATION] use a statistical machine translation model to map the natural language instructions to a path description language to follow directions in a robot navigation problem.', '1707.02387-1-11-3': 'Duvallet et al. [CITATION] exploit imitation learning to train the model via demonstrations of humans following directions.', '1707.02387-1-12-0': 'Our approach is inspired by [CITATION] and [CITATION] with respect to the linguistic parse structure and the probabilistic graphical model.', '1707.02387-1-12-1': 'However, we extend these methods so that they can be directly integrated with optimization-based planners in terms of using appropriate constraints.', '1707.02387-1-13-0': '## Robot Motion Planning in Dynamic Environments', '1707.02387-1-14-0': 'In order to generate collision-free motion plans in dynamic environments, many replanning algorithms have been suggested.', '1707.02387-1-14-1': 'Fox et al. [CITATION] proposed the dynamic window approach that searches for optimal velocity in a short period of time window to avoid dynamic obstacles.', '1707.02387-1-14-2': 'Optimization-based motion planners [CITATION] solve a constrained optimization problem to generate smooth and collision-free robot paths.', '1707.02387-1-14-3': 'These optimization-based planners can be adapted to other planning tasks by adding cost functions and constraints in the optimization formulation.', '1707.02387-1-14-4': 'However, it is hard and tedious to manually tune many parameters and the goal is to generate such constraints and parameters automatically.', '1707.02387-1-15-0': 'There is some work on integrating optimization-based motion planning with NLP in 2D workspaces.', '1707.02387-1-15-1': 'Silver et al. [CITATION].', '1707.02387-1-15-2': 'developed an algorithm of learning navigation cost functions from demonstration.', '1707.02387-1-15-3': 'Howard et al. [CITATION] used probabilistic graphical model to generated motion planning constraints in 2D navigation problem.', '1707.02387-1-15-4': 'As compared to these methods, our approach can handle 3D workspaces and high-dimensional configuration spaces to generate robot motions based on complex NLP commands.', '1707.02387-1-16-0': '## Robot Motion Planning for Human-Robot Interaction', '1707.02387-1-17-0': 'There is considerable work on generating safe motion plans for robots operating next to humans.', '1707.02387-1-17-1': 'Some methods use the notion of social acceptability [CITATION] or legibility [CITATION].', '1707.02387-1-17-2': 'Mainprice and Berenson [CITATION] constructed occupancy map for each voxels in the workspace to explicitly represent whether a voxel is likely to be occupied by humans, so that motion planners can generate collision-free robot paths.', '1707.02387-1-17-3': 'Other techniques focus on efficiency in human-robot collaborative tasks.', '1707.02387-1-17-4': 'Markov Decision Processes (MDP) are widely used to compute the best robot action policies [CITATION].', '1707.02387-1-17-5': 'Optimization-based online replanning [CITATION] has been successfully used for high-DOF robots in dynamically changing environments.', '1707.02387-1-17-6': 'These algorithms interleave planning and trajectory execution timesteps and incrementally compute the solution to perform robot tasks and we use a similar framework.', '1707.02387-1-18-0': '# Overview', '1707.02387-1-19-0': 'We first introduce the notation and terminology used in the paper and give an overview of our natural language processing and motion planning algorithms.', '1707.02387-1-20-0': '## Natural Language Processing', '1707.02387-1-21-0': 'From an input natural language command, we construct a factor graph, as shown in the left column of Fig. [REF], based on parsing of the command.', '1707.02387-1-21-1': 'For each node of the parse tree, we generate three types of node: word phrase node [MATH], grounding node [MATH], and correspondence node [MATH].', '1707.02387-1-22-0': 'The input sentence [MATH] is parsed by NLTK library [CITATION].', '1707.02387-1-22-1': 'Word phrase of each node in the parse tree is denoted as [MATH] for [MATH].', '1707.02387-1-22-2': 'Children of [MATH] are [MATH], [MATH], [MATH].', '1707.02387-1-22-3': 'The root node of parse tree is [MATH].', '1707.02387-1-22-4': 'For example, in Fig. [REF](a), the input sentence is "Put the cup on the table".', '1707.02387-1-22-5': 'The parse tree has the root word phrase [MATH]"Put", and its noun [MATH]"the cup" and the the preposition [MATH]"on" are the children nodes of the root node.', '1707.02387-1-22-6': 'The noun phrase [MATH]"the table" is the child node of [MATH].', '1707.02387-1-22-7': 'Similarly, in Fig. [REF](b), the command "Don\'t put it there" is decomposed into 4 noun phrase nodes.', '1707.02387-1-22-8': 'The word phrase [MATH]"Don\'t" is a negation of the verb, its child node [MATH]"put".', '1707.02387-1-22-9': '[MATH]"it" and [MATH]"there" are the children nodes of [MATH].', '1707.02387-1-22-10': 'Note that the parse tree is different from the parse tree in Fig. [REF](a).', '1707.02387-1-23-0': 'We first compute the groundings [MATH] of each word phrase [MATH].', '1707.02387-1-23-1': 'Grounding of each word phrase is the mapping from the word phrase to the meaning of it in the real world.', '1707.02387-1-23-2': 'Groundings can be objects, locations, motions, tasks or constraints.', '1707.02387-1-23-3': 'In our model, the grounding [MATH] depends on its work phrase [MATH] and children grounding nodes [MATH], [MATH], [MATH], where the tree structure of grounding nodes follow the parse tree.', '1707.02387-1-23-4': 'For example, in Fig. [REF](a), the groundings of [MATH]"the cup" and [MATH]"the table" indicate the objects (the cup and the table) in the environment.', '1707.02387-1-23-5': 'The grounding of [MATH]"on", with consideration of [MATH]the table, indicates the region the surface of the table.', '1707.02387-1-23-6': 'The grounding of [MATH]"Put" is the task of moving the cup on the surface of the table, and it is the task the robot should accomplish.', '1707.02387-1-23-7': 'In Fig. [REF](b), the negation part [MATH]"Don\'t" is interpreted as a constraint that the robot should not do something.', '1707.02387-1-24-0': 'Correspondence node [MATH] indicates the correct matching between the word phrase [MATH] and the grounding [MATH].', '1707.02387-1-24-1': 'It is a binary variable; [MATH] is [MATH] if the word phrase and the grounding correctly matches and [MATH] if not.', '1707.02387-1-24-2': 'For example, in Fig. [REF](a), if the grounding for [MATH]"the cup" is [MATH]the cup in the environment, they match correctly and so [MATH].', '1707.02387-1-24-3': 'However, if the grounding node is predicted incorrectly like [MATH]the table or [MATH](putting task), then it does not correspond to "the cup" so [MATH].', '1707.02387-1-24-4': 'The correspondence nodes are useful in the machine learning framework.', '1707.02387-1-24-5': 'In training step, a ground-truth data sample contains the correct groundings [MATH], thus all correspondence nodes are [MATH].', '1707.02387-1-24-6': 'We can synthesize more data samples by assigning wrong groundings to some [MATH] and set [MATH].', '1707.02387-1-25-0': 'Our primary novel goal is to compute the best cost function parameters that we directly use in the optimization-based motion planning, corresponding to a mapping from the input natural language commands.', '1707.02387-1-25-1': 'We denote [MATH], a real vector of size [MATH], as a collection of cost function parameters, where the size [MATH], the number of cost function parameters, depends on what types of cost functions are used.', '1707.02387-1-25-2': 'We define many types of cost functions, such as the collision avoidance, robot smoothness, robot end-effector speed, target positions and target orientations.', '1707.02387-1-25-3': 'Each cost function has its weight and other cost function parameters if necessary.', '1707.02387-1-25-4': 'For example, the robot end-effector speed cost function has parameters corresponding to the direction and the magnitude of the speed which impose a constraint on the final computed trajectory.', '1707.02387-1-25-5': 'If the weight of end-effector speed cost function is higher than others, then it takes a large portion of the overall object function in the optimization formulation and thus the cost function affects more to the planned path.', '1707.02387-1-25-6': 'If the weight is low, then the end-effector speed cost will be compromised and affect less to the planned path.', '1707.02387-1-25-7': 'We want the robot to avoid collisions in any case, so we set the weight of collision avoidance cost to [MATH], and other cost function weights are normalized by this weight.', '1707.02387-1-25-8': 'The details of cost function parameters are explained in Section [REF].', '1707.02387-1-25-9': 'In Fig. [REF], the resulting constraint based motion planning problems are shown in the middle column.', '1707.02387-1-25-10': 'In (a), we use collision avoidance cost function as default, smoothness cost function and the target location cost function.', '1707.02387-1-25-11': 'The target location, whose 3D coordinates are the cost function parameters, is set on the surface of the table.', '1707.02387-1-25-12': 'The cost function parameter node [MATH] contains the weights of them and the 3D coordinate of target location.', '1707.02387-1-25-13': 'In (b), where a new "Don\'t" command is given, a repulsion cost function is added.', '1707.02387-1-25-14': 'Thus, the cost function weight and the location of repulsion source (below the robot’s end-effector position) are added to [MATH].', '1707.02387-1-26-0': '## Robot Configurations and Motion Planning', '1707.02387-1-27-0': 'We denote a single configuration of the robot as a vector [MATH] that consists of joint-angles.', '1707.02387-1-27-1': 'A configuration at time [MATH], where [MATH], is denoted as [MATH].', '1707.02387-1-27-2': 'We assume [MATH] is twice differentiable, and its derivatives are denoted as [MATH] and [MATH].', '1707.02387-1-27-3': 'The [MATH]-dimensional space of configuration [MATH] is the configuration space [MATH].', '1707.02387-1-27-4': 'We represent each link of the robot as [MATH].', '1707.02387-1-27-5': 'The finite set of bounding box for link [MATH] is [MATH].', '1707.02387-1-27-6': 'The links and bounding boxes at a configuration [MATH] are denoted as [MATH] and [MATH], resp.', '1707.02387-1-28-0': "For a planning task with a given start configuration [MATH] and derivative [MATH], the robot's trajectory is represented by a matrix [MATH], [EQUATION]", '1707.02387-1-28-1': 'The robot trajectory passes through the [MATH] waypoints [MATH], which will be optimized by an objective function under constraints in the motion planning formulation.', '1707.02387-1-28-2': 'Robot configuration at time [MATH] is interpolated from two waypoints.', '1707.02387-1-28-3': 'Formally, for [MATH] such that [MATH], the configuration [MATH] and derivative [MATH] are cubically interpolated using [MATH], [MATH], [MATH] and [MATH].', '1707.02387-1-29-0': 'The [MATH]-th cost functions of the motion planner are [MATH].', '1707.02387-1-29-1': '[MATH] different cost functions we used in this paper are listed in Section [REF].', '1707.02387-1-29-2': 'Our motion planner solves an optimization problem with a non-linear cost functions and linear joint limit constraints to generate robot trajectories for time interval [MATH], [EQUATION]', '1707.02387-1-29-3': 'In the optimization formulation, [MATH] is the [MATH]-th cost function and [MATH] is the weight of the cost function.', '1707.02387-1-29-4': 'The collection of cost function parameters include the cost function weights [MATH], and other required information specified in some cost functions.', '1707.02387-1-29-5': 'For example, [MATH], [MATH] and [MATH] coordinates target position are the parameters we want to find, for the end-effector position cost function [MATH] (explained in Section [REF]).', '1707.02387-1-30-0': '# Dynamic Grounding Graphs', '1707.02387-1-31-0': 'Our work builds on the idea of Generalized Grounding Graphs (G[MATH]) model and Distributed Correspondence Graph (DCG) model [CITATION], and extend it to generate the appropriate learning model.', '1707.02387-1-31-1': 'Our goal of learning natural language commands is to find a mapping from a natural language sentence [MATH] to the cost function parameters [MATH], given the robotic environment [MATH], where the robot is working.', '1707.02387-1-31-2': "[MATH] is a representation for the environment composed of obstacle positions, orientations, and robot's configuration.", '1707.02387-1-31-3': 'From the environment description, feature vectors are constructed in the factor graph.', '1707.02387-1-31-4': '[MATH] is a real-valued vector that contains all cost function parameters used in the optimization-based motion planner.', '1707.02387-1-31-5': 'It also includes the weights of different types of cost functions used in the optimization formulation.', '1707.02387-1-31-6': 'For example, the end-effector position cost function (Equation ([REF])) requires 3D coordinates of target position as parameters.', '1707.02387-1-31-7': 'The repulsion cost function (Equation ([REF])) requires the repulsion source position and the constant in the exponential function.', '1707.02387-1-32-0': 'We use a probabilistic model for [MATH], [MATH] and [MATH].', '1707.02387-1-32-1': 'Finding the best cost parameters is posed as an optimization problem: [EQUATION]', '1707.02387-1-32-2': 'However, modeling the probability function without decomposing the variables and some independence assumptions is hard due to the high-dimensionality of [MATH], [MATH] and [MATH] and the dependencies between them.', '1707.02387-1-32-3': 'To simplify the problem, the natural language sentence is decomposed into [MATH] word phrases based on a parse tree , i.e. [EQUATION]', '1707.02387-1-32-4': 'Like G[MATH], we introduce the intermediate groundings [MATH] of word phrases [MATH], and correspondence variable [MATH].', '1707.02387-1-32-5': 'The correspondence variables [MATH] is a binary random variable.', '1707.02387-1-32-6': 'The value [MATH] indicates that the word phrase [MATH] correctly corresponds to the grounding [MATH].', '1707.02387-1-32-7': '[MATH] means an incorrect correspondence.', '1707.02387-1-33-0': 'We assume conditional independence of the probabilities to construct a factor graph (see left column of Fig. [REF]).', '1707.02387-1-33-1': 'With the independence assumptions, a single factor is connected to a word phrase node and its children grounding nodes containing information about the sub-components.', '1707.02387-1-33-2': 'These independence assumptions simplify the problem and make it solvable by efficiently taking advantage of the tree structure of the probabilistic graphical learning model.', '1707.02387-1-33-3': "Formally, the root grounding node [MATH] contains all the information about robot's motion.", '1707.02387-1-33-4': 'The factor connecting [MATH] and [MATH] means that from the root grounding node, the cost function parameters [MATH] are optimized without any consideration of other nodes.', '1707.02387-1-33-5': 'Other factors connect [MATH], [MATH], [MATH], children grounding nodes [MATH], and the environment [MATH], where parent-child relationship is based on parse tree constructed from the natural language sentence.', '1707.02387-1-33-6': 'This graph representation corresponds to the following equation: [EQUATION].', '1707.02387-1-33-7': 'For the root factor connecting [MATH], [MATH] and [MATH], we formulate the continuous domain of [MATH].', '1707.02387-1-33-8': 'We compute the Gaussian Mixture Model (GMM) on the probability distribution [MATH] and model our probability with non-root factors as following: [EQUATION]', '1707.02387-1-34-0': 'where [MATH] is the normalization factor, [MATH] is the feature function, and [MATH] and [MATH] is the log-linearlization of the feature function.', '1707.02387-1-34-1': 'The function [MATH] generates a feature vector, given a grounding [MATH], a word phrase [MATH], a correspondence [MATH], children groundings [MATH] and the environment [MATH].', '1707.02387-1-35-0': 'As we construct the factor graph, we build a learning model on it and use that for training and inferring the meaning of given commands.', '1707.02387-1-35-1': 'In particular, we use Conditional Random Fields (CRF), a learning model for factor graphs.', '1707.02387-1-35-2': 'During the training step of CRF, we solve the optimization problem of maximizing the probability of the samples in the training dataset over the feature coefficients [MATH] for every parse tree sturcture.', '1707.02387-1-35-3': 'Multiplying Eq. ([REF]) for all training samples, the optimization problem becomes [EQUATION] where superscripts [MATH] mean the indices of the training samples.', '1707.02387-1-35-4': 'This is a tree-structured CRF problem.', '1707.02387-1-36-0': 'At the inference step, we used the trained CRF factor graph models to find the best groundings [MATH] and the cost function parameters [MATH] by solving the CRF maximization problem [EQUATION].', '1707.02387-1-36-1': 'Because the nodes [MATH], [MATH] being optimized has a tree structure in the factor graph, we can solve the optimization problem efficiently using dynamic programming.', '1707.02387-1-36-2': 'Each factor depends on its parent and children varying variables and other fixed variables connected to it.', '1707.02387-1-36-3': 'This implies that we can solve the sub-problems in a bottom-up manner and combine the results to solve the bigger problem corresponding to the root node.', '1707.02387-1-37-0': '# Real-time Trajectory Planning With NLP Input', '1707.02387-1-38-0': 'We present our realtime optimization-based planning algorithm that handle complex NLP commands.', '1707.02387-1-38-1': 'The overall optimization formulation is given in Eqn. [REF] in Sec. [REF].', '1707.02387-1-38-2': 'In order to formulate the constraints, we use the following cost functions:', '1707.02387-1-39-0': 'Collision avoidance: By default, the robot should always avoid obstacles.', '1707.02387-1-39-1': '[EQUATION] where [MATH] is the penetration depth between a robot bounding box [MATH] and an obstacle [MATH].', '1707.02387-1-39-2': 'As it is a default cost function, we set the weight of this cost function 1, and changes the weights of other types of cost functions accordingly.', '1707.02387-1-40-0': "Smoothness: We penalize the magnitude of robot's joint angle speeds to make the trajectory smooth.", '1707.02387-1-40-1': 'This correspond to the integral of first derivative of joint angles over the trajectory duration.', '1707.02387-1-40-2': 'This function is useful when we need to control the speed of the robot.', '1707.02387-1-40-3': "When the robot should operate at a low speed (e.g. when a human is too close), or we don't want abrupt moves (e.g. for human safety), the smoothness cost can have high weights so that the robot moves slowly without jerky motions.", '1707.02387-1-41-0': "End-effector position: Usually, a user specifies the robot's target position so that the robot reaches a goal position.", '1707.02387-1-41-1': "This cost function penalizes the squared distance between robot's end-effector and the target position over the trajectory duration.", '1707.02387-1-41-2': '[EQUATION] where [MATH] is the robot end-effector position at time [MATH], and [MATH] is the target position.', '1707.02387-1-41-3': 'The target position [MATH] is considered as a cost function parameter.', '1707.02387-1-41-4': 'In the NLP algorithm, a position grounding node encodes the target position parameter.', '1707.02387-1-41-5': 'It can be a 3D position or the current object position in the environment.', '1707.02387-1-41-6': 'Typically, the target position is specified by object names in the sentence, such as "pick up the cup" or "move to the box".', '1707.02387-1-41-7': 'In these cases, the grounding nodes for "the cup" and "the box" are interpreted as the current 3D coordinates of target positions, which are the parameters of this cost function.', '1707.02387-1-42-0': 'End-effector orientation: Robotic manipulation tasks are sometimes constrained by the end-effector orientation.', '1707.02387-1-42-1': 'This cost function penalizes the squared angular differences between end-effector orientation and the target orientation over the trajectory duration.', '1707.02387-1-42-2': '[EQUATION]', '1707.02387-1-43-0': "where [MATH] is the quaternion representation of the robot end-effector's orientation at time [MATH], [MATH] is the orientation that we want the robot to maintain its end-effector orientation to be, [MATH] is the normal up-vector of robot's endeffector, and [MATH] is the target up-vector.", '1707.02387-1-43-1': 'Similar to End-effector position cost, the target orientation [MATH] is the cost function parameters.', '1707.02387-1-43-2': 'The target orientation usually depends on the object the robot picked up.', '1707.02387-1-43-3': 'E.g., when the robot is doing a peg-hole insertion task under a given command "insert that into the hole", the orientation of robot\'s end-effector [MATH] should be constrained near the hole.', '1707.02387-1-43-4': 'If the robot arm is holding a cup of water, it should be upright so as to not spill the water.', '1707.02387-1-43-5': 'In this case, [MATH] is set to [MATH].', '1707.02387-1-44-0': "End-effector speed: It penalizes the robot's endeffector speed and direction: [EQUATION] where [MATH] is the robot end-effector speed at time [MATH], and [MATH] is the target speed.", '1707.02387-1-44-1': 'The parameters of this cost function is [MATH].', '1707.02387-1-44-2': "In some cases, it is required to restrict the robot's end-effector velocity.", '1707.02387-1-44-3': "For example, a user wants to pick up a cup filled with water and doesn't want to spill it.", '1707.02387-1-44-4': 'Spilling can be prevented by limiting the end-effector speed to be slow.', '1707.02387-1-44-5': 'Or the user can guide the motion by using natural language commands like "Move left, then move around the obstacle" to compute a better trajectory.', '1707.02387-1-45-0': "Repulsion: The repulsion functions are commonly used as potential fields [EQUATION] where [MATH] is the position where we don't want the robot to move to.", '1707.02387-1-45-1': 'The coefficient [MATH] plays the role of how much is the cost affected by [MATH], the distance between the end-effector position and the repulsion source.', '1707.02387-1-45-2': 'The cost function is maximized at which the end-effector position is exactly at the repulsion source, and decreases as the distance between the end-effector and the repulsion position increases.', '1707.02387-1-45-3': 'For example, if the command is "Don\'t put the cup on the laptop", we can define a repulsion cost with the laptop position as the repulsion source.', '1707.02387-1-45-4': 'The cost function is inversely proportional to the distance between the end-effector and the laptop.', '1707.02387-1-46-0': '# Implementation and Results', '1707.02387-1-47-0': 'We have implemented our algorithm and evaluated its performance in a simulated environment and a 7-DOF Fetch robot.', '1707.02387-1-47-1': 'All the timings are generated on a multi-core PC.', '1707.02387-1-48-0': 'We have evaluated the performance in complex environments composed of multiple objects and local minima.', '1707.02387-1-48-1': 'Based on the NLP commands, the robot decides to pick an appropriate object or is steered towards the goal position in a complex scene (see Fig. [REF].', '1707.02387-1-48-2': 'In particular, the NLP commands are used to guide the robot by the user by using commands such as "move right", "move up", "move left" or "move down".', '1707.02387-1-48-3': 'For each such commands, we compute the appropriate cost functions.', '1707.02387-1-49-0': 'We also integrated our NLP-based planner with ROS and evaluated its performance on the 7-DOF Fetch robot.', '1707.02387-1-49-1': 'In a real-world setting, we test its performance on different tasks corresponding to: (1) move a soda can on the table from one position to the other; (2) not move it over the book.', '1707.02387-1-49-2': 'With a noisy point cloud sensor on the robot, the thin book is not recognized as a separate obstacle by the robot, though the human user wants the robot to avoid it.', '1707.02387-1-49-3': 'Our NLP algorithm evaluates the meaning of the commands and generates appropriate cost function and constraint parameters for the real-time motion planner.', '1707.02387-1-49-4': 'In Fig. [REF], the two sub-tasks are specified in one sentence at the beginning, as "move the can on the table, but don\'t put it on the book".', '1707.02387-1-49-5': "The cost function is used to move the robot's end-effector to the surface of the table.", '1707.02387-1-49-6': 'Another cost function penalizes the distance between the book and the end-effector.', '1707.02387-1-49-7': 'In Fig. [REF], only the first sub-task is given at the beginning.', '1707.02387-1-49-8': 'This results in the robot moving the can on the book.', '1707.02387-1-49-9': 'As the robot gets too close to the book, the person says "stop", then says "don\'t put it there."', '1707.02387-1-49-10': 'The robot recomputes the cost functions and avoids the region around the book.', '1707.02387-1-50-0': '# Analysis and Benefits', '1707.02387-1-51-0': 'In this section, we analyze the performance of our approach and highlight its benefits over prior methods.', '1707.02387-1-51-1': 'We evaluated the end-to-end performance of our approach including the NLP, mapping and the motion planning modules on different benchmarks.', '1707.02387-1-51-2': 'We evaluated the performance based on the following metrics:', '1707.02387-1-52-0': 'Table [REF] shows the results of the experiments with varying number of training data samples on the simulation environment shown in Fig. [REF].', '1707.02387-1-52-1': 'When the number of training data samples increases, the success rate also increases, and the trajectory duration and the trajectory smoothness cost decrease.', '1707.02387-1-52-2': 'Also, the standard deviations also decreases, implying that our planning cost function mapping algorithm becomes more stable.', '1707.02387-1-52-3': 'Table [REF] shows the running time of our algorithm and the distances from the obstacle on the table in the real world settings.', '1707.02387-1-53-0': '## Benefits over Prior Methods', '1707.02387-1-54-0': 'Most prior methods that combine NLP and motion planning have focused on understanding natural language instructions computing robot motion for simple environments and constraints.', '1707.02387-1-54-1': 'In our approach, the goal is to generate appropriate high-DOF motion trajectory in response to complex natural language commands, as shown in Section 6.', '1707.02387-1-54-2': 'Furthermore, our optimization-based formulation is designed to handle challenging scenarios with lot of obstacles and constraints (e.g. dynamics constraints, smoothness constraints), on the resulting trajectories and this results in stable trajectories.', '1707.02387-1-54-3': 'Unlike prior methods, the output of NLP parsing algorithm is directly coupled with the specification of the motion planning problem as a constrained optimization methods.', '1707.02387-1-54-4': 'This enables to handle many complex NLP commands including negation, velocity or position constraints or orientation specification.', '1707.02387-1-54-5': 'As the parsing capabilities of NLP systems improve in the future, we can map more complex commands to our framework.', '1707.02387-1-55-0': '# Limitations, Conclusions and Future Work', '1707.02387-1-56-0': 'We presented a real-time motion planning algorithm that computes appropriate trajectories based on complex NLP commands.', '1707.02387-1-56-1': 'We highlight the performance in simulated and real-world scenes with a 7-DOF manipulator operating next to humans.', '1707.02387-1-56-2': 'The preliminary results are promising and our approach can handle complex scenarios as compared to prior methods.', '1707.02387-1-57-0': 'We use a trajectory optimization algorithm to compute the high-DOF robot trajectory.', '1707.02387-1-57-1': 'It is a high-dimensional optimization problem and the solver get stuck in local minima.', '1707.02387-1-57-2': 'As a result, it is hard to provide rigorous guarantees in terms of satisfying all the constraints or following the intent of the user.', '1707.02387-1-57-3': 'Furthermore, the accuracy of the mapping algorithm varies as a function of the training data.', '1707.02387-1-57-4': 'As future work, we would like to overcome these limitations and evaluate the approach in challenging scenarios with moving obstacles and perform complex robot tasks.', '1707.02387-1-57-5': 'More work is needed to handle the full diversity of a natural language, especially for rare words, complicated grammar styles, as well as hidden intention or emotion in the human speech.', '1707.02387-1-57-6': 'We plan to incorporate stronger natural language processing and machine learning methods such as those based on semantic parsing, neural sequence-to-sequence models, reinforcement learning, and speech-based emotion analysis, and compute the appropriate optimization-based planning formulations.'} | {'1707.02387-2-0-0': '* Generating Realtime Motion Plans from Complex Natural Language Commands Using Dynamic Grounding Graphs', '1707.02387-2-1-0': 'We present an algorithm for combining natural language processing (NLP) and realtime robot motion planning to automatically generate safe robot movements.', '1707.02387-2-1-1': 'We present a novel method to map the complex natural language commands into appropriate cost function and constraint parameters for optimization-based motion planning.', '1707.02387-2-1-2': 'Given NLP commands, we generate a factor graph named Dynamic Grounding Graph (DGG).', '1707.02387-2-1-3': 'The coefficients of this factor graph are learned based on conditional random fields and used to dynamically generate the constraints for motion planning.', '1707.02387-2-1-4': 'We directly map the cost function to the parameters of the motion planner to generate collision-free and smooth paths in complex scenes with moving obstacles, by modeling intricate motion planning parameters such as speed, orientation, position, as well as smoothness, repulsion, and avoidance.', '1707.02387-2-1-5': 'We highlight the performance of our approach in a simulated environment as well as via a human interacting with a 7-DOF Fetch robot with complex NLP commands.', '1707.02387-2-2-0': '# Introduction', '1707.02387-2-3-0': "In human-robot interaction (HRI), natural language processing (NLP) has been used as one of the interfaces to communicate human's intent to a robot [CITATION].", '1707.02387-2-3-1': 'There is substantial work in this area that has focused on simple commands or tasks for robot manipulation, such as pick and place objects.', '1707.02387-2-3-2': 'As robots are increasingly used in complex scenarios and applications, it is important to develop a new generation of motion planning and robot movement techniques that can respond appropriately to diverse and free-form NLP commands for HRI.', '1707.02387-2-3-3': 'This includes automatic parsing of rich and complex natural language commands and generate appropriate robot actions.', '1707.02387-2-4-0': 'For example, humans frequently issue commands that include sentences with orientation-based constraints such as "put a bottle on the table and keep it upright", or "move the knife but don\'t point it towards people", or sentences with velocity-based constraints such as "move slowly when you are human gets close".', '1707.02387-2-4-1': 'In order to generate robot actions and movements in response to such complex natural language commands, we need to deal with two kinds of challenges:', '1707.02387-2-5-0': 'Motion planning has been extensively studied in robotics and related areas for more than four decades.', '1707.02387-2-5-1': 'There is a large body of work to compute collision free paths between two configurations or perform clearly specified tasks, that take into account various constraints corresponding to robot kinematics and dynamics, uncertainty or the environment.', '1707.02387-2-5-2': 'However, current motion planning algorithms used with NLP need to deal with additional constraints that arise due to the use of complex commands corresponding to negation, orientation, velocity, etc.', '1707.02387-2-6-0': 'Main Results: We present a novel approach, called Dynamic Grounding Graphs (DGG), to map natural language commands to appropriate optimization-based motion planning formulation in a dynamic environment.', '1707.02387-2-6-1': 'Our planner uses constrained optimization to compute a high-DOF collision-free trajectory for the robot and we construct parametric constraints based on interpretation of natural language semantics.', '1707.02387-2-6-2': 'Our work takes an important step towards this direction by building models that execute rich natural language commands referring to intricate motion planning parameters such as speed, orientation, position, as well as smoothness, repulsion, and avoidance (via spatial and temporal adjectives, adverbs, superlative and comparative degrees, negations, etc.), by directly mapping such complex instructions to optimization-based planning.', '1707.02387-2-6-3': 'The novel contributions of our work include:', '1707.02387-2-7-0': 'We highlight the performance of our algorithms in a simulated environment as well as on a 7-DOF Fetch robot operating next to a human in a safe manner.', '1707.02387-2-7-1': 'Our approach can handle a rich set of natural language commands and generate appropriate paths in realtime.', '1707.02387-2-7-2': 'These include complex commands such as picking (e.g., "pick up a red object near you"), correcting the motion (e.g., "don\'t pick up that one" or negation "don\'t put it on the book").', '1707.02387-2-8-0': '# Related Work', '1707.02387-2-9-0': 'Most algorithm used to map the natural language instruction to robot actions tends to separate the problem into two parts, the parsing part and motion planning computation.', '1707.02387-2-9-1': 'In this section, we give a brief overview of prior work in these areas.', '1707.02387-2-10-0': '## Natural Language Processing', '1707.02387-2-11-0': 'Kollar et al. [CITATION] present a probabilistic graphical learning model called Generalized Grounding Graphs or G[MATH], by which the robot interprets and grounds natural language commands to the physical world.', '1707.02387-2-11-1': 'The graph is defined based in a syntactic parse structure of the command, enabling the system to associate parts of the command with objects, events, and locations in the external world to which they refer.', '1707.02387-2-11-2': 'Howard et al. [CITATION] reduced the search space by modifying the G[MATH] graph structure by adding all possible grounding nodes (meanings of word phrases) and optimizing the correspondence variables (indicating the word phrase and the grounding matches correctly).', '1707.02387-2-12-0': 'Other related work in language grounding of task instructions includes Branavan et al. [CITATION], who applied reinforcement learning to learn the mapping from natural language instructions to sequences of executable actions.', '1707.02387-2-12-1': 'Reinforcement learning based method requires little annotated data, but the search space can be very large when applied to motion planning for the robot arm.', '1707.02387-2-12-2': 'Matuszek et al. [CITATION] use a statistical machine translation model to map the natural language instructions to a path description language to follow directions in a robot navigation problem.', '1707.02387-2-12-3': 'Duvallet et al. [CITATION] exploit imitation learning to train the model via demonstrations of humans following directions.', '1707.02387-2-13-0': 'Our approach is inspired by [CITATION] and [CITATION] with respect to the linguistic parse structure, the probabilistic graphical model and use of optimization-based motion planning.', '1707.02387-2-14-0': '## Robot Motion Planning in Dynamic Environments', '1707.02387-2-15-0': 'In order to generate collision-free motion plans in dynamic environments, many replanning algorithms have been suggested.', '1707.02387-2-15-1': 'Fox et al. [CITATION] proposed the dynamic window approach that searches for optimal velocity in a short period of time window to avoid dynamic obstacles.', '1707.02387-2-15-2': 'Optimization-based motion planners [CITATION] solve a constrained optimization problem to generate smooth and collision-free robot paths.', '1707.02387-2-15-3': 'These optimization-based planners can be adapted to other planning tasks by adding cost functions and constraints in the optimization formulation.', '1707.02387-2-15-4': 'However, it is hard and tedious to manually tune many parameters and the goal is to generate such constraints and parameters automatically.', '1707.02387-2-16-0': 'There is some work on integrating optimization-based motion planning with NLP in 2D workspaces.', '1707.02387-2-16-1': 'Silver et al. [CITATION].', '1707.02387-2-16-2': 'developed an algorithm of learning navigation cost functions from demonstration.', '1707.02387-2-16-3': 'Howard et al. [CITATION] used probabilistic graphical model to generated motion planning constraints in 2D navigation problem.', '1707.02387-2-16-4': 'As compared to these methods, our approach can handle 3D workspaces and high-dimensional configuration spaces to generate robot motions based on complex NLP commands.', '1707.02387-2-17-0': '## Robot Motion Planning for Human-Robot Interaction', '1707.02387-2-18-0': 'There is considerable work on generating safe motion plans for robots operating next to humans.', '1707.02387-2-18-1': 'Some methods use the notion of social acceptability [CITATION] or legibility [CITATION].', '1707.02387-2-18-2': 'Mainprice and Berenson [CITATION] constructed occupancy map for each voxels in the workspace to explicitly represent whether a voxel is likely to be occupied by humans, so that motion planners can generate collision-free robot paths.', '1707.02387-2-18-3': 'Other techniques focus on efficiency in human-robot collaborative tasks.', '1707.02387-2-18-4': 'Markov Decision Processes (MDP) are widely used to compute the best robot action policies [CITATION].', '1707.02387-2-18-5': 'Optimization-based online replanning [CITATION] has been successfully used for high-DOF robots in dynamically changing environments.', '1707.02387-2-18-6': 'These algorithms interleave planning and trajectory execution timesteps and incrementally compute the solution to perform robot tasks and we use a similar framework.', '1707.02387-2-19-0': '# Overview', '1707.02387-2-20-0': 'We first introduce the notation and terminology used in the paper and give an overview of our natural language processing and motion planning algorithms.', '1707.02387-2-21-0': '## Natural Language Processing', '1707.02387-2-22-0': 'From an input natural language command, we construct a factor graph, as shown in the left column of Fig. [REF], based on parsing of the command.', '1707.02387-2-22-1': 'For each node of the parse tree, we generate three types of node: word phrase node [MATH], grounding node [MATH], and correspondence node [MATH].', '1707.02387-2-23-0': 'The input sentence [MATH] is parsed by NLTK library [CITATION].', '1707.02387-2-23-1': 'Word phrase of each node in the parse tree is denoted as [MATH] for [MATH].', '1707.02387-2-23-2': 'Children of [MATH] are [MATH], [MATH], [MATH].', '1707.02387-2-23-3': 'The root node of parse tree is [MATH].', '1707.02387-2-23-4': 'For example, in Fig. [REF](a), the input sentence is "Put the cup on the table".', '1707.02387-2-23-5': 'The parse tree has the root word phrase [MATH]"Put", and its noun [MATH]"the cup" and the the preposition [MATH]"on" are the children nodes of the root node.', '1707.02387-2-23-6': 'The noun phrase [MATH]"the table" is the child node of [MATH].', '1707.02387-2-23-7': 'Similarly, in Fig. [REF](b), the command "Don\'t put it there" is decomposed into 4 noun phrase nodes.', '1707.02387-2-23-8': 'The word phrase [MATH]"Don\'t" is a negation of the verb, its child node [MATH]"put".', '1707.02387-2-23-9': '[MATH]"it" and [MATH]"there" are the children nodes of [MATH].', '1707.02387-2-23-10': 'Note that the parse tree is different from the parse tree in Fig. [REF](a).', '1707.02387-2-24-0': 'We first compute the groundings [MATH] of each word phrase [MATH].', '1707.02387-2-24-1': 'Grounding of each word phrase is the mapping from the word phrase to the meaning of it in the real world.', '1707.02387-2-24-2': 'Groundings can be objects, locations, motions, tasks or constraints.', '1707.02387-2-24-3': 'In our model, the grounding [MATH] depends on its work phrase [MATH] and children grounding nodes [MATH], [MATH], [MATH], where the tree structure of grounding nodes follow the parse tree.', '1707.02387-2-24-4': 'For example, in Fig. [REF](a), the groundings of [MATH]"the cup" and [MATH]"the table" indicate the objects (the cup and the table) in the environment.', '1707.02387-2-24-5': 'The grounding of [MATH]"on", with consideration of [MATH]the table, indicates the region the surface of the table.', '1707.02387-2-24-6': 'The grounding of [MATH]"Put" is the task of moving the cup on the surface of the table, and it is the task the robot should accomplish.', '1707.02387-2-24-7': 'In Fig. [REF](b), the negation part [MATH]"Don\'t" is interpreted as a constraint that the robot should not do something.', '1707.02387-2-25-0': 'Correspondence node [MATH] indicates the correct matching between the word phrase [MATH] and the grounding [MATH].', '1707.02387-2-25-1': 'It is a binary variable; [MATH] is [MATH] if the word phrase and the grounding correctly matches and [MATH] if not.', '1707.02387-2-25-2': 'For example, in Fig. [REF](a), if the grounding for [MATH]"the cup" is [MATH]the cup in the environment, they match correctly and so [MATH].', '1707.02387-2-25-3': 'However, if the grounding node is predicted incorrectly like [MATH]the table or [MATH](putting task), then it does not correspond to "the cup" so [MATH].', '1707.02387-2-25-4': 'The correspondence nodes are useful in the machine learning framework.', '1707.02387-2-25-5': 'In training step, a ground-truth data sample contains the correct groundings [MATH], thus all correspondence nodes are [MATH].', '1707.02387-2-25-6': 'We can synthesize more data samples by assigning wrong groundings to some [MATH] and set [MATH].', '1707.02387-2-26-0': 'Our primary novel goal is to compute the best cost function parameters that we directly use in the optimization-based motion planning, corresponding to a mapping from the input natural language commands.', '1707.02387-2-26-1': 'We denote [MATH], a real vector of size [MATH], as a collection of cost function parameters, where the size [MATH], the number of cost function parameters, depends on what types of cost functions are used.', '1707.02387-2-26-2': 'We define many types of cost functions, such as the collision avoidance, robot smoothness, robot end-effector speed, target positions and target orientations.', '1707.02387-2-26-3': 'Each cost function has its weight and other cost function parameters if necessary.', '1707.02387-2-26-4': 'For example, the robot end-effector speed cost function has parameters corresponding to the direction and the magnitude of the speed which impose a constraint on the final computed trajectory.', '1707.02387-2-26-5': 'If the weight of end-effector speed cost function is higher than others, then it takes a large portion of the overall object function in the optimization formulation and thus the cost function affects more to the planned path.', '1707.02387-2-26-6': 'If the weight is low, then the end-effector speed cost will be compromised and affect less to the planned path.', '1707.02387-2-26-7': 'We want the robot to avoid collisions in any case, so we set the weight of collision avoidance cost to [MATH], and other cost function weights are normalized by this weight.', '1707.02387-2-26-8': 'The details of cost function parameters are explained in Section [REF].', '1707.02387-2-26-9': 'In Fig. [REF], the resulting constraint based motion planning problems are shown in the middle column.', '1707.02387-2-26-10': 'In (a), we use collision avoidance cost function as default, smoothness cost function and the target location cost function.', '1707.02387-2-26-11': 'The target location, whose 3D coordinates are the cost function parameters, is set on the surface of the table.', '1707.02387-2-26-12': 'The cost function parameter node [MATH] contains the weights of them and the 3D coordinate of target location.', '1707.02387-2-26-13': 'In (b), where a new "Don\'t" command is given, a repulsion cost function is added.', '1707.02387-2-26-14': 'Thus, the cost function weight and the location of repulsion source (below the robot’s end-effector position) are added to [MATH].', '1707.02387-2-27-0': '## Robot Configurations and Motion Planning', '1707.02387-2-28-0': 'We denote a single configuration of the robot as a vector [MATH] that consists of joint-angles.', '1707.02387-2-28-1': 'A configuration at time [MATH], where [MATH], is denoted as [MATH].', '1707.02387-2-28-2': 'We assume [MATH] is twice differentiable, and its derivatives are denoted as [MATH] and [MATH].', '1707.02387-2-28-3': 'The [MATH]-dimensional space of configuration [MATH] is the configuration space [MATH].', '1707.02387-2-28-4': 'We represent each link of the robot as [MATH].', '1707.02387-2-28-5': 'The finite set of bounding box for link [MATH] is [MATH].', '1707.02387-2-28-6': 'The links and bounding boxes at a configuration [MATH] are denoted as [MATH] and [MATH], resp.', '1707.02387-2-29-0': "For a planning task with a given start configuration [MATH] and derivative [MATH], the robot's trajectory is represented by a matrix [MATH], [EQUATION]", '1707.02387-2-29-1': 'The robot trajectory passes through the [MATH] waypoints [MATH], which will be optimized by an objective function under constraints in the motion planning formulation.', '1707.02387-2-29-2': 'Robot configuration at time [MATH] is interpolated from two waypoints.', '1707.02387-2-29-3': 'Formally, for [MATH] such that [MATH], the configuration [MATH] and derivative [MATH] are cubically interpolated using [MATH], [MATH], [MATH] and [MATH].', '1707.02387-2-30-0': 'The [MATH]-th cost functions of the motion planner are [MATH].', '1707.02387-2-30-1': '[MATH] different cost functions we used in this paper are listed in Section [REF].', '1707.02387-2-30-2': 'Our motion planner solves an optimization problem with a non-linear cost functions and linear joint limit constraints to generate robot trajectories for time interval [MATH], [EQUATION]', '1707.02387-2-30-3': 'In the optimization formulation, [MATH] is the [MATH]-th cost function and [MATH] is the weight of the cost function.', '1707.02387-2-30-4': 'The collection of cost function parameters include the cost function weights [MATH], and other required information specified in some cost functions.', '1707.02387-2-30-5': 'For example, [MATH], [MATH] and [MATH] coordinates target position are the parameters we want to find, for the end-effector position cost function [MATH] (explained in Section [REF]).', '1707.02387-2-31-0': '# Dynamic Grounding Graphs', '1707.02387-2-32-0': 'Our work builds on the idea of Generalized Grounding Graphs (G[MATH]) model and Distributed Correspondence Graph (DCG) model [CITATION], and extend it to generate the appropriate learning model.', '1707.02387-2-32-1': 'Our goal of learning natural language commands is to find a mapping from a natural language sentence [MATH] to the cost function parameters [MATH], given the robotic environment [MATH], where the robot is working.', '1707.02387-2-32-2': "[MATH] is a representation for the environment composed of obstacle positions, orientations, and robot's configuration.", '1707.02387-2-32-3': 'From the environment description, feature vectors are constructed in the factor graph.', '1707.02387-2-32-4': '[MATH] is a real-valued vector that contains all cost function parameters used in the optimization-based motion planner.', '1707.02387-2-32-5': 'It also includes the weights of different types of cost functions used in the optimization formulation.', '1707.02387-2-32-6': 'For example, the end-effector position cost function (Equation ([REF])) requires 3D coordinates of target position as parameters.', '1707.02387-2-32-7': 'The repulsion cost function (Equation ([REF])) requires the repulsion source position and the constant in the exponential function.', '1707.02387-2-33-0': 'We use a probabilistic model for [MATH], [MATH] and [MATH].', '1707.02387-2-33-1': 'Finding the best cost parameters is posed as an optimization problem: [EQUATION]', '1707.02387-2-33-2': 'However, modeling the probability function without decomposing the variables and some independence assumptions is hard due to the high-dimensionality of [MATH], [MATH] and [MATH] and the dependencies between them.', '1707.02387-2-33-3': 'To simplify the problem, the natural language sentence is decomposed into [MATH] word phrases based on a parse tree, i.e. [EQUATION]', '1707.02387-2-33-4': 'Like G[MATH], we introduce the intermediate groundings [MATH] of word phrases [MATH], and correspondence variable [MATH].', '1707.02387-2-33-5': 'The correspondence variables [MATH] is a binary random variable.', '1707.02387-2-33-6': 'The value [MATH] indicates that the word phrase [MATH] correctly corresponds to the grounding [MATH].', '1707.02387-2-33-7': '[MATH] means an incorrect correspondence.', '1707.02387-2-34-0': 'We assume conditional independence of the probabilities to construct a factor graph (see left column of Fig. [REF]).', '1707.02387-2-34-1': 'With the independence assumptions, a single factor is connected to a word phrase node and its children grounding nodes containing information about the sub-components.', '1707.02387-2-34-2': 'These independence assumptions simplify the problem and make it solvable by efficiently taking advantage of the tree structure of the probabilistic graphical learning model.', '1707.02387-2-34-3': "Formally, the root grounding node [MATH] contains all the information about robot's motion.", '1707.02387-2-34-4': 'The factor connecting [MATH] and [MATH] means that from the root grounding node, the cost function parameters [MATH] are optimized without any consideration of other nodes.', '1707.02387-2-34-5': 'Other factors connect [MATH], [MATH], [MATH], children grounding nodes [MATH], and the environment [MATH], where parent-child relationship is based on parse tree constructed from the natural language sentence.', '1707.02387-2-34-6': 'This graph representation corresponds to the following equation: [EQUATION].', '1707.02387-2-34-7': 'For the root factor connecting [MATH], [MATH] and [MATH], we formulate the continuous domain of [MATH].', '1707.02387-2-34-8': 'We compute the Gaussian Mixture Model (GMM) on the probability distribution [MATH] and model our probability with non-root factors as following: [EQUATION]', '1707.02387-2-35-0': 'where [MATH] is the normalization factor, [MATH] is the feature function, and [MATH] and [MATH] is the log-linearlization of the feature function.', '1707.02387-2-35-1': 'The function [MATH] generates a feature vector, given a grounding [MATH], a word phrase [MATH], a correspondence [MATH], children groundings [MATH] and the environment [MATH].', '1707.02387-2-36-0': 'As we construct the factor graph, we build a learning model on it and use that for training and inferring the meaning of given commands.', '1707.02387-2-36-1': 'In particular, we use Conditional Random Fields (CRF), a learning model for factor graphs.', '1707.02387-2-36-2': 'During the training step of CRF, we solve the optimization problem of maximizing the probability of the samples in the training dataset over the feature coefficients [MATH] for every parse tree sturcture.', '1707.02387-2-36-3': 'Multiplying Eq. ([REF]) for all training samples, the optimization problem becomes [EQUATION] where superscripts [MATH] mean the indices of the training samples.', '1707.02387-2-36-4': 'This is a tree-structured CRF problem.', '1707.02387-2-37-0': 'At the inference step, we used the trained CRF factor graph models to find the best groundings [MATH] and the cost function parameters [MATH] by solving the CRF maximization problem [EQUATION].', '1707.02387-2-37-1': 'Because the nodes [MATH], [MATH] being optimized has a tree structure in the factor graph, we can solve the optimization problem efficiently using dynamic programming.', '1707.02387-2-37-2': 'Each factor depends on its parent and children varying variables and other fixed variables connected to it.', '1707.02387-2-37-3': 'This implies that we can solve the sub-problems in a bottom-up manner and combine the results to solve the bigger problem corresponding to the root node.', '1707.02387-2-38-0': '# Real-time Trajectory Planning With NLP Input', '1707.02387-2-39-0': 'We present our realtime optimization-based planning algorithm that handle complex NLP commands.', '1707.02387-2-39-1': 'The overall optimization formulation is given in Eqn. [REF] in Sec. [REF].', '1707.02387-2-39-2': 'In order to formulate the constraints, we use the following cost functions:', '1707.02387-2-40-0': 'Collision avoidance: By default, the robot should always avoid obstacles.', '1707.02387-2-40-1': '[EQUATION] where [MATH] is the penetration depth between a robot bounding box [MATH] and an obstacle [MATH].', '1707.02387-2-40-2': 'As it is a default cost function, we set the weight of this cost function 1, and changes the weights of other types of cost functions accordingly.', '1707.02387-2-41-0': "Smoothness: We penalize the magnitude of robot's joint angle speeds to make the trajectory smooth.", '1707.02387-2-41-1': 'This correspond to the integral of first derivative of joint angles over the trajectory duration.', '1707.02387-2-41-2': 'This function is useful when we need to control the speed of the robot.', '1707.02387-2-41-3': "When the robot should operate at a low speed (e.g. when a human is too close), or we don't want abrupt moves (e.g. for human safety), the smoothness cost can have high weights so that the robot moves slowly without jerky motions.", '1707.02387-2-42-0': "End-effector position: Usually, a user specifies the robot's target position so that the robot reaches a goal position.", '1707.02387-2-42-1': "This cost function penalizes the squared distance between robot's end-effector and the target position over the trajectory duration.", '1707.02387-2-42-2': '[EQUATION] where [MATH] is the robot end-effector position at time [MATH], and [MATH] is the target position.', '1707.02387-2-42-3': 'The target position [MATH] is considered as a cost function parameter.', '1707.02387-2-42-4': 'In the NLP algorithm, a position grounding node encodes the target position parameter.', '1707.02387-2-42-5': 'It can be a 3D position or the current object position in the environment.', '1707.02387-2-42-6': 'Typically, the target position is specified by object names in the sentence, such as "pick up the cup" or "move to the box".', '1707.02387-2-42-7': 'In these cases, the grounding nodes for "the cup" and "the box" are interpreted as the current 3D coordinates of target positions, which are the parameters of this cost function.', '1707.02387-2-43-0': 'End-effector orientation: Robotic manipulation tasks are sometimes constrained by the end-effector orientation.', '1707.02387-2-43-1': 'This cost function penalizes the squared angular differences between end-effector orientation and the target orientation over the trajectory duration.', '1707.02387-2-43-2': '[EQUATION]', '1707.02387-2-44-0': "where [MATH] is the quaternion representation of the robot end-effector's orientation at time [MATH], [MATH] is the orientation that we want the robot to maintain its end-effector orientation to be, [MATH] is the normal up-vector of robot's endeffector, and [MATH] is the target up-vector.", '1707.02387-2-44-1': 'Similar to End-effector position cost, the target orientation [MATH] is the cost function parameters.', '1707.02387-2-44-2': 'The target orientation usually depends on the object the robot picked up.', '1707.02387-2-44-3': 'E.g., when the robot is doing a peg-hole insertion task under a given command "insert that into the hole", the orientation of robot\'s end-effector [MATH] should be constrained near the hole.', '1707.02387-2-44-4': 'If the robot arm is holding a cup of water, it should be upright so as to not spill the water.', '1707.02387-2-44-5': 'In this case, [MATH] is set to [MATH].', '1707.02387-2-45-0': "End-effector speed: It penalizes the robot's endeffector speed and direction: [EQUATION] where [MATH] is the robot end-effector speed at time [MATH], and [MATH] is the target speed.", '1707.02387-2-45-1': 'The parameters of this cost function is [MATH].', '1707.02387-2-45-2': "In some cases, it is required to restrict the robot's end-effector velocity.", '1707.02387-2-45-3': "For example, a user wants to pick up a cup filled with water and doesn't want to spill it.", '1707.02387-2-45-4': 'Spilling can be prevented by limiting the end-effector speed to be slow.', '1707.02387-2-45-5': 'Or the user can guide the motion by using natural language commands like "Move left, then move around the obstacle" to compute a better trajectory.', '1707.02387-2-46-0': "Repulsion: The repulsion functions are commonly used as potential fields [EQUATION] where [MATH] is the position where we don't want the robot to move to.", '1707.02387-2-46-1': 'The coefficient [MATH] plays the role of how much is the cost affected by [MATH], the distance between the end-effector position and the repulsion source.', '1707.02387-2-46-2': 'The cost function is maximized at which the end-effector position is exactly at the repulsion source, and decreases as the distance between the end-effector and the repulsion position increases.', '1707.02387-2-46-3': 'For example, if the command is "Don\'t put the cup on the laptop", we can define a repulsion cost with the laptop position as the repulsion source.', '1707.02387-2-46-4': 'The cost function is inversely proportional to the distance between the end-effector and the laptop.', '1707.02387-2-47-0': '# Implementation and Results', '1707.02387-2-48-0': 'We have implemented our algorithm and evaluated its performance in a simulated environment and a 7-DOF Fetch robot.', '1707.02387-2-48-1': 'All the timings are generated on a multi-core PC.', '1707.02387-2-49-0': 'We have evaluated the performance in complex environments composed of multiple objects and local minima.', '1707.02387-2-49-1': 'Based on the NLP commands, the robot decides to pick an appropriate object or is steered towards the goal position in a complex scene (see Fig. [REF].', '1707.02387-2-49-2': 'In particular, the NLP commands are used to guide the robot by the user by using commands such as "move right", "move up", "move left" or "move down".', '1707.02387-2-49-3': 'For each such commands, we compute the appropriate cost functions.', '1707.02387-2-50-0': 'We also integrated our NLP-based planner with ROS and evaluated its performance on the 7-DOF Fetch robot.', '1707.02387-2-50-1': 'In a real-world setting, we test its performance on different tasks corresponding to: (1) move a soda can on the table from one position to the other; (2) not move it over the book.', '1707.02387-2-50-2': 'With a noisy point cloud sensor on the robot, the thin book is not recognized as a separate obstacle by the robot, though the human user wants the robot to avoid it.', '1707.02387-2-50-3': 'Our NLP algorithm evaluates the meaning of the commands and generates appropriate cost function and constraint parameters for the real-time motion planner.', '1707.02387-2-50-4': 'In Fig. [REF], the two sub-tasks are specified in one sentence at the beginning, as "move the can on the table, but don\'t put it on the book".', '1707.02387-2-50-5': "The cost function is used to move the robot's end-effector to the surface of the table.", '1707.02387-2-50-6': 'Another cost function penalizes the distance between the book and the end-effector.', '1707.02387-2-50-7': 'In Fig. [REF], only the first sub-task is given at the beginning.', '1707.02387-2-50-8': 'This results in the robot moving the can on the book.', '1707.02387-2-50-9': 'As the robot gets too close to the book, the person says "stop", then says "don\'t put it there."', '1707.02387-2-50-10': 'The robot recomputes the cost functions and avoids the region around the book.', '1707.02387-2-51-0': '# Analysis and Benefits', '1707.02387-2-52-0': 'In this section, we analyze the performance of our approach and highlight its benefits over prior methods.', '1707.02387-2-52-1': 'We evaluated the end-to-end performance of our approach including the NLP, mapping and the motion planning modules on different benchmarks.', '1707.02387-2-52-2': 'We evaluated the performance based on the following metrics:', '1707.02387-2-53-0': 'Table [REF] shows the results of the experiments with varying number of training data samples on the simulation environment shown in Fig. [REF].', '1707.02387-2-53-1': 'When the number of training data samples increases, the success rate also increases, and the trajectory duration and the trajectory smoothness cost decrease.', '1707.02387-2-53-2': 'Also, the standard deviations also decreases, implying that our planning cost function mapping algorithm becomes more stable.', '1707.02387-2-53-3': 'Table [REF] shows the running time of our algorithm and the distances from the obstacle on the table in the real world settings.', '1707.02387-2-54-0': '## Benefits over Prior Methods', '1707.02387-2-55-0': 'Most prior methods that combine NLP and motion planning have focused on understanding natural language instructions computing robot motion for simple environments and constraints.', '1707.02387-2-55-1': 'In our approach, the goal is to generate appropriate high-DOF motion trajectory in response to complex natural language commands, as shown in Section 6.', '1707.02387-2-55-2': 'Furthermore, our optimization-based formulation is designed to handle challenging scenarios with lot of obstacles and constraints (e.g. dynamics constraints, smoothness constraints), on the resulting trajectories and this results in stable trajectories.', '1707.02387-2-55-3': 'Unlike prior methods, the output of NLP parsing algorithm is directly coupled with the specification of the motion planning problem as a constrained optimization method.', '1707.02387-2-55-4': 'This enables to handle many complex NLP commands including negation, velocity or position constraints or orientation specification.', '1707.02387-2-55-5': 'Our algorithm demonstrates many advantages over [CITATION].', '1707.02387-2-55-6': 'The way we setup the constraints are different from theirs in the sense that our constrained realitime motion algorithm can complex dynamic environments.', '1707.02387-2-55-7': 'In particular, Howard et al. [CITATION] use a discrete set of constraints and their planning algorithm determines whether those constraints are activated or not.', '1707.02387-2-55-8': 'In the worst case, the complexity of the search space can grow exponentially as more constraints are added.', '1707.02387-2-55-9': 'On the other hand, we use appropriate cost functions (can be interpreted as constraints as we manipulate the equation) parameters over the continuous domain.', '1707.02387-2-55-10': 'It is more expressive even with small number of parameters.', '1707.02387-2-55-11': 'Moreover, we highlight the experimental results with a 7-DOF Fetch robot and demonstrate that our approach can handle the uncertainities that arise from a real robot integration including speech recognition, visual sensor errors, control errors, network latency.', '1707.02387-2-55-12': 'In the real robot experiment, the NLP parsing algorithm is successfully integrated with the realtime motion planning algorithm to set up proper parameters in the formulation for our optimization-based motion planning.', '1707.02387-2-55-13': 'As the parsing capabilities of NLP systems improve in the future, we can map more complex commands to our framework.', '1707.02387-2-56-0': '# Limitations, Conclusions and Future Work', '1707.02387-2-57-0': 'We presented a real-time motion planning algorithm that computes appropriate trajectories based on complex NLP commands.', '1707.02387-2-57-1': 'We highlight the performance in simulated and real-world scenes with a 7-DOF manipulator operating next to humans.', '1707.02387-2-57-2': 'The preliminary results are promising and our approach can handle complex scenarios as compared to prior methods.', '1707.02387-2-58-0': 'We use a trajectory optimization algorithm to compute the high-DOF robot trajectory.', '1707.02387-2-58-1': 'It is a high-dimensional optimization problem and the solver get stuck in local minima.', '1707.02387-2-58-2': 'As a result, it is hard to provide rigorous guarantees in terms of satisfying all the constraints or following the intent of the user.', '1707.02387-2-58-3': 'Furthermore, the accuracy of the mapping algorithm varies as a function of the training data.', '1707.02387-2-59-0': 'As future work, we would like to overcome these limitations and evaluate the approach in challenging scenarios with moving obstacles and perform complex robot tasks.', '1707.02387-2-59-1': 'More work is needed to handle the full diversity of a natural language, especially for rare words, complicated grammar styles, as well as hidden intention or emotion in the human speech.', '1707.02387-2-59-2': 'We plan to incorporate stronger natural language processing and machine learning methods such as those based on semantic parsing, neural sequence-to-sequence models, reinforcement learning, and speech-based emotion analysis, and compute the appropriate optimization-based planning formulations.'} | [['1707.02387-1-42-0', '1707.02387-2-43-0'], ['1707.02387-1-42-1', 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['1707.02387-3-43-0', '1707.02387-4-41-3'], ['1707.02387-3-65-1', '1707.02387-4-77-3'], ['1707.02387-3-12-0', '1707.02387-4-12-0'], ['1707.02387-3-12-0', '1707.02387-4-12-2'], ['1707.02387-3-12-3', '1707.02387-4-12-0'], ['1707.02387-3-13-0', '1707.02387-4-12-1'], ['1707.02387-3-13-2', '1707.02387-4-12-2'], ['1707.02387-2-55-1', '1707.02387-3-60-3'], ['1707.02387-2-52-0', '1707.02387-3-60-0'], ['1707.02387-2-55-5', '1707.02387-3-61-0'], ['1707.02387-2-55-9', '1707.02387-3-61-3']] | [['1707.02387-2-18-0', '1707.02387-3-16-4']] | ['1707.02387-1-3-5', '1707.02387-1-5-2', '1707.02387-1-22-2', '1707.02387-1-38-2', '1707.02387-1-42-2', '1707.02387-1-51-2', '1707.02387-2-4-1', '1707.02387-2-6-3', '1707.02387-2-23-2', '1707.02387-2-39-2', '1707.02387-2-43-2', '1707.02387-2-52-2', '1707.02387-3-3-1', '1707.02387-3-4-0', '1707.02387-3-5-0', '1707.02387-3-7-7', '1707.02387-3-21-2', '1707.02387-3-57-0', '1707.02387-4-3-1', '1707.02387-4-4-0', '1707.02387-4-5-0', '1707.02387-4-7-7', '1707.02387-4-19-2', '1707.02387-4-28-1', '1707.02387-4-29-0', '1707.02387-4-61-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1707.02387 | {'1707.02387-3-0-0': 'We present an algorithm for combining natural language processing (NLP) and realtime robot motion planning to automatically generate safe robot movements.', '1707.02387-3-0-1': 'Our formulation uses novel a Dynamic Constraint Mapping to transform the complex, attribute-based natural language instructions into appropriate cost functions and parametric constraints for optimization-based motion planning.', '1707.02387-3-0-2': 'We generate a factor graph called Dynamic Grounding Graph (DGG) from natural language instructions that takes into account the latent parameters.', '1707.02387-3-0-3': 'The coefficients of this factor graph are learned based on conditional random fields (CRFs) and are used to dynamically generate the constraints for motion planning.', '1707.02387-3-0-4': 'We directly map the cost function to the motion parameters of the planner and compute smooth trajectories in dynamic scenes.', '1707.02387-3-0-5': 'We highlight the performance of our approach in a simulated environment as well as via a human interacting with a 7-DOF Fetch robot using intricate language commands that include negation, orientation specification, and distance constraints.', '1707.02387-3-1-0': '# Introduction', '1707.02387-3-2-0': "In the field of human-robot interaction (HRI), natural language has been used as an interface to communicate a human's intent to a robot [CITATION].", '1707.02387-3-2-1': 'Much of the work in this area is related to specifying simple tasks or commands for robot manipulation, such as picking and placing objects.', '1707.02387-3-2-2': 'As robots are increasingly used in complex scenarios and applications, it is important to develop a new generation of motion planning and robot movement techniques that can respond appropriately to diverse, attribute-based NLP instructions for HRI, e.g., those containing negation based phrases or references to position, velocity, and distance constraints.', '1707.02387-3-2-3': 'Furthermore, we need efficient techniques to automatically map the NLP instructions to such motion planners.', '1707.02387-3-3-0': 'Humans frequently issue commands that include sentences with orientation-based or negation constraints such as "put a bottle on the table and keep it upright," or "move the knife but don\'t point it towards people," or sentences with velocity-based constraints such as "move slowly when you get close to a human."', '1707.02387-3-3-1': 'In order to generate robot actions and movements in response to such complex natural language instructions, we need to address two kinds of challenges:', '1707.02387-3-4-0': '1.', '1707.02387-3-4-1': 'Accurate interpretation of attribute-based natural language instructions and their grounded linguistic semantics, especially considering the environment and the context.', '1707.02387-3-4-2': 'For example, a human may say "move a little to the left," or "do not move like this," and the robot planner needs to learn the correct interpretation of these commands with spatial and motion-based adjectives, adverbs, and negation.', '1707.02387-3-5-0': '2.', '1707.02387-3-5-1': 'The realtime motion planner needs to generate appropriate trajectories based on these complex natural language instructions.', '1707.02387-3-5-2': 'This includes appropriately setting up the motion planning problem based on different motion constraints (e.g., orientation, velocity, smoothness, and avoidance) and computing smooth and collision-free paths.', '1707.02387-3-6-0': 'At a high level, natural language instructions can be decomposed into task description and attributes.', '1707.02387-3-6-1': 'Task descriptions are usually verb or noun phrases that describe the underlying task performed by a robot.', '1707.02387-3-6-2': 'The attributes include various adjectives, adverbs, or prepositional phrases are used to specify additional conditions the robot must (or must not) satisfy.', '1707.02387-3-6-3': 'For example, these conditions may specify some information related to the speed, orientation, physical space characteristic, or the distances.', '1707.02387-3-6-4': 'Therefore, it is important to design motion planners that take into account these robotic task descriptions and robot motion constraints.', '1707.02387-3-7-0': 'Main Results: We present an algorithm to generate parameterized constraints for optimization-based motion planning from complex, attribute-based natural language instructions.', '1707.02387-3-7-1': 'We use Dynamic Grounding Graphs (DGG) to parse and interpret the commands and generate the constraints.', '1707.02387-3-7-2': 'Moreover, our formulation includes the latent parameters in the grounding process and that allows us to model many continuous variables in our grounding graph.', '1707.02387-3-7-3': 'Furthermore, we present a new dynamic constraint mapping that takes DCG as the input and computes different constraints and parameters of the motion planner.', '1707.02387-3-7-4': 'The appropriate motion parameters correspond to the speed, orientation, position, smoothness, repulsion, and avoidance.', '1707.02387-3-7-5': 'The final trajectory of the robot is computed using a realtime constraint optimization solver.', '1707.02387-3-7-6': 'Overall, our approach can automatically handle complex natural language instructions corresponding to spatial and temporal adjectives, adverbs, superlative and comparative degrees, negations, etc.', '1707.02387-3-7-7': 'As compared to prior techniques, our overall approach offers the following benefits:', '1707.02387-3-8-0': 'We highlight the performance of our algorithms in a simulated environment as well as on a 7-DOF Fetch robot operating next to a human in a safe manner.', '1707.02387-3-8-1': 'Our approach can handle a rich set of natural language commands and can generate appropriate paths in realtime.', '1707.02387-3-8-2': 'These include complex commands such as picking (e.g., "pick up a red object near you"), correcting the motion (e.g., "don\'t pick up that one"), and negation (e.g., "don\'t put it on the book").', '1707.02387-3-9-0': '# Related Work', '1707.02387-3-10-0': 'Most algorithms used to map natural language instruction to robot actions tend to separate the problem into two parts, the parsing part and the motion planning computation.', '1707.02387-3-10-1': 'In this section, we give a brief overview of prior work in these areas.', '1707.02387-3-11-0': '## Natural Language Processing', '1707.02387-3-12-0': 'Kollar et al. [CITATION] present a probabilistic graphical learning model called Generalized Grounding Graphs or G[MATH], by which the robot interprets and grounds natural language commands to the physical world.', '1707.02387-3-12-1': 'The graph is defined based on the syntactic parse structure of the command, enabling the system to associate parts of the command with objects, events, and locations in the external world.', '1707.02387-3-12-2': 'Howard et al. [CITATION] reduce the search space by modifying the G[MATH] graph structure by adding all possible grounding nodes (meanings of word phrases) and optimizing the correspondence variables (indicating the word phrase and the grounding match correctly).', '1707.02387-3-12-3': 'Duvallet et al. [CITATION] used this algorithm on a ground vehicle for a navigation problem given natural language commands.', '1707.02387-3-12-4': 'Our approach extends the G[MATH] graph structure with respect to the linguistic parse structure, the probabilistic graphical model, and the use of optimization-based motion planning.', '1707.02387-3-13-0': 'Other related work in language grounding of task instructions includes Branavan et al. [CITATION], which uses reinforcement learning to learn the mapping from natural language instructions and apply it to sequences of executable actions.', '1707.02387-3-13-1': 'Reinforcement learning-based methods require little annotated data, but the search space can be very large when applied to motion planning of a robot arm.', '1707.02387-3-13-2': 'Matuszek et al. [CITATION] use a statistical machine translation model to map the natural language instructions to a path description language to follow the directions for robot navigation.', '1707.02387-3-13-3': 'Duvallet et al. [CITATION] use imitation learning to train the model via demonstrations of humans following directions.', '1707.02387-3-13-4': 'Paul et al. [CITATION] proposed the Adaptive Distributed Correspondence Graph (ADCG).', '1707.02387-3-13-5': 'It can handle abstract groundings, providing hierarchical meanings of word phrases from an abstract level to a concrete level.', '1707.02387-3-13-6': 'Arkin et al. [CITATION] further extended DCG, proposing the Hierarchical Distributed Correspondence Graph (HDCG), which defines constraints as discrete inequalities and ground word phrases with corresponding inequalities.', '1707.02387-3-13-7': 'However, their method depends on discretized constraints in a continuous space (e.g., discretized distance constraints) and has not been evaluated on real robots.', '1707.02387-3-13-8': 'Chung et al. [CITATION] use it on ground vehicles for navigation commands and demonstrated performance improvements over G[MATH] in terms of running time, factor evaluations and correctness.', '1707.02387-3-13-9': "Oh et al. [CITATION] integrate HDCG with their navigating robot system and measured performances in terms of completion rates and compared them with humans' behaviors.", '1707.02387-3-13-10': 'Our approach is designed for general purpose tasks and NLP instructions.', '1707.02387-3-14-0': '## Robot Motion Planning in Dynamic Environments', '1707.02387-3-15-0': 'In order to generate collision-free motion plans in dynamic environments, many replanning algorithms have been suggested.', '1707.02387-3-15-1': 'Fox et al. [CITATION] propose the dynamic window approach to compute the optimal velocity in a short time window.', '1707.02387-3-15-2': 'Optimization-based motion planners [CITATION] solve a constrained optimization problem to generate smooth and collision-free robot paths.', '1707.02387-3-15-3': 'These planners can be adapted to other tasks by adding cost functions and constraints in the optimization formulation.', '1707.02387-3-15-4': 'However, it can be difficult and tedious to manually tune many parameters.', '1707.02387-3-15-5': 'We present an automatic scheme to generate the motion planning problem from NLP instructions.', '1707.02387-3-16-0': 'There is some work on integrating optimization-based motion planning with NLP in 2D workspaces.', '1707.02387-3-16-1': 'Silver et al. [CITATION] developed an algorithm for learning navigation cost functions from demonstration.', '1707.02387-3-16-2': 'Howard et al. [CITATION] use a probabilistic graphical model to generate motion planning constraints for a 2D navigation problem.', '1707.02387-3-16-3': 'Compared to these methods, our approach can handle 3D workspaces and high-dimensional configuration spaces to generate robot motions corresponding to complex NLP instructions.', '1707.02387-3-16-4': 'There is considerable work on generating safe motion plans for robots operating next to humans [CITATION], though it is complimentary to our approach.', '1707.02387-3-17-0': '# Overview', '1707.02387-3-18-0': 'We first introduce the notation and terminology used in the paper and give an overview of our natural language processing and motion planning algorithms.', '1707.02387-3-19-0': '## Natural Language Processing', '1707.02387-3-20-0': 'The input to our algorithm is the natural language instruction.', '1707.02387-3-20-1': 'We do not account for any errors due to voice recognition.', '1707.02387-3-20-2': 'From an input natural language command, we construct a factor graph, as shown in Fig. [REF](a), based on the parsing of the command.', '1707.02387-3-20-3': 'For each node of the parse tree, we generate three types of nodes: word phrase node [MATH], grounding node [MATH], and correspondence node [MATH].', '1707.02387-3-21-0': 'The input sentence [MATH] is parsed using the NLTK library [CITATION].', '1707.02387-3-21-1': 'Word phrase of each node in the parse tree is denoted as [MATH] for [MATH].', '1707.02387-3-21-2': 'Children of [MATH] are [MATH], [MATH], [MATH].', '1707.02387-3-21-3': 'The root node of the parse tree is [MATH].', '1707.02387-3-22-0': 'We first compute the groundings [MATH] of each word phrase [MATH].', '1707.02387-3-22-1': 'The grounding of each word phrase is the mapping from the word phrase to its meaning in the real world.', '1707.02387-3-22-2': 'Groundings can be objects, locations, motions, tasks or constraints.', '1707.02387-3-22-3': 'In our model, the grounding [MATH] depends on its work phrase [MATH] and children grounding nodes [MATH], [MATH], [MATH], where the tree structure of the grounding nodes follows the parse tree.', '1707.02387-3-23-0': 'Correspondence node [MATH] indicates the correct matching between the word phrase [MATH] and the grounding [MATH].', '1707.02387-3-23-1': 'It is a binary variable; [MATH] is [MATH] if the word phrase and the grounding correctly matches and [MATH] if not.', '1707.02387-3-24-0': '## Robot Configurations and Motion Plans', '1707.02387-3-25-0': 'We denote a single configuration of the robot as a vector [MATH] that consists of joint-angles or other degrees-of-freedom.', '1707.02387-3-25-1': 'A configuration at time [MATH], where [MATH], is denoted as [MATH].', '1707.02387-3-25-2': 'We assume [MATH] is twice differentiable, and its derivatives are denoted as [MATH] and [MATH].', '1707.02387-3-25-3': 'The [MATH]-dimensional space of configuration [MATH] is the configuration space [MATH].', '1707.02387-3-25-4': 'We represent bounding boxes of each link of the robot as [MATH].', '1707.02387-3-25-5': 'The bounding boxes at a configuration [MATH] are denoted as [MATH].', '1707.02387-3-26-0': "For a planning task with a given start configuration [MATH] and derivative [MATH], the robot's trajectory is represented by a matrix [MATH], whose elements correspond to the waypoints [CITATION].", '1707.02387-3-26-1': 'The robot trajectory passes through the [MATH] waypoints [MATH], which will be optimized by an objective function under constraints in the motion planning formulation.', '1707.02387-3-26-2': 'Robot configuration at time [MATH] is interpolated from two waypoints.', '1707.02387-3-26-3': 'Formally, for [MATH] such that [MATH], the configuration [MATH] and derivative [MATH] are cubically interpolated using [MATH], [MATH], [MATH] and [MATH].', '1707.02387-3-27-0': 'The [MATH]-th cost functions of the motion planner are [MATH].', '1707.02387-3-27-1': 'The [MATH] different cost functions we used in this paper are listed in Section [REF].', '1707.02387-3-27-2': 'Our motion planner solves an optimization problem with non-linear cost functions and linear joint limit constraints to generate robot trajectories for time interval [MATH], [EQUATION]', '1707.02387-3-27-3': 'In the optimization formulation, [MATH] is the [MATH]-th cost function and [MATH] is the weight of the cost function.', '1707.02387-3-28-0': '# Dynamic Grounding Graphs', '1707.02387-3-29-0': 'In this section, we extend the ideas of the Generalized Grounding Graphs (G[MATH]) model and the Distributed Correspondence Graph (DCG) model [CITATION] by including the latent variables in the grounding graph and using them to compute the constraints for motion planning.', '1707.02387-3-29-1': 'We present dynamic grounding graphs and use an appropriate learning model to compute them.', '1707.02387-3-29-2': 'Our goal is to compute a mapping from a natural language sentence [MATH] to the cost function parameters [MATH], given the robotic environment [MATH], where the robot is operating.', '1707.02387-3-29-3': "[MATH] is a representation of the environment composed of obstacle positions, orientations, and the robot's configuration.", '1707.02387-3-29-4': 'Feature vectors are constructed in the factor graph from the environment description.', '1707.02387-3-29-5': '[MATH] is a real-valued vector that contains all cost function parameters used in the optimization-based motion planner.', '1707.02387-3-29-6': 'It also includes the weights of different types of cost functions used in the optimization formulation.', '1707.02387-3-29-7': 'For example, the end-effector position cost function (Eq. ([REF])) requires the 3D coordinates of the target position as parameters.', '1707.02387-3-29-8': 'The repulsion cost function (Eq. ([REF])) requires the repulsion source position and the constant in the exponential function.', '1707.02387-3-30-0': '## Latent Parameters', '1707.02387-3-31-0': 'A key novel component of our approach is inclusion of latent variables in the grounding graph.', '1707.02387-3-31-1': 'Our primary goal is to compute the best cost function parameters [MATH] that we can directly use for optimization-based motion planning.', '1707.02387-3-31-2': 'We denote [MATH], a real vector of size [MATH], as a collection of cost function parameters.', '1707.02387-3-31-3': 'In this case, the size [MATH] and the number of cost function parameters depend on the types of cost functions that are used.', '1707.02387-3-31-4': 'From the predicted groundings [MATH], the cost function parameters in the motion planning formulation (Fig. [REF](b)) is inferred through the latent variable [MATH].', '1707.02387-3-31-5': 'It contains all the cost function parameters (e.g., weights of cost functions, locations and orientations).', '1707.02387-3-31-6': 'The details of the cost function parameters are explained in Section [REF].', '1707.02387-3-32-0': 'In Fig. [REF](b), the resulting constraint-based motion planning problems are shown.', '1707.02387-3-32-1': 'We use the collision avoidance cost function as the default, smoothness cost function and the target location cost function, though weights can vary.', '1707.02387-3-32-2': 'The target location, whose 3D coordinates are the cost function parameters, is set on the surface of the table.', '1707.02387-3-32-3': 'The cost function parameter node [MATH] contains the weights of the parameters and the 3D coordinates of the target location.', '1707.02387-3-32-4': 'In the bottom of Fig. [REF](b), where a new "Don\'t" command is given, a repulsion cost function is added.', '1707.02387-3-32-5': 'Thus, the cost function weight and the location of the repulsion source (below the robot’s end-effector position) are added to [MATH].', '1707.02387-3-33-0': '## Probabilistic Model', '1707.02387-3-34-0': 'We present a new probabilistic model to compute [MATH], [MATH] and [MATH].', '1707.02387-3-34-1': 'We pose the problem of finding the best cost parameters as an optimization problem: [EQUATION]', '1707.02387-3-34-2': 'However, modeling the probability function without decomposing the variables and some independence assumptions is difficult due to the high-dimensionality of [MATH], [MATH] and [MATH] and the dependencies between them.', '1707.02387-3-34-3': 'To simplify the problem, the natural language sentence is decomposed into [MATH] word phrases based on a parse tree, i.e. [EQUATION]', '1707.02387-3-34-4': 'Like G[MATH], we introduce the intermediate groundings [MATH] of word phrases [MATH], and correspondence variables [MATH].', '1707.02387-3-34-5': 'The correspondence variables [MATH] is a binary random variable.', '1707.02387-3-34-6': 'The value [MATH] indicates that the word phrase [MATH] correctly corresponds to the grounding [MATH].', '1707.02387-3-34-7': '[MATH] means an incorrect correspondence.', '1707.02387-3-35-0': 'We assume conditional independence of the probabilities to construct a factor graph (see Fig. [REF](a)).', '1707.02387-3-35-1': 'With the independence assumptions, a single factor is connected to a word phrase node and its children grounding nodes which contain information about the sub-components.', '1707.02387-3-35-2': 'These independence assumptions simplify the problem and make it solvable by efficiently taking advantage of the tree structure of the probabilistic graphical learning model.', '1707.02387-3-35-3': "Formally, the root grounding node [MATH] contains all the information about a robot's motion.", '1707.02387-3-35-4': 'The factor that connects [MATH] and [MATH] implies that, from the root grounding node, the cost function parameters [MATH] are optimized without any consideration of other nodes.', '1707.02387-3-35-5': 'Other factors connect [MATH], [MATH], [MATH], children grounding nodes [MATH], and the environment [MATH], where the parent-child relationship is based on a parse tree constructed from the natural language sentence.', '1707.02387-3-35-6': 'This graphical representation corresponds to the following equation: [EQUATION]', '1707.02387-3-35-7': 'For the root factor connecting [MATH], [MATH] and [MATH], we formulate the continuous domain of [MATH].', '1707.02387-3-35-8': 'We compute the Gaussian Mixture Model (GMM) on the probability distribution [MATH] and model our probability with non-root factors as follows: [EQUATION] where [MATH] is the normalization factor, [MATH] is the feature function, and [MATH] and [MATH] are the log-linearization of the feature function.', '1707.02387-3-35-9': 'The function [MATH] generates a feature vector, given a grounding [MATH], a word phrase [MATH], a correspondence [MATH], children groundings [MATH] and the environment [MATH].', '1707.02387-3-36-0': '## Factor Graph using Conditional Random Fields', '1707.02387-3-37-0': 'We represent our dynamic grounding graph as a factor graph.', '1707.02387-3-37-1': 'We build a factor graph based on the probabilistic model described in Section [REF], and use that for training and inferring the meaning of given commands.', '1707.02387-3-37-2': 'In particular, we use Conditional Random Fields (CRF) [CITATION] as a learning model for factor graphs, because CRFs are a good fit for applying machine learning to our probabilistic graph model with conditional probabilities.', '1707.02387-3-37-3': 'During the training step of CRF, we solve the optimization problem of maximizing the probability of the samples in the training dataset over the feature coefficients [MATH] for every parse tree structure.', '1707.02387-3-37-4': 'By multiplying Eq. ([REF]) for all training samples, the optimization problem becomes [EQUATION] where superscripts [MATH] mean the indices of the training samples.', '1707.02387-3-37-5': 'This is a tree-structured CRF problem.', '1707.02387-3-38-0': 'At the inference step, we used the trained CRF factor graph models to find the best groundings [MATH] and the cost function parameters [MATH] by solving the CRF maximization problem [EQUATION]', '1707.02387-3-38-1': 'Because the nodes [MATH], [MATH] being optimized creates a tree structure in the factor graph, we can solve the optimization problem efficiently using dynamic programming.', '1707.02387-3-38-2': 'Each factor depends on its parent and children varying variables and other fixed variables connected to it.', '1707.02387-3-38-3': 'This implies that we can solve the sub-problems in a bottom-up manner and combine the results to solve the bigger problem corresponding to the root node.', '1707.02387-3-39-0': '# Dynamic Constraint Mapping With NLP Input', '1707.02387-3-40-0': 'In this section, we present our mapping algorithm, Dynamic Constraint Mapping, which maps the word phrase groundings to proper cost function parameters that correspond to natural language instructions.', '1707.02387-3-40-1': 'Our realtime optimization-based planning algorithm [CITATION] solves the cost minimization problem, the function and constraints of which come from DGG, as explained in Sec. [REF].', '1707.02387-3-41-0': 'The overall optimization formulation is given in Eqn. [REF] in Sec. [REF].', '1707.02387-3-41-1': 'In order to formulate the constraints, we use the following cost functions, which are design to account for various attributes in the NLP instructions.', '1707.02387-3-42-0': '## Cost Functions', '1707.02387-3-43-0': 'In our formulation we use many types of cost functions, such as collision avoidance, robot smoothness, robot end-effector speed, target positions and target orientations that are used to handle many attributes of the natural language instructions.', '1707.02387-3-43-1': 'Each cost function has its weight and other cost function parameters, if necessary.', '1707.02387-3-43-2': 'For example, the robot end-effector speed cost function has parameters corresponding to the direction and the magnitude of the speed, which impose a constraint on the final computed trajectory.', '1707.02387-3-43-3': 'If the weight of the end-effector speed cost function is higher than the others, then it contributes more to the overall objective function in the optimization formulation.', '1707.02387-3-43-4': 'If the weight is low, then the end-effector speed cost will be compromised and has lesser impact on the path planner.', '1707.02387-3-43-5': 'We want the robot to avoid collisions in any case, so we set the weight of collision avoidance cost to [MATH], and other cost function weights are normalized by this weight.', '1707.02387-3-44-0': '## Parameterized Constraints', '1707.02387-3-45-0': 'In order to handle various attributes, we use the following parameterized constraints in our optimization formulation.', '1707.02387-3-46-0': 'Collision avoidance: By default, the robot should always avoid obstacles.', '1707.02387-3-46-1': '[EQUATION] where [MATH] is the penetration depth between a robot bounding box [MATH] and an obstacle [MATH].', '1707.02387-3-46-2': 'Because it is a default cost function, we set the weight of this cost function as 1 and change the weights of the other types of cost functions accordingly.', '1707.02387-3-47-0': "Smoothness: We penalize the magnitude of a robot's joint angle speeds to make the trajectory smooth.", '1707.02387-3-47-1': 'This corresponds to the integral of the first derivative of joint angles over the trajectory duration.', '1707.02387-3-47-2': 'This function is useful when we need to control the speed of the robot.', '1707.02387-3-47-3': "When the robot should operate at a low speed (e.g. when a human is too close), or we don't want abrupt movements (e.g., for human safety), the smoothness cost can have high weights so that the robot moves slowly without jerky motions.", '1707.02387-3-48-0': "End-effector position: Usually, a user specifies the robot's target position to make sure that the robot reaches its goal position.", '1707.02387-3-48-1': "This cost function penalizes the squared distance between the robot's end-effector position and the target position over the trajectory duration as [EQUATION] where [MATH] is the robot end-effector position at time [MATH], and [MATH] is the target position.", '1707.02387-3-48-2': 'The target position [MATH] is considered as a cost function parameter.', '1707.02387-3-48-3': 'In the mapping algorithm, a position grounding node encodes the target position parameter.', '1707.02387-3-48-4': 'It can be a 3D position or the current object position in the environment.', '1707.02387-3-48-5': 'Typically, the target position is specified by an object name in the sentence, such as "pick up the cup" or "move to the box".', '1707.02387-3-48-6': 'In these cases, the grounding nodes for "the cup" and "the box" are interpreted as the current 3D coordinates of the target positions, which are the parameters of this cost function.', '1707.02387-3-49-0': 'End-effector orientation: Robotic manipulation tasks are sometimes constrained by the end-effector orientation.', '1707.02387-3-49-1': 'This cost function penalizes the squared angular differences between the end-effector orientation and the target orientation over the trajectory duration.', '1707.02387-3-49-2': "[EQUATION] where [MATH] is the quaternion representation of the robot end-effector's orientation at time [MATH], [MATH] is the end-effector orientation that we want the robot to maintain, [MATH] is the normal up-vector of the robot's end-effector, and [MATH] is the target up-vector.", '1707.02387-3-49-3': 'As with the end-effector position cost, the target orientation [MATH] is the cost function parameter.', '1707.02387-3-49-4': 'The target orientation usually depends on the object the robot picked up.', '1707.02387-3-49-5': 'For example, when the robot is doing a peg-hole insertion task under the command "insert that into the hole," the orientation of the robot\'s end-effector [MATH] should be constrained near the hole.', '1707.02387-3-50-0': "End-effector speed: It penalizes the robot's end effector speed and direction: [EQUATION] where [MATH] is the robot's end-effector speed at time [MATH], and [MATH] is the target speed.", '1707.02387-3-50-1': 'The parameters of this cost function correspond to [MATH].', '1707.02387-3-50-2': "In some cases, we must restrict the robot's end-effector velocity, e.g., if a user wants to pick up a cup filled with water and doesn't want to spill it.", '1707.02387-3-50-3': 'Spilling can be prevented by limiting the end-effector speed, slowing it.', '1707.02387-3-51-0': "Repulsion: The repulsion functions are commonly represented as potential fields [EQUATION] where [MATH] is the position to which we don't want the robot to move.", '1707.02387-3-51-1': 'The coefficient [MATH] suggests how much the cost is affected by [MATH], the distance between the end-effector position and the repulsion source.', '1707.02387-3-51-2': 'The cost function is maximized when the end-effector position is exactly at the repulsion source, and it decreases as the distance between the end-effector and the repulsion position increases.', '1707.02387-3-51-3': 'For example, if the command is "Don\'t put the cup on the laptop," we can define a repulsion cost with the laptop position as the repulsion source.', '1707.02387-3-51-4': 'The cost function is inversely proportional to the distance between the end-effector and the laptop.', '1707.02387-3-52-0': '# Implementation and Results', '1707.02387-3-53-0': 'We have implemented our algorithm and evaluated its performance in a simulated environment and on a 7-DOF Fetch robot.', '1707.02387-3-53-1': 'All the timings are generated on a multi-core PC.with Intel i7-4790 8-core 3.60GHz CPU and a 16GB RAM.', '1707.02387-3-54-0': 'We have evaluated the performance in complex environments composed of multiple objects and local minima.', '1707.02387-3-54-1': 'Based on the NLP commands, the robot decides to pick an appropriate object or is steered towards the goal position in a complex scene.', '1707.02387-3-54-2': 'In particular, the user gives NLP commands such as "move right", "move up", "move left" or "move down" to guide the robot.', '1707.02387-3-54-3': 'For each such command, we compute the appropriate cost functions.', '1707.02387-3-55-0': 'We also integrated our NLP-based planner with ROS and evaluated its performance on the 7-DOF Fetch robot.', '1707.02387-3-55-1': 'In a real-world setting, we tested its performance on different tasks corresponding to: (1) moving a soda can on the table from one position to an other; (2) not moving the soda can over the book.', '1707.02387-3-55-2': 'With a noisy point cloud sensor on the robot, the thin book is not recognized as a separate obstacle by the robot, though the human user wants the robot to avoid it.', '1707.02387-3-55-3': 'All the instructions used in these tasks have different attributes, which makes it hard for prior methods.', '1707.02387-3-55-4': 'In Fig. [REF], the two sub-tasks are specified in one sentence at the beginning, as "move the can on the table, but don\'t put it on the book".', '1707.02387-3-55-5': "The cost function is used to move the robot's end-effector to the surface of the table.", '1707.02387-3-55-6': 'Another cost function penalizes the distance between the book and the end-effector.', '1707.02387-3-55-7': 'In Fig. [REF], only the first sub-task is given at the beginning.', '1707.02387-3-55-8': 'This results in the robot moving the can on the book.', '1707.02387-3-55-9': 'As the robot gets too close to the book, the person says "stop," then says "don\'t put it there."', '1707.02387-3-55-10': 'The robot recomputes the cost functions and avoids the region around the book.', '1707.02387-3-56-0': '## Analysis', '1707.02387-3-57-0': 'We evaluated the performance based on the following metrics:', '1707.02387-3-58-0': 'Table [REF] shows the results on our benchmarks with varying numbers of training data samples on the simulation environment shown in Fig. [REF].', '1707.02387-3-58-1': 'When the number of training data samples increases, the success rate also increases, and the trajectory duration and the trajectory smoothness cost decrease.', '1707.02387-3-58-2': 'The standard deviations also decreases, implying that our dynamic constrain mapping works well.', '1707.02387-3-58-3': 'Table [REF] shows the running time of our algorithm and the distances from the obstacle on the table in the real-world scenarios.', '1707.02387-3-59-0': '# Benefits and Comparisons', '1707.02387-3-60-0': 'In this section, we compare our approach with prior methods and highlight the benefits in terms of handling attribute-based NLP instructions.', '1707.02387-3-60-1': 'Most prior methods that combine NLP and motion planning have focused on understanding natural language instructions to compute robot motion for simple environments and constraints.', '1707.02387-3-60-2': 'Most of these methods are limited to navigation applications [CITATION] or used in simple settings [CITATION] or not evaluated on real robots [CITATION].', '1707.02387-3-60-3': 'In our approach, the goal is to generate appropriate high-DOF motion trajectories in response to attribute-based natural language instructions like negation, distance or orientation constraints, etc.', '1707.02387-3-61-0': 'Our algorithm demonstrates many advantages over Howard et al. [CITATION].', '1707.02387-3-61-1': 'They use a discrete set of constraints and their planning algorithm determines whether those constraints are activated or not.', '1707.02387-3-61-2': 'In the worst case, the complexity of their search space can grow exponentially as more constraints are added.', '1707.02387-3-61-3': 'On the other hand, we use appropriate cost functions as parameters over the continuous domain.', '1707.02387-3-61-4': 'Our constraint setup algorithm based on dynamic constraint mapping is different and our formulation can easily handle complex dynamic environments.', '1707.02387-3-61-5': 'As a result, we can handle complex instructions with attributes in realtime (see Table II).', '1707.02387-3-62-0': 'It may be possible to extend prior methods [CITATION] to handle attribute-based NLP instructions.', '1707.02387-3-62-1': 'For example, distance attributes require a number of constraints in the motion planning formulation.', '1707.02387-3-62-2': 'Lets consider natural language instructions such as: "Pick up the blue block and put it 20 cm to the left of the red block" or "Pick up one of the two blocks on the rightmost, and place it 10 inches away from the block on the leftmost," where the exact distance specifications are the distance attributes.', '1707.02387-3-62-3': 'Prior methods that use G[MATH], DCG and the Hybrid G[MATH]-DCG models have only been evaluated with a small number of attributes (distance, orientation and contact) to solve constrained motion planning problems.', '1707.02387-3-62-4': 'These prior techniques use discretized constraints [CITATION], each of which can be active (i.e. [MATH]), inverted ([MATH]) or ignored (i.e. not included).', '1707.02387-3-62-5': 'Therefore, it is not possible to exactly represent an explicit constraint corresponding to the value of the continuous variable distance in their formulation.', '1707.02387-3-62-6': 'One workaround is to discretize the continuous distance and to create multiple constraints for each discretized values.', '1707.02387-3-62-7': 'However, a large number of constraints increases the runtime overhead of their planner.', '1707.02387-3-62-8': 'However, our model can represent this distance attribute precisely with only one constraint using the latent parameter [MATH] and the corresponding cost function.', '1707.02387-3-63-0': '# Limitations, Conclusions and Future Work', '1707.02387-3-64-0': 'We presented a real-time motion planning algorithm that computes appropriate motion trajectories for a robot based on complex NLP instructions.', '1707.02387-3-64-1': 'Our formulation is based on two novel concepts: dynamic grounding graphs and dynamic constraint mapping.', '1707.02387-3-64-2': 'We highlight the performance in simulated and real-world scenes with a 7-DOF manipulator operating next to humans.', '1707.02387-3-64-3': 'The preliminary results are promising and our approach can handle more complex scenarios than prior methods.', '1707.02387-3-65-0': 'We use a trajectory optimization algorithm to compute the high-DOF robot trajectory.', '1707.02387-3-65-1': 'It is a high-dimensional optimization problem and the solver may get stuck in local minima.', '1707.02387-3-65-2': 'As a result, it is difficult to provide rigorous guarantees in terms of satisfying all the constraints or following the intent of the user.', '1707.02387-3-65-3': 'Furthermore, the accuracy of the mapping algorithm varies as a function of the training data.', '1707.02387-3-66-0': 'As future work, we would like to overcome these limitations and evaluate the approach in challenging scenarios with moving obstacles while performing complex robot tasks.', '1707.02387-3-66-1': 'More work is needed to handle the full diversity of a natural language, especially for rare words, complicated grammar styles, and hidden intention or emotion in human speech.', '1707.02387-3-66-2': 'We plan to incorporate stronger natural language processing and machine learning methods such as those based on semantic parsing, neural sequence-to-sequence models, reinforcement learning, and speech-based emotion analysis, and to compute the appropriate optimization-based planning formulations.'} | {'1707.02387-4-0-0': 'We present an algorithm for combining natural language processing (NLP) and fast robot motion planning to automatically generate robot movements.', '1707.02387-4-0-1': 'Our formulation uses a novel concept called Dynamic Constraint Mapping to transform complex, attribute-based natural language instructions into appropriate cost functions and parametric constraints for optimization-based motion planning.', '1707.02387-4-0-2': 'We generate a factor graph from natural language instructions called the Dynamic Grounding Graph (DGG), which takes latent parameters into account.', '1707.02387-4-0-3': 'The coefficients of this factor graph are learned based on conditional random fields (CRFs) and are used to dynamically generate the constraints for motion planning.', '1707.02387-4-0-4': 'We map the cost function directly to the motion parameters of the planner and compute smooth trajectories in dynamic scenes.', '1707.02387-4-0-5': 'We highlight the performance of our approach in a simulated environment and via a human interacting with a 7-DOF Fetch robot using intricate language commands including negation, orientation specification, and distance constraints.', '1707.02387-4-1-0': '# Introduction', '1707.02387-4-2-0': "In the field of human-robot interaction (HRI), natural language has been used as an interface to communicate a human's intent to a robot [CITATION].", '1707.02387-4-2-1': 'Much of the work in this area is related to specifying simple tasks or commands for robot manipulation, such as picking up and placing objects.', '1707.02387-4-2-2': 'As robots are increasingly used in complex scenarios and applications, it is important to develop a new generation of motion planning and robot movement techniques that can respond appropriately to diverse, attribute-based NLP instructions for HRI, e.g., instructions containing negation based phrases or references to position, velocity, and distance constraints.', '1707.02387-4-2-3': 'Furthermore, we need efficient techniques to automatically map the NLP instructions to such motion planners.', '1707.02387-4-3-0': 'Humans frequently issue commands that include sentences with orientation-based or negation constraints such as "put a bottle on the table and keep it upright" or "move the knife but don\'t point it towards people," or sentences with velocity-based constraints such as "move slowly when you get close to a human."', '1707.02387-4-3-1': 'To generate robot actions and movements in response to such complex natural language instructions, we need to address two kinds of challenges:', '1707.02387-4-4-0': '1.', '1707.02387-4-4-1': 'The accurate interpretation of attribute-based natural language instructions and their grounded linguistic semantics, especially considering the environment and the context.', '1707.02387-4-4-2': 'For example, a human may say "move a little to the left" or "do not move like this," and the robot planner needs to learn the correct interpretation of these commands that include spatial and motion-based adjectives, adverbs, and negation.', '1707.02387-4-5-0': '2.', '1707.02387-4-5-1': 'The motion planner needs to generate appropriate trajectories based on these complex natural language instructions.', '1707.02387-4-5-2': 'This includes appropriately setting up the motion planning problem based on different motion constraints (e.g., orientation, velocity, smoothness, and avoidance) and computing smooth and collision-free paths.', '1707.02387-4-6-0': 'At a high level, natural language instructions can be decomposed into task description and attributes.', '1707.02387-4-6-1': 'Task descriptions are usually verbs or noun phrases that describe the underlying task performed by a robot.', '1707.02387-4-6-2': 'The attributes include various adjectives, adverbs, or prepositional phrases that are used to specify additional conditions the robot must (or must not) satisfy.', '1707.02387-4-6-3': 'For example, these conditions may specify some information related to the movement speed, the orientation, the physical space characteristics, or the distances.', '1707.02387-4-6-4': 'Therefore, it is important to design motion planners that consider these robotic task descriptions and robot motion constraints.', '1707.02387-4-7-0': 'Main Results: We present an algorithm for generating parameterized constraints for optimization-based motion planning from complex, attribute-based natural language instructions.', '1707.02387-4-7-1': 'We use Dynamic Grounding Graphs (DGG) to parse and interpret the commands and to generate the constraints.', '1707.02387-4-7-2': 'Our formulation includes the latent parameters in the grounding process, allowing us to model many continuous variables in our grounding graph.', '1707.02387-4-7-3': 'Furthermore, we present a new dynamic constraint mapping that takes DGG as the input and computes different constraints and parameters for the motion planner.', '1707.02387-4-7-4': 'The appropriate motion parameters are speed, orientation, position, smoothness, repulsion, and avoidance.', '1707.02387-4-7-5': 'The final trajectory of the robot is computed using a constraint optimization solver.', '1707.02387-4-7-6': 'Overall, our approach can automatically handle complex natural language instructions corresponding to spatial and temporal adjectives, adverbs, superlative and comparative degrees, negations, etc.', '1707.02387-4-7-7': 'Compared to prior techniques, our overall approach offers the following benefits:', '1707.02387-4-8-0': 'We highlight the performance of our algorithms in a simulated environment and on a 7-DOF Fetch robot operating next to a human.', '1707.02387-4-8-1': 'Our approach can handle a rich set of natural language commands and can generate appropriate paths.', '1707.02387-4-8-2': 'These include complex commands such as picking (e.g., "pick up a red object near you"), correcting the motion (e.g., "don\'t pick up that one"), and negation (e.g., "don\'t put it on the book").', '1707.02387-4-9-0': '# Related Work', '1707.02387-4-10-0': 'Most algorithms used to map natural language instruction to robot actions tend to separate the problem into two parts: parsing and motion planning computation.', '1707.02387-4-10-1': 'In this section, we give a brief overview of prior work in these areas.', '1707.02387-4-11-0': '## Natural Language Processing', '1707.02387-4-12-0': 'Duvallet et al. [CITATION] use a probabilistic graphical learning model called Generalized Grounding Graphs (G[MATH]) on a ground vehicle for a navigation problem given natural language commands.', '1707.02387-4-12-1': 'Branavan et al. [CITATION] use reinforcement learning to learn a mapping from natural language instructions and then apply it to sequences of executable actions.', '1707.02387-4-12-2': 'Matuszek et al. [CITATION] use a statistical machine translation model to map natural language instructions to path description language, which allows a robot to navigate while following directions.', '1707.02387-4-12-3': 'Duvallet et al. [CITATION] use imitation learning to train the model through demonstrations of humans following directions.', '1707.02387-4-12-4': 'Paul et al. [CITATION] propose the Adaptive Distributed Correspondence Graph (ADCG).', '1707.02387-4-12-5': 'Arkin et al. [CITATION] further extend DCG, proposing the Hierarchical Distributed Correspondence Graph (HDCG), which defines constraints as discrete inequalities and grounds word phrases to corresponding inequalities.', '1707.02387-4-12-6': 'Chung et al. [CITATION] use HDCG on ground vehicles to implement navigation commands and demonstrate performance improvements over G[MATH] in terms of running time, factor evaluations, and correctness.', '1707.02387-4-12-7': 'Oh et al. [CITATION] integrate HDCG with their navigating robot system, measuring performance in terms of completion rates and comparing them to human behaviors.', '1707.02387-4-12-8': 'Scalise et al. [CITATION] collected a corpus of natural language instructions from online crowdsourcing that specify objects of interest for "picking up" command.', '1707.02387-4-12-9': 'The dataset could be used as a training dataset in our method.', '1707.02387-4-13-0': '## Robot Motion Planning in Dynamic Environments', '1707.02387-4-14-0': 'Many replanning algorithms have been suggested to generate collision-free motion plans in dynamic environments.', '1707.02387-4-14-1': 'Fox et al. [CITATION] propose the dynamic window approach to compute optimal velocity in a short time window.', '1707.02387-4-14-2': 'Optimization-based motion planners [CITATION] solve a constrained optimization problem to generate smooth and collision-free robot paths.', '1707.02387-4-14-3': 'We present an automatic scheme that generates the motion planning problem from NLP instructions.', '1707.02387-4-15-0': 'There is some work on integrating optimization-based motion planning with NLP in 2D workspaces.', '1707.02387-4-15-1': 'Silver et al. [CITATION] develop an algorithm for learning navigation cost functions from demonstrations.', '1707.02387-4-15-2': 'Howard et al. [CITATION] use a probabilistic graphical model to generate motion planning constraints for a 2D navigation problem.', '1707.02387-4-15-3': 'Compared to these methods, our approach can handle 3D workspaces and high-dimensional configuration spaces to generate robot motions corresponding to complex NLP instructions.', '1707.02387-4-15-4': 'Other techniques focus on efficiency in human-robot collaborative tasks.', '1707.02387-4-15-5': 'Markov Decision Processes (MDP) are widely used to compute the best robot action policies [CITATION].', '1707.02387-4-15-6': 'These techniques are complementary to our approach.', '1707.02387-4-16-0': '# Dynamic Grounding Graphs', '1707.02387-4-17-0': 'Fig. [REF] shows the basic pipeline of our approach.', '1707.02387-4-17-1': 'When natural language commands are given as input, the NLP module (upper left) creates an optimization problem for a motion planning module (upper middle).', '1707.02387-4-17-2': 'The robot motion trajectory is then computed from the motion planning module (upper right).', '1707.02387-4-17-3': 'As the planned trajectory is executed (bottom right), the result is fed back to the NLP module.', '1707.02387-4-17-4': 'In this section, we present the algorithms used in the NLP module.', '1707.02387-4-18-0': 'We extend the ideas of the Generalized Grounding Graphs (G[MATH]) model and the Distributed Correspondence Graph (DCG) model [CITATION] by including the latent variables in the grounding graph and using them to compute the constraints for motion planning.', '1707.02387-4-18-1': 'The input to our algorithm is the natural language instruction.', '1707.02387-4-18-2': 'We do not account for any errors due to voice recognition.', '1707.02387-4-18-3': 'From a natural language command input, we construct a factor graph, as shown in Fig. [REF](a), which is based on the parsing of the command.', '1707.02387-4-18-4': 'For each node of the parse tree, we generate three types of nodes: word phrase node [MATH], grounding node [MATH], and correspondence node [MATH].', '1707.02387-4-19-0': 'The input sentence [MATH] is parsed using the NLTK library [CITATION].', '1707.02387-4-19-1': 'The word phrase of each node in the parse tree is denoted as [MATH] for [MATH].', '1707.02387-4-19-2': 'Children of [MATH] are [MATH], [MATH], [MATH].', '1707.02387-4-19-3': 'The root node of the parse tree is [MATH].', '1707.02387-4-19-4': 'For example, in Fig. [REF](a), the input sentence is "Put the cup on the table."', '1707.02387-4-19-5': 'The parse tree has the root word phrase [MATH]"Put".', '1707.02387-4-19-6': 'Its noun [MATH]"the cup" and the preposition [MATH]"on," which are the children nodes of the root node.', '1707.02387-4-19-7': 'The noun phrase [MATH]"the table" is the child node of [MATH].', '1707.02387-4-19-8': 'Similarly, in Fig. [REF](b), the command "Don\'t put it there" is decomposed into 4 noun phrase nodes.', '1707.02387-4-19-9': 'The word phrase [MATH]"Don\'t" is a negation of the verb and its child node is [MATH]"put."', '1707.02387-4-19-10': '[MATH]"it" and [MATH]"there" are the children nodes of [MATH].', '1707.02387-4-19-11': 'Note that this parse tree is different from the parse tree in Fig. [REF](a).', '1707.02387-4-20-0': 'Our goal is to compute a mapping from a natural language sentence [MATH] to the cost function parameters [MATH], given the robotic environment [MATH] where the robot is operating.', '1707.02387-4-20-1': "[MATH] is a representation of the environment, which is composed of obstacle positions, orientations, and the robot's configuration.", '1707.02387-4-20-2': 'Feature vectors are constructed in the factor graph from the description of the environment.', '1707.02387-4-20-3': '[MATH] is a real-valued vector that contains all cost function parameters used in the optimization-based motion planner.', '1707.02387-4-20-4': 'It also includes the weights of different types of cost functions used in the optimization formulation.', '1707.02387-4-20-5': 'For example, the end-effector position cost function (Eq. ([REF])) requires the 3D coordinates of the target position as parameters.', '1707.02387-4-20-6': 'The repulsion cost function (Eq. ([REF])) requires the repulsion source position and the constant from the exponential function.', '1707.02387-4-21-0': 'We first compute the groundings [MATH] of each word phrase [MATH].', '1707.02387-4-21-1': 'The grounding of each word phrase is the mapping from the word phrase to its meaning in the real world.', '1707.02387-4-21-2': 'Groundings can be objects, locations, motions, tasks, or constraints.', '1707.02387-4-21-3': 'In our model, the grounding [MATH] depends on its work phrase [MATH] and its children grounding nodes [MATH], [MATH], [MATH], where the tree structure of the grounding nodes follows the parse tree.', '1707.02387-4-21-4': 'Correspondence node [MATH] indicates the correct matching between the word phrase [MATH] and the grounding [MATH].', '1707.02387-4-21-5': 'It is a binary variable; [MATH] is [MATH] if the word phrase and the grounding match and [MATH] if they do not.', '1707.02387-4-22-0': '## Latent Parameters', '1707.02387-4-23-0': 'A key novel component of our approach is the inclusion of latent variables in the grounding graph.', '1707.02387-4-23-1': 'Our primary goal is to compute the best cost function parameters [MATH] to be used directly for optimization-based motion planning.', '1707.02387-4-23-2': 'We denote [MATH], a real vector of size [MATH], as a collection of cost function parameters.', '1707.02387-4-23-3': 'In this case, the size [MATH] and the number of cost function parameters depend on the types of cost functions that are used.', '1707.02387-4-23-4': 'From the predicted groundings [MATH], the cost function parameters in the motion planning formulation (Fig. [REF](b)) are inferred through the latent variable [MATH].', '1707.02387-4-23-5': '[MATH] contains all the cost function parameters (e.g., weights of cost functions, locations, and orientations).', '1707.02387-4-24-0': 'In Fig. [REF](b), the resulting constraint-based motion planning problems are shown.', '1707.02387-4-24-1': 'We use the collision avoidance cost function as the default smoothness cost function and the target location cost function, though weights can vary.', '1707.02387-4-24-2': 'The target location, whose 3D coordinates are the cost function parameters, is set on the surface of the table.', '1707.02387-4-24-3': 'The cost function parameter node [MATH] contains the weights of the parameters and the 3D coordinates of the target location.', '1707.02387-4-24-4': 'In the bottom of Fig. [REF](b), where a new "Don\'t" command is given, a repulsion cost function is added.', '1707.02387-4-24-5': 'Thus, the cost function weight and the location of the repulsion source (below the robot’s end-effector position) are added to [MATH].', '1707.02387-4-25-0': '## Probabilistic Model', '1707.02387-4-26-0': 'We present a new probabilistic model to compute [MATH], [MATH], and [MATH].', '1707.02387-4-26-1': 'We pose the problem of finding the best cost parameters as an optimization problem: [EQUATION]', '1707.02387-4-26-2': 'However, modeling the probability function without decomposing the variables and some assumptions about independence is difficult due to the high-dimensionality of [MATH], [MATH], and [MATH] and the dependencies between them.', '1707.02387-4-26-3': 'To simplify the problem, the natural language sentence is decomposed into [MATH] word phrases based on a parse tree, i.e. [EQUATION]', '1707.02387-4-26-4': 'Like G[MATH], we introduce the intermediate groundings [MATH] of word phrases [MATH] and correspondence variables [MATH].', '1707.02387-4-26-5': 'The correspondence variables [MATH] are binary random variables.', '1707.02387-4-26-6': 'The value [MATH] indicates that the word phrase [MATH] correctly corresponds to the grounding [MATH].', '1707.02387-4-26-7': '[MATH] means an incorrect correspondence.', '1707.02387-4-27-0': 'We assume the conditional independence of the probabilities so that we can construct a factor graph (see Fig. [REF](a)).', '1707.02387-4-27-1': 'With the independence assumptions, a single factor is connected to a word phrase node and its children grounding nodes, which contain information about the sub-components.', '1707.02387-4-27-2': 'These independence assumptions simplify the problem and make it solvable by efficiently taking advantage of the tree structure of the probabilistic graphical learning model.', '1707.02387-4-27-3': "Formally, the root grounding node [MATH] contains all the information about a robot's motion.", '1707.02387-4-27-4': 'The factor that connects [MATH] and [MATH] implies that, from the root grounding node, the cost function parameters [MATH] are optimized without any consideration of other nodes.', '1707.02387-4-27-5': 'Other factors connect [MATH], [MATH], [MATH], children grounding nodes [MATH] and the environment [MATH], where the parent-child relationship is based on a parse tree constructed from the natural language sentence.', '1707.02387-4-27-6': 'This graphical representation corresponds to the following equation: [EQUATION]', '1707.02387-4-27-7': 'For the root factor connecting [MATH], [MATH] and [MATH], we formulate the continuous domain of [MATH].', '1707.02387-4-27-8': 'We compute the Gaussian Mixture Model (GMM) on the probability distribution [MATH] and model our probability with non-root factors as follows: [EQUATION] where [MATH] is the normalization factor, and [MATH] and [MATH] are the log-linearization of the feature function.', '1707.02387-4-27-9': 'The function [MATH] generates a feature vector given a grounding [MATH], a word phrase [MATH], a correspondence [MATH], children groundings [MATH], and the environment [MATH].', '1707.02387-4-27-10': 'The information from the robotic environment is used in the feature function [MATH] and in the log-linearized feature function [MATH].', '1707.02387-4-27-11': 'The attributes of objects in the robotic world such as shapes and colors are encoded as multidimensional binary vectors, which indicate whether the object has a given attribute.', '1707.02387-4-28-0': 'The probability distribution of the latent variable [MATH] is modeled with [MATH] pairs of Gaussian distribution parameters [MATH] and [MATH] with weights [MATH], as follows: [EQUATION]', '1707.02387-4-28-1': 'Word phrases.', '1707.02387-4-28-2': 'The feature vector includes binary-valued vectors for the word and phrase occurrences, and Part of Speech (PoS) tags.', '1707.02387-4-28-3': 'There is a list of words that could be encountered in the training dataset such as put, pick, cup, up, there, [MATH] .', '1707.02387-4-28-4': 'If the word phrase contains the word "put," then the occurrence vector at the first index is set to 1 and the others are set to 0.', '1707.02387-4-28-5': 'If the word phrase is "pick up," then the occurrence vector values at the second, while the fourth is set to 1 and others are set to 0.', '1707.02387-4-28-6': 'This list also includes real-valued word similarities between the word and the pre-defined seed words.', '1707.02387-4-28-7': 'The seed words are the pre-defined words that the users expect to encounter in the natural language instructions.', '1707.02387-4-28-8': 'We used Glove word2vec [CITATION] to measure cosine-similarity (i.e. the inner product of two vectors divided by the lengths of the vectors) between the words.', '1707.02387-4-28-9': 'The measurement indicates that the words are similar if the similarity metric value is near [MATH], that they have opposite meanings if the similarity metric is near -1, and that they have a weak relationship if it is near 0.', '1707.02387-4-28-10': 'This provides more flexibility to our model, especially when it encounters new words that are not trained during the training phase.', '1707.02387-4-29-0': 'Robot states.', '1707.02387-4-29-1': 'From the robot state, we collect the robot joint angles, the velocities, the end-effector position, the end-effector velocity, etc.', '1707.02387-4-29-2': 'This information can affect the cost function parameters even while processing the same natural language commands.', '1707.02387-4-29-3': 'For example, if the robot is too close to a human under the current configuration, then the cost function for end-effector speed [MATH] or smoothness [MATH] will be adjusted so that the robot does not collide with the human.', '1707.02387-4-29-4': 'We also store information about the objects that are close to the robot.', '1707.02387-4-29-5': 'This information includes object type, position, orientation, shape, dimension, etc.', '1707.02387-4-30-0': '## Factor Graph using Conditional Random Fields', '1707.02387-4-31-0': 'We represent our dynamic grounding graph as a factor graph.', '1707.02387-4-31-1': 'We build a factor graph based on the probabilistic model described in Section [REF] and use that for training and for inferring the meaning of given commands.', '1707.02387-4-31-2': 'In particular, we use Conditional Random Fields (CRF) [CITATION] as a learning model for factor graphs because CRFs are a good fit for applying machine learning to our probabilistic graph model with conditional probabilities.', '1707.02387-4-32-0': 'During the training step of CRF, we solve the optimization problem of maximizing the probability of the samples in the training dataset over the feature coefficients [MATH] and the GMM parameters [MATH], [MATH] and [MATH] for every parse tree structure.', '1707.02387-4-32-1': 'By multiplying Eq. ([REF]-[REF]) for all training samples, the optimization problem becomes [EQUATION] where superscripts [MATH] mean the indices of the training samples.', '1707.02387-4-32-2': 'The joint optimization problem Eq. ([REF]-[REF]) of the GMM and the CRF is a hard problem.', '1707.02387-4-32-3': 'So, we separate the problem into two and solve each one separately to maximize the objective.', '1707.02387-4-32-4': 'To solve Eq. ([REF]), the training samples of continuous variable [MATH] is collected under the same conditional variable [MATH].', '1707.02387-4-32-5': "Then, we solve the problem with the collection of [MATH]'s via Expectation Maximization (EM) method.", '1707.02387-4-32-6': 'Eq. ([REF]) is a tree-structured CRF problem.', '1707.02387-4-33-0': 'At the inference step, we used the trained CRF factor graph models to find the best groundings [MATH] and the cost function parameters [MATH] by solving the CRF maximization problem [EQUATION]', '1707.02387-4-33-1': 'When the nodes [MATH], [MATH] are optimized, they create a tree structure in the factor graph, meaning that we can solve the optimization problem efficiently using dynamic programming.', '1707.02387-4-33-2': 'Each factor depends on its parent and children varying variables and other fixed variables connected to it.', '1707.02387-4-33-3': 'This implies that we can solve the sub-problems in a bottom-up manner and combine the results to solve the bigger problem corresponding to the root node.', '1707.02387-4-34-0': '# Dynamic Constraint Mapping With NLP Input', '1707.02387-4-35-0': 'We use an optimization-based algorithm [CITATION] to solve the cost minimization problem.', '1707.02387-4-35-1': 'The function and constraints of this cost minimization problem come from DGG, as explained in Sec. [REF].', '1707.02387-4-35-2': 'In this section, we present our mapping algorithm, Dynamic Constraint Mapping, which maps the word phrase groundings to proper cost function parameters corresponding to natural language instructions.', '1707.02387-4-36-0': '## Robot Configurations and Motion Plans', '1707.02387-4-37-0': 'We denote a single configuration of the robot as a vector [MATH], which consists of joint-angles or other degrees-of-freedom.', '1707.02387-4-37-1': 'A configuration at time [MATH], where [MATH], is denoted as [MATH].', '1707.02387-4-37-2': 'We assume [MATH] is twice differentiable, and its derivatives are denoted as [MATH] and [MATH].', '1707.02387-4-37-3': 'The [MATH]-dimensional space of configuration [MATH] is the configuration space [MATH].', '1707.02387-4-37-4': 'We represent bounding boxes of each link of the robot as [MATH].', '1707.02387-4-37-5': 'The bounding boxes at a configuration [MATH] are denoted as [MATH].', '1707.02387-4-38-0': "For a planning task with a given start configuration [MATH] and derivative [MATH], the robot's trajectory is represented by a matrix [MATH], whose elements correspond to the waypoints [CITATION]: [EQUATION]", '1707.02387-4-38-1': 'The robot trajectory passes through [MATH] waypoints [MATH], which will be optimized by an objective function under constraints in the motion planning formulation.', '1707.02387-4-38-2': 'Robot configuration at time [MATH] is interpolated from two waypoints.', '1707.02387-4-38-3': 'Formally, for [MATH] such that [MATH], the configuration [MATH] and derivative [MATH] are cubically interpolated using [MATH], [MATH], [MATH], and [MATH].', '1707.02387-4-39-0': 'The [MATH]-th cost functions of the motion planner are [MATH].', '1707.02387-4-39-1': 'Our motion planner solves an optimization problem with non-linear cost functions and linear joint limit constraints to generate robot trajectories for time interval [MATH], [EQUATION]', '1707.02387-4-39-2': 'In the optimization formulation, [MATH] is the [MATH]-th cost function and [MATH] is the weight of the cost function.', '1707.02387-4-40-0': '## Cost Functions', '1707.02387-4-41-0': 'The overall optimization formulation is given in Eq. ([REF]).', '1707.02387-4-41-1': 'To formulate the constraints, we use the following cost functions, which are designed to account for various attributes in the NLP instructions.', '1707.02387-4-41-2': 'In our formulation, we use many types of cost functions such as collision avoidance, robot smoothness, robot end-effector speed, target positions, and target orientations.', '1707.02387-4-41-3': 'These cost functions are used to handle many attributes of natural language instructions.', '1707.02387-4-41-4': 'Each cost function has its weight and may also have other cost function parameters, if necessary.', '1707.02387-4-41-5': 'For example, the robot end-effector speed cost function has parameters corresponding to the direction and the magnitude of the speed, which impose a constraint on the final computed trajectory.', '1707.02387-4-41-6': 'If the weight of the end-effector speed cost function is higher than the others, then it contributes more to the overall objective function in the optimization formulation.', '1707.02387-4-41-7': 'If the weight is low, then the end-effector speed cost will be compromised and has a lesser impact on the path planner.', '1707.02387-4-42-0': 'The cost functions [MATH] and the latent parameter [MATH] are closely related.', '1707.02387-4-42-1': '[MATH] is a collection of parameters that describe all types of [MATH] and the weights [MATH].', '1707.02387-4-42-2': 'The cost function parameters of [MATH] and the weights [MATH] are all real-valued.', '1707.02387-4-42-3': 'Those real values are appended to construct the real-valued vector [MATH].', '1707.02387-4-43-0': '## Parameterized Constraints', '1707.02387-4-44-0': 'To handle various attributes, we use the following parameterized constraints in our optimization formulation.', '1707.02387-4-45-0': 'Collision avoidance: By default, the robot should always avoid obstacles.', '1707.02387-4-45-1': '[EQUATION] where [MATH] is the penetration depth between a robot bounding box [MATH] and an obstacle [MATH].', '1707.02387-4-46-0': "Smoothness: We penalize the magnitude of a robot's joint angle speed to make the trajectory smooth.", '1707.02387-4-46-1': 'This corresponds to the integral of the first derivative of joint angles over the trajectory duration, as follows: [EQUATION]', '1707.02387-4-46-2': 'This function is useful when we need to control the speed of the robot.', '1707.02387-4-46-3': "When the robot should operate at a low speed (e.g. when a human is too close), or we don't want abrupt movements (e.g., for human safety), the smoothness cost can have high weights so that the robot moves slowly without jerky motions.", '1707.02387-4-47-0': "End-effector position: A user usually specifies the robot's target position to make sure that the robot reaches its goal.", '1707.02387-4-47-1': "This cost function penalizes the squared distance between the robot's end-effector position and the target position over the trajectory duration as [EQUATION] where [MATH] is the robot end-effector position at time [MATH] and [MATH] is the target position.", '1707.02387-4-47-2': 'The target position [MATH] is considered as a cost function parameter.', '1707.02387-4-47-3': 'In the mapping algorithm, a position grounding node encodes the target position parameter.', '1707.02387-4-47-4': 'This parameter can be a 3D position or the current object position in the environment.', '1707.02387-4-47-5': 'Typically, the target position is specified by an object name in the sentence, such as "pick up the cup" or "move to the box."', '1707.02387-4-47-6': 'In these cases, the grounding nodes for "the cup" and "the box" are interpreted as the current 3D coordinates of the target positions, which are the parameters of this cost function.', '1707.02387-4-48-0': 'End-effector orientation: Robotic manipulation tasks are sometimes constrained by the end-effector orientation.', '1707.02387-4-48-1': 'This cost function penalizes the squared angular differences between the end-effector orientation and the target orientation over the trajectory duration.', '1707.02387-4-48-2': "[EQUATION] where [MATH] is the quaternion representation of the robot end-effector's orientation at time [MATH], [MATH] is the end-effector orientation that we want the robot to maintain, [MATH] is the normal up-vector of the robot's end-effector, and [MATH] is the target up-vector.", '1707.02387-4-48-3': 'As with the end-effector position cost, the target orientation [MATH] is the cost function parameter.', '1707.02387-4-48-4': 'The target orientation usually depends on the object the robot picked up.', '1707.02387-4-48-5': 'For example, when the robot is doing a peg-hole insertion task under the command "insert that into the hole," the orientation of the robot\'s end-effector [MATH] should be constrained near the hole.', '1707.02387-4-48-6': 'If the robot arm is holding a cup of water, it should be upright so it does not spill the water.', '1707.02387-4-48-7': 'In this case, [MATH] is set to [MATH].', '1707.02387-4-49-0': "End-effector speed: This cost function penalizes the robot's end-effector speed and direction: [EQUATION] where [MATH] is the robot's end-effector speed at time [MATH], and [MATH] is the target speed.", '1707.02387-4-49-1': 'The parameters of this cost function correspond to [MATH].', '1707.02387-4-49-2': "In some cases, we must restrict the robot's end-effector velocity, e.g., if a user wants to pick up a cup filled with water and doesn't want to spill it.", '1707.02387-4-49-3': 'Spilling can be prevented by limiting the end-effector speed, making the robot move more slowly.', '1707.02387-4-50-0': "Repulsion: The repulsion functions are commonly represented as potential fields [EQUATION] where [MATH] is the position to which we don't want the robot to move.", '1707.02387-4-50-1': 'The coefficient [MATH] suggests how much the cost is affected by [MATH], the distance between the end-effector position and the repulsion source.', '1707.02387-4-50-2': 'The cost function is maximized when the end-effector position is exactly at the repulsion source, and it decreases as the distance between the end-effector and the repulsion position increases.', '1707.02387-4-50-3': 'For example, if the command is "Don\'t put the cup on the laptop," we can define a repulsion cost with the laptop position as the repulsion source.', '1707.02387-4-50-4': 'The cost function is inversely proportional to the distance between the end-effector and the laptop.', '1707.02387-4-51-0': '# Implementation and Results', '1707.02387-4-52-0': 'We have implemented our algorithm and evaluated its performance in a simulated environment and on a 7-DOF Fetch robot.', '1707.02387-4-52-1': 'All the timings are generated on a multi-core PC.with Intel i7-4790 8-core 3.60GHz CPU and a 16GB RAM.', '1707.02387-4-52-2': 'We use multiple cores for fast evaluation and parallel trajectory search to compute a good solution to the constrained optimization problem [CITATION].', '1707.02387-4-53-0': '## Training DGGs for Demonstrations', '1707.02387-4-54-0': 'We describe how the training dataset for our DGG model was generated.', '1707.02387-4-54-1': 'The training dataset for DGGs requires three components: a natural language sentence, a robotic environment, and the cost function parameters for optimization-based motion planners.', '1707.02387-4-55-0': 'For each demonstration, we write tens of different sentences that specify the take goals the constraints for the motion plans with different nouns, pronouns, adjectives, verbs, adverbs, preposition, etc.', '1707.02387-4-55-1': 'For each sentence, we generate a random robotic environment and an initial state for the robot.', '1707.02387-4-55-2': 'In addition, the robot joint values and joint velocities are randomly set as initial states.', '1707.02387-4-55-3': 'We collect tens or hundreds of random robotic environments.', '1707.02387-4-55-4': 'For a natural language sentence, a random robotic environment, and a random initial state for the robot, the cost function parameters are assigned manually or synthesized from other examples.', '1707.02387-4-55-5': 'Crowdsourcing such as Amazon Mechanical Turk can be alternatively used to assign cost function parameters.', '1707.02387-4-55-6': 'Hundreds of multiple data samples are generated from generated data samples by switching the correspondence variable in the DGG model from [MATH] (true) to [MATH] (false) and changing the grounding variables to the wrong ones to match the false correspondence variable.', '1707.02387-4-55-7': 'The training dataset is created with up to [MATH] samples in our experiments.', '1707.02387-4-55-8': 'When the cost function parameters are determined, the optimization-based motion planner is used to compute a feasible robot trajectory.', '1707.02387-4-55-9': 'In the optimization-based motion planning algorithm, there are some waypoints through which the robot trajectory should pass.', '1707.02387-4-55-10': "For the robot's safety, we check if the robot trajectory with the given cost function parameters is in-collision and appropriately set a higher value of the coefficient of the collision cost and compute a new trajectory.", '1707.02387-4-55-11': 'This process is repeated until the trajectory is collision-free.', '1707.02387-4-55-12': 'The training step took up to an hour with [MATH] training samples in our experimental settings, though the training time can vary depending on the complexity of tasks, environments, and natural language instructions.', '1707.02387-4-56-0': 'We use different training data for each scenario.', '1707.02387-4-56-1': 'For the scenarios shown in Fig. [REF], the initial pose of the robot in front of the table and the positions of the blue and red objects on the table are randomly set.', '1707.02387-4-56-2': 'For "Pick up" commands, appropriate cost function parameters are computed so that the robot picks up a blue or red object depending on the given command.', '1707.02387-4-56-3': 'Similarly, in Fig. [REF], the position and orientation of the laptop is initialized randomly.', '1707.02387-4-56-4': 'Given the "Put" command, we create an end-effector position cost function so that the robot places the object on the table; and a repulsive cost function to avoid the laptop position.', '1707.02387-4-57-0': '## Simulations and Real Robot Demonstrations', '1707.02387-4-58-0': 'We evaluate the performance on optimization problems that occur in complex environments composed of multiple objects.', '1707.02387-4-58-1': 'Based on the NLP commands, the robot decides to pick an appropriate object or is steered towards the goal position in a complex scene.', '1707.02387-4-58-2': 'In particular, the user gives NLP commands such as "move right," "move up," "move left," or "move down" to guide the robot.', '1707.02387-4-58-3': 'For each such command, we compute the appropriate cost functions.', '1707.02387-4-59-0': 'We also integrate our NLP-based planner with ROS and evaluated its performance on the 7-DOF Fetch robot.', '1707.02387-4-59-1': 'In a real-world setting, we test its performance on different tasks corresponding to: (1) moving a soda can on the table from one position to another; (2) not moving the soda can over the book.', '1707.02387-4-59-2': 'With a noisy point cloud sensor on the robot, the thin book is not recognized as a separate obstacle by the robot, though the human user wants the robot to avoid it.', '1707.02387-4-59-3': 'All the instructions used in these tasks have different attributes, which makes it hard for prior methods.', '1707.02387-4-59-4': 'In Fig. [REF], the two sub-tasks are specified in one sentence at the beginning, as "move the can on the table, but don\'t put it on the book".', '1707.02387-4-59-5': "The cost function is used to move the robot's end-effector to the surface of the table.", '1707.02387-4-59-6': 'Another cost function penalizes the distance between the book and the end-effector.', '1707.02387-4-59-7': 'In Fig. [REF], only the first sub-task is given at the beginning.', '1707.02387-4-59-8': 'This results in the robot moving the can on the book.', '1707.02387-4-59-9': 'As the robot gets too close to the book, the person says "stop," then says "don\'t put it there."', '1707.02387-4-59-10': 'The robot recomputes the cost functions and avoids the region around the book.', '1707.02387-4-60-0': '## Analysis', '1707.02387-4-61-0': 'We evaluate the performance based on the following metrics:', '1707.02387-4-62-0': 'Success Rate: The ratio of successful task completion among all trials.', '1707.02387-4-62-1': 'Failure includes colliding with the obstacles due to an incorrect mapping of cost function parameters, violating constraints specified by natural language commands, and not completing the task due to some other reason.', '1707.02387-4-63-0': "Trajectory Duration: The time between the giving of the first NLP command and the robot's successful completion of the task after trajectory computation.", '1707.02387-4-63-1': 'A shorter duration implies a higher performance.', '1707.02387-4-64-0': 'Trajectory Smoothness Cost: A cost based on evaluating the trajectory smoothness according to standard metrics and dividing it by the trajectory duration.', '1707.02387-4-64-1': 'A lower cost implies a smoother and more stable robot trajectory.', '1707.02387-4-65-0': 'Table [REF] shows the results on our benchmarks with varying numbers of training data samples in the simulation environment shown in Fig. [REF].', '1707.02387-4-65-1': 'When the number of training data samples increases, the success rate also increases while the trajectory duration and the trajectory smoothness cost decrease.', '1707.02387-4-65-2': 'Table [REF] shows the running time of our algorithm and the distances from the obstacle on the table in the real-world scenarios.', '1707.02387-4-65-3': 'We use 8 parallel threads for parallel trajectory search in the motion planning module.', '1707.02387-4-66-0': 'Table [REF] and [REF] show the examples of the dataset.', '1707.02387-4-66-1': 'In Table [REF], DGGs with word phrase nodes, grounding nodes and correspondence variables are shown.', '1707.02387-4-66-2': 'The correspondence variables of the graphs on the left column are all true, and the groundings are matched correctly.', '1707.02387-4-66-3': 'Whereas, the correspondence variables on the right column are mixed with true and false.', '1707.02387-4-66-4': 'The groundings are not matched if the correspondence variable is false.', '1707.02387-4-66-5': 'Many data samples are generated by flipping the correspondence variables between true and false to increase the accuracy of the DGG inference step.', '1707.02387-4-66-6': 'In Table [REF], the latent variables are shown for the examples of the grounding graphs and the environment.', '1707.02387-4-66-7': 'The cost function weights and other necessary cost function parameters are manually set in the data generation program.', '1707.02387-4-67-0': '# Benefits and Comparisons', '1707.02387-4-68-0': 'Most prior methods that combine NLP and motion planning have focused on understanding natural language instructions to compute robot motion for simple environments and constraints.', '1707.02387-4-68-1': 'Most of these methods are limited to navigation applications [CITATION] or simple settings [CITATION], or they are not evaluated on real robots [CITATION].', '1707.02387-4-68-2': 'Nyga et al. [CITATION] use probabilistic relation models based on knowledge bases to understand natural language commands that describe visual attributes of objects.', '1707.02387-4-68-3': 'This is complementary to our work.', '1707.02387-4-68-4': 'Broad et al. [CITATION] extedd DCG for a robot manipulator so that it will handle natural language correction for robot motion in realtime.', '1707.02387-4-68-5': 'In our approach, the goal is to generate appropriate high-DOF motion trajectories in response to attribute-based natural language instructions like negation, distance or orientation constraints, etc.', '1707.02387-4-68-6': 'Unlike prior methods, the output of our NLP parsing algorithm is directly coupled with the specification of the motion planning problem as a constrained optimization method.', '1707.02387-4-69-0': 'It may be possible to extend prior methods [CITATION] to handle attribute-based NLP instructions.', '1707.02387-4-69-1': 'For example, distance attributes require a number of constraints in the motion planning formulation.', '1707.02387-4-69-2': 'In natural language instructions such as "Pick up the blue block and put it 20 cm to the left of the red block" or "Pick up one of the two blocks on the rightmost, and place it 10 inches away from the block on the leftmost," the exact distance specifications are the distance attributes.', '1707.02387-4-69-3': 'Prior methods that use G[MATH], DCG, or the Hybrid G[MATH]-DCG models have only been evaluated with a small number of attributes (distance, orientation, and contact) to solve constrained motion planning problems.', '1707.02387-4-69-4': 'These prior techniques use discretized constraints [CITATION], each of which can be active (i.e. [MATH]), inverted ([MATH]), or ignored (i.e. not included).', '1707.02387-4-69-5': 'Therefore, it is not possible to represent an explicit constraint corresponding to the value of the continuous variable distance in their formulation.', '1707.02387-4-70-0': '# Benefits and Comparisons', '1707.02387-4-71-0': 'In this section, we compare our approach with prior methods and highlight the benefits in terms of handling attribute-based NLP instructions.', '1707.02387-4-71-1': 'We evaluated the end-to-end performance of our approach including the NLP, mapping and the motion planning modules on different benchmarks.', '1707.02387-4-71-2': 'Most prior methods that combine NLP and motion planning have focused on understanding natural language instructions to compute robot motion for simple environments and constraints.', '1707.02387-4-71-3': 'Most of these methods are limited to navigation applications [CITATION] or used in simple settings [CITATION] or not evaluated on real robots [CITATION].', '1707.02387-4-71-4': 'In our approach, the goal is to generate appropriate high-DOF motion trajectories in response to attribute-based natural language instructions like negation, distance or orientation constraints, etc.', '1707.02387-4-71-5': 'Unlike prior methods, the output of our NLP parsing algorithm is directly coupled with the specification of the motion planning problem as a constrained optimization method.', '1707.02387-4-72-0': 'Our algorithm demonstrates many advantages over Howard et al. [CITATION].', '1707.02387-4-72-1': 'They use a discrete set of constraints and their planning algorithm determines whether those constraints are activated or not.', '1707.02387-4-72-2': 'In the worst case, the complexity of their search space can grow exponentially as more constraints are added.', '1707.02387-4-72-3': 'On the other hand, we use appropriate cost functions as parameters over the continuous domain.', '1707.02387-4-72-4': 'Our constraint setup algorithm based on dynamic constraint mapping is different and our formulation can easily handle complex dynamic environments.', '1707.02387-4-72-5': 'As a result, we can handle complex instructions with attributes in real-time (see Table II).', '1707.02387-4-73-0': 'It may be possible to extend prior methods [CITATION] to handle attribute-based NLP instructions.', '1707.02387-4-73-1': 'For example, distance attributes require a number of constraints in the motion planning formulation.', '1707.02387-4-73-2': 'Lets consider natural language instructions such as: "Pick up the blue block and put it 20 cm to the left of the red block" or "Pick up one of the two blocks on the rightmost, and place it 10 inches away from the block on the leftmost," where the exact distance specifications are the distance attributes.', '1707.02387-4-74-0': 'Prior methods that use G[MATH], DCG and the Hybrid G[MATH]-DCG models have only been evaluated with a small number of attributes (distance, orientation, and contact) to solve constrained motion planning problems.', '1707.02387-4-74-1': 'These prior techniques use discretized constraints [CITATION], each of which can be active (i.e. [MATH]), inverted ([MATH]) or ignored (i.e. not included).', '1707.02387-4-74-2': 'Therefore, it is not possible to exactly represent an explicit constraint corresponding to the value of the continuous variable distance in their formulation.', '1707.02387-4-74-3': 'One workaround is to discretize the continuous distance and to create multiple constraints for each discretized values.', '1707.02387-4-74-4': 'However, a large number of constraints increases the runtime overhead of their planner.', '1707.02387-4-74-5': 'However, our model can represent this distance attribute precisely with only one constraint using the latent parameter [MATH] and the corresponding cost function.', '1707.02387-4-75-0': 'Moreover, we highlight the experimental results with a 7-DOF Fetch robot and demonstrate that our approach can handle the uncertainties that arise from real robot integration, which include speech recognition, visual sensor errors, and control errors.', '1707.02387-4-75-1': 'In the real robot experiment, the NLP parsing algorithm is successfully integrated with the realtime motion planning algorithm to set up proper parameters in the formulation for our optimization-based motion planning.', '1707.02387-4-76-0': '# Limitations, Conclusions and Future Work', '1707.02387-4-77-0': 'We present a motion planning algorithm that computes appropriate motion trajectories for a robot based on complex NLP instructions.', '1707.02387-4-77-1': 'Our formulation is based on two novel concepts: dynamic grounding graphs and dynamic constraint mapping.', '1707.02387-4-77-2': 'We highlight the performance in simulated and real-world scenes with a 7-DOF manipulator operating next to humans.', '1707.02387-4-77-3': 'We use a high dimensional optimization algorithm and the solver may get stuck in local minima, though we use multiple initializations to solve this problem.', '1707.02387-4-77-4': 'Furthermore, the accuracy of the mapping algorithm varies as a function of the training data.', '1707.02387-4-78-0': 'As future work, we would like to overcome these limitations and evaluate the approach in challenging scenarios with moving obstacles while performing complex robot tasks.', '1707.02387-4-78-1': 'More work is needed to handle the full diversity of a natural language, especially for rare words, complicated grammar styles, and hidden intentions or emotions in human speech.', '1707.02387-4-78-2': 'We plan to incorporate stronger natural language processing and machine learning methods such as those based on semantic parsing, neural sequence-to-sequence models, etc.', '1707.02387-4-78-3': 'We also plan to collect more natural language data from a variety of sources such as recipes or demonstration videos.'} | null | null | null |
1811.05810 | {'1811.05810-1-0-0': 'We investigate photon signatures of general decaying dark-matter particles in halos of primordial black holes.', '1811.05810-1-0-1': 'We derive the halo-profile density and the total decay rate for these combined dark-matter scenarios.', '1811.05810-1-0-2': 'For the case of axion-like particles of masses below [MATH], we find strong bounds on the decay constant which are several orders of magnitude stronger than the strongest existing bounds, for all halo masses above [MATH] solar masses.', '1811.05810-1-0-3': 'Using future X-ray measurements, it will be possible to push these bounds on such combined dark-matter scenarios even further.', '1811.05810-1-1-0': 'Introduction-In the standard model of cosmology, the energy density of the Universe consists of approximately [MATH]% in the form of a pressureless, nearly perfect fluid of non-relativistic objects, so-called (cold) dark matter.', '1811.05810-1-1-1': 'A large number of potential dark-matter candidates have been proposed so far.', '1811.05810-1-1-2': 'The perhaps most well-studied class is constituted by hypothetical particles which only weakly interact with the other standard-model particles.', '1811.05810-1-1-3': 'Amongst this variety, there are so-called WIMPs (cf. [CITATION]), sterile neutrinos (see Ref. [CITATION] for an early discussion on their role as dark-matter components and Refs. [CITATION]), axions [CITATION], and axion-like particles (ALPs) [CITATION].', '1811.05810-1-1-4': 'The latter, whose characteristics we will use in this work, constitutes a class of pseudo Nambu-Goldstone bosons which are coupled to photons.', '1811.05810-1-1-5': 'For photons, their mass and decay constant are related, while this is generally not the case for ALPs.', '1811.05810-1-2-0': 'Besides microscopic candidates like ALPs, dark matter might also be constituted by macroscopic objects such as primordial black holes (PBHs) [CITATION].', '1811.05810-1-2-1': 'These are black holes which have been produced in the very early Universe.', '1811.05810-1-2-2': 'The interest in PBH constituting parts of the dark matter [CITATION] has been revived recently [CITATION], in particular through the gravitational-wave discovery of black-hole binary mergers [CITATION].', '1811.05810-1-2-3': 'The possible PBH formation mechanisms are very diverse and there is a large number of scenarios, which lead to their formation.', '1811.05810-1-2-4': 'All of these have in common that they require some mechanism to generate large overdensities.', '1811.05810-1-3-0': 'Even though most of the emphasis in dark-matter research has been focused on one-component scenarios, models with more than one component have been investigated, including mixed types of both microscopic as well as macroscopic nature.', '1811.05810-1-3-1': 'On the one hand, a small fraction of PBHs could provide seeds for super-massive black holes in the galactic centres [CITATION].', '1811.05810-1-3-2': 'On the other hand, in view of the fact that it appears difficult, although not impossible, to have the entire dark matter in the form of PBHs or UCMHs (cf. Ref. [CITATION] including a summary of relevant constraints), the class of particle dark matter provides a vital supplementary and major candidate.', '1811.05810-1-4-0': 'In all of those combined scenarios, the particles will be gravitationally bound to the PBHs.', '1811.05810-1-4-1': 'This could lead to strong decay [CITATION] and/or annihilation signatures [CITATION].', '1811.05810-1-5-0': 'Halos-As mentioned above, in a combined dark-matter scenario consisting of a large fraction of particles and a small fraction of PBHs, the former will be gravitationally bound to the latter.', '1811.05810-1-5-1': 'For WIMPs, this has been studied by Eroshenko [CITATION] and the authors [CITATION].', '1811.05810-1-5-2': 'However, this formation mechanism, which happens in the radiation-dominated epoch, is not specific to any particular WIMPs model.', '1811.05810-1-5-3': 'In fact, the essential ingredients are the mass and the velocity distribution of the particles.', '1811.05810-1-5-4': 'Hence, we will generalize the results to investigate general halo formation and follow Ref. [CITATION], wherein the technical details can be found.', '1811.05810-1-5-5': 'Figure [REF] presents the halo-profile density as a function of radius [MATH] (in units of the Schwarzschild radius [MATH]) for accreted particles of mass [MATH] around a PBH of a solar mass, assuming a Maxwellian velocity profile.', '1811.05810-1-5-6': 'As expected, lighter particles lead to a more extended halo.', '1811.05810-1-5-7': "Outside of the halo's core, its profile follows [MATH].", '1811.05810-1-6-0': 'Decay-For decay signatures, distinct from annihilations, and unless the halo is extremely close to the telescope, it is practically point-like, and hence its total mass [MATH] matters rather than its concrete density profile.', '1811.05810-1-6-1': 'Given an individual decay rate, the total decay rate is readily obtained using [MATH], with [MATH] being the number of particles within the halo.', '1811.05810-1-6-2': 'For ALPs (see Sec. 111 of Ref. [CITATION] for a recent review), we may write [EQUATION] where [MATH] is the decay constant.', '1811.05810-1-6-3': 'For the QCD axion, Eq. [REF] simply becomes [EQUATION]', '1811.05810-1-6-4': 'For sterile neutrinos, a similar expression holds (see Ref. [CITATION]).', '1811.05810-1-6-5': 'Figure [REF] shows the total decay rate [MATH] as a function of the halo mass [MATH] for different values of the particle mass [MATH], where we assumed for illustrational purpose [MATH].', '1811.05810-1-6-6': 'Furthermore, it holds that [MATH].', '1811.05810-1-7-0': 'Constraints-In Ref. [CITATION], we proposed and investigated a scenario in which the dark matter is constituted by halos of sterile neutrinos around PBHs.', '1811.05810-1-7-1': 'Therein, we studied the possibility that in a certain observational time frame with a certain probability one of those compact objects propagates at a given minimum distance near the telescope.', '1811.05810-1-7-2': "Through the halo's nearness, its radiation may dominate the photon flux from other sources onto the telescope.", '1811.05810-1-7-3': "If the halo's minimum distance is small enough, its signature will be detected.", '1811.05810-1-8-0': 'It is now tempting to generalise this set-up.', '1811.05810-1-8-1': 'Applied to ALPs, we derive new limits on the maximally-allowed decay constant [MATH] by focussing on X-ray data.', '1811.05810-1-8-2': 'Now, Refs. [CITATION] provide bounds on [MATH] as well as instrumental sensitivity flux limits.', '1811.05810-1-8-3': 'Specifically, the data used come from observations of the Large Magellanic Cloud with XMM-Newton (XMM obs ID: 0127720201) [CITATION] (see Ref. [CITATION] for instrumental characteristics).', '1811.05810-1-8-4': 'We compare the observed X-ray fluxes to those originating from the decays of the ALP halos using the methodology of Ref. [CITATION], described in the previous paragraph.', '1811.05810-1-8-5': 'Utilising the relation from the total decay rate to the decay constant [see Eq. [REF]], we obtain constraints for the latter.', '1811.05810-1-9-0': 'Concretely, for a selection of halos with different halo masses ([MATH]), we compare their X-ray fluxes to the one received by the telescope assuming no halos.', '1811.05810-1-9-1': 'Furthermore, we suppose that the local dark-matter density is spatially homogeneous and takes a value of [MATH].', '1811.05810-1-9-2': 'This determines the average distance [MATH] between two halos: [EQUATION]', '1811.05810-1-9-3': 'The velocity distribution of the halos is assumed to be Maxwellian.', '1811.05810-1-9-4': 'As described above, subject to this distribution with a certain probability [MATH], a halo will move near the telescope and shed photons onto it.', '1811.05810-1-9-5': 'As we showed in Ref. [CITATION], [MATH] is approximately given by', '1811.05810-1-10-0': '[EQUATION] where [MATH] and [MATH] are the opening angles of a detector.', '1811.05810-1-10-1': 'Above, [MATH] is the distance from the detector such that a certain flux [MATH] through its effective area [MATH] is observed; it is given by [CITATION] [EQUATION]', '1811.05810-1-10-2': 'For a given observational time, it is then easy to determine the X-ray flux of the halos and compare it to that of the background.', '1811.05810-1-10-3': 'By virtue of Eq. [REF], this can thus be used to constrain [MATH].', '1811.05810-1-11-0': 'As a function of the ALP mass [MATH], our results are depicted in Fig. [REF].', '1811.05810-1-11-1': 'In this figure, it can be observed that an increase of [MATH] leads to a decrease of the constraint line of [MATH].', '1811.05810-1-11-2': 'Also, smaller values of the X-ray energy lead to stronger constraints.', '1811.05810-1-11-3': 'The physical reason for this is that, for a fixed [MATH], the number [MATH] of ALPs within the halo is increasing with decreasing [MATH], whereas the background X-ray flux is given and fixed, the single halo moving in the vicinity of the telescope contains more decaying particles the smaller their mass is.', '1811.05810-1-11-4': 'This extra factor of [MATH] is responsible for the increased detection prospects towards smaller mass.', '1811.05810-1-12-0': 'In Fig. [REF], we observe that for [MATH] below [MATH], we obtain bounds on [MATH] which are several orders of magnitude stronger than the strongest existing bounds, for all [MATH] above [MATH] solar masses.', '1811.05810-1-12-1': 'In particular, for [MATH], we find that the following values of [MATH] will, at least, be excluded [EQUATION]', '1811.05810-1-12-2': 'The obtained bounds in this paper can be regarded a conservative, as we have not included ALP-to-photon conversion due to magnetic fields.', '1811.05810-1-12-3': 'The latter can occur in two distinct situations: internally, for instance for charged and rotating black holes, and externally, for instance from galactic magnetic fields.', '1811.05810-1-12-4': 'Either case only increases the photo emission from the halo objects, and hence strengthens the bounds.', '1811.05810-1-12-5': 'It would be interesting to investigate these instances in the future, but it is beyond the scope of the present paper.', '1811.05810-1-13-0': 'Conclusions-We have investigated decay signatures from a two-component dark-matter scenario in which most of the dark matter is constituted by axion-like particles (ALPs) complemented by primordial black holes (PBHs).', '1811.05810-1-13-1': 'We have studied how the former accrete around the latter and calculated the halo profile (shown in Fig. [REF]).', '1811.05810-1-13-2': 'Then, we have studied the decay signatures (visualised in Fig. [REF]) from which we have derived bounds on the decay constant (depicted in Fig. [REF]).', '1811.05810-1-13-3': 'We have found that this combined scenario leads to detection prospects which, for small ALP masses less than or equal to [MATH] and for halos heavier than [MATH], are far better than the pure ALP scenario.'} | {'1811.05810-2-0-0': 'We investigate photon signatures of general decaying dark-matter particles in halos of primordial black holes.', '1811.05810-2-0-1': 'We derive the halo-profile density and the total decay rate for these combined dark-matter scenarios.', '1811.05810-2-0-2': 'For the case of axion-like particles of masses below [MATH], we find strong bounds on the decay constant which are several orders of magnitude stronger than the strongest existing bounds, for all halo masses above [MATH] solar masses.', '1811.05810-2-0-3': 'Using future X-ray measurements, it will be possible to push these bounds on such combined dark-matter scenarios even further.', '1811.05810-2-1-0': 'Introduction-In the standard model of cosmology, the energy density of the Universe consists of approximately [MATH]% in the form of a pressureless, nearly perfect fluid of non-relativistic objects, so-called (cold) dark matter.', '1811.05810-2-1-1': 'A large number of potential dark-matter candidates have been proposed so far.', '1811.05810-2-1-2': 'The perhaps most well-studied class is constituted by hypothetical particles which only weakly interact with the other standard-model particles.', '1811.05810-2-1-3': 'Amongst this variety, there are so-called WIMPs (cf. Ref. [CITATION]), sterile neutrinos (see Ref. [CITATION] for an early discussion on their role as dark-matter components and Refs. [CITATION]), axions [CITATION], and axion-like particles (ALPs) [CITATION].', '1811.05810-2-1-4': 'The latter, whose characteristics we will use in this work, constitutes a class of pseudo Nambu-Goldstone bosons which are coupled to photons.', '1811.05810-2-1-5': 'For axions, their mass and decay constant are related, while this is generally not the case for ALPs.', '1811.05810-2-2-0': 'Besides microscopic candidates like ALPs, dark matter might also be constituted by macroscopic objects such as primordial black holes (PBHs) [CITATION].', '1811.05810-2-2-1': 'These are black holes which have been produced in the very early Universe.', '1811.05810-2-2-2': 'The interest in PBH constituting parts of the dark matter [CITATION] has been revived recently [CITATION], in particular through the gravitational-wave discovery of black-hole binary mergers [CITATION].', '1811.05810-2-2-3': 'The possible PBH formation mechanisms are very diverse and there is a large number of scenarios, which lead to their formation.', '1811.05810-2-2-4': 'All of these have in common that they require some mechanism to generate large overdensities.', '1811.05810-2-3-0': 'Even though most of the emphasis in dark-matter research has been focused on one-component scenarios, models with more than one component have been investigated, including mixed types of both microscopic as well as macroscopic nature.', '1811.05810-2-3-1': 'On the one hand, a small fraction of PBHs could provide seeds for super-massive black holes in the galactic centres [CITATION].', '1811.05810-2-3-2': 'On the other hand, in view of the fact that it appears difficult, although not impossible, to have the entire dark matter in the form of PBHs or UCMHs (cf. Ref. [CITATION] including a summary of relevant constraints), the class of particle dark matter provides a vital supplementary and major candidate.', '1811.05810-2-4-0': 'In all of those combined scenarios, the particles will be gravitationally bound to the PBHs.', '1811.05810-2-4-1': 'This could lead to strong decay [CITATION] and/or annihilation signatures [CITATION].', '1811.05810-2-5-0': 'Halos-As mentioned above, in a combined dark-matter scenario consisting of a large fraction of particles and a small fraction of PBHs, the former will be gravitationally bound to the latter.', '1811.05810-2-5-1': 'For WIMPs, this has been studied by Eroshenko [CITATION] and the authors [CITATION].', '1811.05810-2-5-2': 'However, this formation mechanism, which happens in the radiation-dominated epoch, is not specific to any particular WIMPs model.', '1811.05810-2-5-3': 'In fact, the essential ingredients are the mass and the velocity distribution of the particles.', '1811.05810-2-5-4': 'Hence, we will generalize the results to investigate general halo formation and follow Ref. [CITATION], wherein the technical details can be found.', '1811.05810-2-5-5': 'Figure [REF] presents the halo-profile density as a function of radius [MATH] (in units of the Schwarzschild radius [MATH]) for accreted particles of mass [MATH] around a PBH of a solar mass [MATH], assuming a Maxwellian velocity profile.', '1811.05810-2-5-6': 'Note that the calculation leading to Fig. [REF] only relies on gravitational dynamics, implying that the halo-profile density solely depends on the particle mass [MATH].', '1811.05810-2-5-7': 'In the case the particle dark matter is constituted by a number of different species (with different masses), the halo-profile density changes in a non-trivial way.', '1811.05810-2-5-8': 'In the following, we will assume that each halo is entirely constituted by a single particle type.', '1811.05810-2-6-0': 'As can be observed from Fig. [REF], lighter particles lead to a more extended halo.', '1811.05810-2-6-1': "Outside of the halo's core, its profile follows [MATH], which can be estimated from Fig. [REF].", '1811.05810-2-6-2': 'The behaviour of the halo-profile density as a function of [MATH] derives from the fact that lighter particles become non-relativistic at a later time than heavier ones, and hence the background density [MATH] will be lower.', '1811.05810-2-6-3': 'The radius of gravitational influence [MATH] of the PBHs scales as [MATH] (see Ref. [CITATION]).', '1811.05810-2-6-4': 'This leads to larger structures for smaller particle masses.', '1811.05810-2-6-5': 'At matter-radiation equality, the mass [MATH] of the halo is comparable to that of the PBH, see the discussion in Ref. [CITATION].', '1811.05810-2-6-6': 'After matter-radiation equality, the growth of the halo is roughly linear in redshift, leading to approximately a factor of 1000 difference in mass.', '1811.05810-2-6-7': 'Thus, essentially all the halo mass is constituted by the sum of the particle masses, meaning that the number [MATH] of particles within the halo is roughly given by [MATH].', '1811.05810-2-6-8': 'We will mainly be interested in the case in which the dark matter is essentially constituted by ALPs, i.e. when [MATH], where [MATH], [MATH], and [MATH] are the energy densities of ALPs, PBHs, and dark matter, respectively.', '1811.05810-2-6-9': 'Furthermore, we will assume that all of the ALPs are bound to the PBHs in the halos, i.e. [MATH].', '1811.05810-2-7-0': 'Decay-For decay signatures, distinct from annihilations, and unless the halo is extremely close to the telescope, it is practically point-like, and hence its total mass matters rather than its concrete density profile.', '1811.05810-2-7-1': 'Given an individual decay rate, the total decay rate is readily obtained using [MATH].', '1811.05810-2-7-2': 'For ALPs (see Sec. 111 of Ref. [CITATION] for a recent review), we may write [EQUATION] where [MATH] is the decay constant.', '1811.05810-2-7-3': 'For the QCD axion, Eq. [REF] simply becomes [EQUATION]', '1811.05810-2-7-4': 'For sterile neutrinos, a similar expression holds (see Ref. [CITATION]).', '1811.05810-2-7-5': 'Figure [REF] shows the total decay rate [MATH] as a function of the halo mass [MATH] for different values of the particle mass [MATH], where we assumed for illustrational purpose [MATH].', '1811.05810-2-7-6': 'Furthermore, we have that [MATH], which holds if [MATH] is fixed.', '1811.05810-2-7-7': 'However, if [MATH] is fixed, we instead have [MATH].', '1811.05810-2-7-8': 'Explicitly for the total decay rate, we obtain [EQUATION] which can easily be computed for, e.g. [EQUATION] that indeed agrees with the results of Fig. [REF].', '1811.05810-2-8-0': 'Constraints-In Ref. [CITATION], we proposed and investigated a scenario in which the dark matter is constituted by halos of sterile neutrinos around PBHs.', '1811.05810-2-8-1': 'Therein, we studied the possibility that in a certain observational time frame with a certain probability one of those compact objects propagates at a given minimum distance near the telescope.', '1811.05810-2-8-2': "Through the halo's nearness, its radiation may dominate the photon flux from other sources onto the telescope.", '1811.05810-2-8-3': "If the halo's minimum distance is small enough, its signature will be detected.", '1811.05810-2-9-0': 'It is now tempting to generalize this set-up.', '1811.05810-2-9-1': 'Applied to ALPs, we derive new limits on the maximally-allowed decay constant [MATH].', '1811.05810-2-9-2': 'Reference [CITATION] provides a variety of bounds for an extended mass range.', '1811.05810-2-9-3': 'We compare the observed photon fluxes to those originating from the decays of the ALP halos using the methodology of Ref. [CITATION], described in the previous paragraph.', '1811.05810-2-9-4': 'Utilizing the relation from the total decay rate to the decay constant [see Eq. [REF]], we obtain constraints for the latter.', '1811.05810-2-10-0': 'Concretely, for a selection of halos with different halo masses ([MATH]), we compare their photon fluxes to the one received by the telescope assuming no halos.', '1811.05810-2-10-1': 'Furthermore, we suppose that the local dark-matter density is spatially homogeneous and takes a value of [MATH].', '1811.05810-2-10-2': 'This determines the average distance [MATH] between two halos: [EQUATION]', '1811.05810-2-10-3': 'The velocity distribution of the halos is assumed to be Maxwellian.', '1811.05810-2-10-4': 'As described above, subject to this distribution with a certain probability [MATH], a halo will move near the telescope and shed photons onto it.', '1811.05810-2-10-5': 'As we showed in Ref. [CITATION], [MATH] is approximately given by [EQUATION] where [MATH] and [MATH] are the opening angles of a detector.', '1811.05810-2-10-6': 'Above, [MATH] is the distance from the detector such that a certain flux [MATH] through its effective area [MATH] is observed; it is given by [CITATION] [EQUATION]', '1811.05810-2-10-7': 'For a given observational time, it is then easy to determine the photon flux of the halos and compare it to that of the background.', '1811.05810-2-10-8': 'By virtue of Eq. [REF], this can thus be used to constrain [MATH].', '1811.05810-2-11-0': 'As a function of the ALP mass [MATH], our results are depicted in Fig. [REF].', '1811.05810-2-11-1': 'In this figure, it can be observed that an increase of [MATH] leads to a decrease of the constraint line of [MATH].', '1811.05810-2-11-2': 'Also, smaller values of the energy lead to stronger constraints.', '1811.05810-2-11-3': 'The physical reason for this is that, for a fixed [MATH], the number [MATH] of ALPs within the halo is increasing with decreasing [MATH], whereas the background ray flux is given and fixed, the single halo moving in the vicinity of the telescope contains more decaying particles the smaller their mass is.', '1811.05810-2-11-4': 'This extra factor of [MATH] is responsible for the increased detection prospects towards smaller mass.', '1811.05810-2-12-0': 'In Fig. [REF], we observe that for [MATH] below [MATH], we obtain bounds on [MATH] which are several orders of magnitude stronger than the strongest existing bounds, for all [MATH] above [MATH] solar masses if one assumes a halo dark-matter fraction of one.', '1811.05810-2-12-1': 'In particular, for [MATH], we find that the following values of [MATH] will, at least, be excluded [EQUATION]', '1811.05810-2-12-2': 'These bounds are slightly relaxed if a lower halo dark-matter fraction is accounted for.', '1811.05810-2-12-3': 'For the three considered masses, the constraints come from the EROS/OGLE microlensing survey [CITATION] (for [MATH], [MATH]) and X-ray emission from accretion gas around PBHs [CITATION] (for [MATH]).', '1811.05810-2-12-4': 'Taken at face value, these yield the maximally allowed halo dark-matter fractions: [MATH], [MATH], and [MATH].', '1811.05810-2-12-5': 'Approximately, the constraint curves in Fig. [REF] scale as [MATH].', '1811.05810-2-13-0': 'The obtained bounds in this paper can be regarded a conservative, as we have not included ALP-to-photon conversion due to magnetic fields.', '1811.05810-2-13-1': 'The latter can occur in two distinct situations: internally, for instance for charged and rotating black holes, and externally, for instance from galactic magnetic fields.', '1811.05810-2-13-2': 'Either case only increases the photo emission from the halo objects, and hence strengthens the bounds.', '1811.05810-2-13-3': 'It would be interesting to investigate these instances in the future, but it is beyond the scope of the present paper.', '1811.05810-2-14-0': 'Conclusions-We have investigated decay signatures from a two-component dark-matter scenario in which most of the dark matter is constituted by axion-like particles (ALPs) complemented by primordial black holes (PBHs).', '1811.05810-2-14-1': 'We have studied how the former accrete around the latter and calculated the halo profile (shown in Fig. [REF]).', '1811.05810-2-14-2': 'Then, we have studied the decay signatures (visualised in Fig. [REF]) from which we have derived bounds on the decay constant (depicted in Fig. [REF]).', '1811.05810-2-14-3': 'We have found that this combined scenario leads to detection prospects which, for small ALP masses less than or equal to [MATH] and for halos heavier than [MATH], are far better than the pure ALP scenario.'} | [['1811.05810-1-2-0', '1811.05810-2-2-0'], ['1811.05810-1-2-1', '1811.05810-2-2-1'], ['1811.05810-1-2-2', '1811.05810-2-2-2'], ['1811.05810-1-2-3', '1811.05810-2-2-3'], ['1811.05810-1-2-4', '1811.05810-2-2-4'], ['1811.05810-1-1-0', '1811.05810-2-1-0'], ['1811.05810-1-1-1', '1811.05810-2-1-1'], ['1811.05810-1-1-2', '1811.05810-2-1-2'], ['1811.05810-1-1-4', '1811.05810-2-1-4'], ['1811.05810-1-0-0', '1811.05810-2-0-0'], ['1811.05810-1-0-1', '1811.05810-2-0-1'], ['1811.05810-1-0-2', '1811.05810-2-0-2'], ['1811.05810-1-0-3', '1811.05810-2-0-3'], ['1811.05810-1-3-0', '1811.05810-2-3-0'], ['1811.05810-1-3-1', '1811.05810-2-3-1'], ['1811.05810-1-3-2', '1811.05810-2-3-2'], ['1811.05810-1-6-2', '1811.05810-2-7-2'], ['1811.05810-1-6-3', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1811.05810 | null | null | null | null | null |
1904.05683 | {'1904.05683-1-0-0': "With a Bayesian Gaussian regression approach, a systematic method for analyzing a storage ring's beam position monitor (BPM) system requirements has been developed.", '1904.05683-1-0-1': 'The ultimate performance of a ring-based accelerator, based on brightness or luminosity, is determined not only by global parameters, but also by local beam properties at some particular points of interest (POI).', '1904.05683-1-0-2': 'BPMs used for monitoring the beam properties, however, can not be located at these points.', '1904.05683-1-0-3': 'Therefore, the underlying and fundamental purpose of a BPM system is to predict whether the beam properties at POIs reach their desired values.', '1904.05683-1-0-4': 'The prediction process is a regression problem with BPM readings as the training data, but containing random noise.', '1904.05683-1-0-5': 'A Bayesian Gaussian regression approach can determine the probability distribution of the predictive errors, which can be used to conversely analyze the BPM system requirements.', '1904.05683-1-0-6': 'This approach is demonstrated by using turn-by-turn data to reconstruct a linear optics model, and predict the brightness degradation for a ring-based light source.', '1904.05683-1-0-7': 'The quality of BPMs was found to be more important than their quantity in mitigating predictive errors.', '1904.05683-1-1-0': '# introduction', '1904.05683-1-2-0': 'The ultimate performance of a ring-based accelerator is determined not only by certain critical global parameters, such as beam emittance, but also by local properties of the beam at particular points of interest (POI).', '1904.05683-1-2-1': 'The capability of diagnosing and controlling local beam parameters at POIs, such as beam size and divergence, is crucial for a machine to achieve its design performance.', '1904.05683-1-2-2': 'Examples of POIs in a dedicated synchrotron light source ring include the undulator locations, from where high brightness X-rays are generated.', '1904.05683-1-2-3': 'In a collider, POIs are reserved for detectors in which the beam-beam luminosity is observed.', '1904.05683-1-2-4': 'However, beam diagnostics elements, such as beam position monitors (BPM) are generally placed outside of the POIs as the POIs are already occupied.', '1904.05683-1-2-5': 'An intuitive, but quantitatively unproven belief, is that the desired beam properties at the POIs can be achieved once the beam properties are well-controlled at the location of the BPMs.', '1904.05683-1-3-0': 'Using observation data at BPMs to indirectly predict the beam properties at POIs is a regression problem and can be treated as a supervised learning process: BPM readings at given locations are used as a training dataset.', '1904.05683-1-3-1': 'Then a ring optics model with a set of quadrupole excitations as its arguments is selected as the hypothesis.', '1904.05683-1-3-2': 'From the dataset, an optics model needs to be generalized first.', '1904.05683-1-3-3': 'Based on the model, the unknown beam properties at POIs can be predicted.', '1904.05683-1-3-4': "However, there exists some systematic error and random uncertainty in the BPMs' readings, and the quantity of BPMs (the dimension of the training dataset) is limited.", '1904.05683-1-3-5': 'Therefore, the parameters in the reconstructed optics model have inherent uncertainties, as do the final beam property predictions at the POIs.', '1904.05683-1-3-6': 'The precision and accuracy of the predictions at the POIs depend on the quantity of BPMs, their physical distribution pattern around the ring, and their calibration, resolution, etc.', '1904.05683-1-3-7': 'When a BPM system is designed for a storage ring, however, it is more important to consider the inverse problem: i.e.', '1904.05683-1-3-8': 'How are the BPM system technical requirements determined in order to observe whether the ring achieves its desired performance?', '1904.05683-1-3-9': 'In this paper, we developed an approach to address this question with Bayesian Gaussian regression.', '1904.05683-1-4-0': 'In statistics, a Bayesian Gaussian regression [CITATION] is a Bayesian approach to multivariate regression, i.e. regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable.', '1904.05683-1-4-1': 'Every finite collection of the data has a normal distribution.', '1904.05683-1-4-2': 'The distribution of generalized arguments of the hypothesis is the joint distribution of all those random variables.', '1904.05683-1-4-3': 'Based on the hypothesis, a prediction can be made for any unknown dataset within a continuous domain.', '1904.05683-1-4-4': "In our case, multiple BPMs' readings are normally distributed around their real values.", '1904.05683-1-4-5': "The standard deviations of the Gaussian distributions are BPM's resolutions.", '1904.05683-1-4-6': "A vector composed of quadrupoles' mis-settings is the argument to be generalized.", '1904.05683-1-4-7': 'The prediction at the POIs is the function of this vector.', '1904.05683-1-4-8': 'The continuous domain is the longitudinal coordinate [MATH] along a storage ring.', '1904.05683-1-5-0': 'To further explain this approach, the remaining sections are outlined as follows: Sect. [REF] introduces the relation between machine performance and beam diagnostics system capabilities.', '1904.05683-1-5-1': 'Sect. [REF] explains the procedure to apply the Bayesian Gaussian regression in the ring optics model reconstruction, and the prediction of local optics properties at POIs.', '1904.05683-1-5-2': 'In Sect. [REF], the National Synchrotron Light Source II (NSLS-II) storage ring and its BPM system are used to illustrate the application of this approach.', '1904.05683-1-5-3': 'Some discussion and a brief summary is given in Sect. [REF].', '1904.05683-1-6-0': '# machine performance and beam diagnostics capability', '1904.05683-1-7-0': 'As mentioned previously, ultimate performance of a ring-based accelerator relies heavily on local beam properties at particular POIs.', '1904.05683-1-7-1': 'Consider a dedicated light source ring.', '1904.05683-1-7-2': 'Its ultimate performance is measured by the brightness of the X-rays generated by undulators.', '1904.05683-1-7-3': 'The brightness of undulator emission is determined by the transverse size of both the electron and photon beam and their angular divergence at their source points [CITATION].', '1904.05683-1-7-4': "Therefore, the undulator brightness performance [MATH] depends on the ring's global emittance and the local transverse optics parameters, [EQUATION]", '1904.05683-1-7-5': 'Here [MATH] are the electron beam emittances, which represent the equilibrium between the quantum excitation and the radiation damping around the whole ring.', '1904.05683-1-7-6': '[MATH] are the Twiss parameters [CITATION], [MATH] are the dispersion and its derivative at the undulators\' locations, [MATH] is the electron beam energy spread [MATH] and [MATH] are the X-ray beam diffraction "waist size" and its natural angular divergence, respectively.', '1904.05683-1-7-7': 'The X-ray wavelength [MATH], is determined based on the requirements of the beam-line experiments, and [MATH] is the undulator periodic length.', '1904.05683-1-7-8': 'The emittance was found to be nearly constant with small [MATH]-beat (see Sect. [REF]).', '1904.05683-1-7-9': "Therefore, monitoring and controlling the local POI's Twiss parameters are crucial.", '1904.05683-1-8-0': 'The final goal of beam diagnostics is to provide sufficient, accurate observations to reconstruct an online accelerator model.', '1904.05683-1-8-1': 'Modern BPM electronics can provide the beam turn-by-turn (TbT) data, which is widely used for the beam optics characterization and the model reconstruction.', '1904.05683-1-8-2': 'Based on the model, we can predict the beam properties not only at the locations of monitors themselves, but more importantly at the POIs.', '1904.05683-1-8-3': 'The capability of indirect prediction of the Twiss parameters at POIs eventually defines the BPM system requirements on TbT data acquisition.', '1904.05683-1-8-4': 'Based on Eq. [REF], how precisely one can predict the bias and the uncertainty of Twiss parameters [MATH] and [MATH] at locations of undulators is the key problem in designing a BPM system.', '1904.05683-1-8-5': 'Therefore, to specify the technical requirements of a BPM system, the following questions need to be addressed: in order to make an accurate and precise prediction of beam properties at POIs, how many BPMs are needed?', '1904.05683-1-8-6': 'How should the BPMs be allocated throughout the accelerator ring, and how precise should the BPM TbT reading be?', '1904.05683-1-9-0': 'In the following section a method of reconstructing the linear optics model, and determining the brightness performance for a ring-based light source will be discussed.', '1904.05683-1-9-1': 'For a collider ring, its luminosity is determined only by the beam sizes at the interaction points [CITATION].', '1904.05683-1-9-2': 'Gaussian regression analysis can therefore be applied to predict its [MATH] and luminosity as well.', '1904.05683-1-10-0': '# Gaussian regression for model reconstruction and prediction', '1904.05683-1-11-0': 'When circulating beam in a storage ring is disturbed, a BPM system can provide its TbT data at multiple longitudinal locations.', '1904.05683-1-11-1': 'TbT data of the BPMs can be represented as an optics model plus some random reading errors, [EQUATION] here [MATH] is the index of turns, [MATH] is a variable dependent on turn number, [MATH] is the envelope function of Twiss parameters at [MATH] location, [MATH] is the betatron tune, [MATH] is the betatron phase, and [MATH] is the BPM reading noise, which generally has a normal distribution.', '1904.05683-1-11-2': 'Based on the accelerator optics model defined in Eq. [REF], we can extract a set of optics Twiss parameters at all BPM locations [CITATION].', '1904.05683-1-11-3': 'Recently, Ref. [CITATION] proposed using a Bayesian approach to infer the mean and uncertainty of Twiss parameters at BPMs simultaneously.', '1904.05683-1-11-4': 'The mean values of [MATH] represent the most likely optics pattern.', '1904.05683-1-11-5': 'The random BPM reading error and the simplification of the optics model can result in some uncertainties, [MATH], in the inference process, [EQUATION] here [MATH] is a vector composed of all normalized quadrupole focusing strengths, and [MATH] is the inference uncertainty.', '1904.05683-1-11-6': 'Unless otherwise stated, bold symbols, such as "[MATH]", are used to denote vectors and matrices throughout this paper.', '1904.05683-1-11-7': 'In accelerator physics, the deviation from the design model [MATH] is often referred to as the [MATH]-beat.', '1904.05683-1-11-8': "From the point of view of model reconstruction, the [MATH]-beat is due to quadrupole excitation errors and can be determined by [EQUATION] where [MATH] represents the quadrupoles' nominal setting and [MATH] is the nominal envelope function along [MATH].", '1904.05683-1-11-9': '[MATH] is the response matrix composed of elements [MATH] observed by the BPMs.', '1904.05683-1-11-10': 'The dependency of [MATH] on [MATH] is not linear in a complete optics model.', '1904.05683-1-11-11': 'However, when quadrupole errors are small enough, the dependence can be approximated as a linear relation as illustrated in Fig. [REF].', '1904.05683-1-11-12': 'The approximation holds for most operational storage rings, and other diffraction limited light sources under design or construction.', '1904.05683-1-11-13': 'A linear approximation allows us to use the linear regression approach for this process.', '1904.05683-1-11-14': 'Eq. [REF] or [REF] is a hypothesis with the unknown arguments [MATH] or [MATH], which need to be generalized from BPM measurement data.', '1904.05683-1-12-0': 'Given a set of inferred optics parameters [MATH]s at multiple locations [MATH] from BPM TbT data, the posterior probability of the quadrupole error distribution can be given according to Bayes theorem, [EQUATION]', '1904.05683-1-12-1': 'Here [MATH] is referred to as the likelihood function, [EQUATION]', '1904.05683-1-12-2': 'Once the mean value of the optics measurement is extracted from the TbT data, a prior quadrupole excitation error distribution [MATH] can be determined by comparing them against the design optics model [CITATION], [EQUATION] in which the variance [MATH] of the prior distribution [MATH] is linearly proportional to the mean value of the measured [MATH]-beat, [EQUATION]', '1904.05683-1-12-3': 'The coefficient [MATH] can be computed based on the optics model either analytically or numerically before carrying out any measurement.', '1904.05683-1-12-4': 'In the NSLS-II ring, [MATH], i.e. a [MATH]-beat corresponds to a distribution of quadrupole errors with the standard deviation [MATH].', '1904.05683-1-13-0': 'Both the likelihood function and the prior distribution are generally normally distributed.', '1904.05683-1-13-1': 'Therefore, the posterior distribution is a normal distribution by summing over the arguments of the exponentials in Eq. [REF] and [REF], [EQUATION]', '1904.05683-1-13-2': 'Here [EQUATION]', '1904.05683-1-13-3': "The identity matrix [MATH] is used in Eq. [REF] because all BPMs' resolutions are assumed to have the same values [MATH].", '1904.05683-1-13-4': "In reality, however, [MATH] needs to be replaced with a diagonal matrix with different elements if the BPMs' resolutions are different.", '1904.05683-1-13-5': "The quadrupoles' error distribution matrix [MATH] needs to be processed in the same way if necessary.", '1904.05683-1-13-6': 'The mean value of the posterior, corresponding to the most likely quadrupole error distribution, can be used to implement the linear optics correction as explained in Ref. [CITATION], [EQUATION] where [MATH].', '1904.05683-1-13-7': 'Adding an extra term [MATH] to prevent overfitting is known as the regularization technique.', '1904.05683-1-13-8': 'The posterior variance represents the uncertainty of quadrupole errors.', '1904.05683-1-13-9': '[EQUATION]', '1904.05683-1-13-10': 'Given [MATH]-beats observed at [MATH], the posterior generalizes an optics model, in which the quadrupoles errors are normally distributed, [EQUATION] with the mean value and the variance given by Eq. [REF] and [REF] respectively.', '1904.05683-1-14-0': 'Thus far, the optics are measured at the locations of the BPMs, and the corresponding quadrupole error distributions are generalized based on the measurements.', '1904.05683-1-14-1': 'To confirm the machine brightness performance, we need to predict the beam properties at POIs.', '1904.05683-1-14-2': 'To do so, the output of all possible posterior quadrupole error distributions must be averaged, [EQUATION]', '1904.05683-1-14-3': "Here [MATH] is the predicted result at POIs' locations [MATH] given the measured [MATH] at [MATH].", '1904.05683-1-14-4': 'The mean values and the variances of the predicted distributions at POIs are [EQUATION] [MATH] is the Jacobian matrix of the optics response to quadrupole errors observed at POIs.', '1904.05683-1-14-5': 'The difference between the mean value [MATH] and the real [MATH] at a POI is referred to as the predicted bias.', '1904.05683-1-14-6': "By substituting the bias and the uncertainty back into Eq. [REF], we can estimate how accurate the brightness could be measured for given BPMs' resolutions.", '1904.05683-1-14-7': 'Based on the desired brightness resolution, we can determine the needed quantity and resolution of BPMs.', '1904.05683-1-15-0': '# Application to NSLS-II ring', '1904.05683-1-16-0': 'In this section, we use the NSLS-II ring and its BPM system TbT data acquisition functionality to demonstrate the application of this approach.', '1904.05683-1-16-1': 'NSLS-II is a [MATH] generation dedicated light source.', '1904.05683-1-16-2': 'All undulator source points (POIs) are located at non-dispersive straights.', '1904.05683-1-16-3': 'A typical photon energy from undulators is around 10 [MATH], with corresponding wavelengths around 0.124 [MATH].', '1904.05683-1-16-4': "The undulators' period length is 20 [MATH].", '1904.05683-1-16-5': 'The horizontal beam emittance is 0.9 [MATH] including the contribution from 3 damping wigglers.', '1904.05683-1-16-6': 'The emittance coupling ratio can be controlled to less than 1%.', '1904.05683-1-16-7': 'At its 15 short straights centers, the Twiss parameters are designed to be as low as [MATH], and [MATH] to generate the desired high brightness x-ray beam from the undulators.', '1904.05683-1-17-0': 'The horizontal emittance growth with an optics distortion was studied by carrying out a lattice simulation.', '1904.05683-1-17-1': 'With [MATH]-beat at a few percent, the corresponding [MATH] and [MATH]-distortions were generated by adding some normally distributed quadrupole errors based on Eq. [REF] and [REF].', '1904.05683-1-17-2': 'The horizontal emittance was found to grow slightly with the average [MATH]-beats as illustrated in Fig. [REF].', '1904.05683-1-17-3': 'When there is about a 1% horizontal [MATH]-beat ([MATH]), the emittance increases by only about 0.1%, which is negligible.', '1904.05683-1-17-4': 'Therefore, in the following calculation, the emittance was represented as a constant.', '1904.05683-1-18-0': 'Degradation of an undulator brightness is determined by its local optics distortion which can be evaluated with Eq. [REF].', '1904.05683-1-18-1': 'Multi-pairs of simulated [MATH] were incorporated into the previously specified undulator parameters to observe the dependence of the X-ray brightness on the [MATH]-beat (see Fig. [REF]).', '1904.05683-1-18-2': 'A change of approximately 1 of the [MATH] in the transverse plane can degrade the brightness by about 1.', '1904.05683-1-18-3': 'In other words, in order to resolve a 1 brightness degradation, the predictive errors of the ring optics (including the bias and uncertainty) at the locations of undulators should be less than 1.', '1904.05683-1-18-4': 'Because multiple undulators are installed around the ring, the predicted performance needs to be evaluated at all POIs simultaneously.', '1904.05683-1-19-0': 'There exist two types of errors in Eq. [REF] which can introduce uncertainties in characterizing the optics parameters at BPMs [CITATION].', '1904.05683-1-19-1': 'First, due to radiation damping, chromatic decoherence and nonlinearity, a disturbed beam trajectory is not a pure linear undamping betatron oscillation.', '1904.05683-1-19-2': 'A reduced model (for example, assuming [MATH] is a constant), will introduce systematic errors.', '1904.05683-1-19-3': 'The second error source is the BPM TbT resolution limit, which results in random noise.', '1904.05683-1-19-4': 'At NSLS-II, the BPM TbT resolution at low beam current ([MATH]) is inferred as [MATH].', '1904.05683-1-19-5': 'When a [MATH] order polynomial function is used to represent the turn-dependent amplitude [MATH], the inferred [MATH] function resolution at BPMs can be reached as low as 0.5%.', '1904.05683-1-20-0': 'First we studied the dependence of predictive errors on the quantity of BPMs.', '1904.05683-1-20-1': 'A comprehensive simulation was set up to compare the Gaussian regression predictive errors with the real errors.', '1904.05683-1-20-2': 'A linear optics simulation code was used to simulate the distorted optics due to a set of quadrupole errors.', '1904.05683-1-20-3': 'The [MATH]-beats observed at the BPMs were marked as the "real" values.', '1904.05683-1-20-4': 'On top the real values, 0.5% random errors were added to simulate one-time measurement uncertainty seen by the BPMs.', '1904.05683-1-20-5': 'A posterior distribution Eq. [REF] and [REF] of the quadrupole errors was obtained by reconstructing the optics model with the likelihood function Eq. [REF], and the prior distribution [REF] and [REF].', '1904.05683-1-20-6': 'The predicted optics parameters with their uncertainties were then calculated based on another likelihood function between quadrupoles and the locations of undulators with Eq. [REF].', '1904.05683-1-21-0': 'The results of comparison are illustrated in Fig. [REF].', '1904.05683-1-21-1': 'As with any regression problem, the training data distribution (i.e. the BPM locations) should be as uniform as possible within the continuous [MATH] domain.', '1904.05683-1-21-2': 'There are 30 cells in the NSLS-II ring, and each cell has 6 BPMs.', '1904.05683-1-21-3': 'Equal numbers of BPMs were selected from each cell to make the training data uniformly distributed.', '1904.05683-1-21-4': 'The goal was to predict all straights optics simultaneously.', '1904.05683-1-21-5': 'The predicted performance was therefore evaluated by averaging at multiple straight centers.', '1904.05683-1-21-6': 'Initially, one BPM was selected per cell.', '1904.05683-1-21-7': 'The number of selected BPMs was then gradually increased to observe the evolution of predictive errors.', '1904.05683-1-21-8': 'It was found that utilizing more BPMs improved the predicted performance, as expected.', '1904.05683-1-21-9': 'Both the bias and uncertainty were reduced with the quantity of BPMs.', '1904.05683-1-21-10': 'However, the improvement became less and less apparent once more than 4 BPMs per cell were used.', '1904.05683-1-22-0': 'Next, we studied the effect of [MATH] measurement resolution on the predictive errors.', '1904.05683-1-22-1': 'A similar analysis was carried out but with different [MATH]-resolution as illustrated in Fig. [REF].', '1904.05683-1-22-2': 'By observing Fig. [REF], several conclusions can be drawn: (1) The degradation of the [MATH] resolution reduced the accuracy of the generalized optics model.', '1904.05683-1-22-3': 'However, this can be improved by applying a more complicated optics model [CITATION].', '1904.05683-1-22-4': 'Thus, the BPM TbT resolution is the final limit on the resolution of [MATH] parameters.', '1904.05683-1-22-5': 'In order to accurately and precisely predict the beam properties at POIs, improving the resolution of BPMs is crucial.', '1904.05683-1-22-6': '(2) After a certain point, the predicted performance is not improved significantly with the quantity of BPMs as seen in both Fig [REF] and [REF].', '1904.05683-1-22-7': 'The advantage of reduction of predictive errors will gradually level out once enough BPMs are used.', '1904.05683-1-22-8': 'Meaning, the improvement in error reduction will eventually become negligible compared to the cost of adding more BPMs.', '1904.05683-1-22-9': 'The higher the resolution each individual BPM has, the less number of BPMs are needed.', '1904.05683-1-22-10': 'There should be a compromise between the required quality and quantity of BPMs to achieve an expected predictive accuracy.', '1904.05683-1-22-11': '(3) The quality (resolution) is much more important than the quantity of BPMs from the point of view of optics characterization.', '1904.05683-1-22-12': 'For example, at NSLS-II, in order to resolve 1% brightness degradation, at least 120 BPMs with a [MATH] resolution better than 1% are needed, or 90 BPMs with a 0.75% resolution, etc.', '1904.05683-1-22-13': 'Having more BPMs than is needed creates no obvious, significant improvement.', '1904.05683-1-22-14': 'Having 60 high precision (0.5% [MATH]-resolution) BPMs yields a better performance than having 180 low precision (1%) BPMs in this example.', '1904.05683-1-23-0': '# Summary', '1904.05683-1-24-0': "A systematic approach has been proposed to analyze a BPM system's technical requirements.", '1904.05683-1-24-1': "The approach is based on the resolution requirements for monitoring a machine's ultimate performance.", '1904.05683-1-24-2': "The Bayesian Gaussian regression is useful in statistical data modelling, such as reconstructing a ring's optics model from beam TbT data.", '1904.05683-1-24-3': 'The optics properties of the ring are contained in a collection of data having a normal distribution.', '1904.05683-1-24-4': 'It is worth noting that our approach is simplified as a linear regression by assuming a known linear dependence of optics distortion on quadrupole errors.', '1904.05683-1-24-5': "If a ring's optics are significantly different from the design model, this assumption is not valid, and the analysis may be not accurate either.", '1904.05683-1-24-6': 'In this case, we need to iteratively calculate the likelihood function [MATH] by incorporating the posterior mean of quadrupole errors Eq. [REF] and compare it to the optics model until the best convergence is reached.', '1904.05683-1-24-7': 'This was not discussed in this paper, however, because our analysis applies to machines whose optics are quite close to their design model.', '1904.05683-1-25-0': 'In designing a beam diagnostics system for a ring, "frequentists" believe in installing as many high resolution BPMs as possible along a ring in order to best characterize beam properties accurately and precisely.', '1904.05683-1-25-1': 'Our analysis shows that having more BPMs does not always significantly improve diagnostics performance and is therefore not necessarily cost-effective for an accelerator design.', '1904.05683-1-25-2': 'Excessive BPMs often only provide redundant information.', '1904.05683-1-25-3': 'The predictive error of beam properties at POIs is not linearly proportional to the number of BPMs present.', '1904.05683-1-25-4': 'In the meantime, too many BPMs could introduce more impedance, making the collective effects worse.', '1904.05683-1-25-5': 'Using the Gaussian regression method, a reasonable compromise can be reached between the quality (resolution) and the quantity of BPMs, which can reduce the burden on the overall scope of an accelerator project.', '1904.05683-1-26-0': 'Other important effects on X-ray brightness, which are not addressed in detail here, are the transverse linear coupling in the electron beam and the residual dispersion at the POIs.', '1904.05683-1-26-1': 'This approach can be applied here also, as the linear coupling can be characterized with beam TbT data [CITATION], and the residual dispersion can be measured by averaging TbT data when beam energy varies.', '1904.05683-1-26-2': 'In a ring-based accelerator, BPMs are used for a multiple other purposes, such as orbit monitoring and optics characterization, etc.', '1904.05683-1-26-3': 'In this paper we only concentrated on a particular use case of TbT data to characterize the linear optics, and then to predict X-ray beam brightness performance.', '1904.05683-1-26-4': 'A similar analysis can be applied to the orbit stability as well.', '1904.05683-1-26-5': "An accelerator's BPM system needs to satisfy several objectives simultaneously.", '1904.05683-1-26-6': 'Therefore the Gaussian regression approach could be extended to a higher dimension parameter space to achieve an optimal compromise among these objectives.', '1904.05683-1-27-0': 'We would like to thank Dr. O. Chubar, Dr. A. He, Dr. D. Hidas and Dr. T. Shaftan (BNL) for discussing the undulator brightness evaluation, and Dr. X. Huang (SLAC) for some fruitful discussion.', '1904.05683-1-27-1': 'This research used resources of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.', '1904.05683-1-27-2': 'This work is also supported by the National Science Foundation under Cooperative Agreement PHY-1102511, the State of Michigan and Michigan State University.'} | {'1904.05683-2-0-0': "With a Bayesian Gaussian regression approach, a systematic method for analyzing a storage ring's beam position monitor (BPM) system requirements has been developed.", '1904.05683-2-0-1': 'The ultimate performance of a ring-based accelerator, based on brightness or luminosity, is determined not only by global parameters, but also by local beam properties at some particular points of interest (POI).', '1904.05683-2-0-2': 'BPMs used for monitoring the beam properties, however, can not be located at these points.', '1904.05683-2-0-3': 'Therefore, the underlying and fundamental purpose of a BPM system is to predict whether the beam properties at POIs reach their desired values.', '1904.05683-2-0-4': 'The prediction process is a regression problem with BPM readings as the training data, but containing random noise.', '1904.05683-2-0-5': 'A Bayesian Gaussian regression approach can determine the probability distribution of the predictive errors, which can be used to conversely analyze the BPM system requirements.', '1904.05683-2-0-6': 'This approach is demonstrated by using turn-by-turn data to reconstruct a linear optics model, and predict the brightness degradation for a ring-based light source.', '1904.05683-2-0-7': 'The quality of BPMs was found to be more important than their quantity in mitigating predictive errors.', '1904.05683-2-1-0': '# introduction', '1904.05683-2-2-0': 'The ultimate performance of a ring-based accelerator is determined not only by certain critical global parameters, such as beam emittance, but also by local properties of the beam at particular points of interest (POI).', '1904.05683-2-2-1': 'The capability of diagnosing and controlling local beam parameters at POIs, such as beam size and divergence, is crucial for a machine to achieve its design performance.', '1904.05683-2-2-2': 'Examples of POIs in a dedicated synchrotron light source ring include the undulator locations, from where high brightness X-rays are generated.', '1904.05683-2-2-3': 'In a collider, POIs are reserved for detectors in which the beam-beam luminosity is observed.', '1904.05683-2-2-4': 'However, beam diagnostics elements, such as beam position monitors (BPM) are generally placed outside of the POIs as the POIs are already occupied.', '1904.05683-2-2-5': 'An intuitive, but quantitatively unproven belief, is that the desired beam properties at the POIs can be achieved once the beam properties are well-controlled at the location of the BPMs.', '1904.05683-2-3-0': 'Using observation data at BPMs to indirectly predict the beam properties at POIs is a regression problem and can be treated as a supervised learning process: BPM readings at given locations are used as a training dataset.', '1904.05683-2-3-1': 'Then a ring optics model with a set of quadrupole excitations as its arguments is selected as the hypothesis.', '1904.05683-2-3-2': 'From the dataset, an optics model needs to be generalized first.', '1904.05683-2-3-3': 'Based on the model, the unknown beam properties at POIs can be predicted.', '1904.05683-2-3-4': "However, there exists some systematic error and random uncertainty in the BPMs' readings, and the quantity of BPMs (the dimension of the training dataset) is limited.", '1904.05683-2-3-5': 'Therefore, the parameters in the reconstructed optics model have inherent uncertainties, as do the final beam property predictions at the POIs.', '1904.05683-2-3-6': 'The precision and accuracy of the predictions at the POIs depend on the quantity of BPMs, their physical distribution pattern around the ring, and their calibration, resolution, etc.', '1904.05683-2-3-7': 'When a BPM system is designed for a storage ring, however, it is more important to consider the inverse problem: i.e.', '1904.05683-2-3-8': 'How are the BPM system technical requirements determined in order to observe whether the ring achieves its desired performance?', '1904.05683-2-3-9': 'In this paper, we developed an approach to address this question with Bayesian Gaussian regression.', '1904.05683-2-4-0': 'In statistics, a Bayesian Gaussian regression [CITATION] is a Bayesian approach to multivariate regression, i.e. regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable.', '1904.05683-2-4-1': 'Every finite collection of the data has a normal distribution.', '1904.05683-2-4-2': 'The distribution of generalized arguments of the hypothesis is the joint distribution of all those random variables.', '1904.05683-2-4-3': 'Based on the hypothesis, a prediction can be made for any unknown dataset within a continuous domain.', '1904.05683-2-4-4': "In our case, multiple BPMs' readings are normally distributed around their real values.", '1904.05683-2-4-5': "The standard deviations of the Gaussian distributions are BPM's resolutions.", '1904.05683-2-4-6': "A vector composed of quadrupoles' mis-settings is the argument to be generalized.", '1904.05683-2-4-7': 'The prediction at the POIs is the function of this vector.', '1904.05683-2-4-8': 'The continuous domain is the longitudinal coordinate [MATH] along a storage ring.', '1904.05683-2-5-0': 'To further explain this approach, the remaining sections are outlined as follows: Sect. [REF] introduces the relation between machine performance and beam diagnostics system capabilities.', '1904.05683-2-5-1': 'Sect. [REF] explains the procedure to apply the Bayesian Gaussian regression in the ring optics model reconstruction, and the prediction of local optics properties at POIs.', '1904.05683-2-5-2': 'In Sect. [REF], the National Synchrotron Light Source II (NSLS-II) storage ring and its BPM system are used to illustrate the application of this approach.', '1904.05683-2-5-3': 'Some discussion and a brief summary is given in Sect. [REF].', '1904.05683-2-6-0': '# machine performance and beam diagnostics capability', '1904.05683-2-7-0': 'As mentioned previously, ultimate performance of a ring-based accelerator relies heavily on local beam properties at particular POIs.', '1904.05683-2-7-1': 'Consider a dedicated light source ring.', '1904.05683-2-7-2': 'Its ultimate performance is measured by the brightness of the X-rays generated by undulators.', '1904.05683-2-7-3': 'The brightness of undulator emission is determined by the transverse size of both the electron and photon beam and their angular divergence at their source points [CITATION].', '1904.05683-2-7-4': "Therefore, the undulator brightness performance [MATH] depends on the ring's global emittance and the local transverse optics parameters, [EQUATION]", '1904.05683-2-7-5': 'Here [MATH] are the electron beam emittances, which represent the equilibrium between the quantum excitation and the radiation damping around the whole ring.', '1904.05683-2-7-6': '[MATH] are the Twiss parameters [CITATION], [MATH] are the dispersion and its derivative at the undulators\' locations, [MATH] is the electron beam energy spread [MATH] and [MATH] are the X-ray beam diffraction "waist size" and its natural angular divergence, respectively.', '1904.05683-2-7-7': 'The X-ray wavelength [MATH], is determined based on the requirements of the beam-line experiments, and [MATH] is the undulator periodic length.', '1904.05683-2-7-8': 'The emittance was found to be nearly constant with small [MATH]-beat (see Sect. [REF]).', '1904.05683-2-7-9': "Therefore, monitoring and controlling the local POI's Twiss parameters are crucial.", '1904.05683-2-8-0': 'The final goal of beam diagnostics is to provide sufficient, accurate observations to reconstruct an online accelerator model.', '1904.05683-2-8-1': 'Modern BPM electronics can provide the beam turn-by-turn (TbT) data, which is widely used for the beam optics characterization and the model reconstruction.', '1904.05683-2-8-2': 'Based on the model, we can predict the beam properties not only at the locations of monitors themselves, but more importantly at the POIs.', '1904.05683-2-8-3': 'The capability of indirect prediction of the Twiss parameters at POIs eventually defines the BPM system requirements on TbT data acquisition.', '1904.05683-2-8-4': 'Based on Eq. [REF], how precisely one can predict the bias and the uncertainty of Twiss parameters [MATH] and [MATH] at locations of undulators is the key problem in designing a BPM system.', '1904.05683-2-8-5': 'Therefore, to specify the technical requirements of a BPM system, the following questions need to be addressed: in order to make an accurate and precise prediction of beam properties at POIs, how many BPMs are needed?', '1904.05683-2-8-6': 'How should the BPMs be allocated throughout the accelerator ring, and how precise should the BPM TbT reading be?', '1904.05683-2-9-0': 'In the following section a method of reconstructing the linear optics model, and determining the brightness performance for a ring-based light source will be discussed.', '1904.05683-2-9-1': 'For a collider ring, its luminosity is determined only by the beam sizes at the interaction points [CITATION].', '1904.05683-2-9-2': 'Gaussian regression analysis can therefore be applied to predict its [MATH] and luminosity as well.', '1904.05683-2-10-0': '# Gaussian regression for model reconstruction and prediction', '1904.05683-2-11-0': 'When circulating beam in a storage ring is disturbed, a BPM system can provide its TbT data at multiple longitudinal locations.', '1904.05683-2-11-1': 'TbT data of the BPMs can be represented as an optics model plus some random reading errors, [EQUATION] here [MATH] is the index of turns, [MATH] is a variable dependent on turn number, [MATH] is the envelope function of Twiss parameters at [MATH] location, [MATH] is the betatron tune, [MATH] is the betatron phase, and [MATH] is the BPM reading noise [CITATION], which generally has a normal distribution.', '1904.05683-2-11-2': 'Based on the accelerator optics model defined in Eq. [REF], we can extract a set of optics Twiss parameters at all BPM locations [CITATION].', '1904.05683-2-11-3': 'Recently, Ref. [CITATION] proposed using a Bayesian approach to infer the mean and uncertainty of Twiss parameters at BPMs simultaneously.', '1904.05683-2-11-4': 'The mean values of [MATH] represent the most likely optics pattern.', '1904.05683-2-11-5': 'The random BPM reading error and the simplification of the optics model can result in some uncertainties, [MATH], in the inference process, [EQUATION] here [MATH] is a vector composed of all normalized quadrupole focusing strengths, and [MATH] is the inference uncertainty.', '1904.05683-2-11-6': 'Unless otherwise stated, bold symbols, such as "[MATH]", are used to denote vectors and matrices throughout this paper.', '1904.05683-2-11-7': 'In accelerator physics, the deviation from the design model [MATH] is often referred to as the [MATH]-beat.', '1904.05683-2-11-8': "From the point of view of model reconstruction, the [MATH]-beat is due to quadrupole excitation errors and can be determined by [EQUATION] where [MATH] represents the quadrupoles' nominal setting and [MATH] is the nominal envelope function along [MATH].", '1904.05683-2-11-9': '[MATH] is the response matrix composed of elements [MATH] observed by the BPMs.', '1904.05683-2-11-10': 'The dependency of [MATH] on [MATH] is not linear in a complete optics model.', '1904.05683-2-11-11': 'However, when quadrupole errors are small enough, the dependence can be approximated as a linear relation as illustrated in Fig. [REF].', '1904.05683-2-11-12': 'The approximation holds for most operational storage rings, and other diffraction limited light sources under design or construction.', '1904.05683-2-11-13': 'A linear approximation allows us to use the linear regression approach for this process.', '1904.05683-2-11-14': 'Eq. [REF] or [REF] is a hypothesis with the unknown arguments [MATH] or [MATH], which need to be generalized from BPM measurement data.', '1904.05683-2-12-0': 'Given a set of inferred optics parameters [MATH]s at multiple locations [MATH] from BPM TbT data, the posterior probability of the quadrupole error distribution can be given according to Bayes theorem, [EQUATION]', '1904.05683-2-12-1': 'Here [MATH] is referred to as the likelihood function, [EQUATION]', '1904.05683-2-12-2': 'Once the mean value of the optics measurement is extracted from the TbT data, a prior quadrupole excitation error distribution [MATH] can be determined by comparing them against the design optics model [CITATION], [EQUATION] in which the variance [MATH] of the prior distribution [MATH] is linearly proportional to the mean value of the measured [MATH]-beat, [EQUATION]', '1904.05683-2-12-3': 'The coefficient [MATH] can be computed based on the optics model either analytically or numerically before carrying out any measurement.', '1904.05683-2-12-4': 'In the NSLS-II ring, [MATH], i.e. a [MATH]-beat corresponds to a distribution of quadrupole errors with the standard deviation [MATH].', '1904.05683-2-13-0': 'Both the likelihood function and the prior distribution are generally normally distributed.', '1904.05683-2-13-1': 'Therefore, the posterior distribution is a normal distribution by summing over the arguments of the exponentials in Eq. [REF] and [REF], [EQUATION]', '1904.05683-2-13-2': 'Here [EQUATION]', '1904.05683-2-13-3': "The identity matrix [MATH] is used in Eq. [REF] because all BPMs' resolutions are assumed to have the same values [MATH].", '1904.05683-2-13-4': "In reality, however, [MATH] needs to be replaced with a diagonal matrix with different elements if the BPMs' resolutions are different.", '1904.05683-2-13-5': "The quadrupoles' error distribution matrix [MATH] needs to be processed in the same way if necessary.", '1904.05683-2-13-6': 'The mean value of the posterior, corresponding to the most likely quadrupole error distribution, can be used to implement the linear optics correction as explained in Ref. [CITATION], [EQUATION] where [MATH].', '1904.05683-2-13-7': 'Adding an extra term [MATH] to prevent overfitting is known as the regularization technique.', '1904.05683-2-13-8': 'The posterior variance represents the uncertainty of quadrupole errors.', '1904.05683-2-13-9': '[EQUATION]', '1904.05683-2-13-10': 'Given [MATH]-beats observed at [MATH], the posterior generalizes an optics model, in which the quadrupoles errors are normally distributed, [EQUATION] with the mean value and the variance given by Eq. [REF] and [REF] respectively.', '1904.05683-2-14-0': 'Thus far, the optics are measured at the locations of the BPMs, and the corresponding quadrupole error distributions are generalized based on the measurements.', '1904.05683-2-14-1': 'To confirm the machine brightness performance, we need to predict the beam properties at POIs.', '1904.05683-2-14-2': 'To do so, the output of all possible posterior quadrupole error distributions must be averaged, [EQUATION]', '1904.05683-2-14-3': "Here [MATH] is the predicted result at POIs' locations [MATH] given the measured [MATH] at [MATH].", '1904.05683-2-14-4': 'The mean values and the variances of the predicted distributions at POIs are [EQUATION] [MATH] is the Jacobian matrix of the optics response to quadrupole errors observed at POIs.', '1904.05683-2-14-5': 'The difference between the mean value [MATH] and the real [MATH] at a POI is referred to as the predicted bias.', '1904.05683-2-14-6': "By substituting the bias and the uncertainty back into Eq. [REF], we can estimate how accurate the brightness could be measured for given BPMs' resolutions.", '1904.05683-2-14-7': 'Based on the desired brightness resolution, we can determine the needed quantity and resolution of BPMs.', '1904.05683-2-15-0': '# Application to NSLS-II ring', '1904.05683-2-16-0': 'In this section, we use the NSLS-II ring and its BPM system TbT data acquisition functionality to demonstrate the application of this approach.', '1904.05683-2-16-1': 'NSLS-II is a [MATH] generation dedicated light source.', '1904.05683-2-16-2': 'All undulator source points (POIs) are located at non-dispersive straights.', '1904.05683-2-16-3': 'A typical photon energy from undulators is around 10 [MATH], with corresponding wavelengths around 0.124 [MATH].', '1904.05683-2-16-4': "The undulators' period length is 20 [MATH].", '1904.05683-2-16-5': 'The horizontal beam emittance is 0.9 [MATH] including the contribution from 3 damping wigglers.', '1904.05683-2-16-6': 'The emittance coupling ratio can be controlled to less than 1%.', '1904.05683-2-16-7': 'At its 15 short straights centers, the Twiss parameters are designed to be as low as [MATH], and [MATH] to generate the desired high brightness x-ray beam from the undulators.', '1904.05683-2-17-0': 'The horizontal emittance growth with an optics distortion was studied by carrying out a lattice simulation.', '1904.05683-2-17-1': 'With [MATH]-beat at a few percent, the corresponding [MATH] and [MATH]-distortions were generated by adding some normally distributed quadrupole errors based on Eq. [REF] and [REF].', '1904.05683-2-17-2': 'The horizontal emittance was found to grow slightly with the average [MATH]-beats as illustrated in Fig. [REF].', '1904.05683-2-17-3': 'When there is about a 1% horizontal [MATH]-beat ([MATH]), the emittance increases by only about 0.1%, which is negligible.', '1904.05683-2-17-4': 'Therefore, in the following calculation, the emittance was represented as a constant.', '1904.05683-2-18-0': 'Degradation of an undulator brightness is determined by its local optics distortion which can be evaluated with Eq. [REF].', '1904.05683-2-18-1': 'Multi-pairs of simulated [MATH] were incorporated into the previously specified undulator parameters to observe the dependence of the X-ray brightness on the [MATH]-beat (see Fig. [REF]).', '1904.05683-2-18-2': 'A change of approximately 1 of the [MATH] in the transverse plane can degrade the brightness by about 1.', '1904.05683-2-18-3': 'In other words, in order to resolve a 1 brightness degradation, the predictive errors of the ring optics (including the bias and uncertainty) at the locations of undulators should be less than 1.', '1904.05683-2-18-4': 'Because multiple undulators are installed around the ring, the predicted performance needs to be evaluated at all POIs simultaneously.', '1904.05683-2-19-0': 'There exist two types of errors in Eq. [REF] which can introduce uncertainties in characterizing the optics parameters at BPMs.', '1904.05683-2-19-1': 'First, due to radiation damping, chromatic decoherence and nonlinearity, a disturbed bunched-beam trajectory is not a pure linear undamping betatron oscillation [CITATION].', '1904.05683-2-19-2': 'A reduced model (for example, assuming [MATH] is a constant), will introduce systematic errors [CITATION].', '1904.05683-2-19-3': 'The second error source is the BPM TbT resolution limit, which results in random noise.', '1904.05683-2-19-4': 'At NSLS-II, the BPM TbT resolution at low beam current ([MATH]) is inferred as [MATH].', '1904.05683-2-19-5': 'When a [MATH] order polynomial function is used to represent the turn-dependent amplitude [MATH], the inferred [MATH] function resolution at BPMs can be reached as low as 0.5% [CITATION].', '1904.05683-2-20-0': 'First we studied the dependence of predictive errors on the quantity of BPMs.', '1904.05683-2-20-1': 'A comprehensive simulation was set up to compare the Gaussian regression predictive errors with the real errors.', '1904.05683-2-20-2': 'A linear optics simulation code was used to simulate the distorted optics due to a set of quadrupole errors.', '1904.05683-2-20-3': 'The [MATH]-beats observed at the BPMs were marked as the "real" values.', '1904.05683-2-20-4': 'On top the real values, 0.5% random errors were added to simulate one-time measurement uncertainty seen by the BPMs.', '1904.05683-2-20-5': 'A posterior distribution Eq. [REF] and [REF] of the quadrupole errors was obtained by reconstructing the optics model with the likelihood function Eq. [REF], and the prior distribution [REF] and [REF].', '1904.05683-2-20-6': 'The predicted optics parameters with their uncertainties were then calculated based on another likelihood function between quadrupoles and the locations of undulators with Eq. [REF].', '1904.05683-2-21-0': 'The results of comparison are illustrated in Fig. [REF].', '1904.05683-2-21-1': 'As with any regression problem, the training data distribution (i.e. the BPM locations) should be as uniform as possible within the continuous [MATH] domain.', '1904.05683-2-21-2': 'There are 30 cells in the NSLS-II ring, and each cell has 6 BPMs.', '1904.05683-2-21-3': 'Equal numbers of BPMs were selected from each cell to make the training data uniformly distributed.', '1904.05683-2-21-4': 'The goal was to predict all straights optics simultaneously.', '1904.05683-2-21-5': 'The predicted performance was therefore evaluated by averaging at multiple straight centers.', '1904.05683-2-21-6': 'Initially, one BPM was selected per cell.', '1904.05683-2-21-7': 'The number of selected BPMs was then gradually increased to observe the evolution of predictive errors.', '1904.05683-2-21-8': 'It was found that utilizing more BPMs improved the predicted performance, as expected.', '1904.05683-2-21-9': 'Both the bias and uncertainty were reduced with the quantity of BPMs.', '1904.05683-2-21-10': 'However, the improvement became less and less apparent once more than 4 BPMs per cell were used.', '1904.05683-2-22-0': 'Next, we studied the effect of [MATH] measurement resolution on the predictive errors.', '1904.05683-2-22-1': 'A similar analysis was carried out but with different [MATH]-resolution as illustrated in Fig. [REF].', '1904.05683-2-22-2': 'By observing Fig. [REF], several conclusions can be drawn: (1) The degradation of the [MATH] resolution reduced the accuracy of the generalized optics model.', '1904.05683-2-22-3': 'However, this can be improved by applying a more complicated optics model [CITATION].', '1904.05683-2-22-4': 'Thus, the BPM TbT resolution is the final limit on the resolution of [MATH] parameters.', '1904.05683-2-22-5': 'In order to accurately and precisely predict the beam properties at POIs, improving the resolution of BPMs is crucial.', '1904.05683-2-22-6': '(2) After a certain point, the predicted performance is not improved significantly with the quantity of BPMs as seen in both Fig [REF] and [REF].', '1904.05683-2-22-7': 'The advantage of reduction of predictive errors will gradually level out once enough BPMs are used.', '1904.05683-2-22-8': 'Meaning, the improvement in error reduction will eventually become negligible compared to the cost of adding more BPMs.', '1904.05683-2-22-9': 'The higher the resolution each individual BPM has, the less number of BPMs are needed.', '1904.05683-2-22-10': 'There should be a compromise between the required quality and quantity of BPMs to achieve an expected predictive accuracy.', '1904.05683-2-22-11': '(3) The quality (resolution) is much more important than the quantity of BPMs from the point of view of optics characterization.', '1904.05683-2-22-12': 'For example, at NSLS-II, in order to resolve 1% brightness degradation, at least 120 BPMs with a [MATH] resolution better than 1% are needed, or 90 BPMs with a 0.75% resolution, etc.', '1904.05683-2-22-13': 'Having more BPMs than is needed creates no obvious, significant improvement.', '1904.05683-2-22-14': 'Having 60 high precision (0.5% [MATH]-resolution) BPMs yields a better performance than having 180 low precision (1%) BPMs in this example.', '1904.05683-2-23-0': '# Summary', '1904.05683-2-24-0': "A systematic approach has been proposed to analyze a BPM system's technical requirements.", '1904.05683-2-24-1': "The approach is based on the resolution requirements for monitoring a machine's ultimate performance.", '1904.05683-2-24-2': "The Bayesian Gaussian regression is useful in statistical data modelling, such as reconstructing a ring's optics model from beam TbT data.", '1904.05683-2-24-3': 'The optics properties of the ring are contained in a collection of data having a normal distribution.', '1904.05683-2-24-4': 'It is worth noting that our approach is simplified as a linear regression by assuming a known linear dependence of optics distortion on quadrupole errors.', '1904.05683-2-24-5': "If a ring's optics are significantly different from the design model, this assumption is not valid, and the analysis may be not accurate either.", '1904.05683-2-24-6': 'In this case, we need to iteratively calculate the likelihood function [MATH] by incorporating the posterior mean of quadrupole errors Eq. [REF] and compare it to the optics model until the best convergence is reached.', '1904.05683-2-24-7': 'This was not discussed in this paper, however, because our analysis applies to machines whose optics are quite close to their design model.', '1904.05683-2-25-0': 'In designing a beam diagnostics system for a ring, "frequentists" believe in installing as many high resolution BPMs as possible along a ring in order to best characterize beam properties accurately and precisely.', '1904.05683-2-25-1': 'Our analysis shows that having more BPMs does not always significantly improve diagnostics performance and is therefore not necessarily cost-effective for an accelerator design.', '1904.05683-2-25-2': 'Excessive BPMs often only provide redundant information.', '1904.05683-2-25-3': 'The predictive error of beam properties at POIs is not linearly proportional to the number of BPMs present.', '1904.05683-2-25-4': 'In the meantime, too many BPMs could introduce more impedance, making the collective effects worse.', '1904.05683-2-25-5': 'Using the Gaussian regression method, a reasonable compromise can be reached between the quality (resolution) and the quantity of BPMs, which can reduce the burden on the overall scope of an accelerator project.', '1904.05683-2-26-0': 'Other important effects on X-ray brightness, which are not addressed in detail here, are the transverse linear coupling in the electron beam and the residual dispersion at the POIs.', '1904.05683-2-26-1': 'This approach can be applied here also, as the linear coupling can be characterized with beam TbT data [CITATION], and the residual dispersion can be measured by averaging TbT data when beam energy varies.', '1904.05683-2-26-2': 'In a ring-based accelerator, BPMs are used for a multiple other purposes, such as orbit monitoring and optics characterization, etc.', '1904.05683-2-26-3': 'In this paper we only concentrated on a particular use case of TbT data to characterize the linear optics, and then to predict X-ray beam brightness performance.', '1904.05683-2-26-4': 'A similar analysis can be applied to the orbit stability, and dynamic aperture reduction due to [MATH]-beat as well.', '1904.05683-2-26-5': "An accelerator's BPM system needs to satisfy several objectives simultaneously.", '1904.05683-2-26-6': 'Therefore the Gaussian regression approach could be extended to a higher dimension parameter space to achieve an optimal compromise among these objectives.', '1904.05683-2-27-0': 'We would like to thank Dr. O. Chubar, Dr. A. He, Dr. D. Hidas and Dr. T. Shaftan (BNL) for discussing the undulator brightness evaluation, and Dr. X. Huang (SLAC) for some fruitful discussion.', '1904.05683-2-27-1': 'This research used resources of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.', '1904.05683-2-27-2': 'This work is also supported by the National Science Foundation under Cooperative Agreement PHY-1102511, the State of Michigan and Michigan State University.'} | [['1904.05683-1-21-0', '1904.05683-2-21-0'], ['1904.05683-1-21-1', '1904.05683-2-21-1'], ['1904.05683-1-21-2', '1904.05683-2-21-2'], ['1904.05683-1-21-3', '1904.05683-2-21-3'], ['1904.05683-1-21-4', '1904.05683-2-21-4'], ['1904.05683-1-21-5', '1904.05683-2-21-5'], ['1904.05683-1-21-6', '1904.05683-2-21-6'], ['1904.05683-1-21-7', '1904.05683-2-21-7'], ['1904.05683-1-21-8', '1904.05683-2-21-8'], ['1904.05683-1-21-9', '1904.05683-2-21-9'], ['1904.05683-1-21-10', '1904.05683-2-21-10'], 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'1904.05683-3-3-0'], ['1904.05683-2-20-4', '1904.05683-3-20-4'], ['1904.05683-2-22-8', '1904.05683-3-23-8'], ['1904.05683-2-24-6', '1904.05683-3-26-2'], ['1904.05683-2-24-7', '1904.05683-3-26-3']] | [] | [['1904.05683-1-26-4', '1904.05683-2-26-4'], ['1904.05683-2-24-0', '1904.05683-3-25-0'], ['1904.05683-2-24-5', '1904.05683-3-26-1']] | [['1904.05683-2-25-5', '1904.05683-3-25-6']] | ['1904.05683-1-13-2', '1904.05683-1-13-9', '1904.05683-1-27-1', '1904.05683-2-13-2', '1904.05683-2-13-9', '1904.05683-2-27-1', '1904.05683-3-13-2', '1904.05683-3-13-9', '1904.05683-3-28-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1904.05683 | {'1904.05683-3-0-0': "With a Bayesian Gaussian regression approach, a systematic method for analyzing a storage ring's beam position monitor (BPM) system requirements has been developed.", '1904.05683-3-0-1': 'The ultimate performance of a ring-based accelerator, based on brightness or luminosity, is determined not only by global parameters, but also by local beam properties at some particular points of interest (POI).', '1904.05683-3-0-2': 'BPMs used for monitoring the beam properties, however, cannot be located at these points.', '1904.05683-3-0-3': 'Therefore, the underlying and fundamental purpose of a BPM system is to predict whether the beam properties at POIs reach their desired values.', '1904.05683-3-0-4': 'The prediction process can be viewed as a regression problem with BPM readings as the training data, but containing random noise.', '1904.05683-3-0-5': 'A Bayesian Gaussian regression approach can determine the probability distribution of the predictive errors, which can be used to conversely analyze the BPM system requirements.', '1904.05683-3-0-6': 'This approach is demonstrated by using turn-by-turn data to reconstruct a linear optics model, and predict the brightness degradation for a ring-based light source.', '1904.05683-3-0-7': 'The quality of BPMs was found to be more important than their quantity in mitigating predictive errors.', '1904.05683-3-1-0': '# introduction', '1904.05683-3-2-0': 'The ultimate performance of a ring-based accelerator is determined not only by certain critical global parameters, such as beam emittance, but also by local properties of the beam at particular points of interest (POI).', '1904.05683-3-2-1': 'The capability of diagnosing and controlling local beam parameters at POIs, such as beam size and divergence, is crucial for a machine to achieve its design performance.', '1904.05683-3-2-2': 'Examples of POIs in a dedicated synchrotron light source ring include the undulator locations, from where high brightness X-rays are generated.', '1904.05683-3-2-3': 'In a collider, POIs are reserved for detectors in which the beam-beam luminosity is observed.', '1904.05683-3-2-4': 'However, beam diagnostics elements, such as beam position monitors (BPM) are generally placed outside of the POIs as the POIs are already occupied.', '1904.05683-3-2-5': 'An intuitive, but quantitatively unproven belief, is that the desired beam properties at the POIs can be achieved once the beam properties are well-controlled at the location of the BPMs.', '1904.05683-3-3-0': 'Using observational data at BPMs to indirectly predict the beam properties at POIs can be viewed as a regression problem and can be treated as a supervised learning process: BPM readings at given locations are used as a training dataset.', '1904.05683-3-3-1': 'Then a ring optics model with a set of quadrupole excitations as its arguments is selected as the hypothesis.', '1904.05683-3-3-2': 'From the dataset, an optics model needs to be generalized first.', '1904.05683-3-3-3': 'Based on the model, the unknown beam properties at POIs can be predicted.', '1904.05683-3-3-4': "However, there exists some systematic error and random uncertainty in the BPMs' readings, and the quantity of BPMs (the dimension of the training dataset) is limited.", '1904.05683-3-3-5': 'Therefore, the parameters in the reconstructed optics model have inherent uncertainties, as do the final beam property predictions at the POIs.', '1904.05683-3-3-6': 'The precision and accuracy of the predictions at the POIs depend on the quantity of BPMs, their physical distribution pattern around the ring, and their calibration, resolution, etc.', '1904.05683-3-3-7': 'When a BPM system is designed for a storage ring, however, it is more important to consider the inverse problem: i.e.', '1904.05683-3-3-8': 'How are the BPM system technical requirements determined in order to observe whether the ring achieves its desired performance?', '1904.05683-3-3-9': 'In this paper, we developed an approach to address this question with Bayesian Gaussian regression.', '1904.05683-3-4-0': 'In statistics, a Bayesian Gaussian regression [CITATION] is a Bayesian approach to multivariate regression, i.e. regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable.', '1904.05683-3-4-1': 'Every finite collection of the data has a normal distribution.', '1904.05683-3-4-2': 'The distribution of generalized arguments of the hypothesis is the joint distribution of all those random variables.', '1904.05683-3-4-3': 'Based on the hypothesis, a prediction can be made for any unknown dataset within a continuous domain.', '1904.05683-3-4-4': "In our case, multiple BPMs' readings are normally distributed around their real values.", '1904.05683-3-4-5': "The standard deviations of the Gaussian distributions are BPM's resolutions.", '1904.05683-3-4-6': "A vector composed of quadrupoles' mis-settings is the argument to be generalized.", '1904.05683-3-4-7': 'The prediction at the POIs is the function of this vector.', '1904.05683-3-4-8': 'The continuous domain is the longitudinal coordinate [MATH] along a storage ring.', '1904.05683-3-5-0': 'To further explain this approach, the remaining sections are outlined as follows: Sect. [REF] introduces the relation between machine performance and beam diagnostics system capabilities.', '1904.05683-3-5-1': 'Sect. [REF] explains the procedure of applying the Bayesian Gaussian regression in the ring optics model reconstruction, and the prediction of local optics properties at POIs.', '1904.05683-3-5-2': 'In Sect. [REF], the National Synchrotron Light Source II (NSLS-II) storage ring and its BPM system are used to illustrate the application of this approach.', '1904.05683-3-5-3': 'Some discussions and a brief summary is given in Sect. [REF].', '1904.05683-3-6-0': '# machine performance and beam diagnostics capability', '1904.05683-3-7-0': 'As mentioned previously, ultimate performance of a ring-based accelerator relies heavily on local beam properties at particular POIs.', '1904.05683-3-7-1': 'Consider a dedicated light source ring.', '1904.05683-3-7-2': 'Its ultimate performance is measured by the brightness of the X-rays generated by undulators.', '1904.05683-3-7-3': 'The brightness of undulator emission is determined by the transverse size of both the electron and photon beam and their angular divergence at their source points [CITATION].', '1904.05683-3-7-4': "Therefore, the undulator brightness performance [MATH] depends on the ring's global emittance and the local transverse optics parameters, [EQUATION]", '1904.05683-3-7-5': 'Here [MATH] are the electron beam emittances, which represent the equilibrium between the quantum excitation and the radiation damping around the whole ring.', '1904.05683-3-7-6': '[MATH] are the Twiss parameters [CITATION], [MATH] are the dispersion and its derivative at the undulators\' locations, [MATH] is the electron beam energy spread [MATH] and [MATH] are the X-ray beam diffraction "waist size" and its natural angular divergence, respectively.', '1904.05683-3-7-7': 'The X-ray wavelength [MATH], is determined based on the requirements of the beam-line experiments, and [MATH] is the undulator periodic length.', '1904.05683-3-7-8': 'The emittance was found to be nearly constant with small [MATH]-beat (see Sect. [REF]).', '1904.05683-3-7-9': "Therefore, monitoring and controlling the local POI's Twiss parameters is crucial.", '1904.05683-3-8-0': 'The final goal of beam diagnostics is to provide sufficient, accurate observations to reconstruct an online accelerator model.', '1904.05683-3-8-1': 'Modern BPM electronics can provide the beam turn-by-turn (TbT) data, which is widely used for the beam optics characterization and the model reconstruction.', '1904.05683-3-8-2': 'Based on the model, we can predict the beam properties not only at the locations of monitors themselves, but more importantly at the POIs.', '1904.05683-3-8-3': 'The capability of indirect prediction of the Twiss parameters at POIs eventually defines the BPM system requirements on TbT data acquisition.', '1904.05683-3-8-4': 'Based on Eq. [REF], how precisely one can predict the bias and the uncertainty of Twiss parameters [MATH] and [MATH] at locations of undulators is the key problem in designing a BPM system.', '1904.05683-3-8-5': 'Therefore, to specify the technical requirements of a BPM system, the following questions need to be addressed: in order to make an accurate and precise prediction of beam properties at POIs, how many BPMs are needed?', '1904.05683-3-8-6': 'How should the BPMs be allocated throughout the accelerator ring, and how precise should the BPM TbT reading be?', '1904.05683-3-9-0': 'In the following section a method of reconstructing the linear optics model, and determining the brightness performance for a ring-based light source will be discussed.', '1904.05683-3-9-1': 'For a collider ring, its luminosity is determined only by the beam sizes at the interaction points [CITATION].', '1904.05683-3-9-2': 'Gaussian regression analysis can therefore be applied to predict its [MATH] and luminosity as well.', '1904.05683-3-10-0': '# Gaussian regression for model reconstruction and prediction', '1904.05683-3-11-0': 'When circulating beam in a storage ring is disturbed, a BPM system can provide its TbT data at multiple longitudinal locations.', '1904.05683-3-11-1': 'TbT data of the BPMs can be represented as an optics model plus some random reading errors, [EQUATION] here [MATH] is the index of turns, [MATH] is a variable dependent on turn number, [MATH] is the envelope function of Twiss parameters at [MATH] location, [MATH] is the betatron tune, [MATH] is the betatron phase, and [MATH] is the BPM reading noise [CITATION], which generally has a normal distribution.', '1904.05683-3-11-2': 'Based on the accelerator optics model defined in Eq. [REF], we can extract a set of optics Twiss parameters at all BPM locations [CITATION].', '1904.05683-3-11-3': 'Recently, Ref. [CITATION] proposed using a Bayesian approach to infer the mean (aka expectation) and uncertainty of Twiss parameters at BPMs simultaneously.', '1904.05683-3-11-4': 'The mean values of [MATH] represent the most likely optics pattern.', '1904.05683-3-11-5': 'The random BPM reading error and the simplification of the optics model can result in some uncertainties, [MATH], in the inference process, [EQUATION] here [MATH] is a vector composed of all normalized quadrupole focusing strengths, and [MATH] is the inference uncertainty.', '1904.05683-3-11-6': 'Unless otherwise stated, bold symbols, such as "[MATH]", are used to denote vectors and matrices throughout this paper.', '1904.05683-3-11-7': 'In accelerator physics, the deviation from the design model [MATH] is often referred to as the [MATH]-beat.', '1904.05683-3-11-8': "From the point of view of model reconstruction, the [MATH]-beat is due to quadrupole excitation errors and can be determined by [EQUATION] where [MATH] represents the quadrupoles' nominal setting and [MATH] is the nominal envelope function along [MATH].", '1904.05683-3-11-9': '[MATH] is the response matrix composed of elements [MATH] observed by the BPMs.', '1904.05683-3-11-10': 'The dependency of [MATH] on [MATH] is not linear in a complete optics model.', '1904.05683-3-11-11': 'However, when quadrupole errors are small enough, the dependence can be approximated as a linear relation as illustrated in Fig. [REF].', '1904.05683-3-11-12': 'The approximation holds for most operational storage rings, and other diffraction limited light sources under design or construction.', '1904.05683-3-11-13': 'A linear approximation allows us to use the linear regression approach for this process.', '1904.05683-3-11-14': 'Eq. [REF] or [REF] is a hypothesis with the unknown arguments [MATH] or [MATH], which need to be generalized from BPM measurement data.', '1904.05683-3-12-0': 'Given a set of measured optics parameters [MATH]s at multiple locations [MATH] from BPM TbT data, the posterior probability of the quadrupole error distribution [MATH] can be given according to Bayes theorem [CITATION], [EQUATION]', '1904.05683-3-12-1': 'Here [MATH] is referred to as the likelihood function, [EQUATION]', '1904.05683-3-12-2': 'Here [MATH] and [MATH] are the expectation value and the variance of the normal distribution of measured [MATH]s.', '1904.05683-3-12-3': 'Once the expectation value of the optics measurement is extracted from the TbT data, a prior quadrupole excitation error distribution [MATH] can be determined by comparing them against the design optics model, [EQUATION] in which the variance [MATH] of the prior distribution [MATH] is linearly proportional to the mean value of the measured [MATH]-beat, [EQUATION]', '1904.05683-3-12-4': 'Here "[MATH]" in Eq. [REF] describes a statistically proportional relationship between [MATH]-beats (in the unit of "m") and quadrupole strength error [MATH] (in units of [MATH]).', '1904.05683-3-12-5': 'The coefficient [MATH] can be computed based on the optics model either analytically or numerically before carrying out any measurements.', '1904.05683-3-12-6': 'In the NSLS-II ring, [MATH], i.e. a [MATH]-beat ([MATH]) corresponds to a distribution of quadrupole errors with the standard deviation [MATH]) as shown in Fig. 1 in Ref. [CITATION].', '1904.05683-3-12-7': 'Qualitatively, the relative [MATH]-beat and quadrupole error, i.e. [MATH] and [MATH] are often used in accelerator literature.', '1904.05683-3-12-8': 'Here the absolute [MATH] and [MATH] are used simply because they were adapted to our quantitative implementation.', '1904.05683-3-13-0': 'Both the likelihood function and the prior distribution are generally normally distributed.', '1904.05683-3-13-1': 'Therefore, the posterior distribution is a normal distribution by summing over the arguments of the exponentials in Eq. [REF] and [REF], [EQUATION]', '1904.05683-3-13-2': 'Here [EQUATION]', '1904.05683-3-13-3': "The identity matrix [MATH] is used in Eq. [REF] because all BPMs' resolutions are assumed to have the same values [MATH].", '1904.05683-3-13-4': "In reality, however, [MATH] needs to be replaced with a diagonal matrix with different elements if the BPMs' resolutions are different.", '1904.05683-3-13-5': "The quadrupoles' error distribution matrix [MATH] needs to be processed in the same way if necessary.", '1904.05683-3-13-6': 'The mean value of the posterior, corresponding to the most likely quadrupole error distribution, can be used to implement the linear optics correction as explained in Ref. [CITATION], [EQUATION] where [MATH].', '1904.05683-3-13-7': 'Adding an extra term [MATH] to prevent overfitting is known as the regularization technique.', '1904.05683-3-13-8': 'The posterior variance represents the uncertainty of quadrupole errors.', '1904.05683-3-13-9': '[EQUATION]', '1904.05683-3-13-10': 'Given [MATH]-beats observed at [MATH], the posterior generalizes an optics model, in which the quadrupoles errors are normally distributed, [EQUATION] with the mean value and the variance given by Eq. [REF] and [REF] respectively.', '1904.05683-3-14-0': 'Thus far, the optics are measured at the locations of the BPMs, and the corresponding quadrupole error distributions are generalized based on the measurements.', '1904.05683-3-14-1': 'To confirm the machine brightness performance, we need to predict the beam properties at POIs.', '1904.05683-3-14-2': 'To do so, the output of all possible posterior quadrupole error distributions must be averaged, [EQUATION]', '1904.05683-3-14-3': "Here [MATH] is the predicted result at POIs' locations [MATH] given the measured [MATH] at [MATH].", '1904.05683-3-14-4': 'The mean values and the variances of the predicted distributions at POIs are [EQUATION] [MATH] is the Jacobian matrix of the optics response to quadrupole errors observed at POIs.', '1904.05683-3-14-5': 'The difference between the mean value [MATH] and the real [MATH] at a POI is referred to as the predicted bias.', '1904.05683-3-14-6': "By substituting the bias and uncertainty back into Eq. [REF], we can estimate how accurate the brightness could be measured for given BPMs' resolutions.", '1904.05683-3-14-7': 'Based on the desired brightness resolution, we can determine the needed quantity and resolution of BPMs.', '1904.05683-3-15-0': '# Application to NSLS-II ring', '1904.05683-3-16-0': 'In this section, we use the NSLS-II ring and its BPM system TbT data acquisition functionality to demonstrate the application of this approach.', '1904.05683-3-16-1': 'NSLS-II is a [MATH] generation dedicated light source.', '1904.05683-3-16-2': 'All undulator source points (POIs) are located at non-dispersive straights.', '1904.05683-3-16-3': 'Typical photon energy from undulators is around 10 [MATH], with corresponding wavelengths around 0.124 [MATH].', '1904.05683-3-16-4': "The undulators' period length is 20 [MATH].", '1904.05683-3-16-5': 'The horizontal beam emittance is 0.9 [MATH] including the contribution from 3 damping wigglers.', '1904.05683-3-16-6': 'The emittance coupling ratio can be controlled to less than 1%.', '1904.05683-3-16-7': 'At its 15 short straight centers, the Twiss parameters are designed to be as low as [MATH], and [MATH] to generate the desired high brightness x-ray beam from the undulators.', '1904.05683-3-17-0': 'The horizontal emittance growth with an optics distortion was studied by carrying out a lattice simulation.', '1904.05683-3-17-1': 'With [MATH]-beat at a few percent, the corresponding [MATH] and [MATH]-distortions were generated by adding some normally distributed quadrupole errors based on Eq. [REF] and [REF].', '1904.05683-3-17-2': 'The horizontal emittance was found to grow slightly with the average [MATH]-beats as illustrated in Fig. [REF].', '1904.05683-3-17-3': 'When there is about a 1% horizontal [MATH]-beat ([MATH]), the emittance increases by only about 0.1%, which is negligible.', '1904.05683-3-17-4': 'Therefore, in the following calculation, the emittance was represented as a constant.', '1904.05683-3-18-0': 'Degradation of an undulatorís brightness is determined by its local optics distortion which can be evaluated with Eq. [REF].', '1904.05683-3-18-1': 'Multi-pairs of simulated [MATH] were incorporated into the previously specified undulator parameters to observe the dependence of the X-ray brightness on the [MATH]-beat (see Fig. [REF]).', '1904.05683-3-18-2': 'A change of approximately 1 of the [MATH] in the transverse plane can degrade the brightness by about 1.', '1904.05683-3-18-3': 'In other words, in order to resolve a 1 brightness degradation, the predictive errors of the ring optics (including the bias and uncertainty) at the locations of undulators should be less than 1.', '1904.05683-3-18-4': 'Because multiple undulators are installed around the ring, the predicted performance needs to be evaluated at all POIs simultaneously.', '1904.05683-3-19-0': 'There exist two types of errors in Eq. [REF] which can introduce uncertainties in characterizing the optics parameters at BPMs.', '1904.05683-3-19-1': 'First, due to radiation damping, chromatic decoherence and nonlinearity, a disturbed bunched-beam trajectory is not a pure linear undamping betatron oscillation [CITATION].', '1904.05683-3-19-2': 'A reduced model (for example, assuming [MATH] is a constant), will introduce systematic errors [CITATION].', '1904.05683-3-19-3': 'The second error source is the BPM TbT resolution limit, which results in random noise.', '1904.05683-3-19-4': 'At NSLS-II, the BPM TbT resolution at low beam current ([MATH]) is inferred as [MATH].', '1904.05683-3-19-5': 'When a [MATH] order polynomial function is used to represent the turn-dependent amplitude [MATH], the inferred [MATH] function resolution at BPMs can be reached as low as 0.5% [CITATION].', '1904.05683-3-20-0': 'First we studied the dependence of predictive errors on the quantity of BPMs.', '1904.05683-3-20-1': 'A comprehensive simulation was set up to compare the Gaussian regression predictive errors with the real errors.', '1904.05683-3-20-2': 'A linear optics simulation code was used to simulate the distorted optics due to a set of quadrupole errors.', '1904.05683-3-20-3': 'The [MATH]-beats observed at the BPMs were marked as the "real" values.', '1904.05683-3-20-4': 'On top of the real values, 0.5% random errors were added to simulate one-time measurement uncertainty seen by the BPMs.', '1904.05683-3-20-5': 'A posterior distribution Eq. [REF] and [REF] of the quadrupole errors was obtained by reconstructing the optics model with the likelihood function Eq. [REF], and the prior distribution [REF] and [REF].', '1904.05683-3-20-6': 'The predicted optics parameters with their uncertainties were then calculated based on another likelihood function between quadrupoles and the locations of undulators with Eq. [REF].', '1904.05683-3-21-0': 'The results of comparison are illustrated in Fig. [REF].', '1904.05683-3-21-1': 'As with any regression problem, the training data distribution (i.e. the BPM locations) should be as uniform as possible within the continuous [MATH] domain.', '1904.05683-3-21-2': 'There are 30 cells in the NSLS-II ring, and each cell has 6 BPMs.', '1904.05683-3-21-3': 'Equal numbers of BPMs were selected from each cell to make the training data uniformly distributed.', '1904.05683-3-21-4': 'The goal was to predict all straight section optics simultaneously.', '1904.05683-3-21-5': 'The predicted performance was therefore evaluated by averaging at multiple straight centers.', '1904.05683-3-21-6': 'Initially, one BPM was selected per cell.', '1904.05683-3-21-7': 'The number of selected BPMs was then gradually increased to observe the evolution of predictive errors.', '1904.05683-3-21-8': 'It was found that utilizing more BPMs improved the predicted performance, as expected.', '1904.05683-3-21-9': 'Both the bias and uncertainty were reduced with the quantity of BPMs.', '1904.05683-3-21-10': 'However, the improvement became less and less apparent once more than 4 BPMs per cell were used.', '1904.05683-3-22-0': 'Since there are 6 BPMs per cell at the NSLS-II ring, we chose different BPM combinations.', '1904.05683-3-22-1': 'We found that some patterns/combinations of BPMs were better used to capture/measure these types of optics distortions.', '1904.05683-3-22-2': 'For example, each end of the straight sections needs one BPM to observe the ID, and at least one BPM needs to be located inside the achromat arc in order to observe the dipoles.', '1904.05683-3-22-3': 'The distribution of the BPMs does not need to be uniform in the longitudinal [MATH] direction, instead, they should be uniform along the betatron phase propagation.', '1904.05683-3-22-4': 'Collider rings would see this effect more clearly due to the existence of interaction points.', '1904.05683-3-22-5': 'However, for most light source rings, including the NSLS-II ring, the phase propagation along the longitudinal direction is mostly quite linear in the longitudinal direction.', '1904.05683-3-23-0': 'Next, we studied the effect of [MATH] measurement resolution on the predictive errors.', '1904.05683-3-23-1': 'A similar analysis was carried out but with different [MATH]-resolution as illustrated in Fig. [REF].', '1904.05683-3-23-2': 'By observing Fig. [REF], several conclusions can be drawn: (1) The degradation of the [MATH] resolution reduced the accuracy of the generalized optics model.', '1904.05683-3-23-3': 'However, this can be improved by applying a more complicated optics model [CITATION].', '1904.05683-3-23-4': 'Thus, the BPM TbT resolution is the final limit on the resolution of [MATH] parameters.', '1904.05683-3-23-5': 'In order to accurately and precisely predict the beam properties at POIs, improving the resolution of BPMs is crucial.', '1904.05683-3-23-6': '(2) After a certain point, the predicted performance is not improved significantly with the quantity of BPMs as seen in both Fig [REF] and [REF].', '1904.05683-3-23-7': 'The advantage of reduction of predictive errors will gradually level out once enough BPMs are used.', '1904.05683-3-23-8': 'Meaning that quantitatively, the improvement in error reduction will eventually become negligible compared to the cost of adding more BPMs.', '1904.05683-3-23-9': 'The higher the resolution each individual BPM has, the less number of BPMs are needed.', '1904.05683-3-23-10': 'There should be a compromise between the required quality and quantity of BPMs to achieve an expected predictive accuracy.', '1904.05683-3-23-11': '(3) The quality (resolution) is much more important than the quantity of BPMs from the point of view of optics characterization.', '1904.05683-3-23-12': 'For example, at NSLS-II, in order to resolve 1% brightness degradation, at least 120 BPMs with a [MATH] resolution better than 1% are needed, or 90 BPMs with a 0.75% resolution, etc.', '1904.05683-3-23-13': 'Having more BPMs than is needed creates no obvious, significant improvement.', '1904.05683-3-23-14': 'Having 60 high precision (0.5% [MATH]-resolution) BPMs yields a better performance than having 180 low precision (1%) BPMs in this example.', '1904.05683-3-24-0': '# Discussion and summary', '1904.05683-3-25-0': 'A systematic approach has been proposed to analyze a BPM systemís technical requirements in this manuscript.', '1904.05683-3-25-1': "The approach is based on the resolution requirements for monitoring a machine's ultimate performance.", '1904.05683-3-25-2': "The Bayesian Gaussian regression is useful in statistical data modelling, such as reconstructing a ring's optics model from beam TbT data.", '1904.05683-3-25-3': 'The optics properties of the ring are contained in a collection of data having a normal distribution.', '1904.05683-3-25-4': 'From past experience in designing and commissioning various accelerators, many will intuitively realize that having more BPMs does not always significantly improve diagnostics performance and is therefore not necessarily cost-effective for an accelerator design.', '1904.05683-3-25-5': 'Using the Gaussian regression method, however, confirmed that quantitatively.', '1904.05683-3-25-6': 'More importantly, a reasonable compromise can be reached between the quality (resolution) and the quantity of BPMs using this method.', '1904.05683-3-26-0': 'It is worth noting that our approach is simplified as a linear regression by assuming a known linear dependence of optics distortion on quadrupole errors.', '1904.05683-3-26-1': "If a ring's optics are significantly different from the design model, this assumption is not valid.", '1904.05683-3-26-2': 'In our case, we needed to iteratively calculate the likelihood function [MATH] by incorporating the posterior mean of quadrupole errors Eq. [REF] and compare it to the optics model until the best convergence was reached.', '1904.05683-3-26-3': 'This was not discussed in this paper, however, because our analysis applies best to machines whose optics are quite close to their design model.', '1904.05683-3-26-4': 'Other important effects on X-ray brightness, such as quadrupolar errors from sextupole feed down, skew quadrupoles, longitudinal misalignments of quadrupoles and BPMs, systematic gain errors in BPMs, magnet fringe field etc. are not addressed in detail here.', '1904.05683-3-26-5': 'These effects are neglected at the NSLS-II ring because either they are small compared with the quadrupole excitation errors and hysteresis, or their effects have been integrated into our optics model.', '1904.05683-3-26-6': 'The Gaussian regression method outlined here, however, can be expanded to take them into account if necessary.', '1904.05683-3-27-0': 'In a ring-based accelerator, BPMs are used for multiple other purposes, such as orbit monitoring and optics characterization, etc.', '1904.05683-3-27-1': 'In this paper we only concentrated on a particular use case of TbT data to characterize the linear optics, and then to predict X-ray beam brightness performance.', '1904.05683-3-27-2': 'A similar analysis can be applied to the orbit stability, and dynamic aperture reduction due to [MATH]-beat as well.', '1904.05683-3-27-3': "An accelerator's BPM system needs to satisfy several objectives simultaneously.", '1904.05683-3-27-4': 'Therefore the Gaussian regression approach could/should be extended to a higher dimension parameter space to achieve an optimal compromise among these objectives.', '1904.05683-3-28-0': 'We would like to thank Dr. O. Chubar, Dr. A. He, Dr. D. Hidas and Dr. T. Shaftan (BNL) for discussing the undulator brightness evaluation, and Dr. X. Huang (SLAC) for other fruitful discussions.', '1904.05683-3-28-1': 'This research used resources of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.', '1904.05683-3-28-2': 'This work is also supported by the National Science Foundation under Cooperative Agreement PHY-1102511, the State of Michigan and Michigan State University.'} | null | null | null | null |
1409.2639 | {'1409.2639-1-0-0': 'The weak decay of the kaon to two pions is studied within the NJL Model.', '1409.2639-1-0-1': 'Using the standard effective weak Hamiltonian, both the decay amplitude arising from an intermediate state [MATH] meson and the direct decay amplitude are calculated.', '1409.2639-1-0-2': 'The effect of final state interactions is also included.', '1409.2639-1-0-3': 'When the matching scale is chosen such that the decay amplitude with isospin [MATH] is close to its experimental value, the [MATH] meson contributes more than 85% of the total [MATH] amplitude.', '1409.2639-1-0-4': 'This supports recent suggestions that the [MATH] meson should play a vital role in explaining the [MATH] rule in this system.', '1409.2639-1-1-0': '# Introduction', '1409.2639-1-2-0': 'The [MATH] rule [CITATION], notably in the [MATH] decay, is one of the major outstanding challenges to our understanding of the hadronic weak interaction.', '1409.2639-1-2-1': 'It has therefore been studied with many different theoretical methods [CITATION].', '1409.2639-1-2-2': 'In recent years these efforts have been extended to include lattice QCD studies, with recent results reported in Ref. [CITATION] and Refs. [CITATION], the latter focussing on decays into the isospin [MATH] channel.', '1409.2639-1-3-0': 'Amongst many quark model studies devoted to this problem, we note that in Ref. [CITATION] the authors calculated the matrix elements up to [MATH] within the framework of the chiral quark model.', '1409.2639-1-3-1': 'Using chiral perturbation theory, Kambor et al. [CITATION] studied the kaon decays to one loop order within SU(3).', '1409.2639-1-3-2': 'Again, within SU(3) chiral perturbation theory, the effect of isospin breaking was included and one-loop results reported in Ref. [CITATION].', '1409.2639-1-3-3': 'Bijnens et al. [CITATION] studied the kaon decays to one loop order within SU(2) chiral perturbation theory.', '1409.2639-1-3-4': 'NLO contributions were considered within the large [MATH] approach in Refs. [CITATION].', '1409.2639-1-3-5': 'The potentially important role of the trace anomaly in weak [MATH]-decays, especially in regard to the [MATH] rule, was discussed in Ref. [CITATION].', '1409.2639-1-4-0': 'The possible role of the charm quark in generating the observed enhancement was discussed in Ref. [CITATION], with the authors presenting there the first results from lattice simulations in the SU(4) flavor limit.', '1409.2639-1-4-1': 'In Ref. [CITATION] the authors studied the problem within the framework of a dual 5-dimensional holographic QCD model.', '1409.2639-1-4-2': 'The possible effect of "new physics", specifically the effect of introducing a heavy colorless [MATH] gauge boson, was discussed by Buras et al. [CITATION].', '1409.2639-1-5-0': 'In a recent report [CITATION], Buras summarized a study of this rule based on the dual representation of QCD using the large [MATH] expansion.', '1409.2639-1-5-1': 'The Wilson coefficients and hadronic matrix elements were evaluated at different energy scales, [MATH], in the early large [MATH] studies, and thus the calculated value of [MATH] was only about 10 of the experimental one.', '1409.2639-1-5-2': 'By evaluating the Wilson coefficients and hadronic matrix elements at the same energy scale, the discrepancy was decreased by about 40.', '1409.2639-1-5-3': 'Moreover, the introduction of QCD penguin operators further decreased the initial discrepancy.', '1409.2639-1-6-0': 'The effect of final state interactions (FSI) was studied in various ways in Refs. [CITATION].', '1409.2639-1-6-1': 'For example, in Ref. [CITATION] the authors directly calculated the relevant Feynman diagrams for the meson rescattering corrections in chiral perturbation theory.', '1409.2639-1-6-2': 'The Omnes approach, which is based on dispersion relations, was used in Refs. [CITATION], while in Refs. [CITATION] the effect of FSI was evaluated within potential models.', '1409.2639-1-7-0': 'Of particular interest to us is the recent work by Crewther and Tunstall [CITATION], which examined the proposal that the [MATH] rule might be resolved if QCD were to have an infrared fixed point.', '1409.2639-1-7-1': 'This suggested that the [MATH] meson would play an especially important role.', '1409.2639-1-7-2': 'While the existence of the [MATH] meson has been controversial for decades, there is now convincing evidence of a pole in the [MATH] scattering amplitude with a mass similar to that of the kaon, albeit with a very large width.', '1409.2639-1-7-3': 'Given that there is a known scalar resonance nearly degenerate with the kaon, it is clear that such a state may well play a significant role in the [MATH] decay.', '1409.2639-1-7-4': 'With this motivation, we use the NJL model, together with the familiar operator product formulation of the non-leptonic weak interaction, to make an explicit calculation of the role of the [MATH] meson in the decay [MATH], with the aim of clarifying its role in the [MATH] rule.', '1409.2639-1-7-5': 'Section [REF] gives details of the calculation of the [MATH] contribution, while the direct decay to pions is found in sect. [REF].', '1409.2639-1-7-6': 'The numerical results and discussion are given in sect. [REF].', '1409.2639-1-8-0': '# Calculation of Kaon Decay including the [MATH] meson', '1409.2639-1-9-0': 'Following the standard conventions we label the [MATH] decay to two pions with isospin zero as [MATH] and with isospin two as [MATH] [CITATION], [EQUATION]', '1409.2639-1-9-1': 'As explained earlier, for the former we calculate the contribution from two different mechanisms; first, the weak transition from [MATH] to a [MATH] meson followed by the decay of the [MATH] to two pions and second, the direct decay to two pions.', '1409.2639-1-9-2': 'For [MATH] only the latter path is available.', '1409.2639-1-10-0': 'For the first contribution to [MATH], illustrated in Fig. [REF],', '1409.2639-1-11-0': 'we write: [EQUATION] where [MATH] is the coupling for the [MATH] transition, [MATH] is the propagator of the [MATH] meson and [MATH] is the [MATH] coupling [CITATION] [EQUATION] and we have neglected the effect of CP-violation.', '1409.2639-1-12-0': 'We employ the NJL model with proper time regularization to describe the structure of these mesons.', '1409.2639-1-12-1': 'The coupling of the [MATH] to the pions is determined from its pole position.', '1409.2639-1-12-2': 'Finally, the effective Hamiltonian describing the non-leptonic weak interaction is obtained using the standard operator product expansion.', '1409.2639-1-12-3': 'We now briefly summarise each of these parts of the calculation.', '1409.2639-1-13-0': '## NJL model', '1409.2639-1-14-0': 'Our work includes the SU(3)-flavour NJL formalism.', '1409.2639-1-14-1': 'After Fierz transformation, the Lagrangian density can be written in the meson channels.', '1409.2639-1-14-2': 'In this form the contributions from the different types of meson can be read directly [CITATION].', '1409.2639-1-14-3': 'Here we are just concerned with the scalar ([MATH]-meson) and pseudocalar (pion and kaon) channels of the Lagrangian density after Fierz transformations, given by: [EQUATION] where the eight Gell-Mann SU(3)-flavor matrices are represented as [MATH].', '1409.2639-1-15-0': 'Since NJL is an effective model, it needs to be regularized.', '1409.2639-1-15-1': 'We chose the proper-time regularization scheme because it has the property of simulating quark confinement [CITATION], in the sense that we do not have thresholds for decays of the color singlets.', '1409.2639-1-15-2': 'We achieve this by including an infrared cutoff, [MATH], that, in analogy with QCD, takes the role of the QCD-scale parameter [MATH].', '1409.2639-1-15-3': 'Therefore, we set [MATH] GeV.', '1409.2639-1-15-4': 'The regularization of the loop integrals includes an ultraviolet cutoff, [MATH].', '1409.2639-1-16-0': 'With [MATH] we follow the standard method of solving the Bethe-Salpeter equations (BSE) for the quark antiquark bound states (mesons) [CITATION].', '1409.2639-1-16-1': 'The diagram describing this BSE in the NJL model is shown in Fig. [REF], and its solutions are given by the following reduced t-matrices: [EQUATION]', '1409.2639-1-16-2': 'Here, the polarization, [MATH], represents the quark-antiquark loops that appear in the diagram for the BSE ([MATH]-meson, pion or kaon).', '1409.2639-1-16-3': 'Their analytic expressions are [EQUATION] and [EQUATION] where [MATH] is a trace in Lorentz indices (the traces over color and flavour having already been taken) and [MATH] are the constituent quark propagators.', '1409.2639-1-16-4': 'For the [MATH] and pion the two propagators contain the same light quark masses, whereas for the kaon case their masses are different.', '1409.2639-1-17-0': 'The pole position of [MATH] corresponds to the mass of each of the mesons, [MATH], which is evident if one examines the expression for [MATH] in pole approximation [CITATION] [EQUATION] where [MATH] is the effective quark-meson coupling, given by [EQUATION]', '1409.2639-1-17-1': 'Here we assume degenerate masses for the constituent light quarks ([MATH]).', '1409.2639-1-17-2': 'Since our treatment of the [MATH]-meson folows the model described in Ref. [CITATION] and summarised in Sec. [REF], we adjust [MATH] to reproduce its "bare" mass ([MATH]).', '1409.2639-1-17-3': 'In addition, [MATH] is fit to reproduce the pion mass.', '1409.2639-1-17-4': 'Finally [MATH] is fit to obtain the pion decay constant [MATH] GeV and the value of [MATH] is fit to ensure the correct kaon mass.', '1409.2639-1-17-5': 'A summary of the parameters used here is given in Table [REF].', '1409.2639-1-18-0': 'The complication associated with such a model, when one needs to match to operators that are defined at some renormalization scale, is that the scale associated with a valence-dominated quark model is typically quite low.', '1409.2639-1-18-1': 'For example, extensive studies of parton distribution functions within the NJL model [CITATION] (as well as other valence-dominated quark models [CITATION]) typically lead to a matching scale of order 0.4-0.5 GeV.', '1409.2639-1-18-2': 'This is rather low and one therefore needs to check the reliability of the effective weak couplings at such a scale.', '1409.2639-1-18-3': 'We address this below.', '1409.2639-1-19-0': '## Effective weak Hamiltonian', '1409.2639-1-20-0': 'Here we need the [MATH] effective Lagrangian of the electroweak interaction [CITATION] [EQUATION] where [MATH] is the relevant CKM matrix element, [MATH] is the Fermi coupling constant and the four-quark operators, [MATH], are: [EQUATION]', '1409.2639-1-20-1': 'The Wilson coefficients, [MATH] and [MATH], have been calculated up to the next to leading order using perturbative QCD [CITATION].', '1409.2639-1-20-2': 'Since [MATH] is relatively small, we will only keep the contribution of the terms with [MATH].', '1409.2639-1-21-0': 'In order to investigate the potential model dependence in matching the renormalization group scale of the operators to the NJL model, in Fig. [REF] we show the variation of the coefficients [MATH] as [MATH] varies from 700 to 450 MeV.', '1409.2639-1-21-1': 'We can see that these Wilson coefficients vary particularly quickly as [MATH] drops below 490 MeV and clearly, if one wants reliable results, one should not choose a scale far below this limit.', '1409.2639-1-22-0': '## Coupling for the [MATH] transition', '1409.2639-1-23-0': 'With the Wilson coefficients and the NJL model explained, we can proceed with the calculation of the weak [MATH] to [MATH] transition amplitude, [MATH] [EQUATION] as illustrated in Fig. [REF].', '1409.2639-1-23-1': 'The corresponding matrix elements are evaluated with dimensional regularization using modified minimal subtraction in order to be consistent with the relevant Wilson coefficients, [MATH].', '1409.2639-1-23-2': 'We find that the contributions of [MATH] to [MATH] vanish, with only [MATH] and [MATH] contributing to [MATH] in our results.', '1409.2639-1-23-3': 'In a more sophisticated model, where the masses of the constituent quarks and the couplings between the mesons and quark pairs were momentum dependent, the operators [MATH]to [MATH] would also contribute to [MATH].', '1409.2639-1-23-4': 'The full expressions for the [MATH] transition amplitude are given in Appendix [REF].', '1409.2639-1-24-0': '## [MATH] model', '1409.2639-1-25-0': 'The problem we are attacking requires an intermediate [MATH] state, therefore we need a model that describes the propagator of this meson.', '1409.2639-1-25-1': 'In Ref. [CITATION] the authors proposed a [MATH] model, which could be used to calculate the propagator for real values of the energy (corresponding to physical [MATH] scattering), while ensuring the correct position of the resonance, corresponding to the [MATH]-meson, in the complex energy plane.', '1409.2639-1-26-0': 'This [MATH]-model is divided in two parts: one where the "bare" mass of the [MATH] is computed from the NJL model (using the formalism of Sec. [REF]); and a second part that includes its correct pole position in the second Riemnann sheet, computed from first principles by Caprini, Colangelo and Leutwyler [CITATION].', '1409.2639-1-26-1': 'This was achieved by including leading order pion loop corrections to the [MATH] propagator.', '1409.2639-1-26-2': 'These were described by the self energy [MATH], depicted in Fig. [REF].', '1409.2639-1-27-0': 'In general, the [MATH] propagator takes the form [EQUATION]', '1409.2639-1-27-1': 'Since [MATH] comes from an effective interaction between the pions and the [MATH] it needs to be regularized.', '1409.2639-1-27-2': 'We employ the original method, including a dipole cut-off with mass [MATH] [CITATION].', '1409.2639-1-28-0': 'Changing [MATH] and keeping the pole position implies variations in the [MATH] to [MATH] coupling, [MATH], as well as the regularization parameter, [MATH].', '1409.2639-1-28-1': 'In Table [REF] we show three parameter sets that were originally presented in Ref. [CITATION], along with the value of the propagator evaluated at the mass of the kaon.', '1409.2639-1-29-0': '# Direct Decay to Pions', '1409.2639-1-30-0': 'The second mechanism contributing to the decay [MATH] proceeds directly to two pions, as illustrated in Fig. [REF].', '1409.2639-1-31-0': 'Since the Wilson coefficients [MATH] and [MATH] are much larger than others, we only consider the contributions of [MATH] and [MATH].', '1409.2639-1-31-1': 'Once again the diagrams are calculated with dimensional regularization and modified minimal subtraction.', '1409.2639-1-31-2': 'After calculation, we find that only Fig. [REF] contributes to our results in the NJL model.', '1409.2639-1-32-0': 'We denote the amplitudes corresponding to diagrams Fig. [REF] without the contribution of the final state interaction as [MATH].', '1409.2639-1-32-1': 'In this case we should also consider the effect of final state interactions(FSI), which we treat using the method of Refs. [CITATION] [EQUATION] where [MATH] is the phase shift for pion-pion scattering with isospin [MATH] and we take the values of [MATH] from Ref. [CITATION].', '1409.2639-1-32-2': 'This yields the result: [EQUATION]', '1409.2639-1-33-0': '# Numerical Results and Discussion', '1409.2639-1-34-0': 'As we have explained, in this work [MATH] contains two contributions, the first, [MATH], involving the coupling to the [MATH] meson and the second, [MATH], involving the direct decay to pions.', '1409.2639-1-34-1': 'Since, in the NJL model, [MATH] involves the weak operators [MATH] and [MATH], while [MATH] iinvolves [MATH] and [MATH], their contributions can be added with no worry about double counting.', '1409.2639-1-34-2': 'The phase shift of [MATH] resulting from the propagator of the [MATH] approximately reproduces the physical [MATH] phase shift appearing in [MATH], so we can set [MATH].', '1409.2639-1-35-0': 'We list the [MATH]-[MATH] coupling and the decay amplitudes, as a function of the matching scale, [MATH], and the dipole cut-off [MATH] in Table [REF].', '1409.2639-1-35-1': 'From this Table one sees that the decay amplitudes are rather more sensitive to [MATH] than [MATH].', '1409.2639-1-35-2': '[h] [MATH]-[MATH] coupling and decay amplitudes with [MATH], [MATH].', '1409.2639-1-35-3': '[MATH] and [MATH] are in units of GeV, [MATH] are in units of [MATH] and [MATH] are in units of eV.', '1409.2639-1-36-0': 'In Refs. [CITATION], the authors used the [MATH] scheme to evolve the Wilson coefficients and hadronic matrix elements to the same energy scale.', '1409.2639-1-36-1': 'In order to match the energy scales, the Wilson coefficients were evolved from [MATH] to [MATH] in the quark-gluon picture, while the hadronic matrix elements were evolved from [MATH] to the same scale [MATH] in the meson picture.', '1409.2639-1-36-2': '[MATH] was found to lie in the range 12.5[MATH]14.9 as [MATH] varied from 0.6[MATH]1 GeV, if only the contributions from [MATH] and [MATH] were included.', '1409.2639-1-37-0': 'We notice that the [MATH] dependence of their results was smaller than what we have found.', '1409.2639-1-37-1': 'Although the Dirac traces and loop integrals in the hadronic matrix elements were evaluated with dimensional regularization and modified minimal subtraction (as were the Wilson coefficients in our approach) the possible [MATH]-dependence of the constituent quark masses and the couplings between mesons and quarks do not naturally appear in the NJL model.', '1409.2639-1-37-2': 'Rather the model is assumed to represent QCD at a scale at which the gluons are effectively frozen out as degrees of freedom and valence quarks interacting through a chiral effective Lagrangian dominate the dynamics.', '1409.2639-1-37-3': 'Thus the best one can do is to match the scale of the effective weak Hamiltonian to the scale at which the NJL model best matches experiment, which seems to be around [MATH] GeV.', '1409.2639-1-38-0': 'We note that, in addition to the processes included here, there are also diagrams which are disconnected if the gluon lines are removed (usually just called disconnected diagrams for short).', '1409.2639-1-38-1': 'While such disconnected diagrams can contribute to [MATH], they do not naturally appear within the NJL model and we omit them here.', '1409.2639-1-38-2': 'Since [MATH] is not contributed by the disconnected diagrams, we use it to fix the energy scale [MATH].', '1409.2639-1-39-0': 'As we already noted earlier, in order that the evolution of the Wilson coefficients is under control, the matching scale, [MATH], should not be lower than about 490 MeV.', '1409.2639-1-39-1': 'This creates some tension as the scale associated with the NJL model, when matching to phenomenological parton distribution functions, tends to be nearer 400-450 MeV.', '1409.2639-1-39-2': 'Fortunately, we see from Table [REF] that if we choose [MATH] to be in the range 0.490[MATH]0.494 GeV, [MATH] (which does not involve the [MATH] meson) actually lies very close to its experimental value, 14.8 eV.', '1409.2639-1-39-3': 'We therefore choose this range for [MATH] in order to calculate [MATH].', '1409.2639-1-40-0': 'From Table [REF], one notices that [MATH] increases by as much as 60%[MATH]70% while [MATH] decreases about only 10% as [MATH] is increased from 0.32 GeV to 0.34 GeV for [MATH] lying in the range (0.490,0.494) GeV.', '1409.2639-1-40-1': 'This is mainly because [MATH], and the variation of the [MATH]-[MATH] coupling and the propagator of the [MATH] meson compensate each other.', '1409.2639-1-41-0': 'With [MATH] fixed in the range where the empirical value of [MATH] is reproduced, one notices that [MATH] lies in the range 320[MATH]370 eV, which is close to the experimental value of 332 eV.', '1409.2639-1-41-1': 'In view of the uncertainties in matching the model scale to the scale of the weak effective Hamiltonian, it is unrealistic to expect to obtain a more precise result.', '1409.2639-1-41-2': 'Nevertheless, our calculation does confirm that the [MATH] meson does indeed play an important role in [MATH], since it contributes more than 85 of the final value.', '1409.2639-1-41-3': 'The direct decay process contributes a mere 10.'} | {'1409.2639-2-0-0': 'The weak decay of the kaon to two pions is studied within the model of Nambu and Jona-Lasinio (NJL Model).', '1409.2639-2-0-1': 'Using the standard effective weak Hamiltonian, both the decay amplitude arising from an intermediate state [MATH] meson and the direct decay amplitude are calculated.', '1409.2639-2-0-2': 'The effect of final state interactions is also included.', '1409.2639-2-0-3': 'When the matching scale is chosen such that the decay amplitude with isospin [MATH] is close to its experimental value, our model including the [MATH] meson contributes up to 80% of the total [MATH] amplitude.', '1409.2639-2-0-4': 'This supports recent suggestions that the [MATH] meson should play a vital role in explaining the [MATH] rule in this system.', '1409.2639-2-1-0': '# Introduction', '1409.2639-2-2-0': 'The [MATH] rule [CITATION], notably in the [MATH] decay, is one of the major outstanding challenges to our understanding of the hadronic weak interaction.', '1409.2639-2-2-1': 'It has therefore been studied with many different theoretical methods [CITATION].', '1409.2639-2-2-2': 'In recent years these efforts have been extended to include lattice QCD studies, with recent results reported in Ref. [CITATION] and Refs. [CITATION], the latter focussing on decays into the isospin [MATH] channel.', '1409.2639-2-3-0': 'Amongst many quark model studies devoted to this problem, we note that in Ref. [CITATION] the authors calculated the matrix elements up to [MATH] within the framework of the chiral quark model.', '1409.2639-2-3-1': 'Using chiral perturbation theory, Kambor et al. [CITATION] studied the kaon decays to one loop order within SU(3).', '1409.2639-2-3-2': 'Again, within SU(3) chiral perturbation theory, the effect of isospin breaking was included and one-loop results reported in Ref. [CITATION].', '1409.2639-2-3-3': 'Bijnens et al. [CITATION] studied the kaon decays to one loop order within SU(2) chiral perturbation theory.', '1409.2639-2-3-4': 'NLO contributions were considered within the large [MATH] approach in Refs. [CITATION].', '1409.2639-2-3-5': 'The potentially important role of the trace anomaly in weak [MATH]-decays, especially in regard to the [MATH] rule, was discussed in Ref. [CITATION].', '1409.2639-2-4-0': 'The possible role of the charm quark in generating the observed enhancement was discussed in Ref. [CITATION], with the authors presenting there the first results from lattice simulations in the SU(4) flavor limit.', '1409.2639-2-4-1': 'In Ref. [CITATION] the authors studied the problem within the framework of a dual 5-dimensional holographic QCD model.', '1409.2639-2-4-2': 'The possible effect of "new physics", specifically the effect of introducing a heavy colorless [MATH] gauge boson, was discussed by Buras et al. [CITATION].', '1409.2639-2-5-0': 'In a recent report [CITATION], Buras summarized a study of this rule based on the dual representation of QCD using the large [MATH] expansion.', '1409.2639-2-5-1': 'The Wilson coefficients and hadronic matrix elements were evaluated at different energy scales, [MATH], in the early large [MATH] studies, and thus the calculated value of [MATH] was only about 10 of the experimental one.', '1409.2639-2-5-2': 'By evaluating the Wilson coefficients and hadronic matrix elements at the same energy scale, the discrepancy was decreased by about 40.', '1409.2639-2-5-3': 'Moreover, the introduction of QCD penguin operators further decreased the initial discrepancy.', '1409.2639-2-6-0': 'The effect of final state interactions (FSI) was studied in various ways in Refs. [CITATION].', '1409.2639-2-6-1': 'For example, in Ref. [CITATION] the authors directly calculated the relevant Feynman diagrams for the meson rescattering corrections in chiral perturbation theory.', '1409.2639-2-6-2': 'The Omnes approach, which is based on dispersion relations, was used in Refs. [CITATION], while in Refs. [CITATION] the effect of FSI was evaluated within potential models.', '1409.2639-2-7-0': 'Of particular interest to us is the recent work by Crewther and Tunstall [CITATION], which examined the proposal that the [MATH] rule might be resolved if QCD were to have an infrared fixed point.', '1409.2639-2-7-1': 'This suggested that the [MATH] meson would play an especially important role.', '1409.2639-2-7-2': 'While the existence of the [MATH] meson has been controversial for decades, there is now convincing evidence of a pole in the [MATH] scattering amplitude with a mass similar to that of the kaon, albeit with a very large width.', '1409.2639-2-7-3': 'Given that there is a known scalar resonance nearly degenerate with the kaon, it is clear that such a state may well play a significant role in the [MATH] decay.', '1409.2639-2-7-4': 'With this motivation, we use the NJL model, together with the familiar operator product formulation of the non-leptonic weak interaction, to make an explicit calculation of the role of the [MATH] meson in the decay [MATH], with the aim of clarifying its role in the [MATH] rule.', '1409.2639-2-7-5': 'Section [REF] gives details of the calculation of the [MATH] contribution, while the direct decay to pions is found in sect. [REF].', '1409.2639-2-7-6': 'The numerical results and discussion are given in sect. [REF].', '1409.2639-2-8-0': '# Calculation of Kaon Decay including the [MATH] meson', '1409.2639-2-9-0': 'Following the standard conventions we label the [MATH] decay to two pions with isospin zero as [MATH] and with isospin two as [MATH] [CITATION], [EQUATION]', '1409.2639-2-9-1': 'As explained earlier, for the former we calculate the contribution from two different mechanisms; first, the weak transition from [MATH] to a [MATH] meson followed by the decay of the [MATH] to two pions and second, the direct decay to two pions.', '1409.2639-2-9-2': 'For [MATH] only the latter path is available.', '1409.2639-2-10-0': 'In the absence of final state interactions (which will be included later), the first contribution to [MATH], as illustrated in Fig. [REF]', '1409.2639-2-11-0': 'is written: [EQUATION] where [MATH] is the coupling for the [MATH] transition, [MATH] is the propagator of the [MATH] meson and [MATH] is the [MATH] coupling [CITATION] [EQUATION] and we have neglected the effect of CP-violation.', '1409.2639-2-12-0': 'We employ the NJL model with dimensional regularization to describe the structure of these mesons.', '1409.2639-2-12-1': 'The coupling of the [MATH] to the pions is also determined within the NJL model.', '1409.2639-2-12-2': 'Finally, the effective Hamiltonian describing the non-leptonic weak interaction is obtained using the standard operator product expansion.', '1409.2639-2-12-3': 'We now briefly summarise each of these parts of the calculation.', '1409.2639-2-13-0': '## NJL model', '1409.2639-2-14-0': 'Our work uses the NJL formalism based upon SU(3)-flavour symmetry.', '1409.2639-2-14-1': 'After Fierz transformation, the Lagrangian density can be written in the meson channels.', '1409.2639-2-14-2': 'In this form the contributions from the different types of meson can be read directly [CITATION].', '1409.2639-2-14-3': 'This has recently been used in the computation of the kaon and pion form factors [CITATION], as well as the study of SU(3)-flavour symmetry in the baryon octet [CITATION].', '1409.2639-2-14-4': 'Those studies included the breaking of SU(3) chiral symmetry with the use of different masses for the constituent light quarks (up and down) and the constituent strange quark.', '1409.2639-2-15-0': 'Here we include different couplings for the scalar ([MATH]) and pseudoscalar mesons (pion and kaon), modifying the NJL Lagrangian density as follows: [EQUATION] where the eight Gell-Mann SU(3)-flavor matrices are represented as [MATH].', '1409.2639-2-15-1': 'This modified NJL lagrangian density preserves [MATH] symmetry.', '1409.2639-2-16-0': 'Since NJL is an effective model, it needs to be regularized.', '1409.2639-2-16-1': 'We chose dimensional regularization for consistency with the computation of the Wilson coefficients when the electroweak interaction is included (Sec. [REF]).', '1409.2639-2-16-2': 'The value of the energy scale [MATH] is constrained by requiring stability of the Wilson coefficients (Fig. [REF]).', '1409.2639-2-16-3': 'With the Lagrangian density of Eq. [REF] the Gap equation for the constituent light quark [MATH] comes from the scalar interaction term: [EQUATION] where [MATH] is the mass of the current light quark.', '1409.2639-2-17-0': 'With [MATH] we follow the standard method of solving the Bethe-Salpeter equations (BSE) for the quark antiquark bound states (mesons) [CITATION].', '1409.2639-2-17-1': 'The diagram describing this BSE in the NJL model is shown in Fig. [REF], and its solutions are given by the following reduced t-matrices: [EQUATION]', '1409.2639-2-17-2': 'Here, the polarization, [MATH], represents the quark-antiquark loops that appear in the diagram for the BSE ([MATH]-meson, pion or kaon).', '1409.2639-2-17-3': 'with the + and - signs corresponding to the pion and [MATH] respectively.', '1409.2639-2-17-4': 'Their analytic expressions are [EQUATION] and [EQUATION] where [MATH] is a trace in Lorentz indices (the traces over color and flavour having already been taken) and [MATH] are the constituent quark propagators.', '1409.2639-2-17-5': 'For the [MATH] and pion the two propagators contain the same light quark masses, whereas for the kaon case their masses are different.', '1409.2639-2-17-6': 'The explicit expressions for [MATH] in dimensional regularization are [EQUATION] and [EQUATION] where [MATH] for the pion, and [MATH] and [MATH] for the kaon.', '1409.2639-2-17-7': 'The integrals [MATH] and [MATH] are given in Appendix [REF].', '1409.2639-2-18-0': 'The pole position of [MATH] corresponds to the mass of each of the mesons, [MATH], which is evident if one examines the expression for [MATH] in pole approximation [CITATION] [EQUATION] where [MATH] is the effective quark-meson coupling, given by [EQUATION]', '1409.2639-2-18-1': 'The - and + signs correspond to the pion (kaon) and the [MATH], respectively, with the sign difference coming from Eq. [REF].', '1409.2639-2-19-0': 'Here we assume degenerate masses for the constituent light quarks ([MATH]).', '1409.2639-2-19-1': 'The mass of the [MATH]-meson ([MATH]) is taken to lie in the range 520 - 600 MeV.', '1409.2639-2-19-2': 'With the gap equation (Eq. [REF]), including a current light quark mass [MATH] of 5 Mev, and the equation for the mass of the [MATH]-meson (pole position in Eq. [REF]), we fit [MATH] and [MATH].', '1409.2639-2-19-3': 'Our result for [MATH] is in reasonable agreement with Ref. [CITATION], where it was shown that [MATH].', '1409.2639-2-19-4': '[MATH] is chosen to reproduce the physical [MATH], and [MATH] to reproduce the kaon mass [MATH].', '1409.2639-2-19-5': 'Finally the effective couplings, [MATH], are computed with Eq. [REF].', '1409.2639-2-19-6': 'The results for [MATH], [MATH] and [MATH] MeV are summarized in Table [REF].', '1409.2639-2-19-7': 'The negative sign of the Lagrangian couplings is a feature of dimensional regularization in the NJL model [CITATION].', '1409.2639-2-19-8': 'We also stress that the difference between [MATH] and [MATH] is of the order of 10%.', '1409.2639-2-20-0': 'The complication associated with such a model, when one needs to match to operators that are defined at some renormalization scale, is that the scale associated with a valence-dominated quark model is typically quite low.', '1409.2639-2-20-1': 'For example, extensive studies of parton distribution functions within the NJL model [CITATION] (as well as other valence-dominated quark models [CITATION]) typically lead to a matching scale of order 0.4-0.5 GeV.', '1409.2639-2-20-2': 'This is rather low and one therefore needs to check the reliability of the effective weak couplings at such a scale.', '1409.2639-2-20-3': 'We address this below.', '1409.2639-2-21-0': '## Coupling between [MATH] and [MATH]-[MATH]', '1409.2639-2-22-0': 'We obtain the coupling, [MATH], between [MATH] and [MATH]-[MATH] within the NJL model.', '1409.2639-2-22-1': 'To that end one should calculate the amplitudes of the [MATH] process at both quark and hadron levels, and match the results.', '1409.2639-2-22-2': 'At quark level, the amplitude can be obtained from Fig. [REF] with the masses and couplings derived within NJL model.', '1409.2639-2-22-3': 'At the hadron level, the amplitude can be easily given from the effective Lagrangian [MATH] in Eq. ([REF]), [EQUATION]', '1409.2639-2-22-4': 'We match both amplitudes at a centre-of-mass energy of the system [MATH], since the coupling [MATH] would be used to study the decay of kaon.', '1409.2639-2-22-5': 'The amplitude from Fig. [REF] at the quark level is energy-scale dependent, and therefore [MATH] also runs as the energy scale [MATH] changes within our model.', '1409.2639-2-22-6': 'However, [MATH] is rather insensitive to [MATH], as we see from the numerical results in Table [REF].', '1409.2639-2-23-0': '## Effective weak Hamiltonian', '1409.2639-2-24-0': 'Here we need the [MATH] effective Lagrangian of the electroweak interaction [CITATION] [EQUATION] where [MATH] is the relevant CKM matrix element, [MATH] is the Fermi coupling constant and the four-quark operators, [MATH], are: [EQUATION]', '1409.2639-2-24-1': 'The Wilson coefficients, [MATH] and [MATH], have been calculated up to the next to leading order using perturbative QCD [CITATION].', '1409.2639-2-24-2': 'Since [MATH] is relatively small, we will only keep the contribution of the terms with [MATH].', '1409.2639-2-25-0': 'In order to investigate the potential model dependence in matching the renormalization group scale of the operators to the NJL model, in Fig. [REF] we show the variation of the coefficients [MATH] as [MATH] varies from 700 to 450 MeV.', '1409.2639-2-25-1': 'We can see that these Wilson coefficients vary particularly quickly as [MATH] drops below 480 MeV and clearly, if one wants reliable results, one should not choose a scale far below this limit.', '1409.2639-2-26-0': '## Coupling for the [MATH] transition', '1409.2639-2-27-0': 'With the Wilson coefficients and the NJL model explained, we can proceed with the calculation of the weak [MATH] to [MATH] transition amplitude, [MATH] [EQUATION] as illustrated in Fig. [REF].', '1409.2639-2-27-1': 'Here we simply assume that the quarks appearing in the QCD operators [MATH] are the same as the NJL quark operators of the corresponding flavors with the energy scale [MATH] lying in some region not yet accurately specified.', '1409.2639-2-27-2': 'Therefore, we first show our results for [MATH] in the range 0.48[MATH]0.70 GeV and then use the numerical results to identify the optimal region.', '1409.2639-2-27-3': 'This is shown in Section [REF].', '1409.2639-2-28-0': 'The corresponding matrix elements are evaluated with dimensional regularization using modified minimal subtraction in order to be consistent with the relevant Wilson coefficients, [MATH].', '1409.2639-2-28-1': 'We find that the contributions of [MATH] to [MATH] vanish, with only [MATH] and [MATH] contributing to [MATH] in our results.', '1409.2639-2-28-2': 'In a more sophisticated model, where the masses of the constituent quarks and the couplings between the mesons and quark pairs were momentum dependent, the operators [MATH]to [MATH] would also contribute to [MATH].', '1409.2639-2-28-3': 'The full expressions for the [MATH] transition amplitude are given in Appendix [REF].', '1409.2639-2-29-0': '# Direct Decay to Pions', '1409.2639-2-30-0': 'The second mechanism contributing to the decay [MATH] proceeds directly to two pions, as illustrated in Fig. [REF].', '1409.2639-2-31-0': 'Since the Wilson coefficients [MATH] and [MATH] are much larger than others, we only consider the contributions of [MATH] and [MATH].', '1409.2639-2-31-1': 'Once again the diagrams are calculated with dimensional regularization and modified minimal subtraction.', '1409.2639-2-31-2': 'After calculation, we find that only Fig. [REF] contributes to our results in the NJL model.', '1409.2639-2-32-0': '# Final State Interaction', '1409.2639-2-33-0': 'We denote the amplitudes corresponding to the diagrams shown in Figs. [REF] and [REF], without the contribution of the final state interaction, as [MATH] and [MATH]).', '1409.2639-2-33-1': 'We must also consider the effect of final state interactions(FSI), which we treat using the method of Refs. [CITATION] [EQUATION] where [MATH] is the phase shift for pion-pion scattering with isospin [MATH] and we take the values of [MATH] from Ref. [CITATION].', '1409.2639-2-33-2': 'This yields the result: [EQUATION]', '1409.2639-2-34-0': '# Numerical Results and Discussion', '1409.2639-2-35-0': 'As we have explained, in this work [MATH] contains two contributions, the first, [MATH], involving the coupling to the [MATH] meson and the second, [MATH], involving the direct decay to pions.', '1409.2639-2-35-1': 'Since, in the NJL model, [MATH] involves the weak operators [MATH] and [MATH], while [MATH] involves [MATH] and [MATH], their contributions can be added with no worry about double counting [MATH].', '1409.2639-2-36-0': 'We list the [MATH]-[MATH] coupling and the decay amplitudes with [MATH], as a function of the matching scale, [MATH], in Tables [REF], [REF], and [REF].', '1409.2639-2-36-1': 'From these Tables one sees that the decay amplitudes are sensitive to both [MATH] and [MATH].', '1409.2639-2-37-0': 'In Refs. [CITATION], the authors used the [MATH] scheme to evolve the Wilson coefficients and hadronic matrix elements to the same energy scale.', '1409.2639-2-37-1': 'In order to match the energy scales, the Wilson coefficients were evolved from [MATH] to [MATH] in the quark-gluon picture, while the hadronic matrix elements were evolved from [MATH] to the same scale [MATH] in the meson picture.', '1409.2639-2-37-2': '[MATH] was found to lie in the range 12.5[MATH]14.9 as [MATH] varied from 0.6[MATH]1 GeV, if only the contributions from [MATH] and [MATH] were included.', '1409.2639-2-38-0': 'We notice that the [MATH] dependence of their results was smaller than what we have found.', '1409.2639-2-38-1': 'Here both the hadronic matrix elements and the Wilson coefficients are evaluated with dimensional regularization and modified minimal subtraction.', '1409.2639-2-38-2': '(As an extension of the present work it would be interesting to attempt to further reduce the [MATH]-dependence by including higher order loop corrections.)', '1409.2639-2-38-3': 'Within the present work, as in many other applications of valence dominated quark models, the model is assumed to represent QCD at a scale at which the gluons are effectively frozen out as degrees of freedom and valence quarks interacting through a chiral effective Lagrangian dominate the dynamics.', '1409.2639-2-38-4': 'Thus the best one can do is to match the scale of the effective weak Hamiltonian to the scale at which the NJL model best matches experiment, which seems to be around [MATH] GeV.', '1409.2639-2-39-0': 'We note that, in addition to the processes included here, there are also diagrams which are disconnected if the gluon lines are removed (usually just called disconnected diagrams for short).', '1409.2639-2-39-1': 'While such disconnected diagrams can contribute to [MATH], they do not naturally appear within the NJL model and we omit them here.', '1409.2639-2-39-2': 'Since [MATH] is not contributed by the disconnected diagrams, we use it to fix the energy scale [MATH].', '1409.2639-2-40-0': 'As we already noted earlier, in order that the evolution of the Wilson coefficients is under control, the matching scale, [MATH], should not be lower than about 480 MeV.', '1409.2639-2-40-1': 'This creates some tension as the scale associated with the NJL model, when matching to phenomenological parton distribution functions, tends to be nearer 400-450 MeV.', '1409.2639-2-40-2': 'Fortunately, we see from Tables [REF], [REF] and [REF] that if we choose [MATH] to be in the range 0.484[MATH]0.488 GeV, [MATH] (which does not involve the [MATH] meson) actually lies very close to its experimental value, 14.8 eV.', '1409.2639-2-40-3': 'We allow a small variation of [MATH] for different values of [MATH] in order to calculate [MATH].', '1409.2639-2-40-4': 'For [MATH], we choose [MATH] to be in the range 0.484[MATH]0.485 GeV, 0.485[MATH]0.486 GeV, and 0.487[MATH]0.488 GeV, respectively.', '1409.2639-2-41-0': 'With [MATH] fixed in the range where the empirical value of [MATH] is reproduced, one notices that [MATH] lies in the range 135[MATH]270 eV, as [MATH] varies over the range 520[MATH]600 MeV.', '1409.2639-2-41-1': '[MATH] is close to the experimental value of 332 eV at [MATH].', '1409.2639-2-41-2': 'From Tables [REF], [REF], and [REF], we notice that [MATH] is sensitive to the choice of [MATH] because [MATH], while [MATH] and [MATH] are not sensitive to it.', '1409.2639-2-41-3': '[MATH] decreases as [MATH] moves away from [MATH].', '1409.2639-2-42-0': 'In view of the uncertainties in matching the model scale to the scale of the weak effective Hamiltonian, it is unrealistic to expect to obtain a prediction for the decay amplitudes.', '1409.2639-2-42-1': 'Nevertheless, our calculation clearly confirms that the [MATH] meson does indeed play an important role in [MATH], since it contributes up to 65% of the final value, while the direct decay process contributes a mere 15%.'} | [['1409.2639-1-2-0', 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'1409.2639-2-20-2'], ['1409.2639-1-18-3', '1409.2639-2-20-3'], ['1409.2639-1-3-0', '1409.2639-2-3-0'], ['1409.2639-1-3-1', '1409.2639-2-3-1'], ['1409.2639-1-3-2', '1409.2639-2-3-2'], ['1409.2639-1-3-3', '1409.2639-2-3-3'], ['1409.2639-1-3-4', '1409.2639-2-3-4'], ['1409.2639-1-3-5', '1409.2639-2-3-5'], ['1409.2639-1-32-2', '1409.2639-2-33-2'], ['1409.2639-1-9-0', '1409.2639-2-9-0'], ['1409.2639-1-9-1', '1409.2639-2-9-1'], ['1409.2639-1-9-2', '1409.2639-2-9-2'], ['1409.2639-1-38-0', '1409.2639-2-39-0'], ['1409.2639-1-38-1', '1409.2639-2-39-1'], ['1409.2639-1-38-2', '1409.2639-2-39-2'], ['1409.2639-1-12-2', '1409.2639-2-12-2'], ['1409.2639-1-12-3', '1409.2639-2-12-3'], ['1409.2639-1-20-0', '1409.2639-2-24-0'], ['1409.2639-1-20-1', '1409.2639-2-24-1'], ['1409.2639-1-20-2', '1409.2639-2-24-2'], ['1409.2639-1-16-0', '1409.2639-2-17-0'], ['1409.2639-1-16-1', '1409.2639-2-17-1'], ['1409.2639-1-16-2', '1409.2639-2-17-2'], ['1409.2639-1-16-3', '1409.2639-2-17-4'], ['1409.2639-1-16-4', '1409.2639-2-17-5'], ['1409.2639-1-39-1', '1409.2639-2-40-1'], ['1409.2639-1-21-0', '1409.2639-2-25-0'], ['1409.2639-1-14-1', '1409.2639-2-14-1'], ['1409.2639-1-14-2', '1409.2639-2-14-2'], ['1409.2639-1-4-0', '1409.2639-2-4-0'], ['1409.2639-1-4-1', '1409.2639-2-4-1'], ['1409.2639-1-4-2', '1409.2639-2-4-2'], ['1409.2639-1-0-1', '1409.2639-2-0-1'], ['1409.2639-1-0-2', '1409.2639-2-0-2'], ['1409.2639-1-0-4', '1409.2639-2-0-4'], ['1409.2639-1-31-0', '1409.2639-2-31-0'], ['1409.2639-1-31-1', '1409.2639-2-31-1'], ['1409.2639-1-31-2', '1409.2639-2-31-2'], ['1409.2639-1-23-0', '1409.2639-2-27-0'], ['1409.2639-1-23-1', '1409.2639-2-28-0'], ['1409.2639-1-23-2', '1409.2639-2-28-1'], ['1409.2639-1-23-3', '1409.2639-2-28-2'], ['1409.2639-1-23-4', '1409.2639-2-28-3'], ['1409.2639-1-17-0', '1409.2639-2-18-0'], ['1409.2639-1-17-1', '1409.2639-2-19-0']] | [['1409.2639-1-37-2', '1409.2639-2-38-3'], ['1409.2639-1-34-1', '1409.2639-2-35-1'], ['1409.2639-1-11-0', '1409.2639-2-11-0'], ['1409.2639-1-32-1', '1409.2639-2-33-1'], ['1409.2639-1-12-0', '1409.2639-2-12-0'], ['1409.2639-1-39-0', '1409.2639-2-40-0'], ['1409.2639-1-39-2', '1409.2639-2-40-2'], ['1409.2639-1-21-1', '1409.2639-2-25-1'], ['1409.2639-1-0-3', '1409.2639-2-0-3'], ['1409.2639-1-41-1', '1409.2639-2-42-0']] | [] | [['1409.2639-1-32-0', '1409.2639-2-33-0'], ['1409.2639-1-35-0', '1409.2639-2-36-0'], ['1409.2639-1-35-1', '1409.2639-2-36-1'], ['1409.2639-1-35-2', '1409.2639-2-36-0'], ['1409.2639-1-35-2', '1409.2639-2-36-1'], ['1409.2639-1-39-3', '1409.2639-2-40-3'], ['1409.2639-1-14-0', '1409.2639-2-14-0'], ['1409.2639-1-0-0', '1409.2639-2-0-0'], ['1409.2639-1-41-0', '1409.2639-2-41-0'], ['1409.2639-1-41-0', '1409.2639-2-41-1'], ['1409.2639-1-41-2', '1409.2639-2-42-1']] | [['1409.2639-1-14-3', '1409.2639-2-15-0']] | ['1409.2639-1-10-0', '1409.2639-2-40-4'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1409.2639 | null | null | null | null | null |
1202.0530 | {'1202.0530-1-0-0': 'We demonstrate through numerical simulations and a mean field calculation that immiscible two-phase flow in a porous medium behaves effectively as a Bingham viscoplastic fluid.', '1202.0530-1-0-1': 'This leads to a generalized Darcy equation where the volumetric flow rate depends quadratically on an excess pressure difference in the range of flow rates where the capillary forces compete with the viscous forces.', '1202.0530-1-0-2': 'At higher rates, the flow is Newtonian.', '1202.0530-1-1-0': 'The simultaneous flow of immiscible fluids in porous media [CITATION] lies at the heart of a wide range of important applications ranging from oil recovery to ground water management.', '1202.0530-1-1-1': 'Within the statistical physics community, there has been considerable interest in this problem since the eighties when the fractal structure of fluid invasion was discovered and explored [CITATION].', '1202.0530-1-1-2': 'Such invasion phenomena are transient and may be characterized by the macroscopic flow parameters, such as injected pore volumes of invading fluids change on the same time scale as those associated with the internal flow.', '1202.0530-1-1-3': 'Much less attention, however, has been offered to steady state flow which occurs when macroscopic parameters change slowly compared to those associated with the internal flow [CITATION].', '1202.0530-1-2-0': 'The steady state has recently been studied by Tallakstad et al. [CITATION] in a two-dimensional Hele-Shaw cell filled with glass beads.', '1202.0530-1-2-1': 'Two opposing edges of the cell, are closed, whereas the third edge is open.', '1202.0530-1-2-2': 'At the edge opposing this, there are 15 inlet evenly spaced tubes.', '1202.0530-1-2-3': 'Air and glycerol are injected at equal rates through every second inlet.', '1202.0530-1-2-4': 'As the two immiscible fluids move along the Hele-Shaw cell, they form interpenetrating clusters separated by interfaces.', '1202.0530-1-2-5': 'This constitutes the steady state.', '1202.0530-1-3-0': 'The steady state is characterized by a number of macroscopic parameters: Capillary number [MATH], viscosity ratio [MATH], total volumetric flow rate [MATH], fractional flow rate [MATH], saturation [MATH] and total pressure [MATH].', '1202.0530-1-3-1': 'The capillary number is the ratio between the typical viscous pressure drop across the pores and the typical capillary force due to the interface between the fluids.', '1202.0530-1-3-2': 'Tallakstad et al. observed that the average pressure gradient [MATH] throughout the system scales as a power law with the capillary number [MATH] as, [EQUATION] where [MATH].', '1202.0530-1-4-0': 'More recently, Rassi et al. [CITATION] have measures a [MATH] in the range [MATH] to [MATH] in steady-state two-phase flow of water and air in a three-dimensional porous medium constructed from glass beads.', '1202.0530-1-5-0': 'These observations have profound implications for the description of multiphase flow in porous media.', '1202.0530-1-5-1': 'An important component in the exploitation of oil reservoirs is the use of reservoir simulators.', '1202.0530-1-5-2': 'These are based on effective transport equations for the fluids - and they assume a linear relation between pressure gradients and flow rates [CITATION].', '1202.0530-1-6-0': 'We will in this Letter demonstrate that the scaling relation ([REF]) is a consequence of the immiscible fluids effectively behaving as a Bingham viscoplastic fluid [CITATION] in the porous medium - i.e. a fluid possessing a yield stress and a constant effective viscosity.', '1202.0530-1-6-1': 'Roux and Herrmann [CITATION] have shown that a Bingham fluid flowing in a porous medium will exhibit a quadratic dependency of the flow rate [MATH] on the excess pressure drop.', '1202.0530-1-6-2': 'Hence, we propose that the steady-state two-phase flow is governed by a generalized Darcy equation [EQUATION] where [MATH] is the cross section of the representative elementary volume, [MATH] its length, [MATH] a constant with units of inverse pressure, [MATH] the effective permeability which depends on the saturation [MATH], the saturation-weighted viscosity [MATH].', '1202.0530-1-6-3': '[MATH] is a threshold pressure which depends on [MATH]: if [MATH] is below this threshold, then [MATH].', '1202.0530-1-6-4': 'We have defined the sign of [MATH] to be the same as [MATH].', '1202.0530-1-6-5': 'Hence, the flow rate [MATH] depends quadratically on [MATH].', '1202.0530-1-6-6': 'This is in contrast to the assumptions made in reservoir simulators.', '1202.0530-1-7-0': 'We present in the following numerical simulations based on a model for steady state two-phase flow [CITATION].', '1202.0530-1-7-1': 'The steady state is implemented in two ways.', '1202.0530-1-7-2': 'The first one is a close emulation of the experimental system studied by Tallakstad et al. [CITATION]: the two fluid species are injected at alternate positions along one edge of the model and they escape at the opposite edge.', '1202.0530-1-7-3': 'Hence, the fractional flow [MATH] is kept constant.', '1202.0530-1-7-4': 'The second way of implementing the steady state is to impose periodic boundary conditions in all directions, i.e. implementing a toroidal topology.', '1202.0530-1-7-5': 'In this case, the saturation [MATH] is kept constant since the system is closed.', '1202.0530-1-8-0': 'We find with both implementations [MATH] for different saturations, fractional flows and viscosity ratios when a non-zero threshold pressure [MATH] is assumed.', '1202.0530-1-8-1': 'By adapting the mean field calculation of Kirkpatrick of the conductivity of heterogeneous conductors [CITATION], we derive the effective Darcy equation ([REF]).', '1202.0530-1-9-0': 'In our numerical studies, we model the porous medium by a two-dimensional network of tubes, forming a square lattice tilted by [MATH] with respect to the imposed pressure gradient.', '1202.0530-1-9-1': 'Two immiscible fluids, one more wetting than the other with respect to the pore walls, flow inside the tubes.', '1202.0530-1-9-2': 'The radius, [MATH], of each tube is chosen from a uniform distribution of random numbers in the range [MATH], which introduces the disorder in the system.', '1202.0530-1-9-3': 'Here [MATH] is the length of the tube, which is equal for the all tubes.', '1202.0530-1-9-4': 'In order to incorporate the shape of the pores in between spherical particles (beads) that introduces the capillary effect in the system, each tube is hour-glass shaped so that the capillary pressure [MATH] is proportional to [MATH] where [MATH] is the surface tension and [MATH] is the position of the meniscus in the tube [CITATION].', '1202.0530-1-10-0': 'The flow is driven by setting up an external global pressure drop.', '1202.0530-1-10-1': 'The local flow rate [MATH] in a tube with a pressure difference [MATH] between the two ends of that tube follows the Washburn equation of capillary flow [CITATION] [EQUATION] where [MATH] is the permeability for cylindrical tubes.', '1202.0530-1-10-2': 'The conical shape of the tubes leads only to an overall geometrical factor.', '1202.0530-1-10-3': '[MATH] is the cross-sectional area of the tube and [MATH] is the volume average of the viscosities of the two phases present inside the tube.', '1202.0530-1-10-4': 'The sum over [MATH] runs over all menisci within the tube.', '1202.0530-1-10-5': '[MATH] is the conductivity of the tube.', '1202.0530-1-10-6': 'This equation together with the Kirchhoff equations balancing the the in and out flow at each node are then solved using Cholesky factorization combined with a conjugate gradient solver.', '1202.0530-1-10-7': 'The system is then integrated in time using an explicit Euler scheme.', '1202.0530-1-10-8': 'Inside a tube all menisci move with a speed determined by [MATH].', '1202.0530-1-10-9': 'When a meniscus reaches the end of a tube, new menisci are formed in the neighboring tubes.', '1202.0530-1-10-10': 'A maximum number of six menisci is allowed in a given tube.', '1202.0530-1-10-11': 'Further details can be found in [CITATION].', '1202.0530-1-11-0': 'Simulations are performed with constant flow rate [MATH] which sets the capillary number [MATH] given by [MATH].', '1202.0530-1-11-1': 'The steady-state condition in the simulation can be implemented in two ways.', '1202.0530-1-11-2': 'The first one is to use bi-periodic boundary conditions (BP), by connecting the inlet and outlet rows so that the network takes a toroidal topology [CITATION].', '1202.0530-1-12-0': 'Bi-periodic boundary condition makes the system infinite but closed, and there is a fixed volume of each of the fluids in the system.', '1202.0530-1-12-1': 'Thus the system is in an ensemble where [MATH] and [MATH] are the independent parameters.', '1202.0530-1-12-2': 'In the laboratory, it not possible to implement bi-periodic boundary conditions.', '1202.0530-1-12-3': 'There, the system has open boundary in the direction of total flow and is closed in the perpendicular direction [CITATION].', '1202.0530-1-12-4': 'A series of inlets are used for fluid injection at one boundary where each alternate inlet injects wetting and non-wetting fluids respectively.', '1202.0530-1-12-5': 'Hence, flow rates of the two fluids may be controlled independently.', '1202.0530-1-12-6': 'The control parameters are in this case the total flow rate [MATH] and the fractional flow [MATH] whereas in the simulation with BP, it is [MATH] and [MATH].', '1202.0530-1-12-7': 'Therefore, in order to compare our simulation results directly with the experiment, we also implement open boundary conditions (OB) with a number of inlet links injecting two fluids alternately.', '1202.0530-1-13-0': 'The steady state is identified from the time evolution of the pressure drop across the system.', '1202.0530-1-13-1': 'In Fig. [REF], the average pressure difference at the inlet (upper curve) and at the middle of the system (lower curve) from the pressure at the open outlet boundary are plotted as a function of time for the OB system.', '1202.0530-1-13-2': 'The simulation starts with injecting the two fluids with constant flow rates by alternate inlets in a system completely saturated with the wetting fluid.', '1202.0530-1-13-3': 'Both drainage and imbibition therefore takes place at the pore level creating new menisci.', '1202.0530-1-13-4': 'This increases resistance caused by the capillary forces.', '1202.0530-1-13-5': 'Consequently, in order to keep the flow-rate constant, the global pressure drop increases with time as seen in the initial part of the pressure curves.', '1202.0530-1-13-6': 'The pressure drop at the middle does not increase until the bubbles reaches there, as seen in the very beginning of the bottom curve.', '1202.0530-1-13-7': 'Soon after both the phases reach the outlet, the system reaches steady state at some characteristic time.', '1202.0530-1-14-0': 'We will now present the central result of this Letter, the scaling of steady-state pressure difference [MATH] with [MATH].', '1202.0530-1-14-1': 'Tallakstad et al. [CITATION] reported the scaling relation ([REF]) with an exponent [MATH].', '1202.0530-1-14-2': 'Rassi et al. [CITATION] also reported a similar law, but with a [MATH] in the range 0.30 to 0.45.', '1202.0530-1-14-3': 'A preliminary numerical study using a model very similar to the present one with BP boundary conditions also gave similar results to those reported experimentally, but with a [MATH] that depended on the saturation [CITATION].', '1202.0530-1-14-4': 'We claim in the following that the correct scaling relation between [MATH] and [MATH] is not Eq. ([REF]), but [EQUATION] where [MATH] is a threshold pressure drop.', '1202.0530-1-14-5': 'Moreover, there are two different flow regimes, a capillary dominated low [MATH] regime with [MATH] and a high [MATH] regime with [MATH].', '1202.0530-1-14-6': 'This is shown in Fig. [REF] for (a) PB and (b) OB systems.', '1202.0530-1-14-7': 'Data for different values of saturation [MATH] for PB system and for different values of fractional flows [MATH] for OB system shows exactly same scaling with different values of the threshold pressure.', '1202.0530-1-14-8': 'The overall threshold pressure [MATH] is due to the random distribution of capillary pressure thresholds that came from the fluid interfaces inside disordered sizes of pore throats inside the porous medium.', '1202.0530-1-14-9': 'These capillary pressure threshold values introduce a lower cutoff in the local pressure differences that required to overcome the local capillary barriers.', '1202.0530-1-14-10': 'We note that in the experimental study by Tallakstad et al. [CITATION] one of the fluids were percolating in the system [CITATION], making [MATH] in this case.', '1202.0530-1-14-11': 'In the numerical studies reported in [CITATION], there is a visible curvature in the scaling plots corresponding to Eq. ([REF]), indicating that a non-zero [MATH] has been ignored and hence resulting in a wandering value for [MATH].', '1202.0530-1-15-0': 'Tallakstad et al. [CITATION] presented a theoretical argument for [MATH] based on the observation that the flow occurs in channels.', '1202.0530-1-15-1': 'The relation between the flow rate in a channel and the pressure drop is assumed to be linear.', '1202.0530-1-15-2': 'They then argued that the number of channels also is proportional to the pressure drop [MATH] - and Eq. ([REF]) follows.', '1202.0530-1-15-3': 'This argument, however, does not predict a non-zero threshold pressure drop [MATH].', '1202.0530-1-15-4': 'It furthermore does not describe well saturations where no channels can be observed.', '1202.0530-1-15-5': 'We note that the experiments were done for one value of [MATH]), and for the range of [MATH] that was studied, open channels were always present.', '1202.0530-1-16-0': 'The numerical results may be expressed through a generalized Darcy equation, Eq. ([REF]).', '1202.0530-1-16-1': 'We now derive this equation in a mean field approximation.', '1202.0530-1-16-2': 'Eq. ([REF]) has the form [MATH].', '1202.0530-1-16-3': 'As the menisci may move to set up capillary forces that counteract [MATH], there will be no flow until [MATH] reaches a threshold value.', '1202.0530-1-16-4': 'This threshold value we define as [MATH] - which may be also be zero or negative depending on whether there are menisci in the tube and the sign of the sum of capillary forces.', '1202.0530-1-16-5': 'Hence, effectively the tubes will behave as [EQUATION]', '1202.0530-1-16-6': 'This is the rheology of a Bingham viscoplastic fluid, but with an effective yield threshold which may be negative.', '1202.0530-1-16-7': 'We also note that the threshold value may change when [MATH] changes sign.', '1202.0530-1-17-0': 'In the following, we base our calculation on the mean-field theory for conduction in percolating systems of Kirkpatrick [CITATION].', '1202.0530-1-17-1': 'Our starting point is a regular lattice with coordination number [MATH].', '1202.0530-1-17-2': 'Each tube in the lattice has a non-linear conductivity [MATH], Eq. ([REF]), drawn from a spatially uncorrelated probability distribution [MATH].', '1202.0530-1-18-0': 'We now set up a pressure difference [MATH] between two neighboring nodes [MATH] and [MATH].', '1202.0530-1-18-1': 'A current [MATH] then flows into the network at [MATH] and leaves the network at [MATH].', '1202.0530-1-18-2': 'The total conductivity between nodes [MATH] and [MATH] is defined as [MATH].', '1202.0530-1-19-0': 'We now pose the following question: can we find an equivalent homogeneous network where all tubes have the same conductivity [MATH] so that the total conductivity between [MATH] and [MATH] in this equivalent network becomes [MATH]?', '1202.0530-1-20-0': 'For symmetry reasons, the current in the equivalent, homogeneous network that flows though the tube connecting nodes [MATH] and [MATH] is [MATH].', '1202.0530-1-20-1': 'Hence, we have the relation [MATH].', '1202.0530-1-20-2': 'We eliminate [MATH] between these two expressions to find [EQUATION]', '1202.0530-1-20-3': 'We now replace the averaged tube between nodes [MATH] and [MATH] carrying a conductance [MATH] by a tube carrying a conductance [MATH] drawn from the probability density [MATH] while keeping the all other tubes in the equivalent homogenized network unchanged.', '1202.0530-1-20-4': 'The conductance between nodes [MATH] and [MATH] then becomes [EQUATION]', '1202.0530-1-20-5': 'Hence, we have [EQUATION]', '1202.0530-1-20-6': 'Rewriting and averaging over this equation gives [EQUATION]', '1202.0530-1-20-7': 'This equation provides a self-consistent expression for the equivalent conductivity.', '1202.0530-1-21-0': 'We now assume that a fraction [MATH] of the tubes have conductance [MATH], the others have zero conductance.', '1202.0530-1-21-1': 'The probability distribution for [MATH] is then [EQUATION]', '1202.0530-1-21-2': 'Combining this distribution with the average in Eq. ([REF]) gives [EQUATION]', '1202.0530-1-21-3': 'This is the central result of Kirkpatrick [CITATION].', '1202.0530-1-21-4': 'The total conductance [MATH] of the network is proportional to [MATH].', '1202.0530-1-22-0': 'At some pressure difference across the system, [MATH], a fraction [MATH] of the tubes conduct.', '1202.0530-1-22-1': 'The others contain interfaces that are locked in place by the capillary forces.', '1202.0530-1-22-2': 'We define a critical pressure [MATH] through the equation [EQUATION]', '1202.0530-1-22-3': 'This is the minimum pressure difference necessary to have flow in the network.', '1202.0530-1-22-4': 'We expand [MATH] for [MATH] and combine with Eq. ([REF]) to find [EQUATION] where [MATH] is a proportionality constant only dependent on the topology of the network.', '1202.0530-1-22-5': 'Integrating the equation [MATH] then gives the final result [EQUATION]', '1202.0530-1-22-6': 'Hence, it is necessary that the higher order terms in the expansion of [MATH] around [MATH] are small.', '1202.0530-1-22-7': 'Eq. ([REF]) is equivalent to Eq. ([REF]).', '1202.0530-1-22-8': 'We note that Eq. ([REF]) is independent of the dimensionality of the porous medium.', '1202.0530-1-23-0': 'To summarize, we have demonstrated numerically and through a mean field calculation that steady-state immiscible two-phase flow in a porous medium behaves as a Bingham viscoplastic fluid.', '1202.0530-1-23-1': 'This leads to a non-linear Darcy equation where the volumetric flow rate depends quadratically on an excess pressure drop at capillary numbers at which the capillary forces compete with the viscous forces.', '1202.0530-1-23-2': 'At higher flow rates, the flow becomes Newtonian.', '1202.0530-1-24-0': 'We thank D. Bedeaux, E. G. Flekkoy, S. Kjelstrup, K. J. Maaloy and L. Talon for useful discussions.', '1202.0530-1-24-1': 'This work was partially supported by the Norwegian Research Council through grant no. 193298.', '1202.0530-1-24-2': 'We thank NOTUR for allocation of computer time.'} | {'1202.0530-2-0-0': 'We demonstrate through numerical simulations and a mean-field calculation that immiscible two-phase flow in a porous medium behaves effectively as a Bingham viscoplastic fluid.', '1202.0530-2-0-1': 'This leads to a generalized Darcy equation where the volumetric flow rate depends quadratically on an excess pressure difference in the range of flow rates where the capillary forces compete with the viscous forces.', '1202.0530-2-0-2': 'At higher rates, the flow is Newtonian.', '1202.0530-2-1-0': 'The simultaneous flow of immiscible fluids in porous media [CITATION] lies at the heart of a wide range of important applications ranging from oil recovery to ground water management.', '1202.0530-2-1-1': 'Within the statistical physics community, there has been considerable interest in this problem since the eighties when the fractal structure of fluid invasion was discovered and explored [CITATION].', '1202.0530-2-1-2': 'Such invasion phenomena are transient and may be characterized by the macroscopic flow parameters, such as injected pore volumes of invading fluids change on the same time scale as those associated with the internal flow.', '1202.0530-2-1-3': 'Much less attention, however, has been offered to steady state flow which occurs when macroscopic parameters change slowly compared to those associated with the internal flow [CITATION].', '1202.0530-2-2-0': 'The steady state has recently been studied experimentally by Tallakstad et al. [CITATION] in a two-dimensional Hele-Shaw cell filled with glass beads.', '1202.0530-2-2-1': 'There are 15 evenly spaced tubes at one edge of the cell.', '1202.0530-2-2-2': 'Air and glycerol are injected at equal rates through each alternate inlet.', '1202.0530-2-2-3': 'Fluids leave at the opposing edge, which is kept open.', '1202.0530-2-2-4': 'The other two edges of the cell, perpendicular to the direction of overall flow, are closed.', '1202.0530-2-2-5': 'As the two immiscible fluids move along the Hele-Shaw cell, they form interpenetrating clusters separated by interfaces.', '1202.0530-2-2-6': 'This constitutes the steady state.', '1202.0530-2-3-0': 'The steady state is characterized by a number of macroscopic parameters: Capillary number [MATH], viscosity ratio [MATH], total volumetric flow rate [MATH], non-wetting fractional flow rate [MATH], non-wetting saturation [MATH] and pressure [MATH].', '1202.0530-2-3-1': 'The capillary number is the ratio between the typical viscous pressure drop across the pores and the typical capillary force due to the interface between the fluids.', '1202.0530-2-3-2': 'Tallakstad et al. observed that the average pressure gradient [MATH] throughout the system scales as a power law with the capillary number [MATH] as, [EQUATION] where [MATH].', '1202.0530-2-4-0': 'More recently, Rassi et al. [CITATION] have measured the exponent [MATH] which varies in the range of [MATH] to [MATH] depending on the saturation, in steady-state two-phase flow of water and air in a three-dimensional porous medium constructed from glass beads.', '1202.0530-2-5-0': 'These observations have profound implications on the description of multiphase flow in porous media.', '1202.0530-2-5-1': 'An important application would be in the reservoir simulators, used in the exploitation of oil reservoirs.', '1202.0530-2-5-2': 'They are based on effective transport equations for the fluids where a linear relation between the pressure gradients and flow rates [CITATION] is assumed so far.', '1202.0530-2-6-0': 'We will in this Letter demonstrate that, in the regime where capillary forces are comparable to the viscous forces (low [MATH]), the capillary effects at the interfaces between the immiscible fluids effectively creates a yield threshold, making the fluids reminiscent of a Bingham viscoplastic fluid [CITATION] in the porous medium - i.e. a fluid possessing a yield stress and a constant effective viscosity.', '1202.0530-2-6-1': 'This introduces a overall threshold pressure [MATH] in the system due to the random distribution of capillary pressure thresholds.', '1202.0530-2-6-2': 'We therefore propose, and will show in the following via numerical simulations and a mean field calculation, that the steady-state two-phase flow in porous media in this flow regime is governed by a generalized Darcy equation [EQUATION] where [MATH] is the sign function.', '1202.0530-2-6-3': 'Here [MATH] is the cross section of the representative elementary volume, [MATH] is its length, [MATH] a constant with units of inverse pressure, [MATH] the effective permeability which depends on the saturation [MATH] and [MATH] is the saturation-weighted viscosity given by [MATH], where [MATH] and [MATH] are the viscosities of wetting and non-wetting fluids respectively.', '1202.0530-2-7-0': 'As a result, the correct scaling relation in general between [MATH] and [MATH] is not Eq. ([REF]), but [EQUATION] with [MATH] for low [MATH].', '1202.0530-2-7-1': 'The relation ([REF]) is just a special case of this relation where [MATH].', '1202.0530-2-8-0': 'There is another flow regime for high [MATH] where the flow is linear with the excess pressure drop which corresponds to [MATH].', '1202.0530-2-8-1': 'This regime is therefore characterized by the standard Darcy equation [MATH].', '1202.0530-2-8-2': 'Here [MATH] and the threshold pressure is not relevant.', '1202.0530-2-9-0': 'The physical existence of the global threshold pressure [MATH] and the quadratic and linear dependence of [MATH] on [MATH] in the two regimes for the system of immiscible fluids can be understood very intuitively following the argument of Roux and Herrmann [CITATION] for networks with link conductances having characteristics like a Bingham fluid: The essential ingredient is a threshold pressure for each link, distributed according to some probability density.', '1202.0530-2-9-1': 'The sum over all the thresholds over a continuous flow path throughout the entire system gives the total threshold pressure along that path and [MATH] corresponds to the minimum sum among all such possible paths.', '1202.0530-2-9-2': 'Now, if we raise the pressure difference across the network by a value [MATH], a number [MATH] of tubes will cross their flow thresholds.', '1202.0530-2-9-3': 'With a reasonably smooth distribution of thresholds, we will have that [MATH].', '1202.0530-2-9-4': 'The conductance of the network [MATH] will then change by an amount [MATH], and [MATH].', '1202.0530-2-9-5': 'An integration over this equation leads to Eq. ([REF]) with the appearance of [MATH].', '1202.0530-2-9-6': 'Moreover, at a very high value of [MATH], when all the links have crossed their individual flow thresholds, each link behaves linearly with excess pressure drop, making [MATH] constant with [MATH] and effectively the overall flow rate becomes linear with the excess pressure drop in the high [MATH] regime.', '1202.0530-2-10-0': 'In the following, we will first present our numerical simulations.', '1202.0530-2-10-1': 'Afterwards, we will derive the effective Darcy equation ([REF]), adapting the mean-field calculation of the conductivity of heterogeneous conductors by Kirkpatrick [CITATION].', '1202.0530-2-11-0': 'In our numerical studies, the porous medium is modeled by a two-dimensional network of tubes, forming a square lattice tilted by [MATH] with respect to the imposed pressure gradient.', '1202.0530-2-11-1': 'Two immiscible fluids, one is more wetting than the other with respect to the pore walls, flow inside the tubes.', '1202.0530-2-11-2': 'Disorder is introduced in the system by choosing the radius [MATH] of each tube randomly from a uniform distribution of random numbers in the range [MATH], where [MATH] is the length of the tubes.', '1202.0530-2-11-3': 'In order to incorporate the shape of the pores in between spherical particles (beads) that introduces the capillary effect in the system, each tube is considered hour-glass shaped so that the capillary pressure [MATH] at a meniscus at position [MATH] is proportional to [MATH] where [MATH] is the surface tension [CITATION].', '1202.0530-2-12-0': 'The flow is driven by setting up an external global pressure drop.', '1202.0530-2-12-1': 'The local flow rate [MATH] in a tube with a pressure difference [MATH] between the two ends of that tube follows the Washburn equation of capillary flow [CITATION] [EQUATION] where [MATH] is the permeability for cylindrical tubes.', '1202.0530-2-12-2': 'The conical shape of the tubes leads only to an overall geometrical factor.', '1202.0530-2-12-3': 'Here [MATH] is the cross-sectional area of the tube and [MATH] is the volume average of the viscosities of the two phases present inside the tube.', '1202.0530-2-12-4': 'Hence, it is a function of the saturation [MATH] in the tube.', '1202.0530-2-12-5': 'The sum over [MATH] runs over all menisci within the tube.', '1202.0530-2-12-6': '[MATH] is thus the tube conductivity.', '1202.0530-2-12-7': 'The set consisting of one equation ([REF]) per tube, together with the Kirchhoff equations balancing the in and out flow at each node are then solved using Cholesky factorization combined with a conjugate gradient solver.', '1202.0530-2-12-8': 'The system is then integrated in time using an explicit Euler scheme.', '1202.0530-2-12-9': 'Inside a tube all menisci move with a speed determined by [MATH].', '1202.0530-2-12-10': 'When a meniscus reaches the end of a tube, new menisci are formed in the neighboring tubes.', '1202.0530-2-12-11': 'Further details of the model and how the menisci are moved can be found in [CITATION].', '1202.0530-2-13-0': 'The steady-state condition in the simulation is achieved in two ways.', '1202.0530-2-13-1': 'The conventional way is to use bi-periodic boundary conditions (BP), by connecting the inlet and outlet rows so that the network takes a toroidal topology [CITATION].', '1202.0530-2-13-2': 'It is then initialized by filling with two fluids randomly or sequentially so that the network attains the desired saturation [MATH].', '1202.0530-2-13-3': 'As the system is closed in this boundary condition, the saturation of the network [MATH] is an independent parameter which remains constant throughout the simulation along with the total flow rate [MATH] whereas the fractional flow [MATH] fluctuates over time.', '1202.0530-2-14-0': 'It is not possible to implement bi-periodic boundary condition in the experiments by Tallakstad et al. [CITATION], where two fluids are injected at one edge of the system through a series of alternate inlets and the opposite edge is kept open.', '1202.0530-2-14-1': 'Flow rates of the two fluids may be controlled independently there.', '1202.0530-2-14-2': 'The control parameters in this case are the total flow rate [MATH] and the fractional flow [MATH] whereas the saturation [MATH] fluctuates.', '1202.0530-2-14-3': 'Therefore, in order to have a close emulation of the experimental system, we also implement open boundary conditions (OB) in our simulation here, controlling the individual flow rates of inlet links.', '1202.0530-2-14-4': 'The simulation starts with injecting the two fluids with constant flow rates in a system completely saturated with the wetting fluid.', '1202.0530-2-14-5': 'Both drainage and imbibition therefore takes place at the pore level creating new menisci.', '1202.0530-2-14-6': 'Away from the inlets, the fluids mix and a steady state is attained as in the experiment.', '1202.0530-2-15-0': 'Simulations are performed with constant flow rate [MATH] which sets the capillary number [MATH] given by [MATH].', '1202.0530-2-15-1': 'A range of [MATH] from [MATH] to [MATH] is considered for a network of [MATH] links and an average over [MATH] different samples is taken for each simulation.', '1202.0530-2-15-2': 'We report results for [MATH] and [MATH] with [MATH] and [MATH] for PB, and [MATH] with [MATH] and [MATH] for OB.', '1202.0530-2-15-3': 'The steady state is identified from the total pressure drop [MATH], which starts to fluctuate over an average value as steady state is reached.', '1202.0530-2-15-4': 'By increasing and then lowering the total flux [MATH] we also verify that it returns to the same steady state.', '1202.0530-2-15-5': 'In order to verify Eq. ([REF]), we now need to calculate [MATH].', '1202.0530-2-15-6': 'Roux and Herrmann [CITATION], for their Bingham system, determined [MATH] by decreasing the external current from a large value and identifying the current paths by a search algorithm.', '1202.0530-2-15-7': 'This procedure is not feasible here as the flow patterns and menisci positions change with global flow rate and time due to fluid instabilities.', '1202.0530-2-15-8': 'Moreover, it is also not possible to follow this in experiments as it would necessitate the knowledge of flow rates at every single pore.', '1202.0530-2-15-9': 'We therefore measure [MATH] with a minimization procedure.', '1202.0530-2-15-10': 'A series of trial [MATH] values are considered, for which the slope ([MATH]) and the least square fit errors, when fitted to Eq. ([REF]), are calculated.', '1202.0530-2-15-11': '[MATH] is then identified corresponding to the minimum value of the error or the best fit.', '1202.0530-2-15-12': 'This is shown in the inset of Fig. [REF](a) for [MATH] and [MATH].', '1202.0530-2-15-13': 'In Table [REF], the absolute values of all [MATH]s, identified by the same procedure, are listed.', '1202.0530-2-15-14': 'In Fig. [REF], [MATH] is then plotted with [MATH] for (a)BP and (b)OB according to Eq. ([REF]).', '1202.0530-2-15-15': 'Interestingly, for different saturations ([MATH]), fractional flows ([MATH]) and boundary conditions, the minimum error corresponds to different values of [MATH] but the same value for [MATH] within error bar for low [MATH] regime.', '1202.0530-2-15-16': 'More surprisingly, [MATH] is found independent to the viscosity ratio which is a strong support towards its intuitive physical description stated before.', '1202.0530-2-15-17': 'As [MATH] is related to the sum of capillary thresholds ([MATH]) over connecting paths and [MATH] does not depend on the viscosities, [MATH] also should not and that is what we found here.', '1202.0530-2-15-18': 'A sharp crossover is also seen for high [MATH] regime with [MATH] where the flow is linear characterized by standard Darcy equation.', '1202.0530-2-16-0': 'In the experiments by Rassi et al. [CITATION], the exponent [MATH], when measured according to Eq. ([REF]) is found to vary from [MATH] to [MATH] depending on the saturation.', '1202.0530-2-16-1': 'A preliminary numerical study [CITATION] assuming Eq. ([REF]) using a very similar model with only BP, also reports similar dependency of [MATH] on saturation with a visible curvature in the scaling plots.', '1202.0530-2-16-2': 'These clearly indicate a non-zero [MATH] that has been ignored hence resulting in a wandering value of [MATH].', '1202.0530-2-16-3': 'Interestingly, the same experimental data by Rassi et al. [CITATION] are found consistent with [MATH] when reanalyzed using Eq. ([REF]) [CITATION].', '1202.0530-2-16-4': 'In the experiment by Tallakstad et al. [CITATION], [MATH] is found as [MATH] with scaling relation ([REF]) which is due to the fact that one of the fluids was percolating in their system [CITATION], making [MATH].', '1202.0530-2-17-0': 'In support of our numerical results, we now derive the generalized Darcy equation, Eq. ([REF]) in a mean field approximation.', '1202.0530-2-17-1': 'Eq. ([REF]) describes instantaneous the flow in a single tube.', '1202.0530-2-17-2': 'Time averaging this equation under steady state conditions leads to an effective flow equation for the single tube [CITATION] [EQUATION] where [MATH] is an effective flow threshold that depends on the shape of the tube.', '1202.0530-2-17-3': 'The effective viscosity that enters into [MATH] is the saturation-weighted sum of the viscosities of each liquid, where the saturation is time averaged over the tube.', '1202.0530-2-18-0': 'The square root singularity near [MATH] is caused by a saddle-node bifurcation and, hence, is a universal feature of the system [CITATION].', '1202.0530-2-19-0': 'The effective conductivity is thus [EQUATION]', '1202.0530-2-20-0': 'In the following, we will derive Eq. ([REF]) based on the mean-field theory originally developed for calculating the conductivity of percolating systems by Kirkpatrick [CITATION].', '1202.0530-2-20-1': 'Our starting point is a regular lattice with coordination number [MATH].', '1202.0530-2-20-2': 'Each tube in the lattice has a non-linear conductivity [MATH], Eq. ([REF]).', '1202.0530-2-20-3': 'The flow thresholds of the tubes are drawn from a spatially uncorrelated probability distribution [MATH].', '1202.0530-2-21-0': 'The mean-field calculation proceeds by focusing on one tube inside the network.', '1202.0530-2-21-1': 'We then replace the rest of the network by an equivalent homogeneous network where all tubes have the same conductivity [MATH] so that the tube we have singled out experiences the same average effect from the homogeneous network as from the original network.', '1202.0530-2-21-2': 'We then average over this last conductance and determine [MATH] in a self-consistent way.', '1202.0530-2-22-0': 'There is one caveat.', '1202.0530-2-22-1': "In Kirkpatrick's original calculation, the conductances were assumed to be linear, whereas in our case, the conductances are highly non-linear, see Eq. ([REF]).", '1202.0530-2-22-2': 'However, the end result of the calculation, [EQUATION] is the same.', '1202.0530-2-22-3': 'This equation provides a self-consistent expression for the equivalent conductivity [MATH].', '1202.0530-2-23-0': 'In terms of the distribution of flow thresholds [MATH], Eq. ([REF]) becomes [EQUATION]', '1202.0530-2-23-1': 'We now combine this expression with Eq. ([REF]), finding [EQUATION] where [MATH] is cumulative probability to find a flow threshold less than or equal to [MATH].', '1202.0530-2-24-0': 'By setting [MATH] in Eq. ([REF]), we determine the effective flow threshold [MATH] for the effective tubes: [EQUATION]', '1202.0530-2-25-0': 'If we now set [MATH] in Eq. ([REF]) and expand to lowest order in [MATH], we find [EQUATION] where we have used that [MATH] to lowest order.', '1202.0530-2-25-1': 'We integrate Eq. ([REF]) and find [EQUATION]', '1202.0530-2-25-2': 'If there are [MATH] tubes in a cross section of the homogeneous network orthogonal to the flow direction, then [MATH], [MATH] and [MATH].', '1202.0530-2-25-3': 'Hence, the generalized Darcy equation ([REF]) follows.', '1202.0530-2-26-0': 'There are corrections associated with larger exponents to Eq. ([REF]) originating from two sources.', '1202.0530-2-26-1': 'The first one comes from solving Eq. ([REF]).', '1202.0530-2-26-2': 'The prefactor of the leading correction from this source is about 7% of the prefactor of the dominating term.', '1202.0530-2-26-3': 'The second source of correction terms comes from the linearization of [MATH] around the value [MATH].', '1202.0530-2-26-4': 'There are no reasons to assume strong non-linear corrections in this region.', '1202.0530-2-27-0': 'To summarize, we have demonstrated numerically and through a mean field calculation that steady-state immiscible two-phase flow in a porous medium behaves similar to a Bingham viscoplastic fluid.', '1202.0530-2-27-1': 'This leads to a non-linear Darcy equation where the volumetric flow rate depends quadratically on an excess pressure drop at capillary numbers at which the capillary forces compete with the viscous forces.', '1202.0530-2-27-2': 'At higher flow rates, the flow becomes Newtonian.', '1202.0530-2-28-0': 'We thank D. Bedeaux, E. G. Flekkoy, S. Kjelstrup, K. J. Maaloy and L. Talon for useful discussions.', '1202.0530-2-28-1': 'This work was partially supported by the Norwegian Research Council through grant no. 193298.', '1202.0530-2-28-2': 'We thank NOTUR for allocation of computer time.'} | [['1202.0530-1-17-1', '1202.0530-2-20-1'], ['1202.0530-1-3-1', '1202.0530-2-3-1'], ['1202.0530-1-3-2', '1202.0530-2-3-2'], ['1202.0530-1-10-0', '1202.0530-2-12-0'], ['1202.0530-1-10-1', '1202.0530-2-12-1'], ['1202.0530-1-10-2', '1202.0530-2-12-2'], ['1202.0530-1-10-4', '1202.0530-2-12-5'], ['1202.0530-1-10-7', '1202.0530-2-12-8'], ['1202.0530-1-10-8', '1202.0530-2-12-9'], ['1202.0530-1-10-9', '1202.0530-2-12-10'], ['1202.0530-1-23-1', '1202.0530-2-27-1'], ['1202.0530-1-23-2', '1202.0530-2-27-2'], ['1202.0530-1-2-4', '1202.0530-2-2-5'], ['1202.0530-1-2-5', '1202.0530-2-2-6'], ['1202.0530-1-1-0', '1202.0530-2-1-0'], ['1202.0530-1-1-1', '1202.0530-2-1-1'], ['1202.0530-1-1-2', '1202.0530-2-1-2'], ['1202.0530-1-1-3', '1202.0530-2-1-3'], ['1202.0530-1-0-1', 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['1202.0530-1-10-3', '1202.0530-2-12-3'], ['1202.0530-1-10-6', '1202.0530-2-12-7'], ['1202.0530-1-23-0', '1202.0530-2-27-0'], ['1202.0530-1-2-0', '1202.0530-2-2-0'], ['1202.0530-1-2-3', '1202.0530-2-2-2'], ['1202.0530-1-0-0', '1202.0530-2-0-0'], ['1202.0530-1-5-0', '1202.0530-2-5-0'], ['1202.0530-1-5-2', '1202.0530-2-5-2'], ['1202.0530-1-8-1', '1202.0530-2-10-1'], ['1202.0530-1-4-0', '1202.0530-2-4-0'], ['1202.0530-1-13-2', '1202.0530-2-14-4']] | [] | [['1202.0530-1-17-0', '1202.0530-2-20-0'], ['1202.0530-1-17-2', '1202.0530-2-20-2'], ['1202.0530-1-17-2', '1202.0530-2-20-3'], ['1202.0530-1-9-2', '1202.0530-2-11-2'], ['1202.0530-1-10-5', '1202.0530-2-12-6'], ['1202.0530-1-10-11', '1202.0530-2-12-11'], ['1202.0530-1-2-2', '1202.0530-2-2-1'], ['1202.0530-1-14-3', '1202.0530-2-16-1'], ['1202.0530-1-14-10', '1202.0530-2-16-4'], ['1202.0530-1-14-11', '1202.0530-2-16-1'], ['1202.0530-1-14-11', '1202.0530-2-16-2'], ['1202.0530-1-6-0', '1202.0530-2-6-0'], ['1202.0530-1-6-2', '1202.0530-2-6-2'], ['1202.0530-1-5-1', '1202.0530-2-5-1'], ['1202.0530-1-19-0', '1202.0530-2-21-1'], ['1202.0530-1-12-5', '1202.0530-2-14-1'], ['1202.0530-1-12-6', '1202.0530-2-14-2'], ['1202.0530-1-12-7', '1202.0530-2-14-3']] | [['1202.0530-1-20-7', '1202.0530-2-22-3']] | ['1202.0530-1-16-2', '1202.0530-1-22-7', '1202.0530-2-19-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1202.0530 | null | null | null | null | null |
astro-ph-0201322 | {'astro-ph-0201322-1-0-0': 'We calculate the Gamma-Ray Burst (GRB) afterglow light-curves from a relativistic jet as seen by observers at a wide range of viewing angles, [MATH], from the jet axis.', 'astro-ph-0201322-1-0-1': 'We describe three increasingly more realistic models and compare the resulting light-curves.', 'astro-ph-0201322-1-0-2': 'An off-axis observer at [MATH] (outside the initial opening of the jet) should see a rising light curve at early times, the flux peaking when the jet Lorentz factor [MATH].', 'astro-ph-0201322-1-0-3': 'After this time the flux is not very different from that seen by an observer along the jet axis ([MATH]).', 'astro-ph-0201322-1-0-4': 'A strong linear polarization ([MATH]) may occur near the peak in the light curve, and slowly decay with time.', 'astro-ph-0201322-1-0-5': 'An observer at [MATH] should see a light curve very similar to that for an on-axis observer ([MATH]).', 'astro-ph-0201322-1-0-6': 'We apply our results to the recently reported observation of a very bright optical transient by the SDSS, whose isotropic luminosity was a factor [MATH] larger than the peak brightness of supernovae.', 'astro-ph-0201322-1-0-7': 'We find that the data for this event are consistent with a GRB afterglow provided that the observer is located off-axis at [MATH], and that the burst occurred [MATH]1.5-3 days before the first SDSS observation.', 'astro-ph-0201322-1-0-8': 'We also discuss the proposed connection between supernova 1998bw and GRB 980425.', 'astro-ph-0201322-1-1-0': '# Introduction', 'astro-ph-0201322-1-2-0': 'Gamma-Ray Bursts (GRBs) are explosions which release roughly 10[MATH] erg in the form of kinetic energy of highly relativistic material (Frail et al. 2001, Panaitescu Kumar 2001).', 'astro-ph-0201322-1-2-1': 'Many GRBs appear to be highly non-spherical explosions, as evidenced by a nearly-achromatic break in the light-curve (e.g. Harrison et al. 1999; Stanek et al. 1999).', 'astro-ph-0201322-1-2-2': 'Highly relativistic jets are "visible" when our line of sight is within the jet aperture ([MATH]), otherwise, because of relativistic beaming of photons away from our line-of-sight, the object is too dim.', 'astro-ph-0201322-1-2-3': 'As the jet decelerates, the relativistic beaming becomes less severe and the emission from the jet becomes detectable to observers at larger viewing angles.', 'astro-ph-0201322-1-2-4': 'Granot et al. (2001) have shown that the light curve seen by an observer located within the initial jet aperture ([MATH]) is very similar to that for an on-axis observer ([MATH]).', 'astro-ph-0201322-1-2-5': 'In this Letter we study the afterglow light-curves for off-axis locations ([MATH]), focusing on observers lying outside of the initial jet opening angle ([MATH]).', 'astro-ph-0201322-1-2-6': 'Dalal et al. (2001) have presented a simple model to calculate the flux in this case.', 'astro-ph-0201322-1-2-7': 'We reanalyze this model (2.1) and consider more realistic models (2.2 2.3) to calculate light-curves.', 'astro-ph-0201322-1-3-0': 'The Sloan Digital Sky Survey (SDSS) team found an extremely bright optical transient (OT) (about 50 times brighter than supernova peak luminosity), at z=0.385, (Vanden Berk et al. 2001).', 'astro-ph-0201322-1-3-1': 'Among other possibilities, they mention that this OT could be a GRB afterglow where the jet was pointing away from us.', 'astro-ph-0201322-1-3-2': 'We investigate this possibility in some detail in 3.', 'astro-ph-0201322-1-4-0': 'A very bright radio supernova with peculiar properties, SN 1998bw, was observed within [MATH] of GRB 980425, the explosions having occurred [MATH]1 day apart.', 'astro-ph-0201322-1-4-1': 'In 4 we analyze the suggestion of Woosley, Eastman, Schmidt (1999) that a relativistic jet emanating from the SN explosion and pointing away from us could explain the GRB observations and optical data.', 'astro-ph-0201322-1-5-0': '# Modeling the off axis emission', 'astro-ph-0201322-1-6-0': 'In this section we calculate the afterglow light curves of jetted GRBs, as seen by observers at different viewing angles, [MATH], w.r.t the symmetry axis of the jet.', 'astro-ph-0201322-1-6-1': 'For simplicity, we consider only a jet propagating into a homogeneous medium.', 'astro-ph-0201322-1-6-2': 'In order to improve our understanding of the underlying physics and in order to check how general the results are, we explore three different models with an increasing level of complexity.', 'astro-ph-0201322-1-7-0': '## Model 1: A Point Source at the Jet Axis', 'astro-ph-0201322-1-8-0': 'We begin with a simple model, where for [MATH] the light curve follows the results of simple jet models (Rhoads 1999; Sari, Piran Halpern 1999, hereafter R-SPH99), and for [MATH] the light curves are calculated assuming the emission is from a point source that moves along the jet axis.', 'astro-ph-0201322-1-8-1': 'A similar model was used by Dalal et al. (2001), however they concentrated on the bolometric luminosity, while we calculate the flux per unit frequency which is more useful for comparison with observations.', 'astro-ph-0201322-1-8-2': 'The on-axis light curve exhibits a jet break at (R-SPH99): [EQUATION] where [MATH] is the isotropic equivalent energy in units of [MATH] erg, [MATH] is the ambient density in [MATH] and [MATH] is the cosmological redshift of the source.', 'astro-ph-0201322-1-8-3': 'At [MATH], [MATH] is taken from Sari, Piran and Narayan (1998), while at [MATH] the temporal scalings of the break frequencies and peak flux change according to R-SPH99.', 'astro-ph-0201322-1-8-4': 'The observed flux density from a point source is [EQUATION] where [MATH] and [MATH] are the spectral luminosity and frequency in the local rest frame of the jet, [MATH] and [MATH] are the angular and luminosity distances to the source, [MATH] is the Lorentz factor of the source and [MATH] is the angle between the direction of motion of the source and the direction to the observer in the observer frame (in our case [MATH]).', 'astro-ph-0201322-1-8-5': 'Since [MATH], where [MATH] and [MATH] are the observed time and frequency, we obtain that [EQUATION] where [MATH] and [MATH] are the observed time and frequency for an observer at [MATH].', 'astro-ph-0201322-1-8-6': 'One therefore obtains that [EQUATION] where, for simplicity, we take [MATH] at [MATH] and [MATH] at [MATH].', 'astro-ph-0201322-1-9-0': 'The light curves obtained using equation [REF] are shown by the dashed lines in Figure [REF].', 'astro-ph-0201322-1-9-1': 'At first [MATH] and [MATH] is the dominant term in equation [REF], giving a sharp rise in the light curve.', 'astro-ph-0201322-1-9-2': 'Once [MATH] becomes [MATH] the flux begins to decay, asymptotically approaching the on-axis light curve.', 'astro-ph-0201322-1-9-3': 'The light curve for off-axis observers peaks when [MATH].', 'astro-ph-0201322-1-10-0': '3.3infigure1.eps [] B-band luminosity for models 1 (dashed lines) and 2 (solid lines), for [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], where [MATH]) is the fraction of the internal energy in the magnetic field (electrons) and [MATH] is the power law index of the electron energy distribution.', 'astro-ph-0201322-1-10-1': 'Model 1 is scaled down by a factor of 2.5, to help compare between the two models.', 'astro-ph-0201322-1-10-2': 'The inset shows the linear polarization for model 2 for the above parameters.', 'astro-ph-0201322-1-11-0': 'The main advantage of this model is that it is very simple, and nevertheless gives reasonable results for [MATH].', 'astro-ph-0201322-1-11-1': 'Its main drawback is that it is not physical for [MATH] at [MATH].', 'astro-ph-0201322-1-11-2': 'This is because for [MATH] the observed radiation is initially dominated by emission from the material within an angle of [MATH] around the line of sight, while in model 1 the emission is always only from along the jet axis, and therefore the dominant contribution to the emission is missing, until the time when [MATH].', 'astro-ph-0201322-1-11-3': 'This problem is overcome by our next model.', 'astro-ph-0201322-1-12-0': '## Model 2: A Homogeneous Jet', 'astro-ph-0201322-1-13-0': 'This model is described in Kumar Panaitescu (2000), and here we briefly point out its main features.', 'astro-ph-0201322-1-13-1': 'The Lorentz factor and energy density per unit solid angle are considered to be independent of angle [MATH] within the jet aperture.', 'astro-ph-0201322-1-13-2': 'The decrease of the Lorentz factor of the jet with time is calculated from the mass and energy conservation equations, and the sideway expansion speed of the jet is taken to be the local sound speed.', 'astro-ph-0201322-1-14-0': 'The radiation calculation includes the synchrotron and inverse Compton processes, and the synchrotron spectrum is taken to be piece-wise power-law with the usual self-absorption, cooling and the synchrotron peak frequencies calculated from the electron spectrum, magnetic field strength and the radiative loss of energy for electrons.', 'astro-ph-0201322-1-14-1': 'The observed flux is obtained by integrating the emissivity over equal arrival time surface (e.g. Kumar Panaitescu 2000).', 'astro-ph-0201322-1-15-0': 'The light curves of model 2 are shown by the solid lines in figure [REF].', 'astro-ph-0201322-1-15-1': 'The flux density in the decaying stage is slightly higher for larger viewing angles [MATH].', 'astro-ph-0201322-1-15-2': 'This effect occurs since at this late stage the whole jet is visible, and for larger [MATH] the radiation from a given radius arrives at the observer at a latter time, on average.', 'astro-ph-0201322-1-15-3': 'Therefor, for a given observed time, larger [MATH] are dominated by emission from smaller radii, resulting in a larger flux density.', 'astro-ph-0201322-1-15-4': 'At a few hundred days, the light curves begin to flatten due to the transition to the non-relativistic regime.', 'astro-ph-0201322-1-16-0': 'The light curves for [MATH] are very different from model 1 (and more realistic).', 'astro-ph-0201322-1-16-1': 'Furthermore, the light curves for [MATH] are very similar to [MATH] in this model.', 'astro-ph-0201322-1-16-2': 'Since the jet is homogeneous, the ratio of the observed flux for [MATH] and [MATH], may be approximated by the ratio of the areas within the jet, that are within an angle of [MATH] around the directions to these two observers (which never decreases below [MATH]).', 'astro-ph-0201322-1-17-0': 'The inset in Figure [REF] shows the linear polarization for model 2, calculated following Ghisellini Lazatti (1999) and using their notations.', 'astro-ph-0201322-1-17-1': 'They assume the magnetic field is strictly in the plane of the shock ([MATH]); for [MATH] the polarization is along the plane containing the line of sight and the jet axis, wile for [MATH] it is rotated by [MATH] (for [MATH] this is reversed, e.g. Sari 1999).', 'astro-ph-0201322-1-17-2': 'A more isotropic magnetic field configuration would result in a smaller degree of polarization, so that the value of the polarization in Figure [REF] ([MATH]) may be viewed as a rough upper limit.', 'astro-ph-0201322-1-17-3': 'For [MATH] the polarization vanishes and reappears rotated by [MATH], around [MATH] (this occurs either once or twice).', 'astro-ph-0201322-1-17-4': 'For [MATH] the polarization has two peaks, the first higher than the second.', 'astro-ph-0201322-1-18-0': 'The main advantage of model 2 is that it provides realistic light curves in a very reasonable computational time, making it very convenient for performing detailed fits to observations (e.g. Panaitescu Kumar 2001,2002).', 'astro-ph-0201322-1-18-1': 'Its main drawback is a relatively simple treatment of the dynamics, which causes some differences in the light curves, compared to our next model.', 'astro-ph-0201322-1-19-0': '## Model 3: 2D Hydrodynamical Simulation', 'astro-ph-0201322-1-20-0': 'This model is described in Granot et al. (2001).', 'astro-ph-0201322-1-20-1': 'The jet dynamics are determined by a 2D hydrodynamical simulation, with initial conditions of a wedge taken from the spherical self similar Blandford-McKee (1976) solution.', 'astro-ph-0201322-1-20-2': 'The light curves for observers at different [MATH] are calculated considering the contribution from all the shocked region, and taking into account the relevant relativistic transformations of the radiation field, and the different photon arrival times to the different observers.', 'astro-ph-0201322-1-21-0': 'Figure [REF] shows the light curves of models 3, while the inset provides the light curves of model 2, for the same set of parameters.', 'astro-ph-0201322-1-21-1': 'In model 3, the peak of the light curves for [MATH] is flatter compared to model 2, and is obtained at a somewhat latter time.', 'astro-ph-0201322-1-21-2': 'The rise before the peak is not as sharp as in models 1 or 2, since in model 3 there is some material at the sides of the jet with a moderate Lorentz factor (Granot et al. 2001; Piran Granot 2001).', 'astro-ph-0201322-1-21-3': 'The emission from this slower material tends to dominate the observed flux at early times for observers at [MATH], resulting in a gentler rise before the peak.', 'astro-ph-0201322-1-21-4': 'The light curves for [MATH] peak at a later time compared to model 2, and the flux during the decay stage grows faster with [MATH], since in model 3 the curvature of the shock front is larger and the emission occurs within a shell of finite width, resulting in a larger photon arrival time, and implying that smaller radii contribute to a given observer time.', 'astro-ph-0201322-1-21-5': 'The light-curves for model 2 3 are quantitatively similar for [MATH].', 'astro-ph-0201322-1-22-0': 'The main advantage of this model is a reliable and rigorous treatment of the jet dynamics, which provides incite on the behavior of the jet and the corresponding light curves.', 'astro-ph-0201322-1-22-1': 'Its main drawback is the long computational time it requires.', 'astro-ph-0201322-1-23-0': '3.3infigure2.eps [] Light curves of model 3, for [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] Hz.', 'astro-ph-0201322-1-23-1': 'The inset shows the same light curves for model 2, where the same traces correspond to the same viewing angles [MATH].', 'astro-ph-0201322-1-24-0': '# The SDSS Orphan Afterglow', 'astro-ph-0201322-1-25-0': 'Vanden Berk et al. (2001) report the SDSS observations of an optical transient whose luminosity exceeds by about two orders of magnitude the peak luminosity of a Type Ia supernovae if the transient is at the redshift [MATH] of the proposed host galaxy.', 'astro-ph-0201322-1-25-1': 'The optical spectrum of the transient is power-law-like, [MATH], with [MATH] and [MATH] at the first and second measurements, respectively, which are 2 days apart.', 'astro-ph-0201322-1-25-2': 'The brightness of the optical transient is roughly constant between the two epochs.', 'astro-ph-0201322-1-26-0': 'Based on the observed spectral energy distribution and high luminosity, Vanden Berk et al. (2001) suggest that the transient could be a GRB afterglow, whose burst has not been observed either by the BATSE or by the IPN.', 'astro-ph-0201322-1-26-1': 'We note that the flatness of the afterglow emission and its spectral softening are not typical of GRB afterglows.', 'astro-ph-0201322-1-26-2': 'GRB afterglows usually exhibit a constant spectral slope [MATH] and a monotonously falling-off optical light-curve, departures from these properties being observed only rarely (e.g. GRB 970508, which exhibited a brightening at [MATH] days, and GRB 000301c, for which a mild brightening and spectral softening has been observed at few days).', 'astro-ph-0201322-1-27-0': 'If the observer was at [MATH], then the SDSS afterglow appeared "orphan" because the burst was intrinsically too dim to be observed.', 'astro-ph-0201322-1-27-1': 'Otherwise, the lack of a detectable GRB emission could be due to an observer location outside the initial GRB jet.', 'astro-ph-0201322-1-27-2': 'The spectral properties and temporal behavior of the SDSS orphan afterglow suggest that the latter case is more likely to have occurred.', 'astro-ph-0201322-1-27-3': 'Given that [MATH] at both epochs, a detectable afterglow dimming should have been observed over 2 days if [MATH], unless some mechanism (e.g. delayed energy injection, gravitational microlensing) brightened the afterglow.', 'astro-ph-0201322-1-28-0': '3.3infigure3.eps [] Fit to the SDSS orphan afterglow using model 2 with parameters [MATH], [MATH], [MATH] erg, [MATH], [MATH], [MATH], [MATH], and [MATH] day.', 'astro-ph-0201322-1-28-1': '[MATH] for 2 dof (10 data points - 8 model parameters).', 'astro-ph-0201322-1-28-2': 'The inset shows the optical afterglow spectrum.', 'astro-ph-0201322-1-29-0': 'Assuming that the slope [MATH] of the electron distribution did not change between the two epochs of observation, the spectral softening requires the passage of a break through the optical domain.', 'astro-ph-0201322-1-29-1': 'In order for a break frequency to evolve substantially over 2 days relative to the width of the observing domain, the delay [MATH] between the unseen burst and the first observation must be [MATH]2 days.', 'astro-ph-0201322-1-29-2': 'Thus we expect that the afterglow light-curve peaked at [MATH] days after the GRB, when [MATH], [MATH] being the jet Lorentz factor.', 'astro-ph-0201322-1-29-3': 'Together with the optical spectral slope [MATH], the afterglow brightness, and the passage of a spectral break through the optical domain at 2-3 days after the GRB, there are 4 constraints that the observations set on the jet models 2 and 3 described above.', 'astro-ph-0201322-1-29-4': 'Thus these models with 8 free parameters (including [MATH]) are underconstrained by the data.', 'astro-ph-0201322-1-30-0': 'Using Model 2 and keeping [MATH] fixed (for simplicity), we find acceptable fits (an example is shown in Figure 3) to the SDSS data for [MATH] and [MATH] days (other parameters have values similar to those found by Panaitescu Kumar 2001 for other GRB afterglows; for [MATH] we find [MATH]).', 'astro-ph-0201322-1-30-1': 'The spectral softening is due to the passage of the [MATH] break associated with the minimum injected electron Lorentz factor, which is slightly below the optical when the first SDSS measurements were made.', 'astro-ph-0201322-1-30-2': 'The cooling frequency [MATH] is below [MATH] and the resulting electron index is between 2.7 and 3.0.', 'astro-ph-0201322-1-30-3': 'Thus the optical spectral slope evolves from [MATH] at [MATH], when [MATH], to [MATH], [MATH] at [MATH], when [MATH].', 'astro-ph-0201322-1-30-4': 'Because the [MATH] break is smooth, the spectral softening is gradual, so that the afterglow spectrum at the two SDSS epochs is well approximated by a power-law of a slope [MATH] between the two asymptotic values [MATH] and [MATH] above.', 'astro-ph-0201322-1-31-0': '# GRB 980425 / SN 1998bw', 'astro-ph-0201322-1-32-0': 'On April 25, 1998, a Gamma-Ray Burst was detected by Beppo SAX and CGRO.', 'astro-ph-0201322-1-32-1': 'The burst consisted of a single wide peak of duration 30 s, peak flux in 24-1820 keV band of [MATH] erg cm[MATH] s[MATH], and fluence of [MATH] erg cm[MATH] (Soffitta et al. 1998, Kippen et al. 1998).', 'astro-ph-0201322-1-32-2': 'The burst had no detectable emission above 300 keV.', 'astro-ph-0201322-1-32-3': 'The burst spectrum was a broken power-law with break at 148[MATH]33 keV, and the high energy power-law photon index of [MATH] (see Galama et al. 1998).', 'astro-ph-0201322-1-32-4': 'These values are not unusual for GRBs.', 'astro-ph-0201322-1-33-0': 'A bright Type Ic supernova, SN 1998bw, located at [MATH], was detected within 8 arc minutes of GRB 980425.', 'astro-ph-0201322-1-33-1': 'From the extrapolation of optical light curves Galama et al. (1998) suggested that the SN went off within a day of the GRB, thereby implying a possible connection between the two events.', 'astro-ph-0201322-1-33-2': 'The probability of this association is strengthened by the uniquely peculiar light curve and spectrum of the SN (e.g., Patat et al 2001).', 'astro-ph-0201322-1-33-3': 'If indeed the two events are associated, then the total isotropic equivalent of energy release in [MATH]-rays for GRB 980425 is [MATH] erg, or a factor of [MATH] smaller than the energy for an average cosmological GRB.', 'astro-ph-0201322-1-34-0': 'Early on, Woosley et al. (1999) gave arguments why SN 1998bw might be a SN exploded by a jet and therefore possibly associated with a GRB.', 'astro-ph-0201322-1-34-1': 'This would arise, for instance, in the collapsar model (Woosley 1993; MacFadyen and Woosley 1999; Paczynski 1998).', 'astro-ph-0201322-1-34-2': 'If GRB 980425 was an ordinary, average GRB, but viewed at a large angle w.r.t. the jet axis, [MATH], it might explain the small [MATH] of the burst.', 'astro-ph-0201322-1-35-0': 'This could happen in two ways.', 'astro-ph-0201322-1-35-1': 'For a GRB made by a jet with [MATH] independent of [MATH] and sharp edges, the observed energy falls off rapidly for [MATH], in fact as [MATH] where [MATH].', 'astro-ph-0201322-1-35-2': 'Moreover, the off-axis observer will find the peak of the spectrum shifted to lower energy by a factor of [MATH], and the burst duration to be longer by the same factor.', 'astro-ph-0201322-1-35-3': 'Taking [MATH], in order to explain the low [MATH] for 980425, we find that the peak of the spectrum and the burst duration, for an observer located inside the jet beam of 980425, are [MATH] MeV and 4[MATH] s, respectively.', 'astro-ph-0201322-1-35-4': 'Moreover, we require [MATH].', 'astro-ph-0201322-1-36-0': 'Another possibility is that the jet does not have sharp edges, but wings of lower energy and Lorentz factor that extend to large [MATH].', 'astro-ph-0201322-1-36-1': 'Such a picture of the jet was suggested by Woosley et al. (1999) and is consistent with the relativistic studies of the collapsar model by Zhang, Woosley, MacFadyen (2002).', 'astro-ph-0201322-1-36-2': 'GRB 980425 would be then be produced by material with [MATH] moving in our direction.', 'astro-ph-0201322-1-36-3': 'But would SN 1998bw still be distinctly visible against the bright optical afterglow of the main GRB?', 'astro-ph-0201322-1-36-4': 'The light curves shown in Figure [REF] are for parameters possibly relevant to GRB 980425 ([MATH], [MATH], [MATH] and a total energy of [MATH] erg in relativistic ejecta).', 'astro-ph-0201322-1-36-5': 'SN 1998bw had a luminosity of approximately 10[MATH] erg 6 days after the explosion and 10[MATH] erg, at peak, 18 days after the explosion (Galama et al. 1998).', 'astro-ph-0201322-1-36-6': 'By 450 days it had declined to 10[MATH] erg s[MATH] (Patat et al. 2001).', 'astro-ph-0201322-1-36-7': 'These data are consistent with Figure [REF] provided [MATH] and suggest the interesting possibility that part of the current light curve of SN 1998bw could be due to the afterglow emission of the main GRB.', 'astro-ph-0201322-1-37-0': '# Discussion', 'astro-ph-0201322-1-38-0': 'We have presented the calculation of light-curves from a relativistic jet for an arbitrary location of the observer; much of the work in this letter is for an observer located outside the initial jet opening, [MATH].', 'astro-ph-0201322-1-38-1': 'We have considered three different jet models of increasing sophistication; the simplest being a point source moving along the jet axis (2.1), and the most sophisticated is 2D hydrodynamical simulation (2.3).', 'astro-ph-0201322-1-38-2': 'The basic qualitative features of the light-curves are similar in all three models, for [MATH].', 'astro-ph-0201322-1-38-3': 'Moreover, the uniform jet model (model 2, 2.2) is in rough quantitative agreement with the hydro-model.', 'astro-ph-0201322-1-39-0': 'We find that "orphan" optical afterglows can be observed within [MATH], provided that the limiting magnitude of the survey is about R=24 and the sampling rate is a few times a day for a few weeks.', 'astro-ph-0201322-1-39-1': 'This estimate is for R[MATH] at [MATH], typical of observed afterglows (which correspond to [MATH]).', 'astro-ph-0201322-1-39-2': 'Therefore, the rate of "orphan" optical transients should exceed the GRB rate by about an order of magnitude.', 'astro-ph-0201322-1-39-3': 'The orphan optical events discussed here can be identified from the initial rise during which the spectral slope is typically [MATH], followed by a decay, on a time scale of [MATH] days, and may show a large degree of linear polarization ([MATH]).', 'astro-ph-0201322-1-39-4': 'The detection of such orphan afterglows may provide a new line of evidence in favor of jetted outflows in GRBs.', 'astro-ph-0201322-1-39-5': 'Huang, Dai and Lu (2001), in a recent work, have considered other possible mechanisms for producing "orphan" afterglows; these should increase the total rate of detection of optical transients.', 'astro-ph-0201322-1-40-0': 'We find that the optical transient reported by the SDSS team (VandenBerk et al. 2001) is consistent with a GRB afterglow, where the observer is located off-axis at [MATH], and the GRB went off [MATH]2 days prior to the first SDSS observation.', 'astro-ph-0201322-1-40-1': 'This provides evidence in favor of a jetted relativistic outflow in this event.', 'astro-ph-0201322-1-41-0': 'We thank Mark Miller for generating the data for model 3.', 'astro-ph-0201322-1-41-1': 'This research was supported by grants NSF PHY 99-79985 and MCA93S025 (computational support), NSF grant PHY-0070928 (JG), Lyman Spitzer, Jr.', 'astro-ph-0201322-1-41-2': 'Fellowship (AP), NAG5-8128 and MIT-292701 (SW).'} | {'astro-ph-0201322-2-0-0': 'We calculate Gamma-Ray Burst (GRB) afterglow light-curves from a relativistic jet of initial opening angle [MATH], as seen by observers at a wide range of viewing angles, [MATH], from the jet axis.', 'astro-ph-0201322-2-0-1': 'We describe three increasingly more realistic models and compare the resulting light-curves.', 'astro-ph-0201322-2-0-2': 'An observer at [MATH] should see a light curve very similar to that for an on-axis observer.', 'astro-ph-0201322-2-0-3': 'An observer at [MATH] should see a rising light curve at early times, the flux peaking when the jet Lorentz factor [MATH].', 'astro-ph-0201322-2-0-4': 'After this time the flux is not very different from that seen by an on-axis observer.', 'astro-ph-0201322-2-0-5': 'A strong linear polarization ([MATH]) may occur near the peak in the light curve, and slowly decay with time.', 'astro-ph-0201322-2-0-6': 'We show that if GRB jets have a universal energy, then orphan afterglows associated with off-axis jets should be seen up to a constant [MATH], therefore the detection rate of orphan afterglows would be proportional to the true GRB rate.', 'astro-ph-0201322-2-0-7': 'We also discuss the proposed connection between supernova 1998bw and GRB 980425.', 'astro-ph-0201322-2-1-0': '# Introduction', 'astro-ph-0201322-2-2-0': 'Gamma-Ray Bursts (GRBs) are explosions which release roughly 10[MATH] erg in the form of kinetic energy of highly relativistic material (Frail et al. 2001, Panaitescu Kumar 2001).', 'astro-ph-0201322-2-2-1': 'Many GRBs appear to be highly non-spherical explosions, as evidenced by a nearly-achromatic break in the light-curve (e.g. Harrison et al. 1999; Stanek et al. 1999).', 'astro-ph-0201322-2-2-2': 'Highly relativistic jets are "visible" when our line of sight is within the jet aperture ([MATH]), otherwise, because of relativistic beaming of photons away from our line-of-sight, the object is too dim.', 'astro-ph-0201322-2-2-3': 'As the jet decelerates, the relativistic beaming becomes less severe and the emission from the jet becomes detectable to observers at larger viewing angles.', 'astro-ph-0201322-2-3-0': 'In this Letter we study the afterglow light-curves for off-axis locations ([MATH]), focusing on observers lying outside of the initial jet opening angle ([MATH]).', 'astro-ph-0201322-2-3-1': 'Granot et al. (2001) have shown that the light curve seen by an observer located within the initial jet aperture ([MATH]) is very similar to that for an on-axis observer ([MATH]).', 'astro-ph-0201322-2-3-2': 'Dalal et al. (2002) and Rossi et al. (2002) have presented simple models to calculate the flux in this case.', 'astro-ph-0201322-2-3-3': 'We reanalyze these models in 2.1 and consider more realistic models in 2.2 2.3.', 'astro-ph-0201322-2-3-4': 'Moderski, Sikora and Bulik (2000) have calculated off-axis light-curves with a more complex model, similar to that presented in 2.2.', 'astro-ph-0201322-2-4-0': 'In 3 we calculate the temporal evolution of the linear polarization for various [MATH].', 'astro-ph-0201322-2-4-1': 'In 4 we analyze the prospects of using the detection rate of orphan afterglows to estimate the collimation of GRB jets.', 'astro-ph-0201322-2-4-2': 'In 5 we analyze the suggestion of Woosley, Eastman, Schmidt (1999) that a relativistic jet emanating from the SN explosion and pointing away from us could explain the observations.', 'astro-ph-0201322-2-5-0': '# Modeling the off axis emission', 'astro-ph-0201322-2-6-0': 'In this section we calculate the afterglow light curves of jetted GRBs, as seen by observers at different viewing angles, [MATH], w.r.t the symmetry axis of the jet.', 'astro-ph-0201322-2-6-1': 'For simplicity, we consider only a jet propagating into a homogeneous medium.', 'astro-ph-0201322-2-6-2': 'In order to improve our understanding of the underlying physics and in order to check how general the results are, we explore three different models with an increasing level of complexity.', 'astro-ph-0201322-2-7-0': '## Model 1: A Point Source at the Jet Axis', 'astro-ph-0201322-2-8-0': 'We begin with a simple model, where for [MATH] the light curve follows the results of simple jet models (Rhoads 1999; Sari, Piran Halpern 1999, hereafter R-SPH99), and for [MATH] the light curves are calculated assuming the emission is from a point source that moves along the jet axis.', 'astro-ph-0201322-2-8-1': 'The on-axis light curve exhibits a jet break at (R-SPH99): [EQUATION] where [MATH] is the isotropic equivalent energy in units of [MATH] erg, [MATH] is the ambient density in [MATH] and [MATH] is the cosmological redshift of the source.', 'astro-ph-0201322-2-8-2': 'At [MATH], [MATH] is taken from Sari, Piran and Narayan (1998), while at [MATH] the temporal scalings of the break frequencies and peak flux change according to R-SPH99.', 'astro-ph-0201322-2-8-3': 'The observed flux density from a point source is [EQUATION] where [MATH] and [MATH] are the spectral luminosity and frequency in the local rest frame of the jet, [MATH] and [MATH] are the angular and luminosity distances to the source, [MATH] is the Lorentz factor of the source and [MATH] is the angle between the direction of motion of the source and the direction to the observer in the observer frame (in our case [MATH]).', 'astro-ph-0201322-2-8-4': 'Since [MATH], where [MATH] and [MATH] are the observed time and frequency, we obtain that [EQUATION] where [MATH] and [MATH] are the observed time and frequency for an observer at [MATH].', 'astro-ph-0201322-2-8-5': 'One therefore obtains that [EQUATION] where, for simplicity, we take [MATH] at [MATH] and [MATH] at [MATH].', 'astro-ph-0201322-2-9-0': 'The light curves obtained using equation [REF] are shown by the dashed lines in Figure [REF].', 'astro-ph-0201322-2-9-1': 'At first [MATH] and [MATH] is the dominant term in equation [REF], giving a sharp rise in the light curve.', 'astro-ph-0201322-2-9-2': 'Once [MATH] becomes [MATH] the flux begins to decay, asymptotically approaching the on-axis light curve.', 'astro-ph-0201322-2-9-3': 'The light curve for off-axis observers peaks when [MATH].', 'astro-ph-0201322-2-10-0': '3.3infig1.eps [] B-band luminosity for models 1 (dashed lines) and 2 (solid lines), for [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], where [MATH]) is the fraction of the internal energy in the magnetic field (electrons) and [MATH] is the power law index of the electron energy distribution.', 'astro-ph-0201322-2-10-1': 'Model 1 is scaled down by a factor of 2.5, to help compare between the two models.', 'astro-ph-0201322-2-11-0': 'The main advantage of this model is that it is very simple, and nevertheless gives reasonable results for [MATH].', 'astro-ph-0201322-2-11-1': 'Its main drawback is that it is not physical for [MATH] at [MATH].', 'astro-ph-0201322-2-11-2': 'This is because for [MATH] the observed radiation is initially dominated by emission from the material within an angle of [MATH] around the line of sight, while in model 1 the emission is always only from along the jet axis, and therefore the dominant contribution to the emission is missing, until the time when [MATH].', 'astro-ph-0201322-2-11-3': 'This problem is overcome by our next model.', 'astro-ph-0201322-2-12-0': '## Model 2: A Homogeneous Jet', 'astro-ph-0201322-2-13-0': 'This model is described in Kumar Panaitescu (2000), and here we briefly point out its main features.', 'astro-ph-0201322-2-13-1': 'The Lorentz factor and energy density per unit solid angle are considered to be independent of angle [MATH] within the jet aperture.', 'astro-ph-0201322-2-13-2': 'The decrease of the Lorentz factor of the jet with time is calculated from the mass and energy conservation equations, and the sideway expansion speed of the jet is taken to be the local sound speed.', 'astro-ph-0201322-2-14-0': 'The radiation calculation includes the synchrotron and inverse Compton processes, and the synchrotron spectrum is taken to be piece-wise power-law with the usual self-absorption, cooling and the synchrotron peak frequencies calculated from the electron spectrum, magnetic field strength and the radiative loss of energy for electrons.', 'astro-ph-0201322-2-14-1': 'The observed flux is obtained by integrating the emissivity over equal arrival time surface (e.g. Kumar Panaitescu 2000).', 'astro-ph-0201322-2-15-0': 'The light curves of model 2 are shown by the solid lines in figure [REF].', 'astro-ph-0201322-2-15-1': 'The flux density in the decaying stage is slightly higher for larger viewing angles [MATH].', 'astro-ph-0201322-2-15-2': 'This effect occurs since at this late stage the whole jet is visible, and for larger [MATH] the radiation from a given radius arrives at the observer at a latter time, on average.', 'astro-ph-0201322-2-15-3': 'Therefore, for a given observed time, larger [MATH] are dominated by emission from smaller radii, resulting in a larger flux density.', 'astro-ph-0201322-2-15-4': 'At a few hundred days, the light curves begin to flatten due to the transition to the non-relativistic regime.', 'astro-ph-0201322-2-16-0': 'The light curves for [MATH] are very different from model 1 (and more realistic).', 'astro-ph-0201322-2-16-1': 'Furthermore, the light curves for [MATH] are very similar to [MATH] in this model.', 'astro-ph-0201322-2-16-2': 'Since the jet is homogeneous, the ratio of the observed flux for [MATH] and [MATH], may be approximated by the ratio of the areas within the jet, that are within an angle of [MATH] around the directions to these two observers (which never decreases below [MATH]).', 'astro-ph-0201322-2-17-0': 'We notice that the lights curve of model 1 for [MATH] are much closer to the light curves of model 2 for [MATH], respectively, than to the light curves for the same viewing angles.', 'astro-ph-0201322-2-17-1': 'This is so because the emission for an observer outside the jet opening angle is dominated by the point in the jet closest to the observer.', 'astro-ph-0201322-2-17-2': 'Therefore, model 1 will become much more realistic (and just as simple) if one would use [MATH], rather than [MATH], in equations [REF] and [REF].', 'astro-ph-0201322-2-18-0': 'The main advantage of model 2 is that it provides realistic light curves in a very reasonable computational time, making it very convenient for performing detailed fits to observations (e.g. Panaitescu Kumar 2001).', 'astro-ph-0201322-2-18-1': 'Its main drawback is a relatively simple treatment of the dynamics, which causes some differences in the light curves, compared to our next model.', 'astro-ph-0201322-2-19-0': '## Model 3: 2D Hydrodynamical Simulation', 'astro-ph-0201322-2-20-0': 'This model is described in Granot et al. (2001).', 'astro-ph-0201322-2-20-1': 'The jet dynamics are determined by a 2D hydrodynamical simulation, with initial conditions of a wedge taken from the spherical self similar Blandford-McKee (1976) solution.', 'astro-ph-0201322-2-20-2': 'The light curves for observers at different [MATH] are calculated considering the contribution from all the shocked region, and taking into account the relevant relativistic transformations of the radiation field, and the different photon arrival times to the different observers.', 'astro-ph-0201322-2-21-0': '3.3infig2.eps', 'astro-ph-0201322-2-22-0': 'Light curves of model 3, for [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH] Hz.', 'astro-ph-0201322-2-22-1': 'The inset shows the same light curves for model 2, where the same traces correspond to the same viewing angles [MATH].', 'astro-ph-0201322-2-23-0': 'Figure [REF] shows the light curves of models 3, while the inset provides the light curves of model 2, for the same set of parameters.', 'astro-ph-0201322-2-23-1': 'In model 3, the peak of the light curves for [MATH] is flatter compared to model 2, and is obtained at a somewhat latter time.', 'astro-ph-0201322-2-23-2': 'The rise before the peak is not as sharp as in models 1 or 2, since in model 3 there is some material at the sides of the jet with a moderate Lorentz factor (Granot et al. 2001; Piran Granot 2001).', 'astro-ph-0201322-2-23-3': 'The emission from this slower material tends to dominate the observed flux at early times for observers at [MATH], resulting in a gentler rise before the peak.', 'astro-ph-0201322-2-23-4': 'The light curves for [MATH] peak at a later time compared to model 2, and the flux during the decay stage grows faster with [MATH], since in model 3 the curvature of the shock front is larger and the emission occurs within a shell of finite width, resulting in a larger photon arrival time, and implying that smaller radii contribute to a given observer time.', 'astro-ph-0201322-2-23-5': 'The light-curves for model 2 3 are quantitatively similar for [MATH].', 'astro-ph-0201322-2-24-0': 'The main advantage of this model is a reliable and rigorous treatment of the jet dynamics, which provides insight on the behavior of the jet and the corresponding light curves.', 'astro-ph-0201322-2-24-1': 'Its main drawback is the long computational time it requires.', 'astro-ph-0201322-2-25-0': '# Linear Polarization', 'astro-ph-0201322-2-26-0': 'While the afterglow emission from a spherical outflow is expected to exhibit little or no linear polarization, as the polarization from the different parts of the afterglow image cancel out, a jetted outflow breaks the circular symmetry of the afterglow image and may exhibit a polarization of up to [MATH] (Ghisellini Lazatti 1999; Sari 1999).', 'astro-ph-0201322-2-26-1': 'One might therefore expect an even larger polarization for an observer at [MATH].', 'astro-ph-0201322-2-27-0': 'An isotropic magnetic field configuration in the local rest frame will produce no linear polarization.', 'astro-ph-0201322-2-27-1': 'However, as the magnetic field is most likely produced at the shock itself, one might expect the magnetic field perpendicular ([MATH]) and parallel ([MATH]) to the shock direction, to have different magnitudes (Gruzinov 1999; Sari 1999).', 'astro-ph-0201322-2-27-2': 'We calculate the linear polarization for model 2 following Ghisellini Lazatti (1999) and using their notations.', 'astro-ph-0201322-2-27-3': 'They assume the magnetic field is strictly in the plane of the shock ([MATH]).', 'astro-ph-0201322-2-28-0': '3.3infig3.eps', 'astro-ph-0201322-2-29-0': 'The linear polarization for model 2 for several viewing angle and for the same parameters as in Figure [REF].', 'astro-ph-0201322-2-30-0': 'Figure [REF] shows the polarization as a function of time for different [MATH] in terms of [MATH].', 'astro-ph-0201322-2-30-1': 'For [MATH] the polarization is along the plane containing the line of sight and the jet axis, wile for [MATH] it is rotated by [MATH] (for [MATH] this is reversed, e.g. Sari 1999).', 'astro-ph-0201322-2-30-2': 'A more isotropic magnetic field configuration would result in a smaller degree of polarization, so the value of the polarization in Figure [REF] ([MATH]) may be viewed as a rough upper limit.', 'astro-ph-0201322-2-30-3': 'For [MATH] the polarization vanishes and reappears rotated by [MATH] around [MATH].', 'astro-ph-0201322-2-30-4': 'This behavior may occur again at a later time, but the subsequent polarization is very low.', 'astro-ph-0201322-2-30-5': 'For [MATH] the polarization has two peaks, the first higher than the second.', 'astro-ph-0201322-2-30-6': 'For [MATH] the polarization is largest near the peak in the light curve, and decreases quite slowly with time, while the peak polarization shows a very weak dependence on [MATH], and is about a factor of [MATH] larger than for [MATH].', 'astro-ph-0201322-2-31-0': '# Orphan Afterglows', 'astro-ph-0201322-2-32-0': 'If GRB jets have well defined edges, the prompt gamma-ray flux drops very sharply outside the opening of the jet, and the prompt burst will be very hard to detect from [MATH].', 'astro-ph-0201322-2-32-1': 'On the other hand, the afterglow emission may be detected out to [MATH], where the exact value of [MATH] depends on the jet parameters (including its redshift), the observed band and the limiting flux for detection.', 'astro-ph-0201322-2-32-2': 'Jetted GRBs with [MATH] are expected to be orphan afterglows (i.e. detectable in the optical but not in gamma-rays).', 'astro-ph-0201322-2-33-0': 'It has been argued by Dalal et al. (2002) that [MATH] for [MATH], so that the detection rate of orphan afterglows [MATH] (associated with off-axis jets) will be a constant [namely [MATH]] times the GRB detection rate [MATH], and thereby a comparison between these two rates will not constrain [MATH] or the true rate of GRBs [MATH].', 'astro-ph-0201322-2-33-1': 'This result was obtained assuming a constant flux, [MATH], at [MATH] for [MATH].', 'astro-ph-0201322-2-33-2': 'However, afterglow observations suggest that the total energy in the jet, [MATH], is roughly constant (Frail et al. 2001, Panaitescu Kumar 2001, Piran et al. 2001) while [MATH] varies over a wider range.', 'astro-ph-0201322-2-33-3': 'In fact, for [MATH], simple jet models (R-SPH99) predict that the hydrodynamical evolution of the jet (and therefore the light curves for all [MATH]) becomes independent of [MATH] once the jet enters the phase of exponential lateral expansion with radius.', 'astro-ph-0201322-2-33-4': 'This corresponds to [MATH] for [MATH], and to [MATH] for [MATH], which includes the time around or after the peak in the light curve.', 'astro-ph-0201322-2-33-5': 'This implies that for [MATH], we have [MATH] (rather than [MATH]) for [MATH].', 'astro-ph-0201322-2-33-6': 'For [MATH] naturally [MATH] is larger [MATH] if the afterglow is detectable from [MATH], and the solid angle between [MATH] and [MATH], [MATH] (in which we have detectable orphan afterglows) remains approximately constant.', 'astro-ph-0201322-2-33-7': 'Furthermore, the distribution of [MATH] inferred from observations (Frail et al. 2001, Panaitescu Kumar 2001) is sharply peaked at low [MATH]).', 'astro-ph-0201322-2-33-8': 'This suggests that most of the orphan afterglows that would be detected should have [MATH].', 'astro-ph-0201322-2-33-9': 'For a reasonable limiting magnitude for detection, this implies [MATH] in most cases, and therefore [MATH].', 'astro-ph-0201322-2-33-10': 'For example, for model 2 with [MATH] ergs (assuming a double sided jet), [MATH], [MATH], [MATH], [MATH] and a limiting magnitude for detection of [MATH] we obtain [MATH] and [MATH] for [MATH].', 'astro-ph-0201322-2-33-11': 'If indeed [MATH], then [MATH] should provide a good estimate of the true GRB rate, [MATH].', 'astro-ph-0201322-2-33-12': 'The average beaming fraction [MATH] is given by [MATH].', 'astro-ph-0201322-2-34-0': '# GRB 980425 / SN 1998bw', 'astro-ph-0201322-2-35-0': 'On April 25, 1998, a Gamma-Ray Burst was detected by Beppo SAX and CGRO.', 'astro-ph-0201322-2-35-1': 'The burst consisted of a single wide peak of duration 30 s, peak flux in 24-1820 keV band of [MATH] erg cm[MATH] s[MATH], and fluence of [MATH] erg cm[MATH] (Soffitta et al. 1998, Kippen et al. 1998).', 'astro-ph-0201322-2-35-2': 'The burst had no detectable emission above 300 keV.', 'astro-ph-0201322-2-35-3': 'The burst spectrum was a broken power-law with break at 148[MATH]33 keV, and the high energy power-law photon index of [MATH] (see Galama et al. 1998).', 'astro-ph-0201322-2-35-4': 'These values are not unusual for GRBs.', 'astro-ph-0201322-2-36-0': 'A bright Type Ic supernova, SN 1998bw, located at [MATH], was detected within 8 arc minutes of GRB 980425.', 'astro-ph-0201322-2-36-1': 'From the extrapolation of optical light curves Galama et al. (1998) suggested that the SN went off within a day of the GRB, thereby implying a possible connection between the two events.', 'astro-ph-0201322-2-36-2': 'The probability of this association is strengthened by the uniquely peculiar light curve and spectrum of the SN (e.g., Patat et al 2001).', 'astro-ph-0201322-2-36-3': 'Early on, Woosley et al. (1999) gave arguments why SN 1998bw might be a SN exploded by a jet and therefore possibly associated with a GRB.', 'astro-ph-0201322-2-36-4': 'This would arise, for instance, in the collapsar model (Woosley 1993; MacFadyen and Woosley 1999; Paczynski 1998).', 'astro-ph-0201322-2-37-0': 'If indeed the two events are associated, then the total isotropic equivalent of energy release in [MATH]-rays for GRB 980425 is [MATH] erg, or a factor of [MATH] smaller than the energy for an average cosmological GRB.', 'astro-ph-0201322-2-37-1': 'This could explained in two ways.', 'astro-ph-0201322-2-38-0': '## Sharp Edged, Homogeneous Jet Seen at [MATH]', 'astro-ph-0201322-2-39-0': 'If GRB 980425 was viewed at [MATH] it might explain its small [MATH].', 'astro-ph-0201322-2-39-1': 'For a GRB arising from a jet with [MATH] independent of [MATH] and sharp edges, the observed energy falls off rapidly for [MATH], in fact as [MATH] where [MATH].', 'astro-ph-0201322-2-39-2': 'Moreover, for an observer at [MATH] the peak of the spectrum is lower by a factor [MATH], and the burst duration longer by the same factor, compared to an observer at [MATH].', 'astro-ph-0201322-2-39-3': 'Therefore the low [MATH] of 980425 implies [MATH] and [MATH].', 'astro-ph-0201322-2-39-4': 'If GRB 980425 were at a cosmological distance and seen from [MATH], the peak of the spectrum and the burst duration would be [MATH] MeV and 4[MATH] s, respectively.', 'astro-ph-0201322-2-40-0': 'A second constraint is set by the condition that the optical afterglow is dimmer than SN 1998bw, which had a luminosity of [MATH] erg at 1 day, rose to [MATH] erg at [MATH] days (Galama et al. 1998), then decayed at [MATH] mag/day (Patat et al. 2001).', 'astro-ph-0201322-2-40-1': 'To compare it with the afterglow luminosity, we shall use the typical properties of the afterglows whose optical light-curves exhibited breaks: [MATH] average jet break [MATH] days in their rest-frame, [MATH] average optical luminosity flux [MATH] erg at [MATH], [MATH] with [MATH] at [MATH], [MATH] with [MATH], at optical frequencies.', 'astro-ph-0201322-2-41-0': 'Using Model 1 described in section 2.1, it can be shown that for an observer at [MATH] the time and optical luminosity at the light-curve peak are [EQUATION]', 'astro-ph-0201322-2-41-1': 'From these equations it can be shown that for [MATH] the peak afterglow luminosity [MATH] is a factor [MATH] lower than [MATH] of SN 1998bw.', 'astro-ph-0201322-2-41-2': 'During the decay phase, the afterglow luminosity remains below that of the SN until about 400 days, when they become comparable.', 'astro-ph-0201322-2-41-3': 'We note that Patat et al. (2001) report a flattening of the SN 1998bw decay after 375 day, to a dimming rate of [MATH] mag/day, which they interpret as the settling in of the [MATH]Co decay or the existence of other sources.', 'astro-ph-0201322-2-42-0': 'Together with the above constraint, [MATH], imposed by the fluence of GRB 980425, the condition that the afterglow emission does not exceed that of the SN 1998bw leads to [MATH] and [MATH].', 'astro-ph-0201322-2-43-0': '## Inhomogeneous Jet Seen Off-Axis', 'astro-ph-0201322-2-44-0': 'Another possibility is that the jet does not have sharp edges, but wings of lower energy and Lorentz factor that extend to large [MATH].', 'astro-ph-0201322-2-44-1': 'Such a picture of the jet was suggested by Woosley et al. (1999) and is consistent with the relativistic studies of the collapsar model by Zhang, Woosley, MacFadyen (2002).', 'astro-ph-0201322-2-44-2': 'GRB 980425 would be then be produced by material with [MATH] moving in our direction.', 'astro-ph-0201322-2-45-0': 'When seen on-axis, an afterglow with the above properties has [MATH] erg at 1 day and [MATH] erg at 16 days, i.e. a "typical" afterglow would be 200 times brighter than SN 1998bw when its first observation was made and several times dimmer when SN 1998bw peaked.', 'astro-ph-0201322-2-45-1': 'All other parameters remaining the same, an afterglow emission dimmer than that of the SN 1998bw at 1 day requires an energy per solid angle [MATH] in the direction toward the observer satisfying [MATH] erg.', 'astro-ph-0201322-2-45-2': 'As in the previous subsection, the observer location satisfies [MATH] so that the optical emission of the main jet of opening [MATH] is below the supernova light-curve.', 'astro-ph-0201322-2-46-0': '# Discussion', 'astro-ph-0201322-2-47-0': 'We have presented the calculation of light-curves from a relativistic jet for an arbitrary location of the observer; much of the work in this letter is for an observer located outside the initial jet opening, [MATH].', 'astro-ph-0201322-2-47-1': 'We have considered three different jet models of increasing sophistication; the simplest being a point source moving along the jet axis (2.1), and the most sophisticated is 2D hydrodynamical simulation (2.3).', 'astro-ph-0201322-2-47-2': 'The basic qualitative features of the light-curves are similar in all three models, for [MATH].', 'astro-ph-0201322-2-47-3': 'Moreover, the uniform jet model (model 2, 2.2) is in rough quantitative agreement with the hydro-model.', 'astro-ph-0201322-2-48-0': 'We find that "orphan" optical afterglows associated with off-axis jets can be observed up to a constant [MATH], rather than a constant [MATH] as suggested by Dalal et al. (2002), if one assumes a constant energy in the jet, rather than a constant flux at the time of the jet break for an on-axis observer.', 'astro-ph-0201322-2-48-1': 'This implies that future surveys for orphan afterglows may provide valuable data for the the distribution of jet opening angles [MATH] and the true event rate of GRBs.', 'astro-ph-0201322-2-48-2': 'The orphan optical events discussed here can be identified from the initial rise during which the spectral slope is typically [MATH], followed by a decay, on a time scale of [MATH] days, and may show a large degree of linear polarization ([MATH]).', 'astro-ph-0201322-2-48-3': 'The detection of such orphan afterglows may provide a new line of evidence in favor of jetted outflows in GRBs.', 'astro-ph-0201322-2-48-4': 'Recently Huang, Dai and Lu (2001) have considered another scenario (failed GRBs) for producing orphan afterglows; this would increase the detection rate of orphan afterglows.', 'astro-ph-0201322-2-48-5': 'A good monitoring of optical transients may help distinguish failed GRBs from jets seen at [MATH], and improve our understanding of them.', 'astro-ph-0201322-2-49-0': 'We thank Mark Miller for generating the data for model 3.', 'astro-ph-0201322-2-49-1': 'This research was supported by grants NSF PHY 99-79985 and MCA93S025 (computational support), NSF grant PHY-0070928 (JG), Lyman Spitzer, Jr.', 'astro-ph-0201322-2-49-2': 'Fellowship (AP), NAG5-8128 and MIT-292701 (SW).'} | [['astro-ph-0201322-1-13-0', 'astro-ph-0201322-2-13-0'], ['astro-ph-0201322-1-13-1', 'astro-ph-0201322-2-13-1'], ['astro-ph-0201322-1-13-2', 'astro-ph-0201322-2-13-2'], ['astro-ph-0201322-1-14-0', 'astro-ph-0201322-2-14-0'], ['astro-ph-0201322-1-14-1', 'astro-ph-0201322-2-14-1'], ['astro-ph-0201322-1-38-0', 'astro-ph-0201322-2-47-0'], ['astro-ph-0201322-1-38-1', 'astro-ph-0201322-2-47-1'], ['astro-ph-0201322-1-38-2', 'astro-ph-0201322-2-47-2'], ['astro-ph-0201322-1-38-3', 'astro-ph-0201322-2-47-3'], 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'astro-ph-0201322-2-23-1'], ['astro-ph-0201322-1-21-2', 'astro-ph-0201322-2-23-2'], ['astro-ph-0201322-1-21-3', 'astro-ph-0201322-2-23-3'], ['astro-ph-0201322-1-21-4', 'astro-ph-0201322-2-23-4'], ['astro-ph-0201322-1-21-5', 'astro-ph-0201322-2-23-5'], ['astro-ph-0201322-1-20-0', 'astro-ph-0201322-2-20-0'], ['astro-ph-0201322-1-20-1', 'astro-ph-0201322-2-20-1'], ['astro-ph-0201322-1-20-2', 'astro-ph-0201322-2-20-2'], ['astro-ph-0201322-1-8-0', 'astro-ph-0201322-2-8-0'], ['astro-ph-0201322-1-8-2', 'astro-ph-0201322-2-8-1'], ['astro-ph-0201322-1-8-3', 'astro-ph-0201322-2-8-2'], ['astro-ph-0201322-1-8-4', 'astro-ph-0201322-2-8-3'], ['astro-ph-0201322-1-8-5', 'astro-ph-0201322-2-8-4'], ['astro-ph-0201322-1-8-6', 'astro-ph-0201322-2-8-5'], ['astro-ph-0201322-1-41-0', 'astro-ph-0201322-2-49-0'], ['astro-ph-0201322-1-41-1', 'astro-ph-0201322-2-49-1'], ['astro-ph-0201322-1-22-1', 'astro-ph-0201322-2-24-1'], ['astro-ph-0201322-1-36-0', 'astro-ph-0201322-2-44-0'], ['astro-ph-0201322-1-36-1', 'astro-ph-0201322-2-44-1'], ['astro-ph-0201322-1-36-2', 'astro-ph-0201322-2-44-2'], ['astro-ph-0201322-1-6-0', 'astro-ph-0201322-2-6-0'], ['astro-ph-0201322-1-6-1', 'astro-ph-0201322-2-6-1'], ['astro-ph-0201322-1-6-2', 'astro-ph-0201322-2-6-2'], ['astro-ph-0201322-1-0-1', 'astro-ph-0201322-2-0-1'], ['astro-ph-0201322-1-0-4', 'astro-ph-0201322-2-0-5'], ['astro-ph-0201322-1-0-8', 'astro-ph-0201322-2-0-7'], ['astro-ph-0201322-1-9-0', 'astro-ph-0201322-2-9-0'], ['astro-ph-0201322-1-9-1', 'astro-ph-0201322-2-9-1'], ['astro-ph-0201322-1-9-2', 'astro-ph-0201322-2-9-2'], ['astro-ph-0201322-1-9-3', 'astro-ph-0201322-2-9-3'], ['astro-ph-0201322-1-23-1', 'astro-ph-0201322-2-22-1'], ['astro-ph-0201322-1-15-0', 'astro-ph-0201322-2-15-0'], ['astro-ph-0201322-1-15-1', 'astro-ph-0201322-2-15-1'], ['astro-ph-0201322-1-15-2', 'astro-ph-0201322-2-15-2'], ['astro-ph-0201322-1-15-4', 'astro-ph-0201322-2-15-4'], ['astro-ph-0201322-1-10-1', 'astro-ph-0201322-2-10-1'], ['astro-ph-0201322-1-2-0', 'astro-ph-0201322-2-2-0'], ['astro-ph-0201322-1-2-1', 'astro-ph-0201322-2-2-1'], ['astro-ph-0201322-1-2-2', 'astro-ph-0201322-2-2-2'], ['astro-ph-0201322-1-2-3', 'astro-ph-0201322-2-2-3'], ['astro-ph-0201322-1-2-4', 'astro-ph-0201322-2-3-1'], ['astro-ph-0201322-1-2-5', 'astro-ph-0201322-2-3-0'], ['astro-ph-0201322-1-39-3', 'astro-ph-0201322-2-48-2'], ['astro-ph-0201322-1-39-4', 'astro-ph-0201322-2-48-3'], ['astro-ph-0201322-1-33-3', 'astro-ph-0201322-2-37-0'], ['astro-ph-0201322-1-33-0', 'astro-ph-0201322-2-36-0'], ['astro-ph-0201322-1-33-1', 'astro-ph-0201322-2-36-1'], ['astro-ph-0201322-1-33-2', 'astro-ph-0201322-2-36-2'], ['astro-ph-0201322-1-34-0', 'astro-ph-0201322-2-36-3'], ['astro-ph-0201322-1-34-1', 'astro-ph-0201322-2-36-4']] | [['astro-ph-0201322-1-4-1', 'astro-ph-0201322-2-4-2'], ['astro-ph-0201322-1-18-0', 'astro-ph-0201322-2-18-0'], ['astro-ph-0201322-1-35-1', 'astro-ph-0201322-2-39-1'], ['astro-ph-0201322-1-22-0', 'astro-ph-0201322-2-24-0'], ['astro-ph-0201322-1-0-0', 'astro-ph-0201322-2-0-0'], ['astro-ph-0201322-1-0-2', 'astro-ph-0201322-2-0-3'], ['astro-ph-0201322-1-0-3', 'astro-ph-0201322-2-0-4'], ['astro-ph-0201322-1-0-5', 'astro-ph-0201322-2-0-2'], ['astro-ph-0201322-1-15-3', 'astro-ph-0201322-2-15-3'], ['astro-ph-0201322-1-10-0', 'astro-ph-0201322-2-10-0'], ['astro-ph-0201322-1-2-6', 'astro-ph-0201322-2-3-2']] | [] | [['astro-ph-0201322-1-35-2', 'astro-ph-0201322-2-39-2'], ['astro-ph-0201322-1-2-7', 'astro-ph-0201322-2-3-3'], ['astro-ph-0201322-1-39-0', 'astro-ph-0201322-2-48-0'], ['astro-ph-0201322-1-39-5', 'astro-ph-0201322-2-48-4']] | [['astro-ph-0201322-1-17-0', 'astro-ph-0201322-2-27-2'], ['astro-ph-0201322-1-17-2', 'astro-ph-0201322-2-30-2'], ['astro-ph-0201322-1-17-3', 'astro-ph-0201322-2-30-3'], ['astro-ph-0201322-1-17-4', 'astro-ph-0201322-2-30-5']] | ['astro-ph-0201322-1-23-0', 'astro-ph-0201322-1-36-6', 'astro-ph-0201322-1-41-2', 'astro-ph-0201322-2-21-0', 'astro-ph-0201322-2-22-0', 'astro-ph-0201322-2-28-0', 'astro-ph-0201322-2-49-2'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/0201322 | null | null | null | null | null |
1401.3528 | {'1401.3528-1-0-0': 'The sensitivity of homodyne timing measurements with femtosecond (fs) lasers is only limited by the amplitude and phase noise .', '1401.3528-1-0-1': 'We describe a novel method to analyze the phase noise of a Ti:Sa oscillator relative to the standard quantum limit.', '1401.3528-1-0-2': 'The broadband passive cavity used to this aim also filters lowest levels of classical noise at sidebands above [MATH]kHz detection frequency.', '1401.3528-1-0-3': 'Leading to quantum limited carrier-envelope-phase noise at [MATH]s-timescales, it can improve the sensitivity of a highly sensitive, homodyne timing jitter measurement by 2 orders of magnitude.', '1401.3528-1-1-0': '140.0140, 270.0270', '1401.3528-1-2-0': ']', '1401.3528-1-3-0': 'Introduction.', '1401.3528-1-3-1': 'Femtosecond optical frequency combs have revolutionized optical metrology given their intrinsic frequency comb structure and the ability to measure and lock the phases of these frequencies.', '1401.3528-1-3-2': 'Recent experiments have demonstrated that they can be extended to massively parallel optical spectroscopy, using a dual comb configuration and an optical cavity to enhance the recorded signal .', '1401.3528-1-3-3': 'Their dual time and frequency structure make them also ideal candidate for ranging or clock synchronization , which has been proven to be optimal using measurement in an homodyne configuration with pulse shaped reference beam .', '1401.3528-1-3-4': 'This concept of projection on the temporal mode carrying the information can be extended to general parameter estimation .', '1401.3528-1-4-0': 'In this context of very high sensitivity metrology, measurements are limited by the low-level intrinsic noise of the laser being of classical or quantum nature.', '1401.3528-1-4-1': 'We demonstrate here that it is possible to efficiently measure filter this noise even close to the quantum limit.', '1401.3528-1-4-2': 'The impact on a highly sensitive, homodyne timing measurement is analyzed.', '1401.3528-1-5-0': 'The paper is organized in the following way: The CEO-phase noise, and for completeness the amplitude fluctuations of a commercial Ti:Sa laser are determined relative to their common quantum limit.', '1401.3528-1-5-1': 'A broadband passive optical cavity is then proposed to filter the remaining fluctuations of the CEO phase.', '1401.3528-1-5-2': 'Besides filtering, it also permits to detect phase noise down to the quantum limit.', '1401.3528-1-5-3': 'By the use of the Ti:Sa oscillator, the resulting realistic sensitivities for the homodyne timing measurement scheme are subsequently discussed at the end of this paper.', '1401.3528-1-6-0': 'Theoretical concept.', '1401.3528-1-6-1': 'We consider a train of fs pulses generated by a commercial mode-locked Ti:Sa oscillator (fig. [REF]).', '1401.3528-1-6-2': 'This laser source can be described as a superposition of equally spaced monochromatic modes of frequencies [MATH], where [MATH] is the repetition rate and [MATH] the carrier-envelope-offset frequency.', '1401.3528-1-6-3': 'But one can also give a time representation of this pulse train.', '1401.3528-1-6-4': 'Introducing the light-cone variable [MATH], one can write the positive frequency component of the electric field as [MATH], where [MATH] is the carrier frequency, [MATH] the CEO phase, [MATH] the pulse to pulse time interval, [MATH] the single pulse photon number, [MATH] a normalization constant and [MATH] is a normalized single pulse mode (non zero in interval [MATH]).', '1401.3528-1-7-0': 'The aim of this paper is to study frequency comb noise properties and noise filtering in view of possible application for ultra sensitive pulse-timing measurements such as those introduced in .', '1401.3528-1-7-1': 'They rely on the homodyne detection of a pulse train in mode [MATH] with a superposition of two modes that are proportional to I: [MATH] and II: [MATH].', '1401.3528-1-7-2': 'Using that reference, the minimum resolvable timing jitter of a fs-pulse train at an analysis frequency [MATH] is in 1Hz resolution bandwidth: [EQUATION]', '1401.3528-1-8-0': 'The normalized variances [MATH](f) of the field quadratures P and Q of the modes I and II are related to: [MATH] the CEO-phase of the signal comb, [MATH] the amplitude quadrature of a mode corresponding to a time-of-flight (TOF) measurement.', '1401.3528-1-8-1': 'Both are equal to unity when the light source noise is simply vacuum quantum noise.', '1401.3528-1-8-2': 'This level is called the Standard Quantum Limit (SQL).', '1401.3528-1-8-3': '[MATH] are the single sideband (SSB) noise power spectral densities (PSD) of the quadratures.', '1401.3528-1-8-4': '[MATH] is the common quantum limited PSD in unities dBc/Hz, [MATH] with the average total comb power [MATH].', '1401.3528-1-8-5': 'The value [MATH] corresponds to the mean squared timing jitter measured with a TOF measurement.', '1401.3528-1-8-6': 'The measurement of both CEO phase noise and TOF jitter is discussed below.', '1401.3528-1-9-0': 'Optical setup.', '1401.3528-1-9-1': 'The setup to access and manipulate the noise properties of an optical frequency comb is drafted in fig. [REF].', '1401.3528-1-9-2': 'It consists of a commercial Femtolasers Ti:Sa oscillator emitting [MATH]fs pulses at [MATH]MHz repetition rate at [MATH]nm.', '1401.3528-1-9-3': 'A passive impedance matched cavity, synchronous with the pump laser (see below) is placed on one arm of a Mach Zehnder-like configuration, closed by a balanced homodyne detection.', '1401.3528-1-9-4': 'A Menlo Systems f-2f interferometer detects the CEO frequency and its fluctuations.', '1401.3528-1-9-5': 'It permits to lock the Ti:Sa CEO with a typical bandwidth below [MATH]kHz.', '1401.3528-1-10-0': 'Noise properties of the free-running laser.', '1401.3528-1-10-1': 'Amplitude and phase noise RF-power spectral densities (PSD) of the free running Ti:Sa oscillator are measured respectively by the balanced detection (when the filtering cavity path of the Mach Zehnder is closed) and by the f-2f interferometer.', '1401.3528-1-10-2': 'They are shown in fig. [REF].', '1401.3528-1-10-3': 'They are compared to the common SQL for 8mW signal.', '1401.3528-1-10-4': 'The relative intensity noise (RIN) reaches the SQL above 2MHz.', '1401.3528-1-10-5': 'The relaxation oscillation peak at 1MHz, which depends on alignment and output power, has been minimized for that measurement.', '1401.3528-1-10-6': 'The locked [MATH] can be considered as free running above the lock-resonance at 30kHz.', '1401.3528-1-10-7': 'The line at 100kHz results from the relaxation oscillation of the pumping VerdiDPSS laser.', '1401.3528-1-10-8': 'This noise follows an approximated [MATH] distribution over more than one decade until the noise level of detection is reached at 700kHz.', '1401.3528-1-10-9': 'It is set by phase-excess noise from the measurement avalanche photodiode.', '1401.3528-1-10-10': 'The [MATH] corresponds to the theoretical prediction of and an additional term from amplitude noise coupling to phase noise.', '1401.3528-1-10-11': 'The levels are more than 60dB above the measured repetition rate phase noise that was measured up to 10kHz sideband frequency within a [MATH] slope (data not shown).', '1401.3528-1-10-12': 'The noise of the amplitude quadrature of the time-of-flight (TOF) mode II is consequently negligible in eq. [MATH].', '1401.3528-1-10-13': 'The first term of the equation, corresponding to a phase measurement, is the dominant one.', '1401.3528-1-11-0': 'It is the CEO noise that dominates the timing measurement sensitivity in this measurement configuration.', '1401.3528-1-11-1': 'A reduction to the SQL can significantly increase the achievable measurement sensitivity at a given measurement frequency.', '1401.3528-1-11-2': 'However even if CEO-phase can be actively locked within a certain bandwidth, sidebands above [MATH]kHz and levels close to the SQL are difficult to reach with active feedback.', '1401.3528-1-11-3': 'We propose the use of a filtering cavity as shown with next section techniques.', '1401.3528-1-12-0': 'The filtering cavity.', '1401.3528-1-12-1': 'An optical cavity, used in transmission, is a well known 2nd order low pass filter acting on both phase and amplitude noise of the input field .', '1401.3528-1-12-2': 'The 3dB cutoff frequency is determined by the speed of light c, the cavity finesse F and its length L to: [MATH].', '1401.3528-1-13-0': 'We have developed an impedance matched passive cavity in a bow-tie geometry consisting in [MATH] zero-dispersion mirrors and contained in a low-vacuum chamber.', '1401.3528-1-13-1': 'Residual dispersion is compensated by an air pressure [MATH]mbar depending on the required spectral shape of transmission.', '1401.3528-1-13-2': 'Input and output couplers have equal reflectivities of [MATH] and the 4 other mirrors are highly reflective.', '1401.3528-1-13-3': 'The cavity length is chosen so that it can be synchronously pumped by the femtosecond laser.', '1401.3528-1-13-4': 'This amounts to have a cavity whose free spectral range is the input laser repetition rate (in our case cavity optical length is then [MATH]m).', '1401.3528-1-13-5': 'A Pound-Drever-Hall scheme is used to lock the cavity on resonance with the input laser.', '1401.3528-1-13-6': 'To avoid the modulated reference to appear in the detected signal, the reflection of a counter-propagating reference-beam is used the generate the error signal.', '1401.3528-1-13-7': 'The CEO-phase of the laser is locked to match the resonance frequencies of the filtering cavity.', '1401.3528-1-13-8': 'When all locks are running, the 45nm FWHM spectrum generated by the Ti:Sa oscillator is entirely (35nm FWHM) transmitted through the cavity as shown in fig [REF].', '1401.3528-1-13-9': 'Comparing input and output spectra, one can conclude that the residual dispersion of the cavity is below [MATH] over the covered spectrum .', '1401.3528-1-13-10': 'The the simulations agree well with the observed spectra.', '1401.3528-1-13-11': 'The pressure dependence of transmission is indicated by the choice of a sightly different pressure for the simulation.', '1401.3528-1-14-0': 'For noise filtering, we measured an effective finesse of [MATH] so that the 3dB cutoff frequency of this cavity is found at [MATH]kHz.', '1401.3528-1-15-0': 'Measurement of relative phase noise.', '1401.3528-1-15-1': 'No measurements have yet been reported quantifying relative phase noise differences close to the SQL.', '1401.3528-1-15-2': 'A shot noise resolving, balanced homodyne detection can do so.', '1401.3528-1-15-3': 'It is used here to evaluate the phase noise difference before and after the filtering cavity.', '1401.3528-1-15-4': 'The scheme is sketched in fig. [REF].', '1401.3528-1-15-5': 'It reveals relative phase fluctuations between the intense local oscillator (LO) and the much smaller signal (S).', '1401.3528-1-15-6': 'Two incident fields [MATH] interfere.', '1401.3528-1-15-7': 'Locking on the phase quadrature [MATH] leads to the homodyne signal [MATH], where [MATH] are the zero mean fluctuations of the relative phase of the incident fields.', '1401.3528-1-15-8': 'In addition, zero mean amplitude fluctuations [MATH] and [MATH] may be present.', '1401.3528-1-15-9': 'Assuming perfect quantum efficiency of the photodetectors, setting the elementary charge to one and neglecting higher order terms, the mean squared homodyne signal becomes : [EQUATION]', '1401.3528-1-15-10': 'It detects relative phase fluctuations [MATH] of both beams and is insensitive to classical intensity noise of the LO.', '1401.3528-1-15-11': 'The over all detected intensity is [MATH].', '1401.3528-1-15-12': 'For constant [MATH] the electronic signal is maximized for equal intensities of signal and LO to [MATH].', '1401.3528-1-15-13': 'This classical signal is always detected relative to the shot noise level [MATH].', '1401.3528-1-15-14': 'The detected phase noise can be expressed relative to the carrier by dBc/Hz=1/2[MATH]rad[MATH]/Hz.', '1401.3528-1-15-15': 'For a signal to noise ratio [MATH] and a detected power P, the minimal resolvable relative phase noise is consequently [EQUATION]', '1401.3528-1-16-0': 'For sufficiently low amplitude noise, the measurement scheme fig. [REF] has quantum limited sensitivity to relative phase noise.', '1401.3528-1-16-1': 'If the cutoff frequency of the cavity [MATH] is sufficiently small, the relative phase noise equals phase noise of the seed.', '1401.3528-1-16-2': 'This is the CEO-phase noise [MATH].', '1401.3528-1-16-3': 'With the [MATH]kHz used here, this relation yields at MHz detection frequencies.', '1401.3528-1-16-4': 'CEO-phase noise becomes consequently measurable down to the SQL microsecond timescales.', '1401.3528-1-17-0': 'Measurement Data.', '1401.3528-1-17-1': 'The measured homodyning signal shown in fig. [REF] arises from the interference of a bright signal from the Ti:Sa oscillator (LO) with the 10dB less intense beam filtered from the cavity (Signal).', '1401.3528-1-17-2': 'The signal from the cavity exhibits intensity excess noise arising from noise-quadrature interconversion by the cavity (not shown).', '1401.3528-1-17-3': 'Nevertheless, its presence does not change the RF-spectral distribution of the homodyning signal shown in fig.[REF].', '1401.3528-1-17-4': 'Classical intensity noise of signal and (LO) cancel in the balanced measurement configuration and only contribute to higher order terms of the signal [MATH] described in eq.[REF].', '1401.3528-1-18-0': 'Phase noise filtering and detection.', '1401.3528-1-18-1': 'The experimental results are plotted in fig. [REF].', '1401.3528-1-18-2': 'The unfiltered laser beams acts as a local oscillator (LO) for the cavity output beam (signal).', '1401.3528-1-18-3': 'The spectral overlap between the two beams is 94%.', '1401.3528-1-18-4': 'The corresponding phase quadrature is selected by locking the relative phase of both ports.', '1401.3528-1-18-5': 'The resulting phase quadrature homodyne signal shows an approximate [MATH] behavior over nearly two decades of RF-frequency.', '1401.3528-1-18-6': 'This matches perfectly the expected value.', '1401.3528-1-18-7': 'Indeed, we measured the CEO phase noise of the Ti:Sa oscillator to follows a [MATH] law, and the filtering power of the cavity is then given by its transfer function which follows a [MATH] distribution.', '1401.3528-1-18-8': 'Consequently, the expected power law for the relative phase noise is the product of both, [MATH].', '1401.3528-1-18-9': 'In conclusion, the cavity filters CEO phase noise significantly, by [MATH]dB per decade.', '1401.3528-1-18-10': 'This confirms, together with the transmitted optical bandwidth, that cavities can not only filter noise of single optical frequencies but also of entire coherent frequency combs.', '1401.3528-1-19-0': 'Similar to the amplitude noise, the homodyne signal [REF] and thus the CEO-phase noise of the oscillator vanishes in shot noise at [MATH]MHz detection frequency.', '1401.3528-1-20-0': 'Conclusions.', '1401.3528-1-20-1': 'The phase noise of a Ti:Sa oscillator is characterized down to the SQL by the use of a broadband passive cavity and shot noise resolving, homodyne detection.', '1401.3528-1-20-2': 'The investigated oscillator is quantum limited in amplitude and phase for above 3MHz detection frequencies.', '1401.3528-1-20-3': 'In this context, filtering the phase noise of an optical frequency comb through a passive cavity has been proved to be applicable to up to [MATH]nm FWHM broad frequency combs.', '1401.3528-1-20-4': 'It potentially increases the sensitivity in metrology experiments limited by high frequency CEO-phase noise.', '1401.3528-1-20-5': 'We illustrate this further by considering its impact on the homodyne measurement of timing-jitter as suggested in : With the amplitude and phase noise from fig. [REF], the realistic sensitivity of such a measurement can be approximated from Eq. [REF] to:[EQUATION]', '1401.3528-1-20-6': 'This expression scales as [MATH] (with [MATH] being the measurement time) and as the square root of the spectral density of CEO-phase fluctuations.', '1401.3528-1-20-7': 'The sensitivity is limited by CEO-phase noise.', '1401.3528-1-20-8': 'For a typical Ti:Sa oscillator this noise is far above the SQL, even down to [MATH] timescales as shown in fig.[REF].', '1401.3528-1-20-9': 'Plotted in fig.[REF], the passive filtering of CEO-phase noise improves the realistic sensitivity by up to 2 orders of magnitude at 1MHz detection frequency.', '1401.3528-1-20-10': 'In addition, by the reach of the SQL, the filtered phase noise may provide for quantum limited measurements above a few MHz frequencies.'} | {'1401.3528-2-0-0': 'It is shown that the sensitivity of a highly sensitive homodyne timing measurement scheme with femtosecond (fs) lasers is limited by carrier-envelope-phase (CEO) noise.', '1401.3528-2-0-1': 'We describe the use of a broadband passive cavity to analyze the phase noise of a Ti:Sapph oscillator relative to the standard quantum limit.', '1401.3528-2-0-2': 'This cavity also filters lowest levels of classical noise at sidebands above [MATH]kHz detection frequency.', '1401.3528-2-0-3': 'Leading to quantum limited CEO-phase noise at [MATH]s-timescales, it can improve the sensitivity of the homodyne pulse timing measurement by 2 orders of magnitude.', '1401.3528-2-1-0': '140.0140, 270.0270', '1401.3528-2-2-0': ']', '1401.3528-2-3-0': 'Introduction.', '1401.3528-2-3-1': 'Femtosecond optical frequency combs have revolutionized optical metrology given their intrinsic frequency comb structure and the ability to measure and lock the phases of these frequencies.', '1401.3528-2-3-2': 'Recent experiments have demonstrated that they can be extended to massively parallel optical spectroscopy, using a dual comb configuration and an optical cavity to enhance the recorded signal .', '1401.3528-2-3-3': 'Their dual time and frequency structure also make them ideal candidate for ranging or clock synchronization .', '1401.3528-2-3-4': 'This measurement has been proven to be optimal if a homodyne configuration with a pulse shaped reference beam is used .', '1401.3528-2-3-5': 'The underlying concept of projection on the temporal mode carrying the information can be extended to general parameter estimation .', '1401.3528-2-4-0': 'In this context of very high sensitivity metrology, measurements are limited by the low-level intrinsic noise of the laser, being of classical or quantum nature.', '1401.3528-2-4-1': 'Although the noise of Ti:Sapph oscillators has been characterized extensively relative to the carrier , no data are available relative to the quantum limit.', '1401.3528-2-4-2': 'We demonstrate here that it is possible to efficiently measure and even filter noise at such low levels.', '1401.3528-2-4-3': 'To this aim we combine shot noise resolving intensity noise detection and the filtering properties of a passive cavity.', '1401.3528-2-5-0': 'The paper is organized in the following way: The carrier-envelope-offset (CEO) phase noise, and for completeness the amplitude fluctuations of a commercial Ti:Sapph laser are determined relative to their common quantum limit.', '1401.3528-2-5-1': 'A broadband passive optical cavity is then proposed to filter the remaining fluctuations of the CEO phase.', '1401.3528-2-5-2': 'Besides filtering, it allows the detection of phase noise down to the quantum limit.', '1401.3528-2-5-3': 'The resulting realistic sensitivities for the homodyne timing measurement scheme are discussed at the end of this paper.', '1401.3528-2-6-0': 'Theoretical concept.', '1401.3528-2-6-1': 'We consider a train of [MATH] pulses generated by a commercial mode-locked Ti:Sapph oscillator (Fig. [REF]).', '1401.3528-2-6-2': 'It can be described as a superposition of equally spaced monochromatic modes of frequencies [MATH], where [MATH] is the repetition rate and [MATH] the CEO frequency.', '1401.3528-2-6-3': 'But one can also give a time representation of this pulse train.', '1401.3528-2-6-4': 'Introducing the light-cone variable [MATH], one can write the positive frequency component of the electric field as [MATH], where [MATH] is the carrier frequency, [MATH] the CEO phase, [MATH] the pulse to pulse time interval, [MATH] the photon number in a single pulse, [MATH] a normalization constant and [MATH] is a normalized single pulse mode (non zero in interval [MATH]).', '1401.3528-2-7-0': 'This paper studies properties of frequency comb noise and the filtering of noise in view of possible application to ultra sensitive pulse-timing measurements, such as those introduced in .', '1401.3528-2-7-1': 'These measurements rely on the homodyne detection of a pulse train in mode [MATH] with a local oscillator in a superposition of two modes that are proportional to I: [MATH] and II: [MATH].', '1401.3528-2-7-2': 'Using this formalism, the minimum resolvable timing jitter of a fs-pulse train is at 1Hz resolution bandwidth and an analysis frequency [MATH]: [EQUATION]', '1401.3528-2-7-3': 'It is a function of the optical bandwidth [MATH] of the signal.', '1401.3528-2-7-4': 'The normalized variances [MATH] of the field quadratures P (phase) and Q (amplitude) of the modes I and II are related to: [MATH] the CEO-phase of the signal comb, [MATH] the amplitude quadrature of a mode corresponding to a time-of-flight (TOF) measurement.', '1401.3528-2-7-5': 'Both are equal to unity when the light source noise is simply vacuum quantum noise.', '1401.3528-2-7-6': 'This level is called the Standard Quantum Limit (SQL).', '1401.3528-2-7-7': 'This theoretical result of can be linked to the experimentally accessible, single sideband (SSB) noise power spectral densities (PSD) [MATH] of the quadratures P and Q of the CEO-phase (I) and the repetition rate (II): [EQUATION]', '1401.3528-2-8-0': '[MATH] is the common quantum limited PSD in unities of dBc/Hz.', '1401.3528-2-8-1': 'Given by [MATH], it is a function of the average total comb power [MATH].', '1401.3528-2-8-2': 'The measurement of both CEO phase noise and TOF jitter is discussed below.', '1401.3528-2-9-0': 'Optical setup.', '1401.3528-2-9-1': 'The setup to access and manipulate the noise properties of an optical frequency comb is drafted in Fig. [REF].', '1401.3528-2-9-2': 'It consists of a commercial Femtolasers Ti:Sa oscillator emitting [MATH]-fs pulses at [MATH]MHz repetition rate, centered at [MATH]nm with an average power of 1W.', '1401.3528-2-9-3': 'A passive, impedance matched cavity, synchronous with the pump laser is placed on one arm of a Mach Zehnder-like configuration, closed by a balanced homodyne detection.', '1401.3528-2-9-4': 'A Menlo Systems f-2f interferometer detects the CEO frequency and its fluctuations after coherent spectral broadening.', '1401.3528-2-9-5': 'It allows to lock the Ti:Sapph CEO frequency with a typical bandwidth of less than [MATH]kHz.', '1401.3528-2-10-0': 'Noise properties of the free-running laser.', '1401.3528-2-10-1': 'Intensity noise is measured using only the balanced detection.', '1401.3528-2-10-2': 'Together with the CEO-phase noise, those data are shown in Fig. [REF].', '1401.3528-2-10-3': 'When compared to the common SQL for a signal of 8mW, the relative intensity noise (RIN) reaches the SQL above 3MHz.', '1401.3528-2-10-4': 'The only slightly distinguishable relaxation oscillation peak at 1.5MHz depends on alignment and output power.', '1401.3528-2-10-5': 'It has been minimized for this measurement.', '1401.3528-2-10-6': 'The locked [MATH] can be considered as free running above the lock-resonance at 30kHz.', '1401.3528-2-10-7': 'The line at 100kHz results from the relaxation oscillation of the pumping Verdi laser, which is also present in the amplitude noise.', '1401.3528-2-10-8': 'The CEO-phase noise follows an approximated [MATH] distribution over more than one decade until the noise level of detection is reached at 700kHz.', '1401.3528-2-10-9': 'It is set by phase-excess noise from the measurement avalanche photodiode.', '1401.3528-2-10-10': 'The [MATH] behavior corresponds to the theoretical prediction of to [MATH] and an additional term from the coupling of amplitude noise to phase noise.', '1401.3528-2-10-11': 'The levels observed are more than 60dB above the repetition rate phase noise.', '1401.3528-2-10-12': 'The latter was observed to follow an [MATH] dependence down to 10kHz sideband frequency (data not shown).', '1401.3528-2-10-13': 'The noise of the amplitude quadrature of the TOF mode (II) is consequently negligible in Eq. [REF].', '1401.3528-2-10-14': 'The first term of the equation, corresponding to a phase measurement, is the dominant one.', '1401.3528-2-10-15': 'From Eq. [REF] and Eq. [REF], the sensitivity of the homodyne timing measurement can be approximated to: [EQUATION]', '1401.3528-2-10-16': 'This expression scales as [MATH] (with [MATH] being the measurement time) and with the square root of the spectral density of CEO-phase fluctuations [MATH].', '1401.3528-2-11-0': 'The sensitivity is therefore limited by CEO-phase noise.', '1401.3528-2-11-1': 'Active locking schemes are typically of kHz-bandwidth.', '1401.3528-2-11-2': 'Sidebands above [MATH]kHz and levels close to the SQL are difficult to reach with active feedback .', '1401.3528-2-11-3': 'A passive cavity can filter phase noise above [MATH] and down to the SQL.', '1401.3528-2-11-4': 'The next section studies this approach.', '1401.3528-2-12-0': 'A filtering cavity.', '1401.3528-2-12-1': 'A transmissive optical cavity is a well known 2nd-order low-pass filter acting on both phase and amplitude noise of the input field .', '1401.3528-2-12-2': 'The 3-dB cutoff frequency [MATH] is determined by the speed of light [MATH], the cavity finesse [MATH], and its length [MATH].', '1401.3528-2-12-3': 'We have developed an impedance-matched passive cavity in a bow-tie geometry consisting of [MATH] zero-dispersion mirrors and contained in a low-vacuum chamber.', '1401.3528-2-12-4': 'Its integration into the experimental setup is depicted in Fig. [REF].', '1401.3528-2-12-5': 'Residual dispersion is compensated by laboratory air.', '1401.3528-2-12-6': 'The constant pressure of [MATH]mbar is chosen depending on the required spectral shape of transmission.', '1401.3528-2-12-7': 'Input and output couplers have equal reflectivities of [MATH] and the 4 other mirrors are highly reflective.', '1401.3528-2-12-8': 'The cavity length is set to the same value as the femtosecond oscillator ([MATH]m).', '1401.3528-2-12-9': 'A Pound-Drever-Hall scheme is used to lock the cavity on resonance with the seed pulse train.', '1401.3528-2-12-10': 'To avoid the modulated reference to appear in the detected signal, we use a counter-propagating reference-beam to generate the error signal.', '1401.3528-2-12-11': 'The CEO-phase of the laser is locked to match the resonance frequencies of the filtering cavity.', '1401.3528-2-12-12': 'When all locks are running, the 45-nm FWHM spectrum generated by the Ti:Sapph oscillator is entirely (35nm FWHM) transmitted through the cavity, see Fig.[REF].', '1401.3528-2-12-13': 'Comparing input and output spectra in Fig.[REF], we estimate that the residual dispersion of the cavity is less than [MATH] over the covered spectrum .', '1401.3528-2-12-14': 'The simulations agree well with the observed spectra.', '1401.3528-2-12-15': 'A sightly different pressure is used for the simulation to illustrate the pressure dependence of the transmitted bandwidth.', '1401.3528-2-12-16': 'Concerning noise filtering, we measured an effective finesse of [MATH].', '1401.3528-2-12-17': 'The 3-dB cutoff frequency of this cavity was measured at [MATH]kHz.', '1401.3528-2-13-0': 'Measurement of relative phase noise.', '1401.3528-2-13-1': 'To quantify the effect of phase noise filtering by the passive cavity, relative phase noise between a filtered and an unfiltered beam have to be measured down to the SQL.', '1401.3528-2-13-2': 'A shot-noise-resolving, balanced homodyne detection can do so.', '1401.3528-2-13-3': 'Its implementation is shown in Fig. [REF].', '1401.3528-2-13-4': 'It measures phase noise difference before and after the filtering cavity.', '1401.3528-2-13-5': 'Two incident fields [MATH] interfere.', '1401.3528-2-13-6': 'Locking on the phase quadrature [MATH] leads to the beating signal [MATH] where [MATH] are the zero mean fluctuations of the relative phase of the interfering fields.', '1401.3528-2-13-7': 'This signal efficiently converts phase- to amplitude fluctuations.', '1401.3528-2-13-8': 'If significant amplitude noise is present, one of both beams is strongly attenuated.', '1401.3528-2-13-9': 'This turns the scheme into a homodyne detection of relative phase noise.', '1401.3528-2-13-10': 'An interference of a bright local oscillator (LO) and an attenuated signal is less sensitive to amplitude fluctuations.', '1401.3528-2-13-11': 'Assuming perfect quantum efficiency of the photodetectors, setting the elementary charge to one and neglecting higher order terms, the mean squared homodyne signal reads : [EQUATION]', '1401.3528-2-13-12': 'It detects relative phase fluctuations [MATH] of the interfering beams.', '1401.3528-2-13-13': 'For a given, over-all detected intensity [MATH], the electronic signal is maximized for equal intensities to [MATH].', '1401.3528-2-13-14': 'This classical signal is always detected relative to the shot noise level [MATH].', '1401.3528-2-13-15': 'The detected phase noise can be expressed relative to the carrier in units dBc/Hz=(1/2)[MATH]rad[MATH]/Hz .', '1401.3528-2-13-16': 'For a signal to noise ratio SNR=[MATH] and a detected power [MATH], the minimal resolvable relative phase noise is [EQUATION]', '1401.3528-2-14-0': 'Consequently, given sufficiently low amplitude noise, the measurement scheme Fig. [REF] has quantum-limited sensitivity to relative phase noise.', '1401.3528-2-14-1': 'For sufficiently high detection frequencies, the phase noise in the filtered arm becomes negligible.', '1401.3528-2-14-2': 'The detected signal is than proportional to the CEO-phase noise [MATH].', '1401.3528-2-14-3': 'With [MATH]kHz used here, this relation holds at MHz detection frequencies.', '1401.3528-2-14-4': 'In conclusion, absolute levels of CEO-phase noise also become measurable down to the SQL.', '1401.3528-2-15-0': 'Measurement Data.', '1401.3528-2-15-1': 'The measured homodyning signal shown in Fig. [REF], trace .5pt-.9pt2.', '1401.3528-2-15-2': 'It arises from the interference of the signal from the Ti:Sapph oscillator (LO) with the 10dB less intense beam filtered from the cavity (Signal).', '1401.3528-2-15-3': 'The signal from the cavity exhibits intensity excess noise arising from noise-quadrature interconversion by the cavity (not shown).', '1401.3528-2-15-4': 'Nevertheless, its presence does not change the RF-spectral distribution of the homodyning signal.', '1401.3528-2-15-5': 'Classical intensity noise of signal and local oscillator cancel in the balanced measurement configuration and only contribute to higher order terms of the signal [MATH] described in Eq. [REF].', '1401.3528-2-16-0': 'Phase noise filtering and detection.', '1401.3528-2-16-1': 'The experimental results are plotted in Fig.[REF].', '1401.3528-2-16-2': 'The interfering beams have a spectral overlap of 94.', '1401.3528-2-16-3': 'The homodyne signal of the phase quadrature follows an approximate [MATH] power law over nearly two decades of detection frequencies.', '1401.3528-2-16-4': 'Indeed, the CEO-phase noise of the Ti:Sapph oscillator was observed to follow an [MATH] dependence.', '1401.3528-2-16-5': 'In addition, the filter-efficiency of the cavity is then given by its transfer function which follows a [MATH] distribution.', '1401.3528-2-16-6': 'Consequently, the expected power law for the relative phase noise is the product of both, [MATH].', '1401.3528-2-16-7': 'This is equivalent to the measured distribution.', '1401.3528-2-16-8': 'It confirms, together with the transmitted optical bandwidth, that cavities can not only filter noise of single optical frequencies but also of entire coherent frequency combs.', '1401.3528-2-16-9': 'The measured amplitude noise of the oscillator is neglectable for the measurement above (see Fig.[REF], trace .5pt-.9pt3).', '1401.3528-2-16-10': 'Similar to the amplitude noise, the homodyne signal and thus the CEO-phase noise of the oscillator vanishes in shot noise at approximately [MATH]MHz detection frequency (see Fig.[REF], trace .5pt-.9pt2).', '1401.3528-2-17-0': 'The consequences of the observed phase noise filtering for the timing measurement discussed with Eq.[REF] are shown in Fig.[REF].', '1401.3528-2-17-1': 'Using the f-2f CEO-phase noise data, the possible precision of the homodyne timing measurement can be calculated for a filtered or an unfiltered beam.', '1401.3528-2-17-2': 'From Fig.[REF] it follows from the interference data that the slope of the CEO-phase noise does not significantly change at microsecond timescales.', '1401.3528-2-17-3': 'The achievable measurement precisions shown in Fig.[REF] can thus be extrapolated down to the SQL.', '1401.3528-2-17-4': 'Using a passive cavity to filter phase noise, the expected sensitivity of the timing measurement could be improved by up to 2 orders of magnitude .', '1401.3528-2-18-0': 'Conclusions.', '1401.3528-2-18-1': 'A broadband resonant, passive cavity has been shown to be a tool for filtering and the detection of CEO-phase noise of a [MATH]-nm FWHM frequency comb.', '1401.3528-2-18-2': 'Together with shot noise resolving balanced detection, it is shown that a commercial Ti:Sapph oscillator is quantum limited in amplitude and phase below 5MHz detection frequency.', '1401.3528-2-18-3': 'Passive filtering of phase noise decreases this frequency.', '1401.3528-2-18-4': 'It potentially improves the sensitivity of a pulse-timing measurement scheme by up to two orders of magnitude.'} | [['1401.3528-1-15-0', '1401.3528-2-13-0'], ['1401.3528-1-15-6', '1401.3528-2-13-5'], ['1401.3528-1-15-13', '1401.3528-2-13-14'], ['1401.3528-1-3-1', 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'1401.3528-2-13-13'], ['1401.3528-1-15-12', '1401.3528-2-13-13'], ['1401.3528-1-15-14', '1401.3528-2-13-15'], ['1401.3528-1-3-3', '1401.3528-2-3-3'], ['1401.3528-1-10-2', '1401.3528-2-10-2'], ['1401.3528-1-10-3', '1401.3528-2-10-3'], ['1401.3528-1-10-5', '1401.3528-2-10-4'], ['1401.3528-1-10-5', '1401.3528-2-10-5'], ['1401.3528-1-10-7', '1401.3528-2-10-7'], ['1401.3528-1-10-11', '1401.3528-2-10-11'], ['1401.3528-1-10-11', '1401.3528-2-10-12'], ['1401.3528-1-10-12', '1401.3528-2-10-13'], ['1401.3528-1-9-2', '1401.3528-2-9-2'], ['1401.3528-1-9-4', '1401.3528-2-9-4'], ['1401.3528-1-9-5', '1401.3528-2-9-5'], ['1401.3528-1-16-2', '1401.3528-2-14-2'], ['1401.3528-1-16-2', '1401.3528-2-14-4'], ['1401.3528-1-16-4', '1401.3528-2-14-4'], ['1401.3528-1-17-1', '1401.3528-2-15-1'], ['1401.3528-1-17-1', '1401.3528-2-15-2'], ['1401.3528-1-5-2', '1401.3528-2-5-2'], ['1401.3528-1-0-1', '1401.3528-2-0-1'], ['1401.3528-1-4-1', '1401.3528-2-4-2'], ['1401.3528-2-10-9', '1401.3528-3-9-9'], ['1401.3528-2-12-14', '1401.3528-3-11-14'], ['1401.3528-2-12-15', '1401.3528-3-11-14'], ['1401.3528-2-11-1', '1401.3528-3-10-1'], ['1401.3528-2-11-2', '1401.3528-3-10-2'], ['1401.3528-1-18-3', '1401.3528-2-16-2'], ['1401.3528-1-18-5', '1401.3528-2-16-3'], ['1401.3528-1-18-7', '1401.3528-2-16-4'], ['1401.3528-1-18-7', '1401.3528-2-16-5'], ['1401.3528-1-19-0', '1401.3528-2-16-10'], ['1401.3528-1-12-2', '1401.3528-2-12-2'], ['1401.3528-1-13-1', '1401.3528-2-12-5'], ['1401.3528-1-13-1', '1401.3528-2-12-6'], ['1401.3528-1-13-3', '1401.3528-2-12-8'], ['1401.3528-1-13-8', '1401.3528-2-12-12'], ['1401.3528-1-13-9', '1401.3528-2-12-13'], ['1401.3528-1-13-11', '1401.3528-2-12-15'], ['1401.3528-1-14-0', '1401.3528-2-12-16'], ['1401.3528-1-7-2', '1401.3528-2-7-2'], ['1401.3528-1-8-4', '1401.3528-2-8-0'], ['1401.3528-1-8-4', '1401.3528-2-8-1'], ['1401.3528-2-14-0', '1401.3528-3-13-6']] | [['1401.3528-1-11-2', '1401.3528-2-11-2']] | ['1401.3528-1-1-0', '1401.3528-1-2-0', '1401.3528-1-3-0', '1401.3528-1-6-0', '1401.3528-1-9-0', '1401.3528-1-12-0', '1401.3528-1-17-0', '1401.3528-1-20-0', '1401.3528-2-1-0', '1401.3528-2-2-0', '1401.3528-2-3-0', '1401.3528-2-6-0', '1401.3528-2-9-0', '1401.3528-2-12-0', '1401.3528-2-15-0', '1401.3528-2-18-0', '1401.3528-3-1-0', '1401.3528-3-2-0', '1401.3528-3-3-0', '1401.3528-3-6-0', '1401.3528-3-8-0', '1401.3528-3-11-0', '1401.3528-3-14-0', '1401.3528-3-17-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1401.3528 | {'1401.3528-3-0-0': 'It is shown that the sensitivity of a highly sensitive homodyne timing measurement scheme with femtosecond (fs) lasers is limited by carrier-envelope-phase (CEO) noise.', '1401.3528-3-0-1': 'We describe the use of a broadband passive cavity to analyze the phase noise of a Ti:Sapph oscillator relative to the standard quantum limit.', '1401.3528-3-0-2': 'This cavity also filters lowest levels of classical noise at sidebands above [MATH]kHz detection frequency.', '1401.3528-3-0-3': 'Leading to quantum limited CEO-phase noise at [MATH]s-timescales, it can improve the sensitivity of the homodyne pulse timing measurement by 2 orders of magnitude.', '1401.3528-3-1-0': '140.0140, 270.0270', '1401.3528-3-2-0': ']', '1401.3528-3-3-0': 'Introduction.', '1401.3528-3-3-1': 'Femtosecond optical frequency combs have revolutionized optical metrology given their intrinsic frequency comb structure and the ability to measure and lock the phases of these frequencies.', '1401.3528-3-3-2': 'Recent experiments have demonstrated that they can be extended to massively parallel optical spectroscopy, using a dual comb configuration and an optical cavity to enhance the recorded signal .', '1401.3528-3-3-3': 'Their dual time and frequency structure also make them ideal candidate for ranging or clock synchronization .', '1401.3528-3-3-4': 'This measurement has been proven to be optimal when used in a homodyne configuration with a pulse shaped reference beam .', '1401.3528-3-3-5': 'The underlying concept of projection on the temporal mode carrying the information can be extended to general parameter estimation .', '1401.3528-3-4-0': 'In this context of very high sensitivity metrology, measurements are limited by the low-level intrinsic noise of the laser, being of classical or quantum nature.', '1401.3528-3-4-1': 'Although the noise of Ti:Sapph oscillators has been characterized extensively relative to the carrier , no data are available relative to the quantum limit.', '1401.3528-3-4-2': 'We demonstrate here that it is possible to efficiently measure and even filter noise at such low levels.', '1401.3528-3-4-3': 'To this aim we combine shot noise resolving intensity noise detection and the filtering properties of a passive cavity.', '1401.3528-3-5-0': 'The paper is organized in the following way: The carrier-envelope-offset (CEO) phase noise, and for completeness the amplitude fluctuations of a commercial Ti:Sapph laser are determined relative to their common quantum limit.', '1401.3528-3-5-1': 'A broadband passive optical cavity is then proposed to filter the remaining fluctuations of the CEO phase.', '1401.3528-3-5-2': 'Besides filtering, it allows the detection of phase noise down to the quantum limit.', '1401.3528-3-5-3': 'The resulting realistic sensitivities for the homodyne timing measurement scheme are discussed at the end of this paper.', '1401.3528-3-6-0': 'Theoretical concept.', '1401.3528-3-6-1': 'We consider a train of [MATH] pulses generated by a commercial mode-locked Ti:Sapph oscillator (Fig. [REF]).', '1401.3528-3-6-2': 'It can be described as a superposition of equally spaced monochromatic modes of frequencies [MATH], where [MATH] is the repetition rate and [MATH] the CEO frequency.', '1401.3528-3-6-3': 'But one can also give a time representation of this pulse train.', '1401.3528-3-6-4': 'Introducing the light-cone variable [MATH], one can write the positive frequency component of the electric field as [MATH], where [MATH] is the carrier frequency, [MATH] the CEO phase, [MATH] the pulse to pulse time interval, [MATH] the photon number in a single pulse, [MATH] a normalization constant and [MATH] is a normalized single pulse mode (non zero in interval [MATH]).', '1401.3528-3-7-0': 'This paper studies properties of frequency comb noise and its filtering in view of possible application to ultra sensitive pulse-timing measurements, such as those introduced in .', '1401.3528-3-7-1': 'These measurements rely on the homodyne detection of a pulse train in mode [MATH] with a local oscillator in a superposition of two modes that are proportional to I: [MATH] and II: [MATH].', '1401.3528-3-7-2': 'Using this formalism, the minimum resolvable timing jitter of a fs-pulse train is at 1Hz resolution bandwidth and an analysis frequency [MATH]: [EQUATION]', '1401.3528-3-7-3': 'It is a function of the optical bandwidth [MATH] of the signal.', '1401.3528-3-7-4': 'The normalized variances [MATH] of the field quadratures P (phase) and Q (amplitude) of the modes I and II are related to: [MATH] the CEO-phase of the signal comb, [MATH] the amplitude quadrature of a mode corresponding to a time-of-flight (TOF) measurement.', '1401.3528-3-7-5': 'Both are equal to unity when the light source noise is simply vacuum quantum noise.', '1401.3528-3-7-6': 'This level is called the Standard Quantum Limit (SQL).', '1401.3528-3-7-7': 'This theoretical result of can be linked to the experimentally accessible, single sideband (SSB) noise power spectral densities (PSD) [MATH] of the quadratures P and Q of the CEO-phase (I) and the repetition rate (II): [EQUATION] [MATH] is the common quantum limited PSD in unities of dBc/Hz.', '1401.3528-3-7-8': 'Given by [MATH], it is a function of the average total comb power [MATH].', '1401.3528-3-7-9': 'The measurement of both CEO phase noise and TOF jitter is discussed below.', '1401.3528-3-8-0': 'Optical setup.', '1401.3528-3-8-1': 'The setup to access and manipulate the noise properties of an optical frequency comb is drafted in Fig. [REF].', '1401.3528-3-8-2': 'It consists of a commercial Femtolasers Ti:Sa oscillator emitting [MATH]-fs pulses at [MATH]MHz repetition rate, centered at [MATH]nm with an average power of 1W.', '1401.3528-3-8-3': 'A passive, impedance matched cavity, synchronous with the pump laser is placed on one arm of a Mach Zehnder-like configuration, closed by a balanced homodyne detection.', '1401.3528-3-8-4': 'A Menlo Systems f-2f interferometer detects the CEO frequency and its fluctuations after coherent spectral broadening.', '1401.3528-3-8-5': 'It allows to lock the Ti:Sapph CEO frequency with a typical bandwidth of less than [MATH]kHz.', '1401.3528-3-9-0': 'Noise properties of the free-running laser.', '1401.3528-3-9-1': 'Intensity noise is measured using only the balanced detection.', '1401.3528-3-9-2': 'Together with the CEO-phase noise, those data are shown in Fig. [REF].', '1401.3528-3-9-3': 'When compared to the common SQL for a signal of 8mW, the relative intensity noise (RIN) reaches the quantum limit above 3MHz.', '1401.3528-3-9-4': 'The tiny relaxation oscillation peak at 1.5MHz depends on alignment and output power.', '1401.3528-3-9-5': 'It has been minimized for this measurement.', '1401.3528-3-9-6': 'The locked [MATH] can be considered as free running above the lock-resonance at 30kHz.', '1401.3528-3-9-7': 'The line at 100kHz results from the relaxation oscillation of the pumping Verdi laser, which is also present in the amplitude noise.', '1401.3528-3-9-8': 'The CEO-phase noise follows an approximated [MATH] distribution over more than one decade until the noise level of detection is reached at 700kHz.', '1401.3528-3-9-9': 'It is set by phase-excess noise from the avalanche photodiode detecting the f-2f beating.', '1401.3528-3-9-10': 'The [MATH] behavior corresponds to the theoretical prediction of to [MATH] and to an additional term from the coupling of amplitude to phase noise.', '1401.3528-3-9-11': 'The levels observed are more than 60dB above the repetition rate phase noise.', '1401.3528-3-9-12': 'The latter was observed to follow a [MATH] dependence down to 10kHz sideband frequency (data not shown).', '1401.3528-3-9-13': 'The noise of the amplitude quadrature of the TOF mode (II) is consequently negligible in Eq. [REF].', '1401.3528-3-9-14': 'The first term of the equation, corresponding to a phase measurement, is the dominant one.', '1401.3528-3-9-15': 'From Eq. [REF] and Eq. [REF], the sensitivity of the homodyne timing measurement can be approximated to: [EQUATION]', '1401.3528-3-9-16': 'This expression scales as [MATH] (with [MATH] being the measurement time) and with the square root of the spectral density of CEO-phase fluctuations [MATH].', '1401.3528-3-10-0': 'The sensitivity is therefore limited by CEO-phase noise.', '1401.3528-3-10-1': 'Active locking schemes typically have a bandwidth of a few kHz.', '1401.3528-3-10-2': 'Sidebands above [MATH]kHz and levels close to the SQL are difficult to reach.', '1401.3528-3-10-3': 'The next section shows that a passive cavity can filter phase noise above [MATH] and down to the SQL.', '1401.3528-3-11-0': 'A filtering cavity.', '1401.3528-3-11-1': 'A transmissive optical cavity is a well known 2nd-order low-pass filter acting on both phase and amplitude noise of the input field .', '1401.3528-3-11-2': 'The 3-dB cutoff frequency [MATH] is determined by the speed of light [MATH], the cavity finesse [MATH], and its length [MATH].', '1401.3528-3-11-3': 'We have developed an impedance-matched passive cavity in a bow-tie geometry consisting of [MATH] zero-dispersion mirrors and contained in a low-vacuum chamber.', '1401.3528-3-11-4': 'Its integration into the experimental setup is depicted in Fig. [REF].', '1401.3528-3-11-5': 'Residual dispersion is compensated by laboratory air.', '1401.3528-3-11-6': 'The constant pressure of [MATH]mbar is chosen depending on the required spectral shape of transmission.', '1401.3528-3-11-7': 'Input and output couplers have equal reflectivities of [MATH] and the 4 other mirrors are highly reflective.', '1401.3528-3-11-8': 'The cavity length is set to the same value as the femtosecond oscillator ([MATH]m).', '1401.3528-3-11-9': 'A Pound-Drever-Hall scheme is used to lock the cavity on resonance with the seed pulse train.', '1401.3528-3-11-10': 'To avoid the modulated reference to appear in the detected signal, we use a counter-propagating reference-beam to generate the error signal.', '1401.3528-3-11-11': 'The CEO-phase of the laser is locked to match the resonance frequencies of the filtering cavity.', '1401.3528-3-11-12': 'When all locks are running, the 45-nm FWHM spectrum generated by the Ti:Sapph oscillator is entirely (35nm FWHM) transmitted through the cavity, see Fig.[REF].', '1401.3528-3-11-13': 'Comparing input and output spectra in Fig.[REF], we estimate that the residual dispersion of the cavity is less than [MATH] over the covered spectrum .', '1401.3528-3-11-14': 'The simulations agree with the observed spectra for a slightly different pressure.', '1401.3528-3-11-15': 'The difference may be attributed to uncertainties in the mirror properties.', '1401.3528-3-11-16': 'Concerning noise filtering, we measured an effective finesse of [MATH].', '1401.3528-3-11-17': 'The 3-dB cutoff frequency of this cavity was measured at [MATH]kHz.', '1401.3528-3-12-0': 'Measurement of relative phase noise.', '1401.3528-3-12-1': 'To quantify the effect of phase noise filtering by the passive cavity, relative phase noise between a filtered and an unfiltered beam have to be measured down to the SQL.', '1401.3528-3-12-2': 'A shot-noise-resolving, balanced homodyne detection can do so.', '1401.3528-3-12-3': 'Its implementation is shown in Fig. [REF].', '1401.3528-3-12-4': 'It measures phase noise differences before and after the filtering cavity.', '1401.3528-3-12-5': 'Two incident fields [MATH] interfere.', '1401.3528-3-12-6': 'Locking on the phase quadrature [MATH] leads to the beating signal [MATH] where [MATH] are the zero mean fluctuations of the relative phase of the interfering fields.', '1401.3528-3-12-7': 'This signal efficiently converts phase- to amplitude fluctuations.', '1401.3528-3-12-8': 'If significant amplitude noise is present, one of both beams is strongly attenuated.', '1401.3528-3-12-9': 'This turns the scheme into a homodyne detection of relative phase noise.', '1401.3528-3-12-10': 'An interference of a bright local oscillator (LO) and an attenuated signal is less sensitive to amplitude fluctuations.', '1401.3528-3-13-0': 'Assuming perfect quantum efficiency of the photodetectors, setting the elementary charge to one and neglecting higher order terms, the mean squared homodyne signal reads : [EQUATION]', '1401.3528-3-13-1': 'It detects relative phase fluctuations [MATH] of the interfering beams.', '1401.3528-3-13-2': 'For a given, over-all detected intensity [MATH], the electronic signal is maximized for equal intensities to [MATH].', '1401.3528-3-13-3': 'This classical signal is always detected relative to the shot noise level [MATH].', '1401.3528-3-13-4': 'The detected phase noise can be expressed relative to the carrier in units dBc/Hz=(1/2)[MATH]rad[MATH]/Hz .', '1401.3528-3-13-5': 'For a signal to noise ratio SNR=[MATH] and a detected power [MATH], the minimal resolvable relative phase noise is [EQUATION]', '1401.3528-3-13-6': 'Consequently, given sufficiently low amplitude noise, the measurement scheme Fig. [REF] of relative phase noise has a quantum limited sensitivity.', '1401.3528-3-13-7': 'For sufficiently high detection frequencies, the phase noise in the filtered arm becomes negligible.', '1401.3528-3-13-8': 'The detected signal is than proportional to the CEO-phase noise [MATH].', '1401.3528-3-13-9': 'With [MATH]kHz used here, this relation holds at MHz detection frequencies.', '1401.3528-3-13-10': 'In conclusion, absolute levels of CEO-phase noise also become measurable down to the SQL.', '1401.3528-3-14-0': 'Measurement Data.', '1401.3528-3-14-1': 'The measured homodyning signal is shown in Fig. [REF], trace .5pt-.9pt2.', '1401.3528-3-14-2': 'It arises from the interference of the signal from the Ti:Sapph oscillator (LO) with the 10dB less intense beam filtered from the cavity (Signal).', '1401.3528-3-14-3': 'The signal from the cavity exhibits intensity excess noise arising from noise-quadrature interconversion by the cavity (not shown).', '1401.3528-3-14-4': 'Nevertheless, its presence does not change the RF-spectral distribution of the homodyning signal.', '1401.3528-3-14-5': 'Classical intensity noise of both the signal and the local oscillator cancel in the balanced measurement configuration and only contribute to higher order terms of the signal [MATH] described in Eq. [REF].', '1401.3528-3-15-0': 'Phase noise filtering and detection.', '1401.3528-3-15-1': 'The experimental results are plotted in Fig.[REF].', '1401.3528-3-15-2': 'The interfering beams have a spectral overlap of 94.', '1401.3528-3-15-3': 'The homodyne signal of the phase quadrature of mode I follows an approximate [MATH] power law over nearly two decades of detection frequencies.', '1401.3528-3-15-4': 'Indeed, the CEO-phase noise of the Ti:Sapph oscillator was observed to follow an [MATH] dependence.', '1401.3528-3-15-5': 'In addition, the filter-efficiency of the cavity is then given by its transfer function which follows a [MATH] distribution.', '1401.3528-3-15-6': 'Consequently, the expected power law for the relative phase noise is the product of both, [MATH].', '1401.3528-3-15-7': 'This is equivalent to the measured distribution.', '1401.3528-3-15-8': 'It confirms, together with the transmitted optical bandwidth, that cavities can not only filter noise of single optical frequencies but also of entire coherent frequency combs.', '1401.3528-3-15-9': 'The measured amplitude noise of the oscillator is neglectable for the measurement above (see Fig.[REF], trace .5pt-.9pt3).', '1401.3528-3-15-10': 'Similar to the amplitude noise, the homodyne signal and thus the CEO-phase noise of the oscillator vanishes in shot noise at approximately [MATH]MHz detection frequency (see Fig.[REF], trace .5pt-.9pt2).', '1401.3528-3-16-0': 'The consequences of the observed phase noise filtering for the timing measurement discussed with Eq.[REF] are shown in Fig.[REF].', '1401.3528-3-16-1': 'Using the f-2f CEO-phase noise data, the possible precision of the homodyne timing measurement can be calculated for a filtered or an unfiltered beam.', '1401.3528-3-16-2': 'From Fig.[REF] it follows from the interference data that the slope of the CEO-phase noise does not significantly change at microsecond timescales.', '1401.3528-3-16-3': 'The achievable measurement precision shown in Fig.[REF] can thus be extrapolated down to the SQL.', '1401.3528-3-16-4': 'Using a passive cavity to filter phase noise, the expected sensitivity of the timing measurement could be improved by up to 2 orders of magnitude .', '1401.3528-3-17-0': 'Conclusions.', '1401.3528-3-17-1': 'A broadband resonant, passive cavity has been shown to be a tool for filtering and the detection of CEO-phase noise of a [MATH]-nm FWHM frequency comb.', '1401.3528-3-17-2': 'Together with shot noise resolving balanced detection, it is shown that a commercial Ti:Sapph oscillator is quantum limited in amplitude and phase below 5MHz detection frequency.', '1401.3528-3-17-3': 'Passive filtering of phase noise decreases this frequency.', '1401.3528-3-17-4': 'It potentially improves the sensitivity of a pulse-timing measurement scheme by up to two orders of magnitude.'} | null | null | null | null |
astro-ph-0205343 | {'astro-ph-0205343-1-0-0': 'Recently there is a suggestion about the star RXJ1856.5[MATH]3754, being a possible candidate for quark star.', 'astro-ph-0205343-1-0-1': 'While from the estimates of the radiation radius, given by the authors, it does not seem to fit with any of the known models for such stars, the mass radius diagram admits the possibility of this being a quark star of mass [MATH], provided the radius mentioned is the actual radius of the star.', 'astro-ph-0205343-1-1-0': 'Subject headings: star: RXJ1856.5[MATH]3754 - strange star - neutron star - equation of state', 'astro-ph-0205343-1-2-0': '# Introduction', 'astro-ph-0205343-1-3-0': 'One of the hot topics in astrophysics of late has been the possibility of the discovery of Strange Stars (stars composed of u, d and s quark matters), through X-ray observations from the Chandra satellite by Drake et al., (2002).', 'astro-ph-0205343-1-3-1': 'Though discussions of the possibility of the existence of such stars dates back to the eighties (Witten, 1984 and later by Alcock et al., 1986, Haensel et al., 1986, etc.), with high resolution data from the X-ray observations, recently observers have improved the accuracy for ascertaining the dimensions of the compact objects associated with the observed emissions (Walter et al., 1996 Walter Mathews 1997 and Pons et al., 2001).', 'astro-ph-0205343-1-3-2': 'During the last few years, there have been several sources, which are considered niether as black holes nor neutron stars, like the Her-X1, 4U 1820[MATH]30 (Dey et al, 1998 and Bombaci, 1997), SAX J1808.4[MATH]3658 (Li et al., 1999a), 4U 1728[MATH]34 (Li et al., 1999b), PSR 0943[MATH]10 (Xu et al., 1999).', 'astro-ph-0205343-1-3-3': 'While it is extremely difficult to come to a conclusion regarding a candidate being a black hole (for masses of the order of few solar masses), it is slightly better for neutron stars, once proper estimates of mass and radius are made along with the evidence for the existence of a hard surface which could produce bursts.', 'astro-ph-0205343-1-4-0': '# Determination of mass-radius relation', 'astro-ph-0205343-1-5-0': 'The method of obtaining the crucial parameters from observational data do depend upon the models adopted.', 'astro-ph-0205343-1-5-1': 'One of the recent candidates, which has evoked a lot of attention in this context is the source RXJ1856.5[MATH]3754, an isolated compact star at a distance of [MATH] 140 pc in the outskirts of the R CrA dark molecular cloud (Drake et al., 2002, Campana et al., 1997 and Pons et al., 2002).', 'astro-ph-0205343-1-5-2': 'According to Drake et al., (2002) this investigation, the implied radius of the star seems to be [MATH] km, where [EQUATION]', 'astro-ph-0205343-1-5-3': 'R being the true radius and M the mass of the star.', 'astro-ph-0205343-1-5-4': "According to their model, R[MATH] called as the 'radiation radius' is determined through the temperature estimates obtained from the observational X-ray data.", 'astro-ph-0205343-1-6-0': 'Formula ([REF]) is purely general relativistic, as obtained through red shift factors of the Schwarzschild solution for a non rotating star of mass M and radius R.', 'astro-ph-0205343-1-6-1': 'It is well known that [MATH] denotes the event horizon in the metric.', 'astro-ph-0205343-1-6-2': 'Further, it is also known that the last circular photon orbit for this geometry is at [MATH] a surface also known for the strange effect of centrifugal force reversal for particles on circular geodesics (Abramowicz Prasanna, 1990).', 'astro-ph-0205343-1-6-3': 'Heyl (2000), using this feature had obtained constraints on neutron star radii for the case of Type I X-ray burst sources.', 'astro-ph-0205343-1-6-4': 'Before considering the authencity of the constraint, one ought to realize that the effect of centrifugal reversal occurs only for purely circular geodesics, whereas the presence of even a small radial velocity for the accreting particles would change its orbit to non circular motion, in which case the effect of centrifugal reversal occurs only if the central star is rotating (Prasanna, 2001).', 'astro-ph-0205343-1-6-5': 'In view of this, the constraint [MATH] may not be effective always.', 'astro-ph-0205343-1-6-6': 'On the other hand, the radiation radius has to be greater than 3M, as the last photon orbit is at [MATH].', 'astro-ph-0205343-1-7-0': 'Consider the equation ([REF]), which may be re-written as [EQUATION]', 'astro-ph-0205343-1-7-1': 'As R represents the true radius of the star of mass M it is imperative that R[MATH] has to be such that the equation will have real roots.', 'astro-ph-0205343-1-7-2': 'As the last term is always positive, one of the roots is always negative.', 'astro-ph-0205343-1-7-3': 'Considering the discriminant [MATH] it is clear that the other two roots, when real, are equal if [MATH] and real but not equal for [MATH].', 'astro-ph-0205343-1-7-4': 'Tables [REF] and [REF] give the locations of the real roots for R for a given M, as also the location of the event horizon R[MATH](black hole case), and the photon orbit R[MATH] for [MATH] and [MATH] respectively.', 'astro-ph-0205343-1-8-0': 'Having thus seen the possible M-R configurations, one can now superpose this configuration space on the configuration space obtained from the models with different equations of states (EOSs) for compact stars, for possible deductions.', 'astro-ph-0205343-1-9-0': 'Fig. ([REF]) shows the mass-radius diagram for compact stars with the solid lines A and B denoting the radii R[MATH] and R[MATH] of table [REF].', 'astro-ph-0205343-1-9-1': 'As may be seen for objects with [MATH] there are objects, within the region covered by the strange stars having equation of states (1) and (2).', 'astro-ph-0205343-1-9-2': 'However Drake et al., (2002) set the radiation radius R[MATH] to be 3.6 to 8.2 kms, which from the table corresponds to masses less than 1M[MATH].', 'astro-ph-0205343-1-9-3': 'Again refering to the figure one finds that the range of true radii of the star R, corresponding to the mentioned range of R[MATH], does not fall anywhere near the M-R relations commensurate with any of the equations of state, so far known for the strange stars.', 'astro-ph-0205343-1-10-0': '# Neutron star and Strange star equation of states', 'astro-ph-0205343-1-11-0': 'There are many neutron star equation of states (EOS) which give mass-radius relation over a wide range when fed into the TOV equation.', 'astro-ph-0205343-1-11-1': 'Almost all of the EOSs are calculated by considering either the relativistic Dirac-Brueckner-Hartree-Fock models, or the relativistic field theoretical models, or the non-relativistic potential models.', 'astro-ph-0205343-1-11-2': 'Also, some models have been considered with the possibility of the stellar core possessing a Bose-Einstien condensate of the negative kaons (Kaplan Nelson, 1986 and Thorsson et al., 1994).', 'astro-ph-0205343-1-11-3': 'We have chosen only four neutron star EOSs just to compare them with the strange star models.', 'astro-ph-0205343-1-11-4': 'Also we saw from literature that there are a wide variety of neutron star EOSs, but none of them come so close to the radius within 8 kms.', 'astro-ph-0205343-1-12-0': 'The curve labelled 7 in Fig. ([REF]) is due to Lorentz, Ravenhall and Pethick (1993).', 'astro-ph-0205343-1-12-1': 'They considered a microscopic Hamiltonian obtained by fitting a Skyrme-like energy density functional to the values of the employed microscopic two body potential V14 and the three body force TNI.', 'astro-ph-0205343-1-12-2': 'These has been earlier used by Friedman and Pandharipande (1981) in hypernated chain techniques considering a range of densities and temperatures.', 'astro-ph-0205343-1-12-3': 'Wiringa Ficks and Fabrocini (1988) calculated an EOS (labelled 5 in Fig. ([REF])) using a Hamiltonian where a two-nucleon potential that fits nucleon-nucleon scattering data and deuteron properties has been employed, and also posseses an explicit three-nucleon interaction term.', 'astro-ph-0205343-1-12-4': 'They calculated the Hamiltonian using five combinations of the Argonne [MATH], Urbana [MATH] two-nucleon potentials and the Urbana VII three-nucleon potential.', 'astro-ph-0205343-1-12-5': 'We have shown here only result with the AV14+UVII.', 'astro-ph-0205343-1-12-6': 'Curve labelled 6 is the EOS as per the Bethe Johnson model (Bethe Johnson, 1974) who used realistic potentials in their calculations, and curve labelled 8 is due to Pandharipande (1971).', 'astro-ph-0205343-1-13-0': 'The idea of existence of exotic stars such as the strange quark stars, has been a long term debate for astrophysicists and particle physicists too.', 'astro-ph-0205343-1-13-1': 'The laboratory scale lifetime for deconfined u, d and s quarks is typically of the order of [MATH] sec, which is far away from the astrophysical scale of stellar lifetime.', 'astro-ph-0205343-1-13-2': 'So, a stable strange quark star model raised questions about the existence of such stars.', 'astro-ph-0205343-1-13-3': 'The scenario has changed very much with the coming up of new generation x-ray satellites.', 'astro-ph-0205343-1-13-4': 'The analysis from observations now supply us new constraints on the mass-radius relation of some of these compact objects.', 'astro-ph-0205343-1-13-5': 'With the advent of time, the new data proved that the stars developed from the neutron star equation of states (EOS) do not match with the observations, and that they are more compact.', 'astro-ph-0205343-1-13-6': 'This leaves the only choice of considering them as the strange quark stars.', 'astro-ph-0205343-1-14-0': 'Ever since the strange star hypothesis was proposed by Witten (1984), various models have been developed.', 'astro-ph-0205343-1-14-1': 'Among all of them, the most common one is that based on the MIT bag model.', 'astro-ph-0205343-1-14-2': 'In this phenomenological model the basic features of QCD, i.e., quark confinement and asymptotic freedom, are built in.', 'astro-ph-0205343-1-14-3': 'However in this model, the deconfinement of quarks at high density is not obvious.', 'astro-ph-0205343-1-14-4': 'Preleminary calculations of strange stars using the bag model has been done by Alcock et al. (1986) and Haensel et al. (1986).', 'astro-ph-0205343-1-14-5': 'A result for the bag model EOS for starnge stars is shown in the figure, where two cases for the EOSs are given, one for the [MATH] and strange quark masss of 0 MeV and 150 MeV for curves labelled 3 and 4 respectively.', 'astro-ph-0205343-1-15-0': 'Alternative to the MIT bag model, Dey et al., in 1998, derived an EOS for strange matter which has asymptotic freedom built in and describes deconfined quarks at high density and confinement at zero density.', 'astro-ph-0205343-1-15-1': 'With a proper choice of the EOS parameters, this model gives absolutely stable strange quark matter.', 'astro-ph-0205343-1-15-2': 'This EOS was used to calculate the structure of static strange stars and the mass-radius relations.', 'astro-ph-0205343-1-15-3': 'Later, it was suggested (Li et al., 1999a) that the millisecond X-ray pulsar SAX1808.4[MATH]3658 is a strange star.', 'astro-ph-0205343-1-15-4': 'This model which also explained the observed properties of some other compact objects like the analysis of semi-empirical mass-radius relations from the QPO observations in 4U 1728-34 (Li et al., 1999b) as also the RXTE observations of Her X-1 and 4U 1820-30 (Dey et al., 1998), leads to the suggestion that these objects host strange stars.', 'astro-ph-0205343-1-15-5': 'In Li et al., (1999a), two sets of EOSs are used for two sets of parameters, namely SS1 and SS2.', 'astro-ph-0205343-1-15-6': 'The maximum gravitational masses are [MATH] for SS1 (curve labelled 2) and [MATH] for SS2 (curve labelled 1).', 'astro-ph-0205343-1-15-7': 'For different values of the mass parameter [MATH] in the D98 model, the stars can have a sequence of masses.', 'astro-ph-0205343-1-15-8': 'The calculations of Dey et al. (1998) has been done considering zero temperature of the strange matter.', 'astro-ph-0205343-1-15-9': 'Ray et al. (2000) calculated the finite temperature effect on the quark stars developed by Dey et al. (1998) and found that it sustains even more mass for a particular radius of the star as compared to the case of cold star.', 'astro-ph-0205343-1-16-0': '# Discussions', 'astro-ph-0205343-1-17-0': 'It is necessary to point out that the patched region in Fig. ([REF]) is from consideration of the fact that the value of R[MATH] is taken higher than that mentioned in Drake et al. (2002) This fits well with at least one type of EOS for strange stars (solid curves 1 2).', 'astro-ph-0205343-1-17-1': "The rest lie far away from the 'allowed zone'.", 'astro-ph-0205343-1-17-2': "One wonders whether the accuracy of measurement of the 'radiation radius' [MATH] is perfect, and if so then one has to really look into some other aspects for modelling the strange star.", 'astro-ph-0205343-1-17-3': "On the other hand if the radius mentioned (3.6 [MATH] 8.2 km) is the true radius R of the star, then it seems that the only model corresponding to curves 1 2, can admit a stable configuration and RXJ1856.5[MATH]3754 is indeed a 'strange star'.", 'astro-ph-0205343-1-18-0': 'We would like to thank Dr. Herman Marshall for a clarification through correspondence.'} | {'astro-ph-0205343-2-0-0': 'Recently there have been controversial claims about the nature of the isolated compact star RXJ1856.5[MATH]3754, with one group claiming it to be a strange star (Drake et al., 2002) while the other asserting it to be a normal neutron star (Walter Lattimer, 2002).', 'astro-ph-0205343-2-0-1': 'The controversy arises mainly due to the distance estimate, which in turn is used to resolve the measured angular diameter, and thus the radiation radius [MATH].', 'astro-ph-0205343-2-0-2': 'In this we discuss the theoretical constraints that appear from analysing the usual mass-radius relation alongwith the redshift factors arising from the strong gravity effects and possible lensing.', 'astro-ph-0205343-2-0-3': 'Unless the distance estimate is confirmed independently, without any uncertainty, it is premature to come to any conclusion regarding the nature of this star.', 'astro-ph-0205343-2-1-0': 'Subject headings: star: RXJ1856.5[MATH]3754 - strange star - neutron star - equation of state', 'astro-ph-0205343-2-2-0': '# Introduction', 'astro-ph-0205343-2-3-0': 'One of the hot topics in astrophysics of late has been the possibility of the discovery of Strange Stars (stars composed of u, d and s quark matter), through X-ray observations of the candidate RX J1856.35[MATH]3754 from the Chandra satellite by Drake et al., (2002).', 'astro-ph-0205343-2-3-1': 'Almost at the same time Walter Lattimer (2002) have given a totally different picture of the same, claiming it to be a normal neutron star.', 'astro-ph-0205343-2-3-2': 'Though discussions of the possibility of the existence of such stars dates back to the eighties (Witten, 1984 and later by Alcock et al., 1986, Haensel et al., 1986, etc.), with high resolution data from the X-ray observations, recently observers have improved the accuracy for ascertaining the dimensions of the compact objects associated with the observed emissions (Walter et al., 1996 Walter Mathews 1997 and Pons et al., 2001).', 'astro-ph-0205343-2-3-3': 'During the last few years, there have been several sources, which are considered neither as black holes nor neutron stars, like the Her-X1, 4U 1820[MATH]30 (Dey et al, 1998 and Bombaci, 1997), SAX J1808.4[MATH]3658 (Li et al., 1999a), 4U 1728[MATH]34 (Li et al., 1999b), PSR 0943[MATH]10 (Xu et al., 1999).', 'astro-ph-0205343-2-3-4': 'The absence of spectral lines in the thermal components of the X-ray compact sources from the observations by Chandra and the XMM-Newton, led Xu (2002) to claim the existence of bare strange stars.', 'astro-ph-0205343-2-3-5': 'While it is extremely difficult to come to a conclusion regarding a candidate being a black hole (for masses of the order of few solar masses), it is slightly better for neutron stars, once proper estimates of mass and radius are made along with the evidence for the existence of a hard surface which could produce bursts.', 'astro-ph-0205343-2-4-0': '# Determination of mass-radius relation', 'astro-ph-0205343-2-5-0': 'The method of obtaining the crucial parameters from observational data does depend upon the models adopted.', 'astro-ph-0205343-2-5-1': 'One of the recent candidates which has evoked a lot of interest in this context is the source RXT1856.3-3754, an isolated compact star at a distance of [MATH] 140 pc in the outskirts of the RCrA dark molecular cloud.', 'astro-ph-0205343-2-5-2': "Incidentally there are two conflicting claims regarding this source with Drake et al. (2002) claiming it to be a 'strange star' with radiation radius [MATH] ranging from 3.8-8.2 kms, and mass [MATH] 1.4 [MATH], whereas Walter Lattimer (2002) claim it to be a neutron star with [MATH] 15 km and [MATH] 1.7 [MATH].", 'astro-ph-0205343-2-5-3': 'In this approach the radiation radius [MATH] is defined through the general relativistic relation [EQUATION] wherein [MATH] and [MATH] are the actual radius and mass of the star.', 'astro-ph-0205343-2-5-4': 'It is apparent that what is measured is [MATH], through observation of the angular diameter of the source, expressed as [MATH] where [MATH] is the distance of the source.', 'astro-ph-0205343-2-5-5': 'Leaving aside the part of ambiguities and uncertainties in the measurement of [MATH], there seems to be very little consensus among observers about the distance measurement in this case.', 'astro-ph-0205343-2-6-0': "However, what needs to be carefully looked into, is the crucial implications of the 'measured' value for determining the mass and radius of the star, which decides whether it is a neutron star or an exotic strange star.", 'astro-ph-0205343-2-6-1': 'As the relevant formulae used takes into account the crucial redshift factor due to strong gravity effects, it is also important to consider other possible general relativistic effects for photon trajectories.', 'astro-ph-0205343-2-6-2': "In this context one should take into account 'the self-lensing effect due to the sources' own gravitational field' of the emitted light rays close to the central star.", 'astro-ph-0205343-2-6-3': "Nollert et al. (1989) have discussed this effect while analyzing the relativistic 'looks' of a neutron star, who clearly points out the relevance of self-lensing while analyzing the data from a compact source.", 'astro-ph-0205343-2-6-4': 'Accordingly, they point out that if [MATH] is the specific intensity of the observed radiation then [MATH] and [MATH] are related through the equation [EQUATION]', 'astro-ph-0205343-2-6-5': 'Hence, if one considers this relation for the specific intensity and evaluate the relation between the radii [MATH] and [MATH] for a black body emission one finds the relation to be [EQUATION] and not the one used by Drake et al. or Walter et al. Before proceeding further with the estimates one needs to consider few other aspects of general relativity.', 'astro-ph-0205343-2-6-6': 'It is well known that [MATH] denotes the event horizon and further [MATH] corresponds to the circular photon orbit, as well as the radius at which the centrifugal force reversal occurs for particles on circular geodesics (Abramowicz and Prasanna, 1990); Heyl, (2000), using this feature had obtained constraints on neutron star radii for the case of Type 1 X-ray burst sources.', 'astro-ph-0205343-2-6-7': 'Before considering the authenticity of the constraint one ought to realize that the effect of centrifugal reversal occurs only for purely circular geodesics, whereas the presence of even a small radial velocity for the accreting particles would change its orbit to non-circular motion in which case the centrifugal reversal occurs only if the central star is rotating (Prasanna, 2001).', 'astro-ph-0205343-2-6-8': 'In view of this, the constraint [MATH] may not be always effective, whereas [MATH] is indeed a must for any observation.', 'astro-ph-0205343-2-7-0': 'Considering now equation ([REF]) which can be written as [EQUATION] a quintic equation in [MATH].', 'astro-ph-0205343-2-7-1': 'Unlike in the case of a cubic, there is no way of expressing a general condition between [MATH] and [MATH] for the existence of real roots.', 'astro-ph-0205343-2-7-2': 'However, one can find numerically that if [MATH] 5.4954[MATH] 3.75[MATH] then there are two positive real roots, for [MATH] one of which certainly lies outside [MATH], the photon circular orbit.', 'astro-ph-0205343-2-8-0': 'Table 1 yields the location of the highest real positive root [MATH] for given [MATH] and [MATH] chosen to be 9m.', 'astro-ph-0205343-2-8-1': 'As these are the only consistent numbers, satisfying the defining equation one has to constrain M-R relation as given by this.', 'astro-ph-0205343-2-8-2': 'For comparison we have also given the photon radius [MATH] for the corresponding mass, and one clearly sees that the actual radius of the star is greater than [MATH] for the chosen [MATH].', 'astro-ph-0205343-2-9-0': 'It may be seen from the table that for the mass [MATH] 1.7 [MATH], and [MATH] = 15.3 kms, the actual stellar radius [MATH] 13.65, a value larger than that obtained by Walter and Lattimer.', 'astro-ph-0205343-2-9-1': "Another important thing to notice here is that the redshift factor '[MATH]' when calculated for the entire range of values of [MATH] and [MATH] as given in Table 1 yields a value [MATH] 0.256, which lies within the range as given by Pons et al. On the other hand, the ranges of [MATH] and [MATH] as obtained, after including the lensing factor takes the star away from the estimates of Drake et al.", 'astro-ph-0205343-2-10-0': 'If lensing is not taken into account then the two radii [MATH] and [MATH] are related through eq. (1), which may be re-written as [EQUATION]', 'astro-ph-0205343-2-10-1': 'As R represents the true radius of the star of mass M it is imperative that R[MATH] has to be such that the equation will have real roots.', 'astro-ph-0205343-2-10-2': 'As the last term is always positive, one of the roots is always negative.', 'astro-ph-0205343-2-10-3': 'Considering the discriminant [MATH] it is clear that the other two roots, when real, are equal if [MATH] and real but not equal for [MATH].', 'astro-ph-0205343-2-11-0': 'Table [REF] gives the locations of the outer real root for R for a given M, and the photon orbit R[MATH] and the Radiation radius [MATH] respectively, for this case.', 'astro-ph-0205343-2-12-0': '# Neutron star and Strange star equation of states', 'astro-ph-0205343-2-13-0': 'There are many neutron star equation of states (EOS) which give mass-radius relation over a wide range when fed into the TOV equation.', 'astro-ph-0205343-2-13-1': 'Almost all of the EOSs are calculated by considering either the relativistic Dirac-Brueckner-Hartree-Fock models, or the relativistic field theoretical models, or the non-relativistic potential models.', 'astro-ph-0205343-2-13-2': 'Also, some models have been considered with the possibility of the stellar core possessing a Bose-Einstien condensate of the negative kaons (Kaplan Nelson, 1986 and Thorsson et al., 1994).', 'astro-ph-0205343-2-13-3': 'We have chosen only four neutron star EOSs just to compare them with the strange star models.', 'astro-ph-0205343-2-13-4': 'Also we saw from literature that there are a wide variety of neutron star EOSs, but none of them come so close to the radius within 8 kms.', 'astro-ph-0205343-2-14-0': 'The curve labelled 7 in Fig. ([REF]) is due to Lorentz, Ravenhall and Pethick (1993).', 'astro-ph-0205343-2-14-1': 'They considered a microscopic Hamiltonian obtained by fitting a Skyrme-like energy density functional to the values of the employed microscopic two body potential V14 and the three body force TNI.', 'astro-ph-0205343-2-14-2': 'These has been earlier used by Friedman and Pandharipande (1981) in hypernated chain techniques considering a range of densities and temperatures.', 'astro-ph-0205343-2-14-3': 'Wiringa Ficks and Fabrocini (1988) calculated an EOS (labelled 5 in Fig. ([REF])) using a Hamiltonian where a two-nucleon potential that fits nucleon-nucleon scattering data and deuteron properties has been employed, and also possesses an explicit three-nucleon interaction term.', 'astro-ph-0205343-2-14-4': 'They calculated the Hamiltonian using five combinations of the Argonne [MATH], Urbana [MATH] two-nucleon potentials and the Urbana VII three-nucleon potential.', 'astro-ph-0205343-2-14-5': 'We have shown here only result with the AV14+UVII.', 'astro-ph-0205343-2-14-6': 'Curve labelled 6 is the EOS as per the Bethe Johnson model (Bethe Johnson, 1974) who used realistic potentials in their calculations, and curve labelled 8 is due to Pandharipande (1971).', 'astro-ph-0205343-2-15-0': 'The idea of existence of exotic stars such as the strange quark stars, has been a long term debate for astrophysicists and particle physicists too.', 'astro-ph-0205343-2-15-1': 'The laboratory scale lifetime for deconfined u, d and s quarks is typically of the order of [MATH] sec, which is far away from the astrophysical scale of stellar lifetime.', 'astro-ph-0205343-2-15-2': 'So, a stable strange quark star model raised questions about the existence of such stars.', 'astro-ph-0205343-2-15-3': 'The scenario has changed very much with the coming up of new generation x-ray satellites.', 'astro-ph-0205343-2-15-4': 'The analysis from observations now supply us new constraints on the mass-radius relation of some of these compact objects.', 'astro-ph-0205343-2-15-5': 'With the advent of time, some of the new data for some sources and their analysis (e.g., Haberl Titarchuk (1995), Bombaci (1997), Dey et al. (1998), Li et al. (1999a 1999b), Xu et al. (1999), Kapoor Shukre (2001), etc,) proved that the stars developed from neutron star EOSs do not match with them, and they are more compact.', 'astro-ph-0205343-2-15-6': 'This leaves the only choice of considering them as the strange quark stars.', 'astro-ph-0205343-2-16-0': 'Ever since the strange star hypothesis was proposed by Witten (1984), various models have been developed.', 'astro-ph-0205343-2-16-1': 'Among all of them, the most common one is that based on the MIT bag model.', 'astro-ph-0205343-2-16-2': 'In this phenomenological model the basic features of QCD, i.e., quark confinement and asymptotic freedom, are built in.', 'astro-ph-0205343-2-16-3': 'However in this model, the deconfinement of quarks at high density is not obvious.', 'astro-ph-0205343-2-16-4': 'Preliminary calculations of strange stars using the bag model has been done by Alcock et al. (1986) and Haensel et al. (1986).', 'astro-ph-0205343-2-16-5': 'A result for the bag model EOS for strange stars is shown in the figure, where two cases for the EOSs are given, one for the [MATH] and strange quark masses of 0 MeV and 150 MeV for curves labelled 3 and 4 respectively.', 'astro-ph-0205343-2-17-0': 'Alternative to the MIT bag model, Dey et al., in 1998, derived an EOS for strange matter which has asymptotic freedom built in and describes deconfined quarks at high density and confinement at zero density.', 'astro-ph-0205343-2-17-1': 'With a proper choice of the EOS parameters, this model gives absolutely stable strange quark matter.', 'astro-ph-0205343-2-17-2': 'This EOS was used to calculate the structure of static strange stars and the mass-radius relations.', 'astro-ph-0205343-2-17-3': 'Later, it was suggested (Li et al., 1999a) that the millisecond X-ray pulsar SAX1808.4[MATH]3658 is a strange star.', 'astro-ph-0205343-2-17-4': 'This model which also explained the observed properties of some other compact objects like the analysis of semi-empirical mass-radius relations from the QPO observations in 4U 1728-34 (Li et al., 1999b) as also the RXTE observations of Her X-1 and 4U 1820-30 (Dey et al., 1998), leads to the suggestion that these objects host strange stars.', 'astro-ph-0205343-2-17-5': 'In Li et al., (1999a), two sets of EOSs are used for two sets of parameters, namely SS1 and SS2.', 'astro-ph-0205343-2-17-6': 'The maximum gravitational masses are [MATH] for SS1 (curve labelled 2) and [MATH] for SS2 (curve labelled 1).', 'astro-ph-0205343-2-17-7': 'For different values of the mass parameter [MATH] in the D98 model, the stars can have a sequence of masses.', 'astro-ph-0205343-2-17-8': 'The calculations of Dey et al. (1998) has been done considering zero temperature of the strange matter.', 'astro-ph-0205343-2-17-9': 'Ray et al. (2000) calculated the finite temperature effect on the quark stars developed by Dey et al. (1998) and found that it sustains even more mass for a particular radius of the star as compared to the case of cold star.', 'astro-ph-0205343-2-18-0': '# Discussions', 'astro-ph-0205343-2-19-0': 'Fig.1 shows the lines of [MATH] relation for the cases with lensing (A) and without lensing (B) alongwith the curves depicting the various equations of state both for neutron stars (solid lines) and strange matter stars (dot dash lines).', 'astro-ph-0205343-2-19-1': 'As is clear there can be possibilities of identifying a star as of either category depending upon the mass estimate and the corresponding radius, once the measured radius [MATH] is certain.', 'astro-ph-0205343-2-19-2': 'Since the existence of the real roots for [MATH] does depend upon the relation between [MATH] and [MATH], it is very important that both these parameters are estimated accurately and only then can one conclude about the nature of the star, provided the associated point in Fig. 1 overlaps the region covered by either of the types of equation of state.', 'astro-ph-0205343-2-19-3': 'It is also equally important to realise, whether the strong gravity effect of lensing is to be taken into account or not.', 'astro-ph-0205343-2-19-4': 'Either way, the regions of overlap between the real roots and that of reasonable equations of state are quite constrained and thus for a final diagnosis of the star in question RXJ1856.35[MATH]3754, one needs to have more unambiguous measurements of botyh [MATH] and [MATH].', 'astro-ph-0205343-2-19-5': 'The theoretical argument put forth above is purely of mathematical and logistic in nature, which cannot be sidelined by any other preferances.'} | [['astro-ph-0205343-1-7-1', 'astro-ph-0205343-2-10-1'], ['astro-ph-0205343-1-7-2', 'astro-ph-0205343-2-10-2'], ['astro-ph-0205343-1-7-3', 'astro-ph-0205343-2-10-3'], ['astro-ph-0205343-1-12-0', 'astro-ph-0205343-2-14-0'], ['astro-ph-0205343-1-12-1', 'astro-ph-0205343-2-14-1'], ['astro-ph-0205343-1-12-2', 'astro-ph-0205343-2-14-2'], ['astro-ph-0205343-1-12-4', 'astro-ph-0205343-2-14-4'], ['astro-ph-0205343-1-12-5', 'astro-ph-0205343-2-14-5'], ['astro-ph-0205343-1-12-6', 'astro-ph-0205343-2-14-6'], ['astro-ph-0205343-1-11-0', 'astro-ph-0205343-2-13-0'], ['astro-ph-0205343-1-11-1', 'astro-ph-0205343-2-13-1'], ['astro-ph-0205343-1-11-2', 'astro-ph-0205343-2-13-2'], ['astro-ph-0205343-1-11-3', 'astro-ph-0205343-2-13-3'], ['astro-ph-0205343-1-11-4', 'astro-ph-0205343-2-13-4'], ['astro-ph-0205343-1-1-0', 'astro-ph-0205343-2-1-0'], ['astro-ph-0205343-1-15-0', 'astro-ph-0205343-2-17-0'], ['astro-ph-0205343-1-15-1', 'astro-ph-0205343-2-17-1'], ['astro-ph-0205343-1-15-2', 'astro-ph-0205343-2-17-2'], ['astro-ph-0205343-1-15-3', 'astro-ph-0205343-2-17-3'], ['astro-ph-0205343-1-15-4', 'astro-ph-0205343-2-17-4'], ['astro-ph-0205343-1-15-5', 'astro-ph-0205343-2-17-5'], ['astro-ph-0205343-1-15-6', 'astro-ph-0205343-2-17-6'], ['astro-ph-0205343-1-15-7', 'astro-ph-0205343-2-17-7'], ['astro-ph-0205343-1-15-8', 'astro-ph-0205343-2-17-8'], ['astro-ph-0205343-1-15-9', 'astro-ph-0205343-2-17-9'], ['astro-ph-0205343-1-13-0', 'astro-ph-0205343-2-15-0'], ['astro-ph-0205343-1-13-1', 'astro-ph-0205343-2-15-1'], ['astro-ph-0205343-1-13-2', 'astro-ph-0205343-2-15-2'], ['astro-ph-0205343-1-13-3', 'astro-ph-0205343-2-15-3'], ['astro-ph-0205343-1-13-4', 'astro-ph-0205343-2-15-4'], ['astro-ph-0205343-1-13-6', 'astro-ph-0205343-2-15-6'], ['astro-ph-0205343-1-3-1', 'astro-ph-0205343-2-3-2'], ['astro-ph-0205343-1-3-3', 'astro-ph-0205343-2-3-5'], ['astro-ph-0205343-1-14-0', 'astro-ph-0205343-2-16-0'], ['astro-ph-0205343-1-14-1', 'astro-ph-0205343-2-16-1'], ['astro-ph-0205343-1-14-2', 'astro-ph-0205343-2-16-2'], ['astro-ph-0205343-1-14-3', 'astro-ph-0205343-2-16-3'], ['astro-ph-0205343-1-12-3', 'astro-ph-0205343-2-14-3'], ['astro-ph-0205343-1-5-0', 'astro-ph-0205343-2-5-0'], ['astro-ph-0205343-1-6-4', 'astro-ph-0205343-2-6-7'], ['astro-ph-0205343-1-3-0', 'astro-ph-0205343-2-3-0'], ['astro-ph-0205343-1-14-4', 'astro-ph-0205343-2-16-4'], ['astro-ph-0205343-1-14-5', 'astro-ph-0205343-2-16-5'], ['astro-ph-0205343-1-6-1', 'astro-ph-0205343-2-6-6'], ['astro-ph-0205343-1-5-1', 'astro-ph-0205343-2-5-1'], ['astro-ph-0205343-1-6-2', 'astro-ph-0205343-2-6-6'], ['astro-ph-0205343-1-6-3', 'astro-ph-0205343-2-6-6'], ['astro-ph-0205343-1-6-5', 'astro-ph-0205343-2-6-8'], ['astro-ph-0205343-1-13-5', 'astro-ph-0205343-2-15-5']] | [['astro-ph-0205343-1-7-1', 'astro-ph-0205343-2-10-1'], ['astro-ph-0205343-1-7-2', 'astro-ph-0205343-2-10-2'], ['astro-ph-0205343-1-7-3', 'astro-ph-0205343-2-10-3'], ['astro-ph-0205343-1-12-0', 'astro-ph-0205343-2-14-0'], ['astro-ph-0205343-1-12-1', 'astro-ph-0205343-2-14-1'], ['astro-ph-0205343-1-12-2', 'astro-ph-0205343-2-14-2'], ['astro-ph-0205343-1-12-4', 'astro-ph-0205343-2-14-4'], ['astro-ph-0205343-1-12-5', 'astro-ph-0205343-2-14-5'], ['astro-ph-0205343-1-12-6', 'astro-ph-0205343-2-14-6'], ['astro-ph-0205343-1-11-0', 'astro-ph-0205343-2-13-0'], ['astro-ph-0205343-1-11-1', 'astro-ph-0205343-2-13-1'], ['astro-ph-0205343-1-11-2', 'astro-ph-0205343-2-13-2'], ['astro-ph-0205343-1-11-3', 'astro-ph-0205343-2-13-3'], ['astro-ph-0205343-1-11-4', 'astro-ph-0205343-2-13-4'], ['astro-ph-0205343-1-1-0', 'astro-ph-0205343-2-1-0'], ['astro-ph-0205343-1-15-0', 'astro-ph-0205343-2-17-0'], ['astro-ph-0205343-1-15-1', 'astro-ph-0205343-2-17-1'], ['astro-ph-0205343-1-15-2', 'astro-ph-0205343-2-17-2'], ['astro-ph-0205343-1-15-3', 'astro-ph-0205343-2-17-3'], ['astro-ph-0205343-1-15-4', 'astro-ph-0205343-2-17-4'], ['astro-ph-0205343-1-15-5', 'astro-ph-0205343-2-17-5'], ['astro-ph-0205343-1-15-6', 'astro-ph-0205343-2-17-6'], ['astro-ph-0205343-1-15-7', 'astro-ph-0205343-2-17-7'], ['astro-ph-0205343-1-15-8', 'astro-ph-0205343-2-17-8'], ['astro-ph-0205343-1-15-9', 'astro-ph-0205343-2-17-9'], ['astro-ph-0205343-1-13-0', 'astro-ph-0205343-2-15-0'], ['astro-ph-0205343-1-13-1', 'astro-ph-0205343-2-15-1'], ['astro-ph-0205343-1-13-2', 'astro-ph-0205343-2-15-2'], ['astro-ph-0205343-1-13-3', 'astro-ph-0205343-2-15-3'], ['astro-ph-0205343-1-13-4', 'astro-ph-0205343-2-15-4'], ['astro-ph-0205343-1-13-6', 'astro-ph-0205343-2-15-6'], ['astro-ph-0205343-1-3-1', 'astro-ph-0205343-2-3-2'], ['astro-ph-0205343-1-3-3', 'astro-ph-0205343-2-3-5'], ['astro-ph-0205343-1-14-0', 'astro-ph-0205343-2-16-0'], ['astro-ph-0205343-1-14-1', 'astro-ph-0205343-2-16-1'], ['astro-ph-0205343-1-14-2', 'astro-ph-0205343-2-16-2'], ['astro-ph-0205343-1-14-3', 'astro-ph-0205343-2-16-3']] | [['astro-ph-0205343-1-12-3', 'astro-ph-0205343-2-14-3'], ['astro-ph-0205343-1-5-0', 'astro-ph-0205343-2-5-0'], ['astro-ph-0205343-1-6-4', 'astro-ph-0205343-2-6-7'], ['astro-ph-0205343-1-3-0', 'astro-ph-0205343-2-3-0'], ['astro-ph-0205343-1-14-4', 'astro-ph-0205343-2-16-4'], ['astro-ph-0205343-1-14-5', 'astro-ph-0205343-2-16-5']] | [['astro-ph-0205343-1-6-1', 'astro-ph-0205343-2-6-6']] | [['astro-ph-0205343-1-5-1', 'astro-ph-0205343-2-5-1'], ['astro-ph-0205343-1-6-2', 'astro-ph-0205343-2-6-6'], ['astro-ph-0205343-1-6-3', 'astro-ph-0205343-2-6-6'], ['astro-ph-0205343-1-6-5', 'astro-ph-0205343-2-6-8'], ['astro-ph-0205343-1-13-5', 'astro-ph-0205343-2-15-5']] | [] | ['astro-ph-0205343-1-3-2', 'astro-ph-0205343-2-3-3'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/0205343 | null | null | null | null | null |
1107.4862 | {'1107.4862-1-0-0': 'For an integral convex polytope [MATH] of dimension [MATH], we call [MATH] its [MATH]-vector and [MATH] its normalized volume.', '1107.4862-1-0-1': 'In this paper, we will establish the new equalities and inequalities on [MATH]-vectors for integral simplices whose normalized volumes are prime.', '1107.4862-1-0-2': 'Moreover, by using those, we will classify all the possible [MATH]-vectors of integral simplices with normalized volume 5 and 7.', '1107.4862-1-1-0': '# Introduction', '1107.4862-1-2-0': 'One of the most fascinating problems on enumerative combinatorics is to characterize the [MATH]-vectors of integral convex polytopes.', '1107.4862-1-3-0': 'Let [MATH] be an integral convex polytope of dimension [MATH], which is a convex polytope any of whose vertices has integer coordinates.', '1107.4862-1-3-1': 'Let [MATH] denote the boundary of [MATH].', '1107.4862-1-3-2': 'Given a positive integer [MATH], we define [EQUATION] where [MATH] and [MATH] is the cardinality of a finite set [MATH].', '1107.4862-1-3-3': 'The enumerative function [MATH] is called the Ehrhart polynomial of [MATH], which was studied originally in the work of Ehrhart [CITATION].', '1107.4862-1-3-4': 'The Ehrhart polynomial has the following fundamental properties:', '1107.4862-1-4-0': 'We refer the reader to [CITATION] and [CITATION] for the introduction to the theory of Ehrhart polynomials.', '1107.4862-1-5-0': 'We define the sequence [MATH] of integers by the formula [EQUATION]', '1107.4862-1-5-1': 'Then, from a fundamental result on generating function ([CITATION]), we know that [MATH] for every [MATH].', '1107.4862-1-5-2': 'We call the integer sequence [EQUATION] which appears in [REF], the [MATH]-vector of [MATH].', '1107.4862-1-5-3': 'In addition, by the reciprocity law, one has [EQUATION]', '1107.4862-1-5-4': 'The [MATH]-vector has the following fundamental properties:', '1107.4862-1-6-0': 'Recently, the [MATH]-vectors of integral convex polytopes have been studied intensively.', '1107.4862-1-6-1': 'For example, see [CITATION], [CITATION] and [CITATION].', '1107.4862-1-7-0': 'There are two well-known inequalities on [MATH]-vectors.', '1107.4862-1-7-1': 'Let [MATH].', '1107.4862-1-7-2': 'One is [EQUATION] which is proved in Stanley [CITATION], and another one is [EQUATION] which appears in Hibi [CITATION].', '1107.4862-1-8-0': 'When [MATH], the above inequalities ([REF]) and ([REF]) characterize the possible [MATH]-vectors completely ([CITATION]).', '1107.4862-1-8-1': 'Moreover, when [MATH], the possible [MATH]-vectors are determined completely ([CITATION]) by ([REF]) and ([REF]) together with an additional condition.', '1107.4862-1-8-2': 'Furthermore, by the proofs of [CITATION] and [CITATION], we know that all the possible [MATH]-vectors can be realized as the [MATH]-vectors of integral simplices when [MATH].', '1107.4862-1-8-3': 'However, unfortunately, it is not true when [MATH].', '1107.4862-1-8-4': '(See [CITATION].)', '1107.4862-1-8-5': 'Therefore, for the further classifications of the [MATH]-vectors with [MATH], it is natural to investigate the [MATH]-vectors of integral simplices.', '1107.4862-1-8-6': 'In this paper, in particular, we establish some new constraints on [MATH]-vectors for integral simplices whose normalized volumes are prime numbers.', '1107.4862-1-8-7': 'The following theorem is our main result of this paper.', '1107.4862-1-9-0': 'Let [MATH] be an integral simplex of dimension [MATH] and [MATH] its [MATH]-vector.', '1107.4862-1-9-1': 'Suppose that [MATH] is an odd prime number.', '1107.4862-1-9-2': 'Let [MATH] be the positive integers such that [MATH] with [MATH].', '1107.4862-1-9-3': 'Then,', '1107.4862-1-10-0': 'We prove Theorem [REF] in Section [REF] via the languages of elementary group theory.', '1107.4862-1-11-0': 'As an application of Theorem [REF], we give a complete characterization of the possible [MATH]-vectors of integral simplices when [MATH] and 7.', '1107.4862-1-12-0': 'Given a finite sequence [MATH] of nonnegative integers, where [MATH] and [MATH], there exists an integral simplex [MATH] of dimension [MATH] whose [MATH]-vector coincides with [MATH] if and only if [MATH] satisfy [MATH] and [MATH] for [MATH] with [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-1-13-0': 'Given a finite sequence [MATH] of nonnegative integers, where [MATH] and [MATH], there exists an integral simplex [MATH] of dimension [MATH] whose [MATH]-vector coincides with [MATH] if and only if [MATH] satisfy [MATH] and [MATH] for [MATH] with [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-1-14-0': 'By virtue of Theorem [REF], the "Only if" parts of Theorem [REF] and [REF] are obvious.', '1107.4862-1-14-1': 'A proof of the "If" part of Theomre [REF] is given in Section [REF] and that of Theorem [REF] is given in Section [REF].', '1107.4862-1-15-0': 'Finally, we note that we cannot characterize the possible [MATH]-vectors of integral simplices with higher prime normalized volumes only by Theorem [REF].', '1107.4862-1-15-1': 'In fact, since the volume of an integral convex polytope containing a unique integer point in its interior has an upper bound, if [MATH] is a sufficiently large prime number, then the integer sequence [MATH] cannot be a [MATH]-vector of some integral simplex of dimension 3, although [MATH] satisfies all the conditions of Theorem [REF].', '1107.4862-1-16-0': '# A proof of Theorem [REF]', '1107.4862-1-17-0': 'The goal of this section is to give a proof of Theorem [REF].', '1107.4862-1-18-0': 'First of all, we recall from [CITATION] the well-known combinatorial technique how to compute the [MATH]-vector of an integral simplex.', '1107.4862-1-19-0': 'Given an integral simplex [MATH] in [MATH] of dimension [MATH] with the vertices [MATH], we set [EQUATION] which is an integral simplex in [MATH] of dimension [MATH] with the vertices [MATH].', '1107.4862-1-19-1': 'Clearly, we have [MATH] for all [MATH].', '1107.4862-1-19-2': 'Let [EQUATION].', '1107.4862-1-19-3': 'Then one has [EQUATION].', '1107.4862-1-19-4': 'Each rational point [MATH] has a unique expression of the form [MATH] with [MATH].', '1107.4862-1-19-5': 'Let [MATH] be the set of all points [MATH] of the form [MATH] where [MATH] with [MATH].', '1107.4862-1-19-6': 'We define the degree of [MATH] with [MATH], i.e., the last coordinate of [MATH].', '1107.4862-1-20-0': 'Let [MATH] be the number of integer points [MATH] with [MATH].', '1107.4862-1-20-1': 'Then, [EQUATION].', '1107.4862-1-21-0': 'Notice that the elements of [MATH] form an abelian group with a unit [MATH].', '1107.4862-1-21-1': 'For [MATH] and [MATH] in [MATH] with [MATH] and [MATH], where [MATH] with [MATH], we define the operation in [MATH] by setting [MATH], where [MATH] denotes the fractional part of a rational number [MATH].', '1107.4862-1-21-2': '(Throughout this paper, in order to distinguish the operation in [MATH] from the usual addition, we use the notation [MATH], which is not a direct sum.)', '1107.4862-1-22-0': 'We prove Theorem [REF] by using the above notations.', '1107.4862-1-23-0': '[A proof of Theorem [REF]] Let [MATH] be the vertices of the integral simplex [MATH] and [MATH] the group appearing above.', '1107.4862-1-23-1': 'Then, since [MATH] is prime, it follows from Lemma [REF] that the order of [MATH] is also prime.', '1107.4862-1-23-2': 'In particular, [MATH] is a cyclic group.', '1107.4862-1-24-0': '(a) Write [MATH] for [MATH] distinct elements with [MATH] for [MATH], that is, [MATH].', '1107.4862-1-24-1': 'Then, for each [MATH], there exists its inverse [MATH] in [MATH].', '1107.4862-1-24-2': 'Let [MATH].', '1107.4862-1-24-3': 'If [MATH] has the expression [MATH], where [MATH] with [MATH], then its inverse has the expression [MATH].', '1107.4862-1-24-4': 'Thus, one has [EQUATION] for all [MATH].', '1107.4862-1-25-0': 'For [MATH] with [MATH], let [MATH] and [MATH].', '1107.4862-1-25-1': 'Since [MATH] is a cyclic group with a prime order, [MATH] generates [MATH], which implies that we can write [MATH] and [MATH] as follows: [EQUATION] for some integer [MATH].', '1107.4862-1-25-2': 'Thus, we have [EQUATION]', '1107.4862-1-25-3': 'Moreover, [MATH] holds.', '1107.4862-1-25-4': 'Thus, we have [MATH] for all [MATH].', '1107.4862-1-25-5': 'Again, since [MATH] is prime, it follows that the denominator of each rational number [MATH] must be [MATH].', '1107.4862-1-25-6': 'Hence, if [MATH] (resp. [MATH]), then [MATH] (resp. [MATH]), so [MATH].', '1107.4862-1-25-7': 'In addition, obviously, if [MATH], then [MATH], so [MATH].', '1107.4862-1-25-8': 'Thus, [MATH], i.e., [MATH].', '1107.4862-1-25-9': 'Hence, we obtain [EQUATION].', '1107.4862-1-25-10': 'Our work is to show that [MATH] for all [MATH].', '1107.4862-1-26-0': 'First, we consider [MATH].', '1107.4862-1-26-1': 'Suppose that [MATH].', '1107.4862-1-26-2': 'Then, there is [MATH] with [MATH].', '1107.4862-1-26-3': 'Thus, it follows that [EQUATION] a contradiction.', '1107.4862-1-26-4': 'Thus, [MATH] must be [MATH].', '1107.4862-1-26-5': 'Next, we consider [MATH].', '1107.4862-1-26-6': 'Since [MATH] and [MATH] we may consider [MATH] among [MATH].', '1107.4862-1-26-7': 'Then, the same discussion can be done.', '1107.4862-1-26-8': 'Hence, [MATH].', '1107.4862-1-26-9': 'Similarly, we have [MATH].', '1107.4862-1-27-0': 'Therefore, we obtain the desired conditions [EQUATION] (b) Write [MATH] for [MATH] distinct elements with [MATH] for [MATH].', '1107.4862-1-27-1': 'Let [MATH].', '1107.4862-1-27-2': 'Then there are [MATH] distinct elements [MATH] in [MATH] with [MATH] for [MATH] satisfying [MATH], where [MATH].', '1107.4862-1-27-3': 'Moreover, for each [MATH], [MATH] satisfies [MATH].', '1107.4862-1-27-4': 'In fact, for [MATH] and [MATH], if they have the expressions [EQUATION] where [MATH] with [MATH], then one has [EQUATION].', '1107.4862-1-27-5': 'Now, Lemma [REF] below guarantees that there exist at least [MATH] elements in [MATH].', '1107.4862-1-27-6': 'In addition, each [MATH] in [MATH] satisfies [MATH].', '1107.4862-1-27-7': 'Thus, we can say that there exist at least [MATH] distinct elements in [MATH] whose degrees are at most [MATH].', '1107.4862-1-27-8': 'From the definition of [MATH], this means that [MATH], as desired.', '1107.4862-1-28-0': 'Let [MATH] be a group with prime order [MATH], where its operation is denoted by [MATH], and let [MATH], where 0 is the unit of [MATH].', '1107.4862-1-28-1': 'We choose two subsets (not subgroups) [MATH] and [MATH] of [MATH] satisfying [MATH] and [MATH] and we set [MATH].', '1107.4862-1-28-2': 'Then one has [EQUATION] where [MATH].', '1107.4862-1-29-0': 'Let [MATH] and [MATH].', '1107.4862-1-29-1': 'We show the assertion by induction on [MATH].', '1107.4862-1-30-0': 'First, we consider [MATH], i.e., [MATH].', '1107.4862-1-30-1': 'Then, [MATH].', '1107.4862-1-30-2': 'For [MATH] let [MATH].', '1107.4862-1-30-3': 'Then we have [EQUATION].', '1107.4862-1-30-4': 'If we suppose that [MATH], then we have [MATH].', '1107.4862-1-30-5': 'Since [MATH] are distinct, one has [MATH].', '1107.4862-1-30-6': 'Thus, [MATH] from the above equality.', '1107.4862-1-30-7': 'However, since [MATH] is prime and [MATH], [MATH] cannot be 0, a contradiction.', '1107.4862-1-30-8': 'Hence, [MATH], which implies that [MATH].', '1107.4862-1-31-0': 'Next, we consider [MATH].', '1107.4862-1-31-1': 'Let [MATH].', '1107.4862-1-31-2': 'Then, by the hypothesis of induction, one has [MATH].', '1107.4862-1-31-3': 'When [MATH], the assertion holds.', '1107.4862-1-31-4': 'Thus, we assume that [MATH].', '1107.4862-1-31-5': 'Let [MATH], where [MATH] are [MATH] distinct elements, [MATH] and [MATH].', '1107.4862-1-31-6': 'Then, again by the hypothesis of induction, one has [MATH].', '1107.4862-1-31-7': 'This implies that there exists at least one element [MATH] in [MATH] such that [MATH] for some [MATH].', '1107.4862-1-31-8': 'When [MATH], then [MATH], which says that the assertion holds.', '1107.4862-1-31-9': 'Hence, we assume that [MATH], say, [MATH].', '1107.4862-1-32-0': 'Now, again by the hypothesis of induction, it is easy to see that we have the following equalities by renumbering [MATH] if necessary: [EQUATION] where [MATH].', '1107.4862-1-32-1': 'Suppose that the inequality [EQUATION] is not satisfied.', '1107.4862-1-32-2': 'From [REF], one has [EQUATION].', '1107.4862-1-32-3': 'Set [MATH].', '1107.4862-1-32-4': 'When [MATH], since [MATH] and [MATH], one has [MATH], which means that [REF] holds.', '1107.4862-1-32-5': 'When [MATH], since [MATH], one has [MATH], which also means that [REF] holds.', '1107.4862-1-32-6': 'Moreover, [MATH] cannot be 0 since [MATH].', '1107.4862-1-32-7': 'In addition, [MATH] cannot be [MATH] since [MATH].', '1107.4862-1-32-8': 'Hence, it must be [MATH], say, [MATH].', '1107.4862-1-32-9': 'Then, again from [REF], [EQUATION].', '1107.4862-1-32-10': 'Set [MATH].', '1107.4862-1-32-11': 'Similarly, when [MATH] or [MATH], [REF] holds.', '1107.4862-1-32-12': 'Moreover, [MATH] cannot be 0, [MATH] and [MATH].', '1107.4862-1-32-13': 'Hence, it must be [MATH], say, [MATH].', '1107.4862-1-32-14': 'By repeating these discussions, we obtain [EQUATION].', '1107.4862-1-32-15': 'Set [MATH].', '1107.4862-1-32-16': 'However, we have [EQUATION] a contradiction.', '1107.4862-1-32-17': 'Thus, the inequality [REF] must be satisfied.', '1107.4862-1-33-0': 'Therefore, we obtain the required inequality [REF].', '1107.4862-1-34-0': '(a) When [MATH], the [MATH]-vector is shifted symmetric.', '1107.4862-1-34-1': 'Shifted symmetric [MATH]-vectors are studied in [CITATION].', '1107.4862-1-34-2': 'Moreover, the theorem [CITATION] says that if [MATH], then we have [MATH].', '1107.4862-1-35-0': '(b) The inequalities [MATH] are not new.', '1107.4862-1-35-1': 'In fact, for example, when [MATH], by [REF], one has [EQUATION].', '1107.4862-1-35-2': 'Thus, we obtain [MATH], i.e., [MATH].', '1107.4862-1-35-3': 'Similarly, one has [EQUATION].', '1107.4862-1-35-4': 'Thus, we obtain [MATH].', '1107.4862-1-35-5': 'Since [MATH], this is equivalent to [MATH].', '1107.4862-1-35-6': 'In the same way, we can obtain all inequalities [MATH].', '1107.4862-1-35-7': 'On the other hand, when [MATH], there are many new inequalities.', '1107.4862-1-36-0': '# The possible [MATH]-vectors of integral simplices with [MATH]', '1107.4862-1-37-0': 'In this section, we give a proof of the "If" part of Theorem [REF], i.e., we classify all the possible [MATH]-vectors of integral simplices whose normalized volume is 5.', '1107.4862-1-38-0': 'Let [MATH] be a nonnegative integer sequence with [MATH] and [MATH] which satisfies [MATH], [MATH] and [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-1-38-1': 'Since [MATH], we notice that [MATH] (resp. [MATH]) is equivalent to [MATH] (resp. [MATH]).', '1107.4862-1-38-2': 'From the conditions [MATH], [MATH] and [MATH], the possible sequences are only the following forms:', '1107.4862-1-39-0': 'Our work is to find integral simplices whose [MATH]-vectors are of the above forms.', '1107.4862-1-40-0': "To construct integral simplices, we define the following integer matrix, which is called the Hermite normal form: [EQUATION] where there are [MATH]'s among the [MATH]'s for [MATH] and the rest of the entries are all 0.", '1107.4862-1-40-1': 'Then, clearly, it must be [MATH] and [MATH].', '1107.4862-1-40-2': 'By determining [MATH], we obtain an integer matrix [MATH] and we define the integral simplex [MATH] from the matrix as follows: [EQUATION] where [MATH] is the [MATH]th row vector of [MATH].', '1107.4862-1-40-3': 'The following lemma enables us to compute [MATH] easily.', '1107.4862-1-41-0': '[[CITATION]] If [MATH], then we have [EQUATION] where [EQUATION]', '1107.4862-1-42-0': '## The case (i)', '1107.4862-1-43-0': 'Let [MATH].', '1107.4862-1-43-1': 'Thus, one has [MATH] and [MATH] from our conditions.', '1107.4862-1-43-2': 'Hence, we can define [MATH].', '1107.4862-1-43-3': 'Then, by Lemma [REF], [MATH] coincides with (i) since [MATH].', '1107.4862-1-44-0': '## The case (ii)', '1107.4862-1-45-0': 'Let [MATH] and [MATH].', '1107.4862-1-45-1': 'Thus, one has [MATH], [MATH] and [MATH].', '1107.4862-1-45-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (ii) since [MATH] and [MATH].', '1107.4862-1-46-0': '## The case (iii)', '1107.4862-1-47-0': 'Let [MATH] and [MATH].', '1107.4862-1-47-1': 'Thus, one has [MATH], [MATH] and [MATH].', '1107.4862-1-47-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iii) since [MATH], [MATH] and [MATH].', '1107.4862-1-48-0': '## The case (iv)', '1107.4862-1-49-0': 'In this case, one has [MATH] and [MATH].', '1107.4862-1-49-1': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iv) since [MATH] and [MATH].', '1107.4862-1-50-0': '(a) The classification of the case (iv) is essentially given in [CITATION].', '1107.4862-1-51-0': '(b) The inequalities [MATH] and [MATH] can be obtained from [REF] as we mentioned in Remark [REF] (b).', '1107.4862-1-51-1': 'Thus, the possible [MATH]-vectors of integral simplices with normalized volume 5 can be essentially characterized only by Theorem [REF] (a) and the inequalities [REF].', '1107.4862-1-52-0': '# The possible [MATH]-vectors of integral simplices with [MATH]', '1107.4862-1-53-0': 'In this section, similarly to the previous one, we give a proof of the "If" part of Theorem [REF], i.e., we classify all the possible [MATH]-vectors of integral simplices whose normalized volume is 7.', '1107.4862-1-54-0': 'Let [MATH] be a nonnegative integer sequence with [MATH] and [MATH] which satisfies [MATH], [MATH] for [MATH] and [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-1-54-1': 'Since [MATH], we need not consider the inequalities [MATH], [MATH], [MATH], [MATH] and [MATH].', '1107.4862-1-54-2': 'From the conditions [MATH], [MATH] and [MATH], the possible sequences are only the following forms:', '1107.4862-1-55-0': "In the same way as the previous section, we define the following integer matrix: [EQUATION] where there are [MATH]'s among the [MATH]'s for [MATH] and the rest of the entries are all 0.", '1107.4862-1-55-1': 'Then it must be [MATH] and [MATH].', '1107.4862-1-55-2': 'By determining [MATH], we obtain the integral simplex [EQUATION] where [MATH] is the [MATH]th row vector of [MATH].', '1107.4862-1-55-3': 'Similarly, the following lemma enables us to compute [MATH] easily.', '1107.4862-1-56-0': '[[CITATION]] If [MATH], then we have [EQUATION] where [EQUATION]', '1107.4862-1-57-0': '## The case (i)', '1107.4862-1-58-0': 'Let [MATH].', '1107.4862-1-58-1': 'Thus, one has [MATH] and [MATH] from our conditions.', '1107.4862-1-58-2': 'Hence, we can define [MATH].', '1107.4862-1-58-3': 'Then, by Lemma [REF], [MATH] coincides with (i) since [MATH].', '1107.4862-1-59-0': '## The case (ii)', '1107.4862-1-60-0': 'Let [MATH] and [MATH].', '1107.4862-1-60-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-60-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (ii) since [MATH] and [MATH].', '1107.4862-1-61-0': '## The case (iii)', '1107.4862-1-62-0': 'Let [MATH] and [MATH].', '1107.4862-1-62-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-62-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iii) since [MATH] and [MATH].', '1107.4862-1-63-0': '## The case (iv)', '1107.4862-1-64-0': 'Let [MATH] and [MATH].', '1107.4862-1-64-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-64-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iv) since [MATH] and [MATH].', '1107.4862-1-65-0': '## The case (v)', '1107.4862-1-66-0': 'Let [MATH] and [MATH].', '1107.4862-1-66-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-66-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (v) since [MATH] and [MATH].', '1107.4862-1-67-0': '## The case (vi)', '1107.4862-1-68-0': 'Let [MATH] and [MATH].', '1107.4862-1-68-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-68-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (vi) since [MATH] and [MATH].', '1107.4862-1-69-0': '## The case (vii)', '1107.4862-1-70-0': 'Let [MATH] and [MATH].', '1107.4862-1-70-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-1-70-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (vii) since [MATH] and [MATH].', '1107.4862-1-71-0': '## The case (viii)', '1107.4862-1-72-0': 'In this case, one has [MATH] and [MATH].', '1107.4862-1-72-1': 'Hence, we can define [MATH] if [MATH] and [MATH].', '1107.4862-1-72-2': 'Moreover, each of [MATH]-vectors of them coincides with (viii) since [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1107.4862-1-73-0': 'When we discuss the cases of (vi) and (viii), we need the new inequality [MATH].', '1107.4862-1-73-1': 'In fact, for example, the sequence [MATH] cannot be the [MATH]-vector of an integral simplex, although this satisfies [MATH].', '1107.4862-1-73-2': 'Similarly, the sequence [MATH] also cannot be the [MATH]-vector of an integral simplex, although this satisfies [MATH].'} | {'1107.4862-2-0-0': 'For an integral convex polytope [MATH] of dimension [MATH], we call [MATH] the [MATH]-vector of [MATH] and [MATH] its normalized volume.', '1107.4862-2-0-1': 'In this paper, we will establish the new equalities and inequalities on [MATH]-vectors for integral simplices whose normalized volumes are prime.', '1107.4862-2-0-2': 'Moreover, by using those, we will classify all the possible [MATH]-vectors of integral simplices with normalized volume 5 and 7.', '1107.4862-2-1-0': '# Introduction', '1107.4862-2-2-0': 'One of the most fascinating problems on enumerative combinatorics is to characterize the [MATH]-vectors of integral convex polytopes.', '1107.4862-2-3-0': 'Let [MATH] be an integral convex polytope of dimension [MATH], which is a convex polytope any of whose vertices has integer coordinates.', '1107.4862-2-3-1': 'Let [MATH] denote the boundary of [MATH].', '1107.4862-2-3-2': 'Given a positive integer [MATH], we define [EQUATION] where [MATH] and [MATH] is the cardinality of a finite set [MATH].', '1107.4862-2-3-3': 'The enumerative function [MATH] has the following fundamental properties, which were studied originally in the work of Ehrhart [CITATION]:', '1107.4862-2-4-0': 'This polynomial [MATH] is called the Ehrhart polynomial of [MATH].', '1107.4862-2-4-1': 'We refer the reader to [CITATION], [CITATION] or [CITATION] for the introduction to the theory of Ehrhart polynomials.', '1107.4862-2-5-0': 'We define the sequence [MATH] of integers by the formula [EQUATION]', '1107.4862-2-5-1': 'Then, from a fundamental result on generating functions ([CITATION]), we know that [MATH] for [MATH].', '1107.4862-2-5-2': 'We call the integer sequence [EQUATION] which appears in [REF], the [MATH]-vector of [MATH].', '1107.4862-2-6-0': 'The [MATH]-vector has the following properties:', '1107.4862-2-7-0': 'Recently, the [MATH]-vectors of integral convex polytopes have been studied intensively.', '1107.4862-2-7-1': 'For example, see [CITATION].', '1107.4862-2-8-0': 'There are two well-known inequalities on [MATH]-vectors.', '1107.4862-2-8-1': 'Let [MATH].', '1107.4862-2-8-2': 'One is [EQUATION] which is proved by Stanley [CITATION], and another one is [EQUATION] which appears in the work of Hibi [CITATION].', '1107.4862-2-9-0': 'On the classification problem on [MATH]-vectors of integral convex polytopes, the above inequalities [REF] and [REF] characterize the possible [MATH]-vectors completely when [MATH] ([CITATION]).', '1107.4862-2-9-1': 'Moreover, when [MATH], the possible [MATH]-vectors are determined completely by [REF] and [REF] together with an additional condition ([CITATION]).', '1107.4862-2-9-2': 'Furthermore, by the proofs of [CITATION] and [CITATION], we know that all the possible [MATH]-vectors can be realized as the [MATH]-vectors of integral simplices when [MATH].', '1107.4862-2-9-3': 'However, unfortunately, this is no longer true when [MATH].', '1107.4862-2-9-4': '(See [CITATION].)', '1107.4862-2-9-5': 'Hence, for the further classifications of [MATH]-vectors with [MATH], it is natural to study [MATH]-vectors of integral simplices at first.', '1107.4862-2-9-6': 'Even for non-simplex cases, since every convex polytope can be triangulated into finitely many simplices and we can compute the [MATH]-vecotor of an integral convex polytope from its triangulation, investigating [MATH]-vectors of integral simplices is an essential and important work.', '1107.4862-2-10-0': 'In this paper, in particular, we establish some new constraints on [MATH]-vectors for integral simplices whose normalized volumes are prime numbers.', '1107.4862-2-10-1': 'The following theorem is our main result of this paper.', '1107.4862-2-11-0': 'Let [MATH] be an integral simplex of dimension [MATH] and [MATH] its [MATH]-vector.', '1107.4862-2-11-1': 'Suppose that [MATH] is an odd prime number.', '1107.4862-2-11-2': 'Let [MATH] be the positive integers such that [MATH] with [MATH].', '1107.4862-2-11-3': 'Then,', '1107.4862-2-12-0': 'We give a proof of Theorem [REF] in Section [REF].', '1107.4862-2-13-0': 'Now, we remark that the part (a) of Theorem [REF] is not a new result in some sense.', '1107.4862-2-13-1': 'In [CITATION], the author proved that for an integral simplex [MATH] with prime normalized volume, if [MATH], then [MATH] is shifted symmetric, i.e., we have [MATH].', '1107.4862-2-13-2': 'Moreover, since every integral simplex with prime normalized volume is either a simplex with [MATH] or a pyramid at height 1 over such simplex and taking such a pyramid does not change the normalized volume and the polynomial [MATH], we also obtain the equalities [MATH] on the case where [MATH].', '1107.4862-2-13-3': 'On the other hand, in this paper, we give an another proof of this statement.', '1107.4862-2-13-4': 'More precisely, we give an elementary proof of Theorem [REF] (a) in terms of some abelian groups associated with integral simplices.', '1107.4862-2-14-0': 'In addition, as an application of Theorem [REF], we give a complete characterization of the possible [MATH]-vectors of integral simplices when [MATH] and 7.', '1107.4862-2-15-0': 'Given a finite sequence [MATH] of nonnegative integers, where [MATH] and [MATH], there exists an integral simplex [MATH] of dimension [MATH] whose [MATH]-vector coincides with [MATH] if and only if [MATH] satisfy [MATH] and [MATH] for [MATH] with [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-2-16-0': 'Given a finite sequence [MATH] of nonnegative integers, where [MATH] and [MATH], there exists an integral simplex [MATH] of dimension [MATH] whose [MATH]-vector coincides with [MATH] if and only if [MATH] satisfy [MATH] and [MATH] for [MATH] with [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-2-17-0': 'By virtue of Theorem [REF], the "Only if" parts of Theorem [REF] and [REF] are obvious.', '1107.4862-2-17-1': 'A proof of the "If" part of Theomre [REF] is given in Section [REF] and that of Theorem [REF] is given in Section [REF].', '1107.4862-2-17-2': 'Moreover, in Section [REF], we note some problems towards the classification of Ehrhart polynomials of integral convex polytopes with general normalized volumes.', '1107.4862-2-18-0': '# A proof of Theorem [REF]', '1107.4862-2-19-0': 'The goal of this section is to give a proof of Theorem [REF].', '1107.4862-2-20-0': 'First of all, we recall the well-known combinatorial technique how to compute the [MATH]-vector of an integral simplex.', '1107.4862-2-20-1': 'Given an integral simplex [MATH] of dimension [MATH] with the vertices [MATH], we set [EQUATION]', '1107.4862-2-20-2': 'We define the degree of [MATH] with [MATH], i.e., the last coordinate of [MATH].', '1107.4862-2-20-3': 'Then we have the following', '1107.4862-2-21-0': 'Let [MATH].', '1107.4862-2-21-1': 'Then each [MATH] is equal to the number of integer points [MATH] with [MATH].', '1107.4862-2-22-0': 'Notice that [MATH] has a structure of an abelian group with a unit [MATH], where [MATH].', '1107.4862-2-22-1': 'For [MATH] and [MATH] in [MATH] with [MATH] and [MATH], where [MATH] with [MATH], we define the operation in [MATH] by setting [MATH], where [MATH] denotes the fractional part of a rational number [MATH].', '1107.4862-2-22-2': '(Throughout this paper, in order to distinguish the operation in [MATH] from the usual addition, we use the notation [MATH], which is not a direct sum.)', '1107.4862-2-23-0': 'We prove Theorem [REF] by using the above notations.', '1107.4862-2-24-0': '[Proof of Theorem [REF]] Let [MATH] be the vertices of the integral simplex [MATH] and [MATH] the abelian group as above.', '1107.4862-2-24-1': 'Then, since [MATH] is prime, it follows from Lemma [REF] that [MATH] is also prime.', '1107.4862-2-24-2': 'In particular, [MATH].', '1107.4862-2-25-0': '(a) Write [MATH] for [MATH] distinct elements belonging to [MATH] with [MATH] for [MATH], that is, [MATH].', '1107.4862-2-25-1': 'Then, for each [MATH], there exists its inverse [MATH] in [MATH].', '1107.4862-2-25-2': 'Let [MATH].', '1107.4862-2-25-3': 'If [MATH] has an expression [MATH], where [MATH] with [MATH], then its inverse has an expression [MATH].', '1107.4862-2-25-4': 'Thus, one has [EQUATION] for all [MATH].', '1107.4862-2-26-0': 'For [MATH], let [MATH] and [MATH].', '1107.4862-2-26-1': 'Since [MATH], [MATH] generates [MATH].', '1107.4862-2-26-2': 'This implies that we can write [MATH] and [MATH] as follows: [EQUATION] for some integer [MATH].', '1107.4862-2-26-3': 'Thus, we have [EQUATION]', '1107.4862-2-26-4': 'Moreover, since [MATH], we have [MATH] for [MATH].', '1107.4862-2-26-5': 'This means that the denominator of each rational number [MATH] must be [MATH].', '1107.4862-2-26-6': 'Hence, if [MATH] (resp. [MATH]), then [MATH] (resp. [MATH]), so [MATH].', '1107.4862-2-26-7': 'In addition, obviously, if [MATH], then [MATH], so [MATH].', '1107.4862-2-26-8': 'Thus, [MATH].', '1107.4862-2-26-9': 'Let [MATH].', '1107.4862-2-26-10': 'Then we obtain [EQUATION].', '1107.4862-2-26-11': 'Our work is to show that [MATH] for [MATH].', '1107.4862-2-27-0': 'First, we consider [MATH].', '1107.4862-2-27-1': 'Suppose that [MATH].', '1107.4862-2-27-2': 'Then, there is [MATH] with [MATH].', '1107.4862-2-27-3': 'Thus, it follows that [EQUATION] a contradiction.', '1107.4862-2-27-4': 'Thus, [MATH] must be [MATH].', '1107.4862-2-27-5': 'Next, we consider [MATH].', '1107.4862-2-27-6': 'Since [MATH] and [MATH] we may consider [MATH] among [MATH].', '1107.4862-2-27-7': 'Then, the same discussion can be done.', '1107.4862-2-27-8': 'Hence, [MATH].', '1107.4862-2-27-9': 'Similarly, we have [MATH].', '1107.4862-2-28-0': 'Therefore, we obtain the desired [EQUATION] (b) Let [MATH] and [MATH] be integers with [MATH] and [MATH].', '1107.4862-2-28-1': 'Write [MATH] for [MATH] distinct elements with [MATH] for [MATH] and set [MATH] and [MATH].', '1107.4862-2-28-2': 'Now, Cauchy-Davenport Theorem (cf [CITATION]) guarantees that [MATH], where [MATH].', '1107.4862-2-28-3': 'Clearly, 0 belongs to [MATH].', '1107.4862-2-28-4': 'Moreover, since [MATH], it follows that [MATH] contains at least [MATH] distinct elements in [MATH].', '1107.4862-2-28-5': 'In addition, for each [MATH], [MATH] satisfies [MATH].', '1107.4862-2-28-6': 'In fact, for non-zero elements [MATH] and [MATH], if they have expressions [MATH] and [MATH] then one has [EQUATION].', '1107.4862-2-29-0': 'Hence, from the definition of [MATH], we obtain the inequalities [MATH] for [MATH] with [MATH], as desired.', '1107.4862-2-30-0': 'It is easy to see that we can reduce the inequalities in Theorem [REF] (b) by using the equalities [MATH] as follows: [EQUATION]', '1107.4862-2-30-1': 'In fact, when [MATH], since [MATH] is prime, we have [MATH].', '1107.4862-2-30-2': 'Thus, [MATH].', '1107.4862-2-30-3': 'By using [MATH], we obtain [MATH], which is [MATH], where [MATH].', '1107.4862-2-30-4': 'Similarly, when [MATH], we have [MATH].', '1107.4862-2-30-5': 'Thus, we can deduce [MATH].', '1107.4862-2-31-0': 'Moreover, some of the inequalities described in Theorem [REF] follow from [REF] and [REF].', '1107.4862-2-32-0': 'Let [MATH] be an integral convex polytope of dimension [MATH] with its [MATH]-vector [MATH] and [MATH] the positive integers such that [MATH] with [MATH], where [MATH].', '1107.4862-2-33-0': '(a) The inequalities [MATH], where [MATH], are equivalent to [REF].', '1107.4862-2-34-0': '(b) The inequalities [MATH], where [MATH], are equivalent to [REF].', '1107.4862-2-35-0': '(a) For each [MATH], the inequality [MATH] follows from [REF].', '1107.4862-2-35-1': 'Then its left-hand side is at least [MATH] by the definition of [MATH].', '1107.4862-2-35-2': 'Thus, in particular, its right-hand side is at least [MATH].', '1107.4862-2-35-3': 'On the other hand, since [MATH], it must be [MATH], which means [MATH].', '1107.4862-2-35-4': 'On the contrary, assume that [MATH].', '1107.4862-2-35-5': 'For each [MATH] with [MATH], there exists a unique [MATH] with [MATH] such that [MATH], where [MATH] and [MATH].', '1107.4862-2-35-6': 'Thus, [EQUATION] (b) For each [MATH], the inequality [MATH] follows from [REF].', '1107.4862-2-35-7': 'Then its right-hand side is at least [MATH].', '1107.4862-2-35-8': 'Thus, it must be [MATH], which means [MATH].', '1107.4862-2-35-9': 'On the contrary, assume that [MATH].', '1107.4862-2-35-10': 'For each [MATH] with [MATH], there exists a unique [MATH] with [MATH] such that [MATH].', '1107.4862-2-35-11': 'Thus, [EQUATION]', '1107.4862-2-35-12': 'As is shown above, the inequalities [MATH] and [MATH] are not new ones.', '1107.4862-2-35-13': 'Howover, the inequalities [MATH] include many new ones.', '1107.4862-2-35-14': 'See Remark [REF] and Example [REF] below.', '1107.4862-2-36-0': '# The possible [MATH]-vectors of integral simplices with [MATH]', '1107.4862-2-37-0': 'In this section, we give a proof of the "If" part of Theorem [REF], namely, we classify all the possible [MATH]-vectors of integral simplices with normalized volume 5.', '1107.4862-2-38-0': 'Let [MATH] be a nonnegative integer sequence with [MATH] and [MATH] which satisfies [MATH], [MATH] and [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-2-38-1': 'By virtue of Theorem [REF], these are necessary conditions for [MATH] to be a [MATH]-vector of some integral simplex.', '1107.4862-2-38-2': 'We notice that [MATH] (resp. [MATH]) is equivalent to [MATH] (resp. [MATH]) since [MATH].', '1107.4862-2-38-3': 'From the conditions [MATH], [MATH] and [MATH], only the possible sequences look like', '1107.4862-2-39-0': 'Our work is to find integral simplices whose [MATH]-vectors are of the above forms.', '1107.4862-2-40-0': "To construct integral simplices, we define the following [MATH] integer matrix: [EQUATION] where [MATH] is a positive integer, there are [MATH]'s among the [MATH]'s for [MATH] and the rest entries are all 0.", '1107.4862-2-40-1': 'Clearly, it must be [MATH] and [MATH].', '1107.4862-2-40-2': 'Thus, by determining [MATH], we obtain an integer matrix [MATH] and we define the integral simplex [MATH] of dimension [MATH] from the matrix [REF] as follows: [EQUATION] where [MATH] is the [MATH]th row vector of [REF].', '1107.4862-2-40-3': 'The following lemma enables us to compute [MATH] easily.', '1107.4862-2-41-0': '[[CITATION]] Let [MATH].', '1107.4862-2-41-1': 'Then [EQUATION] where [EQUATION]', '1107.4862-2-42-0': 'Let [MATH].', '1107.4862-2-42-1': 'In the sequel, in each case of (i) - (iv) above, by giving concrete values of [MATH], we obtain the matrix [MATH] and hence the integral simplex [MATH] whose [MATH]-vector looks like each of (i) - (iv).', '1107.4862-2-42-2': 'The [MATH]-vectors of such simplices can be computed by Lemma [REF].', '1107.4862-2-43-0': '## The case (i)', '1107.4862-2-44-0': 'First, let us consider the case (i), namely, the nonnegative integer sequence like [MATH], which means that [MATH].', '1107.4862-2-44-1': 'Set [MATH].', '1107.4862-2-44-2': 'Then, of course, [MATH].', '1107.4862-2-44-3': 'Moreover, from our conditions, one has [MATH], that is, [MATH].', '1107.4862-2-44-4': 'Hence, we can define [MATH] and calculate [EQUATION].', '1107.4862-2-44-5': 'This implies that [MATH] coincides with [MATH] from Lemma [REF], where [MATH].', '1107.4862-2-44-6': 'Remark that [MATH] should be at most [MATH] by our condition (Theorem [REF] (a)).', '1107.4862-2-45-0': 'Similar discussions can be applied to the rest cases (ii) - (iv).', '1107.4862-2-46-0': '## The case (ii)', '1107.4862-2-47-0': 'In this case, we have [MATH] and [MATH].', '1107.4862-2-47-1': 'Let [MATH] and [MATH].', '1107.4862-2-47-2': 'Thus, one has [MATH], [MATH] and [MATH] from our conditions.', '1107.4862-2-47-3': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (ii) since [MATH] and [MATH].', '1107.4862-2-48-0': '## The case (iii)', '1107.4862-2-49-0': 'Let [MATH] and [MATH].', '1107.4862-2-49-1': 'Thus, one has [MATH], [MATH] and [MATH].', '1107.4862-2-49-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iii) since [MATH], [MATH] and [MATH].', '1107.4862-2-50-0': '## The case (iv)', '1107.4862-2-51-0': 'In this case, one has [MATH] and [MATH].', '1107.4862-2-51-1': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iv) since [MATH] and [MATH].', '1107.4862-2-52-0': '(a) The classification of the case (iv) is essentially given in [CITATION].', '1107.4862-2-53-0': '(b) Since [MATH], the inequalities [MATH] and [MATH] can be obtained from [REF] (see Proposition [REF]).', '1107.4862-2-53-1': 'Thus, the possible [MATH]-vectors of integral simplices with normalized volume 5 can be essentially characterized only by Theorem [REF] (a) and the inequalities [REF].', '1107.4862-2-54-0': '# The possible [MATH]-vectors of integral simplices with [MATH]', '1107.4862-2-55-0': 'In this section, similar to the previous section, we classify all the possible [MATH]-vectors of integral simplices with normalized volume 7.', '1107.4862-2-56-0': 'Let [MATH] be a nonnegative integer sequence with [MATH] and [MATH] which satisfies [MATH], [MATH] for [MATH] and [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-2-56-1': 'Since [MATH], we need not consider the inequalities [MATH], [MATH], [MATH], [MATH] and [MATH].', '1107.4862-2-56-2': 'From the conditions [MATH], [MATH] and [MATH], only the possible sequences look like', '1107.4862-2-57-0': '## The case (i)', '1107.4862-2-58-0': 'Let [MATH].', '1107.4862-2-58-1': 'Thus, one has [MATH] and [MATH] from our conditions.', '1107.4862-2-58-2': 'Hence, we can define [MATH].', '1107.4862-2-58-3': 'By Lemma [REF], [MATH] coincides with (i) since [MATH].', '1107.4862-2-59-0': '## The case (ii)', '1107.4862-2-60-0': 'Let [MATH] and [MATH].', '1107.4862-2-60-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-60-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (ii) since [MATH] and [MATH].', '1107.4862-2-61-0': '## The case (iii)', '1107.4862-2-62-0': 'Let [MATH] and [MATH].', '1107.4862-2-62-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-62-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iii) since [MATH] and [MATH].', '1107.4862-2-63-0': '## The case (iv)', '1107.4862-2-64-0': 'Let [MATH] and [MATH].', '1107.4862-2-64-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-64-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (iv) since [MATH] and [MATH].', '1107.4862-2-65-0': '## The case (v)', '1107.4862-2-66-0': 'Let [MATH] and [MATH].', '1107.4862-2-66-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-66-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (v) since [MATH] and [MATH].', '1107.4862-2-67-0': '## The case (vi)', '1107.4862-2-68-0': 'Let [MATH] and [MATH].', '1107.4862-2-68-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-68-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (vi) since [MATH] and [MATH].', '1107.4862-2-69-0': '## The case (vii)', '1107.4862-2-70-0': 'Let [MATH] and [MATH].', '1107.4862-2-70-1': 'Thus, one has [MATH] and [MATH].', '1107.4862-2-70-2': 'Hence, we can define [MATH] and its [MATH]-vector coincides with (vii) since [MATH] and [MATH].', '1107.4862-2-71-0': '## The case (viii)', '1107.4862-2-72-0': 'In this case, one has [MATH] and [MATH].', '1107.4862-2-72-1': 'Hence, we can define [MATH] if [MATH] and [MATH].', '1107.4862-2-72-2': 'Moreover, each of their [MATH]-vectors coincides with (viii) since [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH].', '1107.4862-2-73-0': 'When we discuss the cases (vi) and (viii), we need the new inequality [MATH].', '1107.4862-2-73-1': 'In fact, for example, the sequence [MATH] cannot be the [MATH]-vector of an integral simplex, although this satisfies [MATH].', '1107.4862-2-73-2': 'Similarly, the sequence [MATH] is also impossible to be the [MATH]-vector of an integral simplex, although this satisfies [MATH].', '1107.4862-2-74-0': 'More generally, the following example shows that many inequalities [MATH] are required to verify whether a given integer sequence is a [MATH]-vector of some integral simplex.', '1107.4862-2-75-0': 'For a prime number [MATH] with [MATH], let [MATH] and [MATH] be positive integers satisfying the condition described in [REF].', '1107.4862-2-75-1': 'Let us consider the integer sequence [EQUATION] where [MATH].', '1107.4862-2-75-2': 'Then [MATH], [MATH] for [MATH] and [MATH], where [MATH] are the positive integers such that [MATH] with [MATH].', '1107.4862-2-75-3': 'Thus, one has [MATH] but [MATH] or [MATH].', '1107.4862-2-75-4': 'In fact, since [EQUATION] we have [MATH] or [MATH].', '1107.4862-2-75-5': 'Hence, this integer sequence satisfies none of the inequalities [MATH] when [MATH] and [MATH] satisfy [MATH] and the condition in [REF].', '1107.4862-2-75-6': 'On the other hand, this satisfies both [MATH] for [MATH] and [MATH] for [MATH].', '1107.4862-2-76-0': 'Remark that since [MATH], if there exists an integral convex polytope of dimension [MATH] whose [MATH]-vector equals this sequence, then it must be a simplex.', '1107.4862-2-76-1': 'Therefore, thanks to Theorem [REF] (b), we can claim that there exists no integral convex polytope whose [MATH]-vector equals this sequence, while we cannot determine whether this integer sequence is a [MATH]-vector of some integral convex polytope only by [REF] and [REF].', '1107.4862-2-77-0': '# Towads the classification of Ehrhart polynomials with general normalized volumes', '1107.4862-2-78-0': 'Finally, we note some future problems on the classification of Ehrhart polynomials of integral convex polytopes.', '1107.4862-2-79-0': '## Higher prime case', '1107.4862-2-80-0': 'Remark that we cannot characterize the possible [MATH]-vectors of integral simplices with higher prime normalized volumes only by Theorem [REF], that is, Theorem [REF] is not sufficient.', '1107.4862-2-80-1': 'In fact, since the volume of an integral convex polytope containing a unique integer point in its interior has an upper bound, if [MATH] is a sufficiently large prime number, then the integer sequence [MATH] cannot be a [MATH]-vector of any integral simplex of dimension 3, although [MATH] satisfies all the conditions of Theorem [REF].', '1107.4862-2-81-0': '## Non-prime case', '1107.4862-2-82-0': 'We also remark that Theorem [REF] is not true when [MATH] is not prime in general.', '1107.4862-2-82-1': 'In fact, for example, there exists an integral simplex of dimension 5 whose [MATH]-vector is [MATH] ([CITATION]), while this satisfies neither [MATH] nor [MATH], where [MATH] and [MATH].', '1107.4862-2-83-0': 'More generally, for a non-prime number integer [MATH], where [MATH] is the least prime divisor of [MATH], let [MATH] and [MATH] with [MATH].', '1107.4862-2-83-1': 'Then, from Lemma [REF], we have [MATH], where [EQUATION]', '1107.4862-2-83-2': 'This [MATH]-vector satisfies neither [MATH] nor [MATH].', '1107.4862-2-84-0': 'On the other hand, Proposition [REF] is true even for non-prime normalized volume case and we also know other analogue of Theorem [REF] for such case as follows.', '1107.4862-2-85-0': 'Let [MATH] be an integral simplex of dimension [MATH] with its [MATH]-vector [MATH] and [MATH] the positive integers such that [MATH] with [MATH], where [MATH] is not prime.', '1107.4862-2-85-1': 'Then one has [EQUATION] where [MATH] is the least prime divisor of [MATH].', '1107.4862-2-86-0': 'By applying [CITATION], a proof of this statement is by with the same as the proof of Theorem [REF] (b).', '1107.4862-2-87-0': 'In fact, it is immediate that the above example satisfies [MATH] for [MATH] with [MATH].'} | [['1107.4862-1-0-1', '1107.4862-2-0-1'], ['1107.4862-1-0-2', '1107.4862-2-0-2'], ['1107.4862-1-25-2', '1107.4862-2-26-3'], ['1107.4862-1-25-7', '1107.4862-2-26-7'], ['1107.4862-1-3-0', '1107.4862-2-3-0'], ['1107.4862-1-3-1', '1107.4862-2-3-1'], ['1107.4862-1-3-2', '1107.4862-2-3-2'], ['1107.4862-1-39-0', '1107.4862-2-39-0'], ['1107.4862-1-60-1', '1107.4862-2-60-1'], ['1107.4862-1-60-2', '1107.4862-2-60-2'], ['1107.4862-1-17-0', '1107.4862-2-19-0'], ['1107.4862-1-5-0', '1107.4862-2-5-0'], ['1107.4862-1-5-2', '1107.4862-2-5-2'], ['1107.4862-1-43-1', '1107.4862-2-58-1'], ['1107.4862-1-43-2', '1107.4862-2-58-2'], ['1107.4862-1-2-0', '1107.4862-2-2-0'], ['1107.4862-1-21-1', '1107.4862-2-22-1'], 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'1107.4862-2-14-0'], ['1107.4862-1-7-2', '1107.4862-2-8-2'], ['1107.4862-1-37-0', '1107.4862-2-37-0'], ['1107.4862-1-73-0', '1107.4862-2-73-0'], ['1107.4862-1-73-2', '1107.4862-2-73-2'], ['1107.4862-1-23-1', '1107.4862-2-24-1'], ['1107.4862-1-8-1', '1107.4862-2-9-1'], ['1107.4862-1-18-0', '1107.4862-2-20-0'], ['1107.4862-1-19-0', '1107.4862-2-20-1']] | [] | [['1107.4862-1-25-1', '1107.4862-2-26-2'], ['1107.4862-1-25-5', '1107.4862-2-26-5'], ['1107.4862-1-25-9', '1107.4862-2-26-10'], ['1107.4862-1-21-0', '1107.4862-2-22-0'], ['1107.4862-1-40-0', '1107.4862-2-40-0'], ['1107.4862-1-6-1', '1107.4862-2-7-1'], ['1107.4862-1-53-0', '1107.4862-2-55-0'], ['1107.4862-1-51-0', '1107.4862-2-53-0'], ['1107.4862-1-20-0', '1107.4862-2-21-1'], ['1107.4862-1-23-0', '1107.4862-2-24-0'], ['1107.4862-1-8-0', '1107.4862-2-9-0'], ['1107.4862-1-8-3', '1107.4862-2-9-3'], ['1107.4862-1-8-5', '1107.4862-2-9-5'], ['1107.4862-1-8-5', '1107.4862-2-9-6']] | [['1107.4862-1-15-0', '1107.4862-2-80-0'], ['1107.4862-1-15-1', '1107.4862-2-80-1'], ['1107.4862-1-28-2', '1107.4862-2-85-1'], ['1107.4862-1-30-3', '1107.4862-2-26-10'], ['1107.4862-1-31-4', '1107.4862-2-30-5'], ['1107.4862-1-32-8', '1107.4862-2-35-8'], ['1107.4862-1-32-13', '1107.4862-2-35-8'], ['1107.4862-1-45-2', '1107.4862-2-47-3'], ['1107.4862-1-55-0', '1107.4862-2-40-0'], ['1107.4862-1-58-1', '1107.4862-2-47-2'], ['1107.4862-1-58-2', '1107.4862-2-44-4']] | ['1107.4862-1-3-4', '1107.4862-1-5-4', '1107.4862-1-7-1', '1107.4862-1-8-4', '1107.4862-1-9-3', '1107.4862-1-19-2', '1107.4862-1-20-1', '1107.4862-1-22-0', '1107.4862-1-24-0', '1107.4862-1-24-1', '1107.4862-1-24-2', '1107.4862-1-24-3', '1107.4862-1-24-4', '1107.4862-1-25-0', '1107.4862-1-25-3', '1107.4862-1-25-6', '1107.4862-1-25-8', '1107.4862-1-26-1', '1107.4862-1-26-8', '1107.4862-1-27-1', '1107.4862-1-29-0', '1107.4862-1-29-1', '1107.4862-1-30-0', '1107.4862-1-30-1', '1107.4862-1-30-2', '1107.4862-1-31-1', '1107.4862-1-32-3', '1107.4862-1-32-10', '1107.4862-1-32-15', '1107.4862-1-33-0', '1107.4862-1-35-2', '1107.4862-1-38-0', '1107.4862-1-38-1', '1107.4862-1-38-2', '1107.4862-1-41-0', '1107.4862-1-43-0', '1107.4862-1-45-0', '1107.4862-1-45-1', '1107.4862-1-47-0', '1107.4862-1-47-1', '1107.4862-1-47-2', '1107.4862-1-54-0', '1107.4862-1-54-1', '1107.4862-1-54-2', '1107.4862-1-56-0', '1107.4862-1-58-0', '1107.4862-1-60-0', '1107.4862-1-62-0', '1107.4862-1-64-0', '1107.4862-1-66-0', '1107.4862-1-68-0', '1107.4862-1-70-0', '1107.4862-1-72-0', '1107.4862-1-72-1', '1107.4862-1-72-2', '1107.4862-2-3-3', '1107.4862-2-6-0', '1107.4862-2-8-1', '1107.4862-2-9-4', '1107.4862-2-11-3', '1107.4862-2-12-0', '1107.4862-2-21-0', '1107.4862-2-23-0', '1107.4862-2-24-2', '1107.4862-2-25-0', '1107.4862-2-25-1', '1107.4862-2-25-2', '1107.4862-2-25-3', '1107.4862-2-25-4', '1107.4862-2-26-0', '1107.4862-2-26-1', '1107.4862-2-26-6', '1107.4862-2-26-8', '1107.4862-2-26-9', '1107.4862-2-27-1', '1107.4862-2-27-8', '1107.4862-2-28-0', '1107.4862-2-28-1', '1107.4862-2-28-2', '1107.4862-2-28-3', '1107.4862-2-28-4', '1107.4862-2-28-5', '1107.4862-2-28-6', '1107.4862-2-30-2', '1107.4862-2-32-0', '1107.4862-2-33-0', '1107.4862-2-34-0', '1107.4862-2-35-11', '1107.4862-2-41-0', '1107.4862-2-41-1', '1107.4862-2-42-0', '1107.4862-2-44-1', '1107.4862-2-47-1', '1107.4862-2-49-0', '1107.4862-2-49-1', '1107.4862-2-49-2', '1107.4862-2-56-0', '1107.4862-2-56-1', '1107.4862-2-56-2', '1107.4862-2-58-0', '1107.4862-2-60-0', '1107.4862-2-62-0', '1107.4862-2-64-0', '1107.4862-2-66-0', '1107.4862-2-68-0', '1107.4862-2-70-0', '1107.4862-2-72-0', '1107.4862-2-72-1', '1107.4862-2-72-2', '1107.4862-2-83-0', '1107.4862-2-83-1', '1107.4862-2-83-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1107.4862 | null | null | null | null | null |
1306.4013 | {'1306.4013-1-0-0': 'We constrain the most likely range of masses for the Milky Way and M31 using an application of the Numerical Action Method (NAM) that optimizes the fit to observed parameters over a large ensemble of NAM-generated solutions.', '1306.4013-1-0-1': 'Our [MATH] confidence level mass ranges, [MATH] for MW and [MATH] for M31, are consistent with the upper range of estimates from other methods and suggests that a larger proportion of the total mass becomes detectable when the peculiar motions of many nearby satellites are taken into account in the dynamical analysis.', '1306.4013-1-0-2': 'We test the method against simulated Local Group catalogs extracted from the Millennium Run to confirm that mass predictions are consistent with actual galaxy halo masses.', '1306.4013-1-1-0': '# Introduction', '1306.4013-1-2-0': 'Estimating the total masses of galaxies, our own in particular, is a continuing challenge of precision cosmology.', '1306.4013-1-2-1': 'Part of the challenge lies in the unknown extent of the dark matter halos within which they are presumably embedded: While the measurement of galaxy rotation curves from coherent stellar motions allows the mass within the visible radius to be inferred, the total mass of the associated dark matter halos predicted in the standard model of cosmology, whose physical extent is not known, is more difficult to estimate.', '1306.4013-1-2-2': 'To probe the total effective gravitational mass the analysis must include the effect on the peculiar motions of nearby galaxies.', '1306.4013-1-3-0': 'The measurement of total galaxy masses from their relative motions was pioneered by Kahn Woltjer 1959.', '1306.4013-1-3-1': 'Their "timing argument" (TA) method, which assumes purely radial infall, indicated a total mass for the MW+M31 system of about [MATH]-a lower bound, since the possibility of transverse motions is excluded.', '1306.4013-1-3-2': 'The total mass of the Local Group (LG) can also be computed from the velocity dispersion of its various member galaxies, asssuming that it is in virial equillibrium and that its velocity ellipsoid is isotropic: Corteau Van den Bergh 1999, using this method, found a LG mass of [MATH].', '1306.4013-1-3-3': 'More recent applications of the TA tend to suggest higher masses.', '1306.4013-1-3-4': 'Li White 2008 confirmed that the TA method used on mock galaxies drawn from the Millennium Run (Springel et al. 2005) systematically underestimates the true mass, and revised the TA method to predict a LG mass of [MATH].', '1306.4013-1-3-5': 'Van der Marel et al. 2012 used the full proper motion of M31 to improve the TA method, estimating a LG mass of about [MATH], somewhat higher than the combined prediction of [MATH] from a Bayesian combination of estimates from different methods.', '1306.4013-1-3-6': 'The TA can also be used with the proper motion of Leo I (Sohn et al. 2012), assuming it is gravitationally bound to the MW, to estimate the mass of the MW alone.', '1306.4013-1-3-7': 'Boylan-Kolchin et al. 2012 combine the TA and other methods in estimating a virial mass for the MW of [MATH] with a 90% confidence interval of [MATH].', '1306.4013-1-4-0': 'Mass estimation methods using the TA are based on the analysis of single-galaxy interactions with the Milky Way.', '1306.4013-1-4-1': 'We show in this paper that the Numerical Action Method (NAM), by taking into account the peculiar motions of a large subset of Local Group satellites, effectively breaks the mass degeneracy in the two-body TA and identifies separate ranges of likely masses for the two principal actors in the LG.', '1306.4013-1-4-2': 'The method avoids the TA assumption that galaxies are gravitationally bound, makes no assumptions about virialization of the LG, and is sensitive to more widely diffused concentrations of dark matter that could remain undetected using other methods.', '1306.4013-1-4-3': 'NAM takes as input the cosmological parameters [MATH] and [MATH], and assumes that linear theory correctly describes velocities at early times and that galaxies and their progenitors back in time can be approximated as simple paths representing the center-of-mass motions of their associated dark matter halos.', '1306.4013-1-5-0': 'Earlier papers on NAM (including Peebles 1989 introducing the method; Peebles 1995; Peebles, et al. 2001; Phelps 2002; Peebles, et al. 2011; Peebles Tully 2013) developed the method in the context of the Local Group, focusing on the analysis of individual solutions or small ensembles.', '1306.4013-1-5-1': 'In what follows we explore the behavior of several thousand independent solutions in order to identify the combinations of MW and M31 masses that yield the best fit to the observational constraints.', '1306.4013-1-5-2': 'Section 1 introduces our implementation of NAM.', '1306.4013-1-5-3': 'In section 2 we apply it to the Local Group, and in section 3 we test NAM predictions in the LG with mock catalogs drawn from the Millennium Run.', '1306.4013-1-5-4': 'A brief discussion including an assessment of future prospects is found in section 4.', '1306.4013-1-6-0': '# The numerical action method', '1306.4013-1-7-0': '## Method and approximations', '1306.4013-1-8-0': 'Our version of NAM is based on the improved algorithm described in Peebles et al. 2011.', '1306.4013-1-8-1': 'We have extended it to include an optional partial canonical transformation of coordinates between the radial distance and the radial velocity (first used in Peebles et al. 2001 and more fully described in Phelps 2002), allowing either the galaxy distance or redshift to be chosen as the boundary condition in the action integration at [MATH], in addition to the two components of the observed galaxy position in the plane of the sky.', '1306.4013-1-8-2': 'This effectively doubles the solution space that can be explored.', '1306.4013-1-8-3': 'Appendix A gives details of the coordinate transformation.', '1306.4013-1-9-0': 'We model galaxies as constant-density spheres with MW radius of 100 kpc and all other galaxy radii scaling proportionately with the cube root of their masses.', '1306.4013-1-9-1': 'From the start of the computation time at [MATH] until [MATH], galaxy paths are interpreted as the mean motions of the coalescing systems of baryonic matter and their associated dark matter halos.', '1306.4013-1-9-2': 'As shown in Peebles et al. 2011, the initial velocities are by construction consistent with linear perturbation theory.', '1306.4013-1-9-3': "This keeps the galaxies well separated from each other at early times, and so passages of galaxies through each other's (nonphysical) cutoff radii are rare.", '1306.4013-1-10-0': 'Galaxy flight paths are reconstructed from initial randomized straight line trial orbits by successive iterations in the direction of the first and second derivatives of the action until a stationary point is reached.', '1306.4013-1-10-1': 'Solutions are verified by comparing them to the solutions to the equations of motion from the same initial timestep in a leapfrog approximation.', '1306.4013-1-10-2': 'With a target of [MATH] in the sum of squares of the gradient of the action, deviations between NAM-generated flight paths and the leapfrog approximation are at most a few kpc at the final timestep.', '1306.4013-1-10-3': 'In the present work we follow Peebles Tully 2013 in excluding several close satellites to the MW, but we use a smaller number of time steps (30, vs. 500 in Peebles et al. 2011 and Peebles Tully 2013), which is still sufficient to produce tightly bending half-orbits.', '1306.4013-1-10-4': 'This significantly reduces computation time and maximizes the number of solutions generated.', '1306.4013-1-11-0': 'NAM solutions are non-unique owing to the mixed boundary conditions: velocities are constrained at the initial time (see discussion in Appendix A, 1.4), while some combination of distances and velocities are fixed at the final time.', '1306.4013-1-11-1': 'As a consequence, different choices of initial trial orbits with the same observational constraints will in general yield a different set of galaxy paths, each of which are solutions to the equations of motion.', '1306.4013-1-11-2': 'However, as there are more constraints than those required for a solution (it is sufficient to specify either distances or redshifts), a [MATH] measure of fit is defined for each observable, [EQUATION] from which a best-fit solution can be selected from an ensemble.', '1306.4013-1-12-0': 'Apart from the initial trial orbits, solutions may also be sensitive functions of the input parameters (distance, redshift, angular position, and mass), particularly for galaxies in close proximity to each other.', '1306.4013-1-12-1': 'These parameters are fixed in the formal NAM computation even though there is an observational uncertainty associated with each.', '1306.4013-1-12-2': 'Since we wish to explore the widest possible range of reasonable physical configurations, we adopt the method, similar to Peebles et al. 2011 and Peebles Tully 2013, of defining a [MATH] measure as a sum over all constraints, fixing the observational constraints as their given values plus a random error within the observational uncertainty: [EQUATION]', '1306.4013-1-12-3': 'We relax the initial trial orbits to a solution to the equations of motion using NAM, and then relax each NAM solution to a minimum in [MATH].', '1306.4013-1-12-4': 'By holding some of the observational constraints relatively fixed, we can repeat this procedure for a large number of NAM solutions to explore an [MATH]-dimensional space of solutions whose minimum in [MATH] gives the most likely values of the desired [MATH] constraints.', '1306.4013-1-12-5': 'In our case we are interested in the masses of the principal actors in the Local Group and so we are looking for a minimum within the two-dimensional space defined by the masses of MW and M31.', '1306.4013-1-12-6': 'Further details of our minimization approach, which is fully general and can be applied to any desired combination of observational parameters, are as follows.', '1306.4013-1-13-0': 'For each solution we assign masses to MW and M31 within a range of [MATH] and [MATH] in a Gaussian random distribution centered around the nominal masses given in the catalog.', '1306.4013-1-13-1': 'This produces higher quality solutions by focusing less attention on combinations of masses which experience has shown are less likely to yield good fits to the constraints.', '1306.4013-1-13-2': 'Random errors are added to all other observables (distances, redshifts, angular positions, and masses), with standard deviations taken generically to be 10% of the distance for distances, [MATH] for redshifts, half a degree for angular positions (allowing the observed galaxies to be offset from the center of mass of their respective halos), and 60% of the nominal catalog masses (giving these quantities the widest possible latitude).', '1306.4013-1-13-3': 'Since the main objective of this study is investigating the masses of MW and M31, and good agreement with M31 constraints is therefore a priority, standard deviations in distances, redshifts, and positions for galaxies beyond 1.5 Mpc are doubled and those for M31 are halved.', '1306.4013-1-13-4': 'Standard deviations in MW and M31 masses are reduced by a factor of 20 relative to their initial randomized guesses, essentially fixing them in the [MATH] relaxation step but allowing them to shift by a small amount if a significantly better solution is found at a slightly different mass.', '1306.4013-1-13-5': 'As large initial velocities are permitted in the solutions (see Appendix A, 1.4), a contribution to [MATH] from the magnitude of the initial velocity is additionally assigned with a standard deviation of [MATH].', '1306.4013-1-13-6': 'This effectively suppresses implausibly large initial velocities in the search for the best solutions.', '1306.4013-1-14-0': 'We find that NAM solutions are efficiently generated by generating a solution for the two dominant galaxies first and then adding the other galaxies, one at a time, in descending order of mass.', '1306.4013-1-14-1': "With each NAM solution we then take each galaxy in turn in the same order, using Powell's method to relax the individual galaxy distance, redshift and angular position to a minimum in its [MATH] since derivatives in [MATH] with respect to these quantities cannot be reliably computed.", '1306.4013-1-14-2': 'Discontinuities in [MATH] may be encountered where a galaxy jumps to a qualitatively different orbit in the descent to the target in the action; these are allowed so long as it leads to an improvement in [MATH].', '1306.4013-1-14-3': "If after relaxing with Powell's method [MATH] remains above a given threshold (100 yielded a good balance between time to solution and quality of solution), we recast the orbit for this galaxy and find another solution.", '1306.4013-1-14-4': 'If, after 50 attempts, [MATH] is still above the threshold, we switch from distance to redshift boundary conditions for this galaxy and allow another 25 attempts.', '1306.4013-1-14-5': 'If [MATH] still remains above the threshold, we take the solution corresponding to the the lowest [MATH] found thus far and move to the next galaxy.', '1306.4013-1-14-6': 'In practice between 25-50% of the galaxies in any given NAM solution will have been fitted to the velocity boundary condition and the rest to the distance boundary condition.', '1306.4013-1-14-7': "Once [MATH] has been minimized in this way, we then jointly relax the galaxy masses, also using Powell's method, to minimize [MATH] still further (we found that relaxing the masses separately from the other quantities gave more rapid convergence to a minimum).", '1306.4013-1-14-8': 'This method of relaxation between the various observational quantities typically reduces [MATH] by an order of magnitude from its initial value; further improvements to this ad hoc procedure are no doubt possible.', '1306.4013-1-15-0': 'We generate four thousand independent solutions using this method, varying the mass of MW and M31 in each solution as described above.', '1306.4013-1-15-1': 'Parallelizing the code to run on 12 supercomputing nodes using 2.40 GHz six-core Xeon processers, each solution typically takes 10 or 20 seconds for catalogs up to a dozen or two particles.', '1306.4013-1-15-2': 'From an ensemble of solutions we then plot a gaussian-smoothed [MATH] map against the MW and M31 masses, dividing the map into 24 x 24 equal bins and keeping the best [MATH] in each bin.', '1306.4013-1-15-3': 'The smoothing is desirable because the solution space can feature many local minima, each corresponding to a qualitatively different configurations of orbits.', '1306.4013-1-16-0': '# Results for the Local Group', '1306.4013-1-17-0': 'Our Local Group catalog, based on Peebles Tully 2013, is listed in Table 1.', '1306.4013-1-17-1': 'The omission of tightly bound satellites of MW and M31 facilitates direct comparison to the simulations, which lack comparable dynamic range, and maximizes the number of solutions which can explored.', '1306.4013-1-17-2': 'The four actors listed immediately after MW and M31 are intended to approximately model the influence of the significant mass concentrations residing immediately beyond the Local Group.', '1306.4013-1-17-3': 'While inclusion of all Local Group actors may hold the promise of producing even better dynamical constraints on the mass of MW and M31, it is offset by the greatly increased effort required to produce acceptable solutions.', '1306.4013-1-18-0': '[MATH] maps for 4000 solutions in four different scenarios, all with [MATH] and [MATH], are shown in Figure 1.', '1306.4013-1-18-1': 'At upper left are the contours in [MATH] generated from a simplified catalog consisting of only MW and M31, to check the consistency of our implementation of NAM with the TA.', '1306.4013-1-18-2': 'As expected from the classic TA, we find a well-defined constraint on the sum [MATH] but are unable to resolve the individual masses.', '1306.4013-1-18-3': 'The bare TA sum indicated by NAM, ([MATH] at [MATH] confidence, is consistent with Li and White (2008), who calibrated the TA against galaxy pairs drawn from the Millennium simulation to find, at [MATH] confidence, [MATH], with a median likelihood estimate of [MATH] (the confidence interval in the latter is larger since it takes cosmic variance into account).', '1306.4013-1-19-0': 'In Figure 1, upper right, we show the results from a reduced version of our catalog which includes the LG actors but excludes the four external groups.', '1306.4013-1-19-1': 'The additional of the additional dynamical actors has broken the degeneracy in the TA, giving independent masses of [MATH] for the MW and [MATH] for M31.', '1306.4013-1-19-2': 'With the addition of the four external groups (Figure 1, lower left), the confidence intervals are broadened and the best mass for the MW increases to [MATH].', '1306.4013-1-19-3': 'This is consistent at the upper end with previous TA measurements of the total LG mass and the individual MW mass (as mentioned in the introduction).', '1306.4013-1-19-4': 'The result at lower right, which also uses the full catalog but with initial assignment of external masses of [MATH] that of the result at lower left, shows the robustness of the minimum and suggests a correlation between the additional of external mass and the broadening of the confidence intervals.', '1306.4013-1-20-0': '# A test of NAM mass predictions in simulations', '1306.4013-1-21-0': 'As a check on the result for the Local Group, we use publicly available data from the Millennium Run (Springel et al. 2005) and follow Li White 2008 in generating mock Local Group catalogs satisfying the following conditions: We select type 0 or 1 galaxies with rotation velocities in the range [MATH] and bulge-to-total luminosity ratio in the range [MATH].', '1306.4013-1-21-1': 'We then sub-select close pairs with comoving separation between 0.5 and 1 Mpc, negative relative peculiar velocity and no massive companion with [MATH] within 2.5 Mpc from the centre-of-mass.', '1306.4013-1-21-2': 'This led to the initial identification of about 100 Local Group candidates.', '1306.4013-1-21-3': 'Since the simulation overproduces satellite galaxies relative to observations (the "missing satellite problem"; see e.g. Bullock 2012), and since the resulting dynamical complexity poses a special difficulty for NAM reconstructions, we additionally exclude catalogs where the gravitational acceleration on mock-MW due to satellites is greater by more than a factor of six than what we expect from the observed distribution (assuming the distance and nominal masses listed in our catalog).', '1306.4013-1-21-4': 'This limited our set of mock catalogs to 32 with reasonably similar dynamics to what expect to be the case with the LG, although in every case but two the dynamical complexity as defined above is greater than in the LG.', '1306.4013-1-21-5': 'To speed up computation time, within each of the 32 selected mock catalogs we include in the NAM computation only those galaxies and satellites within the distance to mock-M31, and all other galaxies out to 7 Mpc which produce an acceleration of at least 5% that of M31.', '1306.4013-1-21-6': 'This reduced the number of particles to those that are most relevant to the dynamics of the principal actors - from as few as 9 to as many as 34, with an average across the catalogs of 19.', '1306.4013-1-22-0': 'Contours in [MATH] for different halo masses of mock-MW and mock-M31 are shown for the four dynamically simplest catalogs in Figure 2.', '1306.4013-1-22-1': 'The predicted values for their masses are consistent with the true galaxy + halo masses from the simulation, within the [MATH] confidence range.', '1306.4013-1-22-2': 'We may expect the NAM prediction to be somewhat higher if it is sensitive to dark matter associated with the two principal actors but beyond their respective cutoff radii (at an overdensity of 200) that define their masses.', '1306.4013-1-22-3': 'A detailed comparison of the underlying dark matter distribution with the halo mass predictions may indicate whether this sensitivity is present.', '1306.4013-1-22-4': 'For the other 28 mock catalogs the mass predictions for both actors were likewise consistent with the [MATH] confidence intervals in all but three cases.', '1306.4013-1-22-5': 'The NAM-based mass predictions from the mock catalogs are thus accurate within statistical expectations.', '1306.4013-1-23-0': '# Discussion', '1306.4013-1-24-0': 'We have shown that the Numerical Action Method offers a promising method of constraining individual masses of the principal actors in the Local Group using an approach that investigates the behavior of [MATH] across a large ensemble of solutions.', '1306.4013-1-24-1': 'It is complementary to earlier work focusing on the detailed dynamical analysis of individual solutions.', '1306.4013-1-25-0': 'The relatively large masses of MW and M31 suggested in this first-of-its-kind implementation of NAM are a point of interest.', '1306.4013-1-25-1': "They may be due to the method's sensitivity to both the extended dark matter halo as well as purely dark concentrations between galaxies, since these otherwise invisible concentrations should leave a signature in galaxy proper motions.", '1306.4013-1-25-2': 'Another possibility is that the result is biased by poor reconstruction of satellite paths, since solutions placing M31 at exactly the observed distance and redshift also tend to place the dwarf galaxies in the near vicinity of the MW at distances much greater than the catalog distance.', '1306.4013-1-26-0': 'We note, as well, that tangential velocities of M31 are a sensitive function of the presence of external groups: in the absence of the former [MATH] are typically less than [MATH], while with external masses [MATH] are typically [MATH] or more.', '1306.4013-1-26-1': 'Recent observational constraints on [MATH] by Van der Marel et al. (2012) indicate [MATH] at [MATH] confidence.', '1306.4013-1-26-2': 'If a tangential velocity constraint for selected galaxies such as M31 were added to [MATH], and if further work on the observational side confirms a low [MATH], this could place significant additional constraints on the solution space.', '1306.4013-1-27-0': 'Further improvements to this method are suggested by the above observations, and can be expected from a number of directions.', '1306.4013-1-27-1': 'On the computational side, increases in the efficiency of the solution finding algorithm will permit a larger number of time steps to be used and potentially allow for more complex orbits, improving in particular the reconstruction of nearby satellites.', '1306.4013-1-27-2': 'It will also permit a larger complement of dwarf and satellite galaxies to be included.', '1306.4013-1-27-3': 'Adding full three-dimensional proper motions of nearby galaxies will certainly further constrain the likely masses.', '1306.4013-1-27-4': 'A comparison of the underlying dark matter distribution with the galaxy halos in the simulations will confirm whether NAM is potentially sensitive to extended distributions of dark matter beyond the nominal halo radii.', '1306.4013-1-28-0': '# Appendix: A faster NAM in redshift space', '1306.4013-1-29-0': 'Sections 1 and 2 are from Peebles et al. 2011.', '1306.4013-1-29-1': 'Section 3 is new to this study and shows how the action can be modified to accommodate redshift boundary conditions.', '1306.4013-1-30-0': '1.', '1306.4013-1-30-1': 'Review of the theory', '1306.4013-1-31-0': '1.1.', '1306.4013-1-31-1': 'Equations of motion', '1306.4013-1-32-0': 'In a cosmologically flat universe the expansion parameter satisfies [EQUATION] with present value [MATH].', '1306.4013-1-32-1': 'The equations of motion in physical length units are [EQUATION]', '1306.4013-1-32-2': 'Changing variables to the comoving coordinates [MATH] used here brings eq. ([REF]) to [EQUATION]', '1306.4013-1-32-3': 'This is derived from the action [EQUATION] when the present positions are fixed, [MATH], and initial conditions satisfy [EQUATION] 1.2.', '1306.4013-1-32-4': 'Discrete representation', '1306.4013-1-33-0': 'In a discrete representation the coordinates are [MATH], where [MATH] labels the particles, [MATH] the Cartesian coordinates, and [MATH] the time steps.', '1306.4013-1-33-1': 'The present positions [MATH] are fixed and given.', '1306.4013-1-33-2': 'The relevant derivatives of the action are [EQUATION]', '1306.4013-1-33-3': 'If [MATH] is close to quadratic in the [MATH] then position shifts [MATH] to a solution at an extremum of [MATH] satisfy [EQUATION]', '1306.4013-1-33-4': 'If the [MATH] are not close to a solution [MATH] is not close to quadratic in the [MATH], but experience shows that coordinate shifts in the direction of [MATH] walk toward a solution.', '1306.4013-1-34-0': 'Approximate the action (4) as [EQUATION]', '1306.4013-1-34-1': 'The times [MATH] interpolate between the time steps at [MATH] and [MATH] in leapfrog fashion.', '1306.4013-1-34-2': 'The approximation to the kinetic energy in eq. ([REF]) is motivated by linear perturbation theory, where [MATH] is nearly independent of time, so [MATH] is a good approximation to [MATH] at [MATH].', '1306.4013-1-34-3': 'The earliest time at which positions are computed is at [MATH].', '1306.4013-1-34-4': 'The leapfrog back in time from [MATH] is to [MATH].', '1306.4013-1-34-5': 'Recall that present positions at [MATH] are given at [MATH].', '1306.4013-1-35-0': 'The derivative of the action with respect to the coordinates [MATH] for [MATH], gives [EQUATION]', '1306.4013-1-35-1': 'When [MATH] this is a discrete approximation to the equation of motion ([REF]).', '1306.4013-1-35-2': 'The common factor [MATH] has been dropped to reduce clutter, which means [MATH].', '1306.4013-1-35-3': '(The asymmetry is in the gravity term.', '1306.4013-1-35-4': 'There still is the symmetry [MATH].)', '1306.4013-1-36-0': 'To simplify eq. ([REF]) and its derivatives wrt [MATH] let [EQUATION]', '1306.4013-1-36-1': 'Note that [EQUATION] follows from [MATH] at [MATH].', '1306.4013-1-36-2': 'Also, write the acceleration (apart from the factor [MATH]) of particle [MATH] due to the other particles [MATH] as [EQUATION]', '1306.4013-1-36-3': 'All this notation brings eq. ([REF]) to [EQUATION]', '1306.4013-1-36-4': 'We need the derivatives of the action with respect to the positions of the particles.', '1306.4013-1-36-5': 'Let the derivative of the acceleration of particle [MATH] wrt the position of particle [MATH] be [EQUATION]', '1306.4013-1-36-6': 'The derivative of the acceleration of particle [MATH] with respect to its own position is [EQUATION]', '1306.4013-1-36-7': 'So the nonzero derivatives of eq. ([REF]) with respect to the coordinates are [EQUATION]', '1306.4013-1-36-8': 'The goal is to use these second derivatives of the action to drive the first derivatives to zero at a stationary point, [MATH].', '1306.4013-1-37-0': '1.3.', '1306.4013-1-37-1': 'Leapfrog integration forward in time', '1306.4013-1-38-0': 'It is worth recording that when the action is at a stationary point, [MATH], a solution of equation ([REF]) is equivalent to a standard leapfrog numerical integration of the equation of motion ([REF]) forward in time.', '1306.4013-1-38-1': 'In this leapfrog, the positions and velocities are computed at interleaved time steps as (in the notation in eq [[REF]]), [EQUATION] and[EQUATION] which in the notation in eq ([REF]) is [EQUATION]', '1306.4013-1-38-2': 'Equations ([REF]) and ([REF]) are equivalent to eq. ([REF]) at [MATH].', '1306.4013-1-39-0': 'The difference from a conventional leapfrog integration is the boundary conditions, which here are the present positions and a condition on the initial velocities, as follows.', '1306.4013-1-40-0': '1.4.', '1306.4013-1-40-1': 'Initial conditions', '1306.4013-1-41-0': 'The representation of the mass distribution in the early universe by galaxy-size particles certainly is crude, but a reasonably useful approximation that motivates the following consideration.', '1306.4013-1-42-0': 'In linear perturbation theory for a continuous pressureless fluid the unwanted decaying mode has peculiar velocity that is decreasing as [MATH].', '1306.4013-1-42-1': 'Eq. ([REF]) formally eliminates this decaying mode.', '1306.4013-1-42-2': 'In the wanted growing mode the coordinate position of a particle is changing with time as [EQUATION]', '1306.4013-1-42-3': 'This implies that in the wanted growing mode the left hand side of eq. ([REF]) is constant, meaning [MATH], where [MATH] is the time measured from [MATH].', '1306.4013-1-42-4': 'This motivates approximating equation ([REF]) at the time [MATH] intermediate between the first two time steps in the leapfrog, [MATH] and [MATH], as [EQUATION]', '1306.4013-1-42-5': 'Recall that the half time step earlier than [MATH] is at [MATH], where [MATH] is supposed to vanish.', '1306.4013-1-42-6': 'Equation ([REF]) agrees with equations ([REF]) and ([REF]) at [MATH].', '1306.4013-1-43-0': 'It will be noted that [MATH] may be much larger than [MATH], meaning the numerical solution commences at modest redshift with small time steps.', '1306.4013-1-43-1': 'But [MATH] is still the time from [MATH] to the time midway between [MATH] and [MATH].', '1306.4013-1-44-0': 'The prescription in eq. ([REF]) allows large peculiar velocities at high redshift.', '1306.4013-1-44-1': 'This is not inconsistent with eq. ([REF]); it corresponds to a large primeval departure from homogeneity.', '1306.4013-1-44-2': 'It does mean that one must select solutions that are judged to have realistic initial peculiar velocities (at [MATH]).', '1306.4013-1-45-0': 'To summarize, the usual initial conditions - position and velocity - in a leapfrog integration of the equation of motion forward in time are replaced by the present position, [MATH], and the relation in equation ([REF]) between the two earliest positions, [MATH] and [MATH], at times [MATH] and [MATH].', '1306.4013-1-45-1': 'If equation ([REF]) is a good approximation this is equivalent to specifying the initial velocity, for then eq. ([REF]) determines [MATH].', '1306.4013-1-45-2': 'That is, the boundary conditions for a solution [MATH] are the present position and the time-variation of the initial velocity.', '1306.4013-1-46-0': '2.', '1306.4013-1-46-1': 'Method of solution', '1306.4013-1-47-0': '2.1.', '1306.4013-1-47-1': 'Single orbit adjustment', '1306.4013-1-48-0': "Since we're adjusting only the orbit of particle [MATH] drop the label [MATH] and write the first derivative of [MATH] as [EQUATION] and write the equation to be solved as [EQUATION]", '1306.4013-1-48-1': 'The nonzero second derivatives are [EQUATION]', '1306.4013-1-48-2': 'Since [EQUATION] eq. ([REF]) is [EQUATION]', '1306.4013-1-48-3': 'Set [MATH] in this equation and rearrange it to [EQUATION]', '1306.4013-1-48-4': 'This gives [MATH] in terms of [MATH] and [MATH].', '1306.4013-1-48-5': 'On iterating we get the form [EQUATION]', '1306.4013-1-48-6': 'At [MATH] this is just [EQUATION]', '1306.4013-1-48-7': 'At the second time step from the start, [MATH], eq. ([REF]) is [EQUATION] because there is no [MATH].', '1306.4013-1-48-8': 'Comparing this with eq. ([REF]) we see that [EQUATION]', '1306.4013-1-48-9': 'At [MATH] the result of substituting the form ([REF]) into eq. ([REF]) is [EQUATION]', '1306.4013-1-48-10': 'So at [MATH] [EQUATION]', '1306.4013-1-48-11': 'Eqs ([REF]) and ([REF]) give the [MATH] and [MATH], [MATH], in terms of the input derivatives of the action.', '1306.4013-1-48-12': 'Then eq. ([REF]) is [MATH] equations for the [MATH], at [MATH], in terms of the [MATH].', '1306.4013-1-48-13': 'We get three more, which fix the [MATH], by setting [MATH] in equation ([REF]) and recalling that [MATH]: [EQUATION]', '1306.4013-1-48-14': 'So write this as [EQUATION] solve this [MATH] set of equations for the [MATH], and then get the rest of the [MATH] from eq. ([REF]).', '1306.4013-1-49-0': '3.', '1306.4013-1-49-1': 'Redshift boundary condition', '1306.4013-1-50-0': 'Solutions to the equations of motion in the method outlined in 2.', '1306.4013-1-50-1': 'above satisfy the constraint that the distances at the present epoch are fixed.', '1306.4013-1-50-2': 'However, since redshifts are known more accurately than the distances, and since we may want to explore a larger space of solutions, it may desirable to recast the problem with fixed redshifts, with the distances at the present epoch emerging as predictions.', '1306.4013-1-50-3': '(note that the choice of boundary condition can be made particle-by-particle.)', '1306.4013-1-50-4': 'This can be accomplished through a partial transformation of coordinates that exchanges radial distances with radial velocities while leaving the angular position coordinates unchanged.', '1306.4013-1-50-5': 'Details are given in Phelps (2002); the procedure is summarized below.', '1306.4013-1-51-0': 'The change to radial velocity coordinates is carried out in the Hamiltonian frame through a canonical transformation of the conjugate variables (positions and momenta).', '1306.4013-1-51-1': 'The generating function of the transformation, which is added to the action outside the integral, is [EQUATION]', '1306.4013-1-51-2': 'Where [MATH] and [MATH] are the conjugate distance and momentum in the radial direction relative to the reference galaxy.', '1306.4013-1-51-3': 'With the addition of the generating function, the problem becomes equivalent to one expressed with a new set of conjugate coordinates [MATH] and [MATH], where [MATH] is the radial momentum and [MATH] is the radial distance.', '1306.4013-1-51-4': 'In these coordinates the boundary term in the variational derivative [MATH] vanishes at [MATH] when the angular positions and the radial velocity vanish.', '1306.4013-1-51-5': 'However, since the transformed Hamiltonian cannot be expressed analytically (the gravitational term cannot be written out in terms of [MATH]), the computation must be carried out in the original coordinate system, expressed in the Lagrangian frame, where the new boundary condition must be imposed by hand to recover the correct equations of motion.', '1306.4013-1-51-6': 'That constraint takes the form of additional terms in the action, as follows.', '1306.4013-1-52-0': '3.1 Modification to the action', '1306.4013-1-53-0': 'The modified action is: [EQUATION]', '1306.4013-1-53-1': 'The gradient at the final time step is: [EQUATION]', '1306.4013-1-53-2': 'The modified second derivatives of the action at the final two timesteps are: [EQUATION]', '1306.4013-1-53-3': 'Note that a new term in the action [MATH] is also present, but it is not used in the computation that follows.', '1306.4013-1-54-0': 'Since the boundary condition is now velocity-limited, [MATH], and we must add a new term in eq. ([REF]) arising from the final two timesteps: [EQUATION]', '1306.4013-1-54-1': 'This is now written as [EQUATION]', '1306.4013-1-54-2': 'As before, this [MATH] set of equations is solved for the [MATH], and we get the rest of the [MATH] from eq. ([REF]).', '1306.4013-1-54-3': 'However, we will find below that the boundary conditions will enable us to replace the [MATH] above with functions of [MATH].', '1306.4013-1-55-0': '3.2 Modification to the position shifts', '1306.4013-1-56-0': 'The change of boundary conditions implies a relationship between the position shifts [MATH] and [MATH] that will further modify the new terms above.', '1306.4013-1-56-1': 'The relationship is simplest to see in the one-dimensional case (setting [MATH]), where [EQUATION] where we continue to drop the particle subscript unless referring to the reference galaxy, [MATH], since we are working on one particle at a time while holding the rest of the catalog fixed.', '1306.4013-1-57-0': 'After the variation, [EQUATION]', '1306.4013-1-57-1': 'Recall that the orbit of the reference galaxy, and thus its contribution to the redshift, is held constant during the variation.', '1306.4013-1-58-0': 'Setting [MATH] gives [EQUATION]', '1306.4013-1-58-1': 'In three dimensions the angular coordinates of the particle must also remain unchanged after the variation, and so now there are three equations constraining the coordinate shifts at the final two timesteps: [EQUATION] where [MATH] and [MATH] are defined relative to the reference galaxy at [MATH]: [EQUATION]', '1306.4013-1-58-2': 'The [MATH] and [MATH] constraints impose a relationship between the three [MATH].', '1306.4013-1-58-3': 'The [MATH] constraint gives: [EQUATION]', '1306.4013-1-58-4': 'The combined [MATH] and [MATH] constraints further give: [EQUATION]', '1306.4013-1-58-5': 'As for the third constraint, the redshift in three dimensions is: [EQUATION] where [EQUATION]', '1306.4013-1-58-6': 'After the position shifts, [EQUATION] where [EQUATION]', '1306.4013-1-58-7': 'Setting [MATH] and using the approximation [MATH], we find, after some algebra, [EQUATION] where [EQUATION]', '1306.4013-1-58-8': 'It can be shown that this reduces to the one-dimensional case when we set two of the three position coordinates to zero.', '1306.4013-1-59-0': 'With the above, the position shifts [MATH] can be replaced in eqs. ([REF]-[REF]) by their equivalent expressions in terms of [MATH]: [EQUATION]', '1306.4013-1-59-1': 'Note that the actual value of the redshift has been nowhere specified and must be arranged by hand, as by choosing the initial trial orbits to have the input redshifts.', '1306.4013-1-59-2': 'However, the approximation [MATH] used to arrive at eq. ([REF]) can fail, particularly in the first few iterations of the action, and so in practice the redshifts may drift away from their input values as the stationary point in the action is reached.', '1306.4013-1-59-3': 'This drift is corrected by periodically adjusting the particle positions at timestep [MATH] according to eq. ([REF]).'} | {'1306.4013-2-0-0': 'We constrain the most likely range of masses for the Milky Way (MW) and M31 using an application of the numerical action method (NAM) that optimizes the fit to observed parameters over a large ensemble of NAM-generated solutions.', '1306.4013-2-0-1': 'Our [MATH] confidence level mass ranges, [MATH] for MW and [MATH] for M31, are consistent with the upper range of estimates from other methods and suggests that a larger proportion of the total mass becomes detectable when the peculiar motions of many nearby satellites are taken into account in the dynamical analysis.', '1306.4013-2-0-2': 'We test the method against simulated Local Group catalogs extracted from the Millennium Run to confirm that mass predictions are consistent with actual galaxy halo masses.', '1306.4013-2-1-0': '# Introduction', '1306.4013-2-2-0': 'Estimating the total masses of galaxies, our own in particular, is a continuing challenge of precision cosmology.', '1306.4013-2-2-1': 'Part of the challenge lies in the unknown extent of the dark matter halos within which they are presumably embedded: while the measurement of galaxy rotation curves from coherent stellar motions allows the mass within the visible radius to be inferred, the total mass of the associated dark matter halos predicted in the standard model of cosmology, whose physical extent is not known, is more difficult to estimate.', '1306.4013-2-2-2': 'To probe the total effective gravitational mass the analysis must include the effect on the peculiar motions of nearby galaxies.', '1306.4013-2-3-0': 'The measurement of total galaxy masses from their relative motions was pioneered by Kahn Woltjer (1959).', '1306.4013-2-3-1': 'Their "timing argument" (TA) method, which assumes purely radial infall, indicated a total mass for the MW+M31 system of about [MATH]-a lower bound, since the possibility of transverse motions is excluded.', '1306.4013-2-3-2': 'The total mass of the Local Group (LG) can also be computed from the velocity dispersion of its various member galaxies, assuming that it is in virial equilibrium and that its velocity ellipsoid is isotropic: Courteau van den Bergh (1999), using this method, found a LG mass of [MATH].', '1306.4013-2-3-3': 'More recent applications of the TA tend to suggest higher masses.', '1306.4013-2-3-4': 'Li White (2008) confirmed that the TA method used on mock galaxies drawn from the Millennium Run (Springel et al. 2005) systematically underestimates the true mass, and revised the TA method to predict a LG mass of [MATH].', '1306.4013-2-3-5': 'Van der Marel et al. (2012) used the full proper motion of M31 to improve the TA method, estimating a LG mass of about [MATH], somewhat higher than the combined prediction of [MATH] from a Bayesian combination of estimates from different methods.', '1306.4013-2-3-6': 'The TA can also be used with the proper motion of Leo I (Sohn et al. 2013), assuming it is gravitationally bound to the Milky Way (MW), to estimate the mass of the MW alone.', '1306.4013-2-3-7': 'Boylan-Kolchin et al. (2013) combine the TA and other methods in estimating a virial mass for the MW of [MATH] with a 90% confidence interval of [MATH].', '1306.4013-2-4-0': 'Mass estimation methods using the TA are based on the analysis of single-galaxy interactions with the Milky Way.', '1306.4013-2-4-1': 'We show in this paper that the numerical action method (NAM), by taking into account the peculiar motions of a large subset of Local Group satellites, effectively breaks the mass degeneracy in the two-body TA and identifies separate ranges of likely masses for the two principal actors in the LG.', '1306.4013-2-4-2': 'The method avoids the TA assumption that galaxies are gravitationally bound, makes no assumptions about virialization of the LG, and is sensitive to more widely diffused concentrations of dark matter that could remain undetected using other methods.', '1306.4013-2-4-3': 'NAM takes as input the cosmological parameters [MATH] and [MATH], and assumes that linear theory correctly describes velocities at early times and that galaxies and their progenitors back in time can be approximated as simple paths representing the center-of-mass motions of their associated dark matter halos.', '1306.4013-2-5-0': 'Earlier papers on NAM (including Peebles 1989 introducing the method; Peebles 1995; Peebles, et al. 2001; Phelps 2002; Peebles, et al. 2011; Peebles Tully 2013) developed the method in the context of the Local Group, focusing on the analysis of individual solutions or small ensembles.', '1306.4013-2-5-1': 'In what follows we explore the behavior of several thousand independent solutions in order to identify the combinations of MW and M31 masses that yield the best fit to the observational constraints.', '1306.4013-2-5-2': 'Section 1 introduces our implementation of NAM.', '1306.4013-2-5-3': 'In Section 2 we apply it to the Local Group, and in Section 3 we test NAM predictions in the LG with mock catalogs drawn from the Millennium Run.', '1306.4013-2-5-4': 'A brief discussion including an assessment of future prospects is found in Section 4.', '1306.4013-2-6-0': '# The numerical action method', '1306.4013-2-7-0': '## Method and Approximations', '1306.4013-2-8-0': 'Our version of NAM is based on the improved algorithm described in Peebles et al. (2011).', '1306.4013-2-8-1': 'We have extended it to include an optional partial canonical transformation of coordinates between the radial distance and the radial velocity (first used in Peebles et al. 2001 and more fully described in Phelps 2002), allowing either the galaxy distance or redshift to be chosen as the boundary condition in the action integration at [MATH], in addition to the two components of the observed galaxy position in the plane of the sky.', '1306.4013-2-8-2': 'This effectively doubles the solution space that can be explored.', '1306.4013-2-8-3': 'The Appendix gives details of the coordinate transformation.', '1306.4013-2-9-0': 'We model galaxies as constant-density spheres with MW radius of 100 kpc and all other galaxy radii scaling proportionately with the cube root of their masses.', '1306.4013-2-9-1': 'From the start of the computation time at [MATH] until [MATH], galaxy paths are interpreted as the mean motions of the coalescing systems of baryonic matter and their associated dark matter halos.', '1306.4013-2-9-2': 'As shown in Peebles et al. (2011), the initial velocities are by construction consistent with linear perturbation theory.', '1306.4013-2-9-3': "This keeps the galaxies well separated from each other at early times, and so passages of galaxies through each other's (nonphysical) cutoff radii are rare.", '1306.4013-2-10-0': 'Galaxy flight paths are reconstructed from initial randomized straight line trial orbits by successive iterations in the direction of the first and second derivatives of the action until a stationary point is reached.', '1306.4013-2-10-1': 'Solutions are verified by comparing them to the solutions to the equations of motion from the same initial timestep in a leapfrog approximation.', '1306.4013-2-10-2': 'With a target of [MATH] in the sum of squares of the gradient of the action, deviations between NAM-generated flight paths and the leapfrog approximation are at most a few kpc at the final timestep.', '1306.4013-2-10-3': 'In the present work we follow Peebles Tully (2013) in excluding several close satellites to the MW, but we use a smaller number of time steps (30, versus 500 in Peebles et al. 2011 and Peebles Tully 2013), which is still sufficient to produce tightly bending half-orbits.', '1306.4013-2-10-4': 'This significantly reduces computation time and maximizes the number of solutions generated.', '1306.4013-2-11-0': 'NAM solutions are non-unique owing to the mixed boundary conditions: velocities are constrained at the initial time (see discussion in the Appendix, A.1.4), while some combination of distances and velocities are fixed at the final time.', '1306.4013-2-11-1': 'As a consequence, different choices of initial trial orbits with the same observational constraints will in general yield a different set of galaxy paths, each of which are solutions to the equations of motion.', '1306.4013-2-11-2': 'However, as there are more constraints than those required for a solution (it is sufficient to specify either distances or redshifts), a [MATH] measure of fit is defined for each observable, [EQUATION] from which a best-fit solution can be selected from an ensemble.', '1306.4013-2-12-0': 'Apart from the initial trial orbits, solutions may also be sensitive functions of the input parameters, particularly for galaxies in close proximity to each other.', '1306.4013-2-12-1': 'These parameters are fixed in the formal NAM computation even though there is an observational uncertainty associated with each.', '1306.4013-2-12-2': 'Since we wish to explore the widest possible range of reasonable physical configurations, we adopt the method, similar to Peebles et al. (2011) and Peebles Tully (2013), of defining a [MATH] measure as a sum over all constraints (distance, redshift, angular position, mass, and, where available, velocity transverse to the line of sight), fixing the observational constraints as their given values plus a random error within the observational uncertainty that differs for each trial solution.', '1306.4013-2-12-3': 'Our per-particle [MATH] is defined as follows: [EQUATION]', '1306.4013-2-12-4': 'We relax the initial trial orbits to a solution to the equations of motion using NAM, and then relax each NAM solution to a minimum in [MATH].', '1306.4013-2-12-5': 'By holding some of the observational constraints relatively fixed, we can repeat this procedure for a large number of NAM solutions to explore an [MATH]-dimensional space of solutions whose minimum in [MATH] gives the most likely values of the desired [MATH] constraints.', '1306.4013-2-12-6': 'In our case we are interested in the masses of the principal actors in the Local Group and so we are looking for a minimum within the two-dimensional space defined by the masses of MW and M31.', '1306.4013-2-12-7': 'Further details of our minimization approach, which is fully general and can be applied to any desired combination of observational parameters, are as follows.', '1306.4013-2-13-0': 'For each solution we assign masses to MW and M31 within a range of [MATH] and [MATH] in a Gaussian random distribution centered around the nominal masses given in the catalog.', '1306.4013-2-13-1': 'This produces higher quality solutions by focusing less attention on combinations of masses which experience has shown are less likely to yield good fits to the constraints.', '1306.4013-2-13-2': 'Random errors are added to all other observables (distances, redshifts, angular positions, and masses), with standard deviations taken generically to be 10% of the distance for distances, [MATH] for redshifts, 60% of the nominal catalog masses (giving these quantities the widest possible latitude), half a degree for angular positions, and the published uncertainties for proper motions (as summarized in Peebles and Tully 2013; these are available for M31, LMC, M33, IC10, and LeoI).', '1306.4013-2-13-3': 'Our relative leniency in angular positions, which we allow owing to the approximate nature of the model, also accounts for the physical possibility that galaxies are offset from the center of mass of their respective dark matter halos.', '1306.4013-2-13-4': 'Since the main objective of this study is investigating the masses of MW and M31, and good agreement with M31 constraints is therefore a priority, standard deviations in distances, redshifts, and positions for galaxies beyond 1.5 Mpc are doubled and those for M31 are halved.', '1306.4013-2-13-5': 'Standard deviations in MW and M31 masses are reduced by a factor of 20 relative to their initial randomized guesses, essentially fixing them in the [MATH] relaxation step but allowing them to shift by a small amount if a significantly better solution is found at a slightly different mass.', '1306.4013-2-13-6': 'As large initial velocities are permitted in the solutions (see Appendix A.1.4), a contribution to [MATH] from the magnitude of the initial velocity (the final term in Equation (2) above) is additionally assigned with a standard deviation of [MATH].', '1306.4013-2-13-7': 'This effectively suppresses implausibly large initial velocities in the search for the best solutions.', '1306.4013-2-14-0': 'We find that NAM solutions are efficiently generated by generating a solution for the two dominant galaxies first and then adding the other galaxies, one at a time, in descending order of mass.', '1306.4013-2-14-1': "With each NAM solution we then take each galaxy in turn in the same order, using Powell's method to relax the individual galaxy distance, redshift and angular position to a minimum in its [MATH] since derivatives in [MATH] with respect to these quantities cannot be reliably computed.", '1306.4013-2-14-2': 'Discontinuities in [MATH] may be encountered where a galaxy jumps to a qualitatively different orbit in the descent to the target in the action; these are allowed so long as it leads to an improvement in [MATH].', '1306.4013-2-14-3': "If after relaxing with Powell's method [MATH] remains above a given threshold (100 yielded a good balance between time to solution and quality of solution), we recast the orbit for this galaxy and find another solution.", '1306.4013-2-14-4': 'If, after 50 attempts, [MATH] is still above the threshold, we switch from distance to redshift boundary conditions for this galaxy and allow another 25 attempts.', '1306.4013-2-14-5': 'If [MATH] still remains above the threshold, we take the solution corresponding to the lowest [MATH] found thus far and move to the next galaxy.', '1306.4013-2-14-6': 'In practice between 25% and 50% of the galaxies in any given NAM solution will have been fitted to the velocity boundary condition and the rest to the distance boundary condition.', '1306.4013-2-14-7': "Once [MATH] has been minimized in this way, we then jointly relax the galaxy masses, also using Powell's method, to minimize [MATH] still further (we found that relaxing the masses separately from the other quantities gave more rapid convergence to a minimum).", '1306.4013-2-14-8': 'This method of relaxation between the various observational quantities typically reduces [MATH] by an order of magnitude from its initial value; further improvements to this ad hoc procedure are no doubt possible.', '1306.4013-2-15-0': 'We generate four thousand independent solutions using this method, varying the mass of MW and M31 in each solution as described above.', '1306.4013-2-15-1': 'Parallelizing the code to run on 12 supercomputing nodes using 2.40 GHz six-core Xeon processors, each solution typically takes 10 or 20 seconds for catalogs up to a dozen or two particles.', '1306.4013-2-15-2': 'From an ensemble of solutions we then plot a Gaussian-smoothed [MATH] map against the MW and M31 masses, dividing the map into [MATH] equal bins and keeping the best [MATH] in each bin.', '1306.4013-2-15-3': 'The smoothing is desirable because the solution space can feature many local minima, each corresponding to a qualitatively different configurations of orbits.', '1306.4013-2-16-0': '# Results for the Local Group', '1306.4013-2-17-0': 'Our Local Group catalog, based on Peebles Tully (2013), is listed in Table 1.', '1306.4013-2-17-1': 'The omission of the smallest, tightly bound satellites of MW and M31 facilitates direct comparison to the simulations, which lack comparable range in mass, and maximizes the number of solutions which can explored.', '1306.4013-2-17-2': 'The four actors listed immediately after MW and M31 are intended to approximately model the influence of the significant mass concentrations residing immediately beyond the Local Group.', '1306.4013-2-17-3': 'While inclusion of all Local Group actors may hold the promise of producing even better dynamical constraints on the mass of MW and M31, it is offset by the greatly increased effort required to produce acceptable solutions.', '1306.4013-2-18-0': '[MATH] maps for 4000 solutions in four different scenarios, all with [MATH] and [MATH], are shown in Figure 1.', '1306.4013-2-18-1': 'At upper left are the contours in [MATH] generated from a simplified catalog consisting of only MW and M31, to check the consistency of our implementation of NAM with the Timing Argument.', '1306.4013-2-18-2': 'As expected from the classic TA, we find a well-defined constraint on the sum [MATH] but are unable to resolve the individual masses.', '1306.4013-2-18-3': 'The bare TA sum indicated by NAM, ([MATH] at [MATH] confidence, is consistent with Li and White (2008), who calibrated the TA against galaxy pairs drawn from the Millennium simulation to find, at [MATH] confidence, [MATH], with a median likelihood estimate of [MATH] (the confidence interval in the latter is larger since it takes cosmic variance into account).', '1306.4013-2-19-0': 'In Figure 1, upper right, we show the results from a reduced version of our catalog which includes the LG actors but excludes the four external groups.', '1306.4013-2-19-1': 'The additional of the additional dynamical actors has broken the degeneracy in the TA, giving independent masses of [MATH] for the MW and [MATH] for M31.', '1306.4013-2-19-2': 'With the addition of the four external groups (Figure 1, lower left), the best mass for the MW increases to [MATH].', '1306.4013-2-19-3': 'This is consistent at the lower end with previous TA measurements of the total LG mass and the individual MW mass.', '1306.4013-2-19-4': 'When the transverse velocity constraints on M31, LMC, M33, IC10, and LeoI are added (Figure 1, lower right), the confidence intervals are broadened and the best-fit mass for MW decreases slightly, to [MATH], reflecting the fact that lower masses for MW are correlated to lower transverse velocities for M31 and other nearby galaxies.', '1306.4013-2-20-0': '# A test of NAM mass predictions in simulations', '1306.4013-2-21-0': 'As a check on the result for the Local Group, we use publicly available data from the Millennium Run (Springel et al. 2005) and follow Li White (2008) in generating mock Local Group catalogs satisfying the following conditions: We select type 0 or 1 galaxies with rotation velocities in the range [MATH] and bulge-to-total luminosity ratio in the range [MATH].', '1306.4013-2-21-1': 'We then sub-select close pairs with comoving separation between 0.5 and 1 Mpc, negative relative peculiar velocity and no massive companion with [MATH] within 2.5 Mpc from the center-of-mass.', '1306.4013-2-21-2': 'This led to the initial identification of about 100 Local Group candidates.', '1306.4013-2-21-3': 'Since the simulation overproduces satellite galaxies relative to observations (the "missing satellite problem"; see, e.g., Bullock 2012), and since the resulting dynamical complexity poses a special difficulty for NAM reconstructions, we additionally exclude catalogs where the gravitational acceleration on mock-MW due to satellites is greater by more than a factor of six than what we expect from the observed distribution (assuming the distance and nominal masses listed in our catalog).', '1306.4013-2-21-4': 'This limited our set of mock catalogs to 32 with reasonably similar dynamics to what expect to be the case with the LG, although in every case but two the dynamical complexity as defined above is greater than in the LG.', '1306.4013-2-21-5': 'To speed up computation time, within each of the 32 selected mock catalogs we include in the NAM computation only those galaxies and satellites within the distance to mock-M31, and all other galaxies out to 7 Mpc which produce an acceleration of at least 5% that of M31.', '1306.4013-2-21-6': 'This reduced the number of particles to those that are most relevant to the dynamics of the principal actors - from as few as 9 to as many as 34, with an average across the catalogs of 19.', '1306.4013-2-22-0': 'Contours in [MATH] for different halo masses of mock-MW and mock-M31 are shown for the four dynamically simplest catalogs in Figure 2.', '1306.4013-2-22-1': 'The predicted values for their masses are consistent with the true galaxy + halo masses from the simulation, within the [MATH] confidence range.', '1306.4013-2-22-2': 'We may expect the NAM prediction to be somewhat higher if it is sensitive to dark matter associated with the two principal actors but beyond their respective cutoff radii (at an overdensity of 200) that define their masses.', '1306.4013-2-22-3': 'A detailed comparison of the underlying dark matter distribution with the halo mass predictions may indicate whether this sensitivity is present.', '1306.4013-2-22-4': 'For the other 28 mock catalogs the mass predictions for both actors were likewise consistent with the [MATH] confidence intervals in all but three cases.', '1306.4013-2-22-5': 'The NAM-based mass predictions from the mock catalogs are thus accurate within statistical expectations.', '1306.4013-2-23-0': '# Discussion', '1306.4013-2-24-0': 'We have shown that the Numerical Action Method offers a promising method of constraining individual masses of the principal actors in the Local Group using an approach that investigates the behavior of [MATH] across a large ensemble of solutions.', '1306.4013-2-24-1': 'It is complementary to earlier work focusing on the detailed dynamical analysis of individual solutions.', '1306.4013-2-25-0': 'The relatively large masses of MW and M31 suggested in this first-of-its-kind implementation of NAM are a point of interest.', '1306.4013-2-25-1': "They may be due to the method's sensitivity to both the extended dark matter halo as well as purely dark concentrations between galaxies, since these otherwise invisible concentrations should leave a signature in galaxy proper motions.", '1306.4013-2-25-2': 'In this case, our result would suggest the presence of mass concentrations larger than previously suspected.', '1306.4013-2-25-3': 'Another possibility is that the result is biased by poor reconstruction of satellite paths, since solutions placing M31 at exactly the observed distance and redshift also tend to place the dwarf galaxies in the near vicinity of the MW at distances greater than the catalog distance.', '1306.4013-2-25-4': 'Related to the above point, our observational constraints assume a value of 220 km/s for the circular velocity of the sun around the MW center; a higher value would reduce the approach velocity of M31, reducing the predicted mass.', '1306.4013-2-26-0': 'Further improvements to this method are suggested by the above observations, and can be expected from a number of directions.', '1306.4013-2-26-1': 'On the computational side, increases in the efficiency of the solution finding algorithm will permit a larger number of time steps to be used and potentially allow for more complex orbits, improving in particular the reconstruction of nearby satellites.', '1306.4013-2-26-2': 'It will also permit a larger complement of dwarf and satellite galaxies to be included.', '1306.4013-2-26-3': 'Adding full three-dimensional proper motions of a larger number of nearby galaxies will certainly further constrain the likely masses.', '1306.4013-2-26-4': 'A comparison of the underlying dark matter distribution with the galaxy halos in the simulations will confirm whether NAM is potentially sensitive to extended distributions of dark matter beyond the nominal halo radii.', '1306.4013-2-27-0': 'We thank Brent Tully for helpful suggestions that improved the presentation of the paper.', '1306.4013-2-27-1': 'This research project was supported at the Technion by the I-CORE Program of the Planning and Budgeting Committee and the ISF.', '1306.4013-2-27-2': 'V. Desjacques acknowledges support by the Swiss National Science Foundation.', '1306.4013-2-28-0': '# Appendix: A faster NAM in redshift space', '1306.4013-2-29-0': 'Sections A.1 and A.2 are from Peebles et al. (2011).', '1306.4013-2-29-1': 'Section A.3 is new to this study and shows how the action can be modified to accommodate redshift boundary conditions.', '1306.4013-2-30-0': 'A.1.', '1306.4013-2-30-1': 'Review of the theory', '1306.4013-2-31-0': 'A.1.1.', '1306.4013-2-31-1': 'Equations of motion', '1306.4013-2-32-0': 'In a cosmologically flat universe the expansion parameter satisfies [EQUATION] with present value [MATH].', '1306.4013-2-32-1': 'The equations of motion in physical length units are [EQUATION]', '1306.4013-2-32-2': 'Changing variables to the comoving coordinates [MATH] used here brings eq. ([REF]) to [EQUATION]', '1306.4013-2-32-3': 'This is derived from the action rCl S & = & _0^t_odt[_i m_i a^2 x_i^22 + 1a(_jiGm_im_jx_i-x_j + 14_i m_iH_o^2 x_i^2)]', '1306.4013-2-33-0': 'when the present positions are fixed, [MATH], and initial conditions satisfy [EQUATION]', '1306.4013-2-33-1': 'A.1.2.', '1306.4013-2-33-2': 'Discrete representation', '1306.4013-2-34-0': 'In a discrete representation the coordinates are [MATH], where [MATH] labels the particles, [MATH] the Cartesian coordinates, and [MATH] the time steps.', '1306.4013-2-34-1': 'The present positions [MATH] are fixed and given.', '1306.4013-2-34-2': 'The relevant derivatives of the action are [EQUATION]', '1306.4013-2-34-3': 'If [MATH] is close to quadratic in the [MATH] then position shifts [MATH] to a solution at an extremum of [MATH] satisfy [EQUATION]', '1306.4013-2-34-4': 'If the [MATH] are not close to a solution [MATH] is not close to quadratic in the [MATH], but experience shows that coordinate shifts in the direction of [MATH] walk toward a solution.', '1306.4013-2-35-0': 'Approximate the action ([REF]) as [EQUATION]', '1306.4013-2-35-1': 'The times [MATH] interpolate between the time steps at [MATH] and [MATH] in leapfrog fashion.', '1306.4013-2-35-2': 'The approximation to the kinetic energy in eq. ([REF]) is motivated by linear perturbation theory, where [MATH] is nearly independent of time, so [MATH] is a good approximation to [MATH] at [MATH].', '1306.4013-2-35-3': 'The earliest time at which positions are computed is at [MATH].', '1306.4013-2-35-4': 'The leapfrog back in time from [MATH] is to [MATH].', '1306.4013-2-35-5': 'Recall that present positions at [MATH] are given at [MATH].', '1306.4013-2-36-0': 'The derivative of the action with respect to the coordinates [MATH] for [MATH], gives [EQUATION]', '1306.4013-2-36-1': 'When [MATH] this is a discrete approximation to the equation of motion ([REF]).', '1306.4013-2-36-2': 'The common factor [MATH] has been dropped to reduce clutter, which means [MATH].', '1306.4013-2-36-3': '(The asymmetry is in the gravity term.', '1306.4013-2-36-4': 'There still is the symmetry [MATH].)', '1306.4013-2-37-0': 'To simplify eq. ([REF]) and its derivatives wrt [MATH] let [EQUATION]', '1306.4013-2-37-1': 'Note that [EQUATION] follows from [MATH] at [MATH].', '1306.4013-2-37-2': 'Also, write the acceleration (apart from the factor [MATH]) of particle [MATH] due to the other particles [MATH] as [EQUATION]', '1306.4013-2-37-3': 'All this notation brings eq. ([REF]) to [EQUATION]', '1306.4013-2-37-4': 'We need the derivatives of the action with respect to the positions of the particles.', '1306.4013-2-37-5': 'Let the derivative of the acceleration of particle [MATH] wrt the position of particle [MATH] be [EQUATION]', '1306.4013-2-37-6': 'The derivative of the acceleration of particle [MATH] with respect to its own position is [EQUATION]', '1306.4013-2-37-7': 'So the nonzero derivatives of eq. ([REF]) with respect to the coordinates are [EQUATION]', '1306.4013-2-37-8': 'The goal is to use these second derivatives of the action to drive the first derivatives to zero at a stationary point, [MATH].', '1306.4013-2-38-0': 'A.1.3.', '1306.4013-2-38-1': 'Leapfrog integration forward in time', '1306.4013-2-39-0': 'It is worth recording that when the action is at a stationary point, [MATH], a solution of equation ([REF]) is equivalent to a standard leapfrog numerical integration of the equation of motion ([REF]) forward in time.', '1306.4013-2-39-1': 'In this leapfrog, the positions and velocities are computed at interleaved time steps as (in the notation in eq [[REF]]), [EQUATION] and[EQUATION] which in the notation in eq ([REF]) is [EQUATION]', '1306.4013-2-39-2': 'Equations ([REF]) and ([REF]) are equivalent to eq. ([REF]) at [MATH].', '1306.4013-2-40-0': 'The difference from a conventional leapfrog integration is the boundary conditions, which here are the present positions and a condition on the initial velocities, as follows.', '1306.4013-2-41-0': 'A.1.4.', '1306.4013-2-41-1': 'Initial conditions', '1306.4013-2-42-0': 'The representation of the mass distribution in the early universe by galaxy-size particles certainly is crude, but a reasonably useful approximation that motivates the following consideration.', '1306.4013-2-43-0': 'In linear perturbation theory for a continuous pressureless fluid the unwanted decaying mode has peculiar velocity that is decreasing as [MATH].', '1306.4013-2-43-1': 'Eq. ([REF]) formally eliminates this decaying mode.', '1306.4013-2-43-2': 'In the wanted growing mode the coordinate position of a particle is changing with time as [EQUATION]', '1306.4013-2-43-3': 'This implies that in the wanted growing mode the left hand side of eq. ([REF]) is constant, meaning [MATH], where [MATH] is the time measured from [MATH].', '1306.4013-2-43-4': 'This motivates approximating equation ([REF]) at the time [MATH] intermediate between the first two time steps in the leapfrog, [MATH] and [MATH], as [EQUATION]', '1306.4013-2-43-5': 'Recall that the half time step earlier than [MATH] is at [MATH], where [MATH] is supposed to vanish.', '1306.4013-2-43-6': 'Equation ([REF]) agrees with equations ([REF]) and ([REF]) at [MATH].', '1306.4013-2-44-0': 'It will be noted that [MATH] may be much larger than [MATH], meaning the numerical solution commences at modest redshift with small time steps.', '1306.4013-2-44-1': 'But [MATH] is still the time from [MATH] to the time midway between [MATH] and [MATH].', '1306.4013-2-45-0': 'The prescription in eq. ([REF]) allows large peculiar velocities at high redshift.', '1306.4013-2-45-1': 'This is not inconsistent with eq. ([REF]); it corresponds to a large primeval departure from homogeneity.', '1306.4013-2-45-2': 'It does mean that one must select solutions that are judged to have realistic initial peculiar velocities (at [MATH]).', '1306.4013-2-46-0': 'To summarize, the usual initial conditions - position and velocity - in a leapfrog integration of the equation of motion forward in time are replaced by the present position, [MATH], and the relation in equation ([REF]) between the two earliest positions, [MATH] and [MATH], at times [MATH] and [MATH].', '1306.4013-2-46-1': 'If equation ([REF]) is a good approximation this is equivalent to specifying the initial velocity, for then eq. ([REF]) determines [MATH].', '1306.4013-2-46-2': 'That is, the boundary conditions for a solution [MATH] are the present position and the time-variation of the initial velocity.', '1306.4013-2-47-0': 'A.2.', '1306.4013-2-47-1': 'Method of solution', '1306.4013-2-48-0': 'A.2.1.', '1306.4013-2-48-1': 'Single orbit adjustment', '1306.4013-2-49-0': "Since we're adjusting only the orbit of particle [MATH] drop the label [MATH] and write the first derivative of [MATH] as [EQUATION] and write the equation to be solved as [EQUATION]", '1306.4013-2-49-1': 'The nonzero second derivatives are [EQUATION]', '1306.4013-2-49-2': 'Since [EQUATION] eq. ([REF]) is [EQUATION]', '1306.4013-2-49-3': 'Set [MATH] in this equation and rearrange it to [EQUATION]', '1306.4013-2-49-4': 'This gives [MATH] in terms of [MATH] and [MATH].', '1306.4013-2-49-5': 'On iterating we get the form [EQUATION]', '1306.4013-2-49-6': 'At [MATH] this is just [EQUATION]', '1306.4013-2-49-7': 'Here and in Equation (25) the Kronecker delta [MATH] is to be distinguished from the position shifts [MATH] introduced in Equation (7).', '1306.4013-2-49-8': 'At the second time step from the start, [MATH], eq. ([REF]) is [EQUATION] because there is no [MATH].', '1306.4013-2-49-9': 'Comparing this with eq. ([REF]) we see that [EQUATION]', '1306.4013-2-49-10': 'At [MATH] the result of substituting the form ([REF]) into eq. ([REF]) is [EQUATION]', '1306.4013-2-49-11': 'So at [MATH] [EQUATION]', '1306.4013-2-49-12': 'Equation (A34), with Equations (A30) and (A32), gives the [MATH] and [MATH] for all time steps in terms of the input derivatives of the action.', '1306.4013-2-49-13': 'We use these to find the position shifts [MATH] in Equation (A29).', '1306.4013-2-49-14': 'All that remains before this can be done is to compute the [MATH], which we find by setting [MATH] in equation (A27) and recalling that [MATH]: [EQUATION]', '1306.4013-2-49-15': 'So write this as [EQUATION] solve this [MATH] set of equations for the [MATH], and then get the rest of the [MATH] from eq. ([REF]).', '1306.4013-2-50-0': 'A.3.', '1306.4013-2-50-1': 'Redshift boundary condition', '1306.4013-2-51-0': 'Solutions to the equations of motion in the method outlined in Appendix A.2 above satisfy the constraint that the distances at the present epoch are fixed.', '1306.4013-2-51-1': 'However, since redshifts are known more accurately than the distances, and since we may want to explore a larger space of solutions, it may desirable to recast the problem with fixed redshifts, with the distances at the present epoch emerging as predictions.', '1306.4013-2-51-2': '(note that the choice of boundary condition can be made particle-by-particle.)', '1306.4013-2-51-3': 'This can be accomplished through a partial transformation of coordinates that exchanges radial distances with radial velocities while leaving the angular position coordinates unchanged.', '1306.4013-2-51-4': 'Details are given in Phelps (2002); the procedure is summarized below.', '1306.4013-2-52-0': 'The change to radial velocity coordinates is carried out in the Hamiltonian frame through a canonical transformation of the conjugate variables (positions and momenta).', '1306.4013-2-52-1': 'The generating function of the transformation, which is added to the action outside the integral, is [EQUATION] where [MATH] and [MATH] are the conjugate distance and momentum in the radial direction relative to the reference galaxy.', '1306.4013-2-52-2': 'With the addition of the generating function, the problem becomes equivalent to one expressed with a new set of conjugate coordinates [MATH] and [MATH], where [MATH] is the radial momentum and [MATH] is the radial distance.', '1306.4013-2-52-3': 'In these coordinates the boundary term in the variational derivative [MATH] vanishes at [MATH] when the angular positions and the radial velocity vanish.', '1306.4013-2-52-4': 'However, since the transformed Hamiltonian cannot be expressed analytically (the gravitational term cannot be written out in terms of [MATH]), the computation must be carried out in the original coordinate system, expressed in the Lagrangian frame, where the new boundary condition must be imposed by hand to recover the correct equations of motion.', '1306.4013-2-52-5': 'That constraint takes the form of additional terms in the action, as follows.', '1306.4013-2-53-0': 'A.3.1 Modification to the action', '1306.4013-2-54-0': 'The modified action is: [EQUATION]', '1306.4013-2-54-1': 'The gradient at the final time step is: [EQUATION]', '1306.4013-2-54-2': 'The modified second derivatives of the action at the final two timesteps are: [EQUATION]', '1306.4013-2-54-3': 'Note that a new term in the action [MATH] is also present, but it is not used in the computation that follows.', '1306.4013-2-55-0': 'Since the boundary condition is now velocity-limited, [MATH], and we must add a new term in eq. ([REF]) arising from the final two timesteps: [EQUATION]', '1306.4013-2-55-1': 'This is now written as [EQUATION]', '1306.4013-2-55-2': 'As before, this [MATH] set of equations is solved for the [MATH], and we get the rest of the [MATH] from eq. ([REF]).', '1306.4013-2-55-3': 'However, we will find below that the boundary conditions will enable us to replace the [MATH] above with functions of [MATH].', '1306.4013-2-56-0': 'A.3.2 Modification to the position shifts', '1306.4013-2-57-0': 'The change of boundary conditions implies a relationship between the position shifts [MATH] and [MATH] that will further modify the new terms above.', '1306.4013-2-57-1': 'The relationship is simplest to see in the one-dimensional case (setting [MATH]), where [EQUATION] where we continue to drop the particle subscript unless referring to the reference galaxy, [MATH], since we are working on one particle at a time while holding the rest of the catalog fixed.', '1306.4013-2-58-0': 'After the variation, [EQUATION]', '1306.4013-2-58-1': 'Recall that the orbit of the reference galaxy, and thus its contribution to the redshift, is held constant during the variation.', '1306.4013-2-59-0': 'Setting [MATH] gives [EQUATION]', '1306.4013-2-59-1': 'In three dimensions the angular coordinates of the particle must also remain unchanged after the variation, and so now there are three equations constraining the coordinate shifts at the final two timesteps: [EQUATION] where [MATH] and [MATH] are defined relative to the reference galaxy at [MATH]: [EQUATION]', '1306.4013-2-59-2': 'The [MATH] and [MATH] constraints impose a relationship between the three [MATH].', '1306.4013-2-59-3': 'The [MATH] constraint gives: [EQUATION]', '1306.4013-2-59-4': 'The combined [MATH] and [MATH] constraints further give: [EQUATION]', '1306.4013-2-59-5': 'As for the third constraint, the redshift in three dimensions is: [EQUATION] where [EQUATION]', '1306.4013-2-59-6': 'After the position shifts, [EQUATION] where [EQUATION]', '1306.4013-2-59-7': 'Setting [MATH] and using the approximation [MATH], we find, after some algebra, [EQUATION] where [EQUATION]', '1306.4013-2-59-8': 'It can be shown that this reduces to the one-dimensional case when we set two of the three position coordinates to zero.', '1306.4013-2-60-0': 'With the above, the position shifts [MATH] can be replaced in eqs. ([REF]-[REF]) by their equivalent expressions in terms of [MATH]: [EQUATION]', '1306.4013-2-60-1': 'Note that the actual value of the redshift has been nowhere specified and must be arranged by hand, as by choosing the initial trial orbits to have the input redshifts.', '1306.4013-2-60-2': 'However, the approximation [MATH] used to arrive at eq. ([REF]) can fail, particularly in the first few iterations of the action, and so in practice the redshifts may drift away from their input values as the stationary point in the action is reached.', '1306.4013-2-60-3': 'This drift is corrected by periodically adjusting the particle positions at timestep [MATH] according to eq. 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'1306.4013-1-31-0', '1306.4013-1-31-1', '1306.4013-1-32-4', '1306.4013-1-37-0', '1306.4013-1-37-1', '1306.4013-1-40-0', '1306.4013-1-40-1', '1306.4013-1-46-0', '1306.4013-1-46-1', '1306.4013-1-47-0', '1306.4013-1-47-1', '1306.4013-1-48-2', '1306.4013-1-48-10', '1306.4013-1-49-0', '1306.4013-1-49-1', '1306.4013-1-52-0', '1306.4013-1-55-0', '1306.4013-2-30-0', '1306.4013-2-30-1', '1306.4013-2-31-0', '1306.4013-2-31-1', '1306.4013-2-32-3', '1306.4013-2-33-1', '1306.4013-2-33-2', '1306.4013-2-38-0', '1306.4013-2-38-1', '1306.4013-2-41-0', '1306.4013-2-41-1', '1306.4013-2-47-0', '1306.4013-2-47-1', '1306.4013-2-48-0', '1306.4013-2-48-1', '1306.4013-2-49-2', '1306.4013-2-49-11', '1306.4013-2-50-0', '1306.4013-2-50-1', '1306.4013-2-53-0', '1306.4013-2-56-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1306.4013 | null | null | null | null | null |
1308.6112 | {'1308.6112-1-0-0': 'Large scale IRAM 30m CO-observations in the giant molecular cloud complex W43 IRAM 30m CO-observations in W43', '1308.6112-1-1-0': 'We aim to give a full description of the distribution and location of dense molecular clouds in the giant molecular cloud complex W43.', '1308.6112-1-1-1': 'It has previously been identified as one of the most massive star-forming regions in our Galaxy.', '1308.6112-1-1-2': 'To trace the moderately dense molecular clouds in the W43 region, we initiated an IRAM 30m large program, named W43-HERO, covering a large dynamic range of scales (from 0.3 to 140 pc).', '1308.6112-1-1-3': 'We obtained on-the-fly-maps in [MATH]CO (2-1) and C[MATH]O (2-1) with a high spectral resolution of 0.1 kms[MATH] and a spatial resolution of 12.', '1308.6112-1-1-4': 'These maps cover an area of [MATH]1.5 square degrees and include the two main clouds of W43, as well as the lower density gas surrounding them.', '1308.6112-1-1-5': 'A comparison with Galactic models and previous distance calculations confirms the location of W43 near the tangential point of the Scutum arm at a distance from the Sun of approximately 6 kpc.', '1308.6112-1-1-6': 'The resulting intensity cubes of the observed region are separated into sub-cubes, centered on single clouds which are then analyzed in detail.', '1308.6112-1-1-7': 'The optical depth, excitation temperature, and H[MATH] column density maps are derived out of the [MATH]CO and C[MATH]O data.', '1308.6112-1-1-8': 'These results are then compared with those derived from Herschel dust maps.', '1308.6112-1-1-9': 'The mass of a typical cloud is several [MATH] while the total mass in the dense molecular gas (> [MATH] cm[MATH]) in W43 is found to be [MATH].', '1308.6112-1-1-10': 'Probability distribution functions obtained from column density maps derived from molecular line data and Herschel imaging show a log-normal distribution for low column densities and a power-law tail for high densities.', '1308.6112-1-1-11': 'A flatter slope for the molecular line data PDF may imply that those selectively show the gravitationally collapsing gas.', '1308.6112-1-2-0': '# Introduction', '1308.6112-1-3-0': 'The formation of high-mass stars is still not fully understood, although they play an important role in the cycle of star-formation as well as in the balance of the interstellar medium.', '1308.6112-1-3-1': 'What we do know is that these stars form in giant molecular clouds .', '1308.6112-1-3-2': 'In order to understand high-mass star-formation it is crucial to understand these GMCs.', '1308.6112-1-3-3': 'One of the most important points to be studied is their formation process.', '1308.6112-1-4-0': 'The region around 30 Galactic longitude was identified as one of the most active star-forming regions in the Galaxy about 10 years ago .', '1308.6112-1-4-1': 'It is heated by a cluster of Wolf-Rayet and OB stars .', '1308.6112-1-4-2': 'Although the object W43 was previously known, [CITATION] were the first to consider it as a Galactic ministarburst region.', '1308.6112-1-5-0': 'Back then, the name W43 was used for the single cloud (G030.8+0.02), that is today known as W43-Main.', '1308.6112-1-5-1': '[CITATION] characterized the complex by analyzing VLA HI continuum emission and the [MATH]CO (1-0) and [MATH]CO (1-0) Galactic plane surveys.', '1308.6112-1-5-2': 'They concluded that W43-Main and G29.96-0.02 (now called W43-South) should be considered as one giant connected complex.', '1308.6112-1-6-0': 'From the position in the Galactic plane and its radial velocity, [CITATION] concluded that W43 is located at the junction point of the Galactic long bar and the Scutum spiral arm at 6 kpc relative to the Sun.', '1308.6112-1-6-1': 'The kinematic distance ambiguity, arising from the Galactic rotation curve, gives relative distances for W43 of [MATH]6 and [MATH]8.5 kpc for the near and the far kinematic distance, respectively.', '1308.6112-1-6-2': 'Although there have been other distances adopted by other authors , most publications favor the near distance.', '1308.6112-1-7-0': 'This position in the Galaxy makes W43 a very interesting object to study the formation of molecular clouds.', '1308.6112-1-7-1': 'Despite its distance, it is possible to analyze the details of this cloud due to its large spatial scale of [MATH]150 pc and the large amount of gas at high density .', '1308.6112-1-8-0': 'In this paper we present initial results of the large IRAM 30m project "W43 Hera/EmiR Observations" (W43 HERO, PIs P. Schilke and F. Motte).', '1308.6112-1-8-1': 'By observing the kinematic structure of this complex, the program aims to draw conclusions about the formation processes of both molecular clouds from atomic gas and high-mass stars from massive clouds and so-called ridges.', '1308.6112-1-9-0': 'One part of the project aimed at mapping the large scale mid density molecular gas ([MATH] cm[MATH]) in the complete W43 region in the [MATH]CO (2-1) and C[MATH]O (2-1) emission lines.', '1308.6112-1-9-1': 'The resulting dataset and a first analysis will be presented in this paper.', '1308.6112-1-9-2': 'The second part of the projects observed several high-density tracers in the densest parts of W43.', '1308.6112-1-9-3': 'This data and its analysis will be published in a separate article (Nguyen Luong et al. in prep).', '1308.6112-1-10-0': 'This paper is structured as follows.', '1308.6112-1-10-1': 'We first give an overview of the observations and the technical details in Sect. [REF].', '1308.6112-1-10-2': 'In Sect. [REF] we present the resulting line integrated and PV-maps and a list of clouds that were separated using the Duchamp Sourcefinder software .', '1308.6112-1-10-3': 'We also show the velocity structure of the complex and determine its position in the Milky Way in Sect. [REF].', '1308.6112-1-10-4': 'Section [REF] describes the calculations that we conducted, including optical depth, excitation temperature and column density of the gas.', '1308.6112-1-10-5': 'We then systematically compare our data to other datasets in Sect. [REF] to further characterize the sources we identified.', '1308.6112-1-10-6': 'In Sect. [REF] we give a more detailed description of the main clumps.', '1308.6112-1-10-7': 'A summary and conclusions are given in Sect. [REF]', '1308.6112-1-11-0': '# Observations', '1308.6112-1-12-0': 'The following data has been observed with the IRAM 30m telescope on Pico Veleta, Spain, between November 2009 and March 2011.', '1308.6112-1-12-1': 'We simultaneously observed the molecular emission lines [MATH]CO (2-1) and C[MATH]O (2-1) at 220.398684 GHz and 219.560358 GHz respectively.', '1308.6112-1-12-2': 'Smaller regions around the two main cloud complexes were additionally observed in high-density tracers such as HCO[MATH] (3-2), H[MATH]CO[MATH] (2-1), N[MATH]H[MATH] (1-0) and C[MATH]S (2-1) (Nguyen Luong et al. in prep.)', '1308.6112-1-13-0': 'This survey spans the whole W43 region, including the two main clouds W43-Main and W43-South and several smaller clouds in their vicinity.', '1308.6112-1-13-1': 'It covers a rectangular map with a size of [MATH]1.4[MATH]1.0 degrees.', '1308.6112-1-13-2': 'This translates to spatial dimensions of [MATH]105 pc, given an estimated distance to the source of about 6 kpc (see Sect. [REF]).', '1308.6112-1-13-3': 'The center of the map lies at 18:46:54.4 -02:14:11 (EQ J2000).', '1308.6112-1-13-4': 'The beam size of the [MATH]CO and C[MATH]O observations is 11[MATH] which corresponds to 0.34 pc at this distance.', '1308.6112-1-14-0': 'For the observations we used HERA, the heterodyne receiver array of the IRAM 30m .', '1308.6112-1-14-1': 'It consists of 3[MATH]3 pixels, separated by 24[MATH] and has two polarizations, pointing at the same location on the sky.', '1308.6112-1-14-2': 'This gave us the possibility to observe both CO isotopologes in one pass, we observed one line per polarization.', '1308.6112-1-14-3': 'HERA can be tuned in the range from 215 to 272 GHz and has a receiver noise temperature of about 100 K at 220 GHz.', '1308.6112-1-14-4': 'Typically, the system temperature of the telescope was in the range of 300 to 400 K during our observations.', '1308.6112-1-15-0': 'We used the VESPA (versatile spectrometer assembly) autocorrelator as a backend which was set to a spectral resolution of 80 kHz per channel with a bandwidth of 80 MHz.', '1308.6112-1-15-1': 'This translates to a resolution of 0.15 kms[MATH] and a bandwidth of [MATH]100 kms[MATH] which was set to cover the velocity range of 30 kms[MATH] to 130 kms[MATH] to cover the complete W43 complex.', '1308.6112-1-16-0': 'A Nyquist sampled on-the-fly mapping mode was used to cover 10[MATH] tiles, taking about 20 minutes each.', '1308.6112-1-16-1': 'Each tile was observed in two orthogonal scanning directions to reduce striping in the results.', '1308.6112-1-16-2': 'The tiles uniformly cover the whole region.', '1308.6112-1-16-3': 'A total of [MATH]3 million spectra in both CO lines was received that way, taking a total observation time of nearly 80 hours.', '1308.6112-1-17-0': 'Calibration scans, pointing and focus were done on a regular basis to assure a correct calibration later.', '1308.6112-1-17-1': 'Calibration scans were done every 10 minutes, a pointing every 60 to 90 minutes.', '1308.6112-1-17-2': 'A focus scan was done every few hours, more often around sunset and sunrise as the atmosphere is less stable then.', '1308.6112-1-17-3': 'For the pointing we used G34.3, a strong nearby ultracompact Hii region.', '1308.6112-1-17-4': 'The calibration was conducted with the MIRA package which is part of the GILDAS software.', '1308.6112-1-17-5': 'We expect the flux calibration to be accurate within error limits of [MATH].', '1308.6112-1-18-0': '## Data reduction', '1308.6112-1-19-0': 'The raw data were processed using the GILDAS[MATH] software package.', '1308.6112-1-19-1': 'The steps taken for data reduction included flagging of bad data (e.g. too high noise level or platforms that could not be removed), platform removal in the spectra, baseline subtraction, and gridding, to create three dimensional data cubes.', '1308.6112-1-19-2': 'About 10 percent of the data had to be flagged due to excessive platforming or strong noise.', '1308.6112-1-19-3': 'Platforming, i.e. an intensity jump in the spectra, is sometimes produced by the VESPA backend and occurs only in specific pixels of the array corresponding to fixed frequencies.', '1308.6112-1-19-4': 'We calculated the intensity offsets by taking several baselines on each side of the jump to remove these effects on the spectra.', '1308.6112-1-20-0': 'Baseline subtraction turned out to be complicated in some regions that are crowded with emission over a large part of the band.', '1308.6112-1-20-1': 'A first order baseline fit was usually adequate, but for a small number of pixels and scans a second order baseline was needed.', '1308.6112-1-20-2': 'We then corrected for the main beam efficiency via [MATH], where [MATH] is the forward efficiency of the IRAM 30m telescope and [MATH] is the main beam efficiency at 210 GHz (no efficiency measurements have been carried out for the IRAM 30m at 220 GHz, but the values should not deviate much from those at 210 GHz).', '1308.6112-1-21-0': 'Finally, the single spectra were gridded to two data cubes, one for each line.', '1308.6112-1-21-1': 'The pixels are separated by half beam steps, 5.9[MATH] in spatial dimension and have channel widths of 0.15 kms[MATH].', '1308.6112-1-21-2': 'This step includes the convolution with a Gaussian of the size of the beam width.', '1308.6112-1-21-3': 'The final cubes have dimensions of 631[MATH]917 data points (RA-DEC-velocity).', '1308.6112-1-22-0': 'The noise of single spectra varied with the weather and also with the pixel of the HERA-array.', '1308.6112-1-22-1': 'Maps, that show the noise level for each spatial point for both lines, are shown in Fig. [REF] in the Appendix.', '1308.6112-1-22-2': 'Typical values are [MATH]1 K, corrected for the main beam efficiency, while several parts of the southern map, observed during worse weather conditions, have rms values of up to 3 K.', '1308.6112-1-22-3': 'In general, the noise level of the C[MATH]O is higher than that of the [MATH]CO line.', '1308.6112-1-22-4': 'Despite our dedicated reduction process, scanning effects are still visible in the resulting maps.', '1308.6112-1-22-5': 'They appear as stripes (see upper part of [MATH]CO map in Fig. [REF]) and tiling patterns (see noise difference of diffuse parts of C[MATH]O in Fig. [REF]).', '1308.6112-1-23-0': '# Results', '1308.6112-1-24-0': 'Integrated intensity maps of the whole W43 region in both [MATH]CO (2-1) and C[MATH]O (2-1) lines are shown in Fig. [REF].', '1308.6112-1-24-1': 'The maps use the whole velocity range from 30 kms[MATH] to 130 kms[MATH] and show a variety of clouds and filaments.', '1308.6112-1-24-2': 'The two main cloud complexes W43-Main in the upper left part of the maps and W43-South in the lower right part are clearly visible.', '1308.6112-1-25-0': 'In Fig. [REF] we show several spectra taken from the data.', '1308.6112-1-25-1': 'The upper plot shows the spectra of [MATH]CO (2-1) and C[MATH]O (2-1) averaged over the complete complex, the center and bottom plots show averaged spectra of the W43-Main and W43-South clouds.', '1308.6112-1-25-2': 'The spectra of the complete cubes show emission nearly across the whole velocity range in [MATH]CO.', '1308.6112-1-25-3': 'Only the components at [MATH] kms[MATH] and velocities higher than 120 kms[MATH] do not show any emission.', '1308.6112-1-25-4': 'The C[MATH]O follows that distribution, although it is not as broad.', '1308.6112-1-25-5': 'We thus can already distinguish two separated velocity components, one between 35 and 55 kms[MATH].', '1308.6112-1-25-6': 'Most of the emission is concentrated in the velocity range between 65 and 120 kms[MATH].', '1308.6112-1-25-7': 'To give an impression of the complexity of some sources we plot several spectra of the W43-Main cloud in Fig. [REF].', '1308.6112-1-26-0': '## Decomposition into sub-cubes', '1308.6112-1-27-0': 'The multitude of sources found in the W43 region complicates the analysis of the complete data cube.', '1308.6112-1-27-1': 'Details get lost when integrating over a too large range in frequency.', '1308.6112-1-27-2': 'We want to examine each source separately so we need to decompose the data cube into sub-cubes only containing one single source each.', '1308.6112-1-27-3': 'This is only done on the [MATH]CO cube, as this is the stronger molecular line.', '1308.6112-1-27-4': 'This breakdown is then copied to the C[MATH]O cube.', '1308.6112-1-28-0': 'We use the Duchamp Sourcefinder software package to automatically find a decomposition.', '1308.6112-1-28-1': 'See [CITATION] for a detailed description of this software.', '1308.6112-1-28-2': 'It finds connected structures in three-dimensional ppv-data cubes by searching for emission that lies above a certain threshold.', '1308.6112-1-28-3': 'The value of this threshold is crucial for the success of the process and needs to be carefully adjusted by hand.', '1308.6112-1-28-4': 'For the decomposition of the [MATH]CO cube we use two different cutoffs.', '1308.6112-1-28-5': 'The lower cutoff of [MATH] per channel is used to identify weaker sources.', '1308.6112-1-28-6': 'A higher cutoff of [MATH] per channel was needed to distinguish sources in the central part of the complex.', '1308.6112-1-29-0': 'We identify a total of 29 clouds (see Table [REF]), 20 in the W43 complex itself and 9 in the fore-/background (see Sect. [REF] for details).', '1308.6112-1-29-1': 'The outcome of this method is not trivial as it is not always clear which parts are still to be considered associated and which are separate structures.', '1308.6112-1-29-2': 'It still needs some correction by hand in some of the very weak sources and the strong complexes.', '1308.6112-1-29-3': 'A few weak sources that have been identified by eye are manually added to our list (e.g. sources 18 and 28).', '1308.6112-1-29-4': 'These are clearly coherent separate structures but are not identified by the algorithm.', '1308.6112-1-29-5': 'On the other hand, a few sources are merged by hand (e.g. source 26) that clearly belong together but are divided into several sub-sources by the software.', '1308.6112-1-29-6': 'Some of these changes are open to interpretation but they show that some adjustment of the software result is needed.', '1308.6112-1-29-7': 'However, the algorithm works quite well and identifies 25 out of 29 clouds on its own.', '1308.6112-1-30-0': 'The resulting detection map is shown in Fig. [REF].', '1308.6112-1-30-1': 'The numbers shown there are color-coded to show the distance of each detected cloud (see Sect. [REF] for details).', '1308.6112-1-30-2': 'Sources are sorted by their peak velocities.', '1308.6112-1-30-3': 'See Table [REF] for positions and dimensions of the clouds, Table [REF] for derived properties, and Sect. [REF] for a detailed description of the main complexes, while plots of all clouds can be found in Appendix [REF].', '1308.6112-1-30-4': 'We see a number of different sizes and shapes from small spherical clouds to expanded filaments and more complex structures.', '1308.6112-1-31-0': 'The resulting data cubes show clouds of different shapes, while the typical spatial scales lie in the range of 10 to 20 pc (see Table [REF]).', '1308.6112-1-31-1': 'We also give the area that the [MATH]CO emission of each source covers which was determined by defining a polygon for each source that contains the [MATH]CO emission.', '1308.6112-1-31-2': 'Thus it accounts for shapes that deviate from spheres or rectangles, which is true for most of our clouds.', '1308.6112-1-31-3': 'Hereafter we will call structures with a size on the order of 10 pc clouds, while on the parsec scale we will call them clumps.', '1308.6112-1-31-4': 'The whole W43 region is considered a cloud complex.', '1308.6112-1-32-0': '## PV-diagram of the region', '1308.6112-1-33-0': 'For a more advanced analysis of the velocity structure we create a position velocity diagram of the [MATH]CO (2-1) line that is averaged along the galactic latitude (see Fig. [REF] (a)).', '1308.6112-1-33-1': 'The distribution of emission across the velocity range that is seen in the averaged spectra in Fig. [REF] can also be identified here, but with additional spatial information along the Galactic longitude.', '1308.6112-1-33-2': 'Note the similarity to the plot of [MATH]CO (1-0) displayed in [CITATION].', '1308.6112-1-33-3': 'However, in our plot we see more details due to the higher angular resolution of our data.', '1308.6112-1-34-0': 'We further analyze the position velocity diagram, shown in Fig. [REF] (a), to separate our cube into several velocity components.', '1308.6112-1-34-1': 'We assume, that these different components are also spatially separated.', '1308.6112-1-35-0': 'Two main velocity complexes can be distinguished: One between 35 and 55 kms[MATH], the other between 65 and 120 kms[MATH].', '1308.6112-1-35-1': 'Both complexes are clearly separated from each other, indicating that they are situated at different positions in the Galaxy.', '1308.6112-1-35-2': 'On second sight, it becomes clear that the complex from 65 to 120 kms[MATH] breaks further down into a narrow component, spanning the range from 65 to 78 kms[MATH], and a broad component between 78 and 120 kms[MATH].', '1308.6112-1-35-3': 'Analyzing the channel maps of the cube verifies that these structures are indeed separated.', '1308.6112-1-35-4': 'See Fig. [REF] for integrated maps of each velocity component.', '1308.6112-1-36-0': 'On the other hand it is clearly visible that all three velocity components span the complete spatial dimension along the Galactic longitude.', '1308.6112-1-36-1': 'The broadest complex at 78-120 kms[MATH] shows two major components at 29.9[MATH] and 30.8[MATH] that coincide with W43-South and W43-Main respectively.', '1308.6112-1-36-2': 'The gap between both complexes is bridged by a smaller clump and all three clumps are surrounded by diffuse gas which forms an envelope around the whole complex.', '1308.6112-1-36-3': 'It is thus suggested to consider W43-Main and W43-South as one giant connected molecular cloud complex.', '1308.6112-1-36-4': 'This connection becomes more clear in the PV-plot than in the spatial map in Fig. [REF] although the blurring from the averaging might merge structures.', '1308.6112-1-37-0': 'The lower velocity complex between 35 and 55 kms[MATH] is a bit more fragmented than the other components.', '1308.6112-1-37-1': 'There is one central object at 30.6[MATH] spanning the whole velocity range.', '1308.6112-1-37-2': 'But to the edges of the map it splits up into two sub-components.', '1308.6112-1-37-3': 'It is hard to tell if we actually see one or two components.', '1308.6112-1-38-0': '## Determination of the distance of W43', '1308.6112-1-39-0': 'We can analyze our data by using a simple rotational model of the Milky Way.', '1308.6112-1-39-1': 'For this model we assume a rotational curve that increases linearly in the inner 3 kpc of the Galaxy, where the bar is situated.', '1308.6112-1-39-2': 'For radii larger than that, we assume a rotation curve that has a value of 254 kms[MATH] at [MATH] and slightly rises with radius at a rate of 2.3 kms[MATH]kpc[MATH] .', '1308.6112-1-39-3': 'The Galactocentric radius of a cloud with a certain relative velocity can be calculated using the formula [EQUATION] as for example used in [CITATION].', '1308.6112-1-39-4': '[MATH] is the galactocentric radius of the Sun, which is assumed to be 8.4 kpc , [MATH] is the Galactic longitude of the source (30[MATH]), [MATH] the radial velocity of the Sun (254 kms[MATH]) and [MATH] the radial velocity of the source.', '1308.6112-1-39-5': '[MATH] is the measured relative velocity between the source and the Sun.', '1308.6112-1-39-6': 'With the knowledge of the radius [MATH] we can then compute the relative distance to the source by the equation: [EQUATION]', '1308.6112-1-39-7': 'So up to the tangent point, for each measured velocity two different possible distances exist, one on the near and one on the far side of the tangent point.', '1308.6112-1-39-8': 'However, this calculation is not entirely accurate as the assumptions of the geometry of the Galaxy bear large errors.', '1308.6112-1-39-9': '[CITATION] state that the errors of the kinematic distance can sum up to a factor as high as 2.', '1308.6112-1-39-10': 'One main reason for uncertainties are streaming motions of molecular clouds relative to the motion of the spiral arms .', '1308.6112-1-39-11': 'Fig. [REF] (b) shows the kinematic distance curve for our case of 30[MATH] Galactic longitude.', '1308.6112-1-39-12': 'This works only for the circular orbits in the spiral arms, not the elliptical orbits in the Galactic bar .', '1308.6112-1-40-0': 'In order to determine the location of each velocity complex we need to break the kinematic distance ambiguity, i.e. for each complex we need to decide if we assume it to be on the near or the far side of the tangent point.', '1308.6112-1-40-1': 'Here, we use the distance estimations by [CITATION], who utilized HI self absorption from the VGPS project .', '1308.6112-1-40-2': 'It is possible to associate several entries of their extensive catalog with clouds we found in our dataset.', '1308.6112-1-40-3': 'Thus we are able to remove the distance ambiguity and attribute distances to these clouds.', '1308.6112-1-40-4': 'In combination with the detailed model of the Milky Way of [CITATION] we are able to fix their position in our Galaxy.', '1308.6112-1-40-5': 'We cannot assign a distance to each single cloud in our dataset as not each was analyzed by [CITATION].', '1308.6112-1-40-6': 'We assume the missing sources to have the same distances as sources nearby.', '1308.6112-1-40-7': 'This may not be exact in all cases but the only unclear assignments are two clouds in the 35 to 55 kms[MATH] velocity component (sources 1 and 3).', '1308.6112-1-40-8': 'The distance of the W43 complex is unambiguous as it is well-determined by the calculations found in [CITATION].', '1308.6112-1-41-0': 'W43 itself (78 to 120 kms[MATH]) is found to lie on the near side of the tangential point, with distances from 5 to 7.3 kpc, increasing with radial velocity.', '1308.6112-1-41-1': 'This places it near the tangential point of the Scutum arm at a Galactocentric radius [MATH] kpc (marker 1 in Fig. [REF] (c)).', '1308.6112-1-41-2': 'For our further analysis we use an average distance of 6 kpc for the whole complex as this is where the mass center is located.', '1308.6112-1-42-0': 'The second velocity component (65 to 78 kms[MATH]) lies in the foreground of the first one, at 4.5 kpc distance to the Sun and [MATH] kpc (marker 2 in Fig. [REF] (c)).', '1308.6112-1-42-1': 'Another indication that these sources are located on the near side of the tangential point is their position above the Galactic plane as seen in Fig. [REF] (b).', '1308.6112-1-42-2': 'The larger the distance from the Sun, the further above the Galactic plane it would be positioned.', '1308.6112-1-42-3': 'This would be difficult to explain since high-mass star-forming regions are typically located within the plane.', '1308.6112-1-42-4': 'It is unclear if this cloud is still situated in the Scutum arm or if it is rather located between spiral arms.', '1308.6112-1-42-5': 'According to the model, it would be placed at the edge of the Scutum arm.', '1308.6112-1-42-6': 'In view of the previously discussed uncertainties it is still possible that this cloud is part of the spiral arm.', '1308.6112-1-43-0': 'The third component between 35 and 55 kms[MATH] is more complex than the other ones, as we find sources to be located on both the near and far side of the tangential point.', '1308.6112-1-43-1': 'The brightest sources in the center of our map are in the background of W43 in the Perseus arm with a distance of 11 to 12 kpc to the Sun and of [MATH] kpc to the (marker 3 in Fig. [REF] (c)).', '1308.6112-1-43-2': 'But several other sources in the North and South are found by [CITATION] to be on the near side at a distance to the Sun of 3.5 to 4 kpc (marker 3[MATH] in Fig. [REF] (c)).', '1308.6112-1-43-3': 'These sources also have a galactocentric radius of 6 kpc.', '1308.6112-1-43-4': 'Table [REF] gives an overview of the distance of each source while Fig. [REF] shows integrated intensity plots of the individual velocity components.', '1308.6112-1-44-0': 'We can now apply this calculation to our data in Fig. [REF] (a) by changing the velocity scale into a distance scale.', '1308.6112-1-44-1': 'The distance scale is inaccurate for those parts of the lowest component that lie on the far side of the tangent point.', '1308.6112-1-44-2': 'But as it is not possible to disentangle near and far clouds in this plot we still show this scale.', '1308.6112-1-44-3': 'These values, taken from the rotation curve, are smaller than the actual distances found by comparing with [CITATION] as we used the newer rotation curve of [CITATION].', '1308.6112-1-44-4': '[CITATION] use the older values from [CITATION], which explains the discrepancy.', '1308.6112-1-44-5': 'But this axis still give an idea of the distribution of the clouds.', '1308.6112-1-44-6': 'Note that for velocities larger than 112 kms[MATH] no distance can be assigned, hence the zero for the 120 kms[MATH] tick in Fig. [REF] (a).', '1308.6112-1-44-7': 'Subplot (d) shows the related modeled PV-diagram from [CITATION].', '1308.6112-1-44-8': 'Our dataset is indicated by the gray box.', '1308.6112-1-45-0': 'Fig. [REF] (c) summarizes our distance determination in a plot taken from [CITATION].', '1308.6112-1-45-1': 'The W43 complex (78-120 kms[MATH], marker 1) lies between 5 and 7 kpc, where the distance increases with velocity, as we found this part to be located on the near branch.', '1308.6112-1-45-2': 'The complex 65 to 78 kms[MATH] (marker 2) is located at the near edge of the Scutum arm while the 35 to 55 kms[MATH] component is marked by 3 and 3[MATH] on both sides of the tangent point.', '1308.6112-1-45-3': 'The far part is located in the Perseus arm at 12 kpc distance, the near part at a distance of 4.5 kpc between the Scutum and the Sagittarius arm.', '1308.6112-1-46-0': 'It may be a bit surprising that no emission from the local part of the Sagittarius arm is seen in our dataset.', '1308.6112-1-46-1': 'The reason is our observed velocity range which only goes down to 30 kms[MATH].', '1308.6112-1-46-2': 'Possible near molecular clouds would have even lower relative velocities of [MATH]20 kms[MATH], which can be seen in the model in Fig. [REF] (d).', '1308.6112-1-46-3': 'In [CITATION] the [MATH]CO (1-0) spectrum, averaged over the W43 complex, shows an additional velocity component at 5 to 15 kms[MATH] which fits to this spiral arm.', '1308.6112-1-47-0': '## Peak velocity and line width', '1308.6112-1-48-0': 'After separating the different velocity components we created moment maps of each component.', '1308.6112-1-48-1': 'For each spectra of the [MATH]CO data cube a Gaussian line profile was fitted.', '1308.6112-1-48-2': 'The first moment resembles the peak velocity, the position of the line peak.', '1308.6112-1-48-3': 'The second moment is the width of the line.', '1308.6112-1-48-4': 'The maps of the W43 complex are shown in Fig. [REF], while the plots of the background components can be seen in the Appendix in Fig. [REF].', '1308.6112-1-48-5': 'Care should be taken in interpretation of these maps.', '1308.6112-1-48-6': 'As some parts of the maps show complex spectra (see Fig. [REF] for some examples) a Gaussian profile is not always a good approximation.', '1308.6112-1-48-7': 'In regions where several velocity components are found, the maps will only give information about the strongest component.', '1308.6112-1-48-8': 'In case of self-absorbed lines the maps may even be misleading.', '1308.6112-1-48-9': 'This especially concerns the southern ridge of W43-Main, called W43-MM2 as defined in .', '1308.6112-1-49-0': 'The line peak velocity map in Fig. [REF] (a) traces a variety of coherent structures.', '1308.6112-1-49-1': 'Most of these correspond to the sources we identified with the Duchamp software.', '1308.6112-1-49-2': 'But as mentioned above, some structures overlap and cannot be defined simply from using this velocity map.', '1308.6112-1-50-0': 'The two main clouds W43-Main and W43-South are again located in the upper left and lower right part of map, respectively.', '1308.6112-1-50-1': 'As in the PV-diagram in Fig. [REF] (a) we note that both clouds are slightly shifted in velocity.', '1308.6112-1-50-2': 'While W43-Main lies in the range of 85 to 100 kms[MATH] W43-South spans velocities from 95 to 105 kms[MATH].', '1308.6112-1-50-3': 'Several smaller sources bridge the gap between the two clouds especially in the higher velocities.', '1308.6112-1-50-4': 'This structure is is also seen in the PV-diagram.', '1308.6112-1-51-0': 'In comparison to the PV-diagram this plot shows the peak velocity distribution in both spatial dimensions.', '1308.6112-1-51-1': 'On the other hand, we lose information of the shape of the lines.', '1308.6112-1-51-2': 'Here we see that the velocity of W43-South is rather homogeneous across the whole cloud.', '1308.6112-1-51-3': 'In contrast, W43-Main shows strong velocity gradients from West to East and from South to North, already seen in [CITATION].', '1308.6112-1-51-4': 'The velocity changes by at least 30 kms[MATH] on a scale of 25 pc.', '1308.6112-1-51-5': 'We can interpret this as mass flows across the cloud, which makes it kinematically much more active than W43-South.', '1308.6112-1-52-0': 'Fig. [REF] (b) shows a map of the FWHM line width of each pixel.', '1308.6112-1-52-1': 'Some parts in W43-Main show unrealistically large values of more than 10 kms[MATH].', '1308.6112-1-52-2': 'This is a line-of-sight effect and originates from several velocity components located at the same point on the sky.', '1308.6112-1-52-3': 'Therefore, it is more accurate to analyze the line width of each source separately.', '1308.6112-1-52-4': 'From these single sources we determine the mean line width, that is given in Table [REF] (7).', '1308.6112-1-53-0': '# Analysis', '1308.6112-1-54-0': '## Calculations', '1308.6112-1-55-0': '### Optical depth', '1308.6112-1-56-0': 'For each identified source we conducted a series of calculations to determine its physical properties.', '1308.6112-1-56-1': 'We did this on a pixel by pixel basis, using maps integrated over the velocity range that is covered by the specific source.', '1308.6112-1-56-2': 'The optical depth of the [MATH]CO gas was calculated from the ratio of the intensities of [MATH]CO (2-1) and C[MATH]O (2-1), assuming that C[MATH]O is optically thin.', '1308.6112-1-56-3': '(This assumption holds for H[MATH] column densities up to [MATH] cm[MATH], but a clear threshold cannot be given.)', '1308.6112-1-56-4': 'Then we computed the excitation temperature of this gas as well as the H[MATH] column density which was then used to estimate the total mass along the line-of-sight.', '1308.6112-1-56-5': 'All these calculations are explained in detail in Appendix [REF].', '1308.6112-1-56-6': 'Example maps for a small filament (source 29) can be seen in Fig. [REF].', '1308.6112-1-57-0': 'We first calculated the [MATH]CO (2-1) optical depth from the ratio of the two CO lines.', '1308.6112-1-57-1': 'Note, that the intrinsic ratios of the different CO isotopologes, used for this calculation, are dependent on the Galactocentric radius.', '1308.6112-1-57-2': 'So we have to use different values for W43 and the fore-/background sources.', '1308.6112-1-57-3': 'An example map of [MATH] of [MATH]CO (2-1) is shown in Fig. [REF] (c).', '1308.6112-1-57-4': 'Typical clouds have optical depths of a fraction of 1 in the outer parts and up to 4 at most in the central cores.', '1308.6112-1-57-5': 'The extreme case is the W43-main cloud, where the [MATH]CO optical depth goes up to 8.', '1308.6112-1-57-6': 'This means that most parts of the clouds are optically thin and we can see through them.', '1308.6112-1-57-7': 'Even at positions where [MATH]CO become optically thick, C[MATH]O still remains optically thin.', '1308.6112-1-57-8': 'Only for the extreme case of the densest part of W43-Main, C[MATH]O starts to become optically thick.', '1308.6112-1-57-9': 'This means that the combination of the two isotopologes reveals most of the information about the medium density CO gas in the W43 complex.', '1308.6112-1-58-0': '### Excitation temperature', '1308.6112-1-59-0': 'The formula used for the computation of the excitation temperatures is explained in Appendix [REF].', '1308.6112-1-59-1': 'The resulting map is shown in Fig. [REF].', '1308.6112-1-59-2': 'There are certain assumptions made.', '1308.6112-1-59-3': 'First, we assumed that [MATH] is the same for the [MATH]CO and the C[MATH]O gas.', '1308.6112-1-59-4': 'This method becomes unrealistic when the temperature distribution along the line-of-sight is not uniform any more.', '1308.6112-1-59-5': 'If there was a temperature gradient we would miss the real ratio of the [MATH]CO to C[MATH]O line intensities and thus either over- or underestimate the temperature.', '1308.6112-1-59-6': 'So the calculated temperature might be incorrect for very large cloud structures that show a complex temperature distribution along the line-of-sight.', '1308.6112-1-59-7': 'This problem is partially circumvented by using the spectral information of our observations.', '1308.6112-1-59-8': 'But for our calculation of the excitation temperature we use intensity maps integrated over at least several kms[MATH], which still leaves room for uncertainties.', '1308.6112-1-59-9': 'This means we do not confuse different clouds but we still average the temperature along the line-of-sight over the complete clouds.', '1308.6112-1-60-0': 'The centers of the two main clouds in the W43 region are candidates for an underestimated excitation temperature.', '1308.6112-1-60-1': 'In case these regions were internally heated a decreasing gradient in the excitation temperature would appear from the inside of the cloud to the outside.', '1308.6112-1-60-2': 'As [MATH]CO is rather optically thick, only the cooler outside of the cloud would be seen by the observer.', '1308.6112-1-60-3': 'In contrast, C[MATH]O would be optically thin, thus also the hot center of the cloud would be observed.', '1308.6112-1-60-4': 'Averaging along the line-of-sight, [MATH](C[MATH]O) would be increased relative to [MATH]CO) which would lead to an overestimated optical depth.', '1308.6112-1-60-5': 'This would then lead to an underestimated calculated excitation temperature.', '1308.6112-1-60-6': 'External heating, on the other hand, would result in an overestimated excitation temperature.', '1308.6112-1-60-7': 'However, regarding the low excitation temperatures we find in some clouds, the first case is more likely.', '1308.6112-1-61-0': 'Another effect, which leads to a reduced excitation temperature, is the beam-filling factor [MATH].', '1308.6112-1-61-1': 'In our calculations we assume it to be 1.', '1308.6112-1-61-2': 'This corresponds to extended clouds that completely fill the telescope beam.', '1308.6112-1-61-3': 'This is not a true representation of molecular clouds, as they are structured on the sub-parsec scale and we would have to use a factor [MATH].', '1308.6112-1-61-4': 'Technically speaking, we calculate the value [MATH], which is smaller than [MATH].', '1308.6112-1-62-0': 'As the C[MATH]O line is much weaker than the [MATH]CO line we cannot use the ratio of them for those pixels where no C[MATH]O is detected, even if [MATH]CO is present.', '1308.6112-1-62-1': 'We find typical temperatures to be between 6 and 25 K, in some rare cases up to 50 K, with a median of 12 K.', '1308.6112-1-63-0': 'Due to the sparsely covered maps (see Fig. [REF]) and the uncertainties described above, we concluded that is was best not to use the excitation temperature maps for the following calculations of the H[MATH] column density.', '1308.6112-1-63-1': 'Instead, we assumed a constant excitation temperature for the complete W43 region.', '1308.6112-1-63-2': 'We chose the value to be 12 K, as this was the median temperature found in the W43 complex.', '1308.6112-1-63-3': 'Assuming a constant temperature value across the cloud is likely not a true representation of the cloud, in particular it does not distinguish between star forming cores and the ambient background.', '1308.6112-1-63-4': 'However, such an assumption is a good first approximation to the temperature in the cloud and is more representative of star forming cores than the aforementioned unrealistically low values.', '1308.6112-1-64-0': '### H[MATH] column density', '1308.6112-1-65-0': 'We also calculated the column density along the line-of-sight of the [MATH]CO gas from the assumed constant excitation temperature, the [MATH]CO integrated emission, and a correction for the opacity (see Appendix [REF] for details).', '1308.6112-1-65-1': 'Assuming a constant ratio between [MATH]CO and H[MATH] it was then possible to find the H[MATH] column density.', '1308.6112-1-65-2': 'Please note, that H[MATH] column densities derived assuming a constant temperature are also subject to the same caveats and accuracies.', '1308.6112-1-66-0': 'All ratios between H[MATH] and CO isotopologes bear errors, as they depend on the Galactocentric radius.', '1308.6112-1-66-1': 'These errors add up with the uncertainty on the assumed excitation temperature.', '1308.6112-1-66-2': 'The final results for column densities and masses must be taken with caution, as there is at least an uncertainty of a factor of 2.', '1308.6112-1-66-3': 'Fig. [REF] shows the calculated H[MATH] column density map of the full W43 complex.', '1308.6112-1-66-4': 'It has been calculated at those points, where the [MATH]CO integrated intensity is higher than 5 Kkms[MATH].', '1308.6112-1-66-5': 'The resulting values range from a few times [MATH] cm[MATH] in the diffuse surrounding gas up to nearly [MATH] cm[MATH] in the center of W43-Main.', '1308.6112-1-67-0': 'The southern ridge of W43-Main, where we calculate high column densities, is the most problematic part of our dataset.', '1308.6112-1-67-1': 'The spectra reveal that [MATH]CO is self-absorbed in this part of the cloud.', '1308.6112-1-67-2': 'We use the integrated intensity ratio to calculate the opacity at each point, which in this case is strongly overestimated.', '1308.6112-1-67-3': 'This leads to both low excitation temperatures and high column densities.', '1308.6112-1-67-4': 'The results for this part of the cloud should be used with caution.', '1308.6112-1-68-0': '### Total mass', '1308.6112-1-69-0': 'From the H[MATH] column density we then determine the total mass of our sources, given in Table [REF].', '1308.6112-1-69-1': 'We find, that the total mass of a typical cloud is in the range of a few [MATH] solar masses.', '1308.6112-1-69-2': 'Of course, this is only the mass seen in the mid- to high-density sources.', '1308.6112-1-69-3': 'The very extended diffuse molecular gas cannot be seen with [MATH]CO, it is generally traced by [MATH]CO lines and accounts for a major fraction of the gas mass .', '1308.6112-1-70-0': 'The total H[MATH] mass as derived from our [MATH]CO (2-1) and C[MATH]O (2-1) observations is found to be [EQUATION] for the W43 complex with about 50% within the clouds we have identified and the rest in the diffuse surrounding gas.', '1308.6112-1-70-1': 'Here we have excluded the foreground sources and only considered the W43 complex itself.', '1308.6112-1-70-2': '[CITATION] used similar areas and velocity ranges ([MATH]190 pc, 80-120 kms[MATH]) and determined a molecular gas mass in W43 clouds from the Galactic Ring Survey of [MATH].', '1308.6112-1-70-3': 'A different estimation of the H[MATH] column density in W43 was done by [CITATION] using Herschel dust emission maps.', '1308.6112-1-70-4': 'Using these maps we find a value of [MATH].', '1308.6112-1-70-5': 'See Sect. [REF] for a discussion of this difference.', '1308.6112-1-71-0': 'We underestimate the real mass where [MATH]CO exists but no [MATH]CO is seen, where C[MATH]O might become optically thick, and where our assumption for the excitation temperature is too high.', '1308.6112-1-71-1': 'On the other hand, we overestimate the real mass where our assumption for the excitation temperature is too low.', '1308.6112-1-71-2': 'In extreme cases of very hot gas the gas mass can be overestimated by 40% at most (cp.', '1308.6112-1-71-3': 'Fig [REF] in Appendix [REF]), while very cold cores can be underestimated by a factor of nearly 10.', '1308.6112-1-71-4': 'Both effects partly cancel out each other, when integrating over the whole region, however, we estimate that the effects which underestimate the real mass are stronger.', '1308.6112-1-71-5': 'Therefore, the mass we calculated should be seen as a lower limit of the real molecular gas mass in the W43 complex.', '1308.6112-1-72-0': '## Shear parameter', '1308.6112-1-73-0': 'Investigating the motion of gas streams in the Galaxy is important to explain how large molecular clouds like W43 can be accumulated.', '1308.6112-1-73-1': 'While Motte et al. in prep.', '1308.6112-1-73-2': 'will investigate streams of [MATH]CO and HI gas in W43 in detail, here we will only consider the aspect of radial shear in this field.', '1308.6112-1-73-3': 'The shear that is created by the differential rotation of the Galaxy at different Galactocentric radii can prevent the formation of dense clouds if it is too strong.', '1308.6112-1-73-4': 'It is possible to calculate a shear parameter [MATH] as described in [CITATION] by taking the Galactocentric radius of a region, its spatial and velocity extent, and its mass into account.', '1308.6112-1-73-5': 'For values of [MATH] higher than 1 the shear is so strong that clouds will get ripped apart, while for values below 1 they are able to form.', '1308.6112-1-74-0': 'The values we use to calculate [MATH] are the total mass from above of [MATH], a velocity extent of 40 kms[MATH], a Galactocentric radius of 4 kpc, and an area of [MATH]pc[MATH].', '1308.6112-1-74-1': 'This is the area that is covered by emission and is smaller than the total size of our map.', '1308.6112-1-74-2': 'These values yield a shear parameter [MATH].', '1308.6112-1-74-3': 'Accordingly, shear forces are not strong enough to disrupt the W43 cloud.', '1308.6112-1-74-4': 'But we have to keep in mind that we probably underestimate the total gas mass, as described in Sect. [REF].', '1308.6112-1-74-5': 'A higher mass would lead to a lower shear parameter.', '1308.6112-1-74-6': 'This calculation is only valid for an axial symmetric potential (i.e. orbits outside the Galactic bar).', '1308.6112-1-74-7': 'Shear forces inside the Galactic bar could be stronger due to the different shape of orbits there.', '1308.6112-1-74-8': 'But as we located W43 at the tip of the bar the calculation still holds.', '1308.6112-1-75-0': 'We can also conduct this calculation for the larger gas mass derived from [MATH]CO (1-0) by [CITATION].', '1308.6112-1-75-1': 'They find a gas mass of [MATH] spread out over an area of [MATH]pc[MATH].', '1308.6112-1-75-2': 'These values lead to a shear parameter of [MATH], which is lower than our value above.', '1308.6112-1-76-0': '## Virial masses', '1308.6112-1-77-0': 'In Table [REF] (7) and (10) we have given the mean line width and the area of our sources.', '1308.6112-1-77-1': 'This allows us to calculate Virial masses by defining an effective region radius by [MATH], where [MATH] is the area of the cloud.', '1308.6112-1-77-2': 'This area cannot be determined exactly as the extent of a cloud depends on the used molecular line.', '1308.6112-1-77-3': 'Here, we use the area of [MATH]CO emission above a certain threshold (20% of the peak intensity).', '1308.6112-1-78-0': 'Virial masses can then be computed using the relation [EQUATION] where [MATH] is the Gaussian velocity dispersion, averaged over the area [MATH], and [MATH] is the gravitational constant.', '1308.6112-1-79-0': 'The resulting virial masses are shown in Table [REF] (8).', '1308.6112-1-79-1': 'We notice that most sources in W43 have masses derived from [MATH]CO that are smaller than their virial masses (sources 4 and 5 show much larger molecular than virial masses which might indicate that their distance was overestimated).', '1308.6112-1-79-2': 'If the sources would be completely virialized we would need bigger masses to produce the observed line widths.', '1308.6112-1-79-3': 'On the other hand, systematic motion of the gas, apart from turbulence, like infall, outflows, or colliding flows would also broaden the lines.', '1308.6112-1-79-4': 'This could be an explanation for the observed large line widths.', '1308.6112-1-80-0': '[CITATION] stated that usually turbulent molecular clouds are not in actual virial equilibrium, as there is a flux of mass, momentum, and energy between the clouds and their environment.', '1308.6112-1-80-1': 'What is normally viewed as virial equilibrium is rather an energy equipartition between self-gravity, kinetic, and magnetic energy.', '1308.6112-1-80-2': 'This energy equipartition is found for most clouds, due to observational limitations.', '1308.6112-1-80-3': 'Clouds out of equilibrium are either not observed due to their short lifetime or not considered clouds at all.', '1308.6112-1-81-0': 'Of course we also need to take the shape of our sources into account.', '1308.6112-1-81-1': 'Non-spherical sources have a more complicated gravitational behavior than spheres.', '1308.6112-1-81-2': 'Therefore, one has to be extremely careful using these results.', '1308.6112-1-81-3': 'In addition, we neglect here the influence of external pressure and magnetic fields on the virial masses.', '1308.6112-1-81-4': 'What we observe in agreement with [CITATION] is that most of our detected clouds show a molecular mass in the order of their virial mass, or up to a factor of 2 higher.', '1308.6112-1-82-0': '# Comparison to other projects', '1308.6112-1-83-0': 'In order to gather more information about the W43 complex we compare the IRAM 30m CO data to other existing datasets.', '1308.6112-1-83-1': 'We pay special attention to three large-scale surveys in this section, the Spitzer GLIMPSE and MIPSGAL projects, the Herschel Hi-GAL survey and the Galactic plane program ATLASGAL, observed with the APEX telescope.', '1308.6112-1-84-0': 'All of these datasets consist of total power maps over certain bands.', '1308.6112-1-84-1': 'They naturally do not contain spectral information, so line-of-sight confusion is considerable since the W43 region is a complex accumulation of different sources.', '1308.6112-1-84-2': 'It can sometimes be complicated to assign the emission of these maps to single sources.', '1308.6112-1-84-3': 'Nevertheless, the additional information is very valuable.', '1308.6112-1-85-0': '## Spitzer GLIMPSE and MIPSGAL', '1308.6112-1-86-0': 'The Spitzer Space Telescope program Galactic Legacy Infrared Mid-Plane Survey Extraordinaire GLIMPSE observed the Galactic plane at several IR wavelengths between 3.6 and 8 [MATH].', '1308.6112-1-86-1': 'It spans the Galactic plane from [MATH] to [MATH] Galactic longitude.', '1308.6112-1-87-0': 'Here we concentrated on the 8 [MATH] band.', '1308.6112-1-87-1': 'It is dominated by UV-excited PAH emission .', '1308.6112-1-87-2': 'These photon dominated regions are heated by young OB stars.', '1308.6112-1-87-3': 'By studying this band in comparison to the IRAM 30m CO maps, we can determine which parts of the molecular clouds contain UV-heated dust.', '1308.6112-1-87-4': 'This is seen as extended emission in the Spitzer maps.', '1308.6112-1-87-5': 'We can also identify nearby UV-heating sources, seen as point sources.', '1308.6112-1-87-6': 'Finally, some parts of specific clouds appear in absorption against the background.', '1308.6112-1-87-7': 'These so called infrared dark clouds show denser dust clouds, not heated by UV-radiation.', '1308.6112-1-87-8': 'That way we are able to determine which sources heat part of the gas and which parts are shielded from UV radiation.', '1308.6112-1-87-9': 'We can also tell if YSOs have already formed inside the clumps we observe and thus estimate the evolutionary stage of the clouds.', '1308.6112-1-87-10': '[CITATION] used this tracer to estimate the star-formation rate of the W43 complex.', '1308.6112-1-88-0': 'MIPSGAL is a Galactic plane survey using the MIPS instrument onboard Spitzer and has created maps at 24 and 70 [MATH]m. Here, we inspect the 24 [MATH]m band which is dominated by the emission of small dust grains.', '1308.6112-1-88-1': 'It also detects proto-stellar cores, although these cores are usually too small, to be resolved at a distance of 6 kpc or more.', '1308.6112-1-89-0': '## APEX ATLASGAL', '1308.6112-1-90-0': 'The APEX telescope large area survey of the galaxy ATLASGAL project used the LABOCA camera, installed at the APEX telescope.', '1308.6112-1-90-1': 'It observed the Galactic plane from [MATH] to [MATH] Galactic longitude at 870 [MATH].', '1308.6112-1-90-2': 'This wavelength traces cold dust and is therefore also a good indicator of dense molecular cloud structures, especially high-mass star-forming clumps.authorschuller2009 also identified hot cores, proto-stars, compact HII regions and young embedded stars by combining their map with other data.', '1308.6112-1-91-0': 'Our project is a direct follow-up of ATLASGAL, from which the idea to observe the W43 region in more detail was born.', '1308.6112-1-92-0': '## Herschel Hi-GAL', '1308.6112-1-93-0': "The Hi-GAL project utilizes Herschel's PACS and SPIRE instruments to observe the Galactic plane from [MATH] to [MATH] Galactic longitude at five wavelengths between 70 and 500 [MATH].", '1308.6112-1-93-1': 'A part of the Hi-GAL maps of the W43 complex are presented in [CITATION].', '1308.6112-1-94-0': 'Fig. [REF] shows one example of the comparison of the different datasets, that we carried out for all identified sources.', '1308.6112-1-94-1': 'It shows source 23, sticking to the notation of Table [REF].', '1308.6112-1-94-2': 'See Appendix.', '1308.6112-1-94-3': '[REF] for an in depth description of the different sources.', '1308.6112-1-95-0': 'In Table [REF] we categorize our sources, whether they have a filamentary shape, consist of cores or show a more complex structure.', '1308.6112-1-95-1': 'We also list the structure of the Spitzer 8 and 24 [MATH]m maps here.', '1308.6112-1-95-2': 'Usually, the two wavelengths are similar.', '1308.6112-1-96-0': 'From the Hi-GAL dust emission it is possible to derive a temperature and a total (gas + dust) H[MATH] column density map .', '1308.6112-1-96-1': 'These calculations were conducted for the W43 region by [CITATION] following the fitting routine detailed in [CITATION], and adapted and applied to Herschel data as in .', '1308.6112-1-96-2': 'This approach uses Hi-GAL data for the calculations where possible.', '1308.6112-1-96-3': 'As on very bright positions Hi-GAL data become saturated or enter the nonlinear response regime, HOBYS data was used to fill the missing data points.', '1308.6112-1-96-4': 'The idea is to fit a modified black body curve to the different wavelengths (in this case the 160 to 350 [MATH]m channels) for each pixel using a dust opacity law of [MATH]cm[MATH]g[MATH] with [MATH].', '1308.6112-1-96-5': 'The final angular resolution of the calculated maps (25 in this case) results from the resolution of the longest wavelength used.', '1308.6112-1-96-6': 'This is the reason the 500 [MATH]m channel has been omitted.', '1308.6112-1-96-7': 'Planck and IRAS offsets were added before calculating the temperature and H[MATH] column density.', '1308.6112-1-97-0': 'This approach assumes the temperature distribution along the line-of-sight to be constant.', '1308.6112-1-97-1': 'As discussed above in Sect [REF] this is not necessarily the case in reality.', '1308.6112-1-97-2': 'Due to temperature gradients along the line-of-sight, the calculated H[MATH] column density might deviate from the real value.', '1308.6112-1-97-3': 'This error is found in the Herschel as well as the CO calculations.', '1308.6112-1-98-0': 'Gas and dust temperatures can deviate in optically thick regions because the volume densities play a key role here.', '1308.6112-1-98-1': 'Only at fairly high densities of more than about [MATH]cm[MATH] does the gas couple to the dust temperature.', '1308.6112-1-98-2': 'At lower densities the dust grains are an excellent coolant, in contrast to the gas.', '1308.6112-1-98-3': 'Therefore, the dust usually shows lower temperatures than the gas.', '1308.6112-1-98-4': 'This is in contrast to our findings and may indeed indicate that we underestimate the [MATH] of the gas as discussed in Sect [REF].', '1308.6112-1-98-5': 'Even in optically thin regions the kinetic temperature and that derived from an SED-fit do not correspond perfectly .', '1308.6112-1-98-6': 'Fig. [REF] shows the results of these calculations for the W43 region.', '1308.6112-1-98-7': 'We show the temperature and H[MATH] column density maps from the photometry data and overlay [MATH]CO contours to indicate the location of the molecular gas clouds.', '1308.6112-1-99-0': 'The dust temperature derived by [CITATION] (Fig. [REF] (a)) lies in the range from 20 to 40 K, the outer parts of the complex show temperatures between 25 and 30 K.', '1308.6112-1-99-1': 'The regions where dense molecular clouds are found are colder (about 20 K) than their surroundings, e.g. the dense ridges in W43-Main are clearly visible.', '1308.6112-1-99-2': 'Some places, where the dust is heated by embedded UV-sources, are hotter (up to 40 K), especially the OB-star cluster in W43 and one core in W43-South catch the eye.', '1308.6112-1-100-0': 'If we compare these results with the excitation temperatures that we calculated above in Sect. [REF], we note that the temperatures derived from CO are lower ([MATH] K) than those using Herschel images.', '1308.6112-1-100-1': 'It is possible that gas and dust are not mixed well and that both could have a different temperature.', '1308.6112-1-100-2': 'Another possibility is that the CO gas is sub-thermally excited and that the excitation temperature is lower than its kinetic temperature.', '1308.6112-1-100-3': 'The Herschel dust temperature map is showing the averaged temperature along the line-of-sight.', '1308.6112-1-100-4': 'Thus lower temperatures of the dense clouds are seen together with hotter diffuse gas around it, which leads to higher averaged temperatures.', '1308.6112-1-100-5': 'As discussed in Sect. [REF], we most probably we underestimate the excitation temperature in our calculations above due to sub-beam clumpiness.', '1308.6112-1-100-6': 'Also, we cannot neglect the fact that the temperatures calculated from Herschel bear large errors on their own.', '1308.6112-1-101-0': 'The H[MATH] column density map derived from Herschel data (Fig. [REF] (b)) nicely traces the distribution of molecular gas, as indicated by the [MATH]CO (2-1) contours.', '1308.6112-1-101-1': 'There is a background level of [MATH] cm[MATH] that is found outside the complex.', '1308.6112-1-101-2': 'The column density rises in the molecular clouds up to a value of [MATH] cm[MATH] in the ridges of W43-Main.', '1308.6112-1-102-0': 'A comparison with the column density values derived from CO (see Sect. [REF] and the plot in Fig. [REF]) reveal certain differences.', '1308.6112-1-102-1': 'In the mid-density regions, the medium-sized clouds and most parts of the two large clouds, both calculations are comparable, after subtracting the background level from the Herschel maps.', '1308.6112-1-102-2': 'These are still systematically higher but difference does rarely exceed a factor of 2.', '1308.6112-1-102-3': 'The same is true for the extended gas between the denser clouds.', '1308.6112-1-102-4': 'The typical values are around a few [MATH] cm[MATH] for the Herschel and CO maps.', '1308.6112-1-102-5': 'However, in the densest parts of W43-Main the Herschel H[MATH] column density reaches values of several [MATH] cm[MATH], with a maximum of [MATH] cm[MATH], while the CO derived values peak at [MATH] cm[MATH].', '1308.6112-1-103-0': 'The offset of [MATH] cm[MATH] (it could be a bit higher but we chose the lower limit) that has to be subtracted from the Herschel map can partly be explained by diffuse cirrus emission along the line-of-sight which is not associated with W43.', '1308.6112-1-103-1': 'This statistical error of the background brightness has been described for the Hi-GAL project by [CITATION].', '1308.6112-1-103-2': '[CITATION] find this offset in agreement with [CITATION].', '1308.6112-1-104-0': 'The total mass that is found for the W43 complex still deviates between both calculations.', '1308.6112-1-104-1': 'The exact value depends on the specific region that we integrated over.', '1308.6112-1-104-2': 'We find a total mass of W43 from [MATH]CO of [MATH].', '1308.6112-1-104-3': 'The Herschel map gives [MATH] if we use the total map with an area of [MATH] pc[MATH] and consider the diffuse cirrus emission that is included by the Herschel data.', '1308.6112-1-104-4': 'This is a factor of 1.4 higher than the CO result, although we did not calculate a value for every single pixel for the CO H[MATH] column density map.', '1308.6112-1-104-5': 'The mean H[MATH] column density is [MATH] cm[MATH]CO) and [MATH] cm[MATH] (Herschel) respectively.', '1308.6112-1-104-6': 'The difference in total mass can be explained when we take into account that the Herschel map covers more points (the difference is reduced to a factor of 1.2 when comparing a smaller region which is covered in both maps, although this might be too small and biased toward the larger clouds).', '1308.6112-1-104-7': 'For a comparison of each Duchamp cloud see Table [REF] (3).', '1308.6112-1-104-8': 'A comparison of the foreground clouds would be complicated due to line-of-sight effects, so we only give numbers for the W43 sources.', '1308.6112-1-104-9': 'Most values lie in the range 1 to 1.5, which affirms that both calculations deviate by about a factor of 1.4.', '1308.6112-1-104-10': 'The difference in H[MATH] column density again depends on the examined region.', '1308.6112-1-105-0': 'As stated in Sect. [REF], the mass derived from CO is a lower limit to the real molecular gas mass.', '1308.6112-1-105-1': 'As we used the lower limit of the Herschel column density offset subtracted in W43 these values are thus an upper limit.', '1308.6112-1-105-2': 'Taking this and the errors still included in both calculations into account, the values are nearly consistent.', '1308.6112-1-106-0': '## Column density histogram - PDF', '1308.6112-1-107-0': 'A detailed investigation of the H[MATH] column density structure of W43 is done by determining a histogram of the H[MATH] column density, normalized to the average column density.', '1308.6112-1-107-1': 'These probability distribution functions (PDFs) are a useful tool to scrutinize between the different physical processes that determine the density structure of a molecular cloud such as turbulence, gravity, feedback, and magnetic fields.', '1308.6112-1-107-2': 'Theoretically, it was shown that isothermal turbulence leads to a log-normal PDF , while gravity and non-isothermality provoke power-law tails at higher densities.', '1308.6112-1-107-3': 'Observationally, power-law tails seen in PDFs obtained from column density maps of visual extinction or Herschel imaging were attributed to self-gravity for low-mass star-forming regions as well as high-mass star-forming regions .', '1308.6112-1-107-4': "Recently, it was shown that feedback processes like the compression of an expanding ionization front lead to a characteristic 'double-peaked' PDF and a significant broadening.", '1308.6112-1-108-0': 'The determination of PDFs from molecular line data was attempted by [CITATION] and [CITATION] but it turned out to be problematic when these lines become optically thick and thus do not correctly reflect the molecular cloud spatial and density structure.', '1308.6112-1-108-1': 'In addition, uncertainties in the abundance can complicate conversion into H[MATH] column density.', '1308.6112-1-108-2': 'On the other hand, molecular lines allow us to significantly reduce line-of-sight (LOS) confusion because PDFs can be determined for selected velocity ranges.', '1308.6112-1-108-3': 'In addition, using molecules with different critical densities in selected velocity ranges thus allows us to make dedicated PDFs focusing on a particular subregion like, e.g., a dense filament.', '1308.6112-1-109-0': 'In this study, we determined the PDFs of W43 in three ways, (i) from a simple conversion of the [MATH]CO (2-1) map into H[MATH] column density using a constant conversion factor and one temperature (5 or 10 K), (ii) from the H[MATH] column density map derived from the [MATH]CO (2-1) emission and including a correction for the optical depth, derived from both CO lines (see Sect. [REF]), and (iii) from the column density map obtained with Herschel using SPIRE and PACS photometry.', '1308.6112-1-109-1': 'Fig. [REF] shows the resulting distributions.', '1308.6112-1-109-2': 'For simplicity, we used the conversion [MATH] for all maps though the Herschel column density map is a mixture of HI and H[MATH] while the CO derived map is most likely dominated by H[MATH].', '1308.6112-1-110-0': "The 'isothermal' PDFs from [MATH]CO without optical depth correction (in black and red) clearly show that there is a cut-off in the PDF at high column densities where the lines become optically thick (A[MATH]20 mag for 10 K and A[MATH]70 mag for 5 K).", '1308.6112-1-110-1': 'Obviously, there is also a strong temperature dependence that shifts the peak of the PDF to lower column densities with increasing temperature.', '1308.6112-1-110-2': 'The assumed gas temperature (see discussion in Sect. [REF]) thus has a strong impact on the resulting PDFs positions but not their shape (not taking into account the uncertainty in the conversion factor).', '1308.6112-1-111-0': 'With the more sophisticated approach to create a column density map out of the [MATH]CO emission and including the information provided by the optically thinner C[MATH]O by correcting for the optical depth [MATH], the PDF (in blue) is more reliable.', '1308.6112-1-111-1': 'It does not show the cut-off at high column densities any more because this effect is compensated by using the optically thin C[MATH]O. Only in those regions where even this line becomes optically thick, the method gives lower limits for the column density.', '1308.6112-1-111-2': 'Therefore it drops below the Herschel PDF for high column densities because the molecular lines underestimate the H[MATH] column densities for very hot gas.', '1308.6112-1-112-0': 'It is remarkable that the PDF derived from [MATH]CO and C[MATH]O shows a log-normal distribution for low column densities and a power-law tail for higher densities, a feature also observed in the Herschel PDF (in green).', '1308.6112-1-112-1': 'The approach to determine a PDF from the cloud/clump distribution with correcting the opacity using C[MATH]O appears to be the right way to get a clearer picture of the distribution of higher column densities.', '1308.6112-1-112-2': 'Note that the absolute scaling in column density for both PDFs - from CO or Herschel - remains problematic due to the uncertainty in the conversion factor (and temperature) for the CO data and the LOS-confusion and opacity variations for the Herschel maps.', '1308.6112-1-113-0': "We observe that the slope of the Herschel power-law tail is steeper than the one obtained from the CO-data and shows a 'double-peak' feature (the low column density part is not strictly log-normally shaped but shows two sub-peaks) as seen in other regions with stellar feedback (Tremblin et al. in prep.)", '1308.6112-1-113-1': 'The column density structure of W43 could thus be explained in a scenario where gravity is the dominating process for the high density range, leading to global cloud collapse and individual core collapse, compression by expanding ionization fronts from embedded HII-regions may lead to an increase in column density, and the lower-density extended emission follows a turbulence dominated log-normal distribution.', '1308.6112-1-113-2': 'This scenario is consistent with what was proposed for other high-mass star-forming regions, i.e. Rosette , RCW36 and M16 (Tremblin et al. in prep.)', '1308.6112-1-113-3': ', and W3 .', '1308.6112-1-114-0': 'Though the overall shape of the PDFs is similar, there are significant differences in the slope of the power-law tails.', '1308.6112-1-114-1': 'The power-law tail of the Herschel PDF is steeper than the CO-PDF.', '1308.6112-1-114-2': "A possible explanation is that the Herschel PDF contains atomic hydrogen in addition to molecular hydrogen (which constitutes mainly the CO PDF) which is less 'participating' in the global collapse of the region and individual clump/core collapse.", '1308.6112-1-114-3': 'In this case our method to derive the H[MATH] column density from [MATH]CO and C[MATH]O turns out to be an efficient tool to identify only the collapsing gas that will end up into a proto-star.', '1308.6112-1-115-0': '# Description of W43-Main and W43-South', '1308.6112-1-116-0': 'Here we give a detailed description of the two most important sources in the W43 complex, W43-Main and W43-South.', '1308.6112-1-116-1': 'Several other interesting sources are described in Appendix [REF].', '1308.6112-1-117-0': '## W43-Main, Source 13', '1308.6112-1-118-0': 'Source 13 (see Fig. [REF] (m)), i.e. W43-Main, is the largest and most prominent of all sources in the W43 complex.', '1308.6112-1-118-1': 'Located in the upper central region of the map, with an extent of roughly 30 by 20 pc, it shows a remarkable Z shape of connected, elongated ridges.', '1308.6112-1-118-2': 'The upper part of this cloud extrudes far to the East with a strong emitting filament, where it curves down South in a weaker extension of this filament.', '1308.6112-1-118-3': 'This structure is especially clear in ATLASGAL and Hi-GAL dust emission.', '1308.6112-1-119-0': 'There are some details hidden in this cloud, that cannot be seen clearly in the complete integrated map.', '1308.6112-1-119-1': 'In the velocity range of 80 to 90 kms[MATH], located in the Southwest of the source, we see a circular, shell like structure surrounding an empty bubble (Fig. [REF] (a)).', '1308.6112-1-119-2': 'This bubble is elliptically shaped with dimensions of 10 times 6 pc, while the molecular shell is about 1.5 pc thick.', '1308.6112-1-119-3': 'It is located where a cluster of young OB stars is situated.', '1308.6112-1-119-4': 'Possibly, this cavity is formed by the radiation of this cluster.', '1308.6112-1-119-5': 'See [CITATION] for a description of the expansion of clouds at the periphery of this (HII) bubble.', '1308.6112-1-120-0': 'The central Z shape of this cloud appears to be monolithic on the first sight.', '1308.6112-1-120-1': 'But as Fig. [REF] (b) shows, the northern and southern part are separated by a gap in the channel maps between 94 and 98 kms[MATH].', '1308.6112-1-120-2': 'Both parts are still connected in channels higher and lower than those velocities.', '1308.6112-1-120-3': 'This chasm we see, is narrow in the center of the cloud, where it has a with of 1 to 2 pc, and opens up to both sides.', '1308.6112-1-120-4': 'The origin of this structure is the cluster of WR and OB stars situated in the very center, that blows the surrounding material out along a plane perpendicular to the line-of-sight.', '1308.6112-1-120-5': 'This agrees with the presence of 4 HCO[MATH] clouds in the 25 kms[MATH] range along the line-of-sight of the Wolf-Rayet cluster .', '1308.6112-1-121-0': 'W43-Main is the most luminous source in our set, with integrated [MATH]CO (2-1) emission of up to 170 Kkms[MATH] and line peaks of up to 23 K at the peak in the northern filament.', '1308.6112-1-121-1': 'Most inner parts of this cloud show integrated intensities of [MATH]90 to 120 Kkms[MATH] and 45 to 60 Kkms[MATH] in the outer parts.', '1308.6112-1-121-2': 'It is also the most optically thick one, with an optical depth of [MATH]CO of up to 8 in the southern arm, while the bulk of the cloud with 2 to 3 is not exceptionally opaque.', '1308.6112-1-121-3': 'It is possible that we overestimated the opacity in the South of this cloud.', '1308.6112-1-121-4': 'The spectra show that [MATH]CO is self-absorbed here, while C[MATH]O is not.', '1308.6112-1-121-5': 'This would lead to an unrealistic ratio of the two isotopologes and an overestimated optical depth.', '1308.6112-1-122-0': 'The cloud shows a velocity gradient across its complete structure (see Fig. [REF]).', '1308.6112-1-122-1': 'Beginning at a relative velocity of [MATH] kms[MATH] at the most southwestern tip, it winds through the Z shape and ends in the eastern extension filament at [MATH] kms[MATH].', '1308.6112-1-122-2': 'This velocity difference of [MATH] kms[MATH], already described in [CITATION] is huge and the largest discovered in the W43 complex.', '1308.6112-1-122-3': 'It could be the sign of a rotation of the cloud.', '1308.6112-1-122-4': 'Comparably impressive are the line widths of the central parts of this clouds.', '1308.6112-1-122-5': 'We find them to be up to 15 kms[MATH] especially in the central parts, indicating very turbulent gas or global motions.', '1308.6112-1-123-0': 'In the most luminous parts in the south we calculate the highest H[MATH] column densities of about [MATH] cm[MATH].', '1308.6112-1-123-1': 'This is due to the high opacities that have been calculated here.', '1308.6112-1-123-2': 'The other central parts of this cloud shows H[MATH] column densities of [MATH] cm[MATH].', '1308.6112-1-123-3': '[CITATION] derived the cloud mass of W43-Main from [MATH]CO (1-0) and find a value of [MATH] M[MATH].', '1308.6112-1-123-4': 'We find a total mass in this source of [MATH] and thus a large fraction of the mass of the complete W43 complex ( 20%).', '1308.6112-1-124-0': 'In the GLIMPSE maps at this position we see strong extended IR emission along the walls of a cavity, just west of the central Z shape, formed by a cluster of young OB stars that is located here (see Fig. [REF]).', '1308.6112-1-124-1': 'This cluster is probably the first result of the star-formation going on here.', '1308.6112-1-124-2': 'The cavity seen in infrared is not the same as the one in CO described above.', '1308.6112-1-124-3': 'But both appear to be connected and the CO channel, maybe due to the strong radiation of the nearby stars.', '1308.6112-1-124-4': 'The northern and southern arm of the Z-shape are seen in absorption in the 8 [MATH] band of GLIMPSE against the infrared background.', '1308.6112-1-124-5': 'This indicates dense cold dust and molecular clouds that are shielded from the stars UV radiation, which is verified by strong emission of cold dust, seen in the ATLASGAL and Hi-GAL maps.', '1308.6112-1-125-0': '## W43-South, Source 20', '1308.6112-1-126-0': 'Source 20, called W43-South, also known as G029.96-0.02 , is the second largest source in the W43 complex and dominates the southwestern part our map.', '1308.6112-1-126-1': 'Fig. [REF] (t) shows a plot of this source.', '1308.6112-1-126-2': 'It has approximately the shape of a tilted ellipse with the dimensions of about 24 times 31 parsec, with several smaller clumps scattered across the cloud.', '1308.6112-1-126-3': 'These clumps emit strongly in [MATH]CO (2-1) up to 150 Kkms[MATH] and are surrounded by less luminous gas, where we see emission between 60 to 90 Kkms[MATH] and down to 30 Kkms[MATH] in the outer parts of the cloud.', '1308.6112-1-126-4': 'Maximum line peaks are 30 K.', '1308.6112-1-126-5': 'This source is less optically thick than W43-Main, the opacity is around 2 to 3 for most parts of the cloud and does not exceed 4 in the dense clumps.', '1308.6112-1-127-0': 'Studying the details of this source, we see that several of the dense clumps are actually shells of gas.', '1308.6112-1-127-1': 'The ringlike structures are clearly recognizable in some channel maps.', '1308.6112-1-127-2': 'Fig. [REF] shows the most intriguing example.', '1308.6112-1-127-3': 'The spectra across the whole ring show infall signatures as discussed in [CITATION] in the optically thick [MATH]CO lines.', '1308.6112-1-127-4': 'However, C[MATH]O, which usually is optically thin shows the same signature as it is still optically thick at this position.', '1308.6112-1-127-5': 'To really trace infall the optically thin line should show a single peak at the position of the position of the absorption feature in the optically thick line.', '1308.6112-1-127-6': 'This is the case for the N[MATH]H[MATH] (1-0) line, also taken at the IRAM 30m during the second part of our program, which is optically thin (although it shows a hyper-fine structure, the strongest peak is centered on the correct velocity).', '1308.6112-1-127-7': 'This could be interpreted as a bubble of gas which is heated from the inside by an embedded UV source, although it is not associated with any ultracompact HII region identified by the CORNISH survey .', '1308.6112-1-127-8': 'The Spitzer 8 [MATH]m also show a heated ring of dust which indicates an embedded heating source.', '1308.6112-1-128-0': 'The relative radial velocity of the gas is more or less constant around 100 kms[MATH] across the whole cloud W43-South.', '1308.6112-1-128-1': 'Small parts in the East are slower, with 95 kms[MATH], while a tip in the Northwest is faster, around 105 kms[MATH].', '1308.6112-1-128-2': 'However, the velocity gradient is not very pronounced.', '1308.6112-1-128-3': 'FWHM line widths show typical values of 5 to 10 kms[MATH] with broad lines found mostly in the eastern part of the cloud.', '1308.6112-1-128-4': 'The bright clumps do not show distinct broad lines.', '1308.6112-1-129-0': 'The calculated H[MATH] column density is around a few times [MATH] cm[MATH] for most of the cloud, but parts in the Northwest and the center peak at about [MATH] cm[MATH].', '1308.6112-1-129-1': 'The total mass of this cloud is [MATH] and it is thus the second most massive source in the W43 complex.', '1308.6112-1-130-0': 'All bright clumps, except one in the central North, are seen in bright emission in the GLIMPSE 8 [MATH]m map.', '1308.6112-1-130-1': 'They are obviously heated from the inside, no external UV sources can be identified.', '1308.6112-1-130-2': 'Apparently YSOs have formed in the dense but separated clumps, that are seen in CO emission, as well as in dust emission.', '1308.6112-1-130-3': 'In the Northeast of the cloud, one slab of cold gas is seen in absorption in the GLIMPSE map, separating some of the bright clumps.', '1308.6112-1-131-0': '# Conclusions', '1308.6112-1-132-0': 'Following the investigations of [CITATION], ATLASGAL and [CITATION] we observed the W43 region in the [MATH]CO (2-1) and C[MATH]O (2-1) emission lines with the IRAM 30m telescope.', '1308.6112-1-132-1': 'We have presented integrated maps of the resulting position-position-velocity cubes in which we identified numerous clouds.', '1308.6112-1-133-0': 'We confirmed that W43 is indeed one connected complex, as described in [CITATION].', '1308.6112-1-133-1': 'The connection between the two main clouds W43-main and W43-south is shown in the PV-diagram in Fig. [REF] (a).', '1308.6112-1-134-0': 'An analysis of the velocity distribution of our dataset and comparison with the Galactic model of [CITATION] reveals emission not only from the W43 complex itself, but also from fore-/background sources.', '1308.6112-1-134-1': 'According to this model, W43 is situated near the tangential point of the Scutum arm, where it meets the Galactic bar.', '1308.6112-1-134-2': 'The low velocity sources are located in the Perseus arm and the space between Sagittarius and Scutum arms (see Fig. [REF] (c)).', '1308.6112-1-134-3': 'The separation of the different components lets us avoid the confusion and line-of-sight effects in our analysis.', '1308.6112-1-135-0': 'We decomposed the data cubes into sub-clouds, using Duchamp Sourcefinder.', '1308.6112-1-135-1': 'We identified a total of 29 clouds, 20 located in the W43 complex, while 9 were found to be foreground and background sources.', '1308.6112-1-136-0': 'We derived physical properties like excitation temperature, H[MATH] column density and total mass, of each sub-cloud of our dataset (see Table [REF]).', '1308.6112-1-136-1': 'Typical smaller sources have spatial scales of 10 to 20 pc and masses of several [MATH].', '1308.6112-1-136-2': 'The two most outstanding sources are W43-Main (source 13) and W43-South (source 20) that have masses of a few [MATH].', '1308.6112-1-137-0': 'We determined the total mass of dense clouds (>[MATH] cm[MATH]) in the W43 complex to be [MATH].', '1308.6112-1-137-1': 'This is a factor of 1.4 lower than the mass derived from Herschel dust emission maps which is a discrepancy that can be explained by the details of both calculations.', '1308.6112-1-138-0': 'The shear parameter of W43 ([MATH]) shows that the accumulation of mass in molecular clouds in this region is not disrupted by shear forces of the Galactic motion.', '1308.6112-1-139-0': 'We created probability distribution functions obtained from column density maps.', '1308.6112-1-139-1': 'We used both the molecular line maps and Herschel imaging data (Fig. [REF]).', '1308.6112-1-139-2': 'Both show a log-normal distribution for low column densities and a power-law tail for high densities.', '1308.6112-1-139-3': 'Still, there are differences seen in peak position and power-law slope.', '1308.6112-1-139-4': 'Possibly the flatter slope of the molecular line data PDFs imply that those could be used to selectively show the gravitationally collapsing gas.', '1308.6112-1-140-0': 'This project is carried out within the Collaborative Research Council 956, sub-project A4, funded by the Deutsche Forschungsgemeinschaft (DFG).', '1308.6112-1-140-1': 'Part of this work was supported by the ANR-11-BS56-010 project "STARFICH".', '1308.6112-1-140-2': 'Part of this work was supported by the French National Agency for Research (ANR) project "PROBeS", number ANR-08-BLAN-0241.', '1308.6112-1-140-3': 'We thank the referee for constructive and valuable comments and ideas.', '1308.6112-1-141-0': '# Noise maps', '1308.6112-1-142-0': 'We create noise maps from both the [MATH]CO (2-1) and the C[MATH]O (2-1) data cubes.', '1308.6112-1-142-1': 'For each spectrum we determine the rms and create maps from these values.', '1308.6112-1-142-2': 'For this purpose we need to calculate the rms from parts of the spectra that are emission free.', '1308.6112-1-142-3': 'We use the velocity range between 120 and 130 kms[MATH] as it is free of emission for the complete region that we mapped.', '1308.6112-1-142-4': 'Typical values are found to be around 1 K or even less, while some parts in the South show values of up to 3 K. All values given here are in [MATH], i.e. corrected for main beam efficiency.', '1308.6112-1-142-5': 'The results can be seen in Fig. [REF].', '1308.6112-1-143-0': 'We find that the structure of the noise is similar for both lines.', '1308.6112-1-143-1': 'The largest differences arise from weather conditions and time of day.', '1308.6112-1-143-2': 'This is seen in the squarish pattern, as each square shows single observations that have been carried out in a small time window.', '1308.6112-1-143-3': 'Still, there is a striped pattern visible, that overlays the whole map.', '1308.6112-1-143-4': 'This stems from the nine different pixels that the HERA receiver is made of.', '1308.6112-1-143-5': 'These pixels have different receiver temperatures, hence the different noise levels.', '1308.6112-1-143-6': 'We also note that observations in the Northern part of the map are usually less noisy than those in the South.', '1308.6112-1-143-7': 'This probably results from the different weather in which the observations have been carried out.', '1308.6112-1-143-8': 'Last, we find, that the C[MATH]O (mean rms of 1.3 K, maximum of 6.7 K) data shows in general a little increase in noise temperature compared to the [MATH]CO line (mean rms of 1.1 K, maximum of 3.7 K).', '1308.6112-1-143-9': 'The latter has been observed with the HERA1 polarization of the HERA receiver, whereas the first has been observed with HERA2, which has an overall higher receiver temperature.', '1308.6112-1-144-0': '# Peak velocity and line width of foreground components', '1308.6112-1-145-0': 'Fig. [REF] shows plots of the peak velocity position and the FWHM line width of the two lower velocity components.', '1308.6112-1-145-1': 'The maps of the W43 complex itself are shown in Fig. [REF] and are described in Sect. [REF].', '1308.6112-1-146-0': '# Calculations', '1308.6112-1-147-0': '## [MATH]CO Optical depth', '1308.6112-1-148-0': 'Assuming a constant abundance ratio of [MATH]CO:[MATH]CO:C[MATH]O we can estimate the optical depth of the [MATH]CO gas .', '1308.6112-1-148-1': 'We compare the intensities of the [MATH]CO and C[MATH]O line emission integrated over the analyzed cloud and solve the equation [EQUATION] for [MATH], where [MATH] is the intrinsic ratio of of the two mapped CO isotopologes.', '1308.6112-1-149-0': 'The isotopic abundance of C and O in the Milky Way is know to depend on the Galactocentric radius.', '1308.6112-1-149-1': 'Often cited values are found in [CITATION].', '1308.6112-1-149-2': 'They find the ratio of [MATH]O and [MATH]O to be 272 at 4 kpc radius, 302 at 4.5 kpc and 390 at 6 kpc, so we take those numbers for the [MATH]CO:C[MATH]O ratio.', '1308.6112-1-149-3': 'Recent values for the C/[MATH]C abundance are given in [CITATION].', '1308.6112-1-149-4': 'Here we take values derived from CO observations and get a [MATH]CO:[MATH]CO ratio of 31 at a Galactocentric radius of 4 kpc for the main component and for the foreground components at a radius of 4.5 and 6 kpc we use a ratio of 43 and 52 respectively.', '1308.6112-1-149-5': 'So in total we use an intrinsic ratio of [MATH]CO:[MATH]CO:C[MATH]O of 1:1/31:1/272 for sources in the main complex and ratios of 1:1/43:1/302 and 1:1/52:1/390 respectively for foreground sources.', '1308.6112-1-150-0': '## Excitation temperature', '1308.6112-1-151-0': 'Once we know the optical depth we can determine the excitation temperature of the CO gas.', '1308.6112-1-152-0': 'For this we use [CITATION] [EQUATION]', '1308.6112-1-152-1': 'With [EQUATION] where we use the line peak intensity [MATH] for [MATH].', '1308.6112-1-152-2': '[MATH] is the cosmic background radiation of 2.7 K, [MATH] the [MATH]CO optical depth and [MATH] the beam filling factor.', '1308.6112-1-152-3': 'This expression can then be solved for [MATH].', '1308.6112-1-153-0': 'Here we assume that the excitation temperature of [MATH]CO and C[MATH]O is the same.', '1308.6112-1-153-1': 'We further assume that the beam filling factor [MATH] is always 1.', '1308.6112-1-153-2': 'It is likely that there is some substructure that we cannot resolve with our beam size.', '1308.6112-1-153-3': 'This would mean that the real [MATH] is lower than 1 and we rather calculate [MATH] than just [MATH].', '1308.6112-1-153-4': 'Thus we underestimate the temperature in cases where there is indeed substructure.', '1308.6112-1-153-5': 'The line peak [MATH] is calculated from fitting a Gaussian to all spectra in the [MATH]CO line emission cube.', '1308.6112-1-154-0': 'We cannot calculate a [MATH] this way for all pixels even if [MATH]CO is present, as the C[MATH]O is much weaker.', '1308.6112-1-154-1': 'As the ratio of [MATH]CO and C[MATH]O is needed a self-consistent temperature can only be computed for points where C[MATH]O is present.', '1308.6112-1-154-2': 'The main uncertainty of the calculation itself is the assumption that the beam filling factor is always 1 and the real value can only be correctly derived where this is true.', '1308.6112-1-154-3': 'This led to the decision not to use these T[MATH] maps for the further calculation of the H[MATH] column density.', '1308.6112-1-154-4': 'We used the calculations to get an idea of the gas temperature and then used a constant value for all further steps of analysis.', '1308.6112-1-154-5': 'We chose this value to be 12 K as this was the median temperature found across the whole W43 region.', '1308.6112-1-155-0': '## Column density and mass', '1308.6112-1-156-0': 'We further compute the H[MATH] column density by using the assumed excitation temperature [MATH] of 12 K and the integrated molecular line emission from our observation through [EQUATION] where the factor containing [MATH] accounts for the effect that we do not see the full gas for optically thicker clouds, with', '1308.6112-1-157-0': '[EQUATION] and the function [EQUATION] where [MATH]s[MATH] is the rotational constant for [MATH]CO, [MATH]D its dipole moment and [MATH] is the upper level of our transition (2 in this case).', '1308.6112-1-157-1': 'We correct all those points for the optical depth where we find an opacity larger than 0.5.', '1308.6112-1-157-2': 'We assume this is the minimum value we can determine correctly as we might confuse emission with noise for lower opacities.', '1308.6112-1-158-0': 'In Fig. [REF] we plot the dependency of the [MATH]CO column density and the excitation temperature.', '1308.6112-1-158-1': 'We note that for values of [MATH] K the column density is nearly independent of the excitation temperature.', '1308.6112-1-158-2': 'On the contrary it rises steeply for temperatures below 10 K.', '1308.6112-1-158-3': 'This is important as we most probably would underestimate the excitation temperature for most clouds as discussed above if we actually used the calculated T[MATH].', '1308.6112-1-158-4': 'We would thus overestimate the column density.', '1308.6112-1-158-5': 'Therefore we can assume that using an excitation temperature of 12 K for our calculation will result in a lower limit for the actual column density.', '1308.6112-1-159-0': 'As we want to calculate the H[MATH] column density, we need to translate [MATH]CO) into [MATH].', '1308.6112-1-159-1': 'The standard factor of [MATH]:[MATH] is [MATH] for local molecular clouds but the ratio varies with sources and also with Galactocentric radius.', '1308.6112-1-159-2': '[CITATION] derive a radius dependent formula for [MATH]:[MATH], using values of [MATH]C/H and [MATH]O/H from [CITATION].', '1308.6112-1-159-3': 'This formula gives a ratio of 6550 for a Galactic radius of 4 kpc 7000 at 4.5 kpc and 8500 at 6 kpc.', '1308.6112-1-159-4': 'Using our ratios of [MATH]CO:[MATH]CO from above we get [MATH] for the W43 complex with a radius of 4 kpc and factors of [MATH] and [MATH] for the fore-/background complexes with radii of 4.5 and 6 kpc components respectively.', '1308.6112-1-159-5': 'These factors are prone to large errors of at least a factor of 2.', '1308.6112-1-160-0': 'To calculate the mass of the observed source from the H[MATH] column density it is necessary to take the relative distance from the Sun to the source into account.', '1308.6112-1-160-1': 'See Sect. [REF] for the distance determination.', '1308.6112-1-160-2': 'We assume the main complex clouds to be 6 kpc away, while the foreground clouds have distances of 3.5, 4, 3.5, and 12 kpc.', '1308.6112-1-160-3': 'Then we just have to count the number of H[MATH] molecules per pixel to receive the mass per pixel in solar masses.', '1308.6112-1-160-4': '[EQUATION]', '1308.6112-1-160-5': 'Here, [MATH] is the distance toward the source, [MATH] the angular extent of one pixel on the sky.', '1308.6112-1-160-6': '[MATH] accounts for higher masses of molecules apart from H[MATH]', '1308.6112-1-161-0': 'The resulting H[MATH] column density map of the full W43 complex is shown in Fig. [REF] which was calculated using the velocity range between 78 and 120 kms[MATH].', '1308.6112-1-161-1': 'It has been calculated for all points, where the integrated [MATH]CO shows an intensity of more than 5 Kkms[MATH].', '1308.6112-1-161-2': 'The integrated [MATH]CO map in Fig. [REF] (a) shows diffuse emission in-between the brighter sources, that were isolated with the Duchamp sourcefinder.', '1308.6112-1-161-3': 'This diffuse emission accounts for about 50% of the the total mass in W43.', '1308.6112-1-162-0': '# Description of important sources', '1308.6112-1-163-0': 'In the following we want to give a description of several important and interesting sources of the W43 complex ([MATH] kpc) found in our datasets.', '1308.6112-1-163-1': 'Information on these sources are listet in Table [REF], while their location is indicated in Fig. [REF].', '1308.6112-1-163-2': 'The corresponding maps can be found in Fig. [REF].', '1308.6112-1-163-3': 'We will discuss the shape, topology and intensity of the maps, as well as fundamental properties like velocity gradients, FWHM line widths, temperature, and column density.', '1308.6112-1-163-4': 'We will also mention conclusions from the comparison with different datasets.', '1308.6112-1-163-5': 'Sources are ordered by their peak velocities.', '1308.6112-1-164-0': '## Source 23', '1308.6112-1-165-0': 'Source 23 (plots can be seen in Fig. [REF] and also Fig. [REF] (w)) consists of one central elliptical clump with one elongated thin extension, protruding from the Southeast, that ends in a hook-like tip, that is curved to the South.', '1308.6112-1-165-1': 'The central clump is elliptically shaped, with a length of the major axes of 5 and 3 parsec and is bound sharply at the southern edge, while it is much more diffuse and more extended in the North.', '1308.6112-1-165-2': 'The extension has a length of 7.5 pc.', '1308.6112-1-165-3': 'We find the maximum integrated intensity of the [MATH]CO (2-1) line to be at about 90 Kkms[MATH] at the peak of the clump, while the filamentary extension lies around 30 to 45 Kkms[MATH].', '1308.6112-1-165-4': 'The line peak intensity rises from 12 K in the filament to 24 K in the clump.', '1308.6112-1-165-5': 'The opacity has typical values of 1 to 2.5, with higher values in the central clump.', '1308.6112-1-166-0': 'We see a gradient in the radial velocity of the cloud from the filament to the center of the source of [MATH] km[MATH].', '1308.6112-1-166-1': 'Which can be interpreted as a flow of gas along the outrigger onto the clump.', '1308.6112-1-166-2': 'The line width (FWHM) changes between 4.5 kms[MATH] in the inner clump and 2 to 2.5 kms[MATH] in the outer parts of the cloud.', '1308.6112-1-167-0': 'The H[MATH] column density, that we calculated rises from [MATH] cm[MATH] in the edges of the cloud to [MATH] cm[MATH] in the center.', '1308.6112-1-167-1': 'The total mass is calculated to be [MATH] and thus resembles a typical total mass of our set of sources.', '1308.6112-1-168-0': 'The CO emission, that we measure in our maps is nearly exactly matched by the dust emission maps of ATLASGAL and Hi-GAL (see Fig. [REF] (b) and (c) respectively).', '1308.6112-1-168-1': 'Both show the strong peak in the central clump and the weaker filament in the Southeast, including the curved tip.', '1308.6112-1-168-2': 'The GLIMPSE map shows very interesting features (see Fig. [REF] (d)).', '1308.6112-1-168-3': 'There is one strong UV point source less than 1 pc off to the South of the CO clump.', '1308.6112-1-169-0': '## Source 25', '1308.6112-1-170-0': 'This filament, seen in Fig. [REF] (y), resides in the central western part of the W43 complex, half-way between W43-Main and W43-South.', '1308.6112-1-170-1': 'It is shaped like an inverted L with two branches, connected by an orthogonal angle.', '1308.6112-1-170-2': 'The vertical branch has a length of 14 pc, the horizontal one is 10 pc long.', '1308.6112-1-170-3': 'The typical width of both branches is between 2 and 3 pc.', '1308.6112-1-170-4': 'One strong clump is seen in the southern part, with an integrated line intensity of the [MATH]CO (2-1) line of 40 Kkms[MATH], while the rest of the filament backbone only reaches 18 to 22 Kkms[MATH].', '1308.6112-1-170-5': 'Line peak intensities range from a few K in the outer parts of the filament up to 15 K in the strong southern clump.', '1308.6112-1-171-0': 'Investigating the line peak velocity map, we realize, that the two branches of this source are actually separated.', '1308.6112-1-171-1': 'The horizontal branch has a constant radial velocity of 108 kms[MATH] across is, while the vertical branch shows a gradient from 110 kms[MATH] in the North to 115 kms[MATH] in the South.', '1308.6112-1-171-2': 'Line widths range between 1 and 2 kms[MATH] in the whole source.', '1308.6112-1-172-0': 'The H[MATH] column density varies between [MATH] cm[MATH] in the outer parts and [MATH] cm[MATH] around the southern core.', '1308.6112-1-172-1': 'The total mass is [MATH].', '1308.6112-1-173-0': 'Comparing this source to the complementary projects is complicated, as the source 17 is located at the same place and overlaps this source.', '1308.6112-1-173-1': 'Most emission, that is seen in the northern part of the source is presumably part of source 17.', '1308.6112-1-173-2': 'Only the embedded core in the South is clearly seen in dust emission and as a compact Spitzer source.', '1308.6112-1-174-0': '## Source 26', '1308.6112-1-175-0': 'Located in the easternmost central part of the W43 complex, lies this filamentary shaped source, whose plot is found in Fig. [REF] (z).', '1308.6112-1-175-1': 'It stretches over a range of 26 pc from Southeast to Northwest.', '1308.6112-1-175-2': 'The filament consists of three sub-sections, that contain several clumps, and has a typical width of 5 pc.', '1308.6112-1-175-3': 'The integrated emission map of the [MATH]CO (2-1) line shows values of up to 35 Kkms[MATH] in the clumps, surrounded by weaker gas.', '1308.6112-1-175-4': 'The strongest clump lies in the southeastern end of the filament while the highest line peak intensities are found in the Northwest with up to 13 K.', '1308.6112-1-176-0': 'The velocity structure of this filament is nearly symmetrical, starting around 106 kms[MATH] in the middle of the filament, increasing toward the tips of it up to 110 kms[MATH] in the West and 112 kms[MATH] in the East.', '1308.6112-1-176-1': 'The width of the lines has nearly homogeneous values around 2 kms[MATH] in the center and Western part of the filament, but showing broad lines with a width of more than 5 kms[MATH] in the eastern clump.', '1308.6112-1-177-0': 'This filament shows a typical distribution of its H[MATH] column density.', '1308.6112-1-177-1': 'Several denser clumps are embedded along the filament.', '1308.6112-1-177-2': 'Column densities vary from a few [MATH] cm[MATH] in the outer parts of the filament up to a maximum of [MATH] cm[MATH] in one clump.', '1308.6112-1-177-3': 'The total mass of this source is [MATH].', '1308.6112-1-178-0': 'The CO emission of this source matches nicely with the dust emission of ALTASGAL and Hi-GAL.', '1308.6112-1-178-1': 'However, the eastern part of the filament is stronger in the 850 [MATH]m map than the shorter wavelengths of Hi-GAL and vice versa in the western part.', '1308.6112-1-178-2': 'Only the West clump of this filament can be seen in the 8 [MATH]m Spitzer data, the East is not traced.', '1308.6112-1-178-3': 'There is one extended source seen in emission in the center part of this source, but this is probably unrelated.', '1308.6112-1-179-0': '## Source 28', '1308.6112-1-180-0': 'Source 28 (see Fig. [REF] (ab)) is located in the Southwest of W43-Main in the central region of the complex.', '1308.6112-1-180-1': 'It is not visible in the total integrated maps of the region, as it is confused with sources of a different relative velocity.', '1308.6112-1-180-2': 'It becomes visible by investigating the channel maps between 110 and 115 kms[MATH] radial velocity.', '1308.6112-1-180-3': 'The source has dimensions of 12 pc in East-West direction and 8 pc in North-South direction.', '1308.6112-1-180-4': 'Its shape is that of a two-armed filament, whose two parts join at an angle of [MATH], where the Eastern arm runs from Southeast to Northwest and the Western arm from East to West.', '1308.6112-1-180-5': 'We see two stronger clumps in the eastern filament, one in the center of it, one in the southeastern tip.', '1308.6112-1-180-6': 'This is a relatively weak source, with an integrated [MATH]CO (2-1) emission of only 25 Kkms[MATH] at the peak in the center of the eastern filament where the maximum line peak is around 13 K.', '1308.6112-1-180-7': 'The integrated intensity goes down to 8 Kkms[MATH] in the outskirts of the filament.', '1308.6112-1-180-8': 'Yet, due to its pronounced filamentary structure and the embedded clumps, it is a valuable source, which is a good candidate for further investigations in filament formation .', '1308.6112-1-181-0': 'The eastern arm is especially interesting.', '1308.6112-1-181-1': 'It has a length of 6 pc and a typical width of 1.5 pc.', '1308.6112-1-181-2': 'We note two embedded clumps embedded in it, as well as a velocity gradient along the filament, starting at 110 kms[MATH] in the North and going to 114 kms[MATH] in the southeastern part of this arm.', '1308.6112-1-181-3': 'The typical line width varies between 1 and 2 kms[MATH], increasing toward the inside of the cloud and reaching the maximum width at the clumps.', '1308.6112-1-182-0': 'As this source is rather weak, we also find H[MATH] column densities to be only around [MATH] cm[MATH] at the maximum around the embedded clumps.', '1308.6112-1-182-1': 'The total derived mass of the molecular gas is [MATH], which makes it one of the less massive sources identified.', '1308.6112-1-183-0': 'Surprisingly, this source is one of the few ones, that is not traced at all in the GLIMPSE 8 [MATH]m map.', '1308.6112-1-183-1': 'It appears that there are no nearby UV sources that could heat the gas.', '1308.6112-1-183-2': 'The two nearest sources were identified to be related to background sources in another Galactic spiral arm.', '1308.6112-1-183-3': 'Also, the gas and the related dust is obviously too faint to appear in absorption.', '1308.6112-1-183-4': 'This is also verified by the ATLASGAL an Hi-GAL maps, which show only weak dust emission in the filament.', '1308.6112-1-184-0': '## Source 29', '1308.6112-1-185-0': 'This source is found in the very center of our region maps, directly south of the W43-Main cloud.', '1308.6112-1-185-1': 'Its shape resembles a crescent moon, opened toward the Southeast, where it is sharply bound.', '1308.6112-1-185-2': 'The outside is more diffuse and shows several outflows away from the center.', '1308.6112-1-185-3': 'See Fig. [REF] (ac) for a plot of the [MATH]CO emission.', '1308.6112-1-185-4': 'The extent of the source is 12 pc from Northeast to Southwest and the filament has a typical width of 2 to 3 pc.', '1308.6112-1-185-5': 'Two stronger clumps with an integrated [MATH]CO intensity of 40 Kkms[MATH] are seen in the center and the Northeastern tip.', '1308.6112-1-185-6': 'The strong backbone of this source has still an integrated intensity of [MATH] Kkms[MATH], where line peak intensity goes up to 18 K.', '1308.6112-1-186-0': 'The central western part of the source moves with a relative radial velocity of 112 kms[MATH] and increases to 118 kms[MATH] toward both ends of the crescent.', '1308.6112-1-186-1': 'The lines shows widths of 2 to 3 kms[MATH] in the central region, decreasing to 1 kms[MATH] in the outer parts.', '1308.6112-1-187-0': 'H[MATH] column densities only reach a few [MATH] cm[MATH] across the inner parts of the structure.', '1308.6112-1-187-1': 'We find a total mass of [MATH].', '1308.6112-1-188-0': 'In the dust emission maps of ATLASGAL and Hi-GAL only the strong backbone of this source can be seen.', '1308.6112-1-188-1': 'The weaker outliers are not traced.', '1308.6112-1-188-2': 'The Spitzer 8 [MATH]m map shows several bright compact sources in the center and some extended emission in the South is most probably related to this source.', '1308.6112-1-188-3': 'However, the northern tip, that shows strong CO emission, is not traced by Spitzer at all.', '1308.6112-1-189-0': '# List of found sources'} | {'1308.6112-2-0-0': 'Large scale IRAM 30m CO-observations in the giant molecular cloud complex W43 IRAM 30m CO-observations in W43', '1308.6112-2-1-0': 'We aim to fully describe the distribution and location of dense molecular clouds in the giant molecular cloud complex W43.', '1308.6112-2-1-1': 'It was previously identified as one of the most massive star-forming regions in our Galaxy.', '1308.6112-2-1-2': 'To trace the moderately dense molecular clouds in the W43 region, we initiated W43-HERO, a large program using the IRAM 30m telescope, which covers a wide dynamic range of scales from 0.3 to 140 pc.', '1308.6112-2-1-3': 'We obtained on-the-fly-maps in [MATH]CO (2-1) and C[MATH]O (2-1) with a high spectral resolution of 0.1 kms[MATH] and a spatial resolution of 12.', '1308.6112-2-1-4': 'These maps cover an area of [MATH]1.5 square degrees and include the two main clouds of W43 and the lower density gas surrounding them.', '1308.6112-2-1-5': 'A comparison to Galactic models and previous distance calculations confirms the location of W43 near the tangential point of the Scutum arm at approximately 6 kpc from the Sun.', '1308.6112-2-1-6': 'The resulting intensity cubes of the observed region are separated into subcubes, which are centered on single clouds and then analyzed in detail.', '1308.6112-2-1-7': 'The optical depth, excitation temperature, and H[MATH] column density maps are derived out of the [MATH]CO and C[MATH]O data.', '1308.6112-2-1-8': 'These results are then compared to those derived from Herschel dust maps.', '1308.6112-2-1-9': 'The mass of a typical cloud is several [MATH] while the total mass in the dense molecular gas (> [MATH] cm[MATH]) in W43 is found to be [MATH].', '1308.6112-2-1-10': 'Probability distribution functions obtained from column density maps derived from molecular line data and Herschel imaging show a log-normal distribution for low column densities and a power-law tail for high densities.', '1308.6112-2-1-11': 'A flatter slope for the molecular line data probability distribution function may imply that those selectively show the gravitationally collapsing gas.', '1308.6112-2-2-0': '# Introduction', '1308.6112-2-3-0': 'The formation of high-mass stars is still not fully understood, although they play an important role in the cycle of star-formation and in the balance of the interstellar medium.', '1308.6112-2-3-1': 'What we do know is that these stars form in giant molecular clouds (GMCs) .', '1308.6112-2-3-2': 'To understand high-mass star-formation, it is crucial to understand these GMCs.', '1308.6112-2-3-3': 'One of the most important points to be studied is their formation process.', '1308.6112-2-4-0': 'The region around 30 Galactic longitude was identified as one of the most active star-forming regions in the Galaxy about 10 years ago .', '1308.6112-2-4-1': 'It was shown to be heated by a cluster of Wolf-Rayet and OB stars .', '1308.6112-2-4-2': 'Although the object W43 was previously known, [CITATION] were the first to consider it as a Galactic ministarburst region.', '1308.6112-2-5-0': 'In the past, the name W43 was used for the single cloud (G030.8+0.02) that is known today as W43-Main.', '1308.6112-2-5-1': '[CITATION] characterized the complex by analyzing atomic hydrogen continuum emission from the Very Large Array (VLA) and the [MATH]CO (1-0) and [MATH]CO (1-0) Galactic plane surveys.', '1308.6112-2-5-2': 'They concluded that W43-Main and G29.96-0.02 (now called W43-South) should be considered as a single connected complex.', '1308.6112-2-6-0': 'From the position in the Galactic plane and its radial velocity, [CITATION] concluded that W43 is located at the junction point of the Galactic long bar and the Scutum spiral arm at 6 kpc relative to the Sun.', '1308.6112-2-6-1': 'The kinematic distance ambiguity, arising from the Galactic rotation curve, gives relative distances for W43 of [MATH]6 and [MATH]8.5 kpc for the near and the far kinematic distance, respectively.', '1308.6112-2-6-2': 'Although there have been other distances adopted by other authors , most publications favor the near kinematic distance.', '1308.6112-2-7-0': 'This position in the Galaxy makes W43 a very interesting object for studying the formation of molecular clouds.', '1308.6112-2-7-1': 'Despite its distance, it is possible to analyze the details of this cloud due to its large spatial scale of [MATH]150 pc and the large amount of gas at high density .', '1308.6112-2-8-0': 'In this paper, we present initial results of the large IRAM 30m project entitled "W43 Hera/EmiR Observations" (W43 HERO, PIs P. Schilke and F. Motte).', '1308.6112-2-8-1': 'By observing the kinematic structure of this complex, the program aims to draw conclusions about the formation processes of both molecular clouds from atomic gas and high-mass stars from massive clouds and so-called ridges.', '1308.6112-2-9-0': 'One part of the project aimed at mapping the large scale mid-density molecular gas ([MATH] cm[MATH]) in the complete W43 region in the [MATH]CO (2-1) and C[MATH]O (2-1) emission lines.', '1308.6112-2-9-1': 'The resulting dataset and a first analysis is presented in this paper.', '1308.6112-2-9-2': 'The second part of the project observed several high-density tracers in the densest parts of W43.', '1308.6112-2-9-3': 'This data and its analysis will be published in a separate article (Nguyen Luong et al. in prep).', '1308.6112-2-10-0': 'This paper is structured as follows.', '1308.6112-2-10-1': 'We first give an overview of the observations and the technical details in Sect. [REF].', '1308.6112-2-10-2': 'In Sect. [REF], we present the resulting line-integrated and PV-maps and a list of clouds that were separated using the Duchamp Sourcefinder software .', '1308.6112-2-10-3': 'We also show the velocity structure of the complex and determine its position in the Milky Way in Sect. [REF].', '1308.6112-2-10-4': 'Section [REF] describes the calculations that we conducted, including optical depth, excitation temperature, and column density of the gas.', '1308.6112-2-10-5': 'We then systematically compare our data to other datasets in Sect. [REF] to further characterize the sources we identified.', '1308.6112-2-10-6': 'In Sect. [REF], we give a more detailed description of the main clumps.', '1308.6112-2-10-7': 'The summary and conclusions are given in Sect. [REF]', '1308.6112-2-11-0': '# Observations', '1308.6112-2-12-0': 'The following data has been observed with the IRAM 30m telescope on Pico Veleta, Spain between November 2009 and March 2011.', '1308.6112-2-12-1': 'We simultaneously observed the molecular emission lines [MATH]CO (2-1) and C[MATH]O (2-1) at 220.398684 GHz and 219.560358 GHz, respectively.', '1308.6112-2-12-2': 'Smaller regions around the two main cloud complexes were additionally observed in high-density tracers, such as HCO[MATH] (3-2), H[MATH]CO[MATH] (2-1), N[MATH]H[MATH] (1-0), and C[MATH]S (2-1) (Nguyen Luong et al. in prep.)', '1308.6112-2-13-0': 'This survey spans the whole W43 region, which includes the two main clouds, W43-Main and W43-South, and several smaller clouds in their vicinity.', '1308.6112-2-13-1': 'It covers a rectangular map with a size of [MATH]1.4[MATH]1.0 degrees.', '1308.6112-2-13-2': 'This translates to spatial dimensions of [MATH]105 pc, given an estimated distance of about 6 kpc to the source (see Sect. [REF]).', '1308.6112-2-13-3': 'The center of the map lies at 18:46:54.4 -02:14:11 (EQ J2000).', '1308.6112-2-13-4': 'The beam size of the [MATH]CO and C[MATH]O observations is 11.7[MATH], which corresponds to 0.34 pc at this distance.', '1308.6112-2-14-0': 'For the observations, we used the heterodyne receiver array (HERA) of the IRAM 30m .', '1308.6112-2-14-1': 'It consists of 3[MATH]3 pixels separated by 24[MATH] and has two polarizations, which point at the same location on the sky.', '1308.6112-2-14-2': 'This gave us the possibility of observing both CO isotopologes in one pass, where we observed one line per polarization.', '1308.6112-2-14-3': 'The instrument HERA can be tuned in the range of 215 to 272 GHz and has a receiver noise temperature of about 100 K at 220 GHz.', '1308.6112-2-14-4': 'Typically, the system temperature of the telescope was in the range of 300 to 400 K during our observations.', '1308.6112-2-15-0': 'We used the versatile spectrometer assembly (VESPA) autocorrelator as a backend, which was set to a spectral resolution of 80 kHz per channel with a bandwidth of 80 MHz.', '1308.6112-2-15-1': 'This translates to a resolution of 0.15 kms[MATH] and a bandwidth of [MATH]100 kms[MATH], which was set to cover the velocity range of 30 kms[MATH] to 130 kms[MATH] to cover the complete W43 complex.', '1308.6112-2-16-0': 'A Nyquist-sampled on-the-fly mapping mode was used to cover 10[MATH] tiles, taking about 20 minutes each.', '1308.6112-2-16-1': 'Each tile was observed in two orthogonal scanning directions to reduce striping in the results.', '1308.6112-2-16-2': 'The tiles uniformly cover the whole region.', '1308.6112-2-16-3': 'A total of [MATH]3 million spectra in both CO lines was received that way, taking a total observation time of nearly 80 hours.', '1308.6112-2-17-0': 'Calibration scans, pointing, and focus were done on a regular basis to assure a correct calibration later.', '1308.6112-2-17-1': 'Calibration scans were done every 10 minutes and a pointing every 60 to 90 minutes.', '1308.6112-2-17-2': 'A focus scan was done every few hours with more scans performed around sunset and sunrise as the atmosphere was less stable then.', '1308.6112-2-17-3': 'For the pointing, we used G34.3, a strong nearby ultracompact Hii region.', '1308.6112-2-17-4': 'The calibration was conducted with the MIRA package, which is part of the GILDAS software.', '1308.6112-2-17-5': 'We expect the flux calibration to be accurate within error limits of [MATH].', '1308.6112-2-18-0': '## Data reduction', '1308.6112-2-19-0': 'The raw data were processed using the GILDAS[MATH] software package.', '1308.6112-2-19-1': 'The steps taken for data reduction included flagging of bad data (e.g., noise level that are too high or platforms that could not be removed), platform removal in the spectra, baseline subtraction, and gridding to create three dimensional data cubes.', '1308.6112-2-19-2': 'About 10 percent of the data had to be flagged due to excessive platforming or strong noise.', '1308.6112-2-19-3': 'Platforming, which is an intensity jump in the spectra, is sometimes produced by the VESPA backend and occurs only in specific pixels of the array that correspond to fixed frequencies.', '1308.6112-2-19-4': 'We calculated the intensity offsets by taking several baselines on each side of the jump to remove these effects on the spectra.', '1308.6112-2-20-0': 'Baseline subtraction turned out to be complicated in some regions that are crowded with emission over a large part of the band.', '1308.6112-2-20-1': 'A first order baseline fit was usually adequate, but a second order baseline was needed for a small number of pixels and scans.', '1308.6112-2-20-2': 'We then corrected for the main beam efficiency via [MATH], where [MATH] was the forward efficiency of the IRAM 30m telescope and [MATH] was the main beam efficiency at 210 GHz (No efficiency measurements had been carried out for the IRAM 30m at 220 GHz, but the values should not deviate much from those at 210 GHz).', '1308.6112-2-21-0': 'Finally, the single spectra were gridded to two data cubes with one for each line.', '1308.6112-2-21-1': 'The pixels are separated by half beam steps, 5.9[MATH] in spatial dimension, and have channel widths of 0.15 kms[MATH].', '1308.6112-2-21-2': 'This step includes the convolution with a Gaussian of beam width.', '1308.6112-2-21-3': 'The final cubes have dimensions of 631[MATH]917 data points (RA-DEC-velocity).', '1308.6112-2-22-0': 'The noise of single spectra varied with the weather and also with the pixel of the HERA-array.', '1308.6112-2-22-1': 'Maps that show the noise level for each spatial point for both lines are shown in Fig. [REF] in the Appendix.', '1308.6112-2-22-2': 'Typical values are [MATH]1 K, which are corrected for the main beam efficiency, while several parts of the southern map that are observed during worse weather conditions have rms values of up to 3 K.', '1308.6112-2-22-3': 'In general, the noise level of the C[MATH]O is higher than that of the [MATH]CO line.', '1308.6112-2-22-4': 'Despite our dedicated reduction process, scanning effects are still visible in the resulting maps.', '1308.6112-2-22-5': 'They appear as stripes (See upper part of [MATH]CO map in Fig. [REF].)', '1308.6112-2-22-6': 'and tiling patterns (See noise difference of diffuse parts of C[MATH]O in Fig. [REF].)', '1308.6112-2-23-0': '# Results', '1308.6112-2-24-0': 'Integrated intensity maps of the whole W43 region in both [MATH]CO (2-1) and C[MATH]O (2-1) lines are shown in Fig. [REF].', '1308.6112-2-24-1': 'The maps use the entire velocity range from 30 kms[MATH] to 130 kms[MATH] and show a variety of clouds and filaments.', '1308.6112-2-24-2': 'The two main cloud complexes, W43-Main in the upper left part of the maps and W43-South in the lower right part, are clearly visible.', '1308.6112-2-25-0': 'In Fig. [REF], we show several spectra taken from the data.', '1308.6112-2-25-1': 'The upper plot shows the spectra of [MATH]CO (2-1) and C[MATH]O (2-1) averaged over the complete complex; the center and bottom plots show averaged spectra of the W43-Main and W43-South clouds.', '1308.6112-2-25-2': 'The spectra of the complete cubes show emission nearly across the whole velocity range in [MATH]CO.', '1308.6112-2-25-3': 'Only the components at [MATH] kms[MATH] and velocities higher than 120 kms[MATH] do not show any emission.', '1308.6112-2-25-4': 'The C[MATH]O follows that distribution, although it is not as broad.', '1308.6112-2-25-5': 'We thus can already distinguish two separated velocity components with one between 35 and 55 kms[MATH].', '1308.6112-2-25-6': 'Most of the emission is concentrated in the velocity range between 65 and 120 kms[MATH].', '1308.6112-2-25-7': 'To give an impression of the complexity of some sources, we plot several spectra of the W43-Main cloud in Fig. [REF].', '1308.6112-2-26-0': '## Decomposition into subcubes', '1308.6112-2-27-0': 'The multitude of sources found in the W43 region complicates the analysis of the complete data cube.', '1308.6112-2-27-1': 'Details get lost when integrating over a range in frequency that is too large.', '1308.6112-2-27-2': 'We want to examine each source separately, so we need to decompose the data cube into subcubes that only contain one single source each.', '1308.6112-2-27-3': 'This is only done on the [MATH]CO cube, as this is the stronger molecular line.', '1308.6112-2-27-4': 'This breakdown is then copied to the C[MATH]O cube.', '1308.6112-2-28-0': 'We use the Duchamp Sourcefinder software package to automatically find a decomposition.', '1308.6112-2-28-1': 'See [CITATION] for a detailed description of this software.', '1308.6112-2-28-2': 'The algorithm finds connected structures in three-dimensional ppv-data cubes by searching for emission that lies above a certain threshold.', '1308.6112-2-28-3': 'The value of this threshold is crucial for the success of the process and needs to be carefully adjusted by hand.', '1308.6112-2-28-4': 'For the decomposition of the [MATH]CO cube, we use two different cutoffs.', '1308.6112-2-28-5': 'The lower cutoff of [MATH] per channel is used to identify weaker sources.', '1308.6112-2-28-6': 'A higher cutoff of [MATH] per channel is needed to distinguish sources in the central part of the complex.', '1308.6112-2-29-0': 'We identify a total of 29 clouds (see Table [REF]), 20 in the W43 complex itself and 9 in the fore-/background (see Sect. [REF] for details).', '1308.6112-2-29-1': 'The outcome of this method is not trivial, as it is not always clear which parts are still to be considered associated and which are separate structures.', '1308.6112-2-29-2': 'It still needs some correction by hand in some of the very weak sources and the strong complexes.', '1308.6112-2-29-3': 'A few weak sources that have been identified by eye are manually added to our list (e.g. sources 18 and 28).', '1308.6112-2-29-4': 'These are clearly coherent separate structures but are not identified by the algorithm.', '1308.6112-2-29-5': 'On the other hand, a few sources are merged by hand (e.g. source 26) that clearly belong together but are divided into several subsources by the software.', '1308.6112-2-29-6': 'Some of these changes are open to interpretation, but they show that some adjustment of the software result is needed.', '1308.6112-2-29-7': 'However, the algorithm works well and identifies 25 out of 29 clouds on its own.', '1308.6112-2-30-0': 'The resulting detection map is shown in Fig. [REF].', '1308.6112-2-30-1': 'The numbers shown there are color-coded to show the distance of each detected cloud (See Sect. [REF] for details.)', '1308.6112-2-30-2': 'Sources are sorted by their peak velocities.', '1308.6112-2-30-3': 'See Table [REF] for positions and dimensions of the clouds, Table [REF] for derived properties, and Sect. [REF] for a detailed description of the main complexes, while plots of all clouds can be found in Appendix [REF].', '1308.6112-2-30-4': 'We see a number of different sizes and shapes that range from small spherical clouds to expanded filaments and more complex structures.', '1308.6112-2-31-0': 'The resulting data cubes show clouds of different shapes, while the typical spatial scales lie in the range of 10 to 20 pc (see Table [REF]).', '1308.6112-2-31-1': 'We also show the area that the [MATH]CO emission of each source covers, which was determined by defining a polygon for each source that contains the [MATH]CO emission.', '1308.6112-2-31-2': 'Thus, it accounts for shapes that deviate from spheres or rectangles, which is true for most of our clouds.', '1308.6112-2-31-3': 'Hereafter we define clouds as objects with a size on the order of 10 pc, while we define clumps as objects on the parsec scale.', '1308.6112-2-31-4': 'The whole W43 region is considered a cloud complex.', '1308.6112-2-32-0': '## PV-diagram of the region', '1308.6112-2-33-0': 'For a more advanced analysis of the velocity structure, we create a position velocity diagram of the [MATH]CO (2-1) line that is averaged along the Galactic latitude (see Fig. [REF] (a)).', '1308.6112-2-33-1': 'The distribution of emission across the velocity range that is seen in the averaged spectra in Fig. [REF] can also be identified here but with additional spatial information along the Galactic longitude.', '1308.6112-2-33-2': 'We note the similarity of this figure to the plot of [MATH]CO (1-0) displayed in [CITATION].', '1308.6112-2-33-3': 'However, we see more details in our plot due to the higher angular resolution of our data.', '1308.6112-2-34-0': 'We further analyze the position velocity diagram, as shown in Fig. [REF] (a), to separate our cube into several velocity components.', '1308.6112-2-34-1': 'We assume that these different components are also spatially separated.', '1308.6112-2-35-0': 'Two main velocity complexes can be distinguished: one between 35 and 55 kms[MATH], the other between 65 and 120 kms[MATH].', '1308.6112-2-35-1': 'Both complexes are clearly separated from each other, indicating that they are situated at different positions in the Galaxy.', '1308.6112-2-35-2': 'On second sight, it becomes clear that the complex from 65 to 120 kms[MATH] breaks down into a narrow component, spanning the range from 65 to 78 kms[MATH] and a broad component between 78 and 120 kms[MATH].', '1308.6112-2-35-3': 'Analyzing the channel maps of the cube verifies that these structures are indeed separated.', '1308.6112-2-35-4': 'See Fig. [REF] for integrated maps of each velocity component.', '1308.6112-2-36-0': 'On the other hand, it is clearly visible that all three velocity components span the complete spatial dimension along the Galactic longitude.', '1308.6112-2-36-1': 'The broadest complex at 78-120 kms[MATH] shows two major components at 29.9[MATH] and 30.8[MATH] that coincide with W43-South and W43-Main, respectively.', '1308.6112-2-36-2': 'The gap between both complexes is bridged by a smaller clump, and all three clumps are surrounded by diffuse gas, which forms an envelope around the whole complex.', '1308.6112-2-36-3': 'It is thus suggested to consider W43-Main and W43-South as one giant connected molecular cloud complex.', '1308.6112-2-36-4': 'This connection becomes more clear in the PV-plot than in the spatial map in Fig. [REF] , although the averaging of values causes blurring which might merge structures.', '1308.6112-2-37-0': 'The lower velocity complex between 35 and 55 kms[MATH] is a bit more fragmented than the other components.', '1308.6112-2-37-1': 'One central object at 30.6[MATH] spans the whole velocity range, but it splits into two subcomponents to the edges of the map.', '1308.6112-2-37-2': 'It is hard to tell if we actually see one or two components.', '1308.6112-2-38-0': '## Determination of the distance of W43', '1308.6112-2-39-0': 'We can analyze our data by using a simple rotational model of the Milky Way.', '1308.6112-2-39-1': 'For this model, we assume a rotational curve that increases linearly in the inner 3 kpc of the Galaxy, where the bar is situated.', '1308.6112-2-39-2': 'For radii larger than that, we assume a rotation curve that has a value of 254 kms[MATH] at [MATH] and slightly rises with the radius at a rate of 2.3 kms[MATH]kpc[MATH] .', '1308.6112-2-39-3': 'The Galactocentric radius of a cloud with a certain relative velocity can be calculated using the formula [EQUATION] as used in [CITATION].', '1308.6112-2-39-4': 'The parameter [MATH] is the Galactocentric radius of the Sun, which is assumed to be 8.4 kpc , [MATH] is the Galactic longitude of the source (30[MATH]), [MATH] the radial velocity of the Sun (254 kms[MATH]), and [MATH] the radial velocity of the source.', '1308.6112-2-39-5': 'The parameter [MATH] is the measured relative velocity between the source and the Sun.', '1308.6112-2-39-6': 'With the knowledge of the radius [MATH], we can then compute the relative distance to the source by the equation: [EQUATION]', '1308.6112-2-39-7': 'Up to the tangent point, two different possible distances exist for each measured velocity: one in front of and one behind the tangent point.', '1308.6112-2-39-8': 'However, this calculation is not entirely accurate as the assumptions of the geometry of the Galaxy bear large errors.', '1308.6112-2-39-9': '[CITATION] state that the errors of the kinematic distance can sum up to a factor as high as 2.', '1308.6112-2-39-10': 'One main reason for uncertainties are the streaming motions of molecular clouds relative to the motion of the spiral arms .', '1308.6112-2-39-11': 'Figure [REF] (b) shows the kinematic distance curve for our case at 30[MATH] Galactic longitude.', '1308.6112-2-39-12': 'This works only for the circular orbits in the spiral arms and not the elliptical orbits in the Galactic bar .', '1308.6112-2-40-0': 'To determine the location of each velocity complex, we need to break the kinematic distance ambiguity; that is we need to decide for each complex if we assume it to be on the near or the far side of the tangent point.', '1308.6112-2-40-1': 'Here, we use the distance estimations by [CITATION], who utilized HI self absorption from the VGPS project .', '1308.6112-2-40-2': 'It is possible to associate several entries of their extensive catalog with clouds we found in our dataset.', '1308.6112-2-40-3': 'Thus, we are able to remove the distance ambiguity and attribute distances to these clouds.', '1308.6112-2-40-4': 'In combination with the detailed model of the Milky Way of [CITATION], we are able to fix their position in our Galaxy.', '1308.6112-2-40-5': 'We cannot assign a distance to each single cloud in our dataset, as each was not analyzed by [CITATION] We assume the missing sources to have the same distances as sources nearby.', '1308.6112-2-40-6': 'This may not be exact in all cases, but the only unclear assignments are two clouds in the 35 to 55 kms[MATH] velocity component (sources 1 and 3).', '1308.6112-2-40-7': 'The distance of the W43 complex is unambiguous as it is well determined by the calculations found in [CITATION].', '1308.6112-2-41-0': 'The object W43 (78 to 120 kms[MATH]) is found to lie on the near side of the tangential point with distances from 5 to 7.3 kpc, which increases with radial velocity.', '1308.6112-2-41-1': 'This places it near the tangential point of the Scutum arm at a Galactocentric radius of [MATH] kpc (marker 1 in Fig. [REF] (c)).', '1308.6112-2-41-2': 'For our analysis, we use an average distance of 6 kpc for the whole complex, since this is where the mass center is located.', '1308.6112-2-42-0': 'The second velocity component (65 to 78 kms[MATH]) lies in the foreground of the first one at a distance of 4.5 kpc to the Sun and [MATH] kpc (marker 2 in Fig. [REF] (c)).', '1308.6112-2-42-1': 'Another indication that these sources are located on the near side of the tangential point is their position above the Galactic plane, as seen in Fig. [REF] (b).', '1308.6112-2-42-2': 'The larger the distance is from the Sun, the further above the Galactic plane it would be positioned.', '1308.6112-2-42-3': 'This would be difficult to explain, since high-mass star-forming regions are typically located within the plane.', '1308.6112-2-42-4': 'It is unclear if this cloud is still situated in the Scutum arm or if it is located between spiral arms.', '1308.6112-2-42-5': 'According to the model, it would be placed at the edge of the Scutum arm.', '1308.6112-2-42-6': 'In light of the previously discussed uncertainties, it is still possible that this cloud is part of the spiral arm.', '1308.6112-2-43-0': 'The third component between 35 and 55 kms[MATH] is more complex than the others, since we find sources to be located on both the near and far side of the tangential point.', '1308.6112-2-43-1': 'The brightest sources in the center of our map are in the background of W43 in the Perseus arm with a distance of 11 to 12 kpc to the Sun and of [MATH] kpc to the (marker 3 in Fig. [REF] (c)).', '1308.6112-2-43-2': 'However, several other sources in the north and south are found by [CITATION] to be near the Sun at 3.5 to 4 kpc (marker 3[MATH] in Fig. [REF] (c)).', '1308.6112-2-43-3': 'These sources also have a Galactocentric radius of 6 kpc.', '1308.6112-2-43-4': 'Table [REF] gives an overview of the distance of each source, while Fig. [REF] shows integrated intensity plots of the individual velocity components.', '1308.6112-2-44-0': 'We can now apply this calculation to our data in Fig. [REF] (a) by changing the velocity scale into a distance scale.', '1308.6112-2-44-1': 'The distance scale is inaccurate for the parts of the lowest component that lie on the far side of the tangent point.', '1308.6112-2-44-2': 'Although it is not possible to disentangle clouds that are nearby and far away in this plot, we still show this scale.', '1308.6112-2-44-3': 'These values, taken from the rotation curve, are smaller than the actual distances found when compared to [CITATION], since we used the newer rotation curve of [CITATION].', '1308.6112-2-44-4': '[CITATION] use the older values from [CITATION], which explains the discrepancy.', '1308.6112-2-44-5': 'However, this axis still gives us an idea of the distribution of the clouds.', '1308.6112-2-44-6': 'We note that no distance can be assigned for velocities larger than 112 kms[MATH], hence the zero for the 120 kms[MATH] tick in Fig. [REF] (a).', '1308.6112-2-44-7': 'Subplot (d) shows the related modeled PV-diagram from [CITATION].', '1308.6112-2-44-8': 'Our dataset is indicated by the gray box.', '1308.6112-2-45-0': 'Figure [REF] (c) summarizes our determination of distance in a plot taken from [CITATION].', '1308.6112-2-45-1': 'The W43 complex (78-120 kms[MATH], marker 1) lies between 5 and 7 kpc, where the distance increases with velocity, which we found to be located on the near branch.', '1308.6112-2-45-2': 'The complex 65 to 78 kms[MATH] (marker 2) is located at the near edge of the Scutum arm, while the 35 to 55 kms[MATH] component is marked by 3 and 3[MATH] on both sides of the tangent point.', '1308.6112-2-45-3': 'The far component is located in the Perseus arm at 12 kpc distance and the near component at a distance of 4.5 kpc between the Scutum and the Sagittarius arm.', '1308.6112-2-46-0': 'It may be a bit surprising that no emission from the local part of the Sagittarius arm is seen in our dataset.', '1308.6112-2-46-1': 'The reason is that our observed velocity range only goes to 30 kms[MATH].', '1308.6112-2-46-2': 'Possible molecular clouds nearby would have even lower relative velocities of [MATH]20 kms[MATH], which can be seen in the model in Fig. [REF] (d).', '1308.6112-2-46-3': 'In [CITATION], the [MATH]CO (1-0) spectrum, which is averaged over the W43 complex, shows an additional velocity component at 5 to 15 kms[MATH] which fits to this spiral arm.', '1308.6112-2-47-0': '## Peak velocity and line width', '1308.6112-2-48-0': 'After separating the different velocity components, we created moment maps of each component.', '1308.6112-2-48-1': 'For each spectra of the [MATH]CO data cube, a Gaussian line profile was fitted.', '1308.6112-2-48-2': 'The first moment resembles the peak velocity, the position of the line peak.', '1308.6112-2-48-3': 'The second moment is the width of the line.', '1308.6112-2-48-4': 'The maps of the W43 complex are shown in Fig. [REF], while the plots of the background components can be seen in the Appendix in Fig. [REF].', '1308.6112-2-48-5': 'Care should be taken in interpretation of these maps.', '1308.6112-2-48-6': 'As some parts of the maps show complex spectra (see Fig. [REF] for some examples), a Gaussian profile is not always a good approximation.', '1308.6112-2-48-7': 'In regions where several velocity components are found, the maps only give information about the strongest component.', '1308.6112-2-48-8': 'In case of self-absorbed lines, the maps may even be misleading.', '1308.6112-2-48-9': 'This especially concerns the southern ridge of W43-Main, called W43-MM2 as defined in .', '1308.6112-2-49-0': 'The line peak velocity map in Fig. [REF] (a) traces a variety of coherent structures.', '1308.6112-2-49-1': 'Most of these correspond to the sources we identified with the Duchamp software.', '1308.6112-2-49-2': 'However, some structures, as mentioned above, overlap and cannot be defined simply from using this velocity map.', '1308.6112-2-50-0': 'The two main clouds, W43-Main and W43-South, are again located in the upper left and lower right part of map, respectively.', '1308.6112-2-50-1': 'As in the PV-diagram in Fig. [REF] (a), we note that both clouds are slightly shifted in velocity.', '1308.6112-2-50-2': 'While W43-Main lies in the range of 85 to 100 kms[MATH], W43-South spans velocities from 95 to 105 kms[MATH].', '1308.6112-2-50-3': 'Several smaller sources bridge the gap between the two clouds, especially in the higher velocities.', '1308.6112-2-50-4': 'This structure is also seen in the PV-diagram.', '1308.6112-2-51-0': 'In comparison to the PV-diagram, this plot shows the peak velocity distribution in both spatial dimensions.', '1308.6112-2-51-1': 'On the other hand, we lose information of the shape of the lines.', '1308.6112-2-51-2': 'Here, we see that the velocity of W43-South is rather homogeneous across the whole cloud.', '1308.6112-2-51-3': 'In contrast, W43-Main shows strong velocity gradients from west to east and from south to north, which are already seen in [CITATION].', '1308.6112-2-51-4': 'The velocity changes by at least 30 kms[MATH] on a scale of 25 pc.', '1308.6112-2-51-5': 'We can interpret this as mass flows across the cloud, which makes it kinematically much more active than W43-South.', '1308.6112-2-52-0': 'Figure [REF] (b) shows a map of the FWHM line width of each pixel.', '1308.6112-2-52-1': 'Some parts in W43-Main show unrealistically large values of more than 10 kms[MATH].', '1308.6112-2-52-2': 'This is a line-of-sight effect and originates in several velocity components located at the same point on the sky.', '1308.6112-2-52-3': 'Therefore, it is more accurate to analyze the line width of each source separately.', '1308.6112-2-52-4': 'From these single sources, we determine the mean line width, which is given in Table [REF] (7).', '1308.6112-2-53-0': '# Analysis', '1308.6112-2-54-0': '## Calculations', '1308.6112-2-55-0': '### Optical depth', '1308.6112-2-56-0': 'For each identified source, we conducted a series of calculations to determine its physical properties.', '1308.6112-2-56-1': 'We did this on a pixel by pixel basis, using maps integrated over the velocity range that is covered by the specific source.', '1308.6112-2-56-2': 'The optical depth of the [MATH]CO gas was calculated from the ratio of the intensities of [MATH]CO (2-1) and C[MATH]O (2-1), assuming that C[MATH]O is optically thin.', '1308.6112-2-56-3': '(This assumption holds for H[MATH] column densities up to [MATH] cm[MATH], but a clear threshold cannot be given.)', '1308.6112-2-56-4': 'Then we computed the excitation temperature of this gas and the H[MATH] column density, which was then used to estimate the total mass along the line-of-sight.', '1308.6112-2-56-5': 'All these calculations are explained in detail in Appendix [REF].', '1308.6112-2-56-6': 'Example maps for a small filament (source 29) can be seen in Fig. [REF].', '1308.6112-2-57-0': 'We first calculated the [MATH]CO (2-1) optical depth from the ratio of the two CO lines.', '1308.6112-2-57-1': 'We note, that the intrinsic ratios of the different CO isotopologes used for this calculation are dependent on the Galactocentric radius, so we have to use different values for W43 and the fore-/background sources.', '1308.6112-2-57-2': 'An example map of [MATH] of [MATH]CO (2-1) is shown in Fig. [REF] (c).', '1308.6112-2-57-3': 'Typical clouds have optical depths of a fraction of 1 in the outer parts and up to 4 at most in the central cores.', '1308.6112-2-57-4': 'The extreme case is the W43-main cloud, where the [MATH]CO optical depth goes up to 8.', '1308.6112-2-57-5': 'This means that most parts of the clouds are optically thin and we can see through them.', '1308.6112-2-57-6': 'Even at positions where [MATH]CO become optically thick, C[MATH]O still remains optically thin.', '1308.6112-2-57-7': 'Only for the extreme case of the densest part of W43-Main, C[MATH]O starts to become optically thick.', '1308.6112-2-57-8': 'This means that the combination of the two isotopologes reveals most of the information about the medium density CO gas in the W43 complex.', '1308.6112-2-58-0': '### Excitation temperature', '1308.6112-2-59-0': 'The formula used for the computation of the excitation temperatures is explained in Appendix [REF].', '1308.6112-2-59-1': 'The resulting map is shown in Fig. [REF].', '1308.6112-2-59-2': 'Certain assumptions are made.', '1308.6112-2-59-3': 'First, we assumed that [MATH] is the same for the [MATH]CO and the C[MATH]O gas.', '1308.6112-2-59-4': 'This method becomes unrealistic when the temperature distribution along the line-of-sight is not uniform anymore.', '1308.6112-2-59-5': 'If there was a temperature gradient, we would miss the real ratio of the [MATH]CO to C[MATH]O line intensities and thus either over- or underestimate the temperature.', '1308.6112-2-59-6': 'Thus the calculated temperature might be incorrect for very large cloud structures that show a complex temperature distribution along the line-of-sight.', '1308.6112-2-59-7': 'This problem is partially circumvented by using the spectral information of our observations, but we use intensity maps integrated over at least several kms[MATH] for our calculation of the excitation temperature, which still leaves room for uncertainties.', '1308.6112-2-59-8': 'This means we do not confuse different clouds, but we still average the temperature along the line-of-sight over the complete clouds.', '1308.6112-2-60-0': 'The centers of the two main clouds in the W43 region are candidates for an underestimated excitation temperature.', '1308.6112-2-60-1': 'In case these regions were internally heated, a decreasing gradient in the excitation temperature would appear from the inside of the cloud to the outside.', '1308.6112-2-60-2': 'As [MATH]CO is rather optically thick, only the cool outside of the cloud would be seen by the observer.', '1308.6112-2-60-3': 'In contrast, C[MATH]O would be optically thin; thus, the hot center of the cloud would also be observed.', '1308.6112-2-60-4': 'Averaging along the line-of-sight, [MATH](C[MATH]O) would be increased relative to [MATH]CO), which would lead to an overestimated optical depth.', '1308.6112-2-60-5': 'This would then lead to an underestimated calculated excitation temperature.', '1308.6112-2-60-6': 'External heating, on the other hand, would result in an overestimated excitation temperature.', '1308.6112-2-60-7': 'However, we find the first case with regard to low excitation temperatures is more likely in some clouds.', '1308.6112-2-61-0': 'Another effect, which leads to a reduced excitation temperature, is the beam-filling factor [MATH].', '1308.6112-2-61-1': 'In our calculations, we assume it to be 1.', '1308.6112-2-61-2': 'This corresponds to extended clouds that completely fill the telescope beam.', '1308.6112-2-61-3': 'This is not a true representation of molecular clouds, as they are structured on the subparsec scale and we would have to use a factor [MATH].', '1308.6112-2-61-4': 'Technically speaking, we calculate the value of [MATH], which is smaller than [MATH].', '1308.6112-2-62-0': 'As the C[MATH]O line is much weaker than the [MATH]CO line, we cannot use the ratio of them for those pixels where no C[MATH]O is detected, even if [MATH]CO is present.', '1308.6112-2-62-1': 'We find typical temperatures to be between 6 and 25 K; in some rare cases, it is up to 50 K with a median of 12 K.', '1308.6112-2-63-0': 'Due to the sparsely covered maps (see Fig. [REF]) and the uncertainties described above, we concluded that is was best not to use the excitation temperature maps for the following calculations of the H[MATH] column density.', '1308.6112-2-63-1': 'Instead, we assumed a constant excitation temperature for the complete W43 region.', '1308.6112-2-63-2': 'We chose the value to be 12 K, since this was the median temperature found in the W43 complex.', '1308.6112-2-63-3': 'Assuming a constant temperature value across the cloud is likely not a true representation of the cloud; in particular, it does not distinguish between star forming cores and the ambient background.', '1308.6112-2-63-4': 'However, such an assumption is a good first approximation to the temperature in the cloud and is more representative of star-forming cores than the aforementioned unrealistically low values.', '1308.6112-2-64-0': '### H[MATH] column density', '1308.6112-2-65-0': 'We also calculated the column density along the line-of-sight of the [MATH]CO gas from the assumed constant excitation temperature, the [MATH]CO integrated emission, and a correction for the opacity (See Appendix [REF] for details.)', '1308.6112-2-65-1': 'Assuming a constant ratio between [MATH]CO and H[MATH], it was then possible to find the H[MATH] column density.', '1308.6112-2-65-2': 'Please note that H[MATH] column densities derived by assuming a constant temperature are also subject to the same caveats and accuracies.', '1308.6112-2-66-0': 'All ratios between H[MATH] and CO isotopologes bear errors, since they depend on the Galactocentric radius.', '1308.6112-2-66-1': 'These errors add up with the uncertainty on the assumed excitation temperature.', '1308.6112-2-66-2': 'The final results for column densities and masses must be taken with caution, because there is at least an uncertainty of a factor of 2.', '1308.6112-2-66-3': 'Fig. [REF] shows the calculated H[MATH] column density map of the full W43 complex.', '1308.6112-2-66-4': 'The column density has been calculated at those points, where the [MATH]CO integrated intensity is higher than 5 Kkms[MATH].', '1308.6112-2-66-5': 'The resulting values range from values that are a few times of [MATH] cm[MATH] in the diffuse surrounding gas up to [MATH] cm[MATH] in the center of W43-Main.', '1308.6112-2-67-0': 'The southern ridge of W43-Main, where we calculate high column densities, is the most problematic part of our dataset.', '1308.6112-2-67-1': 'The spectra reveal that [MATH]CO is self-absorbed in this part of the cloud.', '1308.6112-2-67-2': 'We use the integrated intensity ratio to calculate the opacity at each point, which is strongly overestimated in this case.', '1308.6112-2-67-3': 'This leads to both low excitation temperatures and high column densities.', '1308.6112-2-67-4': 'The results for this part of the cloud should be used with caution.', '1308.6112-2-68-0': '### Total mass', '1308.6112-2-69-0': 'From the H[MATH] column density, we then determine the total mass of our sources, given in Table [REF].', '1308.6112-2-69-1': 'We find that the total mass of a typical cloud is in the range of a few [MATH] solar masses.', '1308.6112-2-69-2': 'Of course, this is only the mass seen in the mid- to high-density sources.', '1308.6112-2-69-3': 'The very extended diffuse molecular gas cannot be seen with [MATH]CO; it is generally traced by [MATH]CO lines and accounts for a major fraction of the gas mass .', '1308.6112-2-70-0': 'The total H[MATH] mass as derived from our [MATH]CO (2-1) and C[MATH]O (2-1) observations is found to be [EQUATION] for the W43 complex with about 50% within the clouds that we have identified and the rest in the diffuse surrounding gas.', '1308.6112-2-70-1': 'Here, we have excluded the foreground sources and only considered the W43 complex itself.', '1308.6112-2-70-2': '[CITATION] used similar areas and velocity ranges ([MATH]190 pc, 80-120 kms[MATH]) and determined a molecular gas mass in W43 clouds from the Galactic Ring Survey of [MATH].', '1308.6112-2-70-3': 'A different estimation of the H[MATH] column density in W43 was done by [CITATION] using Herschel dust emission maps.', '1308.6112-2-70-4': 'Using these maps, we find a value of [MATH].', '1308.6112-2-70-5': 'See Sect. [REF] for a discussion of this difference.', '1308.6112-2-71-0': 'We underestimate the real mass where [MATH]CO exists but no [MATH]CO is seen, where C[MATH]O might become optically thick, and where our assumption for the excitation temperature is too high.', '1308.6112-2-71-1': 'On the other hand, we overestimate the real mass, where our assumption for the excitation temperature is too low.', '1308.6112-2-71-2': 'In extreme cases of very hot gas, the gas mass can be overestimated by 40% at most (cp.', '1308.6112-2-71-3': 'Fig [REF] in Appendix [REF]), while very cold cores can be underestimated by a factor of nearly 10.', '1308.6112-2-71-4': 'Both effects partly cancel out each other, when integrating over the whole region; however, we estimate that the effects, which underestimate the real mass, are stronger.', '1308.6112-2-71-5': 'Therefore, the mass we calculated should be seen as a lower limit of the real molecular gas mass in the W43 complex.', '1308.6112-2-72-0': '## Shear parameter', '1308.6112-2-73-0': 'Investigating the motion of gas streams in the Galaxy is important to explain how large molecular clouds like W43 can be accumulated.', '1308.6112-2-73-1': 'While Motte et al. in prep.', '1308.6112-2-73-2': 'will investigate streams of [MATH]CO and HI gas in W43 in detail, we only consider here the aspect of radial shear in this field.', '1308.6112-2-73-3': 'The shear that is created by the differential rotation of the Galaxy at different Galactocentric radii can prevent the formation of dense clouds if it is too strong.', '1308.6112-2-73-4': 'It is possible to calculate a shear parameter [MATH] as described in [CITATION] by considering the Galactocentric radius of a region, its spatial and velocity extent, and its mass.', '1308.6112-2-73-5': 'For values of [MATH] higher than 1, the shear is so strong that clouds get ripped apart, while they are able to form for values below 1.', '1308.6112-2-74-0': 'The values we use to calculate [MATH] are the total mass, calculated above, of [MATH], a velocity extent of 40 kms[MATH], a Galactocentric radius of 4 kpc, and an area of [MATH]pc[MATH].', '1308.6112-2-74-1': 'This is the area that is covered by emission and is smaller than the total size of our map.', '1308.6112-2-74-2': 'These values yield a shear parameter [MATH].', '1308.6112-2-74-3': 'Accordingly, shear forces are not strong enough to disrupt the W43 cloud.', '1308.6112-2-74-4': 'However, we have to keep in mind that we probably underestimate the total gas mass, as described in Sect. [REF].', '1308.6112-2-74-5': 'A higher mass would lead to a lower shear parameter.', '1308.6112-2-74-6': 'This calculation is only valid for an axial symmetric potential (i.e. orbits outside the Galactic bar).', '1308.6112-2-74-7': 'Shear forces inside the Galactic bar could be stronger due to the different shape of orbits there.', '1308.6112-2-74-8': 'As we, however, located W43 at the tip of the bar, the calculation still holds.', '1308.6112-2-75-0': 'We can also conduct this calculation for the larger gas mass derived from [MATH]CO (1-0) by [CITATION].', '1308.6112-2-75-1': 'They find a gas mass of [MATH] that is spread out over an area of [MATH]pc[MATH].', '1308.6112-2-75-2': 'These values lead to a shear parameter of [MATH], which is lower than our value above.', '1308.6112-2-76-0': '## Virial masses', '1308.6112-2-77-0': 'In Table [REF] (7) and (10), we have given the mean line width and the area of our sources.', '1308.6112-2-77-1': 'This allows us to calculate virial masses by defining an effective region radius by [MATH], where [MATH] is the area of the cloud.', '1308.6112-2-77-2': 'This area cannot be determined exactly, because the extent of a cloud depends on the used molecular line.', '1308.6112-2-77-3': 'Here, we use the area of [MATH]CO emission above a certain threshold (20% of the peak intensity).', '1308.6112-2-78-0': 'Virial masses can then be computed using the relation [EQUATION] where [MATH] is the Gaussian velocity dispersion averaged over the area [MATH] and [MATH] is the gravitational constant.', '1308.6112-2-79-0': 'The resulting virial masses are shown in Table [REF] (8).', '1308.6112-2-79-1': 'We notice that most sources in W43 have masses derived from [MATH]CO that are smaller than their virial masses.', '1308.6112-2-79-2': 'Sources 4 and 5 show much larger molecular than virial masses, which might indicate that their distance was overestimated.', '1308.6112-2-79-3': 'If the sources would be completely virialized, we would need bigger masses to produce the observed line widths.', '1308.6112-2-79-4': 'On the other hand, systematic motion of the gas, apart from turbulence, like infall, outflows, or colliding flows would also broaden the lines.', '1308.6112-2-79-5': 'This could be an explanation for the observed large line widths.', '1308.6112-2-80-0': '[CITATION] stated that usually turbulent molecular clouds are not in actual virial equilibrium, since there is a flux of mass, momentum, and energy between the clouds and their environment.', '1308.6112-2-80-1': 'What is normally viewed as virial equilibrium is an energy equipartition between self-gravity, kinetic, and magnetic energy.', '1308.6112-2-80-2': 'This energy equipartition is found for most clouds due to observational limitations.', '1308.6112-2-80-3': 'Clouds out of equilibrium are either not observed due to their short lifetime or not considered clouds at all.', '1308.6112-2-81-0': 'Of course, we also need to consider the shape of our sources.', '1308.6112-2-81-1': 'Non-spherical sources have a more complicated gravitational behavior than spheres.', '1308.6112-2-81-2': 'Therefore, one has to be extremely careful using these results.', '1308.6112-2-81-3': 'In addition, we neglect here the influence of external pressure and magnetic fields on the virial masses.', '1308.6112-2-81-4': 'What we observe agrees with [CITATION] in that most of our detected clouds show a molecular mass in the order of their virial mass, or up to a factor of 2 higher.', '1308.6112-2-82-0': '# Comparison to other projects', '1308.6112-2-83-0': 'To gather more information about the W43 complex, we compare the IRAM 30m CO data to other existing datasets.', '1308.6112-2-83-1': 'We pay special attention to three large-scale surveys in this section: the Spitzer GLIMPSE and MIPSGAL projects, the Herschel Hi-GAL survey, and the Galactic plane program ATLASGAL as observed with the APEX telescope.', '1308.6112-2-84-0': 'All of these datasets consist of total power maps over certain bands.', '1308.6112-2-84-1': 'They naturally do not contain spectral information, so line-of-sight confusion is considerable, since the W43 region is a complex accumulation of different sources.', '1308.6112-2-84-2': 'It can sometimes be complicated to assign the emission of these maps to single sources.', '1308.6112-2-84-3': 'Nevertheless, the additional information is very valuable.', '1308.6112-2-85-0': '## Spitzer GLIMPSE and MIPSGAL', '1308.6112-2-86-0': 'The Spitzer Space Telescope program, Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) , observed the Galactic plane at several IR wavelengths between 3.6 and 8 [MATH].', '1308.6112-2-86-1': 'It spans the Galactic plane from [MATH] to [MATH] Galactic longitude.', '1308.6112-2-87-0': 'Here, we concentrated on the 8 [MATH] band.', '1308.6112-2-87-1': 'It is dominated by UV-excited PAH emission .', '1308.6112-2-87-2': 'These photon dominated regions (PDRs) are heated by young OB stars.', '1308.6112-2-87-3': 'By studying this band in comparison to the IRAM 30m CO maps, we can determine which parts of the molecular clouds contain UV-heated dust.', '1308.6112-2-87-4': 'This is seen as extended emission in the Spitzer maps.', '1308.6112-2-87-5': 'We can also identify nearby UV-heating sources, as seen as point sources.', '1308.6112-2-87-6': 'Finally, some parts of specific clouds appear in absorption against the background.', '1308.6112-2-87-7': 'These so-called infrared dark clouds (IRDCs) show denser dust clouds that are not heated by UV-radiation.', '1308.6112-2-87-8': 'In this way, we are able to determine which sources heat part of the gas and which parts are shielded from UV radiation.', '1308.6112-2-87-9': 'We can also tell if YSOs have already formed inside the clumps that we have observed and thus estimate the evolutionary stage of the clouds.', '1308.6112-2-87-10': '[CITATION] used this tracer to estimate the star-formation rate of the W43 complex.', '1308.6112-2-88-0': 'MIPSGAL is a Galactic plane survey using the MIPS instrument onboard Spitzer and has created maps at 24 and 70 [MATH]m. Here, we inspect the 24 [MATH]m bandm which is dominated by the emission of small dust grains.', '1308.6112-2-88-1': 'The MIPS instrument also detects proto-stellar cores, although these cores are usually too small to be resolved at a distance of 6 kpc or more.', '1308.6112-2-89-0': '## APEX ATLASGAL', '1308.6112-2-90-0': 'The APEX telescope large area survey of the galaxy (ATLASGAL) used the LABOCA camera, which is installed at the APEX telescope.', '1308.6112-2-90-1': 'It observed the Galactic plane from a Galactic longitude of [MATH] to [MATH] at 870 [MATH].', '1308.6112-2-90-2': 'This wavelength traces cold dust and is therefore also a good indicator of dense molecular cloud structures, especially of high-mass star-forming clumps.authorschuller2009 also identified hot cores, proto-stars, compact HII regions, and young embedded stars by combining their map with other data.', '1308.6112-2-91-0': 'Our project is a direct follow-up of ATLASGAL, from which the idea to observe the W43 region in more detail was born.', '1308.6112-2-92-0': '## Herschel Hi-GAL', '1308.6112-2-93-0': "The Hi-GAL project utilizes Herschel's PACS and SPIRE instruments to observe the Galactic plane from a Galactic longitude of [MATH] to [MATH] at five wavelengths between 70 and 500 [MATH].", '1308.6112-2-93-1': 'A part of the Hi-GAL maps of the W43 complex are presented in [CITATION].', '1308.6112-2-94-0': 'Figure [REF] shows one example of the comparison of the different datasets that we carried out for all identified sources.', '1308.6112-2-94-1': 'It shows source 23, sticking to the notation of Table [REF].', '1308.6112-2-94-2': 'See Appendix [REF] for an in depth description of the different sources.', '1308.6112-2-95-0': 'In Table [REF], we categorize our sources, whether they have a filamentary shape, consist of cores, or show a more complex structure.', '1308.6112-2-95-1': 'We also list the structure of the Spitzer 8 and 24 [MATH]m maps here.', '1308.6112-2-95-2': 'Usually, the two wavelengths are similar.', '1308.6112-2-96-0': 'From the Hi-GAL dust emission, it is possible to derive a temperature and a total (gas + dust) H[MATH] column density map .', '1308.6112-2-96-1': 'These calculations were conducted for the W43 region by [CITATION], following the fitting routine detailed in [CITATION] and adapted and applied to Herschel data as in [CITATION], [CITATION], and [CITATION].', '1308.6112-2-96-2': 'This approach uses Hi-GAL data for the calculations where possible.', '1308.6112-2-96-3': 'As on very bright positions, Hi-GAL data become saturated or enter the nonlinear response regime, HOBYS data was used to fill the missing data points.', '1308.6112-2-96-4': 'The idea is to fit a modified black body curve to the different wavelengths (in this case, the 160 to 350 [MATH]m channels) for each pixel using a dust opacity law of [MATH]cm[MATH]g[MATH] with [MATH].', '1308.6112-2-96-5': 'The final angular resolution of the calculated maps (25 in this case) results from the resolution of the longest wavelength used.', '1308.6112-2-96-6': 'This is the reason that the 500 [MATH]m channel has been omitted.', '1308.6112-2-96-7': 'Planck and IRAS offsets were added before calculating the temperature and H[MATH] column density.', '1308.6112-2-97-0': 'This approach assumes the temperature distribution along the line-of-sight to be constant.', '1308.6112-2-97-1': 'As discussed above in Sect. [REF], this is not necessarily the case in reality.', '1308.6112-2-97-2': 'Due to temperature gradients along the line-of-sight, the calculated H[MATH] column density might deviate from the real value.', '1308.6112-2-97-3': 'This error is found in the Herschel and the CO calculations.', '1308.6112-2-98-0': 'Gas and dust temperatures can deviate in optically thick regions, because the volume densities play a key role here.', '1308.6112-2-98-1': 'Only at fairly high densities of more than about [MATH]cm[MATH] does the gas couple to the dust temperature.', '1308.6112-2-98-2': 'At lower densities, the dust grains are an excellent coolant, in contrast to the gas.', '1308.6112-2-98-3': 'Therefore, the dust usually shows lower temperatures than the gas.', '1308.6112-2-98-4': 'This is in contrast to our findings and may indeed indicate that we underestimate the [MATH] of the gas, as discussed in Sect. [REF].', '1308.6112-2-98-5': 'Even in optically thin regions, the kinetic temperature and the values derived from an SED-fit do not correspond perfectly .', '1308.6112-2-98-6': 'Figure [REF] shows the results of these calculations for the W43 region.', '1308.6112-2-98-7': 'We show the temperature and H[MATH] column density maps from the photometry data and overlay [MATH]CO contours to indicate the location of the molecular gas clouds.', '1308.6112-2-99-0': 'The dust temperature derived by [CITATION] (Fig. [REF] (a)) lies in the range from 20 to 40 K; the outer parts of the complex show temperatures between 25 and 30 K.', '1308.6112-2-99-1': 'The regions where dense molecular clouds are found are colder (about 20 K) than their surroundings; for example the dense ridges in W43-Main are clearly visible.', '1308.6112-2-99-2': 'Some places, where the dust is heated by embedded UV-sources, are hotter (up to 40 K), especially for the OB-star cluster in W43 and one core in W43-South that catches the eye.', '1308.6112-2-100-0': 'If we compare these results with the excitation temperatures that we calculated above in Sect. [REF], we note that the temperatures derived from CO are lower ([MATH] K) than those using Herschel images.', '1308.6112-2-100-1': 'It is possible that gas and dust are not mixed well and that both could have a different temperature.', '1308.6112-2-100-2': 'Another possibility is that the CO gas is subthermally excited and that the excitation temperature is lower than its kinetic temperature.', '1308.6112-2-100-3': 'The Herschel dust temperature map is showing the averaged temperature along the line-of-sight.', '1308.6112-2-100-4': 'Thus, lower temperatures of the dense clouds are seen with hotter diffuse gas around it, which leads to higher averaged temperatures.', '1308.6112-2-100-5': 'As discussed in Sect. [REF], we most probably underestimate the excitation temperature in our calculations above due to subbeam clumpiness.', '1308.6112-2-100-6': 'Also, we cannot neglect that the temperatures calculated from Herschel bear large errors on their own.', '1308.6112-2-101-0': 'The H[MATH] column density map derived from Herschel data (Fig. [REF] (b)) nicely traces the distribution of molecular gas, as indicated by the [MATH]CO (2-1) contours.', '1308.6112-2-101-1': 'There is a background level of [MATH] cm[MATH] that is found outside the complex.', '1308.6112-2-101-2': 'The column density rises in the molecular clouds up to a value of [MATH] cm[MATH] in the ridges of W43-Main.', '1308.6112-2-102-0': 'A comparison to the column density values derived from CO (See Sect. [REF] and the plot in Fig. [REF].)', '1308.6112-2-102-1': 'reveal certain differences.', '1308.6112-2-102-2': 'In the mid-density regions, calculations of both the medium-sized clouds and most parts of the two large clouds are comparable after subtracting the background level from the Herschel maps.', '1308.6112-2-102-3': 'These are still systematically higher, but the difference does rarely exceed a factor of 2.', '1308.6112-2-102-4': 'The same is true for the extended gas between the denser clouds.', '1308.6112-2-102-5': 'The typical values are around a few [MATH] cm[MATH] for the Herschel and CO maps.', '1308.6112-2-102-6': 'However, the Herschel H[MATH] column density reaches values of several [MATH] cm[MATH] in the densest parts of W43-Main, with a maximum of [MATH] cm[MATH], while the CO derived values peak at [MATH] cm[MATH].', '1308.6112-2-103-0': 'The offset of [MATH] cm[MATH] (which could be a bit higher but we chose the lower limit) that has to be subtracted from the Herschel map can partly be explained by diffuse cirrus emission along the line-of-sight, which is not associated with W43.', '1308.6112-2-103-1': 'This statistical error of the background brightness has been described for the Hi-GAL project by [CITATION].', '1308.6112-2-103-2': '[CITATION] find this offset agrees with [CITATION].', '1308.6112-2-104-0': 'The total mass that is found for the W43 complex still deviates between both calculations.', '1308.6112-2-104-1': 'The exact value depends on the specific region that we integrated over.', '1308.6112-2-104-2': 'We find a total mass of W43 from [MATH]CO of [MATH].', '1308.6112-2-104-3': 'The Herschel map gives [MATH], if we use the total map with an area of [MATH] pc[MATH] and consider the diffuse cirrus emission that is included by the Herschel data.', '1308.6112-2-104-4': 'This is a factor of 1.4 higher than the CO result, although we did not calculate a value for every single pixel for the CO H[MATH] column density map.', '1308.6112-2-104-5': 'The mean H[MATH] column density is [MATH] cm[MATH]CO) and [MATH] cm[MATH] (Herschel), respectively.', '1308.6112-2-104-6': 'The difference in total mass can be explained when we consider that the Herschel map covers more points (the difference is reduced to a factor of 1.2, when comparing a smaller region which is covered in both maps, although this might be too small and biased toward the larger clouds).', '1308.6112-2-104-7': 'For a comparison of each Duchamp cloud, see Table [REF] (3).', '1308.6112-2-104-8': 'A comparison of the foreground clouds would be complicated due to line-of-sight effects, so we only give numbers for the W43 sources.', '1308.6112-2-104-9': 'Most values lie in the range 1 to 1.5, which affirms that both calculations deviate by about a factor of 1.4.', '1308.6112-2-104-10': 'The difference in the H[MATH] column density again depends on the examined region.', '1308.6112-2-105-0': 'As stated in Sect. [REF], the mass derived from CO is a lower limit to the real molecular gas mass.', '1308.6112-2-105-1': 'As we used the lower limit of the Herschel column density offset subtracted in W43, these values are thus an upper limit.', '1308.6112-2-105-2': 'Taking this and the errors still included in both calculations into account, the values are nearly consistent.', '1308.6112-2-106-0': '## Column density histogram - PDF', '1308.6112-2-107-0': 'A detailed investigation of the H[MATH] column density structure of W43 is done by determining a histogram of the H[MATH] column density, which is normalized to the average column density.', '1308.6112-2-107-1': 'These probability distribution functions (PDFs) are a useful tool to scrutinize between the different physical processes that determine the density structure of a molecular cloud, such as turbulence, gravity, feedback, and magnetic fields.', '1308.6112-2-107-2': 'Theoretically, it was shown that isothermal turbulence leads to a log-normal PDF , while gravity and non-isothermality provoke power-law tails at higher densities.', '1308.6112-2-107-3': 'Observationally, power-law tails seen in PDFs that are obtained from column density maps of visual extinction or Herschel imaging were attributed to self-gravity for low-mass star-forming regions and high-mass star-forming regions .', '1308.6112-2-107-4': "Recently, it was shown that feedback processes, such as the compression of an expanding ionization front, lead to a characteristic 'double-peaked' PDF and a significant broadening.", '1308.6112-2-108-0': 'The determination of PDFs from molecular line data was attempted by [CITATION] and [CITATION], but it turned out to be problematic when these lines become optically thick and thus do not correctly reflect the molecular cloud spatial and density structure.', '1308.6112-2-108-1': 'In addition, uncertainties in the abundance can complicate conversion into H[MATH] column density.', '1308.6112-2-108-2': 'On the other hand, molecular lines allow us to significantly reduce line-of-sight confusion, because PDFs can be determined for selected velocity ranges.', '1308.6112-2-108-3': 'In addition, using molecules with different critical densities in selected velocity ranges allows us to make dedicated PDFs that focus on a particular subregion like a dense filament.', '1308.6112-2-109-0': 'In this study, we determined the PDFs of W43 in three ways: (i) from a simple conversion of the [MATH]CO (2-1) map into H[MATH] column density by using a constant conversion factor and one temperature (5 or 10 K); (ii) from the H[MATH] column density map derived from the [MATH]CO (2-1) emission, which includes a correction for the optical depth that is derived from both CO lines (see Sect. [REF]); and (iii) from the column density map obtained with Herschel using SPIRE and PACS photometry.', '1308.6112-2-109-1': 'Figure [REF] shows the resulting distributions.', '1308.6112-2-109-2': 'For simplicity, we used the conversion [MATH] for all maps, though the Herschel column density map is a mixture of HI and H[MATH] while the CO derived map is most likely dominated by H[MATH].', '1308.6112-2-110-0': "The 'isothermal' PDFs from [MATH]CO without and optical depth correction (in black and red) clearly show that there is a cut-off in the PDF at high column densities where the lines become optically thick (A[MATH]20 mag for 10 K and A[MATH]70 mag for 5 K).", '1308.6112-2-110-1': 'Obviously, there is also a strong temperature dependence that shifts the peak of the PDF to lower column densities with increasing temperature.', '1308.6112-2-110-2': 'The assumed gas temperature (see discussion in Sect. [REF]) thus has a strong impact on the resulting PDFs positions, but not their shape (not considering the uncertainty in the conversion factor).', '1308.6112-2-111-0': 'With the more sophisticated approach to create a column density map out of the [MATH]CO emission and to include the information provided by the optically thinner C[MATH]O which is corrected for the optical depth [MATH], the PDF (in blue) is more reliable.', '1308.6112-2-111-1': 'It does not show the cut-off at high column densities, because this effect is compensated by using the optically thin C[MATH]O. Only in those regions where this line becomes optically thick, the method gives lower limits for the column density.', '1308.6112-2-111-2': 'Therefore, it drops below the Herschel PDF for high column densities, because the molecular lines underestimate the H[MATH] column densities for very hot gas.', '1308.6112-2-112-0': 'It is remarkable that the PDF derived from [MATH]CO and C[MATH]O shows a log-normal distribution for low column densities and a power-law tail for higher densities.', '1308.6112-2-112-1': 'This feature is also observed in the Herschel PDF.', '1308.6112-2-112-2': 'The approach to determine a PDF from the cloud/clump distribution by correcting the opacity using C[MATH]O appears to be the right way to get a clearer picture of the distribution of higher column densities.', '1308.6112-2-112-3': 'Note that the absolute scaling in column density for both PDFs - from CO or Herschel - remains problematic due to the uncertainty in the conversion factor (and temperature) for the CO data, the line-of-sight confusion, and opacity variations for the Herschel maps.', '1308.6112-2-113-0': "We observe that the slope of the Herschel power-law tail is steeper than the one obtained from the CO-data and shows a 'double-peak' feature (The low column density component is not strictly log-normally shaped but shows two subpeaks.)", '1308.6112-2-113-1': 'as seen in other regions with stellar feedback (Tremblin et al. in prep.)', '1308.6112-2-113-2': 'The column density structure of W43 could thus be explained in a scenario where gravity is the dominating process for the high density range, leading to global cloud collapse and individual core collapse; compression by expanding ionization fronts from embedded HII-regions may lead to an increase in column density, and the lower-density extended emission follows a turbulence dominated log-normal distribution.', '1308.6112-2-113-3': 'This scenario is consistent with what was proposed for other high-mass star-forming regions, such as Rosette , RCW36, M16 (Tremblin et al. in prep.)', '1308.6112-2-113-4': ', and W3 .', '1308.6112-2-114-0': 'Though the overall shape of the PDFs is similar, there are significant differences in the slope of the power-law tails.', '1308.6112-2-114-1': 'The power-law tail of the Herschel PDF is steeper than the CO-PDF.', '1308.6112-2-114-2': "A possible explanation is that the Herschel PDF contains atomic hydrogen in addition to molecular hydrogen (which constitutes mainly the CO PDF), which is less 'participating' in the global collapse of the region and individual clump/core collapse.", '1308.6112-2-114-3': 'In this case, our method to derive the H[MATH] column density from [MATH]CO and C[MATH]O turns out to be an efficient tool to identify only the collapsing gas that ends up into a proto-star.', '1308.6112-2-115-0': '# Description of W43-Main and W43-South', '1308.6112-2-116-0': 'Here, we give a detailed description of the two most important sources in the W43 complex, W43-Main and W43-South.', '1308.6112-2-116-1': 'Several other interesting sources are described in Appendix [REF].', '1308.6112-2-117-0': '## W43-Main, Source 13', '1308.6112-2-118-0': 'Source 13 (see Fig. [REF] (m)), or W43-Main, is the largest and most prominent of all sources in the W43 complex.', '1308.6112-2-118-1': 'Located in the upper central region of the map with an extent of roughly 30 by 20 pc, it shows a remarkable Z-shape of connected, elongated ridges.', '1308.6112-2-118-2': 'The upper part of this cloud extrudes far to the east with a strong emitting filament, where it curves down south in a weaker extension of this filament.', '1308.6112-2-118-3': 'This structure is especially clear in ATLASGAL and Hi-GAL dust emission.', '1308.6112-2-119-0': 'There are some details hidden in this cloud that cannot be seen clearly in the complete integrated map.', '1308.6112-2-119-1': 'In the velocity range of 80 to 90 kms[MATH], which are located in the southwest of the source, we see a circular, shell-like structure surrounding an empty bubble (Fig. [REF] (a)).', '1308.6112-2-119-2': 'This bubble is elliptically shaped with dimensions of 10[MATH]6 pc, while the molecular shell is about 1.5 pc thick.', '1308.6112-2-119-3': 'It is located where a cluster of young OB stars is situated.', '1308.6112-2-119-4': 'Possibly, this cavity is formed by the radiation of this cluster.', '1308.6112-2-119-5': 'See [CITATION] for a description of the expansion of clouds at the periphery of this (HII) bubble.', '1308.6112-2-120-0': 'The central Z-shape of this cloud appears to be monolithic on the first sight.', '1308.6112-2-120-1': 'However, Fig. [REF] (b) shows that the northern and southern parts are separated by a gap in the channel maps between 94 and 98 kms[MATH].', '1308.6112-2-120-2': 'Both parts are still connected in channels higher and lower than those velocities.', '1308.6112-2-120-3': 'This chasm that we see is narrow in the center of the cloud, where it has a width of 1 to 2 pc, and opens up to both sides.', '1308.6112-2-120-4': 'The origin of this structure is the cluster of WR and OB stars situated in the very center that blows the surrounding material out along a plane perpendicular to the line-of-sight.', '1308.6112-2-120-5': 'This agrees with the presence of 4 HCO[MATH] clouds in the 25 kms[MATH] range along the line-of-sight of the Wolf-Rayet cluster .', '1308.6112-2-121-0': 'W43-Main is the most luminous source in our set with integrated [MATH]CO (2-1) emission of up to 170 Kkms[MATH] and line peaks of up to 23 K at the peak in the northern filament.', '1308.6112-2-121-1': 'Most inner parts of this cloud show integrated intensities of [MATH]90 to 120 Kkms[MATH] and 45 to 60 Kkms[MATH] in the outer parts.', '1308.6112-2-121-2': 'It is also the most optically thick with an optical depth of [MATH]CO up to 8 in the southern arm, while the bulk of the cloud with 2 to 3 is not exceptionally opaque.', '1308.6112-2-121-3': 'It is possible that we overestimated the opacity in the south of this cloud.', '1308.6112-2-121-4': 'The spectra show that [MATH]CO is self-absorbed here, while C[MATH]O is not.', '1308.6112-2-121-5': 'This would lead to an unrealistic ratio of the two isotopologes and an overestimated optical depth.', '1308.6112-2-122-0': 'The cloud shows a velocity gradient across its complete structure (see Fig. [REF]).', '1308.6112-2-122-1': 'Beginning at a relative velocity of [MATH] kms[MATH] at the most southwestern tip, it winds through the Z shape and ends in the eastern extension filament at [MATH] kms[MATH].', '1308.6112-2-122-2': 'This velocity difference of [MATH] kms[MATH], already described in [CITATION] is huge and the largest discovered in the W43 complex.', '1308.6112-2-122-3': 'It could be the sign of a rotation of the cloud.', '1308.6112-2-122-4': 'Comparably impressive are the line widths of the central parts of this clouds.', '1308.6112-2-122-5': 'We find them to be up to 15 kms[MATH] especially in the central parts, indicating very turbulent gas or global motions.', '1308.6112-2-123-0': 'In the most luminous parts in the south, we calculate the highest H[MATH] column densities of about [MATH] cm[MATH].', '1308.6112-2-123-1': 'This is due to the high opacities that have been calculated here.', '1308.6112-2-123-2': 'The other central parts of this cloud show H[MATH] column densities of [MATH] cm[MATH].', '1308.6112-2-123-3': '[CITATION] derived the cloud mass of W43-Main from [MATH]CO (1-0) and find a value of [MATH] M[MATH].', '1308.6112-2-123-4': 'We find a total mass in this source of [MATH] and thus a large fraction of the mass of the complete W43 complex ( 20%).', '1308.6112-2-124-0': 'In the GLIMPSE maps at this position, we see strong extended IR emission along the walls of a cavity, which are just west of the central Z-shape, formed by a cluster of young OB stars that is located here (see Fig. [REF]).', '1308.6112-2-124-1': 'This cluster is probably the first result of the star-formation going on here.', '1308.6112-2-124-2': 'The cavity seen in infrared is not the same as the one in CO described above, but both appear to be connected.', '1308.6112-2-124-3': 'The CO channel may be due to the strong radiation of the nearby stars.', '1308.6112-2-124-4': 'The northern and southern arm of the Z-shape are seen in absorption in the 8 [MATH] band of GLIMPSE against the infrared background.', '1308.6112-2-124-5': 'This indicates dense cold dust and molecular clouds that are shielded from the UV radiation of the stars, which is verified by strong emission of cold dust, as seen in the ATLASGAL and Hi-GAL maps.', '1308.6112-2-125-0': '## W43-South, Source 20', '1308.6112-2-126-0': 'Source 20, also known as W43-South and G029.96-0.02 , is the second largest source in the W43 complex and dominates the southwestern part our map.', '1308.6112-2-126-1': 'Figure [REF] (t) shows a plot of this source.', '1308.6112-2-126-2': 'It has approximately the shape of a tilted ellipse with the dimensions of about 24[MATH]31 parsec with several smaller clumps scattered across the cloud.', '1308.6112-2-126-3': 'These clumps emit strongly in [MATH]CO (2-1) up to 150 Kkms[MATH] and are surrounded by less luminous gas, where we see emission between 60 to 90 Kkms[MATH] and down to 30 Kkms[MATH] in the outer parts of the cloud.', '1308.6112-2-126-4': 'Maximum line peaks are 30 K.', '1308.6112-2-126-5': 'This source is less optically thick than W43-Main; the opacity is around 2 to 3 for most parts of the cloud and does not exceed 4 in the dense clumps.', '1308.6112-2-127-0': 'Studying the details of this source, we see that several of the dense clumps are actually shells of gas.', '1308.6112-2-127-1': 'The ringlike structures are clearly recognizable in some channel maps.', '1308.6112-2-127-2': 'Figure [REF] shows the most intriguing example.', '1308.6112-2-127-3': 'The spectra across the whole ring show infall signatures, as discussed in [CITATION] in the optically thick [MATH]CO lines.', '1308.6112-2-127-4': 'However, C[MATH]O, which usually is optically thin shows the same signature as it is still optically thick at this position.', '1308.6112-2-127-5': 'To really trace infall, the optically thin line should show a single peak at the position of the absorption feature in the optically thick line.', '1308.6112-2-127-6': 'This is the case for the N[MATH]H[MATH] (1-0) line, which is also taken at the IRAM 30m telescope during the second part of our program, which is optically thin (although it shows a hyper-fine structure, the strongest peak is centered on the correct velocity).', '1308.6112-2-127-7': 'This could be interpreted as a bubble of gas which is heated from the inside by an embedded UV source, although it is not associated with any ultracompact HII region identified by the CORNISH survey .', '1308.6112-2-127-8': 'The Spitzer 8 [MATH]m also show a heated ring of dust, which indicates an embedded heating source.', '1308.6112-2-128-0': 'The relative radial velocity of the gas is more or less constant around 100 kms[MATH] across the whole cloud W43-South.', '1308.6112-2-128-1': 'Small parts in the east are slower at 95 kms[MATH], while a tip in the northwest is faster with velocities around 105 kms[MATH].', '1308.6112-2-128-2': 'However, the velocity gradient is not very pronounced.', '1308.6112-2-128-3': 'FWHM line widths show typical values of 5 to 10 kms[MATH] with broad lines that are found mostly in the eastern part of the cloud.', '1308.6112-2-128-4': 'The bright clumps do not show distinct broad lines.', '1308.6112-2-129-0': 'The calculated H[MATH] column density is on the order of around a few times of [MATH] cm[MATH] for most of the cloud, but parts in the northwest and the center peak at about [MATH] cm[MATH].', '1308.6112-2-129-1': 'The total mass of this cloud is [MATH], and it is thus the second most massive source in the W43 complex.', '1308.6112-2-130-0': 'All bright clumps, except for the one in the central north, are seen in bright emission in the GLIMPSE 8 [MATH]m map.', '1308.6112-2-130-1': 'They are obviously heated from the inside; no external UV sources can be identified.', '1308.6112-2-130-2': 'Apparently, YSOs have formed in the dense but separated clumps that are seen in CO emission and in dust emission.', '1308.6112-2-130-3': 'In the northeast of the cloud, one slab of cold gas is seen in absorption in the GLIMPSE map, separating some of the bright clumps.', '1308.6112-2-131-0': '# Conclusions', '1308.6112-2-132-0': 'Following the investigations of [CITATION], ATLASGAL, and [CITATION], we observed the W43 region in the [MATH]CO (2-1) and C[MATH]O (2-1) emission lines with the IRAM 30m telescope.', '1308.6112-2-132-1': 'We have presented integrated maps of the resulting position-position-velocity cubes in which we identified numerous clouds.', '1308.6112-2-133-0': 'We have confirmed that W43 is indeed one connected complex, as described in [CITATION].', '1308.6112-2-133-1': 'The connection between the two main clouds W43-main and W43-south is shown in the PV-diagram in Fig. [REF] (a).', '1308.6112-2-134-0': 'An analysis of the velocity distribution of our dataset and comparison to the Galactic model of [CITATION] reveals emission not only from the W43 complex itself but also from fore-/background sources.', '1308.6112-2-134-1': 'According to this model, W43 is situated near the tangential point of the Scutum arm, where it meets the Galactic bar.', '1308.6112-2-134-2': 'The low velocity sources are located in the Perseus arm and the space between the Sagittarius and Scutum arms (see Fig. [REF] (c)).', '1308.6112-2-134-3': 'The separation of the different components lets us avoid the confusion and line-of-sight effects in our analysis.', '1308.6112-2-135-0': 'We decomposed the data cubes into subclouds, using Duchamp Sourcefinder.', '1308.6112-2-135-1': 'We identified a total of 29 clouds whith 20 were located in the W43 complex, while 9 were found to be foreground and background sources.', '1308.6112-2-136-0': 'We have derived physical properties like excitation temperature, H[MATH] column density, and total mass, of each subcloud of our dataset (see Table [REF]).', '1308.6112-2-136-1': 'Typical smaller sources have spatial scales of 10 to 20 pc and masses of several [MATH].', '1308.6112-2-136-2': 'The two most outstanding sources are W43-Main (source 13) and W43-South (source 20) that have masses of a few [MATH].', '1308.6112-2-137-0': 'We have determined the total mass of dense clouds (>[MATH] cm[MATH]) in the W43 complex to be [MATH].', '1308.6112-2-137-1': 'This is a factor of 1.4 lower than the mass derived from Herschel dust emission maps, which is a discrepancy that can be explained by the details of both calculations.', '1308.6112-2-138-0': 'The shear parameter of W43 ([MATH]) shows that the accumulation of mass in molecular clouds in this region is not disrupted by shear forces of the Galactic motion.', '1308.6112-2-139-0': 'We have created probability distribution functions obtained from column density maps.', '1308.6112-2-139-1': 'We use both the molecular line maps and Herschel imaging data (Fig. [REF]).', '1308.6112-2-139-2': 'Both show a log-normal distribution for low column densities and a power-law tail for high densities.', '1308.6112-2-139-3': 'Still, there are differences seen in peak position and a power-law slope.', '1308.6112-2-139-4': 'Possibly, the flatter slope of the molecular line data PDFs imply that those could be used to selectively show the gravitationally collapsing gas.', '1308.6112-2-140-0': 'This project is carried out within the Collaborative Research Council 956, subproject A4, funded by the Deutsche Forschungsgemeinschaft (DFG).', '1308.6112-2-140-1': 'Part of this work was supported by the ANR-11-BS56-010 project "STARFICH".', '1308.6112-2-140-2': 'Part of this work was supported by the French National Agency for Research (ANR) project "PROBeS", number ANR-08-BLAN-0241.', '1308.6112-2-140-3': 'We thank the referee for constructive and valuable comments and ideas.', '1308.6112-2-141-0': '# Noise maps', '1308.6112-2-142-0': 'We create noise maps from both the [MATH]CO (2-1) and the C[MATH]O (2-1) data cubes.', '1308.6112-2-142-1': 'For each spectrum, we determine the rms and create maps from these values.', '1308.6112-2-142-2': 'For this purpose, we need to calculate the root-mean-square (rms) from parts of the spectra that are emission-free.', '1308.6112-2-142-3': 'We use the velocity range between 120 and 130 kms[MATH], because it is free of emission for the complete region that we mapped.', '1308.6112-2-142-4': 'Typical values are found to be around 1 K or even less, while some parts in the south show values of up to 3 K. All values given here are in [MATH], which are corrected for main beam efficiency.', '1308.6112-2-142-5': 'The results can be seen in Fig. [REF].', '1308.6112-2-143-0': 'We find that the structure of the noise is similar for both lines.', '1308.6112-2-143-1': 'The largest differences arise from weather conditions and time of day.', '1308.6112-2-143-2': 'This is seen in the squarish pattern, as each square shows single observations that have been carried out in a small time window.', '1308.6112-2-143-3': 'Still, there is a striped pattern visible that overlays the whole map.', '1308.6112-2-143-4': 'This stems from the nine different pixels that make up the HERA receiver.', '1308.6112-2-143-5': 'These pixels have different receiver temperatures, hence the different noise levels.', '1308.6112-2-143-6': 'We also note that observations in the northern part of the map are usually less noisy than those in the south.', '1308.6112-2-143-7': 'This probably results from the different weather in which the observations have been carried out.', '1308.6112-2-143-8': 'Last, we find that the C[MATH]O (mean rms of 1.3 K, maximum of 6.7 K) data shows a little increase in noise temperature in general compared to the [MATH]CO line (mean rms of 1.1 K, maximum of 3.7 K).', '1308.6112-2-143-9': 'The latter has been observed with the HERA1 polarization of the HERA receiver, whereas the first has been observed with HERA2, which has an overall higher receiver temperature.', '1308.6112-2-144-0': '# Peak velocity and line width of foreground components', '1308.6112-2-145-0': 'Figure [REF] shows plots of the peak velocity position and the FWHM line width of the two lower velocity components.', '1308.6112-2-145-1': 'The maps of the W43 complex itself are shown in Fig. [REF] and are described in Sect. [REF].', '1308.6112-2-146-0': '# Calculations', '1308.6112-2-147-0': '## [MATH]CO Optical depth', '1308.6112-2-148-0': 'Assuming a constant abundance ratio of [MATH]CO:[MATH]CO:C[MATH]O, we can estimate the optical depth of the [MATH]CO gas .', '1308.6112-2-148-1': 'We compare the intensities of the [MATH]CO and C[MATH]O line emission integrated over the analyzed cloud and solve the equation, [EQUATION] for [MATH], where [MATH] is the intrinsic ratio of of the two mapped CO isotopologes.', '1308.6112-2-149-0': 'The isotopic abundance of C and O in the Milky Way is known to depend on the Galactocentric radius.', '1308.6112-2-149-1': 'Often cited values are found in [CITATION].', '1308.6112-2-149-2': 'They find the ratio of [MATH]O and [MATH]O to be 272 at 4 kpc radius, 302 at 4.5 kpc, and 390 at 6 kpc, so we take those numbers for the [MATH]CO:C[MATH]O ratio.', '1308.6112-2-149-3': 'Recent values for the C/[MATH]C abundance are given in [CITATION].', '1308.6112-2-149-4': 'Here, we take values derived from CO observations and get a [MATH]CO:[MATH]CO ratio of 31 at a Galactocentric radius of 4 kpc for the main component and a ratio of 43 and 52 for the foreground components at a radius of 4.5 and 6 kpc, respectively.', '1308.6112-2-149-5': 'In total, we use an intrinsic ratio of [MATH]CO:[MATH]CO:C[MATH]O of 1:1/31:1/272 for sources in the main complex and ratios of 1:1/43:1/302 and 1:1/52:1/390 respectively for foreground sources.', '1308.6112-2-150-0': '## Excitation temperature', '1308.6112-2-151-0': 'Once we know the optical depth, we can determine the excitation temperature of the CO gas.', '1308.6112-2-152-0': 'For this, we use [CITATION]: [EQUATION]', '1308.6112-2-152-1': 'With [EQUATION] we use the line peak intensity [MATH] for [MATH].', '1308.6112-2-152-2': 'The parameter [MATH] is the cosmic background radiation of 2.7 K, [MATH] the [MATH]CO optical depth, and [MATH] the beam filling factor.', '1308.6112-2-152-3': 'This expression can then be solved for [MATH].', '1308.6112-2-153-0': 'Here, we assume that the excitation temperature of [MATH]CO and C[MATH]O is the same.', '1308.6112-2-153-1': 'We then assume that the beam filling factor [MATH] is always 1.', '1308.6112-2-153-2': 'It is likely that there is some substructure that we cannot resolve with our beam size.', '1308.6112-2-153-3': 'This would mean that the real [MATH] is lower than 1, and we calculate [MATH] rather than just [MATH].', '1308.6112-2-153-4': 'Thus, we underestimate the temperature in cases where there is indeed substructure.', '1308.6112-2-153-5': 'The line peak [MATH] is calculated from fitting a Gaussian to all spectra in the [MATH]CO line emission cube.', '1308.6112-2-154-0': 'We cannot calculate a [MATH] in this way for all pixels, even if [MATH]CO is present, as the C[MATH]O is much weaker.', '1308.6112-2-154-1': 'As the ratio of [MATH]CO and C[MATH]O is needed, a self-consistent temperature can only be computed for points where C[MATH]O is present.', '1308.6112-2-154-2': 'The main uncertainty of the calculation itself is the assumption that the beam filling factor is always 1, and the real value can only be correctly derived where this is true.', '1308.6112-2-154-3': 'This led to the decision not to use these T[MATH] maps for the further calculation of the H[MATH] column density.', '1308.6112-2-154-4': 'We used the calculations to get an idea of the gas temperature and then used a constant value for all further steps of analysis.', '1308.6112-2-154-5': 'We chose this value to be 12 K, as this was the median temperature found across the whole W43 region.', '1308.6112-2-155-0': '## Column density and mass', '1308.6112-2-156-0': 'We then compute the H[MATH] column density by using the assumed excitation temperature [MATH] of 12 K and the integrated molecular line emission from our observation through [EQUATION] where the factor containing [MATH] accounts for the effect that the full gas is not seen for optically thicker clouds and', '1308.6112-2-157-0': '[EQUATION] with the function [EQUATION] where [MATH]s[MATH] is the rotational constant for [MATH]CO, [MATH]D is its dipole moment, and [MATH] is the upper level of our transition (2 in this case).', '1308.6112-2-157-1': 'We correct all those points for the optical depth where we find an opacity larger than 0.5.', '1308.6112-2-157-2': 'We assume this is the minimum value we can determine correctly as we might confuse emission with noise for lower opacities.', '1308.6112-2-158-0': 'In Fig. [REF] we plot the dependency of the [MATH]CO column density and the excitation temperature.', '1308.6112-2-158-1': 'We note that the column density for values of [MATH] K is nearly independent of the excitation temperature.', '1308.6112-2-158-2': 'On the contrary, it rises steeply for temperatures below 10 K.', '1308.6112-2-158-3': 'This is important as we most probably would underestimate the excitation temperature for most clouds as discussed above, if we actually used the calculated T[MATH].', '1308.6112-2-158-4': 'We would thus overestimate the column density.', '1308.6112-2-158-5': 'Therefore, we can assume that using an excitation temperature of 12 K for our calculation results in a lower limit for the actual column density.', '1308.6112-2-159-0': 'As we want to calculate the H[MATH] column density, we need to translate [MATH]CO) into [MATH].', '1308.6112-2-159-1': 'The standard factor of [MATH]:[MATH] is [MATH] for local molecular clouds, but the ratio varies with sources and also with the Galactocentric radius.', '1308.6112-2-159-2': '[CITATION] derives a radius dependent formula for [MATH]:[MATH], using values of [MATH]C/H and [MATH]O/H from [CITATION].', '1308.6112-2-159-3': 'This formula gives a ratio of 6550 for a Galactic radius of 4 kpc 7000 at 4.5 kpc and 8500 at 6 kpc.', '1308.6112-2-159-4': 'Using our ratios of [MATH]CO:[MATH]CO from above we get [MATH] for the W43 complex with a radius of 4 kpc and the factors of [MATH] and [MATH] for the fore-/background complexes with radii of 4.5 and 6 kpc components, respectively.', '1308.6112-2-159-5': 'These factors are prone to large errors of at least a factor of 2.', '1308.6112-2-160-0': 'To calculate the mass of the observed source from the H[MATH] column density it is necessary to consider the relative distance from the Sun to the source.', '1308.6112-2-160-1': 'See Sect. [REF] for the distance determination.', '1308.6112-2-160-2': 'We assume the main complex clouds are 6 kpc away, while the foreground clouds have distances of 3.5, 4, 3.5, and 12 kpc.', '1308.6112-2-160-3': 'Then, we just have to count the number of H[MATH] molecules per pixel to receive the mass per pixel in solar masses.', '1308.6112-2-160-4': '[EQUATION]', '1308.6112-2-160-5': 'Here, [MATH] is the distance toward the source, [MATH] the angular extent of one pixel on the sky.', '1308.6112-2-160-6': 'The value of [MATH] accounts for higher masses of molecules, apart from H[MATH] .', '1308.6112-2-161-0': 'The resulting H[MATH] column density map of the full W43 complex is shown in Fig. [REF], which was calculated using the velocity range between 78 and 120 kms[MATH].', '1308.6112-2-161-1': 'It has been calculated for all points, where the integrated [MATH]CO shows an intensity of more than 5 Kkms[MATH].', '1308.6112-2-161-2': 'The integrated [MATH]CO map in Fig. [REF] (a) shows diffuse emission in-between the brighter sources that were isolated with the Duchamp sourcefinder.', '1308.6112-2-161-3': 'This diffuse emission accounts for about 50% of the the total mass in W43.', '1308.6112-2-162-0': '# Description of important sources', '1308.6112-2-163-0': 'In the following, we want to give a description of several important and interesting sources of the W43 complex ([MATH] kpc) found in our datasets.', '1308.6112-2-163-1': 'Information on these sources are listed in Table [REF], while their location is indicated in Fig. [REF].', '1308.6112-2-163-2': 'The corresponding maps can be found in Fig. [REF].', '1308.6112-2-163-3': 'We discuss the shape, topology, and intensity of the maps and fundamental properties like velocity gradients, FWHM line widths, temperature, and column density.', '1308.6112-2-163-4': 'We also mention conclusions from the comparison to different datasets.', '1308.6112-2-163-5': 'Sources are ordered by their peak velocities.', '1308.6112-2-164-0': '## Source 23', '1308.6112-2-165-0': 'Source 23 (plots can be seen in Fig. [REF] and also Fig. [REF] (w)) consists of one central elliptical clump with one elongated thin extension, protruding from the southeast, that ends in a hook-like tip and is curved to the south.', '1308.6112-2-165-1': 'The central clump is elliptically shaped with a length of 5 and 3 parsec on the major axes and is bound sharply at the southern edge, while it is much more diffuse and more extended in the north.', '1308.6112-2-165-2': 'The extension has a length of 7.5 pc.', '1308.6112-2-165-3': 'We find the maximum integrated intensity of the [MATH]CO (2-1) line to be about 90 Kkms[MATH] at the peak of the clump, while the filamentary extension lies around 30 to 45 Kkms[MATH].', '1308.6112-2-165-4': 'The line peak intensity rises from 12 K in the filament to 24 K in the clump.', '1308.6112-2-165-5': 'The opacity has typical values of 1 to 2.5 with higher values in the central clump.', '1308.6112-2-166-0': 'We see a gradient in the radial velocity of the cloud from the filament to the center of the source of [MATH] kms[MATH], which can be interpreted as a flow of gas along the outrigger onto the clump.', '1308.6112-2-166-1': 'The line width (FWHM) changes between 4.5 kms[MATH] in the inner clump and 2 to 2.5 kms[MATH] in the outer parts of the cloud.', '1308.6112-2-167-0': 'The H[MATH] column density that we calculated rises from [MATH] cm[MATH] in the edges of the cloud to [MATH] cm[MATH] in the center.', '1308.6112-2-167-1': 'The total mass is calculated to be [MATH] and thus resembles a typical total mass of our set of sources.', '1308.6112-2-168-0': 'The CO emission that we measure in our maps is nearly exactly matched by the dust emission maps of ATLASGAL and Hi-GAL (see Fig. [REF] (b) and (c), respectively).', '1308.6112-2-168-1': 'Both show the strong peak in the central clump and the weaker filament in the southeast, including that of the curved tip.', '1308.6112-2-168-2': 'The GLIMPSE map shows very interesting features (see Fig. [REF] (d)).', '1308.6112-2-168-3': 'There is one strong UV point source less than 1 pc off to the south of the CO clump.', '1308.6112-2-169-0': '## Source 25', '1308.6112-2-170-0': 'This filament, as seen in Fig. [REF] (y), resides in the central western part of the W43 complex, which is half-way between W43-Main and W43-South.', '1308.6112-2-170-1': 'It is shaped like an inverted L with two branches and connected by an orthogonal angle.', '1308.6112-2-170-2': 'The vertical branch has a length of 14 pc; the horizontal one is 10 pc long.', '1308.6112-2-170-3': 'The typical width of both branches is between 2 and 3 pc.', '1308.6112-2-170-4': 'One strong clump is seen in the southern part with an integrated line intensity of the [MATH]CO (2-1) line of 40 Kkms[MATH], while the rest of the filament backbone only reaches 18 to 22 Kkms[MATH].', '1308.6112-2-170-5': 'Line peak intensities range from a few K in the outer parts of the filament up to 15 K in the strong southern clump.', '1308.6112-2-171-0': 'Investigating the line peak velocity map, we realize that the two branches of this source are actually separated.', '1308.6112-2-171-1': 'The horizontal branch has a constant radial velocity of 108 kms[MATH] across, while the vertical branch shows a gradient from 110 kms[MATH] in the north to 115 kms[MATH] in the south.', '1308.6112-2-171-2': 'Line widths range between 1 and 2 kms[MATH] in the whole source.', '1308.6112-2-172-0': 'The H[MATH] column density varies between [MATH] cm[MATH] in the outer parts and [MATH] cm[MATH] around the southern core.', '1308.6112-2-172-1': 'The total mass is [MATH].', '1308.6112-2-172-2': 'Comparing this source to the complementary projects is complicated, since the source 17 is located at the same place and overlaps this source.', '1308.6112-2-172-3': 'Most emission that is seen in the northern part of the source is presumably part of source 17.', '1308.6112-2-172-4': 'Only the embedded core in the south is clearly seen in dust emission and as a compact Spitzer source.', '1308.6112-2-173-0': '## Source 26', '1308.6112-2-174-0': 'Located in the easternmost central part of the W43 complex lies this filamentary shaped source, whose plot is found in Fig. [REF] (z).', '1308.6112-2-174-1': 'It stretches over a range of 26 pc from southeast to northwest.', '1308.6112-2-174-2': 'The filament consists of three subsections that contain several clumps and has a typical width of 5 pc.', '1308.6112-2-174-3': 'The integrated emission map of the [MATH]CO (2-1) line shows values of up to 35 Kkms[MATH] in the clumps, which is surrounded by weaker gas.', '1308.6112-2-174-4': 'The strongest clump lies in the southeastern end of the filament while the highest line peak intensities are found in the northwest with up to 13 K.', '1308.6112-2-175-0': 'The velocity structure of this filament is nearly symmetrical, starting around 106 kms[MATH] in the middle of the filament and increasing toward its tips up to 110 kms[MATH] in the west and 112 kms[MATH] in the east.', '1308.6112-2-175-1': 'The width of the lines has nearly homogeneous values around 2 kms[MATH] in the center and western part of the filament but shows broad lines with a width of more than 5 kms[MATH] in the eastern clump.', '1308.6112-2-176-0': 'This filament shows a typical distribution of its H[MATH] column density.', '1308.6112-2-176-1': 'Several denser clumps are embedded along the filament.', '1308.6112-2-176-2': 'Column densities vary from a few [MATH] cm[MATH] in the outer parts of the filament up to a maximum of [MATH] cm[MATH] in one clump.', '1308.6112-2-176-3': 'The total mass of this source is [MATH].', '1308.6112-2-177-0': 'The CO emission of this source matches nicely with the dust emission of ALTASGAL and Hi-GAL.', '1308.6112-2-177-1': 'However, the eastern part of the filament is stronger in the 850 [MATH]m map than the shorter wavelengths of Hi-GAL and vice versa in the western part.', '1308.6112-2-177-2': 'Only the clump in the west of this filament can be seen in the 8 [MATH]m Spitzer data, the east is not traced.', '1308.6112-2-177-3': 'There is one extended source seen in emission in the center part of this source, but this is probably unrelated.', '1308.6112-2-178-0': '## Source 28', '1308.6112-2-179-0': 'Source 28 (see Fig. [REF] (ab)) is located in the southwest of W43-Main in the central region of the complex.', '1308.6112-2-179-1': 'It is not visible in the total integrated maps of the region, as it is confused with sources of a different relative velocity.', '1308.6112-2-179-2': 'It becomes visible by investigating the channel maps between 110 and 115 kms[MATH] radial velocity.', '1308.6112-2-179-3': 'The source has dimensions of 12 pc in the east-west direction and 8 pc in the north-south direction.', '1308.6112-2-179-4': 'Its shape is that of a two-armed filament, whose two parts join at an angle of [MATH] where the eastern arm runs from southeast to northwest and the western arm from east to west.', '1308.6112-2-179-5': 'We see two stronger clumps in the eastern filament: one in the center of it and one in the southeastern tip.', '1308.6112-2-179-6': 'This is a relatively weak source with an integrated [MATH]CO (2-1) emission that peaks at only 25 Kkms[MATH] in the center of the eastern filament, where the maximum line peak is around 13 K.', '1308.6112-2-179-7': 'The integrated intensity goes down to 8 Kkms[MATH] in the outskirts of the filament.', '1308.6112-2-179-8': 'Yet, it is a valuable source, due to its pronounced filamentary structure and the embedded clumps, which is a good candidate for more investigations in filament formation .', '1308.6112-2-180-0': 'The eastern arm is especially interesting.', '1308.6112-2-180-1': 'It has a length of 6 pc and a typical width of 1.5 pc.', '1308.6112-2-180-2': 'We note two embedded clumps embedded in it and a velocity gradient along the filament, which starts at 110 kms[MATH] in the north and increases to 114 kms[MATH] in the southeastern part of this arm.', '1308.6112-2-180-3': 'The typical line width varies between 1 and 2 kms[MATH], increasing toward the inside of the cloud and reaching the maximum width at the clumps.', '1308.6112-2-181-0': 'As this source is rather weak, we also find H[MATH] column densities to be only around [MATH] cm[MATH] at the maximum around the embedded clumps.', '1308.6112-2-181-1': 'The total derived mass of the molecular gas is [MATH], which makes it one of the less massive sources identified.', '1308.6112-2-182-0': 'Surprisingly, this source is one of the few that is not traced at all in the GLIMPSE 8 [MATH]m map.', '1308.6112-2-182-1': 'It appears that there are no nearby UV sources that could heat the gas.', '1308.6112-2-182-2': 'The two nearest sources were identified to be related to background sources in another Galactic spiral arm.', '1308.6112-2-182-3': 'Also, the gas and the related dust is obviously too faint to appear in absorption.', '1308.6112-2-182-4': 'This is also verified by the ATLASGAL an Hi-GAL maps, which show only weak dust emission in the filament.', '1308.6112-2-183-0': '## Source 29', '1308.6112-2-184-0': 'This source is found in the very center of our region maps, directly south of the W43-Main cloud.', '1308.6112-2-184-1': 'Its shape resembles a crescent moon, opened toward the southeast, where it is sharply bound.', '1308.6112-2-184-2': 'The outside is more diffuse and shows several outflows away from the center.', '1308.6112-2-184-3': 'See Fig. [REF] (ac) for a plot of the [MATH]CO emission.', '1308.6112-2-184-4': 'The extent of the source is 12 pc from northeast to southwest, and the filament has a typical width of 2 to 3 pc.', '1308.6112-2-184-5': 'Two stronger clumps with an integrated [MATH]CO intensity of 40 Kkms[MATH] are seen in the center and the northeastern tip.', '1308.6112-2-184-6': 'The strong backbone of this source has still an integrated intensity of [MATH] Kkms[MATH], where line peak intensity goes up to 18 K.', '1308.6112-2-185-0': 'The central western part of the source moves with a relative radial velocity of 112 kms[MATH] and increases to 118 kms[MATH] toward both ends of the crescent.', '1308.6112-2-185-1': 'The lines show widths of 2 to 3 kms[MATH] in the central region, decreasing to 1 kms[MATH] in the outer parts.', '1308.6112-2-185-2': 'H[MATH] column densities only reach a few [MATH] cm[MATH] across the inner parts of the structure.', '1308.6112-2-185-3': 'We find a total mass of [MATH].', '1308.6112-2-186-0': 'In the dust emission maps of ATLASGAL and Hi-GAL, only the strong backbone of this source can be seen.', '1308.6112-2-186-1': 'The weaker outliers are not traced.', '1308.6112-2-186-2': 'The Spitzer 8 [MATH]m map shows several bright compact sources in the center, and some extended emission in the south is most probably related to this source.', '1308.6112-2-186-3': 'However, the northern tip that shows strong CO emission is not traced by Spitzer at all.', '1308.6112-2-187-0': '# List of found sources'} | [['1308.6112-1-158-0', '1308.6112-2-158-0'], ['1308.6112-1-158-4', '1308.6112-2-158-4'], ['1308.6112-1-134-1', '1308.6112-2-134-1'], ['1308.6112-1-134-3', '1308.6112-2-134-3'], ['1308.6112-1-70-2', '1308.6112-2-70-2'], ['1308.6112-1-70-3', '1308.6112-2-70-3'], ['1308.6112-1-70-5', '1308.6112-2-70-5'], ['1308.6112-1-9-3', '1308.6112-2-9-3'], ['1308.6112-1-80-3', '1308.6112-2-80-3'], ['1308.6112-1-39-0', '1308.6112-2-39-0'], ['1308.6112-1-39-8', '1308.6112-2-39-8'], ['1308.6112-1-39-9', '1308.6112-2-39-9'], ['1308.6112-1-105-0', '1308.6112-2-105-0'], ['1308.6112-1-105-2', '1308.6112-2-105-2'], ['1308.6112-1-43-1', '1308.6112-2-43-1'], ['1308.6112-1-27-0', '1308.6112-2-27-0'], ['1308.6112-1-27-3', '1308.6112-2-27-3'], ['1308.6112-1-27-4', '1308.6112-2-27-4'], ['1308.6112-1-14-4', '1308.6112-2-14-4'], ['1308.6112-1-66-1', '1308.6112-2-66-1'], ['1308.6112-1-66-3', '1308.6112-2-66-3'], ['1308.6112-1-183-1', '1308.6112-2-182-1'], ['1308.6112-1-183-2', '1308.6112-2-182-2'], ['1308.6112-1-183-3', '1308.6112-2-182-3'], ['1308.6112-1-183-4', '1308.6112-2-182-4'], ['1308.6112-1-143-0', '1308.6112-2-143-0'], ['1308.6112-1-143-1', '1308.6112-2-143-1'], ['1308.6112-1-143-2', '1308.6112-2-143-2'], ['1308.6112-1-143-5', '1308.6112-2-143-5'], ['1308.6112-1-143-7', '1308.6112-2-143-7'], ['1308.6112-1-143-9', '1308.6112-2-143-9'], ['1308.6112-1-145-1', '1308.6112-2-145-1'], ['1308.6112-1-168-2', '1308.6112-2-168-2'], ['1308.6112-1-128-0', '1308.6112-2-128-0'], ['1308.6112-1-128-2', '1308.6112-2-128-2'], ['1308.6112-1-128-4', '1308.6112-2-128-4'], ['1308.6112-1-36-3', '1308.6112-2-36-3'], ['1308.6112-1-79-0', '1308.6112-2-79-0'], ['1308.6112-1-79-3', '1308.6112-2-79-4'], ['1308.6112-1-79-4', '1308.6112-2-79-5'], ['1308.6112-1-44-0', '1308.6112-2-44-0'], ['1308.6112-1-44-4', '1308.6112-2-44-4'], ['1308.6112-1-44-7', '1308.6112-2-44-7'], ['1308.6112-1-44-8', '1308.6112-2-44-8'], ['1308.6112-1-119-3', '1308.6112-2-119-3'], ['1308.6112-1-119-4', '1308.6112-2-119-4'], ['1308.6112-1-119-5', '1308.6112-2-119-5'], ['1308.6112-1-177-0', '1308.6112-2-176-0'], ['1308.6112-1-177-1', '1308.6112-2-176-1'], ['1308.6112-1-177-2', '1308.6112-2-176-2'], ['1308.6112-1-177-3', '1308.6112-2-176-3'], ['1308.6112-1-7-1', '1308.6112-2-7-1'], ['1308.6112-1-74-1', '1308.6112-2-74-1'], ['1308.6112-1-74-2', '1308.6112-2-74-2'], ['1308.6112-1-74-3', '1308.6112-2-74-3'], ['1308.6112-1-74-5', '1308.6112-2-74-5'], ['1308.6112-1-74-6', '1308.6112-2-74-6'], ['1308.6112-1-74-7', '1308.6112-2-74-7'], ['1308.6112-1-21-3', '1308.6112-2-21-3'], ['1308.6112-1-52-1', '1308.6112-2-52-1'], ['1308.6112-1-52-3', '1308.6112-2-52-3'], ['1308.6112-1-102-3', '1308.6112-2-102-4'], ['1308.6112-1-102-4', '1308.6112-2-102-5'], ['1308.6112-1-171-2', '1308.6112-2-171-2'], ['1308.6112-1-20-0', '1308.6112-2-20-0'], ['1308.6112-1-49-0', '1308.6112-2-49-0'], ['1308.6112-1-49-1', '1308.6112-2-49-1'], ['1308.6112-1-103-1', '1308.6112-2-103-1'], ['1308.6112-1-182-0', '1308.6112-2-181-0'], ['1308.6112-1-182-1', '1308.6112-2-181-1'], ['1308.6112-1-133-1', '1308.6112-2-133-1'], ['1308.6112-1-86-1', '1308.6112-2-86-1'], ['1308.6112-1-48-2', '1308.6112-2-48-2'], ['1308.6112-1-48-3', '1308.6112-2-48-3'], ['1308.6112-1-48-4', '1308.6112-2-48-4'], ['1308.6112-1-48-5', '1308.6112-2-48-5'], ['1308.6112-1-48-9', '1308.6112-2-48-9'], ['1308.6112-1-120-2', '1308.6112-2-120-2'], ['1308.6112-1-120-5', '1308.6112-2-120-5'], ['1308.6112-1-63-0', '1308.6112-2-63-0'], ['1308.6112-1-63-1', '1308.6112-2-63-1'], ['1308.6112-1-10-0', '1308.6112-2-10-0'], ['1308.6112-1-10-1', '1308.6112-2-10-1'], ['1308.6112-1-10-3', '1308.6112-2-10-3'], ['1308.6112-1-10-5', '1308.6112-2-10-5'], ['1308.6112-1-61-0', '1308.6112-2-61-0'], ['1308.6112-1-61-2', '1308.6112-2-61-2'], ['1308.6112-1-101-0', '1308.6112-2-101-0'], ['1308.6112-1-101-1', '1308.6112-2-101-1'], ['1308.6112-1-101-2', '1308.6112-2-101-2'], ['1308.6112-1-6-0', '1308.6112-2-6-0'], ['1308.6112-1-6-1', '1308.6112-2-6-1'], ['1308.6112-1-25-2', '1308.6112-2-25-2'], ['1308.6112-1-25-3', '1308.6112-2-25-3'], ['1308.6112-1-25-4', '1308.6112-2-25-4'], ['1308.6112-1-25-6', '1308.6112-2-25-6'], ['1308.6112-1-42-5', '1308.6112-2-42-5'], ['1308.6112-1-167-1', '1308.6112-2-167-1'], ['1308.6112-1-160-1', '1308.6112-2-160-1'], ['1308.6112-1-160-5', '1308.6112-2-160-5'], ['1308.6112-1-100-0', '1308.6112-2-100-0'], ['1308.6112-1-100-1', '1308.6112-2-100-1'], ['1308.6112-1-100-3', '1308.6112-2-100-3'], ['1308.6112-1-59-0', '1308.6112-2-59-0'], ['1308.6112-1-59-1', '1308.6112-2-59-1'], ['1308.6112-1-59-3', '1308.6112-2-59-3'], ['1308.6112-1-87-1', '1308.6112-2-87-1'], ['1308.6112-1-87-3', '1308.6112-2-87-3'], ['1308.6112-1-87-4', '1308.6112-2-87-4'], ['1308.6112-1-87-6', '1308.6112-2-87-6'], ['1308.6112-1-87-10', '1308.6112-2-87-10'], ['1308.6112-1-139-2', '1308.6112-2-139-2'], ['1308.6112-1-1-3', '1308.6112-2-1-3'], ['1308.6112-1-1-7', '1308.6112-2-1-7'], ['1308.6112-1-1-9', '1308.6112-2-1-9'], ['1308.6112-1-1-10', '1308.6112-2-1-10'], ['1308.6112-1-19-0', '1308.6112-2-19-0'], ['1308.6112-1-19-2', '1308.6112-2-19-2'], ['1308.6112-1-19-4', '1308.6112-2-19-4'], ['1308.6112-1-163-2', '1308.6112-2-163-2'], ['1308.6112-1-163-5', '1308.6112-2-163-5'], ['1308.6112-1-180-1', '1308.6112-2-179-1'], ['1308.6112-1-180-2', '1308.6112-2-179-2'], ['1308.6112-1-180-7', '1308.6112-2-179-7'], ['1308.6112-1-22-0', '1308.6112-2-22-0'], ['1308.6112-1-22-3', '1308.6112-2-22-3'], ['1308.6112-1-22-4', '1308.6112-2-22-4'], ['1308.6112-1-118-3', '1308.6112-2-118-3'], ['1308.6112-1-178-0', '1308.6112-2-177-0'], ['1308.6112-1-178-1', '1308.6112-2-177-1'], ['1308.6112-1-178-3', '1308.6112-2-177-3'], ['1308.6112-1-56-1', '1308.6112-2-56-1'], ['1308.6112-1-56-2', '1308.6112-2-56-2'], ['1308.6112-1-56-3', '1308.6112-2-56-3'], ['1308.6112-1-56-5', '1308.6112-2-56-5'], ['1308.6112-1-56-6', '1308.6112-2-56-6'], ['1308.6112-1-165-2', '1308.6112-2-165-2'], ['1308.6112-1-165-4', '1308.6112-2-165-4'], ['1308.6112-1-77-3', '1308.6112-2-77-3'], ['1308.6112-1-166-2', '1308.6112-2-166-1'], ['1308.6112-1-157-1', '1308.6112-2-157-1'], ['1308.6112-1-157-2', '1308.6112-2-157-2'], ['1308.6112-1-93-1', '1308.6112-2-93-1'], ['1308.6112-1-154-3', '1308.6112-2-154-3'], ['1308.6112-1-154-4', '1308.6112-2-154-4'], ['1308.6112-1-159-0', '1308.6112-2-159-0'], ['1308.6112-1-159-3', '1308.6112-2-159-3'], ['1308.6112-1-159-5', '1308.6112-2-159-5'], ['1308.6112-1-114-0', '1308.6112-2-114-0'], ['1308.6112-1-114-1', '1308.6112-2-114-1'], ['1308.6112-1-121-1', '1308.6112-2-121-1'], ['1308.6112-1-121-4', '1308.6112-2-121-4'], ['1308.6112-1-121-5', '1308.6112-2-121-5'], ['1308.6112-1-67-0', '1308.6112-2-67-0'], ['1308.6112-1-67-1', '1308.6112-2-67-1'], ['1308.6112-1-67-3', '1308.6112-2-67-3'], ['1308.6112-1-67-4', '1308.6112-2-67-4'], ['1308.6112-1-107-2', '1308.6112-2-107-2'], ['1308.6112-1-170-3', '1308.6112-2-170-3'], ['1308.6112-1-170-5', '1308.6112-2-170-5'], ['1308.6112-1-84-0', '1308.6112-2-84-0'], ['1308.6112-1-84-2', '1308.6112-2-84-2'], ['1308.6112-1-84-3', '1308.6112-2-84-3'], ['1308.6112-1-29-0', '1308.6112-2-29-0'], ['1308.6112-1-29-2', '1308.6112-2-29-2'], ['1308.6112-1-29-3', '1308.6112-2-29-3'], ['1308.6112-1-29-4', '1308.6112-2-29-4'], ['1308.6112-1-123-1', '1308.6112-2-123-1'], ['1308.6112-1-123-3', '1308.6112-2-123-3'], ['1308.6112-1-123-4', '1308.6112-2-123-4'], ['1308.6112-1-161-1', '1308.6112-2-161-1'], ['1308.6112-1-161-3', '1308.6112-2-161-3'], ['1308.6112-1-69-2', '1308.6112-2-69-2'], ['1308.6112-1-81-1', '1308.6112-2-81-1'], ['1308.6112-1-81-2', '1308.6112-2-81-2'], ['1308.6112-1-81-3', '1308.6112-2-81-3'], ['1308.6112-1-60-0', '1308.6112-2-60-0'], ['1308.6112-1-60-5', '1308.6112-2-60-5'], ['1308.6112-1-60-6', '1308.6112-2-60-6'], ['1308.6112-1-142-0', '1308.6112-2-142-0'], ['1308.6112-1-142-5', '1308.6112-2-142-5'], ['1308.6112-1-75-0', '1308.6112-2-75-0'], ['1308.6112-1-75-2', '1308.6112-2-75-2'], ['1308.6112-1-181-0', '1308.6112-2-180-0'], ['1308.6112-1-181-1', '1308.6112-2-180-1'], ['1308.6112-1-181-3', '1308.6112-2-180-3'], ['1308.6112-1-28-0', '1308.6112-2-28-0'], ['1308.6112-1-28-1', '1308.6112-2-28-1'], ['1308.6112-1-28-3', '1308.6112-2-28-3'], ['1308.6112-1-28-5', '1308.6112-2-28-5'], ['1308.6112-1-46-0', '1308.6112-2-46-0'], ['1308.6112-1-138-0', '1308.6112-2-138-0'], ['1308.6112-1-95-1', '1308.6112-2-95-1'], ['1308.6112-1-95-2', '1308.6112-2-95-2'], ['1308.6112-1-37-0', '1308.6112-2-37-0'], ['1308.6112-1-37-3', '1308.6112-2-37-2'], ['1308.6112-1-96-2', '1308.6112-2-96-2'], ['1308.6112-1-96-5', '1308.6112-2-96-5'], ['1308.6112-1-96-7', '1308.6112-2-96-7'], ['1308.6112-1-4-0', '1308.6112-2-4-0'], ['1308.6112-1-4-2', '1308.6112-2-4-2'], ['1308.6112-1-30-0', '1308.6112-2-30-0'], ['1308.6112-1-30-2', '1308.6112-2-30-2'], ['1308.6112-1-30-3', '1308.6112-2-30-3'], ['1308.6112-1-188-1', '1308.6112-2-186-1'], ['1308.6112-1-0-0', '1308.6112-2-0-0'], ['1308.6112-1-132-1', '1308.6112-2-132-1'], ['1308.6112-1-8-1', '1308.6112-2-8-1'], ['1308.6112-1-17-5', '1308.6112-2-17-5'], ['1308.6112-1-124-1', '1308.6112-2-124-1'], ['1308.6112-1-124-4', '1308.6112-2-124-4'], ['1308.6112-1-98-1', '1308.6112-2-98-1'], ['1308.6112-1-98-3', '1308.6112-2-98-3'], ['1308.6112-1-98-7', '1308.6112-2-98-7'], ['1308.6112-1-57-0', '1308.6112-2-57-0'], ['1308.6112-1-57-3', '1308.6112-2-57-2'], ['1308.6112-1-57-4', '1308.6112-2-57-3'], ['1308.6112-1-57-5', '1308.6112-2-57-4'], ['1308.6112-1-57-6', '1308.6112-2-57-5'], ['1308.6112-1-57-7', '1308.6112-2-57-6'], ['1308.6112-1-57-8', '1308.6112-2-57-7'], ['1308.6112-1-57-9', '1308.6112-2-57-8'], ['1308.6112-1-40-1', '1308.6112-2-40-1'], ['1308.6112-1-40-2', '1308.6112-2-40-2'], ['1308.6112-1-13-1', '1308.6112-2-13-1'], ['1308.6112-1-51-1', '1308.6112-2-51-1'], ['1308.6112-1-51-4', '1308.6112-2-51-4'], ['1308.6112-1-51-5', '1308.6112-2-51-5'], ['1308.6112-1-127-0', '1308.6112-2-127-0'], ['1308.6112-1-127-1', '1308.6112-2-127-1'], ['1308.6112-1-127-4', '1308.6112-2-127-4'], ['1308.6112-1-127-7', '1308.6112-2-127-7'], ['1308.6112-1-108-1', '1308.6112-2-108-1'], ['1308.6112-1-35-1', '1308.6112-2-35-1'], ['1308.6112-1-35-3', '1308.6112-2-35-3'], ['1308.6112-1-35-4', '1308.6112-2-35-4'], ['1308.6112-1-91-0', '1308.6112-2-91-0'], ['1308.6112-1-16-1', '1308.6112-2-16-1'], ['1308.6112-1-16-2', '1308.6112-2-16-2'], ['1308.6112-1-16-3', '1308.6112-2-16-3'], ['1308.6112-1-149-1', '1308.6112-2-149-1'], ['1308.6112-1-149-3', '1308.6112-2-149-3'], ['1308.6112-1-71-0', '1308.6112-2-71-0'], ['1308.6112-1-71-3', '1308.6112-2-71-3'], ['1308.6112-1-71-5', '1308.6112-2-71-5'], ['1308.6112-1-140-1', '1308.6112-2-140-1'], ['1308.6112-1-140-2', '1308.6112-2-140-2'], ['1308.6112-1-140-3', '1308.6112-2-140-3'], ['1308.6112-1-3-3', '1308.6112-2-3-3'], ['1308.6112-1-116-1', '1308.6112-2-116-1'], ['1308.6112-1-104-0', '1308.6112-2-104-0'], ['1308.6112-1-104-1', '1308.6112-2-104-1'], ['1308.6112-1-104-2', '1308.6112-2-104-2'], 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['1308.6112-1-3-2', '1308.6112-2-3-2'], ['1308.6112-1-116-0', '1308.6112-2-116-0'], ['1308.6112-1-12-0', '1308.6112-2-12-0'], ['1308.6112-1-12-2', '1308.6112-2-12-2'], ['1308.6112-1-104-3', '1308.6112-2-104-3'], ['1308.6112-1-104-5', '1308.6112-2-104-5'], ['1308.6112-1-104-6', '1308.6112-2-104-6'], ['1308.6112-1-104-7', '1308.6112-2-104-7'], ['1308.6112-1-104-10', '1308.6112-2-104-10'], ['1308.6112-1-15-0', '1308.6112-2-15-0'], ['1308.6112-1-15-1', '1308.6112-2-15-1'], ['1308.6112-1-90-0', '1308.6112-2-90-0'], ['1308.6112-1-90-1', '1308.6112-2-90-1'], ['1308.6112-1-90-2', '1308.6112-2-90-2'], ['1308.6112-1-24-1', '1308.6112-2-24-1'], ['1308.6112-1-24-2', '1308.6112-2-24-2'], ['1308.6112-1-110-0', '1308.6112-2-110-0'], ['1308.6112-1-110-2', '1308.6112-2-110-2'], ['1308.6112-1-136-0', '1308.6112-2-136-0'], ['1308.6112-1-185-1', '1308.6112-2-184-1'], ['1308.6112-1-185-4', '1308.6112-2-184-4'], ['1308.6112-1-185-5', '1308.6112-2-184-5'], ['1308.6112-1-50-0', '1308.6112-2-50-0'], ['1308.6112-1-50-1', '1308.6112-2-50-1'], ['1308.6112-1-50-2', '1308.6112-2-50-2'], ['1308.6112-1-50-3', '1308.6112-2-50-3'], ['1308.6112-1-50-4', '1308.6112-2-50-4'], ['1308.6112-1-97-1', '1308.6112-2-97-1'], ['1308.6112-1-97-3', '1308.6112-2-97-3'], ['1308.6112-1-111-0', '1308.6112-2-111-0'], ['1308.6112-1-111-1', '1308.6112-2-111-1'], ['1308.6112-1-111-2', '1308.6112-2-111-2'], ['1308.6112-1-73-2', '1308.6112-2-73-2'], ['1308.6112-1-73-4', '1308.6112-2-73-4'], ['1308.6112-1-73-5', '1308.6112-2-73-5'], ['1308.6112-1-126-0', '1308.6112-2-126-0'], ['1308.6112-1-126-1', '1308.6112-2-126-1'], ['1308.6112-1-126-2', '1308.6112-2-126-2'], ['1308.6112-1-126-5', '1308.6112-2-126-5'], ['1308.6112-1-99-0', '1308.6112-2-99-0'], ['1308.6112-1-99-1', '1308.6112-2-99-1'], ['1308.6112-1-99-2', '1308.6112-2-99-2'], ['1308.6112-1-94-0', '1308.6112-2-94-0'], ['1308.6112-1-94-3', '1308.6112-2-94-2'], ['1308.6112-1-83-0', '1308.6112-2-83-0'], ['1308.6112-1-83-1', '1308.6112-2-83-1'], ['1308.6112-1-31-1', '1308.6112-2-31-1'], ['1308.6112-1-31-2', '1308.6112-2-31-2'], ['1308.6112-1-152-1', '1308.6112-2-152-1'], ['1308.6112-1-152-2', '1308.6112-2-152-2'], ['1308.6112-1-153-0', '1308.6112-2-153-0'], ['1308.6112-1-153-1', '1308.6112-2-153-1'], ['1308.6112-1-153-3', '1308.6112-2-153-3'], ['1308.6112-1-153-4', '1308.6112-2-153-4'], ['1308.6112-1-186-1', '1308.6112-2-185-1'], ['1308.6112-1-173-0', '1308.6112-2-172-2'], ['1308.6112-1-173-1', '1308.6112-2-172-3'], ['1308.6112-1-173-2', '1308.6112-2-172-4']] | [] | [['1308.6112-1-39-7', '1308.6112-2-39-7'], ['1308.6112-1-27-1', '1308.6112-2-27-1'], ['1308.6112-1-5-0', '1308.6112-2-5-0'], ['1308.6112-1-5-1', '1308.6112-2-5-1'], ['1308.6112-1-143-4', '1308.6112-2-143-4'], ['1308.6112-1-79-1', '1308.6112-2-79-1'], ['1308.6112-1-79-1', '1308.6112-2-79-2'], ['1308.6112-1-44-2', '1308.6112-2-44-2'], ['1308.6112-1-130-2', '1308.6112-2-130-2'], ['1308.6112-1-74-8', '1308.6112-2-74-8'], ['1308.6112-1-102-0', '1308.6112-2-102-0'], ['1308.6112-1-103-2', '1308.6112-2-103-2'], ['1308.6112-1-160-6', '1308.6112-2-160-6'], ['1308.6112-1-59-7', '1308.6112-2-59-7'], ['1308.6112-1-1-2', '1308.6112-2-1-2'], ['1308.6112-1-113-0', '1308.6112-2-113-0'], ['1308.6112-1-113-0', '1308.6112-2-113-1'], ['1308.6112-1-22-5', '1308.6112-2-22-5'], ['1308.6112-1-22-5', '1308.6112-2-22-6'], ['1308.6112-1-166-0', '1308.6112-2-166-0'], ['1308.6112-1-166-1', '1308.6112-2-166-0'], ['1308.6112-1-81-0', '1308.6112-2-81-0'], ['1308.6112-1-60-7', '1308.6112-2-60-7'], ['1308.6112-1-142-2', '1308.6112-2-142-2'], ['1308.6112-1-37-1', '1308.6112-2-37-1'], ['1308.6112-1-37-2', '1308.6112-2-37-1'], ['1308.6112-1-124-2', '1308.6112-2-124-2'], ['1308.6112-1-124-3', '1308.6112-2-124-3'], ['1308.6112-1-57-1', '1308.6112-2-57-1'], ['1308.6112-1-57-2', '1308.6112-2-57-1'], ['1308.6112-1-40-5', '1308.6112-2-40-5'], ['1308.6112-1-40-6', '1308.6112-2-40-5'], ['1308.6112-1-31-3', '1308.6112-2-31-3'], ['1308.6112-1-152-0', '1308.6112-2-152-0']] | [] | ['1308.6112-1-13-3', '1308.6112-1-94-2', '1308.6112-1-113-2', '1308.6112-1-113-3', '1308.6112-1-160-4', '1308.6112-2-12-1', '1308.6112-2-13-3', '1308.6112-2-102-1', '1308.6112-2-113-3', '1308.6112-2-113-4', '1308.6112-2-160-4'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1308.6112 | null | null | null | null | null |
1411.7041 | {'1411.7041-1-0-0': '[email protected] [email protected] [email protected] We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction.', '1411.7041-1-0-1': 'Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction.', '1411.7041-1-0-2': 'We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue.', '1411.7041-1-0-3': 'Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs.', '1411.7041-1-0-4': 'Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.', '1411.7041-1-1-0': '# Introduction', '1411.7041-1-2-0': 'Almost twenty years after its initial formulation, the AdS/CFT correspondence remains our best-understood example of a precise theory of quantum gravity.', '1411.7041-1-2-1': 'It has shed light on many deep puzzles in quantum gravity, and has also been of practical use in studying the dynamics of strongly interacting quantum field theories.', '1411.7041-1-2-2': 'One aspect that remains mysterious, however, is the emergence of approximate bulk locality.', '1411.7041-1-2-3': 'Locality near the boundary is straightforward.', '1411.7041-1-2-4': 'In the "extrapolate" version of the AdS/CFT dictionary we have a simple relation [CITATION] [EQUATION] between limiting values of a bulk field [MATH] and a conformal field theory operator [MATH]; this dictionary manifestly respects locality in the [MATH] directions since the CFT does.', '1411.7041-1-2-5': 'The radial direction, however, is more subtle.', '1411.7041-1-2-6': 'One way to see this is to observe that naively a local operator in the center of the bulk should commute with every local operator at the boundary on a fixed time slice containing that bulk operator.', '1411.7041-1-2-7': 'This is not consistent, however, with a standard property of quantum field theory; any operator that commutes with all local operators at a fixed time must be proportional to the identity.', '1411.7041-1-2-8': "Bulk locality thus cannot be respected within the CFT at the level of the algebra of operators; we'd then like to know in what sense it is respected.", '1411.7041-1-3-0': 'The basic idea of this paper is that bulk locality is a statement about certain subspaces of states in the CFT.', '1411.7041-1-3-1': 'That these subspaces can be large is a consequence of the large-[MATH] properties of the CFT, but the large degree of non-local entanglement in finite energy states of the CFT also plays an essential role.', '1411.7041-1-3-2': 'Our strategy will be to gradually back away from the [MATH] limit in equation [REF] and study how the the CFT representations of bulk operators spread in spatial support as we do so.', '1411.7041-1-3-3': 'On the bulk side the tool we will mostly use is the AdS-Rindler reconstruction of bulk fields introduced in [CITATION] and refined in [CITATION].', '1411.7041-1-3-4': 'We will observe that this construction has several paradoxical features, which we will illuminate by recasting it on the CFT side in the language of quantum error correcting codes [CITATION].', '1411.7041-1-3-5': 'This language gives a new, more general perspective on the issue of bulk reconstruction, and we believe that it is the natural framework for understanding the idea of "subregion-subregion" duality [CITATION].', '1411.7041-1-3-6': 'In particular, the radial direction in the bulk is realized in the CFT as a measure of how well CFT representations of bulk quantum information are protected from local erasures.', '1411.7041-1-3-7': 'The holographic principle also naturally arises in the guise of the general statement that there is an upper bound on how much quantum information a given code can protect from erasures.', '1411.7041-1-4-0': 'One point that will appear in this analysis is that truncated subalgebras of bulk observables are of interest; these were also advocated in [CITATION] in the context of describing the black hole interior.', '1411.7041-1-4-1': 'Aspects of our proposal are inspired by their construction, but here we do not discuss black hole interiors and we are not violating quantum mechanics [CITATION].', '1411.7041-1-4-2': 'A connection between black holes and quantum error correction was also made in [CITATION], which is essentially an earlier version of the proposal of [CITATION], but again the context was different and our work here should be uncontroversial by comparison.', '1411.7041-1-5-0': 'Before proceeding, let us establish a few conventions used throughout this paper.', '1411.7041-1-5-1': 'We will frequently discuss subspaces and tensor factors of the Hilbert space.', '1411.7041-1-5-2': 'When we say that an operator acts within a subspace we also mean that the same is true for its hermitian conjugate.', '1411.7041-1-5-3': 'If the Hilbert space is a tensor product [MATH], for any operator [MATH] that acts on [MATH] we may trivially form an operator [MATH] that acts on the entire Hilbert space.', '1411.7041-1-5-4': 'We will often drop the identity operator [MATH] and write it simply as [MATH].', '1411.7041-1-5-5': 'The reader should also keep in mind that the CFT regions [MATH] and [MATH] that we will talk about in section [REF] unfortunately correspond to [MATH] and [MATH] respectively in the language of section [REF].', '1411.7041-1-6-0': '# Bulk reconstruction and an AdS-Rindler puzzle', '1411.7041-1-7-0': '## Global AdS reconstruction', '1411.7041-1-8-0': 'We begin by briefly recalling the standard CFT construction of local bulk fields in AdS [CITATION].', '1411.7041-1-8-1': 'We will first work in global coordinates, where the metric asymptotically has the form [EQUATION]', '1411.7041-1-8-2': 'The CFT dual to this system lives on [MATH], with the [MATH] being the time direction.', '1411.7041-1-8-3': 'The Hilbert space of states is the set of field configurations on [MATH].', '1411.7041-1-8-4': 'The idea is then to perturbatively construct operators in the CFT which obey the bulk equations of motion, with the boundary conditions set by the dictionary [REF].', '1411.7041-1-8-5': 'For simplicity we will assume that all bulk interactions are suppressed by inverse powers of a quantity [MATH], which will also set the AdS radius in Planck units.', '1411.7041-1-8-6': 'At leading order in [MATH], this procedure results in a straightforward prescription for the CFT representation of a bulk field [MATH]; we simply have [EQUATION] where the integral is over the conformal boundary and [MATH] is a so-called "smearing function".', '1411.7041-1-8-7': 'The smearing function obeys the bulk wave equation in its [MATH] index, and leads to [REF] as we take [MATH] to the boundary.', '1411.7041-1-8-8': 'It can be chosen to only have support when [MATH] and [MATH] are spacelike separated, which we illustrate for [MATH] in the left diagram of figure [REF]; the point [MATH] is represented by a boundary integral over the green region only.', '1411.7041-1-8-9': 'In the case of empty AdS, where we take [REF] to hold everywhere, explicit representations of the smearing function can be found in [CITATION].', '1411.7041-1-8-10': "[MATH] corrections can be systematically included [CITATION], although we won't really need to discuss them here.", '1411.7041-1-8-11': 'It is clear that this prescription can only work perturbatively about a background, or in other words perturbatively about a specific state in the CFT; otherwise we would run into a contradiction with the argument in the introduction.', '1411.7041-1-8-12': 'At higher orders in this perturbation theory we will need to confront the problem of defining local operators in a diffeomorphism invariant theory, but we postpone discussion of this until section [REF].', '1411.7041-1-9-0': 'Note that once we have a representation of the form [REF], we can use the CFT Hamiltonian to re-express all operators on the right hand side in terms of Heisenberg picture fields on a single Cauchy surface in the CFT, denoted as [MATH] in figure [REF].', '1411.7041-1-9-1': 'This representation is quite nontrivial, in general it involves severely nonlocal and multitrace operators.', '1411.7041-1-9-2': 'It also has the property that if we take [MATH] to be near the boundary but not quite on it, the single-time CFT representation of [MATH] still involves operators with support on all of [MATH].', '1411.7041-1-9-3': 'We might hope to find a representation whose boundary support shrinks as the operator approaches the boundary, and indeed the AdS-Rindler representation does exactly this, as we will now explain.', '1411.7041-1-10-0': '## AdS-Rindler reconstruction', '1411.7041-1-11-0': 'Consider a subregion [MATH] of a CFT Cauchy surface [MATH].', '1411.7041-1-11-1': 'The boundary domain of dependence of [MATH], denoted [MATH], is defined as the set of points on the boundary with the property that every inextendible causal curve, meaning a curve whose tangent vector is never spacelike and which is not part of a larger curve with this property, that passes through it must also intersect [MATH].', '1411.7041-1-11-2': 'This is illustrated for the boundary of [MATH] in the right diagram of figure [REF], where [MATH] is the boundary interval lying between the two vertical hash marks and [MATH] is shaded green.', '1411.7041-1-11-3': 'For any boundary region [MATH], its bulk causal future/past [MATH] is defined as the set of bulk points which can be reached by bulk causal curves evolving from/to the region [MATH].', '1411.7041-1-11-4': 'The causal wedge of a CFT subregion A [CITATION] is defined as [EQUATION]', '1411.7041-1-11-5': 'In the right diagram of figure [REF], [MATH] roughly lies between the dashed lines and [MATH].', '1411.7041-1-11-6': 'The bulk codimension-two surface [MATH] in the figure is the "rim" of the wedge and is commonly referred to as the causal surface of [MATH] [CITATION]; more precisely it is defined as the part of the intersection of the boundaries of [MATH] that does not also intersect the conformal boundary at infinity.', '1411.7041-1-11-7': '[MATH] can also be described as the intersection of the past and future horizons of [MATH].', '1411.7041-1-12-0': 'A simple example of these definitions is where we take the geometry to be pure [MATH], [MATH] to be the [MATH] slice of the boundary, and [MATH] to be one hemisphere of [MATH].', '1411.7041-1-12-1': 'In this case [MATH] becomes what is usually referred to as the AdS-Rindler wedge.', '1411.7041-1-13-0': 'A natural set of bulk coordinates on the AdS-Rindler wedge gives a metric with the form [EQUATION] where the coordinate ranges are [MATH] and the geometry in parentheses is just the [MATH] dimensional hyperbolic disc.', '1411.7041-1-13-1': 'The causal surface [MATH] is given by the limit [MATH] at fixed [MATH], and [MATH] itself is given by [MATH] and [MATH].', '1411.7041-1-13-2': 'We illustrate this for [MATH] in figure [REF].', '1411.7041-1-13-3': 'By acting on this example with bulk isometries (or equivalently boundary conformal transformations), we can arrive at the causal wedge for any round disc in [MATH].', '1411.7041-1-13-4': 'The case of [MATH] is especially simple; all connected boundary regions are intervals and thus can be produced in this way.', '1411.7041-1-14-0': 'The point then is that the construction of CFT representations of bulk fields in the previous subsection can also be implemented purely within the causal wedge [CITATION].', '1411.7041-1-14-1': 'At leading order in [MATH], the claim is that for any [MATH] with [MATH], we can again represent [MATH] via the expression [REF], but with the [MATH] integral now taken only over [MATH].', '1411.7041-1-14-2': 'This is illustrated for [MATH] in the right diagram in figure [REF], where we have allowed for a conformal transformation that changes the size of the boundary interval [MATH].', '1411.7041-1-14-3': 'We review more details of this construction in appendix [REF]; the only major subtlety is that the smearing function [MATH] no longer exists as a function and must be understood as a distribution for integration against CFT expectation values [CITATION].', '1411.7041-1-15-0': 'Thus we see that the AdS-Rindler construction of [MATH] indeed has the property that if [MATH] is close to the boundary, only a small boundary region [MATH] localized near [MATH] is needed to be able to reconstruct [MATH] in [MATH].', '1411.7041-1-15-1': 'Moreover, by making use of the CFT evolution we can again rewrite the expression [REF] entirely in terms of nonlocal Heisenberg operators acting at [MATH], but now they will act only on [MATH].', '1411.7041-1-16-0': '## Overlapping wedges', '1411.7041-1-17-0': 'The AdS-Rindler construction of bulk fields we have just described has the somewhat counter-intuitive property that the same bulk field operator [MATH] lies in multiple causal wedges, and thus can be represented as an operator on distinct regions [MATH] in [MATH].', '1411.7041-1-17-1': 'One consequence of this is shown in the left diagram of figure [REF]; for any bulk field operator [MATH] and any CFT local operator [MATH] such that [MATH] and [MATH] are spacelike separated, we can choose a causal wedge [MATH] such that [MATH] lies in the complement of [MATH] in [MATH].', '1411.7041-1-17-2': 'By CFT locality [MATH] then must exactly commute with our representation of [MATH] in that wedge.', '1411.7041-1-17-3': 'This is coming dangerously close to contradicting the theorem mentioned in the introduction, that is that no nontrivial operator in the CFT can commute with all local CFT operators on [MATH].', '1411.7041-1-18-0': 'To avoid this contradiction it must be the case that the representations of [MATH] in different wedges are not really all the same operator on the CFT Hilbert space.', '1411.7041-1-18-1': 'We can see this in another way by considering the setup of the center diagram in figure [REF], where we have two overlapping wedges [MATH] and [MATH] that both contain the point [MATH] but [MATH] is not contained in [MATH].', '1411.7041-1-18-2': 'For a CFT operator defined with support only on [MATH] to really be equal to a CFT operator defined with support only on [MATH], it must be that the operator really only has support on [MATH].', '1411.7041-1-18-3': 'But given that we have chosen [MATH] to lie outside of [MATH], we do not expect the operator to have such a representation.', '1411.7041-1-18-4': 'In fact in this example the operator has a representation on the complement of [MATH], and we will see in the following section that, as long as we smear the operator over a distance scale that is small compared to its proper distance from [MATH], the no-cloning theorem of quantum mechanics forbids an accurate representation of the operator on [MATH].', '1411.7041-1-19-0': 'We can see the non-equivalence of the operators even more clearly by considering a third example, shown in the right diagram in figure [REF].', '1411.7041-1-19-1': 'Now a bulk field at the point [MATH] lies outside of the causal wedge for any one of the regions, but it can be reconstructed in [MATH], [MATH], or [MATH].', '1411.7041-1-19-2': 'The mutual intersection of these regions is just three points, and if we consider another set of three regions slightly rotated from these we can come up with a set of six possible reconstructions whose mutual intersection is genuinely empty.', '1411.7041-1-19-3': 'There is simply no possible way that they can all be equal as operators.', '1411.7041-1-19-4': 'For future reference we will refer to the three operators as [MATH], [MATH], and [MATH].', '1411.7041-1-20-0': 'We thus need to decide how we are to reconcile these operator inequivalences with the fact that in the bulk theory it seems that the operators are equivalent.', '1411.7041-1-20-1': 'There will clearly be some CFT states where they act quite differently, and we would like to understand the physics of the subset of states where their action is equivalent.', '1411.7041-1-20-2': 'This problem can be nicely understood in the language of quantum error correction, to which we now turn.', '1411.7041-1-21-0': '# Correcting quantum erasures', '1411.7041-1-22-0': 'Say Alice wants to send Bob a quantum state of [MATH] qubits in the mail, but she is worried that some of the qubits might get lost on the way.', '1411.7041-1-22-1': 'Quantum error correction is a procedure that allows her to embed this state into [MATH] qubits in such a way that even if some qubits are lost, Bob can still recover it.', '1411.7041-1-22-2': 'In this section we review some basic facts about this, beginning with an example.', '1411.7041-1-23-0': '## A simple example of erasure correction', '1411.7041-1-24-0': 'The simplest example of quantum error correction actually involves three-state "qutrits" instead of two-state qubits, and it uses three qutrits to send a single-qutrit message [CITATION].', '1411.7041-1-24-1': 'Say Alice wishes to send the state [EQUATION]', '1411.7041-1-24-2': 'The idea is to instead send the state [EQUATION] where [EQUATION]', '1411.7041-1-24-3': 'This protocol has two remarkable properties.', '1411.7041-1-24-4': 'First of all for any state [MATH], the reduced density matrix on any one of the qutrits is maximally mixed.', '1411.7041-1-24-5': 'Thus no single qutrit can be used to acquire any information about the state.', '1411.7041-1-24-6': 'Secondly, from any two of the qutrits Bob can reconstruct the state.', '1411.7041-1-24-7': 'For example, say he has access to only the first two qutrits.', '1411.7041-1-24-8': 'He can make use of the fact that there exists a unitary transformation [MATH] acting only on the first two qutrits that implements [EQUATION]', '1411.7041-1-24-9': 'Acting with this on the encoded message, we see that Bob can recover the state [MATH]: [EQUATION]', '1411.7041-1-24-10': 'Explicitly [MATH] is a permutation that acts as [EQUATION]', '1411.7041-1-24-11': 'Clearly by the symmetry of [REF] a similar construction is also possible if Bob has access only to the second and third, or first and third qutrits.', '1411.7041-1-24-12': 'Thus Bob can correct for the loss of any one of the qutrits; in quantum information terminology one describes this as a quantum error correcting code that can protect against arbitrary single qutrit erasures.', '1411.7041-1-24-13': 'The subspace spanned by [REF] is called the code subspace; the entanglement of the states in the code subspace is essential for the functioning of the protocol.', '1411.7041-1-25-0': 'In our discussion of reconstruction in the previous section we were interested in the action of operators rather than the recovery of states, and we can rephrase the error correction protocol in this language.', '1411.7041-1-25-1': 'Indeed, say that [MATH] is an operator that acts on the single qutrit Hilbert space as [EQUATION]', '1411.7041-1-25-2': 'In quantum computing language, operators that act directly on the code subspace in this manner are called logical operations, since they are the types of things that we want to implement when performing a fault-tolerant quantum computation.', '1411.7041-1-25-3': 'We will use this terminology occasionally.', '1411.7041-1-25-4': 'It is straightforward to see that the operator [EQUATION] acts as [EQUATION] so there is an operator acting only on the first two qutrits that implements the operator [MATH] directly on the code subspace.', '1411.7041-1-25-5': 'Since we can also analogously construct [MATH] or [MATH], we have realized a situation where operators with nontrivial support on different qutrits have the same action on the code subspace.', '1411.7041-1-25-6': 'This should be reminiscent of our discussion of overlapping wedges in the previous section; we will make the connection more explicit soon but first we need to discuss some general properties of quantum erasure correction.', '1411.7041-1-26-0': '## General erasure correction', '1411.7041-1-27-0': 'We now describe a natural generalization of the protocol of the previous subsection.', '1411.7041-1-27-1': 'For familiarity we will describe it using qubits, although none of the results rely on this.', '1411.7041-1-27-2': 'Say that we want to protect a [MATH]-qubit code subspace of an [MATH]-qubit system against the loss of some collection of [MATH] of [MATH] of the qubits.', '1411.7041-1-27-3': 'We define the code subspace [MATH] as the span of the orthonormal states [EQUATION] where [MATH] is called the encoding unitary transformation.', '1411.7041-1-27-4': 'There is a necessary and sufficient condition for the correctability of the erasure of [MATH] [CITATION].', '1411.7041-1-27-5': 'Say that we adjoin to our system a reference system [MATH] of [MATH] additional qubits.', '1411.7041-1-27-6': "We then consider the state [EQUATION] where [MATH] denotes the set of [MATH] qubits that aren't erased.", '1411.7041-1-27-7': 'The code [REF] can correct for the erasure of [MATH] if and only if we have [EQUATION]', '1411.7041-1-27-8': 'Here [MATH], [MATH], etc are the reduced density matrices obtained from [MATH] by partial trace.', '1411.7041-1-27-9': 'This is equivalent to saying that the mutual information [MATH] vanishes, where [MATH] is the Von Neumman entropy, [MATH].', '1411.7041-1-27-10': 'Let us first see that this ensures we can correct the erasure.', '1411.7041-1-27-11': 'The Schmidt decomposition of [MATH], together with [REF], ensures us that there exists a basis [MATH] for [MATH] and a set of orthonormal states [MATH] in [MATH] such that [EQUATION] where [MATH] are some non-negative coefficients obeying [MATH].', '1411.7041-1-27-12': 'In other words there exists a unitary transformation [MATH] acting only on [MATH] such that [EQUATION] where we have denoted the first [MATH] qubits of [MATH] as [MATH] and the rest as [MATH].', '1411.7041-1-27-13': '[MATH] is some state that is independent of [MATH].', '1411.7041-1-27-14': 'This then implies that we must have [EQUATION] which is the analogue of [REF] above and demonstrates that we can use [MATH] to correct the erasure.', '1411.7041-1-27-15': 'If we do not have [REF], then there is nonzero correlation between [MATH] and [MATH], so we can learn about the state of [MATH] by doing measurements on [MATH].', '1411.7041-1-27-16': 'Since any successful protocol must not care about what happens to the qubits we lose, this prevents us from being able to correct the erasure.', '1411.7041-1-27-17': 'We can thus loosely rephrase [REF] as the statement that the erasure of [MATH] is correctable if and only if no information about [MATH] can be obtained from [MATH].', '1411.7041-1-27-18': 'This is related to the no-cloning theorem; if we were able to get the same quantum information about the encoded state [MATH] from both [MATH] and [MATH] then we would have built a machine for cloning that information.', '1411.7041-1-28-0': 'There is a useful reformulation of the condition [REF] as the statement that for any operator [MATH] acting on [MATH], we must have [CITATION] [EQUATION]', '1411.7041-1-28-1': 'In other words we must have the projection of [MATH] onto the code subspace be proportional to the identity.', '1411.7041-1-28-2': 'One immediate consequence of this is that in any state [MATH] in the code subspace, the correlation function of [MATH] with any operator [MATH] that acts within the code subspace must vanish: [EQUATION]', '1411.7041-1-28-3': 'This is another manifestation of the idea that [MATH] has no access to the encoded information.', '1411.7041-1-29-0': 'As in the previous subsection, we can use [MATH] to realize any operator [MATH] acting within the code subspace as an operator [MATH] that acts just on [MATH].', '1411.7041-1-29-1': 'Indeed we have both [EQUATION]', '1411.7041-1-29-2': 'In fact the converse of this statement also holds; if any operator on the code subspace can be realized as an operator on [MATH] as in [REF], then the code is able to correct for the loss of [MATH].', '1411.7041-1-29-3': 'The proof is simple.', '1411.7041-1-29-4': 'Say that the code were not correctable; then as just discussed there must exist an operator [MATH] on [MATH] where [REF] does not hold.', '1411.7041-1-29-5': "By Schur's lemma, there must then exist an operator [MATH] on the code subspace that does not commute with [MATH] on [MATH], that is with [MATH] for some [MATH] and [MATH].", '1411.7041-1-29-6': "But this operator [MATH] can't be realized on [MATH] by an operator [MATH] that acts only on [MATH], since any such operator by definition would commute with [MATH].", '1411.7041-1-30-0': 'We now turn to the question of when we should expect [REF] (or equivalently [REF] or [REF]) to hold.', '1411.7041-1-30-1': 'In situations where we would like our code to be able to correct against a wide variety of erasures, we expect that [MATH] will have full rank.', '1411.7041-1-30-2': 'In that case, in order to be able to have the orthonormal set of states [MATH] we need the dimensionality of [MATH] to be at least as large as dimensionality of [MATH].', '1411.7041-1-30-3': 'In other words we need [EQUATION]', '1411.7041-1-30-4': 'This condition is quite intuitive; wanting to send a larger message or correct larger erasures requires more qubits.', '1411.7041-1-31-0': 'In fact for large systems [REF] is typically not only necessary but sufficient.', '1411.7041-1-31-1': 'Say we take [MATH] to be a random state of [MATH] qubits in the Haar measure.', '1411.7041-1-31-2': "By Page's theorem [CITATION], the density matrix of [MATH] will be exponentially close to maximally mixed provided that [MATH], so by the Schmidt decomposition this is equivalent to choosing a random [MATH]-qubit code subspace of [MATH] qubits.", '1411.7041-1-31-3': "The condition [REF] will hold if [MATH] is maximally mixed, which again by Page's theorem should be true provided that [MATH].", '1411.7041-1-31-4': 'Thus, not only is [REF] necessary for a typical code to correct for the loss of a particular set [MATH] of [MATH] qubits, it is basically sufficient for the code to correct for the loss of any set of [MATH] qubits.', '1411.7041-1-32-0': '## Quantum secret sharing', '1411.7041-1-33-0': 'The three-qutrit example of section [REF] has the interesting property that every collection of qutrits either can perfectly reconstruct the state [MATH] or has no information about it at all.', '1411.7041-1-33-1': 'General error correcting codes do not have this property, since sometimes we can have erasures which can be "partially corrected", but it is interesting to think about the codes that do.', '1411.7041-1-33-2': 'Say that we have a Hilbert space that is a tensor product of [MATH] factors of not necessarily equal size, which in this context we will refer to as shares.', '1411.7041-1-33-3': 'A code subspace [MATH] of this Hilbert space is called a Quantum Secret Sharing Scheme if it has the property that every collection of shares either can distinguish perfectly different elements of [MATH], meaning given access to it we can correct for the erasure of its complement, or it cannot distinguish different elements at all [CITATION].', '1411.7041-1-33-4': 'Collections which enable erasure correction are called authorized and collections which do not are called unauthorized.', '1411.7041-1-33-5': 'We will see a possible application of quantum secret sharing to AdS/CFT in the following section.', '1411.7041-1-34-0': '## Approximate erasure correction', '1411.7041-1-35-0': 'So far we have discussed exact quantum error correction, but in AdS/CFT we only expect the emergence of the bulk to be approximate.', '1411.7041-1-35-1': 'It will thus be important for us to get a sense of how badly we might want to allow our three necessary and sufficient conditions for correctability to be violated.', '1411.7041-1-35-2': 'The simplest way to relax the condition [REF] is to require only [CITATION] [EQUATION]', '1411.7041-1-35-3': 'Here [MATH] is the trace norm of [MATH]; two density matrices whose difference has trace norm [MATH] are "operationally close" in the sense that the probability distributions they predict for arbitrary measurements differ by at most [MATH].', '1411.7041-1-35-4': 'This essentially says that typical states in the code subspace can be reconstructed to accuracy [MATH]; following [CITATION] we take this to be the definition of approximate error correction.', '1411.7041-1-35-5': 'We would like to relate this to our second condition for correctability, [REF], but we need a convenient way to quantify the violation of [REF].', '1411.7041-1-35-6': 'One good choice is to use correlation functions of the form [EQUATION]', '1411.7041-1-35-7': 'Here [MATH] denotes taking the transpose of an operator [MATH] on the code subspace and acting with it on the reference system [MATH]; by construction acting on the state [MATH] this is equivalent to acting with [MATH] on [MATH].', '1411.7041-1-35-8': '[MATH] is essentially the average of the correlation function [REF] over all [MATH]; they become equal in the limit of a large code subspace.', '1411.7041-1-35-9': 'From the right hand side of [REF] it is not difficult to show that [CITATION] [EQUATION] where [MATH] and [MATH] are the largest eigenvalues of their respective operators.', '1411.7041-1-35-10': 'Thus we see that, as one might expect from our discussion around [REF], the presence of nonzero correlation between [MATH] and [MATH] puts a limit on how accurately we can correct for the erasure of [MATH].', '1411.7041-1-35-11': 'This inequality will be very useful in our discussion of AdS/CFT, since after all computing correlation functions in the bulk theory is much easier than computing the trace norm directly.', '1411.7041-1-36-0': '## Operator algebra quantum error correction', '1411.7041-1-37-0': 'In our discussion of AdS/CFT we will soon see that the presence of bulk correlation puts nontrivial restrictions on the correctability of errors via the inequality [REF].', '1411.7041-1-37-1': 'There is, however, a generalized version of quantum error correction, called operator algebra quantum error correction, that is able to accommodate such correlation by requiring that our third necessary and sufficient condition [REF] apply only to a subalgebra of operators on the code subspace [CITATION].', '1411.7041-1-37-2': 'This requirement is greatly illuminated by the following theorem:', '1411.7041-1-38-0': 'Say that we have a code subspace [MATH] and an operator [MATH] that, together with its hermitian conjugate, acts within the code subspace.', '1411.7041-1-38-1': 'In other words we have [EQUATION]', '1411.7041-1-38-2': 'Then there exists an operator [MATH] acting just on [MATH] that obeys [EQUATION] for any [MATH] if and only if [MATH] commutes with the projection of any operator [MATH] onto the code subspace, where [MATH] acts on [MATH].', '1411.7041-1-38-3': 'In other words [EQUATION]', '1411.7041-1-38-4': 'We give a proof of this theorem in appendix [REF].', '1411.7041-1-38-5': "It is clear that the set of [MATH]'s that satisfy the assumptions of the theorem form a unital *-subalgebra [MATH] of the operators on the code subspace, meaning they include the identity and are closed under addition, multiplication, and hermitian conjugation.", '1411.7041-1-38-6': 'If we take [MATH] to be the entire algebra of operators on [MATH] then we recover our condition [REF].', '1411.7041-1-38-7': "Notice, however, that when [MATH] is a proper subalgebra we cannot use our previous argument to derive the condition [REF] from [REF], since the [MATH] that we constructed that doesn't have an [MATH] will not be in [MATH].", '1411.7041-1-38-8': 'This gives a loophole that simultaneously allows correlation between [MATH] and [MATH] and the existence of [MATH].', '1411.7041-1-39-0': 'For example in the two qubit system, consider a code subspace spanned by [MATH] and [MATH].', '1411.7041-1-39-1': 'The operator [MATH] that exchanges these two states can be realized just on the first qubit as the [MATH] operator that flips it, even though in either state this operator is perfectly correlated with the [MATH] operator that flips the second qubit.', '1411.7041-1-39-2': 'This is possible because the [MATH] operator on the code subspace, for which the first state is a [MATH] eigenstate and the second state is a [MATH] eigenstate, cannot be realized as an operator just on the first qubit; this code corrects only the subalgebra generated by [MATH] and [MATH].', '1411.7041-1-40-0': '# AdS/CFT as quantum error correction', '1411.7041-1-41-0': 'We now return to our discussion of bulk reconstruction.', '1411.7041-1-41-1': 'Consider again the right diagram in figure [REF].', '1411.7041-1-41-2': 'We argued using the AdS-Rindler reconstruction that the operator in the center can be represented either as an operator [MATH] with support on [MATH], an operator [MATH] with support on [MATH], or an operator [MATH] with support on [MATH].', '1411.7041-1-41-3': 'By now it should be obvious that this is directly analogous to the situation with [MATH], [MATH], and [MATH] in the three qutrit example, or more generally the existence of the operator [MATH].', '1411.7041-1-41-4': 'The main proposal of this paper is that this is more than an analogy, it is actually how AdS/CFT is reproducing the bulk!', '1411.7041-1-41-5': 'In other words we can think of local bulk operators as logical operations on an encoded subspace, which becomes better and better protected against localized boundary errors as we move the operators inwards in the radial direction.', '1411.7041-1-41-6': 'We illustrate this in figure [REF].', '1411.7041-1-41-7': 'In the remainder of the paper we will spell out this idea in more detail, giving the bulk versions of most of the statements of the previous section.', '1411.7041-1-42-0': '## Defining the code subspace', '1411.7041-1-43-0': 'We begin by defining a candidate code subspace [MATH] for AdS/CFT.', '1411.7041-1-43-1': 'Our proposal is that we should pick some finite set of local bulk operators [MATH], realized in the CFT via the global representation of section [REF].', '1411.7041-1-43-2': "We then define [MATH] as the linear span of states of the form [EQUATION] where we take the range of [MATH], the number of [MATH]'s we act with, and the number of points [MATH] where the operators can be located to be bounded by some fixed finite number.", '1411.7041-1-43-3': 'Here [MATH] is the ground state of the system; we could also do a similar construction around other sufficiently "semiclassical" states, but for rigor we will stick to [MATH] since, as mentioned in section [REF], the existence of appropriate smearing functions has not been completely established in the general case.', '1411.7041-1-43-4': 'We postpone to section [REF] the question of how large [MATH] can be.', '1411.7041-1-43-5': 'It is essential that our definition of the code subspace will be different for different choices of the operators [MATH]; the set of erasures that are correctable will depend on this choice, and we can learn about the way that the bulk theory is realized in the CFT by studying this dependence.', '1411.7041-1-43-6': 'For example, in figure [REF] we see that moving the operators closer to the boundary makes our code subspace less protected against small erasures.', '1411.7041-1-44-0': 'We would like to think of the operators [MATH] as logical operations on this code subspace, but this does not quite work since by construction acting repeatedly with [MATH] will eventually take us out of [MATH].', '1411.7041-1-44-1': 'To get a set of operators that really act within [MATH] we can include projection operators onto [MATH] on both sides of [MATH]; these will be irrelevant except in studying high-point correlation functions, so we will not carry them around explicitly here.', '1411.7041-1-44-2': 'Now consider a decomposition of the boundary Cauchy surface [MATH] into [MATH] and [MATH].', '1411.7041-1-44-3': 'If our code subspace [MATH] can protect against the erasure of [MATH], then by our condition [REF] it must be that we can find a representation of any operator on [MATH] with support only in [MATH].', '1411.7041-1-44-4': "In fact, this is what the AdS-Rindler reconstruction we reviewed in section [REF] provides us; any causal wedge [MATH] which contains the locations of the [MATH]'s used in defining [MATH] will allow a set of operators [MATH] with support only on [MATH] and whose action on [MATH] is the same as that of [MATH].", '1411.7041-1-44-5': 'We now see that in the CFT this is a statement about being able to correct for the erasure of [MATH].', '1411.7041-1-45-0': '## Bulk correlation and smearing', '1411.7041-1-46-0': 'It is illuminating to understand in more detail to what extent the AdS-Rindler reconstruction is consistent with our three equivalent conditions [REF], [REF], [REF] for quantum erasure correction.', '1411.7041-1-46-1': 'We clearly do not expect them to hold exactly, but we might hope for them to hold in the approximate sense of [REF].', '1411.7041-1-46-2': 'As we explained in section [REF], a good diagnostic for approximate quantum erasure correction is that the correlation functions between operators acting within the code subspace and operators acting on the set to be erased are small enough that the inequality [REF] does not preclude [REF] from holding.', '1411.7041-1-47-0': 'In fact it is a basic property of bulk physics that there is correlation between fields in [MATH] and fields in [MATH], as we indicate in figure [REF].', '1411.7041-1-47-1': 'In deciding whether or not this bulk correlation interferes with our interpretation of AdS-Rindler reconstruction as quantum error correction, we need to properly take into account the operator eigenvalues in the denominator of [REF].', '1411.7041-1-47-2': 'Formally these are infinite in a continuum quantum field theory, but every quantum field theorist knows that field operators are not really well-defined until they are integrated against smooth test functions with support over some region of nonzero measure, which we will take to have linear size [MATH].', '1411.7041-1-47-3': 'For simplicity we will take the bulk fields to be massless scalars and take their separation to be small compared to the AdS radius, in which case we have [EQUATION]', '1411.7041-1-47-4': 'Here [MATH] is the spacetime dimension of the boundary theory and [MATH] is the geodesic distance between [MATH] and [MATH].', '1411.7041-1-47-5': 'This formula also holds in other states we produce by acting on [MATH] with smeared operators near [MATH], and thus on average in the code subspace [MATH].', '1411.7041-1-47-6': 'We thus see that the right hand side of [REF] will be small in our case provided that the operators [MATH] used in constructing [MATH] are smeared over a distance which is small compared to their distance to the causal surface [MATH] of the wedge [MATH] in which we are trying to reconstruct them.', '1411.7041-1-48-0': 'This observation does much to justify our interpretation of AdS-Rindler reconstruction as quantum error correction, but it is somewhat unsatisfactory in the sense that the AdS-Rindler reconstruction still seems to work in the situation where we smear the operators over a distance that is comparable to their distance to the bifurcate Rindler horizon [MATH], even though the bulk correlation is then too large to be ignored.', '1411.7041-1-48-1': 'Indeed we interpret this as saying that the conventional quantum error correction of section [REF] does not fully capture the mechanism by which AdS/CFT realizes bulk locality.', '1411.7041-1-48-2': 'The operator algebra quantum error correction introduced in section [REF] however provides precisely the generalization we need to fix this.', '1411.7041-1-48-3': 'Consider for example an operator [MATH] which acts on [MATH] as [MATH], and which annihilates any state orthogonal to [MATH].', '1411.7041-1-48-4': 'This is an operator that acts within the code subspace, but its commutator with an operator [MATH] in [MATH] obeys [EQUATION]', '1411.7041-1-48-5': 'Thus [MATH] clearly cannot have a representation as an operator just on [MATH].', '1411.7041-1-48-6': 'Fortunately there is no reason to expect this operator to have an AdS-Rindler reconstruction, but the broader lesson is that we should really expect AdS-Rindler reconstruction to in general produce only a subalgebra of the operators on [MATH].', '1411.7041-1-48-7': 'We saw in section [REF] that the condition a subalgebra must obey for this to be possible is that the subalgebra must commute with the projection onto [MATH] of any operator on [MATH].', '1411.7041-1-48-8': 'In fact this is precisely the condition that we expect to be true for local operators in [MATH] (and their sums and products), which by bulk causality should commute with operators in [MATH].', '1411.7041-1-48-9': 'That this commutator vanishes with the projections onto [MATH] of all CFT operators in [MATH] is not something we can prove directly, but AdS-Rindler reconstruction requires it.', '1411.7041-1-49-0': 'A second reason to prefer operator algebra quantum error correction is that even when the right hand side of [REF] is small, it will at most be suppressed by some fixed power of [MATH].', '1411.7041-1-49-1': 'This is because we should not smear the operators over distances shorter than the Planck length.', '1411.7041-1-49-2': 'Since we in principle would like a version of AdS-Rindler reconstruction that works to all orders in [MATH], it would be unsatisfying if our error correction interpretation failed at some finite order because of bulk correlation.', '1411.7041-1-50-0': 'We can now state our final proposal: the AdS-Rindler reconstruction of local bulk operators in [CITATION] is dual in the CFT to the operator algebra quantum error correction of [CITATION].', '1411.7041-1-50-1': "An erasure of a region [MATH] is correctable if the [MATH]'s used in defining the code subspace all lie within the causal wedge [MATH].", '1411.7041-1-50-2': 'In cases where the operators we are interested in are well-localized away from the causal surface [MATH] of [MATH], the situation is well-approximated by conventional quantum error correction.', '1411.7041-1-50-3': "Either way, the further the [MATH]'s are from the asymptotic boundary, the better they are protected from CFT erasures.", '1411.7041-1-51-0': 'It is worth emphasizing that in the case where a bulk operator is of order an AdS radius distance from [MATH], our approximate equivalence between conventional and operator algebra quantum error correction requires sub-AdS scale bulk locality.', '1411.7041-1-51-1': 'This is a special property of those CFTs that have local holographic duals, which we have here reformulated in the language of quantum information theory.', '1411.7041-1-52-0': '## Disconnected regions and quantum secret sharing', '1411.7041-1-53-0': 'So far we have only discussed the erasure of connected regions of the boundary.', '1411.7041-1-53-1': 'More general erasures are also interesting.', '1411.7041-1-53-2': 'Consider for example the [MATH] situation depicted in figure [REF].', '1411.7041-1-54-0': 'Here we consider a region [MATH] which is the union of two disjoint intervals; in other words we have erased two disjoint intervals.', '1411.7041-1-54-1': 'Can we choose a code subspace where we can realize the bulk operator in the center as an operator acting on [MATH] or [MATH]?', '1411.7041-1-54-2': 'If the AdS-Rindler reconstruction is the last word on bulk reconstruction [CITATION], then the answer is clearly no; this point lies neither in [MATH] nor in [MATH].', '1411.7041-1-54-3': 'This is possible within the context of quantum error correction, but only if both [MATH] and [MATH] can access partial information about the code subspace.', '1411.7041-1-54-4': 'For example, say that [MATH] had no information whatsoever about which state of the code subspace we are in.', '1411.7041-1-54-5': 'Then by definition [REF] would hold, so we could recover the information from [MATH].', '1411.7041-1-54-6': 'We are not, however, able to determine whether or not such partial information is really present.', '1411.7041-1-55-0': 'In fact there have been recent conjectures in the literature that this operator can still be reconstructed in [MATH] as long as [MATH] is bigger than [MATH]; more generally, the claim is that one can do reconstruction throughout the entanglement wedge, which is defined as the bulk domain of dependence of any bulk spacelike surface whose boundary is the union of [MATH] and the codimension two extremal-area surface of minimal area whose boundary is [MATH] [CITATION].', '1411.7041-1-55-1': 'In the figure, the intersection of the entanglement wedge with a bulk Cauchy surface is shaded blue; the minimal area condition causes a discontinuous change as we increase the size of [MATH].', '1411.7041-1-55-2': 'Is this conjecture compatible with our proposal?', '1411.7041-1-55-3': 'Indeed it is; we saw below equation [REF] that in a generic code subspace any [MATH] which is greater than half of the system can correct for the erasure of its complement [MATH].', '1411.7041-1-55-4': "The sharp jump in correctability as [MATH] surpasses [MATH] in size is consistent with our analysis around [REF], where from Page's theorem we expect that the density matrix of [MATH] together with the reference system will approach being maximally mixed exponentially fast once we cross the transition.", '1411.7041-1-56-0': 'In section [REF] we saw that a division of the CFT into a union of shares with the property that any collection of the shares has either complete information or no information about the encoded state is called a quantum secret sharing scheme; we now see that in the situation of figure [REF] we will be able to reconstruct the operator in the center if and only if our boundary division into four regions gives a quantum secret sharing scheme.', '1411.7041-1-57-0': '## MERA as an error correcting code?', '1411.7041-1-58-0': 'One shortcoming of our work so far is that, although we have laid out a plausible CFT interpretation of AdS-Rindler reconstruction as quantum error correction, we have ultimately relied on the bulk in deriving this reconstruction.', '1411.7041-1-58-1': 'This boils down to the assumption that there exist operators in the CFT that obey the bulk equations of motion and algebra on a subspace.', '1411.7041-1-58-2': 'We then use this assumption to perform the Bogoliubov transformation that relates the global and the Rindler reconstructions.', '1411.7041-1-58-3': 'This assumption is quite plausible, and essentially follows from the assumed large-[MATH] structure of the CFT [CITATION], but it would still be nice if we could explicitly demonstrate the structure of the quantum error correcting code in the CFT.', '1411.7041-1-58-4': 'In particular, in section [REF] we had to use bulk causality to argue that the necessary and sufficient condition [REF] for operator algebra quantum error correction held, and we were not able to check it explicitly for all possible CFT operators on [MATH].', '1411.7041-1-58-5': 'Similarly we were unable to determine whether or not the central point could be reconstructed in the two-interval [MATH] of the previous subsection.', '1411.7041-1-59-0': 'A promising starting point for addressing these issues is the MERA tensor network construction of a discrete version of AdS/CFT [CITATION].', '1411.7041-1-59-1': 'It seems possible that in that fairly controlled setting one could rigorously confirm the quantum error correction structure we have motivated in this paper.', '1411.7041-1-59-2': 'Moreover, one could attempt to determine explicitly whether or not the example of the previous subsection allows reconstruction of the operator in the center; this could be done by using the global construction to make a code subspace, entangling this code subspace with a reference system [MATH] to prepare a state [MATH], and then seeing whether there is mutual information between [MATH] and [MATH].', '1411.7041-1-59-3': 'The state [MATH] would still be prepared by a tensor network, with tensors acting both on the CFT and/or the reference system.', '1411.7041-1-59-4': 'This calculation would go a long way towards settling the "causal wedge vs. entanglement wedge" debate of bulk reconstruction.', '1411.7041-1-59-5': 'We will not attempt this calculation here, but the typicality argument leading to [REF] favors the entanglement wedge; we will say more about this in section [REF].', '1411.7041-1-60-0': '# Backreaction and holography', '1411.7041-1-61-0': 'We now turn to the question of how large we can make the code subspace [MATH].', '1411.7041-1-61-1': 'Each [MATH] that we act with raises the energy of the state, so doing so repeatedly will eventually lead to backreaction becoming important.', '1411.7041-1-61-2': 'When this happens it is clear that the approximation of perturbation theory around a fixed background geometry will break down.', '1411.7041-1-61-3': 'In this section we argue that this is related to a basic property of error correcting codes: the larger the code subspace, the fewer correctable errors.', '1411.7041-1-61-4': 'For erasures we quantified this in equation [REF] above.', '1411.7041-1-62-0': '## Defining local operators', '1411.7041-1-63-0': 'Once we allow nontrivial backreaction, it is no longer possible to ignore the issue of how we define local operators in a diffeomorphism-invariant way.', '1411.7041-1-63-1': 'Following [CITATION], we do this by choosing a cutoff surface at large but finite radius, with induced metric [MATH], and then specifying bulk points by sending in spacelike geodesics from the [MATH] slice of this cutoff surface that start out orthogonal to the [MATH] directions.', '1411.7041-1-63-2': 'We then take the limit as the cutoff surface approaches the boundary.', '1411.7041-1-63-3': 'Points are labeled by a location on [MATH], a renormalized proper distance along the geodesic, and an angle in the radial/temporal plane.', '1411.7041-1-63-4': 'This is illustrated in figure [REF].', '1411.7041-1-64-0': 'These geodesics can be thought of as the "gravitational dressing" of the bulk operator, analogous to the Wilson line one would use to connect a charged operator to the boundary to make it gauge-invariant in electrodynamics.', '1411.7041-1-65-0': 'As in the electromagnetic case, the operators defined in this way will have nonlocal commutators due to their nontrivial Dirac brackets.', '1411.7041-1-65-1': 'This issue has recently been studied in considerable detail in [CITATION], although they send the geodesics from arbitrary times but take them to be orthogonal to the boundary in the temporal direction as well as the [MATH] directions.', '1411.7041-1-65-2': 'In particular they show that in Einstein gravity coupled to matter, the commutators of operators defined in this way are nonlocal only if they involve the metric or its canonical momentum; the matter fields and their canonical momenta still have local commutators.', '1411.7041-1-65-3': 'We can make contact with this result by restricting to operators located at points whose geodesics start out orthogonal to the boundary time direction; these operators are identical to those of [CITATION] and thus have the same commutators.', '1411.7041-1-66-0': 'We can use this result to verify that the nonlocality from the gravitational dressing of operators does not invalidate our previous claims.', '1411.7041-1-66-1': 'For example in the introduction we argued that because in the bulk theory an operator in the center of the space commutes with all local operators at the boundary, the bulk operator algebra is inconsistent with the CFT algebra.', '1411.7041-1-66-2': 'We can now give a version of this argument that includes the gravitational dressing.', '1411.7041-1-66-3': 'Consider a scalar field [MATH] located near the center of AdS and dressed by a geodesic to make it gauge-invariant.', '1411.7041-1-66-4': 'It will have a nonlocal commutator with the metric at infinity, in particular with the ADM energy that is defined as an integral of the boundary stress tensor [MATH] [CITATION].', '1411.7041-1-66-5': 'But using the result of [CITATION] [MATH] will commute with its own canonical momentum at spacelike separation, which extrapolates to the boundary to become [MATH], the time derivative of the CFT operator [MATH] dual to [MATH].', '1411.7041-1-66-6': 'These two statements, however, are inconsistent with the CFT operator algebra; from the conformal Ward identity [MATH] always lies in the operator product expansion of [MATH] with itself at spacelike separation, and can be extracted by an appropriate limit after subtracting off a few relevant operators that are dual to matter fields and thus commute with [MATH].', '1411.7041-1-66-7': 'It is thus impossible for [MATH] to commute with [MATH] but not with [MATH]; again we find that the bulk algebra is inconsistent with the boundary algebra.', '1411.7041-1-66-8': 'The resolution of this contradiction that we are proposing is that the bulk algebra holds only acting on the code subspace.', '1411.7041-1-67-0': 'Similarly we can now revisit our claim that operators in [MATH] commute with operators in [MATH], which was a necessary condition for our interpretation of the AdS-Rindler construction as operator algebra quantum error correction.', '1411.7041-1-67-1': 'For matter fields it clearly works, but we might worry that a matter operator in [MATH] might not commute with a metric operator in [MATH].', '1411.7041-1-67-2': 'We note, however, that as long as the gravitational dressing of an operator at [MATH] also lies entirely in [MATH], meaning that the spatial geodesic connecting [MATH] to the boundary also lies in [MATH], then it will only produce nonlocal commutators with operators that are also located in [MATH]; any operator localized by a geodesic in one wedge will still commute with any operator localized by a geodesic in the complementary wedge.', '1411.7041-1-68-0': 'In this subsection, to allow direct use of the results of [CITATION] we studied only operators attached to geodesics that start out orthogonal to the boundary time direction at [MATH].', '1411.7041-1-68-1': 'It would be interesting to do the analogue of their analysis at arbitrary temporal-radial angle; this amounts to working with boundary conditions that approach the "open-FRW" slicing of AdS [EQUATION] as [MATH].', '1411.7041-1-68-2': 'As explained in [CITATION], this would be a natural bulk construction of gauge-invariant operators on the fixed-time Hilbert space.', '1411.7041-1-69-0': '## Shrinking of the causal wedge', '1411.7041-1-70-0': 'We now return to the question of how backreaction affects causal wedge reconstruction.', '1411.7041-1-70-1': 'Our basic proposal is that adding energy in the bulk causes the causal wedge of a fixed boundary region [MATH] to recede towards the boundary, giving it less access to bulk operators defined at fixed renormalized geodesic distance (for some related discussion see [CITATION]).', '1411.7041-1-71-0': 'Consider for example the AdS-Schwarzschild geometry in [MATH] dimensions.', '1411.7041-1-71-1': '[EQUATION] with [EQUATION] [MATH] is proportional to the ADM mass of this geometry.', '1411.7041-1-71-2': 'Now consider a boundary disc [MATH] of angular size [MATH]; its causal wedge reaches a radius [MATH] in the bulk defined implicitly by [EQUATION]', '1411.7041-1-71-3': 'The proper distance of this radius to a cutoff surface at [MATH] is [EQUATION] so we can subtract [MATH] to define a renormalized proper distance [EQUATION]', '1411.7041-1-71-4': 'We claim that [MATH] is a decreasing function of [MATH] at fixed [MATH], which by differentiating under the integral sign is equivalent to the claim that [EQUATION] for all [MATH] and for all [MATH], where [MATH] is the positive root of [MATH].', '1411.7041-1-71-5': 'This can be shown analytically in various limits, and is easily checked numerically in the general case.', '1411.7041-1-71-6': 'One can also study the asymptotically-[MATH] BTZ black hole, where a similar result holds and all integrals can be done analytically.', '1411.7041-1-71-7': 'Thus we see that indeed the causal wedge has access to fewer and fewer bulk observables as we increase the mass of the matter in the center.', '1411.7041-1-71-8': 'This after all must be the case, since as we keep increasing the mass a point at fixed renormalized geodesic distance from the boundary will eventually go through the horizon.', '1411.7041-1-72-0': 'It is interesting to think about how general this statement is; under what circumstances can the causal wedge move inwards in renormalized geodesic distance as we insert energy?', '1411.7041-1-72-1': 'One might guess that the null energy condition should generically prevent this, but to test that we need a more precise conjecture.', '1411.7041-1-72-2': 'One first guess is that in any geometry obeying the null energy condition the causal wedge of a fixed boundary region can see at most as far in renormalized geodesic distance as it can in the vacuum.', '1411.7041-1-72-3': 'In fact this conjecture is false, we have constructed explicit counterexamples.', '1411.7041-1-72-4': 'Indeed a weaker conjecture, where we replace the null energy condition by the dominant energy condition, still has counterexamples.', '1411.7041-1-72-5': 'One counterexample is given by a small perturbation of [MATH], with the metric [EQUATION] where [EQUATION]', '1411.7041-1-72-6': 'With a small positive [MATH], the causal wedges of certain fixed boundary regions can see farther in renormalized geodesic distance than they can in the vacuum.', '1411.7041-1-72-7': 'These boundary regions include spherical regions whose causal wedges probe deep into the bulk geometry.', '1411.7041-1-73-0': 'Although these counterexamples prevent any straightforward "monoticity of causal wedge recession theorem", we expect that the Schwarzschild calculation we have just discussed captures the general tendency.', '1411.7041-1-73-1': 'It would be nice to prove a more general theorem verifying this, but we have not succeeded in finding one.', '1411.7041-1-74-0': '## Counting states', '1411.7041-1-75-0': 'The recession of the causal wedge has a nice quantum error correction interpretation; as we allow the code subspace to have more and more excited states, a bulk operator localized at some fixed geodesic distance will eventually no longer lie in the causal wedge of a fixed boundary region.', '1411.7041-1-75-1': 'In other words, the code will lose some of its ability to correct erasures; we will need access to more of the boundary to study the same bulk observables.', '1411.7041-1-75-2': 'In this subsection we study this a bit more quantitatively, making contact with the general condition [REF] for typical correctability.', '1411.7041-1-76-0': 'To apply [REF] to AdS/CFT, we need to identify CFT analogues of the quantities [MATH], [MATH], and [MATH].', '1411.7041-1-76-1': '[MATH] is the total number of qubits used in doing the encoding, and should roughly correspond to the total number of CFT degrees of freedom relevant for reconstructing a particular bulk region of interest.', '1411.7041-1-76-2': 'This is somewhat nontrivial; the CFT has an infinite number of degrees of freedom in the UV which are needed to reconstruct bulk operators that are arbitrarily close to the boundary.', '1411.7041-1-76-3': 'To deal with this we take our code subspace to only involve states where we act on the vacuum with operators [MATH] that are all localized within a region [MATH] at the center of the AdS space that has proper size of order the AdS radius.', '1411.7041-1-76-4': 'We will also take them to be smeared over distances that are small compared to their separation from the boundary of [MATH], so that we do not have to worry about the difference between conventional and operator algebra quantum error correction.', '1411.7041-1-76-5': 'The global reconstructions [REF] of these operators involve integrals over functions that vary smoothly on the scale of the radius of curvature of the boundary [MATH], so we can integrate out all CFT degrees of freedom with shorter wavelength.', '1411.7041-1-76-6': 'For concreteness we will consider the case of the [MATH] super Yang-Mills theory in [MATH] boundary dimensions with gauge group [MATH], in which case we have [EQUATION]', '1411.7041-1-76-7': 'Erasing a disc of angular size [MATH] will then correspond to erasing [EQUATION] qubits, where this function is just [MATH] times the ratio of the area of the disc to the area of the [MATH].', '1411.7041-1-77-0': 'Let us first consider the case where the code subspace is small, that is when [MATH].', '1411.7041-1-77-1': 'From [REF] we then expect that we can correct for the erasure as long as [MATH], or in other words [MATH].', '1411.7041-1-77-2': 'But this is exactly what we expect from the AdS-Rindler reconstruction; once [MATH], [MATH] will contain the center of the space.', '1411.7041-1-77-3': 'We can now start increasing [MATH]; nothing interesting will happen until we get [MATH], after which the set of erasures we are able to correct will start decreasing.', '1411.7041-1-77-4': 'But this is exactly the condition for backreaction to become important in the center; with [MATH] the entropy of the code subspace is comparable to that of a black hole filling [MATH] and thus most states in the code subspace must actually be black holes.', '1411.7041-1-77-5': 'So both on the CFT side through equation [REF] and the bulk side via backreaction we arrive at the same conclusion for when correctability should break down.', '1411.7041-1-77-6': 'This is a manifestation of the holographic entropy bound of [CITATION].', '1411.7041-1-78-0': '# Conclusion', '1411.7041-1-79-0': 'In this paper we have provided what we consider to be a new understanding of how the holographic principle is realized in AdS/CFT.', '1411.7041-1-79-1': 'Bulk effective field theory operators emerge as a set of logical operations on an encoded subspace, which is protected against local errors in the boundary CFT.', '1411.7041-1-79-2': 'The bulk algebra is realized only on this subspace, and only if we do not try to describe too many operations at once.', '1411.7041-1-79-3': 'Asking for more causes the error correction procedure to fail, which in the bulk is manifested by the formation of a black hole.', '1411.7041-1-80-0': 'To some extent we have only recast known facts about the AdS-Rindler reconstruction in a new language, but in our view that construction is quite opaque once the operators in the boundary domain of dependence of [MATH] are evolved back to the boundary Cauchy surface [MATH] at [MATH].', '1411.7041-1-80-1': 'Our description in terms of error correction is phrased entirely on this Cauchy surface, and gives what we feel to be a satisfying interpretation of how the AdS-Rindler reconstruction is realized in the CFT that cleanly resolves some of its paradoxical features.', '1411.7041-1-81-0': 'It is of course interesting to ask if there are any implications of this work for the recent controversy on whether or not the interiors of black holes are describable in AdS/CFT; for now we leave this for future study.'} | {'1411.7041-2-0-0': '[email protected] [email protected] [email protected] We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction.', '1411.7041-2-0-1': 'Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction.', '1411.7041-2-0-2': 'We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue.', '1411.7041-2-0-3': 'Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs.', '1411.7041-2-0-4': 'Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.', '1411.7041-2-1-0': '# Introduction', '1411.7041-2-2-0': 'Almost twenty years after its initial formulation, the AdS/CFT correspondence remains our best-understood example of a precise theory of quantum gravity.', '1411.7041-2-2-1': 'It has shed light on many deep puzzles in quantum gravity, and has also been of practical use in studying the dynamics of strongly interacting quantum field theories.', '1411.7041-2-2-2': 'One aspect that remains mysterious, however, is the emergence of approximate bulk locality.', '1411.7041-2-2-3': 'Locality near the boundary is straightforward.', '1411.7041-2-2-4': 'In the "extrapolate" version of the AdS/CFT dictionary we have a simple relation [CITATION] [EQUATION] between limiting values of a bulk field [MATH] and a conformal field theory operator [MATH]; this dictionary manifestly respects locality in the [MATH] directions since the CFT does.', '1411.7041-2-2-5': 'The radial direction, however, is more subtle.', '1411.7041-2-2-6': 'One way to see this is to observe that naively a local operator in the center of the bulk should commute with every local operator at the boundary on a fixed time slice containing that bulk operator.', '1411.7041-2-2-7': 'This is not consistent, however, with a standard property of quantum field theory; any operator that commutes with all local operators at a fixed time must be proportional to the identity.', '1411.7041-2-2-8': "Bulk locality thus cannot be respected within the CFT at the level of the algebra of operators; we'd then like to know in what sense it is respected.", '1411.7041-2-3-0': 'The basic idea of this paper is that bulk locality is a statement about certain subspaces of states in the CFT.', '1411.7041-2-3-1': 'That these subspaces can be large is a consequence of the large-[MATH] properties of the CFT, but the large degree of non-local entanglement in finite energy states of the CFT also plays an essential role.', '1411.7041-2-3-2': 'Our strategy will be to gradually back away from the [MATH] limit in equation [REF] and study how the the CFT representations of bulk operators spread in spatial support as we do so.', '1411.7041-2-3-3': 'On the bulk side the tool we will mostly use is the AdS-Rindler reconstruction of bulk fields introduced in [CITATION] and refined in [CITATION].', '1411.7041-2-3-4': 'We will observe that this construction has several paradoxical features, which we will illuminate by recasting it on the CFT side in the language of quantum error correcting codes [CITATION].', '1411.7041-2-3-5': 'This language gives a new, more general perspective on the issue of bulk reconstruction, and we believe that it is the natural framework for understanding the idea of "subregion-subregion" duality [CITATION].', '1411.7041-2-3-6': 'In particular, the radial direction in the bulk is realized in the CFT as a measure of how well CFT representations of bulk quantum information are protected from local erasures.', '1411.7041-2-3-7': 'The holographic principle also naturally arises in the guise of the general statement that there is an upper bound on how much quantum information a given code can protect from erasures.', '1411.7041-2-4-0': 'One point that will appear in this analysis is that truncated subalgebras of bulk observables are of interest; these were also advocated in [CITATION] in the context of describing the black hole interior.', '1411.7041-2-4-1': 'Aspects of our proposal are inspired by their construction, but here we do not discuss black hole interiors and we are not violating quantum mechanics [CITATION].', '1411.7041-2-4-2': 'A connection between black holes and quantum error correction was also made in [CITATION], which is essentially an earlier version of the proposal of [CITATION], but again the context was different and our work here should be uncontroversial by comparison.', '1411.7041-2-5-0': 'Before proceeding, let us establish a few conventions used throughout this paper.', '1411.7041-2-5-1': 'We will frequently discuss subspaces and tensor factors of the Hilbert space.', '1411.7041-2-5-2': 'When we say that an operator acts within a subspace we also mean that the same is true for its hermitian conjugate.', '1411.7041-2-5-3': 'If the Hilbert space is a tensor product [MATH], for any operator [MATH] that acts on [MATH] we may trivially form an operator [MATH] that acts on the entire Hilbert space.', '1411.7041-2-5-4': 'We will often drop the identity operator [MATH] and write it simply as [MATH].', '1411.7041-2-5-5': 'The reader should also keep in mind that the CFT regions [MATH] and [MATH] that we will talk about in section [REF] unfortunately correspond to [MATH] and [MATH] respectively in the language of section [REF].', '1411.7041-2-6-0': '# Bulk reconstruction and an AdS-Rindler puzzle', '1411.7041-2-7-0': '## Global AdS reconstruction', '1411.7041-2-8-0': 'We begin by briefly recalling the standard CFT construction of local bulk fields in AdS [CITATION].', '1411.7041-2-8-1': 'We will first work in global coordinates, where the metric asymptotically has the form [EQUATION]', '1411.7041-2-8-2': 'The CFT dual to this system lives on [MATH], with the [MATH] being the time direction.', '1411.7041-2-8-3': 'The Hilbert space of states is the set of field configurations on [MATH].', '1411.7041-2-8-4': 'The idea is then to perturbatively construct operators in the CFT which obey the bulk equations of motion, with the boundary conditions set by the dictionary [REF].', '1411.7041-2-8-5': 'For simplicity we will assume that all bulk interactions are suppressed by inverse powers of a quantity [MATH], which will also set the AdS radius in Planck units.', '1411.7041-2-8-6': 'At leading order in [MATH], this procedure results in a straightforward prescription for the CFT representation of a bulk field [MATH]; we simply have [EQUATION] where the integral is over the conformal boundary and [MATH] is a so-called "smearing function".', '1411.7041-2-8-7': 'The smearing function obeys the bulk wave equation in its [MATH] index, and leads to [REF] as we take [MATH] to the boundary.', '1411.7041-2-8-8': 'It can be chosen to only have support when [MATH] and [MATH] are spacelike separated, which we illustrate for [MATH] in the left diagram of figure [REF]; the point [MATH] is represented by a boundary integral over the green region only.', '1411.7041-2-8-9': 'In the case of empty AdS, where we take [REF] to hold everywhere, explicit representations of the smearing function can be found in [CITATION].', '1411.7041-2-8-10': "[MATH] corrections can be systematically included [CITATION], although we won't really need to discuss them here.", '1411.7041-2-8-11': 'At higher orders in this perturbation theory we will need to confront the problem of defining local operators in a diffeomorphism invariant theory, but we postpone discussion of this until section [REF].', '1411.7041-2-9-0': 'It is not obvious from the definition that the operators [REF] have the expected commutators in the bulk; this has been checked perturbatively within low point correlation functions in [CITATION], but must eventually break down in states with enough excitations to avoid a contradiction with the argument in our introduction.', '1411.7041-2-9-1': 'We will argue below that, within the subspace of states that are "perturbatively close" to the vacuum, it breaks down only at the level of non-perturbatively small corrections.', '1411.7041-2-10-0': 'Note that once we have a representation of the form [REF], we can use the CFT Hamiltonian to re-express all operators on the right hand side in terms of Heisenberg picture fields on a single Cauchy surface in the CFT, denoted as [MATH] in figure [REF].', '1411.7041-2-10-1': 'This representation is quite nontrivial, in general it involves severely nonlocal and multitrace operators.', '1411.7041-2-10-2': 'It also has the property that if we take [MATH] to be near the boundary but not quite on it, the single-time CFT representation of [MATH] still involves operators with support on all of [MATH].', '1411.7041-2-10-3': 'We might hope to find a representation whose boundary support shrinks as the operator approaches the boundary, and indeed the AdS-Rindler representation does exactly this, as we will now explain.', '1411.7041-2-11-0': '## AdS-Rindler reconstruction', '1411.7041-2-12-0': 'Consider a subregion [MATH] of a CFT Cauchy surface [MATH].', '1411.7041-2-12-1': 'The boundary domain of dependence of [MATH], denoted [MATH], is defined as the set of points on the boundary with the property that every inextendible causal curve, meaning a curve whose tangent vector is never spacelike and which is not part of a larger curve with this property, that passes through it must also intersect [MATH].', '1411.7041-2-12-2': 'This is illustrated for the boundary of [MATH] in the right diagram of figure [REF], where [MATH] is the boundary interval lying between the two vertical hash marks and [MATH] is shaded green.', '1411.7041-2-12-3': 'For any boundary region [MATH], its bulk causal future/past [MATH] is defined as the set of bulk points which can be reached by bulk causal curves evolving from/to the region [MATH].', '1411.7041-2-12-4': 'The causal wedge of a CFT subregion A [CITATION] (for earlier related definitions see [CITATION]) is defined as [EQUATION]', '1411.7041-2-12-5': 'In the right diagram of figure [REF], [MATH] roughly lies between the dashed lines and [MATH].', '1411.7041-2-12-6': 'The bulk codimension-two surface [MATH] in the figure is the "rim" of the wedge and is commonly referred to as the causal surface of [MATH] [CITATION]; more precisely it is defined as the part of the intersection of the boundaries of [MATH] that does not also intersect the conformal boundary at infinity.', '1411.7041-2-12-7': '[MATH] can also be described as the intersection of the past and future horizons of [MATH].', '1411.7041-2-13-0': 'A simple example of these definitions is where we take the geometry to be pure [MATH], [MATH] to be the [MATH] slice of the boundary, and [MATH] to be one hemisphere of [MATH].', '1411.7041-2-13-1': 'In this case [MATH] becomes what is usually referred to as the AdS-Rindler wedge.', '1411.7041-2-14-0': 'A natural set of bulk coordinates on the AdS-Rindler wedge gives a metric with the form [EQUATION] where the coordinate ranges are [MATH] and the geometry in parentheses is just the [MATH] dimensional hyperbolic disc.', '1411.7041-2-14-1': 'The causal surface [MATH] is given by the limit [MATH] at fixed [MATH], and [MATH] itself is given by [MATH] and [MATH].', '1411.7041-2-14-2': 'We illustrate this for [MATH] in figure [REF].', '1411.7041-2-14-3': 'By acting on this example with bulk isometries (or equivalently boundary conformal transformations), we can arrive at the causal wedge for any round disc in [MATH].', '1411.7041-2-14-4': 'The case of [MATH] is especially simple; all connected boundary regions are intervals and thus can be produced in this way.', '1411.7041-2-15-0': 'The point then is that the construction of CFT representations of bulk fields in the previous subsection can also be implemented purely within the causal wedge [CITATION].', '1411.7041-2-15-1': 'At leading order in [MATH], the claim is that for any [MATH] with [MATH], we can again represent [MATH] via the expression [REF], but with the [MATH] integral now taken only over [MATH].', '1411.7041-2-15-2': 'This is illustrated for [MATH] in the right diagram in figure [REF], where we have allowed for a conformal transformation that changes the size of the boundary interval [MATH].', '1411.7041-2-15-3': 'We review more details of this construction in appendix [REF]; the only major subtlety is that the smearing function [MATH] no longer exists as a function and must be understood as a distribution for integration against CFT expectation values [CITATION] (see also [CITATION] for some related discussion).', '1411.7041-2-16-0': 'Thus we see that the AdS-Rindler construction of [MATH] indeed has the property that if [MATH] is close to the boundary, only a small boundary region [MATH] localized near [MATH] is needed to be able to reconstruct [MATH] in [MATH].', '1411.7041-2-16-1': 'Moreover, by making use of the CFT evolution we can again rewrite the expression [REF] entirely in terms of nonlocal Heisenberg operators acting at [MATH], but now they will act only on [MATH].', '1411.7041-2-17-0': '## Overlapping wedges', '1411.7041-2-18-0': 'The AdS-Rindler construction of bulk fields we have just described has the somewhat counter-intuitive property that the same bulk field operator [MATH] lies in multiple causal wedges, and thus can be represented as an operator on distinct regions [MATH] in [MATH].', '1411.7041-2-18-1': 'One consequence of this is shown in the left diagram of figure [REF]; for any bulk field operator [MATH] and any CFT local operator [MATH] such that [MATH] and [MATH] are spacelike separated, we can choose a causal wedge [MATH] such that [MATH] lies in the complement of [MATH] in [MATH].', '1411.7041-2-18-2': 'By CFT locality [MATH] then must exactly commute with our representation of [MATH] in that wedge.', '1411.7041-2-18-3': 'This is coming dangerously close to contradicting the theorem mentioned in the introduction, that is that no nontrivial operator in the CFT can commute with all local CFT operators on [MATH].', '1411.7041-2-19-0': 'To avoid this contradiction it must be the case that the representations of [MATH] in different wedges are not really all the same operator on the CFT Hilbert space.', '1411.7041-2-19-1': 'We can see this in another way by considering the setup of the center diagram in figure [REF], where we have two overlapping wedges [MATH] and [MATH] that both contain the point [MATH] but [MATH] is not contained in [MATH].', '1411.7041-2-19-2': 'For a CFT operator defined with support only on [MATH] to really be equal to a CFT operator defined with support only on [MATH], it must be that the operator really only has support on [MATH].', '1411.7041-2-19-3': 'But given that we have chosen [MATH] to lie outside of [MATH], we do not expect the operator to have such a representation.', '1411.7041-2-19-4': 'In fact in this example the operator has a representation on the complement of [MATH], and we will see in section [REF] that when this is so a version of the no-cloning theorem of quantum mechanics forbids an accurate representation of the operator on [MATH].', '1411.7041-2-20-0': 'We can see the non-equivalence of the operators even more clearly by considering a third example, shown in the right diagram in figure [REF].', '1411.7041-2-20-1': 'Now a bulk field at the point [MATH] lies outside of the causal wedge for any one of the regions, but it can be reconstructed in [MATH], [MATH], or [MATH].', '1411.7041-2-20-2': 'The mutual intersection of these regions is just three points, and if we consider another set of three regions slightly rotated from these we can come up with a set of six possible reconstructions whose mutual intersection is genuinely empty.', '1411.7041-2-20-3': 'There is simply no possible way that they can all be equal as operators.', '1411.7041-2-20-4': 'For future reference we will refer to the three operators as [MATH], [MATH], and [MATH].', '1411.7041-2-21-0': 'We thus need to decide how we are to reconcile these operator inequivalences with the fact that in the bulk theory it seems that the operators are equivalent.', '1411.7041-2-21-1': 'There will clearly be some CFT states where they act quite differently, and we would like to understand the physics of the subset of states where their action is equivalent.', '1411.7041-2-21-2': 'This problem can be nicely understood in the language of quantum error correction, to which we now turn.', '1411.7041-2-22-0': '# Correcting quantum erasures', '1411.7041-2-23-0': 'Say Alice wants to send Bob a quantum state of [MATH] qubits in the mail, but she is worried that some of the qubits might get lost on the way.', '1411.7041-2-23-1': 'Quantum error correction is a procedure that allows her to embed this state into [MATH] qubits in such a way that even if some qubits are lost, Bob can still recover it.', '1411.7041-2-23-2': 'In this section we review some basic facts about this, beginning with an example.', '1411.7041-2-24-0': '## A simple example of erasure correction', '1411.7041-2-25-0': 'The simplest example of quantum error correction actually involves three-state "qutrits" instead of two-state qubits, and it uses three qutrits to send a single-qutrit message [CITATION].', '1411.7041-2-25-1': 'Say Alice wishes to send the state [EQUATION]', '1411.7041-2-25-2': 'The idea is to instead send the state [EQUATION] where [EQUATION]', '1411.7041-2-25-3': 'This protocol has two remarkable properties.', '1411.7041-2-25-4': 'First of all for any state [MATH], the reduced density matrix on any one of the qutrits is maximally mixed.', '1411.7041-2-25-5': 'Thus no single qutrit can be used to acquire any information about the state.', '1411.7041-2-25-6': 'Secondly, from any two of the qutrits Bob can reconstruct the state.', '1411.7041-2-25-7': 'For example, say he has access to only the first two qutrits.', '1411.7041-2-25-8': 'He can make use of the fact that there exists a unitary transformation [MATH] acting only on the first two qutrits that implements [EQUATION]', '1411.7041-2-25-9': 'Acting with this on the encoded message, we see that Bob can recover the state [MATH]: [EQUATION]', '1411.7041-2-25-10': 'Explicitly [MATH] is a permutation that acts as [EQUATION]', '1411.7041-2-25-11': 'Clearly by the symmetry of [REF] a similar construction is also possible if Bob has access only to the second and third, or first and third qutrits.', '1411.7041-2-25-12': 'Thus Bob can correct for the loss of any one of the qutrits; in quantum information terminology one describes this as a quantum error correcting code that can protect against arbitrary single qutrit erasures.', '1411.7041-2-25-13': 'The subspace spanned by [REF] is called the code subspace; the entanglement of the states in the code subspace is essential for the functioning of the protocol.', '1411.7041-2-26-0': 'In our discussion of reconstruction in the previous section we were interested in the action of operators rather than the recovery of states, and we can rephrase the error correction protocol in this language.', '1411.7041-2-26-1': 'Indeed, say that [MATH] is an operator that acts on the single qutrit Hilbert space as [EQUATION]', '1411.7041-2-26-2': 'For any such [MATH] we can always find a (non-unique) three-qutrit operator [MATH] which implements the same transformation on the code subspace: [EQUATION]', '1411.7041-2-26-3': 'In quantum computing language, operators like [MATH] that act directly on the code subspace in this manner are called logical operations, since they are the types of things that we want to implement when performing a fault-tolerant quantum computation.', '1411.7041-2-27-0': 'For a general code subspace, [MATH] would need to have nontrivial support on all three qutrits.', '1411.7041-2-27-1': 'For the code subspace in question, however, is straightforward to see that the operator [EQUATION] where [MATH] is taken to act on the first qutrit, acts as [EQUATION] [MATH] is thus an [MATH] that has support only on the first two qutrits.', '1411.7041-2-27-2': 'Since we can also analogously construct [MATH] or [MATH], we have realized a situation where operators with nontrivial support on different qutrits have the same action on the code subspace.', '1411.7041-2-27-3': 'This should be reminiscent of our discussion of overlapping wedges in the previous section; we will make the connection more explicit soon but first we need to discuss some general properties of quantum erasure correction.', '1411.7041-2-28-0': 'Before moving on, however, we want to introduce a notational simplification.', '1411.7041-2-28-1': 'So far we have been careful to distinguish the single-qutrit operator [MATH] from its three-qutrit representations [MATH].', '1411.7041-2-28-2': 'We find it convenient, however, to from now on abuse notation by instead thinking of "[MATH]" as an abstract logical operation and using it both cases; which operator we mean should always be clear from the context.', '1411.7041-2-28-3': 'So for example we can write [EQUATION]', '1411.7041-2-29-0': '## General erasure correction', '1411.7041-2-30-0': 'We now describe a natural generalization of the protocol of the previous subsection.', '1411.7041-2-30-1': 'For familiarity we will describe it using qubits, although none of the results rely on this.', '1411.7041-2-30-2': 'Say that we want to protect a [MATH]-qubit code subspace of an [MATH]-qubit system against the loss of some collection of [MATH] of [MATH] of the qubits.', '1411.7041-2-30-3': 'We define the code subspace [MATH] as the span of the orthonormal states [EQUATION] where [MATH] is called the encoding unitary transformation.', '1411.7041-2-30-4': 'There is a necessary and sufficient condition for the correctability of the erasure of [MATH] [CITATION].', '1411.7041-2-30-5': 'Say that we adjoin to our system a reference system [MATH] of [MATH] additional qubits.', '1411.7041-2-30-6': "We then consider the state [EQUATION] where [MATH] denotes the set of [MATH] qubits that aren't erased.", '1411.7041-2-30-7': 'The code [REF] can correct for the erasure of [MATH] if and only if we have [EQUATION]', '1411.7041-2-30-8': 'Here [MATH], [MATH], etc are the reduced density matrices obtained from [MATH] by partial trace.', '1411.7041-2-30-9': 'This is equivalent to saying that the mutual information [MATH] vanishes, where [MATH] is the Von Neumman entropy, [MATH].', '1411.7041-2-30-10': 'Let us first see that this ensures we can correct the erasure.', '1411.7041-2-30-11': 'The Schmidt decomposition of [MATH], together with [REF], ensures us that there exists a basis [MATH] for [MATH] and a set of orthonormal states [MATH] in [MATH] such that [EQUATION] where [MATH] are some non-negative coefficients obeying [MATH].', '1411.7041-2-30-12': 'In other words there exists a unitary transformation [MATH] acting only on [MATH] such that [EQUATION] where we have denoted the first [MATH] qubits of [MATH] as [MATH] and the rest as [MATH].', '1411.7041-2-30-13': '[MATH] is some state that is independent of [MATH].', '1411.7041-2-30-14': 'This then implies that we must have [EQUATION] which is the analogue of [REF] above and demonstrates that we can use [MATH] to correct the erasure.', '1411.7041-2-30-15': 'If we do not have [REF], then there is nonzero correlation between [MATH] and [MATH], so we can learn about the state of [MATH] by doing measurements on [MATH].', '1411.7041-2-30-16': 'Since any successful protocol must not care about what happens to the qubits we lose, this prevents us from being able to correct the erasure.', '1411.7041-2-30-17': 'We can thus loosely rephrase [REF] as the statement that the erasure of [MATH] is correctable if and only if no information about [MATH] can be obtained from [MATH].', '1411.7041-2-30-18': 'This is related to the no-cloning theorem; if we were able to get the same quantum information about the encoded state [MATH] from both [MATH] and [MATH] then we would have built a machine for cloning that information.', '1411.7041-2-31-0': 'There is a useful reformulation of the condition [REF] as the statement that for any operator [MATH] acting on [MATH], we must have [CITATION] [EQUATION]', '1411.7041-2-31-1': 'In other words we must have the projection of [MATH] onto the code subspace be proportional to the identity.', '1411.7041-2-31-2': 'One immediate consequence of this is that in any state [MATH] in the code subspace, the correlation function of [MATH] with any operator [MATH] that acts within the code subspace must vanish: [EQUATION]', '1411.7041-2-31-3': 'This is another manifestation of the idea that [MATH] has no access to the encoded information.', '1411.7041-2-32-0': 'As in the previous subsection, we can use [MATH] to realize any operator [MATH] acting within the code subspace as an operator [MATH] that acts just on [MATH].', '1411.7041-2-32-1': 'Indeed we have both [EQUATION]', '1411.7041-2-32-2': 'In fact the converse of this statement also holds; if any operator on the code subspace can be realized as an operator on [MATH] as in [REF], then the code is able to correct for the loss of [MATH].', '1411.7041-2-32-3': 'The proof is simple.', '1411.7041-2-32-4': 'Say that the code were not correctable; then as just discussed there must exist an operator [MATH] on [MATH] where [REF] does not hold.', '1411.7041-2-32-5': "By Schur's lemma, there must then exist an operator [MATH] on the code subspace that does not commute with [MATH] on [MATH], that is with [MATH] for some [MATH] and [MATH].", '1411.7041-2-32-6': "But this operator [MATH] can't be realized on [MATH] by an operator [MATH] that acts only on [MATH], since any such operator by definition would commute with [MATH].", '1411.7041-2-33-0': 'We now turn to the question of when we should expect [REF] (or equivalently [REF] or [REF]) to hold.', '1411.7041-2-33-1': 'In situations where we would like our code to be able to correct against a wide variety of erasures, we expect that [MATH] will have full rank.', '1411.7041-2-33-2': 'In that case, in order to be able to have the orthonormal set of states [MATH] we need the dimensionality of [MATH] to be at least as large as dimensionality of [MATH].', '1411.7041-2-33-3': 'In other words we need [EQUATION]', '1411.7041-2-33-4': 'This condition is quite intuitive; wanting to send a larger message or correct larger erasures requires more qubits.', '1411.7041-2-34-0': 'In fact for large systems [REF] is typically not only necessary but sufficient.', '1411.7041-2-34-1': 'Say we take [MATH] to be a random state of [MATH] qubits in the Haar measure.', '1411.7041-2-34-2': "By Page's theorem [CITATION], the density matrix of [MATH] will be exponentially close to maximally mixed provided that [MATH], so by the Schmidt decomposition this is equivalent to choosing a random [MATH]-qubit code subspace of [MATH] qubits.", '1411.7041-2-34-3': "The condition [REF] will hold if [MATH] is maximally mixed, which again by Page's theorem should be true provided that [MATH].", '1411.7041-2-34-4': 'Thus, not only is [REF] necessary for a typical code to correct for the loss of a particular set [MATH] of [MATH] qubits, it is basically sufficient for the code to correct for the loss of any set of [MATH] qubits.', '1411.7041-2-35-0': '## Quantum secret sharing', '1411.7041-2-36-0': 'The three-qutrit example of section [REF] has the interesting property that every collection of qutrits either can perfectly reconstruct the state [MATH] or has no information about it at all.', '1411.7041-2-36-1': 'General error correcting codes do not have this property, since sometimes we can have erasures which can be "partially corrected", but it is interesting to think about the codes that do.', '1411.7041-2-36-2': 'Say that we have a Hilbert space that is a tensor product of [MATH] factors of not necessarily equal size, which in this context we will refer to as shares.', '1411.7041-2-36-3': 'A code subspace [MATH] of this Hilbert space is called a Quantum Secret Sharing Scheme if it has the property that every collection of shares either can distinguish perfectly different elements of [MATH], meaning given access to it we can correct for the erasure of its complement, or it cannot distinguish different elements at all [CITATION].', '1411.7041-2-36-4': 'Collections which enable erasure correction are called authorized and collections which do not are called unauthorized.', '1411.7041-2-36-5': 'We will see a possible application of quantum secret sharing to AdS/CFT in section [REF] below.', '1411.7041-2-37-0': '## Approximate erasure correction', '1411.7041-2-38-0': 'So far we have discussed exact quantum error correction, but in AdS/CFT we only expect the emergence of the bulk to be approximate.', '1411.7041-2-38-1': 'It will thus be important for us to get a sense of how badly we might want to allow our three necessary and sufficient conditions for correctability to be violated.', '1411.7041-2-38-2': 'The simplest way to relax the condition [REF] is to require only [CITATION] [EQUATION]', '1411.7041-2-38-3': 'Here [MATH] is the trace norm of [MATH]; two density matrices whose difference has trace norm [MATH] are "operationally close" in the sense that the probability distributions they predict for arbitrary measurements differ by at most [MATH].', '1411.7041-2-38-4': 'This essentially says that typical states in the code subspace can be reconstructed to accuracy [MATH]; following [CITATION] we take this to be the definition of approximate error correction.', '1411.7041-2-38-5': 'We would like to relate this to our second condition for correctability, [REF], but we need a convenient way to quantify the violation of [REF].', '1411.7041-2-38-6': 'One good choice is to use correlation functions of the form [EQUATION]', '1411.7041-2-38-7': 'Here [MATH] denotes taking the transpose of an operator [MATH] on the code subspace and acting with it on the reference system [MATH]; by construction acting on the state [MATH] this is equivalent to acting with [MATH] on [MATH].', '1411.7041-2-38-8': '[MATH] is essentially the average of the correlation function [REF] over all [MATH]; they become equal in the limit of a large code subspace.', '1411.7041-2-38-9': 'From the right hand side of [REF] it is not difficult to show that [CITATION] [EQUATION] where [MATH] and [MATH] are the largest eigenvalues of their respective operators.', '1411.7041-2-38-10': 'Thus we see that, as one might expect from our discussion around [REF], the presence of nonzero correlation between [MATH] and [MATH] puts a limit on how accurately we can correct for the erasure of [MATH].', '1411.7041-2-38-11': 'This inequality will be very useful in our discussion of AdS/CFT, since after all computing correlation functions in the bulk theory is much easier than computing the trace norm directly.', '1411.7041-2-39-0': '## Operator algebra quantum error correction', '1411.7041-2-40-0': 'In our discussion of AdS/CFT we will soon see that the presence of bulk correlation puts nontrivial restrictions on the correctability of errors via the inequality [REF].', '1411.7041-2-40-1': 'There is, however, a generalized version of quantum error correction, called operator algebra quantum error correction, that is able to accommodate such correlation by requiring that our third necessary and sufficient condition [REF] apply only to a subalgebra of operators on the code subspace [CITATION].', '1411.7041-2-40-2': 'This requirement is greatly illuminated by the following theorem:', '1411.7041-2-41-0': 'Say that we have a code subspace [MATH] and an operator [MATH] that, together with its hermitian conjugate, acts within the code subspace.', '1411.7041-2-41-1': 'In other words we have [EQUATION]', '1411.7041-2-41-2': 'Then there exists an operator [MATH] acting just on [MATH] that obeys [EQUATION] for any [MATH] if and only if [MATH] commutes with the projection of any operator [MATH] onto the code subspace, where [MATH] acts on [MATH].', '1411.7041-2-41-3': 'In other words [EQUATION]', '1411.7041-2-41-4': 'We give a proof of this theorem in appendix [REF].', '1411.7041-2-41-5': "It is clear that the set of [MATH]'s that satisfy the assumptions of the theorem form a unital *-subalgebra [MATH] of the operators on the code subspace, meaning they include the identity and are closed under addition, multiplication, and hermitian conjugation.", '1411.7041-2-41-6': 'If we take [MATH] to be the entire algebra of operators on [MATH] then we recover our condition [REF].', '1411.7041-2-41-7': "Notice, however, that when [MATH] is a proper subalgebra we cannot use our previous argument to derive the condition [REF] from [REF], since the [MATH] that we constructed that doesn't have an [MATH] will not be in [MATH].", '1411.7041-2-41-8': 'This gives a loophole that simultaneously allows correlation between [MATH] and [MATH] and the existence of [MATH].', '1411.7041-2-42-0': 'For example in the two qubit system, consider a code subspace spanned by [MATH] and [MATH].', '1411.7041-2-42-1': 'The operator [MATH] that exchanges these two states can be realized just on the first qubit as the [MATH] operator that flips it, even though in either state this operator is perfectly correlated with the [MATH] operator that flips the second qubit.', '1411.7041-2-42-2': 'This is possible because the [MATH] operator on the code subspace, for which the first state is a [MATH] eigenstate and the second state is a [MATH] eigenstate, cannot be realized as an operator just on the first qubit; this code corrects only the subalgebra generated by [MATH] and [MATH].', '1411.7041-2-43-0': 'This example has the perhaps surprising property that the encoded [MATH] and [MATH] operators can be realized on either of the two qubits, which seems in tension with our discussion of the no-cloning theorem above [REF].', '1411.7041-2-43-1': 'This is an artifact, however, of the fact that this subalgebra is abelian, and is thus in some sense classical.', '1411.7041-2-43-2': 'It is easy to prove that as long as the subalgebra is non-abelian, if it can be represented on [MATH] then it cannot be represented on [MATH]; the proof follows immediately by contradiction if we look at the commutator of two non-commuting elements of the algebra, but with one represented on [MATH] and one represented on [MATH].', '1411.7041-2-43-3': 'We used this "algebraic no-cloning theorem" above in our discussion of figure [REF].', '1411.7041-2-44-0': '# AdS/CFT as quantum error correction', '1411.7041-2-45-0': 'We now return to our discussion of bulk reconstruction.', '1411.7041-2-45-1': 'Consider again the right diagram in figure [REF].', '1411.7041-2-45-2': 'We argued using the AdS-Rindler reconstruction that the operator in the center can be represented either as an operator [MATH] with support on [MATH], an operator [MATH] with support on [MATH], or an operator [MATH] with support on [MATH].', '1411.7041-2-45-3': 'By now it should be obvious that this is directly analogous to the situation with [MATH], [MATH], and [MATH] in the three qutrit example, or more generally the existence of the operator [MATH].', '1411.7041-2-45-4': 'The main proposal of this paper is that this is more than an analogy, it is actually how AdS/CFT is reproducing the bulk!', '1411.7041-2-45-5': 'In other words we can think of local bulk operators as logical operations on an encoded subspace, which becomes better and better protected against localized boundary errors as we move the operators inwards in the radial direction.', '1411.7041-2-45-6': 'We illustrate this in figure [REF].', '1411.7041-2-45-7': 'In the remainder of the paper we will spell out this idea in more detail, giving the bulk versions of most of the statements of the previous section.', '1411.7041-2-46-0': '## Defining code subspaces', '1411.7041-2-47-0': 'We begin by defining a set of candidate code subspaces for AdS/CFT.', '1411.7041-2-47-1': 'Our proposal is that we should pick some finite set of local bulk operators [MATH], realized in the CFT via the global representation of section [REF].', '1411.7041-2-47-2': "We then define a code subspace [MATH] as the linear span of states of the form [EQUATION] where we take the range of [MATH], the number of [MATH]'s we act with, and the number of points [MATH] where the operators can be located to be bounded by some fixed finite number.", '1411.7041-2-47-3': 'Here [MATH] is the ground state of the system; we could also do a similar construction around other sufficiently "semiclassical" states, but for rigor we will stick to [MATH] since, as mentioned in section [REF], the existence of appropriate smearing functions has not been completely established in the general case.', '1411.7041-2-47-4': 'We postpone to section [REF] the question of how large [MATH] can be.', '1411.7041-2-47-5': 'It is essential that our definition of the code subspace will be different for different choices of the operators [MATH]; the set of erasures that are correctable will depend on this choice, and we can learn about the way that the bulk theory is realized in the CFT by studying this dependence.', '1411.7041-2-47-6': 'For example, in figure [REF] we see that moving the operators closer to the boundary makes our code subspace less protected against small erasures.', '1411.7041-2-47-7': 'The CFT is not just one error-correcting code, it is many at once!', '1411.7041-2-48-0': 'We would like to think of the operators [MATH] as logical operations on this code subspace, but this does not quite work since by construction acting repeatedly with [MATH] will eventually take us out of [MATH].', '1411.7041-2-48-1': 'To get a set of operators that really act within [MATH] we can include projection operators onto [MATH] on both sides of [MATH]; these will be irrelevant except in studying high-point correlation functions, so we will not carry them around explicitly here.', '1411.7041-2-48-2': 'Now consider a decomposition of the boundary Cauchy surface [MATH] into [MATH] and [MATH].', '1411.7041-2-48-3': 'If our code subspace [MATH] can protect against the erasure of [MATH], then by our condition [REF] it must be that we can find a representation of any operator on [MATH] with support only in [MATH].', '1411.7041-2-48-4': "In fact, this is what the AdS-Rindler reconstruction we reviewed in section [REF] provides us; any causal wedge [MATH] which contains the locations of the [MATH]'s used in defining [MATH] will allow a set of operators [MATH] with support only on [MATH] and whose action on [MATH] is the same as that of [MATH].", '1411.7041-2-48-5': 'We now see that in the CFT this is a statement about being able to correct for the erasure of [MATH].', '1411.7041-2-49-0': 'To avoid confusion, we stress that, just because we do not include some [MATH] in defining the code subspace, we do not mean to imply that its AdS-Rindler reconstruction does not work on that subspace.', '1411.7041-2-49-1': 'We could easily consider a slightly larger subspace where we include it, and we could then interpret its AdS-Rindler reconstruction as arising from quantum error correction.', '1411.7041-2-49-2': 'The only fundamental limitation on the AdS-Rindler reconstruction comes from the backreaction considerations we discuss in section [REF] below.', '1411.7041-2-50-0': '## Bulk correlation and smearing', '1411.7041-2-51-0': 'It is illuminating to understand in more detail to what extent the AdS-Rindler reconstruction is consistent with our three equivalent conditions [REF], [REF], [REF] for quantum erasure correction.', '1411.7041-2-51-1': 'We clearly do not expect them to hold exactly, but we might hope for them to hold in the approximate sense of [REF].', '1411.7041-2-51-2': 'As we explained in section [REF], a good diagnostic for approximate quantum erasure correction is that the correlation functions between operators acting within the code subspace and operators acting on the set to be erased are small enough that the inequality [REF] does not preclude [REF] from holding.', '1411.7041-2-52-0': 'In fact it is a basic property of bulk physics that there is correlation between fields in [MATH] and fields in [MATH], as we indicate in figure [REF].', '1411.7041-2-52-1': 'In deciding whether or not this bulk correlation interferes with our interpretation of AdS-Rindler reconstruction as quantum error correction, we need to properly take into account the operator eigenvalues in the denominator of [REF].', '1411.7041-2-52-2': 'Formally these are infinite in a continuum quantum field theory, but every quantum field theorist knows that field operators are not really well-defined until they are integrated against smooth test functions with support over some region of nonzero measure, which we will take to have linear size [MATH].', '1411.7041-2-52-3': 'For simplicity we will take the bulk fields to be massless scalars and take their separation to be small compared to the AdS radius, in which case we have [EQUATION]', '1411.7041-2-52-4': 'Here [MATH] is the spacetime dimension of the boundary theory and [MATH] is the geodesic distance between [MATH] and [MATH].', '1411.7041-2-52-5': 'This formula also holds in other states we produce by acting on [MATH] with smeared operators near [MATH], and thus on average in the code subspace [MATH].', '1411.7041-2-52-6': 'We thus see that the right hand side of [REF] will be small in our case provided that the operators [MATH] used in constructing [MATH] are smeared over a distance which is small compared to their distance to the causal surface [MATH] of the wedge [MATH] in which we are trying to reconstruct them.', '1411.7041-2-53-0': 'This observation does much to justify our interpretation of AdS-Rindler reconstruction as quantum error correction, but it is somewhat unsatisfactory in the sense that the AdS-Rindler reconstruction still seems to work in the situation where we smear the operators over a distance that is comparable to their distance to the bifurcate Rindler horizon [MATH], even though the bulk correlation is then too large to be ignored.', '1411.7041-2-53-1': 'Indeed we interpret this as saying that the conventional quantum error correction of section [REF] does not fully capture the mechanism by which AdS/CFT realizes bulk locality.', '1411.7041-2-53-2': 'The operator algebra quantum error correction introduced in section [REF], however, provides precisely the generalization we need to fix this.', '1411.7041-2-53-3': 'Consider for example an operator [MATH] which acts on [MATH] as [MATH], and which annihilates any state orthogonal to [MATH].', '1411.7041-2-53-4': 'This is an operator that acts within the code subspace, but its commutator with an operator [MATH] in [MATH] obeys [EQUATION]', '1411.7041-2-53-5': 'Thus [MATH] clearly cannot have a representation as an operator just on [MATH].', '1411.7041-2-53-6': 'Fortunately there is no reason to expect this operator to have an AdS-Rindler reconstruction, but the broader lesson is that we should really expect AdS-Rindler reconstruction to in general produce only a subalgebra of the operators on [MATH].', '1411.7041-2-53-7': 'We saw in section [REF] that the condition a subalgebra must obey for this to be possible is that the subalgebra must commute with the projection onto [MATH] of any operator on [MATH].', '1411.7041-2-53-8': 'In fact this is precisely the condition that we expect to be true for local operators in [MATH] (and their sums and products), which by bulk causality should commute with operators in [MATH].', '1411.7041-2-53-9': 'That this commutator vanishes with the projections onto [MATH] of all CFT operators in [MATH] is not something we can prove directly, but AdS-Rindler reconstruction requires it.', '1411.7041-2-54-0': 'A second reason to prefer operator algebra quantum error correction is that even when the right hand side of [REF] is small, it will at most be suppressed by some fixed power of [MATH].', '1411.7041-2-54-1': 'This is because we should not smear the operators over distances shorter than the Planck length.', '1411.7041-2-54-2': 'Since we in principle would like a version of AdS-Rindler reconstruction that works to all orders in [MATH], it would be unsatisfying if our error correction interpretation failed at some finite order because of bulk correlation.', '1411.7041-2-55-0': 'We can now state our final proposal: the AdS-Rindler reconstruction of local bulk operators in [CITATION] is dual in the CFT to the operator algebra quantum error correction of [CITATION].', '1411.7041-2-55-1': "An erasure of a region [MATH] is correctable if the [MATH]'s used in defining the code subspace all lie within the causal wedge [MATH].", '1411.7041-2-55-2': 'In cases where the operators we are interested in are well-localized away from the causal surface [MATH] of [MATH], the situation is well-approximated by conventional quantum error correction.', '1411.7041-2-55-3': "Either way, the further the [MATH]'s are from the asymptotic boundary, the better they are protected from CFT erasures.", '1411.7041-2-56-0': 'It is worth emphasizing that in the case where a bulk operator is of order an AdS radius distance from [MATH], our approximate equivalence between conventional and operator algebra quantum error correction requires sub-AdS scale bulk locality.', '1411.7041-2-56-1': 'This is a special property of those CFTs that have local holographic duals, which we have here reformulated in the language of quantum information theory.', '1411.7041-2-57-0': '## Disconnected regions and quantum secret sharing', '1411.7041-2-58-0': 'So far we have only discussed the erasure of connected regions of the boundary.', '1411.7041-2-58-1': 'More general erasures are also interesting.', '1411.7041-2-58-2': 'Consider for example the [MATH] situation depicted in figure [REF].', '1411.7041-2-59-0': 'Here we consider a region [MATH] which is the union of two disjoint intervals; in other words we have erased two disjoint intervals.', '1411.7041-2-59-1': 'Can we choose a code subspace where we can realize the bulk operator in the center as an operator acting on [MATH] or [MATH]?', '1411.7041-2-59-2': 'If the AdS-Rindler reconstruction is the last word on bulk reconstruction [CITATION], then the answer is clearly no; this point lies neither in [MATH] nor in [MATH].', '1411.7041-2-59-3': 'This is possible within the context of quantum error correction, but only if both [MATH] and [MATH] can access partial information about the code subspace.', '1411.7041-2-59-4': 'For example, say that [MATH] had no information whatsoever about which state of the code subspace we are in.', '1411.7041-2-59-5': 'Then by definition [REF] would hold, so we could recover the information from [MATH].', '1411.7041-2-59-6': 'We are not, however, able to determine whether or not such partial information is really present.', '1411.7041-2-60-0': 'In fact there have been recent conjectures in the literature that this operator can still be reconstructed in [MATH] as long as [MATH] is bigger than [MATH]; more generally, the claim is that one can do reconstruction throughout the entanglement wedge, which is defined as the bulk domain of dependence of any bulk spacelike surface whose boundary is the union of [MATH] and the codimension two extremal-area surface of minimal area whose boundary is [MATH] [CITATION].', '1411.7041-2-60-1': 'In the figure, the intersection of the entanglement wedge with a bulk Cauchy surface is shaded blue; the minimal area condition causes a discontinuous change as we increase the size of [MATH].', '1411.7041-2-60-2': 'Is this conjecture compatible with our proposal?', '1411.7041-2-60-3': 'Indeed it is; we saw below equation [REF] that in a generic code subspace any [MATH] which is greater than half of the system can correct for the erasure of its complement [MATH].', '1411.7041-2-60-4': "The sharp jump in correctability as [MATH] surpasses [MATH] in size is consistent with our analysis around [REF], where from Page's theorem we expect that the density matrix of [MATH] together with the reference system will approach being maximally mixed exponentially fast once we cross the transition.", '1411.7041-2-61-0': 'In section [REF] we saw that a division of the CFT into a union of shares with the property that any collection of the shares has either complete information or no information about the encoded state is called a quantum secret sharing scheme; we now see that in the situation of figure [REF] we will be able to reconstruct the operator in the center if and only if our boundary division into four regions gives a quantum secret sharing scheme.', '1411.7041-2-62-0': '## MERA as an error correcting code?', '1411.7041-2-63-0': 'One shortcoming of our work so far is that, although we have laid out a plausible CFT interpretation of AdS-Rindler reconstruction as quantum error correction, we have ultimately relied on the bulk in deriving this reconstruction.', '1411.7041-2-63-1': 'This boils down to the assumption that there exist operators in the CFT that obey the bulk equations of motion and algebra on a subspace.', '1411.7041-2-63-2': 'We then use this assumption to perform the Bogoliubov transformation that relates the global and the Rindler reconstructions.', '1411.7041-2-63-3': 'This assumption is quite plausible, and essentially follows from the assumed large-[MATH] structure of the CFT [CITATION], but it would still be nice if we could explicitly demonstrate the structure of the quantum error correcting code in the CFT.', '1411.7041-2-63-4': 'In particular, in section [REF] we had to use bulk causality to argue that the necessary and sufficient condition [REF] for operator algebra quantum error correction held, and we were not able to check it explicitly for all possible CFT operators on [MATH].', '1411.7041-2-63-5': 'Similarly we were unable to determine whether or not the central point could be reconstructed in the two-interval [MATH] of the previous subsection.', '1411.7041-2-64-0': 'A promising starting point for addressing these issues is the MERA tensor network construction of a discrete version of AdS/CFT [CITATION].', '1411.7041-2-64-1': 'It seems possible that in that fairly controlled setting one could rigorously confirm the quantum error correction structure we have motivated in this paper.', '1411.7041-2-64-2': 'Moreover, one could attempt to determine explicitly whether or not the example of the previous subsection allows reconstruction of the operator in the center; this could be done by using the global construction to make a code subspace, entangling this code subspace with a reference system [MATH] to prepare a state [MATH], and then seeing whether there is mutual information between [MATH] and [MATH].', '1411.7041-2-64-3': 'The state [MATH] would still be prepared by a tensor network, with tensors acting both on the CFT and/or the reference system.', '1411.7041-2-64-4': 'This calculation would go a long way towards settling the "causal wedge vs. entanglement wedge" debate of bulk reconstruction.', '1411.7041-2-64-5': 'We will not attempt this calculation here, but the typicality argument leading to [REF] favors the entanglement wedge; we will say more about this in section [REF].', '1411.7041-2-65-0': '# Backreaction and holography', '1411.7041-2-66-0': 'We now turn to the question of how large we can make the code subspace [MATH].', '1411.7041-2-66-1': 'Each [MATH] that we act with raises the energy of the state, so doing so repeatedly will eventually lead to backreaction becoming important.', '1411.7041-2-66-2': 'When this happens it is clear that the approximation of perturbation theory around a fixed background geometry will break down.', '1411.7041-2-66-3': 'In this section we argue that this is related to a basic property of error correcting codes: the larger the code subspace, the fewer correctable errors.', '1411.7041-2-66-4': 'For erasures we quantified this in equation [REF] above.', '1411.7041-2-67-0': '## Defining local operators', '1411.7041-2-68-0': 'Once we allow nontrivial backreaction, it is no longer possible to ignore the issue of how we define bulk local operators in a diffeomorphism-invariant way.', '1411.7041-2-68-1': 'Following [CITATION], we do this by choosing a cutoff surface at large but finite radius, with induced metric [MATH], and then specifying bulk points by sending in spacelike geodesics from the [MATH] slice of this cutoff surface that start out orthogonal to the [MATH] directions.', '1411.7041-2-68-2': 'We then take the limit as the cutoff surface approaches the boundary.', '1411.7041-2-68-3': 'Points are labeled by a location on [MATH], a renormalized proper distance along the geodesic, and an angle in the radial/temporal plane.', '1411.7041-2-68-4': 'This is illustrated in figure [REF].', '1411.7041-2-69-0': 'These geodesics can be thought of as the "gravitational dressing" of the bulk operator, analogous to the Wilson line one would use to connect a charged operator to the boundary to make it gauge-invariant in electrodynamics.', '1411.7041-2-70-0': 'As in the electromagnetic case, the operators defined in this way will have nonlocal commutators due to their nontrivial Dirac brackets.', '1411.7041-2-70-1': 'The study of these commutators was initiated in [CITATION], and more recently elaborated in [CITATION].', '1411.7041-2-70-2': 'A full analysis has not yet been completed, however, and one point that has not yet been addressed is essential for the consistency of the AdS/Rindler reconstruction at higher orders in [MATH]: to all orders in [MATH] perturbation theory around a fixed background, two dressed bulk operators with the property that all points on their dressing geodesics are mutually spacelike separated in that background must commute.', '1411.7041-2-70-3': 'The reason this must be the case is illustrated in figure [REF].', '1411.7041-2-71-0': 'We can use this observation to verify that bulk non-locality from the gravitational dressing of operators does not invalidate some of our previous claims.', '1411.7041-2-71-1': 'In the introduction we argued that, because in the bulk theory a local operator in the center of the space commutes with all local operators at the boundary, the bulk operator algebra is inconsistent with the CFT algebra.', '1411.7041-2-71-2': 'We can now give a version of this argument that includes the gravitational dressing; from figure [REF], we see that we should modify the previous statement to "commutes with all local operators at the boundary except at one point".', '1411.7041-2-71-3': 'Were this to hold as an operator equation in the CFT, it would now not imply that the operator in the center must be trivial in the CFT, but it would imply that this operator can be nontrivial at [MATH] only at the point where the dressing geodesic ends.', '1411.7041-2-71-4': 'This statement, however, is not consistent with bulk causality, as we illustrate in figure [REF].', '1411.7041-2-71-5': 'So we thus indeed find that the bulk operator algebra cannot be realized in the CFT at the level of operator equations.', '1411.7041-2-71-6': 'As already explained, the resolution is that the bulk algebra holds in the CFT only acting on a code subspace of states.', '1411.7041-2-72-0': 'Similarly we can now revisit our claim that bulk operators in [MATH] perturbatively commute with bulk operators in [MATH], which was a necessary condition for our interpretation of the AdS-Rindler construction as operator algebra quantum error correction.', '1411.7041-2-72-1': 'But this is exactly what the argument of figure [REF] accomplishes; as long as the gravitational dressing of an operator at [MATH] also lies entirely in [MATH], meaning that the spatial geodesic connecting [MATH] to the boundary also lies in [MATH], then it will only have non-local commutators with operators that are also located in [MATH]; any operator whose localizing geodesic is entirely in one wedge will still perturbatively commute with any operator whose localizing geodesic is entirely in the complementary wedge.', '1411.7041-2-73-0': 'In this subsection, to connect to the formalism of [CITATION] we studied only operators attached to geodesics that start out orthogonal to the boundary time direction at [MATH].', '1411.7041-2-73-1': 'It would be interesting to do the analogue of their analysis at arbitrary temporal-radial angle; this amounts to working with boundary conditions that approach the "open-FRW" slicing of AdS [EQUATION] as [MATH].', '1411.7041-2-73-2': 'As explained in [CITATION], this would be a natural bulk construction of Schrodinger picture gauge-invariant operators on the fixed-boundary-time Hilbert space.', '1411.7041-2-74-0': '## Shrinking of the causal wedge', '1411.7041-2-75-0': 'We now return to the question of how backreaction affects causal wedge reconstruction.', '1411.7041-2-75-1': 'Our basic proposal is that adding energy in the bulk causes the causal wedge of a fixed boundary region [MATH] to recede towards the boundary, giving it less access to bulk operators defined at fixed renormalized geodesic distance (for some related discussion see [CITATION]).', '1411.7041-2-76-0': 'Consider for example the AdS-Schwarzschild geometry in [MATH] dimensions.', '1411.7041-2-76-1': '[EQUATION] with [EQUATION] [MATH] is proportional to the ADM mass of this geometry.', '1411.7041-2-76-2': 'Now consider a boundary disc [MATH] of angular size [MATH]; its causal wedge reaches a radius [MATH] in the bulk defined implicitly by [EQUATION]', '1411.7041-2-76-3': 'The proper distance of this radius to a cutoff surface at [MATH] is [EQUATION] so we can subtract [MATH] to define a renormalized proper distance [EQUATION]', '1411.7041-2-76-4': 'We claim that [MATH] is a decreasing function of [MATH] at fixed [MATH], which by differentiating under the integral sign is equivalent to the claim that [EQUATION] for all [MATH] and for all [MATH], where [MATH] is the positive root of [MATH].', '1411.7041-2-76-5': 'This can be shown analytically in various limits, and is easily checked numerically in the general case.', '1411.7041-2-76-6': 'One can also study the asymptotically-[MATH] BTZ black hole, where a similar result holds and all integrals can be done analytically.', '1411.7041-2-76-7': 'Thus we see that indeed the causal wedge has access to fewer and fewer bulk observables as we increase the mass of the matter in the center.', '1411.7041-2-76-8': 'This after all must be the case, since as we keep increasing the mass a point at fixed renormalized geodesic distance from the boundary will eventually go through the horizon.', '1411.7041-2-77-0': 'It is interesting to think about how general this statement is; under what circumstances can the causal wedge move inwards in renormalized geodesic distance as we insert energy?', '1411.7041-2-77-1': 'One might guess that the null energy condition should generically prevent this, but to test that we need a more precise conjecture.', '1411.7041-2-77-2': 'One first guess is that in any geometry obeying the null energy condition the causal wedge of a fixed boundary region can see at most as far in renormalized geodesic distance as it can in the vacuum.', '1411.7041-2-77-3': 'In fact this conjecture is false, we have constructed explicit counterexamples.', '1411.7041-2-77-4': 'Indeed a weaker conjecture, where we replace the null energy condition by the dominant energy condition, still has counterexamples.', '1411.7041-2-77-5': 'One counterexample is given by a small perturbation of [MATH], with the metric [EQUATION] where [EQUATION]', '1411.7041-2-77-6': 'With a small positive [MATH], the causal wedges of certain fixed boundary regions can see farther in renormalized geodesic distance than they can in the vacuum.', '1411.7041-2-77-7': 'These boundary regions include spherical regions whose causal wedges probe deep into the bulk geometry.', '1411.7041-2-78-0': 'Although these counterexamples prevent any straightforward "monoticity of causal wedge recession theorem", we expect that the Schwarzschild calculation we have just discussed captures the general tendency.', '1411.7041-2-78-1': 'It would be nice to prove a more general theorem verifying this, but we have not succeeded in finding one.', '1411.7041-2-79-0': '## Counting states', '1411.7041-2-80-0': 'The recession of the causal wedge has a nice quantum error correction interpretation; as we allow the code subspace to have more and more excited states, a bulk operator localized at some fixed geodesic distance will eventually no longer lie in the causal wedge of a fixed boundary region.', '1411.7041-2-80-1': 'In other words, the code will lose some of its ability to correct erasures; we will need access to more of the boundary to study the same bulk observables.', '1411.7041-2-80-2': 'In this subsection we study this a bit more quantitatively, making contact with the general condition [REF] for typical correctability.', '1411.7041-2-81-0': 'To apply [REF] to AdS/CFT, we need to identify CFT analogues of the quantities [MATH], [MATH], and [MATH].', '1411.7041-2-81-1': '[MATH] is the total number of qubits used in doing the encoding, and should roughly correspond to the total number of CFT degrees of freedom relevant for reconstructing a particular bulk region of interest.', '1411.7041-2-81-2': 'This is somewhat nontrivial; the CFT has an infinite number of degrees of freedom in the UV which are needed to reconstruct bulk operators that are arbitrarily close to the boundary.', '1411.7041-2-81-3': 'To deal with this we take our code subspace to only involve states where we act on the vacuum with operators [MATH] that are all localized within a region [MATH] at the center of the AdS space that has proper size of order the AdS radius.', '1411.7041-2-81-4': 'We will also take them to be smeared over distances that are small compared to their separation from the boundary of [MATH], so that we do not have to worry about the difference between conventional and operator algebra quantum error correction.', '1411.7041-2-81-5': 'The global reconstructions [REF] of these operators involve integrals over functions that vary smoothly on the scale of the radius of curvature of the boundary [MATH], so we can integrate out all CFT degrees of freedom with shorter wavelength.', '1411.7041-2-81-6': 'For concreteness we will consider the case of the [MATH] super Yang-Mills theory in [MATH] boundary dimensions with gauge group [MATH], in which case we have [EQUATION]', '1411.7041-2-81-7': 'Erasing a disc of angular size [MATH] will then correspond to erasing [EQUATION] qubits, where this function is just [MATH] times the ratio of the area of the disc to the area of the [MATH].', '1411.7041-2-82-0': 'Let us first consider the case where the code subspace is small, that is when [MATH].', '1411.7041-2-82-1': 'From [REF] we then expect that we can correct for the erasure as long as [MATH], or in other words [MATH].', '1411.7041-2-82-2': 'But this is exactly what we expect from the AdS-Rindler reconstruction; once [MATH], [MATH] will contain the center of the space.', '1411.7041-2-82-3': 'It is interesting to note that the derivation of [REF] applied to an erasure of an arbitrary collection of [MATH] qubits, so this suggests that we should also be able to reconstruct operators in the center on a union of disconnected regions, provided that together they make up more than half of the boundary.', '1411.7041-2-82-4': 'With regards to our discussion of section [REF], this gives support to the entanglement wedge over the causal wedge.', '1411.7041-2-83-0': 'We can now start increasing [MATH]; nothing interesting will happen until we get [MATH], after which the set of erasures we are able to correct will start decreasing.', '1411.7041-2-83-1': 'But this is exactly the condition for backreaction to become important in the center; with [MATH] the entropy of the code subspace is comparable to that of a black hole filling [MATH] and thus most states in the code subspace must actually be black holes.', '1411.7041-2-83-2': 'So both on the CFT side through equation [REF] and the bulk side via backreaction we arrive at the same conclusion for when correctability should break down.', '1411.7041-2-83-3': 'This is a manifestation of the holographic entropy bound of [CITATION].', '1411.7041-2-84-0': '# Conclusion', '1411.7041-2-85-0': 'In this paper we have provided what we consider to be a new understanding of how the holographic principle is realized in AdS/CFT.', '1411.7041-2-85-1': 'Bulk effective field theory operators emerge as a set of logical operations on various encoded subspaces, which are protected against local errors in the boundary CFT.', '1411.7041-2-85-2': 'The bulk algebra is realized only on these subspaces, and only if we do not try to describe too many operations at once.', '1411.7041-2-85-3': 'Asking for more causes the error correction procedure to fail, which in the bulk is manifested by the formation of a black hole.', '1411.7041-2-86-0': 'To some extent we have only recast known facts about the AdS-Rindler reconstruction in a new language, but in our view that construction is quite opaque once the operators in the boundary domain of dependence of [MATH] are evolved back to the boundary Cauchy surface [MATH] at [MATH].', '1411.7041-2-86-1': 'Our description in terms of error correction is phrased entirely on this Cauchy surface, and gives what we feel to be a satisfying interpretation of how the AdS-Rindler reconstruction is realized in the CFT that cleanly resolves some of its paradoxical features.', '1411.7041-2-87-0': 'It is of course interesting to ask if there are any implications of this work for the recent controversy on whether or not the interiors of black holes are describable in AdS/CFT; for now we leave this for future study.'} | [['1411.7041-1-54-0', '1411.7041-2-59-0'], ['1411.7041-1-54-1', '1411.7041-2-59-1'], ['1411.7041-1-54-2', '1411.7041-2-59-2'], ['1411.7041-1-54-3', '1411.7041-2-59-3'], ['1411.7041-1-54-4', '1411.7041-2-59-4'], ['1411.7041-1-54-5', '1411.7041-2-59-5'], ['1411.7041-1-54-6', '1411.7041-2-59-6'], ['1411.7041-1-70-0', '1411.7041-2-75-0'], ['1411.7041-1-70-1', 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['1411.7041-1-76-3', '1411.7041-2-81-3'], ['1411.7041-1-76-4', '1411.7041-2-81-4'], ['1411.7041-1-76-5', '1411.7041-2-81-5'], ['1411.7041-1-76-6', '1411.7041-2-81-6'], ['1411.7041-1-76-7', '1411.7041-2-81-7'], ['1411.7041-1-75-0', '1411.7041-2-80-0'], ['1411.7041-1-75-1', '1411.7041-2-80-1'], ['1411.7041-1-75-2', '1411.7041-2-80-2'], ['1411.7041-1-12-0', '1411.7041-2-13-0'], ['1411.7041-1-12-1', '1411.7041-2-13-1'], ['1411.7041-1-9-0', '1411.7041-2-10-0'], ['1411.7041-1-9-1', '1411.7041-2-10-1'], ['1411.7041-1-9-2', '1411.7041-2-10-2'], ['1411.7041-1-9-3', '1411.7041-2-10-3'], ['1411.7041-1-20-0', '1411.7041-2-21-0'], ['1411.7041-1-20-1', '1411.7041-2-21-1'], ['1411.7041-1-20-2', '1411.7041-2-21-2'], ['1411.7041-1-50-0', '1411.7041-2-55-0'], ['1411.7041-1-50-1', '1411.7041-2-55-1'], ['1411.7041-1-50-2', '1411.7041-2-55-2'], ['1411.7041-1-50-3', '1411.7041-2-55-3'], ['1411.7041-1-17-0', '1411.7041-2-18-0'], ['1411.7041-1-17-1', '1411.7041-2-18-1'], ['1411.7041-1-17-2', '1411.7041-2-18-2'], ['1411.7041-1-17-3', '1411.7041-2-18-3'], ['1411.7041-1-19-0', '1411.7041-2-20-0'], ['1411.7041-1-19-1', '1411.7041-2-20-1'], ['1411.7041-1-19-2', '1411.7041-2-20-2'], ['1411.7041-1-19-3', '1411.7041-2-20-3'], ['1411.7041-1-19-4', '1411.7041-2-20-4'], ['1411.7041-1-4-0', '1411.7041-2-4-0'], ['1411.7041-1-4-1', '1411.7041-2-4-1'], ['1411.7041-1-4-2', '1411.7041-2-4-2'], ['1411.7041-1-8-0', '1411.7041-2-8-0'], ['1411.7041-1-8-1', '1411.7041-2-8-1'], ['1411.7041-1-8-2', '1411.7041-2-8-2'], ['1411.7041-1-8-3', '1411.7041-2-8-3'], ['1411.7041-1-8-4', '1411.7041-2-8-4'], ['1411.7041-1-8-5', '1411.7041-2-8-5'], ['1411.7041-1-8-6', '1411.7041-2-8-6'], ['1411.7041-1-8-7', '1411.7041-2-8-7'], ['1411.7041-1-8-8', '1411.7041-2-8-8'], ['1411.7041-1-8-9', '1411.7041-2-8-9'], ['1411.7041-1-8-10', '1411.7041-2-8-10'], ['1411.7041-1-8-12', '1411.7041-2-8-11'], ['1411.7041-1-43-1', '1411.7041-2-47-1'], ['1411.7041-1-43-3', '1411.7041-2-47-3'], ['1411.7041-1-43-4', 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['1411.7041-2-69-0', '1411.7041-3-69-0'], ['1411.7041-2-64-0', '1411.7041-3-64-0'], ['1411.7041-2-64-1', '1411.7041-3-64-1'], ['1411.7041-2-64-2', '1411.7041-3-64-2'], ['1411.7041-2-64-3', '1411.7041-3-64-3'], ['1411.7041-2-64-4', '1411.7041-3-64-4'], ['1411.7041-2-64-5', '1411.7041-3-64-5'], ['1411.7041-2-28-0', '1411.7041-3-28-0'], ['1411.7041-2-28-1', '1411.7041-3-28-1'], ['1411.7041-2-28-2', '1411.7041-3-28-2'], ['1411.7041-2-28-3', '1411.7041-3-28-3'], ['1411.7041-2-73-0', '1411.7041-3-73-0'], ['1411.7041-2-73-1', '1411.7041-3-73-1'], ['1411.7041-2-73-2', '1411.7041-3-73-2'], ['1411.7041-2-0-0', '1411.7041-3-0-0'], ['1411.7041-2-0-1', '1411.7041-3-0-1'], ['1411.7041-2-0-2', '1411.7041-3-0-2'], ['1411.7041-2-0-3', '1411.7041-3-0-3'], ['1411.7041-2-0-4', '1411.7041-3-0-4'], ['1411.7041-2-14-0', '1411.7041-3-14-0'], ['1411.7041-2-14-1', '1411.7041-3-14-1'], ['1411.7041-2-14-2', '1411.7041-3-14-2'], ['1411.7041-2-14-3', '1411.7041-3-14-3'], ['1411.7041-2-14-4', '1411.7041-3-14-4'], ['1411.7041-2-15-0', '1411.7041-3-15-0'], ['1411.7041-2-15-1', '1411.7041-3-15-1'], ['1411.7041-2-15-2', '1411.7041-3-15-2'], ['1411.7041-2-15-3', '1411.7041-3-15-3'], ['1411.7041-2-41-0', '1411.7041-3-41-0'], ['1411.7041-2-41-1', '1411.7041-3-41-1'], ['1411.7041-2-41-2', '1411.7041-3-41-2'], ['1411.7041-2-41-3', '1411.7041-3-41-3'], ['1411.7041-2-41-4', '1411.7041-3-41-4'], ['1411.7041-2-41-5', '1411.7041-3-41-5'], ['1411.7041-2-41-6', '1411.7041-3-41-6'], ['1411.7041-2-41-7', '1411.7041-3-41-7'], ['1411.7041-2-41-8', '1411.7041-3-41-8'], ['1411.7041-2-76-0', '1411.7041-3-76-0'], ['1411.7041-2-76-1', '1411.7041-3-76-1'], ['1411.7041-2-76-2', '1411.7041-3-76-2'], ['1411.7041-2-76-3', '1411.7041-3-76-3'], ['1411.7041-2-76-4', '1411.7041-3-76-4'], ['1411.7041-2-76-5', '1411.7041-3-76-5'], ['1411.7041-2-76-6', '1411.7041-3-76-6'], ['1411.7041-2-76-7', '1411.7041-3-76-7'], ['1411.7041-2-76-8', '1411.7041-3-76-8'], ['1411.7041-2-75-0', '1411.7041-3-75-0'], ['1411.7041-2-75-1', '1411.7041-3-75-1'], ['1411.7041-2-63-0', '1411.7041-3-63-0'], ['1411.7041-2-63-1', '1411.7041-3-63-1'], ['1411.7041-2-63-2', '1411.7041-3-63-2'], ['1411.7041-2-63-3', '1411.7041-3-63-3'], ['1411.7041-2-63-4', '1411.7041-3-63-4'], ['1411.7041-2-63-5', '1411.7041-3-63-5'], ['1411.7041-1-25-0', '1411.7041-2-26-0'], ['1411.7041-1-25-1', '1411.7041-2-26-1'], ['1411.7041-1-25-5', '1411.7041-2-27-2'], ['1411.7041-1-25-6', '1411.7041-2-27-3'], ['1411.7041-1-77-0', '1411.7041-2-82-0'], ['1411.7041-1-77-1', '1411.7041-2-82-1'], ['1411.7041-1-77-2', '1411.7041-2-82-2'], ['1411.7041-1-77-3', '1411.7041-2-83-0'], ['1411.7041-1-77-4', '1411.7041-2-83-1'], ['1411.7041-1-77-5', '1411.7041-2-83-2'], ['1411.7041-1-77-6', '1411.7041-2-83-3']] | [['1411.7041-1-14-3', '1411.7041-2-15-3'], ['1411.7041-1-68-0', '1411.7041-2-73-0'], ['1411.7041-1-68-2', '1411.7041-2-73-2'], ['1411.7041-1-67-0', '1411.7041-2-72-0'], ['1411.7041-1-63-0', '1411.7041-2-68-0'], ['1411.7041-1-66-0', '1411.7041-2-71-0'], ['1411.7041-1-66-1', '1411.7041-2-71-1'], ['1411.7041-1-43-0', '1411.7041-2-47-0'], ['1411.7041-1-43-2', '1411.7041-2-47-2'], ['1411.7041-1-48-2', '1411.7041-2-53-2'], ['1411.7041-1-79-1', '1411.7041-2-85-1'], ['1411.7041-1-79-2', '1411.7041-2-85-2'], ['1411.7041-1-25-2', '1411.7041-2-26-3']] | [] | [['1411.7041-1-11-4', '1411.7041-2-12-4'], ['1411.7041-1-33-5', '1411.7041-2-36-5'], ['1411.7041-1-18-4', '1411.7041-2-19-4'], ['1411.7041-1-67-2', '1411.7041-2-72-1'], ['1411.7041-1-66-2', '1411.7041-2-71-2'], ['1411.7041-1-66-8', '1411.7041-2-71-6']] | [] | ['1411.7041-1-37-2', '1411.7041-2-40-2', '1411.7041-3-40-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1411.7041 | {'1411.7041-3-0-0': '[email protected] [email protected] [email protected] We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction.', '1411.7041-3-0-1': 'Bulk notions such as Bogoliubov transformations, location in the radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction.', '1411.7041-3-0-2': 'We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a "quantum secret sharing scheme", and suggest a tensor network calculation that may settle the issue.', '1411.7041-3-0-3': 'Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard "operator algebra quantum error correction" of Beny, Kempf, and Kribs.', '1411.7041-3-0-4': 'Our proposal gives a precise formulation of the idea of "subregion-subregion" duality in AdS/CFT, and clarifies the limits of its validity.', '1411.7041-3-1-0': '# Introduction', '1411.7041-3-2-0': 'Almost twenty years after its initial formulation, the AdS/CFT correspondence remains our best-understood example of a precise theory of quantum gravity.', '1411.7041-3-2-1': 'It has shed light on many deep puzzles in quantum gravity, and has also been of practical use in studying the dynamics of strongly interacting quantum field theories.', '1411.7041-3-2-2': 'One aspect that remains mysterious, however, is the emergence of approximate bulk locality.', '1411.7041-3-2-3': 'Locality near the boundary is straightforward.', '1411.7041-3-2-4': 'In the "extrapolate" version of the AdS/CFT dictionary we have a simple relation [CITATION] [EQUATION] between limiting values of a bulk field [MATH] and a conformal field theory operator [MATH]; this dictionary manifestly respects locality in the [MATH] directions since the CFT does.', '1411.7041-3-2-5': 'The radial direction, however, is more subtle.', '1411.7041-3-2-6': 'One way to see this is to observe that naively a local operator in the center of the bulk should commute with every local operator at the boundary on a fixed time slice containing that bulk operator.', '1411.7041-3-2-7': 'This is not consistent, however, with a standard property of quantum field theory; any operator that commutes with all local operators at a fixed time must be proportional to the identity.', '1411.7041-3-2-8': "Bulk locality thus cannot be respected within the CFT at the level of the algebra of operators; we'd then like to know in what sense it is respected.", '1411.7041-3-3-0': 'The basic idea of this paper is that bulk locality is a statement about certain subspaces of states in the CFT.', '1411.7041-3-3-1': 'That these subspaces can be large is a consequence of the large-[MATH] properties of the CFT, but the large degree of non-local entanglement in finite energy states of the CFT also plays an essential role.', '1411.7041-3-3-2': 'Our strategy will be to gradually back away from the [MATH] limit in equation [REF] and study how the the CFT representations of bulk operators spread in spatial support as we do so.', '1411.7041-3-3-3': 'On the bulk side the tool we will mostly use is the AdS-Rindler reconstruction of bulk fields introduced in [CITATION] and refined in [CITATION].', '1411.7041-3-3-4': 'We will observe that this construction has several paradoxical features, which we will illuminate by recasting it on the CFT side in the language of quantum error correcting codes [CITATION].', '1411.7041-3-3-5': 'This language gives a new, more general perspective on the issue of bulk reconstruction, and we believe that it is the natural framework for understanding the idea of "subregion-subregion" duality [CITATION].', '1411.7041-3-3-6': 'In particular, the radial direction in the bulk is realized in the CFT as a measure of how well CFT representations of bulk quantum information are protected from local erasures.', '1411.7041-3-3-7': 'The holographic principle also naturally arises in the guise of the general statement that there is an upper bound on how much quantum information a given code can protect from erasures.', '1411.7041-3-4-0': 'One point that will appear in this analysis is that truncated subalgebras of bulk observables are of interest; these were also advocated in [CITATION] in the context of describing the black hole interior.', '1411.7041-3-4-1': 'Aspects of our proposal are inspired by their construction, but here we do not discuss black hole interiors and we are not violating quantum mechanics [CITATION].', '1411.7041-3-4-2': 'A connection between black holes and quantum error correction was also made in [CITATION], which is essentially an earlier version of the proposal of [CITATION], but again the context was different and our work here should be uncontroversial by comparison.', '1411.7041-3-5-0': 'Before proceeding, let us establish a few conventions used throughout this paper.', '1411.7041-3-5-1': 'We will frequently discuss subspaces and tensor factors of the Hilbert space.', '1411.7041-3-5-2': 'When we say that an operator acts within a subspace we also mean that the same is true for its hermitian conjugate.', '1411.7041-3-5-3': 'If the Hilbert space is a tensor product [MATH], for any operator [MATH] that acts on [MATH] we may trivially form an operator [MATH] that acts on the entire Hilbert space.', '1411.7041-3-5-4': 'We will often drop the identity operator [MATH] and write it simply as [MATH].', '1411.7041-3-5-5': 'The reader should also keep in mind that the CFT regions [MATH] and [MATH] that we will talk about in section [REF] unfortunately correspond to [MATH] and [MATH] respectively in the language of section [REF].', '1411.7041-3-6-0': '# Bulk reconstruction and an AdS-Rindler puzzle', '1411.7041-3-7-0': '## Global AdS reconstruction', '1411.7041-3-8-0': 'We begin by briefly recalling the standard CFT construction of local bulk fields in AdS [CITATION].', '1411.7041-3-8-1': 'We will first work in global coordinates, where the metric asymptotically has the form [EQUATION]', '1411.7041-3-8-2': 'The CFT dual to this system lives on [MATH], with the [MATH] being the time direction.', '1411.7041-3-8-3': 'The Hilbert space of states is the set of field configurations on [MATH].', '1411.7041-3-8-4': 'The idea is then to perturbatively construct operators in the CFT which obey the bulk equations of motion, with the boundary conditions set by the dictionary [REF].', '1411.7041-3-8-5': 'For simplicity we will assume that all bulk interactions are suppressed by inverse powers of a quantity [MATH], which will also set the AdS radius in Planck units.', '1411.7041-3-8-6': 'At leading order in [MATH], this procedure results in a straightforward prescription for the CFT representation of a bulk field [MATH]; we simply have [EQUATION] where the integral is over the conformal boundary and [MATH] is a so-called "smearing function".', '1411.7041-3-8-7': 'The smearing function obeys the bulk wave equation in its [MATH] index, and leads to [REF] as we take [MATH] to the boundary.', '1411.7041-3-8-8': 'It can be chosen to only have support when [MATH] and [MATH] are spacelike separated, which we illustrate for [MATH] in the left diagram of figure [REF]; the point [MATH] is represented by a boundary integral over the green region only.', '1411.7041-3-8-9': 'In the case of empty AdS, where we take [REF] to hold everywhere, explicit representations of the smearing function can be found in [CITATION].', '1411.7041-3-8-10': "[MATH] corrections can be systematically included [CITATION], although we won't really need to discuss them here.", '1411.7041-3-8-11': 'At higher orders in this perturbation theory we will need to confront the problem of defining local operators in a diffeomorphism invariant theory, but we postpone discussion of this until section [REF].', '1411.7041-3-9-0': 'It is not obvious from the definition that the operators [REF] have the expected commutators in the bulk; this has been checked perturbatively within low point correlation functions in [CITATION], but must eventually break down in states with enough excitations to avoid a contradiction with the argument in our introduction.', '1411.7041-3-9-1': 'We will argue below that, within the subspace of states that are "perturbatively close" to the vacuum, it breaks down only at the level of non-perturbatively small corrections.', '1411.7041-3-10-0': 'Note that once we have a representation of the form [REF], we can use the CFT Hamiltonian to re-express all operators on the right hand side in terms of Heisenberg picture fields on a single Cauchy surface in the CFT, denoted as [MATH] in figure [REF].', '1411.7041-3-10-1': 'This representation is quite nontrivial, in general it involves severely nonlocal and multitrace operators.', '1411.7041-3-10-2': 'It also has the property that if we take [MATH] to be near the boundary but not quite on it, the single-time CFT representation of [MATH] still involves operators with support on all of [MATH].', '1411.7041-3-10-3': 'We might hope to find a representation whose boundary support shrinks as the operator approaches the boundary, and indeed the AdS-Rindler representation does exactly this, as we will now explain.', '1411.7041-3-11-0': '## AdS-Rindler reconstruction', '1411.7041-3-12-0': 'Consider a subregion [MATH] of a CFT Cauchy surface [MATH].', '1411.7041-3-12-1': 'The boundary domain of dependence of [MATH], denoted [MATH], is defined as the set of points on the boundary with the property that every inextendible causal curve, meaning a curve whose tangent vector is never spacelike and which is not part of a larger curve with this property, that passes through it must also intersect [MATH].', '1411.7041-3-12-2': 'This is illustrated for the boundary of [MATH] in the right diagram of figure [REF], where [MATH] is the boundary interval lying between the two vertical hash marks and [MATH] is shaded green.', '1411.7041-3-12-3': 'For any boundary region [MATH], its bulk causal future/past [MATH] is defined as the set of bulk points which can be reached by bulk causal curves evolving from/to the region [MATH].', '1411.7041-3-12-4': 'The causal wedge of a CFT subregion A [CITATION] (for earlier related definitions see [CITATION]) is defined as [EQUATION]', '1411.7041-3-12-5': 'In the right diagram of figure [REF], [MATH] roughly lies between the dashed lines and [MATH].', '1411.7041-3-12-6': 'The bulk codimension-two surface [MATH] in the figure is the "rim" of the wedge and is commonly referred to as the causal surface of [MATH] [CITATION]; more precisely it is defined as the part of the intersection of the boundaries of [MATH] that does not also intersect the conformal boundary at infinity.', '1411.7041-3-12-7': '[MATH] can also be described as the intersection of the past and future horizons of [MATH].', '1411.7041-3-13-0': 'A simple example of these definitions is where we take the geometry to be pure [MATH], [MATH] to be the [MATH] slice of the boundary, and [MATH] to be one hemisphere of [MATH].', '1411.7041-3-13-1': 'In this case [MATH] becomes what is usually referred to as the AdS-Rindler wedge.', '1411.7041-3-14-0': 'A natural set of bulk coordinates on the AdS-Rindler wedge gives a metric with the form [EQUATION] where the coordinate ranges are [MATH] and the geometry in parentheses is just the [MATH] dimensional hyperbolic disc.', '1411.7041-3-14-1': 'The causal surface [MATH] is given by the limit [MATH] at fixed [MATH], and [MATH] itself is given by [MATH] and [MATH].', '1411.7041-3-14-2': 'We illustrate this for [MATH] in figure [REF].', '1411.7041-3-14-3': 'By acting on this example with bulk isometries (or equivalently boundary conformal transformations), we can arrive at the causal wedge for any round disc in [MATH].', '1411.7041-3-14-4': 'The case of [MATH] is especially simple; all connected boundary regions are intervals and thus can be produced in this way.', '1411.7041-3-15-0': 'The point then is that the construction of CFT representations of bulk fields in the previous subsection can also be implemented purely within the causal wedge [CITATION].', '1411.7041-3-15-1': 'At leading order in [MATH], the claim is that for any [MATH] with [MATH], we can again represent [MATH] via the expression [REF], but with the [MATH] integral now taken only over [MATH].', '1411.7041-3-15-2': 'This is illustrated for [MATH] in the right diagram in figure [REF], where we have allowed for a conformal transformation that changes the size of the boundary interval [MATH].', '1411.7041-3-15-3': 'We review more details of this construction in appendix [REF]; the only major subtlety is that the smearing function [MATH] no longer exists as a function and must be understood as a distribution for integration against CFT expectation values [CITATION] (see also [CITATION] for some related discussion).', '1411.7041-3-16-0': 'Thus we see that the AdS-Rindler construction of [MATH] indeed has the property that if [MATH] is close to the boundary, only a small boundary region [MATH] localized near [MATH] is needed to be able to reconstruct [MATH] in [MATH].', '1411.7041-3-16-1': 'Moreover, by making use of the CFT evolution we can again rewrite the expression [REF] entirely in terms of nonlocal Heisenberg operators acting at [MATH], but now they will act only on [MATH].', '1411.7041-3-17-0': '## Overlapping wedges', '1411.7041-3-18-0': 'The AdS-Rindler construction of bulk fields we have just described has the somewhat counter-intuitive property that the same bulk field operator [MATH] lies in multiple causal wedges, and thus can be represented as an operator on distinct regions [MATH] in [MATH].', '1411.7041-3-18-1': 'One consequence of this is shown in the left diagram of figure [REF]; for any bulk field operator [MATH] and any CFT local operator [MATH] such that [MATH] and [MATH] are spacelike separated, we can choose a causal wedge [MATH] such that [MATH] lies in the complement of [MATH] in [MATH].', '1411.7041-3-18-2': 'By CFT locality [MATH] then must exactly commute with our representation of [MATH] in that wedge.', '1411.7041-3-18-3': 'This is coming dangerously close to contradicting the theorem mentioned in the introduction, that is that no nontrivial operator in the CFT can commute with all local CFT operators on [MATH].', '1411.7041-3-19-0': 'To avoid this contradiction it must be the case that the representations of [MATH] in different wedges are not really all the same operator on the CFT Hilbert space.', '1411.7041-3-19-1': 'We can see this in another way by considering the setup of the center diagram in figure [REF], where we have two overlapping wedges [MATH] and [MATH] that both contain the point [MATH] but [MATH] is not contained in [MATH].', '1411.7041-3-19-2': 'For a CFT operator defined with support only on [MATH] to really be equal to a CFT operator defined with support only on [MATH], it must be that the operator really only has support on [MATH].', '1411.7041-3-19-3': 'But given that we have chosen [MATH] to lie outside of [MATH], we do not expect the operator to have such a representation.', '1411.7041-3-19-4': 'In fact in this example the operator has a representation on the complement of [MATH], and we will see in section [REF] that when this is so a version of the no-cloning theorem of quantum mechanics forbids an accurate representation of the operator on [MATH].', '1411.7041-3-20-0': 'We can see the non-equivalence of the operators even more clearly by considering a third example, shown in the right diagram in figure [REF].', '1411.7041-3-20-1': 'Now a bulk field at the point [MATH] lies outside of the causal wedge for any one of the regions, but it can be reconstructed in [MATH], [MATH], or [MATH].', '1411.7041-3-20-2': 'The mutual intersection of these regions is just three points, and if we consider another set of three regions slightly rotated from these we can come up with a set of six possible reconstructions whose mutual intersection is genuinely empty.', '1411.7041-3-20-3': 'There is simply no possible way that they can all be equal as operators.', '1411.7041-3-20-4': 'For future reference we will refer to the three operators as [MATH], [MATH], and [MATH].', '1411.7041-3-21-0': 'We thus need to decide how we are to reconcile these operator inequivalences with the fact that in the bulk theory it seems that the operators are equivalent.', '1411.7041-3-21-1': 'There will clearly be some CFT states where they act quite differently, and we would like to understand the physics of the subset of states where their action is equivalent.', '1411.7041-3-21-2': 'This problem can be nicely understood in the language of quantum error correction, to which we now turn.', '1411.7041-3-22-0': '# Correcting quantum erasures', '1411.7041-3-23-0': 'Say Alice wants to send Bob a quantum state of [MATH] qubits in the mail, but she is worried that some of the qubits might get lost on the way.', '1411.7041-3-23-1': 'Quantum error correction is a procedure that allows her to embed this state into [MATH] qubits in such a way that even if some qubits are lost, Bob can still recover it.', '1411.7041-3-23-2': 'In this section we review some basic facts about this, beginning with an example.', '1411.7041-3-24-0': '## A simple example of erasure correction', '1411.7041-3-25-0': 'The simplest example of quantum error correction actually involves three-state "qutrits" instead of two-state qubits, and it uses three qutrits to send a single-qutrit message [CITATION].', '1411.7041-3-25-1': 'Say Alice wishes to send the state [EQUATION]', '1411.7041-3-25-2': 'The idea is to instead send the state [EQUATION] where [EQUATION]', '1411.7041-3-25-3': 'This protocol has two remarkable properties.', '1411.7041-3-25-4': 'First of all for any state [MATH], the reduced density matrix on any one of the qutrits is maximally mixed.', '1411.7041-3-25-5': 'Thus no single qutrit can be used to acquire any information about the state.', '1411.7041-3-25-6': 'Secondly, from any two of the qutrits Bob can reconstruct the state.', '1411.7041-3-25-7': 'For example, say he has access to only the first two qutrits.', '1411.7041-3-25-8': 'He can make use of the fact that there exists a unitary transformation [MATH] acting only on the first two qutrits that implements [EQUATION]', '1411.7041-3-25-9': 'Acting with this on the encoded message, we see that Bob can recover the state [MATH]: [EQUATION]', '1411.7041-3-25-10': 'Explicitly [MATH] is a permutation that acts as [EQUATION]', '1411.7041-3-25-11': 'Clearly by the symmetry of [REF] a similar construction is also possible if Bob has access only to the second and third, or first and third qutrits.', '1411.7041-3-25-12': 'Thus Bob can correct for the loss of any one of the qutrits; in quantum information terminology one describes this as a quantum error correcting code that can protect against arbitrary single qutrit erasures.', '1411.7041-3-25-13': 'The subspace spanned by [REF] is called the code subspace; the entanglement of the states in the code subspace is essential for the functioning of the protocol.', '1411.7041-3-26-0': 'In our discussion of reconstruction in the previous section we were interested in the action of operators rather than the recovery of states, and we can rephrase the error correction protocol in this language.', '1411.7041-3-26-1': 'Indeed, say that [MATH] is an operator that acts on the single qutrit Hilbert space as [EQUATION]', '1411.7041-3-26-2': 'For any such [MATH] we can always find a (non-unique) three-qutrit operator [MATH] which implements the same transformation on the code subspace: [EQUATION]', '1411.7041-3-26-3': 'In quantum computing language, operators like [MATH] that act directly on the code subspace in this manner are called logical operations, since they are the types of things that we want to implement when performing a fault-tolerant quantum computation.', '1411.7041-3-27-0': 'For a general code subspace, [MATH] would need to have nontrivial support on all three qutrits.', '1411.7041-3-27-1': 'For the code subspace in question, however, is straightforward to see that the operator [EQUATION] where [MATH] is taken to act on the first qutrit, acts as [EQUATION] [MATH] is thus an [MATH] that has support only on the first two qutrits.', '1411.7041-3-27-2': 'Since we can also analogously construct [MATH] or [MATH], we have realized a situation where operators with nontrivial support on different qutrits have the same action on the code subspace.', '1411.7041-3-27-3': 'This should be reminiscent of our discussion of overlapping wedges in the previous section; we will make the connection more explicit soon but first we need to discuss some general properties of quantum erasure correction.', '1411.7041-3-28-0': 'Before moving on, however, we want to introduce a notational simplification.', '1411.7041-3-28-1': 'So far we have been careful to distinguish the single-qutrit operator [MATH] from its three-qutrit representations [MATH].', '1411.7041-3-28-2': 'We find it convenient, however, to from now on abuse notation by instead thinking of "[MATH]" as an abstract logical operation and using it both cases; which operator we mean should always be clear from the context.', '1411.7041-3-28-3': 'So for example we can write [EQUATION]', '1411.7041-3-29-0': '## General erasure correction', '1411.7041-3-30-0': 'We now describe a natural generalization of the protocol of the previous subsection.', '1411.7041-3-30-1': 'For familiarity we will describe it using qubits, although none of the results rely on this.', '1411.7041-3-30-2': 'Say that we want to protect a [MATH]-qubit code subspace of an [MATH]-qubit system against the loss of some collection of [MATH] of [MATH] of the qubits.', '1411.7041-3-30-3': 'We define the code subspace [MATH] as the span of the orthonormal states [EQUATION] where [MATH] is called the encoding unitary transformation.', '1411.7041-3-30-4': 'There is a necessary and sufficient condition for the correctability of the erasure of [MATH] [CITATION].', '1411.7041-3-30-5': 'Say that we adjoin to our system a reference system [MATH] of [MATH] additional qubits.', '1411.7041-3-30-6': "We then consider the state [EQUATION] where [MATH] denotes the set of [MATH] qubits that aren't erased.", '1411.7041-3-30-7': 'The code [REF] can correct for the erasure of [MATH] if and only if we have [EQUATION]', '1411.7041-3-30-8': 'Here [MATH], [MATH], etc are the reduced density matrices obtained from [MATH] by partial trace.', '1411.7041-3-30-9': 'This is equivalent to saying that the mutual information [MATH] vanishes, where [MATH] is the Von Neumman entropy, [MATH].', '1411.7041-3-30-10': 'Let us first see that this ensures we can correct the erasure.', '1411.7041-3-30-11': 'The Schmidt decomposition of [MATH], together with [REF], ensures us that there exists a basis [MATH] for [MATH] and a set of orthonormal states [MATH] in [MATH] such that [EQUATION] where [MATH] are some non-negative coefficients obeying [MATH].', '1411.7041-3-30-12': 'In other words there exists a unitary transformation [MATH] acting only on [MATH] such that [EQUATION] where we have denoted the first [MATH] qubits of [MATH] as [MATH] and the rest as [MATH].', '1411.7041-3-30-13': '[MATH] is some state that is independent of [MATH].', '1411.7041-3-30-14': 'This then implies that we must have [EQUATION] which is the analogue of [REF] above and demonstrates that we can use [MATH] to correct the erasure.', '1411.7041-3-30-15': 'If we do not have [REF], then there is nonzero correlation between [MATH] and [MATH], so we can learn about the state of [MATH] by doing measurements on [MATH].', '1411.7041-3-30-16': 'Since any successful protocol must not care about what happens to the qubits we lose, this prevents us from being able to correct the erasure.', '1411.7041-3-30-17': 'We can thus loosely rephrase [REF] as the statement that the erasure of [MATH] is correctable if and only if no information about [MATH] can be obtained from [MATH].', '1411.7041-3-30-18': 'This is related to the no-cloning theorem; if we were able to get the same quantum information about the encoded state [MATH] from both [MATH] and [MATH] then we would have built a machine for cloning that information.', '1411.7041-3-31-0': 'There is a useful reformulation of the condition [REF] as the statement that for any operator [MATH] acting on [MATH], we must have [CITATION] [EQUATION]', '1411.7041-3-31-1': 'In other words we must have the projection of [MATH] onto the code subspace be proportional to the identity.', '1411.7041-3-31-2': 'One immediate consequence of this is that in any state [MATH] in the code subspace, the correlation function of [MATH] with any operator [MATH] that acts within the code subspace must vanish: [EQUATION]', '1411.7041-3-31-3': 'This is another manifestation of the idea that [MATH] has no access to the encoded information.', '1411.7041-3-32-0': 'As in the previous subsection, we can use [MATH] to realize any operator [MATH] acting within the code subspace as an operator [MATH] that acts just on [MATH].', '1411.7041-3-32-1': 'Indeed we have both [EQUATION]', '1411.7041-3-32-2': 'In fact the converse of this statement also holds; if any operator on the code subspace can be realized as an operator on [MATH] as in [REF], then the code is able to correct for the loss of [MATH].', '1411.7041-3-32-3': 'The proof is simple.', '1411.7041-3-32-4': 'Say that the code were not correctable; then as just discussed there must exist an operator [MATH] on [MATH] where [REF] does not hold.', '1411.7041-3-32-5': "By Schur's lemma, there must then exist an operator [MATH] on the code subspace that does not commute with [MATH] on [MATH], that is with [MATH] for some [MATH] and [MATH].", '1411.7041-3-32-6': "But this operator [MATH] can't be realized on [MATH] by an operator [MATH] that acts only on [MATH], since any such operator by definition would commute with [MATH].", '1411.7041-3-33-0': 'We now turn to the question of when we should expect [REF] (or equivalently [REF] or [REF]) to hold.', '1411.7041-3-33-1': 'In situations where we would like our code to be able to correct against a wide variety of erasures, we expect that [MATH] will have full rank.', '1411.7041-3-33-2': 'In that case, in order to be able to have the orthonormal set of states [MATH] we need the dimensionality of [MATH] to be at least as large as dimensionality of [MATH].', '1411.7041-3-33-3': 'In other words we need [EQUATION]', '1411.7041-3-33-4': 'This condition is quite intuitive; wanting to send a larger message or correct larger erasures requires more qubits.', '1411.7041-3-34-0': 'In fact for large systems [REF] is typically not only necessary but sufficient.', '1411.7041-3-34-1': 'Say we take [MATH] to be a random state of [MATH] qubits in the Haar measure.', '1411.7041-3-34-2': "By Page's theorem [CITATION], the density matrix of [MATH] will be exponentially close to maximally mixed provided that [MATH], so by the Schmidt decomposition this is equivalent to choosing a random [MATH]-qubit code subspace of [MATH] qubits.", '1411.7041-3-34-3': "The condition [REF] will hold if [MATH] is maximally mixed, which again by Page's theorem should be true provided that [MATH].", '1411.7041-3-34-4': 'Thus, not only is [REF] necessary for a typical code to correct for the loss of a particular set [MATH] of [MATH] qubits, it is basically sufficient for the code to correct for the loss of any set of [MATH] qubits.', '1411.7041-3-35-0': '## Quantum secret sharing', '1411.7041-3-36-0': 'The three-qutrit example of section [REF] has the interesting property that every collection of qutrits either can perfectly reconstruct the state [MATH] or has no information about it at all.', '1411.7041-3-36-1': 'General error correcting codes do not have this property, since sometimes we can have erasures which can be "partially corrected", but it is interesting to think about the codes that do.', '1411.7041-3-36-2': 'Say that we have a Hilbert space that is a tensor product of [MATH] factors of not necessarily equal size, which in this context we will refer to as shares.', '1411.7041-3-36-3': 'A code subspace [MATH] of this Hilbert space is called a Quantum Secret Sharing Scheme if it has the property that every collection of shares either can distinguish perfectly different elements of [MATH], meaning given access to it we can correct for the erasure of its complement, or it cannot distinguish different elements at all [CITATION].', '1411.7041-3-36-4': 'Collections which enable erasure correction are called authorized and collections which do not are called unauthorized.', '1411.7041-3-36-5': 'We will see a possible application of quantum secret sharing to AdS/CFT in section [REF] below.', '1411.7041-3-37-0': '## Approximate erasure correction', '1411.7041-3-38-0': 'So far we have discussed exact quantum error correction, but in AdS/CFT we only expect the emergence of the bulk to be approximate.', '1411.7041-3-38-1': 'It will thus be important for us to get a sense of how badly we might want to allow our three necessary and sufficient conditions for correctability to be violated.', '1411.7041-3-38-2': 'The simplest way to relax the condition [REF] is to require only [CITATION] [EQUATION]', '1411.7041-3-38-3': 'Here [MATH] is the trace norm of [MATH]; two density matrices whose difference has trace norm [MATH] are "operationally close" in the sense that the probability distributions they predict for arbitrary measurements differ by at most [MATH].', '1411.7041-3-38-4': 'This essentially says that typical states in the code subspace can be reconstructed to accuracy [MATH]; following [CITATION] we take this to be the definition of approximate error correction.', '1411.7041-3-38-5': 'We would like to relate this to our second condition for correctability, [REF], but we need a convenient way to quantify the violation of [REF].', '1411.7041-3-38-6': 'One good choice is to use correlation functions of the form [EQUATION]', '1411.7041-3-38-7': 'Here [MATH] denotes taking the transpose of an operator [MATH] on the code subspace and acting with it on the reference system [MATH]; by construction acting on the state [MATH] this is equivalent to acting with [MATH] on [MATH].', '1411.7041-3-38-8': '[MATH] is essentially the average of the correlation function [REF] over all [MATH]; they become equal in the limit of a large code subspace.', '1411.7041-3-38-9': 'From the right hand side of [REF] it is not difficult to show that [CITATION] [EQUATION] where [MATH] and [MATH] are the largest eigenvalues of their respective operators.', '1411.7041-3-38-10': 'Thus we see that, as one might expect from our discussion around [REF], the presence of nonzero correlation between [MATH] and [MATH] puts a limit on how accurately we can correct for the erasure of [MATH].', '1411.7041-3-38-11': 'This inequality will be very useful in our discussion of AdS/CFT, since after all computing correlation functions in the bulk theory is much easier than computing the trace norm directly.', '1411.7041-3-39-0': '## Operator algebra quantum error correction', '1411.7041-3-40-0': 'In our discussion of AdS/CFT we will soon see that the presence of bulk correlation puts nontrivial restrictions on the correctability of errors via the inequality [REF].', '1411.7041-3-40-1': 'There is, however, a generalized version of quantum error correction, called operator algebra quantum error correction, that is able to accommodate such correlation by requiring that our third necessary and sufficient condition [REF] apply only to a subalgebra of operators on the code subspace [CITATION].', '1411.7041-3-40-2': 'This requirement is greatly illuminated by the following theorem:', '1411.7041-3-41-0': 'Say that we have a code subspace [MATH] and an operator [MATH] that, together with its hermitian conjugate, acts within the code subspace.', '1411.7041-3-41-1': 'In other words we have [EQUATION]', '1411.7041-3-41-2': 'Then there exists an operator [MATH] acting just on [MATH] that obeys [EQUATION] for any [MATH] if and only if [MATH] commutes with the projection of any operator [MATH] onto the code subspace, where [MATH] acts on [MATH].', '1411.7041-3-41-3': 'In other words [EQUATION]', '1411.7041-3-41-4': 'We give a proof of this theorem in appendix [REF].', '1411.7041-3-41-5': "It is clear that the set of [MATH]'s that satisfy the assumptions of the theorem form a unital *-subalgebra [MATH] of the operators on the code subspace, meaning they include the identity and are closed under addition, multiplication, and hermitian conjugation.", '1411.7041-3-41-6': 'If we take [MATH] to be the entire algebra of operators on [MATH] then we recover our condition [REF].', '1411.7041-3-41-7': "Notice, however, that when [MATH] is a proper subalgebra we cannot use our previous argument to derive the condition [REF] from [REF], since the [MATH] that we constructed that doesn't have an [MATH] will not be in [MATH].", '1411.7041-3-41-8': 'This gives a loophole that simultaneously allows correlation between [MATH] and [MATH] and the existence of [MATH].', '1411.7041-3-42-0': 'For example in the two qubit system, consider a code subspace spanned by [MATH] and [MATH].', '1411.7041-3-42-1': 'The operator [MATH] that exchanges these two states can be realized just on the first qubit as the [MATH] operator that flips it, even though in either state this operator is perfectly correlated with the [MATH] operator that flips the second qubit.', '1411.7041-3-42-2': 'This is possible because the [MATH] operator on the code subspace, for which the first state is a [MATH] eigenstate and the second state is a [MATH] eigenstate, cannot be realized as an operator just on the first qubit; this code corrects only the subalgebra generated by [MATH] and [MATH].', '1411.7041-3-43-0': 'This example has the perhaps surprising property that the encoded [MATH] and [MATH] operators can be realized on either of the two qubits, which seems in tension with our discussion of the no-cloning theorem above [REF].', '1411.7041-3-43-1': 'This is an artifact, however, of the fact that this subalgebra is abelian, and is thus in some sense classical.', '1411.7041-3-43-2': 'It is easy to prove that as long as the subalgebra is non-abelian, if it can be represented on [MATH] then it cannot be represented on [MATH]; the proof follows immediately by contradiction if we look at the commutator of two non-commuting elements of the algebra, but with one represented on [MATH] and one represented on [MATH].', '1411.7041-3-43-3': 'We used this "algebraic no-cloning theorem" above in our discussion of figure [REF].', '1411.7041-3-44-0': '# AdS/CFT as quantum error correction', '1411.7041-3-45-0': 'We now return to our discussion of bulk reconstruction.', '1411.7041-3-45-1': 'Consider again the right diagram in figure [REF].', '1411.7041-3-45-2': 'We argued using the AdS-Rindler reconstruction that the operator in the center can be represented either as an operator [MATH] with support on [MATH], an operator [MATH] with support on [MATH], or an operator [MATH] with support on [MATH].', '1411.7041-3-45-3': 'By now it should be obvious that this is directly analogous to the situation with [MATH], [MATH], and [MATH] in the three qutrit example, or more generally the existence of the operator [MATH].', '1411.7041-3-45-4': 'The main proposal of this paper is that this is more than an analogy, it is actually how AdS/CFT is reproducing the bulk!', '1411.7041-3-45-5': 'In other words we can think of local bulk operators as logical operations on an encoded subspace, which becomes better and better protected against localized boundary errors as we move the operators inwards in the radial direction.', '1411.7041-3-45-6': 'We illustrate this in figure [REF].', '1411.7041-3-45-7': 'In the remainder of the paper we will spell out this idea in more detail, giving the bulk versions of most of the statements of the previous section.', '1411.7041-3-46-0': '## Defining code subspaces', '1411.7041-3-47-0': 'We begin by defining a set of candidate code subspaces for AdS/CFT.', '1411.7041-3-47-1': 'Our proposal is that we should pick some finite set of local bulk operators [MATH], realized in the CFT via the global representation of section [REF].', '1411.7041-3-47-2': "We then define a code subspace [MATH] as the linear span of states of the form [EQUATION] where we take the range of [MATH], the number of [MATH]'s we act with, and the number of points [MATH] where the operators can be located to be bounded by some fixed finite number.", '1411.7041-3-47-3': 'Here [MATH] is the ground state of the system; we could also do a similar construction around other sufficiently "semiclassical" states, but for rigor we will stick to [MATH] since, as mentioned in section [REF], the existence of appropriate smearing functions has not been completely established in the general case.', '1411.7041-3-47-4': 'We postpone to section [REF] the question of how large [MATH] can be.', '1411.7041-3-47-5': 'It is essential that our definition of the code subspace will be different for different choices of the operators [MATH]; the set of erasures that are correctable will depend on this choice, and we can learn about the way that the bulk theory is realized in the CFT by studying this dependence.', '1411.7041-3-47-6': 'For example, in figure [REF] we see that moving the operators closer to the boundary makes our code subspace less protected against small erasures.', '1411.7041-3-47-7': 'The CFT is not just one error-correcting code, it is many at once!', '1411.7041-3-48-0': 'We would like to think of the operators [MATH] as logical operations on this code subspace, but this does not quite work since by construction acting repeatedly with [MATH] will eventually take us out of [MATH].', '1411.7041-3-48-1': 'To get a set of operators that really act within [MATH] we can include projection operators onto [MATH] on both sides of [MATH]; these will be irrelevant except in studying high-point correlation functions, so we will not carry them around explicitly here.', '1411.7041-3-48-2': 'Now consider a decomposition of the boundary Cauchy surface [MATH] into [MATH] and [MATH].', '1411.7041-3-48-3': 'If our code subspace [MATH] can protect against the erasure of [MATH], then by our condition [REF] it must be that we can find a representation of any operator on [MATH] with support only in [MATH].', '1411.7041-3-48-4': "In fact, this is what the AdS-Rindler reconstruction we reviewed in section [REF] provides us; any causal wedge [MATH] which contains the locations of the [MATH]'s used in defining [MATH] will allow a set of operators [MATH] with support only on [MATH] and whose action on [MATH] is the same as that of [MATH].", '1411.7041-3-48-5': 'We now see that in the CFT this is a statement about being able to correct for the erasure of [MATH].', '1411.7041-3-49-0': 'To avoid confusion, we stress that, just because we do not include some [MATH] in defining the code subspace, we do not mean to imply that its AdS-Rindler reconstruction does not work on that subspace.', '1411.7041-3-49-1': 'We could easily consider a slightly larger subspace where we include it, and we could then interpret its AdS-Rindler reconstruction as arising from quantum error correction.', '1411.7041-3-49-2': 'The only fundamental limitation on the AdS-Rindler reconstruction comes from the backreaction considerations we discuss in section [REF] below.', '1411.7041-3-50-0': '## Bulk correlation and smearing', '1411.7041-3-51-0': 'It is illuminating to understand in more detail to what extent the AdS-Rindler reconstruction is consistent with our three equivalent conditions [REF], [REF], [REF] for quantum erasure correction.', '1411.7041-3-51-1': 'We clearly do not expect them to hold exactly, but we might hope for them to hold in the approximate sense of [REF].', '1411.7041-3-51-2': 'As we explained in section [REF], a good diagnostic for approximate quantum erasure correction is that the correlation functions between operators acting within the code subspace and operators acting on the set to be erased are small enough that the inequality [REF] does not preclude [REF] from holding.', '1411.7041-3-52-0': 'In fact it is a basic property of bulk physics that there is correlation between fields in [MATH] and fields in [MATH], as we indicate in figure [REF].', '1411.7041-3-52-1': 'In deciding whether or not this bulk correlation interferes with our interpretation of AdS-Rindler reconstruction as quantum error correction, we need to properly take into account the operator eigenvalues in the denominator of [REF].', '1411.7041-3-52-2': 'Formally these are infinite in a continuum quantum field theory, but every quantum field theorist knows that field operators are not really well-defined until they are integrated against smooth test functions with support over some region of nonzero measure, which we will take to have linear size [MATH].', '1411.7041-3-52-3': 'For simplicity we will take the bulk fields to be massless scalars and take their separation to be small compared to the AdS radius, in which case we have [EQUATION]', '1411.7041-3-52-4': 'Here [MATH] is the spacetime dimension of the boundary theory and [MATH] is the geodesic distance between [MATH] and [MATH].', '1411.7041-3-52-5': 'This formula also holds in other states we produce by acting on [MATH] with smeared operators near [MATH], and thus on average in the code subspace [MATH].', '1411.7041-3-52-6': 'We thus see that the right hand side of [REF] will be small in our case provided that the operators [MATH] used in constructing [MATH] are smeared over a distance which is small compared to their distance to the causal surface [MATH] of the wedge [MATH] in which we are trying to reconstruct them.', '1411.7041-3-53-0': 'This observation does much to justify our interpretation of AdS-Rindler reconstruction as quantum error correction, but it is somewhat unsatisfactory in the sense that the AdS-Rindler reconstruction still seems to work in the situation where we smear the operators over a distance that is comparable to their distance to the bifurcate Rindler horizon [MATH], even though the bulk correlation is then too large to be ignored.', '1411.7041-3-53-1': 'Indeed we interpret this as saying that the conventional quantum error correction of section [REF] does not fully capture the mechanism by which AdS/CFT realizes bulk locality.', '1411.7041-3-53-2': 'The operator algebra quantum error correction introduced in section [REF], however, provides precisely the generalization we need to fix this.', '1411.7041-3-53-3': 'Consider for example an operator [MATH] which acts on [MATH] as [MATH], and which annihilates any state orthogonal to [MATH].', '1411.7041-3-53-4': 'This is an operator that acts within the code subspace, but its commutator with an operator [MATH] in [MATH] obeys [EQUATION]', '1411.7041-3-53-5': 'Thus [MATH] clearly cannot have a representation as an operator just on [MATH].', '1411.7041-3-53-6': 'Fortunately there is no reason to expect this operator to have an AdS-Rindler reconstruction, but the broader lesson is that we should really expect AdS-Rindler reconstruction to in general produce only a subalgebra of the operators on [MATH].', '1411.7041-3-53-7': 'We saw in section [REF] that the condition a subalgebra must obey for this to be possible is that the subalgebra must commute with the projection onto [MATH] of any operator on [MATH].', '1411.7041-3-53-8': 'In fact this is precisely the condition that we expect to be true for local operators in [MATH] (and their sums and products), which by bulk causality should commute with operators in [MATH].', '1411.7041-3-53-9': 'That this commutator vanishes with the projections onto [MATH] of all CFT operators in [MATH] is not something we can prove directly, but AdS-Rindler reconstruction requires it.', '1411.7041-3-54-0': 'A second reason to prefer operator algebra quantum error correction is that even when the right hand side of [REF] is small, it will at most be suppressed by some fixed power of [MATH].', '1411.7041-3-54-1': 'This is because we should not smear the operators over distances shorter than the Planck length.', '1411.7041-3-54-2': 'Since we in principle would like a version of AdS-Rindler reconstruction that works to all orders in [MATH], it would be unsatisfying if our error correction interpretation failed at some finite order because of bulk correlation.', '1411.7041-3-55-0': 'We can now state our final proposal: the AdS-Rindler reconstruction of local bulk operators in [CITATION] is dual in the CFT to the operator algebra quantum error correction of [CITATION].', '1411.7041-3-55-1': "An erasure of a region [MATH] is correctable if the [MATH]'s used in defining the code subspace all lie within the causal wedge [MATH].", '1411.7041-3-55-2': 'In cases where the operators we are interested in are well-localized away from the causal surface [MATH] of [MATH], the situation is well-approximated by conventional quantum error correction.', '1411.7041-3-55-3': "Either way, the further the [MATH]'s are from the asymptotic boundary, the better they are protected from CFT erasures.", '1411.7041-3-56-0': 'It is worth emphasizing that in the case where a bulk operator is of order an AdS radius distance from [MATH], our approximate equivalence between conventional and operator algebra quantum error correction requires sub-AdS scale bulk locality.', '1411.7041-3-56-1': 'This is a special property of those CFTs that have local holographic duals, which we have here reformulated in the language of quantum information theory.', '1411.7041-3-57-0': '## Disconnected regions and quantum secret sharing', '1411.7041-3-58-0': 'So far we have only discussed the erasure of connected regions of the boundary.', '1411.7041-3-58-1': 'More general erasures are also interesting.', '1411.7041-3-58-2': 'Consider for example the [MATH] situation depicted in figure [REF].', '1411.7041-3-59-0': 'Here we consider a region [MATH] which is the union of two disjoint intervals; in other words we have erased two disjoint intervals.', '1411.7041-3-59-1': 'Can we choose a code subspace where we can realize the bulk operator in the center as an operator acting on [MATH] or [MATH]?', '1411.7041-3-59-2': 'If the AdS-Rindler reconstruction is the last word on bulk reconstruction [CITATION], then the answer is clearly no; this point lies neither in [MATH] nor in [MATH].', '1411.7041-3-59-3': 'This is possible within the context of quantum error correction, but only if both [MATH] and [MATH] can access partial information about the code subspace.', '1411.7041-3-59-4': 'For example, say that [MATH] had no information whatsoever about which state of the code subspace we are in.', '1411.7041-3-59-5': 'Then by definition [REF] would hold, so we could recover the information from [MATH].', '1411.7041-3-59-6': 'We are not, however, able to determine whether or not such partial information is really present.', '1411.7041-3-60-0': 'In fact there have been recent conjectures in the literature that this operator can still be reconstructed in [MATH] as long as [MATH] is bigger than [MATH]; more generally, the claim is that one can do reconstruction throughout the entanglement wedge, which is defined as the bulk domain of dependence of any bulk spacelike surface whose boundary is the union of [MATH] and the codimension two extremal-area surface of minimal area whose boundary is [MATH] [CITATION].', '1411.7041-3-60-1': 'In the figure, the intersection of the entanglement wedge with a bulk Cauchy surface is shaded blue; the minimal area condition causes a discontinuous change as we increase the size of [MATH].', '1411.7041-3-60-2': 'Is this conjecture compatible with our proposal?', '1411.7041-3-60-3': 'Indeed it is; we saw below equation [REF] that in a generic code subspace any [MATH] which is greater than half of the system can correct for the erasure of its complement [MATH].', '1411.7041-3-60-4': "The sharp jump in correctability as [MATH] surpasses [MATH] in size is consistent with our analysis around [REF], where from Page's theorem we expect that the density matrix of [MATH] together with the reference system will approach being maximally mixed exponentially fast once we cross the transition.", '1411.7041-3-61-0': 'In section [REF] we saw that a division of the CFT into a union of shares with the property that any collection of the shares has either complete information or no information about the encoded state is called a quantum secret sharing scheme; we now see that in the situation of figure [REF] we will be able to reconstruct the operator in the center if and only if our boundary division into four regions gives a quantum secret sharing scheme.', '1411.7041-3-62-0': '## MERA as an error correcting code?', '1411.7041-3-63-0': 'One shortcoming of our work so far is that, although we have laid out a plausible CFT interpretation of AdS-Rindler reconstruction as quantum error correction, we have ultimately relied on the bulk in deriving this reconstruction.', '1411.7041-3-63-1': 'This boils down to the assumption that there exist operators in the CFT that obey the bulk equations of motion and algebra on a subspace.', '1411.7041-3-63-2': 'We then use this assumption to perform the Bogoliubov transformation that relates the global and the Rindler reconstructions.', '1411.7041-3-63-3': 'This assumption is quite plausible, and essentially follows from the assumed large-[MATH] structure of the CFT [CITATION], but it would still be nice if we could explicitly demonstrate the structure of the quantum error correcting code in the CFT.', '1411.7041-3-63-4': 'In particular, in section [REF] we had to use bulk causality to argue that the necessary and sufficient condition [REF] for operator algebra quantum error correction held, and we were not able to check it explicitly for all possible CFT operators on [MATH].', '1411.7041-3-63-5': 'Similarly we were unable to determine whether or not the central point could be reconstructed in the two-interval [MATH] of the previous subsection.', '1411.7041-3-64-0': 'A promising starting point for addressing these issues is the MERA tensor network construction of a discrete version of AdS/CFT [CITATION].', '1411.7041-3-64-1': 'It seems possible that in that fairly controlled setting one could rigorously confirm the quantum error correction structure we have motivated in this paper.', '1411.7041-3-64-2': 'Moreover, one could attempt to determine explicitly whether or not the example of the previous subsection allows reconstruction of the operator in the center; this could be done by using the global construction to make a code subspace, entangling this code subspace with a reference system [MATH] to prepare a state [MATH], and then seeing whether there is mutual information between [MATH] and [MATH].', '1411.7041-3-64-3': 'The state [MATH] would still be prepared by a tensor network, with tensors acting both on the CFT and/or the reference system.', '1411.7041-3-64-4': 'This calculation would go a long way towards settling the "causal wedge vs. entanglement wedge" debate of bulk reconstruction.', '1411.7041-3-64-5': 'We will not attempt this calculation here, but the typicality argument leading to [REF] favors the entanglement wedge; we will say more about this in section [REF].', '1411.7041-3-65-0': '# Backreaction and holography', '1411.7041-3-66-0': 'We now turn to the question of how large we can make the code subspace [MATH].', '1411.7041-3-66-1': 'Each [MATH] that we act with raises the energy of the state, so doing so repeatedly will eventually lead to backreaction becoming important.', '1411.7041-3-66-2': 'When this happens it is clear that the approximation of perturbation theory around a fixed background geometry will break down.', '1411.7041-3-66-3': 'In this section we argue that this is related to a basic property of error correcting codes: the larger the code subspace, the fewer correctable errors.', '1411.7041-3-66-4': 'For erasures we quantified this in equation [REF] above.', '1411.7041-3-67-0': '## Defining local operators', '1411.7041-3-68-0': 'Once we allow nontrivial backreaction, it is no longer possible to ignore the issue of how we define bulk local operators in a diffeomorphism-invariant way.', '1411.7041-3-68-1': 'Following [CITATION], we do this by choosing a cutoff surface at large but finite radius, with induced metric [MATH], and then specifying bulk points by sending in spacelike geodesics from the [MATH] slice of this cutoff surface that start out orthogonal to the [MATH] directions.', '1411.7041-3-68-2': 'We then take the limit as the cutoff surface approaches the boundary.', '1411.7041-3-68-3': 'Points are labeled by a location on [MATH], a renormalized proper distance along the geodesic, and an angle in the radial/temporal plane.', '1411.7041-3-68-4': 'This is illustrated in figure [REF].', '1411.7041-3-69-0': 'These geodesics can be thought of as the "gravitational dressing" of the bulk operator, analogous to the Wilson line one would use to connect a charged operator to the boundary to make it gauge-invariant in electrodynamics.', '1411.7041-3-70-0': 'As in the electromagnetic case, the operators defined in this way will have nonlocal commutators due to their nontrivial Dirac brackets.', '1411.7041-3-70-1': 'The study of these commutators was initiated in [CITATION], and more recently elaborated in [CITATION].', '1411.7041-3-70-2': 'A full analysis has not yet been completed, however, and one point that has not yet been addressed is essential for the consistency of the AdS/Rindler reconstruction at higher orders in [MATH]: to all orders in [MATH] perturbation theory around a fixed background, two dressed bulk operators with the property that all points on their dressing geodesics are mutually spacelike separated in that background must commute.', '1411.7041-3-70-3': 'The reason this must be the case is illustrated in figure [REF].', '1411.7041-3-71-0': 'We can use this observation to verify that bulk non-locality from the gravitational dressing of operators does not invalidate some of our previous claims.', '1411.7041-3-71-1': 'In the introduction we argued that, because in the bulk theory a local operator in the center of the space commutes with all local operators at the boundary, the bulk operator algebra is inconsistent with the CFT algebra.', '1411.7041-3-71-2': 'We can now give a version of this argument that includes the gravitational dressing; from figure [REF], we see that we should modify the previous statement to "commutes with all local operators at the boundary except at one point".', '1411.7041-3-71-3': 'Were this to hold as an operator equation in the CFT, it would now not imply that the operator in the center must be trivial in the CFT, but it would imply that this operator can be nontrivial at [MATH] only at the point where the dressing geodesic ends.', '1411.7041-3-71-4': 'This statement, however, is not consistent with bulk causality, as we illustrate in figure [REF].', '1411.7041-3-71-5': 'So we thus indeed find that the bulk operator algebra cannot be realized in the CFT at the level of operator equations.', '1411.7041-3-71-6': 'As already explained, the resolution is that the bulk algebra holds in the CFT only acting on a code subspace of states.', '1411.7041-3-72-0': 'Similarly we can now revisit our claim that bulk operators in [MATH] perturbatively commute with bulk operators in [MATH], which was a necessary condition for our interpretation of the AdS-Rindler construction as operator algebra quantum error correction.', '1411.7041-3-72-1': 'But this is exactly what the argument of figure [REF] accomplishes; as long as the gravitational dressing of an operator at [MATH] also lies entirely in [MATH], meaning that the spatial geodesic connecting [MATH] to the boundary also lies in [MATH], then it will only have non-local commutators with operators that are also located in [MATH]; any operator whose localizing geodesic is entirely in one wedge will still perturbatively commute with any operator whose localizing geodesic is entirely in the complementary wedge.', '1411.7041-3-73-0': 'In this subsection, to connect to the formalism of [CITATION] we studied only operators attached to geodesics that start out orthogonal to the boundary time direction at [MATH].', '1411.7041-3-73-1': 'It would be interesting to do the analogue of their analysis at arbitrary temporal-radial angle; this amounts to working with boundary conditions that approach the "open-FRW" slicing of AdS [EQUATION] as [MATH].', '1411.7041-3-73-2': 'As explained in [CITATION], this would be a natural bulk construction of Schrodinger picture gauge-invariant operators on the fixed-boundary-time Hilbert space.', '1411.7041-3-74-0': '## Shrinking of the causal wedge', '1411.7041-3-75-0': 'We now return to the question of how backreaction affects causal wedge reconstruction.', '1411.7041-3-75-1': 'Our basic proposal is that adding energy in the bulk causes the causal wedge of a fixed boundary region [MATH] to recede towards the boundary, giving it less access to bulk operators defined at fixed renormalized geodesic distance (for some related discussion see [CITATION]).', '1411.7041-3-76-0': 'Consider for example the AdS-Schwarzschild geometry in [MATH] dimensions.', '1411.7041-3-76-1': '[EQUATION] with [EQUATION] [MATH] is proportional to the ADM mass of this geometry.', '1411.7041-3-76-2': 'Now consider a boundary disc [MATH] of angular size [MATH]; its causal wedge reaches a radius [MATH] in the bulk defined implicitly by [EQUATION]', '1411.7041-3-76-3': 'The proper distance of this radius to a cutoff surface at [MATH] is [EQUATION] so we can subtract [MATH] to define a renormalized proper distance [EQUATION]', '1411.7041-3-76-4': 'We claim that [MATH] is a decreasing function of [MATH] at fixed [MATH], which by differentiating under the integral sign is equivalent to the claim that [EQUATION] for all [MATH] and for all [MATH], where [MATH] is the positive root of [MATH].', '1411.7041-3-76-5': 'This can be shown analytically in various limits, and is easily checked numerically in the general case.', '1411.7041-3-76-6': 'One can also study the asymptotically-[MATH] BTZ black hole, where a similar result holds and all integrals can be done analytically.', '1411.7041-3-76-7': 'Thus we see that indeed the causal wedge has access to fewer and fewer bulk observables as we increase the mass of the matter in the center.', '1411.7041-3-76-8': 'This after all must be the case, since as we keep increasing the mass a point at fixed renormalized geodesic distance from the boundary will eventually go through the horizon.', '1411.7041-3-77-0': 'It is interesting to think about how general this statement is; under what circumstances can the causal wedge move inwards in renormalized geodesic distance as we insert energy?', '1411.7041-3-77-1': 'One might guess that the null energy condition should generically prevent this, but to test that we need a more precise conjecture.', '1411.7041-3-77-2': 'One first guess is that in any geometry obeying the null energy condition the causal wedge of a fixed boundary region can see at most as far in renormalized geodesic distance as it can in the vacuum.', '1411.7041-3-77-3': 'In fact this conjecture is false, we have constructed explicit counterexamples.', '1411.7041-3-77-4': 'Indeed a weaker conjecture, where we replace the null energy condition by the dominant energy condition, still has counterexamples.', '1411.7041-3-77-5': 'One counterexample is given by a small perturbation of [MATH], with the metric [EQUATION] where [EQUATION]', '1411.7041-3-77-6': 'With a small positive [MATH], the causal wedges of certain fixed boundary regions can see farther in renormalized geodesic distance than they can in the vacuum.', '1411.7041-3-77-7': 'These boundary regions include spherical regions whose causal wedges probe deep into the bulk geometry.', '1411.7041-3-78-0': 'Although these counterexamples prevent any straightforward "monoticity of causal wedge recession theorem", we expect that the Schwarzschild calculation we have just discussed captures the general tendency.', '1411.7041-3-78-1': 'It would be nice to prove a more general theorem verifying this, but we have not succeeded in finding one.', '1411.7041-3-79-0': '## Counting states', '1411.7041-3-80-0': 'The recession of the causal wedge has a nice quantum error correction interpretation; as we allow the code subspace to have more and more excited states, a bulk operator localized at some fixed geodesic distance will eventually no longer lie in the causal wedge of a fixed boundary region.', '1411.7041-3-80-1': 'In other words, the code will lose some of its ability to correct erasures; we will need access to more of the boundary to study the same bulk observables.', '1411.7041-3-80-2': 'In this subsection we study this a bit more quantitatively, making contact with the general condition [REF] for typical correctability.', '1411.7041-3-81-0': 'To apply [REF] to AdS/CFT, we need to identify CFT analogues of the quantities [MATH], [MATH], and [MATH].', '1411.7041-3-81-1': '[MATH] is the total number of qubits used in doing the encoding, and should roughly correspond to the total number of CFT degrees of freedom relevant for reconstructing a particular bulk region of interest.', '1411.7041-3-81-2': 'This is somewhat nontrivial; the CFT has an infinite number of degrees of freedom in the UV which are needed to reconstruct bulk operators that are arbitrarily close to the boundary.', '1411.7041-3-81-3': 'To deal with this we take our code subspace to only involve states where we act on the vacuum with operators [MATH] that are all localized within a region [MATH] at the center of the AdS space that has proper size of order the AdS radius.', '1411.7041-3-81-4': 'We will also take them to be smeared over distances that are small compared to their separation from the boundary of [MATH], so that we do not have to worry about the difference between conventional and operator algebra quantum error correction.', '1411.7041-3-81-5': 'The global reconstructions [REF] of these operators involve integrals over functions that vary smoothly on the scale of the radius of curvature of the boundary [MATH], so we can integrate out all CFT degrees of freedom with shorter wavelength.', '1411.7041-3-81-6': 'For concreteness we will consider the case of the [MATH] super Yang-Mills theory in [MATH] boundary dimensions with gauge group [MATH], in which case we have [EQUATION]', '1411.7041-3-81-7': 'Erasing a disc of angular size [MATH] will then correspond to erasing [EQUATION] qubits, where this function is just [MATH] times the ratio of the area of the disc to the area of the [MATH].', '1411.7041-3-82-0': 'Let us first consider the case where the code subspace is small, that is when [MATH].', '1411.7041-3-82-1': 'From [REF] we then expect that we can correct for the erasure as long as [MATH], or in other words [MATH].', '1411.7041-3-82-2': 'But this is exactly what we expect from the AdS-Rindler reconstruction; once [MATH], [MATH] will contain the center of the space.', '1411.7041-3-82-3': 'It is interesting to note that the derivation of [REF] applied to an erasure of an arbitrary collection of [MATH] qubits, so this suggests that we should also be able to reconstruct operators in the center on a union of disconnected regions, provided that together they make up more than half of the boundary.', '1411.7041-3-82-4': 'With regards to our discussion of section [REF], this gives support to the entanglement wedge over the causal wedge.', '1411.7041-3-83-0': 'We can now start increasing [MATH]; nothing interesting will happen until we get [MATH], after which the set of erasures we are able to correct will start decreasing.', '1411.7041-3-83-1': 'But this is exactly the condition for backreaction to become important in the center; with [MATH] the entropy of the code subspace is comparable to that of a black hole filling [MATH] and thus most states in the code subspace must actually be black holes.', '1411.7041-3-83-2': 'So both on the CFT side through equation [REF] and the bulk side via backreaction we arrive at the same conclusion for when correctability should break down.', '1411.7041-3-83-3': 'This is a manifestation of the holographic entropy bound of [CITATION].', '1411.7041-3-84-0': '# Conclusion', '1411.7041-3-85-0': 'In this paper we have provided what we consider to be a new understanding of how the holographic principle is realized in AdS/CFT.', '1411.7041-3-85-1': 'Bulk effective field theory operators emerge as a set of logical operations on various encoded subspaces, which are protected against local errors in the boundary CFT.', '1411.7041-3-85-2': 'The bulk algebra is realized only on these subspaces, and only if we do not try to describe too many operations at once.', '1411.7041-3-85-3': 'Asking for more causes the error correction procedure to fail, which in the bulk is manifested by the formation of a black hole.', '1411.7041-3-86-0': 'To some extent we have only recast known facts about the AdS-Rindler reconstruction in a new language, but in our view that construction is quite opaque once the operators in the boundary domain of dependence of [MATH] are evolved back to the boundary Cauchy surface [MATH] at [MATH].', '1411.7041-3-86-1': 'Our description in terms of error correction is phrased entirely on this Cauchy surface, and gives what we feel to be a satisfying interpretation of how the AdS-Rindler reconstruction is realized in the CFT that cleanly resolves some of its paradoxical features.', '1411.7041-3-87-0': 'It is of course interesting to ask if there are any implications of this work for the recent controversy on whether or not the interiors of black holes are describable in AdS/CFT; for now we leave this for future study.'} | null | null | null | null |
1410.2177 | {'1410.2177-1-0-0': 'The scattering of slow neutron beams provides unique, non-destructive, quantitative information on the structure and dynamics of materials of interest in physics, chemistry, materials science, biology, geology, and other fields.', '1410.2177-1-0-1': 'Liquid hydrogen is a widely-used neutron moderator medium, and an accurate knowledge of its slow neutron cross section is essential for the design and optimization of intense slow neutron sources.', '1410.2177-1-0-2': 'In particular the rapid drop of the slow neutron scattering cross section of liquid parahydrogen below 15 meV, which renders the moderator volume transparent to the neutron energies of most interest for scattering studies, is therefore especially interesting and important.', '1410.2177-1-0-3': 'We have placed an upper bound on the total cross section and the scattering cross section for slow neutrons with energies between 0.43 meV and 16.1 meV on liquid hydrogen at 15.6 K using neutron transmission measurements on the hydrogen target of the NPDGamma collaboration at the Spallation Neutron Source at Oak Ridge National Laboratory.', '1410.2177-1-0-4': 'At 1 meV this new upper bound is a factor of 3 below the data from previous work which has been used in the design of liquid hydrogen moderators at slow neutron sources.', '1410.2177-1-0-5': 'We describe our measurements, compare them with previous work, and discuss the implications for designing more intense slow neutron sources.', '1410.2177-1-1-0': 'The successful development of intense slow neutron sources combined with the increasing phase space acceptance of neutron optical components has enabled a dramatic expansion of the scientific applications of neutron scattering to encompass many fields in science and technology.', '1410.2177-1-1-1': 'The broad applicability of the non-destructive, quantitative information that slow neutron scattering can provide on the internal structure and dynamics of condensed media has motivated the construction of several new neutron scattering facilities over the last decade.', '1410.2177-1-1-2': 'The efficiency of the moderating medium which accepts the relatively high energy neutrons liberated from the nucleus and cools them to the slow neutron energy range below [MATH] meV determines the phase space density of the neutron beams.', '1410.2177-1-1-3': 'New results on physics relevant to the moderation process are therefore of interest to a very broad range of the scientific community.', '1410.2177-1-2-0': 'Many intense neutron sources use liquid hydrogen as a neutron moderator medium.', '1410.2177-1-2-1': 'The near-equality of the neutron and proton mass coupled with the anomalously large s-wave neutron-proton scattering amplitude allow a hydrogen-rich medium to both efficiently lower the incident neutron energy through collisions and also maintain a small neutron mean free path to keep the neutron phase space density high at the source.', '1410.2177-1-2-2': 'In the slow neutron regime, however, the neutron scattering cross section and therefore the mean free path is sensitive to the interference of the scattering amplitudes from neighboring atoms.', '1410.2177-1-3-0': 'In this context, neutron scattering from molecular hydrogen is especially interesting.', '1410.2177-1-3-1': 'Molecular hydrogen has two spin states, labeled orthohydrogen ([MATH]) and parahydrogen ([MATH]).', '1410.2177-1-3-2': 'The lowest orthohydrogen state ([MATH]) lies 15 meV above the lowest parahydrogen state ([MATH]).', '1410.2177-1-3-3': 'The spin singlet state of the protons in the parahydrogen molecule combined with the measured spin dependence of s-wave neutron-proton scattering amplitudes conspire to greatly suppress the total scattering cross section for neutrons on parahydrogen molecules by more than one order of magnitude relative to that from the hydrogen atom.', '1410.2177-1-3-4': 'The total scattering cross section on orthohydrogen is approximately 50 times higher than on parahydrogen (Fig. [REF]) because the destructive interference between the atoms is absent.', '1410.2177-1-3-5': 'Furthermore, this scattering cross section decrease happens for slow neutron energies below the 15 meV energy threshold for the neutron spin flip scattering process which turns parahydrogen into orthohydrogen.', '1410.2177-1-3-6': 'The moderator therefore becomes transparent to the neutrons just as they enter the slow neutron energy range of interest for neutron scattering applications.', '1410.2177-1-3-7': 'Consequently, many studies have shown that the slow neutron intensity from a liquid hydrogen moderator can be greatly increased if the molecules are maintained in the parahydrogen molecular state [CITATION][CITATION][CITATION].', '1410.2177-1-3-8': 'Therefore, the transmission of neutrons through liquid hydrogen is acutely sensitive to the relative concentrations of orthohydrogen and parahydrogen.', '1410.2177-1-4-0': 'In this work, we show that this decrease of the total scattering cross section in liquid parahydrogen in the slow neutron regime is much more rapid than previously realized.', '1410.2177-1-4-1': 'We draw this conclusion from a comparison of our measured upper bound on the total cross section for neutrons with energies between 0.43 meV to 16.1 meV in liquid hydrogen at a temperature of 15.6 K[MATH]0.6 K with previous data [CITATION][CITATION][CITATION].', '1410.2177-1-4-2': 'This measurement was conducted at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory using a 16-liter liquid hydrogen target [CITATION] operated on the Fundamental Neutron Physics Beamline [CITATION] (FnPB) by the NPDGamma collaboration.', '1410.2177-1-5-0': 'Figure [REF] shows the experimental setup for the transmission measurement on the FnPB beamline at the SNS.', '1410.2177-1-5-1': 'The cross sectional area of the neutron beam is 12[MATH]10 cm[MATH] at the exit of the neutron guide.', '1410.2177-1-5-2': 'Neutrons then pass through the normalization monitor, a multi-wire proportional counter [CITATION] with a gas mixture of [MATH]He (15.1 Torr) and N[MATH] (750 Torr) located 15.24 m [MATH] 0.12 m from the moderator.', '1410.2177-1-5-3': 'Neutrons are polarized using a supermirror polarizer [CITATION]; however, the direction of the polarization is reversed at 60 Hz by a spin rotator and pulse averaging analysis results in an effectively unpolarized beam.', '1410.2177-1-5-4': 'Neutrons are incident on the 16 liter liquid hydrogen target [CITATION] centered 17.6 m from the moderator.', '1410.2177-1-5-5': 'Roughly 60% of the neutron beam is captured on the hydrogen in the target with the 2.2 MeV capture gammas relevant to the NPDGamma experiment detected by 48 cesium iodide crystals.', '1410.2177-1-5-6': 'The effective length of liquid hydrogen covered by the neutron beam cross sectional area including beam divergence is 30.065 cm [MATH] 0.005 cm when cold.', '1410.2177-1-5-7': 'The transmitted neutrons exit the downstream end of the target vessel through a 2.5 cm diameter aperture in the [MATH]Li-rich neutron absorber surrounding the target.', '1410.2177-1-5-8': 'The transmitted neutron intensity is measured in a [MATH]He plate ion chamber [CITATION] located 3.44 m[MATH]0.12 m from the normalization monitor.', '1410.2177-1-5-9': 'The charge produced in the monitors is amplified by current-voltage amplifiers [CITATION] with a 10 kHz bandwidth.', '1410.2177-1-6-0': 'The data acquisition system records data in 0.4 ms increments.', '1410.2177-1-6-1': 'In order to avoid contamination from overlapping neutron pulses and to increase the dynamic range of neutron energies for the transmission measurement, data were taken with the two beamline choppers parked open while the SNS was operating at 10 Hz duty cycle rather than the normal 60 Hz.', '1410.2177-1-7-0': 'The target vessel is initially filled with hydrogen gas with 3 orthohydrogen molecules per parahydrogen molecule from equipartition.', '1410.2177-1-7-1': 'The equilibrium parahydrogen concentration increases with decreasing temperature [CITATION].', '1410.2177-1-7-2': 'The slow natural conversion to parahydrogen is accelerated by circulation of the liquid through 150 mL of hydrous iron (III) oxide [CITATION] 30 - 50 mesh powder catalyst in the ortho-para converter (OPC) in the NPDGamma target loop (Fig. [REF]).', '1410.2177-1-7-3': 'The neutron transmission increases with time as hydrogen circulates through the catalyst until a steady-state condition is reached.', '1410.2177-1-7-4': 'Fitting the transmission for 3.42 meV neutrons to an exponential as a function of time (Fig. [REF]) indicates that the parahydrogen concentration in the main target vessel approaches saturation.', '1410.2177-1-7-5': 'This exponential approach to the steady-state condition implies that the conversion is dominated by the first order processes in the OPC as liquid hydrogen circulates through the catalyst.', '1410.2177-1-8-0': 'The conversion process shown in figure [REF] has reached steady-state, where the parahydrogen concentration is near the thermal equilibrium value defined by the temperature of the OPC.', '1410.2177-1-8-1': 'The average temperature of the OPC was 15.4 K[MATH]0.5 K, which corresponds to a thermal equilibrium parahydrogen concentration of 0.99985.', '1410.2177-1-8-2': 'A small amount of para-to-ortho conversion may take place in the liquid in the main vessel, on the walls of the vessel, or the walls of the circulation loop that prevents reaching absolute thermal equilibrium.', '1410.2177-1-8-3': 'Since para-to-ortho conversion is known to be a very slow process it is not expected to limit the ortho-para ratio in the liquid hydrogen, and the liquid hydrogen is expected to be in thermal equilibrium with the catalyst.', '1410.2177-1-8-4': 'However, it was not possible to independently confirm the parahydrogen concentration in this system.', '1410.2177-1-9-0': 'Two different measurements were required in order to measure the empty target and full target transmissions.', '1410.2177-1-9-1': 'The full target measurement was performed over 8 hours with the target vessel at 15.6 K after the target had been in steady-state operation for 4 weeks, which corresponds to 30 conversion time constants.', '1410.2177-1-9-2': 'The empty target measurement was performed 2 weeks later with the target vessel at 16.3 K in order to cancel the temperature dependence of scattering from the aluminum target vessel.', '1410.2177-1-9-3': 'Between these two measurements, the beamline moderator was emptied and refilled with fresh liquid hydrogen, which led to a small change in the moderated neutron spectrum between the two measurements.', '1410.2177-1-9-4': 'A neutron energy dependent correction was applied to the transmission monitor signals using the ratio of the normalization monitor signals to account for this systematic effect.', '1410.2177-1-9-5': 'For each neutron pulse, the signals in each monitor are normalized to the per-pulse beam power by integrating the normalization monitor over peak signal range.', '1410.2177-1-10-0': 'There is not a direct correspondence between time of flight bins in each monitor due to time of flight broadening.', '1410.2177-1-10-1': 'The normalization monitor signal is fit to a cubic spline in order to interpolate for spectrum normalization.', '1410.2177-1-10-2': 'The sharp dips in the pulse shapes in figure [REF] are due to Bragg scattering on aluminum windows along the path of the neutron beam.', '1410.2177-1-10-3': 'These dips are visible at neutron energies of 4.98 meV and 3.74 meV, corresponding to the aluminum (200) and (111) Bragg planes [CITATION], respectively.', '1410.2177-1-10-4': 'The time of flight of the Bragg edges for the normalization monitor is used to determine the distance from the moderator and to convert each time of flight bin to a neutron energy.', '1410.2177-1-10-5': 'The target-full spectrum indicates no measurable neutron flux for energies above 14.5 meV (Fig. [REF]).', '1410.2177-1-10-6': 'This is the minimum energy necessary for the [MATH] spin-flip transition, meaning neutrons with energies above this threshold are scattered out of the beam rather than transmitted through the target.', '1410.2177-1-10-7': 'The data also contain a 240 Hz noise component with an amplitude of a few millivolts.', '1410.2177-1-10-8': 'The amplitude is diminished by averaging pulses over the measurement period and is only visible for small signals.', '1410.2177-1-10-9': 'The transmission monitor signals at long wavelengths are fit to a sinusoidal function corresponding to the 240 Hz noise.', '1410.2177-1-10-10': 'The sinusoidal function is subtracted before extracting the transmission.', '1410.2177-1-10-11': 'After correcting for the pedestal, 240 Hz noise, and moderator spectrum, the final corrected transmission (Fig. [REF]) is given by [EQUATION] where the [MATH] values are monitor signals and [MATH] are monitor gain adjustment factors.', '1410.2177-1-11-0': 'The contamination of the transmission signal by non-forward small angle scattering of neutrons in our geometry was estimated to be less than 0.1% in MCNPX [CITATION] using the ENDF-VII thermal cross sections [CITATION].', '1410.2177-1-11-1': 'The total cross section can then be written as: [EQUATION] where [MATH] is the number density, [MATH] is the hydrogen length, [MATH] is the parahydrogen fraction, [MATH] barns at 2200m/s [CITATION], [MATH] is the total scattering cross section, [MATH] is the orthohydrogen scattering cross section, and [MATH] is the parahydrogen scattering cross section.', '1410.2177-1-12-0': 'The diode temperature sensors have an accuracy of 0.5 K and upward drift due to radiation damage is not worse than 0.3 K, providing a total uncertainty on the temperature of 0.6 K.', '1410.2177-1-12-1': 'The density of the liquid hydrogen in our target is determined from a fit to data compilations of the density of liquid hydrogen as a function of temperature from many sources [CITATION][CITATION][CITATION].', '1410.2177-1-12-2': 'The transmission data include several instrumental effects such as the monitor efficiency, the monitor dead layer, and monitor linearity.', '1410.2177-1-12-3': 'These effects all cancel in the expression above as long as the monitors and preamplifiers are linear and the aluminum components of the experiment were maintained at the same temperature.', '1410.2177-1-12-4': 'The linearity of the transmission monitor was determined from a scan of the bias voltage in order to reduce volume recombination effects in the chambers, with a resulting uncertainty of 0.15 for each monitor.', '1410.2177-1-12-5': 'Controlled current injection was used to measure the linearity of preamplifiers and the gain shift, which are 0.01 and 0.1 respectively.', '1410.2177-1-13-0': 'We have determined the upper limit on the total cross section for liquid hydrogen at 15.6 K from approximately 0.43 meV to 16.1 meV with an uncertainty of approximately 1, or 0.02 barn/atom over the majority of the measurement range (Fig. [REF]).', '1410.2177-1-13-1': 'Because the absorption cross section is well known, we are also able to set an upper limit on the parahydrogen scattering cross section at these energies.', '1410.2177-1-13-2': 'This upper bound is much smaller than the values previously reported in the literature [CITATION][CITATION][CITATION] (Fig. [REF]).', '1410.2177-1-14-0': 'The measurement of the parahydrogen scattering cross section is very sensitive to the orthohydrogen fraction in the target volume because the orthohydrogen cross section is approximately a factor of 50 greater than for parahydrogen.', '1410.2177-1-14-1': 'The upper limit on the parahydrogen scattering cross section from this work along with the Seiffert [CITATION] data and the ENDF-VII parahydrogen kernel evaluated at 20 K [CITATION] are compared in Fig. [REF].', '1410.2177-1-14-2': 'The significant difference in magnitude suggests the presence of unaccounted for orthohydrogen contamination in the Seiffert sample.', '1410.2177-1-14-3': 'Subtraction of an admixture of 0.5% orthohydrogen from Seiffert data brings both results into agreement.', '1410.2177-1-15-0': 'The Squires measurement [CITATION] was performed using a gas mixture with a parahydrogen concentration of 0.9979, which was independently measured using thermal conductivity.', '1410.2177-1-15-1': 'The Seiffert [CITATION] and Celli [CITATION] measurements were both performed using liquid hydrogen in the presence of a catalyst; however, neither experiment independently determined the orthohydrogen concentration but rather inferred that it was either negligible, in the case of Seiffert, or at thermal equilibrium, in the case of Celli.', '1410.2177-1-15-2': 'We therefore treat both the Seiffert and Celli measurements as upper limits as we do for our data.', '1410.2177-1-15-3': 'We conclude that our target system must have less orthohydrogen contamination than these previous two measurements because our observed total cross section is lower.', '1410.2177-1-15-4': 'Of these three measurements in the literature and the measurement in this work, we believe that our measurement has the lowest orthohydrogen contamination and that it provides the most accurate upper limit on the liquid parahydrogen scattering cross section.', '1410.2177-1-16-0': 'These results have important implications for the design of slow neutron sources.', '1410.2177-1-16-1': 'They imply that the intensity of neutron moderators based on liquid hydrogen can be increased if the liquid can be maintained in the parahydrogen molecular state.', '1410.2177-1-16-2': 'Recent simulation work conducted for the European Spallation Source project [CITATION], indicates increased source intensity from liquid parahydrogen neutron moderators incorporated into a realistic target-moderator geometry.', '1410.2177-1-16-3': 'Measurements at J-PARC [CITATION] and LANSCE [CITATION] also show that the moderator intensity for neutrons below 15 meV are highly dependent on the ortho/para ratio.', '1410.2177-1-16-4': 'Our work shows that the parahydrogen cross section is overestimated throughout the slow neutron regime of interest.', '1410.2177-1-16-5': 'This overestimate reaches a factor of 3 at a neutron energy of 1 meV.', '1410.2177-1-16-6': 'The potential for increased slow neutron source intensity from liquid parahydrogen moderators is therefore greater than previously realized.', '1410.2177-1-16-7': 'In order to be able to take full advantage of this potential, however, it would be necessary to maintain the liquid in the parahydrogen state in the presence of the intense radiation environment which inevitably accompanies a intense neutron source [CITATION].', '1410.2177-1-16-8': 'Liquid hydrogen target designs which employ active circulation of the hydrogen through a catalyst coupled with dedicated measurements of the parahydrogen fraction from a liquid hydrogen moderator operated in an intense radiation environment are needed to confirm this potential and demonstrate that it can be realized at an intense neutron source.', '1410.2177-1-17-0': 'We would like to thank Erik Iverson, Phillip Ferguson, Kenneth Herwig, and Franz Gallmeier for productive discussions and encouragement for this experiment.', '1410.2177-1-17-1': 'We thank the management and staff of the Spallation Neutron Source for adapting our measurement to the busy beam delivery schedule.', '1410.2177-1-17-2': 'We gratefully acknowledge the support of the U.S. Department of Energy Office of Nuclear Physics (including Grant No. DE-FG02-03ER41258), the National Science Foundation (including Grant No. PHY-1068712), PAPIIT-UNAM (Grant No. IN111913), and the Indiana University Center for Spacetime Symmetries.'} | {'1410.2177-2-0-0': 'Slow neutron scattering provides quantitative information on the structure and dynamics of materials of interest in physics, chemistry, materials science, biology, geology, and other fields.', '1410.2177-2-0-1': 'Liquid hydrogen is a widely-used neutron moderator medium, and an accurate knowledge of its slow neutron cross section is essential for the design and optimization of intense slow neutron sources.', '1410.2177-2-0-2': 'In particular the rapid drop of the slow neutron scattering cross section of liquid parahydrogen below 14.5 meV is especially interesting and important.', '1410.2177-2-0-3': 'We have measured the total cross section and the scattering cross section for slow neutrons with energies between 0.43 meV and 16.1 meV on liquid hydrogen at 15.6 K using neutron transmission measurements on the hydrogen target of the NPDGamma collaboration at the Spallation Neutron Source at Oak Ridge National Laboratory.', '1410.2177-2-0-4': 'At 1 meV this measurement is a factor of 3 below the data from previous work which has been used in the design of liquid hydrogen moderators at slow neutron sources.', '1410.2177-2-0-5': 'We describe our measurements, compare them with previous work, and discuss the implications for designing more intense slow neutron sources.', '1410.2177-2-1-0': 'The successful development of intense slow neutron sources combined with the increasing phase space acceptance of neutron optical components has enabled a dramatic expansion of the scientific applications of neutron scattering to encompass many fields in science and technology.', '1410.2177-2-1-1': 'The broad applicability of quantitative information that slow neutron scattering can provide on the internal structure and dynamics of condensed media has motivated the construction of several new neutron scattering facilities over the last decade.', '1410.2177-2-1-2': 'The efficiency of the moderating medium which accepts the relatively high energy neutrons liberated from the nucleus and cools them to the slow neutron energy range below [MATH] meV determines the phase space density of the neutron beams.', '1410.2177-2-1-3': 'New results on physics relevant to the moderation process are therefore of interest to a very broad range of the scientific community.', '1410.2177-2-2-0': 'Many intense neutron sources use liquid hydrogen as a neutron moderator medium.', '1410.2177-2-2-1': 'The near-equality of the neutron and proton mass coupled with the anomalously large s-wave neutron-proton scattering amplitude allow a hydrogen-rich medium to both efficiently lower the incident neutron energy through collisions and also maintain a small neutron mean free path to keep the neutron phase space density high at the source.', '1410.2177-2-2-2': 'In the slow neutron regime, however, the neutron scattering cross section and therefore the mean free path is sensitive to the interference of the scattering amplitudes from neighboring atoms.', '1410.2177-2-3-0': 'In this context, neutron scattering from molecular hydrogen is especially interesting.', '1410.2177-2-3-1': 'Molecular hydrogen has two spin states, labeled orthohydrogen ([MATH]) and parahydrogen ([MATH]).', '1410.2177-2-3-2': 'The lowest orthohydrogen state ([MATH]) lies 14.5 meV above the lowest parahydrogen state ([MATH]).', '1410.2177-2-3-3': 'The spin singlet state of the protons in the parahydrogen molecule combined with the measured spin dependence of s-wave neutron-proton scattering amplitudes conspire to greatly suppress the total scattering cross section for neutrons on parahydrogen molecules by more than one order of magnitude relative to that from the hydrogen atom.', '1410.2177-2-3-4': 'The total scattering cross section on orthohydrogen is approximately 50 times higher than on parahydrogen (Fig. [REF]) because the destructive interference between the atoms is absent.', '1410.2177-2-3-5': 'A neutron that scatters from orthohydrogen will be upscattered and gain 14.5 meV, reducing the slow neutron intensity below 14.5 meV from an orthohydrogen rich moderator.', '1410.2177-2-3-6': 'Consequently, many studies have shown that the slow neutron intensity from a liquid hydrogen moderator can be greatly increased if the molecules are maintained in the parahydrogen molecular state [CITATION][CITATION][CITATION].', '1410.2177-2-3-7': 'While absorption ultimately limits the intensity for energies below 2 meV, it is the relative concentrations of orthohydrogen and parahydrogen that is the lever arm available for optimizing the properties of slow neutron moderators.', '1410.2177-2-4-0': 'In this work, we show that this decrease of the total scattering cross section in liquid parahydrogen in the slow neutron regime is much more rapid than previously realized.', '1410.2177-2-4-1': 'We draw this conclusion from a comparison of our measured scattering cross section for neutrons with energies between 0.43 meV to 16.1 meV in liquid hydrogen at a temperature of 15.6 K[MATH]0.6 K with previous data [CITATION][CITATION][CITATION].', '1410.2177-2-4-2': 'This measurement was conducted at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory using a 16-liter liquid hydrogen target [CITATION] operated on the Fundamental Neutron Physics Beamline [CITATION] (FnPB) by the NPDGamma collaboration.', '1410.2177-2-5-0': 'Figure [REF] shows the experimental setup for the transmission measurement on the FnPB beamline at the SNS.', '1410.2177-2-5-1': 'The cross sectional area of the neutron beam is 12[MATH]10 cm[MATH] at the exit of the neutron guide.', '1410.2177-2-5-2': 'Neutrons then pass through the normalization monitor, a multi-wire proportional counter [CITATION] with a gas mixture of [MATH]He (15.1 Torr) and N[MATH] (750 Torr) located 15.24 m [MATH] 0.12 m from the moderator, and are incident on the 16 liter liquid hydrogen target [CITATION] centered 17.6 m from the moderator.', '1410.2177-2-5-3': 'Roughly 60% of the neutron beam is captured on the hydrogen in the target with the 2.2 MeV capture gammas relevant to the NPDGamma experiment detected by 48 cesium iodide crystals.', '1410.2177-2-5-4': 'The effective length of liquid hydrogen covered by the neutron beam cross sectional area including beam divergence is 30.065 cm [MATH] 0.005 cm when cold.', '1410.2177-2-5-5': 'The transmitted neutrons exit the downstream end of the target vessel through a 2.5 cm diameter aperture in the [MATH]Li-rich neutron absorber surrounding the target.', '1410.2177-2-5-6': 'The transmitted neutron intensity is measured in a [MATH]He plate ion chamber [CITATION] located 3.44 m [MATH] 0.02 m from the normalization monitor.', '1410.2177-2-5-7': 'The charge produced in the monitors is amplified by current-voltage amplifiers [CITATION] with a 10 kHz bandwidth.', '1410.2177-2-6-0': 'The data acquisition system records data in 0.4 ms increments.', '1410.2177-2-6-1': 'In order to avoid contamination from overlapping neutron pulses and to increase the dynamic range of neutron energies for the transmission measurement, data were taken with the two beamline choppers parked open while the SNS was operating at 10 Hz duty cycle rather than the normal 60 Hz.', '1410.2177-2-7-0': 'The target vessel is initially filled with hydrogen gas, corresponding to 3 orthohydrogen molecules per parahydrogen molecule from equipartition.', '1410.2177-2-7-1': 'The equilibrium parahydrogen concentration increases with decreasing temperature [CITATION].', '1410.2177-2-7-2': 'The slow natural conversion to parahydrogen is accelerated by circulation of the liquid through 150 mL of hydrous iron (III) oxide [CITATION] 30 - 50 mesh powder catalyst in the ortho-para converter (OPC) in the NPDGamma target loop (Fig. [REF]).', '1410.2177-2-7-3': 'The neutron transmission increases with time as hydrogen circulates through the catalyst until a steady-state condition is reached.', '1410.2177-2-7-4': 'Fitting the transmission for 3.42 meV neutrons to an exponential as a function of time (Fig. [REF]) indicates that the parahydrogen concentration in the main target vessel approaches saturation.', '1410.2177-2-7-5': 'This exponential approach to the steady-state condition implies that the conversion is dominated by the first order processes in the OPC as liquid hydrogen circulates through the catalyst.', '1410.2177-2-8-0': 'The conversion process shown in figure [REF] has reached steady-state, where the parahydrogen concentration is near the thermal equilibrium value defined by the temperature of the OPC.', '1410.2177-2-8-1': 'The average temperature of the OPC was 15.4 K[MATH]0.5 K, which corresponds to a thermal equilibrium parahydrogen concentration of 0.99985.', '1410.2177-2-8-2': 'A small amount of para-to-ortho conversion may take place in the liquid in the main vessel, on the walls of the vessel, or the walls of the circulation loop that prevents reaching absolute thermal equilibrium.', '1410.2177-2-8-3': 'Since para-to-ortho conversion is known to be a very slow process it is not expected to limit the ortho-para ratio in the liquid hydrogen, and the liquid hydrogen is expected to be in thermal equilibrium with the catalyst.', '1410.2177-2-8-4': 'However, it was not possible to independently confirm the parahydrogen concentration in this system.', '1410.2177-2-9-0': 'Two different measurements were required in order to measure the empty target and full target transmissions.', '1410.2177-2-9-1': 'The full target measurement was performed over 8 hours with the target vessel at 15.6 K after the target had been in steady-state operation for 4 weeks, which corresponds to 30 conversion time constants.', '1410.2177-2-9-2': 'The empty target measurement was performed 2 weeks later with the target vessel at 16.3 K in order to cancel the temperature dependence of scattering from the aluminum target vessel.', '1410.2177-2-9-3': 'Between these two measurements, the moderator viewed by the beamline was emptied and refilled with fresh liquid hydrogen, which led to a small change in the moderated neutron spectrum between the two measurements.', '1410.2177-2-9-4': 'A neutron energy dependent correction was applied to the transmission monitor signals using the ratio of the normalization monitor signals to account for this systematic effect.', '1410.2177-2-9-5': 'For each neutron pulse, the signals in each monitor are normalized to the per-pulse beam power by integrating the normalization monitor over peak signal range.', '1410.2177-2-10-0': 'There is not a direct correspondence between time of flight bins in each monitor due to time of flight broadening.', '1410.2177-2-10-1': 'The normalization monitor signal is fit to a cubic spline in order to interpolate for spectrum normalization.', '1410.2177-2-10-2': 'The sharp dips in the pulse shapes in figure [REF] are due to Bragg scattering on aluminum windows along the path of the neutron beam.', '1410.2177-2-10-3': 'These dips are visible at neutron energies of 4.98 meV and 3.74 meV, corresponding to the aluminum (200) and (111) Bragg planes [CITATION], respectively.', '1410.2177-2-10-4': 'The time of flight of the Bragg edges for the normalization monitor is used to determine the distance from the moderator and to convert each time of flight bin to neutron energy.', '1410.2177-2-10-5': 'The uncertainty in time due to these Bragg edges is 0.16 ms, which determines the uncertainty in the normalization monitor position.', '1410.2177-2-10-6': 'The target-full spectrum indicates no measurable neutron flux for energies above 14.5 meV (Fig. [REF]).', '1410.2177-2-10-7': 'This is the minimum energy necessary for the [MATH] spin-flip transition, meaning neutrons with energies above this threshold are scattered out of the beam rather than transmitted through the target.', '1410.2177-2-10-8': 'The data also contain a 240 Hz noise component with an amplitude of a few millivolts.', '1410.2177-2-10-9': 'The amplitude is diminished by averaging pulses over the measurement period and is only visible for small signals.', '1410.2177-2-10-10': 'The transmission monitor signals at long wavelengths are fit to a sinusoidal function corresponding to the 240 Hz noise.', '1410.2177-2-10-11': 'The sinusoidal function is subtracted before extracting the transmission.', '1410.2177-2-10-12': 'After correcting for the pedestal, 240 Hz noise, and moderator spectrum, the final corrected transmission (Fig. [REF]) is given by [EQUATION] where the [MATH] values are monitor signals and [MATH] are monitor gain adjustment factors.', '1410.2177-2-11-0': 'The contamination of the transmission signal by non-forward small angle neutron scattering in our geometry was estimated to be less than 0.1% in MCNPX [CITATION] using the ENDF-VII thermal cross sections [CITATION].', '1410.2177-2-11-1': 'The total cross section can then be written as: [EQUATION] where [MATH] is the number density, [MATH] is the hydrogen length, [MATH] is the parahydrogen fraction, [MATH] barns at 2200m/s [CITATION], [MATH] is the total scattering cross section, [MATH] is the orthohydrogen scattering cross section, and [MATH] is the parahydrogen scattering cross section.', '1410.2177-2-12-0': 'The diode temperature sensors have an accuracy of 0.5 K and upward drift due to radiation damage is not worse than 0.3 K, providing a total uncertainty on the temperature of 0.6 K.', '1410.2177-2-12-1': 'The density of the liquid hydrogen in our target is determined from a fit to data compilations of the density of liquid hydrogen as a function of temperature from many sources [CITATION][CITATION][CITATION].', '1410.2177-2-12-2': 'The transmission data include several instrumental effects such as the monitor efficiency, the monitor dead layer, and monitor linearity.', '1410.2177-2-12-3': 'These effects all cancel in equation [REF] as long as the monitors and preamplifiers are linear and the aluminum components of the experiment were maintained at the same temperature.', '1410.2177-2-12-4': 'The linearity of the transmission monitor was determined from a scan of the bias voltage in order to reduce volume recombination effects in the chambers, with a resulting uncertainty of 0.15 for each monitor.', '1410.2177-2-12-5': 'Controlled current injection was used to measure the linearity of preamplifiers and the gain shift, which are 0.01 and 0.1 respectively.', '1410.2177-2-13-0': 'We have determined the total cross section for liquid hydrogen at 15.6 K from approximately 0.43 meV to 16.1 meV with an uncertainty of approximately 1, or 0.02 barn/atom over the majority of the measurement range (Fig. [REF]).', '1410.2177-2-13-1': 'Because the absorption cross section is well known, we are also able to determine a measurement band for the parahydrogen scattering cross section at these energies.', '1410.2177-2-13-2': 'This measurement band is much smaller than the values previously reported in the literature (Fig. [REF]) [CITATION][CITATION][CITATION], with the Seiffert cross section predicting a transmission for our apparatus that is 2% less than was measured at the lowest energies.', '1410.2177-2-13-3': 'Furthermore, we can set an upper limit on the orthohydrogen concentration in our apparatus by attributing all of the scattering at 0.8 meV to orthohydrogen, which results in an upper limit on the orthohydrogen concentration of 0.0015 using the ENDF-VII orthohydrogen cross section.', '1410.2177-2-13-4': 'At the lowest energies, we cannot distinguish the parahydrogen cross section from zero, however, we can put a band on the parahydrogen cross section at higher energies.', '1410.2177-2-13-5': 'The central value corresponds to the parahydrogen concentration given thermodynamic equilibrium in the OPC, 0.99985.', '1410.2177-2-13-6': 'The upper error bar on this central value is determined by the uncertainties presented in table [REF] and is dominated by the temperature and the time of flight.', '1410.2177-2-13-7': 'The lower error bar is determined by the orthohydrogen upper limit and is determined by the orthohydrogen cross section from ENDF-VII scaled by a factor of 0.0015.', '1410.2177-2-14-0': 'The measurement of the parahydrogen scattering cross section is very sensitive to the orthohydrogen fraction in the target volume because the orthohydrogen cross section is approximately a factor of 50 greater than for parahydrogen.', '1410.2177-2-14-1': 'The parahydrogen scattering cross section from this work along with the Seiffert [CITATION] data and the ENDF-VII parahydrogen kernel evaluated at 20 K [CITATION] are compared in Fig. [REF].', '1410.2177-2-14-2': 'The significant difference in magnitude suggests the presence of unaccounted for orthohydrogen contamination in previous experiments.', '1410.2177-2-14-3': 'Subtraction of an admixture of 0.5% orthohydrogen from Seiffert data brings both results into agreement.', '1410.2177-2-15-0': 'The Squires measurement [CITATION] was performed using a gas mixture with a parahydrogen concentration of 0.9979, which was independently measured using thermal conductivity.', '1410.2177-2-15-1': 'The Seiffert [CITATION] and Celli [CITATION] measurements were both performed using liquid hydrogen in the presence of a catalyst; however, neither experiment independently determined the orthohydrogen concentration but rather inferred that it was either negligible, in the case of Seiffert, or at thermal equilibrium, in the case of Celli.', '1410.2177-2-15-2': 'We therefore treat both the Seiffert and Celli measurements as upper limits.', '1410.2177-2-15-3': 'We conclude that our target system must have less orthohydrogen contamination than these previous two measurements because our observed total cross section is lower.', '1410.2177-2-15-4': 'Of these three measurements in the literature and the measurement in this work, we believe that our measurement has the lowest orthohydrogen contamination and that it provides the most accurate measurement of the liquid parahydrogen scattering cross section.', '1410.2177-2-16-0': 'These results have important implications for the design of slow neutron sources.', '1410.2177-2-16-1': 'Recent simulation work conducted for the European Spallation Source project [CITATION], indicates increased source intensity from liquid parahydrogen neutron moderators incorporated into a realistic target-moderator geometry.', '1410.2177-2-16-2': 'Measurements at J-PARC [CITATION] and LANSCE [CITATION] also show that the moderator intensity for neutrons below 14.5 meV are highly dependent on the ortho/para ratio.', '1410.2177-2-16-3': 'Our work shows that the parahydrogen cross section has been previously overestimated throughout the slow neutron regime of interest.', '1410.2177-2-16-4': 'This overestimate reaches a factor of 3 at a neutron energy of 1 meV.', '1410.2177-2-16-5': 'The potential for increased slow neutron source intensity from liquid parahydrogen moderators is therefore greater than previously realized and impacts the optimal geometry of slow neutron moderators.', '1410.2177-2-16-6': 'In order to be able to take full advantage of this potential, however, it would be necessary to maintain the liquid in the parahydrogen state in the presence of the intense radiation environment accompanying an intense neutron source [CITATION].', '1410.2177-2-16-7': 'Liquid hydrogen target designs which employ active circulation of the hydrogen through a catalyst coupled with dedicated measurements of the parahydrogen fraction from a liquid hydrogen moderator operated in an intense radiation environment are needed to confirm this potential and demonstrate that it can be realized at an intense neutron source.', '1410.2177-2-17-0': 'We would like to thank Erik Iverson, Phillip Ferguson, Kenneth Herwig, and Franz Gallmeier for productive discussions and encouragement for this experiment as well as Michael Mendenhall for thoughtful observations.', '1410.2177-2-17-1': 'We also thank the management and staff of the Spallation Neutron Source for adapting our measurement to the busy beam delivery schedule.', '1410.2177-2-17-2': 'We gratefully acknowledge the support of the U.S. Department of Energy Office of Nuclear Physics (including Grant No. DE-FG02-03ER41258), the National Science Foundation (including Grant No. PHY-1068712), PAPIIT-UNAM (Grant No. IN111913), and the Indiana University Center for Spacetime Symmetries.'} | [['1410.2177-1-8-0', '1410.2177-2-8-0'], ['1410.2177-1-8-1', '1410.2177-2-8-1'], ['1410.2177-1-8-2', '1410.2177-2-8-2'], ['1410.2177-1-8-3', '1410.2177-2-8-3'], ['1410.2177-1-8-4', '1410.2177-2-8-4'], ['1410.2177-1-7-1', '1410.2177-2-7-1'], ['1410.2177-1-7-2', '1410.2177-2-7-2'], ['1410.2177-1-7-3', '1410.2177-2-7-3'], ['1410.2177-1-7-4', '1410.2177-2-7-4'], ['1410.2177-1-7-5', '1410.2177-2-7-5'], ['1410.2177-1-16-0', '1410.2177-2-16-0'], ['1410.2177-1-16-2', '1410.2177-2-16-1'], ['1410.2177-1-16-5', '1410.2177-2-16-4'], ['1410.2177-1-16-8', '1410.2177-2-16-7'], ['1410.2177-1-15-0', '1410.2177-2-15-0'], ['1410.2177-1-15-1', '1410.2177-2-15-1'], ['1410.2177-1-15-3', '1410.2177-2-15-3'], 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['1410.2177-1-9-3', '1410.2177-2-9-3'], ['1410.2177-1-4-1', '1410.2177-2-4-1'], ['1410.2177-1-1-1', '1410.2177-2-1-1'], ['1410.2177-1-17-0', '1410.2177-2-17-0'], ['1410.2177-1-17-1', '1410.2177-2-17-1'], ['1410.2177-1-14-1', '1410.2177-2-14-1'], ['1410.2177-1-14-2', '1410.2177-2-14-2'], ['1410.2177-1-10-4', '1410.2177-2-10-4'], ['1410.2177-1-12-3', '1410.2177-2-12-3'], ['1410.2177-2-0-1', '1410.2177-3-0-6'], ['1410.2177-2-0-3', '1410.2177-3-0-2'], ['1410.2177-2-3-1', '1410.2177-3-3-4']] | [] | [['1410.2177-1-16-1', '1410.2177-2-16-5'], ['1410.2177-1-16-6', '1410.2177-2-16-5'], ['1410.2177-1-15-2', '1410.2177-2-15-2'], ['1410.2177-1-0-2', '1410.2177-2-0-2'], ['1410.2177-1-13-1', '1410.2177-2-13-1'], ['1410.2177-1-13-2', '1410.2177-2-13-2'], ['1410.2177-2-0-4', '1410.2177-3-0-4'], ['1410.2177-2-0-5', '1410.2177-3-0-7'], ['1410.2177-2-4-0', '1410.2177-3-3-8'], ['1410.2177-2-4-1', '1410.2177-3-3-0']] | [] | ['1410.2177-1-17-2', '1410.2177-2-17-2', '1410.2177-3-19-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1410.2177 | {'1410.2177-3-0-0': 'Liquid hydrogen is a dense Bose fluid whose equilibrium properties are both calculable from first principles using various theoretical approaches and of interest for the understanding of a wide range of questions in many body physics.', '1410.2177-3-0-1': 'Unfortunately, the pair correlation function [MATH] inferred from neutron scattering measurements of the differential cross section [MATH] from different measurements reported in the literature are inconsistent.', '1410.2177-3-0-2': 'We have measured the energy dependence of the total cross section and the scattering cross section for slow neutrons with energies between 0.43 meV and 16.1 meV on liquid hydrogen at 15.6 K (which is dominated by the parahydrogen component) using neutron transmission measurements on the hydrogen target of the NPDGamma collaboration at the Spallation Neutron Source at Oak Ridge National Laboratory.', '1410.2177-3-0-3': 'The relationship between the neutron transmission measurement we perform and the total cross section is unambiguous, and the energy range accesses length scales where the pair correlation function is rapidly varying.', '1410.2177-3-0-4': 'At 1 meV our measurement is a factor of 3 below the data from previous work.', '1410.2177-3-0-5': 'We present evidence that these previous measurements of the hydrogen cross section, which assumed that the equilibrium value for the ratio of orthohydrogen and parahydrogen has been reached in the target liquid, were in fact contaminated with an extra non-equilibrium component of orthohydrogen.', '1410.2177-3-0-6': 'Liquid parahydrogen is also a widely-used neutron moderator medium, and an accurate knowledge of its slow neutron cross section is essential for the design and optimization of intense slow neutron sources.', '1410.2177-3-0-7': 'We describe our measurements and compare them with previous work.', '1410.2177-3-1-0': 'The physics of liquid hydrogen is of fundamental importance in quantum many body theory.', '1410.2177-3-1-1': 'It is one of the few examples of a dense Bose fluid available for experimental investigation, and it exhibits behavior which interpolates between dense classical liquids and quantum liquids with Bose condensation such as superfluid helium [CITATION].', '1410.2177-3-1-2': 'Our ability to understand the physics of this liquid at experimentally-accessible densities and temperatures is important for scientists trying to extrapolate this understanding to predict the properties of the interiors of heavy planets like Jupiter [CITATION].', '1410.2177-3-1-3': 'Reliable computational extrapolation to these conditions is thought to require accurate determination of thermodynamic properties of condensed hydrogen at the 1% level [CITATION].', '1410.2177-3-1-4': 'Metallic hydrogen is also a model system for understanding the metal-insulator transition [CITATION].', '1410.2177-3-1-5': 'Accurate calculations of the properties of liquid hydrogen using theoretical approaches such as Path Integral Monte Carlo (PIMC) and Correlated Density Matrix (CDM) techniques are available [CITATION] based on well-established input on hydrogen intermolecular potentials such as the Silvera-Goldman potential [CITATION] and the NWB intermolecular potential [CITATION].', '1410.2177-3-2-0': 'It is therefore disturbing that such a fundamental structural property of liquid hydrogen as the pair correlation function [MATH] (and its Fourier transform partner the static structure factor [MATH]) is not yet well determined experimentally.', '1410.2177-3-2-1': 'The small electron density makes a measurement using X-rays somewhat difficult.', '1410.2177-3-2-2': 'Data on neutron scattering from molecular hydrogen using slow neutrons has been used in the past to help determine [MATH].', '1410.2177-3-2-3': 'In the slow neutron regime the interference scattering from neighboring molecules in the liquid probes a critical region of length scales where the pair correlation function [MATH] is rapidly varying.', '1410.2177-3-2-4': 'Unfortunately, neutron scattering experiments which measure the differential cross section [MATH] and attempt to extract [MATH] [CITATION] are in disagreement.', '1410.2177-3-2-5': 'In neutron measurements, the light mass of the hydrogen gives a larger than usual inelastic contribution to the scattering, and large corrections to the scattering data need to be applied in an attempt to extract [MATH].', '1410.2177-3-3-0': 'In this work we present a new measurement of the energy dependence of the total cross section (and, after subtraction of the well-known neutron-proton absorption cross section, the total scattering cross section) in the slow neutron regime using neutrons with energies between 0.43 meV to 16.1 meV in liquid hydrogen at a temperature of 15.6 K[MATH]0.6 K.', '1410.2177-3-3-1': 'This measurement was conducted at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory using a 16-liter liquid hydrogen target [CITATION] operated on the Fundamental Neutron Physics Beamline [CITATION] (FnPB) by the NPDGamma collaboration.', '1410.2177-3-3-2': 'One of the advantages of the sensitive transmission measurement as a function of neutron energy which we report here is that there is no ambiguity in the extraction of the energy dependence of the total cross section [MATH].', '1410.2177-3-3-3': 'It should therefore be possible to make a more robust comparison of this data with theory.', '1410.2177-3-3-4': 'Recall that molecular hydrogen has two spin states, labeled orthohydrogen ([MATH]) and parahydrogen ([MATH]).', '1410.2177-3-3-5': 'The lowest orthohydrogen state ([MATH]) lies 14.5 meV above the lowest parahydrogen state ([MATH]).', '1410.2177-3-3-6': 'The spin singlet state of the protons in the parahydrogen molecule combined with the measured spin dependence of s-wave neutron-proton scattering amplitudes conspire to greatly suppress the total scattering cross section for neutrons on parahydrogen molecules by more than one order of magnitude relative to that from the hydrogen atom.', '1410.2177-3-3-7': 'The total scattering cross section on orthohydrogen is approximately 50 times higher than on parahydrogen (Fig. [REF]) because the destructive interference between the atoms is absent.', '1410.2177-3-3-8': 'A comparison of our results with previous data [CITATION][CITATION][CITATION] indicates that the decrease of the total scattering cross section in liquid parahydrogen in the slow neutron regime is much more rapid than previously realized.', '1410.2177-3-4-0': 'In addition to the usefulness of this new data for extraction of the pair correlation function in liquid parahydrogen, our results are also of immediate practical interest for slow neutron source development.', '1410.2177-3-4-1': 'The successful development of intense slow neutron sources combined with the increasing phase space acceptance of neutron optical components has enabled a dramatic expansion of the scientific applications of neutron scattering to encompass many fields in science and technology.', '1410.2177-3-4-2': 'The broad applicability of quantitative information that slow neutron scattering can provide on the internal structure and dynamics of condensed media has motivated the construction of several new neutron scattering facilities over the last decade.', '1410.2177-3-4-3': 'The efficiency of the moderating medium which accepts the relatively high energy neutrons liberated from the nucleus and cools them to the slow neutron energy range below [MATH] meV determines the phase space density of the neutron beams.', '1410.2177-3-4-4': 'New results on physics relevant to the moderation process are therefore of interest to a very broad range of the scientific community.', '1410.2177-3-5-0': 'Many intense neutron sources use liquid hydrogen as a neutron moderator medium.', '1410.2177-3-5-1': 'The near-equality of the neutron and proton mass coupled with the anomalously large s-wave neutron-proton scattering amplitude allow a hydrogen-rich medium to both efficiently lower the incident neutron energy through collisions and also maintain a small neutron mean free path to keep the neutron phase space density high at the source.', '1410.2177-3-5-2': 'In the slow neutron regime, however, the neutron scattering cross section and therefore the mean free path is sensitive to the interference of the scattering amplitudes from neighboring atoms.', '1410.2177-3-5-3': 'A neutron that scatters from orthohydrogen will be upscattered and gain 14.5 meV, reducing the slow neutron intensity below 14.5 meV from an orthohydrogen rich moderator.', '1410.2177-3-5-4': 'Consequently, many studies have shown that the slow neutron intensity from a liquid hydrogen moderator can be greatly increased if the molecules are maintained in the parahydrogen molecular state [CITATION][CITATION][CITATION].', '1410.2177-3-5-5': 'While absorption ultimately limits the intensity for energies below 2 meV, it is the relative concentrations of orthohydrogen and parahydrogen that is the lever arm available for optimizing the properties of slow neutron moderators.', '1410.2177-3-5-6': 'Our new results, which show that the neutron scattering cross section from liquid parahydrogen seems to have been overestimated in previous work by as much as a factor of 3 at an energy of 1 meV, is therefore of immediate interest for the designers of bright slow neutron sources.', '1410.2177-3-6-0': 'Differences of the orthohydrogen fraction from that corresponding to thermodynamic equilibrium are an obvious culprit for the disagreement among different measurements of neutron-parahydrogen scattering.', '1410.2177-3-6-1': 'Given the huge orthohydrogen neutron cross section combined with the well-known difficulty of achieving the proportions of parahydrogen and orthohydrogen in the liquid corresponding to thermodynamic equilibrium, one might be concerned about how closely the nominally liquid parahydrogen samples employed in previous measurements have approached the conditions of thermodynamic equilibrium.', '1410.2177-3-6-2': 'Our liquid hydrogen target possessed not only an ortho-para convertor but also a thermosyphon mechanism which forced all of the liquid to pass through the catalyst many times before the neutron transmission measurements were conducted and at a slow but continuous rate during the measurement.', '1410.2177-3-6-3': 'We can take the functional form of our measured neutron cross section as a function of neutron energy on liquid parahydrogen and reproduce previous cross section data by adding to it an extra component of orthohydrogen scattering using the measured neutron energy dependence of scattering on orthohydrogen.', '1410.2177-3-6-4': 'This result strongly suggests to us that the nominally liquid parahydrogen samples used in previous total cross section measurements in fact possessed higher residual orthohydrogen contamination than expected based on thermodynamic equilibrium.', '1410.2177-3-7-0': 'Figure [REF] shows the experimental setup for the transmission measurement on the FnPB beamline at the SNS.', '1410.2177-3-7-1': 'The cross sectional area of the neutron beam is 12[MATH]10 cm[MATH] at the exit of the neutron guide.', '1410.2177-3-7-2': 'Neutrons then pass through the normalization monitor, a multi-wire proportional counter [CITATION] with a gas mixture of [MATH]He (15.1 Torr) and N[MATH] (750 Torr) located 15.24 m [MATH] 0.12 m from the moderator, and are incident on the 16 liter liquid hydrogen target [CITATION] centered 17.6 m from the moderator.', '1410.2177-3-7-3': 'Roughly 60% of the neutron beam is captured on the hydrogen in the target with the 2.2 MeV capture gammas relevant to the NPDGamma experiment detected by 48 cesium iodide crystals.', '1410.2177-3-7-4': 'The effective length of liquid hydrogen covered by the neutron beam cross sectional area including beam divergence is 30.065 cm [MATH] 0.005 cm when cold.', '1410.2177-3-7-5': 'The transmitted neutrons exit the downstream end of the target vessel through a 2.5 cm diameter aperture in the [MATH]Li-rich neutron absorber surrounding the target.', '1410.2177-3-7-6': 'The transmitted neutron intensity is measured in a [MATH]He plate ion chamber [CITATION] located 3.44 m [MATH] 0.02 m from the normalization monitor.', '1410.2177-3-7-7': 'The charge produced in the monitors is amplified by current-voltage amplifiers [CITATION] with a 10 kHz bandwidth.', '1410.2177-3-8-0': 'The data acquisition system records data in 0.4 ms increments.', '1410.2177-3-8-1': 'In order to avoid contamination from overlapping neutron pulses and to increase the dynamic range of neutron energies for the transmission measurement, data were taken with the two beamline choppers parked open while the SNS was operating at 10 Hz duty cycle rather than the normal 60 Hz.', '1410.2177-3-9-0': 'The target vessel is initially filled with hydrogen gas, corresponding to 3 orthohydrogen molecules per parahydrogen molecule from equipartition.', '1410.2177-3-9-1': 'The equilibrium parahydrogen concentration increases with decreasing temperature [CITATION].', '1410.2177-3-9-2': 'The slow natural conversion to parahydrogen is accelerated by circulation of the liquid through 150 mL of hydrous iron (III) oxide [CITATION] 30 - 50 mesh powder catalyst in the ortho-para converter (OPC) in the NPDGamma target loop (Fig. [REF]).', '1410.2177-3-9-3': 'The neutron transmission increases with time as hydrogen circulates through the catalyst until a steady-state condition is reached.', '1410.2177-3-9-4': 'Fitting the transmission for 3.42 meV neutrons to an exponential as a function of time (Fig. [REF]) indicates that the parahydrogen concentration in the main target vessel approaches saturation.', '1410.2177-3-9-5': 'This exponential approach to the steady-state condition implies that the conversion is dominated by the first order processes in the OPC as liquid hydrogen circulates through the catalyst.', '1410.2177-3-10-0': 'The conversion process shown in figure [REF] has reached steady-state, where the parahydrogen concentration is near the thermal equilibrium value defined by the temperature of the OPC.', '1410.2177-3-10-1': 'The average temperature of the OPC was 15.4 K[MATH]0.5 K, which corresponds to a thermal equilibrium parahydrogen concentration of 0.99985.', '1410.2177-3-10-2': 'A small amount of para-to-ortho conversion may take place in the liquid in the main vessel, on the walls of the vessel, or the walls of the circulation loop that prevents reaching absolute thermal equilibrium.', '1410.2177-3-10-3': 'Since para-to-ortho conversion is known to be a very slow process it is not expected to limit the ortho-para ratio in the liquid hydrogen, and the liquid hydrogen is expected to be in thermal equilibrium with the catalyst.', '1410.2177-3-10-4': 'However, it was not possible to independently confirm the parahydrogen concentration in this system.', '1410.2177-3-11-0': 'Two different measurements were required in order to measure the empty target and full target transmissions.', '1410.2177-3-11-1': 'The full target measurement was performed over 8 hours with the target vessel at 15.6 K after the target had been in steady-state operation for 4 weeks, which corresponds to 30 conversion time constants.', '1410.2177-3-11-2': 'The empty target measurement was performed 2 weeks later with the target vessel at 16.3 K in order to cancel the temperature dependence of scattering from the aluminum target vessel.', '1410.2177-3-11-3': 'Between these two measurements, the moderator viewed by the beamline was emptied and refilled with fresh liquid hydrogen, which led to a small change in the moderated neutron spectrum between the two measurements.', '1410.2177-3-11-4': 'A neutron energy dependent correction was applied to the transmission monitor signals using the ratio of the normalization monitor signals to account for this systematic effect.', '1410.2177-3-11-5': 'For each neutron pulse, the signals in each monitor are normalized to the per-pulse beam power by integrating the normalization monitor over peak signal range.', '1410.2177-3-12-0': 'There is not a direct correspondence between time of flight bins in each monitor due to time of flight broadening.', '1410.2177-3-12-1': 'The normalization monitor signal is fit to a cubic spline in order to interpolate for spectrum normalization.', '1410.2177-3-12-2': 'The sharp dips in the pulse shapes in figure [REF] are due to Bragg scattering on aluminum windows along the path of the neutron beam.', '1410.2177-3-12-3': 'These dips are visible at neutron energies of 4.98 meV and 3.74 meV, corresponding to the aluminum (200) and (111) Bragg planes [CITATION], respectively.', '1410.2177-3-12-4': 'The time of flight of the Bragg edges for the normalization monitor is used to determine the distance from the moderator and to convert each time of flight bin to neutron energy.', '1410.2177-3-12-5': 'The uncertainty in time due to these Bragg edges is 0.16 ms, which determines the uncertainty in the normalization monitor position.', '1410.2177-3-12-6': 'The target-full spectrum indicates no measurable neutron flux for energies above 14.5 meV (Fig. [REF]).', '1410.2177-3-12-7': 'This is the minimum energy necessary for the [MATH] spin-flip transition, meaning neutrons with energies above this threshold are scattered out of the beam rather than transmitted through the target.', '1410.2177-3-12-8': 'The data also contain a 240 Hz noise component with an amplitude of a few millivolts.', '1410.2177-3-12-9': 'The amplitude is diminished by averaging pulses over the measurement period and is only visible for small signals.', '1410.2177-3-12-10': 'The transmission monitor signals at long wavelengths are fit to a sinusoidal function corresponding to the 240 Hz noise.', '1410.2177-3-12-11': 'The sinusoidal function is subtracted before extracting the transmission.', '1410.2177-3-12-12': 'After correcting for the pedestal, 240 Hz noise, and moderator spectrum, the final corrected transmission (Fig. [REF]) is given by [EQUATION] where the [MATH] values are monitor signals and [MATH] are monitor gain adjustment factors.', '1410.2177-3-13-0': 'The contamination of the transmission signal by non-forward small angle neutron scattering in our geometry was estimated to be less than 0.1% in MCNPX [CITATION] using the ENDF-VII thermal cross sections [CITATION].', '1410.2177-3-13-1': 'The total cross section can then be written as: [EQUATION] where [MATH] is the number density, [MATH] is the hydrogen length, [MATH] is the parahydrogen fraction, [MATH] barns at 2200m/s [CITATION], [MATH] is the total scattering cross section, [MATH] is the orthohydrogen scattering cross section, and [MATH] is the parahydrogen scattering cross section.', '1410.2177-3-14-0': 'The diode temperature sensors have an accuracy of 0.5 K and upward drift due to radiation damage is not worse than 0.3 K, providing a total uncertainty on the temperature of 0.6 K.', '1410.2177-3-14-1': 'The density of the liquid hydrogen in our target is determined from a fit to data compilations of the density of liquid hydrogen as a function of temperature from many sources [CITATION][CITATION][CITATION].', '1410.2177-3-14-2': 'The transmission data include several instrumental effects such as the monitor efficiency, the monitor dead layer, and monitor linearity.', '1410.2177-3-14-3': 'These effects all cancel in equation [REF] as long as the monitors and preamplifiers are linear and the aluminum components of the experiment were maintained at the same temperature.', '1410.2177-3-14-4': 'The linearity of the transmission monitor was determined from a scan of the bias voltage in order to reduce volume recombination effects in the chambers, with a resulting uncertainty of 0.15 for each monitor.', '1410.2177-3-14-5': 'Controlled current injection was used to measure the linearity of preamplifiers and the gain shift, which are 0.01 and 0.1 respectively.', '1410.2177-3-15-0': 'We have determined the total cross section for liquid hydrogen at 15.6 K from approximately 0.43 meV to 16.1 meV with an uncertainty of approximately 1, or 0.02 barn/atom over the majority of the measurement range (Fig. [REF]).', '1410.2177-3-15-1': 'Because the absorption cross section is well known, we are also able to determine a measurement band for the parahydrogen scattering cross section at these energies.', '1410.2177-3-15-2': 'This measurement band is much smaller than the values previously reported in the literature (Fig. [REF]) [CITATION][CITATION][CITATION], with the Seiffert cross section predicting a transmission for our apparatus that is 2% less than was measured at the lowest energies.', '1410.2177-3-15-3': 'Furthermore, we can set an upper limit on the orthohydrogen concentration in our apparatus by attributing all of the scattering at 0.8 meV to orthohydrogen, which results in an upper limit on the orthohydrogen concentration of 0.0015 using the ENDF-VII orthohydrogen cross section.', '1410.2177-3-15-4': 'At the lowest energies, we cannot distinguish the parahydrogen cross section from zero, however, we can put a band on the parahydrogen cross section at higher energies.', '1410.2177-3-15-5': 'The central value corresponds to the parahydrogen concentration given thermodynamic equilibrium in the OPC, 0.99985.', '1410.2177-3-15-6': 'The upper error bar on this central value is determined by the uncertainties presented in table [REF] and is dominated by the temperature and the time of flight.', '1410.2177-3-15-7': 'The lower error bar is determined by the orthohydrogen upper limit and is determined by the orthohydrogen cross section from ENDF-VII scaled by a factor of 0.0015.', '1410.2177-3-16-0': 'The measurement of the parahydrogen scattering cross section is very sensitive to the orthohydrogen fraction in the target volume because the orthohydrogen cross section is approximately a factor of 50 greater than for parahydrogen.', '1410.2177-3-16-1': 'The parahydrogen scattering cross section from this work along with the Seiffert [CITATION] data and the ENDF-VII parahydrogen kernel evaluated at 20 K [CITATION] are compared in Fig. [REF].', '1410.2177-3-16-2': 'The significant difference in magnitude suggests the presence of unaccounted for orthohydrogen contamination in previous experiments.', '1410.2177-3-16-3': 'Subtraction of an admixture of 0.5% orthohydrogen from Seiffert data brings both results into agreement.', '1410.2177-3-17-0': 'The Squires measurement [CITATION] was performed using a gas mixture with a parahydrogen concentration of 0.9979, which was independently measured using thermal conductivity.', '1410.2177-3-17-1': 'The Seiffert [CITATION] and Celli [CITATION] measurements were both performed using liquid hydrogen in the presence of a catalyst; however, neither experiment independently determined the orthohydrogen concentration but rather inferred that it was either negligible, in the case of Seiffert, or at thermal equilibrium, in the case of Celli.', '1410.2177-3-17-2': 'We therefore treat both the Seiffert and Celli measurements as upper limits.', '1410.2177-3-17-3': 'We conclude that our target system must have less orthohydrogen contamination than these previous two measurements because our observed total cross section is lower.', '1410.2177-3-17-4': 'Of these three measurements in the literature and the measurement in this work, we believe that our measurement has the lowest orthohydrogen contamination and that it provides the most accurate measurement of the liquid parahydrogen scattering cross section.', '1410.2177-3-18-0': 'These results have important implications for the design of slow neutron sources.', '1410.2177-3-18-1': 'Recent simulation work conducted for the European Spallation Source project [CITATION], indicates increased source intensity from liquid parahydrogen neutron moderators incorporated into a realistic target-moderator geometry.', '1410.2177-3-18-2': 'Measurements at J-PARC [CITATION] and LANSCE [CITATION] also show that the moderator intensity for neutrons below 14.5 meV are highly dependent on the ortho/para ratio.', '1410.2177-3-18-3': 'Our work shows that the parahydrogen cross section has been previously overestimated throughout the slow neutron regime of interest.', '1410.2177-3-18-4': 'This overestimate reaches a factor of 3 at a neutron energy of 1 meV.', '1410.2177-3-18-5': 'The potential for increased slow neutron source intensity from liquid parahydrogen moderators is therefore greater than previously realized and impacts the optimal geometry of slow neutron moderators.', '1410.2177-3-18-6': 'In order to be able to take full advantage of this potential, however, it would be necessary to maintain the liquid in the parahydrogen state in the presence of the intense radiation environment accompanying an intense neutron source [CITATION].', '1410.2177-3-18-7': 'Liquid hydrogen target designs which employ active circulation of the hydrogen through a catalyst coupled with dedicated measurements of the parahydrogen fraction from a liquid hydrogen moderator operated in an intense radiation environment are needed to confirm this potential and demonstrate that it can be realized at an intense neutron source.', '1410.2177-3-19-0': 'We would like to thank Erik Iverson, Phillip Ferguson, Kenneth Herwig, and Franz Gallmeier for productive discussions and encouragement for this experiment as well as Michael Mendenhall for thoughtful observations.', '1410.2177-3-19-1': 'We also thank the management and staff of the Spallation Neutron Source for adapting our measurement to the busy beam delivery schedule.', '1410.2177-3-19-2': 'We gratefully acknowledge the support of the U.S. Department of Energy Office of Nuclear Physics (including Grant No. DE-FG02-03ER41258), the National Science Foundation (including Grant No. PHY-1068712), PAPIIT-UNAM (Grant No. IN111913), and the Indiana University Center for Spacetime Symmetries.'} | null | null | null | null |
1002.2603 | {'1002.2603-1-0-0': 'Recent advances in experimental techniques now permit to measure the Casimir force with unprecedented precision.', '1002.2603-1-0-1': 'In order to achieve a comparable precision in the theoretical prediction of the force, it is necessary to accurately determine the electric permittivity of the materials constituting the plates along the imaginary frequency axis.', '1002.2603-1-0-2': 'The latter quantity is not directly accessible to experiments, but it can be determined via dispersion relations from experimental optical data.', '1002.2603-1-0-3': 'In the experimentally important case of conductors, however, a serious drawback of the standard dispersion relations commonly used for this purpose, is their strong dependence on the chosen low-frequency extrapolation of the experimental optical data, which introduces a significant and not easily controllable uncertainty in the result.', '1002.2603-1-0-4': 'In this paper we show that a simple modification of the standard dispersion relations, involving suitable analytic window functions, resolves this difficulty, making it possible to reliably determine the electric permittivity at imaginary frequencies solely using experimental optical data in the frequency interval where they are available, without any need of uncontrolled data extrapolations.', '1002.2603-1-1-0': '# INTRODUCTION', '1002.2603-1-2-0': 'One of the most intriguing predictions of Quantum electrodynamics is the existence of irreducible vacuum fluctuations of the electromagnetic (e.m.) field.', '1002.2603-1-2-1': "It was Casimir's fundamental discovery [CITATION] to realize that this purely quantum phenomenon was not confined to the atomic scale, as in the Lamb shift, but would rather manifest itself also at the macroscopic scale, in the form of a force of attraction between two discharged plates.", '1002.2603-1-2-2': 'For the idealized case of two perfectly reflecting plane-parallel plates at zero temperature, placed at a distance [MATH] in vacuum, Casimir obtained the following remarkably simple estimate of the force per unit area [EQUATION]', '1002.2603-1-2-3': 'An important step forward was made a few years later by Lifshitz and co-workers [CITATION], who obtained a formula for the force between two homogeneous dielectric plane-parallel slabs, at finite temperature.', '1002.2603-1-2-4': 'In this theory, of macroscopic character, the material properties of the slabs were fully characterized in terms of the respective frequency dependent electric permittivities [MATH], accounting for the dispersive and dissipative properties of real materials.', '1002.2603-1-2-5': 'In this way, it was possible for the first time to investigate the influence of material properties on the magnitude of the Casimir force.', '1002.2603-1-3-0': 'Over ten years ago, a series of brilliant experiments [CITATION] exploiting modern experimental techniques provided the definitive demonstration of the Casimir effect.', '1002.2603-1-3-1': 'These now historical experiments spurred enormous interest in the Casimir effect, and were soon followed by many other experiments.', '1002.2603-1-3-2': 'The subsequent experiments were aimed at diverse objectives.', '1002.2603-1-3-3': 'Some of them explored new geometries: while the works [CITATION] used a sphere-plate setup, the original planar geometry investigated by Casimir was adopted in the experiment [CITATION], and a setup with crossed cylinders was considered in [CITATION].', '1002.2603-1-3-4': 'The important issue of the non trivial geometry dependence of the Casimir effect is also being pursued experimentally, using elaborate micro-patterned surfaces [CITATION].', '1002.2603-1-3-5': 'Other experiments aimed at demonstrating new possible uses of the Casimir force, like for example the actuation of micromachines [CITATION], or at demonstrating the possibility of a large modulation of the Casimir force [CITATION], which could also result in interesting technological applications.', '1002.2603-1-3-6': 'There are also experiments using superconducting Casimir cavities, that aim at measuring the change of the Casimir energy across the superconducting phase transition [CITATION].', '1002.2603-1-3-7': 'The experiments performed in the last ten years are just too numerous to mention them all here.', '1002.2603-1-3-8': 'For an updated account we refer the reader to the very recent review paper [CITATION].', '1002.2603-1-4-0': 'Apart from exploring new manifestations of the Casimir effect, a large experimental effort is presently being made also to increase the precision of Casimir force measurements, in simple geometries.', '1002.2603-1-4-1': 'Already in the early experiment [CITATION] a precision upto one percent was obtained.', '1002.2603-1-4-2': 'More recently, a series of experiments with microtorsional oscillators [CITATION] reached an amazing precision of 0.5 percent.', '1002.2603-1-4-3': 'The reader may wonder what is the interest in achieving such a high precision in this kind of experiments.', '1002.2603-1-4-4': 'There are several reasons why this is important.', '1002.2603-1-4-5': 'On one hand, in the theory of dispersion forces puzzling conceptual problems have recently emerged that are connected with the contribution of free charges to the thermal Casimir force, whose resolution crucially depends on the precision of the theory-experiment comparison [CITATION].', '1002.2603-1-4-6': 'On the other hand, the ability to accurately determine the Casimir force is also important for the purpose of obtaining stronger constraints on hypothetical long-range forces predicted by certain theoretical scenarios going beyond the Standard Model of particle physics [CITATION].', '1002.2603-1-5-0': 'The remarkable precision achieved in the most recent experiments poses a challenging demand on the theorist: is it possible to predict the magnitude of the Casimir force with a comparable level of precision, say of one percent?', '1002.2603-1-5-1': 'Assessing the theoretical error affecting present estimates of the Casimir force is a difficult problem indeed, because many different factors must be taken into account [CITATION].', '1002.2603-1-5-2': 'Consider the typical experimental setting of most of the current experiments, where the Casimir force is measured between two bodies covered with gold, placed in vacuum at a distance of a (few) hundred nanometers.', '1002.2603-1-5-3': 'In this separation range, the main factor to consider is the finite penetration depth of electromagnetic fields into the gold layer , resulting from the finite conductivity of gold.', '1002.2603-1-5-4': 'The tool to analyze the influence of such material properties as the conductivity on the Casimir effect is provided by Lifshitz theory [CITATION].', '1002.2603-1-5-5': 'This theory shows that for a separation of 100 nm, the finite conductivity of gold determines a reduction in the magnitude of the Casimir force of about fifty percent in comparison with the perfect metal result [CITATION].', '1002.2603-1-5-6': 'Much smaller corrections, that must nevertheless be considered if the force is to be estimated with percent precision, arise from the finite temperature of the plates and from their surface roughness.', '1002.2603-1-5-7': 'Moreover, geometric effects resulting from the actual shape of the plates should be considered.', '1002.2603-1-5-8': 'We should also mention that the magnitude of residual electrostatic forces between the plates, resulting from contact potentials and patch effects, must be carefully accounted for.', '1002.2603-1-5-9': 'For a discussion of all these issues, which received much attention in the recent literature on the Casimir effect, we again address the reader to Ref. [CITATION].', '1002.2603-1-5-10': 'See also the recent work [CITATION].', '1002.2603-1-6-0': 'In this paper, we focus our attention on the influence of the optical properties of the plates which, as explained above, is by far the most relevant factor to consider.', '1002.2603-1-6-1': 'As we pointed out earlier, in Lifshitz theory the optical properties of the plates enter via the frequency-dependent electric permittivity [MATH] of the material constituting plates.', '1002.2603-1-6-2': 'In order to obtain an accurate prediction of the force, it is therefore of the utmost importance to use accurate data for the electric permittivity.', '1002.2603-1-6-3': 'The common practice adopted in all recent Casimir experiments with gold surfaces is to use tabulated data for gold (most of the times those quoted in Refs. [CITATION]), suitably extrapolated at low frequencies, where optical data are not available, by the simple Drude model of ohmic conductors.', '1002.2603-1-6-4': 'However, already ten years ago Lamoreaux observed [CITATION] that using tabulated data to obtain an accurate prediction of the Casimir force may not be a reliable practice, since optical properties of gold films may vary significantly from sample to sample, depending on the conditions of deposition.', '1002.2603-1-6-5': 'The same author stressed the importance of measuring the optical data of the films actually used in the force measurements, in the frequency range that is relevant for the Casimir force.', '1002.2603-1-6-6': 'The importance of this point was further stressed in [CITATION] and received clear experimental support in a recent paper [CITATION], where the optical properties of several gold films of different thicknesses, and prepared by different procedures, were measured ellipsometrically in a wide range of wavelengths, from 0.14 to 33 microns, and it was found that the frequency dependent electric permittivity changes significantly from sample to sample.', '1002.2603-1-6-7': 'By using the zero-temperature Lifshitz formula, the authors estimated that the observed sample dependence of the electric permittivity implies a variation in the theoretical value of the Casimir force, from one sample to another, easily as large as ten percent, for separations around 100 nm.', '1002.2603-1-6-8': 'It was concluded that in order to achieve a theoretical accuracy better than ten percent in the prediction of the Casimir force, it is necessary to determine the optical properties of the films actually used in the experiment of interest.', '1002.2603-1-7-0': 'The aim of this paper is to improve the mathematical procedure that is actually needed to obtain reliable estimates of the Casimir force, starting from experimental optical data on the material of the plates, like those presented in Ref. [CITATION].', '1002.2603-1-7-1': 'The necessity of such an improvement stems from the very simple and unavoidable fact that experimental optical data are never available in the entire frequency domain, but are always restricted to a finite frequency interval [MATH].', '1002.2603-1-7-2': 'To see why this constitutes problem we recall that Lifshitz formula, routinely used to interpret current experiments, expresses the Casimir force between two parallel plates as an integral over imaginary frequencies [MATH] of a quantity involving the dielectric permittivities of the plates [MATH].', '1002.2603-1-7-3': 'For finite temperature, the continuous frequency integration is replaced by a sum over discrete so-called Matsubara frequencies [MATH], where [MATH], with [MATH] a non-negative integer, and [MATH] the temperature of the plates.', '1002.2603-1-7-4': 'In any case, whatever the temperature, one needs to evaluate the permittivity of the plates for certain imaginary frequencies.', '1002.2603-1-7-5': 'We note that, in principle, recourse to imaginary frequencies is not mandatory because it is possible to rewrite Lifshitz formula in a mathematically equivalent form, involving an integral over the real frequency axis.', '1002.2603-1-7-6': 'In this case however the integrand becomes a rapidly oscillating function of the frequency, which hampers any possibility of numerical evaluation.', '1002.2603-1-7-7': 'In practice, the real-frequency form of Lifshitz formula is never used, and only its imaginary-frequency version is considered.', '1002.2603-1-7-8': 'We remark that occurrence of imaginary frequencies in the expression of the Casimir force, is a general feature of all recent formalisms, extending Lifshitz theory to non-planar geometries [CITATION].', '1002.2603-1-7-9': 'The problem is that the electric permittivity [MATH] at imaginary frequencies cannot be measured directly by any experiment.', '1002.2603-1-7-10': 'The only way to determine it by means of dispersion relations, which allow to express [MATH] in terms of the observable real-frequency electric permittivity [MATH].', '1002.2603-1-7-11': 'In the standard version of dispersion relations [CITATION], adopted so far in all works on the Casimir effect, [MATH] is expressed in terms of an integral of a quantity involving the imaginary part [MATH] of the electric permittivity: [EQUATION]', '1002.2603-1-7-12': 'The above formula shows that, in principle, a determination of [MATH] requires knowledge of [MATH] at all frequencies while, as we said earlier, optical data are available only in some interval [MATH].', '1002.2603-1-7-13': 'In practice, the problem is not so serious on the high frequency side, because the fall-off properties of [MATH] at high frequencies, together with the [MATH] factor in the denominator of the integrand, ensure that the error made by truncating the integral at a suitably large frequency [MATH] is small, provided that [MATH] is large enough.', '1002.2603-1-7-14': 'Typically, an [MATH] larger than, say, [MATH], is good enough for practical purposes.', '1002.2603-1-7-15': 'Things are not so easy though on the low frequency side.', '1002.2603-1-7-16': 'In the case of insulators, optical data are typically available until frequencies [MATH] much smaller than the frequencies of all resonances of the medium.', '1002.2603-1-7-17': 'Because of this, [MATH] is almost zero for [MATH], and therefore the error made by truncating the integral at [MATH] is again negligible.', '1002.2603-1-7-18': 'Problems arise however in the case of ohmic conductors, because then [MATH] has a [MATH] singularity at [MATH].', '1002.2603-1-7-19': 'As a result [MATH] becomes extremely large at low frequencies, in such a way that the integral in Eq. ([REF]) receives a very large contribution from low frequencies.', '1002.2603-1-7-20': 'For typical values of [MATH] that can be reached in practice (for example for gold, the tabulated data in [CITATION] begin at [MATH] meV[MATH], while the data of [CITATION] start at 38 meV[MATH]) truncation of the integral at [MATH] results in a large error.', '1002.2603-1-7-21': 'The traditional remedy to this problem is to make some analytical extrapolation of the data, typically based on Drude model fits of the low-frequency region of data, from [MATH] to zero, and then use the extrapolation to estimate the contribution of the integral in the interval [MATH] where data are not directly available.', '1002.2603-1-7-22': 'It is important to observe that this contribution is usually very large.', '1002.2603-1-7-23': 'For example, even in the case of Ref. [CITATION], the relative contribution of the extrapolation is about fifty percent of the total value of the integral, in the entire range of imaginary frequencies that are needed for estimating the Casimir force.', '1002.2603-1-8-0': 'Clearly, this procedure is not very satisfying.', '1002.2603-1-8-1': 'The use of analytical extrapolations of the data introduces an uncertainty in the obtained values of [MATH], that is not easy to quantify.', '1002.2603-1-8-2': 'The result may in fact depend a lot on the form of the extrapolation, and there is no guarantee that the chosen expression is good enough.', '1002.2603-1-8-3': 'Consider for example Ref. [CITATION], which constitutes the most accurate existing work on this problem.', '1002.2603-1-8-4': 'It was found there that the simple Drude model does not fit so well the data of all samples, making it necessary to improve it by the inclusion of an additional Lorentz oscillator.', '1002.2603-1-8-5': 'Moreover, it was found that for each sample the Drude parameters extracted from the data depended on the used fitting procedure, and were inconsistent which each other within the estimated errors, which is again an indication of the probable inadequacy of the analytical expression chosen for the interpolation.', '1002.2603-1-8-6': 'This state of things led us to investigate if it possible to determine accurately [MATH] solely on the basis of available optical data, without making recourse to data extrapolations.', '1002.2603-1-8-7': 'We shall see below that this is indeed possible, provided that Eq. ([REF]) is suitably modified, in a way that involves multiplying the integrand by an appropriate analytical window function [MATH], which suppresses the contribution of frequencies not belonging to the interval [MATH].', '1002.2603-1-8-8': 'As a result of this modification, the error made by truncating the integral to the frequency range [MATH] can be made negligible at both ends of the integration domain, rendering unnecessary any extrapolation of the optical data outside the interval where they are available.', '1002.2603-1-8-9': 'The procedure outlined in this paper should allow to better evaluate the theoretical uncertainty of Casimir force estimates resulting from experimental errors in the optical data.', '1002.2603-1-9-0': 'The plan of the paper is as follows: in Sec. II we derive a generalized dispersion relation for [MATH], involving analytic window functions [MATH], and we provide a simple choice for the window functions.', '1002.2603-1-9-1': 'In Sec III we present the results of a numerical simulation of our window functions, for the experimentally relevant case of gold, and in Sec IV we estimate numerically the error on the Casimir pressure resulting from the use of our window functions.', '1002.2603-1-9-2': 'Sec IV contains our conclusions and a discussion of the results.', '1002.2603-1-10-0': '# Generalized dispersion relations with window-functions', '1002.2603-1-11-0': 'As it it well known [CITATION], analyticity properties satisfied by the electric permittivity [MATH] of any causal medium (and more in general by any causal response function, the magnetic permeability [MATH] being another example) imply certain integral relations between the real part [MATH] and imaginary part [MATH] of [MATH], known as Kramers-Kronig or dispersion relations.', '1002.2603-1-11-1': 'The dispersion relation of interest to us is the one that permits to express the value [MATH] of the response function at some imaginary frequency [MATH] in terms of an integral along the positive frequency axis, involving [MATH].', '1002.2603-1-11-2': 'It is convenient to briefly review here the simple derivation of this important result, which is an easy exercise in contour integration.', '1002.2603-1-11-3': "For our purposes, it is more convenient to start from an arbitrary complex function [MATH], with the following properties: [MATH] is analytic in the upper complex plane [MATH], fall's off to zero for large [MATH] like some power of [MATH], and admits at most a simple pole at [MATH].", '1002.2603-1-11-4': 'Consider now the closed integration contour [MATH] obtained by closing in the upper complex plane the positively oriented real axis, and let [MATH] be any complex number in [MATH].', '1002.2603-1-11-5': 'It is then a simple matter to verify the identity: [EQUATION]', '1002.2603-1-11-6': 'The assumed fall-off property of [MATH] ensures that the half-circle of infinite radius forming [MATH] contributes nothing to the integral, and then from Eq. ([REF]) we find: [EQUATION]', '1002.2603-1-11-7': 'Consider now a purely imaginary complex number [MATH], and assume in addition that that along the real axis [MATH] satisfies the symmetry property [MATH].', '1002.2603-1-11-8': 'From Eq. ([REF]) we then find: [EQUATION] which is the desired result.', '1002.2603-1-12-0': 'The standard dispersion relation Eq. ([REF]) used to compute the electric permittivity for imaginary frequencies is a special case of the above relation, corresponding to choosing [MATH].', '1002.2603-1-12-1': 'We note that Eq. ([REF]) is valid both for insulators, which have a finite permittivity at zero frequency, as well as for ohmic conductors, whose permittivity has a [MATH] singularity in the origin.', '1002.2603-1-12-2': 'As we explained in the introduction Eq. ([REF]), even though perfectly correct from a mathematical standpoint, has serious drawbacks, when it is used to numerically estimate [MATH] for ohmic conductors, starting from optical data available only in some interval [MATH], because the integral on the r.h.s. of Eq. ([REF]) receives a large contribution from frequencies near zero, where data are not available.', '1002.2603-1-12-3': 'This difficulty can however be overcome in a very simple way, as we now explain.', '1002.2603-1-12-4': 'Consider a window function [MATH], enjoying the following properties: [MATH] is analytic in [MATH], it has no poles in [MATH] except possibly a simple pole at infinity, and satisfies the symmetry property [EQUATION]', '1002.2603-1-12-5': 'Consider now Eq. ([REF]), for [MATH].', '1002.2603-1-12-6': 'Since the permittivity of any medium falls off like [MATH] at infinity [CITATION], the quantity [MATH] falls off at least like [MATH] at infinity, and it satisfies all the properties required for Eq. ([REF]) to hold.', '1002.2603-1-12-7': 'For any [MATH] such that [MATH], we then obtain the following generalized dispersion relation: [EQUATION]', '1002.2603-1-12-8': 'We note that the above relation constitutes an exact result, generalizing the standard dispersion relation Eq. ([REF]), to which it reduces with the choice [MATH].', '1002.2603-1-12-9': 'Another form of dispersion relation, frequently used in the case of conductors or superconductors [CITATION] is obtained by taking [MATH] into Eq. ([REF]).', '1002.2603-1-12-10': 'Recalling the relation [CITATION] [EQUATION] it reads: [EQUATION]', '1002.2603-1-12-11': 'The above form is especially convenient in the case of superconductors, because it avoids the [MATH] singularity characterizing the real part of the conductivity of these materials [CITATION].', '1002.2603-1-13-0': 'We observe now, and this is the key point, that there is no reason to restrict the choice of the function [MATH] to these two possibilities.', '1002.2603-1-13-1': 'Indeed, we can take advantage of the freedom in the choice of [MATH], to suppress the unwanted contribution of low frequencies (as well as of high frequencies), where experimental data on [MATH] are not available.', '1002.2603-1-13-2': 'In order to do that, it is sufficient to choose a window function that goes to zero fast enough for [MATH], as well as for [MATH].', '1002.2603-1-13-3': 'A convenient family of window functions which do the job is the following: [EQUATION] where [MATH] is an arbitrary complex number such that [MATH], and [MATH] and [MATH] are integers such that [MATH].', '1002.2603-1-13-4': 'The constant [MATH] is an irrelevant arbitrary normalization constant, that drops out from the generalized dispersion formula Eq. ([REF]).', '1002.2603-1-13-5': 'As we see, in the limit [MATH], these functions vanish like [MATH], and therefore by taking sufficiently large values for [MATH] we can obtain suppression of low frequencies to any desired level.', '1002.2603-1-13-6': 'On the other hand, for [MATH], [MATH] vanishes like [MATH], and therefore by taking sufficiently large values of [MATH], we can obtain suppression of high frequencies.', '1002.2603-1-13-7': 'Moreover, by suitably choosing the free parameter [MATH], we can also adjust the range of frequencies that effectively contribute to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-1-13-8': 'In Figs. 1 and 2 we plot the real and imaginary parts (in arbitrary units) of our window functions [MATH], versus the frequency [MATH] (expressed in eV).', '1002.2603-1-13-9': 'The two curves displayed correspond to the choices [MATH] (dashed line) and [MATH] (solid line).', '1002.2603-1-13-10': 'In both cases, the parameter [MATH] has the value [MATH].', '1002.2603-1-14-0': 'We observe that along the real frequency axis, our window functions have non-vanishing real and imaginary parts.', '1002.2603-1-14-1': 'This is not a feature of our particular choice of the window functions, but it is an unavoidable consequence of our demand of analyticity on [MATH].', '1002.2603-1-14-2': 'Indeed, for real frequencies [MATH] the real and imaginary parts of [MATH] are related to each other by the usual Kramers-Kronig relations [CITATION] that hold for the boundary values of analytic functions.', '1002.2603-1-14-3': 'In the case when [MATH] vanishes at infinity, they read: [EQUATION] where the symbol [MATH] in front of the integrals denotes the principal value.', '1002.2603-1-14-4': 'These relation show that vanishing of [MATH] implies that of [MATH] and viceversa, and therefore neither [MATH] nor [MATH] can be identically zero.', '1002.2603-1-14-5': 'By virtue of this property of the window functions, it follows from Eq. ([REF]) that both the real and imaginary parts of [MATH] are needed to evaluate [MATH] (unless the standard choices [MATH] or [MATH] are made).', '1002.2603-1-14-6': 'We also note (see Fig. 1 and 2) that the real and imaginary parts of [MATH] do not have a definite sign.', '1002.2603-1-14-7': 'This feature also is a general consequence of our key demand that [MATH] vanishes in the origin, as it can be seen by taking [MATH] in Eqs. ([REF]) and ([REF]).', '1002.2603-1-14-8': 'Since the l.h.s. of both equations are required to vanish, the integrand on the r.h.s. cannot have a definite sign.', '1002.2603-1-14-9': 'Finally, in Fig 3 we show plots of two of our window functions [MATH], versus the imaginary frequency [MATH], expressed in eV, for the same two choices of parameters of Fig. 1 and 2.', '1002.2603-1-14-10': 'It is important to observe that the window functions [MATH] are real along the imaginary axis (as it must be, as a consequence of the symmetry property Eq. ([REF])).', '1002.2603-1-14-11': 'However, the sign of [MATH] is not definite, and as a result of this [MATH] admits zeros along the imaginary axis.', '1002.2603-1-14-12': 'When using Eq. ([REF]) for estimating [MATH] it is then important to choose the window function such that none of its zeroes coincides with the value of [MATH] for which [MATH] is being estimated.', '1002.2603-1-15-0': '# A numerical simulation', '1002.2603-1-16-0': 'In this Section, we perform a simple simulation to test the degree of accuracy with which the quantity [MATH] can be reconstructed using our window functions, starting from data on [MATH] referring to a finite frequency interval.', '1002.2603-1-16-1': 'To do that we can proceed as follows.', '1002.2603-1-17-0': 'According to the standard dispersion relation Eq. ([REF]), the quantity [MATH] is equal to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-1-17-1': 'Following Refs. [CITATION], we can split this integral into three pieces, as follows: [EQUATION] where we set: [EQUATION] and [EQUATION]', '1002.2603-1-17-2': 'By construction, we obviously have: [EQUATION]', '1002.2603-1-17-3': 'An analogous split can be performed in the integral on the r.h.s. of the other standard dispersion relation involving the conductivity Eq. ([REF]): [EQUATION] with an obvious meaning of the symbols.', '1002.2603-1-17-4': 'Again, we have the identity: [EQUATION]', '1002.2603-1-17-5': 'On the other hand, according to our generalized dispersion relation Eq. ([REF]), the quantity [MATH] is also equal to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-1-17-6': 'We can split this integral too in a way analogous to Eq. ([REF]): [EQUATION] where we set: [EQUATION] and [EQUATION]', '1002.2603-1-17-7': 'Then by construction we also have: [EQUATION]', '1002.2603-1-17-8': 'The quantities [MATH], [MATH] and [MATH] evidently represent the contribution of the experimental data.', '1002.2603-1-17-9': 'On the contrary the quantities [MATH], [MATH] and [MATH] can be determined only by extrapolating the data in the low frequency region [MATH], while determination of the quantities [MATH], [MATH] and [MATH] is only possible after we extrapolate the data in the high frequency interval [MATH].', '1002.2603-1-17-10': 'Ideally, we would like to have [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] as small as possible.', '1002.2603-1-18-0': 'To see how things work, we can perform a simple simulation of real experimental data.', '1002.2603-1-18-1': 'We imagine that the electric permittivity of gold is described by the following six-oscillators approximation [CITATION], which is known to provide a rather good description of the permittivity of gold for the frequencies that are relevant to the Casimir effect: [EQUATION]', '1002.2603-1-18-2': 'Here, [MATH] is the plasma frequency and [MATH] is the relaxation frequency for conduction electrons, while the oscillator terms describe core electrons.', '1002.2603-1-18-3': 'The values of the parameters [MATH], [MATH] and [MATH] can be found in the second of Refs. [CITATION].', '1002.2603-1-18-4': 'For [MATH] and [MATH] we use the reference values for crystalline bulk samples, [MATH] eV[MATH] and [MATH] eV[MATH].', '1002.2603-1-18-5': 'Of course with such a simple model for the permittivity of gold, there is no need to use dispersion relations to obtain the expression of [MATH], for this can be simply done by the substitution [MATH] in the r.h.s. of Eq. ([REF]): [EQUATION]', '1002.2603-1-18-6': 'Simulating the real experimental situation, let us pretend however that we know that the optical data of gold are described by Eq. ([REF]) only in some interval [MATH], and assuming that we do not want to make extrapolations of the data outside the experimental interval, let us see how well the quantities [MATH], [MATH] and [MATH] defined earlier reconstruct the exact value of [MATH] given by Eq. ([REF]).', '1002.2603-1-18-7': 'In our simulation we took [MATH] meV[MATH] (representing the minimum frequency value for which data for gold films were measured in [CITATION]) while for [MATH] we choose the value [MATH] eV[MATH].', '1002.2603-1-18-8': 'The chosen value of [MATH] is about thirty times the characteristic frequency [MATH] for a separation [MATH] nm.', '1002.2603-1-18-9': 'The result of our simulation are summarized in Figs. 4 and 5.', '1002.2603-1-19-0': 'In Fig. 4, we report the relative per cent errors [MATH] (black squares) and [MATH] (grey squares) which are made if the quantities [MATH] or [MATH] are used, respectively, as estimators of [MATH].', '1002.2603-1-19-1': 'The integer number on the abscissa labels the Matsubara mode [MATH] K).', '1002.2603-1-19-2': 'Only the first sixty modes are displayed, which are sufficient to estimate the Casimir force at room temperature, for separations larger than 100 nm, with a precision better than one part in ten thousand.', '1002.2603-1-19-3': 'As we see, both [MATH] and [MATH] provide a poor approximation to [MATH], with [MATH] performing somehow better at higher imaginary frequencies, and [MATH] doing better at lower imaginary frequencies.', '1002.2603-1-19-4': 'Indeed, [MATH] and [MATH] suffer from opposite problems.', '1002.2603-1-19-5': 'On one hand the large error affecting [MATH] arises mostly from neglect of the large low-frequency contribution [MATH], and to a much less extent from neglect of the high frequency contribution [MATH] (The magnitude of the high frequency contribution [MATH] is less than two percent of [MATH] for all [MATH]).', '1002.2603-1-19-6': 'The situation is quite the opposite in the case of [MATH].', '1002.2603-1-19-7': 'This difference is of course due to the opposite limiting behaviors of the imaginary parts of the permittivity [MATH] in the limits [MATH], and [MATH], as compared to those of the imaginary part of the conductivity [MATH].', '1002.2603-1-19-8': 'Indeed, for [MATH], [MATH] diverges like [MATH], while [MATH] approaches zero like [MATH].', '1002.2603-1-19-9': 'This explains while the low frequency contribution [MATH] is much larger than [MATH].', '1002.2603-1-19-10': 'On the other hand, in the limit [MATH], [MATH] vanishes like [MATH], while [MATH] vanishes only like [MATH].', '1002.2603-1-19-11': 'This implies that large frequencies are much less of a problem for [MATH] than for [MATH].', '1002.2603-1-19-12': 'The conclusion to be drawn from these considerations is that, if either of the two standard forms Eq. ([REF]) or Eq. ([REF]) of dispersion relations are used, in order to obtain a good estimate of [MATH], one is forced to extrapolate somehow the experimental data both to frequencies less than [MATH], and larger than [MATH].', '1002.2603-1-20-0': 'We can now consider our windowed dispersion relation, Eq. ([REF]), with our choice of the window functions [MATH] in Eq. ([REF]).', '1002.2603-1-20-1': 'In Fig. 5, we display the relative per cent error [MATH] which is made if the quantity [MATH] is used as an estimator of [MATH].', '1002.2603-1-20-2': 'We considered two choices of parameters for our window functions in Eq. ([REF]), i.e. [MATH] (grey squares) and [MATH] (black squares).', '1002.2603-1-20-3': 'In both cases, we took for the parameter [MATH] the constant value [MATH] (See Figs. 1, 2 and 3).', '1002.2603-1-21-0': 'It is apparent from Fig. 5 that both window functions perform very well, for all considered Matsubara modes.', '1002.2603-1-21-1': 'The error made by using [MATH] is less than one percent, in absolute value, while the error made by using [MATH] is less than 0.25 percent.', '1002.2603-1-21-2': 'The jumps displayed by the relative errors in Fig. 5 (around [MATH] for the grey dots, and [MATH] for the black ones) correspond to the approximate positions of the zeroes of the respective window functions [MATH] (see Fig. 3).', '1002.2603-1-21-3': 'Such jumps can be easily avoided, further reducing at the same time the error, by making a different choice of the free parameter [MATH] for each value of [MATH].', '1002.2603-1-21-4': 'We did not do this here for the sake of simplicity.', '1002.2603-1-21-5': 'It is clear that in concrete cases one is free to choose for each value of [MATH], different values of all the parameters [MATH] and [MATH], in such a way that the error is as small as possible.', '1002.2603-1-22-0': '# Simulation of the Casimir force', '1002.2603-1-23-0': 'In this Section, we investigate the performance of our window functions with respect to the determination of the Casimir force.', '1002.2603-1-23-1': 'We consider for simplicity the prototypical case of two identical plane-parallel homogeneous and isotropic gold plates, placed in vacuum at a distance [MATH].', '1002.2603-1-23-2': 'As it is well known, the Casimir force per unit area is given by the following Lifshitz formula: [EQUATION] where the plus sign corresponds to an attraction between the plates.', '1002.2603-1-23-3': 'In this Equation, the prime over the [MATH]-sum means that the [MATH] term has to taken with a weight one half, [MATH] is the temperature, [MATH] denotes the magnitude of the projection of the wave-vector onto the plane of the plates and [MATH], where [MATH] are the Matsubara frequencies.', '1002.2603-1-23-4': 'The quantities [MATH] denote the familiar Fresnel reflection coefficients of the slabs for [MATH]-polarization, evaluated at imaginary frequencies [MATH].', '1002.2603-1-23-5': 'They have the following expressions: [EQUATION] where [MATH].', '1002.2603-1-24-0': 'We have simulated the error made in the estimate of [MATH] if the estimate of [MATH] provided by the window-approximations [MATH] is used: [EQUATION] again assuming the simple six-oscillator model of Eq. ([REF]) for [MATH].', '1002.2603-1-24-1': 'The results are summarized in Fig 6, where we plot the relative error [MATH] in percent, as a function of the separation [MATH] (in microns).', '1002.2603-1-24-2': 'The window functions that have been used are the same as in Fig. 5.', '1002.2603-1-25-0': 'We see from the figure that already with this simple and not-optimized choice of window functions, the error is much less than one part in a thousand in the entire range of separations considered, from 100 nm to one micron.', '1002.2603-1-26-0': '# Conclusions and discussion', '1002.2603-1-27-0': 'In recent years, a lot of efforts have been made to measure accurately the Casimir force.', '1002.2603-1-27-1': 'At the moment of this writing, the most precise experiments using gold-coated micromechanical oscillators claim a precision better than one percent [CITATION].', '1002.2603-1-27-2': 'It is therefore important to see if an analogous level of precision in the prediction of the Casimir force can be obtained at the theoretical level.', '1002.2603-1-27-3': 'A precise determination of the theoretical error is indeed as important as reducing the experimental error, in order to address controversial questions that have emerged in the recent literature on dispersion forces, regarding the influence of free charges on the thermal correction to the Casimir force [CITATION].', '1002.2603-1-28-0': 'Addressing the theoretical error in the magnitude of the Casimir force is indeed difficult, because many physical effect must be accounted for.', '1002.2603-1-28-1': 'However, it has recently been pointed out [CITATION] that perhaps the largest theoretical uncertainty results from incomplete knowledge of the optical data for the surfaces involved in the experiments.', '1002.2603-1-28-2': 'On one hand, the large variability depending on the preparation procedure, of the optical properties of gold coatings, routinely used in Casimir experiments, makes it necessary to accurately characterize the coatings actually used in any experiment.', '1002.2603-1-28-3': 'On the other hand, even when this characterization is done, another problem arises, because for evaluating the Casimir force one needs to determine the electric permittivity [MATH] of the coatings for certain imaginary frequencies [MATH].', '1002.2603-1-28-4': 'This quantity is not directly accessible to any optical measurement, and the only way to determine it is via exploiting dispersion relations, that permit to express [MATH] in terms of the measurable values of the permittivity [MATH] for real frequencies [MATH].', '1002.2603-1-28-5': 'When doing this, one is faced with the difficulty that optical data are necessarily known only in a finite interval of frequencies [MATH].', '1002.2603-1-28-6': 'This practical limitation constitutes a severe problem in the experimentally relevant case of good conductors, because of their large conductivity at low frequencies.', '1002.2603-1-28-7': 'With the standard forms of dispersion relations Eq. ([REF]) and Eq. ([REF]), one finds that for practical values of [MATH] and [MATH], low frequencies less than [MATH] and/or large frequencies larger than [MATH] give a very large contribution to [MATH].', '1002.2603-1-28-8': 'In order to estimate [MATH] accurately, one is then forced to extrapolate available optical data outside the experimental region, on the basis of some theoretical model for [MATH].', '1002.2603-1-28-9': 'Of course, this introduces a further element of uncertainty in the obtained values of [MATH], and the resulting theoretical error is difficult to estimate quantitatively.', '1002.2603-1-29-0': 'In this paper we have shown that this problem can be resolved by suitably modifying the standard dispersion relation used to compute [MATH], in terms of appropriate analytic window functions [MATH] that suppress the contributions both of low and large frequencies.', '1002.2603-1-29-1': 'In this way, it becomes possible to accurately estimate [MATH] solely on the basis of the available optical data, rendering unnecessary any uncontrollable extrapolation of data.', '1002.2603-1-29-2': 'We have checked numerically the performance of simple choices of window functions, by making a numerical simulation based on an analytic fit of the optical properties of gold, that has been used in recent experiments on the Casimir effect [CITATION].', '1002.2603-1-29-3': 'We found that already very simple forms of the window functions permit to estimate the Casimir pressure with an accuracy better than one part in a thousand, on the basis of reasonable intervals of frequencies for the optical data.', '1002.2603-1-29-4': 'It would be interesting to apply these methods to the accurate optical data for thin gold films quoted in Ref. [CITATION].', '1002.2603-1-30-0': 'Before closing the paper, we should note that the relevance of the sample-to-sample dependence of the optical data observed in [CITATION] for the theory of the Casimir effect has been questioned by the authors of Ref. [CITATION], who observed that this dependence mostly originates from relaxation processes of free conduction electrons at infrared and optical frequencies, due for example to different grain sizes in thin films.', '1002.2603-1-30-1': 'The main consequence of these sample-dependent features is the large variability of the Drude parameters, extracted from fits of the low-frequency optical data of the films, which constitutes the basic source of variation of the computed Casimir force reported in Ref. [CITATION].', '1002.2603-1-30-2': 'According to the authors of Ref. [CITATION], relaxation properties of conduction electrons in thin films, described by the fitted values of the Drude parameters, are not relevant for the Casimir effect.', '1002.2603-1-30-3': 'Indeed, according to these authors the quantity [MATH] to be used in Lifshitz formula should not be understood as the actual electric permittivity of the plate, as derived from optical measurements on the sample, but it should be rather regarded as a phenomenological quantity connected to but not identical to the optical electric permittivity of the film.', '1002.2603-1-30-4': 'The ansatz offered by them for [MATH] is dubbed as generalized plasma model, and following Ref. [CITATION] we denote it as [MATH].', '1002.2603-1-30-5': 'This quantity is a semianalytical mathematical construct, defined by the formula: [EQUATION] where [MATH] represents the contribution of core electrons, while the term proportional to the square of the plasma frequency [MATH] describes conduction electrons.', '1002.2603-1-30-6': 'The most striking qualitative feature of this expression is the neglect of ohmic dissipation in the contribution from conduction electrons, but this is not all.', '1002.2603-1-30-7': 'Indeed, the ansatz prescribes that only the core-electron contribution [MATH] should be extracted from optical data of the film.', '1002.2603-1-30-8': 'On the contrary, and more importantly, according to Ref. [CITATION] the value of the plasma frequency [MATH] to be used in Eq. ([REF]) should be the one pertaining to a perfect crystal of the [MATH] material, and not the one obtained by a Drude-model fit of the low-frequency optical data of the film actually used in the experiment.', '1002.2603-1-30-9': 'The justification provided for this choice of the plasma frequency by the authors of Ref. [CITATION] is that the contribution of conduction electrons to the Casimir force should depend only on properties determined by the structure of the crystal cell, which are independent of the sample-to-sample variability determined by the peculiar grain structure of the film, reported in Ref. [CITATION].', '1002.2603-1-30-10': 'It should be noted that for gold, the value of the plasma frequency advocated in [CITATION], [MATH] eV/[MATH], is much higher than the fit values quoted in Ref. [CITATION], which range from 6.8 to 8.4 eV/[MATH].', '1002.2603-1-30-11': 'As a result, the approach advocated in Ref. [CITATION] leads to larger magnitudes of the Casimir force, as compared to the values derived in Ref. [CITATION], with differences ranging, depending on the sample, from 5 [MATH] to 14 [MATH] at 100 nm.', '1002.2603-1-30-12': 'There is no room here to further discuss the merits and faults of these approaches, and we refer the reader to [CITATION] for a thorough analysis.', '1002.2603-1-30-13': 'It is fair to note though that a series of recent experiments by one experimental group [CITATION] appears to favor the generalized plasma approach, and to rule out the more conventional approach based on actual optical data followed in Refs. [CITATION].', '1002.2603-1-31-0': 'The future will tell what is the correct description.', '1002.2603-1-31-1': 'In the meanwhile, we remark that whatever approach is followed, the methods proposed in this paper may prove useful to obtain more reliable estimates of the Casimir force for future experiments.'} | {'1002.2603-2-0-0': 'Recent advances in experimental techniques now permit to measure the Casimir force with unprecedented precision.', '1002.2603-2-0-1': 'In order to achieve a comparable precision in the theoretical prediction of the force, it is necessary to accurately determine the electric permittivity of the materials constituting the plates along the imaginary frequency axis.', '1002.2603-2-0-2': 'The latter quantity is not directly accessible to experiments, but it can be determined via dispersion relations from experimental optical data.', '1002.2603-2-0-3': 'In the experimentally important case of conductors, however, a serious drawback of the standard dispersion relations commonly used for this purpose, is their strong dependence on the chosen low-frequency extrapolation of the experimental optical data, which introduces a significant and not easily controllable uncertainty in the result.', '1002.2603-2-0-4': 'In this paper we show that a simple modification of the standard dispersion relations, involving suitable analytic window functions, resolves this difficulty, making it possible to reliably determine the electric permittivity at imaginary frequencies solely using experimental optical data in the frequency interval where they are available, without any need of uncontrolled data extrapolations.', '1002.2603-2-1-0': '# INTRODUCTION', '1002.2603-2-2-0': 'One of the most intriguing predictions of Quantum electrodynamics is the existence of irreducible vacuum fluctuations of the electromagnetic (e.m.) field.', '1002.2603-2-2-1': "It was Casimir's fundamental discovery [CITATION] to realize that this purely quantum phenomenon was not confined to the atomic scale, as in the Lamb shift, but would rather manifest itself also at the macroscopic scale, in the form of a force of attraction between two discharged plates.", '1002.2603-2-2-2': 'For the idealized case of two perfectly reflecting plane-parallel plates at zero temperature, placed at a distance [MATH] in vacuum, Casimir obtained the following remarkably simple estimate of the force per unit area [EQUATION]', '1002.2603-2-2-3': 'An important step forward was made a few years later by Lifshitz and co-workers [CITATION], who obtained a formula for the force between two homogeneous dielectric plane-parallel slabs, at finite temperature.', '1002.2603-2-2-4': 'In this theory, of macroscopic character, the material properties of the slabs were fully characterized in terms of the respective frequency dependent electric permittivities [MATH], accounting for the dispersive and dissipative properties of real materials.', '1002.2603-2-2-5': 'In this way, it was possible for the first time to investigate the influence of material properties on the magnitude of the Casimir force.', '1002.2603-2-3-0': 'Over ten years ago, a series of brilliant experiments [CITATION] exploiting modern experimental techniques provided the definitive demonstration of the Casimir effect.', '1002.2603-2-3-1': 'These now historical experiments spurred enormous interest in the Casimir effect, and were soon followed by many other experiments.', '1002.2603-2-3-2': 'The subsequent experiments were aimed at diverse objectives.', '1002.2603-2-3-3': 'Some of them explored new geometries: while the works [CITATION] used a sphere-plate setup, the original planar geometry investigated by Casimir was adopted in the experiment [CITATION], and a setup with crossed cylinders was considered in [CITATION].', '1002.2603-2-3-4': 'The important issue of the non trivial geometry dependence of the Casimir effect is also being pursued experimentally, using elaborate micro-patterned surfaces [CITATION].', '1002.2603-2-3-5': 'Other experiments aimed at demonstrating new possible uses of the Casimir force, like for example the actuation of micromachines [CITATION], or at demonstrating the possibility of a large modulation of the Casimir force [CITATION], which could also result in interesting technological applications.', '1002.2603-2-3-6': 'There are also experiments using superconducting Casimir cavities, that aim at measuring the change of the Casimir energy across the superconducting phase transition [CITATION].', '1002.2603-2-3-7': 'The experiments performed in the last ten years are just too numerous to mention them all here.', '1002.2603-2-3-8': 'For an updated account we refer the reader to the very recent review paper [CITATION].', '1002.2603-2-4-0': 'Apart from exploring new manifestations of the Casimir effect, a large experimental effort is presently being made also to increase the precision of Casimir force measurements, in simple geometries.', '1002.2603-2-4-1': 'Already in the early experiment [CITATION] a precision upto one percent was obtained.', '1002.2603-2-4-2': 'More recently, a series of experiments with microtorsional oscillators [CITATION] reached an amazing precision of 0.2 percent.', '1002.2603-2-4-3': 'The reader may wonder what is the interest in achieving such a high precision in this kind of experiments.', '1002.2603-2-4-4': 'There are several reasons why this is important.', '1002.2603-2-4-5': 'On one hand, in the theory of dispersion forces puzzling conceptual problems have recently emerged that are connected with the contribution of free charges to the thermal Casimir force, whose resolution crucially depends on the precision of the theory-experiment comparison [CITATION].', '1002.2603-2-4-6': 'On the other hand, the ability to accurately determine the Casimir force is also important for the purpose of obtaining stronger constraints on hypothetical long-range forces predicted by certain theoretical scenarios going beyond the Standard Model of particle physics [CITATION].', '1002.2603-2-5-0': 'The remarkable precision achieved in the most recent experiments poses a challenging demand on the theorist: is it possible to predict the magnitude of the Casimir force with a comparable level of precision, say of one percent?', '1002.2603-2-5-1': 'Assessing the theoretical error affecting present estimates of the Casimir force is a difficult problem indeed, because many different factors must be taken into account [CITATION].', '1002.2603-2-5-2': 'Consider the typical experimental setting of most of the current experiments, where the Casimir force is measured between two bodies covered with gold, placed in vacuum at a distance of a (few) hundred nanometers.', '1002.2603-2-5-3': 'In this separation range, the main factor to consider is the finite penetration depth of electromagnetic fields into the gold layer , resulting from the finite conductivity of gold.', '1002.2603-2-5-4': 'The tool to analyze the influence of such material properties as the conductivity on the Casimir effect is provided by Lifshitz theory [CITATION].', '1002.2603-2-5-5': 'This theory shows that for a separation of 100 nm, the finite conductivity of gold determines a reduction in the magnitude of the Casimir force of about fifty percent in comparison with the perfect metal result [CITATION].', '1002.2603-2-5-6': 'Much smaller corrections, that must nevertheless be considered if the force is to be estimated with percent precision, arise from the finite temperature of the plates and from their surface roughness.', '1002.2603-2-5-7': 'Moreover, geometric effects resulting from the actual shape of the plates should be considered.', '1002.2603-2-5-8': 'We should also mention that the magnitude of residual electrostatic forces between the plates, resulting from contact potentials and patch effects, must be carefully accounted for.', '1002.2603-2-5-9': 'For a discussion of all these issues, which received much attention in the recent literature on the Casimir effect, we again address the reader to Ref. [CITATION].', '1002.2603-2-5-10': 'See also the recent work [CITATION].', '1002.2603-2-6-0': 'In this paper, we focus our attention on the influence of the optical properties of the plates which, as explained above, is by far the most relevant factor to consider.', '1002.2603-2-6-1': 'As we pointed out earlier, in Lifshitz theory the optical properties of the plates enter via the frequency-dependent electric permittivity [MATH] of the material constituting plates.', '1002.2603-2-6-2': 'In order to obtain an accurate prediction of the force, it is therefore of the utmost importance to use accurate data for the electric permittivity.', '1002.2603-2-6-3': 'The common practice adopted in all recent Casimir experiments with gold surfaces is to use tabulated data for gold (most of the times those quoted in Refs. [CITATION]), suitably extrapolated at low frequencies, where optical data are not available, by simple analytic models (like the Drude model or the so-called generalized plasma model).', '1002.2603-2-6-4': 'However, already ten years ago Lamoreaux observed [CITATION] that using tabulated data to obtain an accurate prediction of the Casimir force may not be a reliable practice, since optical properties of gold films may vary significantly from sample to sample, depending on the conditions of deposition.', '1002.2603-2-6-5': 'The same author stressed the importance of measuring the optical data of the films actually used in the force measurements, in the frequency range that is relevant for the Casimir force.', '1002.2603-2-6-6': 'The importance of this point was further stressed in [CITATION] and received clear experimental support in a recent paper [CITATION], where the optical properties of several gold films of different thicknesses, and prepared by different procedures, were measured ellipsometrically in a wide range of wavelengths, from 0.14 to 33 microns, and it was found that the frequency dependent electric permittivity changes significantly from sample to sample.', '1002.2603-2-6-7': 'By using the zero-temperature Lifshitz formula, the authors estimated that the observed sample dependence of the electric permittivity implies a variation in the theoretical value of the Casimir force, from one sample to another, easily as large as ten percent, for separations around 100 nm.', '1002.2603-2-6-8': 'It was concluded that in order to achieve a theoretical accuracy better than ten percent in the prediction of the Casimir force, it is necessary to determine the optical properties of the films actually used in the experiment of interest.', '1002.2603-2-7-0': 'The aim of this paper is to improve the mathematical procedure that is actually needed to obtain reliable estimates of the Casimir force, starting from experimental optical data on the material of the plates, like those presented in Ref. [CITATION].', '1002.2603-2-7-1': 'The necessity of such an improvement stems from the very simple and unavoidable fact that experimental optical data are never available in the entire frequency domain, but are always restricted to a finite frequency interval [MATH].', '1002.2603-2-7-2': 'To see why this constitutes problem we recall that Lifshitz formula, routinely used to interpret current experiments, expresses the Casimir force between two parallel plates as an integral over imaginary frequencies [MATH] of a quantity involving the dielectric permittivities of the plates [MATH].', '1002.2603-2-7-3': 'For finite temperature, the continuous frequency integration is replaced by a sum over discrete so-called Matsubara frequencies [MATH], where [MATH], with [MATH] a non-negative integer, and [MATH] the temperature of the plates.', '1002.2603-2-7-4': 'In any case, whatever the temperature, one needs to evaluate the permittivity of the plates for certain imaginary frequencies.', '1002.2603-2-7-5': 'We note that, in principle, recourse to imaginary frequencies is not mandatory because it is possible to rewrite Lifshitz formula in a mathematically equivalent form, involving an integral over the real frequency axis.', '1002.2603-2-7-6': 'In this case however the integrand becomes a rapidly oscillating function of the frequency, which hampers any possibility of numerical evaluation.', '1002.2603-2-7-7': 'In practice, the real-frequency form of Lifshitz formula is never used, and only its imaginary-frequency version is considered.', '1002.2603-2-7-8': 'We remark that occurrence of imaginary frequencies in the expression of the Casimir force, is a general feature of all recent formalisms, extending Lifshitz theory to non-planar geometries [CITATION].', '1002.2603-2-7-9': 'The problem is that the electric permittivity [MATH] at imaginary frequencies cannot be measured directly by any experiment.', '1002.2603-2-7-10': 'The only way to determine it by means of dispersion relations, which allow to express [MATH] in terms of the observable real-frequency electric permittivity [MATH].', '1002.2603-2-7-11': 'In the standard version of dispersion relations [CITATION], adopted so far in all works on the Casimir effect, [MATH] is expressed in terms of an integral of a quantity involving the imaginary part [MATH] of the electric permittivity: [EQUATION]', '1002.2603-2-7-12': 'The above formula shows that, in principle, a determination of [MATH] requires knowledge of [MATH] at all frequencies while, as we said earlier, optical data are available only in some interval [MATH].', '1002.2603-2-7-13': 'In practice, the problem is not so serious on the high frequency side, because the fall-off properties of [MATH] at high frequencies, together with the [MATH] factor in the denominator of the integrand, ensure that the error made by truncating the integral at a suitably large frequency [MATH] is small, provided that [MATH] is large enough.', '1002.2603-2-7-14': 'Typically, an [MATH] larger than, say, [MATH], is good enough for practical purposes.', '1002.2603-2-7-15': 'Things are not so easy though on the low frequency side.', '1002.2603-2-7-16': 'In the case of insulators, optical data are typically available until frequencies [MATH] much smaller than the frequencies of all resonances of the medium.', '1002.2603-2-7-17': 'Because of this, [MATH] is almost zero for [MATH], and therefore the error made by truncating the integral at [MATH] is again negligible.', '1002.2603-2-7-18': 'Problems arise however in the case of ohmic conductors, because then [MATH] has a [MATH] singularity at [MATH].', '1002.2603-2-7-19': 'As a result [MATH] becomes extremely large at low frequencies, in such a way that the integral in Eq. ([REF]) receives a very large contribution from low frequencies.', '1002.2603-2-7-20': 'For typical values of [MATH] that can be reached in practice (for example for gold, the tabulated data in [CITATION] begin at [MATH] meV[MATH], while the data of [CITATION] start at 38 meV[MATH]) truncation of the integral at [MATH] results in a large error.', '1002.2603-2-7-21': 'The traditional remedy to this problem is to make some analytical extrapolation of the data, typically based on Drude model fits of the low-frequency region of data, from [MATH] to zero, and then use the extrapolation to estimate the contribution of the integral in the interval [MATH] where data are not directly available.', '1002.2603-2-7-22': 'It is important to observe that this contribution is usually very large.', '1002.2603-2-7-23': 'For example, even in the case of Ref. [CITATION], the relative contribution of the extrapolation is about fifty percent of the total value of the integral, in the entire range of imaginary frequencies that are needed for estimating the Casimir force.', '1002.2603-2-8-0': 'Clearly, this procedure is not very satisfying.', '1002.2603-2-8-1': 'The use of analytical extrapolations of the data introduces an uncertainty in the obtained values of [MATH], that is not easy to quantify.', '1002.2603-2-8-2': 'The result may in fact depend a lot on the form of the extrapolation, and there is no guarantee that the chosen expression is good enough.', '1002.2603-2-8-3': 'Consider for example Ref. [CITATION], which constitutes the most accurate existing work on this problem.', '1002.2603-2-8-4': 'It was found there that the simple Drude model does not fit so well the data of all samples, making it necessary to improve it by the inclusion of an additional Lorentz oscillator.', '1002.2603-2-8-5': 'Moreover, it was found that for each sample the Drude parameters extracted from the data depended on the used fitting procedure, and were inconsistent which each other within the estimated errors, which is again an indication of the probable inadequacy of the analytical expression chosen for the interpolation.', '1002.2603-2-8-6': 'This state of things led us to investigate if it possible to determine accurately [MATH] solely on the basis of available optical data, without making recourse to data extrapolations.', '1002.2603-2-8-7': 'We shall see below that this is indeed possible, provided that Eq. ([REF]) is suitably modified, in a way that involves multiplying the integrand by an appropriate analytical window function [MATH], which suppresses the contribution of frequencies not belonging to the interval [MATH].', '1002.2603-2-8-8': 'As a result of this modification, the error made by truncating the integral to the frequency range [MATH] can be made negligible at both ends of the integration domain, rendering unnecessary any extrapolation of the optical data outside the interval where they are available.', '1002.2603-2-8-9': 'The procedure outlined in this paper should allow to better evaluate the theoretical uncertainty of Casimir force estimates resulting from experimental errors in the optical data.', '1002.2603-2-9-0': 'The plan of the paper is as follows: in Sec. II we derive a generalized dispersion relation for [MATH], involving analytic window functions [MATH], and we provide a simple choice for the window functions.', '1002.2603-2-9-1': 'In Sec III we present the results of a numerical simulation of our window functions, for the experimentally relevant case of gold, and in Sec IV we estimate numerically the error on the Casimir pressure resulting from the use of our window functions.', '1002.2603-2-9-2': 'Sec V contains our conclusions and a discussion of the results.', '1002.2603-2-10-0': '# Generalized dispersion relations with window-functions', '1002.2603-2-11-0': 'As it it well known [CITATION], analyticity properties satisfied by the electric permittivity [MATH] of any causal medium (and more in general by any causal response function, the magnetic permeability [MATH] being another example) imply certain integral relations between the real part [MATH] and imaginary part [MATH] of [MATH], known as Kramers-Kronig or dispersion relations.', '1002.2603-2-11-1': 'The dispersion relation of interest to us is the one that permits to express the value [MATH] of the response function at some imaginary frequency [MATH] in terms of an integral along the positive frequency axis, involving [MATH].', '1002.2603-2-11-2': 'It is convenient to briefly review here the simple derivation of this important result, which is an easy exercise in contour integration.', '1002.2603-2-11-3': "For our purposes, it is more convenient to start from an arbitrary complex function [MATH], with the following properties: [MATH] is analytic in the upper complex plane [MATH], fall's off to zero for large [MATH] like some power of [MATH], and admits at most a simple pole at [MATH].", '1002.2603-2-11-4': 'Consider now the closed integration contour [MATH] obtained by closing in the upper complex plane the positively oriented real axis, and let [MATH] be any complex number in [MATH].', '1002.2603-2-11-5': 'It is then a simple matter to verify the identity: [EQUATION]', '1002.2603-2-11-6': 'The assumed fall-off property of [MATH] ensures that the half-circle of infinite radius forming [MATH] contributes nothing to the integral, and then from Eq. ([REF]) we find: [EQUATION]', '1002.2603-2-11-7': 'Consider now a purely imaginary complex number [MATH], and assume in addition that along the real axis [MATH] satisfies the symmetry property [MATH].', '1002.2603-2-11-8': 'From Eq. ([REF]) we then find: [EQUATION] which is the desired result.', '1002.2603-2-12-0': 'The standard dispersion relation Eq. ([REF]) used to compute the electric permittivity for imaginary frequencies is a special case of the above relation, corresponding to choosing [MATH].', '1002.2603-2-12-1': 'We note that Eq. ([REF]) is valid both for insulators, which have a finite permittivity at zero frequency, as well as for ohmic conductors, whose permittivity has a [MATH] singularity in the origin.', '1002.2603-2-12-2': 'As we explained in the introduction Eq. ([REF]), even though perfectly correct from a mathematical standpoint, has serious drawbacks, when it is used to numerically estimate [MATH] for ohmic conductors, starting from optical data available only in some interval [MATH], because the integral on the r.h.s. of Eq. ([REF]) receives a large contribution from frequencies near zero, where data are not available.', '1002.2603-2-12-3': 'This difficulty can however be overcome in a very simple way, as we now explain.', '1002.2603-2-12-4': 'Consider a window function [MATH], enjoying the following properties: [MATH] is analytic in [MATH], it has no poles in [MATH] except possibly a simple pole at infinity, and satisfies the symmetry property [EQUATION]', '1002.2603-2-12-5': 'Consider now Eq. ([REF]), for [MATH].', '1002.2603-2-12-6': 'Since for any medium [MATH] falls off like [MATH] at infinity [CITATION], the quantity [MATH] falls off at least like [MATH] at infinity, and it satisfies all the properties required for Eq. ([REF]) to hold.', '1002.2603-2-12-7': 'For any [MATH] such that [MATH], we then obtain the following generalized dispersion relation: [EQUATION]', '1002.2603-2-12-8': 'We note that the above relation constitutes an exact result, generalizing the standard dispersion relation Eq. ([REF]), to which it reduces with the choice [MATH].', '1002.2603-2-12-9': 'Another form of dispersion relation, frequently used in the case of conductors or superconductors [CITATION] is obtained by taking [MATH] into Eq. ([REF]).', '1002.2603-2-12-10': 'Recalling the relation [CITATION] [EQUATION] it reads: [EQUATION]', '1002.2603-2-12-11': 'The above form is especially convenient in the case of superconductors, because it avoids the [MATH] singularity characterizing the real part of the conductivity of these materials [CITATION].', '1002.2603-2-13-0': 'We observe now, and this is the key point, that there is no reason to restrict the choice of the function [MATH] to these two possibilities.', '1002.2603-2-13-1': 'Indeed, we can take advantage of the freedom in the choice of [MATH], to suppress the unwanted contribution of low frequencies (as well as of high frequencies), where experimental data on [MATH] are not available.', '1002.2603-2-13-2': 'In order to do that, it is sufficient to choose a window function that goes to zero fast enough for [MATH], as well as for [MATH].', '1002.2603-2-13-3': 'A convenient family of window functions which do the job is the following: [EQUATION] where [MATH] is an arbitrary complex number such that [MATH], and [MATH] and [MATH] are integers such that [MATH].', '1002.2603-2-13-4': 'The constant [MATH] is an irrelevant arbitrary normalization constant, that drops out from the generalized dispersion formula Eq. ([REF]).', '1002.2603-2-13-5': 'As we see, in the limit [MATH], these functions vanish like [MATH], and therefore by taking sufficiently large values for [MATH] we can obtain suppression of low frequencies to any desired level.', '1002.2603-2-13-6': 'On the other hand, for [MATH], [MATH] vanishes like [MATH], and therefore by taking sufficiently large values of [MATH], we can obtain suppression of high frequencies.', '1002.2603-2-13-7': 'Moreover, by suitably choosing the free parameter [MATH], we can also adjust the range of frequencies that effectively contribute to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-2-13-8': 'In Figs. 1 and 2 we plot the real and imaginary parts (in arbitrary units) of our window functions [MATH], versus the frequency [MATH] (expressed in eV).', '1002.2603-2-13-9': 'The two curves displayed correspond to the choices [MATH] (dashed line) and [MATH] (solid line).', '1002.2603-2-13-10': 'In both cases, the parameter [MATH] has the value [MATH].', '1002.2603-2-14-0': 'We observe that along the real frequency axis, our window functions have non-vanishing real and imaginary parts.', '1002.2603-2-14-1': 'This is not a feature of our particular choice of the window functions, but it is an unavoidable consequence of our demand of analyticity on [MATH].', '1002.2603-2-14-2': 'Indeed, for real frequencies [MATH] the real and imaginary parts of [MATH] are related to each other by the usual Kramers-Kronig relations [CITATION] that hold for the boundary values of analytic functions.', '1002.2603-2-14-3': 'In the case when [MATH] vanishes at infinity, they read: [EQUATION] where the symbol [MATH] in front of the integrals denotes the principal value.', '1002.2603-2-14-4': 'These relation show that vanishing of [MATH] implies that of [MATH] and viceversa, and therefore neither [MATH] nor [MATH] can be identically zero.', '1002.2603-2-14-5': 'By virtue of this property of the window functions, it follows from Eq. ([REF]) that both the real and imaginary parts of [MATH] are needed to evaluate [MATH] (unless the standard choices [MATH] or [MATH] are made).', '1002.2603-2-14-6': 'We also note (see Fig. 1 and 2) that the real and imaginary parts of [MATH] do not have a definite sign.', '1002.2603-2-14-7': 'This feature also is a general consequence of our key demand that [MATH] vanishes in the origin, as it can be seen by taking [MATH] in Eqs. ([REF]) and ([REF]).', '1002.2603-2-14-8': 'Since the l.h.s. of both equations are required to vanish, the integrand on the r.h.s. cannot have a definite sign.', '1002.2603-2-14-9': 'Finally, in Fig 3 we show plots of two of our window functions [MATH], versus the imaginary frequency [MATH], expressed in eV, for the same two choices of parameters of Fig. 1 and 2.', '1002.2603-2-14-10': 'It is important to observe that the window functions [MATH] are real along the imaginary axis (as it must be, as a consequence of the symmetry property Eq. ([REF])).', '1002.2603-2-14-11': 'However, the sign of [MATH] is not definite, and as a result of this [MATH] admits zeros along the imaginary axis.', '1002.2603-2-14-12': 'When using Eq. ([REF]) for estimating [MATH] it is then important to choose the window function such that none of its zeroes coincides with the value of [MATH] for which [MATH] is being estimated.', '1002.2603-2-15-0': '# A numerical simulation', '1002.2603-2-16-0': 'In this Section, we perform a simple simulation to test the degree of accuracy with which the quantity [MATH] can be reconstructed using our window functions, starting from data on [MATH] referring to a finite frequency interval.', '1002.2603-2-16-1': 'To do that we can proceed as follows.', '1002.2603-2-17-0': 'According to the standard dispersion relation Eq. ([REF]), the quantity [MATH] is equal to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-2-17-1': 'Following Refs. [CITATION], we can split this integral into three pieces, as follows: [EQUATION] where we set: [EQUATION] and [EQUATION]', '1002.2603-2-17-2': 'By construction, we obviously have: [EQUATION]', '1002.2603-2-17-3': 'An analogous split can be performed in the integral on the r.h.s. of the other standard dispersion relation involving the conductivity Eq. ([REF]): [EQUATION] with an obvious meaning of the symbols.', '1002.2603-2-17-4': 'Again, we have the identity: [EQUATION]', '1002.2603-2-17-5': 'On the other hand, according to our generalized dispersion relation Eq. ([REF]), the quantity [MATH] is also equal to the integral on the r.h.s. of Eq. ([REF]).', '1002.2603-2-17-6': 'We can split this integral too in a way analogous to Eq. ([REF]): [EQUATION] where we set: [EQUATION] and [EQUATION]', '1002.2603-2-17-7': 'Then by construction we also have: [EQUATION]', '1002.2603-2-17-8': 'The quantities [MATH], [MATH] and [MATH] evidently represent the contribution of the experimental data.', '1002.2603-2-17-9': 'On the contrary the quantities [MATH], [MATH] and [MATH] can be determined only by extrapolating the data in the low frequency region [MATH], while determination of the quantities [MATH], [MATH] and [MATH] is only possible after we extrapolate the data in the high frequency interval [MATH].', '1002.2603-2-17-10': 'Ideally, we would like to have [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH] as small as possible.', '1002.2603-2-18-0': 'To see how things work, we can perform a simple simulation of real experimental data.', '1002.2603-2-18-1': 'We imagine that the electric permittivity of gold is described by the following six-oscillators approximation [CITATION], which is known to provide a rather good description of the permittivity of gold for the frequencies that are relevant to the Casimir effect: [EQUATION]', '1002.2603-2-18-2': 'Here, [MATH] is the plasma frequency and [MATH] is the relaxation frequency for conduction electrons, while the oscillator terms describe core electrons.', '1002.2603-2-18-3': 'The values of the parameters [MATH], [MATH] and [MATH] can be found in the second of Refs. [CITATION].', '1002.2603-2-18-4': 'For [MATH] and [MATH] we use the reference values for crystalline bulk samples, [MATH] eV[MATH] and [MATH] eV[MATH].', '1002.2603-2-18-5': 'Of course with such a simple model for the permittivity of gold, there is no need to use dispersion relations to obtain the expression of [MATH], for this can be simply done by the substitution [MATH] in the r.h.s. of Eq. ([REF]): [EQUATION]', '1002.2603-2-18-6': 'Simulating the real experimental situation, let us pretend however that we know that the optical data of gold are described by Eq. ([REF]) only in some interval [MATH], and assuming that we do not want to make extrapolations of the data outside the experimental interval, let us see how well the quantities [MATH], [MATH] and [MATH] defined earlier reconstruct the exact value of [MATH] given by Eq. ([REF]).', '1002.2603-2-18-7': 'In our simulation we took [MATH] eV[MATH] (representing the minimum frequency value for which data for gold films were measured in [CITATION]) while for [MATH] we choose the value [MATH] eV[MATH].', '1002.2603-2-18-8': 'The chosen value of [MATH] is about thirty times the characteristic frequency [MATH] for a separation [MATH] nm.', '1002.2603-2-18-9': 'The result of our simulation are summarized in Figs. 4 and 5.', '1002.2603-2-19-0': 'In Fig. 4, we report the relative per cent errors [MATH] (black squares) and [MATH] (grey triangles) which are made if the quantities [MATH] or [MATH] are used, respectively, as estimators of [MATH].', '1002.2603-2-19-1': 'The integer number on the abscissa labels the Matsubara mode [MATH] K).', '1002.2603-2-19-2': 'Only the first sixty modes are displayed, which are sufficient to estimate the Casimir force at room temperature, for separations larger than 100 nm, with a precision better than one part in ten thousand.', '1002.2603-2-19-3': 'As we see, both [MATH] and [MATH] provide a poor approximation to [MATH], with [MATH] performing somehow better at higher imaginary frequencies, and [MATH] doing better at lower imaginary frequencies.', '1002.2603-2-19-4': 'Indeed, [MATH] and [MATH] suffer from opposite problems.', '1002.2603-2-19-5': 'On one hand the large error affecting [MATH] arises mostly from neglect of the large low-frequency contribution [MATH], and to a much less extent from neglect of the high frequency contribution [MATH] (The magnitude of the high frequency contribution [MATH] is less than two percent of [MATH] for all [MATH]).', '1002.2603-2-19-6': 'The situation is quite the opposite in the case of [MATH].', '1002.2603-2-19-7': 'This difference is of course due to the opposite limiting behaviors of the imaginary parts of the permittivity [MATH] in the limits [MATH], and [MATH], as compared to those of the imaginary part of the conductivity [MATH].', '1002.2603-2-19-8': 'Indeed, for [MATH], [MATH] diverges like [MATH], while [MATH] approaches zero like [MATH].', '1002.2603-2-19-9': 'This explains while the low frequency contribution [MATH] is much larger than [MATH].', '1002.2603-2-19-10': 'On the other hand, in the limit [MATH], [MATH] vanishes like [MATH], while [MATH] vanishes only like [MATH].', '1002.2603-2-19-11': 'This implies that large frequencies are much less of a problem for [MATH] than for [MATH].', '1002.2603-2-19-12': 'The conclusion to be drawn from these considerations is that, if either of the two standard forms Eq. ([REF]) or Eq. ([REF]) of dispersion relations are used, in order to obtain a good estimate of [MATH], one is forced to extrapolate somehow the experimental data both to frequencies less than [MATH], and larger than [MATH].', '1002.2603-2-20-0': 'We can now consider our windowed dispersion relation, Eq. ([REF]), with our choice of the window functions [MATH] in Eq. ([REF]).', '1002.2603-2-20-1': 'In Fig. 5, we display the relative per cent error [MATH] which is made if the quantity [MATH] is used as an estimator of [MATH].', '1002.2603-2-20-2': 'We considered two choices of parameters for our window functions in Eq. ([REF]), i.e. [MATH] (grey triangles) and [MATH] (black squares).', '1002.2603-2-20-3': 'In both cases, we took for the parameter [MATH] the constant value [MATH] (See Figs. 1, 2 and 3).', '1002.2603-2-21-0': 'It is apparent from Fig. 5 that both window functions perform very well, for all considered Matsubara modes.', '1002.2603-2-21-1': 'The error made by using [MATH] is less than one percent, in absolute value, while the error made by using [MATH] is less than 0.25 percent.', '1002.2603-2-21-2': 'The jumps displayed by the relative errors in Fig. 5 (around [MATH] for the grey dots, and [MATH] for the black ones) correspond to the approximate positions of the zeroes of the respective window functions [MATH] (see Fig. 3).', '1002.2603-2-21-3': 'Such jumps can be easily avoided, further reducing at the same time the error, by making a different choice of the free parameter [MATH] for each value of [MATH].', '1002.2603-2-21-4': 'We did not do this here for the sake of simplicity.', '1002.2603-2-21-5': 'It is clear that in concrete cases one is free to choose for each value of [MATH], different values of all the parameters [MATH] and [MATH], in such a way that the error is as small as possible.', '1002.2603-2-22-0': '# Simulation of the Casimir force', '1002.2603-2-23-0': 'In this Section, we investigate the performance of our window functions with respect to the determination of the Casimir force.', '1002.2603-2-23-1': 'We consider for simplicity the prototypical case of two identical plane-parallel homogeneous and isotropic gold plates, placed in vacuum at a distance [MATH].', '1002.2603-2-23-2': 'As it is well known, the Casimir force per unit area is given by the following Lifshitz formula: [EQUATION] where the plus sign corresponds to an attraction between the plates.', '1002.2603-2-23-3': 'In this Equation, the prime over the [MATH]-sum means that the [MATH] term has to taken with a weight one half, [MATH] is the temperature, [MATH] denotes the magnitude of the projection of the wave-vector onto the plane of the plates and [MATH], where [MATH] are the Matsubara frequencies.', '1002.2603-2-23-4': 'The quantities [MATH] denote the familiar Fresnel reflection coefficients of the slabs for [MATH]-polarization, evaluated at imaginary frequencies [MATH].', '1002.2603-2-23-5': 'They have the following expressions: [EQUATION] where [MATH].', '1002.2603-2-24-0': 'We have simulated the error made in the estimate of [MATH] if the estimate of [MATH] provided by the window-approximations [MATH] is used: [EQUATION] again assuming the simple six-oscillator model of Eq. ([REF]) for [MATH].', '1002.2603-2-24-1': 'The results are summarized in Fig 6, where we plot the relative error [MATH] in percent, as a function of the separation [MATH] (in microns).', '1002.2603-2-24-2': 'The window functions that have been used are the same as in Fig. 5.', '1002.2603-2-25-0': 'We see from the figure that already with this simple and not-optimized choice of window functions, the error is much less than one part in a thousand in the entire range of separations considered, from 100 nm to one micron.', '1002.2603-2-26-0': '# Conclusions and discussion', '1002.2603-2-27-0': 'In recent years, a lot of efforts have been made to measure accurately the Casimir force.', '1002.2603-2-27-1': 'At the moment of this writing, the most precise experiments using gold-coated micromechanical oscillators claim a precision better than one percent [CITATION].', '1002.2603-2-27-2': 'It is therefore important to see if an analogous level of precision in the prediction of the Casimir force can be obtained at the theoretical level.', '1002.2603-2-27-3': 'A precise determination of the theoretical error is indeed as important as reducing the experimental error, in order to address controversial questions that have emerged in the recent literature on dispersion forces, regarding the influence of free charges on the thermal correction to the Casimir force [CITATION].', '1002.2603-2-28-0': 'Addressing the theoretical error in the magnitude of the Casimir force is indeed difficult, because many physical effect must be accounted for.', '1002.2603-2-28-1': 'However, it has recently been pointed out [CITATION] that perhaps the largest theoretical uncertainty results from incomplete knowledge of the optical data for the surfaces involved in the experiments.', '1002.2603-2-28-2': 'On one hand, the large variability depending on the preparation procedure, of the optical properties of gold coatings, routinely used in Casimir experiments, makes it necessary to accurately characterize the coatings actually used in any experiment.', '1002.2603-2-28-3': 'On the other hand, even when this characterization is done, another problem arises, because for evaluating the Casimir force one needs to determine the electric permittivity [MATH] of the coatings for certain imaginary frequencies [MATH].', '1002.2603-2-28-4': 'This quantity is not directly accessible to any optical measurement, and the only way to determine it is via exploiting dispersion relations, that permit to express [MATH] in terms of the measurable values of the permittivity [MATH] for real frequencies [MATH].', '1002.2603-2-28-5': 'When doing this, one is faced with the difficulty that optical data are necessarily known only in a finite interval of frequencies [MATH].', '1002.2603-2-28-6': 'This practical limitation constitutes a severe problem in the experimentally relevant case of good conductors, because of their large conductivity at low frequencies.', '1002.2603-2-28-7': 'With the standard forms of dispersion relations Eq. ([REF]) and Eq. ([REF]), one finds that for practical values of [MATH] and [MATH], low frequencies less than [MATH] and/or large frequencies larger than [MATH] give a very large contribution to [MATH].', '1002.2603-2-28-8': 'In order to estimate [MATH] accurately, one is then forced to extrapolate available optical data outside the experimental region, on the basis of some theoretical model for [MATH].', '1002.2603-2-28-9': 'Of course, this introduces a further element of uncertainty in the obtained values of [MATH], and the resulting theoretical error is difficult to estimate quantitatively.', '1002.2603-2-29-0': 'In this paper we have shown that this problem can be resolved by suitably modifying the standard dispersion relation used to compute [MATH], in terms of appropriate analytic window functions [MATH] that suppress the contributions both of low and large frequencies.', '1002.2603-2-29-1': 'In this way, it becomes possible to accurately estimate [MATH] solely on the basis of the available optical data, rendering unnecessary any uncontrollable extrapolation of data.', '1002.2603-2-29-2': 'We have checked numerically the performance of simple choices of window functions, by making a numerical simulation based on an analytic fit of the optical properties of gold, that has been used in recent experiments on the Casimir effect [CITATION].', '1002.2603-2-29-3': 'We found that already very simple forms of the window functions permit to estimate the Casimir pressure with an accuracy better than one part in a thousand, on the basis of reasonable intervals of frequencies for the optical data.', '1002.2603-2-29-4': 'It would be interesting to apply these methods to the accurate optical data for thin gold films quoted in Ref. [CITATION].', '1002.2603-2-30-0': 'Before closing the paper, we should note that the relevance of the sample-to-sample dependence of the optical data observed in [CITATION] for the theory of the Casimir effect has been questioned by the authors of Ref. [CITATION], who observed that this dependence mostly originates from relaxation processes of free conduction electrons at infrared and optical frequencies, due for example to different grain sizes in thin films.', '1002.2603-2-30-1': 'The main consequence of these sample-dependent features is the large variability of the Drude parameters, extracted from fits of the low-frequency optical data of the films, which constitutes the basic source of variation of the computed Casimir force reported in Ref. [CITATION].', '1002.2603-2-30-2': 'According to the authors of Ref. [CITATION], relaxation properties of conduction electrons in thin films, described by the fitted values of the Drude parameters, are not relevant for the Casimir effect.', '1002.2603-2-30-3': 'Indeed, according to these authors the quantity [MATH] to be used in Lifshitz formula should not be understood as the actual electric permittivity of the plate, as derived from optical measurements on the sample, but it should be rather regarded as a phenomenological quantity connected to but not identical to the optical electric permittivity of the film.', '1002.2603-2-30-4': 'The ansatz offered by them for [MATH] is dubbed as generalized plasma model, and following Ref. [CITATION] we denote it as [MATH].', '1002.2603-2-30-5': 'This quantity is a semianalytical mathematical construct, defined by the formula: [EQUATION] where [MATH] represents the contribution of core electrons, while the term proportional to the square of the plasma frequency [MATH] describes conduction electrons.', '1002.2603-2-30-6': 'The most striking qualitative feature of this expression is the neglect of ohmic dissipation in the contribution from conduction electrons, but this is not all.', '1002.2603-2-30-7': 'Indeed, the ansatz prescribes that only the core-electron contribution [MATH] should be extracted from optical data of the film.', '1002.2603-2-30-8': 'On the contrary, and more importantly, according to Ref. [CITATION] the value of the plasma frequency [MATH] to be used in Eq. ([REF]) should be the one pertaining to a perfect crystal of the [MATH] material, and not the one obtained by a Drude-model fit of the low-frequency optical data of the film actually used in the experiment.', '1002.2603-2-30-9': 'The justification provided for this choice of the plasma frequency by the authors of Ref. [CITATION] is that the contribution of conduction electrons to the Casimir force should depend only on properties determined by the structure of the crystal cell, which are independent of the sample-to-sample variability determined by the peculiar grain structure of the film, reported in Ref. [CITATION].', '1002.2603-2-30-10': 'It should be noted that for gold, the value of the plasma frequency advocated in [CITATION], [MATH] eV/[MATH], is much higher than the fit values quoted in Ref. [CITATION], which range from 6.8 to 8.4 eV/[MATH].', '1002.2603-2-30-11': 'As a result, the approach advocated in Ref. [CITATION] leads to larger magnitudes of the Casimir force, as compared to the values derived in Ref. [CITATION], with differences ranging, depending on the sample, from 5 [MATH] to 14 [MATH] at 100 nm.', '1002.2603-2-30-12': 'There is no room here to further discuss the merits and faults of these approaches, and we refer the reader to [CITATION] for a thorough analysis.', '1002.2603-2-30-13': 'It is fair to note though that a series of recent experiments by one experimental group [CITATION] appears to favor the generalized plasma approach, and to rule out the more conventional approach based on actual optical data followed in Refs. [CITATION].', '1002.2603-2-31-0': 'The future will tell what is the correct description.', '1002.2603-2-31-1': 'In the meanwhile, we remark that whatever approach is followed, the methods proposed in this paper may prove useful to obtain more reliable estimates of the Casimir force for future experiments.'} | [['1002.2603-1-7-0', '1002.2603-2-7-0'], ['1002.2603-1-7-1', '1002.2603-2-7-1'], ['1002.2603-1-7-2', '1002.2603-2-7-2'], ['1002.2603-1-7-3', '1002.2603-2-7-3'], ['1002.2603-1-7-4', '1002.2603-2-7-4'], ['1002.2603-1-7-5', '1002.2603-2-7-5'], ['1002.2603-1-7-6', '1002.2603-2-7-6'], ['1002.2603-1-7-7', '1002.2603-2-7-7'], ['1002.2603-1-7-8', '1002.2603-2-7-8'], ['1002.2603-1-7-9', '1002.2603-2-7-9'], ['1002.2603-1-7-10', '1002.2603-2-7-10'], ['1002.2603-1-7-11', '1002.2603-2-7-11'], ['1002.2603-1-7-12', '1002.2603-2-7-12'], ['1002.2603-1-7-13', '1002.2603-2-7-13'], ['1002.2603-1-7-14', '1002.2603-2-7-14'], ['1002.2603-1-7-15', '1002.2603-2-7-15'], ['1002.2603-1-7-16', 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'1002.2603-2-23-2'], ['1002.2603-1-23-3', '1002.2603-2-23-3'], ['1002.2603-1-23-4', '1002.2603-2-23-4'], ['1002.2603-1-23-5', '1002.2603-2-23-5'], ['1002.2603-1-28-0', '1002.2603-2-28-0'], ['1002.2603-1-28-1', '1002.2603-2-28-1'], ['1002.2603-1-28-2', '1002.2603-2-28-2'], ['1002.2603-1-28-3', '1002.2603-2-28-3'], ['1002.2603-1-28-4', '1002.2603-2-28-4'], ['1002.2603-1-28-5', '1002.2603-2-28-5'], ['1002.2603-1-28-6', '1002.2603-2-28-6'], ['1002.2603-1-28-7', '1002.2603-2-28-7'], ['1002.2603-1-28-8', '1002.2603-2-28-8'], ['1002.2603-1-28-9', '1002.2603-2-28-9'], ['1002.2603-1-6-0', '1002.2603-2-6-0'], ['1002.2603-1-6-1', '1002.2603-2-6-1'], ['1002.2603-1-6-2', '1002.2603-2-6-2'], ['1002.2603-1-6-4', '1002.2603-2-6-4'], ['1002.2603-1-6-5', '1002.2603-2-6-5'], ['1002.2603-1-6-6', '1002.2603-2-6-6'], ['1002.2603-1-6-7', '1002.2603-2-6-7'], ['1002.2603-1-6-8', '1002.2603-2-6-8'], ['1002.2603-1-27-0', '1002.2603-2-27-0'], ['1002.2603-1-27-1', '1002.2603-2-27-1'], ['1002.2603-1-27-2', '1002.2603-2-27-2'], ['1002.2603-1-27-3', '1002.2603-2-27-3'], ['1002.2603-1-3-0', '1002.2603-2-3-0'], ['1002.2603-1-3-1', '1002.2603-2-3-1'], ['1002.2603-1-3-2', '1002.2603-2-3-2'], ['1002.2603-1-3-3', '1002.2603-2-3-3'], ['1002.2603-1-3-4', '1002.2603-2-3-4'], ['1002.2603-1-3-5', '1002.2603-2-3-5'], ['1002.2603-1-3-6', '1002.2603-2-3-6'], ['1002.2603-1-3-7', '1002.2603-2-3-7'], ['1002.2603-1-3-8', '1002.2603-2-3-8'], ['1002.2603-1-9-0', '1002.2603-2-9-0'], ['1002.2603-1-9-1', '1002.2603-2-9-1'], ['1002.2603-1-25-0', '1002.2603-2-25-0'], ['1002.2603-1-11-0', '1002.2603-2-11-0'], ['1002.2603-1-11-1', '1002.2603-2-11-1'], ['1002.2603-1-11-2', '1002.2603-2-11-2'], ['1002.2603-1-11-3', '1002.2603-2-11-3'], ['1002.2603-1-11-4', '1002.2603-2-11-4'], ['1002.2603-1-11-5', '1002.2603-2-11-5'], ['1002.2603-1-11-6', '1002.2603-2-11-6'], ['1002.2603-1-11-8', '1002.2603-2-11-8'], ['1002.2603-1-12-0', '1002.2603-2-12-0'], ['1002.2603-1-12-1', '1002.2603-2-12-1'], ['1002.2603-1-12-2', '1002.2603-2-12-2'], ['1002.2603-1-12-3', '1002.2603-2-12-3'], ['1002.2603-1-12-4', '1002.2603-2-12-4'], ['1002.2603-1-12-7', '1002.2603-2-12-7'], ['1002.2603-1-12-8', '1002.2603-2-12-8'], ['1002.2603-1-12-9', '1002.2603-2-12-9'], ['1002.2603-1-12-10', '1002.2603-2-12-10'], ['1002.2603-1-12-11', '1002.2603-2-12-11']] | [['1002.2603-1-19-0', '1002.2603-2-19-0'], ['1002.2603-1-4-2', '1002.2603-2-4-2'], ['1002.2603-1-20-2', '1002.2603-2-20-2'], ['1002.2603-1-18-7', '1002.2603-2-18-7'], ['1002.2603-1-6-3', '1002.2603-2-6-3'], ['1002.2603-1-9-2', '1002.2603-2-9-2'], ['1002.2603-1-11-7', '1002.2603-2-11-7'], ['1002.2603-1-12-6', '1002.2603-2-12-6']] | [] | [] | [] | ['1002.2603-1-12-5', '1002.2603-2-12-5'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1002.2603 | null | null | null | null | null |
1711.00427 | {'1711.00427-1-0-0': 'It has been recently established that the volatility of financial assets is rough.', '1711.00427-1-0-1': 'This means that the behavior of the log-volatility process is similar to that of a fractional Brownian motion with Hurst parameter around 0.1.', '1711.00427-1-0-2': 'Motivated by this finding, we wish to define a natural and relevant limit for the fractional Brownian motion when [MATH] goes to zero.', '1711.00427-1-0-3': 'We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.', '1711.00427-1-1-0': 'Keywords: Fractional Brownian motion, log-correlated random field, rough volatility, multifractal processes.', '1711.00427-1-2-0': '# Introduction', '1711.00427-1-3-0': 'The fractional Brownian motion (fBm for short) is a very popular modeling object in many fields such as hydrology, see for example [CITATION], telecommunications and network traffic, see [CITATION] among others and finance, see the seminal paper [CITATION].', '1711.00427-1-3-1': 'A fBm [MATH] with Hurst parameter [MATH] is a zero-mean Gaussian process with covariance kernel given by [EQUATION].', '1711.00427-1-4-0': 'It has stationary increments and is self-similar with parameter [MATH], that is [MATH] has the same law as [MATH] for any [MATH].', '1711.00427-1-4-1': 'Furthermore, sample paths of fBm have almost surely Holder regularity [MATH] for any [MATH].', '1711.00427-1-5-0': 'One of the main probabilistic features which motivates the use of fBm in the applications mentioned above, is the long memory property of the increments when [MATH].', '1711.00427-1-5-1': 'This means that for [MATH], we have [EQUATION].', '1711.00427-1-5-2': 'Thus the auto-covariance function of the fBm increments decays slowly, which is interesting when modeling persistent phenomena.', '1711.00427-1-6-0': 'However, recently, a new paradigm has been introduced in [CITATION] for the use of fBm in finance.', '1711.00427-1-6-1': 'Indeed, a careful analysis of financial time series suggests that the log-volatility process, that is the intensity of the price variations of an asset, actually behaves like a fBm with Hurst parameter of order [MATH].', '1711.00427-1-6-2': 'Hence various approaches using a fBm with small Hurst parameter have been introduced for volatility modeling.', '1711.00427-1-6-3': 'These models are referred to as rough volatility models, see [CITATION] for more details and practical applications.', '1711.00427-1-7-0': 'Such small estimated values for [MATH] (between 0.05 and 0.2) have been found when studying the volatility process of thousands of assets, see [CITATION].', '1711.00427-1-7-1': 'Consequently, a natural question is the behavior of the fBm in the limiting case when [MATH] is sent to zero.', '1711.00427-1-7-2': 'Of course, putting directly [MATH] in the covariance function does not lead to a relevant process.', '1711.00427-1-7-3': 'Thus, in this work, we wish to build a suitable sequence of normalized fBms and describe its limit as [MATH] goes to zero.', '1711.00427-1-7-4': 'This will lead us to a possible definition of the fractional Brownian motion for [MATH].', '1711.00427-1-7-5': 'Note that several authors already defined some fractional Brownian motion for [MATH], see in particular [CITATION].', '1711.00427-1-7-6': 'This is usually done through a regularization procedure.', '1711.00427-1-7-7': 'Our approach here is quite simple and probably more natural from the financial viewpoint we have in mind.', '1711.00427-1-7-8': 'Instead of regularizing the process, we choose to normalize it in order to get a non-degenerate limit.', '1711.00427-1-7-9': 'Our normalized sequence of processes [MATH] is defined through [EQUATION] where [MATH].', '1711.00427-1-7-10': 'Substracting the integral in the numerator and dividing by [MATH] enables us to get a non-trivial limit for our sequence as [MATH] goes to [MATH].', '1711.00427-1-8-0': 'Our main result is the convergence of [MATH], seen as a random element in the space of tempered distributions, towards an approximately log-correlated Gaussian field.', '1711.00427-1-8-1': 'Denote by [MATH] the real Schwartz space, that is the set of real-valued functions on [MATH] whose derivatives of all orders exist and decay faster than any polynomial at infinity.', '1711.00427-1-8-2': 'We write [MATH] for the dual of [MATH], that is the space of tempered distributions.', '1711.00427-1-8-3': 'We also define the subspace [MATH] of the real Schwartz space, consisting of functions [MATH] from [MATH] with [MATH], and its topological dual [MATH].', '1711.00427-1-8-4': 'A log-correlated Gaussian field (LGF for short) [MATH], is a centered Gaussian field whose covariance kernel satisfies [EQUATION] for any [MATH], see [CITATION] for an overview on LGF.', '1711.00427-1-8-5': 'We show in this paper that the limit of [MATH] as [MATH] goes to zero is "almost" a log-correlated Gaussian field, see Section [REF] for an accurate result.', '1711.00427-1-9-0': 'LGFs are closely related to some multifractal processes pioneered by Mandelbrot (see for example [CITATION]), and further developed in [CITATION], among others.', '1711.00427-1-9-1': 'A process [MATH] is said to be multifractal if for a range of values of [MATH], we have for some [MATH] [EQUATION] where [MATH] is a constant and [MATH] is a non-linear concave function.', '1711.00427-1-9-2': 'In particular, the multifractal random walk model for the log-price of an asset in [CITATION] satisfies such property.', '1711.00427-1-9-3': 'It is defined as [MATH] where [MATH] is a Brownian motion and [EQUATION] with [MATH] and [MATH] a Gaussian process such that for some [MATH] and [MATH] [EQUATION] see [CITATION] for details.', '1711.00427-1-9-4': 'Hence we see that [MATH] formally corresponds to a measure of the form [MATH], where [MATH] is a LGF.', '1711.00427-1-9-5': 'For the precise definition of such measures, see [CITATION] and the generalizations on Gaussian multiplicative chaos in [CITATION] and the references therein.', '1711.00427-1-9-6': 'Finally note that LGFs and more generally the associated theory of Gaussian multiplicative chaos have extensive use in other fields than finance, such as turbulence, see [CITATION], disordered systems, see [CITATION] and Liouville quantum gravity, see [CITATION].', '1711.00427-1-10-0': 'In the following section we introduce our main theorem, that is an accurate statement about the convergence of the normalized fBm towards a LGF as [MATH] goes to zero.', '1711.00427-1-10-1': 'We also discuss the multifractal properties of the limiting LGF in the same section.', '1711.00427-1-10-2': 'The proof of our theorem can be found in Section [REF].', '1711.00427-1-11-0': '# Convergence of the fBM towards a LGF', '1711.00427-1-12-0': '## Main result', '1711.00427-1-13-0': 'We define the weak convergence of elements in [MATH] as in Proposition 12.2 in [CITATION].', '1711.00427-1-13-1': 'We say that [MATH] converges weakly to [MATH] as [MATH] tends to [MATH] if for any [MATH] we have [EQUATION] in law, as [MATH] tends to [MATH].', '1711.00427-1-13-2': 'The main result of our paper is the following.', '1711.00427-1-14-0': 'The sequence [MATH] converges weakly as [MATH] tends to zero towards a centered Gaussian field [MATH] satisfying for any [MATH] [EQUATION] where for [MATH], [MATH] and [MATH] [EQUATION] with [EQUATION].', '1711.00427-1-15-0': 'We see that when [MATH] for some [MATH], then [MATH] is a bounded continuous function.', '1711.00427-1-15-1': 'Hence the covariance kernel exhibits the same type of singularity as that of a LGF.', '1711.00427-1-15-2': 'Consequently, in our framework, the limit when [MATH] goes to zero of a normalized version of the fBm is "almost" a LGF.', '1711.00427-1-16-0': '## Multifractal properties', '1711.00427-1-17-0': 'In [CITATION], the authors study the case of centered Gaussian fields on any domain [MATH] with covariance kernel satisfying [EQUATION] where [MATH] and [MATH] is a bounded continuous function.', '1711.00427-1-17-1': 'Thus, if we restrict [MATH] to the domain [MATH] for some fixed [MATH], then [MATH] is included in the framework of [CITATION].', '1711.00427-1-17-2': 'In particular, their results on the multifractal spectrum of Gaussian fields apply on this restricted domain.', '1711.00427-1-18-0': 'Motivated by the preceding paragraph, we first fix [MATH] and define an approximate volatility measure [MATH] by [EQUATION] for some constant [MATH].', '1711.00427-1-18-1': 'Here we assume that [MATH] vanishes on [MATH].', '1711.00427-1-18-2': 'From the result of Theorem [REF] we deduce the following corollary.', '1711.00427-1-18-3': 'In what follows, convergence in the [MATH] norm stands for the usual convergence of random variables in [MATH].', '1711.00427-1-19-0': 'For [MATH], [MATH] converges as [MATH] approaches zero to a random measure [MATH] in the following sense, [EQUATION].', '1711.00427-1-19-1': 'Moreover, the limiting measure [MATH] is the so-called Gaussian multiplicative chaos.', '1711.00427-1-20-0': 'The proof Corollary [REF] is given in Section [REF].', '1711.00427-1-20-1': 'For the definition and properties of Gaussian multiplicative chaos we refer to a survey paper by Rhodes and Vargas [CITATION].', '1711.00427-1-20-2': 'We will briefly explain some of the properties of [MATH] from the theory of Gaussian multiplicative chaos.', '1711.00427-1-21-0': 'We first describe the behavior of the moments of [MATH].', '1711.00427-1-21-1': 'Note that for [MATH], [MATH] is of the form ([REF]).', '1711.00427-1-21-2': 'From Proposition 2.5 in [CITATION], it follows that for all [MATH] and [MATH], there exists [MATH] such that [EQUATION] where [MATH].', '1711.00427-1-21-3': 'Hence we do obtain a multifractal scaling as described in the introduction.', '1711.00427-1-22-0': 'Finally we consider a quantity closely related to the function [MATH] above: the spectrum of singularities of [MATH].', '1711.00427-1-22-1': 'For any [MATH] and [MATH], we define [EQUATION]', '1711.00427-1-22-2': 'The set [MATH] somehow corresponds to the points [MATH] where the Holder regularity of [MATH] is equal to [MATH].', '1711.00427-1-22-3': 'Let [MATH] denote the Hausdorff dimension of a set [MATH].', '1711.00427-1-22-4': 'Theorem 2.6 in [CITATION] states that [EQUATION]', '1711.00427-1-22-5': 'In particular, we remark that [EQUATION].', '1711.00427-1-22-6': 'This equality means that the Frish-Parisi conjecture relating the scaling exponents of a process to its spectrum of singularities holds in our case, see [CITATION] and [CITATION] for more details on the multifractal formalism.', '1711.00427-1-23-0': 'The rest of this paper is devoted to the proofs of Theorem [REF] and Corollary [REF].', '1711.00427-1-24-0': '# Proof of Theorem [REF] and Corollary [REF].', '1711.00427-1-25-0': 'For [MATH], let [MATH] .', '1711.00427-1-25-1': 'We start the proof with the following important lemma.', '1711.00427-1-26-0': 'For any non-zero [MATH] with [MATH], we have [EQUATION]', '1711.00427-1-26-1': 'We write [MATH], where [EQUATION] and [MATH], where [EQUATION].', '1711.00427-1-26-2': 'We have [EQUATION] so we can assume that [MATH].', '1711.00427-1-27-0': 'Note that for any [MATH], [EQUATION] and therefore [EQUATION].', '1711.00427-1-27-1': 'Next we deal with [MATH], [MATH].', '1711.00427-1-27-2': 'We consider several cases.', '1711.00427-1-28-0': 'Case 1: Assume that [MATH].', '1711.00427-1-28-1': 'We easily get [EQUATION].', '1711.00427-1-28-2': 'For [MATH], note that [EQUATION].', '1711.00427-1-28-3': 'Hence we obtain [EQUATION] and therefore [EQUATION].', '1711.00427-1-28-4': 'Define [EQUATION]', '1711.00427-1-28-5': 'It follows that [EQUATION]', '1711.00427-1-28-6': 'Thus we have [EQUATION]', '1711.00427-1-28-7': 'Consequently, [EQUATION] and [EQUATION].', '1711.00427-1-29-0': 'We also easily get that for [MATH] [EQUATION].', '1711.00427-1-29-1': 'Furthermore, [EQUATION].', '1711.00427-1-29-2': 'Thus we obtain [EQUATION] and we deduce [EQUATION]', '1711.00427-1-29-3': 'Now define [EQUATION]', '1711.00427-1-29-4': 'We have [EQUATION]', '1711.00427-1-29-5': 'Finally we obtain [EQUATION] from which ([REF]) readily follows for [MATH].', '1711.00427-1-30-0': 'Case 2: Assume that [MATH].', '1711.00427-1-30-1': 'We use that [EQUATION] to deduce that ([REF]) follows from the proof of Case 1.', '1711.00427-1-31-0': 'Case 3: Assume that [MATH].', '1711.00427-1-31-1': 'We write [MATH].', '1711.00427-1-31-2': 'Repeating the same steps as in Case 1, we get [EQUATION] and [EQUATION]', '1711.00427-1-31-3': 'It follows that [EQUATION] and we get ([REF]).', '1711.00427-1-31-4': 'Let [MATH].', '1711.00427-1-31-5': 'Since [MATH] and [MATH] are centered Gaussians taking values in [MATH], to prove Theorem [REF], it is enough to show that [EQUATION].', '1711.00427-1-31-6': 'Furthermore, for any [MATH] we have [EQUATION].', '1711.00427-1-31-7': 'Hence Theorem [REF] immediately follows from the next lemma.', '1711.00427-1-32-0': 'For any [MATH], [EQUATION]', '1711.00427-1-32-1': 'First note that for [MATH], [MATH].', '1711.00427-1-32-2': 'Therefore for any [MATH], [EQUATION].', '1711.00427-1-32-3': 'Moreover, [EQUATION]', '1711.00427-1-32-4': 'Hence, from dominated convergence, it follows that [EQUATION].', '1711.00427-1-32-5': 'Let [MATH], where [MATH] and [MATH].', '1711.00427-1-32-6': 'Since [MATH] is concave, for any [MATH], we have [MATH].', '1711.00427-1-32-7': 'Therefore for any [MATH], [EQUATION].', '1711.00427-1-32-8': 'Since [EQUATION] we get from dominated convergence [EQUATION].', '1711.00427-1-32-9': 'It remains to show that [EQUATION].', '1711.00427-1-32-10': 'We again consider several cases.', '1711.00427-1-33-0': 'Case 1: Assume that [MATH].', '1711.00427-1-33-1': 'We obviously have [EQUATION].', '1711.00427-1-33-2': 'Moreover, from a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-1-33-3': 'It follows that for any [MATH], [EQUATION]', '1711.00427-1-33-4': 'Case 1bis: Assume that [MATH].', '1711.00427-1-33-5': 'Since [MATH] we get [EQUATION]', '1711.00427-1-33-6': 'Case 2: Assume that [MATH].', '1711.00427-1-33-7': 'Since [MATH], we have [EQUATION]', '1711.00427-1-33-8': 'Case 3: Assume that [MATH] and [MATH].', '1711.00427-1-33-9': 'Recall that when [MATH], [EQUATION]', '1711.00427-1-33-10': 'Since [MATH] we have [EQUATION]', '1711.00427-1-33-11': 'Using a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-1-33-12': 'It follows that [EQUATION]', '1711.00427-1-33-13': 'Case 4: Assume that [MATH] and [MATH].', '1711.00427-1-33-14': 'Repeating the same lines as in Case 3 but exchanging the roles of [MATH] and [MATH] we get [EQUATION]', '1711.00427-1-33-15': 'Case 5: Assume that [MATH] and [MATH].', '1711.00427-1-33-16': 'Since [MATH], it follows from Case 3 that [EQUATION]', '1711.00427-1-33-17': 'Case 6: Assume that [MATH] and [MATH].', '1711.00427-1-33-18': 'Using again [MATH], we get from Case 4 that [EQUATION]', '1711.00427-1-33-19': 'Finally, we clearly have [EQUATION].', '1711.00427-1-33-20': 'Thus we conclude the proof using the dominated convergence theorem.', '1711.00427-1-34-0': 'Proof of Corollary [REF] From Theorem 25 in [CITATION] it follows that in order to prove Corollary [REF], we need to show ([REF]) and that [MATH] are uniformly integrable.', '1711.00427-1-35-0': 'First note that all [MATH], [MATH], can be constructed on the same probability space, as a convolution of the same Brownian motion with different kernels which depend on [MATH].', '1711.00427-1-35-1': 'Then, a standard computation gives for any compact set [MATH], [EQUATION]', '1711.00427-1-35-2': 'From the proof of Lemma [REF] we have for [MATH], [EQUATION].', '1711.00427-1-35-3': 'Repeating the same argument as in Lemma [REF], we have [EQUATION]', '1711.00427-1-35-4': 'Since [MATH] by our assumption, it follows that [EQUATION] and therefore [MATH] are uniformly integrable.'} | {'1711.00427-2-0-0': 'It has been recently established that the volatility of financial assets is rough.', '1711.00427-2-0-1': 'This means that the behavior of the log-volatility process is similar to that of a fractional Brownian motion with Hurst parameter around 0.1.', '1711.00427-2-0-2': 'Motivated by this finding, we wish to define a natural and relevant limit for the fractional Brownian motion when [MATH] goes to zero.', '1711.00427-2-0-3': 'We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.', '1711.00427-2-1-0': 'Keywords: Fractional Brownian motion, log-correlated random field, rough volatility, multifractal processes.', '1711.00427-2-2-0': '# Introduction', '1711.00427-2-3-0': 'The fractional Brownian motion (fBm for short) is a very popular modeling object in many fields such as hydrology, see for example [CITATION], telecommunications and network traffic, see [CITATION] among others and finance, see the seminal paper [CITATION].', '1711.00427-2-3-1': 'A fBm [MATH] with Hurst parameter [MATH] is a zero-mean Gaussian process with covariance kernel given by [EQUATION].', '1711.00427-2-4-0': 'It has stationary increments and is self-similar with parameter [MATH], that is [MATH] has the same law as [MATH] for any [MATH].', '1711.00427-2-4-1': 'Furthermore, sample paths of fBm have almost surely Holder regularity [MATH] for any [MATH].', '1711.00427-2-5-0': 'One of the main probabilistic features which motivates the use of fBm in the applications mentioned above, is the long memory property of the increments when [MATH].', '1711.00427-2-5-1': 'This means that for [MATH], we have [EQUATION].', '1711.00427-2-5-2': 'Thus the auto-covariance function of the fBm increments decays slowly, which is interesting when modeling persistent phenomena.', '1711.00427-2-6-0': 'However, recently, a new paradigm has been introduced in [CITATION] for the use of fBm in finance.', '1711.00427-2-6-1': 'Indeed, a careful analysis of financial time series suggests that the log-volatility process, that is the intensity of the price variations of an asset, actually behaves like a fBm with Hurst parameter of order [MATH].', '1711.00427-2-6-2': 'Hence various approaches using a fBm with small Hurst parameter have been introduced for volatility modeling.', '1711.00427-2-6-3': 'These models are referred to as rough volatility models, see [CITATION] for more details and practical applications.', '1711.00427-2-7-0': 'Such small estimated values for [MATH] (between 0.05 and 0.2) have been found when studying the volatility process of thousands of assets, see [CITATION].', '1711.00427-2-7-1': 'Consequently, a natural question is the behavior of the fBm in the limiting case when [MATH] is sent to zero.', '1711.00427-2-7-2': 'Of course, putting directly [MATH] in the covariance function does not lead to a relevant process.', '1711.00427-2-7-3': 'Thus, in this work, we wish to build a suitable sequence of normalized fBms and describe its limit as [MATH] goes to zero.', '1711.00427-2-7-4': 'This will lead us to a possible definition of the fractional Brownian motion for [MATH].', '1711.00427-2-7-5': 'Note that several authors already defined some fractional Brownian motion for [MATH], see in particular [CITATION].', '1711.00427-2-7-6': 'This is usually done through a regularization procedure.', '1711.00427-2-7-7': 'Our approach here is quite simple and probably more natural from the financial viewpoint we have in mind.', '1711.00427-2-7-8': 'Instead of regularizing the process, we choose to normalize it in order to get a non-degenerate limit.', '1711.00427-2-7-9': 'Our normalized sequence of processes [MATH] is defined through [EQUATION] where [MATH].', '1711.00427-2-7-10': 'Substracting the integral in the numerator and dividing by [MATH] enables us to get a non-trivial limit for our sequence as [MATH] goes to [MATH].', '1711.00427-2-8-0': 'Our main result is the convergence of [MATH], seen as a random element in the space of tempered distributions, towards an approximately log-correlated Gaussian field.', '1711.00427-2-8-1': 'Denote by [MATH] the real Schwartz space, that is the set of real-valued functions on [MATH] whose derivatives of all orders exist and decay faster than any polynomial at infinity.', '1711.00427-2-8-2': 'We write [MATH] for the dual of [MATH], that is the space of tempered distributions.', '1711.00427-2-8-3': 'We also define the subspace [MATH] of the real Schwartz space, consisting of functions [MATH] from [MATH] with [MATH], and its topological dual [MATH].', '1711.00427-2-8-4': 'A log-correlated Gaussian field (LGF for short) [MATH], is a centered Gaussian field whose covariance kernel satisfies [EQUATION] for any [MATH], see [CITATION] for an overview on LGF.', '1711.00427-2-8-5': 'We show in this paper that the limit of [MATH] as [MATH] goes to zero is "almost" a log-correlated Gaussian field, see Section [REF] for an accurate result.', '1711.00427-2-9-0': 'LGFs are closely related to some multifractal processes pioneered by Mandelbrot (see for example [CITATION]), and further developed in [CITATION], among others.', '1711.00427-2-9-1': 'A process [MATH] is said to be multifractal if for a range of values of [MATH], we have for some [MATH] [EQUATION] where [MATH] is a constant and [MATH] is a non-linear concave function.', '1711.00427-2-9-2': 'In particular, the multifractal random walk model for the log-price of an asset in [CITATION] satisfies such property.', '1711.00427-2-9-3': 'It is defined as [MATH] where [MATH] is a Brownian motion and [EQUATION] with [MATH] and [MATH] a Gaussian process such that for some [MATH] and [MATH] [EQUATION] see [CITATION] for details.', '1711.00427-2-9-4': 'Hence we see that [MATH] formally corresponds to a measure of the form [MATH], where [MATH] is a LGF.', '1711.00427-2-9-5': 'For the precise definition of such measures, see [CITATION] and the generalizations on Gaussian multiplicative chaos in [CITATION] and the references therein.', '1711.00427-2-9-6': 'Finally note that LGFs and more generally the associated theory of Gaussian multiplicative chaos have extensive use in other fields than finance, such as turbulence, see [CITATION], disordered systems, see [CITATION] and Liouville quantum gravity, see [CITATION].', '1711.00427-2-10-0': 'In the following section we introduce our main theorem, that is an accurate statement about the convergence of the normalized fBm towards a LGF as [MATH] goes to zero.', '1711.00427-2-10-1': 'We also discuss the multifractal properties of the limiting LGF in the same section.', '1711.00427-2-10-2': 'The proof of our theorem can be found in Section [REF].', '1711.00427-2-11-0': '# Convergence of the fBM towards a LGF', '1711.00427-2-12-0': '## Main result', '1711.00427-2-13-0': 'We define the weak convergence of elements in [MATH] as in Proposition 12.2 in [CITATION].', '1711.00427-2-13-1': 'We say that [MATH] converges weakly to [MATH] as [MATH] tends to [MATH] if for any [MATH] we have [EQUATION] in law, as [MATH] tends to [MATH].', '1711.00427-2-13-2': 'The main result of our paper is the following.', '1711.00427-2-14-0': 'The sequence [MATH] converges weakly as [MATH] tends to zero towards a centered Gaussian field [MATH] satisfying for any [MATH] [EQUATION] where for [MATH], [MATH] and [MATH] [EQUATION] with [EQUATION].', '1711.00427-2-15-0': 'We see that when [MATH] for some [MATH], then [MATH] is a bounded continuous function.', '1711.00427-2-15-1': 'Hence the covariance kernel exhibits the same type of singularity as that of a LGF.', '1711.00427-2-15-2': 'Consequently, in our framework, the limit when [MATH] goes to zero of a normalized version of the fBm is "almost" a LGF.', '1711.00427-2-16-0': '## Multifractal properties', '1711.00427-2-17-0': 'In [CITATION], the authors study the case of centered Gaussian fields on any domain [MATH] with covariance kernel satisfying [EQUATION] where [MATH] and [MATH] is a bounded continuous function.', '1711.00427-2-17-1': 'Thus, if we restrict [MATH] to the domain [MATH] for some fixed [MATH], then [MATH] is included in the framework of [CITATION].', '1711.00427-2-17-2': 'In particular, their results on the multifractal spectrum of Gaussian fields apply on this restricted domain.', '1711.00427-2-18-0': 'Motivated by the preceding paragraph, we first fix [MATH] and define an approximate volatility measure [MATH] by [EQUATION] for some constant [MATH].', '1711.00427-2-18-1': 'Here we assume that [MATH] vanishes on [MATH].', '1711.00427-2-18-2': 'From the result of Theorem [REF] we deduce the following corollary.', '1711.00427-2-18-3': 'In what follows, convergence in the [MATH] norm stands for the usual convergence of random variables in [MATH].', '1711.00427-2-19-0': 'For [MATH], [MATH] converges as [MATH] approaches zero to a random measure [MATH] in the following sense, [EQUATION].', '1711.00427-2-19-1': 'Moreover, the limiting measure [MATH] is the so-called Gaussian multiplicative chaos.', '1711.00427-2-20-0': 'The proof Corollary [REF] is given in Section [REF].', '1711.00427-2-20-1': 'For the definition and properties of Gaussian multiplicative chaos we refer to a survey paper by Rhodes and Vargas [CITATION].', '1711.00427-2-20-2': 'We will briefly explain some of the properties of [MATH] from the theory of Gaussian multiplicative chaos.', '1711.00427-2-21-0': 'We first describe the behavior of the moments of [MATH].', '1711.00427-2-21-1': 'Note that for [MATH], [MATH] is of the form ([REF]).', '1711.00427-2-21-2': 'From Proposition 2.5 in [CITATION], it follows that for all [MATH] and [MATH], there exists [MATH] such that [EQUATION] where [MATH].', '1711.00427-2-21-3': 'Hence we do obtain a multifractal scaling as described in the introduction.', '1711.00427-2-22-0': 'Finally we consider a quantity closely related to the function [MATH] above: the spectrum of singularities of [MATH].', '1711.00427-2-22-1': 'For any [MATH] and [MATH], we define [EQUATION]', '1711.00427-2-22-2': 'The set [MATH] somehow corresponds to the points [MATH] where the Holder regularity of [MATH] is equal to [MATH].', '1711.00427-2-22-3': 'Let [MATH] denote the Hausdorff dimension of a set [MATH].', '1711.00427-2-22-4': 'Theorem 2.6 in [CITATION] states that [EQUATION]', '1711.00427-2-22-5': 'In particular, we remark that [EQUATION].', '1711.00427-2-22-6': 'This equality means that the Frish-Parisi conjecture relating the scaling exponents of a process to its spectrum of singularities holds in our case, see [CITATION] and [CITATION] for more details on the multifractal formalism.', '1711.00427-2-23-0': 'The rest of this paper is devoted to the proofs of Theorem [REF] and Corollary [REF].', '1711.00427-2-24-0': '# Proof of Theorem [REF] and Corollary [REF].', '1711.00427-2-25-0': 'For [MATH], let [MATH] .', '1711.00427-2-25-1': 'We start the proof with the following important lemma.', '1711.00427-2-26-0': 'For any non-zero [MATH] with [MATH], we have [EQUATION]', '1711.00427-2-26-1': 'We write [MATH], where [EQUATION] and [MATH], where [EQUATION].', '1711.00427-2-26-2': 'We have [EQUATION] so we can assume that [MATH].', '1711.00427-2-27-0': 'Note that for any [MATH], [EQUATION] and therefore [EQUATION].', '1711.00427-2-27-1': 'Next we deal with [MATH], [MATH].', '1711.00427-2-27-2': 'We consider several cases.', '1711.00427-2-28-0': 'Case 1: Assume that [MATH].', '1711.00427-2-28-1': 'We easily get [EQUATION].', '1711.00427-2-28-2': 'For [MATH], note that [EQUATION].', '1711.00427-2-28-3': 'Hence we obtain [EQUATION] and therefore [EQUATION].', '1711.00427-2-28-4': 'Define [EQUATION]', '1711.00427-2-28-5': 'It follows that [EQUATION]', '1711.00427-2-28-6': 'Thus we have [EQUATION]', '1711.00427-2-28-7': 'Consequently, [EQUATION] and [EQUATION].', '1711.00427-2-29-0': 'We also easily get that for [MATH] [EQUATION].', '1711.00427-2-29-1': 'Furthermore, [EQUATION].', '1711.00427-2-29-2': 'Thus we obtain [EQUATION] and we deduce [EQUATION]', '1711.00427-2-29-3': 'Now define [EQUATION]', '1711.00427-2-29-4': 'We have [EQUATION]', '1711.00427-2-29-5': 'Finally we obtain [EQUATION] from which ([REF]) readily follows for [MATH].', '1711.00427-2-30-0': 'Case 2: Assume that [MATH].', '1711.00427-2-30-1': 'We use that [EQUATION] to deduce that ([REF]) follows from the proof of Case 1.', '1711.00427-2-31-0': 'Case 3: Assume that [MATH].', '1711.00427-2-31-1': 'We write [MATH].', '1711.00427-2-31-2': 'Repeating the same steps as in Case 1, we get [EQUATION] and [EQUATION]', '1711.00427-2-31-3': 'It follows that [EQUATION] and we get ([REF]).', '1711.00427-2-31-4': 'Let [MATH].', '1711.00427-2-31-5': 'Since [MATH] and [MATH] are centered Gaussians taking values in [MATH], to prove Theorem [REF], it is enough to show that [EQUATION].', '1711.00427-2-31-6': 'Furthermore, for any [MATH] we have [EQUATION].', '1711.00427-2-31-7': 'Hence Theorem [REF] immediately follows from the next lemma.', '1711.00427-2-32-0': 'For any [MATH], [EQUATION]', '1711.00427-2-32-1': 'First note that for [MATH], [MATH].', '1711.00427-2-32-2': 'Therefore for any [MATH], [EQUATION].', '1711.00427-2-32-3': 'Moreover, [EQUATION]', '1711.00427-2-32-4': 'Hence, from dominated convergence, it follows that [EQUATION].', '1711.00427-2-32-5': 'Let [MATH], where [MATH] and [MATH].', '1711.00427-2-32-6': 'Since [MATH] is concave, for any [MATH], we have [MATH].', '1711.00427-2-32-7': 'Therefore for any [MATH], [EQUATION].', '1711.00427-2-32-8': 'Since [EQUATION] we get from dominated convergence [EQUATION].', '1711.00427-2-32-9': 'It remains to show that [EQUATION].', '1711.00427-2-32-10': 'We again consider several cases.', '1711.00427-2-33-0': 'Case 1: Assume that [MATH].', '1711.00427-2-33-1': 'We obviously have [EQUATION].', '1711.00427-2-33-2': 'Moreover, from a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-2-33-3': 'It follows that for any [MATH], [EQUATION]', '1711.00427-2-33-4': 'Case 1bis: Assume that [MATH].', '1711.00427-2-33-5': 'Since [MATH] we get [EQUATION]', '1711.00427-2-33-6': 'Case 2: Assume that [MATH].', '1711.00427-2-33-7': 'Since [MATH], we have [EQUATION]', '1711.00427-2-33-8': 'Case 3: Assume that [MATH] and [MATH].', '1711.00427-2-33-9': 'Recall that when [MATH], [EQUATION]', '1711.00427-2-33-10': 'Since [MATH] we have [EQUATION]', '1711.00427-2-33-11': 'Using a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-2-33-12': 'It follows that [EQUATION]', '1711.00427-2-33-13': 'Case 4: Assume that [MATH] and [MATH].', '1711.00427-2-33-14': 'Repeating the same lines as in Case 3 but exchanging the roles of [MATH] and [MATH] we get [EQUATION]', '1711.00427-2-33-15': 'Case 5: Assume that [MATH] and [MATH].', '1711.00427-2-33-16': 'Since [MATH], it follows from Case 3 that [EQUATION]', '1711.00427-2-33-17': 'Case 6: Assume that [MATH] and [MATH].', '1711.00427-2-33-18': 'Using again [MATH], we get from Case 4 that [EQUATION]', '1711.00427-2-33-19': 'Finally, we clearly have [EQUATION].', '1711.00427-2-33-20': 'Thus we conclude the proof using the dominated convergence theorem.', '1711.00427-2-34-0': 'Proof of Corollary [REF] From Theorem 25 in [CITATION] it follows that in order to prove Corollary [REF], we need to show ([REF]) and that [MATH] are uniformly integrable.', '1711.00427-2-35-0': 'First note that all [MATH], [MATH], can be constructed on the same probability space, as a convolution of the same Brownian motion with different kernels which depend on [MATH].', '1711.00427-2-35-1': 'Then, a standard computation gives for any compact set [MATH], [EQUATION]', '1711.00427-2-35-2': 'From the proof of Lemma [REF] we have for [MATH], [EQUATION].', '1711.00427-2-35-3': 'Repeating the same argument as in Lemma [REF], we have [EQUATION]', '1711.00427-2-35-4': 'Since [MATH] by our assumption, it follows that [EQUATION] and therefore [MATH] are uniformly integrable.'} | [['1711.00427-1-27-0', '1711.00427-2-27-0'], ['1711.00427-1-27-1', '1711.00427-2-27-1'], ['1711.00427-1-27-2', '1711.00427-2-27-2'], ['1711.00427-1-13-0', '1711.00427-2-13-0'], ['1711.00427-1-13-1', '1711.00427-2-13-1'], ['1711.00427-1-13-2', '1711.00427-2-13-2'], ['1711.00427-1-17-0', '1711.00427-2-17-0'], ['1711.00427-1-17-1', '1711.00427-2-17-1'], ['1711.00427-1-17-2', '1711.00427-2-17-2'], ['1711.00427-1-33-0', '1711.00427-2-33-0'], ['1711.00427-1-33-1', '1711.00427-2-33-1'], ['1711.00427-1-33-2', '1711.00427-2-33-2'], ['1711.00427-1-33-3', '1711.00427-2-33-3'], ['1711.00427-1-33-4', '1711.00427-2-33-4'], ['1711.00427-1-33-5', '1711.00427-2-33-5'], ['1711.00427-1-33-6', '1711.00427-2-33-6'], ['1711.00427-1-33-7', '1711.00427-2-33-7'], ['1711.00427-1-33-8', '1711.00427-2-33-8'], ['1711.00427-1-33-9', '1711.00427-2-33-9'], ['1711.00427-1-33-10', '1711.00427-2-33-10'], ['1711.00427-1-33-11', '1711.00427-2-33-11'], ['1711.00427-1-33-12', '1711.00427-2-33-12'], ['1711.00427-1-33-13', '1711.00427-2-33-13'], ['1711.00427-1-33-14', '1711.00427-2-33-14'], ['1711.00427-1-33-15', '1711.00427-2-33-15'], ['1711.00427-1-33-16', '1711.00427-2-33-16'], ['1711.00427-1-33-17', '1711.00427-2-33-17'], ['1711.00427-1-33-18', '1711.00427-2-33-18'], ['1711.00427-1-33-19', '1711.00427-2-33-19'], ['1711.00427-1-33-20', '1711.00427-2-33-20'], ['1711.00427-1-6-0', '1711.00427-2-6-0'], ['1711.00427-1-6-1', '1711.00427-2-6-1'], ['1711.00427-1-6-2', '1711.00427-2-6-2'], ['1711.00427-1-6-3', '1711.00427-2-6-3'], ['1711.00427-1-1-0', '1711.00427-2-1-0'], ['1711.00427-1-18-0', '1711.00427-2-18-0'], ['1711.00427-1-18-1', '1711.00427-2-18-1'], ['1711.00427-1-18-2', '1711.00427-2-18-2'], ['1711.00427-1-18-3', '1711.00427-2-18-3'], ['1711.00427-1-0-0', '1711.00427-2-0-0'], ['1711.00427-1-0-1', '1711.00427-2-0-1'], ['1711.00427-1-0-2', '1711.00427-2-0-2'], 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'1711.00427-1-29-4', '1711.00427-1-31-1', '1711.00427-1-31-4', '1711.00427-1-32-0', '1711.00427-1-32-3', '1711.00427-1-32-5', '1711.00427-2-14-0', '1711.00427-2-25-0', '1711.00427-2-25-1', '1711.00427-2-26-0', '1711.00427-2-26-1', '1711.00427-2-26-2', '1711.00427-2-28-0', '1711.00427-2-28-1', '1711.00427-2-28-2', '1711.00427-2-28-3', '1711.00427-2-28-4', '1711.00427-2-28-5', '1711.00427-2-28-6', '1711.00427-2-28-7', '1711.00427-2-29-1', '1711.00427-2-29-3', '1711.00427-2-29-4', '1711.00427-2-31-1', '1711.00427-2-31-4', '1711.00427-2-32-0', '1711.00427-2-32-3', '1711.00427-2-32-5', '1711.00427-3-14-0', '1711.00427-3-25-0', '1711.00427-3-25-1', '1711.00427-3-26-0', '1711.00427-3-26-1', '1711.00427-3-26-2', '1711.00427-3-28-0', '1711.00427-3-28-1', '1711.00427-3-28-2', '1711.00427-3-28-3', '1711.00427-3-28-4', '1711.00427-3-28-5', '1711.00427-3-28-6', '1711.00427-3-28-7', '1711.00427-3-29-1', '1711.00427-3-29-3', '1711.00427-3-29-4', '1711.00427-3-31-1', '1711.00427-3-31-4', '1711.00427-3-32-0', '1711.00427-3-32-3', '1711.00427-3-32-5', '1711.00427-3-35-1', '1711.00427-3-36-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1711.00427 | {'1711.00427-3-0-0': 'Rough volatility models are becoming increasingly popular in quantitative finance.', '1711.00427-3-0-1': 'In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1.', '1711.00427-3-0-2': 'Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when [MATH] goes to zero.', '1711.00427-3-0-3': 'We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.', '1711.00427-3-1-0': 'Keywords: Fractional Brownian motion, log-correlated random field, rough volatility, multifractal processes.', '1711.00427-3-2-0': '# Introduction', '1711.00427-3-3-0': 'The fractional Brownian motion (fBm for short) is a very popular modeling object in many fields such as hydrology, see for example [CITATION], telecommunications and network traffic, see [CITATION] among others and finance, see the seminal paper [CITATION].', '1711.00427-3-3-1': 'A fBm [MATH] with Hurst parameter [MATH] is a zero-mean Gaussian process with covariance kernel given by [EQUATION].', '1711.00427-3-4-0': 'It has stationary increments and is self-similar with parameter [MATH], that is [MATH] has the same law as [MATH] for any [MATH].', '1711.00427-3-4-1': 'Furthermore, sample paths of fBm have almost surely Holder regularity [MATH] for any [MATH].', '1711.00427-3-5-0': 'One of the main probabilistic features which motivates the use of fBm in the applications mentioned above, is the long memory property of the increments when [MATH].', '1711.00427-3-5-1': 'This means that for [MATH], we have [EQUATION].', '1711.00427-3-5-2': 'Thus the auto-covariance function of the fBm increments decays slowly, which is interesting when modeling persistent phenomena.', '1711.00427-3-6-0': 'However, recently, a new paradigm has been introduced in [CITATION] for the use of fBm in finance.', '1711.00427-3-6-1': 'Indeed, a careful analysis of financial time series suggests that the log-volatility process, that is the intensity of the price variations of an asset, actually behaves like a fBm with Hurst parameter of order [MATH].', '1711.00427-3-6-2': 'Hence various approaches using a fBm with small Hurst parameter have been introduced for volatility modeling.', '1711.00427-3-6-3': 'These models are referred to as rough volatility models, see [CITATION] for more details and practical applications.', '1711.00427-3-7-0': 'Such small estimated values for [MATH] (between 0.05 and 0.2) have been found when studying the volatility process of thousands of assets, see [CITATION].', '1711.00427-3-7-1': 'Consequently, a natural question is the behavior of the fBm in the limiting case when [MATH] is sent to zero.', '1711.00427-3-7-2': 'Of course, putting directly [MATH] in the covariance function does not lead to a relevant process.', '1711.00427-3-7-3': 'Thus, in this work, we wish to build a suitable sequence of normalized fBms and describe its limit as [MATH] goes to zero.', '1711.00427-3-7-4': 'This will lead us to a possible definition of the fractional Brownian motion for [MATH].', '1711.00427-3-7-5': 'Note that several authors already defined some fractional Brownian motion for [MATH], see in particular [CITATION].', '1711.00427-3-7-6': 'This is usually done through a regularization procedure.', '1711.00427-3-7-7': 'Our approach here is quite simple and probably more natural from the financial viewpoint we have in mind.', '1711.00427-3-7-8': 'Instead of regularizing the process, we choose to normalize it in order to get a non-degenerate limit.', '1711.00427-3-7-9': 'Our normalized sequence of processes [MATH] is defined through [EQUATION] where [MATH].', '1711.00427-3-7-10': 'Substracting the integral in the numerator and dividing by [MATH] enables us to get a non-trivial limit for our sequence as [MATH] goes to [MATH].', '1711.00427-3-8-0': 'Our main result is the convergence of [MATH], seen as a random element in the space of tempered distributions, towards an approximately log-correlated Gaussian field.', '1711.00427-3-8-1': 'Denote by [MATH] the real Schwartz space, that is the set of real-valued functions on [MATH] whose derivatives of all orders exist and decay faster than any polynomial at infinity.', '1711.00427-3-8-2': 'We write [MATH] for the dual of [MATH], that is the space of tempered distributions.', '1711.00427-3-8-3': 'We also define the subspace [MATH] of the real Schwartz space, consisting of functions [MATH] from [MATH] with [MATH], and its topological dual [MATH].', '1711.00427-3-8-4': 'A log-correlated Gaussian field (LGF for short) [MATH], is a centered Gaussian field whose covariance kernel satisfies [EQUATION] for any [MATH], see [CITATION] for an overview on LGF.', '1711.00427-3-8-5': 'We show in this paper that the limit of [MATH] as [MATH] goes to zero is "almost" a log-correlated Gaussian field, see Section [REF] for an accurate result.', '1711.00427-3-9-0': 'LGFs are closely related to some multifractal processes pioneered by Mandelbrot (see for example [CITATION]), and further developed in [CITATION], among others.', '1711.00427-3-9-1': 'A process [MATH] is said to be multifractal if for a range of values of [MATH], we have for some [MATH] [EQUATION] where [MATH] is a constant and [MATH] is a non-linear concave function.', '1711.00427-3-9-2': 'In particular, the multifractal random walk model for the log-price of an asset in [CITATION] satisfies such property.', '1711.00427-3-9-3': 'It is defined as [MATH] where [MATH] is a Brownian motion and [EQUATION] with [MATH] and [MATH] a Gaussian process such that for some [MATH] and [MATH] [EQUATION] see [CITATION] for details.', '1711.00427-3-9-4': 'Hence we see that [MATH] formally corresponds to a measure of the form [MATH], where [MATH] is a LGF.', '1711.00427-3-9-5': 'For the precise definition of such measures, see [CITATION] and the generalizations on Gaussian multiplicative chaos in [CITATION] and the references therein.', '1711.00427-3-9-6': 'Finally note that LGFs and more generally the associated theory of Gaussian multiplicative chaos have extensive use in other fields than finance, such as turbulence, see [CITATION], disordered systems, see [CITATION] and Liouville quantum gravity, see [CITATION].', '1711.00427-3-10-0': 'In the following section we introduce our main theorem, that is an accurate statement about the convergence of the normalized fBm towards a LGF as [MATH] goes to zero.', '1711.00427-3-10-1': 'We also discuss the multifractal properties of the limiting LGF in the same section.', '1711.00427-3-10-2': 'The proof of our theorem can be found in Section [REF].', '1711.00427-3-11-0': '# Convergence of the fBM towards a LGF', '1711.00427-3-12-0': '## Main result', '1711.00427-3-13-0': 'We define the weak convergence of elements in [MATH] as in Proposition 12.2 in [CITATION].', '1711.00427-3-13-1': 'We say that [MATH] converges weakly to [MATH] as [MATH] tends to [MATH] if for any [MATH] we have [EQUATION] in law, as [MATH] tends to [MATH].', '1711.00427-3-13-2': 'The main result of our paper is the following.', '1711.00427-3-14-0': 'The sequence [MATH] converges weakly as [MATH] tends to zero towards a centered Gaussian field [MATH] satisfying for any [MATH] [EQUATION] where for [MATH], [MATH] and [MATH] [EQUATION] with [EQUATION].', '1711.00427-3-15-0': 'We see that when [MATH] for some [MATH], then [MATH] is a bounded continuous function.', '1711.00427-3-15-1': 'Hence the covariance kernel exhibits the same type of singularity as that of a LGF.', '1711.00427-3-15-2': 'Consequently, in our framework, the limit when [MATH] goes to zero of a normalized version of the fBm is "almost" a LGF.', '1711.00427-3-16-0': '## Multifractal properties', '1711.00427-3-17-0': 'In [CITATION], the authors study the case of centered Gaussian fields on any domain [MATH] with covariance kernel satisfying [EQUATION] where [MATH] and [MATH] is a bounded continuous function.', '1711.00427-3-17-1': 'Thus, if we restrict [MATH] to the domain [MATH] for some fixed [MATH], then [MATH] is included in the framework of [CITATION].', '1711.00427-3-17-2': 'In particular, their results on the multifractal spectrum of Gaussian fields apply on this restricted domain.', '1711.00427-3-18-0': 'Motivated by the preceding paragraph, we first fix [MATH] and define an approximate volatility measure [MATH] by [EQUATION] for some constant [MATH].', '1711.00427-3-18-1': 'Here we assume that [MATH] vanishes on [MATH].', '1711.00427-3-18-2': 'From the result of Theorem [REF] we deduce the following corollary.', '1711.00427-3-18-3': 'In what follows, convergence in the [MATH] norm stands for the usual convergence of random variables in [MATH].', '1711.00427-3-19-0': 'For [MATH], [MATH] converges as [MATH] approaches zero to a random measure [MATH] in the following sense, [EQUATION].', '1711.00427-3-19-1': 'Moreover, the limiting measure [MATH] is the so-called Gaussian multiplicative chaos.', '1711.00427-3-20-0': 'The proof Corollary [REF] is given in Section [REF].', '1711.00427-3-20-1': 'For the definition and properties of Gaussian multiplicative chaos we refer to a survey paper by Rhodes and Vargas [CITATION].', '1711.00427-3-20-2': 'We will briefly explain some of the properties of [MATH] from the theory of Gaussian multiplicative chaos.', '1711.00427-3-21-0': 'We first describe the behavior of the moments of [MATH].', '1711.00427-3-21-1': 'Note that for [MATH], [MATH] is of the form ([REF]).', '1711.00427-3-21-2': 'From Proposition 2.5 in [CITATION], it follows that for all [MATH] and [MATH], there exists [MATH] such that [EQUATION] where [MATH].', '1711.00427-3-21-3': 'Hence we do obtain a multifractal scaling as described in the introduction.', '1711.00427-3-22-0': 'Finally we consider a quantity closely related to the function [MATH] above: the spectrum of singularities of [MATH].', '1711.00427-3-22-1': 'For any [MATH] and [MATH], we define [EQUATION]', '1711.00427-3-22-2': 'The set [MATH] somehow corresponds to the points [MATH] where the Holder regularity of [MATH] is equal to [MATH].', '1711.00427-3-22-3': 'Let [MATH] denote the Hausdorff dimension of a set [MATH].', '1711.00427-3-22-4': 'Theorem 2.6 in [CITATION] states that [EQUATION]', '1711.00427-3-22-5': 'In particular, we remark that [EQUATION].', '1711.00427-3-22-6': 'This equality means that the Frish-Parisi conjecture relating the scaling exponents of a process to its spectrum of singularities holds in our case, see [CITATION] and [CITATION] for more details on the multifractal formalism.', '1711.00427-3-23-0': 'The rest of this paper is devoted to the proofs of Theorem [REF] and Corollary [REF].', '1711.00427-3-24-0': '# Proof of Theorem [REF] and Corollary [REF].', '1711.00427-3-25-0': 'For [MATH], let [MATH] .', '1711.00427-3-25-1': 'We start the proof with the following important lemma.', '1711.00427-3-26-0': 'For any non-zero [MATH] with [MATH], we have [EQUATION]', '1711.00427-3-26-1': 'We write [MATH], where [EQUATION] and [MATH], where [EQUATION].', '1711.00427-3-26-2': 'We have [EQUATION] so we can assume that [MATH].', '1711.00427-3-27-0': 'Note that for any [MATH], [EQUATION] and therefore [EQUATION].', '1711.00427-3-27-1': 'Next we deal with [MATH], [MATH].', '1711.00427-3-27-2': 'We consider several cases.', '1711.00427-3-28-0': 'Case 1: Assume that [MATH].', '1711.00427-3-28-1': 'We easily get [EQUATION].', '1711.00427-3-28-2': 'For [MATH], note that [EQUATION].', '1711.00427-3-28-3': 'Hence we obtain [EQUATION] and therefore [EQUATION].', '1711.00427-3-28-4': 'Define [EQUATION]', '1711.00427-3-28-5': 'It follows that [EQUATION]', '1711.00427-3-28-6': 'Thus we have [EQUATION]', '1711.00427-3-28-7': 'Consequently, [EQUATION] and [EQUATION].', '1711.00427-3-29-0': 'We also easily get that for [MATH] [EQUATION].', '1711.00427-3-29-1': 'Furthermore, [EQUATION].', '1711.00427-3-29-2': 'Thus we obtain [EQUATION] and we deduce [EQUATION]', '1711.00427-3-29-3': 'Now define [EQUATION]', '1711.00427-3-29-4': 'We have [EQUATION]', '1711.00427-3-29-5': 'Finally we obtain [EQUATION] from which ([REF]) readily follows for [MATH].', '1711.00427-3-30-0': 'Case 2: Assume that [MATH].', '1711.00427-3-30-1': 'We use that [EQUATION] to deduce that ([REF]) follows from the proof of Case 1.', '1711.00427-3-31-0': 'Case 3: Assume that [MATH].', '1711.00427-3-31-1': 'We write [MATH].', '1711.00427-3-31-2': 'Repeating the same steps as in Case 1, we get [EQUATION] and [EQUATION]', '1711.00427-3-31-3': 'It follows that [EQUATION] and we get ([REF]).', '1711.00427-3-31-4': 'Let [MATH].', '1711.00427-3-31-5': 'Since [MATH] and [MATH] are centered Gaussians taking values in [MATH], to prove Theorem [REF], it is enough to show that [EQUATION].', '1711.00427-3-31-6': 'Furthermore, for any [MATH] we have [EQUATION].', '1711.00427-3-31-7': 'Hence Theorem [REF] immediately follows from the next lemma.', '1711.00427-3-32-0': 'For any [MATH], [EQUATION]', '1711.00427-3-32-1': 'First note that for [MATH], [MATH].', '1711.00427-3-32-2': 'Therefore for any [MATH], [EQUATION].', '1711.00427-3-32-3': 'Moreover, [EQUATION]', '1711.00427-3-32-4': 'Hence, from dominated convergence, it follows that [EQUATION].', '1711.00427-3-32-5': 'Let [MATH], where [MATH] and [MATH].', '1711.00427-3-32-6': 'Since [MATH] is concave, for any [MATH], we have [MATH].', '1711.00427-3-32-7': 'Therefore for any [MATH], [EQUATION].', '1711.00427-3-32-8': 'Since [EQUATION] we get from dominated convergence [EQUATION].', '1711.00427-3-32-9': 'It remains to show that [EQUATION].', '1711.00427-3-32-10': 'We again consider several cases.', '1711.00427-3-33-0': 'Case 1: Assume that [MATH].', '1711.00427-3-33-1': 'We obviously have [EQUATION].', '1711.00427-3-33-2': 'Moreover, from a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-3-33-3': 'It follows that for any [MATH], [EQUATION]', '1711.00427-3-33-4': 'Case 1bis: Assume that [MATH].', '1711.00427-3-33-5': 'Since [MATH] we get [EQUATION]', '1711.00427-3-33-6': 'Case 2: Assume that [MATH].', '1711.00427-3-33-7': 'Since [MATH], we have [EQUATION]', '1711.00427-3-33-8': 'Case 3: Assume that [MATH] and [MATH].', '1711.00427-3-33-9': 'Recall that when [MATH], [EQUATION]', '1711.00427-3-33-10': 'Since [MATH] we have [EQUATION]', '1711.00427-3-33-11': 'Using a Taylor expansion, we get [MATH] and therefore [EQUATION]', '1711.00427-3-33-12': 'It follows that [EQUATION]', '1711.00427-3-33-13': 'Case 4: Assume that [MATH] and [MATH].', '1711.00427-3-33-14': 'Repeating the same lines as in Case 3 but exchanging the roles of [MATH] and [MATH] we get [EQUATION]', '1711.00427-3-33-15': 'Case 5: Assume that [MATH] and [MATH].', '1711.00427-3-33-16': 'Since [MATH], it follows from Case 3 that [EQUATION]', '1711.00427-3-33-17': 'Case 6: Assume that [MATH] and [MATH].', '1711.00427-3-33-18': 'Using again [MATH], we get from Case 4 that [EQUATION]', '1711.00427-3-33-19': 'Finally, we clearly have [EQUATION].', '1711.00427-3-33-20': 'Thus we conclude the proof using the dominated convergence theorem.', '1711.00427-3-34-0': 'Before we prove Corollary [REF], we recall an approximation of [MATH], from Example 2.2 in [CITATION].', '1711.00427-3-34-1': 'For any [MATH] we define the cone [MATH] in [MATH]: [EQUATION].', '1711.00427-3-34-2': 'A direct computation gives [EQUATION].', '1711.00427-3-34-3': 'We approximate [MATH] by the following functions, [EQUATION]', '1711.00427-3-34-4': 'Proof of Corollary [REF] From Theorem 25 in [CITATION] we obtain that Corollary [REF] follows from ([REF]), provided we show that [MATH] are uniformly integrable.', '1711.00427-3-35-0': 'First note that all [MATH], [MATH], can be constructed on the same probability space, as a convolution of the same Brownian motion with different kernels which depend on [MATH] (see [CITATION]).', '1711.00427-3-35-1': 'From ([REF]) and ([REF]) we have for [MATH], [EQUATION]', '1711.00427-3-35-2': 'Using [MATH] for [MATH] we get for all [MATH], [EQUATION]', '1711.00427-3-35-3': 'On the other hand for [MATH] we have [EQUATION]', '1711.00427-3-35-4': 'It follows that [EQUATION]', '1711.00427-3-35-5': 'Let [MATH] be arbitrary small.', '1711.00427-3-35-6': 'Let [MATH] be a centred Gaussian random field with the covariance kernel [MATH], where [MATH] is given in ([REF]), i.e. [EQUATION].', '1711.00427-3-36-0': 'Since [MATH], we get from ([REF]) for [MATH], [EQUATION]', '1711.00427-3-36-1': 'Note that [MATH] is an even function and therefore, [EQUATION].', '1711.00427-3-36-2': 'We define the following of measure [EQUATION]', '1711.00427-3-36-3': 'Recall that [MATH] by our assumption.', '1711.00427-3-36-4': 'Then from the proof of Proposition 3.5 in [CITATION] it follows that there exists [MATH], such that [EQUATION].', '1711.00427-3-36-5': 'By chosing [MATH], we note that [MATH], and from the comparison principal with [MATH] (see Corollary A.2 in [CITATION]), we get [EQUATION].', '1711.00427-3-36-6': 'We therefore conclude that [MATH] are uniformly integrable.'} | null | null | null | null |
0709.2735 | {'0709.2735-1-0-0': 'The quantum kicked prime number rotator (QKPR) is defined as the rotator whose energy levels are prime numbers.', '0709.2735-1-0-1': 'The long time behavior is decided by the kick period [MATH] and kick strength [MATH].', '0709.2735-1-0-2': 'When [MATH] is a irrational number, the state is localized because of the equidistribution theorem.', '0709.2735-1-0-3': 'When [MATH] is rational, the state is localized for small [MATH], because the system seems like a generalized kicked dimer model.', '0709.2735-1-0-4': 'We argue for rational [MATH] QKPR delocalizes for large [MATH].', '0709.2735-1-1-0': 'The kicked prime number rotator is defined as [EQUATION] where [MATH] is the unperturbed Hamiltonian, and [MATH] is the perturbation.', '0709.2735-1-2-0': '[MATH] is a diagonal matrix.', '0709.2735-1-2-1': 'The [MATH]-th eigenvalue [MATH] corresponds to [MATH]-th eigenstate [MATH] is the [MATH]-th prime number [MATH].', '0709.2735-1-2-2': 'When [MATH], [MATH].', '0709.2735-1-2-3': '[MATH].', '0709.2735-1-2-4': 'The diagonal of [MATH] is [MATH].', '0709.2735-1-3-0': '[MATH] is defined as [EQUATION]', '0709.2735-1-3-1': 'The Floquet operator is [MATH].', '0709.2735-1-3-2': 'The matrix elements of [MATH] is [MATH], where [MATH], [MATH] is the Bessel function of the first kind with order [MATH].', '0709.2735-1-3-3': 'We define [MATH].', '0709.2735-1-3-4': '[MATH].', '0709.2735-1-3-5': 'The system is very like the standard quantum kicked rotator (QKR), except its energy levels are now prime numbers.', '0709.2735-1-4-0': 'It seems there is no classical correspondence of this prime number rotator.', '0709.2735-1-4-1': 'Experimental implementation of such a model also seems impossible.', '0709.2735-1-4-2': 'Nevertheless it still has some theoretical interests.', '0709.2735-1-4-3': 'In the paper, we numerically calculate the evolution of the system.', '0709.2735-1-4-4': 'We are interested in the same problem in QKR.', '0709.2735-1-4-5': 'If the particle is in the ground state [MATH] initially, will it diffuse away in the future?', '0709.2735-1-5-0': 'The evolution of the system is calculated by the iterative unitary matrix multiply method (IUMM) [CITATION].', '0709.2735-1-5-1': '[MATH].', '0709.2735-1-5-2': '[MATH].', '0709.2735-1-5-3': '[MATH].', '0709.2735-1-5-4': 'And so on.', '0709.2735-1-5-5': 'In this way, we can calculate [MATH] by [MATH] matrix multiplies.', '0709.2735-1-6-0': 'First, we choose [MATH], [MATH].', '0709.2735-1-6-1': 'The result is displayed in Fig. 1.', '0709.2735-1-6-2': 'The QKPR is localized perfectly.', '0709.2735-1-6-3': 'In our simulation, the exponentially fall of the wave function never changes from [MATH] to [MATH].', '0709.2735-1-6-4': 'The wave function on the [MATH] is [MATH].', '0709.2735-1-6-5': 'From [MATH] to [MATH], the wave function is somewhat curved.', '0709.2735-1-6-6': 'After the first kick, the wave function is the [MATH]-th column of the Floquet matrix [MATH].', '0709.2735-1-6-7': 'The absolute value of [MATH] is [MATH].', '0709.2735-1-6-8': '[MATH] falls to zero faster than exponentially .', '0709.2735-1-6-9': 'This is the reason the curved form of the wave function.', '0709.2735-1-7-0': 'Second, we choose [MATH], [MATH].', '0709.2735-1-7-1': 'The result is displayed in Fig. 3.', '0709.2735-1-7-2': 'The wave function is also localized.', '0709.2735-1-7-3': 'This is expected.', '0709.2735-1-7-4': 'The sequence [MATH] is equidistributed between [MATH], when [MATH] is irrational.', '0709.2735-1-7-5': 'We denote the sequence [MATH] as QKPR[MATH].', '0709.2735-1-7-6': 'In the standard quantum kicked rotator, the sequence [MATH] is also equidistributed between [MATH], for an irrational [MATH].', '0709.2735-1-7-7': 'We denote the sequence [MATH] as (QKR[MATH]).', '0709.2735-1-7-8': 'We can also transform the Floquet eigenstate equation [MATH] into an equation like Anderson localization problem.', '0709.2735-1-7-9': "So from Fishman et al's argument [CITATION], the rotator will localize.", '0709.2735-1-8-0': 'In the upper right of Fig. 4, QKPR[MATH] and QKR[MATH] are displayed.', '0709.2735-1-8-1': 'Though there are apparently some correlations in QKPR[MATH] and QKR[MATH] and the correlation is different between both sequences.', '0709.2735-1-8-2': 'The correlation is surely not strong enough to destroy localization.', '0709.2735-1-8-3': 'If a sequence is periodic with a period [MATH], then the discrete Fourier transform of the sequence is composed by [MATH] modes.', '0709.2735-1-8-4': 'To find whether there is some periodicity in the sequence, we perform a discrete Fourier transform (DFT) of the sequences.', '0709.2735-1-8-5': 'DFT of a sequence [MATH] of length [MATH] is defined as [MATH], where [MATH] runs from 1 to [MATH].', '0709.2735-1-8-6': 'There are some other definitions of DFT with nuanced difference with our definition.', '0709.2735-1-8-7': 'But the difference is irrelevant to our discussion here.', '0709.2735-1-8-8': 'In the lower right of Fig. 4, the DFTs of both sequences are displayed.', '0709.2735-1-8-9': 'There are no rigorous periodicity in both sequences.', '0709.2735-1-8-10': '[MATH] of QKPR[MATH] seems to have a trend to cluster together.', '0709.2735-1-8-11': 'Also it is less uniformly distributed than the [MATH] of QKR[MATH] and tends to be small.', '0709.2735-1-9-0': 'If [MATH] , does QKPR localize?', '0709.2735-1-9-1': 'At first thought, this seems to be a resonant case in the standard QKR and the rotator will delocalize.', '0709.2735-1-9-2': 'The calculation result is in fact it still localizes for small [MATH] such as [MATH].', '0709.2735-1-9-3': 'In Fig. 2, we choose [MATH] and [MATH] and in Fig. 5, [MATH] and [MATH].', '0709.2735-1-9-4': 'QKPR of [MATH] is apparently localized.', '0709.2735-1-9-5': 'Fig. 2 and 5 display wave function at the time as long as [MATH].', '0709.2735-1-9-6': 'At [MATH], the calculation overflows, which also exists in other IUMM applications [CITATION].', '0709.2735-1-10-0': 'The explanation of the localization is QKPR with [MATH] and [MATH] is a kicked pseudo dimer rotator.', '0709.2735-1-10-1': 'The dimer model is defined as every diagonal matrix element is a probability variable which only takes two values [CITATION].', '0709.2735-1-10-2': 'The kicked dimer model can be defined as every diagonal matrix element of [MATH] is a random variable which takes two values.', '0709.2735-1-10-3': 'If it takes more than two values, it is a generalized kicked dimer model.', '0709.2735-1-10-4': 'For [MATH], the sequence [MATH] mainly takes four values.', '0709.2735-1-10-5': 'So it is a generalized kicked pseudo dimer model.', '0709.2735-1-10-6': 'The sequence [MATH] (QKPR[MATH]) is not really random.', '0709.2735-1-10-7': 'But the pseudorandomness is enough to result in localization [CITATION].', '0709.2735-1-10-8': 'To measure how random QKPR[MATH] is, we perform a DFT on it.', '0709.2735-1-10-9': 'In the left of Fig. 4, we compare QKPR[MATH] with a dimer sequence D[MATH], which is defined as [MATH], where every [MATH] is a random variable which takes two values [MATH] and [MATH].', '0709.2735-1-10-10': 'The [MATH]s of D[MATH] and QKPR[MATH] are quite close with each other, except QKPR[MATH] tends to cluster together.', '0709.2735-1-11-0': 'Does QKPR localize for [MATH] and [MATH]?', '0709.2735-1-11-1': 'We think it delocalizes.', '0709.2735-1-11-2': 'There are a series of plateaux in the wave function of QKPR.', '0709.2735-1-11-3': 'The wave function falls abruptly when approaching the boundary (cliff) of a plateau.', '0709.2735-1-11-4': 'Some plateaux disappear at [MATH].', '0709.2735-1-11-5': 'There is quantum wave pass through the cliff, so it disappears intermittently.', '0709.2735-1-12-0': 'The most obvious cliff is from [MATH] to [MATH] or so in Fig.6.', '0709.2735-1-12-1': 'QKPR[MATH] from [MATH] to [MATH] is [MATH].', '0709.2735-1-12-2': 'Note it is periodic from [MATH] to [MATH].', '0709.2735-1-12-3': 'In a perfect periodic potential the wave will always diffuse away.', '0709.2735-1-12-4': 'The quantum wave can not stay at a period potential very long.', '0709.2735-1-12-5': 'Once the wave has propagated into the phase space between [MATH] and [MATH], it diffuses away quickly.', '0709.2735-1-12-6': 'While once the wave propagate the phase space whose neighborhood [MATH] is irregular, the quantum wave is localized there at least temporarily.', '0709.2735-1-13-0': 'When [MATH], [MATH] from [MATH] to [MATH] is a plateau in Fig. 2.', '0709.2735-1-13-1': 'There is a transition from a plateau to a cliff when [MATH] becomes large.', '0709.2735-1-14-0': 'Note in Fig. 4, for lots of [MATH], Lots of [MATH] of QKPR[MATH] and QKPR[MATH] tend to be small.', '0709.2735-1-14-1': 'From Plancherel theorem, there must be some [MATH] tends to very large.', '0709.2735-1-14-2': 'So there is some weak periodicity in the sequence QKPR[MATH].', '0709.2735-1-14-3': 'The sequence from [MATH] to [MATH] increases the periodicity in the sequence QKPR[MATH].', '0709.2735-1-15-0': 'At [MATH], there is a plateau from [MATH] to [MATH] or so.', '0709.2735-1-15-1': 'Even at [MATH], there is a plateau from [MATH] to [MATH].', '0709.2735-1-15-2': 'The localization length of QKR is [MATH] or so [CITATION].', '0709.2735-1-15-3': 'So the distribution length of QKPR is much larger than the localization length.', '0709.2735-1-15-4': 'Even the result of [MATH] is untrustworthy, we think the quantum wave is absolutely not localized in the localization length.', '0709.2735-1-15-5': 'From [MATH] to [MATH] or so, there is apparently unneglectable quantum wave at every [MATH] at for example [MATH].', '0709.2735-1-15-6': 'At [MATH], the quantum wave has already propagated into [MATH].', '0709.2735-1-15-7': 'QKPR with [MATH] and [MATH] is not localized.', '0709.2735-1-16-0': 'In this paper, we apply the iterative unitary matrix multiply method to quantum kicked prime number rotator.', '0709.2735-1-16-1': 'If [MATH] is irrational, the rotator localizes.', '0709.2735-1-16-2': 'If [MATH] is irrational, for small kick strength [MATH], the rotator localizes.', '0709.2735-1-16-3': 'As [MATH] increases, we argue there is a localization-delocalization transition.'} | {'0709.2735-2-0-0': 'The quantum kicked prime number rotator (QKPR) is defined as the rotator whose energy levels are prime numbers.', '0709.2735-2-0-1': 'The long time behavior is decided by the kick period [MATH] and kick strength [MATH].', '0709.2735-2-0-2': 'When [MATH] is a irrational number, the state is localized because of the equidistribution theorem.', '0709.2735-2-0-3': 'When [MATH] is rational, the state is localized for small [MATH], because the system seems like a generalized kicked dimer model.', '0709.2735-2-0-4': 'We argue for rational [MATH] QKPR delocalizes for large [MATH].', '0709.2735-2-1-0': 'The kicked prime number rotator is defined as [EQUATION] where [MATH] is the unperturbed Hamiltonian, and [MATH] is the perturbation.', '0709.2735-2-2-0': '[MATH] is a diagonal matrix.', '0709.2735-2-2-1': 'The [MATH]-th eigenvalue [MATH] corresponding to [MATH]-th eigenstate [MATH] of [MATH] is the [MATH]-th prime number [MATH].', '0709.2735-2-2-2': 'When [MATH], [MATH].', '0709.2735-2-2-3': '[MATH].', '0709.2735-2-2-4': 'The diagonal of [MATH] is [MATH].', '0709.2735-2-3-0': '[MATH] is defined as [EQUATION]', '0709.2735-2-3-1': 'The Floquet operator is [MATH].', '0709.2735-2-3-2': 'The matrix elements of [MATH] is [MATH], where [MATH], [MATH] is the Bessel function of the first kind with order [MATH].', '0709.2735-2-3-3': 'We define [MATH].', '0709.2735-2-3-4': '[MATH].', '0709.2735-2-3-5': 'The system is very like the standard quantum kicked rotator (QKR), except its energy levels are now prime numbers.', '0709.2735-2-4-0': 'It seems there is no classical correspondence of this prime number rotator.', '0709.2735-2-4-1': 'Experimental implementation of such a model also seems impossible.', '0709.2735-2-4-2': 'Nevertheless it still has some theoretical interests.', '0709.2735-2-4-3': 'In the paper, we numerically calculate the evolution of the system.', '0709.2735-2-4-4': 'We are interested in the same problem in QKR.', '0709.2735-2-4-5': 'If the particle is in the ground state [MATH] initially, will it diffuse away in the future?', '0709.2735-2-5-0': 'The evolution of the system is calculated by the iterative unitary matrix multiply method (IUMM) [CITATION].', '0709.2735-2-5-1': '[MATH].', '0709.2735-2-5-2': '[MATH].', '0709.2735-2-5-3': '[MATH].', '0709.2735-2-5-4': 'And so on.', '0709.2735-2-5-5': 'In this way, we can calculate [MATH] by [MATH] matrix multiplies.', '0709.2735-2-6-0': 'First, we choose [MATH], [MATH].', '0709.2735-2-6-1': 'The result is displayed in Fig. 1.', '0709.2735-2-6-2': 'The QKPR is localized perfectly.', '0709.2735-2-6-3': 'In our simulation, the exponentially fall of the wave function never changes from [MATH] to [MATH].', '0709.2735-2-6-4': 'The wave function on the [MATH] is [MATH].', '0709.2735-2-6-5': 'From [MATH] to [MATH], the wave function is somewhat curved.', '0709.2735-2-6-6': 'After the first kick, the wave function is the [MATH]-th column of the Floquet matrix [MATH].', '0709.2735-2-6-7': 'The absolute value of [MATH] is [MATH].', '0709.2735-2-6-8': '[MATH] falls to zero faster than exponentially .', '0709.2735-2-6-9': 'This is the reason the curved form of the wave function.', '0709.2735-2-7-0': 'Second, we choose [MATH], [MATH].', '0709.2735-2-7-1': 'The result is displayed in Fig. 3.', '0709.2735-2-7-2': 'The wave function is also localized.', '0709.2735-2-7-3': 'This is expected.', '0709.2735-2-7-4': 'The sequence [MATH] is equidistributed between [MATH], when [MATH] is irrational.', '0709.2735-2-7-5': 'We denote the sequence [MATH] as QKPR[MATH].', '0709.2735-2-7-6': 'In the standard quantum kicked rotator, the sequence [MATH] is also equidistributed between [MATH], for an irrational [MATH].', '0709.2735-2-7-7': 'We denote the sequence [MATH] as (QKR[MATH]).', '0709.2735-2-7-8': 'We can also transform the Floquet eigenstate equation [MATH] into an equation like Anderson localization problem.', '0709.2735-2-7-9': "So from Fishman et al's argument [CITATION], the rotator will localize.", '0709.2735-2-8-0': 'In the upper right of Fig. 4, QKPR[MATH] and QKR[MATH] are displayed.', '0709.2735-2-8-1': 'Though there are apparently some correlations in QKPR[MATH] and QKR[MATH] and the correlation is different between both sequences.', '0709.2735-2-8-2': 'The correlation is surely not strong enough to destroy localization.', '0709.2735-2-8-3': 'If a sequence is periodic with a period [MATH], then the discrete Fourier transform of the sequence is composed by [MATH] modes.', '0709.2735-2-8-4': 'To find whether there is some periodicity in the sequence, we perform a discrete Fourier transform (DFT) of the sequences.', '0709.2735-2-8-5': 'DFT of a sequence [MATH] of length [MATH] is defined as [MATH], where [MATH] runs from 1 to [MATH].', '0709.2735-2-8-6': 'There are some other definitions of DFT with nuanced difference with our definition.', '0709.2735-2-8-7': 'But the difference is irrelevant to our discussion here.', '0709.2735-2-8-8': 'In the lower right of Fig. 4, the DFTs of both sequences are displayed.', '0709.2735-2-8-9': 'There are no rigorous periodicity in both sequences.', '0709.2735-2-8-10': '[MATH] of QKPR[MATH] seems to have a trend to cluster together.', '0709.2735-2-8-11': 'Also it is less uniformly distributed than the [MATH] of QKR[MATH] and tends to be small.', '0709.2735-2-9-0': 'If [MATH] , does QKPR localize?', '0709.2735-2-9-1': 'At first thought, this seems to be a resonant case in the standard QKR and the rotator will delocalize.', '0709.2735-2-9-2': 'The calculation result is in fact it still localizes for small [MATH] such as [MATH].', '0709.2735-2-9-3': 'In Fig. 2, we choose [MATH] and [MATH] and in Fig. 5, [MATH] and [MATH].', '0709.2735-2-9-4': 'QKPR of [MATH] is apparently localized.', '0709.2735-2-9-5': 'Fig. 2 and 5 display wave function at the time as long as [MATH].', '0709.2735-2-9-6': 'At [MATH], the calculation overflows, which also exists in other IUMM applications [CITATION].', '0709.2735-2-10-0': 'The explanation of the localization is QKPR with [MATH] and [MATH] is a kicked pseudo dimer rotator.', '0709.2735-2-10-1': 'The dimer model is defined as every diagonal matrix element is a probability variable which only takes two values [CITATION].', '0709.2735-2-10-2': 'The kicked dimer model can be defined as every diagonal matrix element of [MATH] is a random variable which takes two values.', '0709.2735-2-10-3': 'If it takes more than two values, it is a generalized kicked dimer model.', '0709.2735-2-10-4': 'For [MATH], the sequence [MATH] mainly takes four values.', '0709.2735-2-10-5': 'So it is a generalized kicked pseudo dimer model.', '0709.2735-2-10-6': 'The sequence [MATH] (QKPR[MATH]) is not really random.', '0709.2735-2-10-7': 'But the pseudorandomness is enough to result in localization [CITATION].', '0709.2735-2-10-8': 'To measure how random QKPR[MATH] is, we perform a DFT on it.', '0709.2735-2-10-9': 'In the left of Fig. 4, we compare QKPR[MATH] with a dimer sequence D[MATH], which is defined as [MATH], where every [MATH] is a random variable which takes two values [MATH] and [MATH].', '0709.2735-2-10-10': 'The [MATH]s of D[MATH] and QKPR[MATH] are quite close with each other, except QKPR[MATH] tends to cluster together.', '0709.2735-2-11-0': 'Does QKPR localize for [MATH] and [MATH]?', '0709.2735-2-11-1': 'We think it delocalizes.', '0709.2735-2-11-2': 'There are a series of plateaux in the wave function of QKPR.', '0709.2735-2-11-3': 'The wave function falls abruptly when approaching the boundary (cliff) of a plateau.', '0709.2735-2-11-4': 'Some plateaux disappear at [MATH].', '0709.2735-2-11-5': 'There is quantum wave pass through the cliff, so it disappears intermittently.', '0709.2735-2-12-0': 'The most obvious cliff is from [MATH] to [MATH] or so in Fig.6.', '0709.2735-2-12-1': 'QKPR[MATH] from [MATH] to [MATH] is [MATH].', '0709.2735-2-12-2': 'Note it is periodic from [MATH] to [MATH].', '0709.2735-2-12-3': 'In a perfect periodic potential the wave will always diffuse away.', '0709.2735-2-12-4': 'The quantum wave can not stay at a period potential very long.', '0709.2735-2-12-5': 'Once the wave has propagated into the phase space between [MATH] and [MATH], it diffuses away quickly.', '0709.2735-2-12-6': 'While once the wave propagate the phase space whose neighborhood [MATH] is irregular, the quantum wave is localized there at least temporarily.', '0709.2735-2-13-0': 'When [MATH], [MATH] from [MATH] to [MATH] is a plateau in Fig. 2.', '0709.2735-2-13-1': 'There is a transition from a plateau to a cliff when [MATH] becomes large.', '0709.2735-2-14-0': 'Note in Fig. 4, for lots of [MATH], Lots of [MATH] of QKPR[MATH] and QKPR[MATH] tend to be small.', '0709.2735-2-14-1': 'From Plancherel theorem, there must be some [MATH] tends to very large.', '0709.2735-2-14-2': 'So there is some weak periodicity in the sequence QKPR[MATH].', '0709.2735-2-14-3': 'The sequence from [MATH] to [MATH] increases the periodicity in the sequence QKPR[MATH].', '0709.2735-2-15-0': 'At [MATH], there is a plateau from [MATH] to [MATH] or so.', '0709.2735-2-15-1': 'Even at [MATH], there is a plateau from [MATH] to [MATH].', '0709.2735-2-15-2': 'The localization length of QKR is [MATH] or so [CITATION].', '0709.2735-2-15-3': 'So the distribution length of QKPR is much larger than the localization length.', '0709.2735-2-15-4': 'Even the result of [MATH] is untrustworthy, we think the quantum wave is absolutely not localized in the localization length.', '0709.2735-2-15-5': 'From [MATH] to [MATH] or so, there is apparently unneglectable quantum wave at every [MATH] at for example [MATH].', '0709.2735-2-15-6': 'At [MATH], the quantum wave has already propagated into [MATH].', '0709.2735-2-15-7': 'QKPR with [MATH] and [MATH] is not localized.', '0709.2735-2-16-0': 'In this paper, we apply the iterative unitary matrix multiply method to quantum kicked prime number rotator.', '0709.2735-2-16-1': 'If [MATH] is irrational, the rotator localizes.', '0709.2735-2-16-2': 'If [MATH] is irrational, for small kick strength [MATH], the rotator localizes.', '0709.2735-2-16-3': 'As [MATH] increases, we argue there is a localization-delocalization transition.'} | [['0709.2735-1-13-0', '0709.2735-2-13-0'], ['0709.2735-1-13-1', '0709.2735-2-13-1'], ['0709.2735-1-14-0', '0709.2735-2-14-0'], ['0709.2735-1-14-1', '0709.2735-2-14-1'], ['0709.2735-1-14-2', '0709.2735-2-14-2'], ['0709.2735-1-14-3', '0709.2735-2-14-3'], ['0709.2735-1-1-0', '0709.2735-2-1-0'], ['0709.2735-1-6-0', '0709.2735-2-6-0'], ['0709.2735-1-6-1', '0709.2735-2-6-1'], ['0709.2735-1-6-2', '0709.2735-2-6-2'], ['0709.2735-1-6-3', '0709.2735-2-6-3'], ['0709.2735-1-6-4', 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['0709.2735-2-8-7', '0709.2735-3-7-7'], ['0709.2735-2-8-9', '0709.2735-3-7-9'], ['0709.2735-2-8-10', '0709.2735-3-7-10'], ['0709.2735-2-8-11', '0709.2735-3-7-11'], ['0709.2735-2-1-0', '0709.2735-3-1-0'], ['0709.2735-2-9-0', '0709.2735-3-8-0'], ['0709.2735-2-9-4', '0709.2735-3-8-4'], ['0709.2735-2-10-0', '0709.2735-3-9-0'], ['0709.2735-2-10-1', '0709.2735-3-9-1'], ['0709.2735-2-10-2', '0709.2735-3-9-2'], ['0709.2735-2-10-3', '0709.2735-3-9-3'], ['0709.2735-2-10-4', '0709.2735-3-9-4'], ['0709.2735-2-10-5', '0709.2735-3-9-5'], ['0709.2735-2-10-6', '0709.2735-3-9-6'], ['0709.2735-2-10-7', '0709.2735-3-9-7'], ['0709.2735-2-10-8', '0709.2735-3-9-8'], ['0709.2735-2-10-10', '0709.2735-3-9-10'], ['0709.2735-2-11-0', '0709.2735-3-10-0'], ['0709.2735-2-11-1', '0709.2735-3-10-1'], ['0709.2735-2-11-2', '0709.2735-3-10-2'], ['0709.2735-2-11-3', '0709.2735-3-10-3'], ['0709.2735-2-11-4', '0709.2735-3-10-4'], ['0709.2735-2-16-0', '0709.2735-3-14-0'], ['0709.2735-2-16-1', '0709.2735-3-14-1'], ['0709.2735-2-16-3', '0709.2735-3-14-3'], ['0709.2735-2-5-5', '0709.2735-3-4-5'], ['0709.2735-2-14-2', '0709.2735-3-12-2'], ['0709.2735-2-12-2', '0709.2735-3-11-2'], ['0709.2735-2-12-3', '0709.2735-3-11-3'], ['0709.2735-2-12-4', '0709.2735-3-11-4'], ['0709.2735-2-12-5', '0709.2735-3-11-5'], ['0709.2735-2-12-6', '0709.2735-3-11-6'], ['0709.2735-2-13-1', '0709.2735-3-11-8']] | [['0709.2735-2-7-1', '0709.2735-3-6-1'], ['0709.2735-2-7-6', '0709.2735-3-6-6'], ['0709.2735-2-7-8', '0709.2735-3-6-8'], ['0709.2735-2-3-5', '0709.2735-3-2-5'], ['0709.2735-2-4-3', '0709.2735-3-3-3'], ['0709.2735-2-0-3', '0709.2735-3-0-3'], ['0709.2735-2-6-1', '0709.2735-3-5-1'], ['0709.2735-2-6-2', '0709.2735-3-5-2'], ['0709.2735-2-8-4', '0709.2735-3-7-4'], ['0709.2735-2-8-8', '0709.2735-3-7-8'], ['0709.2735-2-9-1', '0709.2735-3-8-1'], ['0709.2735-2-9-2', '0709.2735-3-8-2'], ['0709.2735-2-9-3', '0709.2735-3-8-3'], ['0709.2735-2-10-9', '0709.2735-3-9-9'], ['0709.2735-2-16-2', '0709.2735-3-14-2'], ['0709.2735-2-5-0', '0709.2735-3-4-0'], ['0709.2735-2-14-0', '0709.2735-3-12-0'], ['0709.2735-2-14-1', '0709.2735-3-12-1'], ['0709.2735-2-14-3', '0709.2735-3-12-3'], ['0709.2735-2-12-0', '0709.2735-3-11-0'], ['0709.2735-2-12-1', '0709.2735-3-11-1'], ['0709.2735-2-13-0', '0709.2735-3-11-7']] | [] | [['0709.2735-2-7-9', '0709.2735-3-6-9'], ['0709.2735-2-3-2', '0709.2735-3-2-2'], ['0709.2735-2-4-0', '0709.2735-3-3-0'], ['0709.2735-2-0-2', '0709.2735-3-0-2'], ['0709.2735-2-8-0', '0709.2735-3-7-0'], ['0709.2735-2-11-5', '0709.2735-3-10-5']] | [] | ['0709.2735-1-2-0', '0709.2735-1-2-1', '0709.2735-1-2-2', '0709.2735-1-2-3', '0709.2735-1-2-4', '0709.2735-1-3-3', '0709.2735-1-3-4', '0709.2735-1-5-1', '0709.2735-1-5-2', '0709.2735-1-5-3', '0709.2735-1-5-4', '0709.2735-1-7-3', '0709.2735-2-2-0', '0709.2735-2-2-1', '0709.2735-2-2-2', '0709.2735-2-2-3', '0709.2735-2-2-4', '0709.2735-2-3-3', '0709.2735-2-3-4', '0709.2735-2-5-1', '0709.2735-2-5-2', '0709.2735-2-5-3', '0709.2735-2-5-4', '0709.2735-2-7-3', '0709.2735-3-1-3', '0709.2735-3-1-4', '0709.2735-3-2-3', '0709.2735-3-2-4', '0709.2735-3-4-1', '0709.2735-3-4-2', '0709.2735-3-4-3', '0709.2735-3-4-4', '0709.2735-3-6-3'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/0709.2735 | {'0709.2735-3-0-0': 'The quantum kicked prime number rotator (QKPR) is defined as the rotator whose energy levels are prime numbers.', '0709.2735-3-0-1': 'The long time behavior is decided by the kick period [MATH] and kick strength [MATH].', '0709.2735-3-0-2': 'When [MATH] is irrational, QKPR is localized because of the equidistribution theorem.', '0709.2735-3-0-3': 'When [MATH] is rational, QKPR is localized for small [MATH], because the system seems like a generalized kicked dimer model.', '0709.2735-3-0-4': 'We argue for rational [MATH] QKPR delocalizes for large [MATH].', '0709.2735-3-1-0': 'The kicked prime number rotator is defined as [EQUATION] where [MATH] is the unperturbed Hamiltonian, and [MATH] is the perturbation.', '0709.2735-3-1-1': '[MATH] is a diagonal matrix.', '0709.2735-3-1-2': 'The [MATH]-th eigenvalue [MATH] corresponding to [MATH]-th eigenstate [MATH] of [MATH] is the [MATH]-th prime number [MATH].', '0709.2735-3-1-3': 'When [MATH], [MATH].', '0709.2735-3-1-4': '[MATH].', '0709.2735-3-1-5': 'The diagonal of [MATH] is [MATH].', '0709.2735-3-2-0': '[MATH] is defined as [EQUATION]', '0709.2735-3-2-1': 'The Floquet operator is [MATH].', '0709.2735-3-2-2': 'The matrix elements of [MATH] is [MATH], where [MATH], [MATH] is the Bessel function of the first kind.', '0709.2735-3-2-3': 'We set [MATH].', '0709.2735-3-2-4': '[MATH].', '0709.2735-3-2-5': 'The system is very like the quantum kicked rotator (QKR), except its energy levels are now prime numbers.', '0709.2735-3-3-0': 'It seems there is no classical correspondence of QKPR.', '0709.2735-3-3-1': 'Experimental implementation of such a model also seems impossible.', '0709.2735-3-3-2': 'Nevertheless it still has some theoretical interests.', '0709.2735-3-3-3': 'In the paper, we numerically calculate the evolution of QKPR.', '0709.2735-3-3-4': 'We are interested in the same problem in QKR.', '0709.2735-3-3-5': 'If the particle is in the ground state [MATH] initially, will it diffuse away in the future?', '0709.2735-3-4-0': 'The evolution of the system is calculated by the iterative unitary matrix multiply method [CITATION].', '0709.2735-3-4-1': '[MATH].', '0709.2735-3-4-2': '[MATH].', '0709.2735-3-4-3': '[MATH].', '0709.2735-3-4-4': 'And so on.', '0709.2735-3-4-5': 'In this way, we can calculate [MATH] by [MATH] matrix multiplies.', '0709.2735-3-4-6': 'In all our calculation, [MATH] indicates at time [MATH].', '0709.2735-3-4-7': 'For example, the first figure [MATH] means at time [MATH].', '0709.2735-3-4-8': '[MATH] is the [MATH]-th basis [MATH] and [MATH] is the base-10 logarithm of the absolute value of the wave function on the [MATH].', '0709.2735-3-4-9': '[MATH] runs from [MATH] to [MATH] in our calculation.', '0709.2735-3-5-0': 'First, we choose [MATH], [MATH].', '0709.2735-3-5-1': 'The result is displayed in FIG. 1.', '0709.2735-3-5-2': 'QKPR is localized perfectly.', '0709.2735-3-5-3': 'In our simulation, the exponentially fall of the wave function never changes from [MATH] to [MATH].', '0709.2735-3-5-4': 'The wave function on the [MATH] is [MATH].', '0709.2735-3-5-5': 'From [MATH] to [MATH], the wave function is somewhat curved.', '0709.2735-3-5-6': 'After the first kick, the wave function is the [MATH]-th column of the Floquet matrix [MATH].', '0709.2735-3-5-7': 'The absolute value of [MATH] is [MATH].', '0709.2735-3-5-8': '[MATH] falls to zero faster than exponentially .', '0709.2735-3-5-9': 'This is the reason the curved form of the wave function.', '0709.2735-3-6-0': 'Second, we choose [MATH], [MATH].', '0709.2735-3-6-1': 'The result is displayed in FIG. 3.', '0709.2735-3-6-2': 'The wave function is also localized.', '0709.2735-3-6-3': 'This is expected.', '0709.2735-3-6-4': 'The sequence [MATH] is equidistributed between [MATH], when [MATH] is irrational.', '0709.2735-3-6-5': 'We denote the sequence [MATH] as QKPR[MATH].', '0709.2735-3-6-6': 'In QKR, the sequence [MATH] is also equidistributed between [MATH], for an irrational [MATH].', '0709.2735-3-6-7': 'We denote the sequence [MATH] as (QKR[MATH]).', '0709.2735-3-6-8': 'We can also use the inverse Cayley transform method to convert the Floquet eigenstate equation [MATH] into an equation like Anderson localization problem.', '0709.2735-3-6-9': "From Fishman et al's argument [CITATION], QKPR will localize.", '0709.2735-3-7-0': 'In the left of FIG. 2, QKPR[MATH] and QKR[MATH] are displayed.', '0709.2735-3-7-1': 'Though there are apparently some correlations in QKPR[MATH] and QKR[MATH] and the correlation is different between both sequences.', '0709.2735-3-7-2': 'The correlation is surely not strong enough to destroy localization.', '0709.2735-3-7-3': 'If a sequence is periodic with a period [MATH], then the discrete Fourier transform of the sequence is composed by [MATH] modes.', '0709.2735-3-7-4': 'To find whether there is some periodicity in the sequence, we perform a discrete Fourier transform (DFT) on the sequence.', '0709.2735-3-7-5': 'DFT of a sequence [MATH] of length [MATH] is defined as [MATH], where [MATH] runs from 1 to [MATH].', '0709.2735-3-7-6': 'There are some other definitions of DFT with nuanced difference with our definition.', '0709.2735-3-7-7': 'But the difference is irrelevant to our discussion here.', '0709.2735-3-7-8': 'In the right of FIG. 2, the DFTs of both sequences are displayed.', '0709.2735-3-7-9': 'There are no rigorous periodicity in both sequences.', '0709.2735-3-7-10': '[MATH] of QKPR[MATH] seems to have a trend to cluster together.', '0709.2735-3-7-11': 'Also it is less uniformly distributed than the [MATH] of QKR[MATH] and tends to be small.', '0709.2735-3-8-0': 'If [MATH] , does QKPR localize?', '0709.2735-3-8-1': 'At first thought, this seems to be a resonant case in QKR and the rotator will delocalize.', '0709.2735-3-8-2': 'The calculation result is in fact it still localizes for small [MATH].', '0709.2735-3-8-3': 'In FIG. 4, we choose [MATH] and [MATH] and in FIG. 5, [MATH] and [MATH].', '0709.2735-3-8-4': 'QKPR of [MATH] is apparently localized.', '0709.2735-3-9-0': 'The explanation of the localization is QKPR with [MATH] and [MATH] is a kicked pseudo dimer rotator.', '0709.2735-3-9-1': 'The dimer model is defined as every diagonal matrix element is a probability variable which only takes two values [CITATION].', '0709.2735-3-9-2': 'The kicked dimer model can be defined as every diagonal matrix element of [MATH] is a random variable which takes two values.', '0709.2735-3-9-3': 'If it takes more than two values, it is a generalized kicked dimer model.', '0709.2735-3-9-4': 'For [MATH], the sequence [MATH] mainly takes four values.', '0709.2735-3-9-5': 'So it is a generalized kicked pseudo dimer model.', '0709.2735-3-9-6': 'The sequence [MATH] (QKPR[MATH]) is not really random.', '0709.2735-3-9-7': 'But the pseudorandomness is enough to result in localization [CITATION].', '0709.2735-3-9-8': 'To measure how random QKPR[MATH] is, we perform a DFT on it.', '0709.2735-3-9-9': 'In the left of FIG. 4, we compare QKPR[MATH] with a dimer sequence D[MATH], which is defined as [MATH], where every [MATH] is a random variable which takes two values [MATH] and [MATH].', '0709.2735-3-9-10': 'The [MATH]s of D[MATH] and QKPR[MATH] are quite close with each other, except QKPR[MATH] tends to cluster together.', '0709.2735-3-10-0': 'Does QKPR localize for [MATH] and [MATH]?', '0709.2735-3-10-1': 'We think it delocalizes.', '0709.2735-3-10-2': 'There are a series of plateaux in the wave function of QKPR.', '0709.2735-3-10-3': 'The wave function falls abruptly when approaching the boundary (cliff) of a plateau.', '0709.2735-3-10-4': 'Some plateaux disappear at [MATH].', '0709.2735-3-10-5': 'QKPR wave pass through the cliff, so it disappears intermittently.', '0709.2735-3-11-0': 'The most obvious cliff is from [MATH] to [MATH] in FIG.6.', '0709.2735-3-11-1': 'QKPR[MATH] from [MATH] to [MATH] is [EQUATION]', '0709.2735-3-11-2': 'Note it is periodic from [MATH] to [MATH].', '0709.2735-3-11-3': 'In a perfect periodic potential the wave will always diffuse away.', '0709.2735-3-11-4': 'The quantum wave can not stay at a period potential very long.', '0709.2735-3-11-5': 'Once the wave has propagated into the phase space between [MATH] and [MATH], it diffuses away quickly.', '0709.2735-3-11-6': 'While once the wave propagate the phase space whose neighborhood [MATH] is irregular, the quantum wave is localized there at least temporarily.', '0709.2735-3-11-7': 'When [MATH], [MATH] from [MATH] to [MATH] is a plateau in FIG. 2.', '0709.2735-3-11-8': 'There is a transition from a plateau to a cliff when [MATH] becomes large.', '0709.2735-3-12-0': 'Note in FIG. 4, for lots of [MATH], Lots of [MATH] of QKPR[MATH] and QKPR[MATH] tend to be small.', '0709.2735-3-12-1': "From Plancherel's theorem, there must be some [MATH] tends to very large.", '0709.2735-3-12-2': 'So there is some weak periodicity in the sequence QKPR[MATH].', '0709.2735-3-12-3': 'The sequence from [MATH] to [MATH] increases the periodicity of QKPR[MATH].', '0709.2735-3-13-0': 'At [MATH], there is a plateau from [MATH] to [MATH] or so.', '0709.2735-3-13-1': 'Even at [MATH], there is a plateau from [MATH] to [MATH].', '0709.2735-3-13-2': 'The localization length of QKR is [MATH] or so [CITATION].', '0709.2735-3-13-3': 'So the distribution length of QKPR is much larger than the localization length.', '0709.2735-3-13-4': 'Even the result of [MATH] is untrustworthy, we think the quantum wave is absolutely not localized in the localization length.', '0709.2735-3-13-5': 'From [MATH] to [MATH] or so, there is apparently unneglectable quantum wave at every [MATH] at for example [MATH].', '0709.2735-3-13-6': 'At [MATH], the quantum wave has already propagated into [MATH].', '0709.2735-3-13-7': 'QKPR with [MATH] and [MATH] is not localized.', '0709.2735-3-13-8': 'In [CITATION], we point out when the kick strength is larger than [MATH], the inverse Cayley transform method breaks down.', '0709.2735-3-13-9': 'QKPR is apparently an evidence to the failure of the inverse Cayley transform method when [MATH] is large.', '0709.2735-3-14-0': 'In this paper, we apply the iterative unitary matrix multiply method to quantum kicked prime number rotator.', '0709.2735-3-14-1': 'If [MATH] is irrational, the rotator localizes.', '0709.2735-3-14-2': 'If [MATH] is rational, for small kick strength [MATH], the rotator localizes.', '0709.2735-3-14-3': 'As [MATH] increases, we argue there is a localization-delocalization transition.'} | null | null | null | null |
1307.2297 | {'1307.2297-1-0-0': 'The two-sample empirical likelihood is Bartlett correctable (Jing, 1995).', '1307.2297-1-0-1': 'We extend it to the full parameter space and obtain a two-sample extended empirical likelihood which is more accurate than the original and can also achieve the second-order accuracy of the Bartlett correction.', '1307.2297-1-1-0': 'AMS 2000 subject classifications: Primary 62G20; secondary 62E20.', '1307.2297-1-2-0': '# INTRODUCTION', '1307.2297-1-3-0': 'The empirical likelihood introduced by Owen (1988, 1990) is a versatile non-parametric method of inference with many applications (Owen, 2001).', '1307.2297-1-3-1': 'One problem which the empirical likelihood method has been successfully applied to is the two-sample problem (Jing, 1995; Wu and Yan, 2012) where the parameter of interest [MATH] is the difference between the means of two populations.', '1307.2297-1-3-2': 'The well-known Behrens-Fisher problem is a special two-sample problem where the two populations are known to be normally distributed.', '1307.2297-1-3-3': 'Following DiCiccio, Hall and Romano (1991) who showed the surprising result that the (one-sample) empirical likelihood for a smooth function of the mean is Bartlett correctable, Jing (1995) proved that the two-sample empirical likelihood for [MATH] is also Bartlett correctable.', '1307.2297-1-3-4': 'The coverage error of a confidence region based on the original empirical likelihood is [MATH], but that of the Bartlett corrected empirical likelihood is only [MATH].', '1307.2297-1-4-0': 'For a one-sample empirical likelihood, there is a mismatch between its domain and the parameter space in that it is defined on only a part of the parameter space.', '1307.2297-1-4-1': 'This mismatch is a main cause of the undercoverage problem associated with empirical likelihood confidence regions (Tsao, 2013).', '1307.2297-1-4-2': 'The two-sample empirical likelihood for [MATH] also has the mismatch problem as it is defined on a bounded interval but the parameter space is [MATH].', '1307.2297-1-4-3': 'In this paper, we derive an extended version of the original two-sample empirical likelihood (OEL) by mapping its domain onto [MATH] through the composite similarity mapping of Tsao and Wu (2013).', '1307.2297-1-4-4': 'The resulting two-sample extended empirical likelihood (EEL) for [MATH] is defined on the entire [MATH] and hence free from the mismatch problem.', '1307.2297-1-4-5': 'Under mild conditions, this EEL has the same asymptotic properties as the OEL.', '1307.2297-1-4-6': 'It can also attain the second order accuracy of the two-sample Bartlett corrected empirical likelihood (BEL) of Jing (1995).', '1307.2297-1-4-7': 'The first order version of the EEL is more accurate than the OEL.', '1307.2297-1-4-8': 'It is also easy to compute and surprisingly more accurate than the second order BEL.', '1307.2297-1-4-9': 'We recommend it for two-sample empirical likelihood inference.', '1307.2297-1-5-0': '# Two-sample empirical likelihood', '1307.2297-1-6-0': 'Let [MATH] and [MATH] be two independent random samples from two unknown distributions [MATH] and [MATH], respectively.', '1307.2297-1-6-1': 'Let [MATH] and [MATH].', '1307.2297-1-6-2': 'The unknown parameter of interest is the difference in mean [MATH] and the parameter space is [MATH].', '1307.2297-1-6-3': 'Denote by [MATH] and [MATH] two probability vectors satisfying [MATH], [MATH], [MATH] and [MATH].', '1307.2297-1-6-4': 'Let [MATH] and [MATH], and denote by [MATH] their difference, more precisely, [MATH].', '1307.2297-1-6-5': 'For a [MATH], Jing (1995) defined the two-sample empirical likelihood [MATH] as [EQUATION]', '1307.2297-1-6-6': 'The corresponding two-sample empirical log-likelihood ratio for [MATH] is thus [EQUATION]', '1307.2297-1-6-7': 'To differentiate between the [MATH] in ([REF]) and the extended version of [MATH] in the next section, we will refer to the [MATH] in ([REF]) as the original two-sample empirical log-likelihood ratio or simply "OEL [MATH]".', '1307.2297-1-7-0': 'Let [MATH], [MATH] and [MATH].', '1307.2297-1-7-1': 'Without loss of generality, assume that [MATH].', '1307.2297-1-7-2': 'By the method of Lagrangian multipliers, we have [EQUATION] where the multiplier [MATH] satisfies [EQUATION] and [EQUATION]', '1307.2297-1-7-3': 'Under the assumption that [MATH] and [MATH] have finite variances, Jing (1995) showed that [EQUATION]', '1307.2297-1-7-4': 'Hence, the 100([MATH])% OEL confidence interval for [MATH] is [EQUATION] where [MATH] is ([MATH])th quantile of the [MATH] distribution.', '1307.2297-1-7-5': 'The coverage error of [MATH] is [MATH], that is [EQUATION]', '1307.2297-1-7-6': 'Under a stronger assumption that [MATH] and [MATH] have finite fourth moments, Jing (1995) also showed the OEL [MATH] is Bartlett correctable, that is [EQUATION] where [MATH] is the Bartlett corrected empirical likelihood (BEL) confidence interval and [MATH] is the Bartlett correction constant in Theorem 2 of Jing (1995).', '1307.2297-1-7-7': 'Detailed proofs for ([REF]) and ([REF]) are in a supplement to Jing (1995) available from Professor Bing-Yi Jing.', '1307.2297-1-7-8': 'See also Wu and Yan (2012) and Qin (1994) about two-sample empirical likelihood methods.', '1307.2297-1-8-0': '# Two-sample extended empirical likelihood', '1307.2297-1-9-0': 'The two-sample OEL [MATH] also suffers from the mismatch problem between its domain and the parameter space.', '1307.2297-1-9-1': 'To see this, since [MATH] and [MATH], we have [EQUATION].', '1307.2297-1-9-2': 'It follows that the domain of the OEL [MATH], [MATH], is [EQUATION].', '1307.2297-1-9-3': 'Since the parameter space is [MATH], the mismatch can be expressed as [MATH] which is a main cause of the undercoverage problem of empirical likelihood confidence regions (Tsao, 2013; Tsao and Wu, 2013).', '1307.2297-1-9-4': 'To overcome the mismatch, we extend the OEL [MATH] by expanding its domain to the entire [MATH].', '1307.2297-1-10-0': 'For simplicity, we also assume that [MATH] and further that [MATH] so that [MATH], [MATH] and [MATH], for example, are all interchangeable.', '1307.2297-1-10-1': 'A point estimator for [MATH] is [MATH] where [MATH] and [MATH] are the sample means.', '1307.2297-1-10-2': 'It is easy to verify that [MATH] is the maximum empirical likelihood estimator for [MATH].', '1307.2297-1-11-0': 'Following Tsao and Wu (2013), we define the composite similarity mapping [MATH] centred on [MATH] as [EQUATION] where function [MATH] is the expansion factor given by [EQUATION]', '1307.2297-1-11-1': 'Theorem 1 below gives two key properties of mapping [MATH].', '1307.2297-1-12-0': 'Theorem 1.', '1307.2297-1-12-1': 'Suppose the variances of [MATH] and [MATH] are finite and positive.', '1307.2297-1-12-2': 'Then, [MATH] defined by ([REF]) and ([REF]) satisfies (i) it has a unique fixed point at [MATH] and (ii) it is a bijective mapping from [MATH] to [MATH].', '1307.2297-1-13-0': 'Since [MATH] is bijective, it has an inverse function which we denote by [MATH].', '1307.2297-1-13-1': 'For any [MATH], let [MATH].', '1307.2297-1-13-2': 'The extended empirical log-likelihood ratio EEL [MATH] is given by [EQUATION] which is defined for [MATH] values throughout [MATH].', '1307.2297-1-13-3': 'Hence the EEL [MATH] is free from the mismatch problem of the OEL [MATH].', '1307.2297-1-13-4': 'Theorem 2 shows that EEL [MATH] has the same asymptotic chi-square distribution as the OEL [MATH].', '1307.2297-1-14-0': 'Theorem 2.', '1307.2297-1-14-1': 'Suppose the variances of [MATH] and [MATH] are finite and positive.', '1307.2297-1-14-2': 'Then, the EEL [MATH] defined by ([REF]) satisfies [EQUATION]', '1307.2297-1-15-0': 'By Theorem 2, the [MATH] EEL confidence interval for [MATH] is [EQUATION] which has a coverage error of [MATH].', '1307.2297-1-15-1': 'The expansion factor in ([REF]) is a convenient choice which also gives good numerical results.', '1307.2297-1-15-2': 'There are, however, other choices available under which Theorems 1 and 2 also hold.', '1307.2297-1-15-3': 'This provides an opportunity to optimize the choice of expansion factor to obtain the second order accuracy.', '1307.2297-1-15-4': 'Theorem 3 below gives such an optimal choice.', '1307.2297-1-16-0': 'Theorem 3.', '1307.2297-1-16-1': 'Suppose [MATH] and [MATH] have finite and positive fourth moments.', '1307.2297-1-16-2': 'Let [MATH] be the EEL defined by the composite similarity mapping ([REF]) with the following expansion factor [EQUATION] where [MATH] and [MATH] is the Bartlett correction factor for two-sample problem in ([REF]).', '1307.2297-1-16-3': 'Then, we have [EQUATION] and [EQUATION]', '1307.2297-1-17-0': 'Replacing EEL [MATH] in ([REF]) with [MATH] gives an EEL confidence interval which, by ([REF]), has a coverage error of [MATH].', '1307.2297-1-17-1': 'Because of this, we call [MATH] the second order EEL or EEL[MATH].', '1307.2297-1-17-2': 'Correspondingly, we call the EEL [MATH] defined by expansion factor ([REF]) the first order EEL or EEL[MATH].', '1307.2297-1-18-0': 'There are two lemmas that are needed for the proofs of Theorems 1, 2 and 3.', '1307.2297-1-18-1': 'In order to limit the length of the paper, we have not included them here.', '1307.2297-1-18-2': 'The lemmas, their proofs and the proofs of the theorems are given in the appendix of a technical report available on request from the authors.', '1307.2297-1-19-0': '# Numerical examples', '1307.2297-1-20-0': 'We now compare the coverage accuracy of 95 confidence intervals based on the OEL, BEL and EEL with two numerical examples.', '1307.2297-1-20-1': 'Comparisons based on 90 and 99 confidence intervals give similar conclusions and they can be found in the appendix in the aforementioned technical report.', '1307.2297-1-21-0': 'In the first example, [MATH] and [MATH] are both standard normal distributions [MATH].', '1307.2297-1-21-1': 'In the second example, they are [MATH] and [MATH], respectively.', '1307.2297-1-21-2': 'Simulated coverage probabilities for these examples are given in Tables 1 and 2, respectively.', '1307.2297-1-21-3': 'Each simulated probability in the tables is based on 10,000 pairs of random samples whose sizes are indicated by the row and column headings, respectively.', '1307.2297-1-21-4': 'The BEL and EEL[MATH] were computed by using the estimated Bartlett correction factor from page 317 in Jing (1995).', '1307.2297-1-21-5': 'We summarize the tables with the following observations: (1) EEL[MATH] is consistently more accurate than OEL.', '1307.2297-1-21-6': 'Surprisingly, it is also more accurate than the second order BEL and EEL[MATH] for small and moderate sample sizes ([MATH]) and competitive in accuracy when sample sizes are larger.', '1307.2297-1-21-7': '(2) EEL[MATH] is more accurate than OEL and BEL for small and moderate sample size.', '1307.2297-1-21-8': 'It is comparable to BEL when one or both sample sizes are large.', '1307.2297-1-22-0': 'To conclude, the EEL[MATH] is easy-to-compute and is the most accurate overall.', '1307.2297-1-22-1': 'Hence, we recommend EEL[MATH] for two-sample problems.', '1307.2297-1-23-0': 'References'} | {'1307.2297-2-0-0': 'Jing (1995) and Liu et al. (2008) studied the two-sample empirical likelihood and showed it is Bartlett correctable for the univariate and multivariate cases, respectively.', '1307.2297-2-0-1': 'We expand its domain to the full parameter space and obtain a two-sample extended empirical likelihood which is more accurate and can also achieve the second-order accuracy of the Bartlett correction.', '1307.2297-2-1-0': 'AMS 2000 subject classifications: Primary 62G20; secondary 62E20.', '1307.2297-2-2-0': '# Introduction', '1307.2297-2-3-0': 'The empirical likelihood introduced by Owen (1988, 1990) is a versatile non-parametric method of inference with many applications (Owen, 2001).', '1307.2297-2-3-1': 'One problem which the empirical likelihood method has been successfully applied to is the two-sample problem (Jing, 1995; Liu et al. 2008; Wu and Yan, 2012) where the parameter of interest [MATH] is the difference between the means of two populations.', '1307.2297-2-3-2': 'The well-known Behrens-Fisher problem is a special two-sample problem where the two populations are known to be normally distributed.', '1307.2297-2-3-3': 'Following DiCiccio et al. (1991) who showed the surprising result that the (one-sample) empirical likelihood for a smooth function of the mean is Bartlett correctable, Jing (1995) and Liu et al. (2008) proved that the two-sample empirical likelihood for [MATH] is also Bartlett correctable for the univariate and multivariate cases, respectively.', '1307.2297-2-3-4': 'The coverage error of a confidence region based on the original empirical likelihood is [MATH], but that based on the Bartlett corrected empirical likelihood is only [MATH].', '1307.2297-2-4-0': 'For a one-sample empirical likelihood, there is a mismatch between its domain and the parameter space in that it is defined on only a part of the parameter space.', '1307.2297-2-4-1': 'This mismatch is a main cause of the undercoverage problem associated with empirical likelihood confidence regions (Tsao, 2013).', '1307.2297-2-4-2': 'The two-sample empirical likelihood for [MATH] also has the mismatch problem as it is defined on a bounded region but the parameter space is [MATH].', '1307.2297-2-4-3': 'In this paper, we derive an extended version of the original two-sample empirical likelihood (OEL) by expanding its domain into [MATH] through the composite similarity mapping of Tsao and Wu (2013).', '1307.2297-2-4-4': 'The resulting two-sample extended empirical likelihood (EEL) for [MATH] is defined on the entire [MATH] and hence free from the mismatch problem.', '1307.2297-2-4-5': 'Under mild conditions, this EEL has the same asymptotic properties as the OEL.', '1307.2297-2-4-6': 'It can also attain the second order accuracy of the two-sample Bartlett corrected empirical likelihood (BEL) of Jing (1995) and Liu et al. (2008).', '1307.2297-2-4-7': 'The first order version of this EEL is substantially more accurate than the OEL, especially for small sample sizes.', '1307.2297-2-4-8': 'It is also easy to compute and competitive in accuracy to the second order BEL.', '1307.2297-2-4-9': 'We recommend it for two-sample empirical likelihood inference.', '1307.2297-2-5-0': '# Two-sample empirical likelihood', '1307.2297-2-6-0': 'Let [MATH] and [MATH] be independent copies of random vectors [MATH] and [MATH], respectively.', '1307.2297-2-6-1': 'Denote by [MATH] and [MATH] the mean and covariance matrix of [MATH], and by [MATH] and [MATH] the mean and covariance matrix of [MATH], respectively.', '1307.2297-2-6-2': 'The unknown parameter of interest is the difference in means [MATH] and the parameter space is the entire [MATH].', '1307.2297-2-6-3': 'We will need the following three conditions later in the paper:', '1307.2297-2-7-0': 'C1.', '1307.2297-2-7-1': '[MATH] and [MATH] are finite covariance matrix with full rank [MATH];', '1307.2297-2-8-0': 'C2.', '1307.2297-2-8-1': '[MATH] and [MATH];', '1307.2297-2-9-0': 'C3.', '1307.2297-2-9-1': '[MATH] and [MATH].', '1307.2297-2-10-0': 'Denote by [MATH] and [MATH] two probability vectors satisfying [MATH], [MATH], [MATH] and [MATH].', '1307.2297-2-10-1': 'Let [MATH] and [MATH], and denote by [MATH] their difference, that is, [EQUATION].', '1307.2297-2-10-2': 'The original two-sample empirical likelihood for a [MATH], [MATH], is defined as [EQUATION]', '1307.2297-2-10-3': 'The corresponding two-sample empirical log-likelihood ratio for [MATH] is thus [EQUATION]', '1307.2297-2-10-4': 'In order to develop our extended empirical likelihood, it is important to understand the domains of the original empirical likelihood ratio [MATH] and log-likelihood ratio [MATH].', '1307.2297-2-10-5': 'The domain of [MATH] is given by', '1307.2297-2-11-0': '[MATH]p[MATH]q[MATH]_x(p)=_i=1^m p_i X_i[MATH]_y(q)=_j=1^n q_j Y_j[MATH] Since the range of [MATH] and [MATH] are the convex hulls of the [MATH] and [MATH], respectively, [MATH] is a bounded, closed and connected region in [MATH] without voids.', '1307.2297-2-11-1': 'More details about the geometric structure of [MATH] may be found in Lemma 1 and its proof, and one of these is that an interior point of [MATH] can be expressed as [MATH] where all elements of [MATH] and [MATH] are straightly positive.', '1307.2297-2-11-2': 'Correspondingly, a boundary point of [MATH] can only be expressed as [MATH] where one or more elements of [MATH] and [MATH] are zero.', '1307.2297-2-11-3': 'This implies that [MATH] for any boundary point [MATH] of [MATH] and [MATH] for every interior point.', '1307.2297-2-11-4': 'We define the domain of the empirical log-likelihood ratio [MATH] as [EQUATION] which excludes the boundary points of [MATH].', '1307.2297-2-11-5': 'That is, [MATH] is the open set in [MATH] that contains the collection of [MATH] where elements of [MATH] and [MATH] are all straightly positive.', '1307.2297-2-11-6': 'To differentiate between the [MATH] in ([REF]) and the extended version of [MATH] in the next section, we will refer to the [MATH] in ([REF]) as the original two-sample empirical log-likelihood ratio or simply "OEL [MATH]".', '1307.2297-2-11-7': 'The extended version will be referred to as the "EEL [MATH]".', '1307.2297-2-12-0': 'Let [MATH], [MATH] and [MATH].', '1307.2297-2-12-1': 'Without loss of generality, assume that [MATH].', '1307.2297-2-12-2': 'By the method of Lagrangian multipliers, we have [EQUATION] where the multiplier [MATH] satisfies [EQUATION] and [EQUATION]', '1307.2297-2-12-3': 'Under the assumption ([MATH]), Jing (1995) and Liu et al. (2008) showed that [EQUATION]', '1307.2297-2-12-4': 'Hence, the 100([MATH])% OEL confidence interval for [MATH] is [EQUATION] where [MATH] is ([MATH])th quantile of the [MATH] distribution.', '1307.2297-2-12-5': 'The coverage error of [MATH] is [MATH], that is [EQUATION]', '1307.2297-2-12-6': 'Under assumptions ([MATH]), ([MATH]) and ([MATH]), Jing (1995) and Liu et al. (2008) also showed that the OEL [MATH] is Bartlett correctable, that is [EQUATION] where [MATH] is the Bartlett corrected empirical likelihood (BEL) confidence interval and [MATH] is the Bartlett correction constant in Theorem 2 of Liu et al. (2008).', '1307.2297-2-12-7': 'For the one-dimensional case, a formula for this constant was first given in Theorem 2 in Jing (1995) but the formula is incomplete (Liu et al. 2008).', '1307.2297-2-12-8': 'See also Wu and Yan (2012) and Qin (1994) for discussions about two-sample empirical likelihood methods.', '1307.2297-2-13-0': '# Two-sample extended empirical likelihood', '1307.2297-2-14-0': 'Like the one-sample empirical likelihood for the mean, the two-sample OEL [MATH] also suffers from the mismatch problem between its domain and the parameter space since the parameter space is [MATH] but [MATH].', '1307.2297-2-14-1': 'This is a main cause of the undercoverage problem of empirical likelihood confidence regions (Tsao, 2013; Tsao and Wu, 2013).', '1307.2297-2-14-2': 'To overcome the mismatch, we now extend the OEL [MATH] by expanding its domain to the entire [MATH].', '1307.2297-2-15-0': 'For simplicity, in addition to [MATH] we further assume that [MATH] so that [MATH], [MATH] and [MATH], for example, are all interchangeable.', '1307.2297-2-15-1': 'A point estimator for [MATH] is [MATH] where [MATH] and [MATH] are the sample means.', '1307.2297-2-15-2': 'It is easy to verify that [MATH] is the maximum empirical likelihood estimator (MELE) for [MATH].', '1307.2297-2-15-3': 'Following Tsao and Wu (2013), we define the composite similarity mapping [MATH] centred on [MATH] as [EQUATION] where function [MATH] is the expansion factor given by [EQUATION]', '1307.2297-2-15-4': 'To investigate the properties of the composite similarity mapping [MATH], we need Lemma 1 below which gives two properties of the two-sample OEL [MATH].', '1307.2297-2-15-5': 'For convenience, we denote by [MATH] the line segment that connects [MATH] and [MATH] and by [MATH] a boundary point of [MATH].', '1307.2297-2-15-6': 'We have', '1307.2297-2-16-0': 'Lemma 1.', '1307.2297-2-16-1': 'The two-sample OEL [MATH] satisfies: ([MATH]) if [MATH] and [MATH], then [MATH] and ([MATH]) for [MATH], [MATH].', '1307.2297-2-17-0': 'Lemma 1 shows the two-sample OEL [MATH] for the difference of two means behaves exactly like its one-sample counterpart for the mean in terms monotonicity and boundary behaviour: it is "monotone increasing" along each ray originating from the MELE and it goes to infinity as [MATH] approaches a boundary point from within [MATH].', '1307.2297-2-17-1': 'Nevertheless, the two-sample and one-sample cases are not entirely the same; the contours of the two-sample OEL may not be convex but that of the one-sample OEL always are.', '1307.2297-2-17-2': 'Theorem 1 below gives three key properties of composite similarity mapping [MATH].', '1307.2297-2-18-0': 'Theorem 1.', '1307.2297-2-18-1': 'Under the assumption (C1), [MATH] defined by ([REF]) and ([REF]) satisfies (i) it has a unique fixed point at [MATH], (ii) it is a similarity mapping for each individual contour of the OEL [MATH] and (iii) it is a bijective mapping from [MATH] to [MATH].', '1307.2297-2-19-0': 'Since [MATH] is bijective, it has an inverse function which we denote by [MATH].', '1307.2297-2-19-1': 'For any [MATH], let [MATH].', '1307.2297-2-19-2': 'The two-sample extended empirical log-likelihood ratio EEL [MATH] is given by [EQUATION] which is defined for [MATH] values throughout [MATH].', '1307.2297-2-19-3': 'Hence the EEL [MATH] is free from the mismatch problem of the OEL [MATH].', '1307.2297-2-19-4': 'Denote by [MATH] the image of [MATH] under the inverse transformation [MATH], that is [EQUATION]', '1307.2297-2-19-5': 'Then, the EEL [MATH] evaluated at [MATH] is given by [EQUATION]', '1307.2297-2-19-6': 'If [MATH] is very small, then [MATH] will have the same asymptotic distribution as [MATH].', '1307.2297-2-19-7': 'Lemma 2 below shows that [MATH] and that [MATH] is indeed very small.', '1307.2297-2-20-0': 'Lemma 2.', '1307.2297-2-20-1': 'Under assumption (C1), point [MATH] defined by equation ([REF]) satisfies [MATH] and [MATH].', '1307.2297-2-21-0': 'Theorem 2 below shows that EEL [MATH] has the same asymptotic chi-square distribution as the OEL [MATH].', '1307.2297-2-22-0': 'Theorem 2.', '1307.2297-2-22-1': 'Under assumption (C1), the EEL [MATH] defined by ([REF]) satisfies [EQUATION]', '1307.2297-2-23-0': 'By Theorem 2, the [MATH] EEL confidence interval for [MATH] is [EQUATION] which has a coverage error of [MATH].', '1307.2297-2-23-1': 'The expansion factor in ([REF]) is a convenient choice which also gives good numerical results.', '1307.2297-2-23-2': 'There are many other choices available under which Theorems 1 and 2 also hold.', '1307.2297-2-23-3': 'This provides an opportunity to optimize the choice of expansion factor to obtain the second order accuracy.', '1307.2297-2-23-4': 'Theorem 3 below gives such an optimal choice.', '1307.2297-2-24-0': 'Theorem 3.', '1307.2297-2-24-1': 'Under assumptions (C1), (C2) and (C3), and let [MATH] be the EEL defined by the composite similarity mapping ([REF]) with the following expansion factor [EQUATION] where [MATH] and [MATH] is the Bartlett correction factor for the two-sample empirical likelihood in ([REF]).', '1307.2297-2-24-2': 'Then, we have [EQUATION] and [EQUATION]', '1307.2297-2-25-0': 'Replacing EEL [MATH] in ([REF]) with [MATH] gives an EEL confidence interval which, by ([REF]), has a coverage error of [MATH].', '1307.2297-2-25-1': 'Because of this, we call [MATH] the second order EEL or EEL[MATH].', '1307.2297-2-25-2': 'Correspondingly, we call the EEL [MATH] defined by expansion factor ([REF]) the first order EEL or EEL[MATH].', '1307.2297-2-26-0': 'To limit the length of this paper, we have not included the proofs of the above lemmas and theorems here.', '1307.2297-2-26-1': 'They are given in a technical report (Wu and Tsao, 2013) available on request from the authors.', '1307.2297-2-27-0': '# Numerical examples', '1307.2297-2-28-0': 'We now compare the coverage accuracy of 95 confidence intervals based on the OEL, BEL and EEL with two numerical examples.', '1307.2297-2-28-1': 'Comparisons based on 90 and 99 confidence intervals give similar conclusions and are thus not included.', '1307.2297-2-28-2': 'They can also be found in Wu and Tsao (2013).', '1307.2297-2-28-3': 'In the following, [MATH] represents the standard bivariate normal distribution and [MATH], for example, represents a bivariate random vector [MATH] whose two elements are independent [MATH] random variables.', '1307.2297-2-29-0': 'Example 1: [MATH] and [MATH].', '1307.2297-2-30-0': 'Example 2: [MATH] and [MATH].', '1307.2297-2-31-0': 'To see the effect of the composite similarity mapping, Figure 1 compares contours for the OEL [MATH] and the corresponding contours for the EEL[MATH] based on the same pair of [MATH] and [MATH] samples from Example 1.', '1307.2297-2-31-1': 'We see that the contours in the two plots are identical in shape and the contours in both plots are centred on the MELE [MATH] as indicated in Theorem 1.', '1307.2297-2-31-2': 'Further, at any fixed level, the contour of the EEL [MATH] is larger in scale.', '1307.2297-2-32-0': 'Simulated coverage probabilities for the two examples are given in Tables 1 and 2, respectively.', '1307.2297-2-32-1': 'Each simulated probability in the tables is based on 10,000 pairs of random samples whose sizes are indicated by the row and column headings, respectively.', '1307.2297-2-32-2': 'The BEL and EEL[MATH] were computed by using the estimated Bartlett correction factor from page 550 in Liu et al. (2008).', '1307.2297-2-32-3': 'We summarize the tables with the following observations: (1) EEL[MATH] is consistently more accurate than the OEL.', '1307.2297-2-32-4': 'Surprisingly, it is also more accurate than the second order BEL and EEL[MATH] for small and moderate sample sizes ([MATH]) and competitive in accuracy when sample sizes are larger.', '1307.2297-2-32-5': '(2) EEL[MATH] is more accurate than OEL and BEL for small and moderate sample sizes.', '1307.2297-2-32-6': 'It is comparable to BEL when one or both sample sizes are large.', '1307.2297-2-33-0': 'To conclude, EEL[MATH] is easy-to-compute and is the most accurate overall.', '1307.2297-2-33-1': 'Hence, we recommend EEL[MATH] for two-sample problems.', '1307.2297-2-34-0': 'References'} | [['1307.2297-1-3-0', '1307.2297-2-3-0'], ['1307.2297-1-3-2', '1307.2297-2-3-2'], ['1307.2297-1-16-3', '1307.2297-2-24-2'], ['1307.2297-1-4-0', '1307.2297-2-4-0'], ['1307.2297-1-4-1', '1307.2297-2-4-1'], ['1307.2297-1-4-4', '1307.2297-2-4-4'], ['1307.2297-1-4-5', '1307.2297-2-4-5'], ['1307.2297-1-4-9', '1307.2297-2-4-9'], ['1307.2297-1-20-0', '1307.2297-2-28-0'], ['1307.2297-1-21-3', '1307.2297-2-32-1'], ['1307.2297-1-21-6', '1307.2297-2-32-4'], ['1307.2297-1-21-8', '1307.2297-2-32-6'], ['1307.2297-1-7-1', '1307.2297-2-12-1'], ['1307.2297-1-7-2', '1307.2297-2-12-2'], ['1307.2297-1-7-4', '1307.2297-2-12-4'], ['1307.2297-1-7-5', '1307.2297-2-12-5'], ['1307.2297-1-15-0', '1307.2297-2-23-0'], ['1307.2297-1-15-1', '1307.2297-2-23-1'], ['1307.2297-1-15-3', '1307.2297-2-23-3'], ['1307.2297-1-15-4', '1307.2297-2-23-4'], ['1307.2297-1-17-0', '1307.2297-2-25-0'], ['1307.2297-1-17-1', '1307.2297-2-25-1'], ['1307.2297-1-17-2', '1307.2297-2-25-2'], ['1307.2297-1-22-1', '1307.2297-2-33-1'], ['1307.2297-1-13-0', '1307.2297-2-19-0'], ['1307.2297-1-13-3', '1307.2297-2-19-3'], ['1307.2297-1-6-3', '1307.2297-2-10-0'], ['1307.2297-1-6-6', '1307.2297-2-10-3'], ['1307.2297-1-6-7', '1307.2297-2-11-6'], ['1307.2297-1-10-1', '1307.2297-2-15-1'], ['1307.2297-1-11-0', '1307.2297-2-15-3'], ['1307.2297-1-3-1', '1307.2297-2-3-1'], ['1307.2297-1-3-4', '1307.2297-2-3-4'], ['1307.2297-1-16-2', '1307.2297-2-24-1'], ['1307.2297-1-4-2', '1307.2297-2-4-2'], ['1307.2297-1-4-3', '1307.2297-2-4-3'], ['1307.2297-1-4-6', '1307.2297-2-4-6'], ['1307.2297-1-9-4', '1307.2297-2-14-2'], ['1307.2297-1-21-2', '1307.2297-2-32-0'], ['1307.2297-1-21-5', '1307.2297-2-32-3'], ['1307.2297-1-21-7', '1307.2297-2-32-5'], ['1307.2297-1-0-1', '1307.2297-2-0-1'], ['1307.2297-1-7-8', '1307.2297-2-12-8'], ['1307.2297-1-15-2', '1307.2297-2-23-2'], ['1307.2297-1-22-0', '1307.2297-2-33-0'], ['1307.2297-1-13-2', '1307.2297-2-19-2'], ['1307.2297-1-13-4', '1307.2297-2-21-0'], ['1307.2297-1-6-2', '1307.2297-2-6-2'], ['1307.2297-1-10-0', '1307.2297-2-15-0'], ['1307.2297-1-10-2', '1307.2297-2-15-2'], ['1307.2297-1-3-3', '1307.2297-2-3-3'], ['1307.2297-1-4-7', '1307.2297-2-4-7'], ['1307.2297-1-4-8', '1307.2297-2-4-8'], ['1307.2297-1-14-2', '1307.2297-2-22-1'], ['1307.2297-1-20-1', '1307.2297-2-28-1'], ['1307.2297-1-12-2', '1307.2297-2-18-1'], ['1307.2297-1-9-0', '1307.2297-2-14-0'], ['1307.2297-1-9-3', '1307.2297-2-14-1'], ['1307.2297-1-21-4', '1307.2297-2-32-2'], ['1307.2297-1-18-1', '1307.2297-2-26-0'], ['1307.2297-1-0-0', '1307.2297-2-0-0'], ['1307.2297-1-7-3', '1307.2297-2-12-6'], ['1307.2297-1-7-6', '1307.2297-2-12-6'], ['1307.2297-1-6-0', '1307.2297-2-6-0'], ['1307.2297-1-6-4', '1307.2297-2-10-1'], ['1307.2297-1-6-5', '1307.2297-2-10-2'], ['1307.2297-1-11-1', '1307.2297-2-17-2'], ['1307.2297-1-11-1', '1307.2297-2-15-4']] | [['1307.2297-1-3-0', '1307.2297-2-3-0'], ['1307.2297-1-3-2', '1307.2297-2-3-2'], ['1307.2297-1-16-3', '1307.2297-2-24-2'], ['1307.2297-1-4-0', '1307.2297-2-4-0'], ['1307.2297-1-4-1', '1307.2297-2-4-1'], ['1307.2297-1-4-4', '1307.2297-2-4-4'], ['1307.2297-1-4-5', '1307.2297-2-4-5'], ['1307.2297-1-4-9', '1307.2297-2-4-9'], ['1307.2297-1-20-0', '1307.2297-2-28-0'], ['1307.2297-1-21-3', '1307.2297-2-32-1'], ['1307.2297-1-21-6', '1307.2297-2-32-4'], ['1307.2297-1-21-8', '1307.2297-2-32-6'], ['1307.2297-1-7-1', '1307.2297-2-12-1'], ['1307.2297-1-7-2', '1307.2297-2-12-2'], ['1307.2297-1-7-4', '1307.2297-2-12-4'], ['1307.2297-1-7-5', '1307.2297-2-12-5'], ['1307.2297-1-15-0', '1307.2297-2-23-0'], ['1307.2297-1-15-1', '1307.2297-2-23-1'], ['1307.2297-1-15-3', '1307.2297-2-23-3'], ['1307.2297-1-15-4', '1307.2297-2-23-4'], ['1307.2297-1-17-0', '1307.2297-2-25-0'], ['1307.2297-1-17-1', '1307.2297-2-25-1'], ['1307.2297-1-17-2', '1307.2297-2-25-2'], ['1307.2297-1-22-1', '1307.2297-2-33-1'], ['1307.2297-1-13-0', '1307.2297-2-19-0'], ['1307.2297-1-13-3', '1307.2297-2-19-3'], ['1307.2297-1-6-3', '1307.2297-2-10-0'], ['1307.2297-1-6-6', '1307.2297-2-10-3'], ['1307.2297-1-6-7', '1307.2297-2-11-6'], ['1307.2297-1-10-1', '1307.2297-2-15-1'], ['1307.2297-1-11-0', '1307.2297-2-15-3']] | [['1307.2297-1-3-1', '1307.2297-2-3-1'], ['1307.2297-1-3-4', '1307.2297-2-3-4'], ['1307.2297-1-16-2', '1307.2297-2-24-1'], ['1307.2297-1-4-2', '1307.2297-2-4-2'], ['1307.2297-1-4-3', '1307.2297-2-4-3'], ['1307.2297-1-4-6', '1307.2297-2-4-6'], ['1307.2297-1-9-4', '1307.2297-2-14-2'], ['1307.2297-1-21-2', '1307.2297-2-32-0'], ['1307.2297-1-21-5', '1307.2297-2-32-3'], ['1307.2297-1-21-7', '1307.2297-2-32-5'], ['1307.2297-1-0-1', '1307.2297-2-0-1'], ['1307.2297-1-7-8', '1307.2297-2-12-8'], ['1307.2297-1-15-2', '1307.2297-2-23-2'], ['1307.2297-1-22-0', '1307.2297-2-33-0'], ['1307.2297-1-13-2', '1307.2297-2-19-2'], ['1307.2297-1-13-4', '1307.2297-2-21-0'], ['1307.2297-1-6-2', '1307.2297-2-6-2'], ['1307.2297-1-10-0', '1307.2297-2-15-0'], ['1307.2297-1-10-2', '1307.2297-2-15-2']] | [] | [['1307.2297-1-3-3', '1307.2297-2-3-3'], ['1307.2297-1-4-7', '1307.2297-2-4-7'], ['1307.2297-1-4-8', '1307.2297-2-4-8'], ['1307.2297-1-14-2', '1307.2297-2-22-1'], ['1307.2297-1-20-1', '1307.2297-2-28-1'], ['1307.2297-1-12-2', '1307.2297-2-18-1'], ['1307.2297-1-9-0', '1307.2297-2-14-0'], ['1307.2297-1-9-3', '1307.2297-2-14-1'], ['1307.2297-1-21-4', '1307.2297-2-32-2'], ['1307.2297-1-18-1', '1307.2297-2-26-0'], ['1307.2297-1-0-0', '1307.2297-2-0-0'], ['1307.2297-1-7-3', '1307.2297-2-12-6'], ['1307.2297-1-7-6', '1307.2297-2-12-6'], ['1307.2297-1-6-0', '1307.2297-2-6-0'], ['1307.2297-1-6-4', '1307.2297-2-10-1'], ['1307.2297-1-6-5', '1307.2297-2-10-2'], ['1307.2297-1-11-1', '1307.2297-2-17-2'], ['1307.2297-1-11-1', '1307.2297-2-15-4']] | [] | ['1307.2297-1-1-0', '1307.2297-1-6-1', '1307.2297-1-7-0', '1307.2297-1-12-0', '1307.2297-1-13-1', '1307.2297-1-14-0', '1307.2297-1-16-0', '1307.2297-1-23-0', '1307.2297-2-1-0', '1307.2297-2-6-3', '1307.2297-2-7-0', '1307.2297-2-8-0', '1307.2297-2-8-1', '1307.2297-2-9-0', '1307.2297-2-9-1', '1307.2297-2-12-0', '1307.2297-2-12-3', '1307.2297-2-15-6', '1307.2297-2-16-0', '1307.2297-2-16-1', '1307.2297-2-18-0', '1307.2297-2-19-1', '1307.2297-2-20-0', '1307.2297-2-20-1', '1307.2297-2-22-0', '1307.2297-2-24-0', '1307.2297-2-29-0', '1307.2297-2-30-0', '1307.2297-2-34-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1307.2297 | null | null | null | null | null |
1605.07665 | {'1605.07665-1-0-0': 'We use an anti-de Sitter/Quantum Chromodynamics (AdS/QCD) holographic light-front wavefunction for the [MATH] and [MATH] mesons, in conjunction with the Color Glass Condensate (CGC) dipole cross-section whose parameters are fitted to the most recent 2015 high precision HERA data on inclusive Deep Inelastic Scattering (DIS), in order to predict the cross-sections for diffractive [MATH] and [MATH] electroproduction.', '1605.07665-1-0-1': 'Our results suggest that the holographic meson light-front wavefunction is able to give a simultaneous description of [MATH] and [MATH] production data provided we use a set of light quark masses with [MATH] GeV.', '1605.07665-1-1-0': '# Introduction', '1605.07665-1-2-0': 'We use the QCD colour dipole model [CITATION] together with a non-perturbative holographic meson light-front wavefunction [CITATION] to predict the cross-sections for diffractive [MATH] and [MATH] electroproduction measured at the HERA collider [CITATION].', '1605.07665-1-2-1': 'In Ref. [CITATION], successful predictions were obtained for diffractive [MATH] production using the holographic wavefunction for the [MATH] and the CGC dipole cross-section [CITATION] whose parameters were fitted to the 2001 HERA DIS structure function data [CITATION].', '1605.07665-1-2-2': 'In 2015, the latest high precision combined HERA data on inclusive DIS were released [CITATION].', '1605.07665-1-2-3': 'This definitive DIS data set supersedes the earlier ones and is one of the major legacies of the HERA collider.', '1605.07665-1-2-4': 'We shall use these new data here to update the parameters of the CGC dipole cross-section and thus repeat the predictions of Ref. [CITATION].', '1605.07665-1-2-5': 'We shall also extend our predictions to diffractive [MATH] production, thereby testing the holographic wavefunction for the heavier [MATH] meson.', '1605.07665-1-3-0': 'The holographic meson wavefunction is predicted in holographic light-front QCD proposed by Brodsky and de Teramond [CITATION].', '1605.07665-1-3-1': 'A recent review of holographic light-front QCD can be found in Ref. [CITATION].', '1605.07665-1-3-2': 'In a semiclassical approximation of light-front QCD with massless quarks, there is an exact correspondance between the light-front Schrodinger equation for QCD bound states in physical spacetime and the equation of motion of spin-[MATH] modes in the higher dimensional AdS space.', '1605.07665-1-3-3': 'A dilaton field breaking the conformal symmetry of AdS space then dictates the form of the confining potential in physical spacetime.', '1605.07665-1-3-4': 'A phenomenologically successful choice is a dilaton which is quadratic in the fifth dimension of AdS space and this maps onto a light-front harmonic oscillator in physical spacetime.', '1605.07665-1-3-5': 'Remarkably, group theoretical arguments based on the underlying conformality of the classical Lagrangian of QCD reveal that the light-front harmonic potential is unique [CITATION].', '1605.07665-1-4-0': 'A single mass scale, [MATH], appears in the quadratic dilaton field and thus in the light-front harmonic oscillator in physical spacetime.', '1605.07665-1-4-1': 'The holographic light-front Schrodinger equation can then be solved to predict the meson mass spectrum.', '1605.07665-1-4-2': 'The latter has a string model Regge form as is observed experimentally.', '1605.07665-1-4-3': 'The parameter [MATH] can then be fixed to fit the observed slopes of the Regge trajectories for the various meson families.', '1605.07665-1-4-4': 'It is found that for all light mesons, [MATH] GeV [CITATION].', '1605.07665-1-4-5': 'Furthermore, the pion is predicted to be massless, consistent with chiral symmetry.', '1605.07665-1-5-0': 'Accounting for non-zero quark masses goes beyond the AdS/QCD correspondence and in Ref. [CITATION], Brodsky and de Teramond propose an ansatz for including small (on a hadronic scale) quark masses.', '1605.07665-1-5-1': 'The key observation is that the evolution variable for the momentum space light-front wavefunction is the quark-antiquark invariant mass and this can be appropriately modified to account for non-zero quark masses.', '1605.07665-1-5-2': 'With the modified holographic wavefunction, the shift in meson masses can be computed as a first order perturbation.', '1605.07665-1-5-3': "For the pion (and kaon), the mass shift is equal to the meson's physical mass.", '1605.07665-1-5-4': 'This allows the light quark masses to be fixed for a given [MATH].', '1605.07665-1-5-5': 'Ref. [CITATION] reports [MATH] GeV and [MATH] GeV with [MATH] GeV.', '1605.07665-1-5-6': 'The quark masses in holographic light-front QCD are thus effective quark masses, between current and constituent quark masses and they vanish in the chiral limit [CITATION].', '1605.07665-1-5-7': 'Once [MATH] and the quark masses are fixed, the holographic meson wavefunction comes with no free parameters.', '1605.07665-1-6-0': 'In the dipole model, the quark mass acts as an infrared regulator and thus reflects confinement.', '1605.07665-1-6-1': 'In practice, its value is chosen to fit inclusive DIS data.', '1605.07665-1-6-2': 'The typical value of [MATH] GeV, which coincides with the pion mass, was used in early extractions of the dipole cross-section [CITATION] from the inclusive DIS data.', '1605.07665-1-6-3': 'It is worth noting that the predictions in Ref. [CITATION] were generated using a light quark mass of [MATH] GeV, i.e, consistent with the fact the fitted parameters of the CGC dipole cross-section used in Ref. [CITATION] were obtained using that same light quark mass.', '1605.07665-1-6-4': 'The most recent extractions of the dipole cross-section were performed using the 2010 HERA DIS data [CITATION] in Refs. [CITATION].', '1605.07665-1-6-5': 'These authors found that the best fits are obtained using current quark masses [MATH] GeV.', '1605.07665-1-6-6': 'The preference of the DIS data for lower light quark masses was also noted in Ref. [CITATION] although the effective quark masses [MATH] GeV also gave acceptable fits to the 2001 DIS structure function data.', '1605.07665-1-6-7': 'In the recent paper [CITATION], using a new dipole model, both the current quark masses and the effective quark masses [MATH] GeV are found to give equally good fits to the 2010 DIS structure function data [CITATION].', '1605.07665-1-6-8': 'In all cases, SU(3) flavour symmetry is assumed.', '1605.07665-1-7-0': 'We shall start by predicting the vector and tensor coupling constants of the [MATH] and [MATH] mesons using their holographic wavefunctions.', '1605.07665-1-7-1': 'The vector coupling is also referred to as the decay constant since it is related to the measured electronic decay width.', '1605.07665-1-7-2': 'On the other hand, the (scale-dependent) tensor coupling is not extracted from experiment but non perturbative methods like lattice QCD and QCD Sum Rules are able to predict this coupling at a definite scale.', '1605.07665-1-7-3': 'We shall find that we are able to achieve optimal agreement with the decay width data by taking [MATH] GeV.', '1605.07665-1-7-4': 'This upper limit coincides with the light quark mass used in earlier dipole model studies[CITATION].', '1605.07665-1-7-5': 'We are thus led to depart slightly from Ref. [CITATION] by considering two additional sets of quark masses with decreasing strength of SU(3) symmetry breaking: [MATH] GeV and [MATH] GeV.', '1605.07665-1-7-6': 'In all cases, we use [MATH] GeV.', '1605.07665-1-7-7': 'With each set of quark masses, we shall refit the parameters of the dipole cross-section to DIS data and then use the fitted dipole cross-section to predict diffractive [MATH] and [MATH] production without any further adjustment of parameters.', '1605.07665-1-7-8': 'We shall see that the quark mass set with intermediate SU(3) symmetry breaking is necessary to describe the data on the ratio of the [MATH] to [MATH] total cross-sections.', '1605.07665-1-8-0': 'We begin by reviewing the colour dipole model in Section [REF] before discussing the holographic meson wavefunction in Section [REF].', '1605.07665-1-8-1': 'In Section [REF], we report the results of fitting the dipole cross-section to the new 2015 HERA DIS data.', '1605.07665-1-8-2': 'We use the dipole cross-section together with the holographic meson wavefunction to compute diffractive cross-sections for [MATH] and [MATH] in Section [REF].', '1605.07665-1-8-3': 'We conclude in Section [REF].', '1605.07665-1-9-0': '# The dipole model', '1605.07665-1-10-0': 'In the dipole picture, the largeness of the centre-of-mass energy squared, [MATH], guarantees that the scattering amplitude for the diffractive process [MATH] factorizes into an overlap of photon and vector meson light-front wavefunctions and a dipole cross-section [CITATION]: [EQUATION] where [MATH] is the squared momentum transfer at the proton vertex.', '1605.07665-1-10-1': '[MATH] and [MATH] are the light-front wavefunctions of photon and vector meson respectively while [MATH] is the proton-dipole scattering amplitude.', '1605.07665-1-10-2': 'The light-front wavefunctions are the probability amplitudes for the virtual photon or vector meson to fluctuate into a [MATH] color dipole in a given helicity configuration ([MATH] is the helicity of the quark and [MATH] is the helicity of the antiquark) and they depend on the transverse size [MATH] of the [MATH] color dipole and on [MATH], the fraction of light-front momentum of the photon (or vector meson) carried by the quark.', '1605.07665-1-10-3': 'Both wavefunctions are labelled by [MATH] which denotes the polarization of the photon or vector meson.', '1605.07665-1-10-4': "The photon light-front wavefunction is also a function of the photon's virtuality [MATH].", '1605.07665-1-10-5': 'The dipole-proton scattering amplitude is the amplitude for the elastic scattering of the dipole on the proton and it depends on the photon-proton centre-of-mass energy via the modified Bjorken variable [MATH] where [CITATION] [EQUATION]', '1605.07665-1-10-6': 'The dipole-proton scattering amplitude contains all the high energy QCD dynamics of the dipole-proton interaction.', '1605.07665-1-10-7': 'It is a universal object, appearing also in the formula for the fully inclusive DIS process: [MATH].', '1605.07665-1-10-8': 'Indeed, replacing the vector meson by a virtual photon in Eq. [REF], we obtain the amplitude for elastic Compton scattering [MATH], i.e. [EQUATION] where we have introduced the dipole cross-section [EQUATION]', '1605.07665-1-10-9': 'Via the Optical Theorem, the elastic amplitude given by Eq. [REF] is directly related to the inclusive [MATH] total cross-section in DIS: [EQUATION] where now [CITATION] [EQUATION]', '1605.07665-1-10-10': 'This means that one can use the high quality DIS data from HERA to constrain the free parameters of the dipole cross-section section and then use the same dipole cross-section to make predictions for vector meson production and other distinct processes like Deeply Virtual Compton Scattering (DVCS) and Diffractive DIS.', '1605.07665-1-10-11': 'This program has been successfully carried by several authors [CITATION] hinting very strongly at the universality of the dipole cross-section.', '1605.07665-1-11-0': 'Note that the high energy factorization in Eqs. [REF] and [REF] holds beyond the validity of perturbation theory, i.e. for all dipole sizes.', '1605.07665-1-11-1': 'In practice, the expressions for the photon light-front wavefunctions obtained perturbatively in light-front QED are used for all [MATH].', '1605.07665-1-11-2': 'To lowest order in [MATH], the perturbative photon wavefunctions are given by [CITATION]: [EQUATION] where [MATH] and [MATH] is the complex notation for the transverse separation between the quark and anti-quark.', '1605.07665-1-11-3': 'As can be seen, at [MATH] or [MATH], the photon light-front wavefunctions become sensitive to the non-zero quark mass [MATH] which prevents the modified Bessel function [MATH] from diverging, i.e. the quark mass acts as an infrared regulator.', '1605.07665-1-11-4': 'On the other hand, a non-perturbative model for the meson light-front wavefunction is used and assumed to be valid for all [MATH].', '1605.07665-1-12-0': 'To compare with experiment, we compute the differential cross-section in the forward limit, i.e. [EQUATION] and we then assume the [MATH]-dependence to be exponential, i.e. [EQUATION] where the diffractive slope parameter [MATH] is given by [EQUATION] with [MATH] GeV[MATH].', '1605.07665-1-12-1': 'This parametrization of the diffractive slope agrees with the most recent ZEUS data for both [MATH] and [MATH] production [CITATION].', '1605.07665-1-12-2': 'The most recent H1 data for [MATH] production [CITATION] prefer a somewhat larger value of [MATH], but with a larger uncertainty.', '1605.07665-1-13-0': 'Note that Eq. [REF] can be rewritten as [EQUATION] where [MATH] is the ratio of real to imaginary parts of the amplitude.', '1605.07665-1-13-1': 'We estimate [MATH] in the usual way [CITATION] [EQUATION]', '1605.07665-1-13-2': 'We calculate the photo-production cross section after integrating Eq. [REF] over [MATH].', '1605.07665-1-13-3': 'This means that the uncertainty in the diffractive slope [MATH] leads to an uncertainty in the normalization of our predictions for total cross-section.', '1605.07665-1-13-4': 'We shall give predictions for the total cross-section [MATH] where we take [MATH] to compare to the HERA data.', '1605.07665-1-14-0': '# Holographic meson wavefunctions', '1605.07665-1-15-0': 'The vector meson light-front wavefunctions appearing in Eq. [REF] cannot be computed in perturbation theory.', '1605.07665-1-15-1': 'Nevertheless, they can be assumed to have the same spinor and polarization structure as in the photon case, together with an unknown non-perturbative wavefunction [CITATION].', '1605.07665-1-15-2': 'Explicitly, the vector meson light-front wavefunctions can be written as [CITATION] [EQUATION] and [EQUATION]', '1605.07665-1-16-0': 'Various ansatz for the non-perturbative meson wavefunction have been proposed in the literature [CITATION], perhaps the most popular one being the so-called Boosted Gaussian (BG) wavefunction [CITATION] which has been used in the recent studies in Refs. [CITATION] to describe simultaneously the cross-section data on diffractive [MATH] and [MATH] production.', '1605.07665-1-16-1': 'Ref. [CITATION] uses the dipole cross-section extracted in Ref. [CITATION] with the BG wavefunction to predict vector meson production in ultrapheripheral collisions at the LHC.', '1605.07665-1-16-2': 'In Refs. [CITATION], the [MATH] meson wavefunction is extracted from the data using several dipole models which fit the 2001 DIS structure function data.', '1605.07665-1-17-0': 'In recent years, new insights about hadronic light-front wavefunctions based on the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence have been proposed by Brodsky and de Teramond.', '1605.07665-1-17-1': '[CITATION].', '1605.07665-1-17-2': 'These authors found that in a semiclassical approximation of light-front QCD with massless quarks, the meson wavefunction can be written as [CITATION]', '1605.07665-1-18-0': '[EQUATION] where the variable [MATH] is the transverse separation between the quark and the antiquark at equal light-front time.', '1605.07665-1-18-1': 'The transverse wavefunction [MATH] is a solution of the so-called holographic light-front Schrodinger equation: [EQUATION] where [MATH] is the mass of the meson and [MATH] is the confining potential which at present cannot be computed from first-principle in QCD.', '1605.07665-1-18-2': 'On the other hand, making the substitutions [MATH] where [MATH] being the fifth dimension of AdS space, together with [MATH] where [MATH] and [MATH] are the radius of curvature and mass parameter of AdS space respectively, then Eq. [REF] describes the propagation of spin-[MATH] string modes in AdS space.', '1605.07665-1-18-3': 'In this case, the potential is given by [EQUATION] where [MATH] is the dilaton field which breaks the conformal invariance of AdS space.', '1605.07665-1-18-4': 'A quadratic dilaton ([MATH]) profile results in a harmonic oscillator potential in physical spacetime: [EQUATION]', '1605.07665-1-18-5': 'Brodsky, Dosch and de Teramond have shown that the light-front harmonic potential is unique [CITATION].', '1605.07665-1-18-6': 'Solving the holographic Schrodinger equation with this harmonic potential given by Eq. [REF] yields the meson mass spectrum, [EQUATION] with the corresponding normalized eigenfunctions [EQUATION]', '1605.07665-1-18-7': 'To completely specify the holographic wavefunction given by Eq. [REF], the longitudinal wavefunction [MATH] must be determined.', '1605.07665-1-18-8': 'For massless quarks, this is achieved by an exact mapping of the pion electromagnetic form factors in AdS and in physical spacetime resulting in [CITATION].', '1605.07665-1-18-9': '[EQUATION]', '1605.07665-1-18-10': 'Equation [REF] predicts that the mesons lie on linear Regge trajectories as is experimentally observed and thus [MATH] can be chosen to fit the Regge slope.', '1605.07665-1-18-11': 'Ref. [CITATION] reports [MATH] GeV for vector mesons.', '1605.07665-1-18-12': 'Eq. [REF] also predicts that the pion and kaon (with [MATH]) are massless.', '1605.07665-1-18-13': 'To account for their physical masses, non-zero light quark masses have to be introduced.', '1605.07665-1-18-14': 'To do so, we follow the prescription of Brodsky and de Teramond given in Ref. [CITATION] and which we outline below.', '1605.07665-1-19-0': 'For the ground state mesons with [MATH], Eq. [REF] becomes [EQUATION]', '1605.07665-1-19-1': 'A two-dimensional Fourier transform to momentum space yields [EQUATION] where [MATH] is invariant mass of the [MATH] pair is given by [EQUATION]', '1605.07665-1-19-2': 'For non-zero quark masses, this invariant mass becomes [EQUATION]', '1605.07665-1-19-3': 'Inserting Eq. [REF] in Eq. [REF] and Fourier transforming back to configuration space gives [EQUATION] where we have introduced a polarization-dependent normalization constant [MATH].', '1605.07665-1-19-4': 'We fix this normalization constant by requiring that [EQUATION] where [MATH] are given by Eqs. [REF] and [REF].', '1605.07665-1-20-0': 'With the non-zero light quark masses, the meson mass spectrum becomes [CITATION] [EQUATION] where the mass shift is given by [CITATION] [EQUATION]', '1605.07665-1-20-1': 'Hence, Eq. [REF] implies that [EQUATION] which allows to fix the [MATH] (and [MATH]) quark masses for a given [MATH].', '1605.07665-1-20-2': 'Using [MATH] GeV, Ref. [CITATION] reports [MATH] GeV.', '1605.07665-1-20-3': 'To fix the strange quark mass, Ref. [CITATION] uses [MATH] MeV together with [MATH] GeV and obtain [MATH] GeV.', '1605.07665-1-20-4': 'Note that the above Brodsky-de Teramond quark mass prescription is expected to be a good approximation only for light quark masses.', '1605.07665-1-20-5': 'A possible way to account for heavier quark masses is to determine the longitudinal function [MATH] dynamically [CITATION].', '1605.07665-1-21-0': 'Having specified the holographic wavefunction for the vector mesons, we are now able to predict their vector and tensor couplings defined by [CITATION] [EQUATION] and [EQUATION] respectively.', '1605.07665-1-21-1': 'In Eqs. [REF] and [REF], [MATH] and [MATH] are the antiquark and quark fields evaluated at the same spacetime point, [MATH] and [MATH] are the momentum and polarization vectors of the vector meson.', '1605.07665-1-21-2': 'Inserting the Fock expansion of the meson states in the right-hand-side of Eq. [REF] and Eq. [REF], we obtain [CITATION] [EQUATION] and [EQUATION] respectively.', '1605.07665-1-21-3': 'Note that the tensor coupling is dependent on the scale [MATH] although we find that our predictions for [MATH] hardly depend on [MATH] for [MATH].', '1605.07665-1-21-4': 'This means that our predictions are at some unspecified low scale [MATH].', '1605.07665-1-21-5': 'As is obvious from Eq. [REF], the tensor coupling vanishes as [MATH], consistent with the requirement that the tensor current vanishes in the chiral limit.', '1605.07665-1-21-6': 'There is no such requirement for the vector current and indeed we predict a non-vanishing value for the vector coupling as [MATH].', '1605.07665-1-21-7': 'We show the variation of the vector and tensor couplings with the quark mass in Figure [REF].', '1605.07665-1-21-8': 'It is interesting to note that the vector coupling is maximum for [MATH] GeV.', '1605.07665-1-22-0': 'The vector coupling is also referred to as the decay constant as it is related to the measured electronic decay width [MATH] of the vector meson: [EQUATION] where [MATH] and [MATH].', '1605.07665-1-22-1': 'Our results for the electronic decay widths are shown in Table [REF].', '1605.07665-1-22-2': 'Note that we obtain a lower value for the decay width of the [MATH] than that reported in Ref. [CITATION] because we are using a universal [MATH] GeV for both vector mesons compared to [MATH] GeV used in Ref. [CITATION].', '1605.07665-1-22-3': 'We show predictions for the decay width using [MATH] GeV for the [MATH] and [MATH] GeV for the [MATH].', '1605.07665-1-22-4': 'As can be seen in Figure [REF], the vector coupling for the [MATH] meson varies slowly with the quark mass in the range [MATH] GeV and hence our two predictions for the decay width do not differ much from each other with a slight preference for [MATH] GeV.', '1605.07665-1-22-5': 'The variation of [MATH] in the range [MATH] GeV is more important and the lower strange quark mass, [MATH] GeV gives better agreement with the decay width datum.', '1605.07665-1-23-0': 'For both vector mesons, we underestimate the electronic decay width.', '1605.07665-1-23-1': 'But this is also the case with the other non-perturbative methods quoted in Table [REF] for the [MATH].', '1605.07665-1-23-2': 'There are likely perturbative corrections that must be taken into account when predicting the electronic decay width.', '1605.07665-1-24-0': 'Since the optimal agreement (or rather minimal disagreement) with the electronic decay width data is achieved with [MATH] GeV, we choose this quark mass to compare our predictions with QCD Sum Rules, Dyson-Schwinger and lattice predictions as shown in Tables [REF] and [REF].', '1605.07665-1-24-1': 'Recall that our predictions for the transverse decay constant are at [MATH] GeV which prevents an exact comparison with the other predictions all given at a scale [MATH] GeV.', '1605.07665-1-24-2': 'Despite this, it is clear that we predict a smaller transverse decay constant (with [MATH] GeV) than those predicted by the other non-perturbative methods quoted in Table [REF].', '1605.07665-1-25-0': '# Refitting the CGC dipole model', '1605.07665-1-26-0': 'In principle, the dipole-proton scattering amplitude [MATH] can be obtained by solving the Balitsky-Kovchegov (BK) equation [CITATION] which itself can be derived within the Colour Glass Condensate (CGC) formalism [CITATION].', '1605.07665-1-26-1': 'However, work is still in progress to implement in a satisfactory way the impact-parameter dependence in the proton-dipole amplitude [CITATION].', '1605.07665-1-26-2': 'A widely used model is that proposed by Kowalski and Watt [CITATION] where the saturation scale (see below) have a Gaussian dependence on the impact parameter.', '1605.07665-1-26-3': 'However, it has been recently argued in Ref. [CITATION] that the [MATH]-dependence should be exponential.', '1605.07665-1-26-4': 'In any case, considering the [MATH]-dependence introduces an additional parameter which has to be fixed using data on diffractive meson production (for instance [MATH] production) which requires a model for the meson wavefunction.', '1605.07665-1-26-5': 'On the other hand, a simple model for the [MATH]-integrated dipole-proton amplitude, i.e. the dipole cross-section has been proposed long ago in Ref. [CITATION].', '1605.07665-1-26-6': 'This is known as the CGC dipole model and is given by [EQUATION] with [EQUATION] where the saturation scale [MATH] GeV.', '1605.07665-1-26-7': 'The coefficients [MATH] and [MATH] are determined from the condition that the [MATH] and its derivative with respect to [MATH] are continuous at [MATH].', '1605.07665-1-26-8': 'This leads to [EQUATION]', '1605.07665-1-26-9': 'The free parameters of the CGC dipole model are [MATH] and [MATH] which are fixed by a fit to the structure function [MATH] data.', '1605.07665-1-26-10': '[MATH] and [MATH] are fixed at 0.7 and 9.9 (LO BFKL prediction) respectively.', '1605.07665-1-26-11': 'The high quality DIS data from HERA can be used to fix the free parameters of the dipole cross-section.', '1605.07665-1-26-12': 'An earlier fit to the structure function data given in Ref. [CITATION] and used in Ref. [CITATION] to make successful predictions for diffractive [MATH] production are: [MATH] mb, [MATH].', '1605.07665-1-27-0': 'In 2015, the H1 and ZEUS collaborations have released highly precise combined data sets [CITATION] for the reduced cross-section [EQUATION] where [MATH] and [MATH] is the centre of mass energy of the [MATH] system for [MATH] different bins : [MATH] GeV ([MATH] data points), [MATH] GeV ([MATH] data points) and [MATH] GeV ([MATH] data points), [MATH] GeV ([MATH] data points).', '1605.07665-1-27-1': 'The structure functions in Eq. [REF] are given by [EQUATION] and [EQUATION] where in the dipole model, [MATH] is given by equation [REF].', '1605.07665-1-28-0': 'The most recent extraction of the CGC dipole model parameters was performed in Ref. [CITATION] following the release of the combined HERA data in 2010 [CITATION].', '1605.07665-1-28-1': 'In this paper, the authors report a successful fit in the kinematic range [MATH] using very small light quark masses [MATH] GeV and a charm quark mass [MATH] GeV.', '1605.07665-1-28-2': 'The fitted parameters are [MATH] mb, [MATH] with a [MATH].', '1605.07665-1-29-0': 'We start by computing the [MATH] per data point ([MATH]) for the 2015 HERA data using the earlier fitted parameters of Ref. [CITATION] and the most recent fitted parameters of Ref. [CITATION].', '1605.07665-1-29-1': 'We obtain [MATH] and [MATH] respectively.', '1605.07665-1-29-2': 'The latter [MATH] is acceptable and in fact lower than that reported in Ref. [CITATION] for the fit to the 2010 data set.', '1605.07665-1-29-3': 'However, since we are using here effective quark masses and not current quark masses as in Ref. [CITATION], we need to refit the CGC dipole parameters to the new 2015 data.', '1605.07665-1-29-4': 'To obtain the above [MATH] values, we have used [MATH] GeV as in Ref. [CITATION], and we shall also use this same charm mass for our fits.', '1605.07665-1-29-5': 'Departing from Ref. [CITATION], we also include low [MATH] data in our fits, thereby daring to extrapolate the use of the CGC dipole model in the non-perturbative region where the predictions are more sensitive to the quark masses.', '1605.07665-1-30-0': 'Our fitted values for the CGC dipole model parameters together with the resulting [MATH] per degrees of freedom ([MATH]) values are shown in Table [REF].', '1605.07665-1-30-1': 'The first two rows indicate that the fit is not very sensitive to the variation in the strange quark mass.', '1605.07665-1-30-2': 'Comparing the second and third rows, we can see that the data prefer the lower [MATH] and [MATH] quark masses and that increasing them give quite different fit parameters especially for [MATH].', '1605.07665-1-30-3': 'But in all three cases, the [MATH] is less than the value obtained in Ref. [CITATION] ([MATH]) and we regard them as acceptable fits.', '1605.07665-1-30-4': 'This is not to deny that using current quark masses can lead to equally good fits.', '1605.07665-1-30-5': 'In fact, we find that with current quark masses [MATH] GeV, we obtain [MATH], i.e. a fit of similar quality as our best fit with [MATH] GeV and [MATH] GeV.', '1605.07665-1-30-6': 'The fitted parameters are [MATH], [MATH], [MATH] and [MATH] mb, similar to those reported in Ref. [CITATION].', '1605.07665-1-30-7': 'However, we have checked that this quark mass set (with our holographic wavefunction) does not lead to a good agreement with the diffractive cross-section data at low [MATH].', '1605.07665-1-31-0': '# Predicting diffractive cross-sections', '1605.07665-1-32-0': 'Having specified the dipole cross-section and the holographic meson wavefunction, we can now compute cross-sections for diffractive [MATH] and [MATH] production.', '1605.07665-1-32-1': 'We shall show predictions using three sets of the CGC dipole parameters as given in Table [REF].', '1605.07665-1-32-2': 'We shall refer to these three sets of predictions as "max SU(3)" (first row), "int SU(3)" (second row) and "no SU(3)" (third row) respectively.', '1605.07665-1-32-3': 'Recall that all our predictions will be generated using the same holographic wavefunction given by Eq. [REF] and they differ only by the choice of quark masses and the corresponding fitted parameters of the CGC dipole model as given in Table [REF].', '1605.07665-1-33-0': 'We compute the total cross-section as a function of [MATH] in different [MATH] bins as well as a function of [MATH] at fixed [MATH].', '1605.07665-1-33-1': 'We also compute the ratio of longitudinal to transverse cross-sections as a function of [MATH] at fixed [MATH].', '1605.07665-1-33-2': 'Predicting the latter observable is interesting since the normalization uncertainties in the diffractive [MATH]-slope and the dipole cross-section, cancel out, increasing its sensitivity to the meson wavefunction.', '1605.07665-1-34-0': 'For [MATH] production, our predictions for the [MATH] dependence of the total cross-section in different [MATH] bins are shown in Figures [REF] and [REF] while our predictions for the [MATH] dependence of the total cross-section at fixed [MATH] are shown in Figure [REF].', '1605.07665-1-34-1': 'The "max SU(3)" (black solid curves) and the "no SU(3)" (blue dashed curves) are both accommodated by the total cross-section data.', '1605.07665-1-34-2': 'The [MATH] ratio data, shown in Figure [REF], are able to discriminate between them and favour the "no SU(3)" prediction.', '1605.07665-1-34-3': 'Notice that the "no SU(3)" predictions undershoot the data in the two largest [MATH] bins but this is the kinematic region where the holographic wavefunction is expected to be less accurate.', '1605.07665-1-34-4': 'We thus confirm the conclusion of Ref. [CITATION] in which the "no SU(3)" set of quark masses was used.', '1605.07665-1-35-0': 'For [MATH] production, our predictions for the [MATH] dependence of the total cross-section in different [MATH] bins are shown in Figures [REF] and [REF] and for the [MATH] dependence of the total cross-section at fixed [MATH] are shown in Figure [REF].', '1605.07665-1-35-1': 'Here, it is clear that the "max SU(3)" predictions (solid black curves) are not successful.', '1605.07665-1-35-2': 'The data prefer slightly the "int SU(3)" (orange dotted curves) over the "no SU(3)" predictions (blue dashed curves) although the lack of data in the low [MATH] region prevents us from making a definite statement.', '1605.07665-1-35-3': 'At high [MATH], all our predictions tend to undershoot the (ZEUS) data as expected.', '1605.07665-1-35-4': 'Our predictions for the longitudinal to transverse cross-sections ratio for [MATH] production are shown in Fig. [REF].', '1605.07665-1-35-5': 'We can see that the ratio data tend to favour the "max SU(3)" prediction (solid black curve) although they are not precise enough to discard the other two predictions.', '1605.07665-1-36-0': 'In summary, the "no SU(3)" predictions are favoured for [MATH] production and the "int SU(3)" predictions are preferred for [MATH] production.', '1605.07665-1-36-1': 'Notice that in both sets of predictions, we are using exactly the same holographic wavefunction for both [MATH] and [MATH], i.e. a wavefunction with the same [MATH] GeV and [MATH] GeV.', '1605.07665-1-36-2': 'The difference between the two sets of predictions arises from the different electromagnetic couplings and the different fitted parameters of the CGC dipole cross-section.', '1605.07665-1-37-0': 'Finally, we consider the data set on the ratio of the total cross-sections for [MATH] and [MATH] production.', '1605.07665-1-37-1': 'Note if the [MATH] and the [MATH] had identical masses and holographic wavefunctions, this ratio is simply given by the squared ratio of the effective electric charges of the quark-antiquark dipole coupling to the photon: [MATH].', '1605.07665-1-37-2': 'As expected our "no SU(3)" prediction (blue dashed curve) tends to that value at high [MATH].', '1605.07665-1-37-3': 'At lower [MATH], the deviation from [MATH] is due to the vector meson mass entering the modified Bjorken-[MATH] given by Eq. [REF] and the diffractive slope given by Eq. [REF].', '1605.07665-1-37-4': 'However, the data indicate a lower ratio and as can be seen in Figure [REF], the "int SU(3)" prediction (orange dotted curve) is clearly preferred.', '1605.07665-1-37-5': 'This provides evidence for the need to have different quark masses in the holographic wavefunctions of the [MATH] and [MATH].', '1605.07665-1-38-0': 'In view of the above results, we anticipate that a set of quark masses with an even weaker SU(3) symmetry breaking than our "int SU(3)" set, should give the best simultaneous description of both [MATH] and [MATH] diffractive production data.', '1605.07665-1-39-0': '# Conclusion', '1605.07665-1-40-0': 'We have updated the parameters of the CGC dipole model using the definitive 2015 HERA data on inclusive DIS and we have used the fitted dipole cross-section together with a holographic meson wavefunction in order to compute the cross-sections for diffractive [MATH] and [MATH] meson production.', '1605.07665-1-40-1': 'The holographic light-front meson wavefunction is successful to describe simultaneously diffractive [MATH] and [MATH] production with a single universal holographic mass scale [MATH] but with a set of light quark masses with a weaker SU(3) flavour symmetry ([MATH]) breaking than that used in light-front holography ([MATH]) to generate the pion and kaon masses.'} | {'1605.07665-2-0-0': 'We use an anti-de Sitter/Quantum Chromodynamics (AdS/QCD) holographic light-front wavefunction for the [MATH] and [MATH] mesons, in conjunction with the Color Glass Condensate (CGC) dipole cross-section whose parameters are fitted to the most recent 2015 high precision HERA data on inclusive Deep Inelastic Scattering (DIS), in order to predict the cross-sections for diffractive [MATH] and [MATH] electroproduction.', '1605.07665-2-0-1': 'Our results suggest that the holographic meson light-front wavefunction is able to give a simultaneous description of [MATH] and [MATH] production data provided we use a set of light quark masses with [MATH] GeV.', '1605.07665-2-1-0': '# Introduction', '1605.07665-2-2-0': 'We use the QCD colour dipole model [CITATION] together with a non-perturbative holographic meson light-front wavefunction [CITATION] to predict the cross-sections for diffractive [MATH] and [MATH] electroproduction measured at the HERA collider [CITATION].', '1605.07665-2-2-1': 'In Ref. [CITATION], successful predictions were obtained for diffractive [MATH] production using the holographic wavefunction for the [MATH] and the CGC dipole cross-section [CITATION] whose parameters were fitted to the 2001 HERA DIS structure function data [CITATION].', '1605.07665-2-2-2': 'In 2015, the latest high precision combined HERA data on inclusive DIS were released [CITATION].', '1605.07665-2-2-3': 'This definitive DIS data set supersedes the earlier ones and is one of the major legacies of the HERA collider.', '1605.07665-2-2-4': 'We shall use these new data here to update the parameters of the CGC dipole cross-section and thus repeat the predictions of Ref. [CITATION].', '1605.07665-2-2-5': 'We shall also extend our predictions to diffractive [MATH] production, thereby testing the holographic wavefunction for the heavier [MATH] meson.', '1605.07665-2-3-0': 'The holographic meson wavefunction is predicted in holographic light-front QCD proposed by Brodsky and de Teramond [CITATION].', '1605.07665-2-3-1': 'A recent review of holographic light-front QCD can be found in Ref. [CITATION].', '1605.07665-2-3-2': 'In a semiclassical approximation of light-front QCD with massless quarks, there is an exact correspondance between the light-front Schrodinger equation for QCD bound states in physical spacetime and the equation of motion of spin-[MATH] modes in the higher dimensional AdS space.', '1605.07665-2-3-3': 'A dilaton field breaking the conformal symmetry of AdS space then dictates the form of the confining potential in physical spacetime.', '1605.07665-2-3-4': 'A phenomenologically successful choice is a dilaton which is quadratic in the fifth dimension of AdS space and this maps onto a light-front harmonic oscillator in physical spacetime.', '1605.07665-2-3-5': 'Remarkably, group theoretical arguments based on the underlying conformality of the classical Lagrangian of QCD reveal that the light-front harmonic potential is unique [CITATION].', '1605.07665-2-4-0': 'A single mass scale, [MATH], appears in the quadratic dilaton field and thus in the light-front harmonic oscillator in physical spacetime.', '1605.07665-2-4-1': 'The holographic light-front Schrodinger equation can then be solved to predict the meson mass spectrum.', '1605.07665-2-4-2': 'The latter has a string model Regge form as is observed experimentally.', '1605.07665-2-4-3': 'The parameter [MATH] can then be fixed to fit the observed slopes of the Regge trajectories for the various meson families.', '1605.07665-2-4-4': 'It is found that for all light mesons, [MATH] GeV [CITATION].', '1605.07665-2-4-5': 'Furthermore, the pion is predicted to be massless, consistent with chiral symmetry.', '1605.07665-2-5-0': 'Accounting for non-zero quark masses goes beyond the AdS/QCD correspondence and in Ref. [CITATION], Brodsky and de Teramond propose an ansatz for including small (on a hadronic scale) quark masses.', '1605.07665-2-5-1': 'The key observation is that the evolution variable for the momentum space light-front wavefunction is the quark-antiquark invariant mass and this can be appropriately modified to account for non-zero quark masses.', '1605.07665-2-5-2': 'With the modified holographic wavefunction, the shift in meson masses can be computed as a first order perturbation.', '1605.07665-2-5-3': "For the pion (and kaon), the mass shift is equal to the meson's physical mass.", '1605.07665-2-5-4': 'This allows the light quark masses to be fixed for a given [MATH].', '1605.07665-2-5-5': 'Ref. [CITATION] reports [MATH] GeV and [MATH] GeV with [MATH] GeV.', '1605.07665-2-5-6': 'The quark masses in holographic light-front QCD are thus effective quark masses, between current and constituent quark masses and they vanish in the chiral limit [CITATION].', '1605.07665-2-5-7': 'Once [MATH] and the quark masses are fixed, the holographic meson wavefunction comes with no free parameters.', '1605.07665-2-6-0': 'In the dipole model, the quark mass acts as an infrared regulator and thus reflects confinement.', '1605.07665-2-6-1': 'In practice, its value is chosen to fit inclusive DIS data.', '1605.07665-2-6-2': 'The typical value of [MATH] GeV, which coincides with the pion mass, was used in early extractions of the dipole cross-section [CITATION] from the inclusive DIS data.', '1605.07665-2-6-3': 'It is worth noting that the predictions in Ref. [CITATION] were generated using a light quark mass of [MATH] GeV, i.e, consistent with the fact the fitted parameters of the CGC dipole cross-section used in Ref. [CITATION] were obtained using that same light quark mass.', '1605.07665-2-6-4': 'The most recent extractions of the dipole cross-section were performed using the 2010 HERA DIS data [CITATION] in Refs. [CITATION].', '1605.07665-2-6-5': 'These authors found that the best fits are obtained using current quark masses [MATH] GeV.', '1605.07665-2-6-6': 'The preference of the DIS data for lower light quark masses was also noted in Ref. [CITATION] although the effective quark masses [MATH] GeV also gave acceptable fits to the 2001 DIS structure function data.', '1605.07665-2-6-7': 'In the recent paper [CITATION], using a new dipole model, both the current quark masses and the effective quark masses [MATH] GeV are found to give equally good fits to the 2010 DIS structure function data [CITATION].', '1605.07665-2-6-8': 'In all cases, SU(3) flavour symmetry is assumed.', '1605.07665-2-7-0': 'We shall start by predicting the vector and tensor coupling constants of the [MATH] and [MATH] mesons using their holographic wavefunctions.', '1605.07665-2-7-1': 'The vector coupling is also referred to as the decay constant since it is related to the measured electronic decay width.', '1605.07665-2-7-2': 'On the other hand, the (scale-dependent) tensor coupling is not extracted from experiment but non perturbative methods like lattice QCD and QCD Sum Rules are able to predict this coupling at a definite scale.', '1605.07665-2-7-3': 'We shall find that we are able to achieve optimal agreement with the decay width data by taking [MATH] GeV.', '1605.07665-2-7-4': 'This upper limit coincides with the light quark mass used in earlier dipole model studies[CITATION].', '1605.07665-2-7-5': 'We are thus led to depart slightly from Ref. [CITATION] by considering two additional sets of quark masses with decreasing strength of SU(3) symmetry breaking: [MATH] GeV and [MATH] GeV.', '1605.07665-2-7-6': 'In all cases, we use [MATH] GeV.', '1605.07665-2-7-7': 'With each set of quark masses, we shall refit the parameters of the dipole cross-section to DIS data and then use the fitted dipole cross-section to predict diffractive [MATH] and [MATH] production without any further adjustment of parameters.', '1605.07665-2-7-8': 'We shall see that the quark mass set with intermediate SU(3) symmetry breaking is necessary to describe the data on the ratio of the [MATH] to [MATH] total cross-sections.', '1605.07665-2-8-0': 'We begin by reviewing the colour dipole model in Section [REF] before discussing the holographic meson wavefunction in Section [REF].', '1605.07665-2-8-1': 'In Section [REF], we report the results of fitting the dipole cross-section to the new 2015 HERA DIS data.', '1605.07665-2-8-2': 'We use the dipole cross-section together with the holographic meson wavefunction to compute diffractive cross-sections for [MATH] and [MATH] in Section [REF].', '1605.07665-2-8-3': 'We conclude in Section [REF].', '1605.07665-2-9-0': '# The dipole model', '1605.07665-2-10-0': 'In the dipole picture, the largeness of the centre-of-mass energy squared, [MATH], guarantees that the scattering amplitude for the diffractive process [MATH] factorizes into an overlap of photon and vector meson light-front wavefunctions and a dipole cross-section [CITATION]: [EQUATION] where [MATH] is the squared momentum transfer at the proton vertex.', '1605.07665-2-10-1': '[MATH] and [MATH] are the light-front wavefunctions of photon and vector meson respectively while [MATH] is the proton-dipole scattering amplitude.', '1605.07665-2-10-2': 'The light-front wavefunctions are the probability amplitudes for the virtual photon or vector meson to fluctuate into a [MATH] color dipole in a given helicity configuration ([MATH] is the helicity of the quark and [MATH] is the helicity of the antiquark) and they depend on the transverse size [MATH] of the [MATH] color dipole and on [MATH], the fraction of light-front momentum of the photon (or vector meson) carried by the quark.', '1605.07665-2-10-3': 'Both wavefunctions are labelled by [MATH] which denotes the polarization of the photon or vector meson.', '1605.07665-2-10-4': "The photon light-front wavefunction is also a function of the photon's virtuality [MATH].", '1605.07665-2-10-5': 'The dipole-proton scattering amplitude is the amplitude for the elastic scattering of the dipole on the proton and it depends on the photon-proton centre-of-mass energy via the modified Bjorken variable [MATH] where [CITATION] [EQUATION]', '1605.07665-2-10-6': 'The dipole-proton scattering amplitude contains all the high energy QCD dynamics of the dipole-proton interaction.', '1605.07665-2-10-7': 'It is a universal object, appearing also in the formula for the fully inclusive DIS process: [MATH].', '1605.07665-2-10-8': 'Indeed, replacing the vector meson by a virtual photon in Eq. [REF], we obtain the amplitude for elastic Compton scattering [MATH], i.e. [EQUATION] where we have introduced the dipole cross-section [EQUATION]', '1605.07665-2-10-9': 'Via the Optical Theorem, the elastic amplitude given by Eq. [REF] is directly related to the inclusive [MATH] total cross-section in DIS: [EQUATION] where now [CITATION] [EQUATION]', '1605.07665-2-10-10': 'This means that one can use the high quality DIS data from HERA to constrain the free parameters of the dipole cross-section section and then use the same dipole cross-section to make predictions for vector meson production and other distinct processes like Deeply Virtual Compton Scattering (DVCS) and Diffractive DIS.', '1605.07665-2-10-11': 'This program has been successfully carried by several authors [CITATION] hinting very strongly at the universality of the dipole cross-section.', '1605.07665-2-11-0': 'Note that the high energy factorization in Eqs. [REF] and [REF] holds beyond the validity of perturbation theory, i.e. for all dipole sizes.', '1605.07665-2-11-1': 'In practice, the expressions for the photon light-front wavefunctions obtained perturbatively in light-front QED are used for all [MATH].', '1605.07665-2-11-2': 'To lowest order in [MATH], the perturbative photon wavefunctions are given by [CITATION]: [EQUATION] where [MATH] and [MATH] is the complex notation for the transverse separation between the quark and anti-quark.', '1605.07665-2-11-3': 'As can be seen, at [MATH] or [MATH], the photon light-front wavefunctions become sensitive to the non-zero quark mass [MATH] which prevents the modified Bessel function [MATH] from diverging, i.e. the quark mass acts as an infrared regulator.', '1605.07665-2-11-4': 'On the other hand, a non-perturbative model for the meson light-front wavefunction is used and assumed to be valid for all [MATH].', '1605.07665-2-12-0': 'To compare with experiment, we compute the differential cross-section in the forward limit, i.e. [EQUATION] and we then assume the [MATH]-dependence to be exponential, i.e. [EQUATION] where the diffractive slope parameter [MATH] is given by [EQUATION] with [MATH] GeV[MATH].', '1605.07665-2-12-1': 'This parametrization of the diffractive slope agrees with the most recent ZEUS data for both [MATH] and [MATH] production [CITATION].', '1605.07665-2-12-2': 'The most recent H1 data for [MATH] production [CITATION] prefer a somewhat larger value of [MATH], but with a larger uncertainty.', '1605.07665-2-13-0': 'Note that Eq. [REF] can be rewritten as [EQUATION] where [MATH] is the ratio of real to imaginary parts of the amplitude.', '1605.07665-2-13-1': 'We estimate [MATH] in the usual way [CITATION] [EQUATION]', '1605.07665-2-13-2': 'We calculate the photo-production cross section after integrating Eq. [REF] over [MATH].', '1605.07665-2-13-3': 'This means that the uncertainty in the diffractive slope [MATH] leads to an uncertainty in the normalization of our predictions for total cross-section.', '1605.07665-2-13-4': 'We shall give predictions for the total cross-section [MATH] to be compared to the HERA data.', '1605.07665-2-14-0': '# Holographic meson wavefunctions', '1605.07665-2-15-0': 'The vector meson light-front wavefunctions appearing in Eq. [REF] cannot be computed in perturbation theory.', '1605.07665-2-15-1': 'Nevertheless, they can be assumed to have the same spinor and polarization structure as in the photon case, together with an unknown non-perturbative wavefunction [CITATION].', '1605.07665-2-15-2': 'Explicitly, the vector meson light-front wavefunctions can be written as [CITATION] [EQUATION] and [EQUATION]', '1605.07665-2-16-0': 'Various ansatz for the non-perturbative meson wavefunction have been proposed in the literature [CITATION], perhaps the most popular one being the so-called Boosted Gaussian (BG) wavefunction [CITATION] which has been used in the recent studies in Refs. [CITATION] to describe simultaneously the cross-section data on diffractive [MATH] and [MATH] production.', '1605.07665-2-16-1': 'Ref. [CITATION] uses the dipole cross-section extracted in Ref. [CITATION] with the BG wavefunction to predict vector meson production in ultrapheripheral collisions at the LHC.', '1605.07665-2-16-2': 'In Refs. [CITATION], the [MATH] meson wavefunction is extracted from the data using several dipole models which fit the 2001 DIS structure function data.', '1605.07665-2-17-0': 'In recent years, new insights about hadronic light-front wavefunctions based on the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence have been proposed by Brodsky and de Teramond.', '1605.07665-2-17-1': '[CITATION].', '1605.07665-2-17-2': 'These authors found that in a semiclassical approximation of light-front QCD with massless quarks, the meson wavefunction can be written as [CITATION]', '1605.07665-2-18-0': '[EQUATION] where the variable [MATH] is the transverse separation between the quark and the antiquark at equal light-front time.', '1605.07665-2-18-1': 'The transverse wavefunction [MATH] is a solution of the so-called holographic light-front Schrodinger equation: [EQUATION] where [MATH] is the mass of the meson and [MATH] is the confining potential which at present cannot be computed from first-principle in QCD.', '1605.07665-2-18-2': 'On the other hand, making the substitutions [MATH] where [MATH] being the fifth dimension of AdS space, together with [MATH] where [MATH] and [MATH] are the radius of curvature and mass parameter of AdS space respectively, then Eq. [REF] describes the propagation of spin-[MATH] string modes in AdS space.', '1605.07665-2-18-3': 'In this case, the potential is given by [EQUATION] where [MATH] is the dilaton field which breaks the conformal invariance of AdS space.', '1605.07665-2-18-4': 'A quadratic dilaton ([MATH]) profile results in a harmonic oscillator potential in physical spacetime: [EQUATION]', '1605.07665-2-18-5': 'Brodsky, Dosch and de Teramond have shown that the light-front harmonic potential is unique [CITATION].', '1605.07665-2-18-6': 'Solving the holographic Schrodinger equation with this harmonic potential given by Eq. [REF] yields the meson mass spectrum [CITATION] [EQUATION] with the corresponding normalized eigenfunctions [EQUATION]', '1605.07665-2-18-7': 'To completely specify the holographic wavefunction given by Eq. [REF], the longitudinal wavefunction [MATH] must be determined.', '1605.07665-2-18-8': 'For massless quarks, this is achieved by an exact mapping of the pion electromagnetic form factors in AdS and in physical spacetime resulting in [CITATION].', '1605.07665-2-18-9': '[EQUATION]', '1605.07665-2-18-10': 'For meson families with [MATH], Eq. [REF] predicts that the mesons lie on linear Regge trajectories as is experimentally observed and thus [MATH] can be chosen to fit the Regge slope.', '1605.07665-2-18-11': 'Ref. [CITATION] reports [MATH] GeV for vector mesons.', '1605.07665-2-18-12': 'Eq. [REF] also predicts that the pion and kaon (with [MATH]) are massless.', '1605.07665-2-18-13': 'To account for their physical masses, non-zero light quark masses have to be introduced.', '1605.07665-2-18-14': 'To do so, we follow the prescription of Brodsky and de Teramond given in Ref. [CITATION] and which we outline below.', '1605.07665-2-19-0': 'For the ground state mesons with [MATH], Eq. [REF] becomes [EQUATION]', '1605.07665-2-19-1': 'A two-dimensional Fourier transform to momentum space yields [EQUATION] where [MATH] is invariant mass of the [MATH] pair is given by [EQUATION]', '1605.07665-2-19-2': 'For non-zero quark masses, this invariant mass becomes [EQUATION]', '1605.07665-2-19-3': 'Inserting Eq. [REF] in Eq. [REF] and Fourier transforming back to configuration space gives [EQUATION] where we have introduced a polarization-dependent normalization constant [MATH].', '1605.07665-2-19-4': 'We fix this normalization constant by requiring that [EQUATION] where [MATH] are given by Eqs. [REF] and [REF].', '1605.07665-2-20-0': 'With the non-zero light quark masses, the meson mass spectrum becomes [CITATION] [EQUATION] where the mass shift is given by [CITATION] [EQUATION]', '1605.07665-2-20-1': 'Hence, Eq. [REF] implies that [EQUATION] which allows to fix the [MATH] (and [MATH]) quark masses for a given [MATH].', '1605.07665-2-20-2': 'Using [MATH] GeV, Ref. [CITATION] reports [MATH] GeV.', '1605.07665-2-20-3': 'To fix the strange quark mass, Ref. [CITATION] uses [MATH] MeV together with [MATH] GeV and obtain [MATH] GeV.', '1605.07665-2-20-4': 'Note that the above Brodsky-de Teramond quark mass prescription is expected to be a good approximation only for light quark masses.', '1605.07665-2-20-5': 'A possible way to account for heavier quark masses is to determine the longitudinal function [MATH] dynamically [CITATION].', '1605.07665-2-21-0': 'Having specified the holographic wavefunction for the vector mesons, we are now able to predict their vector and tensor couplings defined by [CITATION] [EQUATION] and [EQUATION] respectively.', '1605.07665-2-21-1': 'In Eqs. [REF] and [REF], [MATH] and [MATH] are the antiquark and quark fields evaluated at the same spacetime point, [MATH] and [MATH] are the momentum and polarization vectors of the vector meson.', '1605.07665-2-21-2': 'Inserting the Fock expansion of the meson states in the right-hand-side of Eq. [REF] and Eq. [REF], we obtain [CITATION] [EQUATION] and [EQUATION] respectively.', '1605.07665-2-21-3': 'Note that the tensor coupling is dependent on the scale [MATH] although we find that our predictions for [MATH] hardly depend on [MATH] for [MATH].', '1605.07665-2-21-4': 'This means that our predictions are at some unspecified low scale [MATH].', '1605.07665-2-21-5': 'As is obvious from Eq. [REF], the tensor coupling vanishes as [MATH], consistent with the requirement that the tensor current vanishes in the chiral limit.', '1605.07665-2-21-6': 'There is no such requirement for the vector current and indeed we predict a non-vanishing value for the vector coupling as [MATH].', '1605.07665-2-21-7': 'We show the variation of the vector and tensor couplings with the quark mass in Figure [REF].', '1605.07665-2-21-8': 'It is interesting to note that the vector coupling is maximum for [MATH] GeV.', '1605.07665-2-22-0': 'The vector coupling is also referred to as the decay constant as it is related to the measured electronic decay width [MATH] of the vector meson: [EQUATION] where [MATH] and [MATH].', '1605.07665-2-22-1': 'Our results for the electronic decay widths are shown in Table [REF].', '1605.07665-2-22-2': 'Note that we obtain a lower value for the decay width of the [MATH] than that reported in Ref. [CITATION] because we are using a universal [MATH] GeV for both vector mesons compared to [MATH] GeV used in Ref. [CITATION].', '1605.07665-2-22-3': 'We show predictions for the decay width using [MATH] GeV for the [MATH] and [MATH] GeV for the [MATH].', '1605.07665-2-22-4': 'As can be seen in Figure [REF], the vector coupling for the [MATH] meson varies slowly with the quark mass in the range [MATH] GeV and hence our two predictions for the decay width do not differ much from each other with a slight preference for [MATH] GeV.', '1605.07665-2-22-5': 'The variation of [MATH] in the range [MATH] GeV is more important and the lower strange quark mass, [MATH] GeV gives better agreement with the decay width datum.', '1605.07665-2-23-0': 'For both vector mesons, we underestimate the electronic decay width.', '1605.07665-2-23-1': 'But this is also the case with the other non-perturbative methods quoted in Table [REF] for the [MATH].', '1605.07665-2-23-2': 'There are likely perturbative corrections that must be taken into account when predicting the electronic decay width.', '1605.07665-2-24-0': 'Since the optimal agreement (or rather minimal disagreement) with the electronic decay width data is achieved with [MATH] GeV, we choose this quark mass to compare our predictions with QCD Sum Rules, Dyson-Schwinger and lattice predictions as shown in Tables [REF] and [REF].', '1605.07665-2-24-1': 'Recall that our predictions for the transverse decay constant are at [MATH] GeV which prevents an exact comparison with the other predictions all given at a scale [MATH] GeV.', '1605.07665-2-24-2': 'Despite this, it is clear that we predict a smaller transverse decay constant (with [MATH] GeV) than those predicted by the other non-perturbative methods quoted in Table [REF].', '1605.07665-2-25-0': '# Refitting the CGC dipole model', '1605.07665-2-26-0': 'In principle, the dipole-proton scattering amplitude [MATH] can be obtained by solving the Balitsky-Kovchegov (BK) equation [CITATION] which itself can be derived within the Colour Glass Condensate (CGC) formalism [CITATION].', '1605.07665-2-26-1': 'However, work is still in progress to implement in a satisfactory way the impact-parameter dependence in the proton-dipole amplitude [CITATION].', '1605.07665-2-26-2': 'A widely used model is that proposed by Kowalski and Watt [CITATION] where the saturation scale (see below) have a Gaussian dependence on the impact parameter.', '1605.07665-2-26-3': 'However, it has been recently argued in Ref. [CITATION] that the [MATH]-dependence should be exponential.', '1605.07665-2-26-4': 'In any case, considering the [MATH]-dependence introduces an additional parameter which has to be fixed using data on diffractive meson production (for instance [MATH] production) which requires a model for the meson wavefunction.', '1605.07665-2-26-5': 'On the other hand, a simple model for the [MATH]-integrated dipole-proton amplitude, i.e. the dipole cross-section has been proposed long ago in Ref. [CITATION].', '1605.07665-2-26-6': 'This is known as the CGC dipole model and is given by [EQUATION] with [EQUATION] where the saturation scale [MATH] GeV.', '1605.07665-2-26-7': 'The coefficients [MATH] and [MATH] are determined from the condition that the [MATH] and its derivative with respect to [MATH] are continuous at [MATH].', '1605.07665-2-26-8': 'This leads to [EQUATION]', '1605.07665-2-26-9': 'The free parameters of the CGC dipole model are [MATH] and [MATH] which are fixed by a fit to the structure function [MATH] data.', '1605.07665-2-26-10': '[MATH] and [MATH] are fixed at 0.7 and 9.9 (LO BFKL prediction) respectively.', '1605.07665-2-26-11': 'The high quality DIS data from HERA can be used to fix the free parameters of the dipole cross-section.', '1605.07665-2-26-12': 'An earlier fit to the structure function data given in Ref. [CITATION] and used in Ref. [CITATION] to make successful predictions for diffractive [MATH] production are: [MATH] mb, [MATH].', '1605.07665-2-27-0': 'In 2015, the H1 and ZEUS collaborations have released highly precise combined data sets [CITATION] for the reduced cross-section [EQUATION] where [MATH] and [MATH] is the centre of mass energy of the [MATH] system for [MATH] different bins : [MATH] GeV ([MATH] data points), [MATH] GeV ([MATH] data points) and [MATH] GeV ([MATH] data points), [MATH] GeV ([MATH] data points).', '1605.07665-2-27-1': 'The structure functions in Eq. [REF] are given by [EQUATION] and [EQUATION] where in the dipole model, [MATH] is given by equation [REF].', '1605.07665-2-28-0': 'The most recent extraction of the CGC dipole model parameters was performed in Ref. [CITATION] following the release of the combined HERA data in 2010 [CITATION].', '1605.07665-2-28-1': 'In this paper, the authors report a successful fit in the kinematic range [MATH] using very small light quark masses [MATH] GeV and a charm quark mass [MATH] GeV.', '1605.07665-2-28-2': 'The fitted parameters are [MATH] mb, [MATH] with a [MATH].', '1605.07665-2-29-0': 'We start by computing the [MATH] per data point ([MATH]) for the 2015 HERA data using the earlier fitted parameters of Ref. [CITATION] and the most recent fitted parameters of Ref. [CITATION].', '1605.07665-2-29-1': 'We obtain [MATH] and [MATH] respectively.', '1605.07665-2-29-2': 'The latter [MATH] is acceptable and in fact lower than that reported in Ref. [CITATION] for the fit to the 2010 data set.', '1605.07665-2-29-3': 'However, since we are using here effective quark masses and not current quark masses as in Ref. [CITATION], we need to refit the CGC dipole parameters to the new 2015 data.', '1605.07665-2-29-4': 'To obtain the above [MATH] values, we have used [MATH] GeV as in Ref. [CITATION], and we shall also use this same charm mass for our fits.', '1605.07665-2-29-5': 'Departing from Ref. [CITATION], we also include low [MATH] data in our fits, thereby daring to extrapolate the use of the CGC dipole model in the non-perturbative region where the predictions are more sensitive to the quark masses.', '1605.07665-2-30-0': 'Our fitted values for the CGC dipole model parameters together with the resulting [MATH] per degrees of freedom ([MATH]) values are shown in Table [REF].', '1605.07665-2-30-1': 'The first two rows indicate that the fit is not very sensitive to the variation in the strange quark mass.', '1605.07665-2-30-2': 'Comparing the second and third rows, we can see that the data prefer the lower [MATH] and [MATH] quark masses and that increasing them give quite different fit parameters especially for [MATH].', '1605.07665-2-30-3': 'But in all three cases, the [MATH] is less than the value obtained in Ref. [CITATION] ([MATH]) and we regard them as acceptable fits.', '1605.07665-2-30-4': 'This is not to deny that using current quark masses can lead to equally good fits.', '1605.07665-2-30-5': 'In fact, we find that with current quark masses [MATH] GeV, we obtain [MATH], i.e. a fit of similar quality as our best fit with [MATH] GeV and [MATH] GeV.', '1605.07665-2-30-6': 'The fitted parameters are [MATH], [MATH], [MATH] and [MATH] mb, similar to those reported in Ref. [CITATION].', '1605.07665-2-30-7': 'However, we have checked that this quark mass set (with our holographic wavefunction) does not lead to a good agreement with the diffractive cross-section data at low [MATH].', '1605.07665-2-31-0': '# Predicting diffractive cross-sections', '1605.07665-2-32-0': 'Having specified the dipole cross-section and the holographic meson wavefunction, we can now compute cross-sections for diffractive [MATH] and [MATH] production.', '1605.07665-2-32-1': 'We shall show predictions using three sets of the CGC dipole parameters as given in Table [REF].', '1605.07665-2-32-2': 'We shall refer to these three sets of predictions as "Fit A" (first row), "Fit B" (second row) and "Fit C" (third row) respectively.', '1605.07665-2-32-3': 'Recall that all our predictions will be generated using the same holographic wavefunction given by Eq. [REF] and they differ only by the choice of quark masses and the corresponding fitted parameters of the CGC dipole model as given in Table [REF].', '1605.07665-2-33-0': 'We compute the total cross-section as a function of [MATH] in different [MATH] bins as well as a function of [MATH] at fixed [MATH].', '1605.07665-2-33-1': 'We also compute the ratio of longitudinal to transverse cross-sections as a function of [MATH] at fixed [MATH].', '1605.07665-2-33-2': 'Predicting the latter observable is interesting since the normalization uncertainties in the diffractive [MATH]-slope and the dipole cross-section, cancel out, increasing its sensitivity to the meson wavefunction.', '1605.07665-2-34-0': 'For [MATH] production, our predictions for the [MATH] dependence of the total cross-section in different [MATH] bins are shown in Figures [REF] and [REF] while our predictions for the [MATH] dependence of the total cross-section at fixed [MATH] are shown in Figure [REF].', '1605.07665-2-34-1': 'The "Fit A" (black solid curves) and the "Fit C" (blue dashed curves) are both accommodated by the total cross-section data.', '1605.07665-2-34-2': 'The [MATH] ratio data, shown in Figure [REF], are able to discriminate between them and favour the "Fit C" prediction.', '1605.07665-2-34-3': 'Notice that the "Fit C" predictions undershoot the data in the two largest [MATH] bins but this is the kinematic region where the non-perturbative holographic wavefunction is expected to be less accurate.', '1605.07665-2-34-4': 'We thus confirm the conclusion of Ref. [CITATION] in which the set of equal quark masses was used.', '1605.07665-2-35-0': 'For [MATH] production, our predictions for the [MATH] dependence of the total cross-section in different [MATH] bins are shown in Figures [REF] and [REF] and for the [MATH] dependence of the total cross-section at fixed [MATH] are shown in Figure [REF].', '1605.07665-2-35-1': 'Here, it is clear that the "Fit A" predictions (solid black curves) are not successful.', '1605.07665-2-35-2': 'The data prefer slightly the "Fit B" (orange dotted curves) over the "Fit C" predictions (blue dashed curves) although the lack of data in the low [MATH] region prevents us from making a definite statement.', '1605.07665-2-35-3': 'At high [MATH], all our predictions tend to undershoot the (ZEUS) data as expected.', '1605.07665-2-35-4': 'Our predictions for the longitudinal to transverse cross-sections ratio for [MATH] production are shown in Fig. [REF].', '1605.07665-2-35-5': 'We can see that the ratio data tend to favour the "Fit A" prediction (solid black curve) although they are not precise enough to discard the other two predictions.', '1605.07665-2-36-0': 'In summary, the "Fit C" predictions are favoured for [MATH] production and the "Fit B" predictions are preferred for [MATH] production.', '1605.07665-2-36-1': 'Notice that in both sets of predictions, we are using exactly the same holographic wavefunction for both [MATH] and [MATH], i.e. a wavefunction with the same [MATH] GeV and [MATH] GeV.', '1605.07665-2-36-2': 'The difference between the two sets of predictions arises from the different electromagnetic couplings and the different fitted parameters of the CGC dipole cross-section.', '1605.07665-2-37-0': 'Finally, we consider the data set on the ratio of the total cross-sections for [MATH] and [MATH] production.', '1605.07665-2-37-1': 'Note if the [MATH] and the [MATH] had identical masses and holographic wavefunctions, this ratio is simply given by the squared ratio of the effective electric charges of the quark-antiquark dipole coupling to the photon: [MATH].', '1605.07665-2-37-2': 'As expected our "Fit A" prediction (blue dashed curve) tends to that value at high [MATH].', '1605.07665-2-37-3': 'At lower [MATH], the deviation from [MATH] is due to the vector meson mass entering the modified Bjorken-[MATH] given by Eq. [REF] and the diffractive slope given by Eq. [REF].', '1605.07665-2-37-4': 'However, the data indicate a lower ratio and as can be seen in Figure [REF], the "Fit B" prediction (orange dotted curve) is clearly preferred.', '1605.07665-2-37-5': 'This provides evidence for the need to have different quark masses in the holographic wavefunctions of the [MATH] and [MATH].', '1605.07665-2-38-0': 'In view of the above results, we anticipate that a set of quark masses with an even weaker SU(3) symmetry breaking than our "Fit B" set, should give the best simultaneous description of both [MATH] and [MATH] diffractive production data.', '1605.07665-2-39-0': '# Conclusion', '1605.07665-2-40-0': 'We have updated the parameters of the CGC dipole model using the definitive 2015 HERA data on inclusive DIS and we have used the fitted dipole cross-section together with a holographic meson wavefunction in order to compute the cross-sections for diffractive [MATH] and [MATH] meson production.', '1605.07665-2-40-1': 'The holographic light-front meson wavefunction is successful to describe simultaneously diffractive [MATH] and [MATH] production with a single universal holographic mass scale [MATH] but with a set of light quark masses with a weaker SU(3) flavour symmetry ([MATH]) breaking than that used in light-front holography ([MATH]) to generate the pion and kaon masses.'} | [['1605.07665-1-3-0', '1605.07665-2-3-0'], ['1605.07665-1-3-1', '1605.07665-2-3-1'], ['1605.07665-1-3-2', '1605.07665-2-3-2'], ['1605.07665-1-3-3', '1605.07665-2-3-3'], ['1605.07665-1-3-4', '1605.07665-2-3-4'], ['1605.07665-1-3-5', '1605.07665-2-3-5'], ['1605.07665-1-17-0', '1605.07665-2-17-0'], ['1605.07665-1-17-2', '1605.07665-2-17-2'], ['1605.07665-1-24-0', '1605.07665-2-24-0'], ['1605.07665-1-24-1', '1605.07665-2-24-1'], ['1605.07665-1-24-2', '1605.07665-2-24-2'], ['1605.07665-1-15-0', '1605.07665-2-15-0'], ['1605.07665-1-15-1', '1605.07665-2-15-1'], ['1605.07665-1-15-2', '1605.07665-2-15-2'], ['1605.07665-1-34-0', '1605.07665-2-34-0'], 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['1605.07665-1-18-6', '1605.07665-2-18-6'], ['1605.07665-1-18-10', '1605.07665-2-18-10']] | [] | [['1605.07665-1-36-0', '1605.07665-2-36-0'], ['1605.07665-1-32-2', '1605.07665-2-32-2']] | [] | ['1605.07665-1-17-1', '1605.07665-1-18-9', '1605.07665-2-17-1', '1605.07665-2-18-9'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1605.07665 | null | null | null | null | null |
cond-mat-0006258 | {'cond-mat-0006258-1-0-0': 'We theoretically study the creep of the inhomogeneous vortex density profile (the Bean state) in superconductors.', 'cond-mat-0006258-1-0-1': 'The low temperatures experimental phenomenology, previously interpreted in terms of "quantum tunnelling" of vortices, is reproduced by Monte Carlo simulations of a purely "classical" vortex model.', 'cond-mat-0006258-1-0-2': 'We demonstrate that a non-zero creep rate in the limit of vanishing temperature is to be expected in systems with slow relaxations as a consequence of their evolution in a complex free energy landscape.', 'cond-mat-0006258-1-1-0': 'PACS numbers: 74.60.', 'cond-mat-0006258-1-1-1': 'Ge 47.32.', 'cond-mat-0006258-1-1-2': 'Cc 74.50.', 'cond-mat-0006258-1-1-3': '+r 75.45.', 'cond-mat-0006258-1-1-4': '+j', 'cond-mat-0006258-1-2-0': 'There exist an abundance of experimental evidence that the relaxation rate of the magnetisation in type II superconductors does not vanish as the temperature [MATH] is lowered towards zero (see eg. [CITATION]).', 'cond-mat-0006258-1-2-1': 'This may seem surprising if one assumes that the mechanism allowing the magnetisation to relax is thermal activation over a characteristic energy barrier [MATH].', 'cond-mat-0006258-1-2-2': 'Namely, when [MATH] the relaxation rate should vanish as the Arrhenius factor for thermal activation, [MATH], goes to zero.', 'cond-mat-0006258-1-2-3': 'The experimental observation for a very wide range of superconductors (ranging from conventional low temperature, to organic, heavy fermions and a variety of high temperature superconductors [CITATION]) is, however, that the relaxation rate approaches a temperature independent non-zero constant at low [MATH].', 'cond-mat-0006258-1-2-4': 'The question then arises how relaxation can continue to take place at a finite rate while the thermal activation factor seemingly disappears exponentially rapidly to zero.', 'cond-mat-0006258-1-3-0': 'A number of researchers have suggested that the such a phenomenon is caused by quantum tunnelling of vortices through the barriers in the random pinning potential (for a theoretical review see [CITATION]).', 'cond-mat-0006258-1-3-1': 'The above "quantum" explanation is very intriguing and in good agreement with some experimental results in compounds such as YBCO [CITATION] or BSCCO [CITATION].', 'cond-mat-0006258-1-3-2': 'Also other even more exotic materials, such as organic superconductors [CITATION], give good correspondences.', 'cond-mat-0006258-1-3-3': 'However, in other non conventional systems such as heavy fermions superconductors, the theory of quantum creep is totally unable to describe the observed low [MATH] relaxation [CITATION].', 'cond-mat-0006258-1-3-4': 'In fact, strong discrepancies are found in many other systems ranging from PCeCO crystals to YBCO/PBCO multilayers or YBCO and BSCCO films and crystals [CITATION].', 'cond-mat-0006258-1-3-5': 'One of the problem is [CITATION] that the length of the tunnelling vortex segment [MATH] needed to fit the creep rate data, [MATH], can be orders of magnitude larger that the one theoretically predicted by quantum creep theory [CITATION].', 'cond-mat-0006258-1-3-6': 'Also suspicious is that the experimental temperature dependence of the creep rate, [MATH], is often very different from the one predicted by quantum theory (typically linear instead of quadratic, see for instance [CITATION]).', 'cond-mat-0006258-1-4-0': 'The above contradictory results and the idea that some general simple mechanism might still be present in all the above materials naturally suggests to look for other descriptions of the anomalous low [MATH] magnetic relaxation.', 'cond-mat-0006258-1-4-1': 'It is worth to stress that the observation of a non-vanishing constant creep rate in the limit [MATH] is found under very general circumstances: it does not crucially depend on the thickness of the sample [CITATION] (i.e. on its dimensionality), nor on whether the pinning is caused by columnar defects or random point pins [CITATION].', 'cond-mat-0006258-1-4-2': 'Thus, the mechanism behind the low temperature creep seems to be of a fundamental and basic nature.', 'cond-mat-0006258-1-5-0': 'We demonstrate below that also in "classical" system (i.e., not "quantum") logarithmically slow glassy dynamics can naturally persist even at vanishing temperature and can lead to the experimentally observed phenomenology.', 'cond-mat-0006258-1-5-1': 'This is possible because the low [MATH] limit of glassy dynamics consists of searching, among a very large number, for a few "downhill" or "flat" directions in the free energy landscape.', 'cond-mat-0006258-1-5-2': 'The number of these directions decreases as relaxation proceeds though there always remain some.', 'cond-mat-0006258-1-5-3': 'They can be found only by collective cooperative rearrangements of the system, resulting in a slowing down of relaxation.', 'cond-mat-0006258-1-5-4': 'To illustrate this scenario we make use a simple model which was shown to reproduce a broad range of experimental facts of vortex dynamics.', 'cond-mat-0006258-1-6-0': 'The model - We consider a coarse grained schematic model of the original vortex system.', 'cond-mat-0006258-1-6-1': 'The model was introduced in Ref. [CITATION] and is able to reproduce a large set of essential features of vortex phenomenology.', 'cond-mat-0006258-1-6-2': 'The model correctly predicts the reentrant nature of the equilibrium phase diagram, magnetisation hysteretic loops with "anomalous" second peak, the relative behaviour of the magnetisation and the sweep creep rate, logarithmic relaxation, Bean profiles and more.', 'cond-mat-0006258-1-6-3': '[CITATION].', 'cond-mat-0006258-1-6-4': 'The model also predicts the existence of a "glassy region" at low temperature with strong "aging" effects [CITATION].', 'cond-mat-0006258-1-7-0': 'Here we use the model to study the magnetic relaxation rate, [MATH], in the very low temperature limit.', 'cond-mat-0006258-1-7-1': 'Interestingly, the model reproduces the experimental "anomalous" relaxation rates [CITATION].', 'cond-mat-0006258-1-8-0': 'Our model is an attempt to identify the effective degrees of freedom responsible for the experimental observations.', 'cond-mat-0006258-1-8-1': 'It is called a Multi Occupancy Model (MOM).', 'cond-mat-0006258-1-8-2': 'We imagine to coarse grain the [MATH]-plane of the superconductor by introducing a square lattice of lattice spacing [MATH] of the order of the London penetration length [CITATION].', 'cond-mat-0006258-1-8-3': 'The number of vortices on plaquette number [MATH] is denoted by [MATH].', 'cond-mat-0006258-1-8-4': 'The occupancy of each plaquette is a number between zero and [MATH] (hence the name MOM), where [MATH] is the upper critical magnetic field and [MATH] is the magnetic flux quantum.', 'cond-mat-0006258-1-8-5': 'The vortex interaction is summarised by the following Hamiltonian: [MATH].', 'cond-mat-0006258-1-8-6': 'The first two terms describe the repulsion between the vortices and their self energy.', 'cond-mat-0006258-1-8-7': 'We include on-site and nearest neighbour interactions.', 'cond-mat-0006258-1-8-8': 'That is we ignore the interaction between vortex lines with a separation greater than the London screening length (which by definition is close to [MATH]).', 'cond-mat-0006258-1-8-9': 'We choose [MATH]; [MATH] if [MATH] and [MATH] are nearest neighbours and [MATH] otherwise.', 'cond-mat-0006258-1-8-10': 'The last term in [MATH] describes a delta-distributed random pinning [MATH].', 'cond-mat-0006258-1-8-11': 'Interestingly, the creep phenomena we describe below depend on the presence of pinning in the system, but the general scenario does not depend crucially on the pinning (a fact in correspondence with experimental results [CITATION]).', 'cond-mat-0006258-1-8-12': 'In our model [MATH] sets the energy scale.', 'cond-mat-0006258-1-8-13': 'Below we choose [MATH]; [MATH]; [MATH]; [MATH].', 'cond-mat-0006258-1-9-0': 'The present model is close to a lattice system recently introduced by Bassler and Paczuski [CITATION] to study, at [MATH], vortex avalanches and vortex river flow.', 'cond-mat-0006258-1-9-1': 'Interestingly, such a lattice system can be obtained from ours in the limit [MATH].', 'cond-mat-0006258-1-9-2': 'Our model is explained in all its details in Ref. [CITATION], here we only notice that a theoretical justification of the form of this Hamiltonian consist in the observation that the repulsive interaction between vortices of spatial separation less than [MATH] is included and that the existence of the upper critical field is represented by the constraint [MATH].', 'cond-mat-0006258-1-9-3': 'More compelling is though the fact that mean field calculations and Monte Carlo simulations of the dynamics and thermodynamics of this Hamiltonian are able to reproduce the experimental phenomenology [CITATION].', 'cond-mat-0006258-1-10-0': 'The relaxation of the model is simulated by use of Monte Carlo dynamics on a square lattice in presence of a thermal bath of temperature [MATH].', 'cond-mat-0006258-1-10-1': 'The system is periodic in the [MATH]-direction.', 'cond-mat-0006258-1-10-2': 'The two edges parallel to the [MATH]-direction are in contact with a vortex reservoir, described by the above Hamiltonian without pinning.', 'cond-mat-0006258-1-10-3': 'Particles can enter and exit the system only through the reservoir, which plays the role of the external magnetic field.', 'cond-mat-0006258-1-10-4': 'Hence the reservoir density, [MATH], is used as the external control parameter.', 'cond-mat-0006258-1-10-5': 'We perform the following zero field cooled experiment: at a low temperature [MATH] we increase at constant rate [MATH] the reservoir density from zero to a working value [MATH].', 'cond-mat-0006258-1-10-6': 'We keep [MATH] fixed while we monitor the time dependence of the magnetisation [MATH].', 'cond-mat-0006258-1-10-7': 'Here [MATH] is the vortex density inside the system (of size [MATH]).', 'cond-mat-0006258-1-10-8': 'Below usually [MATH], but we checked our results in the size range [MATH].', 'cond-mat-0006258-1-10-9': 'Time is measured in units of a full lattice update.', 'cond-mat-0006258-1-10-10': 'The data presented below are averaged over 128 realizations of the pinning background.', 'cond-mat-0006258-1-11-0': 'In particular, we investigate the creep rate [EQUATION] as function of [MATH], [MATH] and [MATH].', 'cond-mat-0006258-1-11-1': 'In typical experiments the nature of the [MATH] dependence of [MATH] is such that [MATH] decreases in time.', 'cond-mat-0006258-1-11-2': 'So usually, one deals with an average creep rate, [MATH], in some given temporal window [CITATION].', 'cond-mat-0006258-1-11-3': 'In our model dynamics, [MATH] at low temperatures behaves according to the known logarithmic interpolation formula (see Ref. [CITATION]) found in experiments [CITATION], namely: [MATH].', 'cond-mat-0006258-1-12-0': 'Consistently, we define [MATH] as the average value of [MATH] in the last time decade of our measures (i.e., for [MATH]).', 'cond-mat-0006258-1-12-1': 'The present choice is the most natural one and we stress that the general results presented below do not depend on it.', 'cond-mat-0006258-1-12-2': 'It is worth noting that experimentalists are forced to make similar choices concerning the specific way [MATH] is measured [CITATION].', 'cond-mat-0006258-1-13-0': 'For the reasons explained in the introduction, a very important physical quantity is the distribution, [MATH], of the energy barriers, [MATH], that vortices segments meet during their motion.', 'cond-mat-0006258-1-13-1': 'Since the equilibration times at low temperature are huge, the system is typically off-equilibrium and [MATH] is itself a (logarithmically slow) function of [MATH].', 'cond-mat-0006258-1-13-2': 'Consistently with the above definition of [MATH], we consider the energy barrier distribution averaged over the last time decade of our measurements.', 'cond-mat-0006258-1-14-0': 'Results - When the temperature is very low, the model exhibits the same kind of "anomalous" creep found in the experiments on superconductors.', 'cond-mat-0006258-1-14-1': 'In Fig.[REF], we plot the creep rate, [MATH], as a function of [MATH] in a broad temperature range.', 'cond-mat-0006258-1-14-2': 'For comparison we present equivalent experimental measurements in a BSCCO single crystal (from Ref.[CITATION]) as inset.', 'cond-mat-0006258-1-14-3': 'The numerical values found for [MATH] at low [MATH] in our model and in real samples are interestingly very similar.', 'cond-mat-0006258-1-14-4': 'The temperature scales of the simulations and of real experiments can be compared by considering that the ratio [MATH] in our model is of the same order of magnitude as [MATH] in a real superconductor.', 'cond-mat-0006258-1-14-5': 'This is seen from a comparison of the [MATH] equilibrium phase diagram of our model with the equilibrium temperature-magnetic field, [MATH], phase diagram of, say, a BSCCO superconductor (see Ref. [CITATION]).', 'cond-mat-0006258-1-15-0': 'Apparently, the simulations and the measurements on superconductors exhibit very similar behaviour.', 'cond-mat-0006258-1-15-1': 'In both cases [MATH] approaches a finite value, [MATH], when [MATH].', 'cond-mat-0006258-1-15-2': 'In particular, we find that a linear fit of [MATH] in the low [MATH] regime is very satisfactory (see Fig. [REF]): [EQUATION] where both [MATH] and [MATH] are a function of the applied field [MATH].', 'cond-mat-0006258-1-15-3': 'For a given value of [MATH], the above fit can be improved by including higher powers of [MATH].', 'cond-mat-0006258-1-15-4': 'We also note that in our model [MATH] is non monotonous in [MATH]: in the higher [MATH] region it starts decreasing.', 'cond-mat-0006258-1-15-5': 'This is also a known experimental fact [CITATION].', 'cond-mat-0006258-1-16-0': 'It is interesting that a very similar numerical value for [MATH] is found in both model and experimental data.', 'cond-mat-0006258-1-16-1': 'Experimentally, for many different systems, [MATH] values are typically around the interval [MATH] [CITATION].', 'cond-mat-0006258-1-16-2': 'In our model (for the examined value [MATH]) we find, for instance, a similar range of values by varying the applied field (see Fig. [REF]).', 'cond-mat-0006258-1-16-3': 'In particular, [MATH] seems to decrease on average by increasing the field [MATH].', 'cond-mat-0006258-1-16-4': 'The overall behaviour can be approximately interpolated with a power law: [MATH], where, for [MATH], [MATH] and [MATH].', 'cond-mat-0006258-1-16-5': 'As shown in Fig. [REF], the presence of a small exponent [MATH] implies that [MATH] is slowly varying with [MATH]: sensible variations in [MATH] can be seen only by changing [MATH] of orders of magnitude.', 'cond-mat-0006258-1-16-6': 'Note that the dips in the [MATH] versus [MATH] data in Fig. [REF] at certain values of [MATH] (namely around 3, 13, and 18) are statistically significant.', 'cond-mat-0006258-1-16-7': 'Their origin is currently under investigation.', 'cond-mat-0006258-1-17-0': 'In the inset of Fig. [REF], we show that [MATH] is essentially independent of the ramping rate, [MATH] (the values shown are for [MATH] and [MATH]).', 'cond-mat-0006258-1-17-1': 'This is an other typical experimental observation [CITATION].', 'cond-mat-0006258-1-17-2': 'However, a very small decrease of [MATH] with increasing [MATH] cannot be excluded: we show a fit to the form [MATH], with [MATH].', 'cond-mat-0006258-1-17-3': 'The fact that [MATH] is practically independent on [MATH], far from being a proof of the presence of equilibrium in the system, is due to the fact that at very low temperatures the characteristic equilibration time, [MATH], is enormous (see Ref. [CITATION]).', 'cond-mat-0006258-1-17-4': 'So whenever the driving rate, [MATH], is much larger than [MATH] the off-equilibrium state and dynamics of the system are essentially independent of [MATH].', 'cond-mat-0006258-1-17-5': 'Stronger [MATH] effects have to be expected when [MATH] gets closer to [MATH].', 'cond-mat-0006258-1-17-6': 'In fact, it is well known experimentally that at higher temperatures the systems exhibit strong [MATH] dependent "memory" effects [CITATION], the signature of off-equilibrium dynamics.', 'cond-mat-0006258-1-18-0': 'The above scenario is clarified by the analysis of the energy barrier distribution function, [MATH], recorded during the system evolution at very low [MATH].', 'cond-mat-0006258-1-18-1': 'Such a quantity also clearly shows the simple mechanical origin of the anomalous creep found at very low temperature in the present model.', 'cond-mat-0006258-1-18-2': 'The function [MATH] (where [MATH] is in units of [MATH]), recorded at [MATH], is plotted in Fig. [REF] for two values of the applied field, [MATH].', 'cond-mat-0006258-1-18-3': 'We always find that [MATH] has support also on the negative axis.', 'cond-mat-0006258-1-18-4': 'This is the mark of the off-equilibrium nature of the evolution on the observed time scales, because, by definition, in mechanical equilibrium only non-negative energy barriers should be found.', 'cond-mat-0006258-1-18-5': 'The presence of a [MATH] which extends down to negative values also clearly explains the presence of the recorded relaxation at low [MATH]: in the configuration space the system can still find directions where no positive barriers have to be crossed.', 'cond-mat-0006258-1-18-6': 'The insert in Fig. [REF] clarifies the mechanism behind the relaxation.', 'cond-mat-0006258-1-18-7': 'Here, we plot the signal [MATH] defined, for each single Monte Carlo (MC) step [MATH], in the following way: [MATH] if the MC trial is rejected; [MATH] if the trial is accepted and the energy reduced, i.e. [MATH]; and finally [MATH] when a trial is accepted with [MATH].', 'cond-mat-0006258-1-18-8': 'We plot two sequences of trials.', 'cond-mat-0006258-1-18-9': 'One for [MATH] was measured at the early stage of the relaxation, the second sequence, placed at the interval [MATH], is measured at the late stage of the relaxation.', 'cond-mat-0006258-1-18-10': 'Most trials are rejected ([MATH]) and only once in a while the system does manage to find a route pointing downhill in the energy landscape.', 'cond-mat-0006258-1-18-11': 'Since the temperature is so low the system never manage to be activated over a positive barrier, i.e., [MATH] never occurs.', 'cond-mat-0006258-1-18-12': 'As time proceeds fewer and fewer "negative channels" are available to the relaxation and one observes the decrease in the density of the spikes in [MATH].', 'cond-mat-0006258-1-19-0': 'As the temperature is increased thermal activation over positive energy barriers will become possible as the Arrhenius factor [MATH] assumes a non-vanishing value for an appreciable range of barrier values [MATH].', 'cond-mat-0006258-1-19-1': 'When this happens the relaxation will occur sufficiently fast to allow one, within the experimental time window, to approach very near to the equilibrium configuration where the vortex density profile is more or less flat and relaxation ceases, hence [MATH] goes down, as seen in experiments [CITATION] and in Fig. 1.', 'cond-mat-0006258-1-20-0': 'Finally, we stress that off equilibrium relaxation at very low temperatures (below the "freezing" point) is also observed in Molecular Dynamics simulations of glass forming liquids [CITATION].', 'cond-mat-0006258-1-20-1': 'In that cases too, no activation over barriers occurs and the system simply wanders in its very high dimensional phase space through the few channels where no energy increase is required.', 'cond-mat-0006258-1-21-0': 'Discussion - We have above demonstrated that the phenomenological behaviour of the creep rate at low temperatures can be understood in terms of the off equilibrium nature of the inhomogeneous vortex density profile produced in magnetic creep experiments.', 'cond-mat-0006258-1-21-1': 'In fact, cooperative mechanical rearrangements, possible even at very low [MATH] (where thermal activation over positive barriers can be negligible), dominate the phenomenon.', 'cond-mat-0006258-1-21-2': "In this perspective, it is very important to stress that the system's equilibration time at very low temperature is much larger than any experimentally accessible time window [CITATION].", 'cond-mat-0006258-1-21-3': 'Accordingly, we can say that experimental findings do not enforce the interpretation in terms of macroscopic quantum tunnelling of vortices.', 'cond-mat-0006258-1-22-0': 'Relaxation due to quantum tunnelling might be present along with the mechanical relaxation discussed above.', 'cond-mat-0006258-1-22-1': 'It is then important to ask how compelling the quantum tunnelling interpretation is.', 'cond-mat-0006258-1-22-2': 'The theory of quantum tunnelling assumes a London picture and treats the position of the vortex core as the variable that is able to tunnel.', 'cond-mat-0006258-1-22-3': 'Vortex positions are the loci of the zeros of the Ginzburg-Landau order parameter, which in itself is a mean field approximation to microscopic theories like Gorkov or Bogoliubov-de Gennes theory [CITATION].', 'cond-mat-0006258-1-22-4': 'It is not entirely clear if this is the right level at which to introduce quantum fluctuations, but, more importantly, the quantum tunnelling description assume the existence of a characteristic energy barrier [CITATION].', 'cond-mat-0006258-1-22-5': 'The scale of such a barrier may be estimated from single pining or from collective pinning arguments, but in any case a mean field-like approach is assumed, where the fluctuations in the sizes of the activation barriers, arising from many-body effects in vortex-vortex or vortex-pin interactions, are neglected.', 'cond-mat-0006258-1-22-6': 'Finally, the quantum tunnelling description also tacitly assumes the existence of a static equilibrium state in which barriers are always positive.', 'cond-mat-0006258-1-22-7': 'As we have clearly shown above, this is typically not the case and a dynamical approach is more appropriate.', 'cond-mat-0006258-1-23-0': 'We thank M. Paczuski for stimulating discussions.', 'cond-mat-0006258-1-23-1': 'Work supported by the EPSRC and PRA-INFM-99.'} | {'cond-mat-0006258-2-0-0': 'We theoretically study the creep of the inhomogeneous vortex density profile (the Bean state) in superconductors.', 'cond-mat-0006258-2-0-1': 'The low temperatures experimental phenomenology, previously interpreted in terms of "quantum tunnelling" of vortices, is reproduced by Monte Carlo simulations of a purely "classical" vortex model.', 'cond-mat-0006258-2-0-2': 'We demonstrate that a non-zero creep rate in the limit of vanishing temperature is to be expected in systems with slow relaxations as a consequence of their evolution in a complex free energy landscape.', 'cond-mat-0006258-2-1-0': 'PACS numbers: 74.60.', 'cond-mat-0006258-2-1-1': 'Ge 47.32.', 'cond-mat-0006258-2-1-2': 'Cc 74.50.', 'cond-mat-0006258-2-1-3': '+r 75.45.', 'cond-mat-0006258-2-1-4': '+j', 'cond-mat-0006258-2-2-0': 'There exist an abundance of experimental evidence that the relaxation rate of the magnetisation in type II superconductors does not vanish as the temperature [MATH] is lowered towards zero (see eg. [CITATION]).', 'cond-mat-0006258-2-2-1': 'This may seem surprising if one assumes that the mechanism allowing the magnetisation to relax is thermal activation over a characteristic energy barrier [MATH].', 'cond-mat-0006258-2-2-2': 'Namely, when [MATH] the relaxation rate should vanish as the Arrhenius factor for thermal activation, [MATH], goes to zero.', 'cond-mat-0006258-2-2-3': 'The experimental observation for a very wide range of superconductors (ranging from conventional low temperature, to organic, heavy fermions and a variety of high temperature superconductors [CITATION]) is, however, that the relaxation rate approaches a temperature independent non-zero constant at low [MATH].', 'cond-mat-0006258-2-2-4': 'The question then arises how relaxation can continue to take place at a finite rate while the thermal activation factor seemingly disappears exponentially rapidly to zero.', 'cond-mat-0006258-2-3-0': 'A number of researchers have suggested that the such a phenomenon is caused by quantum tunnelling of vortices through the barriers in the random pinning potential (for a theoretical review see [CITATION]).', 'cond-mat-0006258-2-3-1': 'The above "quantum" explanation is very intriguing and in good agreement with some experimental results in compounds such as YBCO [CITATION] or BSCCO [CITATION].', 'cond-mat-0006258-2-3-2': 'Also other even more exotic materials, such as organic superconductors [CITATION], give good correspondences.', 'cond-mat-0006258-2-3-3': 'However, in other non conventional systems such as heavy fermions superconductors, the theory of quantum creep is totally unable to describe the observed low [MATH] relaxation [CITATION].', 'cond-mat-0006258-2-3-4': 'In fact, strong discrepancies are found in many other systems ranging from PCeCO crystals to YBCO/PBCO multilayers or YBCO and BSCCO films and crystals [CITATION].', 'cond-mat-0006258-2-3-5': 'One of the problem is [CITATION] that the length of the tunnelling vortex segment [MATH] needed to fit the creep rate data, [MATH], can be orders of magnitude larger that the one theoretically predicted by quantum creep theory [CITATION].', 'cond-mat-0006258-2-3-6': 'Also suspicious is that the experimental temperature dependence of the creep rate, [MATH], is often very different from the one predicted by quantum theory (typically linear instead of quadratic, see for instance [CITATION]).', 'cond-mat-0006258-2-4-0': 'The above contradictory results and the idea that some general simple mechanism might still be present in all the above materials naturally suggests to look for other descriptions of the anomalous low [MATH] magnetic relaxation.', 'cond-mat-0006258-2-4-1': 'It is worth to stress that the observation of a non-vanishing constant creep rate in the limit [MATH] is found under very general circumstances: it does not crucially depend on the thickness of the sample [CITATION] (i.e. on its dimensionality), nor on whether the pinning is caused by columnar defects or random point pins [CITATION].', 'cond-mat-0006258-2-4-2': 'Thus, the mechanism behind the low temperature creep seems to be of a fundamental and basic nature.', 'cond-mat-0006258-2-5-0': 'We demonstrate below that also in "classical" system (i.e., not "quantum") logarithmically slow glassy dynamics can naturally persist even at vanishing temperature and can lead to the experimentally observed phenomenology.', 'cond-mat-0006258-2-5-1': 'This is possible because the low [MATH] limit of glassy dynamics consists of searching, among a very large number, for a few "downhill" or "flat" directions in the free energy landscape.', 'cond-mat-0006258-2-5-2': 'The number of these directions decreases as relaxation proceeds though there always remain some.', 'cond-mat-0006258-2-5-3': 'They can be found only by collective cooperative rearrangements of the system, resulting in a slowing down of relaxation.', 'cond-mat-0006258-2-5-4': 'To illustrate this scenario we make use a simple model which was shown to reproduce a broad range of experimental facts of vortex dynamics.', 'cond-mat-0006258-2-6-0': 'The model - We study a finite temperature extension of a Multiple Occupancy cellular-automaton-like Model recently introduced by Bassler and Paczuski (BP) [CITATION].', 'cond-mat-0006258-2-6-1': 'Our extended model is based on an Hamiltonian formulation of vortex interactions and we call it a Restricted Occupancy Model (ROM) [CITATION] since it correctly takes into account the finiteness of the upper critical magnetic field.', 'cond-mat-0006258-2-6-2': 'The model is able to reproduce a large set of essential features of vortex phenomenology; it correctly predicts the reentrant nature of the equilibrium phase diagram, magnetisation hysteretic loops with "anomalous" second peak, the relative behaviour of the magnetisation and the sweep creep rate, logarithmic relaxation, Bean profiles and more [CITATION].', 'cond-mat-0006258-2-6-3': 'The model also predicts the existence of a "glassy region" at low temperature with strong "aging" effects [CITATION].', 'cond-mat-0006258-2-7-0': 'Here we use the ROM model to study the magnetic relaxation rate, [MATH], in the very low temperature limit.', 'cond-mat-0006258-2-7-1': 'Interestingly, the model reproduces the experimental "anomalous" relaxation rates [CITATION].', 'cond-mat-0006258-2-8-0': 'Like the BP model the ROM model is an attempt to identify the essential degrees of freedom responsible for the experimental observations.', 'cond-mat-0006258-2-8-1': 'A system of straight parallel vortex lines is coarse grained in the [MATH]-plane by introducing a square grid of lattice spacing [MATH] of the order of the London penetration length [CITATION].', 'cond-mat-0006258-2-8-2': 'The number of vortices on plaquette number [MATH] is denoted by [MATH].', 'cond-mat-0006258-2-8-3': 'The occupancy of each plaquette is a number larger than zero and, importantly, smaller than [MATH], where [MATH] is the upper critical magnetic field and [MATH] is the magnetic flux quantum.', 'cond-mat-0006258-2-8-4': 'The vortex interaction is summarised by the following Hamiltonian: [MATH].', 'cond-mat-0006258-2-8-5': 'The first two terms describe the repulsion between the vortices and their self energy.', 'cond-mat-0006258-2-8-6': 'On-site and nearest neighbour interactions are included, i.e., the interaction between vortex lines with a separation greater than the London screening length (which by definition is close to [MATH]) is ignored.', 'cond-mat-0006258-2-8-7': 'We choose [MATH]; [MATH] if [MATH] and [MATH] are nearest neighbours and [MATH] otherwise.', 'cond-mat-0006258-2-8-8': 'The last term in [MATH] describes a delta-distributed random pinning [MATH].', 'cond-mat-0006258-2-8-9': 'Interestingly, the creep phenomena we describe below depend on the presence of pinning in the system, but the general scenario does not depend crucially on the pinning (a fact in correspondence with experimental results [CITATION]).', 'cond-mat-0006258-2-8-10': 'In our model [MATH] sets the energy scale.', 'cond-mat-0006258-2-8-11': 'Below we choose [MATH]; [MATH]; [MATH]; [MATH].', 'cond-mat-0006258-2-9-0': 'The ROM model is explained in all its details in Ref. [CITATION], here we only notice that a theoretical justification of the form of this Hamiltonian consist in the observation that the repulsive interaction between vortices of spatial separation less than [MATH] is included and that the existence of the upper critical field is represented by the constraint [MATH].', 'cond-mat-0006258-2-9-1': 'More compelling is though the fact that mean field replica theory and Monte Carlo simulations of the dynamics and thermodynamics of this Hamiltonian are able to reproduce the experimental phenomenology [CITATION].', 'cond-mat-0006258-2-10-0': 'The relaxation of the model is simulated by use of Monte Carlo dynamics on a square lattice in presence of a thermal bath of temperature [MATH].', 'cond-mat-0006258-2-10-1': 'The system is periodic in the [MATH]-direction.', 'cond-mat-0006258-2-10-2': 'The two edges parallel to the [MATH]-direction are in contact with a vortex reservoir, described by the above Hamiltonian without pinning.', 'cond-mat-0006258-2-10-3': 'Particles can enter and exit the system only through the reservoir, which plays the role of the external magnetic field.', 'cond-mat-0006258-2-10-4': 'Hence the reservoir density, [MATH], is used as the external control parameter.', 'cond-mat-0006258-2-10-5': 'We perform the following zero field cooled experiment: at a low temperature [MATH] we increase at constant rate [MATH] the reservoir density from zero to a working value [MATH].', 'cond-mat-0006258-2-10-6': 'We keep [MATH] fixed while we monitor the time dependence of the magnetisation [MATH].', 'cond-mat-0006258-2-10-7': 'Here [MATH] is the vortex density inside the system (of size [MATH]).', 'cond-mat-0006258-2-10-8': 'Below usually [MATH], but we checked our results in the size range [MATH].', 'cond-mat-0006258-2-10-9': 'Time is measured in units of a full lattice update.', 'cond-mat-0006258-2-10-10': 'The data presented below are averaged over 128 realizations of the pinning background.', 'cond-mat-0006258-2-11-0': 'In particular, we investigate the creep rate [EQUATION] as function of [MATH], [MATH] and [MATH].', 'cond-mat-0006258-2-11-1': 'In typical experiments the nature of the [MATH] dependence of [MATH] is such that [MATH] decreases in time.', 'cond-mat-0006258-2-11-2': 'So usually, one deals with an average creep rate, [MATH], in some given temporal window [CITATION].', 'cond-mat-0006258-2-11-3': 'In our model dynamics, [MATH] at low temperatures behaves according to the known logarithmic interpolation formula (see Ref. [CITATION]) found in experiments [CITATION], namely: [MATH].', 'cond-mat-0006258-2-12-0': 'Consistently, we define [MATH] as the average value of [MATH] in the last time decade of our measures (i.e., for [MATH]).', 'cond-mat-0006258-2-12-1': 'The present choice is the most natural one and we stress that the general results presented below do not depend on it.', 'cond-mat-0006258-2-12-2': 'It is worth noting that experimentalists are forced to make similar choices concerning the specific way [MATH] is measured [CITATION].', 'cond-mat-0006258-2-13-0': 'For the reasons explained in the introduction, a very important physical quantity is the distribution, [MATH], of the energy barriers, [MATH], that vortices segments meet during their motion.', 'cond-mat-0006258-2-13-1': 'Since the equilibration times at low temperature are huge, the system is typically off-equilibrium and [MATH] is itself a (logarithmically slow) function of [MATH].', 'cond-mat-0006258-2-13-2': 'Consistently with the above definition of [MATH], we consider the energy barrier distribution averaged over the last time decade of our measurements.', 'cond-mat-0006258-2-14-0': 'Results - When the temperature is very low, the model exhibits the same kind of "anomalous" creep found in the experiments on superconductors.', 'cond-mat-0006258-2-14-1': 'In Fig.[REF], we plot the creep rate, [MATH], as a function of [MATH] in a broad temperature range.', 'cond-mat-0006258-2-14-2': 'For comparison we present equivalent experimental measurements in a BSCCO single crystal (from Ref.[CITATION]) as inset.', 'cond-mat-0006258-2-14-3': 'The numerical values found for [MATH] at low [MATH] in our model and in real samples are interestingly very similar.', 'cond-mat-0006258-2-14-4': 'The temperature scales of the simulations and of real experiments can be compared by considering that the ratio [MATH] in our model is of the same order of magnitude as [MATH] in a real superconductor.', 'cond-mat-0006258-2-14-5': 'This is seen from a comparison of the [MATH] equilibrium phase diagram of our model with the equilibrium temperature-magnetic field, [MATH], phase diagram of, say, a BSCCO superconductor (see Ref. [CITATION]).', 'cond-mat-0006258-2-15-0': 'Apparently, the simulations and the measurements on superconductors exhibit very similar behaviour.', 'cond-mat-0006258-2-15-1': 'In both cases [MATH] approaches a finite value, [MATH], when [MATH].', 'cond-mat-0006258-2-15-2': 'In particular, we find that a linear fit of [MATH] in the low [MATH] regime is very satisfactory (see Fig. [REF]): [EQUATION] where both [MATH] and [MATH] are a function of the applied field [MATH].', 'cond-mat-0006258-2-15-3': 'For a given value of [MATH], the above fit can be improved by including higher powers of [MATH].', 'cond-mat-0006258-2-15-4': 'We also note that in the present model [MATH] is non monotonous in [MATH]: in the higher [MATH] region it starts decreasing.', 'cond-mat-0006258-2-15-5': 'This is also a known experimental fact [CITATION].', 'cond-mat-0006258-2-16-0': 'It is interesting that a very similar numerical value for [MATH] is found in both model and experimental data.', 'cond-mat-0006258-2-16-1': 'Experimentally, for many different systems, [MATH] values are typically around the interval [MATH] [CITATION].', 'cond-mat-0006258-2-16-2': 'In our model (for the examined value [MATH]) we find, for instance, a similar range of values by varying the applied field (see Fig. [REF]).', 'cond-mat-0006258-2-16-3': 'In particular, [MATH] seems to decrease on average by increasing the field [MATH].', 'cond-mat-0006258-2-16-4': 'The overall behaviour can be approximately interpolated with a power law: [MATH], where, for [MATH], [MATH] and [MATH].', 'cond-mat-0006258-2-16-5': 'As shown in Fig. [REF], the presence of a small exponent [MATH] implies that [MATH] is slowly varying with [MATH]: sensible variations in [MATH] can be seen only by changing [MATH] of orders of magnitude.', 'cond-mat-0006258-2-16-6': 'Note that the dips in the [MATH] versus [MATH] data in Fig. [REF] at certain values of [MATH] (namely around 3, 13, and 18) are statistically significant.', 'cond-mat-0006258-2-16-7': 'Their origin is currently under investigation.', 'cond-mat-0006258-2-17-0': 'In the inset of Fig. [REF], we show that [MATH] is essentially independent of the ramping rate, [MATH] (the values shown are for [MATH] and [MATH]).', 'cond-mat-0006258-2-17-1': 'This is an other typical experimental observation [CITATION].', 'cond-mat-0006258-2-17-2': 'However, a very small decrease of [MATH] with increasing [MATH] cannot be excluded: we show a fit to the form [MATH], with [MATH].', 'cond-mat-0006258-2-17-3': 'The fact that [MATH] is practically independent on [MATH], far from being a proof of the presence of equilibrium in the system, is due to the fact that at very low temperatures the characteristic equilibration time, [MATH], is enormous (see Ref. [CITATION]).', 'cond-mat-0006258-2-17-4': 'So whenever the driving rate, [MATH], is much larger than [MATH] the off-equilibrium state and dynamics of the system are essentially independent of [MATH].', 'cond-mat-0006258-2-17-5': 'Stronger [MATH] effects have to be expected when [MATH] gets closer to [MATH].', 'cond-mat-0006258-2-17-6': 'In fact, it is well known experimentally that at higher temperatures the systems exhibit strong [MATH] dependent "memory" effects [CITATION], the signature of off-equilibrium dynamics.', 'cond-mat-0006258-2-18-0': 'The above scenario is clarified by the analysis of the energy barrier distribution function, [MATH], recorded during the system evolution at very low [MATH].', 'cond-mat-0006258-2-18-1': 'Such a quantity also clearly shows the simple mechanical origin of the anomalous creep found at very low temperature in the present model.', 'cond-mat-0006258-2-18-2': 'The function [MATH] (where [MATH] is in units of [MATH]), recorded at [MATH], is plotted in Fig. [REF] for two values of the applied field, [MATH].', 'cond-mat-0006258-2-18-3': 'We always find that [MATH] has support also on the negative axis.', 'cond-mat-0006258-2-18-4': 'This is the mark of the off-equilibrium nature of the evolution on the observed time scales, because, by definition, in mechanical equilibrium only non-negative energy barriers should be found.', 'cond-mat-0006258-2-18-5': 'The presence of a [MATH] which extends down to negative values also clearly explains the presence of the recorded relaxation at low [MATH]: in the configuration space the system can still find directions where no positive barriers have to be crossed.', 'cond-mat-0006258-2-18-6': 'The insert in Fig. [REF] clarifies the mechanism behind the relaxation.', 'cond-mat-0006258-2-18-7': 'Here, we plot the signal [MATH] defined, for each single Monte Carlo (MC) step [MATH], in the following way: [MATH] if the MC trial is rejected; [MATH] if the trial is accepted and the energy reduced, i.e. [MATH]; and finally [MATH] when a trial is accepted with [MATH].', 'cond-mat-0006258-2-18-8': 'We plot two sequences of trials.', 'cond-mat-0006258-2-18-9': 'One for [MATH] was measured at the early stage of the relaxation, the second sequence, placed at the interval [MATH], is measured at the late stage of the relaxation.', 'cond-mat-0006258-2-18-10': 'Most trials are rejected ([MATH]) and only once in a while the system does manage to find a route pointing downhill in the energy landscape.', 'cond-mat-0006258-2-18-11': 'Since the temperature is so low the system never manage to be activated over a positive barrier, i.e., [MATH] never occurs.', 'cond-mat-0006258-2-18-12': 'As time proceeds fewer and fewer "negative channels" are available to the relaxation and one observes the decrease in the density of the spikes in [MATH].', 'cond-mat-0006258-2-19-0': 'As the temperature is increased thermal activation over positive energy barriers will become possible as the Arrhenius factor [MATH] assumes a non-vanishing value for an appreciable range of barrier values [MATH].', 'cond-mat-0006258-2-19-1': 'When this happens the relaxation will occur sufficiently fast to allow one, within the experimental time window, to approach very near to the equilibrium configuration where the vortex density profile is more or less flat and relaxation ceases, hence [MATH] goes down, as seen in experiments [CITATION] and in Fig. 1.', 'cond-mat-0006258-2-20-0': 'Finally, we stress that off equilibrium relaxation at very low temperatures (below the "freezing" point) is also observed in Molecular Dynamics simulations of glass forming liquids [CITATION].', 'cond-mat-0006258-2-20-1': 'In that cases too, no activation over barriers occurs and the system simply wanders in its very high dimensional phase space through the few channels where no energy increase is required.', 'cond-mat-0006258-2-21-0': 'Discussion - We have above demonstrated that the phenomenological behaviour of the creep rate at low temperatures can be understood in terms of the off equilibrium nature of the inhomogeneous vortex density profile produced in magnetic creep experiments.', 'cond-mat-0006258-2-21-1': 'In fact, cooperative mechanical rearrangements, possible even at very low [MATH] (where thermal activation over positive barriers can be negligible), dominate the phenomenon.', 'cond-mat-0006258-2-21-2': "In this perspective, it is very important to stress that the system's equilibration time at very low temperature is much larger than any experimentally accessible time window [CITATION].", 'cond-mat-0006258-2-21-3': 'Accordingly, we can say that experimental findings do not enforce the interpretation in terms of macroscopic quantum tunnelling of vortices.', 'cond-mat-0006258-2-22-0': 'Relaxation due to quantum tunnelling might be present along with the mechanical relaxation discussed above.', 'cond-mat-0006258-2-22-1': 'It is then important to ask how compelling the quantum tunnelling interpretation is.', 'cond-mat-0006258-2-22-2': 'The theory of quantum tunnelling assumes a London picture and treats the position of the vortex core as the variable that is able to tunnel.', 'cond-mat-0006258-2-22-3': 'Vortex positions are the loci of the zeros of the Ginzburg-Landau order parameter, which in itself is a mean field approximation to microscopic theories like Gorkov or Bogoliubov-de Gennes theory [CITATION].', 'cond-mat-0006258-2-22-4': 'It is not entirely clear if this is the right level at which to introduce quantum fluctuations, but, more importantly, the quantum tunnelling description assume the existence of a characteristic energy barrier [CITATION].', 'cond-mat-0006258-2-22-5': 'The scale of such a barrier may be estimated from single pining or from collective pinning arguments, but in any case a mean field-like approach is assumed, where the fluctuations in the sizes of the activation barriers, arising from many-body effects in vortex-vortex or vortex-pin interactions, are neglected.', 'cond-mat-0006258-2-22-6': 'Finally, the quantum tunnelling description also tacitly assumes the existence of a static equilibrium state in which barriers are always positive.', 'cond-mat-0006258-2-22-7': 'As we have clearly shown above, this is typically not the case and a dynamical approach is more appropriate.', 'cond-mat-0006258-2-23-0': 'We thank M. Paczuski for stimulating discussions.', 'cond-mat-0006258-2-23-1': 'Work supported by the EPSRC and PRA-INFM-99.'} | [['cond-mat-0006258-1-11-0', 'cond-mat-0006258-2-11-0'], ['cond-mat-0006258-1-11-1', 'cond-mat-0006258-2-11-1'], ['cond-mat-0006258-1-11-2', 'cond-mat-0006258-2-11-2'], ['cond-mat-0006258-1-11-3', 'cond-mat-0006258-2-11-3'], ['cond-mat-0006258-1-4-0', 'cond-mat-0006258-2-4-0'], ['cond-mat-0006258-1-4-1', 'cond-mat-0006258-2-4-1'], ['cond-mat-0006258-1-4-2', 'cond-mat-0006258-2-4-2'], ['cond-mat-0006258-1-12-0', 'cond-mat-0006258-2-12-0'], ['cond-mat-0006258-1-12-1', 'cond-mat-0006258-2-12-1'], ['cond-mat-0006258-1-12-2', 'cond-mat-0006258-2-12-2'], ['cond-mat-0006258-1-2-0', 'cond-mat-0006258-2-2-0'], ['cond-mat-0006258-1-2-1', 'cond-mat-0006258-2-2-1'], ['cond-mat-0006258-1-2-2', 'cond-mat-0006258-2-2-2'], ['cond-mat-0006258-1-2-3', 'cond-mat-0006258-2-2-3'], ['cond-mat-0006258-1-2-4', 'cond-mat-0006258-2-2-4'], ['cond-mat-0006258-1-17-0', 'cond-mat-0006258-2-17-0'], 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'cond-mat-0006258-1-1-4', 'cond-mat-0006258-1-6-3', 'cond-mat-0006258-1-8-13', 'cond-mat-0006258-2-1-0', 'cond-mat-0006258-2-1-1', 'cond-mat-0006258-2-1-2', 'cond-mat-0006258-2-1-3', 'cond-mat-0006258-2-1-4', 'cond-mat-0006258-2-8-11', 'cond-mat-0006258-3-1-0', 'cond-mat-0006258-3-1-1', 'cond-mat-0006258-3-1-2', 'cond-mat-0006258-3-1-3', 'cond-mat-0006258-3-1-4', 'cond-mat-0006258-3-2-0', 'cond-mat-0006258-3-9-11', 'cond-mat-0006258-3-18-11', 'cond-mat-0006258-3-24-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/cond-mat/0006258 | {'cond-mat-0006258-3-0-0': 'We theoretically study the creep of vortex matter in superconductors.', 'cond-mat-0006258-3-0-1': 'The low temperatures experimental phenomenology, previously interpreted in terms of "quantum tunnelling" of vortices, is reproduced by Monte Carlo simulations of a purely "classical" vortex model.', 'cond-mat-0006258-3-0-2': 'We demonstrate that a non-zero creep rate in the limit of vanishing temperature is to be expected in systems with slow relaxations as a consequence of their off-equilibrium evolution in a complex free energy landscape.', 'cond-mat-0006258-3-1-0': 'PACS numbers: 74.60.', 'cond-mat-0006258-3-1-1': 'Ge 47.32.', 'cond-mat-0006258-3-1-2': 'Cc 74.50.', 'cond-mat-0006258-3-1-3': '+r 75.45.', 'cond-mat-0006258-3-1-4': '+j', 'cond-mat-0006258-3-2-0': ']', 'cond-mat-0006258-3-3-0': 'There exist an abundance of experimental evidence that the relaxation rate of the magnetisation in type II superconductors (ranging from conventional, to organic, heavy fermions and a variety of [MATH] superconductors [CITATION]) does not vanish as the temperature [MATH] is lowered towards zero (see eg. [CITATION]).', 'cond-mat-0006258-3-3-1': 'This may seem surprising if one assumes that the mechanism allowing the magnetisation to relax is thermal activation over a characteristic energy barrier [MATH].', 'cond-mat-0006258-3-3-2': 'Namely, when [MATH] the relaxation rate should vanish since the Arrhenius factor for thermal activation, [MATH], goes to zero.', 'cond-mat-0006258-3-3-3': 'The question then arises how relaxation can take place at a finite rate while the thermal activation factor exponentially approaches zero.', 'cond-mat-0006258-3-4-0': 'A number of researchers have suggested that such a phenomenon is caused by quantum tunnelling of vortices through the barriers in the random pinning potential (for a theoretical review see [CITATION]).', 'cond-mat-0006258-3-4-1': 'The above "quantum" explanation is very intriguing and in good agreement with some experimental results in compounds such as YBCO [CITATION] or BSCCO [CITATION].', 'cond-mat-0006258-3-4-2': 'Also other even more exotic materials, such as organic superconductors [CITATION], give good correspondences.', 'cond-mat-0006258-3-4-3': 'However, in other non conventional systems such as heavy fermion superconductors, the theory of quantum creep is totally unable to describe the observed low [MATH] relaxation [CITATION].', 'cond-mat-0006258-3-4-4': 'In fact, strong discrepancies are found in many other systems ranging from PCeCO crystals to YBCO/PBCO multilayers or YBCO and BSCCO films and crystals [CITATION].', 'cond-mat-0006258-3-4-5': 'One problem is [CITATION] that the length of the tunnelling vortex segment [MATH] needed to fit the creep rate data, [MATH], can be orders of magnitude larger that the one theoretically predicted by quantum creep theory [CITATION].', 'cond-mat-0006258-3-4-6': 'Also the experimental temperature dependence of the creep rate, [MATH], is often very different from the one predicted by quantum theory (see [CITATION]).', 'cond-mat-0006258-3-5-0': 'The above contradictory results suggests to look for additional descriptions of the anomalous low [MATH] magnetic relaxation.', 'cond-mat-0006258-3-5-1': 'It is worth to stress that the observation of a non-vanishing constant creep rate in the limit [MATH] is found under very general circumstances: it does not crucially depend on the thickness of the sample [CITATION] (i.e. on its dimensionality), nor on whether the pinning is caused by columnar defects or random point pins [CITATION].', 'cond-mat-0006258-3-5-2': 'Thus, the mechanism behind the low temperature creep seems to be of a fundamental and basic nature.', 'cond-mat-0006258-3-6-0': 'We demonstrate below that also in a "classical" system (i.e., not "quantum") logarithmically slow glassy dynamics can naturally persist even at vanishing temperatures and can lead to the experimentally observed phenomenology.', 'cond-mat-0006258-3-6-1': 'This is possible because the low [MATH] off-equilibrium dynamics consists of searching, among a very large number, for a few "downhill" or "flat" directions in the free energy landscape.', 'cond-mat-0006258-3-6-2': 'The number of these directions decreases as relaxation proceeds though there always remain some.', 'cond-mat-0006258-3-6-3': 'They can be found only by collective cooperative rearrangements of the system, resulting in a slowing down of relaxation [CITATION].', 'cond-mat-0006258-3-7-0': 'The model - We study a statistical mechanics model for vortex matter called a Restricted Occupancy Model (ROM) [CITATION].', 'cond-mat-0006258-3-7-1': 'In the limit of zero temperature and infinite upper critical field, it reduces to a cellular automaton introduced in [CITATION] to study vortex avalanches.', 'cond-mat-0006258-3-7-2': 'We use Monte Carlo dynamics which enabled the ROM to depict a unified picture of creep and transport phenomena in vortex physics, ranging from magnetisation loops with "anomalous" second peak, logarithmic relaxation, Bean profiles, to history dependent behaviours in vortex flow and I-V characteristics, to the reentrant nature of the equilibrium phase diagram [CITATION].', 'cond-mat-0006258-3-7-3': 'The model also predicts the existence of a "glassy region" at low temperature with strong "aging" effects [CITATION].', 'cond-mat-0006258-3-8-0': 'Here we use the ROM to study the magnetic relaxation rate, [MATH], in the very low temperature limit.', 'cond-mat-0006258-3-8-1': 'Interestingly, the model reproduces the experimental "anomalous" relaxation and the observed behaviour of [MATH] [CITATION].', 'cond-mat-0006258-3-9-0': 'The ROM model is described in full details in Ref. [CITATION].', 'cond-mat-0006258-3-9-1': 'A system of straight parallel vortex lines is coarse grained in the [MATH]-plane by introducing a square grid of lattice spacing [MATH] of the order of the London penetration length, [MATH] [CITATION].', 'cond-mat-0006258-3-9-2': 'The number of vortices on the [MATH]-th coarse grained plaquette is denoted by [MATH].', 'cond-mat-0006258-3-9-3': 'The occupancy of each plaquette is a number larger than zero and, importantly, smaller than [MATH], where [MATH] is the upper critical magnetic field and [MATH] is the magnetic flux quantum.', 'cond-mat-0006258-3-9-4': 'The ROM model is thus defined by the following coarse grained vortex interaction Hamiltonian [CITATION]: [MATH].', 'cond-mat-0006258-3-9-5': 'The first two terms describe the repulsion between the vortices and their self energy.', 'cond-mat-0006258-3-9-6': 'On-site and nearest neighbour interactions are included, i.e., the interaction between vortex lines with a separation greater than the London screening length (which by definition is close to [MATH]) is ignored.', 'cond-mat-0006258-3-9-7': 'We choose [MATH]; [MATH] if [MATH] and [MATH] are nearest neighbours and [MATH] otherwise.', 'cond-mat-0006258-3-9-8': 'The last term in [MATH] describes a delta-distributed random pinning [MATH].', 'cond-mat-0006258-3-9-9': 'Interestingly, the general scenario of creep phenomena we describe below does not depend on the details of pinning in the system (a fact in correspondence with experimental results [CITATION]).', 'cond-mat-0006258-3-9-10': 'In our model [MATH] sets the energy scale.', 'cond-mat-0006258-3-9-11': 'Below we choose [MATH]; [MATH]; [MATH]; [MATH]; [MATH].', 'cond-mat-0006258-3-10-0': 'The relaxation of the model is simulated by use of Monte Carlo dynamics on a square lattice in presence of a thermal bath of temperature [MATH].', 'cond-mat-0006258-3-10-1': 'The system is periodic in the [MATH]-direction.', 'cond-mat-0006258-3-10-2': 'The two edges parallel to the [MATH]-direction are in contact with a vortex reservoir.', 'cond-mat-0006258-3-10-3': 'Particles can enter and exit the system only through the reservoir, which plays the role of the external magnetic field.', 'cond-mat-0006258-3-10-4': 'Hence the reservoir density, [MATH], is used as the external control parameter.', 'cond-mat-0006258-3-10-5': 'We perform the following zero field cooled experiment: at a low temperature [MATH] we increase at constant rate [MATH] the reservoir density from zero to a working value [MATH].', 'cond-mat-0006258-3-10-6': 'We keep [MATH] fixed while we monitor the time dependence of the magnetisation [MATH].', 'cond-mat-0006258-3-10-7': 'Here [MATH] is the vortex density inside the system (of size [MATH] [CITATION]).', 'cond-mat-0006258-3-10-8': 'Time is measured in units of single attempted updates per degrees of freedom of the lattice (see Ref. [CITATION]).', 'cond-mat-0006258-3-10-9': 'The data presented below are averaged over 128 realizations of the pinning background.', 'cond-mat-0006258-3-11-0': 'In particular, we investigate the creep rate [EQUATION] as function of [MATH], [MATH] and [MATH].', 'cond-mat-0006258-3-11-1': 'In typical experiments the nature of the [MATH] dependence of [MATH] is such that [MATH] decreases in time.', 'cond-mat-0006258-3-11-2': 'So usually, one deals with an average creep rate, [MATH], in some given temporal window [CITATION].', 'cond-mat-0006258-3-11-3': 'As shown in the upper inset of Fig. [REF], in our model dynamics [MATH] at low temperatures behaves according to the known logarithmic interpolation formula (see Ref. [CITATION]) found in experiments [CITATION], namely: [MATH].', 'cond-mat-0006258-3-11-4': 'Here, [MATH] is the magnetisation at the time of preparation of the sample ([MATH]), [MATH] its overall variation, the exponent [MATH] is consistent with 1, [MATH] and [MATH] are fit parameters [CITATION].', 'cond-mat-0006258-3-12-0': 'Consistently, we define [MATH] as the average value of [MATH] in the last time decade of our measures (i.e., for [MATH]).', 'cond-mat-0006258-3-12-1': 'The present choice, analogous to those made for experimental data, is the most natural one and the general results presented below do not depend on it.', 'cond-mat-0006258-3-13-0': 'For the reasons explained in the introduction, a very important physical quantity is the distribution, [MATH], of the energy barriers, [MATH], that vortices segments meet during their motion.', 'cond-mat-0006258-3-13-1': 'Since at low [MATH] the system is typically off-equilibrium, [MATH] is itself a (logarithmically slow) function of [MATH] and we consider its average over the last time decade of our measurements.', 'cond-mat-0006258-3-14-0': 'Results - When the temperature is very low, the model exhibits the same kind of "anomalous" creep found in the experiments on superconductors.', 'cond-mat-0006258-3-14-1': 'In Fig.[REF], we plot the creep rate, [MATH], as a function of [MATH] in a broad temperature range.', 'cond-mat-0006258-3-14-2': 'For comparison we present equivalent experimental measurements in a BSCCO single crystal (from Ref.[CITATION]) as inset.', 'cond-mat-0006258-3-14-3': 'The numerical values found for [MATH] at low [MATH] in our model and in real samples are interestingly very similar.', 'cond-mat-0006258-3-14-4': 'The temperature scales of the simulations and of real experiments can be compared by considering that the ratio [MATH] in our model is of the same order of magnitude as [MATH] in a real superconductor.', 'cond-mat-0006258-3-14-5': 'This is seen from a comparison of the [MATH] equilibrium phase diagram of our model with the equilibrium temperature-magnetic field, [MATH], phase diagram of, say, a BSCCO superconductor (see Ref. [CITATION]).', 'cond-mat-0006258-3-15-0': 'In both experiments and simulations, [MATH] approaches a finite value, [MATH], when [MATH].', 'cond-mat-0006258-3-15-1': 'In particular, we find that a linear fit of [MATH] in the low [MATH] regime is very satisfactory (see Fig. [REF]): [EQUATION] where both [MATH] and [MATH] are a function of the applied field [MATH].', 'cond-mat-0006258-3-15-2': 'We also note that in the present model [MATH] is non monotonous in [MATH]: in the higher [MATH] region it starts decreasing.', 'cond-mat-0006258-3-15-3': 'This is also a known experimental fact [CITATION], we discuss it later on.', 'cond-mat-0006258-3-15-4': 'The maximum in [MATH] is just above a characteristic crossover "glassy" temperature, [MATH], defined in [CITATION].', 'cond-mat-0006258-3-16-0': 'In our model by varying the applied field we find a range of values for [MATH] very similar to experimental ones [CITATION] (see Fig. [REF]).', 'cond-mat-0006258-3-16-1': 'In particular, [MATH] seems to decrease on average by increasing the field [MATH].', 'cond-mat-0006258-3-16-2': 'The overall behaviour can be approximately interpolated with a power law: [MATH], where, for [MATH], [MATH] and [MATH].', 'cond-mat-0006258-3-16-3': 'As shown in Fig. [REF], the presence of a small exponent [MATH] implies that sensible variations in [MATH] can be seen only by changing [MATH] of orders of magnitude.', 'cond-mat-0006258-3-16-4': 'Note that the dips in the [MATH] versus [MATH] data in Fig. [REF] at certain values of [MATH] (namely around 3, 13, and 18) are statistically significant.', 'cond-mat-0006258-3-16-5': 'They are related to the low field order-disorder transition, the 2nd peak transition and the reentrant high field order-disorder transition respectively [CITATION].', 'cond-mat-0006258-3-17-0': 'In the lower inset of Fig. [REF], we show that [MATH] is essentially independent of the ramping rate, [MATH] (the values shown are for [MATH] and [MATH]).', 'cond-mat-0006258-3-17-1': 'This is an other typical experimental observation [CITATION].', 'cond-mat-0006258-3-17-2': 'However, a very small decrease of [MATH] with increasing [MATH] cannot be excluded: we show a fit to the form [MATH], with [MATH] and [MATH].', 'cond-mat-0006258-3-17-3': 'The fact that [MATH] is practically independent on [MATH], far from being a proof of the presence of equilibrium in the system, is due to the fact that at very low temperatures the characteristic equilibration time, [MATH], is enormous (see Ref. [CITATION]).', 'cond-mat-0006258-3-17-4': 'So whenever the driving rate, [MATH], is much larger than [MATH] the off-equilibrium state and dynamics of the system are essentially independent of [MATH].', 'cond-mat-0006258-3-17-5': 'Stronger [MATH] effects have to be expected when [MATH] gets closer to [MATH].', 'cond-mat-0006258-3-17-6': 'In fact, it is experimentally well known that at higher temperatures the systems exhibit strong [MATH] dependent "memory" effects [CITATION], the signature of off-equilibrium dynamics.', 'cond-mat-0006258-3-17-7': 'Actually, in the present model at low [MATH], it is possible to show that [MATH] diverges exponentially [CITATION]: [MATH].', 'cond-mat-0006258-3-17-8': 'In that region, the typical observation time windows, [MATH], are such that [MATH], and the system is in the early stage of its off-equilibrium relaxation from its initial state.', 'cond-mat-0006258-3-17-9': 'This is schematically the origin of the flattening of [MATH] at very low [MATH].', 'cond-mat-0006258-3-17-10': 'Notice that, if one could observe the system for an exponentially long time, i.e., if [MATH], then the creep rate, [MATH], would indeed go to zero.', 'cond-mat-0006258-3-18-0': 'The above scenario is clarified by the analysis of the energy barrier distribution function, [MATH], recorded during the system evolution at very low [MATH].', 'cond-mat-0006258-3-18-1': 'Such a quantity also clearly shows the simple mechanical origin of the anomalous creep found at very low temperature in the present model.', 'cond-mat-0006258-3-18-2': 'The function [MATH] (where [MATH] is in units of [MATH]), recorded at [MATH], is plotted in Fig. [REF] for two values of the applied field, [MATH].', 'cond-mat-0006258-3-18-3': 'We always find that [MATH] has support also on the negative axis.', 'cond-mat-0006258-3-18-4': 'This is the mark of the off-equilibrium nature of the evolution on the observed time scales.', 'cond-mat-0006258-3-18-5': 'The presence of a [MATH] which extends down to negative values also explains the presence of the recorded relaxation at low [MATH]: in the configuration space the system can still find directions where no positive barriers have to be crossed.', 'cond-mat-0006258-3-18-6': 'The insert in Fig. [REF] clarifies the mechanism behind the relaxation.', 'cond-mat-0006258-3-18-7': 'Here, we plot the signal [MATH] defined, for each single Monte Carlo (MC) step [MATH], in the following way: [MATH] if the MC trial is rejected; [MATH] if the trial is accepted and the energy reduced, i.e. [MATH]; and finally [MATH] when a trial is accepted with [MATH].', 'cond-mat-0006258-3-18-8': 'We plot two sequences of trials.', 'cond-mat-0006258-3-18-9': 'One for [MATH] was measured at the early stage of the relaxation, the second sequence, placed at the interval [MATH], is measured at the late stage of the relaxation.', 'cond-mat-0006258-3-18-10': 'Most trials are rejected ([MATH]) and only once in a while the system does manage to find a route pointing downhill in the energy landscape.', 'cond-mat-0006258-3-18-11': '[MATH] never occurs.', 'cond-mat-0006258-3-18-12': 'As time proceeds fewer and fewer "negative channels" are available to the relaxation and a decrease in the density of the spikes in [MATH] is observed.', 'cond-mat-0006258-3-19-0': 'As the temperature is increased thermal activation over positive energy barriers will become possible as the Arrhenius factor [MATH] assumes a non-vanishing value for an appreciable range of barrier values [MATH].', 'cond-mat-0006258-3-19-1': 'When this happens the relaxation will occur sufficiently fast to allow one, within the experimental time window, to closely approach the equilibrium configurations where the vortex density profile is more or less flat and relaxation ceases, hence [MATH] goes down, as seen in experiments [CITATION] and in Fig. 1.', 'cond-mat-0006258-3-20-0': 'Finally, we stress that slow off-equilibrium relaxations at very low temperatures are also observed in glass forming liquids [CITATION].', 'cond-mat-0006258-3-20-1': 'In that cases too, no activation over barriers occurs and the system simply wanders in its very high dimensional phase space through the few channels where no energy increase is required.', 'cond-mat-0006258-3-21-0': 'Discussion - We have above demonstrated that the phenomenological behaviour of the creep rate at low temperatures can be understood in terms of the off-equilibrium nature of the inhomogeneous vortex density profile produced in magnetic creep experiments.', 'cond-mat-0006258-3-21-1': 'In fact, cooperative mechanical rearrangements, possible even at very low [MATH] (where thermal activation over positive barriers can be negligible), dominate the phenomenon [CITATION].', 'cond-mat-0006258-3-21-2': "In this perspective, it is very important to stress that the system's equilibration time at very low temperature is much larger than any experimentally accessible time window [CITATION].", 'cond-mat-0006258-3-21-3': 'Accordingly, we can say that experimental findings do not enforce the interpretation in terms of macroscopic quantum tunnelling of vortices.', 'cond-mat-0006258-3-22-0': 'Relaxation due to quantum tunnelling might be present along with the mechanical relaxation discussed above.', 'cond-mat-0006258-3-22-1': 'It is then important to ask how compelling the quantum tunnelling interpretation is.', 'cond-mat-0006258-3-22-2': 'The theory of quantum tunnelling assumes a London picture and treats the position of the vortex core as the variable that is able to tunnel.', 'cond-mat-0006258-3-22-3': 'It is not entirely clear if this is the right level at which to introduce quantum fluctuations, but, more importantly, the quantum tunnelling description assume the existence of a characteristic energy barrier [CITATION], not a time dependent distribution of barrier heights as typically found in many-body systems relaxing off-equilibrium.', 'cond-mat-0006258-3-22-4': 'Finally, the quantum tunnelling description also tacitly assumes the existence of a static equilibrium state in which barriers are always positive.', 'cond-mat-0006258-3-22-5': 'As we have clearly shown above, this is typically not the case and a dynamical approach is more appropriate.', 'cond-mat-0006258-3-23-0': 'The present scenario, where off-equilibrium phenomena dominate the anomalous low [MATH] creep, could be experimentally verified by the discovery of "aging" phenomena [CITATION] like those recently observed in [CITATION].', 'cond-mat-0006258-3-24-0': 'Work supported by EPSRC, INFM-PRA(HOP)/PCI.'} | null | null | null | null |
1211.4847 | {'1211.4847-1-0-0': 'A simple fitting method (gfit) for galaxy shape measurement in weak lensing surveys', '1211.4847-1-1-0': 'It is anticipated that the large sky areas covered by planned wide-field weak lensing surveys will reduce statistical errors to such an extent that systematic errors will instead become the dominant source of uncertainty.', '1211.4847-1-1-1': 'It is therefore crucial to devise numerical methods to measure galaxy shapes with the least possible systematic errors.', '1211.4847-1-1-2': 'We present a simple "forward deconvolution" method, gfit, to measure galaxy shapes given telescope and atmospheric smearings, in the presence of noise.', '1211.4847-1-1-3': 'The method consists in fitting a single 2D elliptical Sersic profile to the data, convolved with the point spread function.', '1211.4847-1-1-4': 'We applied gfit to the data proposed in the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Galaxy Challenge.', '1211.4847-1-1-5': 'In spite of its simplicity, gfit obtained the lowest additive bias ([MATH]) on the shear power spectrum among twelve different methods and the second lowest multiplicative bias ([MATH]).', '1211.4847-1-1-6': 'It remains that gfit is a fitting method and is therefore affected by noise bias.', '1211.4847-1-1-7': 'However, the simplicity of the underlying galaxy model combined with the use of an efficient customized minimization algorithm allow very competitive performances, at least on the GREAT10 data, for a relatively low computing time.', '1211.4847-1-2-0': '# Introduction', '1211.4847-1-3-0': 'Weak gravitational lensing , whereby the gravitational bending of light by structures in the Universe slightly distorts images of distant galaxies, is now recognized as a powerful means to study the history of the Universe and probe the mysterious nature of the dark matter and dark energy .', '1211.4847-1-4-0': 'Since the first detection of weak lensing , a number of methods have been devised and implemented to tackle the inverse problem of recovering the lensing signature from observed, distorted galaxy images .', '1211.4847-1-5-0': 'We describe in this paper gfit, a simple shear measurement method that nevertheless obtained good results in the latest GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Galaxy challenge .', '1211.4847-1-5-1': 'Galaxies are assumed to be well modeled by a seven-parameter, single-component elliptical Sersic profile.', '1211.4847-1-5-2': 'The shape measurement algorithm essentially consist in iteratively shearing and convolving the galaxy model until a sufficiently close match with the observed galaxy is reached.', '1211.4847-1-5-3': 'Instead of an out-the-box minimizer, we employ a custom-developed minimizer well suited to fitting faint and noisy images like those frequently found in weak lensing.', '1211.4847-1-6-0': 'The paper is structured as follows.', '1211.4847-1-6-1': 'We provide in Sect. [REF] a description of the underlying principles, galaxy model and shape measurement algorithm of gfit.', '1211.4847-1-6-2': 'We continue in Sect. [REF] with a presentation of the pipeline we used to participate in the GREAT10 Galaxy challenge and follow with a analysis of the gfit results in Sect. [REF].', '1211.4847-1-6-3': 'We conclude in Sect. [REF].', '1211.4847-1-7-0': '# The gfit shear measurement method', '1211.4847-1-8-0': '## The shear measurement problem', '1211.4847-1-9-0': 'According to the theory of weak gravitational lensing, the light emitted by a galaxy is slightly deviated by the foreground gravitational field, an effect that can be modeled to first order as the combination of two effects, the convergence [MATH] and the shear [MATH], that describe how light bundles emitted by a source are distorted by a potential well.', '1211.4847-1-9-1': 'The convergence models the magnification effect whereby the galaxy image see its apparent size increased without altering its shape, whereas the shear describes a stretching effect where only ellipticity is altered, not size.', '1211.4847-1-10-0': 'All so-called "shear measurement methods" attempt to reconstruct the reduced shear [MATH] which is approximately equals to the shear [MATH] in the weak gravitational limit, where [MATH] and [MATH].', '1211.4847-1-11-0': 'The lensing effect is very subtle, however, and requires measuring the shapes of thousands of faint galaxies.', '1211.4847-1-11-1': 'Moreover, before they reach the observer, the apparent galaxy images undergo a number of additional distortions, unrelated to lensing, that further complicates that task, mainly:', '1211.4847-1-12-0': 'The traditional approach for estimating the shear is to measure the deviation from circularity of a large number of galaxy shapes.', '1211.4847-1-12-1': 'But this technique assumes the shear remains constant across the field of view, which is generally not the case.', '1211.4847-1-12-2': 'Moreover, accurate shear measurement must also account for a spatially varying PSF that must be interpolated at the positions of the galaxies.', '1211.4847-1-13-0': 'A shear measurement pipeline must overcome all the above difficulties, typically going through the following steps:', '1211.4847-1-14-0': 'PSF correction, whose goal is to restore the shape a galaxy had before being convolved with the PSF.', '1211.4847-1-14-1': 'If the spatial variation of the PSF over the field of view is significant, this step also requires interpolating the PSF to the position of the galaxies in the sky.', '1211.4847-1-15-0': 'Shape measurement, that is, the estimation of the galaxy shape after it has been altered by the cosmic shear but before PSF convolution and other subsequent distortions.', '1211.4847-1-15-1': 'In this paper, we call the corresponding shape the sheared galaxy shape.', '1211.4847-1-15-2': 'The real galaxy ellipticity prior to gravitational lensing (i.e. unsheared) is referred to as the intrinsic ellipticity.', '1211.4847-1-16-0': 'Shear measurement, that is, the task of extracting the shear signal from the sheared galaxy shapes estimated in the previous step.', '1211.4847-1-16-1': 'A spatially-varying shear field is commonly described as a power spectrum or a correlation function.', '1211.4847-1-17-0': 'Additional steps may also be performed to correct the images from the effects of pixelation and noise.', '1211.4847-1-18-0': '## Shear measurement with gfit', '1211.4847-1-19-0': 'The gfit "galaxy fitting" method grew from a prototype initially developed by Stephane Paulin-Henrikson on the occasion of the GREAT08 challenge , where it obtained the third and fifth ranks on images with high and low signal to noise ratios respectively .', '1211.4847-1-20-0': 'The GREAT10 gfit code was subsequently made more generic in order to satisfy the more demanding requirements of the Galaxy challenge.', '1211.4847-1-20-1': 'It was also enhanced in several aspects that we describe in subsequent sections of this paper.', '1211.4847-1-21-0': 'It is also worth mentioning that a wavelet-based denoising algorithm described in [CITATION] was also experimented in the Galaxy challenge and proved quite successful.', '1211.4847-1-21-1': 'This denoising scheme is presented in Sect. [REF].', '1211.4847-1-22-0': 'The overall shape measurement procedure is the following:', '1211.4847-1-23-0': 'Application of the denoising algorithm on the galaxy and/or PSF images [optional] Estimation of the galaxy and PSF centroids in all images Application of the PSF correction and shape measurement algorithm Generation of the ellipticity catalogs Production of various statistics and plots for analysis [optional]', '1211.4847-1-24-0': 'We detail in the next sections the shape measurement algorithm along with its underlying models and components.', '1211.4847-1-25-0': '## Modeling the galaxies', '1211.4847-1-26-0': 'gfit is fundamentally a model-fitting method.', '1211.4847-1-26-1': 'We describe here the model used to represent galaxies and cover the fitting-related aspects in Sects. [REF] and [REF].', '1211.4847-1-27-0': 'Galaxies are assumed to have a surface brightness distribution well described by an elliptical Sersic function , defined by: [EQUATION] where:', '1211.4847-1-28-0': 'This model was initially chosen for its simplicity and has the other merit of being relatively easy to fit.', '1211.4847-1-28-1': 'A galaxy represented as the sum of a bulge and a disc would be more realistic, but it is not clear whether such a model would prove more accurate on weak lensing images and worth the additional complexity and computational cost, given degeneracies between parameters.', '1211.4847-1-28-2': 'As regards gfit, the GREAT10 results does not provide a definitive answer (see Sect. [REF]).', '1211.4847-1-29-0': '## Galaxy shape measurement and PSF correction', '1211.4847-1-30-0': 'We describe in this section the shape measurement algorithm of gfit.', '1211.4847-1-30-1': 'It is based on iterative fitting of observed galaxies to the galaxy model described in Sect. [REF].', '1211.4847-1-30-2': 'The basic assumptions of the algorithm are the following:', '1211.4847-1-31-0': 'The fitting algorithm itself is summarized below:', '1211.4847-1-32-0': 'At the end of the iteration cycle, the algorithm yields:', '1211.4847-1-33-0': 'The same procedure is followed for the next galaxy [MATH] in [MATH], until all galaxies have been processed.', '1211.4847-1-34-0': 'Once the shape of all galaxies have been measured, produce a catalog containing the fitted parameters at the positions of the galaxies [MATH] in [MATH] and optionally, additional statistics and plots.', '1211.4847-1-34-1': 'Quantities such as the ellipticity modulus [MATH], the position angle [MATH] or the minor-to-major axis ratio [MATH] can be respectively derived from [MATH] using [MATH], [MATH] and [MATH].', '1211.4847-1-35-0': 'The algorithm just described is not dependent on the galaxy or the PSF model.', '1211.4847-1-35-1': 'It would remain unchanged, for instance, if the centroid of the PSF itself was taken into account during fitting or if a bulge and a disc were incorporated in the galaxy model.', '1211.4847-1-35-2': 'The quality of the whole shape measurement procedure can thus be increased by improving the models.', '1211.4847-1-36-0': 'The shape measurement algorithm has its own strengths and weaknesses, outlined below:', '1211.4847-1-37-0': 'Weaknesses:', '1211.4847-1-38-0': '## The gfit minimizer', '1211.4847-1-39-0': 'The gfit galaxy model expressed by Eq. ([REF]) varies linearly with the parameter [MATH], but non-linearly with the remaining parameters [MATH].', '1211.4847-1-39-1': 'This requires the use of a non-linear minimization algorithm over a seven-dimensional parameter space.', '1211.4847-1-40-0': 'Unlike linear minimizations schemes that only involve a matrix inversion, non-linear optimization requires iterating over the parameter space to find the minimum value of the objective function, which is in our case the [MATH] of the residuals between observed and estimated images.', '1211.4847-1-40-1': 'That minimum is not necessarily the absolute minimum of the [MATH] function but the most relevant from the point of view of the physics of the problem.', '1211.4847-1-40-2': 'In our case, the minimum should coincide with the Sersic model parameters that best fit the galaxy shape.', '1211.4847-1-41-0': 'A good minimizer is essential to any galaxy model fitting algorithm, but finding such a minimum in a reliable manner can prove tricky for a number of reasons:', '1211.4847-1-42-0': 'In this regards, the choice of the galaxy model used by gfit leads to a number of challenges:', '1211.4847-1-43-0': 'Several families of optimization algorithms were experimented on GREAT08 and GREAT10 images: simplex (Nelder-Mead downhill), gradient descent (Powell), Newton quasi-Newton (Newton-CG, BFGS) and Levenberg-Marquardt (LVM).', '1211.4847-1-43-1': 'Descriptions of these algorithms can be found in .', '1211.4847-1-44-0': 'None of these methods proved entirely satisfactory, either failing to converge or yielding insufficient accuracy, especially on low signal to noise ratio (S/N) images.', '1211.4847-1-44-1': 'The LVM implementation from the SciPy library that we used was the fastest and the most accurate.', '1211.4847-1-44-2': 'For these reasons we used it in the GREAT08 version of gfit, but that implementation of LVM:', '1211.4847-1-45-0': 'We also experimented a parameter estimation scheme based on a Bayesian approach and implemented using the pymc Markov Chain Monte Carlo (MCMC) library of [CITATION], but that method produced less accurate estimates while being less computationally efficient.', '1211.4847-1-46-0': 'In an attempt to overcome these issues, we eventually decided to implement a custom minimizer, better suited to fitting noisy, pixelized galaxy images than vanilla minimization algorithms.', '1211.4847-1-46-1': 'We found that a scheme based on an adaptive cyclic coordinate descent algorithm (CCD) was able to produce more accurate estimates while at the same time being more robust:', '1211.4847-1-47-0': 'The CCD algorithm proved suitable for fitting without any single failure the huge number of GREAT10 galaxies.', '1211.4847-1-47-1': 'It nevertheless has a number of drawbacks, namely:', '1211.4847-1-48-0': 'The current gfit implementation can be configured to use either the CCD, LVM or MCMC-based minimizer.', '1211.4847-1-49-0': '## Centroid estimation', '1211.4847-1-50-0': 'gfit does not assume objects to be correctly centered within their postage stamps and accurate estimates for galaxy and PSF objects are required for two main reasons:', '1211.4847-1-51-0': 'gfit relies on centroid estimates obtained from the SExtractor tool .', '1211.4847-1-51-1': 'A catalog is generated with centroid information and additional data such as flux, ellipticities and position angles, that can be optionally used to set guess parameter values.', '1211.4847-1-52-0': '## Denoising', '1211.4847-1-53-0': 'Correcting astronomical images from the effect of noise has always been a challenging task.', '1211.4847-1-53-1': 'This is particularly true for galaxy images captured for weak lensing analysis because noise not only degrades the overall quality of these images but also alters the shapes of the galaxies.', '1211.4847-1-53-2': 'This causes serious difficulties to all existing shear measurement schemes that found their shear extraction algorithms on the accurate measurement of galaxy shapes.', '1211.4847-1-54-0': 'The challenge is then to correct galaxy images from noise without compromising the shear signal they encode.', '1211.4847-1-54-1': 'Unfortunately, popular denoising algorithms based on median filtering , Wiener filtering or discrete wavelet transform (DWT) are not shape preserving and do not meet that requirement.', '1211.4847-1-55-0': 'By default, gfit uses DWT-Wiener, a shape-preserving denoising technique combining DWT and Wiener filtering developed at the laboratory of astrophysics of EPFL by [CITATION].', '1211.4847-1-55-1': 'That algorithm has been experimented during the GREAT10 Galaxy challenge and was able to significantly improve the quality factors of all the shear measurement methods from EPFL that participated in the Galaxy challenge : that was the case for gfit but also for MegaLUT and TVNN .', '1211.4847-1-56-0': 'Interestingly, denoising improves the shape measurement of the three algorithms even though they are fundamentally different from each other.', '1211.4847-1-57-0': 'Beyond shape-preservation, another advantage of the DWT-Wiener algorithm lies in its ability to denoise "in one go" images containing a great number of objects, without having to individually process each object in turn.', '1211.4847-1-57-1': 'In GREAT10, for instance, DWT-Wiener was directly applied to images containing [MATH] PSF or galaxy postage stamps.', '1211.4847-1-58-0': '# Applying gfit to the GREAT10 data', '1211.4847-1-59-0': '## The GREAT10 Galaxy challenge', '1211.4847-1-60-0': 'We describe in this section the specific pipeline we used in the GREAT10 Galaxy Challenge competition that took place between December 2010 and September 2011.', '1211.4847-1-60-1': 'The aim, content and rules of the challenge have been described in the GREAT10 Handbook .', '1211.4847-1-60-2': 'In a nutshell, the main goals of the GREAT Challenges are (i) to test existing weak lensing measurements methods and (ii) to promote the development of new, more accurate, shear measurement techniques.', '1211.4847-1-61-0': 'The data consist of 24 datasets of 200 simulated galaxy images, each containing 10,000 noisy, PSF-convolved [MATH] pixel galaxy postage stamps, arranged on a [MATH] grid (see Fig. [REF]).', '1211.4847-1-61-1': 'The GREAT10 edition includes spatially-varying PSF and shear fields, contrary to its predecessor, the GREAT08 challenge , where these fields were set as constant.', '1211.4847-1-62-0': 'Each of the 24 sets is designed to evaluate the ability of competing methods to deal with galaxy or PSF fields with different properties (e.g. size, signal to noise ratio).', '1211.4847-1-62-1': 'We have reproduced in Table [REF] the main PSF and galaxy characteristic attached to each of the GREAT10 set, as specified in the Galaxy Challenge results paper , Appendix D.', '1211.4847-1-63-0': 'The sets were also classified into "Single epoch", "Multi-epoch" and "Stable single epoch", depending on whether the intrinsic ellipticities and PSF keep the same or change their spatial distribution between images in a set (see Table [REF]).', '1211.4847-1-64-0': 'Participating methods were ranked according to the following metrics :', '1211.4847-1-65-0': 'definecolorFullVariabilityrgb0.824,0.824,0.824', '1211.4847-1-66-0': '## The GREAT10 gfit implementation', '1211.4847-1-67-0': 'The GREAT10 version of gfit only implements the first two steps described in Sect. [REF], that is, PSF correction and galaxy shape measurement.', '1211.4847-1-68-0': 'In GREAT10, the estimation of the shear field (third step in Sect. [REF]) was not mandatory as participants were allowed to supply for each image a catalog of estimated galaxy ellipticities instead of a shear power spectrum : an analysis program was written by the GREAT10 team to calculate a shear power spectrum from user-supplied ellipticity catalogs.', '1211.4847-1-68-1': 'Consequently, like most other competing methods, gfit only provided its estimates in the form of a catalog of estimated ellipticities at requested positions within the images.', '1211.4847-1-68-2': 'Future version of gfit will allow the extraction of a spatially varying shear.', '1211.4847-1-69-0': 'The gfit implementation used in GREAT10 consisted of the following stages:', '1211.4847-1-70-0': 'Optional denoising of the galaxy and PSF images with the DWT-Wiener method presented in Sect. [REF].', '1211.4847-1-71-0': 'Centroid estimation using SExtractor as described in Sect. [REF].', '1211.4847-1-72-0': 'Galaxy shape measurement with the gfit program, configured to use the CCD minimization algorithm described in Sect. [REF].', '1211.4847-1-72-1': 'We used [MATH] pixel cutouts instead of the full [MATH] pixel original galaxy postage stamps.', '1211.4847-1-72-2': 'Similarly, the size of PSF postage stamps was reduced to [MATH] pixels.', '1211.4847-1-72-3': 'That decision was made in order to keep the overall computation time within acceptable limits and avoid picking-up too much noise near the borders of the postage stamps.', '1211.4847-1-73-0': 'Because of the large number of galaxies, running the pipeline on one single processor would not have allowed to meet the GREAT10 deadline.', '1211.4847-1-73-1': 'Even with a processing time per galaxy of 0.5 seconds, it would have taken about one month to complete process the full GREAT10 dataset.', '1211.4847-1-73-2': 'The ability to simultaneously run multiple program instances use of parallelism is thus imperative and all programs (denoising, SExtractor wrapper, gfit) are written to take advantage of of parallel computers through the Message Passing Interface (MPI) .', '1211.4847-1-73-3': 'When only a few processors are required, the same programs can also run on machines with symmetric multiprocessing (SMP) architecture.', '1211.4847-1-73-4': 'It took about 5 days to process the entire GREAT10 challenge images on a 64-processor machine, which corresponds to a processing time between [MATH] and [MATH] seconds per galaxy.', '1211.4847-1-74-0': 'The pipeline is implemented in Python, a programming language known for its power, flexibility and short development cycle.', '1211.4847-1-74-1': 'The usual standard Python libraries are used, notably: NumPy, SciPy, PyFITS and matplotlib.', '1211.4847-1-74-2': 'SciPy is the standard scientific library for Python and most of its functions consist of thin Python wrappers on top of fortran, C and C++ functions.', '1211.4847-1-74-3': 'SciPy takes advantage of installed optimized libraries such as LAPACK (Linear Algebra PACKage) library .', '1211.4847-1-75-0': '# Analysis of the gfit GREAT10 results', '1211.4847-1-76-0': 'We summarize and analyze in this section the main Galaxy Challenge results as far as gfit is concerned.', '1211.4847-1-76-1': 'An overview of the GREAT10 results for available participating shear measurement methods has already been performed in the GREAT10 Galaxy challenge paper .', '1211.4847-1-76-2': 'Our objective here is to provide a more detailed analysis of the gfit results.', '1211.4847-1-77-0': 'We do not, however, analyze the influence of the pixel-denoising and training calibration schemes applied in , which we leave for future investigation.', '1211.4847-1-78-0': '## Overall results', '1211.4847-1-79-0': 'The results of the best [MATH] methods that participated in the GREAT10 Galaxy Challenge are listed in Table 3 of the [CITATION] GREAT10 result paper.', '1211.4847-1-79-1': 'That list aggregates results submitted before the official challenge deadline as well as submissions made during the so-called "Post challenge", a one-week extension to the competition following the deadline.', '1211.4847-1-80-0': 'Two versions of gfit were submitted during the challenge, one named "gfit den cs" that included a denoising step using the DWT-Wiener algorithm described in Sect. [REF] and the other, simply named "gfit", that did not.', '1211.4847-1-80-1': 'The results obtained by both methods are shown in Table [REF].', '1211.4847-1-81-0': 'Although "gfit den cs" obtained superior results and was mentioned in the first versions of the GREAT10 result paper, the final version of the paper mistakenly included the metrics from "gfit" instead of those from "gfit den cs" .', '1211.4847-1-81-1': 'That error was, unfortunately, spotted only after the article was already accepted for publication and could not be corrected.', '1211.4847-1-81-2': 'The "gfit den cs" version is, however, correctly described in Appendix E5 of [CITATION].', '1211.4847-1-81-3': 'The "gfit" version is identical except that no denoising was applied to the data before applying the shape measurement algorithm.', '1211.4847-1-81-4': 'To simplify, we shorten he name "gfit den cs" to "gfit den" in the remainder of this article.', '1211.4847-1-82-0': 'It can be seen from Table [REF] that "gfit den" reaches a raw quality factor [MATH] twice as high as that of "gfit".', '1211.4847-1-82-1': 'This illustrates the gain in accuracy provided by the DWT-Wiener denoising algorithm.', '1211.4847-1-82-2': 'This is further analyzed in Sect. [REF].', '1211.4847-1-82-3': 'When the pixel-level denoising algorithm of [CITATION] is applied, the [MATH] quality factors of both "gfit" and "gfit den" are improved by a factor [MATH], "gfit den" scoring the best [MATH] of all methods ([MATH]).', '1211.4847-1-82-4': 'The training calibration further increases both [MATH] quality factors, especially that of "gfit" (two-fold increase).', '1211.4847-1-83-0': 'We have also included in Table [REF] the average additive and multiplicative biases [MATH] and [MATH] over all [MATH] sets.', '1211.4847-1-83-1': 'Comparing with Table 3 of the [CITATION] result paper, we see that "gfit" reached the lowest average additive bias ([MATH]) and the second lowest average multiplicative bias ([MATH]) of all twelve methods (see also the plot in Figure.', '1211.4847-1-83-2': '1, page 6 of that paper).', '1211.4847-1-84-0': 'We stress that, contrary to what is suggested in Sect. 4.4 of [CITATION], the low overall bias of "gfit" is intrinsic to the method and does not result from the application of a denoising step.', '1211.4847-1-84-1': 'The DWT-Wiener algorithm was only used in "gfit den", not "gfit" and actually, the average multiplicative bias of "gfit den" ([MATH]) is higher than that of "gfit".', '1211.4847-1-84-2': 'Moreover, both methods have similar additive biases.', '1211.4847-1-84-3': 'Drawing a more refined conclusion about these biases requires an analysis at individual set level, which we perform in Sect. [REF].', '1211.4847-1-85-0': '## Method accuracy', '1211.4847-1-86-0': 'In this section we use the quality factor as a measure of accuracy and assess the influence of:', '1211.4847-1-87-0': 'The quality factors scored for each individual image set are plotted on the left-hand side part of Fig. [REF], for each gfit variant.', '1211.4847-1-87-1': 'They are also quoted in Tables [REF] and [REF].', '1211.4847-1-88-0': 'Unfortunately, the shear power spectra attributed to "gfit" in the GREAT10 Galaxy Challenge paper, Figure E9, is that from the "fit2-unfold" method and is thus missing from that paper.', '1211.4847-1-88-1': 'We have therefore included in Appendix the correct "gfit" plot in Fig. [REF] and the corresponding "gfit den" plot in Fig. [REF].', '1211.4847-1-89-0': 'We focus first on the influence of galaxy and PSF features on accuracy.', '1211.4847-1-89-1': 'We leave aside the effects of DWT-Wiener denoising for now and thus base our analysis on the results of the "gfit" variant which is devoid of built-in denoising scheme.', '1211.4847-1-89-2': 'The data of interest are summarized in Figs. [REF] and [REF] and in Tables [REF], [REF].', '1211.4847-1-90-0': '### Influence of Galaxy and PSF characteristics', '1211.4847-1-91-0': '### Effect of denoising on accuracy', '1211.4847-1-92-0': 'We discuss in this section the effect on accuracy of the application of the DWT-Wiener denoising scheme described in Sect. [REF].', '1211.4847-1-92-1': 'The influence of denoising can be clearly observed by comparing the plot of "gfit den" (left) with that of "gfit" in Fig. [REF] (right).', '1211.4847-1-92-2': 'The corresponding scores are also listed in Tables [REF] and [REF].', '1211.4847-1-93-0': 'We find an average two-fold increase in accuracy, the effect being stronger on high S/N images and larger galaxies.', '1211.4847-1-93-1': 'Sets with small galaxies are only slightly improved, however.', '1211.4847-1-93-2': 'We also notice that the plots of "gfit" and "gfit den" show identical quality factors for low S/N sets 4 to 6.', '1211.4847-1-93-3': 'Further investigation showed that denoising was, by mistake, not applied on those sets.', '1211.4847-1-93-4': 'This would probably have improved the overall Q factor of "gfit den".', '1211.4847-1-94-0': 'These results strongly suggest that the DWT-Wiener algorithm really improves the overall accuracy on galaxy shape measurement.', '1211.4847-1-94-1': 'We also note that denoising does not alter the quality factor hierarchy between sets: the sets with best scores in the "gfit" plots remain the same in the "gfit den" plot.', '1211.4847-1-95-0': '## Bias analysis', '1211.4847-1-96-0': 'We investigate in this section how multiplicative and additive biases are affected by galaxy properties and the use of denoising.', '1211.4847-1-97-0': '### Influence of Galaxy and PSF characteristics', '1211.4847-1-98-0': 'To complement the results of Table [REF] relative to bias, we have plotted in Figs. [REF] and [REF] the multiplicative and additive biases of each set.', '1211.4847-1-98-1': 'As noted in Sec. [REF], "gfit den" reached the lowest average additive bias and the second lowest average multiplicative bias of all 12 twelve competing methods.', '1211.4847-1-99-0': 'Focusing on the multiplicative bias and leaving aside the effect of denoising for now, we can make a few observations from the left-hand side plot of Figs. [REF].', '1211.4847-1-100-0': 'As far as additive bias is concerned, we note the following trend:', '1211.4847-1-101-0': '### Effect of denoising on bias', '1211.4847-1-102-0': 'Comparing the plots for "gfit" and "gfit den" in Figs. [REF] and [REF], we find that the additive bias does not change significantly, keeping about the same bias values per set.', '1211.4847-1-102-1': 'In contrast, the structure of the multiplicative plots is significantly altered.', '1211.4847-1-103-0': 'As seen on the "gfit den" plot, denoising tend to introduce some amount of negative multiplicative bias on all sets.', '1211.4847-1-103-1': 'Although the amount of bias on fiducial sets remains roughly the same in absolute value, the DWT-Wiener algorithm clearly impacts the multiplicative bias relative to galaxy size, b/d ratio and turbulence.', '1211.4847-1-104-0': 'As regards galaxy size, even though the multiplicative bias on small galaxies is almost unchanged, that on large galaxies increases about six-fold.', '1211.4847-1-104-1': 'Because of that, the bias on "Smooth" galaxies, which also includes large galaxies, also increases.', '1211.4847-1-105-0': 'DWT-Wiener denoising seems to also improves the resolution of the bulge and disk components, so that "gfit" has more difficulty fitting its underlying single-component Sersic model to images with "uniform" and "offset" b/d.', '1211.4847-1-105-1': 'The effect is stronger on larger galaxies, causing a [MATH] tenfold multiplicative bias increase.', '1211.4847-1-106-0': 'Lastly, the introduction of PSF turbulence result in a [MATH] five-fold multiplicative bias degradation.', '1211.4847-1-107-0': 'Despite the degradation of multiplicative bias on some sets, the accuracy of shape measurements increases two-fold as shown by the corresponding gain in Q factor.', '1211.4847-1-107-1': 'We also note that denoising improves the results on sets that already have a high S/N.', '1211.4847-1-107-2': 'All in all, the use of denoising is thus clearly beneficial.', '1211.4847-1-108-0': '# Conclusions', '1211.4847-1-109-0': 'We have described in this paper the gfit shape measurement method, a model-fitting based on a simple Sersic galaxy model (Sect. [REF]).', '1211.4847-1-109-1': 'The method uses a custom-developed minimizer based on a "coordinate descent" algorithm that finds a local minimum with the lowest [MATH] of the residuals between true and modeled galaxy.', '1211.4847-1-110-0': 'We have also performed an analysis of our results in the GREAT10 Galaxy Challenge (Sect. [REF]).', '1211.4847-1-110-1': 'We participated in the competition with two gfit variants: "gfit den", which applied a denoising step before performing model-fitting, and "gfit", which did not use denoising.', '1211.4847-1-110-2': 'The noise removal technique employed is DWT-Wiener, a wavelet-based, shape-preserving algorithm particularly suitable for shape measurement.', '1211.4847-1-110-3': '(see Sect. [REF]).', '1211.4847-1-111-0': 'We highlight below the main conclusions of your analysis.', '1211.4847-1-112-0': 'It is interesting to see that, despite the simplicity of the galaxy model used, its results in the Galaxy Challenge established gfit as one of the four top-performing methods, both in terms of accuracy and bias.', '1211.4847-1-112-1': 'Given that the results were obtained on simulated data, this raises the question of how important having a realistic galaxy model really matters when measuring galaxy shapes from real data.', '1211.4847-1-112-2': 'Providing more clues on this question is one of the objectives of the forthcoming GREAT3 challenge.', '1211.4847-1-113-0': 'This work is supported by the Swiss National Science Foundation (SNSF).', '1211.4847-1-113-1': 'We thank the GREAT10 Coordination Team for organizing the stimulating challenge and sharing the quality factor calculation codes.', '1211.4847-1-113-2': 'GREAT10 was sponsored by a EU FP7 PASCAL 2 challenge grant.', '1211.4847-1-113-3': 'We also acknowledge support from the International Space Science Institute (ISSI) in Bern, where some of this research has been discussed.', '1211.4847-1-114-0': '# Accuracy and bias perset', '1211.4847-1-115-0': 'Tables [REF] and [REF] respectively quote the actual quality factor and bias values reached by the non-denoised and denoised variants of the gfit shape measurement method.', '1211.4847-1-116-0': 'definecolorS/N10RGB255,236,139 definecolorS/N20RGB193,255,193 definecolorS/N40RGB187,255,255 definecolorALLRGB224,238,238', '1211.4847-1-117-0': '# Sear power spectra', '1211.4847-1-118-0': 'Figs [REF] and [REF] respectively show the shear power spectra of the non-denoised and denoised variants of gfit submitted in the GREAT10 galaxy Challenge.'} | {'1211.4847-2-0-0': 'A simple fitting method (gfit) for galaxy shape measurement in weak lensing surveys', '1211.4847-2-1-0': 'It is anticipated that the large sky areas covered by planned wide-field weak lensing surveys will reduce statistical errors to such an extent that systematic errors will instead become the dominant source of uncertainty.', '1211.4847-2-1-1': 'It is therefore crucial to devise numerical methods to measure galaxy shapes with the least possible systematic errors.', '1211.4847-2-1-2': 'We present a simple "forward deconvolution" method, gfit, to measure galaxy shapes given telescope and atmospheric smearings, in the presence of noise.', '1211.4847-2-1-3': 'The method consists in fitting a single 2D elliptical Sersic profile to the data, convolved with the point spread function.', '1211.4847-2-1-4': 'We applied gfit to the data proposed in the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Galaxy Challenge.', '1211.4847-2-1-5': 'In spite of its simplicity, gfit obtained the lowest additive bias ([MATH]) on the shear power spectrum among twelve different methods and the second lowest multiplicative bias ([MATH]).', '1211.4847-2-1-6': 'It remains that gfit is a fitting method and is therefore affected by noise bias.', '1211.4847-2-1-7': 'However, the simplicity of the underlying galaxy model combined with the use of an efficient customized minimization algorithm allow very competitive performances, at least on the GREAT10 data, for a relatively low computing time.', '1211.4847-2-2-0': '# Introduction', '1211.4847-2-3-0': 'Weak gravitational lensing , whereby the gravitational bending of light by structures in the Universe slightly distorts images of distant galaxies, is now recognized as a powerful means to study the history of the Universe and probe the mysterious nature of the dark matter and dark energy .', '1211.4847-2-4-0': 'Since the first detection of weak lensing , a number of methods have been devised and implemented to tackle the inverse problem of recovering the lensing signature from observed, distorted galaxy images .', '1211.4847-2-5-0': 'We describe in this paper gfit, a simple shear measurement method that nevertheless obtained good results in the latest GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Galaxy challenge .', '1211.4847-2-5-1': 'Galaxies are assumed to be well modeled by a seven-parameter, single-component elliptical Sersic profile.', '1211.4847-2-5-2': 'The shape measurement algorithm essentially consist in iteratively shearing and convolving the galaxy model until a sufficiently close match with the observed galaxy is reached.', '1211.4847-2-5-3': 'Instead of an out-the-box minimizer, we employ a custom-developed minimizer well suited to fitting faint and noisy images like those frequently found in weak lensing.', '1211.4847-2-6-0': 'The paper is structured as follows.', '1211.4847-2-6-1': 'We provide in Sect. [REF] a description of the underlying principles, galaxy model and shape measurement algorithm of gfit.', '1211.4847-2-6-2': 'We continue in Sect. [REF] with a presentation of the pipeline we used to participate in the GREAT10 Galaxy challenge and follow with a analysis of the gfit results in Sect. [REF].', '1211.4847-2-6-3': 'We conclude in Sect. [REF].', '1211.4847-2-7-0': '# The gfit shear measurement method', '1211.4847-2-8-0': '## The shear measurement problem', '1211.4847-2-9-0': 'According to the theory of weak gravitational lensing, the light emitted by a galaxy is slightly deviated by the foreground gravitational field, an effect that can be modeled to first order as the combination of two effects, the convergence [MATH] and the shear [MATH], that describe how light bundles emitted by a source are distorted by a potential well.', '1211.4847-2-9-1': 'The convergence models the magnification effect whereby the galaxy image see its apparent size increased without altering its shape, whereas the shear describes a stretching effect where only ellipticity is altered, not size.', '1211.4847-2-10-0': 'All so-called "shear measurement methods" attempt to reconstruct the reduced shear [MATH] which is approximately equals to the shear [MATH] in the weak gravitational limit, where [MATH] and [MATH].', '1211.4847-2-11-0': 'The lensing effect is very subtle, however, and requires measuring the shapes of thousands of faint galaxies.', '1211.4847-2-11-1': 'Moreover, before they reach the observer, the apparent galaxy images undergo a number of additional distortions, unrelated to lensing, that further complicates that task, mainly:', '1211.4847-2-12-0': 'The traditional approach for estimating the shear is to measure the deviation from circularity of a large number of galaxy shapes.', '1211.4847-2-12-1': 'But this technique assumes the shear remains constant across the field of view, which is generally not the case.', '1211.4847-2-12-2': 'Moreover, accurate shear measurement must also account for a spatially varying PSF that must be interpolated at the positions of the galaxies.', '1211.4847-2-13-0': 'A shear measurement pipeline must overcome all the above difficulties, typically going through the following steps:', '1211.4847-2-14-0': 'PSF correction, whose goal is to restore the shape a galaxy had before being convolved with the PSF.', '1211.4847-2-14-1': 'If the spatial variation of the PSF over the field of view is significant, this step also requires interpolating the PSF to the position of the galaxies in the sky.', '1211.4847-2-15-0': 'Shape measurement, that is, the estimation of the galaxy shape after it has been altered by the cosmic shear but before PSF convolution and other subsequent distortions.', '1211.4847-2-15-1': 'In this paper, we call the corresponding shape the sheared galaxy shape.', '1211.4847-2-15-2': 'The real galaxy ellipticity prior to gravitational lensing (i.e. unsheared) is referred to as the intrinsic ellipticity.', '1211.4847-2-16-0': 'Shear measurement, that is, the task of extracting the shear signal from the sheared galaxy shapes estimated in the previous step.', '1211.4847-2-16-1': 'A spatially-varying shear field is commonly described as a power spectrum or a correlation function.', '1211.4847-2-17-0': 'Additional steps may also be performed to correct the images from the effects of pixelation and noise.', '1211.4847-2-18-0': '## Shear measurement with gfit', '1211.4847-2-19-0': 'The gfit "galaxy fitting" method grew from a prototype initially developed by Stephane Paulin-Henrikson on the occasion of the GREAT08 challenge , where it obtained the third and fifth ranks on images with high and low signal to noise ratios respectively .', '1211.4847-2-20-0': 'The GREAT10 gfit code was subsequently made more generic in order to satisfy the more demanding requirements of the Galaxy challenge.', '1211.4847-2-20-1': 'It was also enhanced in several aspects that we describe in subsequent sections of this paper.', '1211.4847-2-21-0': 'It is also worth mentioning that a wavelet-based denoising algorithm described in [CITATION] was also experimented in the Galaxy challenge and proved quite successful.', '1211.4847-2-21-1': 'This denoising scheme is presented in Sect. [REF].', '1211.4847-2-22-0': 'The overall shape measurement procedure is the following:', '1211.4847-2-23-0': 'Application of the denoising algorithm on the galaxy and/or PSF images [optional] Estimation of the galaxy and PSF centroids in all images Application of the PSF correction and shape measurement algorithm Generation of the ellipticity catalogs Production of various statistics and plots for analysis [optional]', '1211.4847-2-24-0': 'We detail in the next sections the shape measurement algorithm along with its underlying models and components.', '1211.4847-2-25-0': '## Modeling the galaxies', '1211.4847-2-26-0': 'gfit is fundamentally a model-fitting method.', '1211.4847-2-26-1': 'We describe here the model used to represent galaxies and cover the fitting-related aspects in Sects. [REF] and [REF].', '1211.4847-2-27-0': 'Galaxies are assumed to have a surface brightness distribution well described by an elliptical Sersic function , defined by: [EQUATION] where:', '1211.4847-2-28-0': 'This model was initially chosen for its simplicity and has the other merit of being relatively easy to fit.', '1211.4847-2-28-1': 'A galaxy represented as the sum of a bulge and a disc would be more realistic, but it is not clear whether such a model would prove more accurate on weak lensing images and worth the additional complexity and computational cost, given degeneracies between parameters.', '1211.4847-2-28-2': 'As regards gfit, the GREAT10 results does not provide a definitive answer (see Sect. [REF]).', '1211.4847-2-29-0': '## Galaxy shape measurement and PSF correction', '1211.4847-2-30-0': 'We describe in this section the shape measurement algorithm of gfit.', '1211.4847-2-30-1': 'It is based on iterative fitting of observed galaxies to the galaxy model described in Sect. [REF].', '1211.4847-2-30-2': 'The basic assumptions of the algorithm are the following:', '1211.4847-2-31-0': 'The fitting algorithm itself is summarized below:', '1211.4847-2-32-0': 'At the end of the iteration cycle, the algorithm yields:', '1211.4847-2-33-0': 'The same procedure is followed for the next galaxy [MATH] in [MATH], until all galaxies have been processed.', '1211.4847-2-34-0': 'Once the shape of all galaxies have been measured, produce a catalog containing the fitted parameters at the positions of the galaxies [MATH] in [MATH] and optionally, additional statistics and plots.', '1211.4847-2-34-1': 'Quantities such as the ellipticity modulus [MATH], the position angle [MATH] or the minor-to-major axis ratio [MATH] can be respectively derived from [MATH] using [MATH], [MATH] and [MATH].', '1211.4847-2-35-0': 'The algorithm just described is not dependent on the galaxy or the PSF model.', '1211.4847-2-35-1': 'It would remain unchanged, for instance, if the centroid of the PSF itself was taken into account during fitting or if a bulge and a disc were incorporated in the galaxy model.', '1211.4847-2-35-2': 'The quality of the whole shape measurement procedure can thus be increased by improving the models.', '1211.4847-2-36-0': 'The shape measurement algorithm has its own strengths and weaknesses, outlined below:', '1211.4847-2-37-0': 'Weaknesses:', '1211.4847-2-38-0': '## The gfit minimizer', '1211.4847-2-39-0': 'The gfit galaxy model expressed by Eq. ([REF]) varies linearly with the parameter [MATH], but non-linearly with the remaining parameters [MATH].', '1211.4847-2-39-1': 'This requires the use of a non-linear minimization algorithm over a seven-dimensional parameter space.', '1211.4847-2-40-0': 'Unlike linear minimizations schemes that only involve a matrix inversion, non-linear optimization requires iterating over the parameter space to find the minimum value of the objective function, which is in our case the [MATH] of the residuals between observed and estimated images.', '1211.4847-2-40-1': 'That minimum is not necessarily the absolute minimum of the [MATH] function but the most relevant from the point of view of the physics of the problem.', '1211.4847-2-40-2': 'In our case, the minimum should coincide with the Sersic model parameters that best fit the galaxy shape.', '1211.4847-2-41-0': 'A good minimizer is essential to any galaxy model fitting algorithm, but finding such a minimum in a reliable manner can prove tricky for a number of reasons:', '1211.4847-2-42-0': 'In this regards, the choice of the galaxy model used by gfit leads to a number of challenges:', '1211.4847-2-43-0': 'Several families of optimization algorithms were experimented on GREAT08 and GREAT10 images: simplex (Nelder-Mead downhill), gradient descent (Powell), Newton quasi-Newton (Newton-CG, BFGS) and Levenberg-Marquardt (LVM).', '1211.4847-2-43-1': 'Descriptions of these algorithms can be found in .', '1211.4847-2-44-0': 'None of these methods proved entirely satisfactory, either failing to converge or yielding insufficient accuracy, especially on low signal to noise ratio (S/N) images.', '1211.4847-2-44-1': 'The LVM implementation from the SciPy library that we used was the fastest and the most accurate.', '1211.4847-2-44-2': 'For these reasons we used it in the GREAT08 version of gfit, but that implementation of LVM:', '1211.4847-2-45-0': 'We also experimented a parameter estimation scheme based on a Bayesian approach and implemented using the pymc Markov Chain Monte Carlo (MCMC) library of [CITATION], but that method produced less accurate estimates while being less computationally efficient.', '1211.4847-2-46-0': 'In an attempt to overcome these issues, we eventually decided to implement a custom minimizer, better suited to fitting noisy, pixelized galaxy images than vanilla minimization algorithms.', '1211.4847-2-46-1': 'We found that a scheme based on an adaptive cyclic coordinate descent algorithm (CCD) was able to produce more accurate estimates while at the same time being more robust:', '1211.4847-2-47-0': 'The CCD algorithm proved suitable for fitting without any single failure the huge number of GREAT10 galaxies.', '1211.4847-2-47-1': 'It nevertheless has a number of drawbacks, namely:', '1211.4847-2-48-0': 'The current gfit implementation can be configured to use either the CCD, LVM or MCMC-based minimizer.', '1211.4847-2-49-0': '## Centroid estimation', '1211.4847-2-50-0': 'gfit does not assume objects to be correctly centered within their postage stamps and accurate estimates for galaxy and PSF objects are required for two main reasons:', '1211.4847-2-51-0': 'gfit relies on centroid estimates obtained from the SExtractor tool .', '1211.4847-2-51-1': 'A catalog is generated with centroid information and additional data such as flux, ellipticities and position angles, that can be optionally used to set guess parameter values.', '1211.4847-2-52-0': '## Denoising', '1211.4847-2-53-0': 'Correcting astronomical images from the effect of noise has always been a challenging task.', '1211.4847-2-53-1': 'This is particularly true for galaxy images captured for weak lensing analysis because noise not only degrades the overall quality of these images but also alters the shapes of the galaxies.', '1211.4847-2-53-2': 'This causes serious difficulties to all existing shear measurement schemes that found their shear extraction algorithms on the accurate measurement of galaxy shapes.', '1211.4847-2-54-0': 'The challenge is then to correct galaxy images from noise without compromising the shear signal they encode.', '1211.4847-2-54-1': 'Unfortunately, popular denoising algorithms based on median filtering , Wiener filtering or discrete wavelet transform (DWT) are not shape preserving and do not meet that requirement.', '1211.4847-2-55-0': 'By default, gfit uses DWT-Wiener, a shape-preserving denoising technique combining DWT and Wiener filtering developed at the laboratory of astrophysics of EPFL by [CITATION].', '1211.4847-2-55-1': 'That algorithm has been experimented during the GREAT10 Galaxy challenge and was able to significantly improve the quality factors of all the shear measurement methods from EPFL that participated in the Galaxy challenge : that was the case for gfit but also for MegaLUT and TVNN .', '1211.4847-2-56-0': 'Interestingly, denoising improves the shape measurement of the three algorithms even though they are fundamentally different from each other.', '1211.4847-2-57-0': 'Beyond shape-preservation, another advantage of the DWT-Wiener algorithm lies in its ability to denoise "in one go" images containing a great number of objects, without having to individually process each object in turn.', '1211.4847-2-57-1': 'In GREAT10, for instance, DWT-Wiener was directly applied to images containing [MATH] PSF or galaxy postage stamps.', '1211.4847-2-58-0': '# Applying gfit to the GREAT10 data', '1211.4847-2-59-0': '## The GREAT10 Galaxy challenge', '1211.4847-2-60-0': 'We describe in this section the specific pipeline we used in the GREAT10 Galaxy Challenge competition that took place between December 2010 and September 2011.', '1211.4847-2-60-1': 'The aim, content and rules of the challenge have been described in the GREAT10 Handbook .', '1211.4847-2-60-2': 'In a nutshell, the main goals of the GREAT Challenges are (i) to test existing weak lensing measurements methods and (ii) to promote the development of new, more accurate, shear measurement techniques.', '1211.4847-2-61-0': 'The data consist of 24 datasets of 200 simulated galaxy images, each containing 10,000 noisy, PSF-convolved [MATH] pixel galaxy postage stamps, arranged on a [MATH] grid (see Fig. [REF]).', '1211.4847-2-61-1': 'The GREAT10 edition includes spatially-varying PSF and shear fields, contrary to its predecessor, the GREAT08 challenge , where these fields were set as constant.', '1211.4847-2-62-0': 'Each of the 24 sets is designed to evaluate the ability of competing methods to deal with galaxy or PSF fields with different properties (e.g. size, signal to noise ratio).', '1211.4847-2-62-1': 'We have reproduced in Table [REF] the main PSF and galaxy characteristic attached to each of the GREAT10 set, as specified in the Galaxy Challenge results paper , Appendix D.', '1211.4847-2-63-0': 'The sets were also classified into "Single epoch", "Multi-epoch" and "Stable single epoch", depending on whether the intrinsic ellipticities and PSF keep the same or change their spatial distribution between images in a set (see Table [REF]).', '1211.4847-2-64-0': 'Participating methods were ranked according to the following metrics :', '1211.4847-2-65-0': 'definecolorFullVariabilityrgb0.824,0.824,0.824', '1211.4847-2-66-0': '## The GREAT10 gfit implementation', '1211.4847-2-67-0': 'The GREAT10 version of gfit only implements the first two steps described in Sect. [REF], that is, PSF correction and galaxy shape measurement.', '1211.4847-2-68-0': 'In GREAT10, the estimation of the shear field (third step in Sect. [REF]) was not mandatory as participants were allowed to supply for each image a catalog of estimated galaxy ellipticities instead of a shear power spectrum : an analysis program was written by the GREAT10 team to calculate a shear power spectrum from user-supplied ellipticity catalogs.', '1211.4847-2-68-1': 'Consequently, like most other competing methods, gfit only provided its estimates in the form of a catalog of estimated ellipticities at requested positions within the images.', '1211.4847-2-68-2': 'Future version of gfit will allow the extraction of a spatially varying shear.', '1211.4847-2-69-0': 'The gfit implementation used in GREAT10 consisted of the following stages:', '1211.4847-2-70-0': 'Optional denoising of the galaxy and PSF images with the DWT-Wiener method presented in Sect. [REF].', '1211.4847-2-71-0': 'Centroid estimation using SExtractor as described in Sect. [REF].', '1211.4847-2-72-0': 'Galaxy shape measurement with the gfit program, configured to use the CCD minimization algorithm described in Sect. [REF].', '1211.4847-2-72-1': 'We used [MATH] pixel cutouts instead of the full [MATH] pixel original galaxy postage stamps.', '1211.4847-2-72-2': 'Similarly, the size of PSF postage stamps was reduced to [MATH] pixels.', '1211.4847-2-72-3': 'That decision was made in order to keep the overall computation time within acceptable limits and avoid picking-up too much noise near the borders of the postage stamps.', '1211.4847-2-73-0': 'Because of the large number of galaxies, running the pipeline on one single processor would not have allowed to meet the GREAT10 deadline.', '1211.4847-2-73-1': 'Even with a processing time per galaxy of 0.5 seconds, it would have taken about one month to complete process the full GREAT10 dataset.', '1211.4847-2-73-2': 'The ability to simultaneously run multiple program instances use of parallelism is thus imperative and all programs (denoising, SExtractor wrapper, gfit) are written to take advantage of of parallel computers through the Message Passing Interface (MPI) .', '1211.4847-2-73-3': 'When only a few processors are required, the same programs can also run on machines with symmetric multiprocessing (SMP) architecture.', '1211.4847-2-73-4': 'It took about 5 days to process the entire GREAT10 challenge images on a 64-processor machine, which corresponds to a processing time between [MATH] and [MATH] seconds per galaxy.', '1211.4847-2-74-0': 'The pipeline is implemented in Python, a programming language known for its power, flexibility and short development cycle.', '1211.4847-2-74-1': 'The usual standard Python libraries are used, notably: NumPy, SciPy, PyFITS and matplotlib.', '1211.4847-2-74-2': 'SciPy is the standard scientific library for Python and most of its functions consist of thin Python wrappers on top of fortran, C and C++ functions.', '1211.4847-2-74-3': 'SciPy takes advantage of installed optimized libraries such as LAPACK (Linear Algebra PACKage) library .', '1211.4847-2-75-0': '# Analysis of the gfit GREAT10 results', '1211.4847-2-76-0': 'We summarize and analyze in this section the main Galaxy Challenge results as far as gfit is concerned.', '1211.4847-2-76-1': 'An overview of the GREAT10 results for available participating shear measurement methods has already been performed in the GREAT10 Galaxy challenge paper .', '1211.4847-2-76-2': 'Our objective here is to provide a more detailed analysis of the gfit results.', '1211.4847-2-77-0': 'We do not, however, analyze the influence of the pixel-denoising and training calibration schemes applied in , which we leave for future investigation.', '1211.4847-2-78-0': '## Overall results', '1211.4847-2-79-0': 'The results of the best [MATH] methods that participated in the GREAT10 Galaxy Challenge are listed in Table 3 of the [CITATION] GREAT10 result paper.', '1211.4847-2-79-1': 'That list aggregates results submitted before the official challenge deadline as well as submissions made during the so-called "Post challenge", a one-week extension to the competition following the deadline.', '1211.4847-2-80-0': 'Two versions of gfit were submitted during the challenge, one named "gfit den cs" that included a denoising step using the DWT-Wiener algorithm described in Sect. [REF] and the other, simply named "gfit", that did not.', '1211.4847-2-80-1': 'The results obtained by both methods are shown in Table [REF].', '1211.4847-2-81-0': 'The gfit version presented in [CITATION] is what we refer to here as "gfit".', '1211.4847-2-81-1': 'In addition, we also present "gfit den cs", described in Appendix E5 of [CITATION] but whose results were not included in the analysis.', '1211.4847-2-81-2': 'The "gfit" version is identical except that no denoising was applied to the data before applying the shape measurement algorithm.', '1211.4847-2-81-3': 'To simplify, we shorten he name "gfit den cs" to "gfit den" in the remainder of this article.', '1211.4847-2-82-0': 'It can be seen from Table [REF] that "gfit den" reaches a raw quality factor [MATH] twice as high as that of "gfit".', '1211.4847-2-82-1': 'This illustrates the gain in accuracy provided by the DWT-Wiener denoising algorithm.', '1211.4847-2-82-2': 'This is further analyzed in Sect. [REF].', '1211.4847-2-82-3': 'When the pixel-level denoising algorithm of [CITATION] is applied, the [MATH] quality factors of both "gfit" and "gfit den" are improved by a factor [MATH], "gfit den" scoring the best [MATH] of all methods ([MATH]).', '1211.4847-2-82-4': 'The training calibration further increases both [MATH] quality factors, especially that of "gfit" (two-fold increase).', '1211.4847-2-83-0': 'We have also included in Table [REF] the average additive and multiplicative biases [MATH] and [MATH] over all [MATH] sets.', '1211.4847-2-83-1': 'Comparing with Table 3 of the [CITATION] result paper, we see that "gfit" reached the lowest average additive bias ([MATH]) and the second lowest average multiplicative bias ([MATH]) of all twelve methods (see also the plot in Figure.', '1211.4847-2-83-2': '1, page 6 of that paper).', '1211.4847-2-84-0': 'We stress that, contrary to what is suggested in Sect. 4.4 of [CITATION], the low overall bias of "gfit" is intrinsic to the method and does not result from the application of a denoising step.', '1211.4847-2-84-1': 'The DWT-Wiener algorithm was only used in "gfit den", not "gfit" and actually, the average multiplicative bias of "gfit den" ([MATH]) is higher than that of "gfit".', '1211.4847-2-84-2': 'Moreover, both methods have similar additive biases.', '1211.4847-2-84-3': 'Drawing a more refined conclusion about these biases requires an analysis at individual set level, which we perform in Sect. [REF].', '1211.4847-2-85-0': '## Method accuracy', '1211.4847-2-86-0': 'In this section we use the quality factor as a measure of accuracy and assess the influence of:', '1211.4847-2-87-0': 'The quality factors scored for each individual image set are plotted on the left-hand side part of Fig. [REF], for each gfit variant.', '1211.4847-2-87-1': 'They are also quoted in Tables [REF] and [REF].', '1211.4847-2-88-0': 'Due to an editorial mistake, the shear power spectra attributed to "gfit" in the GREAT10 Galaxy Challenge paper, Figure E9, is that from the "fit2-unfold" method.', '1211.4847-2-88-1': 'The correct picture was made available at the time of publication and can be found http://great.roe.ac.uk/data/galaxy_article_figures/here.', '1211.4847-2-88-2': 'We also provide the correct figure in Fig. [REF].', '1211.4847-2-88-3': 'We include the "gfit den" power spectrum in Fig. [REF] as well.', '1211.4847-2-89-0': 'We focus first on the influence of galaxy and PSF features on accuracy.', '1211.4847-2-89-1': 'We leave aside the effects of DWT-Wiener denoising for now and thus base our analysis on the results of the "gfit" variant which is devoid of built-in denoising scheme.', '1211.4847-2-89-2': 'The data of interest are summarized in Figs. [REF] and [REF] and in Tables [REF], [REF].', '1211.4847-2-90-0': '### Influence of Galaxy and PSF characteristics', '1211.4847-2-91-0': '### Effect of denoising on accuracy', '1211.4847-2-92-0': 'We discuss in this section the effect on accuracy of the application of the DWT-Wiener denoising scheme described in Sect. [REF].', '1211.4847-2-92-1': 'The influence of denoising can be clearly observed by comparing the plot of "gfit den" (left) with that of "gfit" in Fig. [REF] (right).', '1211.4847-2-92-2': 'The corresponding scores are also listed in Tables [REF] and [REF].', '1211.4847-2-93-0': 'We find an average two-fold increase in accuracy, the effect being stronger on high S/N images and larger galaxies.', '1211.4847-2-93-1': 'Sets with small galaxies are only slightly improved, however.', '1211.4847-2-93-2': 'We also notice that the plots of "gfit" and "gfit den" show identical quality factors for low S/N sets 4 to 6.', '1211.4847-2-93-3': 'Further investigation showed that denoising was, by mistake, not applied on those sets.', '1211.4847-2-93-4': 'This would probably have improved the overall Q factor of "gfit den".', '1211.4847-2-94-0': 'These results strongly suggest that the DWT-Wiener algorithm really improves the overall accuracy on galaxy shape measurement.', '1211.4847-2-94-1': 'We also note that denoising does not alter the quality factor hierarchy between sets: the sets with best scores in the "gfit" plots remain the same in the "gfit den" plot.', '1211.4847-2-95-0': '## Bias analysis', '1211.4847-2-96-0': 'We investigate in this section how multiplicative and additive biases are affected by galaxy properties and the use of denoising.', '1211.4847-2-97-0': '### Influence of Galaxy and PSF characteristics', '1211.4847-2-98-0': 'To complement the results of Table [REF] relative to bias, we have plotted in Figs. [REF] and [REF] the multiplicative and additive biases of each set.', '1211.4847-2-98-1': 'As noted in Sec. [REF], "gfit den" reached the lowest average additive bias and the second lowest average multiplicative bias of all 12 twelve competing methods.', '1211.4847-2-99-0': 'Focusing on the multiplicative bias and leaving aside the effect of denoising for now, we can make a few observations from the left-hand side plot of Figs. [REF].', '1211.4847-2-100-0': 'As far as additive bias is concerned, we note the following trend:', '1211.4847-2-101-0': '### Effect of denoising on bias', '1211.4847-2-102-0': 'Comparing the plots for "gfit" and "gfit den" in Figs. [REF] and [REF], we find that the additive bias does not change significantly, keeping about the same bias values per set.', '1211.4847-2-102-1': 'In contrast, the structure of the multiplicative plots is significantly altered.', '1211.4847-2-103-0': 'As seen on the "gfit den" plot, denoising tend to introduce some amount of negative multiplicative bias on all sets.', '1211.4847-2-103-1': 'Although the amount of bias on fiducial sets remains roughly the same in absolute value, the DWT-Wiener algorithm clearly impacts the multiplicative bias relative to galaxy size, b/d ratio and turbulence.', '1211.4847-2-104-0': 'As regards galaxy size, even though the multiplicative bias on small galaxies is almost unchanged, that on large galaxies increases about six-fold.', '1211.4847-2-104-1': 'Because of that, the bias on "Smooth" galaxies, which also includes large galaxies, also increases.', '1211.4847-2-105-0': 'DWT-Wiener denoising seems to also improves the resolution of the bulge and disk components, so that "gfit" has more difficulty fitting its underlying single-component Sersic model to images with "uniform" and "offset" b/d.', '1211.4847-2-105-1': 'The effect is stronger on larger galaxies, causing a [MATH] tenfold multiplicative bias increase.', '1211.4847-2-106-0': 'Lastly, the introduction of PSF turbulence result in a [MATH] five-fold multiplicative bias degradation.', '1211.4847-2-107-0': 'Despite the degradation of multiplicative bias on some sets, the accuracy of shape measurements increases two-fold as shown by the corresponding gain in Q factor.', '1211.4847-2-107-1': 'We also note that denoising improves the results on sets that already have a high S/N.', '1211.4847-2-107-2': 'All in all, the use of denoising is thus clearly beneficial.', '1211.4847-2-108-0': '# Conclusions', '1211.4847-2-109-0': 'We have described in this paper the gfit shape measurement method, a model-fitting based on a simple Sersic galaxy model (Sect. [REF]).', '1211.4847-2-109-1': 'The method uses a custom-developed minimizer based on a "coordinate descent" algorithm that finds a local minimum with the lowest [MATH] of the residuals between true and modeled galaxy.', '1211.4847-2-110-0': 'We have also performed an analysis of our results in the GREAT10 Galaxy Challenge (Sect. [REF]).', '1211.4847-2-110-1': 'We participated in the competition with two gfit variants: "gfit den", which applied a denoising step before performing model-fitting, and "gfit", which did not use denoising.', '1211.4847-2-110-2': 'The noise removal technique employed is DWT-Wiener, a wavelet-based, shape-preserving algorithm particularly suitable for shape measurement.', '1211.4847-2-110-3': '(see Sect. [REF]).', '1211.4847-2-111-0': 'We highlight below the main conclusions of your analysis.', '1211.4847-2-112-0': 'It is interesting to see that, despite the simplicity of the galaxy model used, its results in the Galaxy Challenge established gfit as one of the four top-performing methods, both in terms of accuracy and bias.', '1211.4847-2-112-1': 'Given that the results were obtained on simulated data, this raises the question of how important having a realistic galaxy model really matters when measuring galaxy shapes from real data.', '1211.4847-2-112-2': 'Providing more clues on this question is one of the objectives of the forthcoming GREAT3 challenge.', '1211.4847-2-113-0': 'This work is supported by the Swiss National Science Foundation (SNSF).', '1211.4847-2-113-1': 'Many thanks to Tom Kitching for his help and for sharing the shear analysis code.', '1211.4847-2-113-2': 'We also thank the GREAT10 Coordination Team for organizing this stimulating challenge.', '1211.4847-2-113-3': 'GREAT10 was sponsored by a EU FP7 PASCAL 2 challenge grant.', '1211.4847-2-113-4': 'We also acknowledge support from the International Space Science Institute (ISSI) in Bern, where some of this research has been discussed.', '1211.4847-2-114-0': '# Accuracy and bias per set', '1211.4847-2-115-0': 'Tables [REF] and [REF] respectively quote the actual quality factor and bias values reached by the non-denoised and denoised variants of the gfit shape measurement method.', '1211.4847-2-116-0': 'definecolorS/N10RGB255,236,139 definecolorS/N20RGB193,255,193 definecolorS/N40RGB187,255,255 definecolorALLRGB224,238,238', '1211.4847-2-117-0': '# Sear power spectra', '1211.4847-2-118-0': 'Figs [REF] and [REF] respectively show the shear power spectra of the non-denoised and denoised variants of gfit submitted in the GREAT10 galaxy Challenge.'} | [['1211.4847-1-113-0', '1211.4847-2-113-0'], ['1211.4847-1-113-2', '1211.4847-2-113-3'], ['1211.4847-1-113-3', '1211.4847-2-113-4'], ['1211.4847-1-72-0', 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'1211.4847-2-35-1'], ['1211.4847-1-35-2', '1211.4847-2-35-2'], ['1211.4847-1-99-0', '1211.4847-2-99-0'], ['1211.4847-1-60-0', '1211.4847-2-60-0'], ['1211.4847-1-60-1', '1211.4847-2-60-1'], ['1211.4847-1-60-2', '1211.4847-2-60-2'], ['1211.4847-1-3-0', '1211.4847-2-3-0'], ['1211.4847-1-6-0', '1211.4847-2-6-0'], ['1211.4847-1-6-1', '1211.4847-2-6-1'], ['1211.4847-1-6-2', '1211.4847-2-6-2'], ['1211.4847-1-6-3', '1211.4847-2-6-3'], ['1211.4847-1-48-0', '1211.4847-2-48-0']] | [] | [] | [['1211.4847-1-113-1', '1211.4847-2-113-2'], ['1211.4847-1-81-2', '1211.4847-2-81-1'], ['1211.4847-1-88-0', '1211.4847-2-88-0']] | [] | ['1211.4847-1-11-1', '1211.4847-1-13-0', '1211.4847-1-22-0', '1211.4847-1-27-0', '1211.4847-1-30-2', '1211.4847-1-31-0', '1211.4847-1-32-0', '1211.4847-1-36-0', '1211.4847-1-37-0', '1211.4847-1-41-0', '1211.4847-1-42-0', '1211.4847-1-44-2', '1211.4847-1-46-1', '1211.4847-1-47-1', '1211.4847-1-50-0', '1211.4847-1-64-0', '1211.4847-1-65-0', '1211.4847-1-69-0', '1211.4847-1-71-0', '1211.4847-1-86-0', '1211.4847-1-100-0', '1211.4847-1-110-3', '1211.4847-1-111-0', '1211.4847-1-116-0', '1211.4847-2-11-1', '1211.4847-2-13-0', '1211.4847-2-22-0', '1211.4847-2-27-0', '1211.4847-2-30-2', '1211.4847-2-31-0', '1211.4847-2-32-0', '1211.4847-2-36-0', '1211.4847-2-37-0', '1211.4847-2-41-0', '1211.4847-2-42-0', '1211.4847-2-44-2', '1211.4847-2-46-1', '1211.4847-2-47-1', '1211.4847-2-50-0', '1211.4847-2-64-0', '1211.4847-2-65-0', '1211.4847-2-69-0', '1211.4847-2-71-0', '1211.4847-2-86-0', '1211.4847-2-100-0', '1211.4847-2-110-3', '1211.4847-2-111-0', '1211.4847-2-116-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1211.4847 | null | null | null | null | null |
1501.00917 | {'1501.00917-1-0-0': 'In [CITATION] and [CITATION] we showed that a loop in a simply connected compact Lie group [MATH] has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique (what we propose to call) "root subgroup factorization" (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group).', '1501.00917-1-0-1': 'In this sequel our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group [MATH] of inner type.', '1501.00917-1-1-0': 'In the first part of this paper, we consider the finite dimensional situation (i.e. constant loops in [MATH]), and we show that a group element has a Birkhoff factorization if and only if it has a root subgroup factorization.', '1501.00917-1-2-0': 'In the second part of this paper we consider loops in [MATH].', '1501.00917-1-2-1': 'In this case both Birkhoff and root subgroup factorization are far more complicated than for loops in [MATH].', '1501.00917-1-2-2': 'In this noncompact context a root subgroup factorization implies a unique Birkhoff factorization, but there are several obstacles for the converse.', '1501.00917-1-2-3': 'As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.', '1501.00917-1-3-0': '2000 Mathematics Subject Classifications: 22E67', '1501.00917-1-4-0': '# Introduction', '1501.00917-1-5-0': 'Finite dimensional Riemannian symmetric spaces come in dual pairs, one of compact type and one of noncompact type.', '1501.00917-1-5-1': 'Given such a pair, there is a diagram of finite dimensional groups [EQUATION] where [MATH] is the universal covering of the identity component of the isometry group of the compact type symmetric space [MATH], [MATH] is the complexification of [MATH], and [MATH] is a covering of the isometry group for the dual noncompact symmetric space [MATH].', '1501.00917-1-6-0': 'The main purpose of this paper is to investigate Birkhoff (or triangular) factorization and "root subgroup factorization" for [MATH], and for the loop group of [MATH], assuming [MATH] is of inner type.', '1501.00917-1-6-1': 'Birkhoff factorization is investigated in [CITATION] and Chapter 8 of [CITATION], from various points of view.', '1501.00917-1-6-2': 'In particular Birkhoff factorization for [MATH] is developed in Chapter 8 of [CITATION], using the Grassmannian model for the homogeneous space [MATH].', '1501.00917-1-6-3': 'Root subgroup factorization for generic loops in [MATH] appeared more recently in [CITATION] (for [MATH], the rank one case) and in [CITATION].', '1501.00917-1-6-4': 'The Birkhoff decomposition for [MATH], i.e. the intersection of the Birkhoff decomposition for [MATH] with [MATH], is far more complicated than for [MATH].', '1501.00917-1-6-5': 'With respect to root subgroup factorization, beyond loops in a torus (corresponding to imaginary roots), in the compact context the basic building blocks are exclusively spheres (corresponding to real roots), and in the inner noncompact context the building blocks are a combination of spheres and disks.', '1501.00917-1-6-6': 'This introduces additional analytic complications, and perhaps the main point of this paper is to communicate the problems that arise from noncompactness.', '1501.00917-1-7-0': 'For [MATH], the basic fact is that [MATH] has a unique triangular factorization if and only if [MATH] has a unique "root subgroup factorization" (relative to the choice of a reduced sequence of simple reflections in the affine Weyl group).', '1501.00917-1-7-1': 'We will show that this is also true for [MATH] (constant loops).', '1501.00917-1-7-2': 'However this is far from true for loops in [MATH].', '1501.00917-1-8-0': 'Relatively little sophistication is required to state the basic results, and identify the basic obstacles, in the rank one noncompact case.', '1501.00917-1-8-1': 'This is essentially because (in addition to loops in a torus) the basic building blocks are exclusively disks, and there is an essentially unique way to choose a reduced sequence of simple reflections in the affine Weyl group, so that this can be suppressed.', '1501.00917-1-9-0': '## The Rank 1 Case', '1501.00917-1-10-0': 'We consider the data determined by the Riemann sphere and the Poincare disk.', '1501.00917-1-10-1': 'For this pair, the diagram ([REF]) becomes [EQUATION]', '1501.00917-1-10-2': 'Let [MATH] denote the group consisting of maps [MATH] having finite Fourier series, with pointwise multiplication.', '1501.00917-1-10-3': 'The subset of those functions having values in [MATH] is then a subgroup, denoted [MATH].', '1501.00917-1-11-0': 'For each [MATH] and [MATH], the function [MATH] defined by [EQUATION] is in [MATH].', '1501.00917-1-12-0': '[MATH] and [MATH] are dense in the smooth loop groups [MATH] and [MATH], respectively.', '1501.00917-1-12-1': 'This is proven in the compact case in Proposition 3.5.3 of [CITATION], and the argument applies also for [MATH], taking into account the obvious modifications.', '1501.00917-1-13-0': 'For a Laurent series [MATH], let [MATH].', '1501.00917-1-13-1': 'If [MATH] is a domain on the Riemann sphere, we write [MATH] for the vector space of holomorphic scalar valued functions on [MATH].', '1501.00917-1-13-2': 'If [MATH], then [MATH], where [MATH] denotes the open unit disk at [MATH].', '1501.00917-1-14-0': 'Suppose that [MATH] and fix [MATH].', '1501.00917-1-14-1': 'Consider the following three statements:', '1501.00917-1-15-0': '[(I.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are polynomials in [MATH] of order [MATH] and [MATH], respectively, with [MATH].', '1501.00917-1-15-1': '[(I.2)] [MATH] has a "root subgroup factorization" of the form [EQUATION] for some sequence [MATH] in [MATH] and [MATH] is the function in Example [REF].', '1501.00917-1-15-2': '[(I.3)] [MATH] has triangular factorization of the form [EQUATION] where [MATH], the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-1-16-0': 'Statements (I.1) and (I.3) are equivalent.', '1501.00917-1-16-1': '(I.2) implies (I.1) and (I.3).', '1501.00917-1-16-2': 'If [MATH] is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds, i.e. [MATH] has a root subgroup factorization as in (I.2).', '1501.00917-1-17-0': 'There is a similar set of implications for [MATH] and the following statements:', '1501.00917-1-18-0': '[(II.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are polynomials in [MATH] of order [MATH] and [MATH], respectively, with [MATH] and [MATH].', '1501.00917-1-18-1': '[(II.2)] [MATH] has a "root subgroup factorization" of the form [EQUATION] for some sequence [MATH] in [MATH] and [MATH] is the function in Example [REF].', '1501.00917-1-18-2': '[(II.3)] [MATH] has a triangular factorization of the form [EQUATION] where [MATH], and the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-1-19-0': 'When [MATH] and [MATH] have root subgroup factorizations, the scalar entries determining the diagonal factor have the product form [EQUATION]', '1501.00917-1-19-1': 'In general we do not know how to describe the connected component that arises in the first and third conditions.', '1501.00917-1-20-0': 'Consider the case [MATH] and [MATH] is as in II.3 with [MATH], [MATH], [EQUATION] and [EQUATION]', '1501.00917-1-20-1': 'It is straightforward to check that this [MATH] does indeed have values in [MATH].', '1501.00917-1-20-2': 'In order for [MATH], there are two possibilities: the first is that both the numerator and denominator are positive, in which case there is a root subgroup factorization, and the second is that both the top and bottom are negative, in which case root subgroup factorization fails (because when there is a root subgroup factorization, [MATH], and we must have [MATH]).', '1501.00917-1-21-0': 'In order to formulate a general factorization result, we need a [MATH] version of Theorem [REF].', '1501.00917-1-22-0': 'Suppose that [MATH].', '1501.00917-1-22-1': 'The following conditions are equivalent:', '1501.00917-1-23-0': '[(I.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are holomorphic in [MATH] and have [MATH] boundary values, with [MATH].', '1501.00917-1-23-1': '[(I.3)] [MATH] has triangular factorization of the form [EQUATION] where [MATH] is holomorphic in [MATH] with [MATH] boundary values, [MATH], and the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-1-24-0': 'Similarly if [MATH], the following statements are equivalent:', '1501.00917-1-25-0': '[(II.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are holomorphic in [MATH] and have [MATH] boundary values, with [MATH] and [MATH].', '1501.00917-1-25-1': '[(II.3)] [MATH] has a triangular factorization of the form [EQUATION] where [MATH], [MATH] is holomorphic in [MATH] and has [MATH] boundary values, [MATH], and the third factor is a matrix valued function which is holomorphic in [MATH] and has [MATH] boundary values, and is unipotent upper triangular at [MATH].', '1501.00917-1-26-0': 'Let [MATH] denote the anti-holomorphic involution of [MATH] which fixes [MATH]; explicitly [EQUATION].', '1501.00917-1-27-0': 'The following theorem is the analogue of Theorem 0.2 of [CITATION] (the notation in part (b) is taken from Section 1 of [CITATION], and reviewed below the statement of the theorem).', '1501.00917-1-28-0': 'Suppose [MATH], the identity component.', '1501.00917-1-28-1': 'Then [MATH] has a unique "partial root subgroup factorization" of the form [EQUATION] where [MATH] and [MATH] and [MATH] are as in Theorem [REF], if and only [MATH] has a triangular factorization [MATH] (see ([REF]) below) such that the boundary values of [MATH] and [MATH] are [MATH] in magnitude on [MATH].', '1501.00917-1-29-0': 'The following example shows that the unaesthetic condition on the boundary values in part (b) is essential.', '1501.00917-1-30-0': 'Consider [MATH] as in Theorem [REF].', '1501.00917-1-30-1': 'The loop [MATH] (the Hermitian conjugate of [MATH] around the circle) has triangular factorization [EQUATION].', '1501.00917-1-30-2': 'If [MATH], then [MATH] and [MATH], and this loop will often not satisfy the condition [MATH] on [MATH].', '1501.00917-1-30-3': 'In this case [MATH] will not have a partial root subgroup factorization in the sense of Theorem [REF].', '1501.00917-1-31-0': 'The group [MATH] has a Birkhoff decomposition [EQUATION] where [MATH] (an affine Weyl group, and in this case the infinite dihedral group) is a quotient of a discrete group of unitary loops [EQUATION] where [EQUATION] (the reflections corresponding to the two simple roots for the Kac-Moody extension of [MATH]).', '1501.00917-1-31-1': 'The set [MATH] consists of loops which have a (Birkhoff) factorization of the form [EQUATION] where [MATH], [EQUATION] [MATH] has smooth boundary values on [MATH], [MATH], [MATH], [MATH], [MATH], [EQUATION] and [MATH] has smooth boundary values on [MATH].', '1501.00917-1-31-2': 'If [MATH], the generic case, then we say (as in Section 1 of [CITATION]) that [MATH] has a triangular factorization, and in this case the factors are unique.', '1501.00917-1-32-0': 'Next, let [MATH] denote the connected component containing [MATH], and let [EQUATION] and [EQUATION].', '1501.00917-1-32-1': 'Based on finite dimensional intuition (the first part of this paper), and the compact case, one might expect the following to be true:', '1501.00917-1-33-0': '[(1)] Modulo [MATH], the circle subgroup, it should be possible to contract [MATH] down to [MATH]; in particular [MATH] should be empty unless [MATH] is represented by a loop in [MATH], i.e. [MATH] for some [MATH].', '1501.00917-1-33-1': '[(2)] [MATH].', '1501.00917-1-33-2': '[(3)] Each [MATH] should admit a relatively explicit parameterization.', '1501.00917-1-34-0': 'Statements (1) and (2) are definitely false; statement (3) is very elusive, if not doubtful.', '1501.00917-1-35-0': '[MATH]', '1501.00917-1-36-0': '[(a)] [MATH] nonempty does not imply that [MATH] is represented by a loop in [MATH].', '1501.00917-1-36-1': 'For example, [MATH] is nonempty.', '1501.00917-1-36-2': '[(b)] [MATH] is properly contained in [MATH].', '1501.00917-1-37-0': 'To summarize, the set of loops having a root subgroup factorization is properly contained in the set of loops in the identity component which have a triangular factorization which, in turn, is a proper subset of the identity component of [MATH].', '1501.00917-1-37-1': 'The first set has an explicit parametrization.', '1501.00917-1-37-2': 'The second does not not, a severe flaw, and the third is very simple topologically.', '1501.00917-1-37-3': 'This stands in contrast to the compact case of [MATH] where there is only one connected component and every loop admitting a triangular factorization admitted a root subgroup factorization.', '1501.00917-1-37-4': 'This begs the question of why anyone should care about root subgroup factorization in the noncompact case.', '1501.00917-1-38-0': 'To close, we mention an application.', '1501.00917-1-38-1': 'The group [MATH] acts by bounded multiplication operators on the Hilbert space [MATH].', '1501.00917-1-38-2': 'As in chapter 6 of [CITATION], this defines a homomorphism of [MATH] into the restricted general linear group of [MATH] defined relative to the Hardy polarization [MATH], where [MATH] is the subspace of boundary values of functions in [MATH] and [MATH] is the subspace of boundary values of functions in [MATH].', '1501.00917-1-38-3': 'For a loop [MATH], let [MATH] (respectively, [MATH]) denote the corresponding Toeplitz operator, i.e., the compression of multiplication by [MATH] to [MATH] (resp., the shifted Toeplitz operator, i.e. the compression to [MATH]).', '1501.00917-1-38-4': 'It is well known that [MATH] and [MATH] are determinant class operators (i.e., of the form [MATH]).', '1501.00917-1-39-0': 'Suppose that [MATH] has a root subgroup factorization as in part (b) of Theorem [REF].', '1501.00917-1-39-1': 'Then [EQUATION] and if [MATH] is the triangular factorization as in ([REF]) (with [MATH]), then [EQUATION].', '1501.00917-1-40-0': 'When [MATH] and [MATH] are the zero sequences (the abelian case), the first formula specializes to a result of Szego and Widom (see Theorem 7.1 of [CITATION]).', '1501.00917-1-40-1': 'Estelle Basor pointed out to us that this result, for [MATH] as in ([REF]), can be deduced from Theorem 5.1 of [CITATION].', '1501.00917-1-40-2': 'More recently Basor and Torsten Ehrhardt have discovered a more elementary proof of Theorem [REF] (which does not make explicit use of Kac-Moody extensions).', '1501.00917-1-41-0': '## The General Setting of This Paper', '1501.00917-1-42-0': 'The fundamental assumption of this paper is that [EQUATION].', '1501.00917-1-42-1': 'It is well known that this condition is equivalent to a number of other conditions: [MATH] the Cartan involution [MATH] for the pair [MATH] is inner; [MATH] has discrete series unitary representations; [MATH]; the quotients [MATH] and [MATH] are Hermitian symmetric.', '1501.00917-1-43-0': 'This equal rank condition implies the existence of a (symmetric space compatible) triangular decomposition of [MATH] such that each positive root is either of compact or noncompact type.', '1501.00917-1-43-1': 'In the compact case the corresponding root homomorphism induces an embedding [MATH], and complementing the torus we obtain a sphere.', '1501.00917-1-43-2': 'In the noncompact case the corresponding root homomorphism induces an embedding of the rank one diagram ([REF]) into the diagram ([REF]), and complementing the torus we obtain a disk.', '1501.00917-1-43-3': 'This is the origin for the diversity of basic building blocks for the factorization in the inner noncompact case.', '1501.00917-1-44-0': '## Plan of the Paper', '1501.00917-1-45-0': 'This paper consists of two parts.', '1501.00917-1-45-1': 'The first part of the paper concerns Birkhoff and root subgroup factorization for the finite dimensional groups (the constant loops) appearing in ([REF]).', '1501.00917-1-45-2': 'The second part concerns Birkhoff and root subgroup factorization for the corresponding loop groups.', '1501.00917-1-46-0': 'Section [REF] is on background for finite dimensional groups.', '1501.00917-1-47-0': 'Section [REF] concerns factorization for the finite dimensional groups [MATH] and [MATH].', '1501.00917-1-47-1': 'The compact case is relatively well-understood, thanks in large part to Lu (see especially [CITATION]).', '1501.00917-1-47-2': 'We review this, with emphasis on the algorithm for root subgroup factorization (which depends on an ordering of noninverted roots), because this is an important guide in the loop cases.', '1501.00917-1-47-3': 'In finite dimensions the noncompact inner case largely reduces to the compact case, because of the existence of a "block (or coarse) triangular decomposition".', '1501.00917-1-47-4': 'But there is one part of the argument which is indirect, i.e., not algebraic: this is in showing that everything in a component of the Birkhoff decomposition, [MATH], has a root subgroup factorization.', '1501.00917-1-47-5': 'This concludes part one.', '1501.00917-1-48-0': 'Section [REF] is more background, needed for loop groups.', '1501.00917-1-48-1': 'The last subsection describes the basic framework for the remainder of the paper.', '1501.00917-1-49-0': 'In section [REF] we consider the intersection of the Birkhoff decomposition for [MATH] with [MATH].', '1501.00917-1-49-1': 'Unfortunately for loops in [MATH], there does not exist an analogue of "block (coarse) triangular decomposition".', '1501.00917-1-49-2': 'Consequently there does not exist a reduction to the compact type case, as in finite dimensions.', '1501.00917-1-49-3': 'One might still naively expect that there could be a relatively transparent way to parameterize the Birkhoff components intersected with [MATH] (as in the finite dimensional case, and in the case of loops into compact groups, e.g. using root subgroup factorization).', '1501.00917-1-49-4': 'But these intersections turn out to not be so simple topologically.', '1501.00917-1-49-5': 'Most of the section is devoted to rank one examples which illustrate the basic complications.', '1501.00917-1-50-0': 'In Section [REF] we consider root subgroup factorization for generic loops in [MATH].', '1501.00917-1-50-1': 'Our objective in this section is to prove partial analogues of Theorems 4.1, 4.2, and 5.1 of [CITATION], for generic loops in the identity component of (the Kac-Moody central extension of) [MATH] (when [MATH] is of inner type).', '1501.00917-1-50-2': 'As in the rank one case above, all of the statements have to be severely modified.', '1501.00917-1-50-3': 'The structures of the arguments in this noncompact context are roughly the same as in [CITATION].', '1501.00917-1-50-4': 'But there are obviously important differences, and in this paper we will present all of the details in this generic context.'} | {'1501.00917-2-0-0': 'In [CITATION] we showed that a loop in a simply connected compact Lie group [MATH] has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group).', '1501.00917-2-0-1': 'In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group [MATH] of inner type.', '1501.00917-2-0-2': 'In [CITATION] we showed that for an element of [MATH], i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a root subgroup factorization.', '1501.00917-2-0-3': 'However for loops in [MATH], while a root subgroup factorization implies a unique Birkhoff factorization, there are several obstacles to the converse.', '1501.00917-2-0-4': 'As in the compact case, root subgroup factorization is intimately related to factorization of Toeplitz determinants.', '1501.00917-2-1-0': '2000 Mathematics Subject Classifications: 22E67', '1501.00917-2-2-0': '# Introduction', '1501.00917-2-3-0': 'Finite dimensional Riemannian symmetric spaces come in dual pairs, one of compact type and one of noncompact type.', '1501.00917-2-3-1': 'Given such a pair, there is a diagram of finite dimensional groups [EQUATION] where [MATH] is the universal covering of the identity component of the isometry group of the compact type symmetric space [MATH], [MATH] is the complexification of [MATH], and [MATH] is a covering of the isometry group for the dual noncompact symmetric space [MATH].', '1501.00917-2-4-0': 'The main purpose of this paper is to investigate Birkhoff (or triangular) factorization and "root subgroup factorization" for the loop group of [MATH], assuming [MATH] is of inner type.', '1501.00917-2-4-1': 'Birkhoff factorization is investigated in [CITATION] and [CITATION], from various points of view.', '1501.00917-2-4-2': 'In particular Birkhoff factorization for [MATH] is developed in Chapter 8 of [CITATION], using the Grassmannian model for the homogeneous space [MATH].', '1501.00917-2-4-3': 'Root subgroup factorization for generic loops in [MATH] appeared more recently in [CITATION] (for [MATH], the rank one case) and in [CITATION].', '1501.00917-2-4-4': 'The Birkhoff decomposition for [MATH], i.e. the intersection of the Birkhoff decomposition for [MATH] with [MATH], is far more complicated than for [MATH].', '1501.00917-2-4-5': 'With respect to root subgroup factorization, beyond loops in a torus (corresponding to imaginary roots), in the compact context the basic building blocks are exclusively spheres (corresponding to real roots), and in the inner noncompact context the building blocks are a combination of spheres and disks.', '1501.00917-2-4-6': 'This introduces additional analytic complications, and perhaps the main point of this paper is to communicate the problems that arise from noncompactness.', '1501.00917-2-5-0': 'For [MATH], the basic fact is that [MATH] has a unique triangular factorization if and only if [MATH] has a unique "root subgroup factorization" (relative to the choice of a reduced sequence of simple reflections in the affine Weyl group).', '1501.00917-2-5-1': 'This is also true for elements of [MATH] (constant loops); see [CITATION].', '1501.00917-2-5-2': 'However, somewhat to our surprise, this is far from true for loops in [MATH].', '1501.00917-2-6-0': 'Relatively little sophistication is required to state the basic results, and identify the basic obstacles, in the rank one noncompact case.', '1501.00917-2-6-1': 'This is essentially because (in addition to loops in a torus) the basic building blocks are exclusively disks, and there is essentially a unique way to choose a reduced sequence of simple reflections in the affine Weyl group, so that the dependence on this choice can be suppressed.', '1501.00917-2-7-0': '## The Rank 1 Case', '1501.00917-2-8-0': 'We consider the data determined by the Riemann sphere and the Poincare disk.', '1501.00917-2-8-1': 'For this pair, the diagram ([REF]) becomes [EQUATION]', '1501.00917-2-8-2': 'Let [MATH] denote the group consisting of maps [MATH] having finite Fourier series, with pointwise multiplication.', '1501.00917-2-8-3': 'The subset of those functions having values in [MATH] is then a subgroup, denoted [MATH].', '1501.00917-2-9-0': 'For each [MATH] and [MATH], the function [MATH] defined by [EQUATION] is in [MATH].', '1501.00917-2-10-0': '[MATH] and [MATH] are dense in the smooth loop groups [MATH] and [MATH], respectively.', '1501.00917-2-10-1': 'This is proven in the compact case in Proposition 3.5.3 of [CITATION], and the argument applies also for [MATH], taking into account the obvious modifications.', '1501.00917-2-11-0': 'For a Laurent series [MATH], let [MATH].', '1501.00917-2-11-1': 'If [MATH] is a domain on the Riemann sphere, we write [MATH] for the vector space of holomorphic scalar valued functions on [MATH].', '1501.00917-2-11-2': 'If [MATH], then [MATH], where [MATH] denotes the open unit disk at [MATH].', '1501.00917-2-12-0': 'Suppose that [MATH] and fix [MATH].', '1501.00917-2-12-1': 'Consider the following three statements:', '1501.00917-2-13-0': '[(I.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are polynomials in [MATH] of order [MATH] and [MATH], respectively, with [MATH].', '1501.00917-2-13-1': '[(I.2)] [MATH] has a "root subgroup factorization" of the form [EQUATION] for some sequence [MATH] in [MATH] and [MATH] is the function in Example [REF].', '1501.00917-2-13-2': '[(I.3)] [MATH] has triangular factorization of the form [EQUATION] where [MATH], the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-2-14-0': 'Statements (I.1) and (I.3) are equivalent.', '1501.00917-2-14-1': '(I.2) implies (I.1) and (I.3).', '1501.00917-2-14-2': 'If [MATH] is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds, i.e. [MATH] has a root subgroup factorization as in (I.2).', '1501.00917-2-15-0': 'There is a similar set of implications for [MATH] and the following statements:', '1501.00917-2-16-0': '[(II.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are polynomials in [MATH] of order [MATH] and [MATH], respectively, with [MATH] and [MATH].', '1501.00917-2-16-1': '[(II.2)] [MATH] has a "root subgroup factorization" of the form [EQUATION] for some sequence [MATH] in [MATH] and [MATH] is the function in Example [REF].', '1501.00917-2-16-2': '[(II.3)] [MATH] has a triangular factorization of the form [EQUATION] where [MATH], and the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-2-17-0': 'When [MATH] and [MATH] have root subgroup factorizations, the scalar entries determining the diagonal factor have the product form [EQUATION]', '1501.00917-2-17-1': 'In general we do not know how to describe the connected component in the first and third conditions.', '1501.00917-2-17-2': 'The following example shows how disconnectness arises in the simplest nontrivial case.', '1501.00917-2-18-0': 'Consider the case [MATH] and [MATH] is as in II.3 with [MATH], [MATH], [EQUATION] and [EQUATION].', '1501.00917-2-18-1': 'It is straightforward to check that this [MATH] does indeed have values in [MATH].', '1501.00917-2-18-2': 'In order for [MATH], there are two possibilities: the first is that both the numerator and denominator are positive, in which case there is a root subgroup factorization, and the second is that both the top and bottom are negative, in which case root subgroup factorization fails (because when there is a root subgroup factorization, [MATH], and we must have [MATH]).', '1501.00917-2-19-0': 'In order to formulate a general factorization result, we need a [MATH] version of Theorem [REF].', '1501.00917-2-20-0': 'Suppose that [MATH].', '1501.00917-2-20-1': 'The following conditions are equivalent:', '1501.00917-2-21-0': '[(I.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are holomorphic in [MATH] and have [MATH] boundary values, with [MATH].', '1501.00917-2-21-1': '[(I.3)] [MATH] has triangular factorization of the form [EQUATION] where [MATH] is holomorphic in [MATH] with [MATH] boundary values, [MATH], and the third factor is a matrix valued polynomial in [MATH] which is unipotent upper triangular at [MATH].', '1501.00917-2-22-0': 'Similarly if [MATH], the following statements are equivalent:', '1501.00917-2-23-0': '[(II.1)] [MATH] is of the form [EQUATION] where [MATH] and [MATH] are holomorphic in [MATH] and have [MATH] boundary values, with [MATH] and [MATH].', '1501.00917-2-23-1': '[(II.3)] [MATH] has a triangular factorization of the form [EQUATION] where [MATH], [MATH] is holomorphic in [MATH] and has [MATH] boundary values, [MATH], and the third factor is a matrix valued function which is holomorphic in [MATH] and has [MATH] boundary values, and is unipotent upper triangular at [MATH].', '1501.00917-2-24-0': 'Let [MATH] denote the anti-holomorphic involution of [MATH] which fixes [MATH]; explicitly [EQUATION].', '1501.00917-2-25-0': 'The following theorem is the analogue of Theorem 0.2 of [CITATION] (the notation in part (b) is taken from Section 1 of [CITATION], and reviewed below the statement of the theorem).', '1501.00917-2-26-0': 'Suppose [MATH], the identity component.', '1501.00917-2-26-1': 'Then [MATH] has a unique "partial root subgroup factorization" of the form [EQUATION] where [MATH] and [MATH] and [MATH] are as in Theorem [REF], if and only [MATH] has a triangular factorization [MATH] (see ([REF]) below) such that the boundary values of [MATH] and [MATH] are [MATH] in magnitude on [MATH].', '1501.00917-2-27-0': 'The following example shows that the unaesthetic condition on the boundary values in part (b) is essential.', '1501.00917-2-28-0': 'Consider [MATH] as in Theorem [REF].', '1501.00917-2-28-1': 'The loop [MATH] (the Hermitian conjugate of [MATH] around the circle) has triangular factorization [EQUATION].', '1501.00917-2-28-2': 'If [MATH], then [MATH] and [MATH], and this loop will often not satisfy the condition [MATH] on [MATH].', '1501.00917-2-28-3': 'In this case [MATH] will not have a partial root subgroup factorization in the sense of Theorem [REF].', '1501.00917-2-29-0': 'The group [MATH] has a Birkhoff decomposition [EQUATION] where [MATH] (an affine Weyl group, and in this case the infinite dihedral group) is a quotient of a discrete group of unitary loops [EQUATION] where [EQUATION] (the reflections corresponding to the two simple roots for the Kac-Moody extension of [MATH]).', '1501.00917-2-29-1': 'The set [MATH] consists of loops which have a (Birkhoff) factorization of the form [EQUATION] where [MATH], [EQUATION] [MATH] has smooth boundary values on [MATH], [MATH], [MATH], [MATH], [MATH], [EQUATION] and [MATH] has smooth boundary values on [MATH].', '1501.00917-2-29-2': 'If [MATH], the generic case, then we say (as in Section 1 of [CITATION]) that [MATH] has a triangular factorization, and in this case the factors are unique.', '1501.00917-2-30-0': 'Next, let [MATH] denote the connected component containing [MATH], and let [EQUATION] and [EQUATION].', '1501.00917-2-30-1': 'Based on finite dimensional results in [CITATION], and the compact case, one might expect the following to be true:', '1501.00917-2-31-0': '[(1)] Modulo [MATH], the circle subgroup, it should be possible to contract [MATH] down to [MATH]; in particular [MATH] should be empty unless [MATH] is represented by a loop in [MATH], e.g., [MATH] for some [MATH].', '1501.00917-2-31-1': '[(2)] [MATH].', '1501.00917-2-31-2': '[(3)] Each [MATH] should admit a relatively explicit parameterization.', '1501.00917-2-32-0': 'Statements (1) and (2) are definitely false; statement (3) is very elusive, if not doubtful.', '1501.00917-2-33-0': '[MATH]', '1501.00917-2-34-0': '[(a)] [MATH] nonempty does not imply that [MATH] is represented by a loop in [MATH].', '1501.00917-2-34-1': 'For example, [MATH] is nonempty.', '1501.00917-2-34-2': '[(b)] [MATH] is properly contained in [MATH].', '1501.00917-2-35-0': 'To summarize, the set of loops having a root subgroup factorization is properly contained in the set of loops in the identity component which have a triangular factorization which, in turn, is a proper subset of the identity component of [MATH].', '1501.00917-2-35-1': 'The first set has an explicit parametrization.', '1501.00917-2-35-2': 'The second does not, a severe flaw, and the third is very simple topologically.', '1501.00917-2-35-3': 'This stands in contrast to the compact case of [MATH] where there is only one connected component and every loop admitting a triangular factorization admitted a root subgroup factorization.', '1501.00917-2-36-0': '## Toeplitz Determinants', '1501.00917-2-37-0': 'The group [MATH] acts by bounded multiplication operators on the Hilbert space [MATH].', '1501.00917-2-37-1': 'As in chapter 6 of [CITATION], this defines a homomorphism of [MATH] into the restricted general linear group of [MATH] defined relative to the Hardy polarization [MATH], where [MATH] is the subspace of boundary values of functions in [MATH] and [MATH] is the subspace of boundary values of functions in [MATH].', '1501.00917-2-37-2': 'For a loop [MATH], let [MATH] (respectively, [MATH]) denote the corresponding Toeplitz operator, i.e., the compression of multiplication by [MATH] to [MATH] (resp., the shifted Toeplitz operator, i.e. the compression to [MATH]).', '1501.00917-2-37-3': 'It is well known that [MATH] and [MATH] are determinant class operators (i.e., of the form [MATH]).', '1501.00917-2-38-0': 'Suppose that [MATH] has a root subgroup factorization as in part (b) of Theorem [REF].', '1501.00917-2-38-1': 'Then [EQUATION] and if [MATH] is the triangular factorization as in ([REF]) (with [MATH]), then [EQUATION].', '1501.00917-2-39-0': 'When [MATH] and [MATH] are the zero sequences (the abelian case), the first formula specializes to a result of Szego and Widom (see Theorem 7.1 of [CITATION]).', '1501.00917-2-39-1': 'Estelle Basor pointed out to us that this result, for [MATH] as in ([REF]), can be deduced from Theorem 5.1 of [CITATION].', '1501.00917-2-40-0': '## Additional Motivation', '1501.00917-2-41-0': 'There is potentially a useful analogue of root subgroup factorization for the group of homeomorphisms of a circle (see [CITATION]).', '1501.00917-2-41-1': 'This is a very complicated (negatively curved type) group, and we are currently stumbling around in trying to understand what can go wrong.', '1501.00917-2-41-2': 'In this paper our primary contribution is perhaps to identify what can go wrong with Birkhoff and root subgroup factorization for loops into a noncompact target.', '1501.00917-2-41-3': 'There are other variations on root subgroup factorization as well (see [CITATION]).', '1501.00917-2-42-0': 'From another point of view, it is expected that root subgroup factorization is relevant to finding Darboux coordinates for homogeneous Poisson structures on [MATH] and [MATH] (see [CITATION]).', '1501.00917-2-42-1': 'As of this writing, this is an open question.', '1501.00917-2-43-0': '## Plan of the Paper This paper is essentially a sequel to [CITATION] and [CITATION].', '1501.00917-2-43-1': "We will refer to the latter paper as the 'finite dimensional case', and we note the differences as we go along.", '1501.00917-2-44-0': 'Section [REF] is on background for finite dimensional groups (which is identical to [CITATION]) and loop groups.', '1501.00917-2-44-1': 'In section [REF] we consider the intersection of the Birkhoff decomposition for [MATH] with [MATH].', '1501.00917-2-44-2': 'Unfortunately for loops in [MATH], there does not exist an analogue of "block (coarse) triangular decomposition", a key feature of the finite dimensional case.', '1501.00917-2-44-3': 'Consequently there does not exist a reduction to the compact type case, as in finite dimensions.', '1501.00917-2-44-4': 'One might still naively expect that there could be a relatively transparent way to parameterize the Birkhoff components intersected with [MATH] (as in the finite dimensional case, and in the case of loops into compact groups, e.g. using root subgroup factorization).', '1501.00917-2-44-5': 'But these intersections turn out to not be so simple topologically.', '1501.00917-2-44-6': 'Most of the section is devoted to rank one examples which illustrate the basic complications.', '1501.00917-2-45-0': 'In Section [REF] we consider root subgroup factorization for generic loops in [MATH].', '1501.00917-2-45-1': 'Our objective in this section is to prove partial analogues of Theorems 4.1, 4.2, and 5.1 of [CITATION], for generic loops in the identity component of (the Kac-Moody central extension of) [MATH] (when [MATH] is of inner type).', '1501.00917-2-45-2': 'As in the rank one case above, all of the statements have to be severely modified.', '1501.00917-2-45-3': 'The structures of the arguments in this noncompact context are roughly the same as in [CITATION], but there many differences in the details (reflected in the more complicated statements of theorems).'} | [['1501.00917-1-10-0', '1501.00917-2-8-0'], ['1501.00917-1-10-1', '1501.00917-2-8-1'], ['1501.00917-1-10-2', '1501.00917-2-8-2'], ['1501.00917-1-10-3', '1501.00917-2-8-3'], ['1501.00917-1-6-2', '1501.00917-2-4-2'], ['1501.00917-1-6-3', '1501.00917-2-4-3'], ['1501.00917-1-6-4', '1501.00917-2-4-4'], ['1501.00917-1-6-5', '1501.00917-2-4-5'], ['1501.00917-1-6-6', '1501.00917-2-4-6'], 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'1501.00917-2-18-0'], ['1501.00917-1-33-0', '1501.00917-2-31-0'], ['1501.00917-1-0-0', '1501.00917-2-0-0'], ['1501.00917-1-0-1', '1501.00917-2-0-1'], ['1501.00917-1-49-1', '1501.00917-2-44-2'], ['1501.00917-1-7-2', '1501.00917-2-5-1'], ['1501.00917-1-7-2', '1501.00917-2-5-2'], ['1501.00917-1-50-3', '1501.00917-2-45-3'], ['1501.00917-1-1-0', '1501.00917-2-0-2'], ['1501.00917-1-2-2', '1501.00917-2-0-2'], ['1501.00917-1-2-2', '1501.00917-2-0-3']] | [['1501.00917-1-10-0', '1501.00917-2-8-0'], ['1501.00917-1-10-1', '1501.00917-2-8-1'], ['1501.00917-1-10-2', '1501.00917-2-8-2'], ['1501.00917-1-10-3', '1501.00917-2-8-3'], ['1501.00917-1-6-2', '1501.00917-2-4-2'], ['1501.00917-1-6-3', '1501.00917-2-4-3'], ['1501.00917-1-6-4', '1501.00917-2-4-4'], ['1501.00917-1-6-5', '1501.00917-2-4-5'], ['1501.00917-1-6-6', '1501.00917-2-4-6'], ['1501.00917-1-19-0', '1501.00917-2-17-0'], ['1501.00917-1-28-0', '1501.00917-2-26-0'], ['1501.00917-1-28-1', '1501.00917-2-26-1'], ['1501.00917-1-37-0', 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['1501.00917-1-32-0', '1501.00917-2-30-0'], ['1501.00917-1-20-1', '1501.00917-2-18-1'], ['1501.00917-1-20-2', '1501.00917-2-18-2'], ['1501.00917-1-31-0', '1501.00917-2-29-0'], ['1501.00917-1-31-1', '1501.00917-2-29-1'], ['1501.00917-1-31-2', '1501.00917-2-29-2'], ['1501.00917-1-27-0', '1501.00917-2-25-0'], ['1501.00917-1-5-0', '1501.00917-2-3-0'], ['1501.00917-1-5-1', '1501.00917-2-3-1'], ['1501.00917-1-12-0', '1501.00917-2-10-0'], ['1501.00917-1-12-1', '1501.00917-2-10-1'], ['1501.00917-1-15-0', '1501.00917-2-13-0'], ['1501.00917-1-15-1', '1501.00917-2-13-1'], ['1501.00917-1-15-2', '1501.00917-2-13-2'], ['1501.00917-1-23-0', '1501.00917-2-21-0'], ['1501.00917-1-23-1', '1501.00917-2-21-1'], ['1501.00917-1-34-0', '1501.00917-2-32-0'], ['1501.00917-1-29-0', '1501.00917-2-27-0'], ['1501.00917-1-16-0', '1501.00917-2-14-0'], ['1501.00917-1-16-2', '1501.00917-2-14-2'], ['1501.00917-1-33-2', '1501.00917-2-31-2'], ['1501.00917-1-36-0', '1501.00917-2-34-0'], ['1501.00917-1-36-1', '1501.00917-2-34-1'], ['1501.00917-1-36-2', '1501.00917-2-34-2'], ['1501.00917-1-7-0', '1501.00917-2-5-0'], ['1501.00917-1-21-0', '1501.00917-2-19-0'], ['1501.00917-1-38-1', '1501.00917-2-37-0'], ['1501.00917-1-38-2', '1501.00917-2-37-1'], ['1501.00917-1-38-3', '1501.00917-2-37-2'], ['1501.00917-1-38-4', '1501.00917-2-37-3'], ['1501.00917-1-50-0', '1501.00917-2-45-0'], ['1501.00917-1-50-1', '1501.00917-2-45-1'], ['1501.00917-1-50-2', '1501.00917-2-45-2'], ['1501.00917-1-39-0', '1501.00917-2-38-0'], ['1501.00917-1-39-1', '1501.00917-2-38-1'], ['1501.00917-1-2-3', '1501.00917-2-0-4']] | [['1501.00917-1-6-0', '1501.00917-2-4-0'], ['1501.00917-1-6-1', '1501.00917-2-4-1'], ['1501.00917-1-19-1', '1501.00917-2-17-1'], ['1501.00917-1-37-2', '1501.00917-2-35-2'], ['1501.00917-1-8-1', '1501.00917-2-6-1'], ['1501.00917-1-20-0', '1501.00917-2-18-0'], ['1501.00917-1-33-0', '1501.00917-2-31-0'], ['1501.00917-1-0-0', '1501.00917-2-0-0'], ['1501.00917-1-0-1', '1501.00917-2-0-1']] | [] | [['1501.00917-1-49-1', '1501.00917-2-44-2'], ['1501.00917-1-7-2', '1501.00917-2-5-1'], ['1501.00917-1-7-2', '1501.00917-2-5-2'], ['1501.00917-1-50-3', '1501.00917-2-45-3'], ['1501.00917-1-1-0', '1501.00917-2-0-2'], ['1501.00917-1-2-2', '1501.00917-2-0-2'], ['1501.00917-1-2-2', '1501.00917-2-0-3']] | [] | ['1501.00917-1-3-0', '1501.00917-1-11-0', '1501.00917-1-14-0', '1501.00917-1-14-1', '1501.00917-1-16-1', '1501.00917-1-17-0', '1501.00917-1-22-0', '1501.00917-1-22-1', '1501.00917-1-24-0', '1501.00917-1-26-0', '1501.00917-1-32-1', '1501.00917-1-33-1', '1501.00917-1-35-0', '1501.00917-1-46-0', '1501.00917-2-1-0', '1501.00917-2-9-0', '1501.00917-2-12-0', '1501.00917-2-12-1', '1501.00917-2-14-1', '1501.00917-2-15-0', '1501.00917-2-20-0', '1501.00917-2-20-1', '1501.00917-2-22-0', '1501.00917-2-24-0', '1501.00917-2-30-1', '1501.00917-2-31-1', '1501.00917-2-33-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1501.00917 | null | null | null | null | null |
1610.09867 | {'1610.09867-1-0-0': ': In a recent paper [CITATION], the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments.', '1610.09867-1-0-1': 'The present work corresponds to the regularity part of the method.', '1610.09867-1-0-2': 'Precisely we establish that any weakly conformal [MATH] map from a riemann surface [MATH] into a closed oriented sub-manifold [MATH] of an euclidian space [MATH] realizes a stationary varifold if and only if it is a smooth conformal harmonic map form [MATH] into [MATH].', '1610.09867-1-1-0': 'Math.', '1610.09867-1-1-1': 'Class.', '1610.09867-1-1-2': '58E20, 49Q05, 53A10, 49Q15, 49Q20', '1610.09867-1-2-0': '# Introduction', '1610.09867-1-3-0': 'In [CITATION] the author developed a viscosity method in order to produce closed minimal 2 dimensional surfaces into any arbitrary closed oriented sub-manifolds [MATH] of any euclidian spaces [MATH] by min-max type arguments.', '1610.09867-1-3-1': 'The method consists in adding to the area of an immersion [MATH] of a surface [MATH] into [MATH] a more coercive term such as the [MATH] norm of the second fundamental form preceded by a small parameter [MATH]: [EQUATION] where [MATH] is the second fundamental form of the immersion [MATH] and [MATH] is the volume form associated to the induced metric.', '1610.09867-1-3-2': 'For [MATH] and [MATH] one proves that the Lagrangians [MATH] are Palais-Smale in some ad-hoc Finsler bundle of immersions complete for the Palais distance.', '1610.09867-1-3-3': 'By applying the now classical Palais-Smale deformation theory in infinite dimensional space one can then produce critical points [MATH] to [MATH].', '1610.09867-1-3-4': 'It is proved in [CITATION] that, for a sequence of parameters [MATH], the sequence of integer rectifiable varifolds associated to the immersion of [MATH] by [MATH] does not necessarily converge to a stationary integer rectifiable varifold.', '1610.09867-1-3-5': 'However, by applying Struwe\'s monotonicity trick one can always select a sequence [MATH] such that the following additional "entropy estimate" holds [EQUATION]', '1610.09867-1-3-6': 'Assuming this additional estimate, the main achievement of [CITATION] is to prove that the immersion of [MATH] by [MATH] varifold converges to a stationary integer rectifiable varifold given by the image of a smooth riemann surface [MATH] by a weakly conformal [MATH] map [MATH] into [MATH].', '1610.09867-1-3-7': 'The main result of the paper is to prove that such a map is smooth and satisfies the harmonic map equation.', '1610.09867-1-3-8': 'Precisely we define the notion of target harmonic map as follows.', '1610.09867-1-4-0': 'Let [MATH] be a smooth closed riemann surface.', '1610.09867-1-4-1': 'A map [MATH] is target harmonic if for almost every smooth domain and [MATH] and any smooth function [MATH] supported in the complement of an open neighborhood of [MATH] we have [EQUATION] where [MATH] is any metric compatible with the chosen conformal structure on [MATH].', '1610.09867-1-4-2': '[MATH]', '1610.09867-1-5-0': 'One proves that for a weakly conformal [MATH] map into [MATH] the condition for [MATH] to be target harmonic is equivalent to say that the mapping of [MATH] in [MATH] defines a stationary integer rectifiable varifold.', '1610.09867-1-5-1': 'Our main result in the present paper is the following.', '1610.09867-1-6-0': 'Any weakly conformal target harmonic map into an arbitrary closed sub-manifold [MATH] of [MATH] in two dimension is smooth and satisfy the harmonic map equation.', '1610.09867-1-6-1': '[MATH]', '1610.09867-1-7-0': 'It is not known if, without the conformality assumption, a target harmonic map is an harmonic map in the classical sense.', '1610.09867-1-7-1': 'Following the main lines of the proof below one can prove that this is indeed the case in one dimension.'} | {'1610.09867-2-0-0': ': We establish that any weakly conformal [MATH] map from a Riemann surface [MATH] into a closed oriented sub-manifold [MATH] of an euclidian space [MATH] realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form [MATH] into [MATH].', '1610.09867-2-1-0': 'Math.', '1610.09867-2-1-1': 'Class.', '1610.09867-2-1-2': '58E20, 49Q05, 53A10, 49Q15, 49Q20', '1610.09867-2-2-0': '# Introduction', '1610.09867-2-3-0': 'In [CITATION] the author developed a viscosity method in order to produce closed minimal 2 dimensional surfaces into any arbitrary closed oriented sub-manifolds [MATH] of any euclidian spaces [MATH] by min-max type arguments.', '1610.09867-2-3-1': 'The method consists in adding to the area of an immersion [MATH] of a surface [MATH] into [MATH] a more coercive term such as the [MATH] norm of the second fundamental form preceded by a small parameter [MATH]: [EQUATION] where [MATH] is the second fundamental form of the immersion [MATH] and [MATH] is the volume form associated to the induced metric.', '1610.09867-2-3-2': 'For [MATH] and [MATH] one proves that the Lagrangians [MATH] are Palais-Smale in some ad-hoc Finsler bundle of immersions complete for the Palais distance.', '1610.09867-2-3-3': 'By applying the now classical Palais-Smale deformation theory in infinite dimensional space one can then produce critical points [MATH] to [MATH].', '1610.09867-2-3-4': 'It is proved in [CITATION] that, for a sequence of parameters [MATH], the sequence of integer rectifiable varifolds associated to the immersion of [MATH] by [MATH] does not necessarily converge to a stationary integer rectifiable varifold.', '1610.09867-2-3-5': 'However, by applying Struwe\'s monotonicity trick one can always select a sequence [MATH] such that the following additional "entropy estimate" holds [EQUATION]', '1610.09867-2-3-6': 'Assuming this additional estimate, the main achievement of [CITATION] is to prove that the immersion of [MATH] by [MATH] varifold converges to a stationary integer rectifiable varifold given by the image of a smooth Riemann surface [MATH] by a weakly conformal [MATH] map [MATH] into [MATH] equipped by an integer multiplicity.', '1610.09867-2-3-7': 'The main result of the paper is to prove that, when this multiplicity is constant, such a map is smooth and satisfies the harmonic map equation.', '1610.09867-2-3-8': 'To state our main result we need two definitions.', '1610.09867-2-4-0': 'A property is said to hold for almost every smooth domain in [MATH], if for any smooth domain [MATH] and any smooth function [MATH] such that [MATH] and [MATH] on [MATH] then for almost every [MATH] close enough to zero and regular value for [MATH] the property holds for the domain contained in [MATH] or containing [MATH] and bounded by [MATH].', '1610.09867-2-4-1': '[MATH]', '1610.09867-2-5-0': 'Precisely we define the notion of target harmonic map as follows.', '1610.09867-2-6-0': 'Let [MATH] be a smooth closed Riemann surface equipped with a metric compatible with the complex structure.', '1610.09867-2-6-1': 'A map [MATH] is target harmonic if for almost every smooth domain [MATH] and any smooth function [MATH] supported in the complement of an open neighborhood of [MATH] we have [EQUATION] where [MATH] is any metric compatible with the chosen conformal structure on [MATH] and where [MATH] denotes the second fundamental form of [MATH] at the point [MATH] and acting on the pair of vectors [MATH] and by an abuse of notation we write [EQUATION] [MATH]', '1610.09867-2-7-0': 'Observe that the main difference with the general definition of being harmonic is that, for target harmonic one restricts ([REF]) to test functions [MATH] supported in the target while for the definition of harmonic, one requires ([REF]) to hold for any [MATH] test function defined on the domain.', '1610.09867-2-7-1': 'Therefore, being harmonic implies to be target harmonic and the proof of the reverse is the goal of the present work.', '1610.09867-2-7-2': 'For a weakly conformal [MATH] map into [MATH] the condition for [MATH] to be target harmonic is equivalent to saying that the mapping of [MATH] in [MATH] defines a stationary integer rectifiable varifold (see proposition [REF]).', '1610.09867-2-7-3': 'Our main result in the present paper is the following.', '1610.09867-2-8-0': 'Any weakly conformal target harmonic map into an arbitrary closed sub-manifold [MATH] of [MATH] in two dimension is smooth and satisfy the harmonic map equation.', '1610.09867-2-8-1': '[MATH]', '1610.09867-2-9-0': 'It is not known if, without the conformality assumption, a target harmonic map is an harmonic map in the classical sense.', '1610.09867-2-9-1': 'Following the main lines of the proof below one can prove that this is indeed the case in one dimension.'} | [['1610.09867-1-5-1', '1610.09867-2-7-3'], ['1610.09867-1-7-0', '1610.09867-2-9-0'], ['1610.09867-1-7-1', '1610.09867-2-9-1'], ['1610.09867-1-6-0', '1610.09867-2-8-0'], ['1610.09867-1-3-0', '1610.09867-2-3-0'], ['1610.09867-1-3-1', '1610.09867-2-3-1'], ['1610.09867-1-3-2', '1610.09867-2-3-2'], ['1610.09867-1-3-3', '1610.09867-2-3-3'], ['1610.09867-1-3-4', '1610.09867-2-3-4'], ['1610.09867-1-3-5', '1610.09867-2-3-5'], ['1610.09867-1-3-8', '1610.09867-2-5-0'], ['1610.09867-1-5-0', '1610.09867-2-7-2'], ['1610.09867-1-0-2', '1610.09867-2-0-0'], ['1610.09867-1-4-1', '1610.09867-2-6-1'], ['1610.09867-1-3-6', '1610.09867-2-3-6'], ['1610.09867-1-3-7', '1610.09867-2-3-7'], ['1610.09867-1-4-0', '1610.09867-2-6-0']] | [['1610.09867-1-5-1', '1610.09867-2-7-3'], ['1610.09867-1-7-0', '1610.09867-2-9-0'], ['1610.09867-1-7-1', '1610.09867-2-9-1'], ['1610.09867-1-6-0', '1610.09867-2-8-0'], ['1610.09867-1-3-0', '1610.09867-2-3-0'], ['1610.09867-1-3-1', '1610.09867-2-3-1'], ['1610.09867-1-3-2', '1610.09867-2-3-2'], ['1610.09867-1-3-3', '1610.09867-2-3-3'], ['1610.09867-1-3-4', '1610.09867-2-3-4'], ['1610.09867-1-3-5', '1610.09867-2-3-5'], ['1610.09867-1-3-8', '1610.09867-2-5-0']] | [['1610.09867-1-5-0', '1610.09867-2-7-2'], ['1610.09867-1-0-2', '1610.09867-2-0-0'], ['1610.09867-1-4-1', '1610.09867-2-6-1'], ['1610.09867-1-3-6', '1610.09867-2-3-6'], ['1610.09867-1-3-7', '1610.09867-2-3-7']] | [] | [['1610.09867-1-4-0', '1610.09867-2-6-0']] | [] | ['1610.09867-1-1-0', '1610.09867-1-1-1', '1610.09867-1-1-2', '1610.09867-1-4-2', '1610.09867-1-6-1', '1610.09867-2-1-0', '1610.09867-2-1-1', '1610.09867-2-1-2', '1610.09867-2-4-1', '1610.09867-2-8-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1610.09867 | null | null | null | null | null |
1410.3095 | {'1410.3095-1-0-0': 'Bei der Rezeption der Denkpsychologie Piagets durch die Mathematikdidaktik der Bundesrepublik Deutschland Ende der 1960er Jahre wurde der Begriff der Gruppierung ähnlich einer mathematischen Struktur aufgefaßt.', '1410.3095-1-0-1': 'Kann die Berücksichtigung empirischer Theorien zu einem adäquateren Verständnis beitragen?', '1410.3095-1-1-0': "english # Abstract Reviewing Piaget's psychology of reasoning by the mathematical educators of the Federal Republic of Germany late in the 1960[MATH] the concept of grouping has been understood similar to a mathematical structure.", '1410.3095-1-1-1': 'Possibly to consider empirical theories can contribute to an understanding more adequate.', '1410.3095-1-2-0': 'Als Ende der 1960er Jahre die Rezeption der Piagetschen Denkpsychologie durch die Mathematikdidaktik der Bundesrepublik Deutschland in voller Blüte stand, faszinierte vor allem der Begriff der Gruppierung.', '1410.3095-1-2-1': 'Die Elemente einer Gruppierung - die Operationen oder verinnerlichten Handlungen - erfüllen folgende Bedingungen:', '1410.3095-1-3-0': '1.', '1410.3095-1-3-1': 'Komposition: x + x[MATH] = y, y + y[MATH] = z; etc.', '1410.3095-1-4-0': '2.', '1410.3095-1-4-1': 'Reversibilität: y - x = x[MATH] oder y - x[MATH] = x', '1410.3095-1-5-0': '3.', '1410.3095-1-5-1': 'Assoziativität: (x + x[MATH]) + y[MATH] = x + (x[MATH] + y[MATH]) = (z).', '1410.3095-1-6-0': '4.', '1410.3095-1-6-1': 'Allgemeine, identische Operation: x - x = o; y - y = o; etc.', '1410.3095-1-7-0': '5.', '1410.3095-1-7-1': 'Tautologie oder besondere, identische Operation: x + x = x; y + y =', '1410.3095-1-8-0': 'y; etc. (Piaget 1971, S. 49)', '1410.3095-1-9-0': 'Etwas anders formuliert:', '1410.3095-1-10-0': '1.', '1410.3095-1-10-1': '2.', '1410.3095-1-10-2': 'x + x[MATH] = y [MATH] y + (-x) = x[MATH] y + (-x[MATH]) = x', '1410.3095-1-11-0': '3.', '1410.3095-1-11-1': '(x + x[MATH]) + y[MATH] = x + (x[MATH] + y[MATH])', '1410.3095-1-12-0': '4.', '1410.3095-1-12-1': 'x + (-x) = (-x) + x = o', '1410.3095-1-13-0': '5.', '1410.3095-1-13-1': 'x + x = x', '1410.3095-1-14-0': 'In dieser Formulierung sieht man deutlicher, daß es nur eine Verknüpfung (+) für die Operationen gibt und daß es bzgl.', '1410.3095-1-14-1': 'der allgemeinen, identischen Operation (o) zu gewissen Operationen x je eine inverse Operation -x geben kann.', '1410.3095-1-15-0': 'Der Piagetsche Text legte es nahe, den Begriff der Gruppierung so zu verstehen, als besäße eine Gruppierung Modelle, in denen jede dieser Bedingungen erfüllt sei.', '1410.3095-1-15-1': 'Heinz Griesel wies als erster darauf hin, daß eine Gruppierung bei dieser Lesart nur einelementig sein könne, also nur ein triviales Modell habe [Griesel 1970, S. 128].', '1410.3095-1-16-0': 'Wenn man heute - rund 40 Jahre später - diese Diskussion noch einmal nachliest, stellt man erstaunt fest, wie selbstverständlich man annahm, Piagets Formulierungen als mathematisch tragfähig ansehen zu können.', '1410.3095-1-16-1': 'Eine Sichtweise, die allerdings nicht auf deutsche Mathematikdidaktiker beschränkt war.', '1410.3095-1-17-0': '"A grouping incorporates properties from two well-known mathematical structures, the group and the lattice.', '1410.3095-1-17-1': 'A Piagetian grouping is thus a kind of hybrid group-lattice structure" [Flavell 1985, S. 92].', '1410.3095-1-18-0': 'Mathematisch tragfähig erschienen Piagets Formulierungen in dem Sinne, daß die im folgenden genannten Kollegen offenbar davon ausgingen, daß die von ihm angegebenen Bedingungen in der Tat wie ein mathematisches Axiomensystem gelesen werden könnten.', '1410.3095-1-18-1': 'Dazu dürfte die Bezeichnung "Gruppierung" - in Analogie zur (mathematischen) "Gruppe" - und die formale Darstellung der Bedingungen entscheidend beigetragen haben, die aber vermutlich in erster Linie Piagets Neigung zur Mathematik zu danken waren.', '1410.3095-1-18-2': 'Denn die Bedingungen wie ein Axiomensystem zu lesen wirft doch einige wichtige Fragen auf.', '1410.3095-1-19-0': 'Erich Wittmann gab als erster das Axiomensystem einer Struktur an, in der - unter Berücksichtigung der Grieselschen Feststellung - die Bedingungen von Piaget hineininterpretiert werden konnten.', '1410.3095-1-19-1': 'Um die Bedingung der Tautologie zu erfassen führt er neben + eine weitere Verknüpfung ein, indem er mit Hilfe von + eine Quasiordnung unter den Operationen definiert, an die er verbandstheoretische Forderungen stellt [Wittmann 1973].', '1410.3095-1-19-2': 'Hans - Georg Steiner hat das von Wittmann angegebene Axiomensystem aufgegriffen und in eine vereinfachte Form gebracht [Steiner 1973].', '1410.3095-1-19-3': 'Elmar Cohors - Fresenborg beschrieb schließlich die Wittmannsche Gruppierung als ein Semi - Thue - System [Cohors - Fresenborg 1974].', '1410.3095-1-19-4': 'Die von diesen Autoren angegebenen Beispiele für Gruppierungen sind zeitbedingt stark an der "Neuen Mathematik" orientiert und wirken etwas artifiziell.', '1410.3095-1-19-5': 'Natürlich genügen sie dem Axiomensystem und eine gewisse Komplexität der Beispiele ist zwar einleuchtend, da Gruppierungen als Gleichgewichtsform der Operationen/verinnerlichten Handlungen den "Endzustand" der Stufe des konkret - operationalen Denkens beschreiben, auf der sich die Schüler befanden, für die die Beispiele gedacht waren.', '1410.3095-1-19-6': 'Doch erscheint es wenig einleuchtend, daß eine derart komplizierte Gleichgewichtsform, wie sie die Wittmannsche Struktur beschreibt, ohne eine sehr spezifische Förderung erreicht werden kann.', '1410.3095-1-19-7': 'Und dies überzeugt nicht.', '1410.3095-1-19-8': 'Denn nach der Auffassung Piagets erwerben wir die von ihm formulierten Fähigkeiten im Alltag.', '1410.3095-1-19-9': 'Verinnerlicht werden Handlungen, die real durchgeführt werden, eben weit überwiegend im Alltag und nicht in einer Unterrichtssituation.', '1410.3095-1-19-10': 'Es sind in der Regel solche, die auf die WELT ausgeübt werden.', '1410.3095-1-20-0': 'Die Operationen/verinnerlichter Handlungen werden von Piaget wie folgt charakterisiert:', '1410.3095-1-21-0': '"Die spezifische Natur der Operationen besteht, verglichen mit den empirischen Tätigkeiten, gerade in der Tatsache, daß sie niemals in diskontinuierlichem Zustand existieren.', '1410.3095-1-21-1': 'Es ist nur eine gänzlich unerlaubte Abstraktion, wenn man von einer[MATH]Operation spricht; eine vereinzelte Operation kann nicht Operation sein, denn die eigentümlichste Eigenschaft der Operationen liegt gerade darin, daß sie zu Systemen vereinigt sind" [Piaget 1971, S. 41].', '1410.3095-1-22-0': 'Es stellt sich die Frage nach der Organisationsform der real durchgeführten Handlungen, die den Operationen zugrunde liegen.', '1410.3095-1-22-1': 'Denn man kann zwar eine real durchgeführte Handlung als einzelne betrachten, aber aus einer einzelnen Handlung läßt sich kein "operatives Gesamtsystem" (Piaget) ableiten und aus einer Vielzahl von planlos durchgeführten Handlungen ebenfalls nicht.', '1410.3095-1-23-0': 'Das Verhalten von Kindern und damit auch ihr Handeln ist stark regelhaft.', '1410.3095-1-23-1': 'Sie verhalten sich mitunter, als verfügten sie über gewisse Theorien, die ihr Handeln steuern [Gopnik und Meltzoff 1997].', '1410.3095-1-23-2': 'Eine solche Theorie könnte z.B. ihren Umgang mit einem luftgefüllten Ball oder mit einem Luftballon steuern.', '1410.3095-1-23-3': 'Diese Theorien beziehen sich unmittelbar auf die WELT, ihre Begriffe haben eine starke ontologische Bindung.', '1410.3095-1-23-4': 'Man spricht gemeinhin von empirischen Theorien.', '1410.3095-1-23-5': 'Eine etablierte Form zur Darstellung empirischer Theorien ist die strukturalistische Metatheorie, die von Wolfgang Stegmüller und seinem Kreis entwickelt wurde [Stegmüller 1973, 1986], [Balzer 1982].', '1410.3095-1-23-6': 'Ein wesentliches Merkmal dieser Darstellungsform ist, daß sie den Aufbau der Theorie wiedergibt.', '1410.3095-1-24-0': 'Die Darstellung gliedert sich in drei Stufen.', '1410.3095-1-24-1': 'Auf der ersten Stufe werden als sog.', '1410.3095-1-24-2': 'partielle Modelle der Theorie die empirischen Gegebenheiten formuliert, über die die Theorie Aussagen machen kann.', '1410.3095-1-24-3': 'Diese werden aus sog.', '1410.3095-1-24-4': 'paradigmatischen Beispielen abgeleitet.', '1410.3095-1-24-5': 'Im vorliegenden Fall betrifft dies die Objekte, auf die reale Handlungen ausgeübt werden sowie diese Handlungen selbst.', '1410.3095-1-24-6': 'Die nächste Stufe - die der potentiellen Modelle - betrifft die Sprache, in der die Theorie formuliert wird.', '1410.3095-1-24-7': 'Sie soll möglichst präzise und deshalb formalisierbar sein.', '1410.3095-1-24-8': 'Es liegt daher nahe, auf Begriffe der Mathematik, z.B. Relationen oder Funktionen, zurückzugreifen.', '1410.3095-1-24-9': 'Sie erlauben es, die partiellen Modelle formal zu beschreiben.', '1410.3095-1-24-10': 'Während auf der ersten Stufe noch alle Begriffe reale Referenzen haben oder aus bekannten (früher erworbenen) Theorien stammen, bedarf die neu zu formulierende Theorie neuer Begriffe, will sie neues Wissen vermitteln.', '1410.3095-1-24-11': 'Es sind die bzgl.', '1410.3095-1-24-12': 'der in den partiellen Modellen formulierten empirischen Gegebenheiten theoretischen Begriffe, solche, die dort keine Referenzen haben, die erst durch die neu zu formulierende Theorie eine Bedeutung erhalten.', '1410.3095-1-24-13': 'Um diese wird die Sprache der Theorie erweitert.', '1410.3095-1-24-14': 'Auf der dritten Stufe werden schließlich die Axiome formuliert, die die Modelle der Theorie definieren.', '1410.3095-1-24-15': 'Die Formulierung der Modelle ist hier in dem Sinne von besonderem Interesse als auf dieser Stufe die realen Handlungen der ersten Stufe durch mathematische Strukturen beschrieben, d.h. in Systeme eingebunden werden.', '1410.3095-1-24-16': 'Die Systeme der realen Handlungen bilden die Vorlage für die Systeme der verinnerlichten Handlungen.', '1410.3095-1-24-17': 'Man kann die Modelle als die formalen Spiegelbilder der Systeme der Operationen ansehen, die auf der kognitiven Ebene das Verfügen über die empirische Theorie ausmachen.', '1410.3095-1-24-18': 'Dies will sagen, daß eine empirische Theorie zu erwerben auf psychologischer Seite bedeutet, bestimmte Handlungen zu verinnerlichen.', '1410.3095-1-24-19': 'Die Organisationsstruktur der verinnerlichten Handlungen, die man entwickelt, wenn man über eine empirische Theorie zu verfügen lernt, kann einzelne der Piagetschen Bedingungen erfüllen.', '1410.3095-1-24-20': 'Alle Bedingungen einer Gruppierung gleichzeitig zu erfüllen ist - in der Piagetschen Lesart - wegen der Unvereinbarkeit von allgemeiner und besonderer, identischer Operation ohnehin nicht möglich.', '1410.3095-1-25-0': 'Die Eigenschaften einer Gruppierung sind bei dieser Sichtweise ein Extrakt der Eigenschaften real durchgeführter Handlungen, die der einzelne verinnerlicht, wenn er über unterschiedliche empirische Theorien zu verfügen lernt.', '1410.3095-1-25-1': 'Ein so verstandener Gruppierungsbegriff enzieht sich einer mathematischen Modellierung, trifft aber die psychologischen Intentionen, die Piaget verfolgt hat.', '1410.3095-1-25-2': 'Die empirischen Theorien, über die Kinder zu verfügen lernen, sind die natürlichen Zugänge zur Entwicklung von Gruppierungen.', '1410.3095-1-26-0': 'Es kann durchaus sein, daß einzelne der Piagetschen Bedingungen auch in der Formulierung der Modelle auftreten, aber dies wäre eine rein sprachliche Übereinstimmung, denn die Objekte, von denen die Modelle sprechen, sind nicht die Operationen Piagets und die verwendeten Begriffe sind nicht notwendig identisch mit den seinen.', '1410.3095-1-26-1': 'Dies zeigt nicht nur die Beobachtung Griesels.', '1410.3095-1-26-2': 'So betrachtet Piaget z.B. die Gruppierung der asymmetrischen Relationen und schreibt:', '1410.3095-1-27-0': '"Nennen wir a die Relation O < A; b die Relation O < B; c die Relation O < C. Man kann dann die Relation A < B a[MATH] nennen, die Relation B < C b[MATH] etc.', '1410.3095-1-27-1': 'Die umgekehrte Operation besteht aus der Subtraktion einer Relation, was der Addition ihrer Konverse äquivalent ist.', '1410.3095-1-27-2': '.....', '1410.3095-1-27-3': 'Auf der Transitivität, die dieser Seriation eigentümlich ist, gründet sich der Schluß A < B; B < C, also A < C " [Piaget 1971, S. 51].', '1410.3095-1-28-0': 'Es ergibt sich folgende Konsequenz:', '1410.3095-1-29-0': 'a + (-a) = (O < A [MATH] A < O) = (O < O)', '1410.3095-1-30-0': 'Eine (mathematische) asymmetrische Relation ist irreflexiv, was die Piagetsche offensichtlich nicht ist.', '1410.3095-1-30-1': 'Nur aus der sprachlichen Übereinstimmung auch auf eine inhaltliche zwischen Piagets Begriffen und den Begriffen der Mathematik zu schließen, dürfte wenig zweckdienlich sein.', '1410.3095-1-31-0': 'Wie wir wiederholt betont haben, sind wir der Auffassung, daß auch die Inhalte der Elementarmathematik von Schülern im Rahmen empirischer Theorien erworben werden (Ein Standpunkt, der z.B. auch von Griesel geteilt wird [Griesel 2013]).', '1410.3095-1-31-1': 'Elementarmathematik umfaßt dabei mindestens die Inhalte, die auf der konkret - operationalen Stufe oder zuvor erlernt werden, also i.w. alle Inhalte der Klassen 1 bis 10.', '1410.3095-1-31-2': 'Ein zentrales Anliegen des Unterrichts sollte es daher sein, diese Inhalte als Inhalte empirischer Theorien aufzufassen.', '1410.3095-1-31-3': 'Für die Zahlbegriffsentwicklung haben wir an etlichen Beispielen gezeigt, wie dies zu verstehen ist [Burscheid und Struve 2009].', '1410.3095-1-31-4': 'Ein Unterricht, der diese Sichtweise berücksichtigt, trägt ganz selbstverständlich zur Ausbildung von Gruppierungen als Gleichgewichtszuständen des konkret - operationalen Denkens bei.', '1410.3095-1-31-5': 'Den Kindern solche wie "isomorphe Spielhandlungen" (Breidenbach) vorzustellen, weist sicherlich nicht den Weg, auf dem sie diese entwickeln (vgl.', '1410.3095-1-31-6': '[Bussmann 1974]).', '1410.3095-1-31-7': 'Bemerkung: Verzichtet man darauf - wie hier vorgeschlagen - die Piagetschen Bedingungen wie ein Axiomensystem zu lesen, so erledigt sich die Unvereinbarkeit von allgemeiner (4.)', '1410.3095-1-31-8': 'und identischer Operation (5.)', '1410.3095-1-31-9': ', auf die Griesel hingewiesen hatte.', '1410.3095-1-31-10': 'Während in Prozessen, die sich auf - im weitesten Sinne - quantifizierbare Objekte beziehen, einzelne oder alle der Bedingungen 1.', '1410.3095-1-31-11': '- 4.', '1410.3095-1-31-12': 'erfüllt sein mögen, gilt dies nicht unbedingt für solche, die sich auf Qualitäten beziehen.', '1410.3095-1-31-13': 'Betrachtet man z.B. den Prozeß des Rührens eines Teiges, so ist dieser nicht umkehrbar und damit entfällt 4.', '1410.3095-1-31-14': 'Unterbricht man das Rühren und rührt erneut, so ruft dies keine Änderung hervor, d.h. diese Tätigkeit ist in der Sprache Piagets tautologisch.', '1410.3095-1-31-15': 'Daß Piaget solche Beispiele vor Augen gehabt haben muß, belegt folgendes Zitat:', '1410.3095-1-32-0': 'Im Bereich der Zahlen bildet eine zu sich selbst addierte Einheit eine neue Zahl durch Anwendung der Komposition ( ... ).', '1410.3095-1-32-1': 'Es findet eine Iteration statt.', '1410.3095-1-32-2': 'Im Gegensatz dazu verändert sich ein qualitatives Element nicht durch Wiederholung, sondern ergibt eine Tautologie : A + A = A [Piaget 1971, S. 48/49].', '1410.3095-1-33-0': 'Aus der Formulierung wird deutlich, daß Piaget, was er ein qualitatives Element nennt, als etwas nicht Quantifizierbares auffaßt.', '1410.3095-1-33-1': 'Die Elemente einer Gruppierung sind aber Operationen/verinnerlichte Handlungen.', '1410.3095-1-33-2': 'Die Vermutung liegt nahe, daß er solche verinnerlichten Handlungen meinte, deren Realisierungen sich auf Qualitäten beziehen.', '1410.3095-1-34-0': 'german'} | {'1410.3095-2-0-0': 'Bei der Rezeption der Denkpsychologie Piagets durch die Mathematikdidaktik der Bundesrepublik Deutschland Ende der 1960er Jahre wurde der Begriff der Gruppierung ähnlich einer mathematischen Struktur aufgefaßt.', '1410.3095-2-0-1': 'Kann die Berücksichtigung empirischer Theorien zu einem adäquateren Verständnis beitragen?', '1410.3095-2-1-0': "english # Abstract Reviewing Piaget's psychology of reasoning by the mathematical educators of the Federal Republic of Germany late in the 1960[MATH] the concept of grouping has been understood similar to a mathematical structure.", '1410.3095-2-1-1': 'Possibly to consider empirical theories can contribute to an understanding more adequate.', '1410.3095-2-2-0': 'Als Ende der 1960er Jahre die Rezeption der Piagetschen Denkpsychologie durch die Mathematikdidaktik der Bundesrepublik Deutschland in voller Blüte stand, faszinierte vor allem der Begriff der Gruppierung.', '1410.3095-2-2-1': 'Die Elemente einer Gruppierung - die Operationen oder verinnerlichten Handlungen - erfüllen folgende Bedingungen:', '1410.3095-2-3-0': '1.', '1410.3095-2-3-1': 'Komposition: x + x[MATH] = y, y + y[MATH] = z; etc.', '1410.3095-2-4-0': '2.', '1410.3095-2-4-1': 'Reversibilität: y - x = x[MATH] oder y - x[MATH] = x', '1410.3095-2-5-0': '3.', '1410.3095-2-5-1': 'Assoziativität: (x + x[MATH]) + y[MATH] = x + (x[MATH] + y[MATH]) = (z).', '1410.3095-2-6-0': '4.', '1410.3095-2-6-1': 'Allgemeine, identische Operation: x - x = o; y - y = o; etc.', '1410.3095-2-7-0': '5.', '1410.3095-2-7-1': 'Tautologie oder besondere, identische Operation: x + x = x; y + y =', '1410.3095-2-8-0': 'y; etc. (Piaget [1971, S. 49])', '1410.3095-2-9-0': 'Etwas anders formuliert:', '1410.3095-2-10-0': '1.', '1410.3095-2-10-1': '2.', '1410.3095-2-10-2': 'x + x[MATH] = y [MATH] y + (-x) = x[MATH] y + (-x[MATH]) = x', '1410.3095-2-11-0': '3.', '1410.3095-2-11-1': '(x + x[MATH]) + y[MATH] = x + (x[MATH] + y[MATH])', '1410.3095-2-12-0': '4.', '1410.3095-2-12-1': 'x + (-x) = (-x) + x = o', '1410.3095-2-13-0': '5.', '1410.3095-2-13-1': 'x + x = x', '1410.3095-2-14-0': 'In dieser Formulierung sieht man deutlicher, daß es nur eine Verknüpfung (+) für die Operationen gibt und daß es bzgl.', '1410.3095-2-14-1': 'der allgemeinen, identischen Operation (o) zu gewissen Operationen x je eine inverse Operation -x geben kann.', '1410.3095-2-15-0': 'Der Piagetsche Text legte es nahe, den Begriff der Gruppierung so zu verstehen, als besäße eine Gruppierung Modelle, in denen jede dieser Bedingungen erfüllt sei.', '1410.3095-2-15-1': 'Heinz Griesel wies als erster darauf hin, daß eine Gruppierung bei dieser Lesart nur einelementig sein könne, also nur ein triviales Modell habe [1970, S. 128].', '1410.3095-2-16-0': 'Wenn man heute - rund 40 Jahre später - diese Diskussion noch einmal nachliest, stellt man erstaunt fest, wie selbstverständlich man annahm, Piagets Formulierungen als mathematisch tragfähig ansehen zu können.', '1410.3095-2-16-1': 'Eine Sichtweise, die allerdings nicht auf deutsche Mathematikdidaktiker beschränkt war.', '1410.3095-2-17-0': 'A grouping incorporates properties from two well-known mathematical structures, the group and the lattice.', '1410.3095-2-17-1': 'A Piagetian grouping is thus a kind of hybrid group-lattice structure.', '1410.3095-2-17-2': '(Flavell [1985, S. 92])', '1410.3095-2-18-0': 'Mathematisch tragfähig erschienen Piagets Formulierungen in dem Sinne, daß die im folgenden genannten Kollegen offenbar davon ausgingen, daß die von ihm angegebenen Bedingungen in der Tat wie ein mathematisches Axiomensystem gelesen werden könnten.', '1410.3095-2-18-1': 'Dazu dürfte die Bezeichnung Gruppierung - in Analogie zur (mathematischen) Gruppe - und die formale Darstellung der Bedingungen entscheidend beigetragen haben, die aber vermutlich in erster Linie Piagets Neigung zur Mathematik zu danken waren.', '1410.3095-2-18-2': 'Denn die Bedingungen wie ein Axiomensystem zu lesen wirft doch einige wichtige Fragen auf.', '1410.3095-2-19-0': 'Erich Wittmann [1973] gab als erster das Axiomensystem einer Struktur an, in der - unter Berücksichtigung der Grieselschen Feststellung - die Bedingungen von Piaget hineininterpretiert werden konnten.', '1410.3095-2-19-1': 'Um die Bedingung der Tautologie zu erfassen führt er neben + eine weitere Verknüpfung ein, indem er mit Hilfe von + eine Quasiordnung unter den Operationen definiert, an die er verbandstheoretische Forderungen stellt.', '1410.3095-2-19-2': 'Hans - Georg Steiner [1973] hat das von Wittmann angegebene Axiomensystem aufgegriffen und in eine vereinfachte Form gebracht.', '1410.3095-2-19-3': 'Elmar Cohors - Fresenborg [1974] beschrieb schließlich die Wittmannsche Gruppierung als ein Semi - Thue - System.', '1410.3095-2-19-4': 'Die von diesen Autoren angegebenen Beispiele für Gruppierungen sind zeitbedingt stark an der Neuen Mathematik orientiert und wirken heute etwas artifiziell.', '1410.3095-2-19-5': 'Natürlich genügen sie dem Axiomensystem und eine gewisse Komplexität der Beispiele ist zwar einleuchtend, da Gruppierungen als Gleichgewichtsform der Operationen/verinnerlichten Handlungen den "Endzustand" der Stufe des konkret-operationalen Denkens beschreiben, auf der sich die Schüler befanden, für die die Beispiele gedacht waren.', '1410.3095-2-19-6': 'Doch erscheint es wenig einleuchtend, daß eine derart komplizierte Gleichgewichtsform, wie sie die Wittmannsche Struktur beschreibt, ohne eine sehr spezifische Förderung erreicht werden kann.', '1410.3095-2-19-7': 'Und dies überzeugt nicht.', '1410.3095-2-19-8': 'Denn nach der Auffassung Piagets erwerben wir die von ihm formulierten Fähigkeiten im Alltag.', '1410.3095-2-19-9': 'Verinnerlicht werden Handlungen, die real durchgeführt werden, eben weit überwiegend im Alltag und nicht in einer Unterrichtssituation.', '1410.3095-2-19-10': 'Es sind in der Regel solche, die auf die WELT ausgeübt werden.', '1410.3095-2-20-0': 'Die Operationen/verinnerlichter Handlungen werden von Piaget wie folgt charakterisiert:', '1410.3095-2-21-0': 'Die spezifische Natur der Operationen besteht, verglichen mit den empirischen Tätigkeiten, gerade in der Tatsache, daß sie niemals in diskontinuierlichem Zustand existieren.', '1410.3095-2-21-1': 'Es ist nur eine gänzlich unerlaubte Abstraktion, wenn man von einer[MATH]Operation spricht; eine vereinzelte Operation kann nicht Operation sein, denn die eigentümlichste Eigenschaft der Operationen liegt gerade darin, daß sie zu Systemen vereinigt sind.', '1410.3095-2-21-2': '[1971, S. 41]', '1410.3095-2-22-0': 'Es stellt sich die Frage nach der Organisationsform der real durchgeführten Handlungen, die den Operationen zugrunde liegen.', '1410.3095-2-22-1': 'Denn man kann zwar eine real durchgeführte Handlung als einzelne betrachten, aber aus einer einzelnen Handlung läßt sich kein operatives Gesamtsystem (Piaget) ableiten und aus einer Vielzahl von planlos durchgeführten Handlungen ebenfalls nicht.', '1410.3095-2-23-0': 'Das Verhalten von Kindern und damit auch ihr Handeln ist stark regelhaft.', '1410.3095-2-23-1': 'Sie verhalten sich mitunter, als verfügten sie über gewisse Theorien, die ihr Handeln steuern (Gopnik und Meltzoff [1997]).', '1410.3095-2-23-2': 'Eine solche Theorie könnte z.B. ihren Umgang mit einem mit Luft gefüllten Ball oder mit einem Luftballon steuern.', '1410.3095-2-23-3': 'Diese Theorien beziehen sich unmittelbar auf die WELT, ihre Begriffe haben eine starke ontologische Bindung.', '1410.3095-2-23-4': 'Man spricht gemeinhin von empirischen Theorien.', '1410.3095-2-23-5': 'Eine etablierte Form zur Darstellung empirischer Theorien ist die strukturalistische Metatheorie, die von Wolfgang Stegmüller und seinem Kreis entwickelt wurde [1973, 1986], Balzer [1982].', '1410.3095-2-23-6': 'Ein wesentliches Merkmal dieser Darstellungsform ist, daß sie den Aufbau der Theorie wiedergibt.', '1410.3095-2-24-0': 'Die Darstellung gliedert sich in drei Stufen.', '1410.3095-2-24-1': 'Auf der ersten Stufe werden als sog.', '1410.3095-2-24-2': 'partielle Modelle der Theorie die empirischen Gegebenheiten formuliert, über die die Theorie Aussagen machen kann.', '1410.3095-2-24-3': 'Diese werden aus sog.', '1410.3095-2-24-4': 'paradigmatischen Beispielen abgeleitet.', '1410.3095-2-24-5': 'Im vorliegenden Fall betrifft dies die Objekte, auf die reale Handlungen ausgeübt werden sowie diese Handlungen selbst.', '1410.3095-2-24-6': 'Die nächste Stufe - die der potentiellen Modelle - betrifft die Sprache, in der die Theorie formuliert wird.', '1410.3095-2-24-7': 'Sie soll möglichst präzise und deshalb formalisierbar sein.', '1410.3095-2-24-8': 'Es liegt daher nahe, auf Begriffe der Mathematik, z.B. Relationen oder Funktionen, zurückzugreifen.', '1410.3095-2-24-9': 'Sie erlauben es, die partiellen Modelle formal zu beschreiben.', '1410.3095-2-24-10': 'Dies ist in dem Sinne von besonderem Interesse als auf dieser Stufe die realen Handlungen der ersten Stufe durch mathematische Strukturen beschrieben, d.h. in Systeme eingebunden werden.', '1410.3095-2-24-11': 'Die Systeme der realen Handlungen bilden die Vorlage für die Systeme der verinnerlichten Handlungen.', '1410.3095-2-24-12': 'Während auf der ersten Stufe noch alle Begriffe reale Referenzen haben oder aus bekannten (früher erworbenen) Theorien stammen, bedarf die neu zu formulierende Theorie neuer Begriffe, will sie neues Wissen vermitteln.', '1410.3095-2-24-13': 'Es sind die bzgl.', '1410.3095-2-24-14': 'der in den partiellen Modellen formulierten empirischen Gegebenheiten theoretischen Begriffe, solche, die dort keine Referenzen haben, die erst durch die neu zu formulierende Theorie eine Bedeutung erhalten.', '1410.3095-2-24-15': 'Um diese wird die Sprache der Theorie erweitert.', '1410.3095-2-24-16': 'Auf der dritten Stufe werden schließlich die Axiome formuliert, die die Modelle der Theorie definieren.', '1410.3095-2-24-17': 'Man kann die Modelle als die formalen Spiegelbilder der Systeme der Operationen ansehen, die auf der kognitiven Ebene das Verfügen über die empirische Theorie ausmachen.', '1410.3095-2-24-18': 'Dies will sagen, daß eine empirische Theorie zu erwerben - psychologisch betrachtet - bedeutet, bestimmte Handlungen zu verinnerlichen.', '1410.3095-2-25-0': 'Die Organisationsstruktur der verinnerlichten Handlungen, die man entwickelt, wenn man über eine empirische Theorie zu verfügen lernt, kann einzelne der Piagetschen Bedingungen erfüllen.', '1410.3095-2-25-1': 'Alle Bedingungen einer Gruppierung gleichzeitig zu erfüllen ist - in der Piagetschen Lesart - wegen der Unvereinbarkeit von allgemeiner und besonderer, identischer Operation ohnehin nicht möglich.', '1410.3095-2-26-0': 'Die Eigenschaften einer Gruppierung sind bei dieser Sichtweise ein Extrakt der Eigenschaften real durchgeführter Handlungen, die der einzelne verinnerlicht, wenn er über unterschiedliche empirische Theorien zu verfügen lernt.', '1410.3095-2-26-1': 'Ein so verstandener Gruppierungsbegriff enzieht sich einer mathematischen Modellierung, trifft aber die psychologischen Intentionen, die Piaget verfolgt hat.', '1410.3095-2-26-2': 'Die empirischen Theorien, über die Kinder zu verfügen lernen, sind die natürlichen Zugänge zur Entwicklung von Gruppierungen.', '1410.3095-2-27-0': 'Es kann durchaus sein, daß einzelne der Piagetschen Bedingungen auch in der Formulierung der Modelle auftreten, aber dies wäre eine rein sprachliche Übereinstimmung, denn die Objekte, von denen die Modelle sprechen, sind nicht die Operationen Piagets und die verwendeten Begriffe sind nicht notwendig identisch mit den seinen.', '1410.3095-2-27-1': 'Dies zeigt nicht nur die Beobachtung Griesels.', '1410.3095-2-27-2': 'So betrachtet Piaget z.B. die Gruppierung der asymmetrischen Relationen und schreibt:', '1410.3095-2-28-0': 'Nennen wir a die Relation O < A; b die Relation O < B; c die Relation O < C. Man kann dann die Relation A < B a[MATH] nennen, die Relation B < C b[MATH] etc.', '1410.3095-2-28-1': 'Die umgekehrte Operation besteht aus der Subtraktion einer Relation, was der Addition ihrer Konverse äquivalent ist.', '1410.3095-2-28-2': '.....', '1410.3095-2-28-3': 'Auf der Transitivität, die dieser Seriation eigentümlich ist, gründet sich der Schluß A < B; B < C, also A < C. [1971, S. 51]', '1410.3095-2-29-0': 'Es ergibt sich folgende Konsequenz:', '1410.3095-2-30-0': 'a + (-a) = (O < A [MATH] A < O) = (O < O)', '1410.3095-2-31-0': 'Eine (mathematische) asymmetrische Relation ist irreflexiv, was die Piagetsche offensichtlich nicht ist.', '1410.3095-2-31-1': 'Nur aus der sprachlichen Übereinstimmung auch auf eine inhaltliche zwischen Piagets Begriffen und den Begriffen der Mathematik zu schließen, dürfte wenig zweckdienlich sein.', '1410.3095-2-32-0': 'Wie wir wiederholt betont haben, sind wir der Auffassung, daß die Inhalte der Elementarmathematik von Schülern im Rahmen empirischer Theorien erworben werden (Ein Standpunkt, der z.B. auch von Griesel geteilt wird [2013]).', '1410.3095-2-32-1': 'Elementarmathematik umfaßt dabei mindestens die Inhalte, die auf der konkret - operationalen Stufe oder zuvor erlernt werden, also i.w. alle Inhalte der Klassen 1 bis 10.', '1410.3095-2-32-2': 'Ein zentrales Anliegen des Unterrichts sollte es daher sein, diese Inhalte in empirische Theorien zu integrieren.', '1410.3095-2-32-3': 'Für die Zahlbegriffsentwicklung haben wir an etlichen Beispielen gezeigt, wie dies zu verstehen ist [2009].', '1410.3095-2-32-4': 'Ein Unterricht, der diese Sichtweise berücksichtigt, trägt ganz selbstverständlich zur Ausbildung von Gruppierungen als Gleichgewichtszuständen des konkret-operationalen Denkens bei.', '1410.3095-2-32-5': 'Den Kindern solche wie isomorphe Spielhandlungen (Breidenbach) vorzustellen, weist sicherlich nicht den Weg, auf dem sie diese entwickeln (vgl.', '1410.3095-2-32-6': 'Bussmann [1974]).', '1410.3095-2-32-7': 'Bemerkung: Verzichtet man darauf - wie hier vorgeschlagen - die Piagetschen Bedingungen wie ein Axiomensystem zu lesen, so erledigt sich die Unvereinbarkeit von allgemeiner (4.)', '1410.3095-2-32-8': 'und identischer Operation (5.)', '1410.3095-2-32-9': ', auf die Griesel hingewiesen hatte.', '1410.3095-2-32-10': 'Während in Prozessen, die sich auf - im weitesten Sinne - quantifizierbare Objekte beziehen, einzelne oder alle der Bedingungen 1.', '1410.3095-2-32-11': '- 4.', '1410.3095-2-32-12': 'erfüllt sein mögen, gilt dies nicht unbedingt für solche, die sich auf Qualitäten beziehen.', '1410.3095-2-32-13': 'Betrachtet man z.B. den Prozeß des Rührens eines Teiges, so ist dieser nicht umkehrbar und damit entfällt 4.', '1410.3095-2-32-14': 'Unterbricht man das Rühren und rührt erneut, so ruft dies keine Änderung hervor, d.h. diese Tätigkeit ist in der Sprache Piagets tautologisch.', '1410.3095-2-32-15': 'Daß Piaget solche Beispiele vor Augen gehabt haben muß, belegt folgendes Zitat:', '1410.3095-2-33-0': 'Im Bereich der Zahlen bildet eine zu sich selbst addierte Einheit eine neue Zahl durch Anwendung der Komposition ( ... ).', '1410.3095-2-33-1': 'Es findet eine Iteration statt.', '1410.3095-2-33-2': 'Im Gegensatz dazu verändert sich ein qualitatives Element nicht durch Wiederholung, sondern ergibt eine Tautologie : A + A = A. [1971, S. 48/49]', '1410.3095-2-34-0': 'Aus der Formulierung wird deutlich, daß Piaget, was er ein qualitatives Element nennt, als etwas nicht Quantifizierbares auffaßt.', '1410.3095-2-34-1': 'Die Elemente einer Gruppierung sind aber Operationen/verinnerlichte Handlungen.', '1410.3095-2-34-2': 'Die Vermutung liegt nahe, daß er solche verinnerlichten Handlungen meinte, deren Realisierungen sich auf Qualitäten beziehen.', '1410.3095-2-35-0': 'german'} | [['1410.3095-1-27-1', '1410.3095-2-28-1'], ['1410.3095-1-2-0', '1410.3095-2-2-0'], ['1410.3095-1-18-0', '1410.3095-2-18-0'], ['1410.3095-1-18-2', '1410.3095-2-18-2'], ['1410.3095-1-22-0', '1410.3095-2-22-0'], ['1410.3095-1-30-0', '1410.3095-2-31-0'], ['1410.3095-1-30-1', '1410.3095-2-31-1'], ['1410.3095-1-4-1', '1410.3095-2-4-1'], ['1410.3095-1-32-0', '1410.3095-2-33-0'], ['1410.3095-1-32-1', '1410.3095-2-33-1'], ['1410.3095-1-0-0', '1410.3095-2-0-0'], ['1410.3095-1-0-1', '1410.3095-2-0-1'], ['1410.3095-1-25-0', '1410.3095-2-26-0'], 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['1410.3095-1-16-1', '1410.3095-2-16-1'], ['1410.3095-1-26-0', '1410.3095-2-27-0'], ['1410.3095-1-26-1', '1410.3095-2-27-1'], ['1410.3095-1-15-0', '1410.3095-2-15-0'], ['1410.3095-1-31-1', '1410.3095-2-32-1'], ['1410.3095-1-31-7', '1410.3095-2-32-7'], ['1410.3095-1-31-10', '1410.3095-2-32-10'], ['1410.3095-1-31-12', '1410.3095-2-32-12'], ['1410.3095-1-31-13', '1410.3095-2-32-13'], ['1410.3095-1-31-14', '1410.3095-2-32-14'], ['1410.3095-1-1-0', '1410.3095-2-1-0'], ['1410.3095-1-1-1', '1410.3095-2-1-1'], ['1410.3095-1-14-0', '1410.3095-2-14-0'], ['1410.3095-1-14-1', '1410.3095-2-14-1'], ['1410.3095-1-19-6', '1410.3095-2-19-6'], ['1410.3095-1-19-7', '1410.3095-2-19-7'], ['1410.3095-1-19-8', '1410.3095-2-19-8'], ['1410.3095-1-19-9', '1410.3095-2-19-9'], ['1410.3095-1-19-10', '1410.3095-2-19-10'], ['1410.3095-1-6-1', '1410.3095-2-6-1'], ['1410.3095-1-7-1', '1410.3095-2-7-1'], ['1410.3095-1-23-0', '1410.3095-2-23-0'], ['1410.3095-1-23-3', '1410.3095-2-23-3'], ['1410.3095-1-23-4', '1410.3095-2-23-4'], ['1410.3095-1-23-6', '1410.3095-2-23-6'], ['1410.3095-1-24-0', '1410.3095-2-24-0'], ['1410.3095-1-24-1', '1410.3095-2-24-1'], ['1410.3095-1-24-2', '1410.3095-2-24-2'], ['1410.3095-1-24-3', '1410.3095-2-24-3'], ['1410.3095-1-24-5', '1410.3095-2-24-5'], ['1410.3095-1-24-6', '1410.3095-2-24-6'], ['1410.3095-1-24-7', '1410.3095-2-24-7'], ['1410.3095-1-24-8', '1410.3095-2-24-8'], ['1410.3095-1-24-9', '1410.3095-2-24-9'], ['1410.3095-1-24-10', '1410.3095-2-24-12'], ['1410.3095-1-24-11', '1410.3095-2-24-13'], ['1410.3095-1-24-12', '1410.3095-2-24-14'], ['1410.3095-1-24-13', '1410.3095-2-24-15'], ['1410.3095-1-24-14', '1410.3095-2-24-16'], ['1410.3095-1-24-16', '1410.3095-2-24-11'], ['1410.3095-1-24-17', '1410.3095-2-24-17'], ['1410.3095-1-24-19', '1410.3095-2-25-0'], ['1410.3095-1-24-20', '1410.3095-2-25-1']] | [['1410.3095-1-27-0', '1410.3095-2-28-0'], ['1410.3095-1-18-1', '1410.3095-2-18-1'], ['1410.3095-1-17-0', '1410.3095-2-17-0'], ['1410.3095-1-22-1', '1410.3095-2-22-1'], ['1410.3095-1-32-2', '1410.3095-2-33-2'], ['1410.3095-1-15-1', '1410.3095-2-15-1'], ['1410.3095-1-21-0', '1410.3095-2-21-0'], ['1410.3095-1-21-1', '1410.3095-2-21-1'], ['1410.3095-1-31-0', '1410.3095-2-32-0'], ['1410.3095-1-31-3', '1410.3095-2-32-3'], ['1410.3095-1-31-4', '1410.3095-2-32-4'], ['1410.3095-1-31-5', '1410.3095-2-32-5'], ['1410.3095-1-19-0', '1410.3095-2-19-0'], ['1410.3095-1-19-1', '1410.3095-2-19-1'], ['1410.3095-1-19-2', '1410.3095-2-19-2'], ['1410.3095-1-19-3', '1410.3095-2-19-3'], ['1410.3095-1-19-4', '1410.3095-2-19-4'], ['1410.3095-1-19-5', '1410.3095-2-19-5'], ['1410.3095-1-23-1', '1410.3095-2-23-1'], ['1410.3095-1-23-2', '1410.3095-2-23-2'], ['1410.3095-1-23-5', '1410.3095-2-23-5'], ['1410.3095-1-24-15', '1410.3095-2-24-10'], ['1410.3095-1-24-18', '1410.3095-2-24-18']] | [] | [['1410.3095-1-27-3', '1410.3095-2-28-3'], ['1410.3095-1-17-1', '1410.3095-2-17-1'], ['1410.3095-1-31-2', '1410.3095-2-32-2']] | [] | ['1410.3095-1-2-1', '1410.3095-1-3-0', '1410.3095-1-3-1', '1410.3095-1-4-0', '1410.3095-1-5-0', '1410.3095-1-5-1', '1410.3095-1-6-0', '1410.3095-1-7-0', '1410.3095-1-8-0', '1410.3095-1-9-0', '1410.3095-1-10-0', '1410.3095-1-10-1', '1410.3095-1-10-2', '1410.3095-1-11-0', '1410.3095-1-11-1', '1410.3095-1-12-0', '1410.3095-1-12-1', '1410.3095-1-13-0', '1410.3095-1-13-1', '1410.3095-1-20-0', '1410.3095-1-24-4', '1410.3095-1-26-2', '1410.3095-1-27-2', '1410.3095-1-28-0', '1410.3095-1-29-0', '1410.3095-1-31-6', '1410.3095-1-31-8', '1410.3095-1-31-9', '1410.3095-1-31-11', '1410.3095-1-31-15', '1410.3095-1-34-0', '1410.3095-2-2-1', '1410.3095-2-3-0', '1410.3095-2-3-1', '1410.3095-2-4-0', '1410.3095-2-5-0', '1410.3095-2-5-1', '1410.3095-2-6-0', '1410.3095-2-7-0', '1410.3095-2-8-0', '1410.3095-2-9-0', '1410.3095-2-10-0', '1410.3095-2-10-1', '1410.3095-2-10-2', '1410.3095-2-11-0', '1410.3095-2-11-1', '1410.3095-2-12-0', '1410.3095-2-12-1', '1410.3095-2-13-0', '1410.3095-2-13-1', '1410.3095-2-17-2', '1410.3095-2-20-0', '1410.3095-2-21-2', '1410.3095-2-24-4', '1410.3095-2-27-2', '1410.3095-2-28-2', '1410.3095-2-29-0', '1410.3095-2-30-0', '1410.3095-2-32-6', '1410.3095-2-32-8', '1410.3095-2-32-9', '1410.3095-2-32-11', '1410.3095-2-32-15', '1410.3095-2-35-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1410.3095 | null | null | null | null | null |
0710.5616 | {'0710.5616-1-0-0': 'Recent developments in phenomenological analysis of the CLAS data on 2[MATH] electroproduction are presented.', '0710.5616-1-0-1': 'The contributions from isobar channels and [MATH], [MATH] electrocouplings at [MATH] from 0.25 to 0.6 GeV[MATH] were determined from the analysis of comprehensive data on differential and fully integrated 2[MATH] cross sections.', '0710.5616-1-1-0': '# Introduction', '0710.5616-1-2-0': 'Studies of nucleon resonance electrocouplings at various photon virtualities in double charged-pion electroproduction play an important role in the [MATH] program with the CLAS detector [CITATION].', '0710.5616-1-2-1': 'Single and double pion photo and electroproduction are two major exclusive channels, contributing to the total photon-proton cross section in the [MATH] excitation region.', '0710.5616-1-2-2': 'Both of these channels are sensitive to excited states.', '0710.5616-1-2-3': 'Photo and electroproduction of two pions are particularly sensitive to resonances with masses above 1.6 GeV.', '0710.5616-1-2-4': 'Many of these states decay preferentially to final states with two pions.', '0710.5616-1-2-5': 'Furthermore, 1[MATH] and 2[MATH] exclusive channels are strongly coupled by hadronic interactions in the final states (FSI).', '0710.5616-1-2-6': 'Hadronic cross section [MATH] is the second strongest exclusive channel in value amongst exclusive [MATH] cross sections.', '0710.5616-1-2-7': 'Therefore, a combined analysis of at least the two major electroproduction channels is needed to assure the appropriate evaluation of [MATH] electrocouplings.', '0710.5616-1-2-8': 'Eventually other exclusive channels with smaller cross sections may be included.', '0710.5616-1-2-9': 'For these final states, the hadronic interactions with major meson photo or electroproduction channels become even more important.', '0710.5616-1-2-10': 'Therefore, comprehensive information on mechanisms contributing to both 1[MATH] and 2[MATH] electroproduction is of particular interest for the entire [MATH] program.', '0710.5616-1-2-11': 'This information may be obtained in a phenomenological analysis of the CLAS data on meson electroproduction offering valuable input for [MATH] studies in advanced coupled channel approaches, which are currently under development at the Excited Baryon Analysis Center (EBAC) at JLAB [CITATION].', '0710.5616-1-3-0': 'In this proceeding we report results of an analysis of recent CLAS data on double charged-pion electroproduction [CITATION] at [MATH], [MATH]p invariant masses and [MATH] angular distribution [CITATION].', '0710.5616-1-3-1': 'These cross sections were analyzed in the hadronic mass range from 1.41 to 1.89 GeV.', '0710.5616-1-3-2': 'The overall [MATH]-coverage ranges from 0.5 to 1.5 GeV[MATH].', '0710.5616-1-3-3': 'In the 2005 version of this analysis approach (JM05) [CITATION], double charged-pion production was described by the superposition of quasi-two-body channels with the formation and subsequent decay of unstable particles in the intermediate states: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '0710.5616-1-3-4': 'Remaining direct 2[MATH] production mechanisms without formation of unstable intermediate particles were described by a set of exchange terms with the amplitudes as outlined in [CITATION].', '0710.5616-1-3-5': 'The production amplitudes for the first three quasi-two-body intermediate states were treated as sums of [MATH] excitations in the [MATH]-channel and non-resonant mechanisms described in Refs. [CITATION].', '0710.5616-1-3-6': 'All well established resonances with observed decays to the two pion final states were included as well as [MATH] candidate state that was observed in the analysis of 2[MATH] electroproduction data [CITATION].', '0710.5616-1-3-7': 'The production amplitudes for the [MATH], [MATH] and [MATH] intermediate state are described in [CITATION].', '0710.5616-1-4-0': 'In the JM05 approach we succeeded in describing all before mentioned observables in the CLAS data [CITATION].', '0710.5616-1-4-1': 'These results are presented in [CITATION].', '0710.5616-1-5-0': 'In the analysis of the most recent data at [MATH]1.6 GeV and photon virtualities from 0.2 to 0.6 GeV[MATH] [CITATION] for we attempted the first time to fit contributing mechanisms to the set of nine single-differential cross sections in each ([MATH],[MATH]) bin covered by measurements.', '0710.5616-1-5-1': 'In addition to the differential cross sections mentioned above, they also included [MATH] and [MATH] angular distributions and three distributions over angles [MATH] between two planes, composed by the momenta of the initial proton and final hadron (first plane) and two the other final hadrons (a second plane) for three possible combinations amongst these pairs.', '0710.5616-1-5-2': 'We found that the JM05 model reasonably describes the data over all invariant masses and [MATH] angular distributions, as it is shown in Fig. [REF].', '0710.5616-1-5-3': 'All these observables were previously studied in 2[MATH] electroproduction at higher photon virtualities [CITATION].', '0710.5616-1-5-4': 'However, JM05 model version failed in reproducing [MATH] and [MATH] CM angular distributions included in the analysis for the first time.', '0710.5616-1-5-5': 'As it follows from Fig. [REF], where the contributions from various mechanisms are presented, this failure is related to shortcomings in the description of direct 2[MATH] production mechanisms in [CITATION].', '0710.5616-1-6-0': 'In order to achieve a reasonable description of all angular distributions, we modified the dynamics of direct 2[MATH] production mechanisms with respect to those used in the JM05 version.', '0710.5616-1-6-1': 'The mechanisms of [CITATION] were substituted by ladder-type double exchange processes, shown in Fig. [REF].', '0710.5616-1-7-0': 'The amplitudes of these processes are parametrized as Lorentz-invariant contractions between spin-tensors of the initial and final particles, while the propagators for exchange mechanisms are described by exponents.', '0710.5616-1-7-1': 'We refer to this new approach as JM06 model.', '0710.5616-1-7-2': 'In this model we succeeded to describe recent CLAS data [CITATION] in the entire kinematics covered by the measurements [CITATION].', '0710.5616-1-7-3': 'As a typical example, the description of single-differential 2[MATH] cross sections within the framework of the JM06 version is shown in Fig. [REF] together with the contributions from various mechanisms of JM06 approach.', '0710.5616-1-8-0': 'The shapes of cross sections for various contributing mechanisms are substantially different in the observables, but highly correlated by mechanism dynamics.', '0710.5616-1-8-1': 'Therefore, the successful description of all differential cross sections allowed us to pin down all major contributing processes and access their dynamics at the phenomenological level.', '0710.5616-1-9-0': 'To check the reliability of the amplitudes for contributing processes, derived in this phenomenological data analysis, we fixed all JM06 parameters, fitting them to six single-differential cross sections[MATH] all invariant masses, and three final state angular distributions.', '0710.5616-1-9-1': 'The remaining three distributions over the [MATH] angles were calculated, keeping JM06 parameters fixed.', '0710.5616-1-9-2': 'A reasonable description of [MATH] angular distributions was achieved in the entire kinematics covered by measurements.', '0710.5616-1-9-3': 'Therefore, we confirmed the reliability of 2[MATH] electroproduction mechanisms established in phenomenological data analysis within the framework of the JM06 model.', '0710.5616-1-10-0': '# Cross sections for contributing mechanisms and [MATH] electrocouplings.', '0710.5616-1-11-0': 'The contributions from isobar channels to the 2[MATH] electroproduction, and the electrocouplings for [MATH] and [MATH] resonances at [MATH] 0.6 GeV[MATH] were determined within the framework of the JM06 approach.', '0710.5616-1-11-1': 'Electrocouplings of all resonances were varied around their initial values, that were obtained by interpolating previous CLAS and world data.', '0710.5616-1-11-2': 'They were varied randomly according to a normal distribution with a [MATH] of 30 .', '0710.5616-1-11-3': 'Simultaneously, non-resonant mechanism parameters were varied with a [MATH] of 10 .', '0710.5616-1-11-4': 'For each trial set of JM06 parameters we calculated nine differential cross sections in all ([MATH],[MATH]) bins, covered in the CLAS measurements [CITATION].', '0710.5616-1-11-5': 'Normalized to the amount of data points [MATH] were estimated from the comparison between measured and calculated differential cross sections.', '0710.5616-1-11-6': 'Finally, we selected calculated differential cross sections, which were closest to the experimental data applying the restriction for [MATH] listed in Table [REF].', '0710.5616-1-12-0': 'The differential cross sections selected from the fit at [MATH]=1.43 GeV and [MATH]=0.425 GeV[MATH] are shown in Fig. [REF] by various dashed lines.', '0710.5616-1-12-1': 'For each calculated differential cross section, selected in fitting procedure, we estimated the contribution from all isobar channels combined.', '0710.5616-1-12-2': 'The contribution from all isobar channels combined is shown in Fig. [REF] by vertical bars.', '0710.5616-1-12-3': 'The information on the isobar channel contributions to all nine differential 2[MATH] cross sections was obtained in our analysis for the first time.', '0710.5616-1-12-4': 'Fits within the framework of JM06 also enabled us to obtain amplitudes for the superposition of all quasi-two-body processes mentioned in the Section [REF], as well as for any individual isobar channel.', '0710.5616-1-13-0': 'This information is of particular interest for future [MATH] studies in a combined analysis of an 1[MATH] and 2[MATH] exclusive channels within the framework of an advanced coupled channel approach, which is currently under development by EBAC [CITATION].', '0710.5616-1-13-1': 'Moreover, the data on isobar channel differential cross sections and amplitudes open up new opportunities to establish explicit meson-baryon mechanisms contributing to various isobar channels.', '0710.5616-1-13-2': 'Predictions from various models, based on effective meson-baryon Lagrangians [CITATION] may be compared with isobar channel cross sections and amplitudes determined from the CLAS 2[MATH] data analysis.', '0710.5616-1-14-0': 'Fig. [REF] shows the electrocouplings of the [MATH] and [MATH] states.', '0710.5616-1-14-1': 'For the first time, we obtain the [MATH] evolution of electrocouplings for these states from the [MATH] channel at [MATH] from 0.2 to 0.6 GeV[MATH].', '0710.5616-1-15-0': 'These photon virtualities are particularly sensitive to the contributions from [MATH] meson-baryon dressing.', '0710.5616-1-15-1': 'The electrocouplings obtained from this analysis are in reasonable agreement with the results from 1[MATH] exclusive channel [CITATION], as well as from the combined 1[MATH]/2[MATH] analysis [CITATION].', '0710.5616-1-15-2': 'The consistency of the data on the [MATH], [MATH] electrocouplings, obtained from analysis of the two major 1[MATH] and 2[MATH] exclusive channels with substantially different non-resonant processes demonstrates that a reliable evaluation of these fundamental quantities can be obtained from the 1[MATH] and 2[MATH] electroproduction data.', '0710.5616-1-15-3': 'The analysis of the CLAS data on 2[MATH] electroproduction provides compelling evidence for the sign flip of the [MATH] electrocoupling of the [MATH] state at [MATH] in the range from 0.4 to 0.5 GeV[MATH] (Fig. [REF]).', '0710.5616-1-16-0': '# Conclusions and outlook.', '0710.5616-1-17-0': '[MATH].', '0710.5616-1-17-1': 'The work was supported by U.S. DOE contract DE-AC05-060R23177 under which Jefferson Science Associates, LLC, operates the Jefferson Lab.'} | {'0710.5616-2-0-0': 'Recent developments in phenomenological analysis of the CLAS data on 2[MATH] electroproduction are presented.', '0710.5616-2-0-1': 'The contributions from isobar channels and [MATH], [MATH] electrocouplings at [MATH] from 0.25 to 0.6 GeV[MATH] were determined from the analysis of comprehensive data on differential and fully integrated 2[MATH] cross sections.', '0710.5616-2-1-0': '# Introduction', '0710.5616-2-2-0': 'Studies of nucleon resonance electrocouplings at various photon virtualities in double charged-pion electroproduction play an important role in the [MATH] program with the CLAS detector [CITATION].', '0710.5616-2-2-1': 'Single and double pion photo and electroproduction are two major exclusive channels, contributing to the total photon-proton cross section in the [MATH] excitation region.', '0710.5616-2-2-2': 'Both of these channels are sensitive to excited states.', '0710.5616-2-2-3': 'Photo and electroproduction of two pions are particularly sensitive to resonances with masses above 1.6 GeV.', '0710.5616-2-2-4': 'Many of these states decay preferentially to final states with two pions.', '0710.5616-2-2-5': 'Furthermore, 1[MATH] and 2[MATH] exclusive channels are strongly coupled by hadronic interactions in the final states (FSI).', '0710.5616-2-2-6': 'Hadronic cross section [MATH] is the second strongest exclusive channel in value amongst exclusive [MATH] cross sections.', '0710.5616-2-2-7': 'Therefore, a combined analysis of at least the two major electroproduction channels is needed to assure the appropriate evaluation of [MATH] electrocouplings.', '0710.5616-2-2-8': 'Eventually other exclusive channels with smaller cross sections may be included.', '0710.5616-2-2-9': 'For these final states, the hadronic interactions with major meson photo or electroproduction channels become even more important.', '0710.5616-2-2-10': 'Therefore, comprehensive information on mechanisms contributing to both 1[MATH] and 2[MATH] electroproduction is of particular interest for the entire [MATH] program.', '0710.5616-2-2-11': 'This information may be obtained in a phenomenological analysis of the CLAS data on meson electroproduction offering valuable input for [MATH] studies in advanced coupled channel approaches, which are currently under development at the Excited Baryon Analysis Center (EBAC) at JLAB [CITATION].', '0710.5616-2-3-0': 'In this proceeding we report results of an analysis of recent CLAS data on double charged-pion electroproduction [CITATION] at [MATH], [MATH]p invariant masses and [MATH] angular distribution [CITATION].', '0710.5616-2-3-1': 'These cross sections were analyzed in the hadronic mass range from 1.41 to 1.89 GeV.', '0710.5616-2-3-2': 'The overall [MATH]-coverage ranges from 0.5 to 1.5 GeV[MATH].', '0710.5616-2-3-3': 'In the 2005 version of this analysis approach (JM05) [CITATION], double charged-pion production was described by the superposition of quasi-two-body channels with the formation and subsequent decay of unstable particles in the intermediate states: [MATH], [MATH], [MATH], [MATH], [MATH], [MATH].', '0710.5616-2-3-4': 'Remaining direct 2[MATH] production mechanisms without formation of unstable intermediate particles were described by a set of exchange terms with the amplitudes as outlined in [CITATION].', '0710.5616-2-3-5': 'The production amplitudes for the first three quasi-two-body intermediate states were treated as sums of [MATH] excitations in the [MATH]-channel and non-resonant mechanisms described in Refs. [CITATION].', '0710.5616-2-3-6': 'All well established resonances with observed decays to the two pion final states were included as well as [MATH] candidate state that was observed in the analysis of 2[MATH] electroproduction data [CITATION].', '0710.5616-2-3-7': 'The production amplitudes for the [MATH], [MATH] and [MATH] intermediate state are described in [CITATION].', '0710.5616-2-4-0': 'In the JM05 approach we succeeded in describing all before mentioned observables in the CLAS data [CITATION].', '0710.5616-2-4-1': 'These results are presented in [CITATION].', '0710.5616-2-5-0': 'In the analysis of the most recent data at [MATH]1.6 GeV and photon virtualities from 0.2 to 0.6 GeV[MATH] [CITATION] for we attempted the first time to fit contributing mechanisms to the set of nine single-differential cross sections in each ([MATH],[MATH]) bin covered by measurements.', '0710.5616-2-5-1': 'In addition to the differential cross sections mentioned above, they also included [MATH] and [MATH] angular distributions and three distributions over angles [MATH] between two planes, composed by the momenta of the initial proton and final hadron (first plane) and two the other final hadrons (a second plane) for three possible combinations amongst these pairs.', '0710.5616-2-5-2': 'We found that the JM05 model reasonably describes the data over all invariant masses and [MATH] angular distributions, as it is shown in Fig. [REF].', '0710.5616-2-5-3': 'All these observables were previously studied in 2[MATH] electroproduction at higher photon virtualities [CITATION].', '0710.5616-2-5-4': 'However, JM05 model version failed in reproducing [MATH] and [MATH] CM angular distributions included in the analysis for the first time.', '0710.5616-2-5-5': 'As it follows from Fig. [REF], where the contributions from various mechanisms are presented, this failure is related to shortcomings in the description of direct 2[MATH] production mechanisms in [CITATION].', '0710.5616-2-6-0': 'In order to achieve a reasonable description of all angular distributions, we modified the dynamics of direct 2[MATH] production mechanisms with respect to those used in the JM05 version.', '0710.5616-2-6-1': 'The mechanisms of [CITATION] were substituted by ladder-type double exchange processes, shown in Fig. [REF].', '0710.5616-2-7-0': 'The amplitudes of these processes are parametrized as Lorentz-invariant contractions between spin-tensors of the initial and final particles, while the propagators for exchange mechanisms are described by exponents.', '0710.5616-2-7-1': 'We refer to this new approach as JM06 model.', '0710.5616-2-7-2': 'In this model we succeeded to describe recent CLAS data [CITATION] in the entire kinematics covered by the measurements [CITATION].', '0710.5616-2-7-3': 'As a typical example, the description of single-differential 2[MATH] cross sections within the framework of the JM06 version is shown in Fig. [REF] together with the contributions from various mechanisms of JM06 approach.', '0710.5616-2-8-0': 'The shapes of cross sections for various contributing mechanisms are substantially different in the observables, but highly correlated by mechanism dynamics.', '0710.5616-2-8-1': 'Therefore, the successful description of all differential cross sections allowed us to pin down all major contributing processes and access their dynamics at the phenomenological level.', '0710.5616-2-9-0': 'To check the reliability of the amplitudes for contributing processes, derived in this phenomenological data analysis, we fixed all JM06 parameters, fitting them to six single-differential cross sections[MATH] all invariant masses, and three final state angular distributions.', '0710.5616-2-9-1': 'The remaining three distributions over the [MATH] angles were calculated, keeping JM06 parameters fixed.', '0710.5616-2-9-2': 'A reasonable description of [MATH] angular distributions was achieved in the entire kinematics covered by measurements.', '0710.5616-2-9-3': 'Therefore, we confirmed the reliability of 2[MATH] electroproduction mechanisms established in phenomenological data analysis within the framework of the JM06 model.', '0710.5616-2-10-0': '# Cross sections for contributing mechanisms and [MATH] electrocouplings.', '0710.5616-2-11-0': 'The contributions from isobar channels to the 2[MATH] electroproduction, and the electrocouplings for [MATH] and [MATH] resonances at [MATH] 0.6 GeV[MATH] were determined within the framework of the JM06 approach.', '0710.5616-2-11-1': 'Electrocouplings of all resonances were varied around their initial values, that were obtained by interpolating previous CLAS and world data.', '0710.5616-2-11-2': 'They were varied randomly according to a normal distribution with a [MATH] of 30 .', '0710.5616-2-11-3': 'Simultaneously, non-resonant mechanism parameters were varied with a [MATH] of 10 .', '0710.5616-2-11-4': 'For each trial set of JM06 parameters we calculated nine differential cross sections in all ([MATH],[MATH]) bins, covered in the CLAS measurements [CITATION].', '0710.5616-2-11-5': 'Normalized to the amount of data points [MATH] were estimated from the comparison between measured and calculated differential cross sections.', '0710.5616-2-11-6': 'Finally, we selected calculated differential cross sections, which were closest to the experimental data applying the restriction for [MATH] listed in Table [REF].', '0710.5616-2-12-0': 'The differential cross sections selected from the fit at [MATH]=1.43 GeV and [MATH]=0.425 GeV[MATH] are shown in Fig. [REF] by various dashed lines.', '0710.5616-2-12-1': 'For each calculated differential cross section, selected in fitting procedure, we estimated the contribution from all isobar channels combined.', '0710.5616-2-12-2': 'The contribution from all isobar channels combined is shown in Fig. [REF] by vertical bars.', '0710.5616-2-12-3': 'The information on the isobar channel contributions to all nine differential 2[MATH] cross sections was obtained in our analysis for the first time.', '0710.5616-2-12-4': 'Fits within the framework of JM06 also enabled us to obtain amplitudes for the superposition of all quasi-two-body processes mentioned in the Section [REF], as well as for any individual isobar channel.', '0710.5616-2-13-0': 'This information is of particular interest for future [MATH] studies in a combined analysis of an 1[MATH] and 2[MATH] exclusive channels within the framework of an advanced coupled channel approach, which is currently under development by EBAC [CITATION].', '0710.5616-2-13-1': 'Moreover, the data on isobar channel differential cross sections and amplitudes open up new opportunities to establish explicit meson-baryon mechanisms contributing to various isobar channels.', '0710.5616-2-13-2': 'Predictions from various models, based on effective meson-baryon Lagrangians [CITATION] may be compared with isobar channel cross sections and amplitudes determined from the CLAS 2[MATH] data analysis.', '0710.5616-2-14-0': 'Fig. [REF] shows the electrocouplings of the [MATH] and [MATH] states.', '0710.5616-2-14-1': 'For the first time, we obtain the [MATH] evolution of electrocouplings for these states from the [MATH] channel at [MATH] from 0.2 to 0.6 GeV[MATH].', '0710.5616-2-15-0': 'These photon virtualities are particularly sensitive to the contributions from [MATH] meson-baryon dressing.', '0710.5616-2-15-1': 'The electrocouplings obtained from this analysis are in reasonable agreement with the results from 1[MATH] exclusive channel [CITATION], as well as from the combined 1[MATH]/2[MATH] analysis [CITATION].', '0710.5616-2-15-2': 'The consistency of the data on the [MATH], [MATH] electrocouplings, obtained from analysis of the two major 1[MATH] and 2[MATH] exclusive channels with substantially different non-resonant processes demonstrates that a reliable evaluation of these fundamental quantities can be obtained from the 1[MATH] and 2[MATH] electroproduction data.', '0710.5616-2-15-3': 'The analysis of the CLAS data on 2[MATH] electroproduction provides compelling evidence for the sign flip of the [MATH] electrocoupling of the [MATH] state at [MATH] in the range from 0.4 to 0.5 GeV[MATH] (Fig. [REF]).', '0710.5616-2-16-0': '# Conclusions and outlook.', '0710.5616-2-17-0': '[MATH].', '0710.5616-2-17-1': 'The work was supported by U.S. DOE contract DE-AC05-060R23177 under which Jefferson Science Associates, LLC, operates the Jefferson Lab.'} | [['0710.5616-1-14-0', '0710.5616-2-14-0'], ['0710.5616-1-14-1', '0710.5616-2-14-1'], ['0710.5616-1-8-0', '0710.5616-2-8-0'], ['0710.5616-1-8-1', '0710.5616-2-8-1'], ['0710.5616-1-15-0', '0710.5616-2-15-0'], ['0710.5616-1-15-1', '0710.5616-2-15-1'], ['0710.5616-1-15-2', '0710.5616-2-15-2'], ['0710.5616-1-15-3', '0710.5616-2-15-3'], ['0710.5616-1-11-0', '0710.5616-2-11-0'], ['0710.5616-1-11-1', '0710.5616-2-11-1'], ['0710.5616-1-11-2', '0710.5616-2-11-2'], ['0710.5616-1-11-3', '0710.5616-2-11-3'], 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['0710.5616-1-7-3', '0710.5616-2-7-3'], ['0710.5616-1-4-0', '0710.5616-2-4-0'], ['0710.5616-1-4-1', '0710.5616-2-4-1'], ['0710.5616-1-6-0', '0710.5616-2-6-0'], ['0710.5616-1-6-1', '0710.5616-2-6-1'], ['0710.5616-1-9-0', '0710.5616-2-9-0'], ['0710.5616-1-9-1', '0710.5616-2-9-1'], ['0710.5616-1-9-2', '0710.5616-2-9-2'], ['0710.5616-1-9-3', '0710.5616-2-9-3'], ['0710.5616-1-3-0', '0710.5616-2-3-0'], ['0710.5616-1-3-1', '0710.5616-2-3-1'], ['0710.5616-1-3-2', '0710.5616-2-3-2'], ['0710.5616-1-3-3', '0710.5616-2-3-3'], ['0710.5616-1-3-4', '0710.5616-2-3-4'], ['0710.5616-1-3-5', '0710.5616-2-3-5'], ['0710.5616-1-3-6', '0710.5616-2-3-6'], ['0710.5616-1-3-7', '0710.5616-2-3-7'], ['0710.5616-1-0-0', '0710.5616-2-0-0'], ['0710.5616-1-0-1', '0710.5616-2-0-1']] | [] | [] | [] | [] | ['0710.5616-1-17-0', '0710.5616-2-17-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/0710.5616 | null | null | null | null | null |
1310.4021 | {'1310.4021-1-0-0': ': We study the nonparametric estimation of the intensity of the Poisson random measure in continuous-state branching processes with immigration based on the low frequency observations.', '1310.4021-1-0-1': 'This is in terms of the minimization of norms on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-1-0-2': 'We firstly prove the estimators are measurable, then derive the consistency and asymptotic risk bounds of the estimators under some conditions.', '1310.4021-1-1-0': 'Key words and phrases: nonparametric estimation, continuous-state branching process with immigration, stochastic differential equation, ergodicity, consistency, asymptotic risk bounds', '1310.4021-1-2-0': '# Introduction', '1310.4021-1-3-0': 'In this paper we investigate the nonparametric estimation in continuous-state branching processes with immigration.', '1310.4021-1-3-1': 'Branching processes have been a popular approach used in biology as well as in financial world.', '1310.4021-1-3-2': 'For example, Galton-Watson branching processes with immigration (GWI-processes) are used to study the evolution of different species.', '1310.4021-1-3-3': 'Continuous-state branching processes (CB-processes) were first introduced by Jirina (1958).', '1310.4021-1-3-4': 'In particular, a continuous CB-process can be obtained as the unique solution of a stochastic equation system driven by Brownian motion.', '1310.4021-1-3-5': 'Kawazu and Watanabe (1971) constructed continuous-state branching processes with immigration (CBI-processes).', '1310.4021-1-3-6': 'In view of Dawson and Li (2006), a general single-type CBI-process is the unique strong solution of a stochastic equation driven by Brownian motions and Poisson random measures.', '1310.4021-1-4-0': 'The Cox-Ingersoll-Ross model (CIR model) introduced by Cox et al. (1985) has been applied widely in the financial world.', '1310.4021-1-4-1': 'This model has many appealing advantages.', '1310.4021-1-4-2': 'For example, it is mean-reverting and remains positive.', '1310.4021-1-4-3': 'Let [MATH], [MATH] and [MATH] be given constants.', '1310.4021-1-4-4': 'The classical CIR model is a positive diffusion process [MATH] defined by [EQUATION] where [MATH] is a standard Brownian motion.', '1310.4021-1-4-5': 'The process defined by ([REF]) has continuous sample paths.', '1310.4021-1-5-0': 'However, it is well-known that interest rate is influenced not only by the market, but also by the governmental policies in some countries.', '1310.4021-1-5-1': 'The standard CIR model does not consider the influence of the policy, so there exits errors and it cannot satisfy the need in practice.', '1310.4021-1-5-2': 'Then we must develop a new model that can interpret these factors.', '1310.4021-1-5-3': 'But the idea of the CIR model should not be abandoned, since it is of great importance in some specific field.', '1310.4021-1-5-4': 'Thus we should make some modifications on the basis of the CIR model.', '1310.4021-1-5-5': 'A much easier way is adding a jump to the CIR model, i.e. [EQUATION] where [MATH] is a Poisson random measure on [MATH] with intensity [MATH], [MATH] is a finite measure on [MATH].', '1310.4021-1-5-6': 'In this paper we always assume [MATH] is absolutely continuous with respect to Lebesgue measure, i.e. according to the Radon-Nikodym Theorem, there exits a non-negative function [MATH] satisfying [MATH].', '1310.4021-1-6-0': 'However, before using ([REF]) to solve practical problems, we need to estimate the parameters [MATH] and the characteristic measure [MATH] in the equation using the historical information.', '1310.4021-1-6-1': 'The parameter estimations have been given by Huang et al. (2011) as a general case below: [EQUATION] where [MATH] and [MATH] is a Poisson random measure on [MATH] with intensity [MATH], [MATH] is a finite measure on [MATH].', '1310.4021-1-6-2': 'If [MATH], this equation is same as ([REF]).', '1310.4021-1-6-3': 'So we just need to investigate the estimation of [MATH] in ([REF]) with [MATH], [MATH] and [MATH] known.', '1310.4021-1-7-0': 'For the standard CIR-model, there are a lot of approaches to estimate the parameters, such as Long-staff and Schwartz (1992) and Bibby and Sorensen (1995).', '1310.4021-1-7-1': 'Overbeck and Ryden (1997) also gave the conditional least squares estimators (CLSEs).', '1310.4021-1-7-2': 'Li and Ma (2013) studied the asymptotics of CLSEs and WCLSEs of a stable CIR-model.', '1310.4021-1-7-3': 'Huang et al. (2011) approached weighted conditional lest squares estimators (WCLSEs) for the CBI processes and the asymptotic distribution of WCLSEs.', '1310.4021-1-7-4': 'The CLSEs and WCLSEs for the two-type CBI processes and the asymptotic distribution of CLSEs and WCLSEs were introduced in Xu (2013).', '1310.4021-1-7-5': 'It is well-known that the CBI-processes are special examples of the affine Markov processes studied in Duffie et al. (2003).', '1310.4021-1-7-6': 'The ergodicity and estimation of some different two-dimensional affine processes were studied in Barczy et al. (2013a, 2013b, 2013c).', '1310.4021-1-8-0': 'Unfortunately, limited work has been done in the nonparametric estimation for CBI-processes compared with which in Levy processes and Ornstein-Uhlenbeck(O-U) processes.', '1310.4021-1-8-1': 'For example, Watteel and Kulperger (2003) proposed and implemented an approach for estimating the jump distribution of the Levy processes by fixed spectral cut-off procedure.', '1310.4021-1-8-2': 'The penalized projection method was applied in Figueroa-Lopez and Houdre (2006) to estimate the Levy density on a compact interval separated from the origin, based on a continuous time observation of the sample path throughout a time interval [MATH].', '1310.4021-1-8-3': 'Moreover, Figueroa-Lopez (2009) used the projection method for discrete observations and provided minimax risks of estimation for smooth Levy densities, as well as estimated on a compact interval separated from the origin.', '1310.4021-1-8-4': 'Comte and Genon-Catalot (2009, 2010) used a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global [MATH]- risk with both of high frequency data (2009) and low frequency data (2010).', '1310.4021-1-8-5': 'Jongbloed et al. (2005) considered a related low-frequency problem for the canonical function in Levy-Ornstein-Uhlenbeck processes, where a consistent estimator has been constructed.', '1310.4021-1-8-6': 'Roberts et al. (2004) developed estimation for O-U processes considering Bayesian estimation for parametric models.', '1310.4021-1-9-0': 'In this work, we give two nonparametric estimators of [MATH] in ([REF]) based on the low frequency observations at equidistant time points [MATH] of a single realization [MATH].', '1310.4021-1-9-1': 'For simplicity, we take [MATH], but all the results presented below can be generalized to the general case.', '1310.4021-1-9-2': 'This is based on the minimization of the norm on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-1-9-3': 'The approach has been developed in Jongbloed et al. (2005), who applied their results to O-U processes.', '1310.4021-1-9-4': 'Then, like Comte and Genon-Catalot (2009, 2010), we explore the consistency and asymptotic risk bounds of the estimators.', '1310.4021-1-10-0': 'Ths paper is organized as follows.', '1310.4021-1-10-1': 'In Section 2, we give the ergodicity and strong mixing property of CBI-process, which is prerequisite for the study of estimators.', '1310.4021-1-10-2': 'Section 3 is devoted to the study of the nonparametric estimators of [MATH] and proves an important theorem to make sure that the estimators are well defined and measurable.', '1310.4021-1-10-3': 'The consistency and asymptotic risk bounds of the estimators are given in the Section 4.', '1310.4021-1-10-4': 'All the proofs are showed in Section 5.', '1310.4021-1-11-0': '# CBI-processes and ergodicity', '1310.4021-1-12-0': 'In this section, we give some simple properties of CBI processes.', '1310.4021-1-12-1': 'In particular, we show that the solution of ([REF]) is a two-type CBI-process and prove a subcritical CBI process is ergodic and strongly mixing.', '1310.4021-1-12-2': 'These results are useful in the study of the consistence of the estimators and the convergent rate of the risk bound.', '1310.4021-1-12-3': 'We start with an important special case of those processes.', '1310.4021-1-12-4': 'Let [MATH] and [MATH] be constants.', '1310.4021-1-12-5': 'For [MATH] set [EQUATION]', '1310.4021-1-12-6': 'A Markov process with state space [MATH] is called a continuous-state branching process (CB process) with branching mechanism [MATH] if it has transition semigroup [MATH] given by [EQUATION] where [MATH] is the unique positive solution of [EQUATION]', '1310.4021-1-12-7': 'By solving ([REF]), we get [EQUATION]', '1310.4021-1-12-8': 'Let [MATH] be a constant and [MATH] be a finite measure on [MATH].', '1310.4021-1-12-9': 'For any [MATH], Set [EQUATION]', '1310.4021-1-12-10': 'A Markov process with state space [MATH] is called a CBI process with branching mechanism [MATH] and immigration rate [MATH] if it has transition semigroup [MATH] given by [EQUATION]', '1310.4021-1-12-11': 'Actually, the solution of ([REF]) exists uniquely see Fu and Li (2012).', '1310.4021-1-12-12': 'Furthermore, it is a CBI process with branching mechanism [MATH] and immigration mechanism [MATH] defined above.', '1310.4021-1-12-13': 'See Theorem 3.1 of Dawson and Li (2012) or Theorem 2.1 of Li and Ma (2008).', '1310.4021-1-12-14': 'For more detail about CBI processes, readers can refer to Li (2011).', '1310.4021-1-12-15': 'Before proving the erogidicity and the mixing of the CBI processes, we recall Theorem 3.20 in Li (2011), which gives a necessary and sufficient condition for the ergodicity of the transition semigroup [MATH].', '1310.4021-1-12-16': 'In Li and Ma (2013), they prove that SCIR processes have exponential ergodicity, which is very strong.', '1310.4021-1-13-0': '(Li, 2011, p.66) Suppose that [MATH] and [MATH] for [MATH].', '1310.4021-1-13-1': 'Then for any [MATH], [MATH] conerges to a probability measure [MATH] on [MATH] as [MATH] if and only if [EQUATION]', '1310.4021-1-13-2': 'If ([REF]) holds, the laplace transform of [MATH] is given by [EQUATION]', '1310.4021-1-13-3': 'Under the conditions of Theorem [REF], for any finite set [MATH] we can define the probability measure [MATH] on [MATH] by [EQUATION]', '1310.4021-1-13-4': 'It is easy to see that [MATH] is a consistent family.', '1310.4021-1-13-5': "By Kolmogorov's theorem, there is a stochastic process [MATH] with finite-dimensional distributions given by ([REF]).", '1310.4021-1-13-6': 'This process is a stationary Markov process with one-dimensional marginal distribution [MATH] and transition semigroup [MATH].', '1310.4021-1-13-7': 'Since [MATH] is a Feller semigroup, the process [MATH] has a cadlag modification.', '1310.4021-1-14-0': 'labelt2.2 Suppose that the conditions in Theorem [REF] hold.', '1310.4021-1-14-1': 'Let [MATH] be a Markov process with finite-dimensional distributions given by ([REF]).', '1310.4021-1-14-2': 'Then it is strongly mixing, i.e. as [MATH], [EQUATION]', '1310.4021-1-14-3': "By Theorem [REF] and Birkhoff's ergodic theorem in Durrett (2005), we have", '1310.4021-1-15-0': 'Under the conditions of Theorem [REF].', '1310.4021-1-15-1': '[MATH] is ergodic.', '1310.4021-1-16-0': 'With the conclusions above, in this paper, we always assume that [MATH] defined by ([REF]) is a stationary and ergodic process, but by a fairly simple (continuous time) coupling argument it can be seen that the conclusions given in this paper are valid for arbitrary initial distributions.', '1310.4021-1-17-0': '# Estimators and measurability', '1310.4021-1-18-0': 'Recall [MATH].', '1310.4021-1-18-1': 'Let [MATH].', '1310.4021-1-18-2': 'In this section, firstly we will give some theorems to make sure the estimators are well defined and measurable.', '1310.4021-1-18-3': 'Then we will construct the estimators by minimizing the norms of the elements of a closed and convex subset in a special [MATH] space.', '1310.4021-1-19-0': 'Before the important theorem, we first recall a conclusion in functional analysis without proof, for the proof can be found in many books.', '1310.4021-1-20-0': 'If [MATH] is a Hilbert space with norm [MATH], [MATH] is a nonempty, closed, convex subset of [MATH], then [MATH] contains a unique element of smallest norm.', '1310.4021-1-21-0': '[(1)] In a Banach space [MATH], for any [MATH] and [MATH], [MATH] is convex.', '1310.4021-1-21-1': '[(2)] Every subspace is convex.', '1310.4021-1-21-2': '[(3)] If [MATH] is convex and [MATH], then [MATH] is convex.', '1310.4021-1-22-0': 'With this lemma we will give the most important theorem, which will guarantee the measurability of the following estimators.', '1310.4021-1-23-0': 'Let [MATH] be a probability space and [MATH] be a separable Hilbert space with the Borel [MATH]-algebra [MATH].', '1310.4021-1-23-1': '[MATH] is a nonempty, closed and convex subset, [MATH] is a measurable function: [MATH].', '1310.4021-1-23-2': 'Let [EQUATION]', '1310.4021-1-23-3': 'Then [MATH] is well defined and [MATH] is measurable with respect to [MATH].', '1310.4021-1-24-0': '[(1)] LSE, CLSE and WCLSE used widely in parameter estimation are just the special cases of this theorem.', '1310.4021-1-24-1': '[(2)] [MATH] space as a special case satisfies the conditions of this theorem.', '1310.4021-1-25-0': 'Recall that [MATH].', '1310.4021-1-25-1': 'Let [MATH].', '1310.4021-1-25-2': 'Then [MATH].', '1310.4021-1-25-3': 'So we just need to find [MATH] in the set of functions that are integrable with respect to [MATH], thus define [EQUATION].', '1310.4021-1-26-0': 'The total variation of a real-valued function [MATH], defined on an interval [MATH] is the quantity [EQUATION] where the supremum is taken over the set [MATH] is a partition of [MATH] of all partitions of the interval considered.', '1310.4021-1-27-0': 'A real-valued function [MATH] on the real line is said to be of locally bounded variation, if its total variation on any compact subset of [MATH] is finite, i.e. for any compact subset [MATH], we have [MATH].', '1310.4021-1-28-0': 'In order to construct a suitable subset, we define [EQUATION].', '1310.4021-1-28-1': 'Then [MATH] is a convex subset of [MATH].', '1310.4021-1-28-2': 'However, [MATH] is not a closed set.', '1310.4021-1-28-3': 'In order to satisfy the conditions in Theorem [REF], we need to define [EQUATION] and a mapping [MATH].', '1310.4021-1-28-4': 'We can see [MATH] is an onto mapping.', '1310.4021-1-29-0': 'Let [EQUATION]', '1310.4021-1-29-1': 'By the erogidicity of [MATH], for any [MATH] we can prove [EQUATION] and [EQUATION] where [EQUATION]', '1310.4021-1-29-2': 'According to the Lemma 7.6 in Sato (1999), the following definitions are reasonable: [EQUATION]', '1310.4021-1-29-3': 'By ([REF]), ([REF]) and the Mapping theorem, we have [EQUATION]', '1310.4021-1-29-4': 'In order to satisfy the conditions of Theorem [REF], we need to find a suitable Hilbert space.', '1310.4021-1-29-5': 'Define [EQUATION] where [MATH] is a weighted function, which be discussed later.', '1310.4021-1-29-6': 'We can easily prove [MATH] is a Hilbert Space if we define the inner product [MATH], where [MATH].', '1310.4021-1-29-7': 'With the preparation above we can construct the estimators of [MATH] as follows: [EQUATION] and [EQUATION]', '1310.4021-1-29-8': 'However in order to make the infimum is well defined, we need the integrations in ([REF]) and ([REF]) not always to be infinite.', '1310.4021-1-29-9': 'Thus we need to find a suitable weighted function [MATH], i.e. [MATH] satisfies [EQUATION] where [MATH].', '1310.4021-1-29-10': 'Here, we show how to determine [MATH] when [MATH].', '1310.4021-1-29-11': 'Firstly by the definitions of [MATH] and [MATH].', '1310.4021-1-29-12': '[EQUATION]', '1310.4021-1-29-13': 'Secondly, [EQUATION]', '1310.4021-1-29-14': 'Indeed, [EQUATION] and [EQUATION]', '1310.4021-1-29-15': 'For [EQUATION] so [EQUATION]', '1310.4021-1-29-16': 'Thus [EQUATION]', '1310.4021-1-29-17': 'From the conclusion above, we can get [EQUATION]', '1310.4021-1-29-18': 'Thus [EQUATION].', '1310.4021-1-29-19': 'From this and ([REF]), [MATH] needs to satisfy [EQUATION].', '1310.4021-1-29-20': 'For example, we can choose [MATH] or any bounded function with compact support.', '1310.4021-1-29-21': 'In this paper, we will always assume [MATH] is a bounded and non-negative function with compact support, denote by [MATH], and there exist [MATH] such that [MATH].', '1310.4021-1-30-0': 'However, since [MATH] is not closed and we need to minimize [MATH] in [MATH], so ([REF]) and ([REF]) may be not well defined or we cannot get the estimators directly from them.', '1310.4021-1-30-1': 'In another way, we can consider the following case: [EQUATION]', '1310.4021-1-30-2': 'Before discussing properties of [MATH], we give a very important conclusion about the mapping [MATH].', '1310.4021-1-31-0': '[MATH] is a one-to-one, onto and continuous mapping, where the topologies of [MATH] and [MATH] are inducted from their initial space respectively.', '1310.4021-1-32-0': 'In order to make sure [MATH] is well defined, we need to find a closed and convex subset in [MATH] and discuss the measurability of [MATH].', '1310.4021-1-32-1': 'Unfortunately, we can easily prove that [MATH] is not a closed subset.', '1310.4021-1-32-2': 'So we need to make some adjustment.', '1310.4021-1-32-3': 'In the following we find that the inverse of [MATH] and the continuity of its inverse are needed.', '1310.4021-1-32-4': 'So we should find a new subset [MATH] s.t. [MATH] is well defined and continuous.', '1310.4021-1-32-5': 'To obtain an appropriate closed subset, for any [MATH], we first pick up an integrable function [MATH] arbitrarily satisfying that [MATH] and define [EQUATION].', '1310.4021-1-32-6': 'We can choose [MATH] such that [MATH], as [MATH].', '1310.4021-1-32-7': 'Then define [EQUATION].', '1310.4021-1-32-8': 'Next we will prove that [MATH] and [MATH] satisfy our requirement.', '1310.4021-1-33-0': '[(1)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-1-33-1': '[(2)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-1-34-0': 'Since [MATH] is a one-to-one, onto and continuous mapping, furthermore [MATH] and [MATH] are compact, we can easily get the following conclusion.', '1310.4021-1-35-0': 'The inverse operator of [MATH], [MATH] is continuous.', '1310.4021-1-36-0': '[MATH] is a closed subset of [MATH] and [MATH] is a closed subset of [MATH].', '1310.4021-1-37-0': 'By Theorem [REF], Corollary [REF] and Remark [REF], we can give the well defined estimators in the following theorem.', '1310.4021-1-38-0': 'Let [EQUATION]', '1310.4021-1-38-1': 'Then [MATH] and [MATH] are well defined, measurable and exist uniquely.', '1310.4021-1-38-2': 'By the continuity of [MATH], [MATH] and [MATH] uniquely exist and are measurable.', '1310.4021-1-39-0': '[(1)] All the right-continuous and monotone functions are included in [MATH].', '1310.4021-1-39-1': '[(2)] Since [MATH], we usually choose [EQUATION] [(3)] If [MATH] is a finite dimensional subspace, then the nonparametric estimation turns to be parameter estimation.', '1310.4021-1-39-2': 'This is just a special case of Theorem [REF].', '1310.4021-1-40-0': '# Consistence and asymptotic risk bound', '1310.4021-1-41-0': 'In this section, we will introduce the strong consistency of [MATH] and [MATH].', '1310.4021-1-41-1': 'At the same time we prove that [MATH] and [MATH] are consistent.', '1310.4021-1-41-2': 'Although we can give the locally asymptotic risk bound of [MATH], we do not give any information about the asymptotic risk bound of [MATH], since the mapping [MATH] is not good enough.', '1310.4021-1-41-3': 'In order to prove the strong consistency of [MATH] and [MATH], we give a simple conclusion in probability theory.', '1310.4021-1-42-0': '[MATH] uniformly on any compact subset, i.e. for any [MATH] is a compact subset of [MATH], we have [EQUATION].', '1310.4021-1-43-0': 'If the real function [MATH] for some [MATH], then [EQUATION] and [EQUATION] [(1)] According to the proof of Theorem [REF], we can find that it is very important to assume [MATH] has a compact support.', '1310.4021-1-43-1': '[(2)] Since [MATH] can be seen as a Laplace transform of some probability measure, maybe we can modify it according to the idea of convolution kernel estimate and the asymptotic risk bound of [MATH] can also be gotten.', '1310.4021-1-44-0': 'The following theorem gives the asymptotic risk bound of [MATH].', '1310.4021-1-44-1': 'The consistency of [MATH] and [MATH] can be easily gotten from it.', '1310.4021-1-45-0': 'Assume [MATH] for any [MATH].', '1310.4021-1-45-1': 'Then [EQUATION] where [MATH] is a constant.', '1310.4021-1-46-0': '[(1)] The inequality in this theorem shows that [MATH] is consistent.', '1310.4021-1-46-1': 'By the continuous of [MATH], the consistence of [MATH] can also be proved.', '1310.4021-1-46-2': '[(2)] The rate of the convergence in this theorem is about [MATH], so the convergent rate of [MATH] is about [MATH], which is very good.', '1310.4021-1-47-0': '# Proof', '1310.4021-1-48-0': 'Proof of Theorem [REF]', '1310.4021-1-49-0': 'Proof.', '1310.4021-1-49-1': 'Let [MATH] and [MATH].', '1310.4021-1-49-2': 'By the Markov property, there is a function [MATH] so that [MATH] and [MATH].', '1310.4021-1-49-3': 'then by Theorem [REF], we have [EQUATION]', '1310.4021-1-49-4': 'The right-side of the final inequality converges to [MATH] independently on [MATH] and [MATH], as [MATH].', '1310.4021-1-49-5': 'Thus [MATH] is strongly mixing.', '1310.4021-1-49-6': '[MATH] Proof of Theorem [REF]', '1310.4021-1-50-0': 'Proof.', '1310.4021-1-50-1': 'Let [MATH].', '1310.4021-1-50-2': 'Firstly, we will show [MATH] is measurable w.r.t [MATH].', '1310.4021-1-50-3': 'For [MATH] is separable and [MATH] is convex, so [MATH] also is separable.', '1310.4021-1-50-4': 'Suppose [MATH] is a subset of [MATH] satisfying that [MATH], then [EQUATION].', '1310.4021-1-50-5': 'For [MATH] is measurable and [MATH] is measurable, so [MATH] is measurable from [MATH] to [MATH].', '1310.4021-1-51-0': 'Now we prove [MATH] is measurable: [MATH].', '1310.4021-1-52-0': 'Let [EQUATION].', '1310.4021-1-52-1': 'Firstly, we prove [MATH] is a [MATH]-algebra .', '1310.4021-1-53-0': '[(1)] [MATH].', '1310.4021-1-53-1': '[(2)] If [MATH], then [EQUATION] [(3)] For any [MATH], then [EQUATION]', '1310.4021-1-53-2': 'So [MATH] is a [MATH]-algebra.', '1310.4021-1-54-0': 'For any [MATH] and [MATH], let [EQUATION].', '1310.4021-1-54-1': 'Secondly, we will prove [MATH].', '1310.4021-1-54-2': 'We just need to prove [EQUATION].', '1310.4021-1-54-3': 'For [MATH] is separable, then for any [MATH], there exit [MATH] s.t. for any [MATH] there exists a [MATH], we have [MATH].', '1310.4021-1-55-0': 'Let [EQUATION]', '1310.4021-1-55-1': 'Next we will prove [MATH].', '1310.4021-1-56-0': '[(1)] [MATH].', '1310.4021-1-57-0': 'For any [MATH], there exists [MATH] such that [MATH].', '1310.4021-1-57-1': 'Then for any [MATH], there exists [MATH] satifying [MATH], thus [EQUATION]', '1310.4021-1-57-2': 'So [MATH].', '1310.4021-1-57-3': '[(2)] [MATH].', '1310.4021-1-58-0': 'For any [MATH] and [MATH], there exists [MATH] s.t. [EQUATION].', '1310.4021-1-58-1': 'Then [EQUATION].', '1310.4021-1-58-2': 'So there exists [MATH] such that[MATH].', '1310.4021-1-58-3': 'Since [MATH] is a Hilbert space and [MATH] is a closed and convex ball, so by Theorem 2.1, we have [MATH] exists uniquely.', '1310.4021-1-58-4': 'For [EQUATION] and [EQUATION] we have [EQUATION].', '1310.4021-1-58-5': 'So [MATH] and [MATH].', '1310.4021-1-59-0': 'For [MATH] is obvious, so [MATH] and [MATH] is measurable [MATH] Proof of Theorem [REF]', '1310.4021-1-60-0': 'Before giving the proof this theorem, we prove a lemma.', '1310.4021-1-61-0': 'For any two probability measures [MATH] and [MATH], denote their Laplace transform by [MATH] and [MATH].', '1310.4021-1-61-1': 'If [MATH] on some interval [MATH], where [MATH], then [MATH] for any [MATH] and [MATH].', '1310.4021-1-62-0': 'Proof.', '1310.4021-1-62-1': 'Let [MATH] and [MATH], then [MATH] and [MATH] are analytic on this strip [MATH].', '1310.4021-1-62-2': 'By the assumption in this lemma and theorem in complex analysis, we have [MATH] on this strip.', '1310.4021-1-62-3': 'Thus let [MATH] we have [MATH] for any [MATH].', '1310.4021-1-62-4': 'By the one-to-one correspondences between probability measures and their Laplace transforms, we have [MATH].', '1310.4021-1-62-5': '[MATH] Now let us prove the this theorem.', '1310.4021-1-63-0': 'Proof.', '1310.4021-1-63-1': 'Firstly, the definition of [MATH], we can easily get that [MATH] is a onto mapping.', '1310.4021-1-63-2': 'Secondly, we will prove that it is also a one-to-one mapping.', '1310.4021-1-63-3': 'For any [MATH], satisfying [EQUATION]', '1310.4021-1-63-4': 'We just need to prove that [EQUATION]', '1310.4021-1-63-5': 'By ([REF]), we have for any [MATH], [EQUATION]', '1310.4021-1-63-6': 'So [EQUATION]', '1310.4021-1-63-7': 'By the definition of CBI processes, thus must exists two probability [MATH] and [MATH], such that [EQUATION] and [EQUATION].', '1310.4021-1-64-0': 'By lemma [REF], [MATH], thus [EQUATION] so [MATH] is a one-to-one mapping.', '1310.4021-1-65-0': 'Finally, let prove [MATH] is a continuous mapping.', '1310.4021-1-65-1': 'For any [MATH], [EQUATION]', '1310.4021-1-65-2': "Where [MATH] doesn't depend on [MATH] and [MATH].", '1310.4021-1-65-3': 'Thus [MATH] is continuous.', '1310.4021-1-65-4': '[MATH] Proof of Theorem [REF]', '1310.4021-1-66-0': 'Proof.', '1310.4021-1-66-1': '(1) Obviously, [MATH] is convex.', '1310.4021-1-66-2': 'We just need to prove that [MATH] is compact.', '1310.4021-1-67-0': 'Let [MATH] be a sequence in [MATH].', '1310.4021-1-67-1': 'Since [MATH] is integrable, so there must exists [MATH], such that [MATH].', '1310.4021-1-67-2': 'Since [MATH], then for any [MATH] we have [MATH].', '1310.4021-1-68-0': 'Suppose for some [MATH], there is [MATH], s.t. [MATH].', '1310.4021-1-68-1': 'Since [MATH], so [MATH], we have [MATH].', '1310.4021-1-68-2': 'So for any [MATH], since [MATH], then [MATH], we have [MATH].', '1310.4021-1-68-3': 'Similarly, for any [MATH] and [MATH], we have [EQUATION]', '1310.4021-1-68-4': 'For [MATH], then [EQUATION]', '1310.4021-1-68-5': 'Thus [MATH] are uniformly bounded and uniformly bounded variation on [MATH], so there exits two nonnegative, monotone increasing and right-continuous functions [MATH] and [MATH] s.t. [MATH].', '1310.4021-1-68-6': 'By ([REF]) we have [EQUATION]', '1310.4021-1-68-7': 'Without loss of generality, we assume that [MATH], so [MATH].', '1310.4021-1-68-8': 'By ([REF]), we have [EQUATION]', '1310.4021-1-68-9': 'Furthermore, [MATH] and [MATH] are determined by all strictly positive rational points, we can use a diagonalization argument to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH].', '1310.4021-1-68-10': 'For any [MATH] define [EQUATION] and [EQUATION].', '1310.4021-1-68-11': 'Define [MATH], then [MATH] at all continuity points of [MATH] on [MATH].', '1310.4021-1-68-12': 'Let [MATH], for [MATH], then [MATH].', '1310.4021-1-68-13': 'For [MATH] and [MATH] are right-continuous and nonnegative monotone increasing functions, so is [MATH] and the number of discontinuity points of [MATH] is at most countable.', '1310.4021-1-68-14': 'Thus [MATH] on [MATH].', '1310.4021-1-68-15': 'Then we can use a diagonalization argument again to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH], then repeat the program above again and we get [MATH] such that [MATH] on [MATH].', '1310.4021-1-68-16': 'Obviously, [MATH] on [MATH].', '1310.4021-1-68-17': 'we repeat the process above again and again.', '1310.4021-1-69-0': 'Finally, we use a diagonalization argument again with respect to [MATH] to find a subsequence [MATH] of [MATH] such that [MATH] converge to some function [MATH] almost surely on [MATH].', '1310.4021-1-69-1': 'For [MATH], then by dominated convergence theorem we have [MATH].', '1310.4021-1-69-2': 'Hence, [MATH] is compact.', '1310.4021-1-70-0': '(2) For [MATH] is continuous, then [MATH] is compact too.', '1310.4021-1-70-1': 'The convexity of [MATH] follows from the convexity of [MATH].', '1310.4021-1-70-2': '[MATH] Proof of Corollary [REF]', '1310.4021-1-71-0': 'Proof.', '1310.4021-1-71-1': 'For [MATH] and [MATH] are Laplace transforms of some measures, we assume them to be [MATH] and [MATH].', '1310.4021-1-72-0': 'For [MATH], [MATH], so [MATH].', '1310.4021-1-72-1': 'By Lemma 7.6 in Sato(1999), we can get [EQUATION] uniformly on any compact subset, we get the conclusion.', '1310.4021-1-72-2': '[MATH] Proof of Theorem [REF]', '1310.4021-1-73-0': 'Proof.', '1310.4021-1-73-1': 'From Corollary [REF], [EQUATION] thus by the theorem 7.6.3 in Chung, [EQUATION].', '1310.4021-1-73-2': 'So by ([REF]) [EQUATION]', '1310.4021-1-73-3': 'By the continuity of [MATH] and ([REF]), we have [EQUATION] [MATH] Proof of Theorem [REF]', '1310.4021-1-74-0': 'Proof.', '1310.4021-1-74-1': "Let [EQUATION] and [MATH], it's easily to prove [MATH] is a martingale with respect to [MATH].", '1310.4021-1-74-2': 'Let [EQUATION]', '1310.4021-1-74-3': 'Thus [EQUATION]', '1310.4021-1-74-4': 'By the stationarity and ergodicity of [MATH] and the assumption in the theorem, we have [EQUATION]', '1310.4021-1-74-5': 'By the martingale central limitation theorem, we have [EQUATION]', '1310.4021-1-74-6': 'By (3.12), we have [EQUATION]', '1310.4021-1-74-7': 'So [EQUATION]', '1310.4021-1-74-8': 'Furthermore, by the definition of [MATH] and [MATH], we have [EQUATION]', '1310.4021-1-74-9': 'For [EQUATION].', '1310.4021-1-74-10': 'It is easily to prove that [MATH] is bounded on any compact set, so when [MATH] is large enough, we have [EQUATION] where [MATH].', '1310.4021-1-74-11': 'Thus [EQUATION] and [EQUATION]', '1310.4021-1-74-12': 'Furthermore, [EQUATION]', '1310.4021-1-74-13': 'By the definition of [MATH], we have [EQUATION]', '1310.4021-1-74-14': 'Assume [MATH], then [EQUATION]', '1310.4021-1-74-15': 'By the Markov property, we have [EQUATION]', '1310.4021-1-74-16': 'For [EQUATION] so [EQUATION]', '1310.4021-1-74-17': 'Thus [EQUATION] and [EQUATION].', '1310.4021-1-74-18': 'When [MATH] is large enough, we have [EQUATION] so [EQUATION] [MATH]'} | {'1310.4021-2-0-0': ': We study the nonparametric estimation of the intensity of the Poisson random measure in continuous-state branching processes with immigration based on the low frequency observations.', '1310.4021-2-0-1': 'This is given in terms of the minimization of norms on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-2-0-2': 'We establish the measurability of the estimators and derive their consistency and asymptotic risk bounds under some conditions.', '1310.4021-2-1-0': 'Key words and phrases: nonparametric estimation, continuous-state branching process with immigration, stochastic differential equation, ergodicity, consistency, asymptotic risk bounds', '1310.4021-2-2-0': '# Introduction', '1310.4021-2-3-0': 'In this paper we investigate the nonparametric estimation in continuous-state branching processes with immigration (CBI-processes).', '1310.4021-2-3-1': 'Branching processes have been a popular approach used in biology as well as in financial world.', '1310.4021-2-3-2': 'For example, Galton-Watson branching processes with immigration (GWI-processes) are used to study the evolution of different species.', '1310.4021-2-3-3': 'Continuous-state branching processes (CB-processes) were first introduced by Jirina [CITATION].', '1310.4021-2-3-4': 'In particular, a continuous CB-process can be obtained as the unique solution of a stochastic equation driven by a Brownian motion.', '1310.4021-2-3-5': 'Kawazu and Watanabe [CITATION] constructed Continuous-state branching processes with immigration (CBI-processes).', '1310.4021-2-3-6': 'In view of the result of Dawson and Li [CITATION], a general single-type CBI-process is the unique strong solution of a stochastic equation driven by Brownian motions and Poisson random measures.', '1310.4021-2-4-0': 'The Cox-Ingersoll-Ross model (CIR model) introduced by Cox et al. [CITATION] has been applied widely in the financial world.', '1310.4021-2-4-1': 'This model has many appealing advantages.', '1310.4021-2-4-2': 'For example, it is mean-reverting and remains positive.', '1310.4021-2-4-3': 'Let [MATH], [MATH] and [MATH] be given constants.', '1310.4021-2-4-4': 'The classical CIR model is a positive diffusion process [MATH] defined by [EQUATION] where [MATH] is a standard Brownian motion.', '1310.4021-2-4-5': 'The process defined by ([REF]) has continuous sample paths.', '1310.4021-2-5-0': 'However, it is well-known that interest rate is influenced not only by the market, but also by sudden events such as changes of governmental policies and so on.', '1310.4021-2-5-1': 'The standard CIR model does not consider the influence of those events, so it sometimes cannot satisfy the need in practice.', '1310.4021-2-5-2': 'For this reason, we need to develop a new model that can interpret these factors.', '1310.4021-2-5-3': 'But the basic idea of the CIR model should not be abandoned, since it is of great importance in some specific field.', '1310.4021-2-5-4': 'Thus we should make some modifications on the basis of the CIR model.', '1310.4021-2-5-5': 'A much easy way is to add jumps to the CIR model, i.e. [EQUATION] where [MATH] is a Poisson random measure on [MATH] with intensity [MATH], [MATH] is a finite measure on [MATH].', '1310.4021-2-5-6': 'In this paper we always assume [MATH] is absolutely continuous with respect to Lebesgue measure, i.e. according to the Radon-Nikodym theorem, there exits a non-negative function [MATH] satisfying [MATH].', '1310.4021-2-6-0': 'However, before using ([REF]) to solve practical problems, we need to estimate the parameters [MATH] and the characteristic measure [MATH] in the equation.', '1310.4021-2-6-1': 'The parameter estimations for [MATH], [MATH] and [MATH] have been given by Huang et al. (2011) in a slightly more general model.', '1310.4021-2-6-2': 'So we just need to investigate the estimation of [MATH] in ([REF]) with [MATH], [MATH] and [MATH] known.', '1310.4021-2-7-0': 'For the standard CIR-model, there are a lot of approaches to estimate the parameters, such as Long-staff and Schwartz [CITATION] and Bibby and Sorensen [CITATION].', '1310.4021-2-7-1': 'Overbeck and Ryden [CITATION] also gave the conditional least squares estimators (CLSEs).', '1310.4021-2-7-2': 'Li and Ma [CITATION] studied the asymptotic properties of CLSEs and WCLSEs in a stable CIR-model.', '1310.4021-2-7-3': 'Huang et al. [CITATION] approached weighted conditional lest squares estimators (WCLSEs) for the CBI-processes and the asymptotic distribution of WCLSEs.', '1310.4021-2-7-4': 'The CLSEs and WCLSEs for the two-type CBI-processes and the asymptotic distribution of CLSEs and WCLSEs were introduced in Xu [CITATION].', '1310.4021-2-7-5': 'It is well-known that the CBI-processes are special examples of the affine Markov processes studied in Duffie et al. [CITATION].', '1310.4021-2-7-6': 'The ergodicity and estimation of some different two-dimensional affine processes were studied in Barczy et al. [CITATION].', '1310.4021-2-8-0': 'Unfortunately, limited work has been done in the nonparametric estimation in CBI-processes compared with that in Levy processes and Ornstein-Uhlenbeck processes (O-U processes).', '1310.4021-2-8-1': 'For example, Watteel and Kulperger [CITATION] proposed and implemented an approach for estimating the jump distribution of the Levy processes by fixed spectral cut-off procedure.', '1310.4021-2-8-2': 'The penalized projection method was applied in Figueroa-Lopez and Houdre [CITATION] to estimate the Levy density on a compact interval separated from the origin, based on a continuous time observation of the sample path throughout a time interval [MATH].', '1310.4021-2-8-3': 'Moreover, Figueroa-Lopez [CITATION] used the projection method for discrete observations and provided minimum risks of estimation for smooth Levy densities, as well as estimated on a compact interval separated from the origin.', '1310.4021-2-8-4': 'Comte and Genon-Catalot used a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global [MATH]- risk with both of high frequency data [CITATION] and low frequency data [CITATION], the method used in this work is a truncated version analogous to the one used in Neumann and Reiss [CITATION] which studied the nonparametric estimation for Levy processes based on the empirical characteristic function.', '1310.4021-2-8-5': 'Jongbloed et al. [CITATION] considered a related low-frequency problem for the canonical function in Levy-Ornstein-Uhlenbeck processes, where a consistent estimator has been constructed.', '1310.4021-2-8-6': 'Roberts et al. [CITATION] developed estimation for O-U processes considering Bayesian estimation for parametric models.', '1310.4021-2-9-0': 'In this work, we give two nonparametric estimators of [MATH] in ([REF]) based on the low frequency observations at equidistant time points [MATH] of a single realization [MATH].', '1310.4021-2-9-1': 'For simplicity, we take [MATH], but all the results presented below can be extended to the general case.', '1310.4021-2-9-2': 'This is based on the minimization of the norm on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-2-9-3': 'The approach has been developed in Jongbloed et al. [CITATION], who applied their results to O-U processes.', '1310.4021-2-9-4': 'Then, like Comte and Genon-Catalot [CITATION], we explore the consistency and asymptotic risk bounds of the estimators.', '1310.4021-2-10-0': 'This paper is organized as follows.', '1310.4021-2-10-1': 'In Section 2, we give the ergodicity and strong mixing property of CBI-processes, which is prerequisite for the study of estimators.', '1310.4021-2-10-2': 'Section 3 is devoted to the study of the nonparametric estimators of [MATH] and proves an fundamental theorem to make sure that the estimators are well defined and measurable.', '1310.4021-2-10-3': 'The consistency and asymptotic risk bounds of the estimators are given in the Section 4.', '1310.4021-2-10-4': 'All the proofs are presented in Section 5.', '1310.4021-2-11-0': 'Notation: In this paper, [MATH], [MATH] and [MATH] mean converge almost surely, in law and weakly, respectively.', '1310.4021-2-11-1': 'Similarly, [MATH], [MATH] and [MATH] mean equal almost surely, in law and weakly,', '1310.4021-2-12-0': '# CBI-processes and ergodicity', '1310.4021-2-13-0': 'In this section, we give some simple properties of CBI-processes.', '1310.4021-2-13-1': 'In particular, we provide that a subcritical CBI-process is ergodic and strongly mixing.', '1310.4021-2-13-2': 'These results are useful in the study of the consistence of the estimators and the convergent rate of the risk bound.', '1310.4021-2-13-3': 'We start with an important special case of those processes.', '1310.4021-2-13-4': 'Let [MATH] and [MATH] be constants.', '1310.4021-2-13-5': 'For [MATH] set [EQUATION]', '1310.4021-2-13-6': 'A Markov process with state space [MATH] is called a continuous-state branching process (CB-process) with branching mechanism [MATH] if it has transition semigroup [MATH] given by [EQUATION] where [MATH] is the unique positive solution to [EQUATION]', '1310.4021-2-13-7': 'By solving ([REF]), we get [EQUATION]', '1310.4021-2-13-8': 'Let [MATH] be a constant and [MATH] be a finite measure on [MATH].', '1310.4021-2-13-9': 'For any [MATH], Set [EQUATION]', '1310.4021-2-13-10': 'A Markov process with state space [MATH] is called a CBI-process with branching mechanism [MATH] and immigration rate [MATH] if it has transition semigroup [MATH] given by [EQUATION]', '1310.4021-2-13-11': 'Actually, the solution of ([REF]) exists uniquely see Fu and Li [CITATION].', '1310.4021-2-13-12': 'Furthermore, it is a CBI-process with branching mechanism [MATH] and immigration mechanism [MATH] defined above, see, e.g. Theorem 3.1 of Dawson and Li [CITATION] or Theorem 2.1 of Li and Ma [CITATION].', '1310.4021-2-13-13': 'For more detail about CBI-processes, readers can refer to Li [CITATION].', '1310.4021-2-13-14': 'Before proving the erogidicity and the mixing of the CBI processes, we recall Theorem 3.20 in Li [CITATION], which gives a necessary and sufficient condition for the ergodicity of the transition semigroup [MATH].', '1310.4021-2-13-15': 'In Li and Ma [CITATION], they prove that stable CIR processes have exponential ergodicity, which is very strong.', '1310.4021-2-14-0': '(Li, 2011, p.66) Suppose that [MATH] and [MATH] for [MATH].', '1310.4021-2-14-1': 'Then for any [MATH], [MATH] conerges to a probability measure [MATH] on [MATH] as [MATH] if and only if [EQUATION]', '1310.4021-2-14-2': 'If ([REF]) holds, the laplace transform of [MATH] is given by [EQUATION]', '1310.4021-2-14-3': 'Under the conditions of Theorem [REF], for any finite set [MATH] we can define the probability measure [MATH] on [MATH] by [EQUATION]', '1310.4021-2-14-4': 'It is easy to see that [MATH] is a consistent family.', '1310.4021-2-14-5': "By Kolmogorov's theorem, there is a stochastic process [MATH] with finite-dimensional distributions given by ([REF]).", '1310.4021-2-14-6': 'This process is a stationary Markov process with one-dimensional marginal distribution [MATH] and transition semigroup [MATH].', '1310.4021-2-14-7': 'Since [MATH] is a Feller semigroup, the process [MATH] has a cadlag modification.', '1310.4021-2-15-0': 'labelt2.2 Suppose that the conditions in Theorem [REF] hold.', '1310.4021-2-15-1': 'Let [MATH] be a Markov process with finite-dimensional distributions given by ([REF]).', '1310.4021-2-15-2': 'Then it is strongly mixing, i.e. as [MATH], [EQUATION]', '1310.4021-2-15-3': "By Theorem [REF] and Birkhoff's ergodic theorem in Durrett [CITATION], we have", '1310.4021-2-16-0': 'Under the conditions of Theorem [REF].', '1310.4021-2-16-1': '[MATH] is ergodic.', '1310.4021-2-17-0': 'With the conclusions above, in this paper, we always assume that [MATH] defined by ([REF]) is a stationary and ergodic process, but by a fairly simple (continuous time) coupling argument it can be seen that the conclusions given in this paper are valid for arbitrary initial distributions.', '1310.4021-2-18-0': '# Estimators and measurability', '1310.4021-2-19-0': 'Recall [MATH].', '1310.4021-2-19-1': 'Let [MATH].', '1310.4021-2-19-2': 'In this section, firstly we will give some theorems to make sure the estimators are well defined and measurable.', '1310.4021-2-19-3': 'Then we will construct estimators by minimizing the norms of the elements of a closed and convex subset in a special [MATH] space.', '1310.4021-2-20-0': 'Before the important theorem, we recall a conclusion in functional analysis without proof, since the proof can be found in many books.', '1310.4021-2-21-0': 'If [MATH] is a Hilbert space with norm [MATH], [MATH] is a nonempty, closed, convex subset of [MATH], then [MATH] contains a unique element of smallest norm.', '1310.4021-2-22-0': '[(1)] In a Banach space [MATH], for any [MATH] and [MATH], [MATH] is convex.', '1310.4021-2-22-1': '[(2)] Every subspace is convex.', '1310.4021-2-22-2': '[(3)] If [MATH] is convex and [MATH], then [MATH] is convex.', '1310.4021-2-23-0': 'With this lemma we will give the most important theorem, which will guarantee the measurability of the estimators.', '1310.4021-2-24-0': 'Let [MATH] be a probability space and [MATH] be a separable Hilbert space with the Borel [MATH]-algebra [MATH].', '1310.4021-2-24-1': '[MATH] is a nonempty, closed and convex subset, [MATH] is a measurable function: [MATH].', '1310.4021-2-24-2': 'Let [EQUATION]', '1310.4021-2-24-3': 'Then [MATH] is well defined and measurable with respect to [MATH].', '1310.4021-2-25-0': '[(1)] LSE, CLSE and WCLSE used widely in parameter estimation are just special cases of this theorem.', '1310.4021-2-25-1': '[(2)] [MATH] space as a special case satisfies the conditions of this theorem.', '1310.4021-2-26-0': 'The total variation of a real-valued function [MATH], defined on an interval [MATH] is the quantity [EQUATION] where the supremum is taken over the set [MATH] and [MATH] is a partition of [MATH] of all partitions of the interval considered.', '1310.4021-2-27-0': 'A real-valued function [MATH] on the real line is said to be of locally bounded variation, if its total variation on any compact subset of [MATH] is finite, i.e. for any compact subset [MATH], we have [MATH].', '1310.4021-2-28-0': 'Recall that [MATH].', '1310.4021-2-28-1': 'Assume [MATH] is right continuous and local variation.', '1310.4021-2-28-2': 'Let [MATH].', '1310.4021-2-28-3': 'Then [MATH].', '1310.4021-2-28-4': 'So we need to find a set of functions that are integrable with respect to [MATH] and includes [MATH].', '1310.4021-2-28-5': 'Thus define [EQUATION].', '1310.4021-2-28-6': 'In order to construct a suitable subset, we define [EQUATION].', '1310.4021-2-28-7': 'Then [MATH] is a convex subset of [MATH].', '1310.4021-2-28-8': 'However, [MATH] is not a closed set.', '1310.4021-2-28-9': 'In order to satisfy the conditions in Theorem [REF], we need to find a closed and convex set.', '1310.4021-2-28-10': 'Define [EQUATION] and a mapping [MATH].', '1310.4021-2-28-11': 'We can see [MATH] is an onto mapping.', '1310.4021-2-29-0': 'Let [EQUATION]', '1310.4021-2-29-1': 'By the erogidicity of [MATH], for any [MATH] we can prove [EQUATION] and [EQUATION] where [EQUATION]', '1310.4021-2-29-2': 'According to the Lemma 7.6 in Sato [CITATION], the following definitions are reasonable: [EQUATION]', '1310.4021-2-29-3': 'By ([REF]), ([REF]) and the Mapping theorem, we have [EQUATION]', '1310.4021-2-29-4': 'In order to satisfy the conditions of Theorem [REF], we need to find a suitable Hilbert space.', '1310.4021-2-29-5': 'Define [EQUATION] where [MATH] is a weighted function, which be discussed later.', '1310.4021-2-29-6': 'We can easily prove [MATH] is a Hilbert Space if we define the inner product [MATH], where [MATH].', '1310.4021-2-29-7': 'With the preparation above we can construct the estimators of [MATH] as follows: [EQUATION] and [EQUATION]', '1310.4021-2-29-8': 'However in order to make the minimum is well defined, we need the integrations in ([REF]) and ([REF]) not always to be infinite.', '1310.4021-2-29-9': 'So in this paper, we will always assume [MATH] is a bounded and non-negative function with compact support, denote by [MATH], and there exist [MATH] such that [MATH].', '1310.4021-2-29-10': 'Thus [MATH] satisfies [EQUATION] where [MATH].', '1310.4021-2-30-0': 'However, since [MATH] is not closed and we need to minimize [MATH] in [MATH], so ([REF]) and ([REF]) may be not well defined or we cannot get the estimators directly from them.', '1310.4021-2-30-1': 'In another way, we can consider the following case: [EQUATION]', '1310.4021-2-30-2': 'Before discussing properties of [MATH], we give a very important conclusion about the mapping [MATH].', '1310.4021-2-31-0': '[MATH] is a one-to-one, onto and continuous mapping, where the topologies of [MATH] and [MATH] are inducted from their initial space respectively.', '1310.4021-2-32-0': 'In order to make sure [MATH] is well defined, we need to find a closed and convex subset in [MATH] and discuss the measurability of [MATH].', '1310.4021-2-32-1': 'Unfortunately, we can easily prove that [MATH] is not a closed subset.', '1310.4021-2-32-2': 'So we need to make some adjustment.', '1310.4021-2-32-3': 'In the following we find that the continuity of the inverse of [MATH] is needed.', '1310.4021-2-32-4': 'So we should find a new subset [MATH] s.t. [MATH] is well defined and continuous.', '1310.4021-2-32-5': 'To obtain an appropriate closed subset, for any [MATH], we first pick up an integrable function [MATH] arbitrarily satisfying that [MATH] and define [EQUATION].', '1310.4021-2-32-6': 'We can choose [MATH] such that [MATH], as [MATH].', '1310.4021-2-32-7': 'Then define [EQUATION].', '1310.4021-2-32-8': 'Next we will prove that [MATH] and [MATH] satisfy our requirements.', '1310.4021-2-33-0': '[(1)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-2-33-1': '[(2)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-2-34-0': 'Since [MATH] is a one-to-one, onto and continuous mapping, furthermore [MATH] and [MATH] are compact, we can easily get the following conclusion.', '1310.4021-2-35-0': 'The inverse operator of [MATH], [MATH] is continuous.', '1310.4021-2-36-0': '[MATH] is a closed subset of [MATH] and [MATH] is a closed subset of [MATH].', '1310.4021-2-37-0': 'By Theorem [REF], Corollary [REF] and Remark [REF], we can give the well defined estimators in the following theorem.', '1310.4021-2-38-0': 'Let [EQUATION]', '1310.4021-2-38-1': 'Then [MATH] and [MATH] are well defined, measurable and exist uniquely.', '1310.4021-2-38-2': 'By the continuity of [MATH], [MATH] and [MATH] uniquely exist and are measurable.', '1310.4021-2-39-0': '[(1)] All the right-continuous and monotone functions are included in [MATH].', '1310.4021-2-40-0': '[(2)] Since [MATH], we usually choose [EQUATION] [(3)] If [MATH] is a finite dimensional subspace, then the nonparametric estimation turns to be parameter estimation.', '1310.4021-2-40-1': 'This is just a special case of Theorem [REF].', '1310.4021-2-41-0': '[(4)] Actually, if [MATH] is just measurable, since [MATH] can be approached by right continuous and local variation functions, so we can also find a approximate estimation for [MATH] in [MATH] for some [MATH].', '1310.4021-2-42-0': '# Consistence and asymptotic risk bound', '1310.4021-2-43-0': 'In this section, we will introduce the strong consistency of [MATH] and [MATH].', '1310.4021-2-43-1': 'At the same time we prove that [MATH] and [MATH] are consistent.', '1310.4021-2-43-2': 'Although we give the asymptotic risk bound of [MATH], we can not give any information about the asymptotic risk bound of [MATH], since the mapping [MATH] is not good enough.', '1310.4021-2-43-3': 'In order to prove the strong consistency of [MATH] and [MATH], we give a simple conclusion in probability theory.', '1310.4021-2-44-0': '[MATH] uniformly on any compact subset, i.e. for any [MATH] is a compact subset of [MATH], we have [EQUATION].', '1310.4021-2-45-0': 'If the real function [MATH] for some [MATH], then [EQUATION] and [EQUATION] [(1)] According to the proof of Theorem [REF], we can find that it is very important to assume [MATH] has a compact support.', '1310.4021-2-46-0': '[(2)] Since [MATH] can be seen as a Laplace transform of some probability measure, maybe we can modify it according to the idea of convolution kernel estimate and the asymptotic risk bound of [MATH] can also be gotten.', '1310.4021-2-47-0': 'The following theorem gives the asymptotic risk bound of [MATH].', '1310.4021-2-47-1': 'The consistency of [MATH] and [MATH] can be easily gotten from it.', '1310.4021-2-48-0': 'Assume [MATH] is a finite measure and [MATH] for any [MATH].', '1310.4021-2-48-1': 'Then [EQUATION]', '1310.4021-2-49-0': '[(1)] The inequality in this theorem shows that [MATH] is consistent.', '1310.4021-2-49-1': 'By the continuous of [MATH], the consistence of [MATH] can also be proved.', '1310.4021-2-50-0': '[(2)] The rate of the convergence in this theorem is about [MATH], so the convergent rate of [MATH] is about [MATH], which is very good.', '1310.4021-2-51-0': '[(3)] According to the proof of Theorem [REF], the condition in the theorem is need to keep [MATH].', '1310.4021-2-51-1': 'However, if we know for some [MATH] such that for any [MATH], [MATH].', '1310.4021-2-51-2': 'Then this condition can be deleted by choosing a proper weighted function [MATH] with [MATH], where [MATH] satisfies that [MATH].', '1310.4021-2-52-0': '# Proof', '1310.4021-2-53-0': 'Proof of Theorem [REF]', '1310.4021-2-54-0': 'Proof.', '1310.4021-2-54-1': 'Let [MATH] and [MATH].', '1310.4021-2-54-2': 'By the Markov property, there is a function [MATH] so that [MATH] and [MATH].', '1310.4021-2-54-3': 'then by Theorem [REF], we have [EQUATION]', '1310.4021-2-54-4': 'The right-side of the final inequality converges to [MATH] independently on [MATH] and [MATH], as [MATH].', '1310.4021-2-54-5': 'Thus [MATH] is strongly mixing.', '1310.4021-2-54-6': '[MATH] Proof of Theorem [REF]', '1310.4021-2-55-0': 'Proof.', '1310.4021-2-55-1': 'Let [MATH].', '1310.4021-2-55-2': 'Firstly, we will show [MATH] is measurable w.r.t [MATH].', '1310.4021-2-55-3': 'For [MATH] is separable, so [MATH] also is separable.', '1310.4021-2-55-4': 'Suppose [MATH] is a subset of [MATH] satisfying that [MATH], then [EQUATION].', '1310.4021-2-55-5': 'For [MATH] is measurable and [MATH] is measurable, so [MATH] is measurable from [MATH] to [MATH].', '1310.4021-2-56-0': 'Now we prove [MATH] is measurable: [MATH].', '1310.4021-2-57-0': 'Let [EQUATION].', '1310.4021-2-57-1': 'Firstly, we prove [MATH] is a [MATH]-algebra .', '1310.4021-2-58-0': '[(1)] [MATH].', '1310.4021-2-58-1': '[(2)] If [MATH], then [EQUATION] [(3)] For any [MATH], then [EQUATION]', '1310.4021-2-58-2': 'So [MATH] is a [MATH]-algebra.', '1310.4021-2-59-0': 'For any [MATH] and [MATH], let [EQUATION].', '1310.4021-2-59-1': 'Secondly, we will prove [MATH].', '1310.4021-2-59-2': 'We just need to prove [EQUATION].', '1310.4021-2-59-3': 'For [MATH] is separable, then for any [MATH], there exit [MATH] s.t. for any [MATH] there exists a [MATH], we have [MATH].', '1310.4021-2-60-0': 'Let [EQUATION]', '1310.4021-2-60-1': 'Next we will prove [MATH].', '1310.4021-2-61-0': '[(1)] [MATH].', '1310.4021-2-62-0': 'For any [MATH], there exists [MATH] such that [MATH].', '1310.4021-2-62-1': 'Then for any [MATH], there exists [MATH] satifying [MATH], thus [EQUATION]', '1310.4021-2-62-2': 'So [MATH].', '1310.4021-2-62-3': '[(2)] [MATH].', '1310.4021-2-63-0': 'For any [MATH] and [MATH], there exists [MATH] s.t. [EQUATION].', '1310.4021-2-63-1': 'Then [EQUATION].', '1310.4021-2-63-2': 'So there exists [MATH] such that[MATH].', '1310.4021-2-63-3': 'Since [MATH] is a Hilbert space and [MATH] is a closed and convex ball, so by Theorem 2.1, we have [MATH] exists uniquely.', '1310.4021-2-63-4': 'For [EQUATION] and [EQUATION] we have [EQUATION].', '1310.4021-2-63-5': 'So [MATH] and [MATH].', '1310.4021-2-64-0': 'For [MATH] is obvious, so [MATH] and [MATH] is measurable [MATH] Proof of Theorem [REF]', '1310.4021-2-65-0': 'Before giving the proof this theorem, we prove a lemma.', '1310.4021-2-66-0': 'For any two probability measures [MATH] and [MATH], denote their Laplace transform by [MATH] and [MATH].', '1310.4021-2-66-1': 'If [MATH] on some interval [MATH], where [MATH], then [MATH] for any [MATH] and [MATH].', '1310.4021-2-67-0': 'Proof.', '1310.4021-2-67-1': 'Let [MATH] and [MATH], then [MATH] and [MATH] are analytic on this strip [MATH].', '1310.4021-2-67-2': 'By the assumption in this lemma and theorem in complex analysis, we have [MATH] on this strip.', '1310.4021-2-67-3': 'Thus let [MATH] we have [MATH] for any [MATH].', '1310.4021-2-67-4': 'By the one-to-one correspondences between probability measures and their Laplace transforms, we have [MATH].', '1310.4021-2-67-5': '[MATH] Now let us prove the this theorem.', '1310.4021-2-68-0': 'Proof.', '1310.4021-2-68-1': 'Firstly, the definition of [MATH], we can easily get that [MATH] is a onto mapping.', '1310.4021-2-68-2': 'Secondly, we will prove that it is also a one-to-one mapping.', '1310.4021-2-68-3': 'For any [MATH], satisfying [EQUATION]', '1310.4021-2-68-4': 'By the definition of CBI processes, thus must exists two probability [MATH] and [MATH], such that [EQUATION] and [EQUATION].', '1310.4021-2-68-5': 'By ([REF]), we have for any [MATH], [EQUATION]', '1310.4021-2-68-6': 'So [MATH] on [MATH].', '1310.4021-2-69-0': 'By lemma [REF], we have [MATH] and [MATH].', '1310.4021-2-69-1': 'Thus [EQUATION] so [MATH] is a one-to-one mapping.', '1310.4021-2-70-0': 'Finally, let prove [MATH] is a continuous mapping.', '1310.4021-2-70-1': 'For any [MATH], [EQUATION]', '1310.4021-2-70-2': 'Since [EQUATION] and [EQUATION] so [EQUATION]', '1310.4021-2-70-3': 'Thus [EQUATION]', '1310.4021-2-70-4': "Where [MATH] doesn't depend on [MATH] and [MATH].", '1310.4021-2-70-5': 'Thus [MATH] is continuous.', '1310.4021-2-70-6': '[MATH] Proof of Theorem [REF]', '1310.4021-2-71-0': 'Proof.', '1310.4021-2-71-1': '(1) Obviously, [MATH] is convex.', '1310.4021-2-71-2': 'We just need to prove that [MATH] is compact.', '1310.4021-2-72-0': 'Let [MATH] be a sequence in [MATH].', '1310.4021-2-72-1': 'Since [MATH] is integrable, so there must exists [MATH], such that [MATH] as [MATH].', '1310.4021-2-72-2': 'Since [MATH], then for any [MATH] we have [MATH] as [MATH].', '1310.4021-2-73-0': 'Suppose for some [MATH], there is [MATH], s.t. [MATH].', '1310.4021-2-73-1': 'Since [MATH], so for any [MATH], we have [MATH].', '1310.4021-2-73-2': 'Similarly, for any [MATH], since [MATH], then for any [MATH], we have [MATH].', '1310.4021-2-73-3': 'Similarly, for any [MATH] and [MATH], we have [EQUATION]', '1310.4021-2-73-4': 'For [MATH], then [EQUATION]', '1310.4021-2-73-5': 'Thus [MATH] are uniformly bounded and uniformly bounded variation on [MATH], so there exits two nonnegative, monotone increasing and right-continuous functions [MATH] and [MATH] such that [MATH].', '1310.4021-2-73-6': 'By ([REF]) we have [EQUATION]', '1310.4021-2-73-7': 'Without loss of generality, we assume that [MATH], so [MATH].', '1310.4021-2-73-8': 'By ([REF]), we have [EQUATION]', '1310.4021-2-73-9': 'Furthermore, [MATH] and [MATH] are determined by all strictly positive rational points, we can use a diagonalization argument to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH].', '1310.4021-2-73-10': 'For any [MATH] define [EQUATION] and [EQUATION].', '1310.4021-2-73-11': 'Define [MATH], then [MATH] at all continuity points of [MATH] on [MATH].', '1310.4021-2-73-12': 'Let [MATH], for [MATH], then [MATH].', '1310.4021-2-73-13': 'For [MATH] and [MATH] are right-continuous and nonnegative monotone increasing functions, so is [MATH] and the number of discontinuity points of [MATH] is at most countable.', '1310.4021-2-73-14': 'Thus [MATH] on [MATH].', '1310.4021-2-73-15': 'Then we can use a diagonalization argument again to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH], then repeat the program above again and we get [MATH] such that [MATH] on [MATH].', '1310.4021-2-73-16': 'Obviously, [MATH] on [MATH].', '1310.4021-2-73-17': 'we repeat the process above again and again.', '1310.4021-2-74-0': 'Finally, we use a diagonalization argument again with respect to [MATH] to find a subsequence [MATH] of [MATH] such that [MATH] converge to some function [MATH] almost surely on [MATH].', '1310.4021-2-74-1': 'For [MATH], then by dominated convergence theorem we have [MATH].', '1310.4021-2-74-2': 'Hence, [MATH] is compact.', '1310.4021-2-75-0': '(2) For [MATH] is continuous, then [MATH] is compact too.', '1310.4021-2-75-1': 'The convexity of [MATH] follows from the convexity of [MATH].', '1310.4021-2-75-2': '[MATH] Proof of Corollary [REF]', '1310.4021-2-76-0': 'Proof.', '1310.4021-2-76-1': 'For [MATH] and [MATH] are Laplace transforms of some measures, we assume them to be [MATH] and [MATH].', '1310.4021-2-77-0': 'For [MATH], [MATH], so [MATH].', '1310.4021-2-77-1': 'By Lemma 7.6 in Sato [CITATION], we can get [EQUATION] uniformly on any compact subset, we get the conclusion.', '1310.4021-2-77-2': '[MATH] Proof of Theorem [REF]', '1310.4021-2-78-0': 'Proof.', '1310.4021-2-78-1': 'From Corollary [REF], [EQUATION] thus by the theorem 7.6.3 in Chung, [EQUATION].', '1310.4021-2-78-2': 'So by ([REF]) [EQUATION]', '1310.4021-2-78-3': 'By the continuity of [MATH] and ([REF]), we have [EQUATION] [MATH] Proof of Theorem [REF]', '1310.4021-2-79-0': 'Proof.', '1310.4021-2-79-1': "Let [EQUATION] and [MATH], it's easily to prove [MATH] is a martingale with respect to [MATH].", '1310.4021-2-79-2': 'Let [EQUATION] where [EQUATION]', '1310.4021-2-79-3': 'Thus [EQUATION]', '1310.4021-2-79-4': 'By the stationarity and ergodicity of [MATH] and the assumption in the theorem, we have [EQUATION]', '1310.4021-2-79-5': 'It is easily to prove that [MATH] is continuous and bounded on any compact set.', '1310.4021-2-79-6': 'By the martingale central limitation theorem, we have [EQUATION]', '1310.4021-2-79-7': 'By (3.12), we have [EQUATION]', '1310.4021-2-79-8': 'Since [MATH] is a finite measure, by Ito formula or ([REF]), it is easily to prove that [MATH].', '1310.4021-2-79-9': 'So for any [MATH], [EQUATION]', '1310.4021-2-79-10': 'Moreover, [MATH] is continuous with respect to [MATH].', '1310.4021-2-79-11': 'So [EQUATION].', '1310.4021-2-79-12': 'By Tonelli theorem [EQUATION]', '1310.4021-2-79-13': 'Furthermore, by the definition of [MATH] and [MATH], we have [EQUATION] where [EQUATION].', '1310.4021-2-79-14': 'From ([REF]), we have [MATH].', '1310.4021-2-79-15': "By Skorohod's theorem, see, e.g. Ikeda and Watanabe (1989), there exist [MATH] and [MATH] defined on [MATH], where [MATH] is Lebesgue measure, such that for any [MATH], [MATH] and [MATH].", '1310.4021-2-79-16': 'Define [EQUATION] so [MATH].', '1310.4021-2-79-17': 'Then [EQUATION] thus [MATH] as [MATH] and [EQUATION]', '1310.4021-2-79-18': 'By dominate convergence theorem, for any [MATH], we have [EQUATION]', '1310.4021-2-79-19': 'Since [MATH] is arbitrage, then [EQUATION]', '1310.4021-2-79-20': 'Since [MATH] is local bounded, so [MATH].', '1310.4021-2-79-21': 'Thus [EQUATION] [MATH]'} | [['1310.4021-1-58-0', '1310.4021-2-63-0'], ['1310.4021-1-58-2', '1310.4021-2-63-2'], ['1310.4021-1-58-3', 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'1310.4021-3-24-1'], ['1310.4021-2-24-3', '1310.4021-3-24-3'], ['1310.4021-2-75-0', '1310.4021-3-74-0'], ['1310.4021-2-75-1', '1310.4021-3-74-1'], ['1310.4021-2-75-2', '1310.4021-3-74-2'], ['1310.4021-2-77-1', '1310.4021-3-76-1'], ['1310.4021-2-77-2', '1310.4021-3-76-2'], ['1310.4021-2-36-0', '1310.4021-3-36-0'], ['1310.4021-2-27-0', '1310.4021-3-27-0'], ['1310.4021-2-11-0', '1310.4021-3-11-0'], ['1310.4021-2-44-0', '1310.4021-3-44-0'], ['1310.4021-2-51-2', '1310.4021-3-50-2'], ['1310.4021-2-15-0', '1310.4021-3-15-0'], ['1310.4021-2-15-1', '1310.4021-3-15-1'], ['1310.4021-2-15-2', '1310.4021-3-15-2'], ['1310.4021-2-15-3', '1310.4021-3-15-3'], ['1310.4021-2-37-0', '1310.4021-3-37-0'], ['1310.4021-2-4-0', '1310.4021-3-4-0'], ['1310.4021-2-4-1', '1310.4021-3-4-1'], ['1310.4021-2-4-2', '1310.4021-3-4-2'], ['1310.4021-2-4-3', '1310.4021-3-4-3'], ['1310.4021-2-4-4', '1310.4021-3-4-4'], ['1310.4021-2-4-5', '1310.4021-3-4-5'], ['1310.4021-2-14-2', '1310.4021-3-14-2'], ['1310.4021-2-14-3', '1310.4021-3-14-3'], ['1310.4021-2-14-4', '1310.4021-3-14-4'], ['1310.4021-2-14-5', '1310.4021-3-14-5'], ['1310.4021-2-14-6', '1310.4021-3-14-6'], ['1310.4021-2-14-7', '1310.4021-3-14-7'], ['1310.4021-2-26-0', '1310.4021-3-26-0'], ['1310.4021-2-73-0', '1310.4021-3-72-0'], ['1310.4021-2-73-1', '1310.4021-3-72-1'], ['1310.4021-2-73-2', '1310.4021-3-72-2'], ['1310.4021-2-73-3', '1310.4021-3-72-3'], ['1310.4021-2-73-5', '1310.4021-3-72-5'], ['1310.4021-2-73-7', '1310.4021-3-72-7'], ['1310.4021-2-73-9', '1310.4021-3-72-9'], ['1310.4021-2-73-10', '1310.4021-3-72-10'], ['1310.4021-2-73-11', '1310.4021-3-72-11'], ['1310.4021-2-73-13', '1310.4021-3-72-13'], ['1310.4021-2-73-15', '1310.4021-3-72-15'], ['1310.4021-2-73-16', '1310.4021-3-72-16'], ['1310.4021-2-73-17', '1310.4021-3-72-17'], ['1310.4021-2-29-1', '1310.4021-3-29-1'], ['1310.4021-2-29-2', '1310.4021-3-29-2'], ['1310.4021-2-29-3', '1310.4021-3-29-3'], ['1310.4021-2-29-4', '1310.4021-3-29-4'], ['1310.4021-2-29-5', '1310.4021-3-29-5'], ['1310.4021-2-29-6', '1310.4021-3-29-6'], ['1310.4021-2-29-7', '1310.4021-3-29-7'], ['1310.4021-2-29-8', '1310.4021-3-29-8'], ['1310.4021-2-29-9', '1310.4021-3-29-9'], ['1310.4021-2-29-10', '1310.4021-3-29-10'], ['1310.4021-2-31-0', '1310.4021-3-31-0'], ['1310.4021-2-19-2', '1310.4021-3-19-2'], ['1310.4021-2-19-3', '1310.4021-3-19-3'], ['1310.4021-2-50-0', '1310.4021-3-49-0'], ['1310.4021-2-0-0', '1310.4021-3-0-0'], ['1310.4021-2-0-1', '1310.4021-3-0-1'], ['1310.4021-2-0-2', '1310.4021-3-0-2'], ['1310.4021-2-9-0', '1310.4021-3-9-0'], ['1310.4021-2-9-1', '1310.4021-3-9-1'], ['1310.4021-2-9-2', '1310.4021-3-9-2'], ['1310.4021-2-9-3', '1310.4021-3-9-3'], ['1310.4021-2-9-4', '1310.4021-3-9-4'], ['1310.4021-2-74-0', '1310.4021-3-73-0'], ['1310.4021-2-74-1', '1310.4021-3-73-1'], ['1310.4021-2-74-2', '1310.4021-3-73-2'], ['1310.4021-2-79-1', '1310.4021-3-78-1'], ['1310.4021-2-79-2', '1310.4021-3-78-2'], ['1310.4021-2-79-4', '1310.4021-3-78-4'], ['1310.4021-2-79-5', '1310.4021-3-78-5'], 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'1310.4021-2-54-0', '1310.4021-2-54-1', '1310.4021-2-55-0', '1310.4021-2-55-1', '1310.4021-2-56-0', '1310.4021-2-57-0', '1310.4021-2-57-1', '1310.4021-2-58-0', '1310.4021-2-58-1', '1310.4021-2-58-2', '1310.4021-2-60-0', '1310.4021-2-60-1', '1310.4021-2-61-0', '1310.4021-2-62-0', '1310.4021-2-62-1', '1310.4021-2-62-2', '1310.4021-2-62-3', '1310.4021-2-63-1', '1310.4021-2-63-5', '1310.4021-2-64-0', '1310.4021-2-65-0', '1310.4021-2-66-0', '1310.4021-2-66-1', '1310.4021-2-67-0', '1310.4021-2-68-0', '1310.4021-2-68-5', '1310.4021-2-68-6', '1310.4021-2-69-0', '1310.4021-2-69-1', '1310.4021-2-70-0', '1310.4021-2-70-1', '1310.4021-2-70-2', '1310.4021-2-70-3', '1310.4021-2-70-4', '1310.4021-2-70-5', '1310.4021-2-70-6', '1310.4021-2-71-0', '1310.4021-2-72-0', '1310.4021-2-72-1', '1310.4021-2-72-2', '1310.4021-2-73-4', '1310.4021-2-73-6', '1310.4021-2-73-8', '1310.4021-2-73-12', '1310.4021-2-73-14', '1310.4021-2-76-0', '1310.4021-2-77-0', '1310.4021-2-78-0', '1310.4021-2-78-2', '1310.4021-2-79-0', '1310.4021-2-79-3', '1310.4021-2-79-7', '1310.4021-2-79-11', '1310.4021-2-79-14', '1310.4021-2-79-21', '1310.4021-3-11-1', '1310.4021-3-14-0', '1310.4021-3-16-0', '1310.4021-3-16-1', '1310.4021-3-19-0', '1310.4021-3-19-1', '1310.4021-3-24-2', '1310.4021-3-28-0', '1310.4021-3-28-2', '1310.4021-3-28-3', '1310.4021-3-28-5', '1310.4021-3-29-0', '1310.4021-3-32-7', '1310.4021-3-35-0', '1310.4021-3-38-0', '1310.4021-3-39-0', '1310.4021-3-52-0', '1310.4021-3-53-0', '1310.4021-3-53-1', '1310.4021-3-54-0', '1310.4021-3-54-1', '1310.4021-3-55-0', '1310.4021-3-56-0', '1310.4021-3-56-1', '1310.4021-3-57-0', '1310.4021-3-57-1', '1310.4021-3-57-2', '1310.4021-3-59-0', '1310.4021-3-59-1', '1310.4021-3-60-0', '1310.4021-3-61-0', '1310.4021-3-61-1', '1310.4021-3-61-2', '1310.4021-3-61-3', '1310.4021-3-62-1', '1310.4021-3-62-5', '1310.4021-3-63-0', '1310.4021-3-64-0', '1310.4021-3-65-0', '1310.4021-3-65-1', '1310.4021-3-66-0', '1310.4021-3-67-0', '1310.4021-3-67-5', '1310.4021-3-67-6', '1310.4021-3-68-0', '1310.4021-3-68-1', '1310.4021-3-69-0', '1310.4021-3-69-1', '1310.4021-3-69-2', '1310.4021-3-69-3', '1310.4021-3-69-4', '1310.4021-3-69-5', '1310.4021-3-69-6', '1310.4021-3-70-0', '1310.4021-3-71-0', '1310.4021-3-71-1', '1310.4021-3-71-2', '1310.4021-3-72-4', '1310.4021-3-72-6', '1310.4021-3-72-8', '1310.4021-3-72-12', '1310.4021-3-72-14', '1310.4021-3-75-0', '1310.4021-3-76-0', '1310.4021-3-77-0', '1310.4021-3-77-2', '1310.4021-3-78-0', '1310.4021-3-78-3', '1310.4021-3-78-7', '1310.4021-3-78-11', '1310.4021-3-78-14', '1310.4021-3-78-21', '1310.4021-4-1-0', '1310.4021-4-9-1', '1310.4021-4-17-0', '1310.4021-4-17-8', '1310.4021-4-17-9', '1310.4021-4-17-14', '1310.4021-4-17-23', '1310.4021-4-19-0', '1310.4021-4-20-0', '1310.4021-4-20-5', '1310.4021-4-20-7', '1310.4021-4-21-0', '1310.4021-4-22-0', '1310.4021-4-22-22', '1310.4021-4-24-1', '1310.4021-4-24-4', '1310.4021-4-27-1', '1310.4021-4-27-11', '1310.4021-4-28-1', '1310.4021-4-28-2', '1310.4021-4-28-5', '1310.4021-4-28-15', '1310.4021-4-29-4'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1310.4021 | {'1310.4021-3-0-0': ': We study the nonparametric estimation of the intensity of the Poisson random measure in continuous-state branching processes with immigration based on the low frequency observations.', '1310.4021-3-0-1': 'This is given in terms of the minimization of norms on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-3-0-2': 'We establish the measurability of the estimators and derive their consistency and asymptotic risk bounds under some conditions.', '1310.4021-3-1-0': 'Key words and phrases: nonparametric estimation, continuous-state branching process with immigration, stochastic differential equation, ergodicity, consistency, asymptotic risk bounds', '1310.4021-3-2-0': '# Introduction', '1310.4021-3-3-0': 'In this paper we investigate the nonparametric estimation in continuous-state branching processes with immigration (CBI-processes).', '1310.4021-3-3-1': 'Branching processes have been a popular approach used in biology as well as in financial world.', '1310.4021-3-3-2': 'For example, Galton-Watson branching processes with immigration (GWI-processes) are used to study the evolution of different species.', '1310.4021-3-3-3': 'Continuous-state branching processes (CB-processes) were first introduced by Jirina [CITATION].', '1310.4021-3-3-4': 'In particular, a continuous CB-process can be obtained as the unique solution of a stochastic equation driven by a Brownian motion.', '1310.4021-3-3-5': 'Kawazu and Watanabe [CITATION] constructed Continuous-state branching processes with immigration (CBI-processes).', '1310.4021-3-3-6': 'In view of the result of Dawson and Li [CITATION], a general single-type CBI-process is the unique strong solution of a stochastic equation driven by Brownian motions and Poisson random measures.', '1310.4021-3-4-0': 'The Cox-Ingersoll-Ross model (CIR model) introduced by Cox et al. [CITATION] has been applied widely in the financial world.', '1310.4021-3-4-1': 'This model has many appealing advantages.', '1310.4021-3-4-2': 'For example, it is mean-reverting and remains positive.', '1310.4021-3-4-3': 'Let [MATH], [MATH] and [MATH] be given constants.', '1310.4021-3-4-4': 'The classical CIR model is a positive diffusion process [MATH] defined by [EQUATION] where [MATH] is a standard Brownian motion.', '1310.4021-3-4-5': 'The process defined by ([REF]) has continuous sample paths.', '1310.4021-3-5-0': 'However, it is well-known that interest rate is influenced not only by the market, but also by sudden events such as changes of governmental policies and so on.', '1310.4021-3-5-1': 'The standard CIR model does not consider the influence of those events, so it sometimes cannot satisfy the need in practice.', '1310.4021-3-5-2': 'For this reason, we need to develop a new model that can interpret these factors.', '1310.4021-3-5-3': 'But the basic idea of the CIR model should not be abandoned, since it is of great importance in some specific field.', '1310.4021-3-5-4': 'Thus we should make some modifications on the basis of the CIR model.', '1310.4021-3-5-5': 'A much easy way is to add jumps to the CIR model, i.e. [EQUATION] where [MATH] is a Poisson random measure on [MATH] with intensity [MATH], [MATH] is a finite measure on [MATH].', '1310.4021-3-5-6': 'In this paper we always assume [MATH] is absolutely continuous with respect to Lebesgue measure, i.e. according to the Radon-Nikodym theorem, there exits a non-negative function [MATH] satisfying [MATH].', '1310.4021-3-6-0': 'However, before using ([REF]) to solve practical problems, we need to estimate the parameters [MATH] and the characteristic measure [MATH] in the equation.', '1310.4021-3-6-1': 'The parameter estimations for [MATH], [MATH] and [MATH] have been given by Huang et al. (2011) in a slightly more general model.', '1310.4021-3-6-2': 'So we just need to investigate the estimation of [MATH] in ([REF]) with [MATH], [MATH] and [MATH] known.', '1310.4021-3-7-0': 'For the standard CIR-model, there are a lot of approaches to estimate the parameters, such as Long-staff and Schwartz [CITATION] and Bibby and Sorensen [CITATION].', '1310.4021-3-7-1': 'Overbeck and Ryden [CITATION] also gave the conditional least squares estimators (CLSEs).', '1310.4021-3-7-2': 'Li and Ma [CITATION] studied the asymptotic properties of CLSEs and WCLSEs in a stable CIR-model.', '1310.4021-3-7-3': 'Huang et al. [CITATION] approached weighted conditional lest squares estimators (WCLSEs) for the CBI-processes and the asymptotic distribution of WCLSEs.', '1310.4021-3-7-4': 'The CLSEs and WCLSEs for the two-type CBI-processes and the asymptotic distribution of CLSEs and WCLSEs were introduced in Xu [CITATION].', '1310.4021-3-7-5': 'It is well-known that the CBI-processes are special examples of the affine Markov processes studied in Duffie et al. [CITATION].', '1310.4021-3-7-6': 'The ergodicity and estimation of some different two-dimensional affine processes were studied in Barczy et al. [CITATION].', '1310.4021-3-8-0': 'Unfortunately, limited work has been done in the nonparametric estimation in CBI-processes compared with that in Levy processes and Ornstein-Uhlenbeck processes (O-U processes).', '1310.4021-3-8-1': 'For example, Watteel and Kulperger [CITATION] proposed and implemented an approach for estimating the jump distribution of the Levy processes by fixed spectral cut-off procedure.', '1310.4021-3-8-2': 'The penalized projection method was applied in Figueroa-Lopez and Houdre [CITATION] to estimate the Levy density on a compact interval separated from the origin, based on a continuous time observation of the sample path throughout a time interval [MATH].', '1310.4021-3-8-3': 'Moreover, Figueroa-Lopez [CITATION] used the projection method for discrete observations and provided minimum risks of estimation for smooth Levy densities, as well as estimated on a compact interval separated from the origin.', '1310.4021-3-8-4': 'Comte and Genon-Catalot used a Fourier approach to construct an adaptive nonparametric estimator and to provide a bound for the global [MATH]- risk with both of high frequency data [CITATION] and low frequency data [CITATION], the method used in this work is a truncated version analogous to the one used in Neumann and Reiss [CITATION] which studied the nonparametric estimation for Levy processes based on the empirical characteristic function.', '1310.4021-3-8-5': 'Jongbloed et al. [CITATION] considered a related low-frequency problem for the canonical function in Levy driven O-U processes, where a consistent estimator has been constructed.', '1310.4021-3-8-6': 'Roberts et al. [CITATION] developed estimation for O-U processes considering Bayesian estimation for parametric models.', '1310.4021-3-9-0': 'In this work, we give two nonparametric estimators of [MATH] in ([REF]) based on the low frequency observations at equidistant time points [MATH] of a single realization [MATH].', '1310.4021-3-9-1': 'For simplicity, we take [MATH], but all the results presented below can be extended to the general case.', '1310.4021-3-9-2': 'This is based on the minimization of the norm on a nonempty, closed and convex subset in a special Hilbert space.', '1310.4021-3-9-3': 'The approach has been developed in Jongbloed et al. [CITATION], who applied their results to O-U processes.', '1310.4021-3-9-4': 'Then, like Comte and Genon-Catalot [CITATION], we explore the consistency and asymptotic risk bounds of the estimators.', '1310.4021-3-10-0': 'This paper is organized as follows.', '1310.4021-3-10-1': 'In Section 2, we give the ergodicity and strong mixing property of CBI-processes, which is prerequisite for the study of estimators.', '1310.4021-3-10-2': 'Section 3 is devoted to the study of the nonparametric estimators of [MATH] and proves an fundamental theorem to make sure that the estimators are well defined and measurable.', '1310.4021-3-10-3': 'The consistency and asymptotic risk bounds of the estimators are given in the Section 4.', '1310.4021-3-10-4': 'All the proofs are presented in Section 5.', '1310.4021-3-11-0': 'Notation: In this paper, [MATH], [MATH] and [MATH] mean converge almost surely, in law and weakly, respectively.', '1310.4021-3-11-1': 'Similarly, [MATH], [MATH] and [MATH] mean equal almost surely, in law and weakly,', '1310.4021-3-12-0': '# CBI-processes and ergodicity', '1310.4021-3-13-0': 'In this section, we give some simple properties of CBI-processes.', '1310.4021-3-13-1': 'In particular, we provide that a subcritical CBI-process is ergodic and strongly mixing.', '1310.4021-3-13-2': 'These results are useful in the study of the consistence of the estimators and the convergent rate of the risk bound.', '1310.4021-3-13-3': 'We start with an important special case of those processes.', '1310.4021-3-13-4': 'Let [MATH] and [MATH] be constants.', '1310.4021-3-13-5': 'For [MATH] set [EQUATION]', '1310.4021-3-13-6': 'A Markov process with state space [MATH] is called a continuous-state branching process (CB-process) with branching mechanism [MATH] if it has transition semigroup [MATH] given by [EQUATION] where [MATH] is the unique positive solution to [EQUATION]', '1310.4021-3-13-7': 'By solving ([REF]), we get [EQUATION]', '1310.4021-3-13-8': 'Let [MATH] be a constant and [MATH] be a finite measure on [MATH].', '1310.4021-3-13-9': 'For any [MATH], Set [EQUATION]', '1310.4021-3-13-10': 'A Markov process with state space [MATH] is called a CBI-process with branching mechanism [MATH] and immigration rate [MATH] if it has transition semigroup [MATH] given by [EQUATION]', '1310.4021-3-13-11': 'Actually, the solution of ([REF]) exists uniquely see Fu and Li [CITATION].', '1310.4021-3-13-12': 'Furthermore, it is a CBI-process with branching mechanism [MATH] and immigration mechanism [MATH] defined above, see, e.g. Theorem 3.1 of Dawson and Li [CITATION] or Theorem 2.1 of Li and Ma [CITATION].', '1310.4021-3-13-13': 'For more detail about CBI-processes, readers can refer to Li [CITATION].', '1310.4021-3-13-14': 'Before proving the erogidicity and the mixing of the CBI-processes, we recall Theorem 3.20 in Li [CITATION], which gives a necessary and sufficient condition for the ergodicity of the transition semigroup [MATH].', '1310.4021-3-13-15': 'In Li and Ma [CITATION], they prove that stable CIR model have exponential ergodicity, which is very strong.', '1310.4021-3-14-0': '(Li, 2011, p.66) Suppose that [MATH] and [MATH] for [MATH].', '1310.4021-3-14-1': 'Then for any [MATH], [MATH] converges to a probability measure [MATH] on [MATH] as [MATH] if and only if [EQUATION]', '1310.4021-3-14-2': 'If ([REF]) holds, the laplace transform of [MATH] is given by [EQUATION]', '1310.4021-3-14-3': 'Under the conditions of Theorem [REF], for any finite set [MATH] we can define the probability measure [MATH] on [MATH] by [EQUATION]', '1310.4021-3-14-4': 'It is easy to see that [MATH] is a consistent family.', '1310.4021-3-14-5': "By Kolmogorov's theorem, there is a stochastic process [MATH] with finite-dimensional distributions given by ([REF]).", '1310.4021-3-14-6': 'This process is a stationary Markov process with one-dimensional marginal distribution [MATH] and transition semigroup [MATH].', '1310.4021-3-14-7': 'Since [MATH] is a Feller semigroup, the process [MATH] has a cadlag modification.', '1310.4021-3-15-0': 'labelt2.2 Suppose that the conditions in Theorem [REF] hold.', '1310.4021-3-15-1': 'Let [MATH] be a Markov process with finite-dimensional distributions given by ([REF]).', '1310.4021-3-15-2': 'Then it is strongly mixing, i.e. as [MATH], [EQUATION]', '1310.4021-3-15-3': "By Theorem [REF] and Birkhoff's ergodic theorem in Durrett [CITATION], we have", '1310.4021-3-16-0': 'Under the conditions of Theorem [REF].', '1310.4021-3-16-1': '[MATH] is ergodic.', '1310.4021-3-17-0': 'With the conclusions above, in this paper, we always assume that [MATH] defined by ([REF]) is a stationary and ergodic process, but by a fairly simple (continuous time) coupling argument it can be seen that the conclusions given in this paper are valid for arbitrary initial distributions.', '1310.4021-3-18-0': '# Estimators and measurability', '1310.4021-3-19-0': 'Recall [MATH].', '1310.4021-3-19-1': 'Let [MATH].', '1310.4021-3-19-2': 'In this section, firstly we will give some theorems to make sure the estimators are well defined and measurable.', '1310.4021-3-19-3': 'Then we will construct estimators by minimizing the norms of the elements of a closed and convex subset in a special [MATH] space.', '1310.4021-3-20-0': 'Before the important theorem, we recall a conclusion in functional analysis without proof, since the proof can be found in many books.', '1310.4021-3-21-0': 'If [MATH] is a Hilbert space with norm [MATH], [MATH] is a nonempty, closed, convex subset of [MATH], then [MATH] contains a unique element of smallest norm.', '1310.4021-3-22-0': '[(1)] In a Banach space [MATH], for any [MATH] and [MATH], [MATH] is convex.', '1310.4021-3-22-1': '[(2)] Every subspace is convex.', '1310.4021-3-22-2': '[(3)] If [MATH] is convex and [MATH], then [MATH] is convex.', '1310.4021-3-23-0': 'With this lemma we will give the most important theorem, which will guarantee the measurability of the estimators.', '1310.4021-3-24-0': 'Let [MATH] be a probability space and [MATH] be a separable Hilbert space with the Borel [MATH]-algebra [MATH].', '1310.4021-3-24-1': '[MATH] is a nonempty, closed and convex subset, [MATH] is a measurable function: [MATH].', '1310.4021-3-24-2': 'Let [EQUATION]', '1310.4021-3-24-3': 'Then [MATH] is well defined and measurable with respect to [MATH].', '1310.4021-3-25-0': '[(1)] LSE, CLSE and WCLSE used widely in parameter estimation are just special cases of this theorem.', '1310.4021-3-25-1': '[(2)] [MATH] space as a special case satisfies the conditions of this theorem.', '1310.4021-3-26-0': 'The total variation of a real-valued function [MATH], defined on an interval [MATH] is the quantity [EQUATION] where the supremum is taken over the set [MATH] and [MATH] is a partition of [MATH] of all partitions of the interval considered.', '1310.4021-3-27-0': 'A real-valued function [MATH] on the real line is said to be of locally bounded variation, if its total variation on any compact subset of [MATH] is finite, i.e. for any compact subset [MATH], we have [MATH].', '1310.4021-3-28-0': 'Recall that [MATH].', '1310.4021-3-28-1': 'Assume [MATH] is right-continuous and of locally bounded variation.', '1310.4021-3-28-2': 'Let [MATH].', '1310.4021-3-28-3': 'Then [MATH].', '1310.4021-3-28-4': 'So we need to find a set of functions that are integrable with respect to [MATH] and includes [MATH].', '1310.4021-3-28-5': 'Thus define [EQUATION].', '1310.4021-3-28-6': 'In order to construct a suitable subset, we define [EQUATION].', '1310.4021-3-28-7': 'Then [MATH] is a convex subset of [MATH].', '1310.4021-3-28-8': 'However, [MATH] is not a closed set.', '1310.4021-3-28-9': 'In order to satisfy the conditions in Theorem [REF], we need to find a closed and convex set.', '1310.4021-3-28-10': 'Define [EQUATION] and a mapping [MATH].', '1310.4021-3-28-11': 'We can see [MATH] is an onto mapping.', '1310.4021-3-29-0': 'Let [EQUATION]', '1310.4021-3-29-1': 'By the erogidicity of [MATH], for any [MATH] we can prove [EQUATION] and [EQUATION] where [EQUATION]', '1310.4021-3-29-2': 'According to the Lemma 7.6 in Sato [CITATION], the following definitions are reasonable: [EQUATION]', '1310.4021-3-29-3': 'By ([REF]), ([REF]) and the Mapping theorem, we have [EQUATION]', '1310.4021-3-29-4': 'In order to satisfy the conditions of Theorem [REF], we need to find a suitable Hilbert space.', '1310.4021-3-29-5': 'Define [EQUATION] where [MATH] is a weighted function, which be discussed later.', '1310.4021-3-29-6': 'We can easily prove [MATH] is a Hilbert Space if we define the inner product [MATH], where [MATH].', '1310.4021-3-29-7': 'With the preparation above we can construct the estimators of [MATH] as follows: [EQUATION] and [EQUATION]', '1310.4021-3-29-8': 'However in order to make the minimum is well defined, we need the integrations in ([REF]) and ([REF]) not always to be infinite.', '1310.4021-3-29-9': 'So in this paper, we will always assume [MATH] is a bounded and non-negative function with compact support, denote by [MATH], and there exist [MATH] such that [MATH].', '1310.4021-3-29-10': 'Thus [MATH] satisfies [EQUATION] where [MATH].', '1310.4021-3-30-0': 'However, since [MATH] is not closed and we need to minimize [MATH] in [MATH], so ([REF]) and ([REF]) may be not well defined or we cannot get the estimators directly from them.', '1310.4021-3-30-1': 'In another way, we can consider the following case: [EQUATION]', '1310.4021-3-30-2': 'Before discussing properties of [MATH], we give a very important conclusion about the mapping [MATH].', '1310.4021-3-31-0': '[MATH] is a one-to-one, onto and continuous mapping, where the topologies of [MATH] and [MATH] are inducted from their initial space respectively.', '1310.4021-3-32-0': 'In order to make sure [MATH] is well defined, we need to find a closed and convex subset in [MATH] and discuss the measurability of [MATH].', '1310.4021-3-32-1': 'Unfortunately, we can easily prove that [MATH] is not a closed subset.', '1310.4021-3-32-2': 'So we need to make some adjustment.', '1310.4021-3-32-3': 'In the following we find that the continuity of the inverse of [MATH] is needed.', '1310.4021-3-32-4': 'So we should find a new subset [MATH] s.t. [MATH] is well defined and continuous.', '1310.4021-3-32-5': 'To obtain an appropriate closed subset, for any [MATH], we first pick up an integrable function [MATH] arbitrarily satisfying that [MATH] and define [EQUATION].', '1310.4021-3-32-6': 'We can choose [MATH] such that [MATH], as [MATH].', '1310.4021-3-32-7': 'Then define [EQUATION].', '1310.4021-3-32-8': 'Next we will prove that [MATH] and [MATH] satisfy our requirements.', '1310.4021-3-33-0': '[(1)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-3-33-1': '[(2)] [MATH] is a compact, convex subset of [MATH].', '1310.4021-3-34-0': 'Since [MATH] is a one-to-one, onto and continuous mapping, furthermore [MATH] and [MATH] are compact, we can easily get the following conclusion.', '1310.4021-3-35-0': 'The inverse operator of [MATH], [MATH] is continuous.', '1310.4021-3-36-0': '[MATH] is a closed subset of [MATH] and [MATH] is a closed subset of [MATH].', '1310.4021-3-37-0': 'By Theorem [REF], Corollary [REF] and Remark [REF], we can give the well defined estimators in the following theorem.', '1310.4021-3-38-0': 'Let [EQUATION]', '1310.4021-3-38-1': 'Then [MATH] and [MATH] are well defined, measurable and exist uniquely.', '1310.4021-3-38-2': 'By the continuity of [MATH], [MATH] and [MATH] uniquely exist and are measurable.', '1310.4021-3-39-0': '[(1)] [MATH] contains all the right-continuous and monotone functions.', '1310.4021-3-40-0': '[(2)] Since [MATH], we usually choose [EQUATION] [(3)] If [MATH] is a finite dimensional subspace, then the nonparametric estimation turns to be parameter estimation.', '1310.4021-3-40-1': 'This is just a special case of Theorem [REF].', '1310.4021-3-41-0': '[(4)] Actually, if [MATH] is just measurable, since [MATH] can be approached by right continuous functions with locally bounded variation , so we can also find a approximate estimation for [MATH] in [MATH] for some [MATH].', '1310.4021-3-42-0': '# Consistency and asymptotic risk bound', '1310.4021-3-43-0': 'In this section, we introduce the strong consistency of [MATH] and [MATH].', '1310.4021-3-43-1': 'At the same time we prove that [MATH] and [MATH] are consistent.', '1310.4021-3-43-2': 'Although we give the asymptotic risk bound of [MATH], we can not give any information about the asymptotic risk bound of [MATH], since the mapping [MATH] is not good enough.', '1310.4021-3-43-3': 'In order to prove the strong consistency of [MATH] and [MATH], we give a simple conclusion in probability theory.', '1310.4021-3-44-0': '[MATH] uniformly on any compact subset, i.e. for any [MATH] is a compact subset of [MATH], we have [EQUATION].', '1310.4021-3-45-0': 'If the real function [MATH] for some [MATH], then [EQUATION] and [EQUATION] [(1)] According to the proof of Theorem [REF], we can find that it is very important to assume [MATH] has a compact support.', '1310.4021-3-46-0': '[(2)] Since [MATH] can be seen as a Laplace transform of some probability measure, maybe we can modify it according to the idea of convolution kernel estimate, then the consistency and asymptotic risk bound of [MATH] can also be gotten.', '1310.4021-3-47-0': 'The following theorem gives the asymptotic risk bound of [MATH].', '1310.4021-3-47-1': 'The consistency of [MATH] and [MATH] can be easily gotten from it.', '1310.4021-3-48-0': 'Assume [MATH] is a finite measure and [MATH] for any [MATH].', '1310.4021-3-48-1': 'Then [EQUATION] [(1)] The inequality in this theorem shows that [MATH] is consistent.', '1310.4021-3-48-2': 'By the continuous of [MATH], the consistency of [MATH] can also be proved.', '1310.4021-3-49-0': '[(2)] The rate of the convergence in this theorem is about [MATH], so the convergent rate of [MATH] is about [MATH], which is very good.', '1310.4021-3-50-0': '[(3)] According to the proof of Theorem [REF], the condition in the theorem is needed to keep [MATH].', '1310.4021-3-50-1': 'However, if we know for some [MATH] such that [MATH] for any [MATH].', '1310.4021-3-50-2': 'Then this condition can be deleted by choosing a proper weighted function [MATH] with [MATH], where [MATH] satisfies that [MATH].', '1310.4021-3-51-0': '# Proof', '1310.4021-3-52-0': 'Proof of Theorem [REF]', '1310.4021-3-53-0': 'Proof.', '1310.4021-3-53-1': 'Let [MATH] and [MATH].', '1310.4021-3-53-2': 'By the Markov property, there is a function [MATH] so that [MATH] and [MATH].', '1310.4021-3-53-3': 'then by Theorem [REF], we have [EQUATION]', '1310.4021-3-53-4': 'The right-side of the final inequality converges to [MATH] independently on [MATH] and [MATH], as [MATH].', '1310.4021-3-53-5': 'Thus [MATH] is strongly mixing.', '1310.4021-3-53-6': '[MATH] Proof of Theorem [REF]', '1310.4021-3-54-0': 'Proof.', '1310.4021-3-54-1': 'Let [MATH].', '1310.4021-3-54-2': 'Firstly, we will show [MATH] is measurable w.r.t [MATH].', '1310.4021-3-54-3': 'For [MATH] is separable, so [MATH] also is separable.', '1310.4021-3-54-4': 'Suppose [MATH] is a subset of [MATH] satisfying that [MATH], then [EQUATION].', '1310.4021-3-54-5': 'For [MATH] is measurable and [MATH] is measurable, so [MATH] is measurable from [MATH] to [MATH].', '1310.4021-3-55-0': 'Now we prove [MATH] is measurable: [MATH].', '1310.4021-3-56-0': 'Let [EQUATION].', '1310.4021-3-56-1': 'Firstly, we prove [MATH] is a [MATH]-algebra .', '1310.4021-3-57-0': '[(1)] [MATH].', '1310.4021-3-57-1': '[(2)] If [MATH], then [EQUATION] [(3)] For any [MATH], then [EQUATION]', '1310.4021-3-57-2': 'So [MATH] is a [MATH]-algebra.', '1310.4021-3-58-0': 'For any [MATH] and [MATH], let [EQUATION].', '1310.4021-3-58-1': 'Secondly, we will prove [MATH].', '1310.4021-3-58-2': 'We just need to prove [EQUATION].', '1310.4021-3-58-3': 'For [MATH] is separable, then for any [MATH], there exit [MATH] s.t. for any [MATH] there exists a [MATH], we have [MATH].', '1310.4021-3-59-0': 'Let [EQUATION]', '1310.4021-3-59-1': 'Next we will prove [MATH].', '1310.4021-3-60-0': '[(1)] [MATH].', '1310.4021-3-61-0': 'For any [MATH], there exists [MATH] such that [MATH].', '1310.4021-3-61-1': 'Then for any [MATH], there exists [MATH] satifying [MATH], thus [EQUATION]', '1310.4021-3-61-2': 'So [MATH].', '1310.4021-3-61-3': '[(2)] [MATH].', '1310.4021-3-62-0': 'For any [MATH] and [MATH], there exists [MATH] s.t. [EQUATION].', '1310.4021-3-62-1': 'Then [EQUATION].', '1310.4021-3-62-2': 'So there exists [MATH] such that[MATH].', '1310.4021-3-62-3': 'Since [MATH] is a Hilbert space and [MATH] is a closed and convex ball, so by Theorem 2.1, we have [MATH] exists uniquely.', '1310.4021-3-62-4': 'For [EQUATION] and [EQUATION] we have [EQUATION].', '1310.4021-3-62-5': 'So [MATH] and [MATH].', '1310.4021-3-63-0': 'For [MATH] is obvious, so [MATH] and [MATH] is measurable [MATH] Proof of Theorem [REF]', '1310.4021-3-64-0': 'Before giving the proof this theorem, we prove a lemma.', '1310.4021-3-65-0': 'For any two probability measures [MATH] and [MATH], denote their Laplace transform by [MATH] and [MATH].', '1310.4021-3-65-1': 'If [MATH] on some interval [MATH], where [MATH], then [MATH] for any [MATH] and [MATH].', '1310.4021-3-66-0': 'Proof.', '1310.4021-3-66-1': 'Let [MATH] and [MATH], then [MATH] and [MATH] are analytic on this strip [MATH].', '1310.4021-3-66-2': 'By the assumption in this lemma and theorem in complex analysis, we have [MATH] on this strip.', '1310.4021-3-66-3': 'Thus let [MATH] we have [MATH] for any [MATH].', '1310.4021-3-66-4': 'By the one-to-one correspondences between probability measures and their Laplace transforms, we have [MATH].', '1310.4021-3-66-5': '[MATH] Now let us prove the this theorem.', '1310.4021-3-67-0': 'Proof.', '1310.4021-3-67-1': 'Firstly, the definition of [MATH], we can easily get that [MATH] is a onto mapping.', '1310.4021-3-67-2': 'Secondly, we will prove that it is also a one-to-one mapping.', '1310.4021-3-67-3': 'For any [MATH], satisfying [EQUATION]', '1310.4021-3-67-4': 'By the definition of CBI processes, thus must exists two probability [MATH] and [MATH], such that [EQUATION] and [EQUATION].', '1310.4021-3-67-5': 'By ([REF]), we have for any [MATH], [EQUATION]', '1310.4021-3-67-6': 'So [MATH] on [MATH].', '1310.4021-3-68-0': 'By lemma [REF], we have [MATH] and [MATH].', '1310.4021-3-68-1': 'Thus [EQUATION] so [MATH] is a one-to-one mapping.', '1310.4021-3-69-0': 'Finally, let prove [MATH] is a continuous mapping.', '1310.4021-3-69-1': 'For any [MATH], [EQUATION]', '1310.4021-3-69-2': 'Since [EQUATION] and [EQUATION] so [EQUATION]', '1310.4021-3-69-3': 'Thus [EQUATION]', '1310.4021-3-69-4': "Where [MATH] doesn't depend on [MATH] and [MATH].", '1310.4021-3-69-5': 'Thus [MATH] is continuous.', '1310.4021-3-69-6': '[MATH] Proof of Theorem [REF]', '1310.4021-3-70-0': 'Proof.', '1310.4021-3-70-1': '(1) Obviously, [MATH] is convex.', '1310.4021-3-70-2': 'We just need to prove that [MATH] is compact.', '1310.4021-3-71-0': 'Let [MATH] be a sequence in [MATH].', '1310.4021-3-71-1': 'Since [MATH] is integrable, so there must exists [MATH], such that [MATH] as [MATH].', '1310.4021-3-71-2': 'Since [MATH], then for any [MATH] we have [MATH] as [MATH].', '1310.4021-3-72-0': 'Suppose for some [MATH], there is [MATH], s.t. [MATH].', '1310.4021-3-72-1': 'Since [MATH], so for any [MATH], we have [MATH].', '1310.4021-3-72-2': 'Similarly, for any [MATH], since [MATH], then for any [MATH], we have [MATH].', '1310.4021-3-72-3': 'Similarly, for any [MATH] and [MATH], we have [EQUATION]', '1310.4021-3-72-4': 'For [MATH], then [EQUATION]', '1310.4021-3-72-5': 'Thus [MATH] are uniformly bounded and uniformly bounded variation on [MATH], so there exits two nonnegative, monotone increasing and right-continuous functions [MATH] and [MATH] such that [MATH].', '1310.4021-3-72-6': 'By ([REF]) we have [EQUATION]', '1310.4021-3-72-7': 'Without loss of generality, we assume that [MATH], so [MATH].', '1310.4021-3-72-8': 'By ([REF]), we have [EQUATION]', '1310.4021-3-72-9': 'Furthermore, [MATH] and [MATH] are determined by all strictly positive rational points, we can use a diagonalization argument to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH].', '1310.4021-3-72-10': 'For any [MATH] define [EQUATION] and [EQUATION].', '1310.4021-3-72-11': 'Define [MATH], then [MATH] at all continuity points of [MATH] on [MATH].', '1310.4021-3-72-12': 'Let [MATH], for [MATH], then [MATH].', '1310.4021-3-72-13': 'For [MATH] and [MATH] are right-continuous and nonnegative monotone increasing functions, so is [MATH] and the number of discontinuity points of [MATH] is at most countable.', '1310.4021-3-72-14': 'Thus [MATH] on [MATH].', '1310.4021-3-72-15': 'Then we can use a diagonalization argument again to find a subsequence [MATH] from [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH], then repeat the program above again and we get [MATH] such that [MATH] on [MATH].', '1310.4021-3-72-16': 'Obviously, [MATH] on [MATH].', '1310.4021-3-72-17': 'we repeat the process above again and again.', '1310.4021-3-73-0': 'Finally, we use a diagonalization argument again with respect to [MATH] to find a subsequence [MATH] of [MATH] such that [MATH] converge to some function [MATH] almost surely on [MATH].', '1310.4021-3-73-1': 'For [MATH], then by dominated convergence theorem we have [MATH].', '1310.4021-3-73-2': 'Hence, [MATH] is compact.', '1310.4021-3-74-0': '(2) For [MATH] is continuous, then [MATH] is compact too.', '1310.4021-3-74-1': 'The convexity of [MATH] follows from the convexity of [MATH].', '1310.4021-3-74-2': '[MATH] Proof of Corollary [REF]', '1310.4021-3-75-0': 'Proof.', '1310.4021-3-75-1': 'For [MATH] and [MATH] are Laplace transforms of some measures, we assume them to be [MATH] and [MATH].', '1310.4021-3-76-0': 'For [MATH], [MATH], so [MATH].', '1310.4021-3-76-1': 'By Lemma 7.6 in Sato [CITATION], we can get [EQUATION] uniformly on any compact subset, we get the conclusion.', '1310.4021-3-76-2': '[MATH] Proof of Theorem [REF]', '1310.4021-3-77-0': 'Proof.', '1310.4021-3-77-1': 'From Corollary [REF], [EQUATION] thus by the theorem 7.6.3 in Chung, [EQUATION].', '1310.4021-3-77-2': 'So by ([REF]) [EQUATION]', '1310.4021-3-77-3': 'By the continuity of [MATH] and ([REF]), we have [EQUATION] [MATH] Proof of Theorem [REF]', '1310.4021-3-78-0': 'Proof.', '1310.4021-3-78-1': "Let [EQUATION] and [MATH], it's easily to prove [MATH] is a martingale with respect to [MATH].", '1310.4021-3-78-2': 'Let [EQUATION] where [EQUATION]', '1310.4021-3-78-3': 'Thus [EQUATION]', '1310.4021-3-78-4': 'By the stationarity and ergodicity of [MATH] and the assumption in the theorem, we have [EQUATION]', '1310.4021-3-78-5': 'It is easily to prove that [MATH] is continuous and bounded on any compact set.', '1310.4021-3-78-6': 'By the martingale central limitation theorem, we have [EQUATION]', '1310.4021-3-78-7': 'By (3.12), we have [EQUATION]', '1310.4021-3-78-8': 'Since [MATH] is a finite measure, by Ito formula or ([REF]), it is easily to prove that [MATH].', '1310.4021-3-78-9': 'So for any [MATH], [EQUATION]', '1310.4021-3-78-10': 'Moreover, [MATH] is continuous with respect to [MATH].', '1310.4021-3-78-11': 'So [EQUATION].', '1310.4021-3-78-12': 'By Tonelli theorem [EQUATION]', '1310.4021-3-78-13': 'Furthermore, by the definition of [MATH] and [MATH], we have [EQUATION] where [EQUATION].', '1310.4021-3-78-14': 'From ([REF]), we have [MATH].', '1310.4021-3-78-15': "By Skorohod's theorem, see, e.g. Ikeda and Watanabe (1989), there exist [MATH] and [MATH] defined on [MATH], where [MATH] is Lebesgue measure, such that for any [MATH], [MATH] and [MATH].", '1310.4021-3-78-16': 'Define [EQUATION] so [MATH].', '1310.4021-3-78-17': 'Then [EQUATION] thus [MATH] as [MATH] and [EQUATION]', '1310.4021-3-78-18': 'By dominate convergence theorem, for any [MATH], we have [EQUATION]', '1310.4021-3-78-19': 'Since [MATH] is arbitrage, then [EQUATION]', '1310.4021-3-78-20': 'Since [MATH] is local bounded, so [MATH].', '1310.4021-3-78-21': 'Thus [EQUATION] [MATH]'} | {'1310.4021-4-0-0': ': We study the nonparametric estimation for the intensity of Poisson random measure in jump-diffusion CIR model based on the low frequency observations.', '1310.4021-4-0-1': 'This is given in terms of the minimization of norms on a nonempty, closed and convex subset of some special Hilbert space.', '1310.4021-4-0-2': 'We establish the measurability of the estimator and derive its consistency and asymptotic risk bound.', '1310.4021-4-1-0': 'Mathematics Subject Classification (2010): Primary 62G05, 90A19, 60J75; Secondary 62G20, 60J85, 90A16', '1310.4021-4-2-0': 'Keywords: Nonparametric estimation, jump-diffusion, CIR-model, branching process, ergodicity, consistency, asymptotic risk bounds', '1310.4021-4-3-0': '# Introduction and main results', '1310.4021-4-4-0': 'The Cox-Ingersoll-Ross model (CIR model) defined by the following stochastic differential equation (SDE) was firstly introduced by Cox et al. (1985) in the study of term structure of interest rate: [EQUATION] where [MATH] are given constants and [MATH] is a standard Brownian motion.', '1310.4021-4-4-1': 'Motivated by the study of jump risks which can not be ignored in the pricing of assets, Ahn and Thompson (1988) studied the following jump-diffusion process by adding a jump component into ([REF]): [EQUATION] where [MATH] is a Poisson random measure on [MATH] with Levy intensity [MATH] and [MATH] is a finite measure on [MATH]; more details can be seen Duffie et al. (2000, 2003).', '1310.4021-4-4-2': 'In this paper we always assume [MATH] is absolutely continuous with respect to Lebesgue measure, i.e. there exits a non-negative function [MATH] satisfying [MATH].', '1310.4021-4-5-0': 'Actually, by Theorem 2.5 in Fu and Li (2010), the nonnegative solution [MATH] to ([REF]) uniquely exists and is a continuous-sate branching process with immigration (CBI processes) with transition semigroup [MATH] given by [EQUATION] where [EQUATION] and [EQUATION]', '1310.4021-4-5-1': 'Otherwise, from Theorem 3.20 in Li (2011, p66) we have the semigroup [MATH] is ergodic, i.e. for any [MATH], [MATH] converges to a probability measure [MATH] on [MATH] as [MATH] and the Laplace transform of [MATH] is given by [EQUATION]', '1310.4021-4-5-2': 'For any finite set [MATH] we can define the probability measure [MATH] on [MATH] by [EQUATION]', '1310.4021-4-5-3': 'Since [MATH] is a consistent family, there is a stationary Markov process [MATH] with finite-dimensional distributions given by ([REF]) and one-dimensional marginal distribution [MATH].', '1310.4021-4-5-4': "From Remark 2.6 in Li and Ma (2015) and Birkhoff's ergodic theorem, we have [MATH] is ergodic.", '1310.4021-4-5-5': 'By a fairly simple (continuous time) coupling argument, with out loss of generality we always assume [MATH] defined by ([REF]) is a stationary and ergodic process.', '1310.4021-4-6-0': 'Before applying ([REF]) into practical problems, the key preparation is estimating [MATH] and [MATH].', '1310.4021-4-6-1': 'Since estimations for [MATH] have been given by Huang et al. (2011), we just need to found some suitable estimations of [MATH] with other parameters known.', '1310.4021-4-6-2': 'There are a lot of works about parameter estimations for the standard CIR-model and a review had been given in Xu (2014) including the conditional least squares estimators (CLSEs) and the maximum likelihood estimators (MLEs) given by Overbeck and Ryden (1997) and Overbeck (1998).', '1310.4021-4-6-3': 'Here we only give a summary of some known works about nonparametric estimation for jump-diffusion processes.', '1310.4021-4-6-4': 'Unfortunately, limited works have been done in the nonparametric estimation in jump-diffusion CIR models compared with Levy processes.', '1310.4021-4-6-5': 'Watteel and Kulperger (2003) proposed and implemented an approach for estimating the jump distribution of the Levy processes by fixed spectral cut-off procedure.', '1310.4021-4-6-6': 'The penalized projection method was applied in Figueroa-Lopez and Houdre (2006) to estimate the Levy density on a compact interval separated from the origin, based on a continuous time observation of the sample path throughout a time interval [MATH].', '1310.4021-4-6-7': 'Moreover, Figueroa-Lopez (2009) used the projection method for discrete observations and provided minimum risks of estimation for smooth Levy densities, as well as estimated on a compact interval separated from the origin.', '1310.4021-4-6-8': 'Comte and Genon-Catalot used a Fourier approach to construct an adaptive nonparametric estimators and provide bounds for the global [MATH]- risk with high frequency data and low frequency data respectively; see Comte and Genon-Catalot (2009, 2010).', '1310.4021-4-6-9': 'Neumann and Reiss (2009) studied the nonparametric estimation for Levy processes based on the empirical characteristic function.', '1310.4021-4-6-10': 'Jongbloed et al. (2005) considered a related low-frequency problem for the canonical function in Ornstein-Uhlenbeck processes driven by Levy processes and a consistent estimator has been constructed.', '1310.4021-4-7-0': 'In this work, based on the low frequency observations at equidistant time points [MATH] of a single realization, we establish some nonparametric estimators for the Levy density [MATH] by minimazing the norms of the elements of a closed and convex subset in some special Hilbert space.', '1310.4021-4-7-1': 'For simplicity, we take [MATH] and denote the observation be [MATH] but all the results presented below can be extended to the general case.', '1310.4021-4-7-2': 'We always assume all functions below are defined on [MATH].', '1310.4021-4-7-3': 'Let [MATH] and [EQUATION]', '1310.4021-4-7-4': 'We say a real-valued function [MATH], if for any [MATH], [EQUATION] where [MATH].', '1310.4021-4-7-5': 'We define the following convex subset of [MATH] [EQUATION]', '1310.4021-4-7-6': 'Otherwise, a linear operator [MATH] is defined by [EQUATION]', '1310.4021-4-7-7': 'Let [EQUATION] where [MATH] and [MATH].', '1310.4021-4-7-8': 'By the erogidicity of [MATH] and the continuous mapping theorem, for any [MATH] we have [EQUATION]', '1310.4021-4-7-9': 'Let [MATH] be a increasing sequence of functions satisfying that [MATH].', '1310.4021-4-7-10': 'For any [MATH] define [EQUATION].', '1310.4021-4-7-11': 'Here for any fixed [MATH], we establish the following estimator for [MATH]: [EQUATION] where [MATH] is a bounded and non-negative weighted function with compact support, denote by [MATH], and there exist [MATH] such that [MATH].', '1310.4021-4-7-12': 'We give the main results in the following three theorems.', '1310.4021-4-8-0': 'For any [MATH], we have [MATH] is well defined, i.e. [MATH] exists uniquely and is measurable.', '1310.4021-4-9-0': 'If [MATH] for some [MATH], then the estimator [MATH] is strongly consistent.', '1310.4021-4-9-1': 'In details, [EQUATION]', '1310.4021-4-9-2': 'Suppose [MATH] is a finite measure and [MATH] for any [MATH].', '1310.4021-4-9-3': 'If [MATH] for some [MATH], then there exists a constant [MATH] such that for [MATH] large enough have [EQUATION]', '1310.4021-4-9-4': 'Conditions in Theorem [REF] can be weakened; i.e., if [MATH] for some [MATH], we can choose a weighted function [MATH] with [MATH].', '1310.4021-4-10-0': 'This paper is organized as follows.', '1310.4021-4-10-1': 'In Section 2, we will prove Theorem [REF].', '1310.4021-4-10-2': 'The consistency and asymptotic risk bound of estimator (Theorem [REF] and [REF]) will be proved in Section 3.', '1310.4021-4-11-0': 'Notation: In this paper, we denote [MATH] and [MATH] be the set of all retional number.', '1310.4021-4-11-1': 'Moreover, [MATH] denotes the Laplace transform of the measure [MATH], [MATH] and [MATH] mean converge almost surely and in distribution respectively.', '1310.4021-4-11-2': 'Similarly, [MATH] and [MATH] mean equal almost surely and in distribution.', '1310.4021-4-12-0': '# Existence, uniqueness and measurability', '1310.4021-4-13-0': 'In this section, we will prove Theorem [REF] by identifying the estimator defined by ([REF]) exists uniquely and is measurable.', '1310.4021-4-13-1': 'Firstly, we recall a conclusion which can be found in many books about functional analysis.', '1310.4021-4-14-0': 'If [MATH] is a Banach space with norm [MATH], [MATH] is a nonempty, closed, convex subset of [MATH], then [MATH] contains a unique element of smallest norm.', '1310.4021-4-15-0': 'With this lemma we will give the most important theorem, which will guarantee the measurability of the estimators.', '1310.4021-4-15-1': 'Actually, least squares estimators (LSEs) and maximum likelihood estimators (MLEs) are just special cases of this theorem.', '1310.4021-4-16-0': 'Suppose [MATH] is a probability space and [MATH] is a separable Banach space with norm [MATH] and Borel [MATH]-algebra [MATH].', '1310.4021-4-16-1': 'Let [MATH] be a measurable mapping from [MATH] to [MATH] and [MATH] be a nonempty, closed and convex subset.', '1310.4021-4-16-2': 'Then [EQUATION] is well defined and [MATH]-measurable.', '1310.4021-4-17-0': 'Proof.', '1310.4021-4-17-1': 'From Lemma [REF], we have [MATH] exists uniquely.', '1310.4021-4-17-2': 'Now we prove it is [MATH]-measurable.', '1310.4021-4-17-3': 'Let [MATH] which is [MATH]-measurable.', '1310.4021-4-17-4': 'Indeed, since [MATH] is separable, there exists a countable subset of [MATH], denoted by [MATH] such that [EQUATION].', '1310.4021-4-17-5': 'Since [MATH] and [MATH] are measurable for any [MATH], we have [MATH] is [MATH]-measurable.', '1310.4021-4-17-6': 'Let [MATH] and [MATH], both of them are [MATH]-algebras.', '1310.4021-4-17-7': 'We just prove [MATH] is a [MATH]-algebra.', '1310.4021-4-17-8': 'Obviously, [MATH].', '1310.4021-4-17-9': 'If [MATH], then [EQUATION]', '1310.4021-4-17-10': 'For any [MATH], then [EQUATION]', '1310.4021-4-17-11': 'For any [MATH] and [MATH], let [MATH].', '1310.4021-4-17-12': 'The desired result follows if we prove for any [MATH] have [EQUATION]', '1310.4021-4-17-13': 'Since [MATH] is separable, for any [MATH] there exits a subset [MATH] such that for any [MATH] there exists an element [MATH] with [MATH] .', '1310.4021-4-17-14': 'Let [EQUATION]', '1310.4021-4-17-15': 'So it suffices to prove [MATH].', '1310.4021-4-17-16': 'Actually, for any [MATH], there exists [MATH] such that [MATH].', '1310.4021-4-17-17': 'Then for any [MATH], there exists [MATH] satifying [MATH].', '1310.4021-4-17-18': 'Thus [EQUATION] which means [MATH] and [MATH].', '1310.4021-4-17-19': 'Otherwise, for any [MATH] and [MATH], there exists [MATH] such that [EQUATION] which means [MATH] as [MATH].', '1310.4021-4-17-20': 'So there exists [MATH] such that [MATH].', '1310.4021-4-17-21': 'Since [MATH] is a Hilbert space and [MATH] is a closed and convex ball, from Theorem 2.1 we have [MATH] exists uniquely.', '1310.4021-4-17-22': 'Otherwise, since [EQUATION] we have [EQUATION].', '1310.4021-4-17-23': 'So [MATH] and [MATH].', '1310.4021-4-17-24': 'Here we have finished this proof.', '1310.4021-4-17-25': '[MATH] Define [EQUATION] which is a Hilbert Space with inner product [MATH] for any [MATH].', '1310.4021-4-17-26': 'Let [MATH], it is easy to see [MATH].', '1310.4021-4-17-27': 'Indeed, for any [MATH] we have [EQUATION]', '1310.4021-4-17-28': 'Since [MATH] is compact, there exists [MATH] such that [MATH].', '1310.4021-4-17-29': 'We have [EQUATION] where [MATH] is a constant independent to [MATH].', '1310.4021-4-18-0': 'For any two probability measures [MATH] and [MATH] on [MATH], we have [MATH] if and only if [MATH] on some interval [MATH] with [MATH].', '1310.4021-4-19-0': 'Proof.', '1310.4021-4-19-1': 'Sufficiency is obvious, we just need to prove necessity.', '1310.4021-4-19-2': 'Since [MATH] and [MATH] are analytic on this strip [MATH].', '1310.4021-4-19-3': 'By the assumption in this lemma and theorem in Brown and Churchill (2009, p.84), we have [MATH] on this strip.', '1310.4021-4-19-4': 'The desired result follows from [MATH] when we choose [MATH].', '1310.4021-4-19-5': '[MATH] is a continuous bijection.', '1310.4021-4-20-0': 'Proof.', '1310.4021-4-20-1': 'It is easy to see [MATH] is a one-to-one mapping.', '1310.4021-4-20-2': 'Indeed, for any [MATH] satisfying [MATH] we have [MATH] for any [MATH].', '1310.4021-4-20-3': 'Moreover, by the definition of CBI processes, there exist two probabilities [MATH] and [MATH] on [MATH] such that for i=1,2 [EQUATION]', '1310.4021-4-20-4': 'By lemma [REF], we have [MATH] and [MATH] almost everywhere.', '1310.4021-4-20-5': 'Thus [MATH].', '1310.4021-4-20-6': 'Since [MATH] is a linear operator, its continuity follows directly from its boundedness which have been proved in ([REF]).', '1310.4021-4-20-7': '[MATH]', '1310.4021-4-21-0': 'For any [MATH], [MATH] and [MATH] are compact, convex subsets of [MATH] and [MATH] respectively.', '1310.4021-4-22-0': 'Proof.', '1310.4021-4-22-1': 'Since [MATH] is continuous, [MATH] is compact and its convexity follows from the convexity of [MATH] which is obvious.', '1310.4021-4-22-2': 'It suffices to prove [MATH] is compact.', '1310.4021-4-22-3': 'Since [MATH] is integrable, so there must exists [MATH], such that [MATH] as [MATH].', '1310.4021-4-22-4': 'For any fixed sequence [MATH] in [MATH], we have [MATH] as [MATH].', '1310.4021-4-22-5': 'There exist [MATH] and [MATH] such that [MATH] for for any [MATH].', '1310.4021-4-22-6': 'Without loss of generality, we assume [MATH].', '1310.4021-4-22-7': 'For any [MATH], since [MATH], we have [MATH] for any [MATH].', '1310.4021-4-22-8': 'Similarly, since [MATH], we have [MATH] for any [MATH].', '1310.4021-4-22-9': 'Thus [MATH] are uniformly bounded and have bounded variation on [MATH], which means there exit two sequences of nonnegative, monotone increasing and right-continuous functions [MATH] and [MATH] such that [MATH].', '1310.4021-4-22-10': 'Obviously, we have [EQUATION]', '1310.4021-4-22-11': 'Without loss of generality, we assume that [MATH], so [MATH] and [EQUATION]', '1310.4021-4-22-12': 'Applying the diagonalization argument to [MATH] and [MATH], there exists a subsequence [MATH] such that [MATH] and [MATH] converge to some functions [MATH] and [MATH] on [MATH] respectively.', '1310.4021-4-22-13': 'For any and [MATH], define [EQUATION]', '1310.4021-4-22-14': 'Define [MATH], then [MATH] and [MATH] at all continuity points of [MATH] as [MATH].', '1310.4021-4-22-15': 'Since [MATH] and [MATH] are right-continuous and nonnegative monotone increasing functions, so is [MATH] and the number of discontinuity points of [MATH] is at most countable.', '1310.4021-4-22-16': 'Thus [MATH] almost everywhere as [MATH].', '1310.4021-4-22-17': 'Applying the diagonalization argument again, for [MATH] we can find a subsequence of [MATH], denoted by [MATH], such that the new subsequences converges to some functions [MATH] on [MATH], then repeat the program above again and we get [MATH] on [MATH].', '1310.4021-4-22-18': 'Obviously, [MATH] on [MATH].', '1310.4021-4-22-19': 'Thus [MATH] converges to some function [MATH] almost everywhere on [MATH].', '1310.4021-4-22-20': 'Applying the diagonalization argument again and the dominated convergence theorem, we have [MATH] as [MATH].', '1310.4021-4-22-21': 'Here we have proved that [MATH] is compact.', '1310.4021-4-22-22': '[MATH]', '1310.4021-4-23-0': 'For any [MATH], the inverse operator of [MATH], [MATH], is continuous.', '1310.4021-4-24-0': 'Proof of Theorem [REF].', '1310.4021-4-24-1': 'Define [EQUATION]', '1310.4021-4-24-2': 'From Lemma [REF] and Theorem [REF], we have [MATH] is well defined, i.e. it exists uniquely and is [MATH]-measurable.', '1310.4021-4-24-3': 'The desired results follows from [MATH] and Corollary [REF].', '1310.4021-4-24-4': '[MATH]', '1310.4021-4-25-0': '# Consistency and asymptotic risk bound', '1310.4021-4-26-0': 'We will prove Theorem [REF] in this section.', '1310.4021-4-26-1': 'The consistency of [MATH] comes directly from the following result.', '1310.4021-4-27-0': '[MATH] uniformly on any compact subset, i.e. for any compact subset [MATH] of [MATH], we have [EQUATION]', '1310.4021-4-27-1': 'Proof.', '1310.4021-4-27-2': 'Obviously, [MATH] is Laplace transform of the distribution [MATH] of [MATH], where [MATH] is a CBI processes defined by ([REF]) with [MATH] and [MATH].', '1310.4021-4-27-3': 'For any [MATH], [MATH] is Laplace transform of some measure denoted by [MATH].', '1310.4021-4-27-4': 'Since [MATH] for any [MATH], we have [MATH] weakly.', '1310.4021-4-27-5': 'The desired result follows from Lemma 7.6 in Sato (1999), i.e. [MATH] uniformly on any compact subset.', '1310.4021-4-27-6': '[MATH] Proof of Theorem [REF].', '1310.4021-4-27-7': 'Recall [MATH] defined in the proof of Theorem [REF].', '1310.4021-4-27-8': 'From Corollary [REF] and Theorem 7.6.3 in Chung (2001), we have [EQUATION].', '1310.4021-4-27-9': 'By the definition of [MATH] in ([REF]), [EQUATION]', '1310.4021-4-27-10': 'The desired result follows from this result and Corollary [REF].', '1310.4021-4-27-11': '[MATH]', '1310.4021-4-28-0': 'Suppose conditions in Theorem [REF] hold, then there exists a constant [MATH] such that for [MATH] large enough have [EQUATION]', '1310.4021-4-28-1': 'Proof.', '1310.4021-4-28-2': 'Let [MATH], where [EQUATION].', '1310.4021-4-28-3': "It's easily to prove [MATH] is a [MATH]-martingale.", '1310.4021-4-28-4': 'Obviously, we have [EQUATION]', '1310.4021-4-28-5': 'Define [MATH].', '1310.4021-4-28-6': 'From ([REF]) and ergodicity of [MATH] we have [EQUATION] where [EQUATION]', '1310.4021-4-28-7': 'It is easy to identify that [MATH] is continuous and bounded on any compact set.', '1310.4021-4-28-8': 'By the martingale central limitation theorem (see Durrett, 2010), we have [EQUATION]', '1310.4021-4-28-9': 'Furthermore, by the definition of [MATH] [EQUATION]', '1310.4021-4-28-10': "By the Skorohod's representation theorem, see Billingsley (1999, p.70), there exist [MATH] and [MATH] defined on [MATH], where [MATH] is Lebesgue measure, such that [MATH] for any [MATH], [MATH] and [MATH].", '1310.4021-4-28-11': 'Define [MATH], then [EQUATION] thus [MATH] as [MATH] and [EQUATION]', '1310.4021-4-28-12': "By Tonelli's theorem and Fatou's theorem, we have [EQUATION]", '1310.4021-4-28-13': 'From this and([REF]), there exists a constant [MATH] such that [EQUATION].', '1310.4021-4-28-14': 'Here we have finished this proof.', '1310.4021-4-28-15': '[MATH]', '1310.4021-4-29-0': 'Proof of Theorem [REF].', '1310.4021-4-29-1': 'Obviously, we have [EQUATION]', '1310.4021-4-29-2': 'For any fixed [MATH], since [MATH] is a continuous and linear bijection from [MATH] to [MATH], from Corollary 2.12(c) in Rudin (1991, p49) we have [EQUATION] where [MATH] is a constant determined by the norm of [MATH].', '1310.4021-4-29-3': 'The desired result follows from this and Lemma [REF].', '1310.4021-4-29-4': '[MATH]'} | null | null | null |
1003.3720 | {'1003.3720-1-0-0': 'In a region free of currents, magnetostatics can be described by the Laplace equation of a scalar magnetic potential, and one can apply the same methods commonly used in electrostatics.', '1003.3720-1-0-1': 'Here we show how to calculate the general vector field inside a real (finite) solenoid, using only the magnitude of the field along the symmetry axis.', '1003.3720-1-0-2': 'Our method does not require integration or knowledge of the current distribution, and is presented through practical examples, including a non-uniform finite solenoid used to produce cold atomic beams via laser cooling.', '1003.3720-1-0-3': 'These examples allow educators to discuss the non-trivial calculation of fields off-axis using concepts familiar to most students, while offering the opportunity to introduce important advancements of current modern research.', '1003.3720-1-1-0': '# Introduction', '1003.3720-1-2-0': 'Magnetic fields produced by solenoids and axially symmetric coils are ubiquitous, and the ability to calculate them is an integral part of training in physics.', '1003.3720-1-2-1': 'Time constraints, however, tend to focus the attention of most introductory electromagnetism (EM) courses to the analytical solution of only a few highly symmetrical cases, such as the field along the axis of a circular coil or inside an infinite solenoid[CITATION].', '1003.3720-1-2-2': 'Nevertheless, many applications require at least an estimate of the full vector field in regions away from the axis [CITATION], which involve mathematical tools often not discussed at the introductory level.', '1003.3720-1-2-3': 'On the other hand, most EM courses already dedicate a fair amount of time teaching students to identify and solve electrostatic problems using the Laplace equation.', '1003.3720-1-2-4': 'In some cases the same methods can be applied to magnetostatic problems, sometimes leading to useful insights.', '1003.3720-1-3-0': 'Sadly, most students do not usually appreciate the similarities between the two classes of problems [CITATION] due to a limited exposure to practical examples involving the magnetostatic potential.', '1003.3720-1-3-1': 'We feel that this ability is useful[CITATION], particularly because scalar potentials are generally more intuitive and easier to visualize.', '1003.3720-1-3-2': 'Besides, a unified treatment could be pedagogically relevant in generalizing the discussion of the multipole expansions[CITATION].', '1003.3720-1-3-3': 'Therefore, the primary goal here is to present a couple of pedagogical examples illustrating the application of the magnetostatic potential method to real solenoids.', '1003.3720-1-4-0': 'In addition, these examples also offer the opportunity to discuss in the classroom axisymmetric fields evaluated off-axis, without the need to introduce the formalism of elliptic integrals.', '1003.3720-1-4-1': 'Although other methods for finding off-axis magnetic fields have been mentioned earlier in the literature [CITATION], to our knowledge this has not been presented from such a simple and intuitive viewpoint.', '1003.3720-1-5-0': 'Moreover, as further motivation, we have chosen an example that brings a real and practical application from the cutting edge of research into the classroom: a non-uniform solenoid used in many research laboratories to produce beams of slow (cold) atoms.', '1003.3720-1-5-1': 'This solenoid, called Zeeman-slower[CITATION], is used in conjunction with appropriately prepared laser beams to slow down and cool neutral atoms, from hundreds of Kelvin to milliKelvin temperatures, by combining the action of radiation pressure with the Zeeman effect.', '1003.3720-1-5-2': 'This device is one of the staple developments in the area of laser cooling[CITATION], and one of the enabling technologies leading to the 1997 Nobel prize in Physics[CITATION].', '1003.3720-1-5-3': 'The techniques for laser cooling and trapping of atoms have produced many dramatic advancements in our understanding of quantum physics[CITATION], including the achievement of Bose-Einstein condensation[CITATION], which was recognized with another Nobel prize[CITATION] in 2001.', '1003.3720-1-5-4': 'In both cases, magnetic fields have been an important part of experimental design and data interpretation.', '1003.3720-1-5-5': 'Educators can astutely use the solenoid discussed here, as well as the references herein, to introduce and discuss some of these modern developments in quantum physics, making the subject even more interesting to students.', '1003.3720-1-6-0': '# Reviewing some basic concepts', '1003.3720-1-7-0': 'We begin here by recalling the fundamental equation of magnetostatics : [MATH], where [MATH] is the magnetic field and [MATH] the current density.', '1003.3720-1-7-1': 'Typically [MATH] is related to the magnetic induction field [MATH] by some constitutive relation expressing the properties of a particular material.', '1003.3720-1-7-2': 'For linear and isotropic materials, with a magnetic permeability [MATH], [MATH] and in a current-free region [MATH], implying that [MATH].', '1003.3720-1-7-3': "Since Maxwell's equations also state that [MATH], this results in [MATH], which is Laplace's equation for the magnetic potential [MATH], in any current-free region.", '1003.3720-1-8-0': "Although Laplace's equation is only typically valid in a region free of charges or currents, these are allowed to exist on or outside a surface [MATH] surrounding that region.", '1003.3720-1-8-1': "The solutions of Laplace's equation present three important properties: superposition, smoothness and uniqueness.", '1003.3720-1-8-2': "The property of superposition results from the fact that Laplace's equation is a linear equation.", '1003.3720-1-8-3': 'Smoothness implies that no solution in a region [MATH] of space, bounded by a surface [MATH], can present either a maximum or a minimum within [MATH] (extreme values can occur only at the surface [MATH]).', '1003.3720-1-8-4': 'The third property is the one most relevant to us here, as it states [CITATION] that if one finds a solution [MATH], in a region of space consistent with the prescribed boundary conditions, that solution is unique up to an additive constant.', '1003.3720-1-8-5': 'Therefore, it does not matter what particular method is used to find the solution.', '1003.3720-1-8-6': 'Once an appropriate solution is found, it is uniquely valid.', '1003.3720-1-9-0': 'However, despite the obvious similarities between the electrostatic and magnetostatic potentials, there are indeed reasons why the analogy can only be taken so far[CITATION], and is not widely explored further in textbooks.', '1003.3720-1-9-1': 'The first one arises whenever [MATH], in which case it is not trivial to write a relation between [MATH] and [MATH].', '1003.3720-1-9-2': 'The second complication occurs due to the fact that the scalar potential is generally a multiply valued function, requiring a prescription specifying where it can be used.', '1003.3720-1-9-3': 'However, as it has been shown by Bronzan[CITATION], these complications can be overcome, permitting one to exploit the advantages of the concept of a scalar magnetic potential.', '1003.3720-1-10-0': '# The magnetic field of a finite uniform solenoid', '1003.3720-1-11-0': 'We start by considering the field in the interior of a finite uniform solenoid carrying a current [MATH], as illustrated in Fig. [REF].', '1003.3720-1-11-1': 'For generality and convenience, we describe the problem using spherical coordinates.', '1003.3720-1-11-2': 'In this geometry it is easy to note that due to the axial symmetry of the problem the field [MATH] depends only on [MATH].', '1003.3720-1-11-3': 'The magnetostatic potential can be found using for a boundary condition the magnitude of the field along [MATH], which is readily available through simple summation formulas over the approximately circular coils forming the solenoid.', '1003.3720-1-12-0': 'In spherical coordinates, the solution of the axisymmetric scalar potential [MATH] can be written in the form: [EQUATION] where [MATH] and [MATH] are coefficients to be determined and the [MATH] represents a Legendre polynomial of order [MATH].', '1003.3720-1-13-0': 'Because we are mainly interested in the values of the field inside the solenoid, we set [MATH] to avoid the singularity at [MATH].', '1003.3720-1-13-1': 'As a result, the potential takes the simpler form: [EQUATION]', '1003.3720-1-13-2': 'For points along the [MATH]-axis, we have [MATH] and Eq. ([REF]) becomes [EQUATION]', '1003.3720-1-13-3': 'Incidentally, we can in general also expand the scalar potential in a Taylor series around some point [MATH], [EQUATION] and comparing it to Eq. ([REF]), for [MATH], we obtain the coefficients [MATH] in terms of the series expansion [EQUATION]', '1003.3720-1-13-4': 'In this way, the full scalar potential becomes analytically determinable, allowing us to evaluate [MATH] at any point in space where Eq. ([REF]) applies.', '1003.3720-1-14-0': 'As a first example, let us now consider the case of a finite solenoid of length [MATH] and radius [MATH], carrying a uniform current [MATH].', '1003.3720-1-14-1': 'If the solenoid has [MATH] turns per unit length, the magnetic field along the [MATH]-axis can be easily calculated by integrating the expression for the axial field of a circular current loop [CITATION], resulting in: [EQUATION] where [MATH], and [MATH] in SI units.', '1003.3720-1-14-2': 'Now, since [EQUATION] we can write [EQUATION]', '1003.3720-1-14-3': 'Using Eq. ([REF]) in Eq. ([REF]) we obtain the general form of the potential for the finite solenoid along the axis: [EQUATION]', '1003.3720-1-14-4': 'Expanding Eq. ([REF]) around [MATH], as in Eq. ([REF]), we get the various coefficients for [MATH].', '1003.3720-1-14-5': 'Using these coefficients and introducing the expression for the Legendre polynomials [MATH], while keeping terms up to third order, we finally get: [EQUATION]', '1003.3720-1-14-6': 'From the last equation, one can calculate the components [MATH]and [MATH] of the magnetic field by simply taking the gradient of the potential: [EQUATION] and [EQUATION]', '1003.3720-1-14-7': 'It is interesting to note that these results give approximate analytical results for the magnetic field everywhere inside the solenoid, with their precision limited by the number of terms included in the power series expansion.', '1003.3720-1-14-8': 'One can test these results by comparing the expressions ([REF]) and ([REF]) with those presented in Chapter 5 of ref. [CITATION], where a different method was used to evaluate the field components.', '1003.3720-1-14-9': 'In particular, we will show that if one keeps only the first order in the expansion, the result simplifies to the approximate solution of problem 5.2 of reference [CITATION].', '1003.3720-1-14-10': 'For that we recall the relations [EQUATION] from which we obtain, up to third order, [EQUATION]', '1003.3720-1-14-11': 'Finally, using [MATH] and [MATH], in the limit [MATH], Eq. ([REF]) yields [EQUATION] which is expressed here in CGS units, with [MATH], to facilitate direct comparison with the result presented in ref. [CITATION].', '1003.3720-1-15-0': 'To show a practical application of the method, we now compare our results to a realistic numerical calculation of a finite solenoid.', '1003.3720-1-15-1': 'Figure [REF] shows the numerical results for the magnitude of the magnetic field and its components.', '1003.3720-1-15-2': 'The calculation assumes that the solenoid is composed of a series of single circular coils, and performs a direct summation over the exact analytical expression, based on elliptic integrals, for each individual coil.', '1003.3720-1-15-3': 'The numerical result was determined to be very close to the experimental results, as at this scale the error introduced by approximating the actual helical winding by a sequence of circular single coils is typically negligible.', '1003.3720-1-15-4': 'The relevant physical parameters are given in the captions of Fig. [REF].', '1003.3720-1-16-0': 'Further, in Fig. [REF], we compare the analytical results obtained by keeping the first eight terms in the expansion (corresponding to the fifteenth order in [MATH]) against the numerical calculations shown in Fig. [REF].', '1003.3720-1-16-1': 'In addition, Fig. [REF] also shows the good partial agreement obtained using the third order approximation, extending to distances up to about half the size of the solenoid.', '1003.3720-1-16-2': 'Note that, for a real finite system, the disagreement increases rapidly after some point.', '1003.3720-1-16-3': 'That can be improved significantly by including higher order terms, allowing for a much better approximation near the edges, as shown in Fig. [REF].', '1003.3720-1-16-4': 'However, due to some of the simplifications made, the approximate analytical result still does not contain all the physics of the problem.', '1003.3720-1-16-5': 'For instance, it does not accurately describe the field outside the solenoid.', '1003.3720-1-16-6': 'Nevertheless, the magnetostatic potential method still provides a reasonable representation of the internal fields up to the very end of the solenoid.', '1003.3720-1-17-0': '# The Zeeman-slower: an inhomogeneous finite solenoid', '1003.3720-1-18-0': 'Now we will consider another very interesting and practical problem, familiar to many atomic physics laboratories, namely the design of a solenoid capable of producing an axial field with a parabolic profile, as in [EQUATION] where [MATH], [MATH] and [MATH] are constants.', '1003.3720-1-18-1': 'Such a field is suitable for slowing atomic beams using laser light [CITATION].', '1003.3720-1-18-2': 'The field of Eq. ([REF]) causes a spatially varying Zeeman effect that compensates for the changing Doppler shift of the moving atoms, thus keeping them in resonance with the light as they decelerate along the beam path.', '1003.3720-1-18-3': 'This technique is called Zeeman slowing[CITATION], and the parabolic shape is chosen to keep the radiation pressure constant, typically at a rate of [MATH], throughout the Zeeman solenoid[CITATION] shown in Figure [REF].', '1003.3720-1-19-0': 'In general, the atomic beam encompasses a certain solid angle as it traverses the solenoid and most atoms follow trajectories which do not lie exactly on the axis.', '1003.3720-1-19-1': 'Since the resonance condition with the laser depends on both the magnitude (via the detuning) and direction (via the polarization) of the magnetic field, the knowledge of the off-axis field is important in understanding how light interacts with atoms at different points inside the solenoid.', '1003.3720-1-20-0': 'The magnetic potential along the z-axis, in this case, takes the form [EQUATION] where the constant of integration has been suppressed.', '1003.3720-1-21-0': 'Following the same steps in section [REF], and after calculating the derivatives and solving for the coefficients [MATH], we obtain the general form of the magnetic potential for the Zeeman solenoid: [EQUATION] where [MATH] is the gamma function.', '1003.3720-1-21-1': 'Now we can calculate the spherical components of the magnetic field, [EQUATION] and [EQUATION]', '1003.3720-1-21-2': 'The transverse and axial components can be easily obtained from Eq. (13), with the shorthand [MATH]: [EQUATION] and [EQUATION]', '1003.3720-1-21-3': 'Notice that the transverse component of the magnetic field inside the Zeeman-slower does not depend on [MATH].', '1003.3720-1-21-4': 'Also, note that [MATH] at [MATH] (on-axis), as expected.', '1003.3720-1-21-5': 'It can be verified that the on-axis field sums back to the exact expression of Eq. ([REF]).', '1003.3720-1-21-6': "Although caution may be necessary when evaluating the field for [MATH]), the careful use of L'Hospital's rule ensures finite and correct answers.", '1003.3720-1-22-0': 'Now, to compare these analytical approximations to the experimental field shown in Fig. [REF], we will follow a different approach here.', '1003.3720-1-22-1': 'The motivation is to mimic a situation where the current distribution that generates the field may not be known exactly, but the axial field can be measured directly in the laboratory.', '1003.3720-1-22-2': 'This could be the case in a real practical application, where imperfections in the winding pattern often are not considered in the ideal model.', '1003.3720-1-22-3': 'From the experimental data one can then build a mathematical model, using either a fitting function (if the functional form is known or can be easily guessed), or by using an interpolating function, such as a polynomial, to represent the data in a limited region of space.', '1003.3720-1-22-4': 'Here, since the approximate functional form of the axial field is known, we will extract the model parameters by numerically fitting the data in Fig. [REF], to [MATH] in Eq. ([REF]), and substituting them in the expressions (21) and (22).', '1003.3720-1-22-5': 'Note that the limitations of this type of modeling may result in some inaccuracies, particularly close to the edges, where fringe effects are important.', '1003.3720-1-22-6': 'In any practical situation one may need to explore different approaches to find a mathematical model accurate enough in the region of interest.', '1003.3720-1-23-0': 'After following these steps to model the data, we show in Fig. [REF] a comparison between our analytical approximations and the experimental field.', '1003.3720-1-23-1': 'There is a reasonable agreement between the solid lines, representing equations (21) and (22) and the data points, representing the experimentally determined field.', '1003.3720-1-23-2': 'Note that, in contrast to the uniform finite solenoid where the power series was used to approximate the exact solution, here the power series simply approximates our model[CITATION] function.', '1003.3720-1-23-3': 'Therefore, increasing the order[CITATION] of the expansion only improves the agreement with the model (fitting) function, which represents the data only over a limited region and still does not contain all the information about the fields in the problem.', '1003.3720-1-23-4': 'This is clearly visible in Fig. [REF] where good agreement is found only in the range [MATH] (0.4 to 1.2) m.', '1003.3720-1-24-0': '# Conclusion', '1003.3720-1-25-0': 'Using the simple concept of the magnetostatic scalar potential, and only the knowledge of the field along the symmetry axis, we have shown how to determine the vector magnetic field anywhere inside an inhomogeneous finite solenoid, without explicitly integrating (or even knowing) the current distribution.', '1003.3720-1-25-1': 'In cases where the current distribution is known, but the expression for the field off-axis is non-trivial (for instance, given by elliptical integrals), one can still gain some insight by using the method described here.', '1003.3720-1-25-2': 'This simple analysis follows from a straightforward analogy with the electrostatic boundary value problem, and can be useful in determining field inhomogeneities in various practical experiments involving solenoids.', '1003.3720-1-25-3': 'In the present article we have used an example from contemporary atomic physics experiments to demonstrate the method.', '1003.3720-1-25-4': 'However, we believe that a simplified version of this discussion (e.g.: the uniform finite solenoid) could be used in a regular classroom setting, as a practical example of a calculation of off-axis magnetic fields, for undergraduate courses and teaching laboratories in physics and engineering.'} | {'1003.3720-2-0-0': 'In a region free of currents, magnetostatics can be described by the Laplace equation of a scalar magnetic potential, and one can apply the same methods commonly used in electrostatics.', '1003.3720-2-0-1': 'Here we show how to calculate the general vector field inside a real (finite) solenoid, using only the magnitude of the field along the symmetry axis.', '1003.3720-2-0-2': 'Our method does not require integration or knowledge of the current distribution, and is presented through practical examples, including a non-uniform finite solenoid used to produce cold atomic beams via laser cooling.', '1003.3720-2-0-3': 'These examples allow educators to discuss the non-trivial calculation of fields off-axis using concepts familiar to most students, while offering the opportunity to introduce important advancements of current modern research.', '1003.3720-2-1-0': '# Introduction', '1003.3720-2-2-0': 'Magnetic fields produced by solenoids and axially symmetric coils are ubiquitous, and the ability to calculate them is an integral part of training in physics.', '1003.3720-2-2-1': 'Time constraints, however, tend to focus the attention of most introductory electromagnetism (EM) courses to the analytical solution of only a few highly symmetrical cases, such as the field along the axis of a circular coil or inside an infinite solenoid[CITATION].', '1003.3720-2-2-2': 'Nevertheless, many applications require at least an estimate of the full vector field in regions away from the axis [CITATION], which involve mathematical tools often not discussed at the introductory level.', '1003.3720-2-2-3': 'On the other hand, most EM courses already dedicate a fair amount of time teaching students to identify and solve electrostatic problems using the Laplace equation.', '1003.3720-2-2-4': 'In some cases the same methods can be applied to magnetostatic problems, sometimes leading to useful insights.', '1003.3720-2-3-0': 'Sadly, most students do not usually appreciate the similarities between the two classes of problems [CITATION] due to a limited exposure to practical examples involving the magnetostatic potential.', '1003.3720-2-3-1': 'We feel that this ability is useful[CITATION], particularly because scalar potentials are generally more intuitive and easier to visualize.', '1003.3720-2-3-2': 'Besides, a unified treatment could be pedagogically relevant in generalizing the discussion of the multipole expansions[CITATION].', '1003.3720-2-3-3': 'Therefore, the primary goal here is to present a couple of pedagogical examples illustrating the application of the magnetostatic potential method to real solenoids.', '1003.3720-2-4-0': 'In addition, these examples also offer the opportunity to discuss in the classroom axisymmetric fields evaluated off-axis, without the need to introduce the formalism of elliptic integrals.', '1003.3720-2-4-1': 'Although other methods for finding off-axis magnetic fields have been mentioned earlier in the literature [CITATION], to our knowledge this has not been presented from such a simple and intuitive viewpoint.', '1003.3720-2-5-0': 'Moreover, as further motivation, we have chosen an example that brings a real and practical application from the cutting edge of research into the classroom: a non-uniform solenoid used in many research laboratories to produce beams of slow (cold) atoms.', '1003.3720-2-5-1': 'This solenoid, called a Zeeman-slower[CITATION], is used in conjunction with appropriately prepared laser beams to slow down and cool neutral atoms, from hundreds of Kelvin to milliKelvin temperatures, by combining the action of radiation pressure with the Zeeman effect.', '1003.3720-2-5-2': 'This device is one of the staple developments in the area of laser cooling[CITATION], and one of the enabling technologies leading to the 1997 Nobel prize in Physics[CITATION].', '1003.3720-2-5-3': 'The techniques for laser cooling and trapping of atoms have produced many dramatic advancements in our understanding of quantum physics[CITATION], including the achievement of Bose-Einstein condensation[CITATION], which was recognized with another Nobel prize[CITATION] in 2001.', '1003.3720-2-5-4': 'In both cases, magnetic fields have been an important part of experimental design and data interpretation.', '1003.3720-2-5-5': 'Educators can astutely use the solenoid discussed here, as well as the references herein, to introduce and discuss some of these modern developments in quantum physics, making the subject even more interesting to students.', '1003.3720-2-6-0': '# Reviewing some basic concepts', '1003.3720-2-7-0': 'We begin here by recalling the fundamental equation of magnetostatics : [MATH], where [MATH] is the magnetic field and [MATH] the current density.', '1003.3720-2-7-1': 'Typically [MATH] is related to the magnetic induction field [MATH] by some constitutive relation expressing the properties of a particular material.', '1003.3720-2-7-2': 'For linear and isotropic materials, with a magnetic permeability [MATH], [MATH] and in a current-free region [MATH], implying that [MATH].', '1003.3720-2-7-3': "Since Maxwell's equations also state that [MATH], this results in [MATH], which is Laplace's equation for the magnetic potential [MATH], in any current-free region.", '1003.3720-2-8-0': "Although Laplace's equation is only typically valid in a region free of charges or currents, these are allowed to exist on or outside a surface [MATH] surrounding that region.", '1003.3720-2-8-1': "The solutions of Laplace's equation present three important properties: superposition, smoothness and uniqueness.", '1003.3720-2-8-2': "The property of superposition results from the fact that Laplace's equation is a linear equation.", '1003.3720-2-8-3': 'Smoothness implies that no solution in a region [MATH] of space, bounded by a surface [MATH], can present either a maximum or a minimum within [MATH] (extreme values can occur only at the surface [MATH]).', '1003.3720-2-8-4': 'The third property is the one most relevant to us here, as it states [CITATION] that if one finds a solution [MATH], in a region of space consistent with the prescribed boundary conditions, that solution is unique up to an additive constant.', '1003.3720-2-8-5': 'Therefore, it does not matter what particular method is used to find the solution.', '1003.3720-2-8-6': 'Once an appropriate solution is found, it is uniquely valid.', '1003.3720-2-9-0': 'However, despite the obvious similarities between the electrostatic and magnetostatic potentials, there are indeed reasons why the analogy can only be taken so far[CITATION], and is not widely explored further in textbooks.', '1003.3720-2-9-1': 'The first one arises whenever [MATH], in which case it is not trivial to write a relation between [MATH] and [MATH].', '1003.3720-2-9-2': 'The second complication occurs due to the fact that the scalar potential is generally a multiply valued function, requiring a prescription specifying where it can be used.', '1003.3720-2-9-3': 'However, as it has been shown by Bronzan[CITATION], these complications can be overcome, permitting one to exploit the advantages of the concept of a scalar magnetic potential.', '1003.3720-2-10-0': '# The magnetic field of a finite uniform solenoid', '1003.3720-2-11-0': 'We start by considering the field in the interior of a finite uniform solenoid carrying a current [MATH], as illustrated in Fig. [REF].', '1003.3720-2-11-1': 'For generality and convenience, we describe the problem using spherical coordinates.', '1003.3720-2-11-2': 'In this geometry it is easy to note that due to the axial symmetry of the problem the field [MATH] depends only on [MATH].', '1003.3720-2-11-3': 'The magnetostatic potential can be found using for a boundary condition the magnitude of the field along [MATH], which is readily available through simple summation formulas over the approximately circular coils forming the solenoid.', '1003.3720-2-12-0': 'In spherical coordinates, the solution of the axisymmetric scalar potential [MATH] can be written in the form: [EQUATION] where [MATH] and [MATH] are coefficients to be determined and the [MATH] represents a Legendre polynomial of order [MATH].', '1003.3720-2-13-0': 'Because we are mainly interested in the values of the field inside the solenoid, we set [MATH] to avoid the singularity at [MATH].', '1003.3720-2-13-1': 'As a result, the potential takes the simpler form: [EQUATION]', '1003.3720-2-13-2': 'For points along the [MATH]-axis, we have [MATH] and Eq. ([REF]) becomes [EQUATION]', '1003.3720-2-13-3': 'Incidentally, we can in general also expand the scalar potential in a Taylor series around some point [MATH], [EQUATION] and comparing it to Eq. ([REF]), for [MATH], we obtain the coefficients [MATH] in terms of the series expansion [EQUATION]', '1003.3720-2-13-4': 'In this way, the full scalar potential becomes analytically determinable, allowing us to evaluate [MATH] at any point in space where equations ([REF]) and ([REF]) appliy.', '1003.3720-2-14-0': 'As a first example, let us now consider the case of a finite solenoid of length [MATH] and radius [MATH], carrying a uniform current [MATH].', '1003.3720-2-14-1': 'If the solenoid has [MATH] turns per unit length, the magnetic field along the [MATH]-axis can be easily calculated by integrating the expression for the axial field of a circular current loop [CITATION], resulting in: [EQUATION] where [MATH], and [MATH] in SI units.', '1003.3720-2-14-2': 'Now, since [EQUATION] we can write [EQUATION]', '1003.3720-2-14-3': 'Using Eq. ([REF]) in Eq. ([REF]) we obtain the general form of the potential for the finite solenoid along the axis: [EQUATION]', '1003.3720-2-14-4': 'Expanding Eq. ([REF]) around [MATH], as in Eq. ([REF]), we get the various coefficients for [MATH].', '1003.3720-2-14-5': 'Using these coefficients and introducing the expression for the Legendre polynomials [MATH], while keeping terms up to third order, we finally get: [EQUATION]', '1003.3720-2-14-6': 'From the last equation, one can calculate the components [MATH]and [MATH] of the magnetic field by simply taking the gradient of the potential: [EQUATION] and [EQUATION]', '1003.3720-2-14-7': 'It is interesting to note that these are approximate analytical results for the magnetic field inside the solenoid, provided it is within the radius of convergence of the power series and away from the current paths (wires), with their precision limited by the number of terms included in the expansion.', '1003.3720-2-15-0': 'One can test these results by comparing the expressions ([REF]) and ([REF]) with those presented in Chapter 5 of ref. [CITATION], where a different method was used to evaluate the field components.', '1003.3720-2-15-1': 'In particular, we will show that if one keeps only the first order in the expansion, the result simplifies to the approximate solution of problem 5.2 (b) in the 2nd ed.', '1003.3720-2-15-2': 'of ref. [CITATION].', '1003.3720-2-15-3': 'For that we recall the relations [EQUATION] from which we obtain, up to third order, [EQUATION]', '1003.3720-2-15-4': 'Finally, using [MATH] and [MATH], in the limit [MATH], Eq. ([REF]) yields [EQUATION] which is expressed here in CGS (Gaussian) units, with [MATH], to facilitate a direct comparison with the result presented in the second edition of reference [CITATION].', '1003.3720-2-16-0': 'To show a practical application of the method, we now compare our results to a realistic numerical calculation of a finite solenoid.', '1003.3720-2-16-1': 'Figure [REF] shows the numerical results for the magnitude of the magnetic field and its components.', '1003.3720-2-16-2': 'The calculation assumes that the solenoid is composed of a series of circular coils, and performs a direct summation over the exact analytical expression, based on elliptic integrals, for each individual coil.', '1003.3720-2-16-3': 'The numerical result were verified to accurately represent the field of a physical solenoid, through measurements with a Hall probe along the axis.', '1003.3720-2-16-4': 'This was expected since the error introduced by approximating the actual helical winding by a sequence of circular coils is typically negligible at this scale.', '1003.3720-2-16-5': 'The relevant physical parameters are given in the captions of Fig. [REF].', '1003.3720-2-17-0': 'Further, in Fig. [REF], we compare the analytical results obtained by keeping the first eight terms in the expansion (corresponding to the fifteenth order in [MATH]) against the numerical calculations shown in Fig. [REF].', '1003.3720-2-17-1': 'In addition, Fig. [REF] also shows the good partial agreement obtained using the third order approximation, extending to distances up to about half the size of the solenoid.', '1003.3720-2-17-2': 'Note that, for a real finite system, the disagreement increases rapidly after some point.', '1003.3720-2-17-3': 'That can be improved significantly by including higher order terms, allowing for a much better approximation near the edges, as shown in Fig. [REF].', '1003.3720-2-17-4': 'However, due to the simplifications made, the approximate analytical result does not contain all the physics of the problem.', '1003.3720-2-17-5': 'For instance, it does not accurately describe the field outside the solenoid, and it will most likely fail outside the radius of convergence of the power series expansion.', '1003.3720-2-17-6': 'Nevertheless, the magnetostatic potential method still provides a reasonable representation of the internal fields up to the very end of the solenoid.', '1003.3720-2-18-0': '# The Zeeman-slower: an inhomogeneous finite solenoid', '1003.3720-2-19-0': 'Now we will consider another interesting and very practical problem, familiar to many atomic physics laboratories, namely the design of a solenoid capable of producing an axial field with a parabolic profile, as in [EQUATION] where [MATH], [MATH] and [MATH] are constants.', '1003.3720-2-19-1': 'Such a field is suitable for slowing atomic beams using laser light [CITATION].', '1003.3720-2-19-2': 'The field of Eq. ([REF]) causes a spatially varying Zeeman effect that compensates for the changing Doppler shift of the moving atoms, thus keeping them in resonance with the light as they decelerate along the beam path.', '1003.3720-2-19-3': 'This technique is called Zeeman slowing[CITATION], and the parabolic shape is chosen to keep the radiation pressure constant, typically at a rate of [MATH], throughout the Zeeman solenoid[CITATION] shown in Figure [REF].', '1003.3720-2-20-0': 'In general, the atomic beam encompasses a certain solid angle as it traverses the solenoid and most atoms follow trajectories which do not lie exactly on the axis.', '1003.3720-2-20-1': 'Since the resonance condition with the laser depends on both the magnitude (via the detuning) and direction (via the polarization) of the magnetic field, the knowledge of the off-axis field is important in understanding how light interacts with atoms at different points inside the solenoid.', '1003.3720-2-21-0': 'The magnetic potential along the z-axis, in this case, takes the form [EQUATION] where the constant of integration has been suppressed.', '1003.3720-2-22-0': 'Following the same steps in section [REF], and after calculating the derivatives and solving for the coefficients [MATH], we obtain the general form of the magnetic potential for the Zeeman solenoid: [EQUATION] where [MATH] is the gamma function.', '1003.3720-2-22-1': 'Now we can calculate the spherical components of the magnetic field, [EQUATION] and', '1003.3720-2-23-0': '[EQUATION]', '1003.3720-2-23-1': 'The transverse and axial components can be easily obtained from Eq. (13), with the shorthand [MATH]: [EQUATION] and', '1003.3720-2-24-0': '[EQUATION]', '1003.3720-2-24-1': 'Notice that the transverse component of the magnetic field inside the Zeeman-slower does not depend on [MATH].', '1003.3720-2-24-2': 'Also, note that [MATH] at [MATH] (on-axis), as expected.', '1003.3720-2-24-3': 'It can be verified that the on-axis field sums back to the exact expression of Eq. ([REF]).', '1003.3720-2-24-4': "Although caution may be necessary when evaluating the field for [MATH]), the careful use of L'Hospital's rule ensures correct answers.", '1003.3720-2-25-0': 'Now, to compare these analytical approximations to the experimental field represented in Fig. [REF], we will follow a different approach.', '1003.3720-2-25-1': 'The motivation here is to mimic a situation where the current distribution that generates the field may not be known exactly, but the axial field can be measured directly in the laboratory.', '1003.3720-2-25-2': 'This could be the case in a real practical application, where imperfections in the winding pattern often are not considered in the ideal model.', '1003.3720-2-25-3': 'From the experimental data one can then build a mathematical model, using either a fitting function (if the functional form is known or could be easily guessed), or by using an interpolating function, such as a polynomial, to represent the data in a limited region of space.', '1003.3720-2-25-4': 'Here, since the approximate functional form of the axial field is known, we will extract the model parameters by numerically fitting the data in Fig. [REF], to [MATH] in Eq. ([REF]), and substituting them in the expressions (21) and (22).', '1003.3720-2-25-5': 'Note that the limitations of this type of modeling may result in some inaccuracies, particularly close to the edges.', '1003.3720-2-25-6': 'In any practical situation one may need to explore different approaches to find a mathematical model accurate enough in the region of interest.', '1003.3720-2-26-0': 'After following these steps to model the data, we show in Fig. [REF] a comparison between our analytical approximations and the numerical result (Fig. [REF]), that represent very accurately the experimental field, as determined by measurements in our laboratory.', '1003.3720-2-26-1': 'There is a reasonable agreement between the solid lines, representing equations (21) and (22) and the data points, representing the numerically calculated field.', '1003.3720-2-26-2': 'Note that, in contrast to the uniform finite solenoid where the power series was used to approximate the exact solution, here the power series simply approximates our model[CITATION] function.', '1003.3720-2-26-3': 'Therefore, increasing the order[CITATION] of the expansion only improves the agreement with the model (fitting) function, which represents the data only over a limited region and does not contain all the information about the fields in the problem.', '1003.3720-2-26-4': 'This is clearly visible in Fig. [REF] where good agreement is found only in the range [MATH] (0.4 to 1.2) m.', '1003.3720-2-27-0': '# Conclusion', '1003.3720-2-28-0': 'Using the simple concept of the magnetostatic scalar potential, and only the knowledge of the field along the symmetry axis, we have shown how to determine the vector magnetic field anywhere inside an inhomogeneous finite solenoid, without explicitly integrating (or even knowing) the current distribution.', '1003.3720-2-28-1': 'In cases where the current distribution is known, but the expression for the field off-axis is non-trivial (for instance, given by elliptical integrals), one can still gain some insight by using the method described here.', '1003.3720-2-28-2': 'This simple analysis follows from a straightforward analogy with the electrostatic boundary value problem, and can be useful in determining field inhomogeneities in various practical experiments involving solenoids.', '1003.3720-2-28-3': 'In the present article we have used an example from contemporary atomic physics experiments to demonstrate the method.', '1003.3720-2-28-4': 'However, we believe that a simplified version of this discussion (e.g.: the uniform finite solenoid) could be used in a regular classroom setting, as a practical example of a calculation of off-axis magnetic fields, for undergraduate courses and teaching laboratories in physics and engineering.'} | [['1003.3720-1-19-0', '1003.3720-2-20-0'], ['1003.3720-1-19-1', '1003.3720-2-20-1'], ['1003.3720-1-9-0', '1003.3720-2-9-0'], ['1003.3720-1-9-1', '1003.3720-2-9-1'], ['1003.3720-1-9-2', '1003.3720-2-9-2'], 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'1003.3720-2-26-0']] | [['1003.3720-1-19-0', '1003.3720-2-20-0'], ['1003.3720-1-19-1', '1003.3720-2-20-1'], ['1003.3720-1-9-0', '1003.3720-2-9-0'], ['1003.3720-1-9-1', '1003.3720-2-9-1'], ['1003.3720-1-9-2', '1003.3720-2-9-2'], ['1003.3720-1-9-3', '1003.3720-2-9-3'], ['1003.3720-1-13-0', '1003.3720-2-13-0'], ['1003.3720-1-13-1', '1003.3720-2-13-1'], ['1003.3720-1-13-2', '1003.3720-2-13-2'], ['1003.3720-1-13-3', '1003.3720-2-13-3'], ['1003.3720-1-0-0', '1003.3720-2-0-0'], ['1003.3720-1-0-1', '1003.3720-2-0-1'], ['1003.3720-1-0-2', '1003.3720-2-0-2'], ['1003.3720-1-0-3', '1003.3720-2-0-3'], ['1003.3720-1-22-2', '1003.3720-2-25-2'], ['1003.3720-1-22-4', '1003.3720-2-25-4'], ['1003.3720-1-22-6', '1003.3720-2-25-6'], ['1003.3720-1-25-0', '1003.3720-2-28-0'], ['1003.3720-1-25-1', '1003.3720-2-28-1'], ['1003.3720-1-25-2', '1003.3720-2-28-2'], ['1003.3720-1-25-3', '1003.3720-2-28-3'], ['1003.3720-1-25-4', '1003.3720-2-28-4'], ['1003.3720-1-4-0', '1003.3720-2-4-0'], ['1003.3720-1-4-1', '1003.3720-2-4-1'], ['1003.3720-1-3-0', '1003.3720-2-3-0'], ['1003.3720-1-3-1', '1003.3720-2-3-1'], ['1003.3720-1-3-2', '1003.3720-2-3-2'], ['1003.3720-1-3-3', '1003.3720-2-3-3'], ['1003.3720-1-12-0', '1003.3720-2-12-0'], ['1003.3720-1-15-0', '1003.3720-2-16-0'], ['1003.3720-1-15-1', '1003.3720-2-16-1'], ['1003.3720-1-15-4', '1003.3720-2-16-5'], ['1003.3720-1-7-0', '1003.3720-2-7-0'], ['1003.3720-1-7-1', '1003.3720-2-7-1'], ['1003.3720-1-7-2', '1003.3720-2-7-2'], ['1003.3720-1-7-3', '1003.3720-2-7-3'], ['1003.3720-1-18-1', '1003.3720-2-19-1'], ['1003.3720-1-18-2', '1003.3720-2-19-2'], ['1003.3720-1-18-3', '1003.3720-2-19-3'], ['1003.3720-1-5-0', '1003.3720-2-5-0'], ['1003.3720-1-5-2', '1003.3720-2-5-2'], ['1003.3720-1-5-3', '1003.3720-2-5-3'], ['1003.3720-1-5-4', '1003.3720-2-5-4'], ['1003.3720-1-5-5', '1003.3720-2-5-5'], ['1003.3720-1-16-0', '1003.3720-2-17-0'], ['1003.3720-1-16-1', '1003.3720-2-17-1'], ['1003.3720-1-16-2', '1003.3720-2-17-2'], ['1003.3720-1-16-3', '1003.3720-2-17-3'], ['1003.3720-1-16-6', '1003.3720-2-17-6'], ['1003.3720-1-23-2', '1003.3720-2-26-2'], ['1003.3720-1-23-4', '1003.3720-2-26-4'], ['1003.3720-1-11-0', '1003.3720-2-11-0'], ['1003.3720-1-11-1', '1003.3720-2-11-1'], ['1003.3720-1-11-2', '1003.3720-2-11-2'], ['1003.3720-1-11-3', '1003.3720-2-11-3'], ['1003.3720-1-20-0', '1003.3720-2-21-0'], ['1003.3720-1-2-0', '1003.3720-2-2-0'], ['1003.3720-1-2-1', '1003.3720-2-2-1'], ['1003.3720-1-2-2', '1003.3720-2-2-2'], ['1003.3720-1-2-3', '1003.3720-2-2-3'], ['1003.3720-1-2-4', '1003.3720-2-2-4'], ['1003.3720-1-8-0', '1003.3720-2-8-0'], ['1003.3720-1-8-1', '1003.3720-2-8-1'], ['1003.3720-1-8-2', '1003.3720-2-8-2'], ['1003.3720-1-8-3', '1003.3720-2-8-3'], ['1003.3720-1-8-4', '1003.3720-2-8-4'], ['1003.3720-1-8-5', '1003.3720-2-8-5'], ['1003.3720-1-8-6', '1003.3720-2-8-6'], ['1003.3720-1-14-0', '1003.3720-2-14-0'], ['1003.3720-1-14-1', '1003.3720-2-14-1'], ['1003.3720-1-14-2', '1003.3720-2-14-2'], ['1003.3720-1-14-3', '1003.3720-2-14-3'], ['1003.3720-1-14-4', '1003.3720-2-14-4'], ['1003.3720-1-14-5', '1003.3720-2-14-5'], ['1003.3720-1-14-6', '1003.3720-2-14-6'], ['1003.3720-1-14-8', '1003.3720-2-15-0'], ['1003.3720-1-14-10', '1003.3720-2-15-3'], ['1003.3720-1-21-0', '1003.3720-2-22-0'], ['1003.3720-1-21-3', '1003.3720-2-24-1'], ['1003.3720-1-21-4', '1003.3720-2-24-2'], ['1003.3720-1-21-5', '1003.3720-2-24-3']] | [['1003.3720-1-13-4', '1003.3720-2-13-4'], ['1003.3720-1-22-0', '1003.3720-2-25-0'], ['1003.3720-1-22-1', '1003.3720-2-25-1'], ['1003.3720-1-22-3', '1003.3720-2-25-3'], ['1003.3720-1-15-2', '1003.3720-2-16-2'], ['1003.3720-1-18-0', '1003.3720-2-19-0'], ['1003.3720-1-5-1', '1003.3720-2-5-1'], ['1003.3720-1-16-4', '1003.3720-2-17-4'], ['1003.3720-1-23-1', '1003.3720-2-26-1'], ['1003.3720-1-23-3', '1003.3720-2-26-3'], ['1003.3720-1-14-7', '1003.3720-2-14-7'], ['1003.3720-1-14-9', '1003.3720-2-15-1'], ['1003.3720-1-14-11', '1003.3720-2-15-4'], ['1003.3720-1-21-1', '1003.3720-2-22-1'], ['1003.3720-1-21-2', '1003.3720-2-23-1'], ['1003.3720-1-21-6', '1003.3720-2-24-4']] | [] | [['1003.3720-1-22-5', '1003.3720-2-25-5'], ['1003.3720-1-15-3', '1003.3720-2-16-4'], ['1003.3720-1-16-5', '1003.3720-2-17-5'], ['1003.3720-1-23-0', '1003.3720-2-26-0']] | [] | ['1003.3720-2-15-2', '1003.3720-2-23-0', '1003.3720-2-24-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1003.3720 | null | null | null | null | null |
astro-ph-9912007 | {'astro-ph-9912007-1-0-0': 'Heavy particles in galaxy clusters tend to be more centrally concentrated than light ones according to the Boltzmann distribution.', 'astro-ph-9912007-1-0-1': 'An estimate of the drift velocity suggests that there exists a possibility that the helium nuclei may have entirely or partially sedimented into the cluster core within the Hubble time.', 'astro-ph-9912007-1-0-2': 'We demonstrate the scenario using the NFW profile as the dark matter distribution of clusters and assuming that the intracluster gas is isothermal and in hydrostatic equilibrium.', 'astro-ph-9912007-1-0-3': 'We find that a greater fraction of baryonic matter is distributed at small radii than at large radii, which challenges the prevailing claim that the baryon fraction increases monotonically with cluster radius.', 'astro-ph-9912007-1-0-4': 'It shows that the conventional mass estimate using X-ray measurements of intracluster gas along with a constant mean molecular weight may have underestimated the total cluster mass by [MATH], which in turn leads to an overestimate of the total baryon fraction by the same percentage.', 'astro-ph-9912007-1-0-5': 'Additionally, it is pointed out that the sedimentation of helium nuclei toward cluster cores may at least partially account for the sharp peaks in the central X-ray emissions observed in some clusters.', 'astro-ph-9912007-1-1-0': '# Introduction', 'astro-ph-9912007-1-2-0': "An accurate determination of the total gravitating mass of galaxy clusters is crucial for the 'direct' measurements of the cosmic mass density parameter [MATH] through the mass-to-light technique (e.g. Bahcall, Lubin Dorman 1995) and the baryon fraction method (e.g. White et al. 1993).", 'astro-ph-9912007-1-2-1': 'The latter has received much attention in recent years because of the rapid progress in X-ray astronomy and particularly the large spatial extension of the hot and diffuse X-ray emitting gas in clusters.', 'astro-ph-9912007-1-2-2': 'In the conventional treatment, the volume-averaged baryon fraction [MATH], defined as the ratio of the gas mass [MATH] to the total mass [MATH] of a cluster, is obtained by assuming a thermal bremsstrahlung emission and hydrostatic equilibrium for the intracluster gas.', 'astro-ph-9912007-1-2-3': 'While such an exercise has been done for almost every X-ray selected cluster with known temperature, the resultant baryon fractions show rather a large dispersion among different clusters.', 'astro-ph-9912007-1-2-4': 'In particular, [MATH] appears to be a monotonically increasing function of cluster radius, and thereby cannot, in principle, be used for the determination of the cosmic density parameter [MATH] since the asymptotic form of [MATH] at large radii does not approach a universal value (White Fabian 1995; David et al. 1995; Markevitch Vikhlinin 1997a,b; White et al. 1997; Ettori Fabian 1999; Nevalainen et al. 1999; Markevitch et al. 1999; Wu Xue 1999).', 'astro-ph-9912007-1-2-5': 'This leads to an uncomfortable situation that a greater fraction of dark matter is distributed at small scales than at large scales.', 'astro-ph-9912007-1-2-6': 'It is commonly believed that these puzzles have arisen from our poor understanding of the local dynamics inside clusters such as cooling/non-cooling flows, substructures, mergers, etc.', 'astro-ph-9912007-1-2-7': 'Yet, a satisfactory explanation has not been achieved.', 'astro-ph-9912007-1-3-0': 'The conventional cluster mass estimate from X-ray measurement of intracluster gas assumes a constant mean molecular weight [MATH].', 'astro-ph-9912007-1-3-1': 'That is, all the chemical elements of the X-ray emitting gas obey exactly the same spatial distribution.', 'astro-ph-9912007-1-3-2': 'Under this assumption, the total mass of the cluster [MATH], while the mass in gas [MATH].', 'astro-ph-9912007-1-3-3': 'Consequently, the cluster baryon fraction [MATH] depends sensitively on the value of [MATH].', 'astro-ph-9912007-1-3-4': 'It appears that our estimates of total mass and baryon fraction of a cluster would be seriously affected if its chemical elements do not share the same spatial distribution.', 'astro-ph-9912007-1-4-0': 'The Boltzmann distribution of particles in a gravitational field [MATH] follows [MATH], where [MATH] and [MATH] are the number density and the individual mass of the particles, respectively.', 'astro-ph-9912007-1-4-1': 'The heavier a particle is, the slower its thermal motion will be.', 'astro-ph-9912007-1-4-2': 'So, heavy ions in the intracluster gas will have a tendency to drift toward the cluster center.', 'astro-ph-9912007-1-4-3': 'As a consequence of this sedimentation of heavy ions toward the central region, given that the cluster has survived for a sufficiently long time in the Universe, [MATH] will no longer be a constant in the cluster.', 'astro-ph-9912007-1-4-4': 'If the intracluster gas is entirely composed of hydrogen and helium with their primordial abundances, the value of [MATH] will be higher at cluster center while asymptotically reaches 0.5 at large radii for a fully ionized hydrogen gas.', 'astro-ph-9912007-1-5-0': 'The chemically inhomogeneous distribution in galaxy clusters was initiated by Fabian Pringle (1977), who studied the sedimentation of iron nuclei.', 'astro-ph-9912007-1-5-1': 'They found that the iron nuclei in the X-ray gas may settle into the cluster core within a Hubble time.', 'astro-ph-9912007-1-5-2': 'Taking into account the collisions of iron nuclei with helium nuclei, Rephaeli (1978) argued that a much longer sedimentation time than the Hubble time was required so that the iron nuclei would not have enough time to sediment into the core.', 'astro-ph-9912007-1-5-3': 'In this Letter we wish to focus on helium nuclei instead of iron nuclei.', 'astro-ph-9912007-1-5-4': 'Since the drift velocity [MATH] (e.g. Fabian Pringle 1977), where [MATH] and [MATH] are the atomic weight and the charge of the ion, respectively, the helium nuclei will settle much faster than the iron nuclei, and thus the sedimentation may eventually take place.', 'astro-ph-9912007-1-5-5': 'Indeed, Abramopoulos, Chana Ku (1981) have calculated the equilibrium distribution of the elements in the Coma cluster, assuming an analytic King potential, and found that helium and other heavy elements are strongly concentrated to the cluster core.', 'astro-ph-9912007-1-5-6': 'In particular, Gilfanov Sunyaev (1984) have demonstrated that the diffusion of elements in the X-ray gas may significantly enhance the deuterium, helium, and lithium abundances in the core regions of rich clusters of galaxies.', 'astro-ph-9912007-1-5-7': 'For simplicity, we assume that the intracluster gas consists of hydrogen and helium, and then demonstrate their equilibrium distributions in the gravitational potential characterized by the universal density profile (Navarro, Frenk White 1995, 1997; hereafter NFW).', 'astro-ph-9912007-1-5-8': 'As will be shown, under the scenario of helium sedimentation, the baryonic matter in clusters can be more centrally distributed than the dark matter, in contradiction to what is commonly believed.', 'astro-ph-9912007-1-5-9': 'This may open a possibility to solve the puzzle for an increasing baryon fraction with cluster radius.', 'astro-ph-9912007-1-6-0': '# Sedimentation of helium nuclei in galaxy clusters', 'astro-ph-9912007-1-7-0': 'Our working model is based on the assumption that the intracluster gas composed of hydrogen and helium is in hydrostatic equilibrium at a temperature [MATH] with the underlying gravitational potential.', 'astro-ph-9912007-1-7-1': 'This can be justified by a simple estimate of their relaxation time [MATH] (Spitzer 1978) [EQUATION] where [MATH], [MATH], [MATH] and [MATH] are the masses and the charges of proton and helium nucleus, respectively.', 'astro-ph-9912007-1-7-2': '[MATH] is the proton number density, and the Coulomb logarithm is taken to be [MATH].', 'astro-ph-9912007-1-7-3': 'Obviously, [MATH] is much shorter than the present age of the Universe, and hence, the protons and the helium nuclei are readily in hydrostatic equilibrium at the same temperature.', 'astro-ph-9912007-1-8-0': 'According to the Boltzmann distribution of particles in a gravitational field, heavy particles tend to be more centrally distributed in galaxy clusters than light particles.', 'astro-ph-9912007-1-8-1': 'Therefore, with respect to protons, helium nuclei will tend to drift toward the cluster center.', 'astro-ph-9912007-1-8-2': 'An immediate question is: Is the drift velocity sufficiently large for the helium nuclei to have settled into the cluster core within the Hubble time ?', 'astro-ph-9912007-1-8-3': 'The drift velocity [MATH] of helium nuclei can be estimated through (Fabian Pringle 1977) [EQUATION] where [MATH] is the gravitational acceleration.', 'astro-ph-9912007-1-8-4': 'Eq. (2) indicates that within the Hubble time the helium nuclei can drift a distance [EQUATION] which is indeed comparable to cluster scales.', 'astro-ph-9912007-1-8-5': 'Moreover, as [MATH], the value of [MATH] (and [MATH]) increases rapidly with cluster radius [MATH].', 'astro-ph-9912007-1-8-6': 'Therefore, Eq. (3) suggests that the majority of the helium nuclei have probably sedimented into the cluster core within the Hubble time.', 'astro-ph-9912007-1-8-7': 'Note that, due to the requirement of electrical neutrality, the electrons of the same charge will simultaneously sediment along with the helium nuclei.', 'astro-ph-9912007-1-8-8': 'Here, we have not considered the effects of magnetic fields and subcluster mergers, which may somewhat retard the sedimentation of helium nuclei (Rephaeli 1978; Gilfanov Sunyaev 1984).', 'astro-ph-9912007-1-9-0': 'The sedimentation of helium nuclei in galaxy clusters will lead to a dramatic change of baryonic matter distribution.', 'astro-ph-9912007-1-9-1': 'Consequently, the determination of gas and total mass of a cluster will be affected through the mean molecular weight [MATH].', 'astro-ph-9912007-1-9-2': 'Another significant effect is that a sharp peak in the X-ray emission concentrated in cluster core will be expected due to the electron - helium nucleus radiation.', 'astro-ph-9912007-1-9-3': "This provides an alternative scenario for the 'cooling flows' seen in some clusters, for which a detailed investigation will be presented elsewhere.", 'astro-ph-9912007-1-9-4': 'In the present Letter, we only focus on the dynamical effect.', 'astro-ph-9912007-1-9-5': 'In the extreme case, the X-ray gas will be helium-dominated at cluster center, while hydrogen-dominated at large radii.', 'astro-ph-9912007-1-9-6': 'The mean molecular weight [MATH], which is commonly used as a constant, will be a decreasing function of the cluster radius.', 'astro-ph-9912007-1-9-7': 'At cluster center, [MATH] reaches [MATH], the value for a fully ionized helium gas, while at large radii, [MATH] approaches [MATH], the value for a fully ionized hydrogen gas.', 'astro-ph-9912007-1-10-0': 'The conventional mass estimate from X-ray measurement assumes a constant mean molecular weight of [MATH].', 'astro-ph-9912007-1-10-1': 'As the total dynamical mass [MATH] of a cluster is uniquely determined by the intracluster gas at large radii, the difference between [MATH] (the value from the conventional method with [MATH]) and [MATH] as a result of helium sedimentation is simply [EQUATION]', 'astro-ph-9912007-1-10-2': 'This indicates that the conventional method using X-ray measurement of intracluster gas, together with a constant mean molecular weight, may have underestimated the total cluster mass by [MATH], which in turn, results in an overestimate of the total baryon fraction by the same percentage.', 'astro-ph-9912007-1-10-3': 'While the effect of the helium concentration toward cluster center can alter the gas distribution, the total mass in gas of the whole cluster remains unaffected because of the mass conservation.', 'astro-ph-9912007-1-11-0': '# Gas distribution under the NFW potential', 'astro-ph-9912007-1-12-0': 'We now demonstrate how hydrogen and helium are distributed in clusters described by the NFW profile [EQUATION]', 'astro-ph-9912007-1-12-1': 'As has been shown by Makino et al. (1998), such a potential results in an analytic form of gas number density: [EQUATION] where [MATH], and [MATH].', 'astro-ph-9912007-1-12-2': 'Except for the small core radius, [MATH] is well approximated by the conventional [MATH] model.', 'astro-ph-9912007-1-12-3': 'If we neglect the interaction between protons and helium nuclei, then [EQUATION] where [MATH].', 'astro-ph-9912007-1-12-4': 'Eqs (7) and (8) give [MATH], which differs from the prediction by Gilfanov Sunyaev (1984), [MATH].', 'astro-ph-9912007-1-12-5': 'The discrepancy is probably due to the fact that we have not accounted for the diffusion-induced electric fields.', 'astro-ph-9912007-1-13-0': 'We display in Fig.1 the radial distributions of protons and helium nuclei as well as their combined result for a typical nearby cluster with [MATH] keV, [MATH] and [MATH] Mpc (e.g. Ettori Fabian 1999).', 'astro-ph-9912007-1-13-1': 'It is immediate that the intracluster gas is dominated by different elements at different radius ranges: Within the core radius of a few tenth of [MATH], the number density of helium is about four times larger than that of hydrogen because of the sedimentation of helium nuclei, giving rise to a significant excess of both mass in gas (see Fig.2) and X-ray emission in the central region of the cluster relative to the conventional model.', 'astro-ph-9912007-1-13-2': 'Outside the core radius, helium profile shows a sharp drop and protons become to be the major component of the gas.', 'astro-ph-9912007-1-13-3': 'It has been claimed that the total gas distribution can be approximated by the [MATH] model (Makino et al. 1998).', 'astro-ph-9912007-1-13-4': 'In fact, it is easy to show that both the narrow (helium) and extended (hydrogen) components of intracluster gas can be fitted by the [MATH] models with different [MATH] parameters.', 'astro-ph-9912007-1-13-5': 'This provides a natural explanation for the double [MATH] model advocated recently for the cooling flow clusters (e.g. Ikebe et al. 1996; Xu et al. 1998; Mohr, Mathiesen Evrard 1999; etc.).', 'astro-ph-9912007-1-13-6': 'Also plotted in Fig.1 is the mean molecular weight calculated by [MATH].', 'astro-ph-9912007-1-13-7': 'The constant mean molecular weight is now replaced by a decreasing function of radius.', 'astro-ph-9912007-1-13-8': 'The asymptotic values of [MATH] at small and large radii are [MATH] and [MATH], respectively.', 'astro-ph-9912007-1-14-0': 'fig1', 'astro-ph-9912007-1-15-0': 'We present in Fig.2 a comparison of gas masses determined by the invariant and variant mean molecular weights using the same cluster parameters as in Fig.1.', 'astro-ph-9912007-1-15-1': 'To facilitate the comparison, we require that the total particles within [MATH] remains unchanged, where [MATH] is the radius within which the mean cluster mass density is 200 times the critical mass density of the Universe ([MATH]).', 'astro-ph-9912007-1-15-2': 'It appears that the sedimentation of helium nuclei leads to a remarkable concentration of baryonic matter towards cluster center, which challenges the conventional prediction that the baryon fraction increases monotonically with radius (Fig.3).', 'astro-ph-9912007-1-15-3': 'This opens a possibility that the asymptotic baryon fraction at large radii may match the universal value defined by the Big Bang Nucleosysthesis, although a detailed investigation will still be needed.', 'astro-ph-9912007-1-16-0': 'fig2 fig3', 'astro-ph-9912007-1-17-0': '# Conclusions', 'astro-ph-9912007-1-18-0': 'Intracluster gas is mainly composed of hydrogen and helium.', 'astro-ph-9912007-1-18-1': 'Their average abundances over a whole cluster should be of the cosmic mixture.', 'astro-ph-9912007-1-18-2': 'However, their spatial distributions are entirely determined by the underlying gravitational potential of the cluster, and thus follow the Boltzmann distribution.', 'astro-ph-9912007-1-18-3': 'On the other hand, clusters are believed to have formed at redshift [MATH].', 'astro-ph-9912007-1-18-4': 'Therefore, the helium nuclei in clusters may have entirely or partially sedimented into the central cores of clusters today.', 'astro-ph-9912007-1-18-5': 'This will lead to a significant change of the radial distributions of gas and baryon fraction.', 'astro-ph-9912007-1-18-6': 'In the present Letter, we have only discussed the impact on the dynamical aspect of clusters.', 'astro-ph-9912007-1-18-7': 'We will present elsewhere the effect on the cluster X-ray emission (e.g., cooling flows, the double [MATH] model, etc.) as a result of the sedimentation of helium nuclei.', 'astro-ph-9912007-1-19-0': 'In the conventional treatment where the mean molecular weight is assumed to be constant, one may have underestimated the total dynamical mass of clusters by [MATH].', 'astro-ph-9912007-1-19-1': 'Using a more vigorous way in which the NFW profile is taken as the background gravitational field of clusters, we have studied the hydrogen and helium distributions.', 'astro-ph-9912007-1-19-2': 'Indeed, the sedimentation of helium nuclei toward cluster centers has significantly changed the distribution of intracluster gas, with gas being more centrally concentrated than dark matter.', 'astro-ph-9912007-1-19-3': 'This may open a possibility to resolve the puzzle that the baryon fraction increases monotonically with radius predicted by the conventional model.', 'astro-ph-9912007-1-19-4': 'In a word, a number of cosmological applications of the dynamical properties of clusters will be affected by the sedimentation of helium nuclei if it has really taken place during the evolution of clusters, which includes the determination of [MATH] through [MATH], the constraints on the cosmological models through X-ray luminosity - temperature relation, etc.', 'astro-ph-9912007-1-19-5': 'A detailed theoretical study, together with the observational constraints, will be made in subsequent work.', 'astro-ph-9912007-1-20-0': 'We thank an anonymous referee for helpful comments.', 'astro-ph-9912007-1-20-1': 'This work was supported by the National Science Foundation of China, under Grant No. 1972531.', 'astro-ph-9912007-1-21-0': 'Distribution of intracluster gas tracing the gravitational potential by the NFW profile for a typical nearby cluster, [MATH] keV, [MATH] and [MATH] Mpc.', 'astro-ph-9912007-1-21-1': 'The saturated distributions of hydrogen and helium as well as their combined result are shown by dotted lines and solid line, respectively.', 'astro-ph-9912007-1-21-2': 'All the profiles are scaled by the central number density of hydrogen.', 'astro-ph-9912007-1-21-3': 'The corresponding mean molecular weight is displayed in the bottom panel by solid line.', 'astro-ph-9912007-1-21-4': 'Also plotted is the conventionally adopted value [MATH] (dotted line).', 'astro-ph-9912007-1-22-0': 'Different mass components plotted against cluster radius for the same cluster in Fig.1.', 'astro-ph-9912007-1-22-1': 'The dashed line shows the mass in gas obtained in the conventional model, in which the central gas density is assumed to be [MATH] cm[MATH].', 'astro-ph-9912007-1-22-2': 'The conservation of total baryonic particles (or mass) within a radius of [MATH] Mpc is required for both with and without the sedimentation of helium nuclei.', 'astro-ph-9912007-1-23-0': 'A comparison of baryon fractions between the conventional model (dotted line) and the scenario with the sedimentation of helium nuclei (solid line).', 'astro-ph-9912007-1-23-1': 'The cluster parameters are the same as in Fig.2.'} | {'astro-ph-9912007-2-0-0': 'Heavy particles in galaxy clusters tend to be more centrally concentrated than light ones according to the Boltzmann distribution.', 'astro-ph-9912007-2-0-1': 'An estimate of the drift velocity suggests that it is possible that the helium nuclei may have entirely or partially sedimented into the cluster core within the Hubble time.', 'astro-ph-9912007-2-0-2': 'We demonstrate the scenario using the Navarro-Frenk-White profile as the dark matter distribution of clusters and assuming that the intracluster gas is isothermal and in hydrostatic equilibrium.', 'astro-ph-9912007-2-0-3': 'We find that a greater fraction of baryonic matter is distributed at small radii than at large radii, which challenges the prevailing claim that the baryon fraction increases monotonically with cluster radius.', 'astro-ph-9912007-2-0-4': 'It shows that the conventional mass estimate using X-ray measurements of intracluster gas along with a constant mean molecular weight may have underestimated the total cluster mass by [MATH], which in turn leads to an overestimate of the total baryon fraction by the same percentage.', 'astro-ph-9912007-2-0-5': 'Additionally, it is pointed out that the sedimentation of helium nuclei toward cluster cores may at least partially account for the sharp peaks in the central X-ray emissions observed in some clusters.', 'astro-ph-9912007-2-1-0': '# Introduction', 'astro-ph-9912007-2-2-0': "An accurate determination of the total gravitating mass of galaxy clusters is crucial for the 'direct' measurements of the cosmic mass density parameter [MATH] through the mass-to-light technique (e.g. Bahcall, Lubin Dorman 1995) and the baryon fraction method (e.g. White et al. 1993).", 'astro-ph-9912007-2-2-1': 'The latter has received much attention in recent years because of the rapid progress in X-ray astronomy and particularly the large spatial extension of the hot and diffuse X-ray emitting gas in clusters.', 'astro-ph-9912007-2-2-2': 'In the conventional treatment, the volume-averaged baryon fraction [MATH], defined as the ratio of the gas mass [MATH] to the total mass [MATH] of a cluster, is obtained by assuming a thermal bremsstrahlung emission and hydrostatic equilibrium for the intracluster gas.', 'astro-ph-9912007-2-2-3': 'While such an exercise has been done for almost every X-ray selected cluster with known temperature, the resultant baryon fractions show rather a large dispersion among different clusters.', 'astro-ph-9912007-2-2-4': 'In particular, [MATH] appears to be a monotonically increasing function of cluster radius, and thereby cannot, in principle, be used for the determination of the cosmic density parameter [MATH] since the asymptotic form of [MATH] at large radii does not approach a universal value (White Fabian 1995; David et al. 1995; Markevitch Vikhlinin 1997a,b; White et al. 1997; Ettori Fabian 1999; Nevalainen et al. 1999; Markevitch et al. 1999; Wu Xue 1999).', 'astro-ph-9912007-2-2-5': 'This leads to an uncomfortable situation that a greater fraction of dark matter is distributed at small scales than at large scales.', 'astro-ph-9912007-2-2-6': 'It is commonly believed that these puzzles have arisen from our poor understanding of the local dynamics inside clusters such as cooling/non-cooling flows, substructures, mergers, etc.', 'astro-ph-9912007-2-2-7': 'Yet, a satisfactory explanation has not been achieved.', 'astro-ph-9912007-2-3-0': 'The conventional cluster mass estimate from X-ray measurement of intracluster gas assumes a constant mean molecular weight [MATH].', 'astro-ph-9912007-2-3-1': 'That is, all the chemical elements of the X-ray emitting gas obey exactly the same spatial distribution.', 'astro-ph-9912007-2-3-2': 'Under this assumption, the total mass of the cluster [MATH], while the mass in gas [MATH].', 'astro-ph-9912007-2-3-3': 'Consequently, the cluster baryon fraction [MATH] depends sensitively on the value of [MATH].', 'astro-ph-9912007-2-3-4': 'It appears that our estimates of total mass and baryon fraction of a cluster would be seriously affected if its chemical elements do not share the same spatial distribution.', 'astro-ph-9912007-2-4-0': 'The Boltzmann distribution of particles in a gravitational field [MATH] follows [MATH], where [MATH] and [MATH] are the number density and the individual mass of the particles, respectively.', 'astro-ph-9912007-2-4-1': 'The heavier a particle is, the slower its thermal motion will be.', 'astro-ph-9912007-2-4-2': 'So, heavy ions in the intracluster gas will have a tendency to drift toward the cluster center.', 'astro-ph-9912007-2-4-3': 'As a consequence of this sedimentation of heavy ions toward the central region, given that the cluster has survived for a sufficiently long time in the Universe, [MATH] will no longer be a constant in the cluster.', 'astro-ph-9912007-2-4-4': 'If the intracluster gas is entirely composed of hydrogen and helium with their primordial abundances, the value of [MATH] will be higher at cluster center while asymptotically reaches 0.5 at large radii for a fully ionized hydrogen gas.', 'astro-ph-9912007-2-5-0': 'The chemically inhomogeneous distribution in galaxy clusters was initiated by Fabian Pringle (1977), who studied the sedimentation of iron nuclei.', 'astro-ph-9912007-2-5-1': 'They found that the iron nuclei in the X-ray gas may settle into the cluster core within a Hubble time.', 'astro-ph-9912007-2-5-2': 'Taking into account the collisions of iron nuclei with helium nuclei, Rephaeli (1978) argued that a much longer sedimentation time than the Hubble time was required so that the iron nuclei would not have enough time to sediment into the core.', 'astro-ph-9912007-2-5-3': 'In this Letter we wish to focus on helium nuclei instead of iron nuclei.', 'astro-ph-9912007-2-5-4': 'Since the drift velocity [MATH] (e.g. Fabian Pringle 1977), where [MATH] and [MATH] are the atomic weight and the charge of the ion, respectively, the helium nuclei will settle much faster than the iron nuclei, and thus the sedimentation may eventually take place.', 'astro-ph-9912007-2-5-5': 'Indeed, Abramopoulos, Chana Ku (1981) have calculated the equilibrium distribution of the elements in the Coma cluster, assuming an analytic King potential, and found that helium and other heavy elements are strongly concentrated to the cluster core.', 'astro-ph-9912007-2-5-6': 'In particular, Gilfanov Sunyaev (1984) have demonstrated that the diffusion of elements in the X-ray gas may significantly enhance the deuterium, helium, and lithium abundances in the core regions of rich clusters of galaxies.', 'astro-ph-9912007-2-5-7': 'For simplicity, we assume that the intracluster gas consists of hydrogen and helium, and then demonstrate their equilibrium distributions in the gravitational potential characterized by the universal density profile (Navarro, Frenk White 1995, 1997; hereafter NFW).', 'astro-ph-9912007-2-5-8': 'As will be shown, under the scenario of helium sedimentation, the baryonic matter in clusters can be more centrally distributed than the dark matter, in contradiction to what is commonly believed.', 'astro-ph-9912007-2-5-9': 'This may open a possibility to solve the puzzle for an increasing baryon fraction with cluster radius.', 'astro-ph-9912007-2-6-0': '# Sedimentation of helium nuclei in galaxy clusters', 'astro-ph-9912007-2-7-0': 'Our working model is based on the assumption that the intracluster gas composed of hydrogen and helium is in hydrostatic equilibrium at a temperature [MATH] with the underlying gravitational potential.', 'astro-ph-9912007-2-7-1': 'This can be justified by a simple estimate of their relaxation time [MATH] (Spitzer 1978) [EQUATION] where [MATH], [MATH], [MATH], and [MATH] are the masses and the charges of proton and helium nucleus, respectively.', 'astro-ph-9912007-2-7-2': '[MATH] is the proton number density, and the Coulomb logarithm is taken to be [MATH].', 'astro-ph-9912007-2-7-3': 'Obviously, [MATH] is much shorter than the present age of the Universe, and hence, the protons and the helium nuclei are readily in hydrostatic equilibrium at the same temperature.', 'astro-ph-9912007-2-8-0': 'According to the Boltzmann distribution of particles in a gravitational field, heavy particles tend to be more centrally distributed in galaxy clusters than light particles.', 'astro-ph-9912007-2-8-1': 'Therefore, with respect to protons, helium nuclei will tend to drift toward the cluster center.', 'astro-ph-9912007-2-8-2': 'An immediate question is: Is the drift velocity sufficiently large for the helium nuclei to have settled into the cluster core within the Hubble time ?', 'astro-ph-9912007-2-8-3': 'The drift velocity [MATH] of helium nuclei can be estimated through (Fabian Pringle 1977) [EQUATION] where [MATH] is the gravitational acceleration.', 'astro-ph-9912007-2-8-4': 'Eq. (2) indicates that within the Hubble time the helium nuclei can drift a distance [EQUATION] which is indeed comparable to cluster scales.', 'astro-ph-9912007-2-8-5': 'Moreover, as [MATH], the value of [MATH] (and [MATH]) increases rapidly with cluster radius [MATH].', 'astro-ph-9912007-2-8-6': 'Therefore, Eq. (3) suggests that the majority of the helium nuclei have probably sedimented into the cluster core within the Hubble time.', 'astro-ph-9912007-2-8-7': 'Note that, due to the requirement of electrical neutrality, the electrons of the same charge will simultaneously sediment along with the helium nuclei.', 'astro-ph-9912007-2-8-8': 'Here, we have not considered the effects of magnetic fields and subcluster mergers, which may somewhat retard the sedimentation of helium nuclei (Rephaeli 1978; Gilfanov Sunyaev 1984).', 'astro-ph-9912007-2-9-0': 'The sedimentation of helium nuclei in galaxy clusters will lead to a dramatic change of baryonic matter distribution.', 'astro-ph-9912007-2-9-1': 'Consequently, the determination of gas and total mass of a cluster will be affected through the mean molecular weight [MATH].', 'astro-ph-9912007-2-9-2': 'Another significant effect is that a sharp peak in the X-ray emission concentrated in cluster core will be expected due to the electron - helium nucleus radiation.', 'astro-ph-9912007-2-9-3': "This provides an alternative scenario for the 'cooling flows' seen in some clusters, for which a detailed investigation will be presented elsewhere.", 'astro-ph-9912007-2-9-4': 'In the present Letter, we only focus on the dynamical effect.', 'astro-ph-9912007-2-9-5': 'In the extreme case, the X-ray gas will be helium-dominated at cluster center, while hydrogen-dominated at large radii.', 'astro-ph-9912007-2-9-6': 'The mean molecular weight [MATH], which is commonly used as a constant, will be a decreasing function of the cluster radius.', 'astro-ph-9912007-2-9-7': 'At cluster center, [MATH] reaches [MATH], the value for a fully ionized helium gas, while at large radii, [MATH] approaches [MATH], the value for a fully ionized hydrogen gas.', 'astro-ph-9912007-2-10-0': 'The conventional mass estimate from X-ray measurement assumes a constant mean molecular weight of [MATH].', 'astro-ph-9912007-2-10-1': 'As the total dynamical mass [MATH] of a cluster is uniquely determined by the intracluster gas at large radii, the difference between [MATH] (the value from the conventional method with [MATH]) and [MATH] as a result of helium sedimentation is simply [EQUATION]', 'astro-ph-9912007-2-10-2': 'This indicates that the conventional method using X-ray measurement of intracluster gas, together with a constant mean molecular weight, may have underestimated the total cluster mass by [MATH], which in turn, results in an overestimate of the total baryon fraction by the same percentage.', 'astro-ph-9912007-2-10-3': 'While the effect of the helium concentration toward cluster center can alter the gas distribution, the total mass in gas of the whole cluster remains unaffected because of the mass conservation.', 'astro-ph-9912007-2-11-0': '# Gas distribution under the NFW potential', 'astro-ph-9912007-2-12-0': 'We now demonstrate how hydrogen and helium are distributed in clusters described by the NFW profile [EQUATION]', 'astro-ph-9912007-2-12-1': 'As has been shown by Makino et al. (1998), such a potential results in an analytic form of gas number density: [EQUATION] where [MATH], and [MATH].', 'astro-ph-9912007-2-12-2': 'Except for the small core radius, [MATH] is well approximated by the conventional [MATH] model.', 'astro-ph-9912007-2-12-3': 'If we neglect the interaction between protons and helium nuclei, then [EQUATION] where [MATH].', 'astro-ph-9912007-2-12-4': 'Eqs (7) and (8) give [MATH], which differs from the prediction by Gilfanov Sunyaev (1984), [MATH].', 'astro-ph-9912007-2-12-5': 'The discrepancy is probably due to the fact that we have not accounted for the diffusion-induced electric fields.', 'astro-ph-9912007-2-13-0': 'We display in Fig.1 the radial distributions of protons and helium nuclei as well as their combined result for a typical nearby cluster with [MATH] keV, [MATH] and [MATH] Mpc (e.g. Ettori Fabian 1999).', 'astro-ph-9912007-2-13-1': 'It is apparent that the intracluster gas is dominated by different elements at different radius ranges: Within the core radius of a few tenth of [MATH], the number density of helium is about four times larger than that of hydrogen because of the sedimentation of helium nuclei, giving rise to a significant excess of both mass in gas (see Fig.2) and X-ray emission in the central region of the cluster relative to the conventional model.', 'astro-ph-9912007-2-13-2': 'Outside the core radius, helium profile shows a sharp drop and protons become the major component of the gas.', 'astro-ph-9912007-2-13-3': 'It has been claimed that the total gas distribution can be approximated by the [MATH] model (Makino et al. 1998).', 'astro-ph-9912007-2-13-4': 'In fact, it is easy to show that both the narrow (helium) and extended (hydrogen) components of intracluster gas can be fitted by the [MATH] models with different [MATH] parameters.', 'astro-ph-9912007-2-13-5': 'This provides a natural explanation for the double [MATH] model advocated recently for the cooling flow clusters (e.g. Ikebe et al. 1996; Xu et al. 1998; Mohr, Mathiesen Evrard 1999; etc.).', 'astro-ph-9912007-2-13-6': 'Also plotted in Fig.1 is the mean molecular weight calculated by [MATH].', 'astro-ph-9912007-2-13-7': 'The constant mean molecular weight is now replaced by a decreasing function of radius.', 'astro-ph-9912007-2-13-8': 'The asymptotic values of [MATH] at small and large radii are [MATH] and [MATH], respectively.', 'astro-ph-9912007-2-14-0': 'fig1', 'astro-ph-9912007-2-15-0': 'We present in Fig.2 a comparison of gas masses determined by the invariant and variant mean molecular weights using the same cluster parameters as in Fig.1.', 'astro-ph-9912007-2-15-1': 'To facilitate the comparison, we require that the total particles within [MATH] remains unchanged, where [MATH] is the radius within which the mean cluster mass density is 200 times the critical mass density of the Universe ([MATH]).', 'astro-ph-9912007-2-15-2': 'It appears that the sedimentation of helium nuclei leads to a remarkable concentration of baryonic matter towards cluster center, which challenges the conventional prediction that the baryon fraction increases monotonically with radius (Fig.3).', 'astro-ph-9912007-2-15-3': 'This opens a possibility that the asymptotic baryon fraction at large radii may match the universal value defined by the Big Bang Nucleosysthesis, although a detailed investigation will still be needed.', 'astro-ph-9912007-2-16-0': 'fig2 fig3', 'astro-ph-9912007-2-17-0': '# Conclusions', 'astro-ph-9912007-2-18-0': 'Intracluster gas is mainly composed of hydrogen and helium.', 'astro-ph-9912007-2-18-1': 'Their average abundances over a whole cluster should be of the cosmic mixture.', 'astro-ph-9912007-2-18-2': 'However, their spatial distributions are entirely determined by the underlying gravitational potential of the cluster, and thus follow the Boltzmann distribution.', 'astro-ph-9912007-2-18-3': 'On the other hand, clusters are believed to have formed at redshift [MATH].', 'astro-ph-9912007-2-18-4': 'Therefore, the helium nuclei in clusters may have entirely or partially sedimented into the central cores of clusters today.', 'astro-ph-9912007-2-18-5': 'This will lead to a significant change of the radial distributions of gas and baryon fraction.', 'astro-ph-9912007-2-18-6': 'In the present Letter, we have only discussed the impact on the dynamical aspect of clusters.', 'astro-ph-9912007-2-18-7': 'We will present elsewhere the effect on the cluster X-ray emission (e.g., cooling flows, the double [MATH] model, etc.) as a result of the sedimentation of helium nuclei.', 'astro-ph-9912007-2-19-0': 'In the conventional treatment where the mean molecular weight is assumed to be constant, one may have underestimated the total dynamical mass of clusters by [MATH].', 'astro-ph-9912007-2-19-1': 'Using a more vigorous way in which the NFW profile is taken as the background gravitational field of clusters, we have studied the hydrogen and helium distributions.', 'astro-ph-9912007-2-19-2': 'Indeed, the sedimentation of helium nuclei toward cluster centers has significantly changed the distribution of intracluster gas, with gas being more centrally concentrated than dark matter.', 'astro-ph-9912007-2-19-3': 'This may open a possibility to resolve the puzzle that the baryon fraction increases monotonically with radius predicted by the conventional model.', 'astro-ph-9912007-2-19-4': 'In a word, a number of cosmological applications of the dynamical properties of clusters will be affected by the sedimentation of helium nuclei if it has really taken place during the evolution of clusters, which includes the determination of [MATH] through [MATH], the constraints on the cosmological models through X-ray luminosity - temperature relation, etc.', 'astro-ph-9912007-2-19-5': 'A detailed theoretical study, together with the observational constraints, will be made in subsequent work.', 'astro-ph-9912007-2-20-0': 'We thank an anonymous referee for helpful comments.', 'astro-ph-9912007-2-20-1': 'This work was supported by the National Science Foundation of China, under Grant No. 1972531.', 'astro-ph-9912007-2-21-0': 'Distribution of intracluster gas tracing the gravitational potential by the NFW profile for a typical nearby cluster, [MATH] keV, [MATH] and [MATH] Mpc.', 'astro-ph-9912007-2-21-1': 'The saturated distributions of hydrogen and helium as well as their combined result are shown by dotted lines and solid line, respectively.', 'astro-ph-9912007-2-21-2': 'All the profiles are scaled by the central number density of hydrogen.', 'astro-ph-9912007-2-21-3': 'The corresponding mean molecular weight is displayed in the bottom panel by solid line.', 'astro-ph-9912007-2-21-4': 'Also plotted is the conventionally adopted value [MATH] (dotted line).', 'astro-ph-9912007-2-22-0': 'Different mass components plotted against cluster radius for the same cluster in Fig.1.', 'astro-ph-9912007-2-22-1': 'The dashed line shows the mass in gas obtained in the conventional model, in which the central gas density is assumed to be [MATH] cm[MATH].', 'astro-ph-9912007-2-22-2': 'The conservation of total baryonic particles (or mass) within a radius of [MATH] Mpc is required for both with and without the sedimentation of helium nuclei.', 'astro-ph-9912007-2-23-0': 'A comparison of baryon fractions between the conventional model (dotted line) and the scenario with the sedimentation of helium nuclei (solid line).', 'astro-ph-9912007-2-23-1': 'The cluster parameters are the same as in Fig.2.'} | [['astro-ph-9912007-1-5-0', 'astro-ph-9912007-2-5-0'], ['astro-ph-9912007-1-5-1', 'astro-ph-9912007-2-5-1'], ['astro-ph-9912007-1-5-2', 'astro-ph-9912007-2-5-2'], ['astro-ph-9912007-1-5-3', 'astro-ph-9912007-2-5-3'], ['astro-ph-9912007-1-5-4', 'astro-ph-9912007-2-5-4'], ['astro-ph-9912007-1-5-5', 'astro-ph-9912007-2-5-5'], ['astro-ph-9912007-1-5-6', 'astro-ph-9912007-2-5-6'], ['astro-ph-9912007-1-5-7', 'astro-ph-9912007-2-5-7'], 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'astro-ph-9912007-2-22-0'], ['astro-ph-9912007-1-22-1', 'astro-ph-9912007-2-22-1'], ['astro-ph-9912007-1-22-2', 'astro-ph-9912007-2-22-2'], ['astro-ph-9912007-1-4-0', 'astro-ph-9912007-2-4-0'], ['astro-ph-9912007-1-4-1', 'astro-ph-9912007-2-4-1'], ['astro-ph-9912007-1-4-2', 'astro-ph-9912007-2-4-2'], ['astro-ph-9912007-1-4-3', 'astro-ph-9912007-2-4-3'], ['astro-ph-9912007-1-4-4', 'astro-ph-9912007-2-4-4'], ['astro-ph-9912007-1-0-0', 'astro-ph-9912007-2-0-0'], ['astro-ph-9912007-1-0-3', 'astro-ph-9912007-2-0-3'], ['astro-ph-9912007-1-0-4', 'astro-ph-9912007-2-0-4'], ['astro-ph-9912007-1-0-5', 'astro-ph-9912007-2-0-5'], ['astro-ph-9912007-1-15-0', 'astro-ph-9912007-2-15-0'], ['astro-ph-9912007-1-15-1', 'astro-ph-9912007-2-15-1'], ['astro-ph-9912007-1-15-2', 'astro-ph-9912007-2-15-2'], ['astro-ph-9912007-1-15-3', 'astro-ph-9912007-2-15-3'], ['astro-ph-9912007-1-9-0', 'astro-ph-9912007-2-9-0'], ['astro-ph-9912007-1-9-1', 'astro-ph-9912007-2-9-1'], ['astro-ph-9912007-1-9-2', 'astro-ph-9912007-2-9-2'], ['astro-ph-9912007-1-9-3', 'astro-ph-9912007-2-9-3'], ['astro-ph-9912007-1-9-4', 'astro-ph-9912007-2-9-4'], ['astro-ph-9912007-1-9-5', 'astro-ph-9912007-2-9-5'], ['astro-ph-9912007-1-9-6', 'astro-ph-9912007-2-9-6'], ['astro-ph-9912007-1-9-7', 'astro-ph-9912007-2-9-7'], ['astro-ph-9912007-1-2-0', 'astro-ph-9912007-2-2-0'], ['astro-ph-9912007-1-2-1', 'astro-ph-9912007-2-2-1'], ['astro-ph-9912007-1-2-2', 'astro-ph-9912007-2-2-2'], ['astro-ph-9912007-1-2-3', 'astro-ph-9912007-2-2-3'], ['astro-ph-9912007-1-2-4', 'astro-ph-9912007-2-2-4'], ['astro-ph-9912007-1-2-5', 'astro-ph-9912007-2-2-5'], ['astro-ph-9912007-1-2-6', 'astro-ph-9912007-2-2-6'], ['astro-ph-9912007-1-2-7', 'astro-ph-9912007-2-2-7'], ['astro-ph-9912007-1-23-0', 'astro-ph-9912007-2-23-0'], ['astro-ph-9912007-1-23-1', 'astro-ph-9912007-2-23-1']] | [['astro-ph-9912007-1-7-1', 'astro-ph-9912007-2-7-1'], ['astro-ph-9912007-1-13-1', 'astro-ph-9912007-2-13-1'], ['astro-ph-9912007-1-13-2', 'astro-ph-9912007-2-13-2'], ['astro-ph-9912007-1-0-1', 'astro-ph-9912007-2-0-1'], ['astro-ph-9912007-1-0-2', 'astro-ph-9912007-2-0-2']] | [] | [] | [] | ['astro-ph-9912007-1-14-0', 'astro-ph-9912007-1-16-0', 'astro-ph-9912007-2-14-0', 'astro-ph-9912007-2-16-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/astro-ph/9912007 | null | null | null | null | null |
cs-0605104 | {'cs-0605104-1-0-0': '# Introduction', 'cs-0605104-1-1-0': 'Programming is the enterprise of fitting the infinitely subtle subjects of algorithms and interfaces into the rigid confines of a formal language defined by a few unyielding rules-is it any wonder that this process can be so difficult.', 'cs-0605104-1-1-1': 'The first step in this process it the selection of the language.', 'cs-0605104-1-1-2': 'As we go along in this enterprise, we might find that our selected language is inadequate for the task at hand; at which point we can: forge ahead with an imperfect language, we can attempt to address the problematic section in a different language, or we can jettison the language for another with its own limitations, thereby duplicating the effort already put into writing the program in the first language.', 'cs-0605104-1-1-3': 'With ever larger, more complex programs, we increasingly find that no single language is especially well-suited-yet if we try to use multiple languages, we face significant hurdles in integrating the languages, with rare exceptions.', 'cs-0605104-1-1-4': 'A fourth possibility presents itself: we could create a new programming language that contains all of the features we will ever need in any section of the program; aside from the fact that creating a general programming language is a monumental effort in and of itself, the resulting programming language will likely be a cumbersome monster.', 'cs-0605104-1-1-5': 'What we seek is a language that is at once general enough to suffice for very large programs, while also having specific features for each portion of the program.', 'cs-0605104-1-2-0': 'There are a great deal of mature programming languages in existence, each with its own advantages and disadvantages.', 'cs-0605104-1-2-1': 'None of these are the language we seek.', 'cs-0605104-1-2-2': 'Ideally, we would like to be able to take an existing programming language and-without having to duplicate the tremendous amount of effort which went into its creation and development, not to mention our own effort in learning it-mold it to our needs.', 'cs-0605104-1-3-0': 'We do not have time to survey the major languages, but programs in these languages do fit a general mold: programs must be syntactically well-formed; then they must be semantically well-meaning; and finally, they must specify a program that is free from run-time errors.', 'cs-0605104-1-3-1': 'Moving from the source code for a program to a run-time executable involves three phases: syntax analysis, semantic analysis, and code generation.', 'cs-0605104-1-3-2': 'The first two analysis phases are not separated in practice, but are performed in concert by a parser which is generated by a parser generator.', 'cs-0605104-1-3-3': 'The parser generator takes a grammar describing the syntax of the programming language, in addition to the semantic value of each production in the grammar; from these it produces a parser.', 'cs-0605104-1-3-4': 'If we had the source code to the compiler, we could change it to suit our purposes, producing a defnderived language.', 'cs-0605104-1-3-5': 'However, we must be careful if we do this, for changes to the code generator could produce binaries that lack compatibility with existing binaries.', 'cs-0605104-1-4-0': 'The aforementioned approach is not terribly common: its most glaring problem is that a program written in a derived language cannot be compiled by a "normal" compiler.', 'cs-0605104-1-4-1': 'An alternative is to make a new compiler wholesale-one that, rather than outputting a binary, outputs source code in an existing programming language; such a compiler is known as a defnsource-to-source translator.', 'cs-0605104-1-4-2': 'The practice of creating source-to-source translators is much more common that the practice of creating derived languages; two examples are the cfront compiler for C++ and a WSDL compiler for SOAP.', 'cs-0605104-1-4-3': 'These two examples illustrate an interesting point: the new language can share much with the target language, as is the case with cfront; or, the new language can share nothing with the target language, as is the case with a WSDL compiler.', 'cs-0605104-1-5-0': 'Creating a derived language is an attractive concept because we can directly leverage an existing implementation of a base language, but modifying any large program-the compiler, in this case-in an ad-hoc manner is not exactly an easy task.', 'cs-0605104-1-5-1': 'This approach becomes decidedly less attractive if we seek to radically alter the language: we will likely find that the code generator is tightly coupled with the parser, and that the facilities for creating abstract syntax trees have no more generality than is necessary for the original language.', 'cs-0605104-1-5-2': 'Creating a source-to-source translator, on the other hand, is an attractive concept because we can make a language that departs from the target language as much or as little as we want; however, perhaps too much information is lost in the conversion to the target language: data such as debugging information, higher-order typing, optimization hints, and details necessary for proper error handling are just a few of the things which might get lost.', 'cs-0605104-1-5-3': 'Another problem is what I refer to as the "language tower problem": say we start with a language L, then we create a source-to-source translator from L++ to L, then we create a source-to-source translator for Aspect-L++, then we create a source-to-source translator for Visual Aspect-L++, ad nauseam-in short, we end up with far too many parsers.', 'cs-0605104-1-6-0': 'We will take an approach somewhere between these two.', 'cs-0605104-1-6-1': 'We would like to develop a defnbase language that is general purpose enough to serve us in its unmodified form, yet can be modified at our pleasure.', 'cs-0605104-1-6-2': 'In light of our consideration of derived languages, we will create a general-purpose framework for abstract syntax trees that both the base language, and any derived language, can use to capture the full range of the semantics of a program.', 'cs-0605104-1-6-3': 'Also, we will not require someone wishing to create a derived language to create an entire grammar: we will allow modifications of the existing grammar.', 'cs-0605104-1-6-4': 'We have avoided most of the problems associated with source-to-source translation as well: the data which would not get translated to the target program can be readily stored in the abstract syntax tree.', 'cs-0605104-1-6-5': 'Since we are making it easy to modify the language, we would expect that the language tower problem would be exacerbated, but this is hardly the case, for there is only ever a singular parser.', 'cs-0605104-1-7-0': 'We pause to note that there must be some way of specifying the semantic actions of a production.', 'cs-0605104-1-7-1': 'We can assume that these actions are specified in a programming language, probably the base language itself, and that the parser has an interpreter for that language.', 'cs-0605104-1-8-0': 'The code generator only understands so much of what is potentially in an abstract syntax tree.', 'cs-0605104-1-8-1': 'Everything else-the debugging, type, optimization, and error data-which gets added to the tree must be, to a large extent, ignored by the code generator.', 'cs-0605104-1-8-2': 'However, these data-we will call them defnextended semantics data-are not valueless; thus, we will allow additional analysis phases to be performed on the extended semantics data between that parsing and the code generation phases.', 'cs-0605104-1-8-3': 'Here again, an interpreter embedded in the parser will be invaluable.', 'cs-0605104-1-9-0': 'How might a language like this be used?', 'cs-0605104-1-9-1': 'We can use it to add gross language features, for example object-oriented or aspect-oriented support.', 'cs-0605104-1-9-2': 'Or we could add more behind-the-scenes features, improving for example, the optimizer.', 'cs-0605104-1-9-3': 'If we know that we are using a particular library, we can give first-class syntactic support to common patterns-for example, we could support monitors, as Java does with the synchronized keyword.', 'cs-0605104-1-9-4': 'Finally, we could create a modified grammar to eliminate repetitive code, using the modifiability of the language as a sort of macro processor.', 'cs-0605104-1-10-0': 'In order to use our parser as a macro processor, we must consider the scope of a modification to the parser.', 'cs-0605104-1-10-1': 'If we have a file, or other, similar unit of a program, which contains directives to modify the parser, and we require that the modification to the parser not take place at least until after parsing is complete, then our parser will do everything we want of it, except function as a macro processor.', 'cs-0605104-1-10-2': 'Will we thus require that the parser can be modified during parsing.', 'cs-0605104-1-10-3': 'From here on, we will assume that a parser operates strictly left-to-right.', 'cs-0605104-1-10-4': 'No longer can we treat the syntax analysis and semantic analysis phases as entirely separate, even conceptually, for some part of a file may define the syntax and semantics of the remainder of the file.', 'cs-0605104-1-11-0': 'The study of formal languages has produces many interesting classes of languages: regular, context-free, context-sensitive, and recursive-enumerable being the best known.', 'cs-0605104-1-11-1': 'If [MATH] is a class of languages, then the set of defntransformative [MATH] languages are those languages whose strings [MATH] can be decomposed as [MATH], such that [MATH] is a substring of an element of one of the languages in [MATH], which we term the [MATH] defninstantaneous language; further, [MATH] specifies the instantaneous language [MATH].', 'cs-0605104-1-12-0': 'Our goal in the present work will be to develop a method of parsing a useful class of transformative languages.', 'cs-0605104-1-12-1': 'Our parser will operate much like a classical parser, except that, as it moves over the boundary between [MATH] and [MATH], it will modify itself-more precisely, it will modify its grammar, and then its parsing tables.', 'cs-0605104-1-12-2': 'Since we are dealing with a self-modifying parser, we would run into problems if the parser were to backtrack, from [MATH] to [MATH]-not insurmountable problems, to be sure, but we will find a satisfactory non-backtracking method of parsing that does not have these problems.', 'cs-0605104-1-13-0': 'Our goal in the present work will be to develop a method of parsing a useful class of transformative languages-such a language was described as an defnextensible language.', 'cs-0605104-1-13-1': 'Our parser will operate much like a classical parser, except that, as it moves over the boundary between', 'cs-0605104-1-14-0': 'The idea of a transformative language is a refinement of an old idea about programming languages: the idea that we could define new algebraic and abstract data types, operations, and control structures by instructing the parser to modify itself.', 'cs-0605104-1-14-1': 'Our method of self-modifying parsers has also been presented as a way of parsing non-context-free languages.', 'cs-0605104-1-14-2': 'Extensible languages went out of style with the introduction of operator overloading and abstract data types, as in languages like C++, and the introduction of generic programming.', 'cs-0605104-1-15-0': '## Application of Transformative Parsing', 'cs-0605104-1-16-0': 'Let us say that we need to write a graphical program in a language much like Java which can access both a web service and a database; let us assume that we do not have any visual rapid-development tools.', 'cs-0605104-1-16-1': 'We must write a lot of GUI code like "make a window, put a layout in the window, put the following controls in the layout: , add a toolbar to the window, add an item to the toolbar with the label \'x\', set the callback object to \'y"\' etc.', 'cs-0605104-1-16-2': 'We must write a lot of database like "parse a query, bind the following variables (), execute the query, create a cursor, advance the cursor, get the first column, get the second column" etc.', 'cs-0605104-1-16-3': 'We must write a lot of web service code like "create a procedure call, marshal the input, call the procedure, demarshall the output, handle any exceptions" etc.', 'cs-0605104-1-16-4': 'The GUI, database, and web service functionality is most likely handled by a library.', 'cs-0605104-1-16-5': 'Would that each library added syntax constructs for the operations it provides.', 'cs-0605104-1-16-6': 'We could declare the GUI with code like', 'cs-0605104-1-17-0': "window layout...; toolbar item label = 'x'; action = y", 'cs-0605104-1-18-0': 'The interesting thing is that the -y- identifier is bound to the correct lexical scope.', 'cs-0605104-1-18-1': 'We could process our query with code like', 'cs-0605104-1-19-0': 'query(select col1, col2 from t1 where col=[MATH]k[MATH]k0[MATH]k[MATH]k[MATH]k 1[MATH](,N,P,S)[MATH][MATH]N[MATH]P[MATH]S[MATH][MATH]G=(,N,P,S)[MATH][MATH]Ntimes(N)^*[MATH]P[MATH]A[MATH](A,)P[MATH][MATH]^*[MATH]G=(,N,P,S)[MATH](N)^*[MATH]FIRST_k()[MATH]y^*[MATH]y=k[MATH]_G^* yx[MATH]x^*[MATH]FIRST()FIRST_1()[MATH]FIRST()[MATH][MATH]G=(N,,P,S)[MATH]G\'=(N\',0[MATH]S\' _G\' rm^* A w _G\' rm w,[MATH]S\' _G\' rm^* B x _G\' rm y,[MATH]FIRST_k(w)=FIRST_k(y)[MATH]A y=B x[MATH]xG[MATH]x[MATH]xG[MATH]x[MATH]G[MATH]K[MATH]K[MATH]I_K[MATH](N)^*[MATH]a[MATH]a[MATH]k[MATH]action=shift m[MATH]a[MATH]m[MATH]action=reduce " A"[MATH][MATH]k\'[MATH]goto[MATH]action=error[MATH]action=accept[MATH]G=(,N,P,S)[MATH][MATH]S\'N[MATH][A,a][MATH][MATH]a[MATH][MATH][MATH][MATH][S\'S, [MATH]][MATH]I_0[MATH][AB,a][MATH]B[MATH][B,b][MATH]b[MATH]FIRST(a)[MATH]I_k[MATH]I_m[MATH]X(N)[MATH]I_k[MATH]X[MATH]I_m[MATH]m=goto[MATH][AX,a]I_k[MATH][AX,a]I_m[MATH]I_k[MATH]a[MATH][Aa ,b]I_k[MATH]action=shift m[MATH]m=goto[MATH][A,a]I_k[MATH]action=reduce "A"[MATH]goto[MATH]action[MATH]action[MATH]error[MATH]G=(,N,P,S)[MATH]x^*[MATH]xG[MATH]S[MATH]X[MATH]Y_1,Y_2,,Y_n[MATH]XY_1Y_2Y_n[MATH][MATH][MATH]x[MATH]x[MATH]x[MATH]xG[MATH][MATH] M_1, M_2,, M_m[MATH]S( N)=[MATH]C( N)=S( N)[MATH][MATH][MATH]x[MATH]x[MATH]G_T[MATH]_T[MATH]L(G_T)[MATH][MATH][MATH][MATH][MATH]M[MATH]_1[MATH]_2[MATH]_3[MATH]_1[MATH]_1,_3_T[MATH]w_1^*[MATH]z_2^*[MATH]M[MATH]wL(G_T)[MATH]M[MATH][MATH]^*[MATH](,N,P,S,T,M)[MATH][MATH][MATH]G=(,N,P,S,T,M)[MATH]G[MATH]H[MATH]H=(,N,P,S)[MATH]_H^*[MATH]_G nt^*[MATH]G=(,N,P,S,T,M)[MATH](N,P_+,P_-)[MATH]P_- P_+ = [MATH]P_-[MATH]P_+[MATH][MATH]GT(G,)[MATH]G[MATH] A[MATH][MATH]G[MATH]G[MATH][MATH]G=(,N,P,S,T,M)[MATH]M[MATH]_1[MATH]_2[MATH]w_2^*[MATH]w[MATH]g T ^*[MATH]G[MATH]L(G T)[MATH],(N)^*[MATH]x,z^*[MATH]^* _2^* _3 T _2^*[MATH](x,y,g)[MATH](,z)[MATH][MATH]M[MATH]z[MATH]M[MATH]M[MATH](x,y,g)[MATH]z=S[MATH]z _G nt^* x[MATH]G\'=G[MATH]u=w[MATH]z=beta\' z_G nt^*y z=x[MATH][MATH]y^*[MATH]FIRST(w)=FIRST(z)[MATH] w[MATH]G\'=G[MATH](,u)=(y,w,g)[MATH]z[MATH]w[MATH]x[MATH]G[MATH]u[MATH]G[MATH]G\'[MATH]x[MATH]z[MATH]w[MATH]u[MATH]( x,u,G)(z,w,G\')[MATH]G[MATH]x[MATH]_i[MATH]G_i[MATH]_i[MATH]G_i[MATH]G[MATH]c,d[MATH]S,A,B,C,D[MATH]S[MATH][MATH]M[MATH]G_1[MATH]G_2[MATH]G[MATH]_0[MATH]_1[MATH]_2[MATH][MATH]M[MATH]G[MATH]M[MATH]_0[MATH]G_1[MATH]M[MATH]_1[MATH]G_2[MATH]M[MATH]_2[MATH]c[MATH]u_i[MATH][MATH]i[MATH]c[MATH]G[MATH]c[MATH]G[MATH][MATH]M[MATH]M[MATH]n[MATH]n[MATH][MATH][MATH][MATH][MATH][MATH]G=(,N,P,S,T,M)[MATH] AB[MATH] A= B[MATH] A< B[MATH](N)^*[MATH]^*[MATH][MATH]B[MATH]B a x[MATH](N)^*[MATH]a[MATH]x^*[MATH]a[MATH]B[MATH]y^*[MATH]y[MATH]B a x^*y[MATH]B[MATH]a[MATH]B a x[MATH]N_( P)[MATH]N_( P) = n+1[MATH]n[MATH]_1, [MATH]_2,,[MATH]_n[MATH]_i[MATH]N_( P)=i[MATH] B< P< A[MATH]N_( P)=n+1[MATH]n[MATH]N_( P) = -1[MATH]P[MATH]V_[MATH][MATH]y[MATH]y[MATH]y\'[MATH]Bax ^* y[MATH]Bax^* y\'[MATH]y[MATH]y\'[MATH]ax[MATH]ax[MATH]B[MATH]B[MATH]N_[MATH]V_[MATH]y[MATH](N)^*[MATH]V_B()=-1[MATH][MATH][MATH]V_B()=i[MATH][MATH]A[MATH]0 < i [MATH]i[MATH][MATH][MATH][MATH]V_B()=i[MATH][MATH]A [MATH]i = + 1 [MATH][MATH][MATH]G[MATH][MATH](N)^*[MATH]^*[MATH][MATH]P[MATH]BY_1 Y_2 Y_m[MATH][MATH][MATH]P[MATH][MATH]AX_1 X_2 X_n[MATH]V_()n[MATH]1iV_()[MATH]X_i=Y_i[MATH]A=B[MATH]_ [MATH]V_()=n+1[MATH]phi[MATH](N)^*[MATH]G[MATH]a x[MATH]a[MATH]x^*[MATH][MATH]a[MATH]G=(,N,P,S,T,M)[MATH]G[MATH] G=(,N_G, P_G, S, T,M)[MATH]A[MATH]a[MATH](N)^*[MATH]BN[MATH]a[MATH]x^*[MATH]A a x[MATH]G[MATH]P_G[MATH]A a x[MATH]P[MATH][MATH]B a[MATH]G[MATH]B a[MATH]V_G(B a)[MATH]_V_G(B a)[MATH][S\'S,[MATH]][MATH][A,a][MATH]a[MATH]C[MATH]C[MATH]B[MATH]a[MATH]B[MATH][MATH]a[MATH]B[MATH][MATH]a[MATH]B[MATH]a[MATH][MATH][MATH]G=(,N,P,S,T,M)[MATH]M[MATH]_1[MATH]_2[MATH]w_2^*[MATH]w[MATH]g T[MATH]G[MATH]L(G T)[MATH],(N)^*[MATH]x,z^*[MATH]^*_2^* T^* T _2^*[MATH](x,y,g)[MATH](,z)[MATH][MATH]M[MATH]z[MATH]M[MATH]M[MATH](x,y,g)[MATH]z=S[MATH]z _G nt^* x[MATH]G\'=G[MATH]u=w[MATH]z=beta\' z_G nt^* y z=x[MATH][MATH]y^*[MATH]FIRST(w)=FIRST(z)[MATH](,u)=(y,w,g)[MATH]w[MATH]G\'=G[MATH]G\'[MATH][MATH]a[MATH]G[MATH]a=FIRST(z)[MATH]z[MATH]w[MATH]x[MATH]G[MATH]u[MATH]G[MATH]G\'[MATH]x[MATH]z[MATH]w[MATH]u[MATH]( x,u,G)(z,w,G\')[MATH]G_0=(,N,P,S,T,M)[MATH]x^*[MATH]w_0=[MATH]_0=x[MATH]k>0[MATH]_k=S[MATH]xG[MATH]x[MATH]G[MATH]L ^*[MATH]L[MATH]T[MATH]L[MATH]G=(,N,P,S,T,M)[MATH]L[MATH]a_1, a_2, , a_n[MATH]N=S, A, B[MATH]T=S A[MATH]M[MATH]M[MATH]x ^*[MATH]x L[MATH]_[MATH]x L[MATH], P)[MATH]x L[MATH]x G[MATH]x L[MATH]x G[MATH][MATH]G[MATH]A[MATH]c[MATH]V_G(A c)[MATH]V_G(B c)[MATH]G=(,N_G, P_ G, S, T)[MATH]x^*[MATH]B c x[MATH][MATH]B c x[MATH]G[MATH][MATH]Bcx[MATH][MATH]P_G[MATH]_B c xphi[MATH][MATH]A c x[MATH]y^*[MATH]y[MATH]B[MATH]c[MATH] B < P < C[MATH]A[MATH] B < Q < C[MATH] P U[MATH] P W[MATH]L( P)[MATH]Ac[MATH] P W[MATH]L ( P)[MATH]Ac[MATH]N_Ac( P)=N_Bc( P)[MATH] PW[MATH]N_Ac( P)=N_Bc( P)[MATH] P W[MATH]L( P)[MATH]N_Ac( P)=N_Bc( P)[MATH]L( P)[MATH]N_Ac( P)=N_Bc( P)[MATH] UW[MATH] PU[MATH]N_Bc( P)>0[MATH]N_Ac( P)=N_Bc( P)[MATH]N_Bc( P)[MATH]N_Ac > 0[MATH] PU[MATH][MATH]Bcx[MATH]V_G(A c)[MATH]P_G[MATH]_A c xphi[MATH]_B c xphi[MATH]V_G(B y)[MATH]FIRST(y)=FIRST(w)[MATH]C ^*_G A z ^*_G B y z[MATH]C ^*_G A z[MATH]C _G A z[MATH][MATH][MATH]G[MATH]x^*[MATH]^*_G x[MATH]C ^*_G A x [MATH]n[MATH]n>1[MATH]C _G^* A z[MATH]n[MATH]D_G^* A u[MATH]n[MATH]V_G( B y)[MATH][MATH]G[MATH]t^*[MATH]_G^* t[MATH]G[MATH]V_G(B)[MATH]A _G B[MATH][MATH]G[MATH]G[MATH]n 1[MATH]n+1[MATH]x[MATH]x[MATH]B x[MATH]x=[MATH][MATH]G[MATH]G[MATH]n[MATH]C_G^*B[MATH]V_G(Aa)[MATH]A x _G A x[MATH][MATH]G[MATH]= B[MATH][MATH]A[MATH]x[MATH]A x[MATH][MATH][MATH]G[MATH][MATH]G[MATH]G[MATH]V_G(A)[MATH]AB_G Aax[MATH][MATH]G[MATH]a[MATH][MATH]G[MATH]_G^* z[MATH]n[MATH]n+1[MATH][MATH]a[MATH]Aa x _G^* A a x[MATH][MATH]a[MATH][MATH]G[MATH]w[MATH]_G^* w[MATH]a[MATH][MATH]C[MATH][MATH]G[MATH][MATH]H[MATH] AC_G^* Aat[MATH][MATH][MATH] E[MATH]A Ephi[MATH]n[MATH]AB_G^*Aax[MATH]V_G(Aa)[MATH]AB_ G^* Aay[MATH]AB_GAax[MATH][MATH]G[MATH][MATH] G[MATH]y^*[MATH]_G^* y[MATH]AB_G^* A a y[MATH]n[MATH]AB_G^*Aax[MATH]n+1[MATH]a[MATH][MATH]a[MATH][MATH]a [MATH][MATH]G[MATH]w[MATH]_G^* w[MATH]a[MATH][MATH][MATH]C [MATH][MATH]C[MATH][MATH]G[MATH]v[MATH]_G^* v\'[MATH]A C_G^*A a[MATH]n[MATH][MATH][MATH]E[MATH]V_G(A E a)[MATH]V_G( A E a)[MATH]G[MATH][MATH]a[MATH]G[MATH]a[MATH]n[MATH][MATH]_1,_2,,_n(N)^*[MATH]_0=[MATH]_n a[MATH]0 k < n[MATH]_k a[MATH][MATH]G=(,N,P,S)[MATH],(N)^*[MATH]beta[MATH][MATH][MATH]alpha[MATH]^*[MATH]^*[MATH]G[MATH]A[MATH]a[MATH]V_G(A a)[MATH]A[MATH]a[MATH]G[MATH]G[MATH]x^*[MATH]Aax[MATH]G[MATH]w^*[MATH]Aaw[MATH]G[MATH]Aax[MATH]A[MATH]a[MATH]A[MATH]a[MATH]PC X [MATH]X=a[MATH]X[MATH]Aax[MATH]P z[MATH]V_G(Aa)[MATH]PC X [MATH]G[MATH]X^*_Gau[MATH]u^*[MATH]P v[MATH]v^*[MATH]P v[MATH]X=a[MATH]X[MATH]X=D[MATH]C[MATH]C E[MATH]V_G( Ea)[MATH]G[MATH]B[MATH]V_G(Ba)[MATH][MATH]a[MATH]G[MATH]a[MATH]n[MATH][MATH]G[MATH]a[MATH]n-1[MATH]B[MATH]G[MATH]G\'[MATH]H[MATH]G[MATH]G[MATH]n[MATH]n=1[MATH]_G\' B[MATH]a[MATH]B a[MATH]G\'[MATH]B a[MATH]H[MATH]n 1[MATH]n+1[MATH]n[MATH]B _G\'^* [MATH]n 1[MATH]V_G(C a)[MATH]a[MATH]n-1[MATH]B[MATH]H[MATH]x[MATH]G[MATH]G[MATH]G[MATH]G=(,N,P,S,T,M)[MATH]x[MATH]N^*[MATH]e[MATH]xG[MATH]G[MATH]a[MATH]x[MATH]s[MATH]action=shift[MATH]a=EOF[MATH]action=shift[MATH]a[MATH]goto[MATH]a[MATH]action=reduce [MATH]T[MATH][MATH]G[MATH]s\'[MATH]A[MATH]goto[MATH]action=error[MATH]G[MATH]G=(,N,P,S,T,M)[MATH][MATH][MATH][MATH]G[MATH][MATH][MATH][MATH][MATH]Delta G [MATH]G=G[MATH]G[MATH][MATH]a[MATH]s[MATH]goto[MATH]G[MATH][MATH]goto[MATH][MATH]G[MATH]g()[MATH]I_g()[MATH][MATH][MATH]g()=0[MATH][S\'S, [MATH]][MATH]I_0[MATH]g[MATH]n[MATH]n0[MATH]X[MATH]n+1[MATH]x[MATH]X x[MATH]X x[MATH][MATH]I_g()[MATH][MATH]m+1A_m[MATH]M[MATH]A_m[MATH] A_M[MATH]M[MATH][MATH]A_m+1A_m _m[MATH]0 m < M[MATH]I_g()[MATH]I_g(X)[MATH]g[MATH]G[MATH][MATH]a[MATH]a[MATH]n[MATH]G[MATH]a[MATH]n[MATH][MATH]a[MATH][MATH]M[MATH]n[MATH]n[MATH]k0[MATH]n=k+1[MATH]G[MATH]a[MATH]k[MATH]A[MATH]A[MATH]a[MATH]G[MATH]a[MATH]k[MATH]a[MATH]n[MATH]A[MATH][MATH]M[MATH]A[MATH]G[MATH]a[MATH]k[MATH]G[MATH]G=(,N,P,S,T,M)[MATH][MATH]M[MATH]G[MATH]x^*[MATH](N)^*[MATH]x[MATH]x1[MATH]x=[MATH][MATH]G[MATH]n[MATH]n0[MATH]n[MATH]k1[MATH]x=k+1[MATH]x=ax\'[MATH]a[MATH]x\'^*[MATH]G[MATH]a[MATH]n[MATH]n0[MATH]a[MATH]n[MATH][MATH]x\'=k[MATH]G=(,N,P,S,T,M)[MATH][MATH]M[MATH]G[MATH]x^*[MATH]N,P,S,T,M)[MATH]X_1 X_2 X_m[MATH]a[MATH]X_m=B[MATH]I_0,I_1,,I_n[MATH]s_0,s_1,,s_m+1[MATH]V_()[MATH](,k)[MATH]V_something[MATH]k[MATH][MATH]G=(,N,P,S,T,M)[MATH]X_2 X_m[MATH](N)^*[MATH]s_0,s_1,,s_m+1[MATH]a[MATH]V_P[MATH](,i)[MATH]P[MATH]i [MATH]B ,b][MATH]I_s[MATH]G[MATH][MATH]j[MATH]a[MATH]V_F[MATH]f[MATH]V_T = V_TV_F[MATH]f[MATH]G[MATH][MATH]j[MATH]a[MATH]V_A[MATH]V_T = V_TV_A[MATH]P[MATH]V_x[MATH]V_P=(,V_x()P[MATH]V_P[MATH][MATH][MATH][MATH][MATH]^*[MATH]V_F[MATH]f[MATH]V_F[MATH]f[MATH]G=(,N,P,S,T,M)[MATH]X_2 X_m[MATH](N)^*[MATH]s_0,s_1,,s_m+1[MATH]a[MATH]j=[AB,b][MATH]V_F[MATH](,i)[MATH]P[MATH]i [MATH]Y_2 Y_n[MATH][MATH]AB [MATH]i[MATH]n[MATH]Y_i=a[MATH](,+1+i)[MATH]V_Z[MATH]V_F[MATH]Y_i[MATH]V_F[MATH]Y_i[MATH]G[MATH]Y_i[MATH]a[MATH][MATH]V_E[MATH]f_N[MATH]e[MATH][MATH]e[MATH]f_N[MATH]V_Z=V_ZV_E[MATH]f_N[MATH](,+1+i)[MATH]V_Z[MATH]V_F[MATH]V_Z=[MATH]f[MATH]e[MATH]V_F[MATH]f[MATH](,B+1)[MATH]V_Z[MATH]+1[MATH]s[MATH]J=j_0[MATH]j_0[MATH][AB,b][MATH]I_s[MATH]J[MATH][[MATH],c][MATH]J[MATH][[MATH],d][MATH]I_s[MATH]J[MATH]J[MATH][[MATH],e][MATH]J[MATH][MATH](,+1)[MATH]V_Z[MATH]G[MATH]a[MATH][[MATH],e] [MATH]j[MATH]V_P[MATH]f_P[MATH]f_P[MATH]f[MATH]V_Z=V_ZV_P[MATH]V_Z=[MATH]V_F[MATH]f[MATH]j[MATH]j[MATH]j[MATH]G=(,N,P,S,T,M)[MATH]X_2 X_m[MATH](N)^*[MATH]s_0,s_1,,s_m+1[MATH]a[MATH]j=[AB,b][MATH]V_A[MATH](,i)[MATH]P[MATH]i [MATH]j[MATH]J[MATH][[MATH],c][MATH]J[MATH][[MATH],d][MATH]I_s[MATH]J[MATH]J[MATH]k=[[MATH],d][MATH]J[MATH][MATH](,+1)[MATH]V_A[MATH][MATH]G[MATH]a[MATH]k[MATH]j[MATH]V_A\'[MATH]V_A=V_AV_A\'[MATH]V_A[MATH]V_S[MATH][MATH]e[MATH]X^*[MATH]f[MATH]X^*ax[MATH]x^*[MATH]G[MATH]G=(,N,P,S,T,M)[MATH]X[MATH]N[MATH][MATH]a[MATH]V_S[MATH](,n)[MATH][MATH]e[MATH]f[MATH]e[MATH]f[MATH]X[MATH]V_S=[MATH]X=a[MATH]f[MATH]V_S[MATH][MATH]f[MATH]e[MATH][MATH]X[MATH][MATH][MATH][MATH][MATH][MATH]G[MATH][MATH][MATH]a[MATH]V_*[MATH]_*[MATH]f_*[MATH]k[MATH]k=+1[MATH]e[MATH]V_*[MATH](,k)[MATH]V_S[MATH]V_S[MATH][MATH]f[MATH]e[MATH]V_*[MATH][MATH]k[MATH]^*[MATH]k=+1[MATH]^*ax[MATH]x^*[MATH]1k[MATH]k=-1[MATH]^*[MATH]^*ay[MATH]y^*[MATH]k=+1[MATH]f[MATH]G[MATH]G=(,N,P,S,T,M)[MATH][MATH](N)^*[MATH][MATH]a[MATH]V_*[MATH](,n)[MATH][MATH]f[MATH]k[MATH]-1k+1[MATH]k=-1[MATH]f[MATH]X_2 X_n[MATH]i[MATH]n[MATH]G[MATH]X_i[MATH][MATH]V_S[MATH]_S[MATH]e[MATH]f_S[MATH]V_S[MATH]e[MATH]V_*[MATH][MATH]f[MATH]n + 1[MATH]f[MATH]V_*=[MATH]k=-1[MATH]V_*[MATH][MATH]f[MATH]k[MATH][MATH][MATH]I[MATH]k[MATH]x^*[MATH]x=k[MATH]B[MATH]i[MATH]j[MATH]I[MATH]i[MATH][[MATH],y][MATH]j[MATH][[MATH],z][MATH]x[MATH]y[MATH]z[MATH]j[MATH]i[MATH]L[MATH]I[MATH][[MATH],w][MATH]L[MATH]k[MATH]x[MATH]I[MATH][MATH][MATH]G[MATH]x^*[MATH]X_1,X_2,,X_m[MATH]s_0,s_1,,s_m[MATH]1 pm[MATH]i=[ A , x][MATH]I_s_p[MATH]j[MATH]j[MATH]I_s_p[MATH]j[MATH]i[MATH]j[MATH][ X , x][MATH]I_s_p-1[MATH]X[MATH]j[MATH]x[MATH]i[MATH]i[MATH]j[MATH]j[MATH]i[MATH]G[MATH]x^*[MATH]j_1,j_2,,j_n[MATH]j_1[MATH][ A , x][MATH]j_k[MATH]j_k+1[MATH]1k<n[MATH]n<1[MATH]j_1 [MATH]j_2[MATH]p[MATH]q[MATH]j_p=j_q[MATH]ph< q[MATH]j_h [MATH]j_h+1[MATH]p<r<q[MATH]j_p=j_r[MATH]j_q=j_r[MATH]1in[MATH]s(j_i)[MATH]j_i[MATH]x[MATH]s(j_n)[MATH]s(j_1)[MATH]G[MATH]B[MATH]a[MATH]X_1,X_2,,X_p[MATH]s_0, s_1, , s_p[MATH]J=(j_1,j_2,,j_n)[MATH][MATH]s(j_n)[MATH]s(j_0)[MATH]I=(i_1,i_2,,i_m)[MATH]a[MATH]s(i_n)[MATH]s(i_m)[MATH]s=[[MATH]][MATH]j_1=[[MATH]][MATH]m=0[MATH]s[MATH][[MATH]][MATH][MATH] s[MATH]s(j_n)=p-1[MATH]m>0[MATH]i_1=[ C , a][MATH]^* a x[MATH]x^*[MATH]_h^*[MATH]h1[MATH]I[MATH]J[MATH]s[MATH]a[MATH]a[MATH]B[MATH]I[MATH]J[MATH]s[MATH]B[MATH]I=(i_1, i_2, , i_n)[MATH]x[MATH]P=p_1, p_2, , p_m[MATH]p_1 < p_2 < < p_m[MATH]1h m[MATH]j_p_h[MATH][[MATH]][MATH][MATH]h_0P[MATH]j_h_0[MATH][[MATH]][MATH]j_p_h=[[MATH]][MATH]1 h n_p[MATH]X_h[MATH]X_1 X_2 X_n_p[MATH][MATH]I[MATH]B[MATH]a[MATH]I[MATH]J[MATH]s[MATH]I[MATH]J[MATH]s[MATH]_J[MATH]J[MATH]I=0[MATH]s[MATH][ A B][MATH]_J B[MATH]I>0[MATH]_J _I[MATH]_I[MATH]I[MATH][MATH](I, J, s)[MATH]B[MATH]a[MATH]U[MATH]J[MATH]s[MATH]I[MATH]J[MATH]s[MATH](U, J, s)[MATH]B[MATH]s_1, s_2, , s_n)[MATH]I_1, I_2, , I_m[MATH]G[MATH]U=(u_1, u_2, , u_p)[MATH]a[MATH]s(u_p)[MATH]s(u_1)[MATH]J=(j_1, j_2, , j_q)[MATH][MATH]s(j_q)[MATH]s(j_1)[MATH]X_2 X_n[MATH]r[MATH]U=0[MATH]r=s[MATH]U>0[MATH]u_1=[ A_U _U _U, a][MATH]r=[ A_U _U _U][MATH]J[MATH]B[MATH]s(j_q)[MATH]s(j_q)[MATH]s(j_q)=0[MATH]j J[MATH]j[MATH][ A_j _j][MATH]J[MATH][MATH]J[MATH][MATH]s(j_q)[MATH]s(j_q)=n_i[MATH]n_i 0[MATH]s(j_q)=n_i + 1[MATH]q_g[MATH]s(j_q_g)=n_i[MATH]s(j_q_i)=n_i + 1[MATH]q_g < q_i q[MATH]j_q_g + 1[MATH][ A_g _g X_g _g][MATH]J\'=(j_1, j_2, , j_q_g)[MATH]J[MATH]U=0[MATH]s(j_q)=n-1[MATH]s[MATH]= + 1[MATH]J[MATH][MATH](U, J, s)[MATH]B[MATH]U 0[MATH]U > 0[MATH]t=s(u_p) - s(j_q)[MATH]t[MATH]t=1[MATH]s(u_h)=s(j_q) + 1[MATH]1 h p[MATH]u_1[MATH]1 < h p[MATH]u_h[MATH]u_p[MATH]U=1[MATH]s(j_q)=n - 1[MATH]U[MATH]B[MATH]J[MATH][MATH](U, J, s)[MATH]B[MATH]U[MATH]t=k[MATH]k 1[MATH]t=k+1[MATH]u_1[MATH]u_1=[ A_u_1 _u_1 _u_1, a][MATH]u_f=[ A_u_1 _u_1 _u_1][MATH]s(j_p)=s(u_1) - 1[MATH]j_p [MATH]u_f[MATH]u_1[MATH][ A_u_1 _u_1\' X_u_1 _u_1, a][MATH]_u_1\' X_u_1=_u_1[MATH]u_1[MATH]I_s(u_1)[MATH]X_u_1=X_n-t+2[MATH]U[MATH]J[MATH]_J=X_1 X_2 X_s(j_q)[MATH]U>0[MATH]U[MATH]_U=X_s(u_1) X_s(u_2) X_s(u_p-1) B[MATH]s(u_1)=s(j_q) + 1[MATH]s(u_p)=n[MATH]_J _U=B[MATH]a[MATH][MATH]^*a x[MATH]x^*[MATH]X_m[MATH]k[MATH]y^*[MATH]k[MATH][MATH]a[MATH][MATH]k=1[MATH]X_1 X_2 X_k-1[MATH][MATH]a[MATH][MATH]^*a x[MATH]x^*[MATH]X_m[MATH]k[MATH]X_k=a[MATH]a[MATH][MATH]L[MATH]a[MATH][MATH]k[MATH]X_n[MATH]L_d[MATH]a[MATH]_d[MATH]L_d[MATH]L[MATH] X_k _d[MATH]L_t[MATH]a[MATH]_t[MATH]L_t[MATH]L[MATH] X_k _t[MATH]L_1[MATH]L_2[MATH] X_k _d[MATH] X_k _t[MATH]P(L_1,L_2)[MATH]L_1, L_2, , L_n[MATH]L_i=(_i,a)[MATH](_i,k_i)[MATH]_i[MATH]a[MATH]i<n[MATH](_i,k_i)[MATH]_i[MATH]a[MATH]i=n[MATH]1j<n[MATH]1k<n[MATH]jk[MATH]P(L_j,L_j+1)=P(L_k,L_k+1)[MATH]_1[MATH]a[MATH]I[MATH]J[MATH]s[MATH]I[MATH]J[MATH]s[MATH]a[MATH]s=[[MATH]][MATH]D[MATH][MATH]a[MATH](I,J,s,D)[MATH][MATH]a[MATH][MATH]a[MATH]G=(,N,P,S,T,M)[MATH]x[MATH]B[MATH]a[MATH][[MATH],b][MATH]a[MATH][MATH]V_S[MATH]V_*[MATH][MATH]J[MATH]J[MATH]J[MATH]J[MATH]a[MATH][MATH][MATH]a[MATH]a[MATH]j_1,j_2,,j_n[MATH]a[MATH]V_P[MATH]V_P[MATH][MATH]V_P()[MATH]i_2, , i_n)[MATH]1kn[MATH]J=(j_1,j_2,,j_m)[MATH]D=(L_1, L_2, , L_p)[MATH]V_PZ[MATH]V_, PZ[MATH]V_PZ[MATH]V_P Z[MATH]_h=X_h,1 X_h,2 X_h,q_h[MATH]V_, P Z[MATH]s=[ D_s _s _s][MATH]V_P Z[MATH]L_V, PZ[MATH]L_V[MATH]V_P[MATH]L_V[MATH]L_V[MATH]N[MATH]N[MATH]N[MATH]a[MATH]N[MATH]G=(,N,P,S,T,M)[MATH]B a x[MATH]G[MATH](N)^*[MATH]x^*[MATH]B[MATH]a[MATH]y[MATH]G[MATH]^*y[MATH]y[MATH] A_1, A_2,, A_n[MATH][MATH]B[MATH]a[MATH][MATH] T\'[MATH] Y= A_i[MATH]1in[MATH] Y= B[MATH] Y= A_i[MATH]1in[MATH] Y= B[MATH] X > A[MATH] T\'[MATH]B a x[MATH]B a x[MATH]B[MATH]a[MATH] C_1, C_2, , C_n[MATH][MATH]1in[MATH] C_i[MATH] WU[MATH] DW[MATH]i[MATH] C_i[MATH] W= M_1, M_2,, M_m[MATH] M_1< M_2<< M_m[MATH]0im[MATH]0im[MATH] A_VV_i[MATH] A_V[MATH] A_1, A_2V_i[MATH] A_1A_V[MATH] A_2A_V[MATH]P( A_1)=P( A_2)[MATH] A_3, A_4V_i[MATH] A_3> A_V[MATH] A_4> A_V[MATH]P( A_3)=P( A_4)[MATH] A_VV_i[MATH] A_V[MATH] A_5[MATH] A_6[MATH]P( A_5)=P( A_6)[MATH]B a x[MATH]B[MATH]a[MATH] D_1, D_2X[MATH]L( D_1)=L( D_2)[MATH]B a x[MATH]B[MATH]a[MATH]B ax[MATH]G=(,N,P,S,T,M)[MATH]B[MATH]a[MATH][MATH]B[MATH]x^*[MATH]a x [MATH]a[MATH]a[MATH]G=(,N,P,S,T,M)[MATH]T_Ba[MATH]N[MATH]Ba[MATH]T_Ba[MATH] TT_Ba[MATH]B[MATH]a[MATH] C_1, C_2,, C_n[MATH][MATH] V_i[MATH][MATH]m= W=[MATH]0im[MATH] V_i 4P[MATH] V_i[MATH] U[MATH]4mP[MATH] X[MATH]2P[MATH]4mP + 2P[MATH]a[MATH][MATH]a[MATH]F_a[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH] TT_a[MATH] PT[MATH] A_1, A_2,, A_n[MATH]P[MATH] TF_a[MATH]a[MATH]a[MATH]P_0[MATH]a() -1[MATH]P_0[MATH]a[MATH]P[MATH]Y_a[MATH]Z_a[MATH]P_0[MATH]a[MATH]P[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH][MATH] TF_a[MATH]H_ T( P)T\'[MATH] P\'[MATH] T\'[MATH]B[MATH]B[MATH] T F_a[MATH]H_ T( P)W= M_1, M_2,, M_m[MATH]0im[MATH] V_i[MATH] V_A[MATH] U[MATH] A[MATH][MATH]0jm[MATH] N_1, N_2, N_3V_j[MATH] N_1[MATH] N_2[MATH] N_3[MATH]j[MATH]j[MATH][MATH]j[MATH] N_1[MATH] N_2[MATH] N_3[MATH]j[MATH] M_1[MATH] M_2[MATH] M_3[MATH]j[MATH] M_1< N_1[MATH] Q_1[MATH] Q_2[MATH] N_1[MATH] N_2[MATH] N_3[MATH] Q_3[MATH] N_3[MATH] Q_3[MATH][MATH] N_2< PN_3[MATH] N_2[MATH] Q_2[MATH] N_1[MATH] Q_2[MATH] PN_2[MATH] N_3[MATH] Q_3[MATH] N_2[MATH] Q_3[MATH] N_3[MATH] Q_3[MATH] N_1[MATH] Q_3[MATH] Q_3[MATH] Q_3[MATH][MATH][MATH]a[MATH][MATH] N_1[MATH][MATH]p[MATH]^p T=^p+1 T[MATH]^p T[MATH]cXN[MATH] K_1, K_2, K_3X[MATH]a[MATH][MATH]a[MATH][MATH]a[MATH] K_1, K_2, K_3[MATH]a[MATH]a[MATH] J_1, J_2, J_3[MATH] J_1< K_1[MATH] R_1[MATH] R_2[MATH] R_3[MATH] K_1[MATH] K_2[MATH] K_3[MATH] P_A[MATH] P_A[MATH] P_A= P[MATH][MATH] K_2< P_0K_3[MATH] K_2[MATH] R_2[MATH] K_1[MATH] R_2[MATH] P_0K_2[MATH] K_3[MATH] R_3[MATH] K_2[MATH] R_3[MATH] K_3[MATH] R_3[MATH] K_1[MATH] R_3[MATH][MATH]q[MATH]^q T=^q+1 T[MATH]^q T[MATH][MATH] P\'[MATH]T[MATH]H_T,a( P\')=H_ T,a( P)[MATH] P"[MATH]T[MATH]H_T,a( P")=H_ T,a( P)[MATH] T_p=^p^q T[MATH] P_p[MATH] T_p[MATH]H_ T_p,a( P_p)=H_ T( P)[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH][MATH]P_0[MATH]P_0[MATH]a[MATH]P[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH][MATH]G[MATH] G=(, N_G, P_G, S, T, M)[MATH]V_G(a)[MATH]P_G[MATH]a[MATH]P[MATH]V_G(a)[MATH][MATH]x^*[MATH]ax[MATH]V_ ax()=-1[MATH]x[MATH]ax[MATH]P( P)=[MATH][MATH]x^*[MATH]ax[MATH]B_ax()=n[MATH]nay()>V_ ax()[MATH]P( P)=[MATH]n^th[MATH] T_sF_a[MATH] P_0 T_s[MATH]H_ T_s,a( P)=n[MATH]Y_a( P)=n[MATH] T_1F_ a[MATH] T_1[MATH]n_0^th[MATH]n_0^th[MATH]a[MATH]a[MATH]x_1^*[MATH]ax[MATH]z^*[MATH] ax_1^* z[MATH] T_z[MATH]z[MATH]qT_1T_z[MATH] T_1[MATH]q( P)[MATH] T_z[MATH]N_ax_1( P)=n_0[MATH]Y_F_a,a( P)=n[MATH]ax[MATH]V_ax()=n+1[MATH]y^*[MATH] ax^*y[MATH]y[MATH]N_ax( P)=n+1[MATH]Y_F_a,a( P)=n+1[MATH]P_G[MATH]a[MATH]P[MATH]ax()>Y_a()[MATH]x[MATH]y^*[MATH]ax^*y[MATH]y[MATH]N_ax( P)>-1[MATH] B< P< A[MATH] T_0F_a[MATH] P_0[MATH] T_0[MATH]P( P)=[MATH][MATH]Y_a()=n+1[MATH]P_ G[MATH]P_G[MATH]a[MATH]P[MATH]T_B,a,G[MATH]G[MATH]U_B,a,G[MATH]A_B,a,G[MATH]D_B,a,G[MATH]S_B,a,G[MATH]P_B,a,G[MATH]G[MATH][MATH]a[MATH] T[MATH]T[MATH] P[MATH] T[MATH] P[MATH] P_A[MATH] P[MATH] P[MATH] P_A= P[MATH] P_A[MATH]Q x I x T[MATH]Q x(I)[MATH]I=(i_1, i_2, , i_n)[MATH]i_1=[[MATH], x][MATH]I=1[MATH]_1=[MATH]Q x (I) T i[MATH] T 1[MATH] A_1_1[MATH]I>1[MATH]_1 [MATH]_1=X[MATH]X (N)[MATH] T= Q x(i_1, i_2, , i_n-1)[MATH] P[MATH] T[MATH] T\'[MATH] T[MATH]X[MATH] P[MATH] P[MATH] T[MATH]Q x(I) T\'[MATH]I>1[MATH]_1 = [MATH] V= Q x(i_1, i_2, , i_n-1)[MATH] Q[MATH] V[MATH] V[MATH] V[MATH][MATH] P[MATH] Q[MATH] V[MATH]Q x(I) V[MATH]Q_x[MATH]x[MATH]a[MATH]Q_U[MATH][MATH]Q_A[MATH]q_U[MATH]q_U[MATH] T[MATH]I[MATH]I[MATH][[MATH],a][MATH][[MATH],a][MATH]q_U[MATH]I[MATH]I[MATH]I[MATH]Q_U[MATH]T^*[MATH]T[MATH]q_D (D T^*) (D T^*)[MATH]DD[MATH](L_1, L_2, , L_n)[MATH] Y=( T_1, T_2, , T_m)[MATH]q_D(D, Y)[MATH]D[MATH]q_D(D, Y)(D, Y)[MATH]D[MATH]L_m[MATH]L_m=(,k)[MATH]X_2 X_p[MATH]1 ik-1[MATH] V_i[MATH]X_i^*[MATH] V_k[MATH]a[MATH] Y_0=( V_1, V_2, , V_k)[MATH]D_0=(L_1, L_2, , L_n-1)[MATH]q_D(D, Y)(D_0, Y_0)[MATH]D[MATH]L_m[MATH]a[MATH]L_m=(,k)[MATH]X_2 X_p[MATH]1ik-1[MATH] V_i[MATH]X_i^*[MATH] Y[MATH][MATH] V_k[MATH] Y[MATH][MATH] Y_0=( V_1, V_2, , V_k)[MATH]D_0=(L_1, L_2, , L_n-1)[MATH]q_D(D, Y)(D_0, Y_0)[MATH]Q_DD T^*[MATH]Q_D(D)[MATH] Y T^*[MATH](D_, Y)[MATH]q_D[MATH](D, Y_)[MATH]D_[MATH] Y_[MATH]Q_D[MATH] T[MATH]G[MATH][MATH] M[MATH] T[MATH][MATH] N[MATH] N[MATH] P_1, P_2, , P_m[MATH] P_1 < P_2 < < P_m[MATH]y( N) ( P_1) ( P_2) ( P_m)[MATH] N[MATH]C( N)[MATH]S( N)[MATH]F_UTT[MATH](I,J,s,D)[MATH]B[MATH]a[MATH]RP T[MATH]R(I,J,s,D)[MATH] T_U=Q_U(I)[MATH] Y=Q_D(D)[MATH] T_A=Q_A(J)[MATH] T_U,0[MATH] T_U[MATH] N\'[MATH]C( N\') S( N\')[MATH]A_1, A_2, , A_n[MATH]A_1 ^* [MATH] N\'[MATH]A_2 ^* [MATH] N\'[MATH]I=0[MATH] T_0[MATH]B[MATH] T[MATH] Y[MATH] T[MATH]I>0[MATH] T_0[MATH] T_U,0[MATH] T[MATH] Y[MATH] T[MATH]R(I,J,s,D)= T_0[MATH]R[MATH] T[MATH]G[MATH][MATH] L=( K_1, K_2, , K_n)[MATH] K_1 < K_2 < < K_n[MATH]L( K_i) N[MATH]1 i n[MATH][MATH] T[MATH] T[MATH]G[MATH][MATH] N[MATH] T[MATH] A [MATH] N[MATH][MATH] N[MATH]R(U, J, s, D)[MATH]P_x[MATH]x[MATH]Q_x(P_x)[MATH]P_x=(p_1, p_2, , p_n)[MATH]n[MATH]n=0[MATH]n=1[MATH]Q_x(P_x)[MATH]n=2[MATH]q_x[MATH]p_2[MATH][ A , x][MATH]q_x[MATH]k 2[MATH]Q_x[MATH]n=k+1[MATH]k[MATH]q_x[MATH]q_x[MATH] T[MATH] P[MATH] P[MATH] P[MATH] T[MATH] P[MATH] P[MATH] P[MATH]Q_U(U)[MATH]Q_A(J)[MATH] Y=Q_D(D)[MATH] Y[MATH] Y[MATH]D=1[MATH] Y[MATH] Y=Q_D(D\')[MATH]D\'=k[MATH]D=k+1[MATH]D=(d_1, d_2, , d_m_D)[MATH]d_1[MATH]x[MATH]k[MATH]q_D[MATH] Y[MATH]q_D[MATH] Y_0[MATH] Y_0[MATH]y_D - 1[MATH] T_D[MATH] T_D[MATH] Y[MATH]q_D[MATH] Y[MATH] Y_0[MATH]R(U, J, s, D)[MATH] T_U[MATH] T_A[MATH] Y[MATH] T_U[MATH] T_A[MATH] Y[MATH](U, J, s, D)[MATH]R(U, J, s, D)[MATH]P_x=(p_1, p_2, , p_n)[MATH]x[MATH]n[MATH]Q_x(P_x)[MATH]P_x < 2[MATH]P_x=2[MATH]P_x[MATH] T_Q=Q_x(P_x)[MATH] A_2 _2[MATH]A_2[MATH]A_2 _1[MATH]P_x\'[MATH]x[MATH]P_x\'=k 2[MATH]Q_x(P_x\')[MATH]P_x=k+1[MATH] T_k[MATH]k[MATH]q_x[MATH]P_x[MATH]p_k+1[MATH]p_k+1[MATH][ A X , x][MATH] P_A[MATH] A X [MATH] P_A[MATH][MATH]q_x[MATH]X[MATH] P_A[MATH] T_0[MATH] P_A[MATH] T_0[MATH]X[MATH]X [MATH]p_k+1[MATH][ C , x][MATH]q_x[MATH][ D_A _A C _A, x][MATH] P_A[MATH]_A[MATH]q_x[MATH]_A C[MATH]_A C _A[MATH]Q_U(U)[MATH]Q_A(J)[MATH]D=(L_1, L_2, , L_n)[MATH]n=1[MATH]q_x[MATH]Q_x[MATH] Y=( T_1, T_2, , T_m)[MATH] T_1, T_2, , T_m-1[MATH] T_m[MATH]Q_D(D\')[MATH]D\'=k[MATH]D=k+1[MATH]k[MATH]q_D[MATH](,k_)[MATH] Y=( V_1, V_2, , V_m)[MATH] Y[MATH]n^th[MATH]q_D[MATH]p\'[MATH]p\'-1[MATH](,k_)[MATH]D[MATH]P(L_1,L_2)[MATH]q_D[MATH] V\'[MATH] Y[MATH] V\'[MATH] C_1, C_2, , C_k_[MATH]P(L_1, L_2)[MATH]Q_D(D)[MATH]Q_A(J)[MATH]Q_U(U)[MATH]Q_D(D)[MATH]R(U, J, s, D)[MATH]s=[ A_ _X_ _][MATH]U=0[MATH]X_= [ X_ _1, a][MATH] R(Q_U(U)) = X_[MATH]Q_U(U)[MATH] X_ _1[MATH] P_0,A[MATH]Q_A(J)[MATH] A_ _X__[MATH]Q_A(J)[MATH]_[MATH] P_0,A[MATH]_ X_[MATH]L_1[MATH]_[MATH]Q_D(D)[MATH]X_k_[MATH] V_k_[MATH]L_1[MATH] V_k_[MATH]X_k_=a[MATH] V_k_[MATH]P(L_1, L_2)[MATH] X [MATH]L(R( V_k_))=X_k_[MATH]R(U, J, s, D)[MATH]B a[MATH]Q_D(D)=( V_1, V_2, , V_n_D)[MATH] V_1, V_2, , V_n_D - 1[MATH][MATH] V_n_D[MATH]a[MATH]P_x=(p_1, p_2, , p_n)[MATH]x[MATH] T=Q_x(P_x)[MATH]q_x[MATH]p_i_1, p_i_2, , p_i_m[MATH]i_1 < i_2 < < i_m[MATH]Q_x(P_x)[MATH]X_1 X_2 X_m[MATH]P_x[MATH]Q_A(J)[MATH]Q_U(U)[MATH][MATH]R(U, J, s, D)[MATH]Q_A(J)[MATH]Q_U(U)[MATH] V_n_D[MATH] a[MATH]R(U, J, s, D)[MATH]R(U, J, s, D)[MATH]y L(G)[MATH]y[MATH] T=R(U, J, s, D)[MATH]N[MATH] T[MATH]A[MATH] F_A[MATH]z_A ^*[MATH] T_1[MATH] T[MATH] N N[MATH] F_L( N)[MATH]D[MATH] J_0[MATH] T_1[MATH] J_0[MATH]S( J_0)[MATH] J_0[MATH]S( J_0)=_0 Z_1 Z_2 Z_m[MATH]_0=C( J_0)[MATH] Z_ J_0[MATH]( F_Z_1, F_Z_2, , F_Z_m)[MATH] T_2[MATH] J_f D[MATH] Z_ J_f[MATH] J_f[MATH]z[MATH] T_2[MATH]z[MATH] T_2[MATH] T_2[MATH] T_2[MATH] N_z[MATH] N_z[MATH][MATH] N_z[MATH]L( N_z)[MATH] T_2[MATH] M_z[MATH]a[MATH] M_z > A[MATH] A[MATH]a[MATH] T_1[MATH] T_2[MATH] H_x[MATH] T_1[MATH] T_2[MATH] H_z[MATH]Q_A(J)[MATH]Q_A(J)[MATH] H_J[MATH]H_z[MATH]Q_A(J)[MATH] H_J[MATH]Q_D(D)[MATH]i^th[MATH] H_J[MATH]i[MATH] H_z[MATH] Q_D(D)[MATH] H_z[MATH] Q_D(D)[MATH] H_D[MATH] A_D[MATH]Q_D(D)[MATH]a[MATH]S( H_D) C( H_D)[MATH] H_D[MATH] A_D[MATH] A_D[MATH] H_D[MATH] T_2[MATH] T_2[MATH]Q_U(U)[MATH]F_U(Q_U(U))[MATH] T_1[MATH] T[MATH][MATH] N_[MATH]L( N_)[MATH] T_2[MATH]L( N_)[MATH] N_[MATH] T_2[MATH] T[MATH]R(U, J, s, D)[MATH]a[MATH](U, J, s, D)[MATH]R(U, J, s, D)[MATH] T=R(U, J, s, D)[MATH]D=(L_1, L_2, , L_n)[MATH]U=(u_1, u_2, , u_p)[MATH] A[MATH] B[MATH]R(U, J, s, D)[MATH]B[MATH]a[MATH] T[MATH] A[MATH] B[MATH] T[MATH] A[MATH] P_A,1[MATH] P_A,2[MATH] A[MATH] B[MATH] L( P_A,1)=L( P_A,2)[MATH]L_i_1[MATH]L_i_2[MATH] P_A,1[MATH] P_A,2[MATH] P_A,1[MATH] P_A,2[MATH]P(L_i_1, L_i_1+1)[MATH]P(L_i_2, L_i_2+1)[MATH] T[MATH] A[MATH]P_x=(p_1, p_2, , p_m)[MATH]x[MATH] T_x = Q_x(P_x)[MATH] T_x[MATH]P_x[MATH] U[MATH] T_x[MATH] W U[MATH] T_x[MATH] W= M_1, M_2, , M_p[MATH] M_1< M_2< < M_p[MATH]1 i m[MATH]A_1, A_2, A_3 V_i[MATH]i[MATH] A_1[MATH]q_x[MATH][ A_1 _1, x][MATH] A_2[MATH] A_3[MATH]A_1[MATH]A_2[MATH]A_3[MATH]u_r_1[MATH]u_r_2[MATH]u_r_3[MATH]r_3 < q < r_1[MATH]u_q [MATH]u_q+1[MATH]u_r_3=u_q=u_r_1[MATH]q=r_2[MATH] T[MATH] B[MATH]Q_U(U)[MATH]Q_A(J)[MATH] T[MATH] A[MATH]a[MATH]G=(, N, P, S, T, M)[MATH] (N)^*[MATH]B N[MATH]B[MATH]a[MATH]R[MATH]R[MATH] T_s[MATH]B[MATH]a[MATH] A[MATH] B[MATH]a[MATH]B[MATH]B[MATH] S[MATH] T_s[MATH] S[MATH] B[MATH] A[MATH] R_B[MATH] S[MATH] B[MATH] B[MATH] T_J[MATH] R_B[MATH] T_s[MATH] S[MATH] R_B[MATH] T\'_U[MATH] R_B[MATH] V_1, V_2, , V_n_v[MATH] S[MATH] R_B[MATH] Y=( V_1, V_2, , V_n_v)[MATH] T U[MATH] T U\'[MATH] N_[MATH] N_[MATH][MATH]_D[MATH]L( S)[MATH]q_D^-1colon (T^* (N)^*) D [MATH]q_D^-1( Y_D, _D)[MATH] Y_D=( V_D,1, V_D,2, , V_D,n_D)[MATH] V_D,n_D[MATH]L[MATH]_D[MATH]a[MATH](_D, n_D)[MATH]q_D^-1( Y_D, _D) (L)[MATH] V_D,n_D[MATH]L\'[MATH]_D[MATH]A[MATH](_D, n_D)[MATH] W_1, W_2, , W_m_D[MATH]L( R( V_D,n_D))[MATH]-1(( W_1, W_2, , W_m_D), = (L_1, L_2, , L_p_D)[MATH]q_D^-1( Y_D, _D) (L\', L_1, L_2, , L_p_D)[MATH] Y T^*[MATH] Y=( V_1, V_2, , V_m_I)[MATH] V_1, V_2, , V_m_I-1[MATH][MATH] V_m_I[MATH]a[MATH][MATH] V_m_I[MATH] V_m_I h_V[MATH]h_V = 1[MATH] V_m_I[MATH]a[MATH]q_D^-1( Y, )[MATH](, m_I)[MATH]q_D(q_D^-1( Y, ), ())=((), Y\')[MATH] Y\'=( Q_1, Q_2, , Q_m_I\')[MATH] Q_m_I\'=m_I[MATH]m_I-1[MATH] Y\'[MATH] V_m_I[MATH]a[MATH]h_V=1[MATH]h_V =k_V[MATH]k_V1[MATH]h_V=k_V+1[MATH]k_V[MATH]q_D^-1[MATH]q_D^-1( Y_h, _h)[MATH] Y_h[MATH]k_V[MATH]h_V[MATH](L_1, L_2, , L_h_V)[MATH]L_2[MATH][MATH] T_[MATH]q_D[MATH]( Y_h, _h)[MATH]k_V[MATH]q_D[MATH]q_D[MATH](L")[MATH] Y"[MATH]L_1[MATH]q_D[MATH]k_V[MATH] Y"= Y_h[MATH]q_D[MATH]L_1[MATH][MATH]a[MATH](I, m I)[MATH]X_2 X_[MATH] V_1\', V_2\', , V_m I - 1\'[MATH] V_m I\'[MATH] Y_h[MATH] X_m I [MATH] Y[MATH]L_0^*[MATH][MATH]T P T[MATH]q x ^-1 T P L_0^*[MATH] T x T P[MATH] T x[MATH] T x[MATH] A x x[MATH] T x \'[MATH] T x[MATH](i_1, i_2, , i_n r) = q x ^-1 ( T x \')[MATH] T x[MATH] N x[MATH]X x[MATH]m x - 1[MATH] N x[MATH] T x "[MATH] T x[MATH] N x[MATH] E Z_1 Z_2 Z_s x,[MATH] T y[MATH] T y[MATH]n y[MATH]n y[MATH]n y = 1[MATH] A y X y y[MATH]q x ^-1[MATH]q x[MATH]q x[MATH] A y X y y[MATH] T y[MATH]k y[MATH]k y 1[MATH]n y = k y + 1[MATH] T y[MATH] F y[MATH]L( F y)= C y y[MATH] T y\'[MATH] T y[MATH] F y[MATH]q x ^-1[MATH]q x ^-1[MATH] T y \'[MATH]Q x(q x ^-1 ( T y \')) = T y \'[MATH]q x ^-1( T y)[MATH][ y, x][MATH]q x ^-1 ( T y \')[MATH]Q x[MATH]Q x(q x ^-1 ( T y ))[MATH]Q x[MATH]q x ^-1 ( T y \')[MATH]Q x[MATH] T y\'[MATH]Q x[MATH][ y, x][MATH] T y\'[MATH] y[MATH]Q x(q x ^-1 ( T y ))= T y[MATH]L( F y)=X[MATH]X N[MATH] F y[MATH] D y y X y [MATH] F y[MATH]y[MATH] T y"[MATH] T y[MATH] F y[MATH]Q x(q x ^-1( T y))[MATH] T y"[MATH][ D y y X y , x][MATH] T y[MATH]Q A ^-1[MATH] T a[MATH]Q u ^-1[MATH]D q D ^-1 ( Y)[MATH]J Q A ^-1 ( T J)[MATH]L( S)[MATH] S[MATH] T s[MATH] S[MATH] B[MATH] B[MATH]i b[MATH]F [MATH]=i b[MATH]s=[ F ][MATH]U[MATH] T U[MATH]U ()[MATH] T U[MATH]U Q U ^-1( T U)[MATH] V R(U, J, s, D)[MATH] V s= T s[MATH]R[MATH] T J = Q A (J)[MATH] Y = Q D (D)[MATH] T U[MATH]B[MATH] T J[MATH] T U[MATH]B[MATH] T U[MATH] T U = Q U (U)[MATH] T U,0[MATH] T U \'[MATH]R(U, J, s, D)[MATH] T U \'[MATH] T J[MATH] Y[MATH] T U \'[MATH] T s[MATH]R[MATH] P_1=(U_1, J_1, s_1, D_1)[MATH]P_2=(U_2, J_2, s_2, D_2)[MATH]P_1[MATH]P_2[MATH]P_w[MATH]R_w[MATH]w[MATH]Q_w(P_w) = Q_w(R_w)[MATH]P_w = R_w[MATH]P_w R_w[MATH]n P[MATH]Q_w(P_w)[MATH]n R[MATH]Q_w(R_w)[MATH]n P = P_w[MATH]n R = R_w[MATH]P_w = R_w[MATH]P_w=(p_1, p_2, , p_n P)[MATH]R_w=(r_1, r_2, , r_n R)[MATH]i z[MATH]p_i z r_i z[MATH]p_k = [ A_P,k _P,k _P,k, w][MATH]r_k = [ A_ P,k _P,k _P,k, w][MATH]1 k n R[MATH]i z = 1[MATH]P = R = [MATH]Q_w[MATH](p_1)[MATH](r_1)[MATH]A_P, 1 _P, 1 = A_R, 1 _R, 1[MATH]i z > 1[MATH]_P, i z [MATH]p_i z - 1 = r_i z - 1[MATH]_P, i z = [MATH]p_i z[MATH]p_k z[MATH]k z < i z[MATH] N_P[MATH]Q_w(P_w)[MATH] N_R[MATH]Q_w(R_w)[MATH]p_k z[MATH]r_k z[MATH] N_P[MATH] N_R[MATH]p_i z = r_i z[MATH]_P, i z [MATH]p_i z - 1[MATH]r_i z - 1[MATH]p_i z - 1[MATH]p_i z - 1[MATH]p_i z[MATH]Y[MATH]r_i z - 1[MATH]p_i z[MATH]p_i z = r_i z[MATH]P_w=R_w[MATH]U_1=U_2[MATH]J_1=J_2[MATH]D_1=D_2[MATH]Q D(D_1) = Q D(D_2)[MATH]H[MATH]Q D(D_1)[MATH]D_1=(L_1, L_2, , L_H)[MATH]D_2=(L_1\', L_2\', , L_H\')[MATH]k d[MATH]L_k d L_k d\'[MATH]k d = 1[MATH]L_1[MATH]L_1\'[MATH]L[MATH]L_1=(L, Q D(D_1))[MATH]L_1\'=(L, Q D(D_2))[MATH] Q D(D_1)= Q D(D_2)[MATH]L_1=L_1\'[MATH]k d > 1[MATH] T d[MATH] T d\'[MATH]Q D(D_1)[MATH]Q D(D_2)[MATH] N d[MATH] T d[MATH] N d[MATH]D_1 - k d[MATH] N d\'[MATH] T d\'[MATH] N d[MATH] N d\'[MATH]L_k d - 1[MATH]L_k d - 1\'[MATH] A L L[MATH]L_k d[MATH]L_k d\'[MATH]L[MATH] N d[MATH]n d[MATH] N d\'[MATH]n d\'[MATH]n d=n d\'[MATH]L_k d = L_k d\'[MATH]D_1=D_2[MATH]s_1 s_2[MATH] P s[MATH]Q A(J_1)[MATH]L( P s) = A s s X s s[MATH] P s[MATH]s[MATH] P s[MATH] s[MATH]L_1[MATH]D_1[MATH]D_2[MATH]s[MATH](U_1, J_1, s_1, D_1)=(U_2, J_2, s_2, D_2)[MATH]R[MATH]G=(, N, P, S, T, M)[MATH]B[MATH]a[MATH]P[MATH]B[MATH][MATH]P=(U, J, s, D)[MATH]U=(u_1, u_2, , u_n)[MATH]J=(j_1, j_2, , j_m)[MATH]s=[ E ][MATH]D=(L_1, L_2, , L_p)[MATH]1 i_n n[MATH]u_i n [ A_i n _i n _i n, a][MATH]1 i_m m[MATH]j_i m [ C_i m _i m _i m][MATH]1 i_p p[MATH]L_i p (_i p, k_i p)[MATH] T[MATH]B[MATH]a[MATH] T U = Q U(U)[MATH] T J = Q A(J)[MATH] Y = Q D(D)[MATH]A [MATH]L_ V, P()=n V[MATH]n V = -1[MATH]M(B a, T, ) n V[MATH]n V -1[MATH]V () = n V[MATH]u_i u[MATH]U[MATH]l u=i u - _i u[MATH]_l u = [MATH]Q U[MATH]u_l u[MATH][MATH] T u[MATH] N u[MATH] N u[MATH] B[MATH] A[MATH]M(B a, T, ) = _l u + 1[MATH]V_,() = n V[MATH][MATH]1 i_n[MATH][MATH] T U[MATH] T U,0[MATH] N U,0[MATH] N U,0[MATH] T J[MATH] B[MATH] A[MATH]M(B a, T, ) = + 1[MATH]V_() = n V[MATH]1 k J m[MATH] C_k J _k J _k J = [MATH] T J[MATH] B[MATH] B[MATH]k\'=k J - _k J[MATH]Q A[MATH]j_k\'[MATH][MATH] T J[MATH] N J[MATH]_k J[MATH] N J[MATH]M(B a, T, ) = _k J + 1[MATH]V_() = n V[MATH]1 h V p[MATH]Q D[MATH]L_h V[MATH] T D[MATH][MATH]k_h V + 1[MATH]V_,() = n V[MATH]1 h V\' p[MATH]_h V\'= X_1 X_2 X_N D[MATH][MATH]1 l D N D[MATH]Q D[MATH]L_h V\'[MATH] P_D,[MATH][MATH] P_D,[MATH] B[MATH] A[MATH] B < P_D, < A[MATH]V_() = n V[MATH]E [MATH] T J[MATH] T U[MATH] Y[MATH] T U[MATH] - 1[MATH] T J[MATH] T U[MATH] Y[MATH] Y[MATH] A[MATH] > 1[MATH] P s[MATH] Q A[MATH] P s[MATH] A[MATH] + Y[MATH] A[MATH][MATH]M( B a, T, )=n T[MATH]n T = -1[MATH]n T > -1[MATH] P_[MATH] T[MATH]L( P_)=[MATH] P_[MATH] B[MATH] A[MATH] P_[MATH] B[MATH] P_[MATH] A[MATH] T U,0[MATH] P_[MATH][MATH]H_ T( P_)[MATH] + 1[MATH]u B[MATH]U[MATH] P_[MATH]V_() = + 1[MATH] P_[MATH] A[MATH] P_[MATH](U, J, s, D)[MATH] Y[MATH] P_[MATH] P Y[MATH] Y[MATH] T Y[MATH] P Y[MATH] T Y[MATH]h a[MATH]Q D[MATH]D[MATH] P Y\'[MATH]Q D[MATH]L_h a - 1[MATH] P Y[MATH] P T\'[MATH]Q D[MATH]L_h a - 1[MATH] P T\'[MATH]k_k a[MATH]P(L_h a, L_h a + 1)=[MATH] P_[MATH]J[MATH] q[MATH] P_[MATH] q[MATH] A[MATH] A[MATH]J[MATH] P_[MATH]Q A[MATH] P A[MATH]Q A(J)[MATH] P_[MATH] P A[MATH]n p[MATH] T J[MATH] P A[MATH] T[MATH]Q A[MATH]j_m - n p[MATH]j_m - n p[MATH]j_m[MATH]J > 1[MATH]J X J s[MATH] P_[MATH] P A[MATH] P c[MATH] P c[MATH]B[MATH] P c[MATH]Q U(U)[MATH] Y[MATH] Y=k_1[MATH]J[MATH] P_[MATH] P_[MATH]n T - 1[MATH]J J = [MATH] P_[MATH] B[MATH] A[MATH] B < P < A[MATH] P_[MATH] B[MATH] A[MATH] B < P < A[MATH] Q B[MATH] Q B[MATH]Q U(U)[MATH] T U, 0[MATH] T U, 0[MATH] P_[MATH] P U, 0[MATH] R B[MATH] Q B[MATH] B[MATH] T U, 0[MATH] R B[MATH]1 k_0 n[MATH] P_[MATH] B[MATH] A[MATH] Y=( V_1, V_2, , V_k_1)[MATH] P_[MATH] P Y, [MATH] Y[MATH] P Y, [MATH] V_1, V_2, , V_k_1 - 1[MATH] V_h Y[MATH] P Y, [MATH][MATH] V_h Y[MATH] P_[MATH] V_k_1[MATH] V_k_1[MATH]a[MATH] P Y, [MATH] A[MATH] P Y,[MATH][MATH]L V()= + 1[MATH]P[MATH]T P[MATH]R[MATH]k[MATH]k[MATH]k[MATH]k=1[MATH][MATH]V[MATH]V[MATH][MATH][MATH][MATH][MATH]'} | {'cs-0605104-2-0-0': '# Introduction', 'cs-0605104-2-1-0': 'Programming is the enterprise of fitting the infinitely subtle subjects of algorithms and interfaces into the rigid confines of a formal language defined by a few unyielding rules-is it any wonder that this process can be so difficult?', 'cs-0605104-2-1-1': 'The first step in this process it the selection of the language.', 'cs-0605104-2-1-2': 'As we go along in this enterprise, we might find that our selected language is inadequate for the task at hand; at which point we can: forge ahead with an imperfect language, we can attempt to address the problematic section in a different language, or we can jettison the language for another with its own limitations, thereby duplicating the effort already put into writing the program in the first language.', 'cs-0605104-2-1-3': 'With ever larger, more complex programs, we increasingly find that no single language is especially well-suited-yet if we try to use multiple languages, we face significant hurdles in integrating the languages, with rare exceptions.', 'cs-0605104-2-1-4': 'A fourth possibility presents itself: we could create a new programming language that contains all of the features we will ever need in any section of the program; aside from the fact that creating a general programming language is a monumental effort in and of itself, the resulting programming language will likely be a cumbersome monster.', 'cs-0605104-2-1-5': 'What we seek is a language that is at once general enough to suffice for very large programs, while also having specific features for each portion of the program.', 'cs-0605104-2-2-0': 'There are a great deal of mature programming languages in existence, each with its own advantages and disadvantages.', 'cs-0605104-2-2-1': 'None of these are the language we seek.', 'cs-0605104-2-2-2': 'Ideally, we would like to be able to take an existing programming language and-without having to duplicate the tremendous amount of effort which went into its creation and development, not to mention our own effort in learning it-mold it to our needs.', 'cs-0605104-2-3-0': 'We do not have time to survey the major languages, but programs in these languages do fit a general mold: programs must be syntactically well-formed, then they must be semantically well-meaning, and finally, they must specify a program that is free from run-time errors.', 'cs-0605104-2-3-1': 'Moving from the source code for a program to a run-time executable involves three phases: syntax analysis, semantic analysis, and code generation.', 'cs-0605104-2-3-2': 'The first two analysis phases are not separated in practice, but are performed in concert by a parser which is generated by a parser generator.', 'cs-0605104-2-3-3': 'The parser generator takes a grammar describing the syntax of the programming language, in addition to the semantic value of each production in the grammar, from which it produces a parser.', 'cs-0605104-2-3-4': 'If we had the source code to the compiler, we could change it to suit our purposes, producing a defnderived language.', 'cs-0605104-2-3-5': 'However, we must be careful if we do this, for changes to the code generator could produce binaries that lack compatibility with existing binaries.', 'cs-0605104-2-4-0': 'The aforementioned approach is not terribly common: its most glaring problem is that a program written in a derived language cannot be compiled by a "normal" compiler.', 'cs-0605104-2-4-1': 'An alternative is to make a new compiler wholesale-one that, rather than outputting a binary, outputs source code in an existing programming language; such a compiler is known as a defnsource-to-source translator.', 'cs-0605104-2-4-2': 'The practice of creating source-to-source translators is much more common that the practice of creating derived languages; two examples are the cfront compiler for C++ and a WSDL compiler for SOAP.', 'cs-0605104-2-4-3': 'These two examples illustrate an interesting point: the new language can share much with the target language, as is the case with cfront; or, the new language can share nothing with the target language, as is the case with a WSDL compiler.', 'cs-0605104-2-5-0': 'Creating a derived language is an attractive concept because we can directly leverage an existing implementation of a base language, but modifying any large program-the compiler, in this case-in an ad-hoc manner is not exactly an easy task.', 'cs-0605104-2-5-1': 'This approach becomes decidedly less attractive if we seek to radically alter the language: we will likely find that the code generator is tightly coupled with the parser, and that the facilities for creating abstract syntax trees have no more generality than is necessary for the original language.', 'cs-0605104-2-5-2': 'Creating a source-to-source translator, on the other hand, is an attractive concept because we can make a language that departs from the target language as much or as little as we want; however, perhaps too much information is lost in the conversion to the target language: data such as debugging information, higher-order typing, optimization hints, and details necessary for proper error handling are just a few of the things which might get lost.', 'cs-0605104-2-5-3': 'Another problem is what I refer to as the "language tower problem": say we start with a language L, then we create a source-to-source translator from L++ to L, then we create a source-to-source translator for Aspect-L++, then we create a source-to-source translator for Visual Aspect-L++, ad nauseam-in short, we end up with far too many parsers.', 'cs-0605104-2-6-0': 'We will take an approach somewhere between these two.', 'cs-0605104-2-6-1': 'We would like to develop a defnbase language that is general purpose enough to serve us in its unmodified form, yet can be modified at our pleasure.', 'cs-0605104-2-6-2': 'In light of our consideration of derived languages, we will create a general-purpose framework for abstract syntax trees that both the base language and any derived language can use to capture the full range of the semantics of a program.', 'cs-0605104-2-6-3': 'Also, we will not require someone wishing to create a derived language to create an entire grammar: we will allow modifications of the existing grammar.', 'cs-0605104-2-6-4': 'We have avoided most of the problems associated with source-to-source translation as well: the data.', 'cs-0605104-2-6-5': 'As an example consider: run-time metadata (like profiling and/or debugging data), data type, optimization hints, and error messages; none of these are likely to be present in the binary if source-to-source translators are used.', 'cs-0605104-2-6-6': 'Since we are making it easy to modify the language, we would expect that the language tower problem would be exacerbated, but this is hardly the case, for there is only ever a singular parser.', 'cs-0605104-2-7-0': 'We pause to note that there must be some way of specifying the semantic actions of a production.', 'cs-0605104-2-7-1': 'We can assume that these actions are specified in a programming language, probably the base language itself, and that the parser has an interpreter for that language included in its implementation.', 'cs-0605104-2-8-0': 'The code generator only understands so much of what is potentially in an abstract syntax tree.', 'cs-0605104-2-8-1': 'Everything else-the debugging, type, optimization, and error data-which gets added to the tree must be, to a large extent, ignored by the code generator.', 'cs-0605104-2-8-2': 'However, these data-we will call them defnextended semantics data-are not valueless; thus, we will allow additional analysis phases to be performed on the abstract syntax tree between that parsing and the code generation phases.', 'cs-0605104-2-8-3': 'Here again, an interpreter embedded in the parser will be invaluable.', 'cs-0605104-2-9-0': 'How might a language like this be used?', 'cs-0605104-2-9-1': 'We can use it to add gross language features, for example object-oriented or aspect-oriented support.', 'cs-0605104-2-9-2': 'Or we could add more behind-the-scenes features, improving for example, the optimizer.', 'cs-0605104-2-9-3': 'If we know that we are using a particular library, we can give first-class syntactic support to common patterns-for example, we could support monitors, as Java does with the synchronized keyword.', 'cs-0605104-2-9-4': 'Finally, we could create a modified grammar to eliminate repetitive code, using the modifiability of the language as a sort of macro processor.', 'cs-0605104-2-10-0': 'We will allow the parser to modify itself during parsing.', 'cs-0605104-2-10-1': 'From here on, we will assume that a parser operates strictly left-to-right.', 'cs-0605104-2-10-2': 'No longer can we treat the syntax analysis and semantic analysis phases as entirely separate, even conceptually, for some part of a file may define the syntax and semantics of the remainder of the file.', 'cs-0605104-2-11-0': 'The study of formal languages has produced many interesting classes of languages: regular, context-free, context-sensitive, and recursive-enumerable being the best known.', 'cs-0605104-2-11-1': 'If [MATH] is a class of languages, then the set of defntransformative [MATH] languages are those languages whose strings [MATH] can be decomposed as [MATH], such that [MATH] is a substring of an element of one of the languages in [MATH], which we term the [MATH] defninstantaneous language; further, [MATH] specifies the instantaneous language [MATH].', 'cs-0605104-2-12-0': 'Our goal in the present work will be to develop a method of parsing a useful class of transformative languages.', 'cs-0605104-2-12-1': 'Our parser will operate much like a classical parser, except that, as it moves over the boundary between [MATH] and [MATH], it will modify itself-more precisely, it will modify its grammar, and then its parsing tables.', 'cs-0605104-2-12-2': 'Since we are dealing with a self-modifying parser, we would run into problems if the parser were to backtrack from [MATH] to [MATH]-not insurmountable problems, to be sure, but we will find a satisfactory non-backtracking method of parsing that does not have these problems.', 'cs-0605104-2-13-0': '## Applications of Transformative Parsing', 'cs-0605104-2-14-0': 'Let us say that we need to write a graphical program in a language much like Java which can access both a web service and a database; let us assume that we do not have any visual rapid-development tools.', 'cs-0605104-2-14-1': 'We must write a lot of GUI code like "make a window, put a layout in the window, put the following controls in the layout: , add a toolbar to the window, add an item to the toolbar with the label \'x\', set the callback object to \'y"\' etc.', 'cs-0605104-2-14-2': 'We must write a lot of database like "parse a query, bind the following variables (), execute the query, create a cursor, advance the cursor, get the first column, get the second column" etc.', 'cs-0605104-2-14-3': 'We must write a lot of web service code like "create a procedure call, marshal the input, call the procedure, demarshall the output, handle any exceptions" etc.', 'cs-0605104-2-14-4': 'The GUI, database, and web service functionality is most likely handled by a library.', 'cs-0605104-2-14-5': 'Would that each library added syntax constructs for the operations it provides.', 'cs-0605104-2-14-6': 'We could declare the GUI with code like', 'cs-0605104-2-15-0': "window layout...; toolbar item label = 'x'; action = y", 'cs-0605104-2-16-0': 'The interesting thing is that the -y- identifier is bound to the correct lexical scope.', 'cs-0605104-2-17-0': 'We could process our query with code like', 'cs-0605104-2-18-0': 'query(select col1, col2 from t1 where col=[MATH](, N, P, S)[MATH](, N, P,S,T,M)[MATH]k[MATH]k0[MATH]k[MATH]k[MATH]k 1[MATH](,N,P,S)[MATH][MATH]N[MATH]P[MATH]S[MATH][MATH]G=(,N,P,S)[MATH][MATH]Ntimes(N)^*[MATH]P[MATH]A[MATH](A,)P[MATH][MATH]^*[MATH]G=(,N,P,S)[MATH](N)^*[MATH]FIRST_k()[MATH]y^*[MATH]y=k[MATH]_G^* yx[MATH]x^*[MATH]FIRST()[MATH]FIRST_1()[MATH]FIRST()[MATH][MATH]G=(, N, P, S)[MATH]S\'[MATH]N[MATH](, N S\', P S\' S , S\')[MATH]G[MATH]k[MATH]G=(N,,P,S)[MATH]G\'=(N\',,P\',S\')[MATH]G[MATH]k[MATH]k 0[MATH]S\' _G\'^* A w _G\' w,[MATH]S\' _G\'^* B x _G\' y,[MATH]FIRST_k(w)=FIRST_k(y)[MATH]A y=B x[MATH]Z_k (N )[MATH]K[MATH]x G[MATH]x[MATH]x G[MATH]x[MATH]S\' S[MATH]a[MATH]a[MATH]k[MATH]action=shift m[MATH]a[MATH](m, a)[MATH]action=reduce " A"[MATH][MATH](k\', X)[MATH](goto, A)[MATH]action=error[MATH]action=accept[MATH]G=(,N,P,S)[MATH][A,a][MATH][MATH]a[MATH][MATH][MATH][MATH]a[MATH][MATH][MATH]a[MATH][S\'S, [MATH]][MATH]I_0[MATH][AB,a][MATH]B[MATH][B,b][MATH]b[MATH]FIRST(a)[MATH]I_k[MATH]I_m[MATH]X(N)[MATH]I_k[MATH]X[MATH]I_m[MATH]m=goto[MATH][AX,a]I_k[MATH][AX,a]I_m[MATH]I_k[MATH]a[MATH][Aa ,b]I_k[MATH]action=shift m[MATH]m=goto[MATH][A,a]I_k[MATH]action=reduce "A"[MATH]S\' S[MATH]goto[MATH]action[MATH]action[MATH]error[MATH]G=(,N,P,S)[MATH]x^*[MATH]xG[MATH]S[MATH]X[MATH]Y_1,Y_2,,Y_n[MATH]XY_1Y_2Y_n[MATH][MATH][MATH]x[MATH]x[MATH]x[MATH]xG[MATH][MATH]( T)[MATH]B [MATH] M_1, M_2, , M_m[MATH]( N)=[MATH] ( N)[MATH][MATH][MATH]x[MATH]x[MATH] N_1[MATH] N_2[MATH] N_1[MATH] N_2[MATH]T[MATH] AB[MATH] A= B[MATH]C T D[MATH]G=(a, b, S, P, S)[MATH]P=[MATH][MATH]aabb[MATH]G T[MATH]T[MATH]L(G T)[MATH][MATH][MATH][MATH][MATH]M[MATH]_1[MATH]_2[MATH]_3[MATH]_1, _2 [MATH]_3 T[MATH]w_2^*[MATH]z_2^*[MATH]M[MATH]w(G T)[MATH]M[MATH][MATH]^*[MATH][MATH](w, )[MATH][MATH]z[MATH](,N,P,S,T,M)[MATH]G=(,N,P,S,T,M)[MATH]G_0=(, N, P, S)[MATH], (N)^*[MATH] _G nt^* [MATH]_G t^* [MATH]_G[MATH]_G nt[MATH]G=(, N, P, S, T, M)[MATH](, N, P, S)[MATH]G[MATH]G=(,N,P,S,T,M)[MATH](N,P_+,P_-)[MATH]P_-[MATH]P_+[MATH]G[MATH] A[MATH][MATH]G[MATH]T[MATH]T P[MATH]P_- T = [MATH]_e=(, , )[MATH](N)^*[MATH]G[MATH]a x[MATH]a[MATH]x^*[MATH][MATH]a[MATH]G=(, N, P, S, T, M)[MATH]G\'=(, N\', P\', S, T, M)[MATH]M[MATH]_1[MATH]_2[MATH]_3[MATH]g T ^*[MATH]G[MATH]L(G T)[MATH], (N)^*[MATH]x, z ^*[MATH]u, w _2^*[MATH](z, w, G\')[MATH](x, u, G)[MATH](x, u, G) (z, w, G\')[MATH]z=S[MATH]z _G nt^* x[MATH]G\' = G[MATH]z = B z G z G^* yz = x,[MATH]y ^*[MATH](N)^*[MATH][MATH]b[MATH]G\'[MATH]b=FIRST(z)[MATH]M[MATH](y, u, g)[MATH](w, )[MATH]G\' = G[MATH]G[MATH]x[MATH]G[MATH]c,d[MATH]S,A,B,C,D[MATH]S[MATH][MATH]M[MATH]G_1[MATH]G_2[MATH]G[MATH]_0[MATH]_1[MATH]_2[MATH][MATH]M[MATH]G[MATH]M[MATH]_0[MATH]G_1[MATH]M[MATH]_1[MATH]G_2[MATH]M[MATH]_2[MATH]c[MATH]u_i[MATH][MATH]i[MATH]c[MATH]G[MATH]n[MATH]n[MATH][MATH][MATH][MATH][MATH]G=(,N,P,S,T,M)[MATH](N)^*[MATH]^*[MATH][MATH]B[MATH]B a x[MATH](N)^*[MATH]a[MATH]x^*[MATH]a[MATH]B[MATH]y ^*[MATH]B a x^*y[MATH]y^*[MATH]B[MATH]a[MATH]B a x[MATH]n[MATH]N_, T( P)[MATH]N_, T( P) = n+1[MATH]_1, [MATH]_2,,[MATH]_n[MATH]_i[MATH]N_, T( P)=i[MATH] B< P< A[MATH]N_, T( P)=n+1[MATH]N_, T( P) = -1[MATH]P[MATH]V_[MATH][MATH][MATH]a[MATH]V_a()[MATH]V_a x()[MATH]x[MATH]a x[MATH](N)^*[MATH]V_()=-1[MATH][MATH][MATH]V_()=i[MATH][MATH]A [MATH]0 < i [MATH]i[MATH][MATH][MATH][MATH]V_()=i[MATH][MATH]A[MATH]i = + 1 [MATH][MATH][MATH]G=(,N,P,S,T,M)[MATH][MATH](N)^*[MATH]^*[MATH][MATH]P[MATH]BY_1 Y_2 Y_m[MATH][MATH][MATH]P[MATH][MATH]P[MATH]AX_1 X_2 X_n[MATH]V_()n[MATH]1iV_()[MATH]X_i=Y_i[MATH]A=B[MATH]_ [MATH]V_()=n+1[MATH]phi[MATH]P\'[MATH][MATH]N\' N[MATH]P\'[MATH]P[MATH]_ [MATH]P\'[MATH][MATH]P[MATH]G=(,N,P,S,T,M)[MATH]G[MATH] G=(,N_G, P_G, S, T,M)[MATH]A[MATH]a[MATH](N)^*[MATH]BN[MATH]a[MATH]x^*[MATH]A a x[MATH]G[MATH]P_G[MATH]A a x[MATH]P[MATH][MATH]B a[MATH]G[MATH]B a[MATH]V_G(B a)[MATH]_e=(,,)[MATH]V_G(B a)[MATH][S\'S,[MATH]][MATH][A,a][MATH]a[MATH][Aa,b][MATH][Aa,b][MATH]a[MATH][A B , c][MATH]C [MATH]C[MATH]C[MATH]B[MATH]a[MATH]B[MATH][MATH]a[MATH]B[MATH][MATH]a[MATH]B[MATH]a[MATH][MATH]G=(, N, P, S, T, M)[MATH]G\'=(, N\', P\', S, T, M)[MATH]M[MATH]_1[MATH]_2[MATH]_3[MATH]g T ^*[MATH]G[MATH]L(G T)[MATH], (N)^*[MATH]x, z ^*[MATH]u, w _2^*[MATH](z, w, G\')[MATH](x, u, G)[MATH](x, u, G) (z, w, G\')[MATH]z=S[MATH]z _G nt^* x[MATH]G\' = G[MATH]z = B z G z G^* yz = x,[MATH]y ^*[MATH](N)^*[MATH][MATH]b[MATH]G\'[MATH]b=FIRST(z)[MATH]M[MATH](y, u, g)[MATH](w, )[MATH]G\' = G[MATH] G[MATH]G_0=(,N,P,S,T,M)[MATH]x^*[MATH]w_0=[MATH]_0=x[MATH]k>0[MATH]xG[MATH]x[MATH]G[MATH]L ^*[MATH]L[MATH]T[MATH]L[MATH]G=(,N,P,S,T,M)[MATH]L[MATH]a_1, a_2, , a_n[MATH]N=S, A, B[MATH]T=S A[MATH]M[MATH]M[MATH]x ^*[MATH]x L[MATH]_e[MATH]x L[MATH], P)[MATH]x L[MATH]x G[MATH]x L[MATH]x G[MATH][MATH]G[MATH]A[MATH]c[MATH]V_G(A c)[MATH]V_G(B c)[MATH]G=(,N_G, P_ G, S, T, M)[MATH]V_G(B c)[MATH]x^*[MATH]B c x[MATH][MATH]B c x[MATH]G[MATH][MATH] Bcx[MATH][MATH]P_G[MATH] _B c x [MATH][MATH]A c x[MATH]y^*[MATH] T_y[MATH]y[MATH]B[MATH]c[MATH] B < P < C[MATH]A[MATH] B < Q < C[MATH] P U[MATH] P W[MATH]( P)[MATH]Acx[MATH] PW[MATH] ( P)[MATH]Acx[MATH]N_Acx, T_y( P)=N_Bcx( P)[MATH] P W[MATH]N_Acx, T_y( P)=N_Bcx( P)[MATH] P W[MATH]( P)[MATH]( P)[MATH]N_Acx, T_y( P)=N_B cx, T_y( P)[MATH] UW[MATH] PU[MATH]N_Bcx, T_y( P)>0[MATH]N_Acx, T_y( P)=N_Bcx, T_y( P)[MATH]N_Acx, T_y > 0[MATH] PU[MATH][MATH]Bcx[MATH]V_G(A c)[MATH]P_G[MATH]_A c xphi[MATH]_B c xphi[MATH]V_G(B y)[MATH]FIRST(y)=FIRST(w)[MATH]C ^*_G A z ^*_G B y z[MATH]C ^*_G A z[MATH]C _G A z[MATH][MATH][MATH]G[MATH]x^*[MATH]^*_G x[MATH]C ^*_G A x [MATH]n[MATH]n>1[MATH]C _G^* A z[MATH]n[MATH]D_G^* A u[MATH]n[MATH]V_G( B y)[MATH][MATH]G[MATH]t^*[MATH]_G^* t[MATH]V_G(Aa)[MATH]A x _G A x[MATH][MATH]G[MATH]n+1[MATH]= C[MATH][MATH][MATH]A[MATH]x[MATH][MATH][MATH]G[MATH][MATH]G[MATH]G[MATH]V_G(B a)[MATH]A a _G B a[MATH][MATH]G[MATH]G[MATH]n 1[MATH]n+1[MATH][MATH]A a _G C a[MATH]G[MATH]y ^*[MATH]V_G(A a)[MATH]AB_G Aax[MATH][MATH]G[MATH]a[MATH][MATH]G[MATH]z ^*[MATH]_G^* z[MATH]n[MATH]n 1[MATH]n+1[MATH][MATH]a[MATH]a [MATH][MATH]a[MATH][MATH]G[MATH]w[MATH]_G^* w[MATH]a[MATH][MATH]C[MATH]a[MATH][MATH]G[MATH]C _G^* at[MATH]t ^*[MATH] AC_G^* Aat[MATH]t\', s\' ^*[MATH][MATH]u, v ^*[MATH][MATH]E[MATH]= A V_G(E a)[MATH][MATH][MATH]_E aphi[MATH]n[MATH]G=(, N, P, S)[MATH][MATH]a[MATH]G[MATH]a[MATH]n[MATH][MATH]_1,_2,,_n(N)^*[MATH]_n a[MATH]0 k < n[MATH]_k a[MATH][MATH]G=(,N,P,S)[MATH],(N)^*[MATH]beta[MATH][MATH][MATH]alpha[MATH]^*[MATH]^* [MATH]G[MATH] G[MATH] G[MATH] G[MATH] G[MATH]G[MATH]A[MATH]a[MATH]V_G(A a)[MATH]A[MATH]a[MATH]G[MATH]G[MATH]x^*[MATH]Aax[MATH]G[MATH]w^*[MATH]Aaw[MATH]G[MATH]t ^*[MATH]A a y _G^* t[MATH]t[MATH]A[MATH]a[MATH]A[MATH]a[MATH]PC X [MATH]X=a[MATH]X[MATH]Aax[MATH]P z[MATH] _G^* A[MATH] _G^*[MATH]V_G(Aa)[MATH]PC X [MATH]G[MATH]X^*_Gau[MATH]u^*[MATH]P v[MATH]v^*[MATH]X=a[MATH]X[MATH]X=D[MATH]C[MATH]C E[MATH]V_G(Ea)[MATH]G[MATH]B [MATH]G[MATH]V_G(Ba)[MATH][MATH]a[MATH]G[MATH]a[MATH]n[MATH][MATH]G[MATH]a[MATH]n-1[MATH]B[MATH]G[MATH]n[MATH]n=1[MATH]_G B[MATH]a[MATH]B a[MATH]G[MATH]B a[MATH]G[MATH]k 1[MATH]k=n + 1[MATH]n[MATH]B _G^* [MATH]V_G(C a)[MATH]a[MATH]n-1[MATH]B[MATH]G[MATH]x[MATH]G[MATH]G[MATH]G[MATH]G=(,N,P,S,T,M)[MATH]x[MATH]N^*[MATH]e[MATH]xG[MATH]w_)[MATH]a[MATH]x[MATH]s[MATH]action=shift[MATH]a=[MATH][MATH]action=shift[MATH](goto,a)[MATH]w_a[MATH]a[MATH]action=reduce [MATH]T[MATH]G[MATH]w_goto, A)[MATH]action=error[MATH]G[MATH]G=(,N,P,S,T,M)[MATH][MATH]w_[MATH]^*[MATH]z_[MATH][MATH]G[MATH][MATH]z_[MATH][MATH][MATH](w_, z_, G)[MATH](w_, z_, )[MATH]Delta G [MATH]G=G[MATH]G[MATH][MATH](s, X)[MATH]i[MATH][MATH]X[MATH]i^th[MATH][MATH]s[MATH](goto, a)[MATH]G[MATH][MATH]z_[MATH]goto[MATH]G=(, N, P, S)[MATH]X_i (N)[MATH]1 i n[MATH]g (N)^* Z_k[MATH][MATH]G[MATH]g()[MATH]I_g()[MATH][MATH][MATH]g()=0[MATH][S\'S, [MATH]][MATH]I_0[MATH]I_0[MATH]g[MATH]n[MATH]n0[MATH] = n + 1[MATH]X[MATH]x[MATH]X x[MATH]X x[MATH][MATH]I_g(= X y z[MATH]A_0 = X[MATH]w_0 = y[MATH]w_m ^*[MATH]A_M w_M z[MATH]= m+1A_m _m[MATH]0 m < M[MATH]I_g(g(X)[MATH]g[MATH]G=(, N, P, S)[MATH]X_i[MATH]1 i n[MATH]X_1 X_2 X_n[MATH]G[MATH]P ( N)^* ( Z_k^* ( N )) ^*[MATH]G=(, N, P, S)[MATH]= X_1 X_2 X_n[MATH]G[MATH]1 i n[MATH]X_i N[MATH]X_i + 1 X_i + 2 X_n ^*[MATH]X_1 X_2 X_i[MATH]x = X_i + 1 X_i + 2 X_n[MATH][MATH][MATH]x[MATH][MATH]\'[MATH]P()[MATH]G_0=(, N, P, S, T, M)[MATH]x G[MATH]x[MATH]1 i n[MATH]_n[MATH]B[MATH][MATH]x[MATH][MATH]x[MATH]G_i[MATH]n - i = 0[MATH][MATH][MATH][S\' S , [MATH]][MATH]n - i > 0[MATH]n - i[MATH]n - i = 1[MATH]y ^*[MATH]x _G_i^* y[MATH]y[MATH][MATH]x[MATH]j^th[MATH]j > 0[MATH]n - i = j + 1[MATH]_G_i^* i + 1[MATH][MATH][MATH]G_i + 1[MATH][MATH]G=(, N, P, S, T, M)[MATH][MATH]a[MATH](, N, P, S)[MATH]a[MATH]n[MATH][MATH]a[MATH]a[MATH]n[MATH][MATH]a[MATH]n[MATH]n=0[MATH]a[MATH]k 0[MATH]n=k+1[MATH]G[MATH]G=(, N, P, S, T, M)[MATH] G[MATH]x ^*[MATH]x G[MATH]G[MATH][MATH]x[MATH]i 0[MATH]_i = B y[MATH](N)^*[MATH]B N[MATH]y ^*[MATH]B [MATH]y[MATH]y=1[MATH]y[MATH][MATH]m[MATH]y[MATH]m[MATH]k 1[MATH]y = k + 1[MATH]y[MATH]k[MATH]x G[MATH] G[MATH][MATH]V_()[MATH](,k)[MATH]V_something[MATH]k[MATH][MATH]G=(,N,P,S,T,M)[MATH]), (s_1, X_1), , (s_m, X_m))[MATH]a[MATH]V P[MATH](C , i)[MATH]C P[MATH]i[MATH]G[MATH]I_0, I_1, , I_p[MATH]V T[MATH]V_P[MATH](s, B)[MATH]j=[AB ,b][MATH]I_s[MATH]G[MATH]j[MATH]a[MATH]V F[MATH]f[MATH]V T = V T V F[MATH]f[MATH]G[MATH]j[MATH]a[MATH]V A[MATH]V T = V TV A[MATH]P[MATH]V x[MATH]V P=(,V x() P[MATH]V P[MATH][MATH][MATH][MATH][MATH]^*[MATH]V F[MATH]f[MATH]V F[MATH]f[MATH]G=(,N,P,S,T,M)[MATH]), (s_1, X_1), , (s_m, X_m))[MATH]a[MATH]j=[AB,b][MATH]V F[MATH](C , i)[MATH]C P[MATH]i[MATH]f[MATH]G[MATH]I_0, I_1, , I_p[MATH]f[MATH]V Z[MATH]V F[MATH]Y_2 Y_n[MATH][MATH]AB [MATH]i[MATH]n[MATH]Y_i=a[MATH](,+1+i)[MATH]V Z[MATH]V F[MATH]Y_i[MATH]V F[MATH]Y_i[MATH]G[MATH]Y_i[MATH]a[MATH][MATH]V E[MATH]f_N[MATH]e[MATH][MATH]e[MATH]f_N[MATH]V Z=V ZV E[MATH]f_N[MATH](, +1+i)[MATH]V Z[MATH]V F[MATH]V Z=[MATH]f[MATH]e[MATH]V F[MATH]f[MATH](,B+1)[MATH]V Z[MATH]+1[MATH]s[MATH]J=j_0[MATH]j_0[MATH][AB,b][MATH]I_s[MATH]J[MATH][[MATH],c][MATH]J[MATH][[MATH],d][MATH]I_s[MATH]J[MATH]J[MATH][[MATH],e][MATH]J[MATH][MATH](,+1)[MATH]V Z[MATH]G[MATH]a[MATH][[MATH],e] [MATH]j[MATH]V P[MATH]f P[MATH]f P[MATH]f[MATH]V Z=V ZV P[MATH]V Z=[MATH]V F[MATH]f[MATH]j[MATH]j[MATH]j[MATH]G=(,N,P,S,T,M)[MATH]), (s_1, X_1), , (s_m, X_m))[MATH]a[MATH]j=[AB,b][MATH]V A[MATH](C ,i)[MATH]C P[MATH]i[MATH]G[MATH]I_0, I_1, , I_p[MATH]+1[MATH]s[MATH]J=j[MATH]J[MATH][[MATH],c][MATH]J[MATH][[MATH],d][MATH]I_s[MATH]J[MATH]J[MATH]k=[[MATH],d][MATH]J[MATH][MATH](,+1)[MATH]V A[MATH][MATH]G[MATH]a[MATH]k[MATH]j[MATH]V A\'[MATH]V A=V AV A\'[MATH]V A[MATH]V S[MATH][MATH]e[MATH]X^*[MATH]f[MATH]X^*ax[MATH]x^*[MATH]G=(,N,P,S,T,M)[MATH]X[MATH]N[MATH][MATH]a[MATH]V S[MATH](,n)[MATH][MATH]e[MATH]f[MATH]e[MATH]f[MATH]X[MATH]V S=[MATH]X=a[MATH]f[MATH]V S[MATH][MATH]f[MATH]e[MATH]X [MATH]X [MATH]X [MATH][MATH]G[MATH][MATH][MATH]a[MATH]V_*[MATH]_*[MATH]f_*[MATH]k[MATH]k=+1[MATH]e[MATH]S=V SV_*[MATH](,k)[MATH]V S[MATH]V S[MATH][MATH]f[MATH]e[MATH]V_*[MATH][MATH]k[MATH]^*[MATH]k=+1[MATH]^*ax[MATH]x^*[MATH]1k[MATH]k=-1[MATH]^*[MATH]^*ay[MATH]y^*[MATH]k=+1[MATH]f[MATH]G=(,N,P,S,T,M)[MATH][MATH](N)^*[MATH][MATH]a[MATH]V_*[MATH](,n)[MATH][MATH]f[MATH]k[MATH]-1k+1[MATH]k=-1[MATH]f[MATH]X_2 X_n[MATH]i[MATH]n[MATH]G[MATH]X_i[MATH][MATH]V S[MATH]S[MATH]e[MATH]f S[MATH]S[MATH]f S[MATH]e[MATH]k=i[MATH]f S[MATH]f[MATH]V_*=V_*V S[MATH]e[MATH]V_*[MATH][MATH]f[MATH]n + 1[MATH]f[MATH]V_*=[MATH]k=-1[MATH]V_*[MATH][MATH]f[MATH]k[MATH][MATH][MATH]G=(, N, P, S, T, M)[MATH](, N, P, S)[MATH]k[MATH]I[MATH]i=[[MATH], y][MATH]j=[[MATH], z][MATH]I[MATH]j [MATH]i[MATH]G=(, N, P, S, T, M)[MATH](, N, P, S)[MATH]k[MATH]G[MATH]I_0, I_1, , I_n[MATH]i[MATH]j[MATH]1 p m[MATH]i I_s_p[MATH]j [MATH]i[MATH]j[MATH]I_s_p - 1[MATH][A X p , z][MATH]i[MATH][X p , y][MATH]j [MATH]i[MATH]G=(, N, P, S, T, M)[MATH](, N, P, S)[MATH]k[MATH]G[MATH]I_0, I_1, , I_n[MATH]j_1, j_2, , j_q[MATH]_1, _2, , _q 0, 1, , n[MATH]1 i < q[MATH] _i + 1 - _i[MATH](j_1, j_2, , j_q;_1, _2, , _q)[MATH]j_q I_s[MATH]s=s__q[MATH]1r < q[MATH]j_1[MATH][C , x][MATH]x=k[MATH]p[MATH]q[MATH]j_p = j_q[MATH]_p = _q[MATH]p h < q[MATH]j_h [MATH]j_h + 1[MATH]p < r < q[MATH]j_p = j_r[MATH]j_q = j_r[MATH]_1, _2, , _q[MATH]G=(, N, P, S, T, M)[MATH]B=X_1 X_2 X_p[MATH]a[MATH]((s_0, ), (s_1, X_1),,(s_p,X_p))[MATH]J=(j_1, j_2, , j_n; _1, _2, , _n)[MATH]U=(u_1, u_2, , u_m; _1, _2, , _m)[MATH]s=[[MATH]][MATH]u_i=[ C_i _i _i, a][MATH]1 i m[MATH]j_1=[[MATH]][MATH]m=0[MATH]s[MATH][[MATH]][MATH][MATH] s[MATH]_n=p-1[MATH]m>0[MATH]^* a x[MATH]x^*[MATH]_h^*[MATH]h 1[MATH]U[MATH]J[MATH]s[MATH]a[MATH]a[MATH]B[MATH]U[MATH]J[MATH]s[MATH]B[MATH]U=(u_1, u_2, , u_n)[MATH]P=p_1, p_2, , p_m[MATH]p_1 < p_2 < < p_m[MATH]1 h m[MATH]j_p_h[MATH][[MATH]][MATH][MATH]h_0P[MATH]j_h_0[MATH][[MATH]][MATH]j_p_h=[[MATH]][MATH]1 h n_p[MATH]X_h[MATH]X_1 X_2 X_m[MATH][MATH]U[MATH]B[MATH]a[MATH]U[MATH]J[MATH]s[MATH]_J[MATH]J[MATH]U=0[MATH]s[MATH][ A B][MATH]_J B[MATH]U>0[MATH]_J _U[MATH]_U[MATH]U[MATH][MATH](U, J, s)[MATH]B[MATH]a[MATH]U[MATH]J[MATH]s[MATH](U, J, s)[MATH]B[MATH]s_1, s_2, , s_n)[MATH]I_0, I_1, I_2, , I_m[MATH]G[MATH]U=(u_1, u_2, , u_p; _1, _2, , _p)[MATH]J=(j_1, j_2, , j_q; _1, _2, , _q)[MATH] X_2 X_n[MATH]r[MATH]U=0[MATH]r=s[MATH]U>0[MATH]u_1=[ A U U U, a][MATH]r=[ A U U U][MATH]J[MATH]B[MATH]_q[MATH]_q[MATH]_q=0[MATH]j J[MATH]j[MATH][ A j j][MATH]J[MATH][MATH]J[MATH]B[MATH]_q[MATH]J=n j[MATH]n j 0[MATH]_q=n j + 1[MATH]q g[MATH]_q_g=n j[MATH]_q g=n j + 1[MATH]j_q_g + 1[MATH][ A g g X g g][MATH]J\'=(j_1, j_2, , j_q g)[MATH]J[MATH]U=0[MATH]_q = n - 1[MATH]s[MATH]J[MATH][MATH](U, J, s)[MATH] B[MATH]U > 0[MATH]t=_p - _q[MATH]t[MATH]t=1[MATH]_h = _q + 1[MATH]1 h p[MATH]u_1[MATH]1 < h p[MATH]u_h[MATH]u_p[MATH]U=1[MATH]s(j_q)=n - 1[MATH]U[MATH]B[MATH]J[MATH][MATH](U, J, s)[MATH]B[MATH]U[MATH]t=k[MATH]k 1[MATH]t=k+1[MATH]u_1[MATH]u_1=[ A_u_1 _u_1 _u_1, a][MATH]u f=[ A_u_1 _u_1 _u_1][MATH]s(j_p)=s(u_1) - 1[MATH]j_p [MATH]u f[MATH]u_1[MATH][ A_u_1 _u_1\' X_u_1 _u_1, a][MATH]_u_1\' X_u_1=_u_1[MATH]u_1[MATH]I__1[MATH]X_u_1=X_n-t+2[MATH]U[MATH]J[MATH]_J=X_1 X_2 X__q[MATH]U>0[MATH]U[MATH]_U=X__1 X__2 X__p-1 B[MATH]_1=_q + 1[MATH]_p=n[MATH]_J _U=B[MATH]G=(, N, P, S, T, M)[MATH]a[MATH][MATH]^*a x[MATH]x^*[MATH]X_m[MATH]k[MATH]y^*[MATH](k, )[MATH]a[MATH][MATH]k=1[MATH]X_1 X_2 X_k-1[MATH][MATH]G=(, N, P, S, T, M)[MATH]a[MATH][MATH]^*a x[MATH]x^*[MATH]X_m[MATH]k[MATH]X_k=a[MATH](k, )[MATH]a[MATH][MATH]L[MATH]a[MATH][MATH]k[MATH]X_n[MATH]L_d[MATH]a[MATH]_d[MATH]L_d[MATH]L[MATH] X_k _d[MATH]L_t[MATH]a[MATH]_t[MATH]L_t[MATH]L[MATH] X_k _t[MATH]L_1[MATH]L_2[MATH] X_k _d[MATH] X_k _t[MATH]P(L_1,L_2)[MATH]L_1, L_2, , L_n[MATH]L_i=(_i,a)[MATH](_i,k_i)[MATH]_i[MATH]a[MATH]i<n[MATH](_i,k_i)[MATH]_i[MATH]a[MATH]i=n[MATH]1j<n[MATH]1k<n[MATH]jk[MATH]P(L_j,L_j+1)=P(L_k,L_k+1)[MATH]_1[MATH]a[MATH]G=(, N, P, S, T, M)[MATH][MATH]a[MATH]U[MATH]J[MATH]s[MATH]s=[[MATH]][MATH]D[MATH][MATH]a[MATH]U=(u_1, u_2, , u_n ; _1, _2, , _n)[MATH]1 i n[MATH]u_i[MATH]_i ^* [MATH]1 i n[MATH](U,J,s,D)[MATH][MATH]a[MATH][MATH]a[MATH]G=(,N,P,S,T,M)[MATH]x[MATH]B[MATH]a[MATH][[MATH],b][MATH]a[MATH][MATH]V S[MATH]V_*[MATH][MATH]J[MATH]J[MATH]J[MATH]J[MATH]a[MATH][MATH][MATH]a[MATH]a[MATH]a[MATH]V P[MATH]V P[MATH][MATH]V P()[MATH]P=(U,J,s,D)[MATH][MATH]a[MATH]U=(u_1, u_2, , u_n)[MATH]1kn[MATH]J=(j_1,j_2,,j_m)[MATH]1 h m[MATH]D=(L_1, L_2, , L_p)[MATH]1 i p[MATH]V_PZ[MATH]V_, PZ[MATH]V_PZ[MATH]V_P Z[MATH]_h=X_h,1 X_h,2 X_h,q_h[MATH]V_, P Z[MATH]s=[ D s s s][MATH]V_P Z[MATH]L_V, PPZ[MATH]L_V, P[MATH]V P[MATH]L V[MATH]L V[MATH]N_, T[MATH]N_, T[MATH]N_, T[MATH]a[MATH]N_, T[MATH]G=(,N,P,S,T,M)[MATH] B a x[MATH]G[MATH](N)^*[MATH]x^*[MATH]B[MATH]a[MATH]y[MATH]G[MATH]^*y[MATH]y[MATH] A_1, A_2,, A_n[MATH]B[MATH]B[MATH]a[MATH][MATH][MATH] Y= A_i[MATH]1in[MATH] Y= A_i[MATH]1in[MATH] X > A[MATH]T[MATH]T[MATH]C [MATH]C[MATH]T[MATH]B a[MATH][MATH]G=(, N, P, S)[MATH] T[MATH]G[MATH][MATH] N[MATH] T[MATH] A [MATH] N[MATH][MATH] N[MATH]B a x[MATH]B[MATH]a[MATH] C Z[MATH] C[MATH] U[MATH] W U[MATH] DW[MATH] C Z[MATH] C[MATH]( M_0, M_1, , M_m)[MATH]0im[MATH]( V_i)_i=0^m[MATH]0im[MATH] A_1[MATH] A_2[MATH] V_i[MATH]( A_1)=[MATH] ( A_2)[MATH]B a[MATH]B[MATH]a[MATH] D_1, D_2X[MATH]( D_1)=[MATH]( D_2)[MATH]B a [MATH]B[MATH]a[MATH]B ax[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH]B[MATH]a[MATH]a[MATH]G=(,N,P,S)[MATH][MATH]a[MATH] TT_a[MATH] PT[MATH] A_1, A_2,, A_n[MATH]P[MATH] TF_a[MATH]a[MATH]a[MATH]0, 1[MATH]f[MATH]n[MATH]Z_n - 1[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]B[MATH]a[MATH][MATH][MATH]_,[MATH]( T)=[MATH]_,[MATH]( T)=( A, B, C)[MATH] P_ A[MATH] P_ B[MATH] P_ C[MATH] P_ A[MATH]_,[MATH]( T, A, B, C) = 0[MATH] A[MATH] B[MATH] P_ A[MATH] P_ A[MATH]( T, A, B, C)=1[MATH] P_ B[MATH] P_ B[MATH]_, [MATH]X[MATH]X X X X^n[MATH](a_1, a_2, , a_n), (b_1, b_2, , b_n) X^n[MATH](a_1, a_2, , a_n) (b_1, b_2, , b_n)[MATH]a_i = b_i[MATH]1 i < j n[MATH]a_j < b_j[MATH]a_i = b_i[MATH]1 i n[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]B[MATH]a[MATH] W=( M_0, M_1,, M_m)[MATH]( V_i )_i=0^n[MATH]0jm[MATH] N_1, N_2, N_3V_j[MATH]( N_1, N_2, N_3)[MATH] V_j[MATH]j[MATH]B( T)=[MATH]j[MATH]( N_1, N_2, N_3)[MATH]V_j[MATH]( M_1, M_2, M_3)[MATH]( N_1, N_2, N_3)[MATH] B( T) = ( N_1, N_2, N_3)[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]B[MATH]a[MATH] C_1, C_2, C_3 X[MATH]( C_1, C_2, C_3)[MATH]( C_1, C_2, C_3)[MATH]( D_1, D_2, D_3)[MATH]( C_1, C_2, C_3)[MATH]A( T)=[MATH]( C_1, C_2, C_3)[MATH]A( T)=( C_1, C_2, C_3)[MATH]_0[MATH]_0( T, A, B, C)=0[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]B[MATH]a[MATH]P[MATH]= A [MATH]1 n + 1[MATH] C_1[MATH] C_2,[MATH] C_3[MATH] C_1[MATH] C_2[MATH] C_3[MATH]_, n( T, C_1, C_2, C_3)[MATH] C_2[MATH]_, n( T, C_1, C_2, C_3)=0[MATH]_, n( T, C_1, C_2, C_3)=1[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]B[MATH]a[MATH][MATH]_, B[MATH]_, A[MATH]p[MATH]q[MATH]_^p + 1 T=^p T[MATH]_^q + 1 T=^q T[MATH]p[MATH]q[MATH][MATH][MATH]_[MATH]_[MATH]__^q _^p[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]_ T, P[MATH]_[MATH], n[MATH]P)[MATH]n=H_ T( P)[MATH]_0[MATH]__0[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]_^p T[MATH]p[MATH][MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B N[MATH]B a[MATH][MATH]p[MATH][MATH]_[MATH]( V_i)_i=0^n[MATH]p[MATH]p = 0[MATH] B_1[MATH] B_2[MATH]( B_1)=[MATH]( B_2)[MATH]_T = T[MATH]_T[MATH]_^p Z Z[MATH][MATH]p Z[MATH]p Z 0[MATH]p = p Z + 1[MATH]( B_1, B_2, B_3)[MATH]( B_1)=[MATH]( B_2)=[MATH]( B_3)[MATH]_T[MATH][MATH]_[MATH]_T[MATH]p - 1[MATH]_^p - 1 _T[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B a[MATH]_^q T[MATH]q[MATH][MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B N[MATH]B a[MATH][MATH]q[MATH][MATH]_[MATH]q[MATH]q = 0[MATH] D[MATH] D\'[MATH]( D)=[MATH]( D\')[MATH]_T = T[MATH]_T [MATH][MATH]_[MATH]q Y 0[MATH]_^q Y Y[MATH]q=q Y + 1[MATH]( C_1, C_2, C_3)[MATH]( C_1)=[MATH]( C_2)=[MATH]( C_3)[MATH]_ T[MATH][MATH]_[MATH]_T[MATH]q - 1[MATH]_^q - 1 _T[MATH]G=(, N, P, S)[MATH]x=qjqjqjccccbkrrhhhr[MATH]x L(G)[MATH]x[MATH]qjqjqjccccB[MATH]k[MATH]Z_a[MATH]T_a[MATH]G=(,N,P,S,T,M)[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B N[MATH]T_B a[MATH] T T_B a[MATH]B[MATH]a[MATH]( V_i)_i=0^n[MATH]D N[MATH]D\' N[MATH]D\' D[MATH]D\' ^* [MATH]N_D[MATH] B a[MATH] B a[MATH] Q A_Q [MATH]N_D[MATH] Y A[MATH] X A[MATH] Y A[MATH] X A 2 P[MATH] Y A[MATH] X A + 1[MATH]N_D[MATH] Y A[MATH] X A + N_D[MATH] Z B[MATH] B a[MATH][MATH] Z B[MATH]N_ N[MATH]C N_[MATH]C ^* [MATH]N_C_1, C_2, , C_m[MATH]h_i[MATH]C_i ^* [MATH]M B = h_1, h_2, , h_m [MATH] Z B[MATH]M B[MATH](N)^*[MATH]= B[MATH] V_i[MATH]0 i [MATH]2 P[MATH] T S[MATH] T S[MATH]2P [MATH]G=(, N, P, S, T, M)[MATH]P_0[MATH][MATH]N\' N[MATH]a() -1[MATH]P_0[MATH]a[MATH]P[MATH]Y_a[MATH]Z_a[MATH]P_0[MATH]a[MATH]P[MATH]G=(,N,P,S)[MATH][MATH]a[MATH][MATH] TF_a[MATH]H_ T( P)T\' T_a[MATH] P\'[MATH] T\'[MATH]B[MATH]B[MATH] T F_a[MATH]H_ T( P) W=( M_0, M_1,, M_m)[MATH]( V_i )_i=0^n[MATH]_ T, P[MATH]_ T, P[MATH] P\'[MATH]T[MATH]H_T( P\')=H_ T( P)[MATH]T[MATH] P p[MATH]T[MATH]H_ T( P p)=H_ T( P)[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH][MATH]P_0[MATH]P_0[MATH]a[MATH]P[MATH]G=(,N,P,S,T,M)[MATH][MATH]a[MATH][MATH]G[MATH] G=(, N_G, P_G, S, T, M)[MATH]V_G(a)[MATH]P_G[MATH]a[MATH]P[MATH]V_G(a)[MATH]x ^*[MATH]a x[MATH]y ^*[MATH]a x ^* y[MATH]y[MATH]B[MATH]a[MATH][MATH]V_ a()=-1[MATH]( P)=[MATH][MATH]V_a()=n[MATH]-1 n [MATH]( P)=[MATH] n[MATH][MATH] T[MATH] P_0[MATH]H_ T s( P_0)=n[MATH]Y_ a() > n[MATH]Y_a() = n_0 > n[MATH] T_1F_a[MATH] B_1[MATH] A_1[MATH]B[MATH]a[MATH] T_1[MATH]n_0^th[MATH] A_1[MATH][MATH]a[MATH]x_1^*[MATH]a x_1[MATH]z^*[MATH]a x_1^* z[MATH] T_z[MATH]z[MATH]qT_1T_z[MATH] T_1[MATH]q( F)[MATH] T_z[MATH]N_a x_1, T_z(q( P))=n_0[MATH]Y_ a()=n[MATH] P_G[MATH]P_G[MATH]a[MATH]P[MATH]P_G[MATH]a[MATH]P[MATH]a() > Y_a()[MATH]x[MATH]y^*[MATH] ax^*y[MATH]y[MATH]N_ax, T( P)>-1[MATH] B< P< A[MATH][MATH] T_0F_a[MATH] P_0[MATH] T_0[MATH]( P)=[MATH][MATH]Y_a()=n+1[MATH]P_ G[MATH]P_G[MATH]a[MATH]P[MATH]G[MATH]T_G[MATH]G[MATH][MATH]a[MATH]P_B,a,G[MATH]G[MATH][MATH]a[MATH]G[MATH][MATH]a[MATH]U_0[MATH]U_1[MATH]J[MATH]U[MATH] T[MATH]T[MATH] P[MATH] T[MATH] P[MATH] P A[MATH] P[MATH] P[MATH] P A= P[MATH] P A[MATH]G=(, N, P, S, T, M)[MATH][MATH]a[MATH]k 0, 1[MATH]x ^*[MATH]x=k[MATH]Q_k U_k T[MATH]Q_k(U)[MATH]U=(u_1, u_2, , u_n)[MATH]u_n=[[MATH], x][MATH]U=1[MATH] (U) = T i[MATH] T i[MATH] A[MATH]U>1[MATH][MATH]X[MATH]X (N)[MATH] T= Q_k(u_1, u_2, , u_n-1)[MATH] P[MATH] T[MATH] T\'[MATH] T[MATH]X[MATH] P[MATH] P[MATH] T[MATH]Q_k(U) = T\'[MATH]U>1[MATH] V= Q_k(u_1, u_2, , u_n-1)[MATH] Q[MATH] V[MATH] V\'[MATH] V[MATH][MATH] Q[MATH] Q[MATH] V\'[MATH]Q_k(U) = V\'[MATH]Q_k[MATH]k[MATH]Q U[MATH]Q A[MATH]G[MATH][MATH]a[MATH]T^*[MATH]T[MATH]Q D D T^*[MATH]D D[MATH](L_1, L_2, , L_n)[MATH]Q D (D)[MATH]n=1[MATH]L_1=(X_1 X_2 X_p, k)[MATH]1 i < k[MATH] V_i[MATH]X_i ^* [MATH] V_k[MATH]X_k[MATH]n > 1[MATH]L_1=(X_1 X_2 X_p, k)[MATH]1 i < k[MATH] V_i[MATH]X_i ^* [MATH]Q D(L_2, L_3, , L_n)=( Y_1, Y_2, , Y_h)[MATH] V_k[MATH]Q D(D)=( V_1, V_2, , V_k)[MATH]Q D[MATH]G[MATH]( N)=[MATH]( N)[MATH]G=(, N, P, S)[MATH]F U TT[MATH]F U( T)[MATH] N[MATH]( N) [MATH]( N)[MATH]A_1, A_2, , A_n N^*[MATH]( N) = [MATH]( N) A_1 A_2 A_n[MATH] V_1, V_2,, V_n[MATH]1 i n[MATH] V_i[MATH]A_i ^* [MATH](U, J, s, D)[MATH]B[MATH]a[MATH]RP T[MATH]R(U, J, s, D)[MATH] T U=F U(Q U(U))[MATH] Y=Q D(D)[MATH] T J=Q A(J)[MATH]U=0[MATH] T_0[MATH]B[MATH] T J[MATH] Y[MATH] T J[MATH]U>0[MATH] T_0[MATH] T U[MATH] T J[MATH] Y[MATH] T J[MATH]R(I,J,s,D)= T_0[MATH]R[MATH](U, J, s, D)[MATH]R(U, J, s, D)[MATH]k 0, 1[MATH]x ^*[MATH]x=k[MATH]P_k I_k[MATH]P_k=(p_1, p_2, , p_n)[MATH]n[MATH]Q_k(P_k)[MATH]P_k < 2[MATH]P_k=2[MATH]P_k[MATH] T Q=Q_k(P_k)[MATH] A_2 _2[MATH]A_2[MATH]A_2 _1[MATH]P_k\' I_k[MATH]P_k\'2[MATH]Q_k(P_k)[MATH]P_x=P_k\'+1[MATH] T P[MATH]P_k\'[MATH]Q_k[MATH]p_n[MATH]p_n[MATH][ A X , x][MATH] P A[MATH] A X [MATH] P A[MATH][MATH]Q k[MATH]X[MATH] P A[MATH] T_0[MATH] P_A[MATH] T_0[MATH]X[MATH]X [MATH]p_n[MATH][ C , x][MATH] T P[MATH][ D A A C A, x][MATH] P A[MATH]A[MATH]Q k[MATH]_A C[MATH] _A C A[MATH]Q U(U)[MATH]Q A(J)[MATH]D=(L_1, L_2, , L_p)[MATH]p=1[MATH]Q D[MATH] Y=( T_1, T_2, , T_m)[MATH] T_1, T_2, , T_m-1[MATH] T_m[MATH]Q D(D\')[MATH]D\' 1[MATH]p=D\' + 1[MATH]p - 1[MATH]Q D[MATH](,k_)[MATH] Y=( V_1, V_2, , V_q)[MATH] Y[MATH]Q D[MATH]k_[MATH]k_- 1[MATH]P(L_1,L_2)[MATH]Q D[MATH] V\'[MATH] Y[MATH] V\'[MATH] C_1, C_2, , C_k_[MATH]P(L_1, L_2)[MATH]Q D(D)[MATH]Q A(J)[MATH]Q U(U)[MATH]Q D(D)[MATH]R(U, J, s, D)[MATH]s=[ A_ _X_ _][MATH]U=0[MATH]X_> 0[MATH]u_1 = [ X_ _1, a][MATH](Q U(U)) = X_[MATH]Q U(U)[MATH] X_ _1[MATH] P_0, A[MATH]Q A(J)[MATH] A_ _X__[MATH]Q A(J)[MATH]_[MATH] P_0, A[MATH]_ X_[MATH]L_1[MATH]_[MATH]Q D(D)[MATH]X_k_[MATH] V_k_[MATH]L_1[MATH] V_k_[MATH]X_k_=a[MATH] V_k_[MATH]P(L_1, L_2)[MATH] X [MATH]( V_k_))=X_k_[MATH]G[MATH] T[MATH] L=( K_1, K_2, , K_n)[MATH] K_1 < K_2 < < K_n[MATH]( K_i) N [MATH]1 i n[MATH][MATH] T[MATH]R(U, J, s, D)[MATH]B a[MATH]Q D(D)=( V_1, V_2, , V_n d)[MATH] V_1, V_2, , V_n d - 1[MATH][MATH] V_n d[MATH]a[MATH]k 0, 1[MATH]x ^*[MATH]x=k[MATH]P_k=(p_1, p_2, , p_n)[MATH] T=Q_k(P_k)[MATH]Q_k[MATH]p_i_1, p_i_2, , p_i_m[MATH]i_1 < i_2 < < i_m[MATH]1 h m[MATH]Q_k(P_k)[MATH]X_1 X_2 X_m[MATH]P_k[MATH]Q A(J)[MATH]Q U(U)[MATH]B[MATH]R(U, J, s, D)[MATH]Q A(J)[MATH]Q U(U)[MATH] V_n d[MATH]B a[MATH]R(U, J, s, D)[MATH]R(U, J, s, D)[MATH]y G[MATH]y[MATH]_0 T=R(U, J, s, D)[MATH]N[MATH]_0 T[MATH]A[MATH] F_A[MATH]z_A ^*[MATH] T_1[MATH] T[MATH] N N[MATH] F_[MATH]( N)[MATH]D[MATH] J_0[MATH] T_1[MATH] ( J_0) [MATH]( J_0)[MATH] J_0[MATH]( J_0)=_0 Z_1 Z_2 Z_m[MATH]_0=[MATH]( J_0)[MATH] Z_ J_0[MATH]( F_Z_1, F_Z_2, , F_Z_m)[MATH] T_2[MATH] J f D[MATH] Z_ J f[MATH] J f[MATH]z[MATH] T_2[MATH]z[MATH] T_2[MATH][MATH]T_2[MATH] T_2[MATH][MATH] N z[MATH] N z[MATH][MATH] N z[MATH]( N z)[MATH][MATH] M z[MATH]a[MATH] M z > A[MATH] A[MATH]a[MATH] T_1[MATH] T_2[MATH] H x[MATH] T_1[MATH] T_2[MATH] H z[MATH]Q A(J)[MATH]Q A(J)[MATH] H J[MATH]H z[MATH]Q A(J)[MATH] H J[MATH]Q D(D)[MATH]i^th[MATH] H J[MATH]i[MATH] H z[MATH]Q D(D)[MATH] H z[MATH]Q D(D)[MATH] H D[MATH] A D[MATH]Q D(D)[MATH]a[MATH]( H D) [MATH]( H D)[MATH] H D[MATH] A D[MATH] H D[MATH] T_2[MATH][MATH] T_2[MATH] T_2[MATH]Q U(U)[MATH]F U(Q U(U))[MATH] T_1[MATH] T[MATH][MATH] N_[MATH]( N_)[MATH][MATH] T_2[MATH]( N_)[MATH] N_[MATH][MATH] T_2[MATH] T[MATH]R(U, J, s, D)[MATH]a[MATH](U, J, s, D)[MATH]R(U, J, s, D)[MATH]G=(, N, P, S)[MATH] B[MATH]a[MATH]B N[MATH](U, J, s, D) P_B, a, G[MATH] T=R(U, J, s, D)[MATH] A[MATH] B[MATH] T[MATH]B[MATH]a[MATH] T[MATH] A[MATH] B[MATH]D=(L_1, L_2, , L_n)[MATH] P A,1[MATH] P A,2[MATH] P A,3[MATH] A[MATH] B[MATH]( P A,1)=[MATH]( P A,2)=[MATH]( P A,3)[MATH] P A,1 < P A,2 < P A,3[MATH]L_i_1[MATH]L_i_2[MATH]L_i_3[MATH]i_1 < i_2 < i_3[MATH] P A,1[MATH] P A,2[MATH] P A,3[MATH]P(L_i_1, L_i_1 + 1)=P(L_i_2, L_i_2 + 1)=P(L_i_3, L_i_3 + 1)[MATH] T[MATH] A[MATH] T[MATH]B a[MATH]( V_i)_i=0^q[MATH]i[MATH] V_i[MATH] C_1[MATH] C_2[MATH] C_3[MATH] V_i[MATH]k[MATH]x[MATH]I[MATH]I=(r_1, r_2, , r_m)[MATH] C_1[MATH]Q_k[MATH][ A_1 _1, x][MATH] C_2[MATH] C_3[MATH] C_1[MATH] C_2[MATH] C_3[MATH]r_t_1[MATH]r_t_2[MATH]r_t_3[MATH]t_1 < q t< t_3[MATH]r_q t [MATH]r_q t + 1[MATH]r_t_1=r_q t=r_t_2[MATH]q t=t_2[MATH] T[MATH] B[MATH]Q U(U)[MATH]Q A(J)[MATH] T[MATH] A[MATH]a[MATH]G=(, N, P, S, T, M)[MATH] (N)^*[MATH]B N[MATH]B[MATH]a[MATH]R[MATH]R[MATH] T s[MATH]B[MATH]a[MATH] A[MATH] B[MATH]a[MATH]B[MATH] S[MATH] T s[MATH] S[MATH] B[MATH] A[MATH] R_B[MATH] S[MATH] B[MATH] T J[MATH] R_B[MATH] T s[MATH] S[MATH] R_B[MATH] T\' U[MATH] R_B[MATH] V_1, V_2, , V_n v[MATH] S[MATH] R_B[MATH] Y=( V_1, V_2, , V_n v)[MATH] T U[MATH] T U\'[MATH] N_[MATH] N_[MATH][MATH]D[MATH]( S)[MATH]Q D^-1colon (T^* ( N)^*) D [MATH]Q D^-1( Y D, D)[MATH] Y D=( V D,1, V D,2, , V_ D,n D)[MATH] V_D,n D[MATH]L[MATH]D[MATH]a[MATH](D, n D)[MATH]Q D^-1( Y D, D) = (L)[MATH] V_D,n D[MATH]L\'[MATH]D[MATH]a[MATH](D, n D)[MATH] V_D, n D[MATH] W_1, W_2, , W_m D[MATH]( V_D,n D))[MATH]L_2, , L_p D)=Q D^-1(( W_1, W_2, , W_m D), D^-1( Y D, D) = (L\', L_1, L_2, , L_p D)[MATH] Y T^*[MATH] Y=( V_1, V_2, , V_m_I)[MATH] V_1, V_2, , V_m I-1[MATH][MATH] V_m I[MATH]a[MATH][MATH] V_m I[MATH] V_m I h V[MATH]h V = 1[MATH] V_m I[MATH]a[MATH]Q D^-1( Y, )[MATH](, m I)[MATH]Q D(Q D^-1( Y, )= Y\'[MATH] Y\'=( Q_1, Q_2, , Q_m I\')[MATH]Q_m I\' I[MATH]m I - 1[MATH] Y\'[MATH] V_m I[MATH]a[MATH]h V=1[MATH]h V =k V[MATH]k V1[MATH]h V=k V+1[MATH]Q D^-1[MATH] Y[MATH]Q D^-1[MATH]( Y h, h)[MATH] Y h[MATH]k V[MATH]Q D(Q D^-1( Y h, h)= Y_h[MATH]Q D[MATH]L_1[MATH][MATH]a[MATH](I, m I)[MATH]X_2 X_[MATH] V_1\', V_2\', , V_m I - 1\'[MATH] V_m I\'[MATH] Y_h[MATH] X_m I [MATH] Y[MATH]k 0, 1[MATH]y ^*[MATH]y=k[MATH]T P T[MATH][MATH]Q_k,y ^-1 T P U_k[MATH] T x T P[MATH] T x[MATH] T x[MATH] A x x[MATH] T x \'[MATH] T x[MATH](i_1, i_2, , i_n r) = Q_k,y ^-1 ( T x \')[MATH] T x[MATH] N x[MATH]X x[MATH] N x[MATH]E x[MATH]m x - 1[MATH] N x[MATH] T x "[MATH] T x[MATH] N x[MATH]x = Z_1 Z_2 Z_s x,[MATH]k 0, 1[MATH]z ^*[MATH]z=k[MATH] T y[MATH] T y[MATH]n y[MATH]n y[MATH]n y = 1[MATH] A y y[MATH]Q_k,z ^-1[MATH]Q_k[MATH]Q_k[MATH] A y y[MATH] T y[MATH]k y[MATH]k y 1[MATH]n y = k y + 1[MATH] T y[MATH] F y[MATH]( F y)= C y y[MATH] T y\'[MATH] T y[MATH] F y[MATH]Q_k,z^-1[MATH] T y[MATH]Q_k,z[MATH] T y\'[MATH]Q_k(Q_k,z ^-1 ( T y \')) = T y \'[MATH]Q_k,z ^-1( T y)[MATH][ y, z][MATH]Q_k,z ^-1 ( T y \')[MATH]Q_k[MATH]Q_k(Q_k,z ^-1 ( T y ))[MATH]Q_k[MATH]Q_k,z ^-1 ( T y \')[MATH] T y\'[MATH]Q_k[MATH][ y, z][MATH] T y\'[MATH] y[MATH]Q_k(Q_k,z ^-1 ( T y ))= T y[MATH]( F y)=X[MATH]X N[MATH] F y[MATH] D y y X y [MATH] F y[MATH]y [MATH] T y"[MATH] T y[MATH] F y[MATH]Q_k(Q_k,z ^-1( T y))[MATH] T y"[MATH][ D y y X y , z][MATH] T y[MATH]Q A ^-1[MATH] T a[MATH]Q u ^-1[MATH]D = Q D ^-1 ( Y)[MATH]J = Q A ^-1 ( T J)[MATH]S)[MATH] S[MATH] T s[MATH] S[MATH] B[MATH]i b[MATH]F [MATH]=i b[MATH]s=[ F ][MATH]U[MATH] T U[MATH]U ()[MATH] T U[MATH]U Q U ^-1( T U)[MATH] V R(U, J, s, D)[MATH] V s= T s[MATH]R[MATH] T J = Q A (J)[MATH] Y = Q D (D)[MATH] T U[MATH]B[MATH] T J[MATH] T U[MATH]B[MATH] T U[MATH] T U = Q U (U)[MATH] T U,0[MATH] T U \'[MATH]R(U, J, s, D)[MATH] T U \'[MATH] T J[MATH] Y[MATH] T U \'[MATH] T s[MATH]R[MATH](U_1, J_1, s_1, D_1)[MATH](U_2, J_2, s_2, D_2)[MATH]P_k[MATH]R_k[MATH]Q_k(P_k) = Q_k(R_k)[MATH]w ^*[MATH]w=k[MATH]P_k = R_k[MATH]P_k R_k[MATH]Q_k(P_k)[MATH]P_k[MATH]Q_k(P_k)=Q_k(R_k)[MATH]P_k=R_k[MATH]P_k=(p_1, p_2, , p_n P)[MATH]R_k=(r_1, r_2, , r_n R)[MATH]i z[MATH]p_i z r_i z[MATH]p_h = [ A_P,h _P,h _P,h, w][MATH]r_k = [ A_P,h _P,h _P,h, w][MATH]1 h n R[MATH]i z = 1[MATH]P = R = [MATH]Q_k[MATH](p_1)[MATH](r_1)[MATH]A_P, 1 _P, 1 = A_R, 1 _R, 1[MATH]i z > 1[MATH]p_i z - 1 = r_i z - 1[MATH]_P, i z = [MATH]p_i z[MATH]p_i z -1 [MATH] N_P[MATH]Q_k(P_k)[MATH] N_R[MATH]Q_k(R_k)[MATH]p_i z - 1[MATH]r_i z - 1[MATH] N_P[MATH] N_R[MATH]p_i z = r_i z[MATH]_P, i z [MATH]p_i z - 1[MATH]r_i z - 1[MATH]p_i z - 1[MATH]p_i z - 1[MATH]p_i z[MATH]Y[MATH]r_i z - 1[MATH]p_i z[MATH]r_i z[MATH]p_i z = r_i z[MATH]P_k=R_k[MATH]U_1=U_2[MATH]J_1=J_2[MATH]D_1 D_2[MATH]Q D(D_1) = Q D(D_2)[MATH]H[MATH]Q D(D_1)[MATH]D_1=(L_1, L_2, , L_H)[MATH]D_2=(L_1\', L_2\', , L_H\')[MATH]k d[MATH]L_k d L_k d\'[MATH]k d = 1[MATH]Q D(D_1)=(Y L,1, Y L,2, , Y_L, n Y)[MATH]L_1=(L, Q D(D_1))[MATH]L_1\'=( L, Q D(D_2))[MATH] Q D(D_1)= Q D(D_2)[MATH]L_1=L_1\'[MATH]k d > 1[MATH] T d[MATH] T d\'[MATH]Q D(D_1)[MATH]Q D(D_2)[MATH] N d[MATH] T d[MATH] N d[MATH]D_1 - k d[MATH] N d\'[MATH] T d\'[MATH] N d[MATH] N d\'[MATH]L_k d - 1[MATH]L_k d - 1\'[MATH] A L L[MATH]L_k d[MATH]L_k d\'[MATH]L[MATH] N d[MATH]n d[MATH] N d\'[MATH]L_k d = L_k d\'[MATH]D_1=D_2[MATH]s_1 s_2[MATH] P s[MATH]Q A(J_1)[MATH]( P s) = A s s X s s[MATH] P s[MATH]s[MATH] P s[MATH] s[MATH]L_1[MATH]D_1[MATH]D_2[MATH]s[MATH](U_1, J_1, s_1, D_1)=(U_2, J_2, s_2, D_2)[MATH]R[MATH]G=(, N, P, S)[MATH]B[MATH]a[MATH]B N[MATH]P[MATH]B[MATH][MATH]P=(U, J, s, D)[MATH]U=(u_1, u_2, , u_n)[MATH]J=(j_1, j_2, , j_m)[MATH]s=[ E ][MATH]D=(L_1, L_2, , L_p)[MATH]1 i_n n[MATH]u_i n [ A_i n _i n _i n, a][MATH]1 i_m m[MATH]j_i m [ C_i m _i m _i m][MATH]1 i_p p[MATH]L_i p (_i p, k_i p)[MATH] T[MATH]B[MATH]a[MATH] T U = Q U(U)[MATH] T J = Q A(J)[MATH] Y = Q D(D)[MATH]A [MATH]L_ V, P()=n V[MATH]n V = -1[MATH]M( T, ) n V[MATH]n V -1[MATH]V () = n V[MATH]i u[MATH]l u=i u - _i u[MATH]_l u = [MATH]Q U[MATH]u_l u[MATH][MATH]Q U((u_1, u_2, , u_l u - 1))[MATH] N u[MATH] N u[MATH] B[MATH] A[MATH]M( T, ) = _l u + 1[MATH]V_,() = n V[MATH][MATH]1 i_n[MATH][MATH] T U[MATH] T U,0[MATH] N U,0[MATH] N U,0[MATH] T J[MATH] B[MATH] A[MATH]M( T, ) = + 1[MATH]V_() = n V[MATH]1 k J m[MATH] C_k J _k J _k J = [MATH] T J[MATH] B[MATH]k\'=k J - _k J[MATH]Q A[MATH]j_k\'[MATH][MATH] T J[MATH] N J[MATH]_k J[MATH] N J[MATH]M( T, ) = _k J + 1[MATH]V_() = n V[MATH]1 h V p[MATH]Q D[MATH]L_h V[MATH] T D[MATH][MATH]k_h V + 1[MATH]V_,() = n V[MATH]1 h V\' p[MATH]_h V\'= X_1 X_2 X_N D[MATH][MATH]1 l D N D[MATH]Q D[MATH]L_h V\'[MATH] P_D,[MATH][MATH] P_D,[MATH] B[MATH] A[MATH] B < P_D, < A[MATH]V_() = n V[MATH]E [MATH] T J[MATH] T U[MATH] Y[MATH] T U[MATH] - 1[MATH] T J[MATH] T U[MATH] Y[MATH] Y[MATH] A[MATH] > 1[MATH] P s[MATH] Q A[MATH] P s[MATH] A[MATH] + Y[MATH] A[MATH][MATH]M( T, )=n T[MATH]n T = -1[MATH]n T > -1[MATH] P_[MATH] T[MATH]( P_)=[MATH] P_[MATH] B[MATH] A[MATH] P_[MATH] B[MATH] P_[MATH] A[MATH] T U,0[MATH] P_[MATH][MATH]H_ T( P_)[MATH] + 1[MATH]u B[MATH]U[MATH] P_[MATH]V_() = + 1[MATH] P_[MATH] A[MATH] P_[MATH](U, J, s, D)[MATH] Y[MATH] P_[MATH] P Y[MATH] Y[MATH] T Y[MATH] P Y[MATH] T Y[MATH]h a[MATH]Q D[MATH]D[MATH] P Y\'[MATH] P Y[MATH]Q D[MATH]L_h a + 1[MATH] P T\'[MATH] T Y[MATH]Q D[MATH]L_h a + 1[MATH] P T\'[MATH]k_k a + 1[MATH]P(L_h a + 1, L_h a + 2)=[MATH] P [MATH] T J[MATH] P A[MATH]T J[MATH] P_[MATH] P A[MATH]n T - 1[MATH]T J[MATH] P A[MATH] T J[MATH]Q A[MATH]j_m - n T + 1[MATH]j_m - n T + 1[MATH]j_m[MATH]J > 1[MATH]J X J s[MATH] P_[MATH] P A[MATH] P c[MATH] P c[MATH]B[MATH] P c[MATH]Q U(U)[MATH] Y[MATH] Y=k_1[MATH]J[MATH] P_[MATH] P_[MATH]n T - 1[MATH]J J = [MATH] P_[MATH] B[MATH] A[MATH] B < P < A[MATH] P_[MATH] B[MATH] A[MATH] B < P < A[MATH] Q B[MATH] Q B[MATH]Q U(U)[MATH] T U, 0[MATH] T U, 0[MATH] P_[MATH] P U, 0[MATH] R B[MATH] Q B[MATH] B[MATH] T U, 0[MATH] R B[MATH]1 k_0 n[MATH] P_[MATH] B[MATH] A[MATH] Y=( V_1, V_2, , V_k_1)[MATH] P_[MATH] P Y, [MATH] Y[MATH] P Y, [MATH] V_1, V_2, , V_k_1 - 1[MATH] V_h Y[MATH] P Y, [MATH][MATH] V_h Y[MATH] P_[MATH] V_k_1[MATH] V_k_1[MATH]a[MATH] P Y, [MATH] A[MATH] P Y,[MATH][MATH]P[MATH]T P[MATH]R[MATH]k[MATH]k[MATH]k[MATH]k=1[MATH][MATH]V[MATH]V[MATH][MATH][MATH][MATH][MATH]'} | [['cs-0605104-1-3-1', 'cs-0605104-2-3-1'], ['cs-0605104-1-3-2', 'cs-0605104-2-3-2'], ['cs-0605104-1-3-4', 'cs-0605104-2-3-4'], ['cs-0605104-1-3-5', 'cs-0605104-2-3-5'], ['cs-0605104-1-8-0', 'cs-0605104-2-8-0'], ['cs-0605104-1-8-1', 'cs-0605104-2-8-1'], ['cs-0605104-1-8-3', 'cs-0605104-2-8-3'], ['cs-0605104-1-1-1', 'cs-0605104-2-1-1'], ['cs-0605104-1-1-2', 'cs-0605104-2-1-2'], ['cs-0605104-1-1-3', 'cs-0605104-2-1-3'], ['cs-0605104-1-1-4', 'cs-0605104-2-1-4'], ['cs-0605104-1-1-5', 'cs-0605104-2-1-5'], ['cs-0605104-1-2-0', 'cs-0605104-2-2-0'], ['cs-0605104-1-2-1', 'cs-0605104-2-2-1'], ['cs-0605104-1-2-2', 'cs-0605104-2-2-2'], ['cs-0605104-1-11-1', 'cs-0605104-2-11-1'], ['cs-0605104-1-7-0', 'cs-0605104-2-7-0'], ['cs-0605104-1-9-0', 'cs-0605104-2-9-0'], ['cs-0605104-1-9-1', 'cs-0605104-2-9-1'], ['cs-0605104-1-9-2', 'cs-0605104-2-9-2'], ['cs-0605104-1-9-3', 'cs-0605104-2-9-3'], ['cs-0605104-1-9-4', 'cs-0605104-2-9-4'], ['cs-0605104-1-10-3', 'cs-0605104-2-10-1'], ['cs-0605104-1-10-4', 'cs-0605104-2-10-2'], ['cs-0605104-1-6-0', 'cs-0605104-2-6-0'], ['cs-0605104-1-6-1', 'cs-0605104-2-6-1'], ['cs-0605104-1-6-3', 'cs-0605104-2-6-3'], ['cs-0605104-1-6-5', 'cs-0605104-2-6-6'], ['cs-0605104-1-5-0', 'cs-0605104-2-5-0'], ['cs-0605104-1-5-1', 'cs-0605104-2-5-1'], ['cs-0605104-1-5-2', 'cs-0605104-2-5-2'], ['cs-0605104-1-5-3', 'cs-0605104-2-5-3'], ['cs-0605104-1-18-0', 'cs-0605104-2-16-0'], ['cs-0605104-1-4-0', 'cs-0605104-2-4-0'], ['cs-0605104-1-4-1', 'cs-0605104-2-4-1'], ['cs-0605104-1-4-2', 'cs-0605104-2-4-2'], ['cs-0605104-1-4-3', 'cs-0605104-2-4-3'], ['cs-0605104-1-16-0', 'cs-0605104-2-14-0'], ['cs-0605104-1-16-1', 'cs-0605104-2-14-1'], ['cs-0605104-1-16-2', 'cs-0605104-2-14-2'], ['cs-0605104-1-16-3', 'cs-0605104-2-14-3'], ['cs-0605104-1-16-4', 'cs-0605104-2-14-4'], ['cs-0605104-1-16-5', 'cs-0605104-2-14-5'], ['cs-0605104-1-16-6', 'cs-0605104-2-14-6'], ['cs-0605104-1-12-0', 'cs-0605104-2-12-0'], ['cs-0605104-1-12-1', 'cs-0605104-2-12-1'], ['cs-0605104-1-3-0', 'cs-0605104-2-3-0'], ['cs-0605104-1-3-3', 'cs-0605104-2-3-3'], ['cs-0605104-1-8-2', 'cs-0605104-2-8-2'], ['cs-0605104-1-1-0', 'cs-0605104-2-1-0'], ['cs-0605104-1-11-0', 'cs-0605104-2-11-0'], ['cs-0605104-1-7-1', 'cs-0605104-2-7-1'], ['cs-0605104-1-6-2', 'cs-0605104-2-6-2'], ['cs-0605104-1-12-2', 'cs-0605104-2-12-2'], ['cs-0605104-1-10-2', 'cs-0605104-2-10-0'], ['cs-0605104-1-6-4', 'cs-0605104-2-6-4']] | [['cs-0605104-1-3-1', 'cs-0605104-2-3-1'], ['cs-0605104-1-3-2', 'cs-0605104-2-3-2'], ['cs-0605104-1-3-4', 'cs-0605104-2-3-4'], ['cs-0605104-1-3-5', 'cs-0605104-2-3-5'], ['cs-0605104-1-8-0', 'cs-0605104-2-8-0'], ['cs-0605104-1-8-1', 'cs-0605104-2-8-1'], ['cs-0605104-1-8-3', 'cs-0605104-2-8-3'], ['cs-0605104-1-1-1', 'cs-0605104-2-1-1'], ['cs-0605104-1-1-2', 'cs-0605104-2-1-2'], ['cs-0605104-1-1-3', 'cs-0605104-2-1-3'], ['cs-0605104-1-1-4', 'cs-0605104-2-1-4'], ['cs-0605104-1-1-5', 'cs-0605104-2-1-5'], ['cs-0605104-1-2-0', 'cs-0605104-2-2-0'], ['cs-0605104-1-2-1', 'cs-0605104-2-2-1'], ['cs-0605104-1-2-2', 'cs-0605104-2-2-2'], ['cs-0605104-1-11-1', 'cs-0605104-2-11-1'], ['cs-0605104-1-7-0', 'cs-0605104-2-7-0'], ['cs-0605104-1-9-0', 'cs-0605104-2-9-0'], ['cs-0605104-1-9-1', 'cs-0605104-2-9-1'], ['cs-0605104-1-9-2', 'cs-0605104-2-9-2'], ['cs-0605104-1-9-3', 'cs-0605104-2-9-3'], ['cs-0605104-1-9-4', 'cs-0605104-2-9-4'], ['cs-0605104-1-10-3', 'cs-0605104-2-10-1'], ['cs-0605104-1-10-4', 'cs-0605104-2-10-2'], ['cs-0605104-1-6-0', 'cs-0605104-2-6-0'], ['cs-0605104-1-6-1', 'cs-0605104-2-6-1'], ['cs-0605104-1-6-3', 'cs-0605104-2-6-3'], ['cs-0605104-1-6-5', 'cs-0605104-2-6-6'], ['cs-0605104-1-5-0', 'cs-0605104-2-5-0'], ['cs-0605104-1-5-1', 'cs-0605104-2-5-1'], ['cs-0605104-1-5-2', 'cs-0605104-2-5-2'], ['cs-0605104-1-5-3', 'cs-0605104-2-5-3'], ['cs-0605104-1-18-0', 'cs-0605104-2-16-0'], ['cs-0605104-1-4-0', 'cs-0605104-2-4-0'], ['cs-0605104-1-4-1', 'cs-0605104-2-4-1'], ['cs-0605104-1-4-2', 'cs-0605104-2-4-2'], ['cs-0605104-1-4-3', 'cs-0605104-2-4-3'], ['cs-0605104-1-16-0', 'cs-0605104-2-14-0'], ['cs-0605104-1-16-1', 'cs-0605104-2-14-1'], ['cs-0605104-1-16-2', 'cs-0605104-2-14-2'], ['cs-0605104-1-16-3', 'cs-0605104-2-14-3'], ['cs-0605104-1-16-4', 'cs-0605104-2-14-4'], ['cs-0605104-1-16-5', 'cs-0605104-2-14-5'], ['cs-0605104-1-16-6', 'cs-0605104-2-14-6'], ['cs-0605104-1-12-0', 'cs-0605104-2-12-0'], ['cs-0605104-1-12-1', 'cs-0605104-2-12-1']] | [['cs-0605104-1-3-0', 'cs-0605104-2-3-0'], ['cs-0605104-1-3-3', 'cs-0605104-2-3-3'], ['cs-0605104-1-8-2', 'cs-0605104-2-8-2'], ['cs-0605104-1-1-0', 'cs-0605104-2-1-0'], ['cs-0605104-1-11-0', 'cs-0605104-2-11-0'], ['cs-0605104-1-7-1', 'cs-0605104-2-7-1'], ['cs-0605104-1-6-2', 'cs-0605104-2-6-2'], ['cs-0605104-1-12-2', 'cs-0605104-2-12-2']] | [] | [['cs-0605104-1-10-2', 'cs-0605104-2-10-0'], ['cs-0605104-1-6-4', 'cs-0605104-2-6-4']] | [] | ['cs-0605104-1-17-0', 'cs-0605104-1-19-0', 'cs-0605104-2-15-0', 'cs-0605104-2-17-0', 'cs-0605104-2-18-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/cs/0605104 | null | null | null | null | null |
1210.7062 | {'1210.7062-1-0-0': 'We consider a model for a one-sided limit order book proposed by Lakner et al. [CITATION] and show that it can be coupled with a branching random walk.', '1210.7062-1-0-1': 'We then use the coupling to answer non-trivial questions about the long-term behavior of the price.', '1210.7062-1-0-2': 'The coupling relies on a classical idea of enriching the state-space by artificially creating a filiation, in this context between orders of the book, that we believe has the potential of being useful for a broader class of models.', '1210.7062-1-1-0': '# Introduction', '1210.7062-1-2-0': 'Limit order books.', '1210.7062-1-2-1': 'A limit order book is a financial trading mechanism that keeps track of orders made by traders, and allows to execute them in the future.', '1210.7062-1-2-2': 'Typically, a trader places an order to buy a commodity at a certain level [MATH].', '1210.7062-1-2-3': 'If the current price of the commodity is larger than [MATH] when the order is placed, then the order is kept in the book and may be fulfilled later in the future, as the price of the commodity fluctuates and falls below [MATH].', '1210.7062-1-2-4': 'Similarly, traders may place sell orders, which gives rise to two-sided order books.', '1210.7062-1-2-5': 'Because of the importance of limit order books in financial markets, there has been a lot of research on these models, see for instance the survey by Gould et al. [CITATION].', '1210.7062-1-3-0': 'There are many variants of limit order books, depending for instance on which information of the book traders have access to.', '1210.7062-1-3-1': 'Typically, traders have only access to the current so-called bid and ask prices, that correspond to the lowest sell order and the highest buy order, respectively.', '1210.7062-1-3-2': 'This gives an incentive for traders to place orders in the vicinity of these prices, thus creating an intricate dynamic where the state of the book influences its evolution.', '1210.7062-1-3-3': 'Stochastic models capturing this dynamic have for instance been proposed in Cont et al. [CITATION], Lakner et al. [CITATION] and Yudovina [CITATION].', '1210.7062-1-3-4': 'In the present paper we study the one-sided limit order book model of Lakner et al. [CITATION], and our goal is to show how some properties of this model can be efficiently studied thanks to a coupling with a branching random walk.', '1210.7062-1-4-0': 'From a high-level perspective, this coupling adds a new dimension to the initial limit order book model by creating a filiation between the orders.', '1210.7062-1-4-1': 'Such ideas have been extremely successful in queueing theory, see for instance Kendall [CITATION], and we believe they can also be useful beyond the context of the model proposed here.', '1210.7062-1-4-2': 'For instance, the model proposed by Yudovina [CITATION] is also amenable to a tree representation, but the corresponding dynamic on trees is more challenging to analyze than the one here.', '1210.7062-1-5-0': 'Branching random walks.', '1210.7062-1-5-1': 'The Galton Watson process is the simplest model of branching process.', '1210.7062-1-5-2': 'It represents the size of a population that evolves in discrete time, where at every time step each individual dies and is replaced by a random number of offspring, see for instance Athreya and Ney [CITATION] for more details.', '1210.7062-1-5-3': 'A branching random walk is an extension of a Galton Watson process that adds a spatial component to the model.', '1210.7062-1-5-4': "In addition to the genealogical structure given by the Galton Watson process, each individual has some location, say on the real line [MATH], that is given by a random displacement of her parent's location.", '1210.7062-1-5-5': "Branching random walks can therefore be represented by trees with labels on the edges: the structure of the tree represents the genealogy of the underlying Galton Watson process, and the labels on the edges represent the displacement of the child with respect to her parent's location.", '1210.7062-1-5-6': 'In this paper we will consider the simplest model of branching random walks, where labels on the edges are i.i.d., and will use results by Biggins [CITATION] and Biggins et al. [CITATION] to study the limit order book model.'} | {'1210.7062-2-0-0': 'We consider a model for a one-sided limit order book proposed by Lakner et al. [CITATION].', '1210.7062-2-0-1': 'We show that it can be coupled with a branching random walk and use this coupling to answer a non-trivial question about the long-term behavior of the price.', '1210.7062-2-0-2': 'The coupling relies on a classical idea of enriching the state-space by artificially creating a filiation, in this context between orders of the book, that we believe has the potential of being useful for a broader class of models.', '1210.7062-2-1-0': '# Introduction', '1210.7062-2-2-0': 'Limit order books.', '1210.7062-2-2-1': 'A limit order book is a financial trading mechanism that keeps track of orders made by traders, and allows to execute them in the future.', '1210.7062-2-2-2': 'Typically, a trader places an order to buy a security at a certain level [MATH].', '1210.7062-2-2-3': 'If the price of the security is larger than [MATH] when the order is placed, then the order is kept in the book and may be fulfilled later in the future, as the price of the security fluctuates and falls below [MATH].', '1210.7062-2-2-4': 'Similarly, traders may place sell orders, which gives rise to two-sided order books.', '1210.7062-2-2-5': 'Because of the importance of limit order books in financial markets, there has been a lot of research on these models, see for instance the survey by Gould et al. [CITATION].', '1210.7062-2-3-0': 'There are many variants of limit order books, depending for instance on which information of the book traders have access to.', '1210.7062-2-3-1': 'For instance, traders may only have access to the current so-called bid and ask prices, that correspond to the lowest sell order and the highest buy order.', '1210.7062-2-3-2': 'In this case, traders have an incentive to place orders in the vicinity of these prices.', '1210.7062-2-3-3': 'More generally, the dynamic of a limit order book is intricate because its current state influences its future evolution.', '1210.7062-2-3-4': 'Stochastic models capturing this dynamic have for instance been proposed in Cont et al. [CITATION], Lakner et al. [CITATION] and Yudovina [CITATION].', '1210.7062-2-3-5': 'In the present paper we study the one-sided limit order book model of Lakner et al. [CITATION], and our goal is to show how some properties of this model can be efficiently studied thanks to a coupling with a branching random walk.', '1210.7062-2-4-0': 'From a high-level perspective, the coupling we introduce adds a new dimension to the initial limit order book model by creating a filiation between the orders.', '1210.7062-2-4-1': 'Such ideas have been extremely successful in queueing theory, see for instance Kendall [CITATION], and we believe they can also be useful beyond the context of the model proposed here.', '1210.7062-2-4-2': 'For instance, the model proposed by Yudovina [CITATION] is also amenable to a tree representation, but the corresponding dynamic on trees is more challenging to analyze than the one here.', '1210.7062-2-5-0': 'Branching random walks.', '1210.7062-2-5-1': 'The Galton Watson process is the simplest model of a branching process.', '1210.7062-2-5-2': 'It represents the size of a population that evolves in discrete time, where at every time step each individual dies and is replaced by a random number of offspring, see for instance Athreya and Ney [CITATION] for more details.', '1210.7062-2-5-3': 'A branching random walk is an extension of a Galton Watson process that adds a spatial component to the model.', '1210.7062-2-5-4': "In addition to the genealogical structure given by the Galton Watson process, each individual has some location, say on the real line [MATH], that is given by a random displacement of her parent's location.", '1210.7062-2-5-5': "Branching random walks can therefore be represented by trees with labels on the edges: the structure of the tree represents the genealogy of the underlying Galton Watson process, and the labels on the edges represent the displacement of the child with respect to her parent's location.", '1210.7062-2-5-6': 'In this paper we will consider the simplest model of branching random walks, where labels on the edges are i.i.d., and will use results by Biggins [CITATION] and Biggins et al. [CITATION] to study the limit order book model.'} | [['1210.7062-1-4-1', '1210.7062-2-4-1'], ['1210.7062-1-4-2', '1210.7062-2-4-2'], ['1210.7062-1-0-2', '1210.7062-2-0-2'], ['1210.7062-1-3-0', '1210.7062-2-3-0'], ['1210.7062-1-3-3', '1210.7062-2-3-4'], ['1210.7062-1-3-4', '1210.7062-2-3-5'], ['1210.7062-1-2-1', '1210.7062-2-2-1'], ['1210.7062-1-2-4', '1210.7062-2-2-4'], ['1210.7062-1-2-5', '1210.7062-2-2-5'], ['1210.7062-1-5-2', '1210.7062-2-5-2'], ['1210.7062-1-5-3', '1210.7062-2-5-3'], ['1210.7062-1-5-4', '1210.7062-2-5-4'], ['1210.7062-1-5-5', '1210.7062-2-5-5'], ['1210.7062-1-5-6', '1210.7062-2-5-6'], ['1210.7062-2-2-2', '1210.7062-3-2-2'], ['1210.7062-2-2-3', '1210.7062-3-2-3'], ['1210.7062-2-2-4', '1210.7062-3-2-4'], ['1210.7062-2-2-5', '1210.7062-3-2-5'], ['1210.7062-2-5-1', '1210.7062-3-5-1'], ['1210.7062-2-5-2', '1210.7062-3-5-2'], ['1210.7062-2-5-3', '1210.7062-3-5-3'], ['1210.7062-2-5-4', '1210.7062-3-5-4'], ['1210.7062-2-5-5', '1210.7062-3-5-5'], ['1210.7062-2-5-6', '1210.7062-3-5-6'], ['1210.7062-2-3-1', '1210.7062-3-3-1'], ['1210.7062-2-3-2', '1210.7062-3-3-2'], ['1210.7062-2-3-3', '1210.7062-3-3-3'], ['1210.7062-2-3-4', '1210.7062-3-3-4'], ['1210.7062-2-3-5', '1210.7062-3-3-5'], ['1210.7062-2-0-0', '1210.7062-3-0-0'], ['1210.7062-2-0-1', '1210.7062-3-0-1'], ['1210.7062-2-0-2', '1210.7062-3-0-2'], ['1210.7062-2-4-0', '1210.7062-3-4-0'], ['1210.7062-2-4-1', '1210.7062-3-4-1'], ['1210.7062-2-4-2', '1210.7062-3-4-2'], ['1210.7062-1-4-0', '1210.7062-2-4-0'], ['1210.7062-1-3-1', '1210.7062-2-3-1'], ['1210.7062-1-2-2', '1210.7062-2-2-2'], ['1210.7062-1-2-3', '1210.7062-2-2-3'], ['1210.7062-1-5-1', '1210.7062-2-5-1'], ['1210.7062-2-2-1', '1210.7062-3-2-1'], ['1210.7062-1-0-0', '1210.7062-2-0-0'], ['1210.7062-1-0-0', '1210.7062-2-0-1'], ['1210.7062-1-0-1', '1210.7062-2-0-1'], ['1210.7062-1-3-2', '1210.7062-2-3-2'], ['1210.7062-2-3-0', '1210.7062-3-3-0']] | [['1210.7062-1-4-1', '1210.7062-2-4-1'], ['1210.7062-1-4-2', '1210.7062-2-4-2'], ['1210.7062-1-0-2', '1210.7062-2-0-2'], ['1210.7062-1-3-0', '1210.7062-2-3-0'], ['1210.7062-1-3-3', '1210.7062-2-3-4'], ['1210.7062-1-3-4', '1210.7062-2-3-5'], ['1210.7062-1-2-1', '1210.7062-2-2-1'], ['1210.7062-1-2-4', '1210.7062-2-2-4'], ['1210.7062-1-2-5', '1210.7062-2-2-5'], ['1210.7062-1-5-2', '1210.7062-2-5-2'], ['1210.7062-1-5-3', '1210.7062-2-5-3'], ['1210.7062-1-5-4', '1210.7062-2-5-4'], ['1210.7062-1-5-5', '1210.7062-2-5-5'], ['1210.7062-1-5-6', '1210.7062-2-5-6'], ['1210.7062-2-2-2', '1210.7062-3-2-2'], ['1210.7062-2-2-3', '1210.7062-3-2-3'], ['1210.7062-2-2-4', '1210.7062-3-2-4'], ['1210.7062-2-2-5', '1210.7062-3-2-5'], ['1210.7062-2-5-1', '1210.7062-3-5-1'], ['1210.7062-2-5-2', '1210.7062-3-5-2'], ['1210.7062-2-5-3', '1210.7062-3-5-3'], ['1210.7062-2-5-4', '1210.7062-3-5-4'], ['1210.7062-2-5-5', '1210.7062-3-5-5'], ['1210.7062-2-5-6', '1210.7062-3-5-6'], ['1210.7062-2-3-1', '1210.7062-3-3-1'], ['1210.7062-2-3-2', '1210.7062-3-3-2'], ['1210.7062-2-3-3', '1210.7062-3-3-3'], ['1210.7062-2-3-4', '1210.7062-3-3-4'], ['1210.7062-2-3-5', '1210.7062-3-3-5'], ['1210.7062-2-0-0', '1210.7062-3-0-0'], ['1210.7062-2-0-1', '1210.7062-3-0-1'], ['1210.7062-2-0-2', '1210.7062-3-0-2'], ['1210.7062-2-4-0', '1210.7062-3-4-0'], ['1210.7062-2-4-1', '1210.7062-3-4-1'], ['1210.7062-2-4-2', '1210.7062-3-4-2']] | [['1210.7062-1-4-0', '1210.7062-2-4-0'], ['1210.7062-1-3-1', '1210.7062-2-3-1'], ['1210.7062-1-2-2', '1210.7062-2-2-2'], ['1210.7062-1-2-3', '1210.7062-2-2-3'], ['1210.7062-1-5-1', '1210.7062-2-5-1'], ['1210.7062-2-2-1', '1210.7062-3-2-1']] | [] | [['1210.7062-1-0-0', '1210.7062-2-0-0'], ['1210.7062-1-0-0', '1210.7062-2-0-1'], ['1210.7062-1-0-1', '1210.7062-2-0-1'], ['1210.7062-1-3-2', '1210.7062-2-3-2'], ['1210.7062-2-3-0', '1210.7062-3-3-0']] | [] | ['1210.7062-1-2-0', '1210.7062-1-5-0', '1210.7062-2-2-0', '1210.7062-2-5-0', '1210.7062-3-2-0', '1210.7062-3-5-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1210.7062 | {'1210.7062-3-0-0': 'We consider a model for a one-sided limit order book proposed by Lakner et al. [CITATION].', '1210.7062-3-0-1': 'We show that it can be coupled with a branching random walk and use this coupling to answer a non-trivial question about the long-term behavior of the price.', '1210.7062-3-0-2': 'The coupling relies on a classical idea of enriching the state-space by artificially creating a filiation, in this context between orders of the book, that we believe has the potential of being useful for a broader class of models.', '1210.7062-3-1-0': '# Introduction', '1210.7062-3-2-0': 'Limit order books.', '1210.7062-3-2-1': 'A limit order book is a financial trading mechanism that keeps track of orders made by traders, and allows them to execute trades in the future.', '1210.7062-3-2-2': 'Typically, a trader places an order to buy a security at a certain level [MATH].', '1210.7062-3-2-3': 'If the price of the security is larger than [MATH] when the order is placed, then the order is kept in the book and may be fulfilled later in the future, as the price of the security fluctuates and falls below [MATH].', '1210.7062-3-2-4': 'Similarly, traders may place sell orders, which gives rise to two-sided order books.', '1210.7062-3-2-5': 'Because of the importance of limit order books in financial markets, there has been a lot of research on these models, see for instance the survey by Gould et al. [CITATION].', '1210.7062-3-3-0': 'There are many variants of information of the book which traders have access to.', '1210.7062-3-3-1': 'For instance, traders may only have access to the current so-called bid and ask prices, that correspond to the lowest sell order and the highest buy order.', '1210.7062-3-3-2': 'In this case, traders have an incentive to place orders in the vicinity of these prices.', '1210.7062-3-3-3': 'More generally, the dynamic of a limit order book is intricate because its current state influences its future evolution.', '1210.7062-3-3-4': 'Stochastic models capturing this dynamic have for instance been proposed in Cont et al. [CITATION], Lakner et al. [CITATION] and Yudovina [CITATION].', '1210.7062-3-3-5': 'In the present paper we study the one-sided limit order book model of Lakner et al. [CITATION], and our goal is to show how some properties of this model can be efficiently studied thanks to a coupling with a branching random walk.', '1210.7062-3-4-0': 'From a high-level perspective, the coupling we introduce adds a new dimension to the initial limit order book model by creating a filiation between the orders.', '1210.7062-3-4-1': 'Such ideas have been extremely successful in queueing theory, see for instance Kendall [CITATION], and we believe they can also be useful beyond the context of the model proposed here.', '1210.7062-3-4-2': 'For instance, the model proposed by Yudovina [CITATION] is also amenable to a tree representation, but the corresponding dynamic on trees is more challenging to analyze than the one here.', '1210.7062-3-5-0': 'Branching random walks.', '1210.7062-3-5-1': 'The Galton Watson process is the simplest model of a branching process.', '1210.7062-3-5-2': 'It represents the size of a population that evolves in discrete time, where at every time step each individual dies and is replaced by a random number of offspring, see for instance Athreya and Ney [CITATION] for more details.', '1210.7062-3-5-3': 'A branching random walk is an extension of a Galton Watson process that adds a spatial component to the model.', '1210.7062-3-5-4': "In addition to the genealogical structure given by the Galton Watson process, each individual has some location, say on the real line [MATH], that is given by a random displacement of her parent's location.", '1210.7062-3-5-5': "Branching random walks can therefore be represented by trees with labels on the edges: the structure of the tree represents the genealogy of the underlying Galton Watson process, and the labels on the edges represent the displacement of the child with respect to her parent's location.", '1210.7062-3-5-6': 'In this paper we will consider the simplest model of branching random walks, where labels on the edges are i.i.d., and will use results by Biggins [CITATION] and Biggins et al. [CITATION] to study the limit order book model.'} | null | null | null | null |
1908.06119 | {'1908.06119-1-0-0': 'Atomically thin films of Pb on Si(111) provide an experimentally tunable system comprising a highly structured electronic density of states.', '1908.06119-1-0-1': 'The lifetime of excited electrons in these states is limited by both electron-electron (e-e) and electron-phonon (e-ph) scattering.', '1908.06119-1-0-2': 'We employ the description by a Master equation for the electronic occupation numbers to analyze the relative importance of both scattering mechanisms.', '1908.06119-1-0-3': 'The electronic and phononic band structures, as well as the matrix elements for electron-phonon coupling within deformation potential theory were obtained from density functional calculations, thus taking into account quantum confinement effects.', '1908.06119-1-0-4': 'For the relaxation dynamics, the contribution of impact ionization processes to the lifetime is estimated from the imaginary part of the electronic self-energy calculated in the [MATH] approximation.', '1908.06119-1-0-5': 'By numerically solving rate equations for the occupations of the Pb-derived electronic states coupled to a phononic heat bath, we are able to follow the distribution of the electronic excitation energy to the various modes of Pb lattice vibrations.', '1908.06119-1-0-6': 'While e-e scattering is the dominant relaxation mechanism, we demonstrate that the e-ph scattering is highly phonon-mode-specific, with a large contribution from surface phonons.', '1908.06119-1-0-7': "At electron energies of about 0.3 eV above the Fermi surface, a 'phonon bottleneck' characteristic of relaxation in nanostructures with well-separated electronic states is observed.", '1908.06119-1-0-8': 'The time scales extracted from the simulations are compared to data from pump-probe experiments using time-resolved two-photon photoemission.', '1908.06119-1-1-0': '# Introduction', '1908.06119-1-2-0': 'The thermalization of hot carriers in metals after optical excitation is accomplished both by the Coulomb scattering among the carriers (electron-electron (e-e) interactions) and by the scattering of electrons and holes by lattice vibrations (electron-phonon (e-ph) interaction).', '1908.06119-1-2-1': 'In a well-established picture [CITATION], the relaxation can be understood as a two-step process: at early times ([MATH] fs), e-e scattering dominates and brings the electrons to a thermal (or possibly non-thermal) distribution.', '1908.06119-1-2-2': 'At later times ([MATH] ps) the e-ph interaction establishes equilibrium between the electronic distribution and the lattice temperature.', '1908.06119-1-2-3': 'In this second stage, the high density of excited carriers close to the Fermi energy (within an energy interval corresponding to few phonon quanta) is thought to be responsible for most of the energy flow between the electronic and the phononic system.', '1908.06119-1-2-4': 'If so, the e-ph coupling inferred from thermalization experiments should relate directly to the microscopic e-ph coupling constant that governs electric resistivity or the superconducting transition temperature [CITATION].', '1908.06119-1-2-5': 'In this prevailing view, the role of e-ph interactions already in the early stages of relaxation is usually ignored.', '1908.06119-1-2-6': 'However, this simple picture is questioned by studies, both experimental and theoretical [CITATION], suggesting overlapping timescales of e-e and e-ph-driven thermalization.', '1908.06119-1-2-7': 'Moreover, there is little knowledge how the electrons far above the Fermi level (several tenth of eV) interact with the phonons, for instance long-lived states far above the Fermi energy.', '1908.06119-1-2-8': 'Population of such states, e.g. at the Pb-covered Si(111) surface, has been observed in photoemission experiments [CITATION].', '1908.06119-1-2-9': 'The situation at high energies is in contrast to the e-ph interaction in close vicinity to the Fermi surface, which is crucial for a variety of physical phenomena such as electrical resistivity or superconductivity induced by electron-phonon coupling in thin films [CITATION], and is quite well explored utilizing the concept of the Eliashberg function (for an overview, see Ref. [CITATION]).', '1908.06119-1-3-0': 'In this paper, we attempt to obtain a better understanding of the relative importance of e-e and e-ph interaction in highly excited states of a metal and their respective contributions to the early stage of relaxation.', '1908.06119-1-3-1': 'To introduce our approach, we have chosen thin multilayer Pb films on Si(111).', '1908.06119-1-3-2': 'The fact that this materials system shows a highly structured electronic density of states due to confinement effects [CITATION] has been a great advantage for analyzing the energy-dependent lifetime of the excited electrons using time-resolved pump-probe spectroscopy.[', '1908.06119-1-3-3': "[CITATION] The experimental results were explained in terms of e-e interaction only, and it was concluded that the electronic lifetime closely follows the behavior expected from Landau's theory of Fermi liquids.[", '1908.06119-1-3-4': '[CITATION] Yet, a contribution of e-ph scattering to the lifetime cannot be excluded completely based on the achieved level of agreement between experiment and theory.', '1908.06119-1-4-0': 'In a previous paper [CITATION] by us, we have worked out a realistic atomistic description for multilayer Pb films on Si(111) and have carried out first-principles calculations of the electronic and phononic band structure and of e-ph coupling in electronic states far away from the Fermi level.', '1908.06119-1-4-1': 'While the e-ph interaction in bulk solids has become accessible to first-principles calculations [CITATION], thin films on a substrate are still difficult to treat on a microscopic level because the adequate supercell typically contains tens to hundreds of atoms and computational costs are high.', '1908.06119-1-4-2': 'For the Pb films on Si(111), for instance, the complex phase diagram [CITATION] results in various reconstructions requiring large supercells for their description [CITATION].', '1908.06119-1-4-3': 'In the present work, we constructed a [MATH] unit cell of Si(111) matched with a [MATH] unit cell of Pb(111) to describe the atomic structure consisting of 40 Pb and 30 Si atoms.', '1908.06119-1-4-4': '[CITATION] While the two-dimensional Brillouin zone of reconstructed surface plus interface is smaller than the Brillouin zone of a bulk material, the supercell contains a large number of bands, both in the electronic and phononic spectra.', '1908.06119-1-4-5': 'Therefore, a thoughtful selection of bands will be required to arrive at a tractable model for e-ph coupling.', '1908.06119-1-5-0': 'In this paper, building upon the knowledge of our previous work [CITATION], we elaborate on the consequences of these microscopic data for the e-ph scattering rate using a kinetic Master equation.', '1908.06119-1-5-1': 'The detailed modeling of e-ph scattering is combined with a description of the e-e interaction at the level of Fermi liquid theory.', '1908.06119-1-5-2': 'This combination allows us to simulate the temporal evolution of electronic populations on the relevant scales and to make contact with experimental observations.', '1908.06119-1-6-0': '# Theory', '1908.06119-1-7-0': 'The general problem of an excited electronic system coupled to lattice degrees of freedom can be cast into the form of a model Hamiltonian written down in the language of second quantization.', '1908.06119-1-7-1': 'In this work, we emphasize the quantum nature of the phonons, while the weak coupling of the electrons to phonons and to external fields can be treated in first-order perturbation theory.', '1908.06119-1-7-2': 'Thus, the total Hamiltonian takes the form [MATH] with [MATH] being the ground-state Hamiltonian with phonons described in the harmonic approximation, [EQUATION]', '1908.06119-1-7-3': 'The creation and annihilation operators [MATH] and [MATH] obey the usual anticommutator relations for fermions and commutator relations for bosons, respectively.', '1908.06119-1-7-4': 'The first, integer index [MATH] specifies the band, while the second index [MATH] describes the crystal momentum in the form of a two-dimensional vector within the Brillouin zone of a thin slab.', '1908.06119-1-7-5': 'Capital letters are used to index phonon modes, whereas small letters refer to electronic bands.', '1908.06119-1-8-0': 'We are interested in a realistic description of the ground state of a particular system.', '1908.06119-1-8-1': 'For this reason, all the band energies and phonon frequencies entering [MATH] are determined by density-functional theory calculations.', '1908.06119-1-8-2': 'The VASP code [CITATION] with the settings described in Ref. [CITATION] has been employed for this purpose.', '1908.06119-1-8-3': 'The electronic single-particle energies [MATH] are taken to be equal to the Kohn-Sham eigenvalues obtained with the GGA-PBE exchange-correlation functional [CITATION].', '1908.06119-1-8-4': 'The phonon frequencies [MATH] and the corresponding eigenmodes are obtained from DFT calculations using the method of finite atomic displacements within a supercell, as detailed in Ref. [CITATION].', '1908.06119-1-9-0': 'The interaction Hamiltonian [MATH] contains any further interactions required to describe the problem at hand.', '1908.06119-1-9-1': 'These interactions could e.g. be the electron-electron interactions beyond the effective mean-field description of density functional theory (see below).', '1908.06119-1-9-2': 'Moreover, the interaction with an external electromagnetic field, e.g. of a laser pulse, can be considered as part of [MATH].', '1908.06119-1-9-3': 'Most importantly for the present study, [MATH] contains a term [MATH] describing in linear order the coupling of the electrons to quantized phonons, [EQUATION]', '1908.06119-1-9-4': 'The term in parentheses is linear in each phonon coordinate.', '1908.06119-1-10-0': 'In principle it is possible to describe the quantum non-equilibrium dynamics under the action of [MATH] exactly by a density matrix.', '1908.06119-1-10-1': 'Schemes for evolving the density matrix in time have been worked out [CITATION], and applications to surfaces and low-dimensional systems can be found in the literature.', '1908.06119-1-10-2': '[CITATION] However, since the system we want to describe is quite complex, we resort to a simpler description of the dynamics which is appropriate if the coherent excitation by an optical pulse and the subsequent relaxation take place on separable time scales.', '1908.06119-1-10-3': 'While quantum coherence is important during the interaction of the system with the light field, electron-electron scattering usually leads to a fast loss of coherence.', '1908.06119-1-10-4': 'In the limit of vanishing coherence, only the diagonal elements of the density matrix, the populations [MATH] of states indexed by [MATH] and the wave vector [MATH], are important.', '1908.06119-1-10-5': 'For the investigation of the ultrafast population dynamics in our system, the quantities which we have to look at are the electronic occupation numbers [MATH] and the phononic occupation numbers [MATH].', '1908.06119-1-10-6': 'For the latter, we employ a bath approximation [EQUATION].', '1908.06119-1-10-7': 'In the numerical calculations presented below, we will use different baths, one for each high-lying optical mode of the Pb film ([MATH] THz) with temperature [MATH], and a common one for all low-frequency phonons of the Pb film ([MATH] THz) with temperature [MATH].', '1908.06119-1-10-8': 'More details are given in the appendix.', '1908.06119-1-11-0': 'Using the Markov approximation and the second-order Born approximation for the transitions, it is possible to derive from the density-matrix equations a set of coupled differential equations that can be cast into the form of a Master equation (cf. Ref. [CITATION]).', '1908.06119-1-11-1': '[EQUATION]', '1908.06119-1-11-2': 'The expressions for the rates, both for scattering into and out of the state [MATH], are made up of an electronic and a phononic contribution each, i.e., [MATH].', '1908.06119-1-11-3': 'This hold for both [MATH] and [MATH] that both consist two terms owing to electron-electron scattering and electron-phonon scattering: [EQUATION]', '1908.06119-1-11-4': "Exploiting conservation of crystal momentum parallel to the film, [MATH] with the sign depending on phonon emission or absorption, the electron-phonon scattering rates originating from the Hamiltonian ([REF]) can be expressed according to Fermi's golden rule as [EQUATION]", '1908.06119-1-11-5': 'These expressions include processes where the electron absorbs a phonon as well as phonon emission processes.', '1908.06119-1-11-6': 'This is denoted by the [MATH] signs in the equations, where the minus sign stands for absorption, and the plus sign for both spontaneous and induced emission, proportional to [MATH].', '1908.06119-1-12-0': '## Initial condition', '1908.06119-1-13-0': 'Density matrix dynamics allows to treat the optical excitation of the system as part of the dynamics by including an external field in the Hamiltonian [MATH].', '1908.06119-1-13-1': 'In the velocity gauge, the light field is described by a time-dependent vector potential [MATH] in the dipole approximation, it couples to the electrons via a term proportional to [MATH] in the Hamiltonian, where [MATH] is the momentum operator.', '1908.06119-1-13-2': 'The interaction of the light field with the electrons drives a coherent dynamics.', '1908.06119-1-13-3': 'In the present work, we make the simplifying assumption that the excitation and the relaxation of the system take place subsequently, i.e., we bypass the time interval of coherent dynamics and start directly with an initial condition [MATH] for the incoherent relaxation processes described by the Master equation.', '1908.06119-1-13-4': 'The initial distribution is chosen such that it describes the response of our specific system, multilayers of Pb on Si(111) to a short optical pulse with frequency centered around [MATH] eV.', '1908.06119-1-13-5': 'This corresponds to the photon energy of the pump laser used in the experiment [CITATION].', '1908.06119-1-13-6': 'The polarization of the electric field, denoted by the unit vector [MATH], is chosen parallel to the Pb film surface.', '1908.06119-1-13-7': 'Before the laser pulse arrives, the system is described by a Fermi-Dirac distribution with low temperature, [MATH]; hence [MATH].', '1908.06119-1-13-8': 'To be specific, we evaluate the expression [EQUATION]', '1908.06119-1-13-9': 'In the numerical evaluation, a broadening of the [MATH]-function by 0.02 eV is used.', '1908.06119-1-13-10': 'The dipole matrix elements are delivered by the VASP code using the keyword LOPTICS [CITATION].', '1908.06119-1-13-11': 'The proportionality factor [MATH] is chosen such that the energy of excited electrons and holes deposited in the Pb films amounts to [MATH] eV per supercell area, equivalent to 3.7 [MATH]J/cm[MATH].', '1908.06119-1-14-0': '## Model for e-e scattering', '1908.06119-1-15-0': 'In principle, the part of the perturbation Hamiltonian [MATH] responsible for e-e scattering can be written down exactly in the language of second-order quantization.', '1908.06119-1-15-1': 'It essentially consists of those parts of the screened Coulomb interaction between electrons not covered by the exchange-correlation potential of DFT.', '1908.06119-1-15-2': 'Due to the strong screening in the metallic film, this interaction is short-ranged, and it is assumed here that it can be evaluated in the bulk metal, rather than for the film, without loss of information.', '1908.06119-1-15-3': 'Within the same approximations as made already in deriving the Master equation, the lifetime of hot electrons due to e-e scattering can be described by a self-energy formalism, as discussed in Ref. [CITATION].', '1908.06119-1-15-4': 'The loss term [MATH] in eq. ([REF]) is given by [EQUATION]', '1908.06119-1-15-5': 'The self-energy [MATH] is obtained from a [MATH] calculation of bulk Pb.', '1908.06119-1-15-6': 'Here, [MATH] stands for the electronic Green function, and [MATH] for the screened Coulomb interaction.', '1908.06119-1-15-7': 'These quantities are calculated from the DFT wave functions and Kohn-Sham eigenvalues using the built-in capabilities of VASP [CITATION].', '1908.06119-1-15-8': 'To be specific, a [MATH] k-point mesh is used, and the denominator in [MATH] is evaluated with a small shift of the transition energy away from the real axis, [MATH] eV, much smaller than typical values used in [MATH] calculations of band structures.', '1908.06119-1-15-9': "The result obtained for [MATH] in the conduction band is fitted to the [MATH] dependence expected from Landau's theory of the Fermi liquid.", '1908.06119-1-15-10': 'Our result [MATH] (eV)[MATH] is in excellent agreement with earlier [MATH] calculations of bulk Pb [CITATION].', '1908.06119-1-15-11': 'Finally, we obtain the expression [EQUATION] in words, an electron at one eV above the Fermi level has a lifetime of 30 fs.', '1908.06119-1-15-12': 'This dependence is plotted in Fig. [REF].', '1908.06119-1-16-0': 'Although the scattering-in term [MATH] of eq. ([REF]) can be obtained from the Master equation [CITATION] as well, we choose for computational convenience a simpler treatment in our present study.', '1908.06119-1-16-1': 'The gain term is assumed to factorize into an energy-dependent and a time-dependent factor, [EQUATION]', '1908.06119-1-16-2': 'The distribution function [MATH] describes the secondary electrons and holes produced via impact ionization by a relaxing high-energy electron.', '1908.06119-1-16-3': 'Following the work of Baranov and Kabanov [CITATION], we use for [MATH] a stationary solution of the Boltzmann equation with a Coulomb scattering kernel, [EQUATION] where the electronic temperature [MATH] K was chosen in accordance with the overall energy of 0.1 eV deposited by the laser and the electronic heat capacity of Pb.', '1908.06119-1-16-4': 'The time-dependent factor [MATH] for creation of secondary electrons and holes is determined by energy conservation in the e-e scattering.', '1908.06119-1-16-5': 'Our simplified treatment assumes that the initial electron in state [MATH] ends up at the Fermi energy, transferring all its initial energy to secondary electron-hole pairs.', '1908.06119-1-16-6': 'This motivates the choice [EQUATION] where [MATH] is the electronic density of states of the particular Pb film under study, [EQUATION]', '1908.06119-1-16-7': 'Both [MATH] and [MATH] are evaluated numerically using as input the DFT band structure of slab models for Pb/Si(111)[MATH] multilayer films.', '1908.06119-1-17-0': '## Model for e-ph scattering', '1908.06119-1-18-0': 'It is our goal to calculate the contribution of e-ph scattering, in addition to e-e scattering, to the lifetime of specific quantum well states in Pb/Si(111) films.', '1908.06119-1-18-1': 'As described in the experimental paper [CITATION], there are significant differences between the lifetimes in films with an even and an odd number of Pb layers.', '1908.06119-1-18-2': 'Therefore, we study two representative systems, a Pb film with 4 monolayers (ML) and one with 5 ML on Si(111).', '1908.06119-1-18-3': 'Side views of the corresponding slabs are depicted in Fig. [REF].', '1908.06119-1-18-4': 'Motivated by the experimental focus on excited electrons in unoccupied bands, we include e-ph scattering rates for the electrons excited into quantum well states.', '1908.06119-1-18-5': 'Since the population of the valence bands was not analyzed in these experiments, the hole states are treated in less detail, and and only Coulomb scattering, as described in the previous Section [REF], is considered among the holes.', '1908.06119-1-18-6': 'To solve the rate equations, we need explicit expressions for the quantities [MATH] and [MATH] in eq. ([REF]) entering the decay rates [MATH] and [MATH].', '1908.06119-1-18-7': 'Both quantities depend on the phonon branches [MATH].', '1908.06119-1-18-8': 'Of all phonon modes [MATH] of the supercell obtained with our first-principles approach [CITATION], those with Pb character are taken into account, see Fig. 3 and 4 in Ref. [CITATION].', '1908.06119-1-18-9': 'This amounts to [MATH] for the 4 ML Pb slab and [MATH] for the 5 ML Pb film on Si(111).', '1908.06119-1-18-10': 'To keep the number of individual scattering processes at a tractable level, we also restrict ourselves to a subspace of the electronic bands: Since we are interested in the electron-phonon coupling in quantum well states in Pb, only those electronic bands that have a significant overlap with the Pb [MATH] orbitals, as indicated by the VASP calculation, are retained in the Hamiltonian [MATH] in eq. ([REF]).', '1908.06119-1-18-11': 'The electronic states belonging to a specific Pb-derived band are grouped together into subsets indexed by [MATH] of all band indices [MATH].', '1908.06119-1-18-12': 'To be specific, we used the five (six) lowest-lying conduction bands with appreciable Pb [MATH] character for the 4 ML and 5 ML Pb film, respectively, i.e. [MATH].', '1908.06119-1-18-13': 'These 5 (6) bands, selected from the full band structure shown in [CITATION], are reproduced in Fig. [REF] for the 4 (5) ML Pb films.', '1908.06119-1-18-14': 'Due to the use of a supercell and backfolding of the bands, these bandstructures are different from the bandstructure of Pb(111)[MATH] slabs that had been used in previous discussions [CITATION].', '1908.06119-1-19-0': 'For evaluating the electron-phonon scattering rates, eq. ([REF]), we use techniques based on deformation potential theory that allows us to obtain [MATH] from first-principles calculations of the phonon spectrum, the electronic wavefunctions, and Kohn-Sham eigenvalues, with only few approximations.', '1908.06119-1-19-1': 'As the most significant one, we neglect of the [MATH]-dependence of the deformation potential, while keeping its dependence on band index [MATH] and crystal momentum [MATH].', '1908.06119-1-19-2': 'This is a good approximation for optical phonons and corresponds to keeping the leading (constant) term in an expansion in powers of [MATH], cf. Ref. [CITATION].', '1908.06119-1-19-3': 'In the energy-conserving [MATH]-function in eq. ([REF]), we retain the finite phonon energy [MATH], but neglect the dispersion of the optical phonon branches.', '1908.06119-1-19-4': 'This is justified since the dispersion remains small (cf. Fig. 6 in Ref. [CITATION]) due to the large real-space unit cell, and hence small Brillouin zone, of the Pb films.', '1908.06119-1-19-5': 'Within these approximations, the matrix element for electron-phonon scattering in eq. ([REF]) can be replaced by [EQUATION]', '1908.06119-1-19-6': 'Here [MATH] is the area of the Si(111)[MATH] supercell used to model the Pb/Si(111) film, [MATH] and [MATH] are the atomic mass and atomic volume of Pb.', '1908.06119-1-19-7': '[MATH] is the deformation potential of the [MATH] electronic band under the phonon mode [MATH].', '1908.06119-1-19-8': 'The [MATH] have been obtained from DFT calculations [CITATION] by evaluating the electronic eigenvalue shift under finite displacements of the atomic positions given by the corresponding mode eigenvector of the phonon.', '1908.06119-1-19-9': 'The [MATH]-symbols reflect conservation of crystal momentum in e-ph scattering, and the projection of [MATH] to the finite electronic subspace, as described above.', '1908.06119-1-19-10': 'The matrix elements [MATH] account for the difference between intra-band ([MATH]) and interband scattering ([MATH]), and for the dependence on both the initial and final electron momenta [MATH] and [MATH].', '1908.06119-1-19-11': 'They are obtained from the overlap of the corresponding DFT wave functions.', '1908.06119-1-19-12': 'More details of the derivation are given in the appendix.', '1908.06119-1-19-13': 'In summary, this approach allows us to arrive at a simplified and computationally tractable, yet parameter-free description of e-ph scattering even for such a complex systems as an overlayer on a substrate.', '1908.06119-1-20-0': '# Results', '1908.06119-1-21-0': '## Lifetime of Quantum Well States in Pb films', '1908.06119-1-22-0': 'In this Section, we compare simulation results for Pb films of 4 ML and 5 ML thickness as representatives of films with an even and odd number of layers studied experimentally in Ref. [CITATION].', '1908.06119-1-22-1': 'By solving the Master eq. ([REF]) numerically, we are able to follow the relaxation of the excited electrons in real time.', '1908.06119-1-22-2': 'We define an energy and time dependent population [EQUATION]', '1908.06119-1-22-3': 'Fig. [REF] shows on a logarithmic scale the energy distribution [MATH] of the excited electrons for various times [MATH] after the excitation.', '1908.06119-1-22-4': 'For plotting the results, the [MATH]-function in eq. ([REF]) has been replaced by a rectangle with a width of 0.06 eV.', '1908.06119-1-22-5': 'The results were obtained for excitation by a laser pulse polarized parallel to the surface with photons of 1.9 eV, as delivered by the experimentally used pump laser [CITATION].', '1908.06119-1-23-0': 'We start with a discussion of the initial distribution, shown by the thick black line, calculated according to the transition dipole strength, eq. ([REF]).', '1908.06119-1-23-1': 'For the 4 ML Pb film (Fig. [REF](a)), the distribution is highly structured with a sharp maximum at 0.58 eV and a broad peak around 1.21 eV.', '1908.06119-1-23-2': 'For the 5ML Pb film (thick black line in Fig. [REF](b)), only the peak at 1.21 eV (and possibly a short-lived peak at higher energies) remain visible, while the low-energy peak is much less pronounced.', '1908.06119-1-23-3': 'These results are in excellent agreement with the experimental observations of Ref. [CITATION].', '1908.06119-1-23-4': 'In this work, a high-energy peak in the range of 1.1 to 1.2 eV was observed for films with an odd number of Pb monolayers, whereas the peak at 0.6 eV was dominant in Pb films with an even number of layers.', '1908.06119-1-23-5': 'Note that, due to experimental limitations of the probe laser energy, excited electrons with energies lower than [MATH] eV could not be detected in Ref. [CITATION].', '1908.06119-1-24-0': 'Next, we analyze the relaxation of the energy distributions for later times.', '1908.06119-1-24-1': 'From Fig. [REF] it is obvious that all distributions develop a low-energy part corresponding a quasi-thermal distribution of secondary electrons, showing up as an exponentially decreasing function of energy.', '1908.06119-1-24-2': 'The high-energy part of the initial spectrum decays mainly due to e-e scattering, thereby creating secondary electrons via impact ionization.', '1908.06119-1-24-3': 'Therefore, the high-energy tails decay quickly, simultaneously accompanied by an increasing weight of the secondary-electron distribution.', '1908.06119-1-24-4': 'Now turning to longer time scales, we observe that the low-energy part in the 4 ML Pb film and the population in the energy window between 0.4 and 0.6 eV decay more slowly.', '1908.06119-1-24-5': 'At the same time, a broad shoulder on top of the secondary electron distribution builds up at [MATH] eV (marked by the arrow in Fig. [REF](a) and magnified in the inset).', '1908.06119-1-24-6': "Both phenomena can be traced back to the effect of e-ph scattering: Excited electrons of the initial distribution at energies of 0.4 to 0.6 eV 'glide down' the electronic band structure (cf. Fig. [REF](a) ), thereby emitting phonons.", '1908.06119-1-24-7': "The relatively slow rate at which this occurs leads to a 'phonon bottleneck', i.e., to the build-up of the shoulder centered at [MATH] eV.", '1908.06119-1-24-8': 'This effect, related to the discreteness of electronic states, is quite common for the e-ph relaxation in nanostructures, and has been observed e.g. in quantum dots [CITATION] as well as in two-dimensional layered semiconductors [CITATION].', '1908.06119-1-24-9': "While there is still continuous energy dissipation to the crystal lattice by very low-energy secondary electrons, the 'phonon bottleneck' affects electrons at higher energies and results in additional, but delayed production of phonons emitted by these electrons 'gliding down' the conduction bands.", '1908.06119-1-24-10': "A similar, but weaker 'phonon bottleneck' can be observed on top of the secondary electron distribution in the simulations for the 5 ML Pb film in Fig. [REF](b), marked by the arrow and magnified in the inset.", '1908.06119-1-25-0': 'In Fig. [REF], we analyze the time scales associated with the electronic population decay at selected energies where peaks had been found in the populations in Fig. [REF].', '1908.06119-1-25-1': 'The lines without symbols in Fig. [REF] show the decay according to the full relaxation dynamics, including both e-e and e-ph scattering.', '1908.06119-1-25-2': 'All populations show a nearly exponential decay, albeit with different decay rates.', '1908.06119-1-25-3': 'The lifetimes for the population maxima at 1.21 eV and 0.58 eV, extracted via exponential fits, are 21 fs and 101 fs, respectively.', '1908.06119-1-25-4': 'For the lowest energy of 0.46 eV, we find an initial rise of the population due to scattering-in from electrons at higher energies, followed by a population decay after about 30 fs, corresponding to a lifetime of 183 fs.', '1908.06119-1-25-5': 'The rather broad (in time) maximum of the 0.46 eV curve results from a compensation of the rates of incoming electrons from higher energies, mostly originating from e-e scattering at these high energies, and losses due to both e-e and e-ph scattering, the latter one gaining in relative importance as we go to lower energies.', '1908.06119-1-25-6': 'Interestingly, the described broad maximum of the population evolution at low energies is also seen in the experimental data [CITATION].', '1908.06119-1-25-7': 'In the computer simulation, we can deliberately turn off the e-e scattering channel after a very short initial time interval of 6 fs.', '1908.06119-1-25-8': 'In the very early stages, the e-e scattering is still permitted since it is required to establish a somewhat smoothened electron distribution including an appropriate low-energy secondary-electron part.', '1908.06119-1-25-9': 'The result of these runs are displayed in Fig. [REF] by the lines with circular symbols.', '1908.06119-1-25-10': 'If the relaxation after 6 fs proceeds by e-ph scattering only, the population at 0.46 eV initially decays on a time scale of 350 fs (green symbols in Fig. [REF](a) ), which can be taken as an estimate of the e-ph scattering rate in this energy range.', '1908.06119-1-25-11': 'This initial decay is followed by a much slower decay over several picoseconds.', '1908.06119-1-26-0': 'A similar analysis has been carried out for the 5 ML Pb film, see Fig. [REF](b).', '1908.06119-1-26-1': 'For the peak energies at 1.21, 0.81 and 0.52 eV, overall lifetimes of 21, 47 and 126 fs are obtained from exponential fits to the full relaxation dynamics.', '1908.06119-1-26-2': "Again, it is possible to estimate the relative importance of e-ph scattering by watching the population decay after the e-e scattering has been 'turned off'.", '1908.06119-1-26-3': 'From the slopes of the curves marked by the circular symbols in Fig. [REF](b), characteristic times of 24 ps and 4.1 ps are obtained for 0.81 and 0.52 eV electron energy, respectively.', '1908.06119-1-27-0': 'From this analysis, we learn that the contribution of e-ph scattering to the total lifetime of the peaks at energies larger than 0.5 eV is much smaller compared to the e-e contribution.', '1908.06119-1-27-1': 'This finding confirms the original analysis of the experimental data by Kirchmann et al. [CITATION] where e-ph scattering had been disregarded.', '1908.06119-1-27-2': 'Analyzing the experimental data for many Pb film thicknesses, they concluded that the low-energy peak is clearly observed in films with an even number of atomic layers and has a lifetime of [MATH] fs, while the high-energy peak is visible only in the odd-layer films and has an energy-dependent lifetime which turns out to be [MATH] fs for 5 ML Pb.', '1908.06119-1-27-3': 'Our simulation results of 101 fs and 21 fs are in reasonable agreement with their experimental findings, in particular if it is taken into account that the e-e scattering rate is very sensitive to the precise energetic position of the peak.', '1908.06119-1-27-4': 'The lifetime extracted from our simulations are summarized by the circular symbols in Fig. [REF].', '1908.06119-1-27-5': 'Despite the additional decay channel of e-ph scattering being taken into account, the simulated lifetimes lie above the lifetime of isolated electrons due to e-e scattering alone.', '1908.06119-1-27-6': "This is because the simulations describe a realistic distribution of excited electrons, and the incoming flux from higher-lying electronic states effectively 'conserves' the population of the lower lying states over longer times.", '1908.06119-1-28-0': '## Excitation of lattice vibrations', '1908.06119-1-29-0': 'Although the contribution of e-ph scattering to the lifetime of the quantum well states was found to be small, the low-energy states populated by the secondary electrons couple significantly to the lattice vibrations.', '1908.06119-1-29-1': 'At these low energies, the e-ph scattering as loss mechanism even dominates over e-e scattering, since the lifetime due to e-e interactions rises above 300 fs for electrons below 0.33 eV according to eq. ([REF]).', '1908.06119-1-29-2': 'With the help of the simulations, it is possible to follow the energy transferred from the electrons to each of the phonon modes separately.', '1908.06119-1-29-3': 'Fig. [REF] shows the increase in time of the excess vibrational energy (in addition to the thermal energy corresponding to the initial substrate temperature) in the various Pb vibrational modes.', '1908.06119-1-29-4': 'In our simulations, we explicitly allowed for the possibility of conversion between different vibrational modes (see appendix), but it should be noted that vibrational coupling takes place on a much longer time scale of at least 30 ps and is hardly relevant for the observations in Fig. [REF].', '1908.06119-1-29-5': 'It is seen that the energy transfer is highly mode-selective.', '1908.06119-1-29-6': 'One particular surface phonon mode receives a major part of the energy.', '1908.06119-1-29-7': 'The dominance of this single mode depends on film thickness; in the 4 ML film it is clearly more pronounced than in the 5 ML film where several modes participate in the energy uptake.', '1908.06119-1-29-8': 'In the 4 ML Pb film, this is a mode with frequency 2.26 THz (labeled 47 in our previous publication [CITATION]), while in the 5 ML Pb film it is a phonon mode at 2.03 THz.', '1908.06119-1-29-9': 'In experiments using a high laser fluence, the strong coupling to a specific mode leading to coherently driven vibrations has indeed been observed; in the 5 ML Pb film, it was detected experimentally by a periodic shift of the quantum well energy with a frequency of [MATH] THz [CITATION], which is in excellent agreement with the frequency of 2.03 THz identified in our simulation.', '1908.06119-1-29-10': 'For both film thicknesses, the modes at frequencies below 2 THz, which are similar to phonon modes in bulk Pb, receive only a much smaller amount of energy on average.', '1908.06119-1-29-11': 'As seen from Fig. [REF], a sub-linear increase of the energy over long time scales, e.g. over 4 ps, is observed which may eventually go into saturation at even longer time scales.', '1908.06119-1-29-12': 'We attribute this delayed response to the rather long time required by the electrons to relax from high energies down to [MATH] via multiple phonon emission processes.', '1908.06119-1-29-13': "Due to the 'phonon bottleneck' described above, this may take even longer than expected from estimates [CITATION] solely based on the electron-phonon coupling constant of bulk materials as determined from ultrafast reflectivity measurements [CITATION].", '1908.06119-1-29-14': 'Experimentally, the excitation of low-lying Pb modes results in thermal displacements of the atomic positions on medium to large length scale that can be followed in a time-resolved electron diffraction experiment.', '1908.06119-1-29-15': 'Preliminary experimental data for Pb films [CITATION] indicate that the low-energy phonons are indeed getting excited on the time scale of a few picoseconds.', '1908.06119-1-29-16': 'Interestingly, recent diffraction experiments were able demonstrate the mode-selective energy transfer to phonons even for bulk materials such as aluminium [CITATION] or nickel [CITATION].', '1908.06119-1-29-17': 'This seems to indicate that phonon excitation by strongly excited charge carriers opens up the exploration of a new class of non-equilibrium phenomena, not only in bulk metals, but also in semiconductors [CITATION] and nanostructures.', '1908.06119-1-29-18': 'Due to the widely different timescales of e-ph interaction and phonon-phonon interaction, the non-equilibrium distribution of phonons created by the hot carriers can persist over a timespan of several picoseconds.', '1908.06119-1-29-19': 'This appears to be a more wide-spread phenomenon than previously thought, and further research along these lines could be fruitful.', '1908.06119-1-30-0': '# Conclusion', '1908.06119-1-31-0': 'Simulations of electronic relaxation have been performed for a realistic system, metallic multilayer Pb films on Si(111), with the help of a parameter-free approach based on density functional theory.', '1908.06119-1-31-1': 'Electron-electron (e-e) and electron-phonon (e-ph) scattering were both included in the Master equation.', '1908.06119-1-31-2': 'Not surprisingly, e-e scattering was found to dominate over e-ph scattering for short times and highly excited electrons more than 0.5 eV above the Fermi level.', '1908.06119-1-31-3': 'Our simulation results thus justify the neglect of e-ph scattering in the analysis of experimental data [CITATION] in this regime.', '1908.06119-1-31-4': 'The simulated lifetimes for the quantum well states at 0.58 eV and 1.21 eV of 100 fs and 21 fs, respectively, are in reasonable agreement with the experimental findings.', '1908.06119-1-31-5': 'The importance of e-ph scattering shows up in the simulations at lower electron energies where the e-e scattering rate decreases strongly.', '1908.06119-1-31-6': 'Taking the fastest e-ph relaxation found in the present simulations with a characteristic time of 350 fs as a marker for the cross-over between e-e and e-ph scattering, we conclude that e-ph scattering significantly contributes to the relaxation of electrons in Pb at energies below 0.3 eV.', '1908.06119-1-31-7': 'Indeed, the simulations show a population pile-up around 0.3 eV due to the cross-over of the e-e and e-ph scattering time scales at this energy.', '1908.06119-1-31-8': "After 300 fs, it is fair to describe the electronic population by a thermal distribution, however, only up to an excess population of electrons getting stuck in the 'phonon bottleneck'.", '1908.06119-1-31-9': 'Remarkably, this does by no means imply that the phonon populations could be described by a temperature as well.', '1908.06119-1-31-10': 'Contrarily, the simulations show that even up to 4 ps after excitation high-frequency surface vibrational modes are preferentially excited by strongly mode-selective phonon emission.', '1908.06119-1-32-0': 'In summary, our simulations enable us to disentangle the contributions of e-e and e-ph scattering at short times, [MATH] ps after optical excitation.', '1908.06119-1-32-1': 'Although a first glance at the data is compatible with electronic thermalization by e-e scattering, a contribution of e-ph scattering can be observed already in this early stage, in particular at low electron energies.', '1908.06119-1-32-2': 'The phonon system requires much longer (several picoseconds) to equilibrate, and additional simulations beyond the scope of the present work are desirable to gain an improved understanding of the energy transfer between subsystems in the later stages of relaxation.'} | {'1908.06119-2-0-0': 'Atomically thin films of Pb on Si(111) provide an experimentally tunable system comprising a highly structured electronic density of states.', '1908.06119-2-0-1': 'The lifetime of excited electrons in these states is limited by both electron-electron (e-e) and electron-phonon (e-ph) scattering.', '1908.06119-2-0-2': 'We employ the description by a Master equation for the electronic occupation numbers to analyze the relative importance of both scattering mechanisms.', '1908.06119-2-0-3': 'The electronic and phononic band structures, as well as the matrix elements for electron-phonon coupling within deformation potential theory were obtained from density functional calculations, thus taking into account quantum confinement effects.', '1908.06119-2-0-4': 'For the relaxation dynamics, the contribution of impact ionization processes to the lifetime is estimated from the imaginary part of the electronic self-energy calculated in the [MATH] approximation.', '1908.06119-2-0-5': 'By numerically solving rate equations for the occupations of the Pb-derived electronic states coupled to a phononic heat bath, we are able to follow the distribution of the electronic excitation energy to the various modes of Pb lattice vibrations.', '1908.06119-2-0-6': 'While e-e scattering is the dominant relaxation mechanism, we demonstrate that the e-ph scattering is highly phonon-mode-specific, with a large contribution from surface phonons.', '1908.06119-2-0-7': "At electron energies of about 0.3 eV above the Fermi surface, a 'phonon bottleneck' characteristic of relaxation in nanostructures with well-separated electronic states is observed.", '1908.06119-2-0-8': 'The time scales extracted from the simulations are compared to data from pump-probe experiments using time-resolved two-photon photoemission.', '1908.06119-2-1-0': '# Introduction', '1908.06119-2-2-0': 'The thermalization of hot carriers in metals after optical excitation is accomplished both by the Coulomb scattering among the carriers (electron-electron (e-e) interactions) and by the scattering of electrons and holes by lattice vibrations (electron-phonon (e-ph) interaction).', '1908.06119-2-2-1': 'In a well-established picture [CITATION], the relaxation can be understood as a two-step process: at early times ([MATH] ps), e-e scattering dominates and brings the electrons to a thermal (or possibly non-thermal) distribution.', '1908.06119-2-2-2': 'At later times ([MATH] ps) the e-ph interaction establishes equilibrium between the electronic distribution and the lattice temperature.', '1908.06119-2-2-3': 'In this second stage, the high density of excited carriers close to the Fermi energy (within an energy interval corresponding to few phonon quanta) is thought to be responsible for most of the energy flow between the electronic and the phononic system.', '1908.06119-2-2-4': 'If so, the e-ph coupling inferred from thermalization experiments should relate directly to the microscopic e-ph coupling constant that governs electric resistivity or the superconducting transition temperature [CITATION].', '1908.06119-2-2-5': 'In this prevailing view, the role of e-ph interactions already in the early stages of relaxation is usually ignored.', '1908.06119-2-2-6': 'However, this simple picture is questioned by studies, both experimental and theoretical [CITATION], suggesting overlapping timescales of e-e and e-ph-driven thermalization.', '1908.06119-2-2-7': 'Moreover, there is little knowledge how the electrons far above the Fermi level (several tenth of eV) interact with the phonons.', '1908.06119-2-2-8': 'For instance, long-lived population of such states, e.g. at the Pb-covered Si(111) surface, has been observed in photoemission experiments [CITATION].', '1908.06119-2-2-9': 'The situation at high energies is in contrast to the e-ph interaction in close vicinity to the Fermi surface, which is crucial for a variety of physical phenomena such as electrical resistivity or superconductivity induced by electron-phonon coupling in thin films [CITATION], and is quite well explored utilizing the concept of the Eliashberg function (for an overview, see Ref. [CITATION]).', '1908.06119-2-2-10': 'In conclusion, there is a need for more studies of the e-ph interaction at energies further away from the Fermi energy.', '1908.06119-2-3-0': 'In this paper, we attempt to obtain a better understanding of the relative importance of e-e and e-ph interaction in highly excited states of a metal and their respective contributions to the early stage of relaxation.', '1908.06119-2-3-1': 'To introduce our approach, we have chosen thin multilayer Pb films on Si(111).', '1908.06119-2-3-2': 'The fact that this materials system shows a highly structured electronic density of states due to confinement effects [CITATION] has been a great advantage for analyzing the energy-dependent lifetime of the excited electrons using time-resolved pump-probe spectroscopy.[', '1908.06119-2-3-3': "[CITATION] The experimental results were rationalized in Ref. Kirchmann2010 in terms of e-e interaction only, and it was concluded that the electronic lifetime closely follows the behavior expected from Landau's theory of Fermi liquids.[", '1908.06119-2-3-4': '[CITATION] Yet, a contribution of e-ph scattering to the lifetime cannot be excluded completely based on the achieved level of agreement between experiment and theory.', '1908.06119-2-3-5': 'Therefore, we aim at a detailled analysis of the role of e-ph scattering for the features observed in photoemission.', '1908.06119-2-4-0': 'Since ample experimental and computational data are available for the Pb/Si(111) films, we consider this system a good test case for quantitative studies of electronic relaxation dynamics.', '1908.06119-2-4-1': 'In a previous paper [CITATION] by us, we have worked out a realistic atomistic description for multilayer Pb films on Si(111) and have carried out first-principles calculations of the electronic and phononic band structure and of e-ph coupling in electronic states far away from the Fermi level.', '1908.06119-2-4-2': 'While the e-ph interaction in bulk solids has become accessible to first-principles calculations by using density functional perturbation theory together with Wannier interpolation methods to enhance the number of reciprocal-space sampling points [CITATION], thin films on a substrate are still difficult to treat on a microscopic level because the adequate supercell typically contains tens to hundreds of atoms and computational costs are high.', '1908.06119-2-4-3': 'For the Pb films on Si(111), for instance, the complex phase diagram [CITATION] results in various reconstructions requiring large supercells for their description [CITATION].', '1908.06119-2-4-4': 'In the present work, we constructed a [MATH] unit cell of Si(111) matched with a [MATH] unit cell of Pb(111) to describe the atomic structure consisting of 40 Pb and 30 Si atoms.', '1908.06119-2-4-5': '[CITATION] While the two-dimensional Brillouin zone of reconstructed surface plus interface is smaller than the Brillouin zone of a bulk material, the supercell contains a large number of bands, both in the electronic and phononic spectra.', '1908.06119-2-4-6': 'Therefore, a thoughtful selection of bands will be required to arrive at a tractable model for e-ph coupling.', '1908.06119-2-4-7': 'The approach via density functional perturbation theory and the calculation of the Eliashberg function would be too cumbersome for large supercells.', '1908.06119-2-5-0': 'In this paper, building upon the knowledge of our previous work [CITATION], we elaborate on the consequences of these microscopic data for the e-ph scattering rate using a kinetic Master equation.', '1908.06119-2-5-1': 'The detailed modeling of e-ph scattering is combined with a description of the e-e interaction at the level of Fermi liquid theory.', '1908.06119-2-5-2': 'This combination allows us to simulate the temporal evolution of electronic populations on the relevant scales and to make contact with experimental observations.', '1908.06119-2-6-0': '# Theory', '1908.06119-2-7-0': 'The general problem of an excited electronic system coupled to lattice degrees of freedom can be approached from various perspectives.', '1908.06119-2-7-1': 'If one is satisfied with a classical description of the atomic positions and velocities and their dynamics can be described in the trajectory approximation, carrying out non-adiabatic molecular dynamics simulations (e.g. with the methodology described in [CITATION]) is the method of choice.', '1908.06119-2-7-2': 'As an advantage, this approach can handle large deviations of the atomic positions from their ground state, and the forces acting on the atoms are calculated directly within the first-principles electronic structure framework.', '1908.06119-2-7-3': 'Thus, it is suitable for systems with very strong and non-linear electron-phonon coupling, as encountered e.g. in two-dimensional materials [CITATION].', '1908.06119-2-7-4': 'In this work, we emphasize the quantum nature of the phonons, while the weak coupling of the electrons to phonons and to external fields can be treated in first-order perturbation theory.', '1908.06119-2-7-5': 'Casting the problem into the form of a model Hamiltonian, it reads [MATH] with [MATH] being the ground-state Hamiltonian with phonons described in the harmonic approximation, [EQUATION]', '1908.06119-2-7-6': 'The creation and annihilation operators [MATH] and [MATH] obey the usual anticommutator relations for fermions and commutator relations for bosons, respectively.', '1908.06119-2-7-7': 'The first, integer index [MATH] specifies the band, while the second index [MATH] describes the crystal momentum in the form of a two-dimensional vector within the Brillouin zone of a thin slab.', '1908.06119-2-7-8': 'Capital letters are used to index phonon modes, whereas small letters refer to electronic bands.', '1908.06119-2-7-9': 'In contrast to the molecular dynamics approach mentioned at the beginning of this paragraph, the full quantum treatment is best suited when the coupling terms in the interaction Hamiltonian [MATH] are weak, and the model Hamiltonian [MATH] provides already a good starting point for the coupled dynamics.', '1908.06119-2-8-0': 'Utilizing model Hamiltonians for describing electronic dynamics is a well-established technique in the field of ultrafast soild-state optics, see e.g. [CITATION].', '1908.06119-2-8-1': 'In semiconductor bulk materials and quantum wells, the dispersion [MATH] entering the Hamiltonian can be approximated as being quadratic (and sometimes as being linear, e.g. for graphene [CITATION]), and a full solution of the relaxation dynamics for various scattering mechanisms has been achieved in these cases.', '1908.06119-2-8-2': 'Here, we are interested in a realistic description of the ground state of a particular system.', '1908.06119-2-8-3': 'For this reason, all the band energies and phonon frequencies entering [MATH] are determined by density-functional theory calculations.', '1908.06119-2-8-4': 'The VASP code [CITATION] with the settings described in Ref. [CITATION] has been employed for this purpose.', '1908.06119-2-8-5': 'The electronic single-particle energies [MATH] are taken to be equal to the Kohn-Sham eigenvalues obtained with the GGA-PBE exchange-correlation functional [CITATION].', '1908.06119-2-8-6': 'The phonon frequencies [MATH] and the corresponding eigenmodes are obtained from DFT calculations using the method of finite atomic displacements within a supercell, as detailed in Ref. [CITATION].', '1908.06119-2-8-7': 'In case of the Pb/Si(111) films, such a detailed first-principles description is considered necessary in view of the experimental findings: The two-photon photoemission spectra show peaks at certain intermediate-state energies of the electrons that are referred to as quantum well states (QWS).', '1908.06119-2-8-8': 'These are energies where the electronic density of states is high and/or where the excited electrons are long-lived.', '1908.06119-2-8-9': 'For a correct prediction of the energetic position of the QWS, the [MATH] periodicity of a free-standing Pb(111) films is not sufficiently accurate.', '1908.06119-2-8-10': '[CITATION].', '1908.06119-2-8-11': 'It is required to take the larger [MATH] periodicity enforced by the Si(111) substrate into account.', '1908.06119-2-8-12': 'As major achievement of the first-principles calculations in Ref. [CITATION], we were able to reproduce quantitatively the dependence of the energetic position of the QWS on the number of Pb layers in the film, as well as the very small dispersion of the occupied QWS in the films with an odd number of Pb layers.', '1908.06119-2-8-13': 'On this basis, the present work is addressing the role of the electronic lifetime in the QWS for the experimentally detected peaks.', '1908.06119-2-9-0': 'The interaction Hamiltonian [MATH] contains any further interactions required to describe the problem at hand.', '1908.06119-2-9-1': 'These interactions could e.g. be the electron-electron interactions beyond the effective mean-field description of density functional theory (see below).', '1908.06119-2-9-2': 'Moreover, the interaction with an external electromagnetic field, e.g. of a laser pulse, can be considered as part of [MATH].', '1908.06119-2-9-3': 'Most importantly for the present study, [MATH] contains a term [MATH] describing in linear order the coupling of the electrons to quantized phonons, [EQUATION]', '1908.06119-2-9-4': 'The term in parentheses is linear in each phonon coordinate.', '1908.06119-2-10-0': 'In principle it is possible to describe the quantum non-equilibrium dynamics under the action of [MATH] exactly by a density matrix.', '1908.06119-2-10-1': 'Schemes for evolving the density matrix in time have been worked out [CITATION], and applications to surfaces and low-dimensional systems can be found in the literature.', '1908.06119-2-10-2': '[CITATION] However, since the system we want to describe is quite complex, we resort to a simpler description of the dynamics which is appropriate if the coherent excitation by an optical pulse and the subsequent relaxation take place on separable time scales.', '1908.06119-2-10-3': 'While quantum coherence is important during the interaction of the system with the light field, electron-electron scattering usually leads to a fast loss of coherence.', '1908.06119-2-10-4': '[CITATION] For Pb films, an example of calculations taking the effects of coherence into account can be found in [CITATION].', '1908.06119-2-10-5': 'In the limit of vanishing coherence, only the diagonal elements of the density matrix, the populations [MATH] of states indexed by [MATH] and the wave vector [MATH], are important.', '1908.06119-2-10-6': 'For the investigation of the ultrafast population dynamics in our system, the quantities which we have to look at are the electronic occupation numbers [MATH] and the phononic occupation numbers [MATH].', '1908.06119-2-10-7': 'For the latter, we employ a bath approximation [EQUATION].', '1908.06119-2-10-8': 'In the numerical calculations presented below, we will use different baths, one for each high-lying optical mode of the Pb film ([MATH] THz) with temperature [MATH], and a common one for all low-frequency phonons of the Pb film ([MATH] THz) with temperature [MATH].', '1908.06119-2-10-9': 'More details are given in the appendix.', '1908.06119-2-11-0': 'Using the Markov approximation and the second-order Born approximation for the transitions, it is possible to derive from the density-matrix equations a set of coupled differential equations that can be cast into the form of a Master equation (cf. Ref. [CITATION]).', '1908.06119-2-11-1': '[EQUATION]', '1908.06119-2-11-2': 'The expressions for the rates, both for scattering into and out of the state [MATH], are made up of an electronic and a phononic contribution each, i.e., [MATH].', '1908.06119-2-11-3': 'This hold for both [MATH] and [MATH] that both consist two terms owing to electron-electron scattering and electron-phonon scattering: [EQUATION]', '1908.06119-2-12-0': "Exploiting conservation of crystal momentum parallel to the film, [MATH] with the sign depending on phonon emission or absorption, the electron-phonon scattering rates originating from the Hamiltonian ([REF]) can be expressed according to Fermi's golden rule as [EQUATION]", '1908.06119-2-13-0': 'These expressions include processes where the electron absorbs a phonon as well as phonon emission processes.', '1908.06119-2-13-1': 'This is denoted by the [MATH] signs in the equations, where the minus sign stands for absorption, and the plus sign for both spontaneous and induced emission, proportional to [MATH].', '1908.06119-2-14-0': 'It is our goal to calculate the contribution of e-ph scattering to the lifetime of specific quantum well states (QWS) in Pb/Si(111) films.', '1908.06119-2-14-1': 'In a very simple picture, the conduction band electrons of Pb with crystal momentum normal to the surface or interface of the Pb(111) films are confined, similar to the quantum-mechanical particle-in-a-box problem.', '1908.06119-2-14-2': 'In an atomistic picture, these conduction band states are derived from the [MATH] orbitals of the Pb atoms and their wavefunctions extend both above the surface and into the Si(111) substrate, see Ref. [CITATION].', '1908.06119-2-14-3': 'As described in the experimental paper [CITATION], there are significant differences between the lifetimes in films with an even and an odd number of Pb layers.', '1908.06119-2-14-4': 'Therefore, we study two representative systems, a Pb film with 4 monolayers (ML) and one with 5 ML on Si(111).', '1908.06119-2-14-5': 'Side views of the corresponding slabs are depicted in Fig. [REF].', '1908.06119-2-14-6': 'Motivated by the experimental focus on excited electrons in unoccupied bands, we include e-ph scattering rates for the electrons excited into QWS.', '1908.06119-2-14-7': 'Since the population of the valence bands was not analyzed in these experiments, the hole states are treated in less detail, and and only Coulomb scattering, as described in Section [REF], will be considered among the holes.', '1908.06119-2-14-8': 'To solve the rate equations, we need explicit expressions for the quantities [MATH] and [MATH] in eq. ([REF]) entering the decay rates [MATH] and [MATH].', '1908.06119-2-14-9': 'Both quantities depend on the phonon branches [MATH].', '1908.06119-2-14-10': 'Of all phonon modes [MATH] of the supercell obtained with our first-principles approach [CITATION], those with Pb character are taken into account, see Fig. 6 in Ref. [CITATION].', '1908.06119-2-14-11': 'This amounts to [MATH] for the 4 ML Pb slab and [MATH] for the 5 ML Pb film on Si(111).', '1908.06119-2-14-12': 'To keep the number of individual scattering processes at a tractable level, we also restrict ourselves to a subspace of the electronic bands: Since we are interested in the electron-phonon coupling in QWS in Pb, only those electronic bands that have a significant overlap with the Pb [MATH] orbitals, as indicated by the VASP calculation, are retained in the Hamiltonian [MATH] in eq. ([REF]).', '1908.06119-2-14-13': 'The electronic states belonging to a specific Pb-derived band are grouped together into subsets indexed by [MATH] of all band indices [MATH].', '1908.06119-2-14-14': 'To be specific, we used the five (six) lowest-lying conduction bands with appreciable Pb [MATH] character for the 4 ML and 5 ML Pb film, respectively, i.e. [MATH].', '1908.06119-2-14-15': 'These 5 (6) bands are displayed in Fig. [REF] by the thick red lines and symbols, together with the full band structure (dashed lines) that is also shown (over a wider range of energies and wavevectors) in Fig. 3 and 4 of Ref. [CITATION].', '1908.06119-2-14-16': 'Due to the use of a supercell and backfolding of the bands, these bandstructures are different from the bandstructure of Pb(111)[MATH] slabs that had been used previously[CITATION] in the experimental data analysis.', '1908.06119-2-15-0': 'For evaluating the electron-phonon scattering rates, eq. ([REF]), we use techniques based on deformation potential theory that allows us to obtain [MATH] from first-principles calculations of the phonon spectrum, the electronic wavefunctions, and Kohn-Sham eigenvalues, with only few approximations.', '1908.06119-2-15-1': 'As the most significant one, we neglect of the [MATH]-dependence of the deformation potential, while keeping its dependence on band index [MATH] and crystal momentum [MATH].', '1908.06119-2-15-2': 'This is a good approximation for optical phonons and corresponds to keeping the leading (constant) term in an expansion in powers of [MATH], cf. Ref. [CITATION].', '1908.06119-2-15-3': 'In the energy-conserving [MATH]-function in eq. ([REF]), we retain the finite phonon energy [MATH], but neglect the dispersion of the optical phonon branches.', '1908.06119-2-15-4': 'This is justified since the dispersion remains small (cf. Fig. 6 in Ref. [CITATION]) due to the large real-space unit cell, and hence small Brillouin zone, of the Pb films.', '1908.06119-2-15-5': 'Within these approximations (see appendix for the derivation), the matrix element for electron-phonon scattering in eq. ([REF]) can be replaced by [EQUATION]', '1908.06119-2-15-6': 'Here [MATH] is the area of the Si(111)[MATH] supercell used to model the Pb/Si(111) film, [MATH] and [MATH] are the atomic mass and atomic volume of Pb.', '1908.06119-2-15-7': '[MATH] is the deformation potential of the [MATH] electronic band under the phonon mode [MATH].', '1908.06119-2-15-8': 'The [MATH] have been obtained from DFT calculations [CITATION] by evaluating the electronic eigenvalue shift under finite displacements of the atomic positions given by the corresponding mode eigenvector of the phonon.', '1908.06119-2-15-9': 'The two [MATH]-symbols reflect conservation of crystal momentum in e-ph scattering, and the projection of [MATH] to the finite electronic subspace, as described above.', '1908.06119-2-15-10': 'The matrix elements [MATH] account for the difference between intra-band ([MATH]) and interband scattering ([MATH]), and for the dependence on both the initial and final electron momenta [MATH] and [MATH].', '1908.06119-2-15-11': 'They are obtained from the overlap of the corresponding DFT wave functions.', '1908.06119-2-15-12': 'More details of the derivation are given in the appendix.', '1908.06119-2-15-13': 'In summary, this approach allows us to arrive at a simplified and computationally tractable, yet parameter-free description of e-ph scattering even for such a complex systems as an overlayer on a substrate.', '1908.06119-2-16-0': '# Results', '1908.06119-2-17-0': 'In metals, e-ph and e-e scattering are closely intertwined, since the vast majority of phonons is emitted by secondary electrons and holes rather than by the charge carriers initially excited by the light pulse.', '1908.06119-2-17-1': 'This is because e-e scattering quickly generates an avalanche of secondary electron-hole pairs with small energies around the Fermi level.', '1908.06119-2-17-2': 'Since these secondary electrons and holes are produced with high density and their energy still exceeds typical phonon energies, they play a major role in determining the rate at which the energy is dissipated from the electronic system into the lattice.', '1908.06119-2-17-3': 'Nevertheless, we start our discussion by considering the contribution of both e-ph and e-e scattering separately.', '1908.06119-2-18-0': '## Relaxation due to e-ph scattering', '1908.06119-2-19-0': 'First we investigate how the population of a QWS decays under the sole effect of e-ph scattering.', '1908.06119-2-19-1': 'For this purpose, we initially populate a single QWS at the [MATH]-point and let the population evolve according to the master equation ([REF]) using only the rates [MATH] and [MATH].', '1908.06119-2-19-2': 'The results are shown in Fig. [REF](a).', '1908.06119-2-19-3': 'At comparable energies of the QWS of [MATH]eV, the decay is much faster in the 4 ML than in the 5 ML Pb film.', '1908.06119-2-19-4': 'This is to be expected from the different size of the deformation potentials in the two films reported in Ref. [CITATION].', '1908.06119-2-19-5': 'The relaxation rate increases with the temperature of the phonon heat bath, which is indicative of the role of stimulated emission of phonons.', '1908.06119-2-19-6': 'by decreasing the phonon temperature from 400 K and 100 K, the lifetime of the QWS in the 4 ML film increases from 1.3 to 2.7 ps.', '1908.06119-2-19-7': 'For the 5 ML film, the lifetimes fall between 13 and 37 ps.', '1908.06119-2-20-0': '## Relaxation due to e-e scattering', '1908.06119-2-21-0': 'The lifetime of hot electrons due to e-e scattering can be described by a self-energy formalism, as discussed in Ref. [CITATION].', '1908.06119-2-21-1': 'The loss term [MATH] in eq. ([REF]) is given by [MATH].', '1908.06119-2-21-2': 'The self-energy [MATH] is obtained from a [MATH] calculation of bulk Pb.', '1908.06119-2-21-3': 'Here, [MATH] stands for the electronic Green function, and [MATH] for the screened Coulomb interaction.', '1908.06119-2-21-4': 'These quantities are calculated from the DFT wave functions and Kohn-Sham eigenvalues using the built-in capabilities of VASP [CITATION].', '1908.06119-2-21-5': 'To be specific, a [MATH] k-point mesh is used, and the denominator in [MATH] is evaluated with a small shift of the transition energy away from the real axis, [MATH] eV, much smaller than typical values used in [MATH] calculations of band structures.', '1908.06119-2-21-6': "The result obtained for [MATH] in the conduction band is fitted to the [MATH] dependence expected from Landau's theory of the Fermi liquid.", '1908.06119-2-21-7': 'Our result [MATH] (eV)[MATH] is in excellent agreement with earlier [MATH] calculations of bulk Pb [CITATION].', '1908.06119-2-21-8': 'Finally, we obtain the expression [EQUATION] plotted in Fig. [REF](b).', '1908.06119-2-22-0': 'In the relaxation of highly excited electrons and holes, the energy is dissipated to secondary electron-hole pairs.', '1908.06119-2-22-1': 'This process is very efficient in metals, since, in contrast to semiconductors, there is no energy gap preventing the generation of secondary particles.', '1908.06119-2-22-2': 'These effects are included in the the scattering-in term [MATH] of eq. ([REF]).', '1908.06119-2-22-3': 'Although this term can be obtained from the Master equation [CITATION] as well, we choose for computational convenience a simpler treatment in our present study.', '1908.06119-2-22-4': 'The gain term is assumed to factorize into an energy-dependent and a time-dependent factor, [MATH].', '1908.06119-2-22-5': 'The distribution function [MATH] describes the secondary electrons and holes produced via impact ionization by a relaxing high-energy electron.', '1908.06119-2-22-6': 'Following the work of Baranov and Kabanov [CITATION], we use for [MATH] a stationary solution of the Boltzmann equation with a Coulomb scattering kernel, [MATH], where the electronic temperature [MATH] K was chosen in accordance with the energy of [MATH] deposited by the laser and the electronic heat capacity of Pb [MATH].', '1908.06119-2-22-7': 'The time-dependent factor [MATH] for creation of secondary electrons and holes is determined by energy conservation in the e-e scattering.', '1908.06119-2-22-8': 'Our simplified treatment assumes that the initial electron in state [MATH] ends up at the Fermi energy, transferring all its initial energy to secondary electron-hole pairs.', '1908.06119-2-22-9': 'This motivates the choice [EQUATION] where [MATH] is the electronic density of states.', '1908.06119-2-22-10': 'Both [MATH] and [MATH] are evaluated numerically using as input the DFT band structure of slab models for Pb/Si(111)[MATH] multilayer films.', '1908.06119-2-23-0': '## Competition between e-e and e-ph scattering', '1908.06119-2-24-0': 'In this Section, we compare simulation results for Pb films of 4 ML and 5 ML thickness as representatives of films with an even and odd number of layers studied by optical pump-probe experiments in Ref. [CITATION].', '1908.06119-2-24-1': 'While these experiments measure the total probability for two-photon photoemission, our simulations model the population of the intermediate electronic states that are reached by the electron after applying the pump pulse and subsequent relaxation.', '1908.06119-2-24-2': 'The second step of the two-photon photoemission, which kicks the electron into the vacuum, is not modeled.', '1908.06119-2-24-3': 'Provided that the probability of ionization by the probe pulse is a smooth function of energy, the measured yield can be considered approximately proportional to the population of the intermediate state.', '1908.06119-2-25-0': 'The initial distribution is chosen such that it describes the response of our specific system, multilayers of Pb on Si(111), to a short optical pulse with frequency centered around [MATH] eV.', '1908.06119-2-25-1': 'This corresponds to the photon energy of the pump laser used in the experiment [CITATION].', '1908.06119-2-25-2': 'The polarization of the electric field, denoted by the unit vector [MATH], is chosen parallel to the Pb film surface.', '1908.06119-2-25-3': 'Before the laser pulse arrives, the system is described by a Fermi-Dirac distribution with low temperature, [MATH]; hence [MATH].', '1908.06119-2-25-4': 'To be specific, we evaluate dipole matrix elements[CITATION] [EQUATION]', '1908.06119-2-25-5': 'In the numerical evaluation , a broadening of the [MATH]-function by 0.02 eV is used.', '1908.06119-2-25-6': 'The proportionality factor [MATH] is chosen such that the energy of excited electrons and holes deposited in the Pb films amounts to [MATH] eV per supercell area, equivalent to 3.7 [MATH]J/cm[MATH].', '1908.06119-2-26-0': 'By solving the Master eq. ([REF]) numerically, we are able to follow the relaxation of the excited electrons in real time.', '1908.06119-2-26-1': 'We define an energy and time dependent population of the intermediate state [EQUATION]', '1908.06119-2-26-2': 'Fig. [REF] shows on a logarithmic scale the energy distribution [MATH] of the excited electrons for various times [MATH] after the excitation.', '1908.06119-2-26-3': 'For plotting the results, the [MATH]-function in eq. ([REF]) has been replaced by a rectangle with a width of 0.06 eV.', '1908.06119-2-27-0': 'We start with a discussion of the initial distribution, shown by the thick black line, calculated according to the transition dipole strength, eq. ([REF]).', '1908.06119-2-27-1': 'For the 4 ML Pb film (Fig. [REF](a)), the distribution is highly structured with a sharp maximum at 0.58 eV and a broad peak around 1.21 eV.', '1908.06119-2-27-2': 'For the 5 ML Pb film (thick black line in Fig. [REF](b)), only the peak at 1.21 eV (and possibly a short-lived peak at higher energies) remain visible, while the low-energy peak is much less pronounced.', '1908.06119-2-27-3': 'These results are in excellent agreement with the experimental observations of Ref. [CITATION].', '1908.06119-2-27-4': 'In this work, a high-energy peak in the range of 1.1 to 1.2 eV was observed for films with an odd number of Pb monolayers, whereas the peak at 0.6 eV was dominant in Pb films with an even number of layers.', '1908.06119-2-27-5': 'Note that, due to experimental limitations of the probe laser energy, excited electrons with energies lower than [MATH] eV could not be detected in Ref. [CITATION].', '1908.06119-2-28-0': 'Next, we analyze the relaxation of the energy distributions for later times.', '1908.06119-2-28-1': 'From Fig. [REF] it is obvious that all distributions develop a low-energy part corresponding a quasi-thermal distribution of secondary electrons, showing up as an exponentially decreasing function of energy.', '1908.06119-2-28-2': 'The high-energy part of the initial spectrum decays mainly due to e-e scattering, thereby creating secondary electrons via impact ionization.', '1908.06119-2-28-3': 'Therefore, the high-energy tails decay quickly, simultaneously accompanied by an increasing weight of the secondary-electron distribution.', '1908.06119-2-28-4': 'Now turning to longer time scales, we observe that the low-energy part in the 4 ML Pb film and the population in the energy window between 0.4 and 0.6 eV decay more slowly.', '1908.06119-2-28-5': 'At the same time, a broad shoulder on top of the secondary electron distribution builds up at [MATH] eV (marked by the arrow in Fig. [REF](a) and magnified in the inset).', '1908.06119-2-28-6': "Both phenomena can be traced back to the effect of e-ph scattering: Excited electrons of the initial distribution at energies of 0.4 to 0.6 eV 'glide down' the electronic band structure (cf. Fig. [REF](a) ), thereby emitting phonons.", '1908.06119-2-28-7': "The relatively slow rate at which this occurs leads to a 'phonon bottleneck', i.e., to the build-up of the shoulder centered at [MATH] eV.", '1908.06119-2-28-8': 'This effect, related to the discreteness of electronic states, is quite common for the e-ph relaxation in nanostructures, and has been observed e.g. in quantum dots [CITATION] as well as in two-dimensional layered semiconductors [CITATION].', '1908.06119-2-28-9': "While there is still continuous energy dissipation to the crystal lattice by very low-energy secondary electrons, the 'phonon bottleneck' affects electrons at higher energies and results in additional, but delayed production of phonons emitted by these electrons 'gliding down' the conduction bands.", '1908.06119-2-28-10': "A similar, but weaker 'phonon bottleneck' can be observed on top of the secondary electron distribution in the simulations for the 5 ML Pb film in Fig. [REF](b), marked by the arrow and magnified in the inset.", '1908.06119-2-29-0': 'In Fig. [REF], we analyze the time scales associated with the electronic population decay at selected energies where peaks had been found in the populations in Fig. [REF].', '1908.06119-2-29-1': 'The lines without symbols in Fig. [REF] show the decay according to the full relaxation dynamics, including both e-e and e-ph scattering.', '1908.06119-2-29-2': 'All populations show a nearly exponential decay, albeit with different decay rates.', '1908.06119-2-29-3': 'The lifetimes for the population maxima at 1.21 eV and 0.58 eV, extracted via exponential fits, are 21 fs and 101 fs, respectively.', '1908.06119-2-29-4': 'For the lowest energy of 0.46 eV, we find an initial rise of the population due to scattering-in from electrons at higher energies, followed by a population decay after about 30 fs, corresponding to a lifetime of 183 fs.', '1908.06119-2-29-5': 'The rather broad (in time) maximum of the 0.46 eV curve results from a compensation of the rates of incoming electrons from higher energies, mostly originating from e-e scattering at these high energies, and losses due to both e-e and e-ph scattering, the latter one gaining in relative importance as we go to lower energies.', '1908.06119-2-29-6': 'Interestingly, the described broad maximum of the population evolution at low energies is also seen in the experimental data [CITATION].', '1908.06119-2-29-7': 'In the computer simulation, we can deliberately turn off the e-e scattering channel after a very short initial time interval.', '1908.06119-2-29-8': 'The e-e scattering was permitted only in the very early times, somewhat arbitrarily chosen to be less than 6 fs, since some mechanism is required to establish a realistic smoothened electron distribution including an appropriate low-energy secondary-electron part.', '1908.06119-2-29-9': 'The result of these runs are displayed in Fig. [REF] by the lines with circular symbols.', '1908.06119-2-29-10': 'If the relaxation after 6 fs proceeds by e-ph scattering only, the population at 0.46 eV initially decays on a time scale of 350 fs (green symbols in Fig. [REF](a)), which can be taken as an estimate of the e-ph scattering rate in this energy range.', '1908.06119-2-29-11': 'This initial decay is followed by a much slower decay over several picoseconds.', '1908.06119-2-29-12': 'At the higher electron energies of 0.58, 0.81 and 1.21 eV, the scattering-in events of electrons from higher energies are equally probable or even more frequent than the scattering-out events, and hence a net contribution of e-ph scattering to the decay is not detectable on the time scale shown in Fig. [REF].', '1908.06119-2-30-0': 'A similar analysis has been carried out for the 5 ML Pb film, see Fig. [REF](b).', '1908.06119-2-30-1': 'For the peak energies at 1.21, 0.81 and 0.52 eV, overall lifetimes of 21, 47 and 126 fs are obtained from exponential fits to the full relaxation dynamics.', '1908.06119-2-30-2': "Again, it is possible to estimate the relative importance of e-ph scattering by watching the population decay after the e-e scattering has been 'turned off'.", '1908.06119-2-30-3': 'From the slopes of the curves marked by the circular symbols in Fig. [REF](b), characteristic times of 24 ps and 4.1 ps are obtained for 0.81 and 0.52 eV electron energy, respectively.', '1908.06119-2-30-4': 'At the highest electron energy of 0.58 eV, again we find that the contribution of e-ph scattering is too small to be detectable on the time scale shown in Fig. [REF].', '1908.06119-2-31-0': 'From this analysis, we learn that the contribution of e-ph scattering to the total lifetime of the peaks at energies larger than 0.5 eV is much smaller compared to the e-e contribution.', '1908.06119-2-31-1': 'This finding confirms the original analysis of the experimental data by Kirchmann et al. [CITATION] where e-ph scattering had been disregarded.', '1908.06119-2-31-2': 'Analyzing the experimental data for many Pb film thicknesses, they concluded that the low-energy peak is clearly observed in films with an even number of atomic layers and has a lifetime of [MATH] fs, while the high-energy peak is visible only in the odd-layer films and has an energy-dependent lifetime which turns out to be [MATH] fs for 5 ML Pb.', '1908.06119-2-31-3': 'Our simulation results of 101 fs and 21 fs are in reasonable agreement with their experimental findings, in particular if it is taken into account that the e-e scattering rate is very sensitive to the precise energetic position of the peak.', '1908.06119-2-31-4': 'The lifetimes extracted from our simulations are summarized by the circular symbols in Fig. [REF](b).', '1908.06119-2-31-5': 'Despite the additional decay channel of e-ph scattering being taken into account, the simulated lifetimes lie above the lifetime of isolated electrons due to e-e scattering alone.', '1908.06119-2-31-6': "This is because the simulations describe a realistic distribution of excited electrons, and the incoming flux from higher-lying electronic states effectively 'conserves' the population of the lower lying states over longer times.", '1908.06119-2-32-0': '## Excitation of lattice vibrations', '1908.06119-2-33-0': 'Although the contribution of e-ph scattering to the lifetime of the quantum well states was found to be small, the low-energy states populated by the secondary electrons couple significantly to the lattice vibrations.', '1908.06119-2-33-1': 'At these low energies, the e-ph scattering as loss mechanism even dominates over e-e scattering, since the lifetime due to e-e interactions rises above 300 fs for electrons below 0.33 eV according to eq. ([REF]).', '1908.06119-2-33-2': 'With the help of the simulations, it is possible to follow the energy transferred from the electrons to each of the phonon modes separately.', '1908.06119-2-33-3': 'Fig. [REF] shows the increase in time of the excess vibrational energy (in addition to the thermal energy corresponding to the initial substrate temperature) in the various Pb vibrational modes.', '1908.06119-2-33-4': 'Summation over all modes yields a total energy transfer between the electronic and the lattice degrees of freedom of 8.2 meV/ps for the 4 ML Pb and 0.79 meV/ps the 5 ML Pb film, respectively.', '1908.06119-2-33-5': 'The smallness of these quantities (a few meV compared to 0.1 eV electronic energy in the film) gives an a posteriori justification for the perturbative expression used for the interaction Hamiltonian [MATH].', '1908.06119-2-33-6': 'It is seen from Fig. [REF] that the energy transfer is highly mode-selective.', '1908.06119-2-33-7': 'One particular surface phonon mode receives a major part of the energy.', '1908.06119-2-33-8': 'The dominance of this single mode depends on film thickness; in the 4 ML film it is clearly more pronounced than in the 5 ML film where several modes participate in the energy uptake.', '1908.06119-2-33-9': 'In the 4 ML Pb film, this is a mode with frequency 2.26 THz (labeled 47 in our previous publication [CITATION]), while in the 5 ML Pb film it is a phonon mode at 2.03 THz.', '1908.06119-2-33-10': 'In experiments using a high laser fluence, the strong coupling to a specific mode has indeed been observed; in the 5 ML Pb film, it was detected experimentally by a periodic shift of the quantum well energy with a frequency of [MATH] THz [CITATION], which is in excellent agreement with the frequency of 2.03 THz identified in our simulation.', '1908.06119-2-33-11': 'For both film thicknesses, the modes at frequencies below 2 THz, which are similar to phonon modes in bulk Pb, receive only a much smaller amount of energy on average.', '1908.06119-2-34-0': 'As seen from Fig. [REF], the increase of the energy over long time scales, e.g. over 4 ps, is sub-linear and saturation is expected at even longer times.', '1908.06119-2-34-1': 'We attribute this slow response to the rather long time required by the electrons to relax from high energies down to [MATH] via multiple phonon emission processes.', '1908.06119-2-34-2': "Due to the 'phonon bottleneck' described above, this takes longer than expected from Allan's formula[CITATION].", '1908.06119-2-34-3': 'This formula requires as sole input the electron-phonon coupling constant of the bulk material that can e.g. be determined from ultrafast reflectivity measurements [CITATION].', '1908.06119-2-35-0': 'Eventually, we comment on the possibility to directly observe the excitation of the lattice experimentally.', '1908.06119-2-35-1': 'The excitation of low-lying Pb modes results in irregular displacements of the atomic positions on medium to large length scale that can be followed in the Debye-Waller factor of a time-resolved electron diffraction experiment.', '1908.06119-2-35-2': 'Preliminary experimental data for Pb films [CITATION] indicate that the low-energy phonons are indeed getting excited on the time scale of a few picoseconds.', '1908.06119-2-35-3': 'Interestingly, recent diffraction experiments were able demonstrate the mode-selective energy transfer to phonons even for bulk materials such as aluminium [CITATION] or nickel [CITATION].', '1908.06119-2-35-4': 'Apparently, phonon excitation by strongly excited charge carriers opens up the exploration of a new class of non-equilibrium phenomena, not only in bulk metals, but also in semiconductors [CITATION] and nanostructures.', '1908.06119-2-35-5': 'Due to the widely different timescales of e-ph interaction and phonon-phonon interaction, the non-equilibrium distribution of phonons created by the hot carriers can persist over a time span of several picoseconds.', '1908.06119-2-35-6': 'Thus, non-thermal phonon distributions could be a more wide-spread phenomenon than previously thought, and further research along these lines could be fruitful.', '1908.06119-2-36-0': '# Conclusion', '1908.06119-2-37-0': 'Simulations of electronic relaxation have been performed for a realistic system, metallic multilayer Pb films on Si(111), with the help of a parameter-free approach based on density functional theory.', '1908.06119-2-37-1': 'Electron-electron (e-e) and electron-phonon (e-ph) scattering were both included in the Master equation.', '1908.06119-2-37-2': 'Not surprisingly, e-e scattering was found to dominate over e-ph scattering for short times and highly excited electrons more than 0.5 eV above the Fermi level.', '1908.06119-2-37-3': 'Our simulation results thus justify the neglect of e-ph scattering in the analysis of experimental data [CITATION] in this regime.', '1908.06119-2-37-4': 'The simulated lifetimes of 101 fs and 21 fs for the quantum well states at 0.58 eV and 1.21 eV, respectively, are in reasonable agreement with the experimental findings.', '1908.06119-2-37-5': 'The importance of e-ph scattering shows up in the simulations at lower electron energies where the e-e scattering rate decreases strongly.', '1908.06119-2-37-6': 'Taking the fastest e-ph relaxation found in the present simulations with a characteristic time of 350 fs as a marker for the cross-over between e-e and e-ph scattering, we conclude that e-ph scattering significantly contributes to the relaxation of electrons in Pb at energies below 0.3 eV.', '1908.06119-2-37-7': 'Indeed, the simulations show a population pile-up around 0.3 eV due to the cross-over of the e-e and e-ph scattering time scales at this energy.', '1908.06119-2-37-8': "After 300 fs, it is fair to describe the electronic population by a thermal distribution, however, only up to an excess population of electrons getting stuck in the 'phonon bottleneck'.", '1908.06119-2-37-9': 'Remarkably, this does by no means imply that the phonon populations could be described by a temperature as well.', '1908.06119-2-37-10': 'Contrarily, the simulations show that even up to 4 ps after excitation high-frequency surface vibrational modes are preferentially excited by strongly mode-selective phonon emission.', '1908.06119-2-38-0': 'In summary, our simulations enable us to disentangle the contributions of e-e and e-ph scattering at short times, [MATH] ps after optical excitation.', '1908.06119-2-38-1': 'Although a first glance at the data is compatible with electronic thermalization by e-e scattering, a contribution of e-ph scattering can be observed already in this early stage, in particular at low electron energies.', '1908.06119-2-38-2': 'The phonon system requires much longer (several picoseconds) to equilibrate.', '1908.06119-2-38-3': 'Additional simulations beyond the scope of the present work are desirable to gain an improved understanding of the energy transfer between the two subsystems in the later stages of relaxation.'} | [['1908.06119-1-19-0', '1908.06119-2-15-0'], ['1908.06119-1-19-1', '1908.06119-2-15-1'], ['1908.06119-1-19-2', '1908.06119-2-15-2'], ['1908.06119-1-19-3', '1908.06119-2-15-3'], ['1908.06119-1-19-4', '1908.06119-2-15-4'], ['1908.06119-1-19-6', '1908.06119-2-15-6'], ['1908.06119-1-19-7', '1908.06119-2-15-7'], ['1908.06119-1-19-8', '1908.06119-2-15-8'], ['1908.06119-1-19-10', '1908.06119-2-15-10'], ['1908.06119-1-19-11', '1908.06119-2-15-11'], ['1908.06119-1-19-12', '1908.06119-2-15-12'], ['1908.06119-1-19-13', '1908.06119-2-15-13'], ['1908.06119-1-32-0', '1908.06119-2-38-0'], ['1908.06119-1-32-1', '1908.06119-2-38-1'], ['1908.06119-1-0-0', '1908.06119-2-0-0'], ['1908.06119-1-0-1', '1908.06119-2-0-1'], ['1908.06119-1-0-2', '1908.06119-2-0-2'], ['1908.06119-1-0-3', '1908.06119-2-0-3'], ['1908.06119-1-0-4', '1908.06119-2-0-4'], ['1908.06119-1-0-5', '1908.06119-2-0-5'], ['1908.06119-1-0-6', '1908.06119-2-0-6'], ['1908.06119-1-0-7', '1908.06119-2-0-7'], 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'1908.06119-2-27-1'], ['1908.06119-1-23-3', '1908.06119-2-27-3'], ['1908.06119-1-23-4', '1908.06119-2-27-4'], ['1908.06119-1-23-5', '1908.06119-2-27-5'], ['1908.06119-1-5-0', '1908.06119-2-5-0'], ['1908.06119-1-5-1', '1908.06119-2-5-1'], ['1908.06119-1-5-2', '1908.06119-2-5-2'], ['1908.06119-1-26-0', '1908.06119-2-30-0'], ['1908.06119-1-26-1', '1908.06119-2-30-1'], ['1908.06119-1-26-2', '1908.06119-2-30-2'], ['1908.06119-1-26-3', '1908.06119-2-30-3'], ['1908.06119-1-11-0', '1908.06119-2-11-0'], ['1908.06119-1-11-2', '1908.06119-2-11-2'], ['1908.06119-1-11-3', '1908.06119-2-11-3'], ['1908.06119-1-11-4', '1908.06119-2-12-0'], ['1908.06119-1-11-5', '1908.06119-2-13-0'], ['1908.06119-1-11-6', '1908.06119-2-13-1'], ['1908.06119-1-22-1', '1908.06119-2-26-0'], ['1908.06119-1-22-3', '1908.06119-2-26-2'], ['1908.06119-1-22-4', '1908.06119-2-26-3'], ['1908.06119-1-29-0', '1908.06119-2-33-0'], ['1908.06119-1-29-1', '1908.06119-2-33-1'], ['1908.06119-1-29-2', '1908.06119-2-33-2'], 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'1908.06119-2-29-7'], ['1908.06119-1-25-10', '1908.06119-2-29-10'], ['1908.06119-1-2-1', '1908.06119-2-2-1'], ['1908.06119-1-2-8', '1908.06119-2-2-8'], ['1908.06119-1-18-0', '1908.06119-2-14-0'], ['1908.06119-1-18-4', '1908.06119-2-14-6'], ['1908.06119-1-18-5', '1908.06119-2-14-7'], ['1908.06119-1-18-8', '1908.06119-2-14-10'], ['1908.06119-1-18-10', '1908.06119-2-14-12'], ['1908.06119-1-18-14', '1908.06119-2-14-16'], ['1908.06119-1-23-2', '1908.06119-2-27-2'], ['1908.06119-1-22-0', '1908.06119-2-24-0'], ['1908.06119-1-29-5', '1908.06119-2-33-6'], ['1908.06119-1-29-9', '1908.06119-2-33-10'], ['1908.06119-1-29-12', '1908.06119-2-34-1'], ['1908.06119-1-29-14', '1908.06119-2-35-1'], ['1908.06119-1-29-17', '1908.06119-2-35-4'], ['1908.06119-1-29-18', '1908.06119-2-35-5']] | [] | [['1908.06119-1-32-2', '1908.06119-2-38-2'], ['1908.06119-1-32-2', '1908.06119-2-38-3'], ['1908.06119-1-16-6', '1908.06119-2-22-9'], ['1908.06119-1-15-11', '1908.06119-2-21-8'], ['1908.06119-1-7-0', '1908.06119-2-7-0'], ['1908.06119-1-7-2', '1908.06119-2-7-5'], ['1908.06119-1-13-8', '1908.06119-2-25-4'], ['1908.06119-1-25-8', '1908.06119-2-29-8'], ['1908.06119-1-2-7', '1908.06119-2-2-7'], ['1908.06119-1-18-13', '1908.06119-2-14-15'], ['1908.06119-1-22-2', '1908.06119-2-26-1'], ['1908.06119-1-29-11', '1908.06119-2-34-0'], ['1908.06119-1-29-13', '1908.06119-2-34-2'], ['1908.06119-1-29-13', '1908.06119-2-34-3'], ['1908.06119-1-29-19', '1908.06119-2-35-6']] | [] | ['1908.06119-1-11-1', '1908.06119-2-8-10', '1908.06119-2-11-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1908.06119 | null | null | null | null | null |
1905.05690 | {'1905.05690-1-0-0': 'It is investigated the gravitational waves phenomena in the geometric scalar theory of gravity (GSG), a class of theories such that gravity is described by a single scalar field.', '1905.05690-1-0-1': 'The associated physical metric describing the spacetime is constructed from a disformal transformation of Minkowski geometry.', '1905.05690-1-0-2': 'In this theory, a weak field approximation gives rise to a description similar to that one obtained in general relativity, although the gravitational waves in GSG have a characteristic longitudinal polarization mode, besides others modes that are observer dependent.', '1905.05690-1-0-3': 'We also analyze the energy carried by the gravitational waves as well as how their emission affects the orbital period of a binary system.', '1905.05690-1-1-0': '# Introduction', '1905.05690-1-2-0': 'Although general relativity (GR) has been a very successful gravitational theory during the last century, many proposals for modification of Einstein original formulation appeared in the literature over the past decades.', '1905.05690-1-2-1': 'Most of these ideas come up within the cosmological scenario, where GR only works if unknown components, like dark matter or dark energy, are introduced.', '1905.05690-1-2-2': "Such alternative descriptions are basically variations of Einstein's theory, either assuming most general Lagrangians for the gravitational field or adding new fields together with the metric.", '1905.05690-1-3-0': 'Unlike those variations of GR, it was recently proposed a theory of gravity in the realm of purely scalar theories, introducing some crucial modifications from the previous attempts that took place before the emergence of GR [CITATION].', '1905.05690-1-3-1': 'It represents the gravitational field with a single scalar function [MATH], that obeys a non-linear dynamics.', '1905.05690-1-3-2': 'Interaction with matter fields is given only trough a minimal coupling to the physical metric [MATH], constructed from a disformal transformation of a auxiliary and flat metric [MATH], namely [EQUATION] with, [EQUATION] and the short notation [MATH].', '1905.05690-1-3-3': 'A complete theory can only be set if one defines the functions [MATH] and [MATH], and also the Lagrangian of the scalar field.', '1905.05690-1-3-4': 'Then, a field equation, characterizing the theory, can be derived.', '1905.05690-1-3-5': 'We refer to this class of gravitational theories as geometric scalar gravity (GSG).', '1905.05690-1-4-0': 'In early communications on GSG, it was explored a specific set of those functions defining the theory, which shows that it is possible to go further in representing the gravitational field as a single scalar, giving realistic descriptions of the solar system and cosmology [CITATION].', '1905.05690-1-4-1': 'An analysis of GSG within the so called parametrized post-Newtonian formalism was also made and, although the theory is not covered by the formalism, a limited situation indicate a good agreement with the observations [CITATION].', '1905.05690-1-4-2': 'Intending to improve the understanding of how GSG deals with gravitational interaction, the present work develops the theoretical description and characterization of gravitational waves (GW).', '1905.05690-1-5-0': 'The direct detections of GW by LIGO and Virgo collaborations initiated a new era of testing gravitational theories.', '1905.05690-1-5-1': "It enables to construct constraints over a series of theoretical mechanisms associated with GW's physics, but the crucial point relies on the observed waveform and how a theory can reproduce it [CITATION].", '1905.05690-1-5-2': 'Notwithstanding, this is not the scope of this work.', '1905.05690-1-5-3': 'We are mainly focused in analyzing the GW fundamentals on the perspective of GSG, studying their propagation, polarization modes and defining an appropriated tensor to describe the energy and momentum carried by the waves.', '1905.05690-1-5-4': 'As an example, we analyze the binary system, deriving an expression for the orbital variation that should be caused by the loss of energy due to gravitational radiation.', '1905.05690-1-5-5': 'An expected confrontation with observational data from binary pulsars is still not possible since a post-Keplerian analysis is needed in order to determine the masses of the system components according to GSG.', '1905.05690-1-5-6': 'This is a distinct treatment of what is present here and, therefore, is left to a future work.', '1905.05690-1-6-0': 'The paper is organized as follows.', '1905.05690-1-6-1': 'In section [REF] is presented a brief overview of GSG in order to introduce to the reader the main features of this theory.', '1905.05690-1-6-2': "The following section describes the theory's weak field approximation.", '1905.05690-1-6-3': 'In section [REF] the study of the propagation and vibration modes associated to gravitational waves is made.', '1905.05690-1-6-4': 'The definition of a energy-momentum tensor for the linear waves is treated in section [REF].', '1905.05690-1-6-5': 'Generation of waves, including the computation of the orbital variation of binary systems due to the emission of GW, is discussed in section [REF] and the last section presents our concluding remarks.', '1905.05690-1-6-6': 'Also, two appendices were introduced in order to complement the middle steps of calculations present in section [REF].', '1905.05690-1-7-0': '# Overview of geometric scalar gravity', '1905.05690-1-8-0': 'GSG is a class of gravitational theories which identifies the gravitational field to a single real scalar function [MATH], satisfying a non-linear dynamic described by the action [EQUATION] where [MATH] is the determinant of the Minkowski metric and [MATH] is defined in eq. [REF].', '1905.05690-1-8-1': 'Metric signature convention is [MATH].', '1905.05690-1-8-2': 'The physical metric is constructed from the gravitational field according to the expression [REF] and its contravariant form is obtained from the definition of an inverse metric, [MATH], namely, [EQUATION] where, [EQUATION]', '1905.05690-1-8-3': "In order to describe the interaction of the scalar gravitational field with matter, GSG makes the fundamental hypothesis, according to Einstein's proposal, that gravity is a geometric phenomenon.", '1905.05690-1-8-4': 'Thus, it is assumed that the interaction with [MATH] is given only through a minimal coupling with the gravitational metric [MATH].', '1905.05690-1-8-5': 'The matter action in GSG is then described as [EQUATION]', '1905.05690-1-8-6': "A complete theory should specify the metric's functions [MATH] and [MATH] together with the Lagrangian of the scalar field [MATH], in order to be possible to derive its field equation.", '1905.05690-1-8-7': 'Up to now in the literature, it has been explored the case in which the following choice is made, [EQUATION] with [EQUATION]', '1905.05690-1-8-8': 'Using the standard definition of the energy momentum tensor in terms of a metric structure, we set [EQUATION]', '1905.05690-1-8-9': "Then, the dynamics of the scalar field is described by the equation [EQUATION] where the [MATH] indicates the d'Alembertian operator constructed with the physical metric [MATH], [MATH] is a coupling constant and the source term [MATH] is provided by [EQUATION] where '[MATH]' indicates a covariant derivative with respect to the physical metric, [MATH] and [EQUATION]", '1905.05690-1-8-10': 'The choices made in [REF]-[REF] are such that the resulting theory satisfies the Newtonian limit, the classical gravitational tests and the spherically symmetric vacuum solution is given by the Schwarzschild geometry.', '1905.05690-1-8-11': 'Moreover, in the absence of any matter fields, [MATH] is a free wave propagating in the metric [MATH] [CITATION].', '1905.05690-1-8-12': 'More details concerning the fundamentals of GSG and how this specific model can successfully describe the solar system physics and cosmology can be found in [CITATION].', '1905.05690-1-8-13': 'In the present work we will consider only this model.', '1905.05690-1-8-14': 'To work with different functions [MATH], [MATH] and [MATH], all the process of constructing the field equation of the theory has to be redone, as well as it should be checked the feasibility of the new theory.', '1905.05690-1-9-0': '# Weak field approximation', '1905.05690-1-10-0': 'To discuss linear gravitational waves we should consider an isolated system, distant from any source, embedded in a homogenous and isotropic universe.', '1905.05690-1-10-1': 'At a particular moment of time and specific distance from the isolated system, the background metric can be transformed to assume a flat Minkowskian form, resulting in a geometric structure given by, [EQUATION] where [MATH] and [MATH] represents the first order perturbations.', '1905.05690-1-11-0': 'In this sense, the weak-field approximation of GSG consists in a small deviation of a cosmological solution [MATH].', '1905.05690-1-11-1': 'Thus, we set [EQUATION]', '1905.05690-1-11-2': 'In order to construct the geometric structure as in [REF], for simplicity, we start with a coordinate system [MATH], where the auxiliary metric [MATH] assumes the usual diagonal form indicated as [MATH], and we expand the kinetic term and the metric coefficients as follows, [EQUATION]', '1905.05690-1-11-3': 'The subindex "[MATH]" identifies quantities constructed with [MATH] according to basic expressions given in the previous section.', '1905.05690-1-11-4': 'The gravitational metric takes the form [EQUATION] where, [EQUATION] and, [EQUATION]', '1905.05690-1-11-5': 'With the following coordinate transformation [EQUATION] the desired structure is achieved, [EQUATION] where [EQUATION]', '1905.05690-1-11-6': 'In this new coordinate system, [MATH] is equivalente to the cosmological time and [MATH] (in units where [MATH]), where [MATH] is the Hubble parameter (please see [CITATION] for more details on GSG cosmology).', '1905.05690-1-11-7': 'Then, the perturbed metric becomes, [EQUATION]', '1905.05690-1-11-8': 'The corresponding covariant expression for [REF] is obtained from the definition [MATH].', '1905.05690-1-11-9': 'It reads [EQUATION] where, [EQUATION]', '1905.05690-1-11-10': 'Equations [REF] and [REF] shows that in the weak field limit the indices are lowered and raised by the Minkowski background metric.', '1905.05690-1-12-0': 'Note that, the perturbed metric [REF] can also be derived from the expansion of the exact form given in [REF] starting already with the coordinates [MATH], where the auxiliary metric [MATH] takes the form [EQUATION]', '1905.05690-1-12-1': 'The resulting covariant expression can be written as [EQUATION] with [EQUATION]', '1905.05690-1-12-2': 'In reference [CITATION] a distinct weak field approximation was made where the scalar field was expanded around a vanishing background value.', '1905.05690-1-12-3': 'Although consistent, that scheme is not suitable for the description of GW, due to a term [MATH] that is present in [MATH].', '1905.05690-1-12-4': 'Oscillatory solutions would then lead to a singular behavior of the metric, evidencing that the background cosmological scenario can not be neglected.', '1905.05690-1-13-0': '## The cosmological backgroung', '1905.05690-1-14-0': 'Before proceeding in the analysis of GW in GSG, let us clarify important points of the cosmological background described by [MATH].', '1905.05690-1-14-1': 'To a more detailed discussion about the cosmology in GSG we refer to the reader the analysis present in [CITATION].', '1905.05690-1-14-2': 'By considering the scalar field as a function of coordinate [MATH] only, the metric arising is of Friedman-Robertson-Walker type with a flat spatial section.', '1905.05690-1-14-3': 'The cosmological time is achieved by the time transformation given by the first expression in [REF] and the scale factor, said [MATH], is related with the [MATH] as follows, [EQUATION]', '1905.05690-1-14-4': 'The dynamical equation [REF] contains two regimes classified by the term [MATH], a consequence of the particular choice of the scalar field Lagrangian.', '1905.05690-1-14-5': 'The case where [MATH], with a barotropic fluid as source, describes a eternal universe without singularities.', '1905.05690-1-14-6': 'The universe has a bouncing, followed by a early accelerated phase and a final decelerated expansion.', '1905.05690-1-14-7': 'The problematic value [MATH] is unattainable, in other words, the minimal value of the scale factor [MATH] is always greater than [MATH].', '1905.05690-1-14-8': 'A distinct behavior occurs for the solutions with barotropic fluids in the region where [MATH]; the universe starts from a initial singularity, it expands to a certain maximum value of the scale factor, smaller than [MATH], and then returns to a final singular point.', '1905.05690-1-14-9': 'This two regions are then disjoint classes of cosmological solutions.', '1905.05690-1-14-10': 'In the present work, we will consider only the case where [EQUATION] since it represents a class of more realistic descriptions of the universe.', '1905.05690-1-15-0': '# Propagation and polarization of gravitational waves', '1905.05690-1-16-0': 'At the level of the dynamical equation we can consider [MATH] as a constant, since its timescale variation is longer compared to the dynamical timescale for the local system.', '1905.05690-1-16-1': "Expanding the left hand side of Eq. [REF] and neglecting second order terms, one has, [EQUATION] where we refer to Minkowskian d'Alembertian operator as [MATH].", '1905.05690-1-16-2': 'Thus, without the presence of sources, one has [EQUATION]', '1905.05690-1-16-3': 'The perturbed scalar field has oscillatory solutions which propagates at the speed of light.', '1905.05690-1-16-4': 'Once the metric is constructed with the field and its first derivatives, such solutions yields oscillations as GW in the geometric structure of the spacetime.', '1905.05690-1-16-5': 'Moreover, it is verified that [EQUATION] thus gravitational waves in GSG propagates with the speed of light.', '1905.05690-1-17-0': '## Polarization states', '1905.05690-1-18-0': 'The most general (weak) gravitational wave that any metric theory of gravity is able to predict can contain six modes of polarization.', '1905.05690-1-18-1': 'Considering plane null waves propagating in a given direction, these modes are related to tetrad components of the irreducible parts of the Riemann tensor, or the Newmann-Penrose quantities (NPQ): [MATH] and [MATH] and [MATH] are complex quantities and each one represents two modes of polarization) [CITATION].', '1905.05690-1-18-2': 'The others NPQ are negligible by the weak field approximation, or are described in terms of these four ones.', '1905.05690-1-19-0': 'The linearized dynamical equations of a gravitational theory can automatically vanish some of these NPQ, specifying then the predicted number of polarization states.', '1905.05690-1-19-1': 'For instance, in GR only [MATH] is not identically zero, which characterizes two transversal polarization modes, called "[MATH]" and "[MATH]" states.', '1905.05690-1-19-2': 'In general, the six polarization modes can not be specified in a observer-independent way because of their behavior under Lorentz transformations.', '1905.05690-1-19-3': 'Nevertheless, if we restrict our attention to a set of specific observers, which agree with the GW on the frequency and on the direction of propagation, then is possible to make some observer-invariant statements about the NPQ.', '1905.05690-1-19-4': 'Such assertions are on the basis of the so called E(2)-classification of gravitational theories, introduced in ref. [CITATION]:', '1905.05690-1-20-0': 'The [MATH]-classification of GSG is easily obtained by noticing that the Ricci scalar is not identically null.', '1905.05690-1-20-1': 'Actually, from the weak field approximation, one has [EQUATION] with [MATH] and, using relations [REF] together with linearized vacuum field equation [REF], it is verified that, [EQUATION]', '1905.05690-1-20-2': 'This result implies [MATH] (cf. equation (A4) of [CITATION]) and GSG is from the class II[MATH].', '1905.05690-1-20-3': 'This [MATH] represents a pure longitudinal polarization state (see figure [REF]) that is always present in the GW, although other modes can be detected depending on the observer.', '1905.05690-1-21-0': 'GSG belongs to the most general class of theories with respect to the [MATH]-classification, where is always possible to find an observer that measures all six gravitational wave modes.', '1905.05690-1-21-1': 'The authors in [CITATION] already pointed out the fact that the number of polarization states predicted by a gravitational theory does not necessarily match the numbers of degrees of freedom inside the theory.', '1905.05690-1-21-2': 'They also give an example of this with the so called stratified theories.', '1905.05690-1-21-3': 'Other examples of theories also classified as [MATH] is the well know [MATH] extensions of general relativity [CITATION].', '1905.05690-1-22-0': 'Thus, the description of GW by GSG carries a substancial distinction from GR, as it predicts the presence of a longitudinal polarization mode.', '1905.05690-1-22-1': 'Up to now, the recent detections of GW can not exclude the existence of any one of the six modes of polarization [CITATION].', '1905.05690-1-22-2': 'But in the future, with the appropriated network of detectors, with different orientations, this information can be used to restrict gravitational theories.', '1905.05690-1-23-0': '# Energy of the gravitational wave', '1905.05690-1-24-0': 'In order to associate an energy-momentum tensor to the gravitational waves in GSG we follow a standard procedure, identifying the relation between the second and the first order perturbations of the gravitational field [CITATION].', '1905.05690-1-24-1': 'First note that, without approximations, the following relation holds, [EQUATION]', '1905.05690-1-24-2': 'Thus, taking [MATH], where the subindexes indicates the order, and computing the second order vacuum field equation, it yields [EQUATION] with [MATH].', '1905.05690-1-24-3': 'The right hand side of this equation contains only the derivatives of the first order field [MATH], thus it can be interpreted as the source for the second order field generated by the linear waves.', '1905.05690-1-25-0': 'From the general structure of the field equation of GSG, the influence of any energy-momentum tensor enters in the equation of motion uniquely through the quantity [MATH] [cf. equation [REF]].', '1905.05690-1-25-1': 'Thus, the energy-momentum tensor of the GW, said [MATH], must be consistent with, [EQUATION] where [MATH] means the second order approximation of the source term calculated with the energy-momentum tensor of the gravitational field [MATH], instead of [MATH].', '1905.05690-1-25-2': "Therefore, we write [EQUATION] which has the same general structure of GSG's field equation.", '1905.05690-1-26-0': 'To describe the energy and momentum carried by the linear waves, the second-order approximation of [MATH] must be quadratic in the first derivatives of [MATH].', '1905.05690-1-26-1': 'This lead us to a specific form for it, [EQUATION] with [MATH] and [MATH] being arbitrary constants.', '1905.05690-1-26-2': 'The condition [REF] returns the relation [EQUATION]', '1905.05690-1-26-3': 'Any tensor, described like in Eq. [REF] and satisfying the above relation, can be used as the energy-momentum tensor of the linear GW in GSG.', '1905.05690-1-26-4': 'This ambiguity already appeared in reference [CITATION], where the authors show how to construct the energy-momentum tensor of the gravitational field in GSG, without using approximate methods.', '1905.05690-1-26-5': 'In that occasion, they fixed the functions defining the energy tensor by requiring that [MATH] can be derived from the Lagrangian.', '1905.05690-1-26-6': 'As expected, their results are consistent with the relation above and are recovered (inside the approximation method) if [MATH] and [MATH].', '1905.05690-1-26-7': 'In what follows we will proceed with the generic expression for [MATH] and look for a specific example, the orbital variations in binary systems, to see how this ambiguity can influence in a observed phenomenon.', '1905.05690-1-27-0': '# Orbital variation of a binary system', '1905.05690-1-28-0': 'This section focus on deriving an expression for the orbital variation of a binary system, due to the emission of gravitational waves, as it is predicted by GSG.', '1905.05690-1-28-1': 'In order to obtain the energy rate emitted by the system one should consider the influence of the source into the dynamics in the linear approximation.', '1905.05690-1-28-2': "Since the left hand side of Eq. [REF] reduces to a Minkowskian d'Alembertian when linearized (c.f. eq. [REF]), from the method of Green functions, we immediately write down the general solution as, [EQUATION] where [MATH] attends to the first order approximation of the source term [cf.[REF]] and [MATH], with [MATH], is the retarded time.", '1905.05690-1-29-0': "By considering that the source is far away from the point where we calculate the scalar field [MATH], where [MATH], and [MATH] is the typical distances between the source's components[MATH], it is possible to make a multipole expansion.", '1905.05690-1-29-1': 'Further assuming that the typical velocities of the source components are non relativistic, it is also possible to expand the time dependent terms of the integrand in a Taylor series.', '1905.05690-1-29-2': 'For our purpose here it is sufficient to take only the first term of this expansion.', '1905.05690-1-29-3': 'Thus, one has [EQUATION] where [MATH] and we have neglected terms of order [MATH].', '1905.05690-1-30-0': 'Most of terms in the above integration contains the scalar field [MATH], explicitly.', '1905.05690-1-30-1': 'To solve them, we have to expand these terms using the correspondent post-Newtonian approximation of the field in the near-zone region [CITATION].', '1905.05690-1-30-2': 'However, to keep the final result up to order [MATH], it is only necessary the Newtonian approximation of the near-zone scalar field, namely [MATH].', '1905.05690-1-30-3': "By the viral theorem, we know that, for slow motions, [MATH], where [MATH], [MATH] and [MATH] are the typical velocity, mass and distances in the source's components, respectively.", '1905.05690-1-30-4': 'Thus, [MATH], [MATH] and [MATH] (see appendix [REF] for more details).', '1905.05690-1-30-5': 'The energy-momentum tensor also depends on source velocities with [MATH] and [MATH].', '1905.05690-1-30-6': 'Thus, keeping terms up to order [MATH] (since [MATH]) and using the perturbed metric expressions in [REF], one gets [EQUATION]', '1905.05690-1-30-7': 'The [MATH] term does not contribute by Gauss law.', '1905.05690-1-30-8': 'Also, to derive the above expressions we take into account that [EQUATION]', '1905.05690-1-30-9': 'Specifying the source for the case of a binary system, we have [EQUATION] where summation is over the two particles of the system, i.e. [MATH].', '1905.05690-1-30-10': 'With these expressions, all the integrals in [REF] can be analytically calculated (more details in Appendix [REF]) to give [EQUATION] where the dot indicates a derivative with respect to retarded time, [MATH] attends to constant terms that does not contribute to the radiation, [MATH] is the Newtonian gravitational constant as measured today (see Appendix [REF]) and the notation was shortened by the definitions below, [EQUATION]', '1905.05690-1-30-11': 'Also, we are adopting the usual center of mass notation such that, [EQUATION] with [MATH] and [MATH].', '1905.05690-1-31-0': 'Once we are dealing with a binary system as the source of the gravitational field, we can use the Keplerian orbital parameters to simplify the above expression [CITATION].', '1905.05690-1-31-1': 'The distance between the two masses are, [EQUATION] where [MATH] is the semimajor axis and [MATH] is the eccentricity of the orbit.', '1905.05690-1-31-2': 'They are related with the total energy [MATH] and the angular momentum [MATH] by [EQUATION] with [MATH].', '1905.05690-1-31-3': 'The fact that [MATH], allow us to derive the following relation, [EQUATION]', '1905.05690-1-31-4': 'Then, in [REF], all time derivatives can be expressed in terms of [MATH], yielding [EQUATION]', '1905.05690-1-31-5': 'To calculate the energy-flux that is carried off by GW we use the gravitational energy-momentum tensor presented in the previous section.', '1905.05690-1-31-6': 'The flux in the radial direction will be [MATH] thus, the energy radiated per unit time that is passing through a sphere of radius [MATH], is given by [EQUATION] where we have used the fact that [EQUATION]', '1905.05690-1-31-7': 'At this point, we go further in the approximation scheme in order to get a more treatable expression for the rate of energy loss.', '1905.05690-1-31-8': 'Let us consider that the background field is too small, i.e. [MATH], and take only the leading order terms.', '1905.05690-1-31-9': 'This is realistic since it is always expected that the cosmological influence on local systems are minimal.', '1905.05690-1-31-10': 'Expression [REF] can be then simplified, reading [EQUATION] where [MATH] is given by [EQUATION]', '1905.05690-1-31-11': 'Averaging the energy loss over an orbital period [MATH], where [EQUATION] we have, [EQUATION]', '1905.05690-1-31-12': 'The above integral is directly solved, yielding [EQUATION] with [EQUATION]', '1905.05690-1-31-13': 'To finish, we derive how this loss of energy changes the orbital period of the system.', '1905.05690-1-31-14': 'From [REF], one gets that [EQUATION]', '1905.05690-1-31-15': 'The result has the same proportionality with the constants [MATH] and [MATH], as in GR, but has a rather more involved dependence on the masses and the eccentricity of the orbit.', '1905.05690-1-32-0': 'Note that equation [REF] must be negative, otherwise it would imply that the masses are moving away from each other.', '1905.05690-1-32-1': 'In other words, the system would be increasing their energy by the emission of GW, an unrealistic situation.', '1905.05690-1-32-2': 'The function [MATH] is positive, as it can be verified by comparison between the term [MATH] and the part involved by the round brackets multiplying [MATH] (the only part that could be negative), [EQUATION]', '1905.05690-1-32-3': 'Since [MATH] for elliptical orbits, it follows that [MATH] is always positive.', '1905.05690-1-32-4': 'Thus to guarantee [MATH], we must have [MATH].', '1905.05690-1-33-0': 'The Keplerian and post-Keplerian orbital parameters of a binary system can be extracted from the timing pulsar observation in a theory-independent way, but the determination of the masses of the pulsar and its companion is only obtained by making use of specific equations relating them to that set of parameters.', '1905.05690-1-33-1': 'These relations are particular for each gravitational theory [CITATION].', '1905.05690-1-33-2': 'Thus, a confrontation between the orbital variation of a binary system, as predicted by GSG, and the observational data is only possible after obtaining the so called post-Keplerian parametrization of the theory to extract the mass values according to GSG.', '1905.05690-1-33-3': 'We will leave this task to be addressed in a future work.', '1905.05690-1-34-0': '# Concluding remarks', '1905.05690-1-35-0': 'We have presented a discussion on gravitational waves (GW) in the context of the geometric scalar gravity (GSG), a class of theories describing the effects of gravity as a consequence of a modification of spacetime metric in terms of a single scalar field.', '1905.05690-1-35-1': 'GSG overcomes the serious drawbacks present in all previous attempts to formulate a scalar theory of gravity.', '1905.05690-1-35-2': 'Its fundamental idea rests on the proposal that the geometrical structure of the spacetime is described by a disformal transformation of a conformal flat metric.', '1905.05690-1-35-3': 'The model analyzed here has already showed several advances within the scalar gravity program, featuring a good representation of the gravitational phenomena both in the solar system domains as well as in cosmology.', '1905.05690-1-36-0': 'Initially it was shown the procedure used to construct the weak field limit in GSG considering an expansion of the scalar field over a background cosmological solution.', '1905.05690-1-36-1': 'Within this approximation scheme it was shown that the scalar dynamical equations assumes oscillatory solutions that represent GW in the spacetime structure propagating with light velocity.', '1905.05690-1-36-2': 'An important distinction appears in the polarization states of the waves, which is characterized by the presence of a longitudinal mode in GSG.', '1905.05690-1-36-3': 'Within the [MATH]-classification of gravitational theories, GSG is of the type [MATH], since [MATH].', '1905.05690-1-36-4': 'This is the most general class, where the detection of all the other five polarization modes depends on the observer.', '1905.05690-1-37-0': 'It was also discussed how to define an energy-momentum tensor for the linear GW following a field theoretical point of view.', '1905.05690-1-37-1': 'An ambiguity emerges since GSG fundamental equation includes a non trivial interaction between matter/energy and the scalar field, leading to non unique expression for the approximated gravitational energy-momentum tensor.', '1905.05690-1-37-2': 'As an example fo application of the previous results, an expression for the orbital variation of a binary system, due to the emission of GW, was derived and, in principle, it could be useful to provide the more appropriate energy-momentum tensor for the gravitational waves.', '1905.05690-1-37-3': 'However, a confrontation with observational data coming from pulsars can only be performed in posses of the post-Keplerian parameters as predicted by GSG, in order to determine the masses of a binary system components.', '1905.05690-1-37-4': 'We hope to come back to this question in the near future.', '1905.05690-1-38-0': 'I wish to thank J.C. Fabris and T.R.P. Carames for dedicated reviews of this work.', '1905.05690-1-38-1': 'This research is supported by FAPES and CAPES through the PROFIX program.', '1905.05690-1-39-0': '# The near zone scalar field', '1905.05690-1-40-0': 'The linearized dynamical equation of GSG is a traditional wave equation, which posses some properties depending whether [MATH] (the point where the field is being calculated) is larger or smaller than the typical wavelength [MATH] of the solution [CITATION].', '1905.05690-1-40-1': 'In the wave zone, where [MATH], the difference between the retarded time [MATH] and [MATH] is large, so the time derivative of the field is comparable to the spatial derivative.', '1905.05690-1-40-2': 'This is the region where the radiation effects are influent in determining the metric.', '1905.05690-1-40-3': 'On the other hand, in the region where [MATH], called near zone, the difference between the [MATH] and [MATH] are small and the time derivatives becomes irrelevant in front of the spatial derivatives.', '1905.05690-1-41-0': "The near zone region is covered by the post-Newtonian approximation of the gravitational field, expanding it in orders of [MATH], where [MATH] is the typical velocities of the source's components, and considering also slow motion.", '1905.05690-1-41-1': 'This is the approximation required for the scalar field when integrating the wave equation.', '1905.05690-1-41-2': 'Once the scalar field aways appears multiplied by [MATH] in the integrand, we only need to know its leading order, i.e. its Newtonian approximation.', '1905.05690-1-41-3': 'Thus, equation [REF] reduces to [EQUATION] where only the zeroth-order terms is considered in the above integral.', '1905.05690-1-42-0': "From [REF] and [REF], and using a multipole expansion, one can easily sees that the metric assumes the form [EQUATION] where [EQUATION] and [MATH] attends for the Newton's gravitational constant as measured today and it is a redefinition of the theory's coupling constant, [EQUATION]", '1905.05690-1-42-1': 'Note that the dependence of [MATH] with the cosmological background field implies its change as a result of the evolution of the universe.', '1905.05690-1-42-2': "This effect has not been evident in the previous analysis of GSG's newtonian limit since the cosmological influence was neglect in those works [CITATION].", '1905.05690-1-42-3': 'We will not discuss its implications in the present work, but it certainly shows the importance of take into account the cosmological background when analyzing the Newtonian and post-Newtonian limits of GSG.', '1905.05690-1-43-0': '# More detailed calculations', '1905.05690-1-44-0': 'In this section we aim to be more clear on the calculation of the integrals of the quantities appearing in expressions [REF], [REF] and [REF].', '1905.05690-1-44-1': 'We start with the linearized conservation law, [MATH], from where is possible to derive the following expressions, [EQUATION] where [MATH] represents the second momenta of mass distribution.', '1905.05690-1-44-2': 'Only the trace of the quadrupole moment enters in the field equation and it is directly calculated, [EQUATION]', '1905.05690-1-44-3': 'The time derivatives can be now easily calculated.', '1905.05690-1-45-0': 'The remaining integrals does contain the newtonian limit of the scalar field explicitly.', '1905.05690-1-45-1': 'For the specific case of binary system as a source, the solution [REF] becomes [EQUATION]', '1905.05690-1-45-2': 'When calculating the Newtonian gravitational potential in the position of one of the particles of the system we have to neglect the infinity self potential, thus [EQUATION] where the summation above is taken excluding terms when [MATH].', '1905.05690-1-45-3': 'This can be interpreted as a mass renormalization [CITATION].', '1905.05690-1-45-4': 'Using this we can integrate expression [REF] to give [EQUATION]', '1905.05690-1-45-5': 'For the remaining integrals it is needed the derivatives of the Newtonian scalar field, namely [EQUATION] and the kinect term, [EQUATION]', '1905.05690-1-45-6': 'In the above expressions [MATH], [MATH], and [MATH].', '1905.05690-1-45-7': 'The sub-indexes [MATH] are summation indices assuming the values [MATH] or [MATH].', '1905.05690-1-45-8': 'The upper-indexes [MATH] refer to the usual components of a three-vector and they run from [MATH] to [MATH].', '1905.05690-1-46-0': 'Let us calculate one of the integrals explicitly, [EQUATION] where the symbol [MATH] means the product of two summations, one in [MATH] and other in [MATH], with both never assuming the value of [MATH].', '1905.05690-1-46-1': 'Using that [EQUATION] and [EQUATION] we have [EQUATION]', '1905.05690-1-46-2': 'In the last equality we used the relations [REF].', '1905.05690-1-46-3': 'The procedure is similar for the other integrals and, paying attention that [MATH], it follows [EQUATION]', '1905.05690-1-46-4': 'Putting all these terms together, following [REF], we obtain the relation, [EQUATION]', '1905.05690-1-46-5': 'In the integral of [REF] the following terms will appear, [EQUATION] and, with a time derivative, we obtain the last terms remaining to get the expression [REF].'} | {'1905.05690-2-0-0': 'It is investigated the gravitational waves phenomena in the geometric scalar theory of gravity (GSG), a class of theories such that gravity is described by a single scalar field.', '1905.05690-2-0-1': 'The associated physical metric describing the spacetime is constructed from a disformal transformation of Minkowski geometry.', '1905.05690-2-0-2': 'In this theory, a weak field approximation gives rise to a description similar to that one obtained in general relativity, with the gravitational waves propagating at the same speed as the light, although they have a characteristic longitudinal polarization mode, besides others modes that are observer dependent.', '1905.05690-2-0-3': 'We also analyze the energy carried by the gravitational waves as well as how their emission affects the orbital period of a binary system.', '1905.05690-2-0-4': 'Observational data coming from Hulse and Taylor binary pulsar is then used to constraint the theory parameter.', '1905.05690-2-1-0': '# Introduction', '1905.05690-2-2-0': 'Although general relativity (GR) has been a very successful gravitational theory during the last century, many proposals for modification of Einstein original formulation appeared in the literature over the past decades.', '1905.05690-2-2-1': 'Most of these ideas come up within the cosmological scenario, where GR only works if unknown components, like dark matter or dark energy, are introduced.', '1905.05690-2-2-2': "Such alternative descriptions are basically variations of Einstein's theory, either assuming most general Lagrangians for the gravitational field or adding new fields together with the metric.", '1905.05690-2-3-0': 'Unlike those variations of GR, it was recently proposed a theory of gravity in the realm of purely scalar theories, introducing some crucial modifications from the previous attempts that took place before the emergence of GR [CITATION].', '1905.05690-2-3-1': 'It represents the gravitational field with a single scalar function [MATH], that obeys a non-linear dynamics.', '1905.05690-2-3-2': 'Interaction with matter fields is given only trough a minimal coupling to the physical metric [MATH], constructed from a disformal transformation of a auxiliary and flat metric [MATH], namely [EQUATION] with, [EQUATION] and the short notation [MATH].', '1905.05690-2-3-3': 'Disformal transformations in gravity has been discussed early by Bekenstein [CITATION] and the role of disformal couplings in gravitational theories have attracted interest recently, as in Refs. [CITATION].', '1905.05690-2-3-4': 'Moreover, general ways to deform the spacetime has been considered to enlighten many gravitational problems and metric disformations are also embedded in this class of transformations (see [CITATION] and the references therein).', '1905.05690-2-3-5': 'In this sense, the structure in [REF] can be seen as a natural way to introduce a general metric structure through a scalar field.', '1905.05690-2-4-0': 'A complete theory can only be set if one defines the functions [MATH] and [MATH], and also the Lagrangian of the scalar field.', '1905.05690-2-4-1': 'Then, a field equation, characterizing the theory, can be derived.', '1905.05690-2-4-2': 'We refer to this class of gravitational theories as geometric scalar gravity (GSG).', '1905.05690-2-4-3': 'In early communications on GSG, it was explored a specific set of those functions defining the theory, which shows that it is possible to go further in representing the gravitational field as a single scalar, giving realistic descriptions of the solar system and cosmology [CITATION].', '1905.05690-2-4-4': 'An analysis of GSG within the so called parametrized post-Newtonian formalism was also made and, although the theory is not covered by the formalism, a limited situation indicate a good agreement with the observations [CITATION].', '1905.05690-2-4-5': 'Intending to improve the understanding of how GSG deals with gravitational interaction, the present work develops the theoretical description and characterization of gravitational waves (GW).', '1905.05690-2-5-0': 'The direct detections of GW by LIGO and Virgo collaborations initiated a new era of testing gravitational theories.', '1905.05690-2-5-1': "It enables to construct constraints over a series of theoretical mechanisms associated with GW's physics, but crucial point relies on the observed waveform and how a theory can reproduce it [CITATION].", '1905.05690-2-5-2': 'Notwithstanding, this is not the scope of this work.', '1905.05690-2-5-3': 'We are mainly focused in analyzing the GW fundamentals on the perspective of GSG, studying their propagation, polarization modes and defining an appropriated tensor to describe the energy and momentum carried by the waves.', '1905.05690-2-5-4': 'The velocity of propagation of GW has been measured with good precision indeed, but this data does not constraint GSG once the gravitational signal travels in vacuum with the same speed of light, as it will be shown later.', '1905.05690-2-5-5': 'However, the theory can be constrained by observational data from pulsars through its prediction for the orbital variation of a binary system that should be caused by the loss of energy due to gravitational radiation.', '1905.05690-2-6-0': 'The paper is organized as follows.', '1905.05690-2-6-1': 'In section [REF] is presented a brief overview of GSG in order to introduce to the reader the main features of this theory.', '1905.05690-2-6-2': "The following section describes the theory's weak field approximation.", '1905.05690-2-6-3': 'In section [REF] the study of the propagation and vibration modes associated to gravitational waves is made.', '1905.05690-2-6-4': 'The definition of a energy-momentum tensor for the linear waves is treated in section [REF].', '1905.05690-2-6-5': 'Generation of waves, including the computation of the orbital variation of binary systems due to the emission of GW, is discussed in section [REF] and the last section presents our concluding remarks.', '1905.05690-2-6-6': 'Also, two appendices were introduced in order to complement the middle steps of calculations present in section [REF].', '1905.05690-2-7-0': '# Overview of geometric scalar gravity', '1905.05690-2-8-0': 'GSG is a class of gravitational theories which identifies the gravitational field to a single real scalar function [MATH], satisfying a non-linear dynamic described by the action [EQUATION] where [MATH] is the determinant of the Minkowski metric and [MATH] is defined in eq. [REF].', '1905.05690-2-8-1': 'Metric signature convention is [MATH].', '1905.05690-2-8-2': 'The physical metric is constructed from the gravitational field according to the expression [REF] and its contravariant form is obtained from the definition of an inverse metric, [MATH], namely, [EQUATION] where, [EQUATION]', '1905.05690-2-8-3': "In order to describe the interaction of the scalar gravitational field with matter, GSG makes the fundamental hypothesis, according to Einstein's proposal, that gravity is a geometric phenomenon.", '1905.05690-2-8-4': 'Thus, it is assumed that the interaction with [MATH] is given only through a minimal coupling with the gravitational metric [MATH].', '1905.05690-2-8-5': 'The matter action in GSG is then described as [EQUATION]', '1905.05690-2-8-6': "A complete theory should specify the metric's functions [MATH] and [MATH] together with the Lagrangian of the scalar field [MATH], in order to be possible to derive its field equation.", '1905.05690-2-8-7': 'Up to now in the literature, it has been explored the case in which the following choice is made, [EQUATION] with [EQUATION]', '1905.05690-2-8-8': 'Using the standard definition of the energy momentum tensor in terms of a metric structure, we set [EQUATION]', '1905.05690-2-8-9': "Then, the dynamics of the scalar field is described by the equation [EQUATION] where the [MATH] indicates the d'Alembertian operator constructed with the physical metric [MATH], [MATH] is a coupling constant and the source term [MATH] is provided by [EQUATION] where '[MATH]' indicates a covariant derivative with respect to the physical metric, [MATH] and [EQUATION]", '1905.05690-2-8-10': 'The choices made in [REF]-[REF] are such that the resulting theory satisfies the Newtonian limit, the classical gravitational tests and the spherically symmetric vacuum solution is given by the Schwarzschild geometry.', '1905.05690-2-8-11': 'Moreover, in the absence of any matter fields, [MATH] is a free wave propagating in the metric [MATH] [CITATION].', '1905.05690-2-8-12': 'More details concerning the fundamentals of GSG and how this specific model can successfully describe the solar system physics and cosmology can be found in [CITATION].', '1905.05690-2-8-13': 'In the present work we will consider only this model.', '1905.05690-2-8-14': 'To work with different functions [MATH], [MATH] and [MATH], all the process of constructing the field equation of the theory has to be redone, as well as it should be checked the feasibility of the new theory.', '1905.05690-2-9-0': '# Weak field approximation', '1905.05690-2-10-0': 'To discuss linear gravitational waves we should consider an isolated system, distant from any source, embedded in a homogenous and isotropic universe.', '1905.05690-2-10-1': 'At a particular moment of time and specific distance from the isolated system, the background metric can be transformed to assume a flat Minkowskian form, resulting in a geometric structure given by, [EQUATION] where [MATH] and [MATH] represents the first order perturbations.', '1905.05690-2-11-0': 'In this sense, the weak-field approximation of GSG consists in a small deviation of a cosmological solution [MATH].', '1905.05690-2-11-1': 'Thus, we set [EQUATION]', '1905.05690-2-11-2': 'In order to construct the geometric structure as in [REF], for simplicity, we start with a coordinate system [MATH], where the auxiliary metric [MATH] assumes the usual diagonal form indicated as [MATH], and we expand the kinetic term and the metric coefficients as follows, [EQUATION]', '1905.05690-2-11-3': 'The subindex "[MATH]" identifies quantities constructed with [MATH] according to basic expressions given in the previous section.', '1905.05690-2-11-4': 'The gravitational metric takes the form [EQUATION] where, [EQUATION] and, [EQUATION]', '1905.05690-2-11-5': 'With the following coordinate transformation [EQUATION] the desired structure is achieved, [EQUATION] where [EQUATION]', '1905.05690-2-11-6': 'In this new coordinate system, [MATH] is equivalente to the cosmological time and [MATH] (in units where [MATH]), where [MATH] is the Hubble parameter (please see [CITATION] for more details on GSG cosmology).', '1905.05690-2-11-7': 'Then, the perturbed metric becomes, [EQUATION]', '1905.05690-2-11-8': 'The corresponding covariant expression for [REF] is obtained from the definition [MATH].', '1905.05690-2-11-9': 'It reads [EQUATION] where, [EQUATION]', '1905.05690-2-11-10': 'Equations [REF] and [REF] shows that in the weak field limit the indices are lowered and raised by the Minkowski background metric.', '1905.05690-2-12-0': 'Note that, the perturbed metric [REF] can also be derived from the expansion of the exact form given in [REF] starting already with the coordinates [MATH], where the auxiliary metric [MATH] takes the form [EQUATION]', '1905.05690-2-12-1': 'The resulting covariant expression can be written as [EQUATION] with [EQUATION]', '1905.05690-2-12-2': 'In reference [CITATION] a distinct weak field approximation was made where the scalar field was expanded around a vanishing background value.', '1905.05690-2-12-3': 'Although consistent, that scheme is not suitable for the description of GW, due to a term [MATH] that is present in [MATH].', '1905.05690-2-12-4': 'Oscillatory solutions would then lead to a singular behavior of the metric, evidencing that the background cosmological scenario can not be neglected.', '1905.05690-2-13-0': '## The cosmological backgroung', '1905.05690-2-14-0': 'Before proceeding in the analysis of GW in GSG, let us clarify important points of the cosmological background described by [MATH].', '1905.05690-2-14-1': 'To a more detailed discussion about the cosmology in GSG we refer to the reader the analysis present in [CITATION].', '1905.05690-2-14-2': 'By considering the scalar field as a function of coordinate [MATH] only, the metric arising is of Friedman-Robertson-Walker type with a flat spatial section.', '1905.05690-2-14-3': 'The cosmological time is achieved by the time transformation given by the first expression in [REF] and the scale factor, said [MATH], is related with the [MATH] as follows, [EQUATION]', '1905.05690-2-14-4': 'The dynamical equation [REF] contains two regimes classified by the term [MATH], a consequence of the particular choice of the scalar field Lagrangian.', '1905.05690-2-14-5': 'The case where [MATH], with a barotropic fluid as source, describes a eternal universe without singularities.', '1905.05690-2-14-6': 'The universe has a bouncing, followed by a early accelerated phase and a final decelerated expansion.', '1905.05690-2-14-7': 'The problematic value [MATH] is unattainable, in other words, the minimal value of the scale factor [MATH] is always greater than [MATH].', '1905.05690-2-14-8': 'A distinct behavior occurs for the solutions with barotropic fluids in the region where [MATH]; the universe starts from a initial singularity, it expands to a certain maximum value of the scale factor, smaller than [MATH], and then returns to a final singular point.', '1905.05690-2-14-9': 'This two regions are then disjoint classes of cosmological solutions.', '1905.05690-2-14-10': 'In the present work, we will consider only the case where [EQUATION] since it represents a class of more realistic descriptions of the universe.', '1905.05690-2-15-0': '# Propagation and polarization of gravitational waves', '1905.05690-2-16-0': 'At the level of the dynamical equation we can consider [MATH] as a constant, since its timescale variation is longer compared to the dynamical timescale for the local system.', '1905.05690-2-16-1': "Expanding the left hand side of Eq. [REF] and neglecting second order terms, one has, [EQUATION] where we refer to Minkowskian d'Alembertian operator as [MATH].", '1905.05690-2-16-2': 'Thus, without the presence of sources, one has [EQUATION]', '1905.05690-2-16-3': 'The perturbed scalar field has oscillatory solutions which propagates at the speed of light.', '1905.05690-2-16-4': 'Once the metric is constructed with the field and its first derivatives, such solutions yields oscillations as GW in the geometric structure of the spacetime.', '1905.05690-2-16-5': 'Moreover, it is verified that [EQUATION] thus gravitational waves in GSG does propagate with the speed of light, showing no deviation with respect to GR in this aspect.', '1905.05690-2-16-6': 'Consequently, GSG is also supported by the combined data of the GW event GW170817 and the gamma-ray burst GRB 170817A, which constraint the velocity of propagation of GW to be the same as the speed of light within deviations of order [MATH] [CITATION].', '1905.05690-2-17-0': '## Polarization states', '1905.05690-2-18-0': 'The most general (weak) gravitational wave that any metric theory of gravity is able to predict can contain six modes of polarization.', '1905.05690-2-18-1': 'Considering plane null waves propagating in a given direction, these modes are related to tetrad components of the irreducible parts of the Riemann tensor, or the Newmann-Penrose quantities (NPQ): [MATH] and [MATH] and [MATH] are complex quantities and each one represents two modes of polarization) [CITATION].', '1905.05690-2-18-2': 'The others NPQ are negligible by the weak field approximation, or are described in terms of these four ones.', '1905.05690-2-19-0': 'The linearized dynamical equations of a gravitational theory can automatically vanish some of these NPQ, specifying then the predicted number of polarization states.', '1905.05690-2-19-1': 'For instance, in GR only [MATH] is not identically zero, which characterizes two transversal polarization modes, called "[MATH]" and "[MATH]" states.', '1905.05690-2-19-2': 'In general, the six polarization modes can not be specified in a observer-independent way because of their behavior under Lorentz transformations.', '1905.05690-2-19-3': 'Nevertheless, if we restrict our attention to a set of specific observers, which agree with the GW on the frequency and on the direction of propagation, then is possible to make some observer-invariant statements about the NPQ.', '1905.05690-2-19-4': 'Such assertions are on the basis of the so called E(2)-classification of gravitational theories, introduced in ref. [CITATION]:', '1905.05690-2-20-0': 'The [MATH]-classification of GSG is easily obtained by noticing that the Ricci scalar is not identically null.', '1905.05690-2-20-1': 'Actually, from the weak field approximation, one has [EQUATION] with [MATH] and, using relations [REF] together with linearized vacuum field equation [REF], it is verified that, [EQUATION]', '1905.05690-2-20-2': 'This result implies [MATH] (cf. equation (A4) of [CITATION]) and GSG is from the class II[MATH].', '1905.05690-2-20-3': 'This [MATH] represents a pure longitudinal polarization state (see figure [REF]) that is always present in the GW, although other modes can be detected depending on the observer.', '1905.05690-2-21-0': 'GSG belongs to the most general class of theories with respect to the [MATH]-classification, where is always possible to find an observer that measures all six gravitational wave modes.', '1905.05690-2-21-1': 'The authors in [CITATION] already pointed out the fact that the number of polarization states predicted by a gravitational theory does not necessarily match the numbers of degrees of freedom inside the theory.', '1905.05690-2-21-2': 'They also give an example of this with the so called stratified theories.', '1905.05690-2-21-3': 'Other examples of theories also classified as [MATH] is the well know [MATH] extensions of general relativity [CITATION].', '1905.05690-2-22-0': 'Thus, the description of GW by GSG carries a substancial distinction from GR, as it predicts the presence of a longitudinal polarization mode.', '1905.05690-2-22-1': 'Up to now, the recent detections of GW can not exclude the existence of any one of the six modes of polarization [CITATION].', '1905.05690-2-22-2': 'But in the future, with the appropriated network of detectors, with different orientations, this information can be used to restrict gravitational theories.', '1905.05690-2-23-0': '# Energy of the gravitational wave', '1905.05690-2-24-0': 'In order to associate an energy-momentum tensor to the gravitational waves in GSG we follow a standard procedure, identifying the relation between the second and the first order perturbations of the gravitational field [CITATION].', '1905.05690-2-24-1': 'First note that, without approximations, the following relation holds, [EQUATION]', '1905.05690-2-24-2': 'Thus, taking [MATH], where the subindexes indicates the order, and computing the second order vacuum field equation, it yields [EQUATION] with [MATH].', '1905.05690-2-24-3': 'The right hand side of this equation contains only the derivatives of the first order field [MATH], thus it can be interpreted as the source for the second order field generated by the linear waves.', '1905.05690-2-25-0': 'From the general structure of the field equation of GSG, the influence of any energy-momentum tensor enters in the equation of motion uniquely through the quantity [MATH] [cf. equation [REF]].', '1905.05690-2-25-1': 'Thus, the energy-momentum tensor of the GW, said [MATH], must be consistent with, [EQUATION] where [MATH] means the second order approximation of the source term calculated with the energy-momentum tensor of the gravitational field [MATH], instead of [MATH].', '1905.05690-2-25-2': "Therefore, we write [EQUATION] which has the same general structure of GSG's field equation.", '1905.05690-2-26-0': 'To describe the energy and momentum carried by the linear waves, the second-order approximation of [MATH] must be quadratic in the first derivatives of [MATH].', '1905.05690-2-26-1': 'This lead us to a specific form for it, [EQUATION] with [MATH] and [MATH] being arbitrary constants.', '1905.05690-2-26-2': 'The condition [REF] returns the relation [EQUATION]', '1905.05690-2-26-3': 'Any tensor, described like in Eq. [REF] and satisfying the above relation, can be used as the energy-momentum tensor of the linear GW in GSG.', '1905.05690-2-26-4': 'This ambiguity already appeared in reference [CITATION], where the authors show how to construct the energy-momentum tensor of the gravitational field in GSG, without using approximate methods.', '1905.05690-2-26-5': 'In that occasion, they fixed the functions defining the energy tensor by requiring that [MATH] can be derived from the Lagrangian.', '1905.05690-2-26-6': 'As expected, their results are consistent with the relation above and are recovered (inside the approximation method) if [MATH] and [MATH].', '1905.05690-2-26-7': 'In what follows we will proceed with the generic expression for [MATH] and look for a specific example, the orbital variations in binary systems, to see how this ambiguity can influence in a observed phenomenon.', '1905.05690-2-27-0': '# Orbital variation of a binary system', '1905.05690-2-28-0': 'This section focus on deriving an expression for the orbital variation of a binary system, due to the emission of gravitational waves, as it is predicted by GSG.', '1905.05690-2-28-1': 'In order to obtain the energy rate emitted by the system one should consider the influence of the source into the dynamics in the linear approximation.', '1905.05690-2-28-2': "Since the left hand side of Eq. [REF] reduces to a Minkowskian d'Alembertian when linearized (c.f. eq. [REF]), from the method of Green functions, we immediately write down the general solution as, [EQUATION] where [MATH] attends to the first order approximation of the source term [cf.[REF]] and [MATH], with [MATH], is the retarded time.", '1905.05690-2-29-0': "By considering that the source is far away from the point where we calculate the scalar field [MATH], where [MATH], and [MATH] is the typical distances between the source's components[MATH], it is possible to make a multipole expansion.", '1905.05690-2-29-1': 'Further assuming that the typical velocities of the source components are non relativistic, it is also possible to expand the time dependent terms of the integrand in a Taylor series.', '1905.05690-2-29-2': 'For our purpose here it is sufficient to take only the first term of this expansion.', '1905.05690-2-29-3': 'Thus, one has [EQUATION] where [MATH] and we have neglected terms of order [MATH].', '1905.05690-2-30-0': 'Most of terms in the above integration contains the scalar field [MATH], explicitly.', '1905.05690-2-30-1': 'To solve them, we have to expand these terms using the correspondent post-Newtonian approximation of the field in the near-zone region [CITATION].', '1905.05690-2-30-2': 'However, to keep the final result up to order [MATH], it is only necessary the Newtonian approximation of the near-zone scalar field, namely [MATH].', '1905.05690-2-30-3': "By the viral theorem, we know that, for slow motions, [MATH], where [MATH], [MATH] and [MATH] are the typical velocity, mass and distances in the source's components, respectively.", '1905.05690-2-30-4': 'Thus, [MATH], [MATH] and [MATH] (see appendix [REF] for more details).', '1905.05690-2-30-5': 'The energy-momentum tensor also depends on source velocities with [MATH] and [MATH].', '1905.05690-2-30-6': 'Thus, keeping terms up to order [MATH] (since [MATH]) and using the perturbed metric expressions in [REF], one gets [EQUATION]', '1905.05690-2-30-7': 'The [MATH] term does not contribute by Gauss law.', '1905.05690-2-30-8': 'Also, to derive the above expressions we take into account that [EQUATION]', '1905.05690-2-30-9': 'Specifying the source for the case of a binary system, we have [EQUATION] where summation is over the two particles of the system, i.e. [MATH].', '1905.05690-2-30-10': 'With these expressions, all the integrals in [REF] can be analytically calculated (more details in Appendix [REF]) to give [EQUATION] where the dot indicates a derivative with respect to retarded time, [MATH] attends to constant terms that does not contribute to the radiation, [MATH] is the Newtonian gravitational constant as measured today (see Appendix [REF]) and the notation was shortened by the definitions below, [EQUATION]', '1905.05690-2-30-11': 'Also, we are adopting the usual center of mass notation such that, [EQUATION] with [MATH] and [MATH].', '1905.05690-2-31-0': 'Once we are dealing with a binary system as the source of the gravitational field, we can use the Keplerian orbital parameters to simplify the above expression [CITATION].', '1905.05690-2-31-1': 'The distance between the two masses are, [EQUATION] where [MATH] is the semimajor axis and [MATH] is the eccentricity of the orbit.', '1905.05690-2-31-2': 'They are related with the total energy [MATH] and the angular momentum [MATH] by [EQUATION] with [MATH].', '1905.05690-2-31-3': 'The fact that [MATH], allow us to derive the following relation, [EQUATION]', '1905.05690-2-31-4': 'Then, in [REF], all time derivatives can be expressed in terms of [MATH], yielding [EQUATION]', '1905.05690-2-31-5': 'To calculate the energy-flux that is carried off by GW we use the gravitational energy-momentum tensor presented in the previous section.', '1905.05690-2-31-6': 'The flux in the radial direction will be [MATH] thus, the energy radiated per unit time that is passing through a sphere of radius [MATH], is given by [EQUATION] where we have used the fact that [EQUATION]', '1905.05690-2-31-7': 'At this point, we go further in the approximation scheme in order to get a more treatable expression for the rate of energy loss.', '1905.05690-2-31-8': 'Let us consider that the background field is too small, i.e. [MATH], and take only the leading order terms.', '1905.05690-2-31-9': 'This is realistic since it is always expected that the cosmological influence on local systems are minimal.', '1905.05690-2-31-10': 'Expression [REF] can be then simplified, reading [EQUATION] where [MATH] is given by [EQUATION]', '1905.05690-2-31-11': 'Averaging the energy loss over an orbital period [MATH], where [EQUATION] we have, [EQUATION]', '1905.05690-2-31-12': 'The above integral is directly solved, yielding [EQUATION] with [EQUATION]', '1905.05690-2-31-13': 'To finish, we derive how this loss of energy changes the orbital period of the system.', '1905.05690-2-31-14': 'From [REF], one gets that [EQUATION]', '1905.05690-2-31-15': 'The result has the same proportionality with the constants [MATH] and [MATH], as in GR, but has a rather more involved dependence on the masses and the eccentricity of the orbit.', '1905.05690-2-32-0': 'Note that equation [REF] must be negative, otherwise it would imply that the masses are moving away from each other.', '1905.05690-2-32-1': 'In other words, the system would be increasing their energy by the emission of GW, an unrealistic situation.', '1905.05690-2-32-2': 'The function [MATH] is positive, as it can be verified by comparison between the term [MATH] and the part involved by the round brackets multiplying [MATH] (the only part that could be negative), [EQUATION]', '1905.05690-2-32-3': 'Since [MATH] for elliptical orbits, it follows that [MATH] is always positive.', '1905.05690-2-32-4': 'Thus to guarantee [MATH], we must have [MATH].', '1905.05690-2-33-0': 'We can use the data from the so called Hulse and Taylor pulsar, PSR 1913+16, to constraint [MATH] demanding that GSG prediction is in agreement with observations.', '1905.05690-2-33-1': 'The data comes from the measurements of time-of-arrivals during [MATH] years [CITATION].', '1905.05690-2-33-2': 'They are collected in Table [REF].', '1905.05690-2-34-0': 'First, we rewrite equation [REF] in a more appropriated form to use the numerical values.', '1905.05690-2-34-1': 'Using [REF], one has [EQUATION]', '1905.05690-2-34-2': 'Substituting the numerical values, with the appropriated propagation of errors, we estimate [MATH] and [MATH] (through condition [REF]), constraining the parameters entering in the GW energy momentum tensor.', '1905.05690-2-35-0': 'It is worth to note that the orbital parameters of the binary system are extracted from the timing pulsar observations in a theory-independent way, but the determination of the masses of the pulsar and its companion are model dependent [CITATION].', '1905.05690-2-35-1': 'The mass values in Table [REF] are from GR but its usage here is reasonable due to the satisfactory agreement of GSG in the Solar System tests at the post-Newtonian level.', '1905.05690-2-35-2': 'However, any modification on these values will lead to a distinct estimation of [MATH] but not an invalidation of GSG by pulsars data.', '1905.05690-2-36-0': '# Concluding remarks', '1905.05690-2-37-0': 'We have presented a discussion on gravitational waves (GW) in the context of the geometric scalar gravity (GSG), a class of theories describing the effects of gravity as a consequence of a modification of spacetime metric in terms of a single scalar field.', '1905.05690-2-37-1': 'GSG overcomes the serious drawbacks present in all previous attempts to formulate a scalar theory of gravity.', '1905.05690-2-37-2': 'Its fundamental idea rests on the proposal that the geometrical structure of the spacetime is described by a disformal transformation of a conformal flat metric.', '1905.05690-2-37-3': 'The model analyzed here has already showed several advances within the scalar gravity program, featuring a good representation of the gravitational phenomena both in the solar system domains as well as in cosmology.', '1905.05690-2-38-0': 'Initially it was shown the procedure used to construct the weak field limit in GSG considering an expansion of the scalar field over a background cosmological solution.', '1905.05690-2-38-1': 'Within this approximation scheme the scalar dynamical equations assumes oscillatory solutions that represent GW in the spacetime structure propagating with light velocity, which is in agreement with recent data from GW and gamma-ray burst detections from the merge of a binary neutron star system.', '1905.05690-2-39-0': 'An important distinction appears in the polarization states of the waves, which is characterized by the presence of a longitudinal mode in GSG.', '1905.05690-2-39-1': 'Within the [MATH]-classification of gravitational theories, GSG is of the type [MATH], since [MATH].', '1905.05690-2-39-2': 'This is the most general class, where the detection of all the other five polarization modes depend on the observer.', '1905.05690-2-39-3': 'The detection of extra polarization states (or the absence of them) shall be a decisive test to alternative theories of gravity [CITATION].', '1905.05690-2-39-4': 'Model-independent tools that allow to see how polarization modes affect the response function in GW detectors has been recently developed and must be also applied in GSG [CITATION].', '1905.05690-2-39-5': 'This procedure should be considered in the future.', '1905.05690-2-40-0': 'It was also discussed how to define an energy-momentum tensor for the linear GW, following a field theoretical point of view.', '1905.05690-2-40-1': 'An ambiguity emerges since GSG fundamental equation includes a non trivial interaction between matter/energy and the scalar field, leading to non unique expression for the approximated gravitational energy-momentum tensor.', '1905.05690-2-40-2': 'This freedom is encoded in the constant parameter [MATH], which has directly influence in the energy-loss rate when emitting gravitational waves.', '1905.05690-2-40-3': "Consequently, GSG prediction for the orbital variation of a binary system can be used to constraint the theory's parameter with observational data coming from PSR 1913+16.", '1905.05690-2-40-4': 'This numerical computation was performed using GR mass values as a first estimation since GSG is in agreement with classical tests and should not present strong deviations on these values.', '1905.05690-2-40-5': 'It is then expected that, after analyzing the so called post-Keplerian parametrization of the theory to extract the mass values of a binary system according to GSG, the theory can be more properly constrained.', '1905.05690-2-40-6': 'This task will be addressed in a future work.', '1905.05690-2-41-0': 'I wish to thank J.C. Fabris and T.R.P. Carames for dedicated reviews of this work.', '1905.05690-2-41-1': 'This research is supported by FAPES and CAPES through the PROFIX program.', '1905.05690-2-42-0': '# The near zone scalar field', '1905.05690-2-43-0': 'The linearized dynamical equation of GSG is a traditional wave equation, which posses some properties depending whether [MATH] (the point where the field is being calculated) is larger or smaller than the typical wavelength [MATH] of the solution [CITATION].', '1905.05690-2-43-1': 'In the wave zone, where [MATH], the difference between the retarded time [MATH] and [MATH] is large, so the time derivative of the field is comparable to the spatial derivative.', '1905.05690-2-43-2': 'This is the region where the radiation effects are influent in determining the metric.', '1905.05690-2-43-3': 'On the other hand, in the region where [MATH], called near zone, the difference between the [MATH] and [MATH] are small and the time derivatives becomes irrelevant in front of the spatial derivatives.', '1905.05690-2-44-0': "The near zone region is covered by the post-Newtonian approximation of the gravitational field, expanding it in orders of [MATH], where [MATH] is the typical velocities of the source's components, and considering also slow motion.", '1905.05690-2-44-1': 'This is the approximation required for the scalar field when integrating the wave equation.', '1905.05690-2-44-2': 'Once the scalar field aways appears multiplied by [MATH] in the integrand, we only need to know its leading order, i.e. its Newtonian approximation.', '1905.05690-2-44-3': 'Thus, equation [REF] reduces to [EQUATION] where only the zeroth-order terms is considered in the above integral.', '1905.05690-2-45-0': "From [REF] and [REF], and using a multipole expansion, one can easily sees that the metric assumes the form [EQUATION] where [EQUATION] and [MATH] attends for the Newton's gravitational constant as measured today and it is a redefinition of the theory's coupling constant, [EQUATION]", '1905.05690-2-45-1': 'Note that the dependence of [MATH] with the cosmological background field implies its change as a result of the evolution of the universe.', '1905.05690-2-45-2': "This effect has not been evident in the previous analysis of GSG's newtonian limit since the cosmological influence was neglect in those works [CITATION].", '1905.05690-2-45-3': 'We will not discuss its implications in the present work, but it certainly shows the importance of take into account the cosmological background when analyzing the Newtonian and post-Newtonian limits of GSG.', '1905.05690-2-46-0': '# More detailed calculations', '1905.05690-2-47-0': 'In this section we aim to be more clear on the calculation of the integrals of the quantities appearing in expressions [REF], [REF] and [REF].', '1905.05690-2-47-1': 'We start with the linearized conservation law, [MATH], from where is possible to derive the following expressions, [EQUATION] where [MATH] represents the second momenta of mass distribution.', '1905.05690-2-47-2': 'Only the trace of the quadrupole moment enters in the field equation and it is directly calculated, [EQUATION]', '1905.05690-2-47-3': 'The time derivatives can be now easily calculated.', '1905.05690-2-48-0': 'The remaining integrals does contain the newtonian limit of the scalar field explicitly.', '1905.05690-2-48-1': 'For the specific case of binary system as a source, the solution [REF] becomes [EQUATION]', '1905.05690-2-48-2': 'When calculating the Newtonian gravitational potential in the position of one of the particles of the system we have to neglect the infinity self potential, thus [EQUATION] where the summation above is taken excluding terms when [MATH].', '1905.05690-2-48-3': 'This can be interpreted as a mass renormalization [CITATION].', '1905.05690-2-48-4': 'Using this we can integrate expression [REF] to give [EQUATION]', '1905.05690-2-48-5': 'For the remaining integrals it is needed the derivatives of the Newtonian scalar field, namely [EQUATION] and the kinect term, [EQUATION]', '1905.05690-2-48-6': 'In the above expressions [MATH], [MATH], and [MATH].', '1905.05690-2-48-7': 'The sub-indexes [MATH] are summation indices assuming the values [MATH] or [MATH].', '1905.05690-2-48-8': 'The upper-indexes [MATH] refer to the usual components of a three-vector and they run from [MATH] to [MATH].', '1905.05690-2-49-0': 'Let us calculate one of the integrals explicitly, [EQUATION] where the symbol [MATH] means the product of two summations, one in [MATH] and other in [MATH], with both never assuming the value of [MATH].', '1905.05690-2-49-1': 'Using that [EQUATION] and [EQUATION] we have [EQUATION]', '1905.05690-2-49-2': 'In the last equality we used the relations [REF].', '1905.05690-2-49-3': 'The procedure is similar for the other integrals and, paying attention that [MATH], it follows [EQUATION]', '1905.05690-2-49-4': 'Putting all these terms together, following [REF], we obtain the relation, [EQUATION]', '1905.05690-2-49-5': 'In the integral of [REF] the following terms will appear, [EQUATION] and, with a time derivative, we obtain the last terms remaining to get the expression [REF].'} | [['1905.05690-1-21-0', '1905.05690-2-21-0'], ['1905.05690-1-21-1', '1905.05690-2-21-1'], ['1905.05690-1-21-2', '1905.05690-2-21-2'], ['1905.05690-1-21-3', '1905.05690-2-21-3'], ['1905.05690-1-26-0', '1905.05690-2-26-0'], ['1905.05690-1-26-1', '1905.05690-2-26-1'], ['1905.05690-1-26-2', '1905.05690-2-26-2'], ['1905.05690-1-26-3', '1905.05690-2-26-3'], ['1905.05690-1-26-4', '1905.05690-2-26-4'], ['1905.05690-1-26-5', '1905.05690-2-26-5'], ['1905.05690-1-26-6', '1905.05690-2-26-6'], ['1905.05690-1-26-7', '1905.05690-2-26-7'], ['1905.05690-1-22-0', '1905.05690-2-22-0'], ['1905.05690-1-22-1', '1905.05690-2-22-1'], ['1905.05690-1-22-2', '1905.05690-2-22-2'], ['1905.05690-1-18-0', '1905.05690-2-18-0'], ['1905.05690-1-18-1', '1905.05690-2-18-1'], ['1905.05690-1-18-2', '1905.05690-2-18-2'], 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1905.05690 | null | null | null | null | null |
1908.02116 | {'1908.02116-1-0-0': 'Facial landmark detection aims to localize the anatomically defined points of human faces.', '1908.02116-1-0-1': 'In this paper, we study facial landmark detection from partially labeled facial images.', '1908.02116-1-0-2': "A typical approach is to (1) train a detector on the labeled images; (2) generate new training samples using this detector's prediction as pseudo labels of unlabeled images; (3) retrain the detector on the labeled samples and partial pseudo labeled samples.", '1908.02116-1-0-3': 'In this way, the detector can learn from both labeled and unlabeled data to become robust.', '1908.02116-1-1-0': 'In this paper, we propose an interaction mechanism between a teacher and two students to generate more reliable pseudo labels for unlabeled data, which are beneficial to semi-supervised facial landmark detection.', '1908.02116-1-1-1': 'Specifically, the two students are instantiated as dual detectors.', '1908.02116-1-1-2': 'The teacher learns to judge the quality of the pseudo labels generated by the students and filter out unqualified samples before the retraining stage.', '1908.02116-1-1-3': 'In this way, the student detectors get feedback from their teacher and are retrained by premium data generated by itself.', '1908.02116-1-1-4': 'Since the two students are trained by different samples, a combination of their predictions will be more robust as the final prediction compared to either prediction.', '1908.02116-1-1-5': 'Extensive experiments on 300-W and AFLW benchmarks show that the interactions between teacher and students contribute to better utilization of the unlabeled data and achieves state-of-the-art performance.', '1908.02116-1-2-0': '# Introduction', '1908.02116-1-3-0': 'Facial landmark detection aims to find some pre-defined anatomical keypoints of human faces [CITATION].', '1908.02116-1-3-1': 'These keypoints include the corners of a mouth, the boundary of eyes, the tip of a nose, etc [CITATION].', '1908.02116-1-3-2': 'It is usually a prerequisite of a large number of computer vision tasks.', '1908.02116-1-3-3': 'For example, facial landmark coordinates are required to align faces to ease the visualization for users when people would like to sort their faces by time and see the changes over time [CITATION].', '1908.02116-1-3-4': 'Other examples include face morphing [CITATION], face replacement [CITATION], etc.', '1908.02116-1-4-0': 'The main challenge in recent landmark detection literatures is how to obtain abundant facial landmark labels.', '1908.02116-1-4-1': 'The annotation challenge comes from two perspectives.', '1908.02116-1-4-2': 'First, a large number of keypoints are required for a single face image, e.g., 68 keypoints for each face in the 300-W dataset [CITATION].', '1908.02116-1-4-3': 'To precisely depict the facial features for a whole dataset, millions of keypoints are usually required.', '1908.02116-1-4-4': 'Second, different annotators have a semantic gap.', '1908.02116-1-4-5': 'There is no universal standard for the annotation of the keypoints, so different annotators give different positions for the same keypoints.', '1908.02116-1-4-6': 'A typical way to reduce such semantic deviations among various annotators is to merge the labels from several annotators.', '1908.02116-1-4-7': 'This will further increase the costs of the whole annotation work.', '1908.02116-1-5-0': 'Semi-supervised landmark detection can to some extent alleviate the expensive and sophisticated annotations by utilizing the unlabeled images.', '1908.02116-1-5-1': 'Typical approaches [CITATION] for semi-supervised learning use self-training or similar paradigms to utilize the unlabeled samples.', '1908.02116-1-5-2': 'For example, the authors of [CITATION] adopt a heuristic unsupervised criterion to select the pseudo labeled data for the retraining procedure.', '1908.02116-1-5-3': 'This criterion is the loss of each pseudo labeled data, where its predicted pseudo label is treated as the ground truth to calculate the loss [CITATION].', '1908.02116-1-5-4': 'Since no extra supervision is given to train the criterion function, this unsupervised loss criterion has a high possibility of passing inaccurate pseudo labeled data to the retraining stage.', '1908.02116-1-5-5': 'In this way, these inaccurate data will mislead the optimization of the detector and make it easier to trap into a local minimum.', '1908.02116-1-5-6': 'A straightforward solution to this problem is to use multiple models and regularize each other by the co-training strategy [CITATION].', '1908.02116-1-5-7': 'Unfortunately, even if co-training performs well in simple tasks such as classification [CITATION], in more complex scenarios such as detection, co-training requires extremely sophisticated design and careful tuning of many additional hyper-parameters [CITATION], e.g., more than 10 hyper-parameters for three models in [CITATION].', '1908.02116-1-6-0': 'To better utilize the pseudo labeled data as well as avoid the complicated model tuning for landmark detection, we propose Teacher Supervises StudentS (TS[MATH]).', '1908.02116-1-6-1': 'As illustrated in fig:high-level, TS[MATH] is an interaction mechanism between one teacher network and two (or multiple) student networks.', '1908.02116-1-6-2': 'Two student detection networks learn to generate pseudo labels for unlabeled images.', '1908.02116-1-6-3': 'The teacher network learns to judge the quality of the pseudo labels generated from students.', '1908.02116-1-6-4': 'Consequently, the teacher can select qualified pseudo labeled samples and use them to retrain the students.', '1908.02116-1-6-5': 'TS[MATH] applies these steps in an iterative manner, where students gradually become more robust, and the teacher is adaptively updated with the improved students.', '1908.02116-1-6-6': 'Besides, two students can also encourage each other to advance their performances in two ways.', '1908.02116-1-6-7': 'First, predictions from two students can be ensembled to further improve the quality of pseudo labels.', '1908.02116-1-6-8': 'Second, two students can regularize each other by training on different samples.', '1908.02116-1-6-9': 'The interactions between the teacher and students as well as the students themselves help to provide more accurate pseudo labeled samples for retraining and the model does not need careful hyper-parameter tuning.', '1908.02116-1-7-0': 'To highlight our contribution, we propose an easy-to-train interaction mechanism between teacher and students (TS[MATH]) to provide more reliable pseudo labeled samples in semi-supervised facial landmark detection.', '1908.02116-1-7-1': 'To validate the performance of our TS[MATH], we do experiments on 300-W, 300-VW, and AFLW benchmarks.', '1908.02116-1-7-2': 'TS[MATH] achieves state-of-the-art semi-supervised performance on all three benchmarks.', '1908.02116-1-7-3': 'In addition, using only 30% labels, our TS[MATH] achieves competitive results compared to supervised methods using all labels on 300-W and AFLW.', '1908.02116-1-8-0': '# Related Work', '1908.02116-1-9-0': 'We will first introduce some supervised facial landmark algorithms in sec:relate-supervised.', '1908.02116-1-9-1': 'Then, we will compare our algorithm with semi-supervised learning algorithms and semi-supervised facial landmark algorithm in sec:relate-semi.', '1908.02116-1-9-2': 'Lastly, we explain our algorithm in a meta learning perspective in sec:relate-meta.', '1908.02116-1-10-0': '## Supervised Facial Landmark Detection', '1908.02116-1-11-0': 'Supervised facial landmark detection algorithms can be categorized into linear regression based methods [CITATION] and heatmap regression based methods [CITATION].', '1908.02116-1-11-1': 'Linear regression based methods learn a function that maps the input face image to the normalized landmark coordinates [CITATION].', '1908.02116-1-11-2': 'Heatmap regression based methods produce one heatmap for each landmark, where the coordinate is the location of the highest response on this heatmap [CITATION].', '1908.02116-1-11-3': 'All above algorithms can be readily integrated into our framework, serving as different student detectors.', '1908.02116-1-12-0': 'These supervised algorithms require a large amount of data to train deep neural networks.', '1908.02116-1-12-1': 'However, it is tedious to annotate the precise facial landmarks, which need to average different annotations from multiple different annotators.', '1908.02116-1-12-2': 'Therefore, to reduce the annotation cost, it is necessary to investigate the semi-supervised facial landmark detection.', '1908.02116-1-13-0': '## Semi-supervised Facial Landmark Detection', '1908.02116-1-14-0': 'Some early semi-supervised learning algorithms are difficult to handle large scale datasets due to the high complexity [CITATION].', '1908.02116-1-14-1': 'Others exploit pseudo-labels of unlabeled data in the semi-supervised scenario [CITATION].', '1908.02116-1-14-2': 'Since most of these algorithms studied their effect on small-scale datasets [CITATION], a question remains open: can they be used to improve large-scale semi-supervised landmark detection?', '1908.02116-1-14-3': 'In addition, those self-training or co-training approaches [CITATION] simply leverage the confidence score or an unsupervised loss to select qualified samples.', '1908.02116-1-14-4': 'For example, Dong et al. [CITATION] proposed a model communication mechanism to select reliable pseudo labeled samples based on loss and score.', '1908.02116-1-14-5': 'However, such selection criterion does not reflect the real quality of a pseudo labeled sample.', '1908.02116-1-14-6': 'In contrast, our teacher directly learns to model the quality, and selected samples are thus more reliable.', '1908.02116-1-15-0': 'There are only few of researchers study the semi-supervised facial landmark detection algorithms.', '1908.02116-1-15-1': 'A recent work [CITATION] presented two techniques to improve landmark localization from partially annotated face images.', '1908.02116-1-15-2': 'The first technique is to jointly train facial landmark network with an attribute network, which predicts the emotion, head pose, etc.', '1908.02116-1-15-3': 'In this multi-task framework, the gradient from the attribute network can benefit the landmark prediction.', '1908.02116-1-15-4': 'The second technique is a kind of supervision without the need of manual labels, which enables the transformation invariant of landmark prediction.', '1908.02116-1-15-5': 'Compared to using the supervision from transformation, our approach leverages a progressive paradigm to learn facial shape information from unlabeled data.', '1908.02116-1-15-6': 'In this way, our approach is orthogonal to [CITATION], and these two techniques can complement our approach to further boost the performance.', '1908.02116-1-16-0': 'Radosavovic et al. [CITATION] applied the data augmentation to improve the quality of generated pseudo landmark labels.', '1908.02116-1-16-1': 'For an unlabeled image, they ensemble predictions from multiple transformations, such as flipping and rotation.', '1908.02116-1-16-2': 'This strategy can also be used to improve the accuracy of our pseudo labels and complement our approach.', '1908.02116-1-16-3': 'Since the data augmentation is not the focus of this paper, we did not apply their algorithms in our approach.', '1908.02116-1-16-4': 'Dong et al. [CITATION] proposed a self-supervised loss by exploiting the temporal consistence on unlabeled videos to enhance the detector.', '1908.02116-1-16-5': 'This is a video-based approach and not the focus of our work.', '1908.02116-1-16-6': 'Therefore, we do not discuss more with those video-based approach [CITATION].', '1908.02116-1-17-0': '## Meta Learning', '1908.02116-1-18-0': 'In a meta learning perspective, our TS[MATH] learns a teacher network to learn which pseudo labeled samples are helpful to train student detectors.', '1908.02116-1-18-1': 'In this sense, we are related to some recent literature in "learning to learn" [CITATION].', '1908.02116-1-18-2': 'For example, Ren et al. [CITATION] learn to re-weight samples based on gradients of a model on the clean validation set.', '1908.02116-1-18-3': 'Xu et al. [CITATION] suggest using meta-learning to tune the optimization schedule of alternative optimization problems.', '1908.02116-1-18-4': 'Jiang et al. [CITATION] propose an architecture to learn data-driven curriculum on corrupted labels.', '1908.02116-1-18-5': 'Fan et al. [CITATION] leverage reinforcement learning to learn a policy to select good training samples for a single student model.', '1908.02116-1-18-6': 'These algorithms are designed in the supervised scenarios and can not easily be modified in semi-supervised scenario.', '1908.02116-1-19-0': 'Difference with other teacher-student frameworks and generative adversarial networks (GAN).', '1908.02116-1-19-1': 'Our learns to utilize the output (pseudo labels) of the student model qualified by the teacher model to do semi-supervised learning.', '1908.02116-1-19-2': 'Other teacher-student methods [CITATION] aim to fit the output of the student model to that of the teacher model.', '1908.02116-1-19-3': 'The student and teacher in our work do similar jobs as the generator and discriminator in GAN [CITATION], while we aim to predict/generate qualified pseudo labels in semi-supervised learning using a different training strategy.', '1908.02116-1-20-0': '# Methodology', '1908.02116-1-21-0': 'In this section, we will first introduce the scenario of the semi-supervised facial landmark detection in sec:method-setting.', '1908.02116-1-21-1': 'We explain how to design our student detectors and the teacher network in sec:teacher-student.', '1908.02116-1-21-2': 'Lastly, we demonstrate our overall algorithm in sec:method-algo.', '1908.02116-1-22-0': '## The Semi-Supervised Scenario', '1908.02116-1-23-0': 'We introduce some necessary notations for the presentation of the proposed method.', '1908.02116-1-23-1': 'Let [MATH] be the labeled data in the training set and [MATH] be the unlabeled data in the training set, where [MATH] denotes the [MATH]-th image, and [MATH] denotes the ground-truth landmark label of [MATH].', '1908.02116-1-23-2': '[MATH] is the number of the facial landmarks, and the [MATH]-th column of [MATH] indicates the coordinate of the [MATH]-th landmark.', '1908.02116-1-23-3': '[MATH] and [MATH] denote the number of labeled data and unlabeled data, respectively.', '1908.02116-1-23-4': 'The semi-supervised facial landmark detection aims to learn robust detectors from both [MATH] and [MATH].', '1908.02116-1-24-0': '## Teacher and Students Design', '1908.02116-1-25-0': 'The Student Detectors.', '1908.02116-1-25-1': 'We choose the convolutional pose machine (CPM) [CITATION] and stacked hourglass (HG) [CITATION] models as our student detectors.', '1908.02116-1-25-2': 'These two landmark detection architectures are the cornerstone of many facial landmark detection algorithms [CITATION].', '1908.02116-1-25-3': 'Moreover, their architectures are quite different, and can thus complement each other to achieve a better detection performance compared to using two similar neural architectures.', '1908.02116-1-25-4': 'Therefore, we integrate these two detectors in our TS[MATH] approach.', '1908.02116-1-25-5': 'In this paragraph, we will give a brief overview of these two facial landmark detectors.', '1908.02116-1-25-6': 'We illustrate the structures of CPM and HG in fig:network.', '1908.02116-1-25-7': 'Both CPM and HG are the heatmap regression based methods and utilize the cascaded structure.', '1908.02116-1-25-8': 'Formally, suppose there are [MATH] convolutional stages in CPM, the output of CPM is:', '1908.02116-1-26-0': '[EQUATION]', '1908.02116-1-27-0': 'where [MATH] indicates the CPM student detector whose parameters are [MATH].', '1908.02116-1-27-1': '[MATH] is the RGB image of the [MATH]-th data-point and [MATH] indicates the heatmap prediction of the [MATH]-th stage.', '1908.02116-1-27-2': '[MATH] and [MATH] denote the spatial height and width of the heatmap.', '1908.02116-1-27-3': 'Similarly, we use [MATH] indicates the HG student detector whose parameters are [MATH].', '1908.02116-1-27-4': 'The detection loss function of the CPM student is:', '1908.02116-1-28-0': '[EQUATION] where [MATH] is a function taking the label [MATH] as inputs to generate the the ideal heatmap [MATH].', '1908.02116-1-28-1': 'Details of [MATH] can be found in [CITATION].', '1908.02116-1-28-2': 'During the evaluation, we take the argmax results over the first [MATH] channel of the last heatmap [MATH] as the coordinates of landmarks, and the [MATH]-th channel corresponding to the background will be omitted.', '1908.02116-1-29-0': 'The Teacher Network.', '1908.02116-1-29-1': 'Since our student detectors are based on heatmap, the pseudo label is in the form of heatmap and ground truth label is the ideal heatmap.', '1908.02116-1-29-2': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-1-29-3': 'As shown in fig:value-network, the input of this teacher network is the concatenation of a face image and its heatmap prediction [MATH].', '1908.02116-1-29-4': 'The output of this teacher network is a scalar representing the quality of a pseudo labeled facial image.', '1908.02116-1-29-5': 'Since we train the teacher on the trustworthy labeled data, we could obtain a supervised detection loss by calculating [MATH].', '1908.02116-1-29-6': 'We consider the negative value of this detection loss as the ground truth label of the quality, because a high negative value of the detection loss indicates a high similarity between the predicted heatmap and the ideal heatmap.', '1908.02116-1-29-7': 'In another word, a higher quality scalar corresponds to a more accurate pseudo label.', '1908.02116-1-30-0': '[EQUATION] where the parameters of the teacher is [MATH]. "[', '1908.02116-1-30-1': '[MATH]" first resizes the tensor [MATH] into the same spatial shape as [MATH] and then concatenates the resized tensor with [MATH] to get a new tensor.', '1908.02116-1-30-2': 'This new tensor is regarded as pseudo labeled image and will be qualified by the teacher later.', '1908.02116-1-30-3': 'The teacher outputs a scalar [MATH] representing the quality of the [MATH]-th sample associated with its pseudo label [MATH].', '1908.02116-1-30-4': 'We optimize the teacher on the trustworthy labeled data by minimizing eq:teacher-loss.', '1908.02116-1-31-0': '[t!]', '1908.02116-1-31-1': 'The Algorithm Description of Our TS[MATH]', '1908.02116-1-32-0': '[1] Labeled data [MATH]', '1908.02116-1-33-0': 'Unlabeled data [MATH]', '1908.02116-1-34-0': 'Two student detectors [MATH] with [MATH] and [MATH] with [MATH]', '1908.02116-1-35-0': 'The teacher network [MATH] with parameters [MATH]', '1908.02116-1-36-0': 'The selection ratio [MATH] and the maximum step [MATH]', '1908.02116-1-37-0': 'Initialize the [MATH] and [MATH] by minimizing eq:cpm-hg-loss on [MATH]; [MATH]; [MATH]++', '1908.02116-1-38-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the first student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with prediction [MATH] and ground truth label [MATH] Compute the quality scalar of each sample in [MATH] using the optimized teacher via eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-1-39-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the second student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with [MATH] and [MATH] Compute the quality scalar of each sample in [MATH] using eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-1-40-0': 'Students with optimized parameters [MATH] and [MATH]', '1908.02116-1-41-0': '## The TS[MATH] Algorithm', '1908.02116-1-42-0': 'Our TS[MATH] aims to progressively improve the performance of the student detector.', '1908.02116-1-42-1': 'The key idea is to learn a teacher network that can teach students which pseudo labeled sample is reliable and can be used for training.', '1908.02116-1-42-2': 'In this procedure, we define the pseudo label of a facial image is as follows:', '1908.02116-1-43-0': '[EQUATION] where [MATH] indicates the heatmap prediction from the first student at the [MATH]-th stage for the [MATH]-th sample.', '1908.02116-1-43-1': '[MATH] in eq:ensemble indicates the ensemble result from both two students detection networks.', '1908.02116-1-43-2': 'It will be used as the prediction during the inference procedure.', '1908.02116-1-44-0': 'We show our overall algorithm in alg:SSFLD.', '1908.02116-1-44-1': 'We first initialize the two detectors [MATH] and [MATH] on the labeled facial images [MATH].', '1908.02116-1-44-2': 'Then, in the first round, our algorithm applies the following procedures: (1) generate pseudo labels on [MATH] via eq:ensemble and train the teacher network from scratch with these pseudo labels; (2) generate pseudo labels on [MATH] and estimate the quality of these pseudo labeled using the learned teacher; (3) select some high-quality pseudo labeled samples to retrain one student network from scratch.', '1908.02116-1-44-3': '(4) repeat the first three steps to update another student detection network.', '1908.02116-1-44-4': 'In the next rounds, each student can be improved and generate more accurate pseudo labels.', '1908.02116-1-44-5': 'In this way, we will select more pseudo labeled samples when retraining the students.', '1908.02116-1-44-6': 'As the rounds go, students will gradually become better, and the teacher will also be adaptive with the improved students.', '1908.02116-1-44-7': 'Our interaction mechanism helps to obtain more accurate pseudo labels and select more reliable pseudo labeled samples.', '1908.02116-1-44-8': 'As a result, our algorithm achieves better performance in the semi-supervised facial landmark detection.', '1908.02116-1-45-0': '## Discussion', '1908.02116-1-46-0': 'Can this algorithm generalize to other tasks?', '1908.02116-1-46-1': 'Our algorithm relies on the design of the teacher network.', '1908.02116-1-46-2': 'It requires the input pseudo label to be a structured prediction.', '1908.02116-1-46-3': 'Therefore, our algorithm is possible to be applied to tasks with structured predictions, such as segmentation and pose estimation, but is not suitable other tasks like classification.', '1908.02116-1-47-0': 'Limitation.', '1908.02116-1-47-1': 'It is challenging for a teacher to judge the quality of a pseudo label for an image, especially when the spatial shape of this image becomes large.', '1908.02116-1-47-2': 'Therefore, in this paper, we use an input size of 64[MATH]64.', '1908.02116-1-47-3': 'If we increase the input size to 256[MATH]256, the teacher will fail and need to be modified accordingly.', '1908.02116-1-47-4': 'There are two main reasons: (1) the larger resolution requires a deeper architecture or dilated convolutions for the teacher network and (2) the high-resolution faces bring high-dimensional inputs, and consequently, the teacher needs much more training data.', '1908.02116-1-47-5': 'This drawback limits the extension of our algorithm to high-resolution tasks, such as segmentation.', '1908.02116-1-47-6': 'We will explore to solve this problem in the future.', '1908.02116-1-48-0': 'Further improvements.', '1908.02116-1-48-1': '(1) In our algorithm, during the retraining procedure, a part of unlabeled samples are not involved during retraining.', '1908.02116-1-48-2': 'To utilize these unlabeled facial images, we could use self-supervised techniques such as [CITATION] to improve the detectors.', '1908.02116-1-48-3': '(2) In this framework, we use only two student detectors, while it is easy to integrate more student detectors.', '1908.02116-1-48-4': 'More student detectors are likely to improve the prediction accuracy, but this will introduce more computation costs.', '1908.02116-1-48-5': '(3) The specifically designed data augmentation [CITATION] is another direction to improve the accuracy and precision of the pseudo labels.', '1908.02116-1-49-0': 'Will the teacher network over-fit to the labeled data?', '1908.02116-1-50-0': "In alg:SSFLD, since labeled data set [MATH] is used to optimize both teacher and students, the teacher's judgment could suffer from the over-fitting problem.", '1908.02116-1-50-1': "Most of the students' predictions on the labeled data can be similar to the ground truth labels.", '1908.02116-1-50-2': 'In other words, most pseudo labeled samples on [MATH] are "correctly" labeled samples.', '1908.02116-1-50-3': 'If the teacher is optimized on [MATH] with those pseudo labels, it might only learn what a good pseudo labeled sample is, but overlook what a bad one is.', '1908.02116-1-50-4': 'It would be more reasonable to let students predict on the unseen validation set, and then train the teacher on this validation set.', '1908.02116-1-50-5': 'However, having an additional validation set during training is different from the typical setting of previous semi-supervised facial landmark detection.', '1908.02116-1-50-6': 'We would explore this problem in our future work.', '1908.02116-1-51-0': '# Empirical Studies', '1908.02116-1-52-0': 'We perform experiments on three benchmark datasets to investigate the behavior of the proposed method.', '1908.02116-1-52-1': 'The datasets and experiment settings are introduced in sec:datasets and sec:setting.', '1908.02116-1-52-2': 'We first compare the proposed semi-supervised facial landmark algorithm with other state-of-the-art algorithms in Sec. [REF].', '1908.02116-1-52-3': 'We then perform ablation studies in Sec. [REF] and visualize our results at last.', '1908.02116-1-53-0': '## Datasets', '1908.02116-1-54-0': 'The 300-W dataset [CITATION] annotates 68 landmarks from five facial landmark datasets, i.e., LFPW, AFW, HELEN, XM2VTS, and IBUG.', '1908.02116-1-54-1': 'Following the common settings [CITATION], we regard all the training samples from LFPW, HELEN and the full set of AFW as the training set, in which there is 3148 training images.', '1908.02116-1-54-2': 'The common test subset consists of 554 test images from LFPW and HELEN.', '1908.02116-1-54-3': 'The challenging test subset consists of 135 images from IBUG to construct .', '1908.02116-1-54-4': 'The full test set the union of the common and challenging subsets, 689 images in total.', '1908.02116-1-55-0': 'The AFLW dataset [CITATION] contains 21997 real-world images with 25993 faces in total.', '1908.02116-1-55-1': 'They provide at most 21 landmark coordinates for each face, but they exclude invisible landmarks.', '1908.02116-1-55-2': 'Faces in AFLW usually have a different head pose, expression, occlusion or illumination, and therefore it causes difficulties to train a robust detector.', '1908.02116-1-55-3': 'Following the same setting as in [CITATION], we do not use the landmarks of two ears.', '1908.02116-1-55-4': 'There are two types of AFLW splits, i.e., AFLW-Full and AFLW-Frontal following [CITATION].', '1908.02116-1-55-5': 'AFLW-Full contains 20000 training samples and 4386 test samples.', '1908.02116-1-55-6': 'AFLW-Front uses the same training samples as in AFLW-Full, but only use the 1165 samples with the frontal face as the test set.', '1908.02116-1-56-0': 'The 300-VW dataset [CITATION] is a video-based facial landmark benchmark.', '1908.02116-1-56-1': 'It contains 50 training videos with 95192 frames.', '1908.02116-1-56-2': 'Following [CITATION], we report the results for the 49 inner points on the category C subset of the 300-VW test set, which has 26338 frames.', '1908.02116-1-57-0': '## Experimental Settings', '1908.02116-1-58-0': 'Training student detection networks.', '1908.02116-1-58-1': 'The first student detector is CPM [CITATION].', '1908.02116-1-58-2': 'We follow the same model configuration as the base detector used in [CITATION], and the number of cascaded stages is set as three.', '1908.02116-1-58-3': 'To train CPM, we apply the SGD optimizer with the momentum of 0.9 and the weight decay of 0.0005.', '1908.02116-1-58-4': 'For each stage, we train the CPM for 50 epochs in total.', '1908.02116-1-58-5': 'We start the learning rate of 0.00005, and reduce it by 0.5 at 20-th, 25-th, 30-th, and 40-th epoch.', '1908.02116-1-59-0': 'The second student detector is HG [CITATION].', '1908.02116-1-59-1': 'We follow the same model configuration as [CITATION] but use the number of cascaded stages of three to build our HG model.', '1908.02116-1-59-2': 'To train HG, we apply the RMSprop optimizer with the alpha of 0.99.', '1908.02116-1-59-3': 'For each stage, we train the HG for 110 epochs in total.', '1908.02116-1-59-4': 'We start the learning rate of 0.00025, and reduce it by 0.5 at 50-th, 70-th, 90-th, and 100-th.', '1908.02116-1-60-0': 'For both of these two detectors, we use the batch size of eight on two GPUs.', '1908.02116-1-60-1': 'To generate the heatmap ground truth labels, we apply the Gaussian distribution with the sigma of 3.', '1908.02116-1-60-2': 'Each face image is first resized into the size of 64[MATH]64, and then randomly resized between the scale of 0.9 and 1.1.', '1908.02116-1-60-3': 'After the random resize operation, the face image will be randomly rotated with the maximum degree of 30, and then randomly cropped with the size of 64[MATH]64.', '1908.02116-1-60-4': 'We set selection ratio [MATH] as [MATH] and the maximum step [MATH] as [MATH] based on cross-validation.', '1908.02116-1-61-0': 'Training the teacher network.', '1908.02116-1-61-1': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-1-61-2': 'Given a 64[MATH]64 face image, we first resize the predicted heatmap into the same spatial size of 64[MATH]64.', '1908.02116-1-61-3': 'We use the Adam to train this teacher network.', '1908.02116-1-61-4': 'The initial learning rate is 0.01, and the batch size is 128.', '1908.02116-1-61-5': 'Random flip, random rotation, random scale and crop are applied as data argumentation.', '1908.02116-1-62-0': 'Evaluation.', '1908.02116-1-62-1': 'Normalized Mean Error (NME) is usually applied to evaluate the performance for facial landmark predictions [CITATION].', '1908.02116-1-62-2': 'For the 300-W dataset, we use the inter-ocular distance to normalize mean error following the same setting as in [CITATION].', '1908.02116-1-62-3': 'For the AFLW dataset, we use the face size to normalize mean error [CITATION].', '1908.02116-1-62-4': 'Area Under the Curve (AUC) @ 0.08 error is also employed for evaluation [CITATION].', '1908.02116-1-62-5': 'When training on the partially labeled data, the sets of [MATH] and [MATH] are randomly sampled.', '1908.02116-1-62-6': 'During evaluation, we use eq:ensemble to obtain the final heatmap and follow [CITATION] to generate the coordinate of each landmark.', '1908.02116-1-62-7': 'We repeat each experiment three times and report the mean result.', '1908.02116-1-62-8': 'The codes will be public available upon the acceptance.', '1908.02116-1-63-0': '## Comparison with state-of-the-art', '1908.02116-1-64-0': 'Comparisons on 300-W.', '1908.02116-1-64-1': 'We compare our algorithm with several state-of-the-art algorithms [CITATION], as shown in table:300W-ALL.', '1908.02116-1-64-2': 'In this table, [CITATION] are very recent methods, which represent the state-of-the-art supervised facial landmark algorithms.', '1908.02116-1-64-3': 'By using 100% facial landmark labels on 300-W training set and unlabeled AFLW, our algorithm achieves competitive 3.49 NME on the 300-W common test set, which is competitive to other state-of-the-art algorithms.', '1908.02116-1-64-4': 'In addition, even though our approach utilizes two detectors, the number of parameters is much lower than SAN [CITATION].', '1908.02116-1-64-5': 'The robust detection performance of ours can be mainly caused by two reasons.', '1908.02116-1-64-6': 'First, the proposed teacher network can effectively sample the qualified pseudo labeled data, which enables the model to exploit more useful information.', '1908.02116-1-64-7': 'Second, our framework leverages two advanced CNN architectures, which can complement each other.', '1908.02116-1-65-0': 'We also compare our with a recent work on semi-supervised facial landmark detection [CITATION] in table:300W-ALL.', '1908.02116-1-65-1': 'When using 10 of labels, our obtains a lower NME result on the challenging test set than RCN[MATH] [CITATION] (5.64 NME vs. 6.32 NME).', '1908.02116-1-65-2': 'When using 20 of labels, our is also superior to it (5.03 NME vs. 5.88 NME).', '1908.02116-1-65-3': 'Note that [CITATION] utilizes a transformation invariant auxiliary loss function.', '1908.02116-1-65-4': 'This auxiliary loss can also be easily integrated into our framework.', '1908.02116-1-65-5': 'Therefore, [CITATION] is orthogonal to our work, combining two methods can potentially achieve a better performance.', '1908.02116-1-66-0': 'Comparisons on AFLW.', '1908.02116-1-66-1': 'We also show the NME comparison on the AFLW dataset in table:aflw.', '1908.02116-1-66-2': 'Compared to semi-supervised facial landmark detection algorithm [CITATION], we achieve a similar performance.', '1908.02116-1-66-3': 'RCN[MATH] [CITATION] can learn transformation invariant information from a large amount of unlabeled images, while ours does not consider this information as it is not our focus.', '1908.02116-1-66-4': 'On the AFLW-Full test set, using 20 annotation, our framework achieves 1.99 NME, which is competitive to other supervised algorithms.', '1908.02116-1-66-5': 'On the AFLW-Front test set, using only 10 annotation, our framework achieves competitive NME results to [CITATION].', '1908.02116-1-66-6': 'The above results demonstrate our framework can train a robust detector with much less annotation effort.', '1908.02116-1-67-0': 'Comparisons on 300-VW.', '1908.02116-1-67-1': 'We experiment our algorithm to leverage a large amount of unlabeled facial video frames on 300-VW.', '1908.02116-1-67-2': 'We use the labeled 300-W training set and the unlabeled 300-VW training set to train our .', '1908.02116-1-67-3': 'We evaluate the learned detectors on the 300-VW C test subset w.r.t. AUC @ 0.08.', '1908.02116-1-67-4': 'Some video-based facial landmark detection algorithms [CITATION] utilize the labeled 300-VW training data to improve the base detectors.', '1908.02116-1-67-5': 'Compared with them, without using any label on 300-VW, our obtains a higher AUC result than them, i.e., 59.65 vs. 59.39, as shown in table:300VW-C.', '1908.02116-1-68-0': '## Ablation Study', '1908.02116-1-69-0': 'The key contribution of our lies on two components: (1) the teacher supervising the training data selection of students.', '1908.02116-1-69-1': '(2) the complementary effect of two students.', '1908.02116-1-69-2': 'In this subsection, we validate the contribution of these two components to the final detection performance.', '1908.02116-1-70-0': 'The effect of the teacher.', '1908.02116-1-70-1': 'Compared to other progressive pseudo label generation strategies [CITATION], our designed teacher can sample pseudo labeled with higher quality.', '1908.02116-1-70-2': 'In fig:compare, we show the detection results after the first five training rounds (only 10% labels are used).', '1908.02116-1-70-3': 'We use SPL [CITATION] to separately train CPM and HG, and then ensemble them together as eq:ensemble.', '1908.02116-1-70-4': 'We use SPaCo [CITATION] to jointly optimize CPM and HG in a co-training strategy.', '1908.02116-1-70-5': 'To make a fair comparison, at each round, we control the number of pseudo labels is the same across these three algorithms.', '1908.02116-1-70-6': 'From fig:compare, several conclusions can be made: (1) obtains the lowest NME, because the quality of selected pseudo labels is better than others.', '1908.02116-1-70-7': '(2) SPL falls into a local trap at round[MATH] and results in a higher error at round[MATH], whereas SPaCo and our not.', '1908.02116-1-70-8': 'This could be caused by that the interaction between two students can help regularize each other.', '1908.02116-1-70-9': '(3) Our converges faster than SPaCo and achieves better results.', '1908.02116-1-70-10': 'The pseudo labeled data selection in SPaco is a heuristic unsupervised criterion, whereas our criterion is a supervised teacher.', '1908.02116-1-70-11': 'Since no extra supervision is given in SPaCo, their criterion might induce inaccurate pseudo labeled samples.', '1908.02116-1-71-0': 'The effect of the interaction between students.', '1908.02116-1-71-1': 'From Table [REF], we show the ablative studies on the complementary effect of multiple students.', '1908.02116-1-71-2': 'In these experiments, we use the same teacher structure, while "CPM" and "HG" are trained without the interaction between students.', '1908.02116-1-71-3': 'Using 10 labels, CPM achieves 8.28 NME, and HG achieves 6.25 NME on 300-W.', '1908.02116-1-71-4': 'Leveraging from their mutual benefits, our can boost the performance to 5.64, which is higher than CPM by about 30 and than HG by 9.', '1908.02116-1-71-5': 'Under different portion of annotations, we can conclude similar observations.', '1908.02116-1-71-6': 'This ablation study demonstrates the contribution of student interaction to the final performance.', '1908.02116-1-71-7': 'Note that, our algorithm can be readily applied to multiple students without introducing additional hyper-parameters.', '1908.02116-1-71-8': 'In contrast, the number of hyper-parameters in other co-training strategies [CITATION] is quadratic to the number of detectors.', '1908.02116-1-72-0': '## Qualitative Analysis', '1908.02116-1-73-0': 'On the 300-W training set, we train our using only 10% labeled facial images, and we show some qualitative results of the 300-W test set in fig:results.', '1908.02116-1-73-1': 'The first row shows seven raw input facial images.', '1908.02116-1-73-2': 'The second row shows the ground truth background heatmaps, and the third row shows the faces with ground truth landmarks of these images.', '1908.02116-1-73-3': 'We visualize the predicted background heatmap in the fourth row and the predicted coordinates in the fifth row.', '1908.02116-1-73-4': 'As we can see, the predicted landmarks of our are very close to the ground truth.', '1908.02116-1-73-5': 'These predictions are already robust enough, and human may not be able to distinguish the difference between our predictions (the third line) and the ground truth (the fifth line).', '1908.02116-1-74-0': '# Conclusion', '1908.02116-1-75-0': 'In this paper, we propose an interaction mechanism between a teacher and multiple students for semi-supervised facial landmark detection.', '1908.02116-1-75-1': 'The students learn to generate pseudo labels for the unlabeled data, while the teacher learns to judge the quality of these pseudo labeled data.', '1908.02116-1-75-2': 'After that, the teacher can filter out unqualified samples; and the students get feedback from the teacher and improve itself by the qualified samples.', '1908.02116-1-75-3': 'The teacher is adaptive along with the improved students.', '1908.02116-1-75-4': 'Besides, multiple students can not only regularize each other but also be ensembled to predict more accurate pseudo labels.', '1908.02116-1-75-5': 'We empirically demonstrate that the proposed interaction mechanism achieves state-of-the-art performance on three facial landmark benchmarks.'} | {'1908.02116-2-0-0': 'Facial landmark detection aims to localize the anatomically defined points of human faces.', '1908.02116-2-0-1': 'In this paper, we study facial landmark detection from partially labeled facial images.', '1908.02116-2-0-2': "A typical approach is to (1) train a detector on the labeled images; (2) generate new training samples using this detector's prediction as pseudo labels of unlabeled images; (3) retrain the detector on the labeled samples and partial pseudo labeled samples.", '1908.02116-2-0-3': 'In this way, the detector can learn from both labeled and unlabeled data to become robust.', '1908.02116-2-1-0': 'In this paper, we propose an interaction mechanism between a teacher and two students to generate more reliable pseudo labels for unlabeled data, which are beneficial to semi-supervised facial landmark detection.', '1908.02116-2-1-1': 'Specifically, the two students are instantiated as dual detectors.', '1908.02116-2-1-2': 'The teacher learns to judge the quality of the pseudo labels generated by the students and filter out unqualified samples before the retraining stage.', '1908.02116-2-1-3': 'In this way, the student detectors get feedback from their teacher and are retrained by premium data generated by itself.', '1908.02116-2-1-4': 'Since the two students are trained by different samples, a combination of their predictions will be more robust as the final prediction compared to either prediction.', '1908.02116-2-1-5': 'Extensive experiments on 300-W and AFLW benchmarks show that the interactions between teacher and students contribute to better utilization of the unlabeled data and achieves state-of-the-art performance.', '1908.02116-2-2-0': '# Introduction', '1908.02116-2-3-0': 'Facial landmark detection aims to find some pre-defined anatomical keypoints of human faces [CITATION].', '1908.02116-2-3-1': 'These keypoints include the corners of a mouth, the boundary of eyes, the tip of a nose, etc [CITATION].', '1908.02116-2-3-2': 'It is usually a prerequisite of a large number of computer vision tasks [CITATION].', '1908.02116-2-3-3': 'For example, facial landmark coordinates are required to align faces to ease the visualization for users when people would like to sort their faces by time and see the changes over time [CITATION].', '1908.02116-2-3-4': 'Other examples include face morphing [CITATION], face replacement [CITATION], etc.', '1908.02116-2-4-0': 'The main challenge in recent landmark detection literatures is how to obtain abundant facial landmark labels.', '1908.02116-2-4-1': 'The annotation challenge comes from two perspectives.', '1908.02116-2-4-2': 'First, a large number of keypoints are required for a single face image, e.g., 68 keypoints for each face in the 300-W dataset [CITATION].', '1908.02116-2-4-3': 'To precisely depict the facial features for a whole dataset, millions of keypoints are usually required.', '1908.02116-2-4-4': 'Second, different annotators have a semantic gap.', '1908.02116-2-4-5': 'There is no universal standard for the annotation of the keypoints, so different annotators give different positions for the same keypoints.', '1908.02116-2-4-6': 'A typical way to reduce such semantic deviations among various annotators is to merge the labels from several annotators.', '1908.02116-2-4-7': 'This will further increase the costs of the whole annotation work.', '1908.02116-2-5-0': 'Semi-supervised landmark detection can to some extent alleviate the expensive and sophisticated annotations by utilizing the unlabeled images.', '1908.02116-2-5-1': 'Typical approaches [CITATION] for semi-supervised learning use self-training or similar paradigms to utilize the unlabeled samples.', '1908.02116-2-5-2': 'For example, the authors of [CITATION] adopt a heuristic unsupervised criterion to select the pseudo labeled data for the retraining procedure.', '1908.02116-2-5-3': 'This criterion is the loss of each pseudo labeled data, where its predicted pseudo label is treated as the ground truth to calculate the loss [CITATION].', '1908.02116-2-5-4': 'Since no extra supervision is given to train the criterion function, this unsupervised loss criterion has a high possibility of passing inaccurate pseudo labeled data to the retraining stage.', '1908.02116-2-5-5': 'In this way, these inaccurate data will mislead the optimization of the detector and make it easier to trap into a local minimum.', '1908.02116-2-5-6': 'A straightforward solution to this problem is to use multiple models and regularize each other by the co-training strategy [CITATION].', '1908.02116-2-5-7': 'Unfortunately, even if co-training performs well in simple tasks such as classification [CITATION], in more complex scenarios such as detection, co-training requires extremely sophisticated design and careful tuning of many additional hyper-parameters [CITATION], e.g., more than 10 hyper-parameters for three models in [CITATION].', '1908.02116-2-6-0': 'To better utilize the pseudo labeled data as well as avoid the complicated model tuning for landmark detection, we propose Teacher Supervises StudentS (TS[MATH]).', '1908.02116-2-6-1': 'As illustrated in fig:high-level, TS[MATH] is an interaction mechanism between one teacher network and two (or multiple) student networks.', '1908.02116-2-6-2': 'Two student detection networks learn to generate pseudo labels for unlabeled images.', '1908.02116-2-6-3': 'The teacher network learns to judge the quality of the pseudo labels generated from students.', '1908.02116-2-6-4': 'Consequently, the teacher can select qualified pseudo labeled samples and use them to retrain the students.', '1908.02116-2-6-5': 'TS[MATH] applies these steps in an iterative manner, where students gradually become more robust, and the teacher is adaptively updated with the improved students.', '1908.02116-2-6-6': 'Besides, two students can also encourage each other to advance their performances in two ways.', '1908.02116-2-6-7': 'First, predictions from two students can be ensembled to further improve the quality of pseudo labels.', '1908.02116-2-6-8': 'Second, two students can regularize each other by training on different samples.', '1908.02116-2-6-9': 'The interactions between the teacher and students as well as the students themselves help to provide more accurate pseudo labeled samples for retraining and the model does not need careful hyper-parameter tuning.', '1908.02116-2-7-0': 'To highlight our contribution, we propose an easy-to-train interaction mechanism between teacher and students (TS[MATH]) to provide more reliable pseudo labeled samples in semi-supervised facial landmark detection.', '1908.02116-2-7-1': 'To validate the performance of our TS[MATH], we do experiments on 300-W, 300-VW, and AFLW benchmarks.', '1908.02116-2-7-2': 'TS[MATH] achieves state-of-the-art semi-supervised performance on all three benchmarks.', '1908.02116-2-7-3': 'In addition, using only 30% labels, our TS[MATH] achieves competitive results compared to supervised methods using all labels on 300-W and AFLW.', '1908.02116-2-8-0': '# Related Work', '1908.02116-2-9-0': 'We will first introduce some supervised facial landmark algorithms in sec:relate-supervised.', '1908.02116-2-9-1': 'Then, we will compare our algorithm with semi-supervised learning algorithms and semi-supervised facial landmark algorithm in sec:relate-semi.', '1908.02116-2-9-2': 'Lastly, we explain our algorithm in a meta learning perspective in sec:relate-meta.', '1908.02116-2-10-0': '## Supervised Facial Landmark Detection', '1908.02116-2-11-0': 'Supervised facial landmark detection algorithms can be categorized into linear regression based methods [CITATION] and heatmap regression based methods [CITATION].', '1908.02116-2-11-1': 'Linear regression based methods learn a function that maps the input face image to the normalized landmark coordinates [CITATION].', '1908.02116-2-11-2': 'Heatmap regression based methods produce one heatmap for each landmark, where the coordinate is the location of the highest response on this heatmap [CITATION].', '1908.02116-2-11-3': 'All above algorithms can be readily integrated into our framework, serving as different student detectors.', '1908.02116-2-12-0': 'These supervised algorithms require a large amount of data to train deep neural networks.', '1908.02116-2-12-1': 'However, it is tedious to annotate the precise facial landmarks, which need to average different annotations from multiple different annotators.', '1908.02116-2-12-2': 'Therefore, to reduce the annotation cost, it is necessary to investigate the semi-supervised facial landmark detection.', '1908.02116-2-13-0': '## Semi-supervised Facial Landmark Detection', '1908.02116-2-14-0': 'Some early semi-supervised learning algorithms are difficult to handle large scale datasets due to the high complexity [CITATION].', '1908.02116-2-14-1': 'Others exploit pseudo-labels of unlabeled data in the semi-supervised scenario [CITATION].', '1908.02116-2-14-2': 'Since most of these algorithms studied their effect on small-scale datasets [CITATION], a question remains open: can they be used to improve large-scale semi-supervised landmark detection?', '1908.02116-2-14-3': 'In addition, those self-training or co-training approaches [CITATION] simply leverage the confidence score or an unsupervised loss to select qualified samples.', '1908.02116-2-14-4': 'For example, Dong et al. [CITATION] proposed a model communication mechanism to select reliable pseudo labeled samples based on loss and score.', '1908.02116-2-14-5': 'However, such selection criterion does not reflect the real quality of a pseudo labeled sample.', '1908.02116-2-14-6': 'In contrast, our teacher directly learns to model the quality, and selected samples are thus more reliable.', '1908.02116-2-15-0': 'There are only few of researchers study the semi-supervised facial landmark detection algorithms.', '1908.02116-2-15-1': 'A recent work [CITATION] presented two techniques to improve landmark localization from partially annotated face images.', '1908.02116-2-15-2': 'The first technique is to jointly train facial landmark network with an attribute network, which predicts the emotion, head pose, etc.', '1908.02116-2-15-3': 'In this multi-task framework, the gradient from the attribute network can benefit the landmark prediction.', '1908.02116-2-15-4': 'The second technique is a kind of supervision without the need of manual labels, which enables the transformation invariant of landmark prediction.', '1908.02116-2-15-5': 'Compared to using the supervision from transformation, our approach leverages a progressive paradigm to learn facial shape information from unlabeled data.', '1908.02116-2-15-6': 'In this way, our approach is orthogonal to [CITATION], and these two techniques can complement our approach to further boost the performance.', '1908.02116-2-16-0': 'Radosavovic et al. [CITATION] applied the data augmentation to improve the quality of generated pseudo landmark labels.', '1908.02116-2-16-1': 'For an unlabeled image, they ensemble predictions from multiple transformations, such as flipping and rotation.', '1908.02116-2-16-2': 'This strategy can also be used to improve the accuracy of our pseudo labels and complement our approach.', '1908.02116-2-16-3': 'Since the data augmentation is not the focus of this paper, we did not apply their algorithms in our approach.', '1908.02116-2-16-4': 'Dong et al. [CITATION] proposed a self-supervised loss by exploiting the temporal consistence on unlabeled videos to enhance the detector.', '1908.02116-2-16-5': 'This is a video-based approach and not the focus of our work.', '1908.02116-2-16-6': 'Therefore, we do not discuss more with those video-based approach [CITATION].', '1908.02116-2-17-0': '## Meta Learning', '1908.02116-2-18-0': 'In a meta learning perspective, our TS[MATH] learns a teacher network to learn which pseudo labeled samples are helpful to train student detectors.', '1908.02116-2-18-1': 'In this sense, we are related to some recent literature in "learning to learn" [CITATION].', '1908.02116-2-18-2': 'For example, Ren et al. [CITATION] learn to re-weight samples based on gradients of a model on the clean validation set.', '1908.02116-2-18-3': 'Xu et al. [CITATION] suggest using meta-learning to tune the optimization schedule of alternative optimization problems.', '1908.02116-2-18-4': 'Jiang et al. [CITATION] propose an architecture to learn data-driven curriculum on corrupted labels.', '1908.02116-2-18-5': 'Fan et al. [CITATION] leverage reinforcement learning to learn a policy to select good training samples for a single student model.', '1908.02116-2-18-6': 'These algorithms are designed in the supervised scenarios and can not easily be modified in semi-supervised scenario.', '1908.02116-2-19-0': 'Difference with other teacher-student frameworks and generative adversarial networks (GAN).', '1908.02116-2-19-1': 'Our learns to utilize the output (pseudo labels) of the student model qualified by the teacher model to do semi-supervised learning.', '1908.02116-2-19-2': 'Other teacher-student methods [CITATION] aim to fit the output of the student model to that of the teacher model.', '1908.02116-2-19-3': 'The student and teacher in our work do similar jobs as the generator and discriminator in GAN [CITATION], while we aim to predict/generate qualified pseudo labels in semi-supervised learning using a different training strategy.', '1908.02116-2-20-0': '# Methodology', '1908.02116-2-21-0': 'In this section, we will first introduce the scenario of the semi-supervised facial landmark detection in sec:method-setting.', '1908.02116-2-21-1': 'We explain how to design our student detectors and the teacher network in sec:teacher-student.', '1908.02116-2-21-2': 'Lastly, we demonstrate our overall algorithm in sec:method-algo.', '1908.02116-2-22-0': '## The Semi-Supervised Scenario', '1908.02116-2-23-0': 'We introduce some necessary notations for the presentation of the proposed method.', '1908.02116-2-23-1': 'Let [MATH] be the labeled data in the training set and [MATH] be the unlabeled data in the training set, where [MATH] denotes the [MATH]-th image, and [MATH] denotes the ground-truth landmark label of [MATH].', '1908.02116-2-23-2': '[MATH] is the number of the facial landmarks, and the [MATH]-th column of [MATH] indicates the coordinate of the [MATH]-th landmark.', '1908.02116-2-23-3': '[MATH] and [MATH] denote the number of labeled data and unlabeled data, respectively.', '1908.02116-2-23-4': 'The semi-supervised facial landmark detection aims to learn robust detectors from both [MATH] and [MATH].', '1908.02116-2-24-0': '## Teacher and Students Design', '1908.02116-2-25-0': 'The Student Detectors.', '1908.02116-2-25-1': 'We choose the convolutional pose machine (CPM) [CITATION] and stacked hourglass (HG) [CITATION] models as our student detectors.', '1908.02116-2-25-2': 'These two landmark detection architectures are the cornerstone of many facial landmark detection algorithms [CITATION].', '1908.02116-2-25-3': 'Moreover, their architectures are quite different, and can thus complement each other to achieve a better detection performance compared to using two similar neural architectures.', '1908.02116-2-25-4': 'Therefore, we integrate these two detectors in our TS[MATH] approach.', '1908.02116-2-25-5': 'In this paragraph, we will give a brief overview of these two facial landmark detectors.', '1908.02116-2-25-6': 'We illustrate the structures of CPM and HG in fig:network.', '1908.02116-2-25-7': 'Both CPM and HG are the heatmap regression based methods and utilize the cascaded structure.', '1908.02116-2-25-8': 'Formally, suppose there are [MATH] convolutional stages in CPM, the output of CPM is:', '1908.02116-2-26-0': '[EQUATION]', '1908.02116-2-27-0': 'where [MATH] indicates the CPM student detector whose parameters are [MATH].', '1908.02116-2-27-1': '[MATH] is the RGB image of the [MATH]-th data-point and [MATH] indicates the heatmap prediction of the [MATH]-th stage.', '1908.02116-2-27-2': '[MATH] and [MATH] denote the spatial height and width of the heatmap.', '1908.02116-2-27-3': 'Similarly, we use [MATH] indicates the HG student detector whose parameters are [MATH].', '1908.02116-2-27-4': 'The detection loss function of the CPM student is:', '1908.02116-2-28-0': '[EQUATION] where [MATH] is a function taking the label [MATH] as inputs to generate the the ideal heatmap [MATH].', '1908.02116-2-28-1': 'Details of [MATH] can be found in [CITATION].', '1908.02116-2-28-2': 'During the evaluation, we take the argmax results over the first [MATH] channel of the last heatmap [MATH] as the coordinates of landmarks, and the [MATH]-th channel corresponding to the background will be omitted.', '1908.02116-2-29-0': 'The Teacher Network.', '1908.02116-2-29-1': 'Since our student detectors are based on heatmap, the pseudo label is in the form of heatmap and ground truth label is the ideal heatmap.', '1908.02116-2-29-2': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-2-29-3': 'As shown in fig:value-network, the input of this teacher network is the concatenation of a face image and its heatmap prediction [MATH].', '1908.02116-2-29-4': 'The output of this teacher network is a scalar representing the quality of a pseudo labeled facial image.', '1908.02116-2-29-5': 'Since we train the teacher on the trustworthy labeled data, we could obtain a supervised detection loss by calculating [MATH].', '1908.02116-2-29-6': 'We consider the negative value of this detection loss as the ground truth label of the quality, because a high negative value of the detection loss indicates a high similarity between the predicted heatmap and the ideal heatmap.', '1908.02116-2-29-7': 'In another word, a higher quality scalar corresponds to a more accurate pseudo label.', '1908.02116-2-30-0': '[EQUATION] where the parameters of the teacher is [MATH]. "[', '1908.02116-2-30-1': '[MATH]" first resizes the tensor [MATH] into the same spatial shape as [MATH] and then concatenates the resized tensor with [MATH] to get a new tensor.', '1908.02116-2-30-2': 'This new tensor is regarded as pseudo labeled image and will be qualified by the teacher later.', '1908.02116-2-30-3': 'The teacher outputs a scalar [MATH] representing the quality of the [MATH]-th sample associated with its pseudo label [MATH].', '1908.02116-2-30-4': 'We optimize the teacher on the trustworthy labeled data by minimizing eq:teacher-loss.', '1908.02116-2-31-0': '[t!]', '1908.02116-2-31-1': 'The Algorithm Description of Our TS[MATH]', '1908.02116-2-32-0': '[1] Labeled data [MATH]', '1908.02116-2-33-0': 'Unlabeled data [MATH]', '1908.02116-2-34-0': 'Two student detectors [MATH] with [MATH] and [MATH] with [MATH]', '1908.02116-2-35-0': 'The teacher network [MATH] with parameters [MATH]', '1908.02116-2-36-0': 'The selection ratio [MATH] and the maximum step [MATH]', '1908.02116-2-37-0': 'Initialize the [MATH] and [MATH] by minimizing eq:cpm-hg-loss on [MATH]; [MATH]; [MATH]++', '1908.02116-2-38-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the first student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with prediction [MATH] and ground truth label [MATH] Compute the quality scalar of each sample in [MATH] using the optimized teacher via eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-2-39-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the second student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with [MATH] and [MATH] Compute the quality scalar of each sample in [MATH] using eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-2-40-0': 'Students with optimized parameters [MATH] and [MATH]', '1908.02116-2-41-0': '## The TS[MATH] Algorithm', '1908.02116-2-42-0': 'Our TS[MATH] aims to progressively improve the performance of the student detector.', '1908.02116-2-42-1': 'The key idea is to learn a teacher network that can teach students which pseudo labeled sample is reliable and can be used for training.', '1908.02116-2-42-2': 'In this procedure, we define the pseudo label of a facial image is as follows:', '1908.02116-2-43-0': '[EQUATION] where [MATH] indicates the heatmap prediction from the first student at the [MATH]-th stage for the [MATH]-th sample.', '1908.02116-2-43-1': '[MATH] in eq:ensemble indicates the ensemble result from both two students detection networks.', '1908.02116-2-43-2': 'It will be used as the prediction during the inference procedure.', '1908.02116-2-44-0': 'We show our overall algorithm in alg:SSFLD.', '1908.02116-2-44-1': 'We first initialize the two detectors [MATH] and [MATH] on the labeled facial images [MATH].', '1908.02116-2-44-2': 'Then, in the first round, our algorithm applies the following procedures: (1) generate pseudo labels on [MATH] via eq:ensemble and train the teacher network from scratch with these pseudo labels; (2) generate pseudo labels on [MATH] and estimate the quality of these pseudo labeled using the learned teacher; (3) select some high-quality pseudo labeled samples to retrain one student network from scratch.', '1908.02116-2-44-3': '(4) repeat the first three steps to update another student detection network.', '1908.02116-2-44-4': 'In the next rounds, each student can be improved and generate more accurate pseudo labels.', '1908.02116-2-44-5': 'In this way, we will select more pseudo labeled samples when retraining the students.', '1908.02116-2-44-6': 'As the rounds go, students will gradually become better, and the teacher will also be adaptive with the improved students.', '1908.02116-2-44-7': 'Our interaction mechanism helps to obtain more accurate pseudo labels and select more reliable pseudo labeled samples.', '1908.02116-2-44-8': 'As a result, our algorithm achieves better performance in the semi-supervised facial landmark detection.', '1908.02116-2-45-0': '## Discussion', '1908.02116-2-46-0': 'Can this algorithm generalize to other tasks?', '1908.02116-2-46-1': 'Our algorithm relies on the design of the teacher network.', '1908.02116-2-46-2': 'It requires the input pseudo label to be a structured prediction.', '1908.02116-2-46-3': 'Therefore, our algorithm is possible to be applied to tasks with structured predictions, such as segmentation and pose estimation, but is not suitable other tasks like classification.', '1908.02116-2-47-0': 'Limitation.', '1908.02116-2-47-1': 'It is challenging for a teacher to judge the quality of a pseudo label for an image, especially when the spatial shape of this image becomes large.', '1908.02116-2-47-2': 'Therefore, in this paper, we use an input size of 64[MATH]64.', '1908.02116-2-47-3': 'If we increase the input size to 256[MATH]256, the teacher will fail and need to be modified accordingly.', '1908.02116-2-47-4': 'There are two main reasons: (1) the larger resolution requires a deeper architecture or dilated convolutions for the teacher network and (2) the high-resolution faces bring high-dimensional inputs, and consequently, the teacher needs much more training data.', '1908.02116-2-47-5': 'This drawback limits the extension of our algorithm to high-resolution tasks, such as segmentation.', '1908.02116-2-47-6': 'We will explore to solve this problem in the future.', '1908.02116-2-48-0': 'Further improvements.', '1908.02116-2-48-1': '(1) In our algorithm, during the retraining procedure, a part of unlabeled samples are not involved during retraining.', '1908.02116-2-48-2': 'To utilize these unlabeled facial images, we could use self-supervised techniques such as [CITATION] to improve the detectors.', '1908.02116-2-48-3': '(2) In this framework, we use only two student detectors, while it is easy to integrate more student detectors.', '1908.02116-2-48-4': 'More student detectors are likely to improve the prediction accuracy, but this will introduce more computation costs.', '1908.02116-2-48-5': '(3) The specifically designed data augmentation [CITATION] is another direction to improve the accuracy and precision of the pseudo labels.', '1908.02116-2-49-0': 'Will the teacher network over-fit to the labeled data?', '1908.02116-2-50-0': "In alg:SSFLD, since labeled data set [MATH] is used to optimize both teacher and students, the teacher's judgment could suffer from the over-fitting problem.", '1908.02116-2-50-1': "Most of the students' predictions on the labeled data can be similar to the ground truth labels.", '1908.02116-2-50-2': 'In other words, most pseudo labeled samples on [MATH] are "correctly" labeled samples.', '1908.02116-2-50-3': 'If the teacher is optimized on [MATH] with those pseudo labels, it might only learn what a good pseudo labeled sample is, but overlook what a bad one is.', '1908.02116-2-50-4': 'It would be more reasonable to let students predict on the unseen validation set, and then train the teacher on this validation set.', '1908.02116-2-50-5': 'However, having an additional validation set during training is different from the typical setting of previous semi-supervised facial landmark detection.', '1908.02116-2-50-6': 'We would explore this problem in our future work.', '1908.02116-2-51-0': '# Empirical Studies', '1908.02116-2-52-0': 'We perform experiments on three benchmark datasets to investigate the behavior of the proposed method.', '1908.02116-2-52-1': 'The datasets and experiment settings are introduced in sec:datasets and sec:setting.', '1908.02116-2-52-2': 'We first compare the proposed semi-supervised facial landmark algorithm with other state-of-the-art algorithms in Sec. [REF].', '1908.02116-2-52-3': 'We then perform ablation studies in Sec. [REF] and visualize our results at last.', '1908.02116-2-53-0': '## Datasets', '1908.02116-2-54-0': 'The 300-W dataset [CITATION] annotates 68 landmarks from five facial landmark datasets, i.e., LFPW, AFW, HELEN, XM2VTS, and IBUG.', '1908.02116-2-54-1': 'Following the common settings [CITATION], we regard all the training samples from LFPW, HELEN and the full set of AFW as the training set, in which there is 3148 training images.', '1908.02116-2-54-2': 'The common test subset consists of 554 test images from LFPW and HELEN.', '1908.02116-2-54-3': 'The challenging test subset consists of 135 images from IBUG to construct .', '1908.02116-2-54-4': 'The full test set the union of the common and challenging subsets, 689 images in total.', '1908.02116-2-55-0': 'The AFLW dataset [CITATION] contains 21997 real-world images with 25993 faces in total.', '1908.02116-2-55-1': 'They provide at most 21 landmark coordinates for each face, but they exclude invisible landmarks.', '1908.02116-2-55-2': 'Faces in AFLW usually have a different head pose, expression, occlusion or illumination, and therefore it causes difficulties to train a robust detector.', '1908.02116-2-55-3': 'Following the same setting as in [CITATION], we do not use the landmarks of two ears.', '1908.02116-2-55-4': 'There are two types of AFLW splits, i.e., AFLW-Full and AFLW-Frontal following [CITATION].', '1908.02116-2-55-5': 'AFLW-Full contains 20000 training samples and 4386 test samples.', '1908.02116-2-55-6': 'AFLW-Front uses the same training samples as in AFLW-Full, but only use the 1165 samples with the frontal face as the test set.', '1908.02116-2-56-0': 'The 300-VW dataset [CITATION] is a video-based facial landmark benchmark.', '1908.02116-2-56-1': 'It contains 50 training videos with 95192 frames.', '1908.02116-2-56-2': 'Following [CITATION], we report the results for the 49 inner points on the category C subset of the 300-VW test set, which has 26338 frames.', '1908.02116-2-57-0': '## Experimental Settings', '1908.02116-2-58-0': 'Training student detection networks.', '1908.02116-2-58-1': 'The first student detector is CPM [CITATION].', '1908.02116-2-58-2': 'We follow the same model configuration as the base detector used in [CITATION], and the number of cascaded stages is set as three.', '1908.02116-2-58-3': 'To train CPM, we apply the SGD optimizer with the momentum of 0.9 and the weight decay of 0.0005.', '1908.02116-2-58-4': 'For each stage, we train the CPM for 50 epochs in total.', '1908.02116-2-58-5': 'We start the learning rate of 0.00005, and reduce it by 0.5 at 20-th, 25-th, 30-th, and 40-th epoch.', '1908.02116-2-59-0': 'The second student detector is HG [CITATION].', '1908.02116-2-59-1': 'We follow the same model configuration as [CITATION] but use the number of cascaded stages of three to build our HG model.', '1908.02116-2-59-2': 'To train HG, we apply the RMSprop optimizer with the alpha of 0.99.', '1908.02116-2-59-3': 'For each stage, we train the HG for 110 epochs in total.', '1908.02116-2-59-4': 'We start the learning rate of 0.00025, and reduce it by 0.5 at 50-th, 70-th, 90-th, and 100-th.', '1908.02116-2-60-0': 'For both of these two detectors, we use the batch size of eight on two GPUs.', '1908.02116-2-60-1': 'To generate the heatmap ground truth labels, we apply the Gaussian distribution with the sigma of 3.', '1908.02116-2-60-2': 'Each face image is first resized into the size of 64[MATH]64, and then randomly resized between the scale of 0.9 and 1.1.', '1908.02116-2-60-3': 'After the random resize operation, the face image will be randomly rotated with the maximum degree of 30, and then randomly cropped with the size of 64[MATH]64.', '1908.02116-2-60-4': 'We set selection ratio [MATH] as [MATH] and the maximum step [MATH] as [MATH] based on cross-validation.', '1908.02116-2-61-0': 'Training the teacher network.', '1908.02116-2-61-1': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-2-61-2': 'Given a 64[MATH]64 face image, we first resize the predicted heatmap into the same spatial size of 64[MATH]64.', '1908.02116-2-61-3': 'We use the Adam to train this teacher network.', '1908.02116-2-61-4': 'The initial learning rate is 0.01, and the batch size is 128.', '1908.02116-2-61-5': 'Random flip, random rotation, random scale and crop are applied as data argumentation.', '1908.02116-2-62-0': 'Evaluation.', '1908.02116-2-62-1': 'Normalized Mean Error (NME) is usually applied to evaluate the performance for facial landmark predictions [CITATION].', '1908.02116-2-62-2': 'For the 300-W dataset, we use the inter-ocular distance to normalize mean error following the same setting as in [CITATION].', '1908.02116-2-62-3': 'For the AFLW dataset, we use the face size to normalize mean error [CITATION].', '1908.02116-2-62-4': 'Area Under the Curve (AUC) @ 0.08 error is also employed for evaluation [CITATION].', '1908.02116-2-62-5': 'When training on the partially labeled data, the sets of [MATH] and [MATH] are randomly sampled.', '1908.02116-2-62-6': 'During evaluation, we use eq:ensemble to obtain the final heatmap and follow [CITATION] to generate the coordinate of each landmark.', '1908.02116-2-62-7': 'We repeat each experiment three times and report the mean result.', '1908.02116-2-62-8': 'The codes will be public available upon the acceptance.', '1908.02116-2-63-0': '## Comparison with state-of-the-art', '1908.02116-2-64-0': 'Comparisons on 300-W.', '1908.02116-2-64-1': 'We compare our algorithm with several state-of-the-art algorithms [CITATION], as shown in table:300W-ALL.', '1908.02116-2-64-2': 'In this table, [CITATION] are very recent methods, which represent the state-of-the-art supervised facial landmark algorithms.', '1908.02116-2-64-3': 'By using 100% facial landmark labels on 300-W training set and unlabeled AFLW, our algorithm achieves competitive 3.49 NME on the 300-W common test set, which is competitive to other state-of-the-art algorithms.', '1908.02116-2-64-4': 'In addition, even though our approach utilizes two detectors, the number of parameters is much lower than SAN [CITATION].', '1908.02116-2-64-5': 'The robust detection performance of ours can be mainly caused by two reasons.', '1908.02116-2-64-6': 'First, the proposed teacher network can effectively sample the qualified pseudo labeled data, which enables the model to exploit more useful information.', '1908.02116-2-64-7': 'Second, our framework leverages two advanced CNN architectures, which can complement each other.', '1908.02116-2-65-0': 'We also compare our with a recent work on semi-supervised facial landmark detection [CITATION] in table:300W-ALL.', '1908.02116-2-65-1': 'When using 10 of labels, our obtains a lower NME result on the challenging test set than RCN[MATH] [CITATION] (5.64 NME vs. 6.32 NME).', '1908.02116-2-65-2': 'When using 20 of labels, our is also superior to it (5.03 NME vs. 5.88 NME).', '1908.02116-2-65-3': 'Note that [CITATION] utilizes a transformation invariant auxiliary loss function.', '1908.02116-2-65-4': 'This auxiliary loss can also be easily integrated into our framework.', '1908.02116-2-65-5': 'Therefore, [CITATION] is orthogonal to our work, combining two methods can potentially achieve a better performance.', '1908.02116-2-66-0': 'Comparisons on AFLW.', '1908.02116-2-66-1': 'We also show the NME comparison on the AFLW dataset in table:aflw.', '1908.02116-2-66-2': 'Compared to semi-supervised facial landmark detection algorithm [CITATION], we achieve a similar performance.', '1908.02116-2-66-3': 'RCN[MATH] [CITATION] can learn transformation invariant information from a large amount of unlabeled images, while ours does not consider this information as it is not our focus.', '1908.02116-2-66-4': 'On the AFLW-Full test set, using 20 annotation, our framework achieves 1.99 NME, which is competitive to other supervised algorithms.', '1908.02116-2-66-5': 'On the AFLW-Front test set, using only 10 annotation, our framework achieves competitive NME results to [CITATION].', '1908.02116-2-66-6': 'The above results demonstrate our framework can train a robust detector with much less annotation effort.', '1908.02116-2-67-0': 'Comparisons on 300-VW.', '1908.02116-2-67-1': 'We experiment our algorithm to leverage a large amount of unlabeled facial video frames on 300-VW.', '1908.02116-2-67-2': 'We use the labeled 300-W training set and the unlabeled 300-VW training set to train our .', '1908.02116-2-67-3': 'We evaluate the learned detectors on the 300-VW C test subset w.r.t. AUC @ 0.08.', '1908.02116-2-67-4': 'Some video-based facial landmark detection algorithms [CITATION] utilize the labeled 300-VW training data to improve the base detectors.', '1908.02116-2-67-5': 'Compared with them, without using any label on 300-VW, our obtains a higher AUC result than them, i.e., 59.65 vs. 59.39, as shown in table:300VW-C.', '1908.02116-2-68-0': '## Ablation Study', '1908.02116-2-69-0': 'The key contribution of our lies on two components: (1) the teacher supervising the training data selection of students.', '1908.02116-2-69-1': '(2) the complementary effect of two students.', '1908.02116-2-69-2': 'In this subsection, we validate the contribution of these two components to the final detection performance.', '1908.02116-2-70-0': 'The effect of the teacher.', '1908.02116-2-70-1': 'Compared to other progressive pseudo label generation strategies [CITATION], our designed teacher can sample pseudo labeled with higher quality.', '1908.02116-2-70-2': 'In fig:compare, we show the detection results after the first five training rounds (only 10% labels are used).', '1908.02116-2-70-3': 'We use SPL [CITATION] to separately train CPM and HG, and then ensemble them together as eq:ensemble.', '1908.02116-2-70-4': 'We use SPaCo [CITATION] to jointly optimize CPM and HG in a co-training strategy.', '1908.02116-2-70-5': 'To make a fair comparison, at each round, we control the number of pseudo labels is the same across these three algorithms.', '1908.02116-2-70-6': 'From fig:compare, several conclusions can be made: (1) obtains the lowest NME, because the quality of selected pseudo labels is better than others.', '1908.02116-2-70-7': '(2) SPL falls into a local trap at round[MATH] and results in a higher error at round[MATH], whereas SPaCo and our not.', '1908.02116-2-70-8': 'This could be caused by that the interaction between two students can help regularize each other.', '1908.02116-2-70-9': '(3) Our converges faster than SPaCo and achieves better results.', '1908.02116-2-70-10': 'The pseudo labeled data selection in SPaco is a heuristic unsupervised criterion, whereas our criterion is a supervised teacher.', '1908.02116-2-70-11': 'Since no extra supervision is given in SPaCo, their criterion might induce inaccurate pseudo labeled samples.', '1908.02116-2-70-12': 'Besides, as discussed in sec:discussion, our can utilize validation set to further improve the performance by avoid over-fitting, but the compared methods may not effectively utilize validation set.', '1908.02116-2-71-0': 'The effect of the interaction between students.', '1908.02116-2-71-1': 'From Table [REF], we show the ablative studies on the complementary effect of multiple students.', '1908.02116-2-71-2': 'In these experiments, we use the same teacher structure, while "CPM" and "HG" are trained without the interaction between students.', '1908.02116-2-71-3': 'Using 10 labels, CPM achieves 8.28 NME, and HG achieves 6.25 NME on 300-W.', '1908.02116-2-71-4': 'Leveraging from their mutual benefits, our can boost the performance to 5.64, which is higher than CPM by about 30 and than HG by 9.', '1908.02116-2-71-5': 'Under different portion of annotations, we can conclude similar observations.', '1908.02116-2-71-6': 'This ablation study demonstrates the contribution of student interaction to the final performance.', '1908.02116-2-71-7': 'Note that, our algorithm can be readily applied to multiple students without introducing additional hyper-parameters.', '1908.02116-2-71-8': 'In contrast, the number of hyper-parameters in other co-training strategies [CITATION] is quadratic to the number of detectors.', '1908.02116-2-72-0': '## Qualitative Analysis', '1908.02116-2-73-0': 'On the 300-W training set, we train our using only 10% labeled facial images, and we show some qualitative results of the 300-W test set in fig:results.', '1908.02116-2-73-1': 'The first row shows seven raw input facial images.', '1908.02116-2-73-2': 'The second row shows the ground truth background heatmaps, and the third row shows the faces with ground truth landmarks of these images.', '1908.02116-2-73-3': 'We visualize the predicted background heatmap in the fourth row and the predicted coordinates in the fifth row.', '1908.02116-2-73-4': 'As we can see, the predicted landmarks of our are very close to the ground truth.', '1908.02116-2-73-5': 'These predictions are already robust enough, and human may not be able to distinguish the difference between our predictions (the third line) and the ground truth (the fifth line).', '1908.02116-2-74-0': '# Conclusion', '1908.02116-2-75-0': 'In this paper, we propose an interaction mechanism between a teacher and multiple students for semi-supervised facial landmark detection.', '1908.02116-2-75-1': 'The students learn to generate pseudo labels for the unlabeled data, while the teacher learns to judge the quality of these pseudo labeled data.', '1908.02116-2-75-2': 'After that, the teacher can filter out unqualified samples; and the students get feedback from the teacher and improve itself by the qualified samples.', '1908.02116-2-75-3': 'The teacher is adaptive along with the improved students.', '1908.02116-2-75-4': 'Besides, multiple students can not only regularize each other but also be ensembled to predict more accurate pseudo labels.', '1908.02116-2-75-5': 'We empirically demonstrate that the proposed interaction mechanism achieves state-of-the-art performance on three facial landmark benchmarks.'} | [['1908.02116-1-14-0', '1908.02116-2-14-0'], ['1908.02116-1-14-1', '1908.02116-2-14-1'], ['1908.02116-1-14-2', '1908.02116-2-14-2'], ['1908.02116-1-14-3', '1908.02116-2-14-3'], ['1908.02116-1-14-4', '1908.02116-2-14-4'], ['1908.02116-1-14-5', '1908.02116-2-14-5'], ['1908.02116-1-14-6', '1908.02116-2-14-6'], ['1908.02116-1-11-0', '1908.02116-2-11-0'], ['1908.02116-1-11-1', '1908.02116-2-11-1'], ['1908.02116-1-11-2', '1908.02116-2-11-2'], ['1908.02116-1-11-3', '1908.02116-2-11-3'], ['1908.02116-1-54-0', 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'1908.02116-3-66-0', '1908.02116-3-67-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1908.02116 | {'1908.02116-3-0-0': 'Facial landmark detection aims to localize the anatomically defined points of human faces.', '1908.02116-3-0-1': 'In this paper, we study facial landmark detection from partially labeled facial images.', '1908.02116-3-0-2': "A typical approach is to (1) train a detector on the labeled images; (2) generate new training samples using this detector's prediction as pseudo labels of unlabeled images; (3) retrain the detector on the labeled samples and partial pseudo labeled samples.", '1908.02116-3-0-3': 'In this way, the detector can learn from both labeled and unlabeled data to become robust.', '1908.02116-3-1-0': 'In this paper, we propose an interaction mechanism between a teacher and two students to generate more reliable pseudo labels for unlabeled data, which are beneficial to semi-supervised facial landmark detection.', '1908.02116-3-1-1': 'Specifically, the two students are instantiated as dual detectors.', '1908.02116-3-1-2': 'The teacher learns to judge the quality of the pseudo labels generated by the students and filter out unqualified samples before the retraining stage.', '1908.02116-3-1-3': 'In this way, the student detectors get feedback from their teacher and are retrained by premium data generated by itself.', '1908.02116-3-1-4': 'Since the two students are trained by different samples, a combination of their predictions will be more robust as the final prediction compared to either prediction.', '1908.02116-3-1-5': 'Extensive experiments on 300-W and AFLW benchmarks show that the interactions between teacher and students contribute to better utilization of the unlabeled data and achieves state-of-the-art performance.', '1908.02116-3-2-0': '# Introduction', '1908.02116-3-3-0': 'Facial landmark detection aims to find some pre-defined anatomical keypoints of human faces [CITATION].', '1908.02116-3-3-1': 'These keypoints include the corners of a mouth, the boundary of eyes, the tip of a nose, etc [CITATION].', '1908.02116-3-3-2': 'It is usually a prerequisite of a large number of computer vision tasks [CITATION].', '1908.02116-3-3-3': 'For example, facial landmark coordinates are required to align faces to ease the visualization for users when people would like to sort their faces by time and see the changes over time [CITATION].', '1908.02116-3-3-4': 'Other examples include face morphing [CITATION], face replacement [CITATION], etc.', '1908.02116-3-4-0': 'The main challenge in recent landmark detection literatures is how to obtain abundant facial landmark labels.', '1908.02116-3-4-1': 'The annotation challenge comes from two perspectives.', '1908.02116-3-4-2': 'First, a large number of keypoints are required for a single face image, e.g., 68 keypoints for each face in the 300-W dataset [CITATION].', '1908.02116-3-4-3': 'To precisely depict the facial features for a whole dataset, millions of keypoints are usually required.', '1908.02116-3-4-4': 'Second, different annotators have a semantic gap.', '1908.02116-3-4-5': 'There is no universal standard for the annotation of the keypoints, so different annotators give different positions for the same keypoints.', '1908.02116-3-4-6': 'A typical way to reduce such semantic deviations among various annotators is to merge the labels from several annotators.', '1908.02116-3-4-7': 'This will further increase the costs of the whole annotation work.', '1908.02116-3-5-0': 'Semi-supervised landmark detection can to some extent alleviate the expensive and sophisticated annotations by utilizing the unlabeled images.', '1908.02116-3-5-1': 'Typical approaches [CITATION] for semi-supervised learning use self-training or similar paradigms to utilize the unlabeled samples.', '1908.02116-3-5-2': 'For example, the authors of [CITATION] adopt a heuristic unsupervised criterion to select the pseudo labeled data for the retraining procedure.', '1908.02116-3-5-3': 'This criterion is the loss of each pseudo labeled data, where its predicted pseudo label is treated as the ground truth to calculate the loss [CITATION].', '1908.02116-3-5-4': 'Since no extra supervision is given to train the criterion function, this unsupervised loss criterion has a high possibility of passing inaccurate pseudo labeled data to the retraining stage.', '1908.02116-3-5-5': 'In this way, these inaccurate data will mislead the optimization of the detector and make it easier to trap into a local minimum.', '1908.02116-3-5-6': 'A straightforward solution to this problem is to use multiple models and regularize each other by the co-training strategy [CITATION].', '1908.02116-3-5-7': 'Unfortunately, even if co-training performs well in simple tasks such as classification [CITATION], in more complex scenarios such as detection, co-training requires extremely sophisticated design and careful tuning of many additional hyper-parameters [CITATION], e.g., more than 10 hyper-parameters for three models in [CITATION].', '1908.02116-3-6-0': 'To better utilize the pseudo labeled data as well as avoid the complicated model tuning for landmark detection, we propose Teacher Supervises StudentS (TS[MATH]).', '1908.02116-3-6-1': 'As illustrated in fig:high-level, TS[MATH] is an interaction mechanism between one teacher network and two (or multiple) student networks.', '1908.02116-3-6-2': 'Two student detection networks learn to generate pseudo labels for unlabeled images.', '1908.02116-3-6-3': 'The teacher network learns to judge the quality of the pseudo labels generated from students.', '1908.02116-3-6-4': 'Consequently, the teacher can select qualified pseudo labeled samples and use them to retrain the students.', '1908.02116-3-6-5': 'TS[MATH] applies these steps in an iterative manner, where students gradually become more robust, and the teacher is adaptively updated with the improved students.', '1908.02116-3-6-6': 'Besides, two students can also encourage each other to advance their performances in two ways.', '1908.02116-3-6-7': 'First, predictions from two students can be ensembled to further improve the quality of pseudo labels.', '1908.02116-3-6-8': 'Second, two students can regularize each other by training on different samples.', '1908.02116-3-6-9': 'The interactions between the teacher and students as well as the students themselves help to provide more accurate pseudo labeled samples for retraining and the model does not need careful hyper-parameter tuning.', '1908.02116-3-7-0': 'To highlight our contribution, we propose an easy-to-train interaction mechanism between teacher and students (TS[MATH]) to provide more reliable pseudo labeled samples in semi-supervised facial landmark detection.', '1908.02116-3-7-1': 'To validate the performance of our TS[MATH], we do experiments on 300-W, 300-VW, and AFLW benchmarks.', '1908.02116-3-7-2': 'TS[MATH] achieves state-of-the-art semi-supervised performance on all three benchmarks.', '1908.02116-3-7-3': 'In addition, using only 30% labels, our TS[MATH] achieves competitive results compared to supervised methods using all labels on 300-W and AFLW.', '1908.02116-3-8-0': '# Related Work', '1908.02116-3-9-0': 'We will first introduce some supervised facial landmark algorithms in sec:relate-supervised.', '1908.02116-3-9-1': 'Then, we will compare our algorithm with semi-supervised learning algorithms and semi-supervised facial landmark algorithm in sec:relate-semi.', '1908.02116-3-9-2': 'Lastly, we explain our algorithm in a meta learning perspective in sec:relate-meta.', '1908.02116-3-10-0': '## Supervised Facial Landmark Detection', '1908.02116-3-11-0': 'Supervised facial landmark detection algorithms can be categorized into linear regression based methods [CITATION] and heatmap regression based methods [CITATION].', '1908.02116-3-11-1': 'Linear regression based methods learn a function that maps the input face image to the normalized landmark coordinates [CITATION].', '1908.02116-3-11-2': 'Heatmap regression based methods produce one heatmap for each landmark, where the coordinate is the location of the highest response on this heatmap [CITATION].', '1908.02116-3-11-3': 'All above algorithms can be readily integrated into our framework, serving as different student detectors.', '1908.02116-3-12-0': 'These supervised algorithms require a large amount of data to train deep neural networks.', '1908.02116-3-12-1': 'However, it is tedious to annotate the precise facial landmarks, which need to average different annotations from multiple different annotators.', '1908.02116-3-12-2': 'Therefore, to reduce the annotation cost, it is necessary to investigate the semi-supervised facial landmark detection.', '1908.02116-3-13-0': '## Semi-supervised Facial Landmark Detection', '1908.02116-3-14-0': 'Some early semi-supervised learning algorithms are difficult to handle large scale datasets due to the high complexity [CITATION].', '1908.02116-3-14-1': 'Others exploit pseudo-labels of unlabeled data in the semi-supervised scenario [CITATION].', '1908.02116-3-14-2': 'Since most of these algorithms studied their effect on small-scale datasets [CITATION], a question remains open: can they be used to improve large-scale semi-supervised landmark detection?', '1908.02116-3-14-3': 'In addition, those self-training or co-training approaches [CITATION] simply leverage the confidence score or an unsupervised loss to select qualified samples.', '1908.02116-3-14-4': 'For example, Dong et al. [CITATION] proposed a model communication mechanism to select reliable pseudo labeled samples based on loss and score.', '1908.02116-3-14-5': 'However, such selection criterion does not reflect the real quality of a pseudo labeled sample.', '1908.02116-3-14-6': 'In contrast, our teacher directly learns to model the quality, and selected samples are thus more reliable.', '1908.02116-3-15-0': 'There are only few of researchers study the semi-supervised facial landmark detection algorithms.', '1908.02116-3-15-1': 'A recent work [CITATION] presented two techniques to improve landmark localization from partially annotated face images.', '1908.02116-3-15-2': 'The first technique is to jointly train facial landmark network with an attribute network, which predicts the emotion, head pose, etc.', '1908.02116-3-15-3': 'In this multi-task framework, the gradient from the attribute network can benefit the landmark prediction.', '1908.02116-3-15-4': 'The second technique is a kind of supervision without the need of manual labels, which enables the transformation invariant of landmark prediction.', '1908.02116-3-15-5': 'Compared to using the supervision from transformation, our approach leverages a progressive paradigm to learn facial shape information from unlabeled data.', '1908.02116-3-15-6': 'In this way, our approach is orthogonal to [CITATION], and these two techniques can complement our approach to further boost the performance.', '1908.02116-3-16-0': 'Radosavovic et al. [CITATION] applied the data augmentation to improve the quality of generated pseudo landmark labels.', '1908.02116-3-16-1': 'For an unlabeled image, they ensemble predictions from multiple transformations, such as flipping and rotation.', '1908.02116-3-16-2': 'This strategy can also be used to improve the accuracy of our pseudo labels and complement our approach.', '1908.02116-3-16-3': 'Since the data augmentation is not the focus of this paper, we did not apply their algorithms in our approach.', '1908.02116-3-16-4': 'Dong et al. [CITATION] proposed a self-supervised loss by exploiting the temporal consistence on unlabeled videos to enhance the detector.', '1908.02116-3-16-5': 'This is a video-based approach and not the focus of our work.', '1908.02116-3-16-6': 'Therefore, we do not discuss more with those video-based approach [CITATION].', '1908.02116-3-17-0': '## Meta Learning', '1908.02116-3-18-0': 'In a meta learning perspective, our TS[MATH] learns a teacher network to learn which pseudo labeled samples are helpful to train student detectors.', '1908.02116-3-18-1': 'In this sense, we are related to some recent literature in "learning to learn" [CITATION].', '1908.02116-3-18-2': 'For example, Ren et al. [CITATION] learn to re-weight samples based on gradients of a model on the clean validation set.', '1908.02116-3-18-3': 'Xu et al. [CITATION] suggest using meta-learning to tune the optimization schedule of alternative optimization problems.', '1908.02116-3-18-4': 'Jiang et al. [CITATION] propose an architecture to learn data-driven curriculum on corrupted labels.', '1908.02116-3-18-5': 'Fan et al. [CITATION] leverage reinforcement learning to learn a policy to select good training samples for a single student model.', '1908.02116-3-18-6': 'These algorithms are designed in the supervised scenarios and can not easily be modified in semi-supervised scenario.', '1908.02116-3-19-0': 'Difference with other teacher-student frameworks and generative adversarial networks (GAN).', '1908.02116-3-19-1': 'Our learns to utilize the output (pseudo labels) of the student model qualified by the teacher model to do semi-supervised learning.', '1908.02116-3-19-2': 'Other teacher-student methods [CITATION] aim to fit the output of the student model to that of the teacher model.', '1908.02116-3-19-3': 'The student and teacher in our work do similar jobs as the generator and discriminator in GAN [CITATION], while we aim to predict/generate qualified pseudo labels in semi-supervised learning using a different training strategy.', '1908.02116-3-20-0': '# Methodology', '1908.02116-3-21-0': 'In this section, we will first introduce the scenario of the semi-supervised facial landmark detection in sec:method-setting.', '1908.02116-3-21-1': 'We explain how to design our student detectors and the teacher network in sec:teacher-student.', '1908.02116-3-21-2': 'Lastly, we demonstrate our overall algorithm in sec:method-algo.', '1908.02116-3-22-0': '## The Semi-Supervised Scenario', '1908.02116-3-23-0': 'We introduce some necessary notations for the presentation of the proposed method.', '1908.02116-3-23-1': 'Let [MATH] be the labeled data in the training set and [MATH] be the unlabeled data in the training set, where [MATH] denotes the [MATH]-th image, and [MATH] denotes the ground-truth landmark label of [MATH].', '1908.02116-3-23-2': '[MATH] is the number of the facial landmarks, and the [MATH]-th column of [MATH] indicates the coordinate of the [MATH]-th landmark.', '1908.02116-3-23-3': '[MATH] and [MATH] denote the number of labeled data and unlabeled data, respectively.', '1908.02116-3-23-4': 'The semi-supervised facial landmark detection aims to learn robust detectors from both [MATH] and [MATH].', '1908.02116-3-24-0': '## Teacher and Students Design', '1908.02116-3-25-0': 'The Student Detectors.', '1908.02116-3-25-1': 'We choose the convolutional pose machine (CPM) [CITATION] and stacked hourglass (HG) [CITATION] models as our student detectors.', '1908.02116-3-25-2': 'These two landmark detection architectures are the cornerstone of many facial landmark detection algorithms [CITATION].', '1908.02116-3-25-3': 'Moreover, their architectures are quite different, and can thus complement each other to achieve a better detection performance compared to using two similar neural architectures.', '1908.02116-3-25-4': 'Therefore, we integrate these two detectors in our TS[MATH] approach.', '1908.02116-3-25-5': 'In this paragraph, we will give a brief overview of these two facial landmark detectors.', '1908.02116-3-25-6': 'We illustrate the structures of CPM and HG in fig:network.', '1908.02116-3-25-7': 'Both CPM and HG are the heatmap regression based methods and utilize the cascaded structure.', '1908.02116-3-25-8': 'Formally, suppose there are [MATH] convolutional stages in CPM, the output of CPM is:', '1908.02116-3-26-0': '[EQUATION]', '1908.02116-3-27-0': 'where [MATH] indicates the CPM student detector whose parameters are [MATH].', '1908.02116-3-27-1': '[MATH] is the RGB image of the [MATH]-th data-point and [MATH] indicates the heatmap prediction of the [MATH]-th stage.', '1908.02116-3-27-2': '[MATH] and [MATH] denote the spatial height and width of the heatmap.', '1908.02116-3-27-3': 'Similarly, we use [MATH] indicates the HG student detector whose parameters are [MATH].', '1908.02116-3-27-4': 'The detection loss function of the CPM student is:', '1908.02116-3-28-0': '[EQUATION] where [MATH] is a function taking the label [MATH] as inputs to generate the the ideal heatmap [MATH].', '1908.02116-3-28-1': 'Details of [MATH] can be found in [CITATION].', '1908.02116-3-28-2': 'During the evaluation, we take the argmax results over the first [MATH] channel of the last heatmap [MATH] as the coordinates of landmarks, and the [MATH]-th channel corresponding to the background will be omitted.', '1908.02116-3-29-0': 'The Teacher Network.', '1908.02116-3-29-1': 'Since our student detectors are based on heatmap, the pseudo label is in the form of heatmap and ground truth label is the ideal heatmap.', '1908.02116-3-29-2': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-3-29-3': 'As shown in fig:value-network, the input of this teacher network is the concatenation of a face image and its heatmap prediction [MATH].', '1908.02116-3-29-4': 'The output of this teacher network is a scalar representing the quality of a pseudo labeled facial image.', '1908.02116-3-29-5': 'Since we train the teacher on the trustworthy labeled data, we could obtain a supervised detection loss by calculating [MATH].', '1908.02116-3-29-6': 'We consider the negative value of this detection loss as the ground truth label of the quality, because a high negative value of the detection loss indicates a high similarity between the predicted heatmap and the ideal heatmap.', '1908.02116-3-29-7': 'In another word, a higher quality scalar corresponds to a more accurate pseudo label.', '1908.02116-3-30-0': '[EQUATION] where the parameters of the teacher is [MATH]. "[', '1908.02116-3-30-1': '[MATH]" first resizes the tensor [MATH] into the same spatial shape as [MATH] and then concatenates the resized tensor with [MATH] to get a new tensor.', '1908.02116-3-30-2': 'This new tensor is regarded as pseudo labeled image and will be qualified by the teacher later.', '1908.02116-3-30-3': 'The teacher outputs a scalar [MATH] representing the quality of the [MATH]-th sample associated with its pseudo label [MATH].', '1908.02116-3-30-4': 'We optimize the teacher on the trustworthy labeled data by minimizing eq:teacher-loss.', '1908.02116-3-31-0': '[t!]', '1908.02116-3-31-1': 'The Algorithm Description of Our TS[MATH]', '1908.02116-3-32-0': '[1] Labeled data [MATH]', '1908.02116-3-33-0': 'Unlabeled data [MATH]', '1908.02116-3-34-0': 'Two student detectors [MATH] with [MATH] and [MATH] with [MATH]', '1908.02116-3-35-0': 'The teacher network [MATH] with parameters [MATH]', '1908.02116-3-36-0': 'The selection ratio [MATH] and the maximum step [MATH]', '1908.02116-3-37-0': 'Initialize the [MATH] and [MATH] by minimizing eq:cpm-hg-loss on [MATH]; [MATH]; [MATH]++', '1908.02116-3-38-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the first student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with prediction [MATH] and ground truth label [MATH] Compute the quality scalar of each sample in [MATH] using the optimized teacher via eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-3-39-0': 'Predict [MATH] on both [MATH] and [MATH] using eq:ensemble, and denote [MATH] with its pseudo labels as [MATH] update the second student Optimize teacher with [MATH] by minimizing eq:teacher-loss on [MATH] with [MATH] and [MATH] Compute the quality scalar of each sample in [MATH] using eq:teacher Pickup the top [MATH] samples from [MATH], named as [MATH] Retrain [MATH] on [MATH] by minimizing eq:cpm-hg-loss', '1908.02116-3-40-0': 'Students with optimized parameters [MATH] and [MATH]', '1908.02116-3-41-0': '## The TS[MATH] Algorithm', '1908.02116-3-42-0': 'Our TS[MATH] aims to progressively improve the performance of the student detector.', '1908.02116-3-42-1': 'The key idea is to learn a teacher network that can teach students which pseudo labeled sample is reliable and can be used for training.', '1908.02116-3-42-2': 'In this procedure, we define the pseudo label of a facial image is as follows:', '1908.02116-3-43-0': '[EQUATION] where [MATH] indicates the heatmap prediction from the first student at the [MATH]-th stage for the [MATH]-th sample.', '1908.02116-3-43-1': '[MATH] in eq:ensemble indicates the ensemble result from both two students detection networks.', '1908.02116-3-43-2': 'It will be used as the prediction during the inference procedure.', '1908.02116-3-44-0': 'We show our overall algorithm in alg:SSFLD.', '1908.02116-3-44-1': 'We first initialize the two detectors [MATH] and [MATH] on the labeled facial images [MATH].', '1908.02116-3-44-2': 'Then, in the first round, our algorithm applies the following procedures: (1) generate pseudo labels on [MATH] via eq:ensemble and train the teacher network from scratch with these pseudo labels; (2) generate pseudo labels on [MATH] and estimate the quality of these pseudo labeled using the learned teacher; (3) select some high-quality pseudo labeled samples to retrain one student network from scratch.', '1908.02116-3-44-3': '(4) repeat the first three steps to update another student detection network.', '1908.02116-3-44-4': 'In the next rounds, each student can be improved and generate more accurate pseudo labels.', '1908.02116-3-44-5': 'In this way, we will select more pseudo labeled samples when retraining the students.', '1908.02116-3-44-6': 'As the rounds go, students will gradually become better, and the teacher will also be adaptive with the improved students.', '1908.02116-3-44-7': 'Our interaction mechanism helps to obtain more accurate pseudo labels and select more reliable pseudo labeled samples.', '1908.02116-3-44-8': 'As a result, our algorithm achieves better performance in the semi-supervised facial landmark detection.', '1908.02116-3-45-0': '## Discussion', '1908.02116-3-46-0': 'Can this algorithm generalize to other tasks?', '1908.02116-3-46-1': 'Our algorithm relies on the design of the teacher network.', '1908.02116-3-46-2': 'It requires the input pseudo label to be a structured prediction.', '1908.02116-3-46-3': 'Therefore, our algorithm is possible to be applied to tasks with structured predictions, such as segmentation and pose estimation, but is not suitable other tasks like classification.', '1908.02116-3-47-0': 'Limitation.', '1908.02116-3-47-1': 'It is challenging for a teacher to judge the quality of a pseudo label for an image, especially when the spatial shape of this image becomes large.', '1908.02116-3-47-2': 'Therefore, in this paper, we use an input size of 64[MATH]64.', '1908.02116-3-47-3': 'If we increase the input size to 256[MATH]256, the teacher will fail and need to be modified accordingly.', '1908.02116-3-47-4': 'There are two main reasons: (1) the larger resolution requires a deeper architecture or dilated convolutions for the teacher network and (2) the high-resolution faces bring high-dimensional inputs, and consequently, the teacher needs much more training data.', '1908.02116-3-47-5': 'This drawback limits the extension of our algorithm to high-resolution tasks, such as segmentation.', '1908.02116-3-47-6': 'We will explore to solve this problem in the future.', '1908.02116-3-48-0': 'Further improvements.', '1908.02116-3-48-1': '(1) In our algorithm, during the retraining procedure, a part of unlabeled samples are not involved during retraining.', '1908.02116-3-48-2': 'To utilize these unlabeled facial images, we could use self-supervised techniques such as [CITATION] to improve the detectors.', '1908.02116-3-48-3': '(2) In this framework, we use only two student detectors, while it is easy to integrate more student detectors.', '1908.02116-3-48-4': 'More student detectors are likely to improve the prediction accuracy, but this will introduce more computation costs.', '1908.02116-3-48-5': '(3) The specifically designed data augmentation [CITATION] is another direction to improve the accuracy and precision of the pseudo labels.', '1908.02116-3-49-0': 'Will the teacher network over-fit to the labeled data?', '1908.02116-3-50-0': "In alg:SSFLD, since labeled data set [MATH] is used to optimize both teacher and students, the teacher's judgment could suffer from the over-fitting problem.", '1908.02116-3-50-1': "Most of the students' predictions on the labeled data can be similar to the ground truth labels.", '1908.02116-3-50-2': 'In other words, most pseudo labeled samples on [MATH] are "correctly" labeled samples.', '1908.02116-3-50-3': 'If the teacher is optimized on [MATH] with those pseudo labels, it might only learn what a good pseudo labeled sample is, but overlook what a bad one is.', '1908.02116-3-50-4': 'It would be more reasonable to let students predict on the unseen validation set, and then train the teacher on this validation set.', '1908.02116-3-50-5': 'However, having an additional validation set during training is different from the typical setting of previous semi-supervised facial landmark detection.', '1908.02116-3-50-6': 'We would explore this problem in our future work.', '1908.02116-3-51-0': '# Empirical Studies', '1908.02116-3-52-0': 'We perform experiments on three benchmark datasets to investigate the behavior of the proposed method.', '1908.02116-3-52-1': 'The datasets and experiment settings are introduced in sec:datasets and sec:setting.', '1908.02116-3-52-2': 'We first compare the proposed semi-supervised facial landmark algorithm with other state-of-the-art algorithms in Sec. [REF].', '1908.02116-3-52-3': 'We then perform ablation studies in Sec. [REF] and visualize our results at last.', '1908.02116-3-53-0': '## Datasets', '1908.02116-3-54-0': 'The 300-W dataset [CITATION] annotates 68 landmarks from five facial landmark datasets, i.e., LFPW, AFW, HELEN, XM2VTS, and IBUG.', '1908.02116-3-54-1': 'Following the common settings [CITATION], we regard all the training samples from LFPW, HELEN and the full set of AFW as the training set, in which there is 3148 training images.', '1908.02116-3-54-2': 'The common test subset consists of 554 test images from LFPW and HELEN.', '1908.02116-3-54-3': 'The challenging test subset consists of 135 images from IBUG to construct .', '1908.02116-3-54-4': 'The full test set the union of the common and challenging subsets, 689 images in total.', '1908.02116-3-55-0': 'The AFLW dataset [CITATION] contains 21997 real-world images with 25993 faces in total.', '1908.02116-3-55-1': 'They provide at most 21 landmark coordinates for each face, but they exclude invisible landmarks.', '1908.02116-3-55-2': 'Faces in AFLW usually have a different head pose, expression, occlusion or illumination, and therefore it causes difficulties to train a robust detector.', '1908.02116-3-55-3': 'Following the same setting as in [CITATION], we do not use the landmarks of two ears.', '1908.02116-3-55-4': 'There are two types of AFLW splits, i.e., AFLW-Full and AFLW-Frontal following [CITATION].', '1908.02116-3-55-5': 'AFLW-Full contains 20000 training samples and 4386 test samples.', '1908.02116-3-55-6': 'AFLW-Front uses the same training samples as in AFLW-Full, but only use the 1165 samples with the frontal face as the test set.', '1908.02116-3-56-0': 'The 300-VW dataset [CITATION] is a video-based facial landmark benchmark.', '1908.02116-3-56-1': 'It contains 50 training videos with 95192 frames.', '1908.02116-3-56-2': 'Following [CITATION], we report the results for the 49 inner points on the category C subset of the 300-VW test set, which has 26338 frames.', '1908.02116-3-57-0': '## Experimental Settings', '1908.02116-3-58-0': 'Training student detection networks.', '1908.02116-3-58-1': 'The first student detector is CPM [CITATION].', '1908.02116-3-58-2': 'We follow the same model configuration as the base detector used in [CITATION], and the number of cascaded stages is set as three.', '1908.02116-3-58-3': 'Its number of parameters is 16.70 MB and its FLOPs is 1720.98 M. To train CPM, we apply the SGD optimizer with the momentum of 0.9 and the weight decay of 0.0005.', '1908.02116-3-58-4': 'For each stage, we train the CPM for 50 epochs in total.', '1908.02116-3-58-5': 'We start the learning rate of 0.00005, and reduce it by 0.5 at 20-th, 25-th, 30-th, and 40-th epoch.', '1908.02116-3-59-0': 'The second student detector is HG [CITATION].', '1908.02116-3-59-1': 'We follow the same model configuration as [CITATION] but use the number of cascaded stages of four to build our HG model, where the number of parameters is 24.97 MB and FLOPs is 1600.85 M. To train HG, we apply the RMSprop optimizer with the alpha of 0.99.', '1908.02116-3-59-2': 'For each stage, we train the HG for 110 epochs in total.', '1908.02116-3-59-3': 'We start the learning rate of 0.00025, and reduce it by 0.5 at 50-th, 70-th, 90-th, and 100-th.', '1908.02116-3-60-0': 'For both of these two detectors, we use the batch size of eight on two GPUs.', '1908.02116-3-60-1': 'To generate the heatmap ground truth labels, we apply the Gaussian distribution with the sigma of 3.', '1908.02116-3-60-2': 'Each face image is first resized into the size of 64[MATH]64, and then randomly resized between the scale of 0.9 and 1.1.', '1908.02116-3-60-3': 'After the random resize operation, the face image will be randomly rotated with the maximum degree of 30, and then randomly cropped with the size of 64[MATH]64.', '1908.02116-3-60-4': 'We set selection ratio [MATH] as [MATH] and the maximum step [MATH] as [MATH] based on cross-validation.', '1908.02116-3-61-0': 'Training the teacher network.', '1908.02116-3-61-1': 'We build our teacher network using the structure of discriminators adopted in CycleGAN [CITATION].', '1908.02116-3-61-2': 'Given a 64[MATH]64 face image, we first resize the predicted heatmap into the same spatial size of 64[MATH]64.', '1908.02116-3-61-3': 'We use the Adam to train this teacher network.', '1908.02116-3-61-4': 'The initial learning rate is 0.01, and the batch size is 128.', '1908.02116-3-61-5': 'Random flip, random rotation, random scale and crop are applied as data argumentation.', '1908.02116-3-62-0': 'Evaluation.', '1908.02116-3-62-1': 'Normalized Mean Error (NME) is usually applied to evaluate the performance for facial landmark predictions [CITATION].', '1908.02116-3-62-2': 'For the 300-W dataset, we use the inter-ocular distance to normalize mean error following the same setting as in [CITATION].', '1908.02116-3-62-3': 'For the AFLW dataset, we use the face size to normalize mean error [CITATION].', '1908.02116-3-62-4': 'Area Under the Curve (AUC) @ 0.08 error is also employed for evaluation [CITATION].', '1908.02116-3-62-5': 'When training on the partially labeled data, the sets of [MATH] and [MATH] are randomly sampled.', '1908.02116-3-62-6': 'During evaluation, we use eq:ensemble to obtain the final heatmap and follow [CITATION] to generate the coordinate of each landmark.', '1908.02116-3-62-7': 'We repeat each experiment three times and report the mean result.', '1908.02116-3-62-8': 'The codes will be public available upon the acceptance.', '1908.02116-3-63-0': '## Comparison with state-of-the-art', '1908.02116-3-64-0': 'Comparisons on 300-W.', '1908.02116-3-64-1': 'We compare our algorithm with several state-of-the-art algorithms [CITATION], as shown in table:300W-ALL.', '1908.02116-3-64-2': 'In this table, [CITATION] are very recent methods, which represent the state-of-the-art supervised facial landmark algorithms.', '1908.02116-3-64-3': 'By using 100% facial landmark labels on 300-W training set and unlabeled AFLW, our algorithm achieves competitive 3.49 NME on the 300-W common test set, which is competitive to other state-of-the-art algorithms.', '1908.02116-3-64-4': 'In addition, even though our approach utilizes two detectors, the number of parameters is much lower than SAN [CITATION].', '1908.02116-3-64-5': 'The robust detection performance of ours can be mainly caused by two reasons.', '1908.02116-3-64-6': 'First, the proposed teacher network can effectively sample the qualified pseudo labeled data, which enables the model to exploit more useful information.', '1908.02116-3-64-7': 'Second, our framework leverages two advanced CNN architectures, which can complement each other.', '1908.02116-3-65-0': 'We also compare our with a recent work on semi-supervised facial landmark detection [CITATION] in table:300W-ALL.', '1908.02116-3-65-1': 'When using 10 of labels, our obtains a lower NME result on the challenging test set than RCN[MATH] [CITATION] (5.64 NME vs. 6.32 NME).', '1908.02116-3-65-2': 'When using 20 of labels, our is also superior to it (5.03 NME vs. 5.88 NME).', '1908.02116-3-65-3': 'Note that [CITATION] utilizes a transformation invariant auxiliary loss function.', '1908.02116-3-65-4': 'This auxiliary loss can also be easily integrated into our framework.', '1908.02116-3-65-5': 'Therefore, [CITATION] is orthogonal to our work, combining two methods can potentially achieve a better performance.', '1908.02116-3-66-0': 'Comparisons on AFLW.', '1908.02116-3-66-1': 'We also show the NME comparison on the AFLW dataset in table:aflw.', '1908.02116-3-66-2': 'Compared to semi-supervised facial landmark detection algorithm [CITATION], we achieve a similar performance.', '1908.02116-3-66-3': 'RCN[MATH] [CITATION] can learn transformation invariant information from a large amount of unlabeled images, while ours does not consider this information as it is not our focus.', '1908.02116-3-66-4': 'On the AFLW-Full test set, using 20 annotation, our framework achieves 1.99 NME, which is competitive to other supervised algorithms.', '1908.02116-3-66-5': 'On the AFLW-Front test set, using only 10 annotation, our framework achieves competitive NME results to [CITATION].', '1908.02116-3-66-6': 'The above results demonstrate our framework can train a robust detector with much less annotation effort.', '1908.02116-3-67-0': 'Comparisons on 300-VW.', '1908.02116-3-67-1': 'We experiment our algorithm to leverage a large amount of unlabeled facial video frames on 300-VW.', '1908.02116-3-67-2': 'We use the labeled 300-W training set and the unlabeled 300-VW training set to train our .', '1908.02116-3-67-3': 'We evaluate the learned detectors on the 300-VW C test subset w.r.t. AUC @ 0.08.', '1908.02116-3-67-4': 'Some video-based facial landmark detection algorithms [CITATION] utilize the labeled 300-VW training data to improve the base detectors.', '1908.02116-3-67-5': 'Compared with them, without using any label on 300-VW, our obtains a higher AUC result than them, i.e., 59.65 vs. 59.39, as shown in table:300VW-C.', '1908.02116-3-68-0': '## Ablation Study', '1908.02116-3-69-0': 'The key contribution of our lies on two components: (1) the teacher supervising the training data selection of students.', '1908.02116-3-69-1': '(2) the complementary effect of two students.', '1908.02116-3-69-2': 'In this subsection, we validate the contribution of these two components to the final detection performance.', '1908.02116-3-70-0': 'The effect of the teacher.', '1908.02116-3-70-1': 'Compared to other progressive pseudo label generation strategies [CITATION], our designed teacher can sample pseudo labeled with higher quality.', '1908.02116-3-70-2': 'In fig:compare, we show the detection results after the first five training rounds (only 10% labels are used).', '1908.02116-3-70-3': 'We use SPL [CITATION] to separately train CPM and HG, and then ensemble them together as eq:ensemble.', '1908.02116-3-70-4': 'We use SPaCo [CITATION] to jointly optimize CPM and HG in a co-training strategy.', '1908.02116-3-70-5': 'To make a fair comparison, at each round, we control the number of pseudo labels is the same across these three algorithms.', '1908.02116-3-70-6': 'From fig:compare, several conclusions can be made: (1) obtains the lowest NME, because the quality of selected pseudo labels is better than others.', '1908.02116-3-70-7': '(2) SPL falls into a local trap at round[MATH] and results in a higher error at round[MATH], whereas SPaCo and our not.', '1908.02116-3-70-8': 'This could be caused by that the interaction between two students can help regularize each other.', '1908.02116-3-70-9': '(3) Our converges faster than SPaCo and achieves better results.', '1908.02116-3-70-10': 'The pseudo labeled data selection in SPaco is a heuristic unsupervised criterion, whereas our criterion is a supervised teacher.', '1908.02116-3-70-11': 'Since no extra supervision is given in SPaCo, their criterion might induce inaccurate pseudo labeled samples.', '1908.02116-3-70-12': 'Besides, as discussed in sec:discussion, our can utilize validation set to further improve the performance by avoid over-fitting, but the compared methods may not effectively utilize validation set.', '1908.02116-3-71-0': 'The effect of the interaction between students.', '1908.02116-3-71-1': 'From Table [REF], we show the ablative studies on the complementary effect of multiple students.', '1908.02116-3-71-2': 'In these experiments, we use the same teacher structure, while "CPM" and "HG" are trained without the interaction between students.', '1908.02116-3-71-3': 'Using 10 labels, CPM achieves 8.28 NME, and HG achieves 6.25 NME on 300-W.', '1908.02116-3-71-4': 'Leveraging from their mutual benefits, our can boost the performance to 5.64, which is higher than CPM by about 30 and than HG by 9.', '1908.02116-3-71-5': 'Under different portion of annotations, we can conclude similar observations.', '1908.02116-3-71-6': 'This ablation study demonstrates the contribution of student interaction to the final performance.', '1908.02116-3-71-7': 'Note that, our algorithm can be readily applied to multiple students without introducing additional hyper-parameters.', '1908.02116-3-71-8': 'In contrast, the number of hyper-parameters in other co-training strategies [CITATION] is quadratic to the number of detectors.', '1908.02116-3-72-0': '## Qualitative Analysis', '1908.02116-3-73-0': 'On the 300-W training set, we train our using only 10% labeled facial images, and we show some qualitative results of the 300-W test set in fig:results.', '1908.02116-3-73-1': 'The first row shows seven raw input facial images.', '1908.02116-3-73-2': 'The second row shows the ground truth background heatmaps, and the third row shows the faces with ground truth landmarks of these images.', '1908.02116-3-73-3': 'We visualize the predicted background heatmap in the fourth row and the predicted coordinates in the fifth row.', '1908.02116-3-73-4': 'As we can see, the predicted landmarks of our are very close to the ground truth.', '1908.02116-3-73-5': 'These predictions are already robust enough, and human may not be able to distinguish the difference between our predictions (the third line) and the ground truth (the fifth line).', '1908.02116-3-74-0': '# Conclusion', '1908.02116-3-75-0': 'In this paper, we propose an interaction mechanism between a teacher and multiple students for semi-supervised facial landmark detection.', '1908.02116-3-75-1': 'The students learn to generate pseudo labels for the unlabeled data, while the teacher learns to judge the quality of these pseudo labeled data.', '1908.02116-3-75-2': 'After that, the teacher can filter out unqualified samples; and the students get feedback from the teacher and improve itself by the qualified samples.', '1908.02116-3-75-3': 'The teacher is adaptive along with the improved students.', '1908.02116-3-75-4': 'Besides, multiple students can not only regularize each other but also be ensembled to predict more accurate pseudo labels.', '1908.02116-3-75-5': 'We empirically demonstrate that the proposed interaction mechanism achieves state-of-the-art performance on three facial landmark benchmarks.'} | null | null | null | null |
1804.03392 | {'1804.03392-1-0-0': 'All-sky surveys for isolated continuous gravitational waves present a significant data-analysis challenge.', '1804.03392-1-0-1': 'Semicoherent search methods are commonly used to efficiently perform the computationally-intensive task of searching for these weak signals in the noisy data of gravitational-wave detectors such as LIGO and Virgo.', '1804.03392-1-0-2': 'We present a new implementation of a semicoherent search method, Weave, that for the first time makes full use of a parameter-space metric to generate search templates in the correct resolution, combined with optimal lattices to minimize the required number of templates and hence the computational cost of the search.', '1804.03392-1-0-3': 'We describe the implementation of Weave and associated design choices, and characterize its behavior using semi-analytic models.', '1804.03392-1-1-0': '# Introduction', '1804.03392-1-2-0': 'The detections of short-duration gravitational-wave events from the inspiral and merger of binary black holes [CITATION] and binary neutron stars [CITATION] are enabling advances across astronomy, astrophysics, and cosmology.', '1804.03392-1-2-1': 'As the gravitational-wave detectors LIGO [CITATION], Virgo [CITATION] improve in sensitivity in the coming years, and as new detectors KAGRA [CITATION] and LIGO India [CITATION] come online, it may become possible to detect gravitational radiation from other astrophysical phenomena.', '1804.03392-1-2-2': 'Rapidly-spinning, non-axisymmetrically-deformed neutron stars will emit gravitational waves in the form of continuous quasi-sinusoidal signals, and remain an intriguing prospect for detection with advanced instruments.', '1804.03392-1-2-3': 'Searches for continuous gravitational waves in contemporary LIGO and Virgo data are ongoing [CITATION].', '1804.03392-1-3-0': 'Since the maximum non-axisymmetric deformation of neutron stars is expected to be small [CITATION], continuous waves are expected to be weak relative to the sensitivity of the current generation of interferometric detectors.', '1804.03392-1-3-1': 'Consequentially there has accumulated a significant body of research devoted to the data analysis challenge of extracting such weak signals from the gravitational-wave detector data.', '1804.03392-1-3-2': 'Early results [CITATION] focused on the method of matched filtering the entire dataset against the known continuous-wave signal model; while theoretically optimal (in the Neyman–Pearson sense), this method quickly becomes computationally intractable if some or all of the model parameters are unknown.', '1804.03392-1-3-3': 'Such is the case if one wished to target an interesting sky direction e.g. associated with a supernova remnant [CITATION] or a low-mass X-ray binary [CITATION], or perform an all-sky survey for isolated continuous-wave sources unassociated with known pulsars [CITATION].', '1804.03392-1-3-4': 'It is the latter type of search that is the subject of this paper.', '1804.03392-1-4-0': 'The additional challenge of a practical upper limit on the computational cost of all-sky searches has spurred the development of various sub-optimal but computationally-tractable hierarchical or semicoherent algorithms [CITATION].', '1804.03392-1-4-1': 'They share a common approach: the dataset (which for this example we assume is contiguous) with timespan [MATH] is partitioned into [MATH] segments, each with timespan [MATH].', '1804.03392-1-4-2': 'A fully-coherent matched filter search is then performed individually for each segment.', '1804.03392-1-4-3': 'Most methods then combine segments by incoherently summing the power from [MATH] filters, one from each segment, which together follow a consistent frequency evolution as dictated by the continuous-wave signal model.', '1804.03392-1-4-4': 'The phase evolution need not be continuous over the [MATH] filters, however; nor need the gravitational-wave amplitudes in each segment be consistent.', '1804.03392-1-4-5': 'This loss of complete signal self-consistency comes, however, with a computational benefit: while the computational cost of a fully-coherent matched filter search of the entire dataset scales as [MATH] with [MATH] a high power [MATH] to 6, the cost of a semicoherent method typically scales as [MATH] with [MATH] [CITATION].', '1804.03392-1-4-6': 'The strain sensitivities of a fully-coherent and semicoherent search typically scale as [MATH] and [MATH] respectively, with [MATH] [CITATION]; for the loss of a factor [MATH] in sensitivity, a semicoherent method is able to gain by being able to analyze large (e.g. [MATH] year) datasets, whereas a fully-coherent search would be computationally restricted to a much shorter (e.g. [MATH] year) subset.', '1804.03392-1-5-0': 'An important early advance in the development of semicoherent methods was the adaption of the Hough transform [CITATION], originally created to analyze tracks in bubble chamber photographs, to instead track the frequency evolution of a continuous gravitational-wave signal [CITATION].', '1804.03392-1-5-1': 'A number of variations of the Hough transform have been implemented, which map the signal track in the time-frequency plane to either its sky position at a fixed reference frequency and frequency derivative [CITATION], or conversely to its reference frequency and frequency derivative at a fixed sky position [CITATION].', '1804.03392-1-5-2': 'The detection statistic computed, the number count, sums either zero or one from each segment depending on whether the significance of a filter exceeds a set threshold.', '1804.03392-1-5-3': 'Some variations use short-duration ([MATH]s) segments and incoherently sum power above threshold from each segment; others analyze longer segments, and set a threshold on the [MATH]-statistic [CITATION] which computes the matched filter analytically maximized over the gravitational-wave amplitudes.', '1804.03392-1-5-4': 'Another modification is to weigh each segment by the antenna response function of the detector, and to sum these weights instead of zero or one [CITATION].', '1804.03392-1-6-0': 'Two semicoherent methods which use short-duration segments but which, unlike the Hough transform methods, sum power without thresholding are the StackSlide [CITATION] and PowerFlux [CITATION] methods.', '1804.03392-1-6-1': 'The StackSlide method builds a time-frequency plane, where each column represents a segment.', '1804.03392-1-6-2': 'For each choice of signal parameters, it "slides" each column up and down in frequency so that a signal with those parameters would follow a horizontal line, and then "stacks" (i.e. sums) the columns horizontally to accumulate the signal power over time for each frequency bin.', '1804.03392-1-6-3': '(Due to this intuitive representation of a semicoherent search method, the term StackSlide is often used to refer to semicoherent methods in general [CITATION].)', '1804.03392-1-6-4': "The PowerFlux method follows a similar methodology, and in addition weights the power from each segment by that segment's noise level and antenna response function, so that segments containing transient instrumental noise and/or where the response of the detector is weak are deweighted.", '1804.03392-1-6-5': 'A "loosely coherent" adaption to PowerFlux allows the degree of phase consistency imposed at the semicoherent stage to be controlled explicitly [CITATION].', '1804.03392-1-6-6': 'A third semicoherent method [CITATION] was developed based on the observance of global correlations between search parameters [CITATION] and uses longer segments analyzed with the [MATH]-statistic.', '1804.03392-1-6-7': 'A comprehensive comparison of many of the all-sky search methods described above is performed in [CITATION].', '1804.03392-1-7-0': 'Aside from developments in semicoherent search techniques, two other ideas have played an important role in the development of continuous gravitational-wave data analysis.', '1804.03392-1-7-1': 'First is the use of a parameter-space metric [CITATION], which is used to determine the appropriate resolution of the bank of template signals such that the mismatch, or fractional loss in signal-to-noise ratio between any signal present in the data and its nearest template, never exceeds a prescribed maximum.', '1804.03392-1-7-2': 'The metric of the [MATH]-statistic for continuous-wave signals was first studied rigorously in [CITATION].', '1804.03392-1-7-3': 'An approximate form of the metric was utilized in semicoherent search methods developed by [CITATION], and a related approximation was used in [CITATION].', '1804.03392-1-7-4': 'The latter approximation, however, lead to an underestimation of the number of required templates in the sky parameter space when analyzing long data stretches; an improved approximate metric developed in [CITATION] addresses this limitation.', '1804.03392-1-7-5': 'It was also later realized that a further approximation fundamental to the metric derivation - namely that the prescribed maximum mismatch (as measured by the metric) could be assumed small - generally does not hold under realistic restrictions on computational cost.', '1804.03392-1-7-6': 'This issue was addressed in [CITATION] which computed an empirical relation between the metric-measured mismatch and the true mismatch of the [MATH]-statistic.', '1804.03392-1-8-0': 'A second important idea is the borrowing of results from lattice theory [CITATION] to optimize the geometric placement of templates within the search parameter space, so as to fulfill the maximum prescribed mismatch criteria described above with the smallest possible density of templates [CITATION].', '1804.03392-1-8-1': 'Practical algorithms for generating template banks for continuous-wave searches, using both the parameter-space metric and optimal lattices, were proposed in [CITATION].', '1804.03392-1-8-2': 'An alternative idea studied in [CITATION] is to instead place templates at random, using the parameter-space metric only as a guide as to the relative density of templates; this idea has found utility in searches for radio [CITATION] and X-ray [CITATION] pulsars.', '1804.03392-1-9-0': 'In this paper we present Weave, an implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-1-9-1': 'This implementation brings together, for the first time, several strands of previous research: the use of a semicoherent method to combine data segments analyzed with the [MATH]-statistic, combined with optimal template placement using the parameter-space metric of [CITATION] and optimal lattices [CITATION].', '1804.03392-1-9-2': 'After a review of relevant background information in Section [REF], the Weave implementation is presented in Section [REF].', '1804.03392-1-9-3': 'In Section [REF] we demonstrate that important behaviors of the Weave implementation can be modeled semi-analytically, thereby enabling characterization and optimization of a search setup without, in the first instance, the need to resort to time-consuming Monte-Carlo simulations.', '1804.03392-1-9-4': 'In Section [REF] we discuss ideas for further improvement and extension.', '1804.03392-1-10-0': '# Background', '1804.03392-1-11-0': 'This section presents background material pertaining to the continuous-wave signal model, parameter-space metric, and template bank generation.', '1804.03392-1-12-0': '## Continuous-wave signals', '1804.03392-1-13-0': 'The phase of a continuous-wave signal [MATH] at time [MATH] at the detector is given by, neglecting relativistic corrections [CITATION], [EQUATION]', '1804.03392-1-13-1': 'The first term on the right-hand side primarily encodes the loss of rotational energy of the neutron star as observed from the Solar System barycenter: [MATH] is the gravitational-wave frequency; and the spindowns [MATH], [MATH], etc. are the 1st-order, 2nd-order, etc. rates of change of the gravitational-wave frequency with time.', '1804.03392-1-13-2': 'All [MATH] parameters are given with respect to a reference time [MATH].', '1804.03392-1-13-3': 'The second term on the right-hand side describes the Doppler modulation of the gravitational waves due to the motion of an Earth-based detector: [MATH] is the detector position relative to the Solar System barycenter, thereby including both the sidereal and orbital motions of the Earth; and [MATH] is a unit vector pointing from the Solar System barycenter to the continuous-wave source.', '1804.03392-1-13-4': 'The value of [MATH] is chosen conservatively to be the maximum of [MATH] over the timespan of the analyzed data.', '1804.03392-1-14-0': 'Together the phase evolution parameters [MATH] parameterize the continuous-wave signal template; additional amplitude parameters [MATH] are analytically maximized over when computing the [MATH]-statistic [CITATION].', '1804.03392-1-14-1': 'In noise the [MATH]-statistic is a central [MATH] statistic with 4 degrees of freedom; when in the vicinity of a signal, the noncentrality parameter [MATH] of the noncentral [MATH] distribution scales as [MATH], where [MATH] is the gravitational-wave amplitude, [MATH] the amount of analyzed data, and [MATH] is the noise power spectral density in the vicinity of the signal frequency [MATH].', '1804.03392-1-15-0': '## Parameter-space metric', '1804.03392-1-16-0': 'The parameter-space metric [MATH] of the [MATH]-statistic is defined by a 2nd-order Taylor expansion of the noncentrality parameter: [EQUATION] with implicit summation over [MATH], and where [EQUATION]', '1804.03392-1-16-1': 'Here [MATH] is the noncentrality parameter of the [MATH]-statistic when perfectly matched to a signal with parameters [MATH], and [MATH] is the noncentrality parameter when computed at some mismatched parameters [MATH].', '1804.03392-1-16-2': 'The mismatch is defined to be [EQUATION]', '1804.03392-1-16-3': 'A very useful approximation to Eq. [REF] is the phase metric [CITATION]; it discards the amplitude modulation of the signal, and thereby the dependence on the known parameters [MATH], retaining only dependence on the phase evolution parameters: [EQUATION]', '1804.03392-1-17-0': '## Optimal template placement', '1804.03392-1-18-0': 'Template placement using optimal lattices is an example of a sphere covering [CITATION]: a collection of lattice-centered [MATH]-dimensional spheres of equal radius.', '1804.03392-1-18-1': 'The radius is chosen to be the smallest value that satisfies the property that each point in the [MATH]-dimensional parameter space is contained in at least one sphere.', '1804.03392-1-18-2': 'A lattice where the ratio of the volume of the sphere to the volume of a lattice cell is minimized generates a minimal sphere covering, i.e. the minimal number of points required to cover a parameter space, which is exactly the property desired for template banks.', '1804.03392-1-18-3': '(For example, in two dimensions the minimal sphere covering is generated by the hexagonal lattice.)', '1804.03392-1-18-4': 'We identify the covering spheres with the metric ellipsoids [MATH], where [MATH] is the prescribed maximum; it follows that the radii of the covering spheres is [MATH].', '1804.03392-1-18-5': 'A matrix transform [MATH] can then be constructed [CITATION] which takes integers in [MATH] to template parameters [MATH] to generate the template bank: [EQUATION] where [MATH] is a function of the metric [MATH], and [MATH] is particular to the lattice being used.', '1804.03392-1-18-6': 'If [MATH] is a lower triangular matrix, an efficient algorithm [CITATION] can be found for generating the template bank.', '1804.03392-1-19-0': '## Reduced supersky metric', '1804.03392-1-20-0': 'In order for Eq. [REF] to preserve the sphere covering property, however, it must be independent of the template parameters [MATH].', '1804.03392-1-20-1': 'Since [MATH] is a function of the metric, we require a metric which is also independent of [MATH]: [MATH].', '1804.03392-1-20-2': 'The phase metric of Eq. [REF] is independent of the frequency and spindown parameters [MATH], but retains a dependence on sky position parameters, e.g. [MATH] in terms of right ascension [MATH] and declination [MATH].', '1804.03392-1-20-3': 'The question of how to derive a useful metric which is independent of the sky position parameters, i.e. [MATH], has stimulated numerous approaches [CITATION].', '1804.03392-1-20-4': 'In [CITATION], a useful [MATH] is derived through the following procedure: [(i)]', '1804.03392-1-21-0': '[MATH] is expressed in terms of the 3 components of [MATH], instead of 2 parameters such as [MATH].', '1804.03392-1-21-1': 'The 3 components of [MATH] are taken to be independent; geometrically this is equivalent to embedding [MATH] into a 3-dimensional supersky parameter space, instead of being restricted to the 2-sphere defined by [MATH].', '1804.03392-1-21-2': 'In the supersky parameter space, [MATH] is independent of the sky position parameters, i.e. we have the desired [MATH], but with the addition of a 3rd unwanted parameter-space dimension.', '1804.03392-1-22-0': 'A linear coordinate transform [MATH] is derived which satisfies: [MATH] is diagonal in the sky position parameters [MATH], i.e. [MATH]; [MATH]; and [MATH].', '1804.03392-1-22-1': 'The last two properties imply that the metric ellipsoids are much longer along the [MATH] axis than along the [MATH] and [MATH] axes.', '1804.03392-1-22-2': 'In computing the coordinate transform, use is made of the well-known correlation between the sky and frequency/spindown parameters of the continuous-wave signal [CITATION].', '1804.03392-1-22-3': 'The correlations arise because, on sufficiently short timescales, the change in phase due to the cyclical sidereal and orbital motions of the Earth may be Taylor expanded as linear, quadratic, etc. changes in phase with time, and thereby are equivalent to changes in the frequency ([MATH]), 1st spindown ([MATH]), etc. parameters.', '1804.03392-1-23-0': 'Since, in the new coordinates [MATH] the mismatch [MATH] is only weakly dependent on [MATH], a useful approximate metric [MATH] is found by discarding the [MATH] dimension.', '1804.03392-1-23-1': 'Geometrically this corresponds to projecting the 3-dimensional supersky parameter space and metric onto the 2-dimensional [MATH]-[MATH] plane.', '1804.03392-1-23-2': 'The resultant reduced supersky parameter-space metric [MATH] and associated coordinates [MATH] has reduced the sky parameter space dimensionality back to 2, while retaining the property that [MATH] is parameter-independent.', '1804.03392-1-24-0': '# Weave Implementation', '1804.03392-1-25-0': 'This section describes the Weave implementation of the semicoherent search method, a schematic of which is shown in Figure [REF].', '1804.03392-1-25-1': 'The implementation is freely available as part of the LALSuite [CITATION] gravitational-wave data analysis library.', '1804.03392-1-26-0': '## Overview', '1804.03392-1-27-0': 'In step 1 the user runs a precursor program lalappsWeaveSetup, which takes as an argument a list of [MATH] segments [MATH] into which the dataset is to be partitioned.', '1804.03392-1-27-1': 'The program computes in step 2 the [MATH] coherent parameter-space metrics [MATH] used to construct template banks within each segment, and the semicoherent parameter-space metric [MATH] used to incoherently combine segments.', '1804.03392-1-27-2': 'The metrics are written to a setup file in the FITS format [CITATION].', '1804.03392-1-27-3': 'Due to the numerical ill-conditionedness of the parameter-space metric [CITATION], this computation involves a bootstrapping process, whereby successively better-conditioned iterations of the supersky metric are computed, before then computing the reduced supersky metric as outlined in Section [REF].', '1804.03392-1-27-4': 'Since this bootstrapping process can be time-consuming for large [MATH], and may give slightly different results on different computer hardware, precomputing the metrics both saves computing time and adds robustness against numerical errors.', '1804.03392-1-27-5': 'Note that, by Eq. [REF], the sky components of the metrics will scale with [MATH]; since its value depends on the search frequency parameter space, which is not known by lalappsWeaveSetup, an arbitrary fiducial value [MATH] is used, and the sky components of the metrics are later rescaled by [MATH].', '1804.03392-1-28-0': 'In step 3 the user runs the main search program lalappsWeave.', '1804.03392-1-28-1': 'The principle arguments to this program are the setup file output by lalappsWeaveSetup, the search parameter space, and the prescribed maximum mismatches [MATH] and [MATH] for the coherent and semicoherent template banks respectively.', '1804.03392-1-28-2': 'The frequency and spindown parameter space is specified by ranges [MATH], where [MATH], 1, etc. as required.', '1804.03392-1-28-3': 'The sky search parameter space may be specified either as a rectangular patch in right ascension and declination [MATH], or alternatively partitioned into [MATH] patches containing approximately equal number of templates (see Appendix [REF]), and a patch selected by an index [MATH], [MATH].', '1804.03392-1-28-4': 'In step 4 various preparatory tasks are performed, such as loading the gravitational-wave detector data into memory, before beginning the main search loop.', '1804.03392-1-29-0': 'The main search loop of a semicoherent search method may be structured in two complementary ways, which differ in the memory each requires to store intermediate results: [(i)]', '1804.03392-1-30-0': 'The semicoherent template bank [MATH] is stored in memory, and the [MATH] segments are processed in sequence.', '1804.03392-1-30-1': 'For each segment [MATH], every coherent template [MATH] is mapped back to the semicoherent template bank, i.e. [MATH].', '1804.03392-1-30-2': 'Because the semicoherent template bank must track the continuous-wave signal over a larger timespan [MATH] than the coherent template banks, it will contain a greater density of templates; the ratio of semicoherent to coherent template bank densities is the refinement factor [MATH] [CITATION].', '1804.03392-1-30-3': 'It follows that the mapping [MATH] will be one-to-many.', '1804.03392-1-31-0': 'As the [MATH] segments are processed, any semicoherent detection statistic associated with [MATH] is then updated based on the corresponding coherent detection statistic associated with [MATH].', '1804.03392-1-31-1': 'For example, it is common to compute the summed [MATH]-statistic [MATH]; here we would then have [MATH].', '1804.03392-1-31-2': 'Once every segment has been processed, computed [MATH] for every [MATH] will exist in memory.', '1804.03392-1-31-3': 'The memory usage of the main search loop will therefore be proportional to the number of semicoherent templates [MATH], where [MATH] is the average number of templates in a coherent template bank.', '1804.03392-1-32-0': 'The [MATH] coherent template banks [MATH] are stored in memory, and the semicoherent template bank is processed in sequence.', '1804.03392-1-32-1': 'Each semicoherent template [MATH] is mapped back the coherent template bank in each segment [MATH], i.e. [MATH]; since [MATH] in each segment this mapping will be many-to-one.', '1804.03392-1-32-2': 'With these [MATH] mappings in hand, the semicoherent detection statistics may be immediately computed in full, e.g. [MATH].', '1804.03392-1-32-3': 'The memory usage of the main search loop will therefore be proportional to [MATH].', '1804.03392-1-33-0': 'For the parameter-space metric for all-sky searches, [MATH] [CITATION], and therefore the latter structuring given above will have the lower memory requirement; the Weave implementation uses this structuring of the main search loop.', '1804.03392-1-33-1': 'The semicoherent template bank [MATH] is generated one template at a time using the algorithm described in [CITATION].', '1804.03392-1-33-2': 'For each coherent template bank, an efficient lookup table [CITATION] is constructed for the mapping [MATH].', '1804.03392-1-34-0': 'We note an important distinction between the definition of the Weave template banks and the traditional StackSlide picture of a semicoherent search method.', '1804.03392-1-34-1': 'In the latter picture, the frequency and spindown template banks of each segment are defined with respect to individual reference times [MATH], typically the midtime of each segment.', '1804.03392-1-34-2': 'When combining segments, therefore, the frequency and spindown parameters of each coherent template must be adjusted so as to bring the parameters of all segments to a common reference time [MATH]; this is the "sliding" step.', '1804.03392-1-34-3': 'The Weave implementation, however, defines the frequency and spindown templates banks of all segments at the same reference time [MATH], which is also the reference time of the semicoherent bank.', '1804.03392-1-34-4': 'Consequentially, there is no analogy to the "sliding" step of StackSlide.', '1804.03392-1-34-5': 'Instead, the orientation of the metric ellipses in the [MATH] plane changes from segment to segment, as illustrated in Figure [REF].', '1804.03392-1-34-6': 'As the absolute difference [EQUATION] between the midtime of each segment [MATH] and [MATH] increases, both the extent of the ellipses in [MATH] and the correlation between [MATH] and [MATH] also increase.', '1804.03392-1-35-0': 'Steps 5-16 comprise the main search loop; which performs two key tasks: the computation and output of the detection statistics over the semicoherent template bank (steps 5, 6, and 12-17), and the management of an internal cache of required detection statistics computed on each coherent template bank (steps 7-11).', '1804.03392-1-35-1': 'These two tasks are described more fully in the following two sections, and with reference to a diagram of their operation in Figure [REF].', '1804.03392-1-36-0': 'In this section and in Figure [REF] we focus for simplicity on the computation of the semicoherent [MATH]-statistics [MATH] and [MATH].', '1804.03392-1-36-1': 'The computation of other detection statistics is also possible: in particular a family of Bayesian statistics has been developed which weigh the likelihood of a continuous wave signal against that of an instrumental line which appears in all segments [CITATION], or a transient instrumental line which appears only in one segment [CITATION].', '1804.03392-1-36-2': 'Computation of the former statistic, denoted [MATH], is also illustrated in Figure [REF]; it takes as input the multi-detector [MATH] which uses data from all gravitational-wave detectors, as well as the per-detector [MATH] which are computed from each detector [MATH] individually.', '1804.03392-1-37-0': '## Computation of semicoherent statistics', '1804.03392-1-38-0': 'In steps 5 and 16 (Figure [REF]), the main loop of the search method generates successive points [MATH] in the semicoherent template bank.', '1804.03392-1-38-1': 'An example of such a point is indicated in Figure [REF].', '1804.03392-1-38-2': 'Next, in steps 6 and 13, each segment [MATH] is visited and the mapping [MATH] is performed.', '1804.03392-1-38-3': 'The mapping used by Weave is nearest-neighbor interpolation: the [MATH] is expressed in the coherent metric coordinates of the [MATH]th segment, and the nearest (with respect to the metric) coherent template in the respective bank [MATH] is determined.', '1804.03392-1-38-4': 'If the template bank is constructed on a lattice, efficient algorithms exist for determining the nearest point [CITATION].', '1804.03392-1-38-5': 'In Figure [REF], example nearest coherent templates are labeled [MATH], [MATH], and [MATH].', '1804.03392-1-39-0': 'As each nearest point is determined, the coherent [MATH]-statistic in the respective segment is computed (steps 7-11, see Section [REF]), and the value of the semicoherent statistic [MATH] is updated (step 12).', '1804.03392-1-39-1': 'Once all segments have been processed (step 13), additional semicoherent statistics such as [MATH] are computed (step 14), and a candidate comprising the signal parameters together with the computed semicoherent statistics is added (step 15) to one or more toplists which ranks each candidate by a chosen semicoherent statistic.', '1804.03392-1-39-2': 'The size of the toplists is generally of a fixed user-determined size so that only a fixed number of the most promising candidates will be returned.', '1804.03392-1-40-0': 'Once the semicoherent template bank is exhausted (step 16) the toplists are written to an output file in the FITS format, and the search concludes (step 17).', '1804.03392-1-41-0': '## Management of cache of coherent statistics', '1804.03392-1-42-0': 'It is important that the main search loop minimizes its memory usage as much as possible.', '1804.03392-1-42-1': 'Even though in Section [REF] we chose a structuring of the main search loop so as to reduce memory usage, a naive implementation which stores [MATH] coherent statistics would still require a prohibitive amount of memory, given that both [MATH] and [MATH] are typically large.', '1804.03392-1-42-2': 'We therefore implement a per-segment cache which stores only those coherent statistics associated with coherent templates [MATH] accessible from the unprocessed portion of the semicoherent template bank via the mapping [MATH].', '1804.03392-1-42-3': 'Put another way, if a [MATH] can no longer be mapped to by any [MATH] remaining in [MATH], then [MATH] can be safely removed from the cache.', '1804.03392-1-43-0': 'In order to devise a cache management algorithm with the above desired properties, we first define an operator called relevance, denoted [MATH].', '1804.03392-1-43-1': 'The relevance operates on both coherent and semicoherent templates, and should satisfy the following property: [EQUATION]', '1804.03392-1-43-2': 'A definition of [MATH] satisfying this property is derived as follows.', '1804.03392-1-44-0': 'First, take any coherent template (e.g. [MATH] in Figure [REF]) and surround it by its metric ellipsoid at mismatch [MATH].', '1804.03392-1-44-1': 'Then surround the metric ellipsoid in turn by its bounding box, the smallest coordinate box which contains the ellipsoid [CITATION]; the metric ellipse bounding box centered on [MATH] is also shown in Figure [REF].', '1804.03392-1-44-2': 'Now, transform the bounding box into the semicoherent parameter space; practically this may be achieved by expressing the coordinates of each vertex of the bounding box in the semicoherent metric coordinates.', '1804.03392-1-44-3': 'See Figure [REF] for the transformed bounding box of [MATH] in the semicoherent parameter space, which is centered on [MATH].', '1804.03392-1-45-0': 'Note that, by definition, any semicoherent template [MATH] outside of the transformed bounding box centered on [MATH] cannot map to [MATH] under [MATH].', '1804.03392-1-45-1': 'Thus, to determine whether [MATH] is accessible by [MATH], we can compute whether [MATH] is within the transformed bounding box of [MATH].', '1804.03392-1-45-2': 'To be conservative, however, we also surround [MATH] by its bounding box as shown in Figure [REF], and instead compute whether the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-1-46-0': 'To simplify the bounding box intersection calculation, we compare just the coordinates of the bounding boxes of [MATH] and [MATH] in one dimension; for reasons that will soon be apparent, we choose the lowest-dimensional coordinate, [MATH].', '1804.03392-1-46-1': 'First, we define the relevance [MATH] for both coherent and semicoherent templates: [EQUATION]', '1804.03392-1-46-2': 'We now compute [MATH] and [MATH]; in Figure [REF], [MATH] is the [MATH] coordinate of the right-most edge of the transformed bounding box of [MATH], and [MATH] is the [MATH] coordinate of the left-most edge of the bounding box of [MATH].', '1804.03392-1-46-3': 'In this example, [MATH], and it follows from the definition of [MATH] in Eqs. [REF] that the bounding boxes of [MATH] and [MATH] cannot intersect.', '1804.03392-1-47-0': 'On the other hard, let us choose another coherent template [MATH], and examine its relevance [MATH]; here we have [MATH] (see Figure [REF]).', '1804.03392-1-47-1': 'From the simplified bounding box intersection calculation, we conclude that the bounding boxes of [MATH] and [MATH] could potentially intersect, since at least in the [MATH] dimension the bounding boxes overlap (although in this example the bounding boxes do not overlap in the [MATH] dimension).', '1804.03392-1-48-0': 'Finally, if for some [MATH] we have [MATH], then this condition is guaranteed to remain true for all remaining [MATH] in the template bank.', '1804.03392-1-48-1': 'This is simply a consequence of the algorithm used to generate the semicoherent template bank [CITATION], which operates as follows: first, values of [MATH] are generated in a constant range [MATH]; then, for each value of [MATH], values of [MATH] are generated in ranges [MATH] dependent on [MATH], and so on.', '1804.03392-1-48-2': 'It follows that the value of [MATH] can only increase during the generation of the semicoherent template bank, and since [MATH] is defined in terms of [MATH], it too can only increase.', '1804.03392-1-49-0': 'To summarize, the relevance operator [MATH] defined by Eqs. [REF] satisfies the desired property given by Eq. [REF].', '1804.03392-1-49-1': 'In Figure [REF], since [MATH], the cache management algorithm would discard any coherent statistics associated with [MATH] from memory, since they cannot be accessed by [MATH] nor any remaining semicoherent template.', '1804.03392-1-49-2': 'On the other hard, the algorithm would retain any coherent statistics associated with [MATH], since they could still be needed for future semicoherent templates; indeed in Figure [REF] it is clear that the next semicoherent template in the bank, labeled [MATH], could require coherent statistics associated with [MATH], since the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-1-50-0': 'The cache management algorithm described above is implemented in the main search loop in steps 7-11 (Figure [REF]).', '1804.03392-1-50-1': 'In step 7 the cache is interrogated for a required [MATH]-statistic value [MATH]: if it is in the cache, it is retrieved and utilized (step 8), otherwise it is computed and inserted into the cache (step 9).', '1804.03392-1-50-2': 'In the latter case, the cache is also checked to see if any cache items can be discarded.', '1804.03392-1-50-3': 'Starting with step 10, cache items indexed by [MATH] are retrieved in order of ascending [MATH].', '1804.03392-1-50-4': 'If [MATH], the cache items are discarded (step 11).', '1804.03392-1-50-5': 'Only one cache item is removed at any one time, and therefore the memory usage of the cache will either remain constant, or increase by one item per main search loop iteration.', '1804.03392-1-50-6': 'The cache is implemented using two data structures [CITATION]: a binary heap to rank cache items by [MATH], and a hash table to find cache items indexed by [MATH].', '1804.03392-1-51-0': '# Models of Weave Behavior', '1804.03392-1-52-0': 'This section presents semi-analytic models of the Weave implementation.', '1804.03392-1-52-1': 'It greatly facilitates the practical usage of any search method if its behavior can be characterized a priori as much as possible using a computationally-cheap model.', '1804.03392-1-52-2': 'For example, a model of the distribution of [MATH]-statistic mismatches (Section [REF]) permits the estimation of the sensitivity of a particular search setup [CITATION] which in turn allows the setup to be optimized so as to maximize sensitivity [CITATION].', '1804.03392-1-52-3': 'Similarly, models of the number of coherent and semicoherent templates (Section [REF]) and computational cost (Section [REF]) allow the parameters of the optimal search setup to be estimated [CITATION].', '1804.03392-1-52-4': 'The memory usage (Section [REF]) and input data bandwidth (Section [REF]) required by the implementation are also important properties when implementing a search pipeline.', '1804.03392-1-53-0': 'Each model presented in this section is implemented as an Octave [CITATION] script, and is freely available as part of the OctApps [CITATION] script library.', '1804.03392-1-54-0': '## Distribution of [MATH]-statistic mismatches', '1804.03392-1-55-0': 'The distribution of the mismatch between the [MATH]-statistic computed at an exact signal location, and at the nearest point in the Weave semicoherent template bank, gives an idea of the expected loss in signal-to-noise ratio due to the necessary coarseness of the template bank.', '1804.03392-1-55-1': 'Figure [REF] plots the predicted means and standard deviations of Weave [MATH]-statistic mismatch distributions, against their measured values, for a variety of setups given in Table [REF].', '1804.03392-1-55-2': 'The distributions were measured using software injection studies, where relatively strong ([MATH]) simulated signals are added to Gaussian-distributed noise and then searched for using lalappsWeave.', '1804.03392-1-56-0': 'The predicted means and standard deviations are from the model presented in [CITATION], and are generally conservative: Figure [REF] shows that the model generally overestimates the mean [MATH]-statistic mismatch by [MATH] (Figure [REF]) to [MATH] (Figure [REF]); and the predicted standard deviations imply slightly broader distributions than were measured.', '1804.03392-1-56-1': 'As explored in [CITATION], the relationship between the maximum mismatches of the coherent and semicoherent template banks (which are inputs to lalappsWeave) and the [MATH]-statistic mismatch distribution (which is output by lalappsWeave) is difficult to model when the former are large e.g. [MATH].', '1804.03392-1-57-0': 'In addition, an optimization implemented in Weave but not accounted for in the model of [CITATION] complicates the picture: the coherent and semicoherent template banks are constructed to have equally-spaced templates in the frequency parameter [MATH].', '1804.03392-1-57-1': 'This permits (in step 9 of Figure [REF]) the simultaneous computation of a series of [MATH] values at equally-spaced values of [MATH] across the frequency parameter space, which can be performed efficiently using Fast Fourier Transform-based algorithms (see Section [REF]).', '1804.03392-1-57-2': 'The construction of equal-frequency-spacing coherent and semicoherent template banks is performed by first constructing each bank independently, and then reducing the frequency spacing in all banks to that of the smallest frequency spacing in any bank.', '1804.03392-1-57-3': 'This construction will always reduce the maximum possible mismatch in each grid, but never increase it, and so we would expect the mean [MATH]-statistic mismatch measured by Weave to be smaller than that predicted by the model of [CITATION].', '1804.03392-1-58-0': 'The model of [CITATION] is implemented in the OctApps script WeaveFstatMismatch.m.', '1804.03392-1-59-0': '## Number of templates', '1804.03392-1-60-0': 'Since the Weave coherent and semicoherent template banks are constructed using lattices (see Section [REF]), the number of templates in each is estimated starting from the formula [CITATION] [EQUATION] where [MATH] is the volume of the [MATH]-dimensional parameter space, [MATH] the parameter-space metric, and [MATH] the maximum mismatch.', '1804.03392-1-60-1': 'The normalized thickness [MATH] is a property of the particular lattice used to generate the template bank [CITATION].', '1804.03392-1-61-0': 'The parameter-space volume is given explicitly by the following expressions: [EQUATION]', '1804.03392-1-61-1': 'Here, [MATH] is the vector whose components are the extents of the bounding box of [MATH] in each dimension; it is used to ensure that the volume of the parameter space in each dimension is not smaller than the extent of a single template.', '1804.03392-1-61-2': 'In Eq. [REF], the volume of the sky parameter space may be specified either by a rectangular patch [MATH], or by the number [MATH] of equal-size sky patches (see Section [REF]).', '1804.03392-1-62-0': 'Finally, the total number of coherent and semicoherent templates, [MATH] and [MATH] respectively, are given by: [EQUATION]', '1804.03392-1-62-1': 'The numerical prefactor on the right-hand side of Eq. [REF] is chosen to better match [MATH] to the number of coherent templates actually computed by lalappsWeave: the coherent parameter space is augmented with additional padding along its boundaries to ensure that it encloses the semicoherent parameter space, i.e. that it includes a nearest neighbor for every [MATH].', '1804.03392-1-63-0': 'Equations [REF] and [REF] are used to predict the number of templates computed by lalappsWeave for a variety of search setups detailed in Table [REF].', '1804.03392-1-63-1': 'Figure [REF] plots the predicted [MATH] and [MATH] against the values measured by running lalappsWeave.', '1804.03392-1-63-2': 'Reasonable agreement is achieved between predicted and measured [MATH] (Figure [REF]): while Eq. [REF] sometimes underestimates the number of coherent templates, it rarely does so by more than a factor of a few.', '1804.03392-1-63-3': 'Better agreement is seen between predicted and measured [MATH] (Figure [REF]).', '1804.03392-1-64-0': 'Equations [REF] and [REF] are implemented in the OctApps script WeaveTemplateCount.m.', '1804.03392-1-65-0': '## Computational cost', '1804.03392-1-66-0': 'The total computational cost [MATH] of a particular search setup may be modeled in terms of the number of coherent [MATH] and semicoherent [MATH] templates (see Section [REF]), the number of segments [MATH] and number of detectors [MATH].', '1804.03392-1-66-1': 'Following [CITATION] we write [EQUATION] where [MATH] and [MATH] denote the computational cost of the coherent and semicoherent stages of the search method respectively, and [MATH] denotes any unmodeled computational costs.', '1804.03392-1-67-0': 'The computational cost model takes as input fundamental timing constants which give the time taken to complete certain fundamental computations.', '1804.03392-1-67-1': 'Their values are highly dependent on various properties of the computer hardware used to run lalappsWeave, such as the processor speed and cache sizes, as well as what other programs were using the computer hardware at the same time as lalappsWeave.', '1804.03392-1-67-2': 'Some values are also specific to the search setups detailed in Table [REF].', '1804.03392-1-67-3': 'For the interest of the reader, Table [REF] lists representative values of the fundamental timing constants obtained on a particular computer cluster.', '1804.03392-1-68-0': 'The coherent cost [MATH] is simply the cost of computing the [MATH]-statistic (step 9 of Figure [REF]): [EQUATION]', '1804.03392-1-68-1': 'The fundamental timing constant [MATH] gives the time taken to compute the [MATH]-statistic per template and per detector, and is further described in [CITATION].', '1804.03392-1-68-2': 'Its value depends primarily upon the range of the frequency parameter space [MATH], the coherent segment length [MATH], and the algorithm used to compute the [MATH]-statistic.', '1804.03392-1-68-3': 'Choices for the latter are the Fast Fourier Transform-based resampling algorithm [CITATION], or the demodulation algorithm of [CITATION].', '1804.03392-1-68-4': 'The additional cost of managing the cache of computed [MATH]-statistic values (steps 8, 10, and 11) are amortized into [MATH].', '1804.03392-1-69-0': 'The semicoherent cost [EQUATION] has a number of components: [(i)] [MATH] is the cost of iterating over the semicoherent template bank (steps 5 and 16 of Figure [REF]); [MATH] is the cost of finding the nearest templates in the coherent template banks (step 6 and 13) and of interrogating the cache of computed [MATH]-statistic values (step 7); [MATH] is the cost of computing [MATH] and, if required, [MATH] (step 12); [MATH] is the cost of computing [MATH] (step 14); [MATH] is the cost of computing [MATH], if required (step 14); and [MATH] is the cost of adding candidates to toplists (step 15).', '1804.03392-1-70-0': 'These components of [MATH] are further defined in terms of [MATH], [MATH], [MATH], and various fundamental timing constants (see Table [REF]) as follows: [EQUATION]', '1804.03392-1-70-1': 'Figure [REF] compares the computational cost model of Eqs. [REF]-[REF] against the measured computational cost of lalappsWeave (see Table [REF]), using the search setups detailed in Table [REF].', '1804.03392-1-70-2': 'The total computational cost of lalappsWeave is generally well-modeled (Figure [REF]) and the unmodeled component of the measured computational cost is low (Figure [REF]).', '1804.03392-1-70-3': 'The coherent computational cost [MATH] of Eq. [REF] and the components of the semicoherent cost [MATH] of Eq. [REF] are also in good agreement (Figures [REF]-[REF]).', '1804.03392-1-71-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveRunTime.m.', '1804.03392-1-72-0': '## Memory usage', '1804.03392-1-73-0': 'The memory usage [MATH] of lalappsWeave is modeled by [EQUATION]', '1804.03392-1-73-1': 'The first term on the right-hand side, [MATH], is the memory usage of the [MATH]-statistic algorithm (which includes the gravitational-wave detector data) and is further described in [CITATION].', '1804.03392-1-73-2': 'The second term, [MATH], is the memory usage of the cache of computed [MATH]-statistic values, and is further given by [EQUATION] where [MATH] is the maximum size of the cache (across all segments), and [MATH] MiB is the memory required to store one [MATH] value as a 4-byte single precision floating-point number.', '1804.03392-1-73-3': 'The maximum cache [MATH] cannot easily be predicted from first principles, i.e. given the search setup, parameter space, and other input arguments to lalappsWeave.', '1804.03392-1-73-4': 'Instead, it is measured by running lalappsWeave in a special mode which simulates the performance of the cache but without computing any [MATH]-statistic or derived values; essentially it follows Figure [REF] but with the first part of step 9, step 12, and step 14 omitted.', '1804.03392-1-74-0': 'Figure [REF] plots the predicted memory usage of Eqs. [REF] and [REF] against the measured memory usage of lalappsWeave, using the search setups detailed in Table [REF].', '1804.03392-1-74-1': 'The [MATH]-statistic is computed using both the resampling and demodulation algorithms: in the former case, both [MATH] and [MATH] are computed, thereby triggering the first case in Eq. [REF]; in the latter case, only [MATH] is computed, thereby triggering the second case in Eq. [REF].', '1804.03392-1-74-2': 'Good agreement between predicted and measured memory usage is seen for both algorithms.', '1804.03392-1-75-0': 'Equations [REF] and [REF] are also implemented in the OctApps script WeaveRunTime.m.', '1804.03392-1-76-0': '## Input data bandwidth', '1804.03392-1-77-0': 'Our final Weave model concerns what bandwidth of the input gravitational-wave detector data is required to search a given frequency range.', '1804.03392-1-77-1': 'For most continuous-wave search pipelines, short (typically 1800 s) contiguous segments of gravitational-wave strain data are Fourier transformed, and the resulting complex spectra stored as Short Fourier Transform (SFT) files.', '1804.03392-1-77-2': 'A continuous-wave search of a large frequency parameter space will generally be divided into smaller jobs, with each job searching a smaller partition of the whole frequency parameter space.', '1804.03392-1-77-3': 'Each job therefore requires that only a small bandwidth out of the full SFT spectra be read into memory.', '1804.03392-1-78-0': 'Given an input frequency parameter space [MATH] and spindown parameter space [MATH], predicting the bandwidth of the SFT spectra required by lalappsWeave proceeds in several steps.', '1804.03392-1-78-1': 'First, the input parameter spaces are augmented to account for extra padding of the Weave template banks: [EQUATION] where [MATH] and [MATH] are empirically chosen.', '1804.03392-1-78-2': 'Next, the maximum frequency range [MATH] is found by evolving the frequency-spindown parameter space [MATH] from the reference time [MATH] to the start and end times of each segment, [MATH] and [MATH] respectively: [EQUATION]', '1804.03392-1-78-3': 'Finally, the SFT bandwidth [MATH] of the SFT spectra which is required by lalappsWeave is given by: [EQUATION]', '1804.03392-1-78-4': 'The [MATH] enlarges [MATH] to account for the maximum frequency-dependent Doppler modulation of a continuous-wave signal due to the sidereal and orbital motions of the Earth, and is given by [EQUATION] where [MATH] is the speed of light, [MATH] is the Earth-Sun distance and [MATH] the radius of the Earth.', '1804.03392-1-78-5': 'Additional padding of [MATH] is also required for use by the chosen [MATH]-statistic algorithm, and is given by [MATH] [CITATION].', '1804.03392-1-79-0': 'Figure [REF] compares the model of Eqs. [REF]-[REF] against the behavior of lalappsWeave when run with the search setups detailed in Table [REF].', '1804.03392-1-79-1': 'Note that the model satisfies [EQUATION] i.e. all circles plotted in Figure [REF] are below the horizontal axis, and [EQUATION] i.e. all crosses plotted in Figure [REF] are above the horizontal axis.', '1804.03392-1-79-2': 'The model is therefore conservative, i.e. it may predict a slightly larger SFT bandwidth than required, but should never predict a smaller SFT bandwidth, which would cause a fatal error in lalappsWeave.', '1804.03392-1-79-3': "The model is generally more conservative at higher frequencies, where the Doppler modulation due to the Earth's motion is higher.", '1804.03392-1-80-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveInputSFTBand.m.', '1804.03392-1-81-0': '# Discussion', '1804.03392-1-82-0': 'This paper details the Weave implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-1-82-1': 'It focuses on all-sky surveys for isolated continuous-wave sources, for which the parameter space is the sky position and frequency evolution of the source.', '1804.03392-1-82-2': 'We note, however, that the implementation is in fact indifferent to the parameter space being searched, as long as the relevant constant parameter-space metric is available.', '1804.03392-1-82-3': 'The implementation could therefore be adapted to search other parameter spaces for continuous-wave sources such as known low-mass X-ray binaries, for which the parameter space includes the evolution parameters of the binary orbit, using the metric of [CITATION].', '1804.03392-1-83-0': 'There is scope to improve the semi-analytic models of the behavior of lalappsWeave presented in Section [REF].', '1804.03392-1-83-1': 'In particular, a more accurate model of the distribution of [MATH]-statistic mismatches than that presented in Section [REF] would allow the sensitivity of a search to be more accurately estimated without resorting to software injection studies.', '1804.03392-1-83-2': 'The memory model of Section [REF] would also be improved if the maximum cache size [MATH] could be predicted from first principles.', '1804.03392-1-84-0': 'In a forthcoming paper [CITATION] we plan to more fully characterize the performance of the Weave implementation, and compare it to an implementation of the method of [CITATION] using a mock data challenge.', '1804.03392-1-85-0': 'We thank Bruce Allen and Heinz-Bernd Eggenstein for valuable discussions.', '1804.03392-1-85-1': 'KW is supported by ARC CE170100004.', '1804.03392-1-85-2': 'Numerical simulations were performed on the Atlas computer cluster of the Max Planck Institute for Gravitational Physics.', '1804.03392-1-85-3': 'This paper has document number LIGO-P1800074-v2.', '1804.03392-1-86-0': '# Properties of equal-area sky patches', '1804.03392-1-87-0': 'The search program lalappsWeave allows the sky search parameter space to be partitioned into [MATH] patches, and a patch selected by an index [MATH].', '1804.03392-1-87-1': 'Tests of this feature found that, provided [MATH] (the number of templates with just one patch), the variation in the number of templates between patches [MATH] is generally small and well-approximated by [EQUATION]', '1804.03392-1-87-2': 'The ratio [MATH] of the number of templates in all [MATH] patches to the number of templates with just one patch is generally [MATH]%.', '1804.03392-1-87-3': 'The union of all templates in a set of [MATH] patches also faithfully reproduces the unpartitioned template bank, i.e. with just one patch.'} | {'1804.03392-2-0-0': 'All-sky surveys for isolated continuous gravitational waves present a significant data-analysis challenge.', '1804.03392-2-0-1': 'Semicoherent search methods are commonly used to efficiently perform the computationally-intensive task of searching for these weak signals in the noisy data of gravitational-wave detectors such as LIGO and Virgo.', '1804.03392-2-0-2': 'We present a new implementation of a semicoherent search method, Weave, that for the first time makes full use of a parameter-space metric to generate banks of search templates at the correct resolution, combined with optimal lattices to minimize the required number of templates and hence the computational cost of the search.', '1804.03392-2-0-3': 'We describe the implementation of Weave and associated design choices, and characterize its behavior using semi-analytic models.', '1804.03392-2-1-0': '# Introduction', '1804.03392-2-2-0': 'The detections of short-duration gravitational-wave events from the inspiral and merger of binary black holes [CITATION] and binary neutron stars [CITATION] are enabling advances across astronomy, astrophysics, and cosmology.', '1804.03392-2-2-1': 'As the gravitational-wave detectors LIGO [CITATION], Virgo [CITATION] improve in sensitivity in the coming years, and as new detectors KAGRA [CITATION] and LIGO India [CITATION] come online, it may become possible to detect gravitational radiation from other astrophysical phenomena.', '1804.03392-2-2-2': 'Rapidly-spinning, non-axisymmetrically-deformed neutron stars will emit gravitational waves in the form of continuous quasi-sinusoidal signals, and remain an intriguing prospect for detection with advanced instruments.', '1804.03392-2-2-3': 'Searches for continuous gravitational waves in contemporary LIGO and Virgo data are ongoing [CITATION].', '1804.03392-2-3-0': 'Since the maximum non-axisymmetric deformation of neutron stars is expected to be small [CITATION], continuous waves are expected to be weak relative to the sensitivity of the current generation of interferometric detectors.', '1804.03392-2-3-1': 'Consequentially there has accumulated a significant body of research devoted to the data analysis challenge of extracting such weak signals from the gravitational-wave detector data.', '1804.03392-2-3-2': 'Early results [CITATION] focused on the method of matched filtering the entire dataset against the known continuous-wave signal model; while theoretically optimal (in the Neyman–Pearson sense), this method quickly becomes computationally intractable if some or all of the model parameters are unknown.', '1804.03392-2-3-3': 'Such is the case if one wished to target an interesting sky direction e.g. associated with a supernova remnant [CITATION] or a low-mass X-ray binary [CITATION], or perform an all-sky survey for isolated continuous-wave sources unassociated with known pulsars [CITATION].', '1804.03392-2-3-4': 'It is the latter type of search that is the subject of this paper.', '1804.03392-2-4-0': 'The additional challenge of a practical upper limit on the computational cost of all-sky searches has spurred the development of various sub-optimal but computationally-tractable hierarchical or semicoherent algorithms [CITATION].', '1804.03392-2-4-1': 'They share a common approach: the dataset (which for this example we assume is contiguous) with timespan [MATH] is partitioned into [MATH] segments, each with timespan [MATH].', '1804.03392-2-4-2': 'A fully-coherent matched filter search is then performed individually for each segment.', '1804.03392-2-4-3': 'Most methods then combine segments by incoherently summing the power from [MATH] filters, one from each segment, which together follow a consistent frequency evolution as dictated by the continuous-wave signal model.', '1804.03392-2-4-4': 'The phase evolution need not be continuous over the [MATH] filters, however; nor need the gravitational-wave amplitudes in each segment be consistent.', '1804.03392-2-4-5': 'This loss of complete signal self-consistency comes, however, with a computational benefit: while the computational cost of a fully-coherent matched filter search of the entire dataset scales as [MATH] with [MATH] a high power [MATH] to 6, the cost of a semicoherent method typically scales as [MATH] with [MATH] [CITATION].', '1804.03392-2-4-6': 'The strain sensitivities of a fully-coherent and semicoherent search typically scale as [MATH] and [MATH] respectively, with [MATH] [CITATION]; for the loss of a factor [MATH] in sensitivity, a semicoherent method is able to gain by being able to analyze large (e.g. [MATH] year) datasets, whereas a fully-coherent search would be computationally restricted to a much shorter (e.g. [MATH] year) subset.', '1804.03392-2-5-0': 'An important early advance in the development of semicoherent methods was the adaption of the Hough transform [CITATION], originally created to analyze tracks in bubble chamber photographs, to instead track the frequency evolution of a continuous gravitational-wave signal [CITATION].', '1804.03392-2-5-1': 'A number of variations of the Hough transform have been implemented, which map the signal track in the time-frequency plane to either its sky position at a fixed reference frequency and frequency derivative [CITATION], or conversely to its reference frequency and frequency derivative at a fixed sky position [CITATION].', '1804.03392-2-5-2': 'The detection statistic computed, the number count, sums either zero or one from each segment depending on whether the significance of a filter exceeds a set threshold.', '1804.03392-2-5-3': 'Some variations use short-duration ([MATH]s) segments and incoherently sum power above threshold from each segment; others analyze longer segments, and set a threshold on the [MATH]-statistic [CITATION] which computes the matched filter analytically maximized over the gravitational-wave amplitudes.', '1804.03392-2-5-4': 'Another modification is to weigh each segment by the antenna response function of the detector, and to sum these weights instead of zero or one [CITATION].', '1804.03392-2-6-0': 'Two semicoherent methods which use short-duration segments but which, unlike the Hough transform methods, sum power without thresholding are the StackSlide [CITATION] and PowerFlux [CITATION] methods.', '1804.03392-2-6-1': 'The StackSlide method builds a time-frequency plane, where each column represents a segment.', '1804.03392-2-6-2': 'For each choice of signal parameters, it "slides" each column up and down in frequency so that a signal with those parameters would follow a horizontal line, and then "stacks" (i.e. sums) the columns horizontally to accumulate the signal power over time for each frequency bin.', '1804.03392-2-6-3': '(Due to this intuitive representation of a semicoherent search method, the term StackSlide is often used to refer to semicoherent methods in general [CITATION].)', '1804.03392-2-6-4': "The PowerFlux method follows a similar methodology, and in addition weights the power from each segment by that segment's noise level and antenna response function, so that segments containing transient instrumental noise and/or where the response of the detector is weak are deweighted.", '1804.03392-2-6-5': 'A "loosely coherent" adaption to PowerFlux allows the degree of phase consistency imposed at the semicoherent stage to be controlled explicitly [CITATION].', '1804.03392-2-6-6': 'A third semicoherent method [CITATION] was developed based on the observance of global correlations between search parameters [CITATION] and uses longer segments analyzed with the [MATH]-statistic.', '1804.03392-2-6-7': 'A comprehensive comparison of many of the all-sky search methods described above is performed in [CITATION].', '1804.03392-2-7-0': 'Aside from developments in semicoherent search techniques, two other ideas have played an important role in the development of continuous gravitational-wave data analysis.', '1804.03392-2-7-1': 'First is the use of a parameter-space metric [CITATION], which is used to determine the appropriate resolution of the bank of template signals such that the mismatch, or fractional loss in signal-to-noise ratio between any signal present in the data and its nearest template, never exceeds a prescribed maximum.', '1804.03392-2-7-2': 'The metric of the [MATH]-statistic for continuous-wave signals was first studied rigorously in [CITATION].', '1804.03392-2-7-3': 'An approximate form of the metric was utilized in semicoherent search methods developed by [CITATION], and a related approximation was used in [CITATION].', '1804.03392-2-7-4': 'The latter approximation, however, lead to an underestimation of the number of required templates in the sky parameter space when analyzing long data stretches; an improved approximate metric developed in [CITATION] addresses this limitation.', '1804.03392-2-7-5': 'It was also later realized that a further approximation fundamental to the metric derivation - namely that the prescribed maximum mismatch (as measured by the metric) could be assumed small - generally does not hold under realistic restrictions on computational cost.', '1804.03392-2-7-6': 'This issue was addressed in [CITATION] which computed an empirical relation between the metric-measured mismatch and the true mismatch of the [MATH]-statistic.', '1804.03392-2-8-0': 'A second important idea is the borrowing of results from lattice theory [CITATION] to optimize the geometric placement of templates within the search parameter space, so as to fulfill the maximum prescribed mismatch criteria described above with the smallest possible density of templates [CITATION].', '1804.03392-2-8-1': 'Practical algorithms for generating template banks for continuous-wave searches, using both the parameter-space metric and optimal lattices, were proposed in [CITATION].', '1804.03392-2-8-2': 'An alternative idea studied in [CITATION] is to instead place templates at random, using the parameter-space metric only as a guide as to the relative density of templates; this idea has found utility in searches for radio [CITATION] and X-ray [CITATION] pulsars.', '1804.03392-2-9-0': 'The number of computations that must be performed during an all-sky search, even when utilizing an efficient semicoherent search method, remains formidable.', '1804.03392-2-9-1': 'For example, a recent all-sky search [CITATION] of data from the first Advanced LIGO observing run divided the data into [MATH] segments of timespan [MATH] hours, performed [MATH] matched-filtering operations per segment, and finally performed [MATH] incoherent summations to combine filter power from each segment.', '1804.03392-2-9-2': 'The total computational cost of the search was [MATH] CPU days, although this was distributed over [MATH] computers volunteered through the Einstein@Home distributed computing project [CITATION].', '1804.03392-2-9-3': 'Nevertheless, the significant number of filtering/incoherent summation operations that must be performed during a typical all-sky search emphasizes the need to optimize the construction of the template banks, and thereby minimize the computational cost of the search, as much as practicable.', '1804.03392-2-10-0': 'In this paper we present Weave, an implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-2-10-1': 'This implementation brings together, for the first time, several strands of previous research: the use of a semicoherent method to combine data segments analyzed with the [MATH]-statistic, combined with optimal template placement using the parameter-space metric of [CITATION] and optimal lattices [CITATION].', '1804.03392-2-10-2': 'After a review of relevant background information in Section [REF], the Weave implementation is presented in Section [REF].', '1804.03392-2-10-3': 'In Section [REF] we demonstrate that important behaviors of the Weave implementation can be modeled semi-analytically, thereby enabling characterization and optimization of a search setup without, in the first instance, the need to resort to time-consuming Monte-Carlo simulations.', '1804.03392-2-10-4': 'In Section [REF] we discuss ideas for further improvement and extension.', '1804.03392-2-11-0': '# Background', '1804.03392-2-12-0': 'This section presents background material pertaining to the continuous-wave signal model, parameter-space metric, and template bank generation.', '1804.03392-2-13-0': '## Continuous-wave signals', '1804.03392-2-14-0': 'The phase of a continuous-wave signal [MATH] at time [MATH] at the detector is given by, neglecting relativistic corrections [CITATION], [EQUATION]', '1804.03392-2-14-1': 'The first term on the right-hand side primarily encodes the loss of rotational energy of the neutron star as observed from the Solar System barycenter: [MATH] is the gravitational-wave frequency; and the spindowns [MATH], [MATH], etc. are the 1st-order, 2nd-order, etc. rates of change of the gravitational-wave frequency with time.', '1804.03392-2-14-2': 'All [MATH] parameters are given with respect to a reference time [MATH].', '1804.03392-2-14-3': 'The second term on the right-hand side describes the Doppler modulation of the gravitational waves due to the motion of an Earth-based detector: [MATH] is the detector position relative to the Solar System barycenter, thereby including both the sidereal and orbital motions of the Earth; and [MATH] is a unit vector pointing from the Solar System barycenter to the continuous-wave source.', '1804.03392-2-14-4': 'The value of [MATH] is chosen conservatively to be the maximum of [MATH] over the timespan of the analyzed data.', '1804.03392-2-15-0': 'Together the phase evolution parameters [MATH] parameterize the continuous-wave signal template; additional amplitude parameters [MATH] are analytically maximized over when computing the [MATH]-statistic [CITATION].', '1804.03392-2-15-1': 'In noise the [MATH]-statistic is a central [MATH] statistic with 4 degrees of freedom; when in the vicinity of a signal, the noncentrality parameter [MATH] of the noncentral [MATH] distribution scales as [MATH], where [MATH] is the gravitational-wave amplitude, [MATH] the amount of analyzed data, and [MATH] is the noise power spectral density in the vicinity of the signal frequency [MATH].', '1804.03392-2-16-0': '## Parameter-space metric', '1804.03392-2-17-0': 'The parameter-space metric [MATH] of the [MATH]-statistic is defined by a 2nd-order Taylor expansion of the noncentrality parameter: [EQUATION] with implicit summation over [MATH], and where [EQUATION]', '1804.03392-2-17-1': 'Here [MATH] is the noncentrality parameter of the [MATH]-statistic when perfectly matched to a signal with parameters [MATH], and [MATH] is the noncentrality parameter when computed at some mismatched parameters [MATH].', '1804.03392-2-17-2': 'The mismatch is defined to be [EQUATION]', '1804.03392-2-17-3': 'A very useful approximation to Eq. [REF] is the phase metric [CITATION]; it discards the amplitude modulation of the signal, and thereby the dependence on the known parameters [MATH], retaining only dependence on the phase evolution parameters: [EQUATION]', '1804.03392-2-18-0': '## Optimal template placement', '1804.03392-2-19-0': 'Template placement using optimal lattices is an example of a sphere covering [CITATION]: a collection of lattice-centered [MATH]-dimensional spheres of equal radius.', '1804.03392-2-19-1': 'The radius is chosen to be the smallest value that satisfies the property that each point in the [MATH]-dimensional parameter space is contained in at least one sphere.', '1804.03392-2-19-2': 'A lattice where the ratio of the volume of the sphere to the volume of a lattice cell is minimized generates a minimal sphere covering, i.e. the minimal number of points required to cover a parameter space, which is exactly the property desired for template banks.', '1804.03392-2-19-3': '(For example, in two dimensions the minimal sphere covering is generated by the hexagonal lattice.)', '1804.03392-2-19-4': 'We identify the covering spheres with the metric ellipsoids [MATH], where [MATH] is the prescribed maximum; it follows that the radii of the covering spheres is [MATH].', '1804.03392-2-19-5': 'A matrix transform [MATH] can then be constructed [CITATION] which takes integers in [MATH] to template parameters [MATH] to generate the template bank: [EQUATION] where [MATH] is a function of the metric [MATH], and [MATH] is particular to the lattice being used.', '1804.03392-2-19-6': 'If [MATH] is a lower triangular matrix, an efficient algorithm [CITATION] can be found for generating the template bank.', '1804.03392-2-20-0': '## Reduced supersky metric', '1804.03392-2-21-0': 'In order for Eq. [REF] to preserve the sphere covering property, however, it must be independent of the template parameters [MATH].', '1804.03392-2-21-1': 'Since [MATH] is a function of the metric, we require a metric which is also independent of [MATH]: [MATH].', '1804.03392-2-21-2': 'The phase metric of Eq. [REF] is independent of the frequency and spindown parameters [MATH], but retains a dependence on sky position parameters, e.g. [MATH] in terms of right ascension [MATH] and declination [MATH].', '1804.03392-2-21-3': 'The question of how to derive a useful metric which is independent of the sky position parameters, i.e. [MATH], has stimulated numerous approaches [CITATION].', '1804.03392-2-21-4': 'In [CITATION], a useful [MATH] is derived through the following procedure: [(i)]', '1804.03392-2-22-0': '[MATH] is expressed in terms of the 3 components of [MATH], instead of 2 parameters such as [MATH].', '1804.03392-2-22-1': 'The 3 components of [MATH] are taken to be independent; geometrically this is equivalent to embedding [MATH] into a 3-dimensional supersky parameter space, instead of being restricted to the 2-sphere defined by [MATH].', '1804.03392-2-22-2': 'In the supersky parameter space, [MATH] is independent of the sky position parameters, i.e. we have the desired [MATH], but with the addition of a 3rd unwanted parameter-space dimension.', '1804.03392-2-23-0': 'A linear coordinate transform [MATH] is derived which satisfies: [MATH] is diagonal in the sky position parameters [MATH], i.e. [MATH]; [MATH]; and [MATH].', '1804.03392-2-23-1': 'The last two properties imply that the metric ellipsoids are much longer along the [MATH] axis than along the [MATH] and [MATH] axes.', '1804.03392-2-23-2': 'In computing the coordinate transform, use is made of the well-known correlation between the sky and frequency/spindown parameters of the continuous-wave signal [CITATION].', '1804.03392-2-23-3': 'The correlations arise because, on sufficiently short timescales, the change in phase due to the cyclical sidereal and orbital motions of the Earth may be Taylor expanded as linear, quadratic, etc. changes in phase with time, and thereby are equivalent to changes in the frequency ([MATH]), 1st spindown ([MATH]), etc. parameters.', '1804.03392-2-24-0': 'Since, in the new coordinates [MATH] the mismatch [MATH] is only weakly dependent on [MATH], a useful approximate metric [MATH] is found by discarding the [MATH] dimension.', '1804.03392-2-24-1': 'Geometrically this corresponds to projecting the 3-dimensional supersky parameter space and metric onto the 2-dimensional [MATH]-[MATH] plane.', '1804.03392-2-24-2': 'The resultant reduced supersky parameter-space metric [MATH] and associated coordinates [MATH] has reduced the sky parameter space dimensionality back to 2, while retaining the property that [MATH] is parameter-independent.', '1804.03392-2-25-0': '# Weave Implementation', '1804.03392-2-26-0': 'This section describes the Weave implementation of the semicoherent search method, a schematic of which is shown in Figure [REF].', '1804.03392-2-26-1': 'The implementation is freely available as part of the LALSuite [CITATION] gravitational-wave data analysis library.', '1804.03392-2-27-0': '## Overview', '1804.03392-2-28-0': 'In step 1 the user runs a precursor program lalappsWeaveSetup, which takes as an argument a list of [MATH] segments [MATH] into which the dataset is to be partitioned.', '1804.03392-2-28-1': 'The program computes in step 2 the [MATH] coherent parameter-space metrics [MATH] used to construct template banks within each segment, and the semicoherent parameter-space metric [MATH] used to incoherently combine segments.', '1804.03392-2-28-2': 'The metrics are written to a setup file in the FITS format [CITATION].', '1804.03392-2-28-3': 'Due to the numerical ill-conditionedness of the parameter-space metric [CITATION], this computation involves a bootstrapping process, whereby successively better-conditioned iterations of the supersky metric are computed, before then computing the reduced supersky metric as outlined in Section [REF].', '1804.03392-2-28-4': 'Since this bootstrapping process can be time-consuming for large [MATH], and may give slightly different results on different computer hardware, precomputing the metrics both saves computing time and adds robustness against numerical errors.', '1804.03392-2-28-5': 'Note that, by Eq. [REF], the sky components of the metrics will scale with [MATH]; since its value depends on the search frequency parameter space, which is not known by lalappsWeaveSetup, an arbitrary fiducial value [MATH] is used, and the sky components of the metrics are later rescaled by [MATH].', '1804.03392-2-29-0': 'In step 3 the user runs the main search program lalappsWeave.', '1804.03392-2-29-1': 'The principle arguments to this program are the setup file output by lalappsWeaveSetup, the search parameter space, and the prescribed maximum mismatches [MATH] and [MATH] for the coherent and semicoherent template banks respectively.', '1804.03392-2-29-2': 'The frequency and spindown parameter space is specified by ranges [MATH], where [MATH], 1, etc. as required.', '1804.03392-2-29-3': 'The sky search parameter space may be specified either as a rectangular patch in right ascension and declination [MATH], or alternatively partitioned into [MATH] patches containing approximately equal number of templates (see Appendix [REF]), and a patch selected by an index [MATH], [MATH].', '1804.03392-2-29-4': 'In step 4 various preparatory tasks are performed, such as loading the gravitational-wave detector data into memory, before beginning the main search loop.', '1804.03392-2-30-0': 'The main search loop of a semicoherent search method may be structured in two complementary ways, which differ in the memory each requires to store intermediate results: [(i)]', '1804.03392-2-31-0': 'The semicoherent template bank [MATH] is stored in memory, and the [MATH] segments are processed in sequence.', '1804.03392-2-31-1': 'For each segment [MATH], every coherent template [MATH] is mapped back to the semicoherent template bank, i.e. [MATH].', '1804.03392-2-31-2': 'Because the semicoherent template bank must track the continuous-wave signal over a larger timespan [MATH] than the coherent template banks, it will contain a greater density of templates; the ratio of semicoherent to coherent template bank densities is the refinement factor [MATH] [CITATION].', '1804.03392-2-31-3': 'It follows that the mapping [MATH] will be one-to-many.', '1804.03392-2-32-0': 'As the [MATH] segments are processed, any semicoherent detection statistic associated with [MATH] is then updated based on the corresponding coherent detection statistic associated with [MATH].', '1804.03392-2-32-1': 'For example, it is common to compute the summed [MATH]-statistic [MATH]; here we would then have [MATH].', '1804.03392-2-32-2': 'Once every segment has been processed, computed [MATH] for every [MATH] will exist in memory.', '1804.03392-2-32-3': 'The memory usage of the main search loop will therefore be proportional to the number of semicoherent templates [MATH], where [MATH] is the average number of templates in a coherent template bank.', '1804.03392-2-33-0': 'The [MATH] coherent template banks [MATH] are stored in memory, and the semicoherent template bank is processed in sequence.', '1804.03392-2-33-1': 'Each semicoherent template [MATH] is mapped back the coherent template bank in each segment [MATH], i.e. [MATH]; since [MATH] in each segment this mapping will be many-to-one.', '1804.03392-2-33-2': 'With these [MATH] mappings in hand, the semicoherent detection statistics may be immediately computed in full, e.g. [MATH].', '1804.03392-2-33-3': 'The memory usage of the main search loop will therefore be proportional to [MATH].', '1804.03392-2-34-0': 'For the parameter-space metric for all-sky searches, [MATH] [CITATION], and therefore the latter structuring given above will have the lower memory requirement; the Weave implementation uses this structuring of the main search loop.', '1804.03392-2-34-1': 'The semicoherent template bank [MATH] is generated one template at a time using the algorithm described in [CITATION].', '1804.03392-2-34-2': 'For each coherent template bank, an efficient lookup table [CITATION] is constructed for the mapping [MATH].', '1804.03392-2-35-0': 'We note an important distinction between the definition of the Weave template banks and the traditional StackSlide picture of a semicoherent search method.', '1804.03392-2-35-1': 'In the latter picture, the frequency and spindown template banks of each segment are defined with respect to individual reference times [MATH], typically the midtime of each segment.', '1804.03392-2-35-2': 'When combining segments, therefore, the frequency and spindown parameters of each coherent template must be adjusted so as to bring the parameters of all segments to a common reference time [MATH]; this is the "sliding" step.', '1804.03392-2-35-3': 'The Weave implementation, however, defines the frequency and spindown templates banks of all segments at the same reference time [MATH], which is also the reference time of the semicoherent bank.', '1804.03392-2-35-4': 'Consequentially, there is no analogy to the "sliding" step of StackSlide.', '1804.03392-2-35-5': 'Instead, the orientation of the metric ellipses in the [MATH] plane changes from segment to segment, as illustrated in Figure [REF].', '1804.03392-2-35-6': 'As the absolute difference [EQUATION] between the midtime of each segment [MATH] and [MATH] increases, both the extent of the ellipses in [MATH] and the correlation between [MATH] and [MATH] also increase.', '1804.03392-2-36-0': 'Steps 5-16 comprise the main search loop; which performs two key tasks: the computation and output of the detection statistics over the semicoherent template bank (steps 5, 6, and 12-17), and the management of an internal cache of required detection statistics computed on each coherent template bank (steps 7-11).', '1804.03392-2-36-1': 'These two tasks are described more fully in the following two sections, and with reference to a diagram of their operation in Figure [REF].', '1804.03392-2-37-0': 'In this section and in Figure [REF] we focus for simplicity on the computation of the semicoherent [MATH]-statistics [MATH] and [MATH].', '1804.03392-2-37-1': 'The computation of other detection statistics is also possible: in particular a family of Bayesian statistics has been developed which weigh the likelihood of a continuous wave signal against that of an instrumental line which appears in all segments [CITATION], or a transient instrumental line which appears only in one segment [CITATION].', '1804.03392-2-37-2': 'Computation of the former statistic, denoted [MATH], is also illustrated in Figure [REF]; it takes as input the multi-detector [MATH] which uses data from all gravitational-wave detectors, as well as the per-detector [MATH] which are computed from each detector [MATH] individually.', '1804.03392-2-38-0': '## Computation of semicoherent statistics', '1804.03392-2-39-0': 'In steps 5 and 16 (Figure [REF]), the main loop of the search method generates successive points [MATH] in the semicoherent template bank.', '1804.03392-2-39-1': 'An example of such a point is indicated in Figure [REF].', '1804.03392-2-39-2': 'Next, in steps 6 and 13, each segment [MATH] is visited and the mapping [MATH] is performed.', '1804.03392-2-39-3': 'The mapping used by Weave is nearest-neighbor interpolation: the [MATH] is expressed in the coherent metric coordinates of the [MATH]th segment, and the nearest (with respect to the metric) coherent template in the respective bank [MATH] is determined.', '1804.03392-2-39-4': 'If the template bank is constructed on a lattice, efficient algorithms exist for determining the nearest point [CITATION].', '1804.03392-2-39-5': 'In Figure [REF], example nearest coherent templates are labeled [MATH], [MATH], and [MATH].', '1804.03392-2-40-0': 'As each nearest point is determined, the coherent [MATH]-statistic in the respective segment is computed (steps 7-11, see Section [REF]), and the value of the semicoherent statistic [MATH] is updated (step 12).', '1804.03392-2-40-1': 'Once all segments have been processed (step 13), additional semicoherent statistics such as [MATH] are computed (step 14), and a candidate comprising the signal parameters together with the computed semicoherent statistics is added (step 15) to one or more toplists which ranks each candidate by a chosen semicoherent statistic.', '1804.03392-2-40-2': 'The size of the toplists is generally of a fixed user-determined size so that only a fixed number of the most promising candidates will be returned.', '1804.03392-2-41-0': 'Once the semicoherent template bank is exhausted (step 16) the toplists are written to an output file in the FITS format, and the search concludes (step 17).', '1804.03392-2-42-0': '## Management of cache of coherent statistics', '1804.03392-2-43-0': 'It is important that the main search loop minimizes its memory usage as much as possible.', '1804.03392-2-43-1': 'Even though in Section [REF] we chose a structuring of the main search loop so as to reduce memory usage, a naive implementation which stores [MATH] coherent statistics would still require a prohibitive amount of memory, given that both [MATH] and [MATH] are typically large.', '1804.03392-2-43-2': 'We therefore implement a per-segment cache which stores only those coherent statistics associated with coherent templates [MATH] accessible from the unprocessed portion of the semicoherent template bank via the mapping [MATH].', '1804.03392-2-43-3': 'Put another way, if a [MATH] can no longer be mapped to by any [MATH] remaining in [MATH], then [MATH] can be safely removed from the cache.', '1804.03392-2-44-0': 'In order to devise a cache management algorithm with the above desired properties, we first define an operator called relevance, denoted [MATH].', '1804.03392-2-44-1': 'The relevance operates on both coherent and semicoherent templates, and should satisfy the following property: [EQUATION]', '1804.03392-2-44-2': 'A definition of [MATH] satisfying this property is derived as follows.', '1804.03392-2-45-0': 'First, take any coherent template (e.g. [MATH] in Figure [REF]) and surround it by its metric ellipsoid at mismatch [MATH].', '1804.03392-2-45-1': 'Then surround the metric ellipsoid in turn by its bounding box, the smallest coordinate box which contains the ellipsoid [CITATION]; the metric ellipse bounding box centered on [MATH] is also shown in Figure [REF].', '1804.03392-2-45-2': 'Now, transform the bounding box into the semicoherent parameter space; practically this may be achieved by expressing the coordinates of each vertex of the bounding box in the semicoherent metric coordinates.', '1804.03392-2-45-3': 'See Figure [REF] for the transformed bounding box of [MATH] in the semicoherent parameter space, which is centered on [MATH].', '1804.03392-2-46-0': 'Note that, by definition, any semicoherent template [MATH] outside of the transformed bounding box centered on [MATH] cannot map to [MATH] under [MATH].', '1804.03392-2-46-1': 'Thus, to determine whether [MATH] is accessible by [MATH], we can compute whether [MATH] is within the transformed bounding box of [MATH].', '1804.03392-2-46-2': 'To be conservative, however, we also surround [MATH] by its bounding box as shown in Figure [REF], and instead compute whether the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-2-47-0': 'To simplify the bounding box intersection calculation, we compare just the coordinates of the bounding boxes of [MATH] and [MATH] in one dimension; for reasons that will soon be apparent, we choose the lowest-dimensional coordinate, [MATH].', '1804.03392-2-47-1': 'First, we define the relevance [MATH] for both coherent and semicoherent templates: [EQUATION]', '1804.03392-2-47-2': 'We now compute [MATH] and [MATH]; in Figure [REF], [MATH] is the [MATH] coordinate of the right-most edge of the transformed bounding box of [MATH], and [MATH] is the [MATH] coordinate of the left-most edge of the bounding box of [MATH].', '1804.03392-2-47-3': 'In this example, [MATH], and it follows from the definition of [MATH] in Eqs. [REF] that the bounding boxes of [MATH] and [MATH] cannot intersect.', '1804.03392-2-48-0': 'On the other hard, let us choose another coherent template [MATH], and examine its relevance [MATH]; here we have [MATH] (see Figure [REF]).', '1804.03392-2-48-1': 'From the simplified bounding box intersection calculation, we conclude that the bounding boxes of [MATH] and [MATH] could potentially intersect, since at least in the [MATH] dimension the bounding boxes overlap (although in this example the bounding boxes do not overlap in the [MATH] dimension).', '1804.03392-2-49-0': 'Finally, if for some [MATH] we have [MATH], then this condition is guaranteed to remain true for all remaining [MATH] in the template bank.', '1804.03392-2-49-1': 'This is simply a consequence of the algorithm used to generate the semicoherent template bank [CITATION], which operates as follows: first, values of [MATH] are generated in a constant range [MATH]; then, for each value of [MATH], values of [MATH] are generated in ranges [MATH] dependent on [MATH], and so on.', '1804.03392-2-49-2': 'It follows that the value of [MATH] can only increase during the generation of the semicoherent template bank, and since [MATH] is defined in terms of [MATH], it too can only increase.', '1804.03392-2-50-0': 'To summarize, the relevance operator [MATH] defined by Eqs. [REF] satisfies the desired property given by Eq. [REF].', '1804.03392-2-50-1': 'In Figure [REF], since [MATH], the cache management algorithm would discard any coherent statistics associated with [MATH] from memory, since they cannot be accessed by [MATH] nor any remaining semicoherent template.', '1804.03392-2-50-2': 'On the other hard, the algorithm would retain any coherent statistics associated with [MATH], since they could still be needed for future semicoherent templates; indeed in Figure [REF] it is clear that the next semicoherent template in the bank, labeled [MATH], could require coherent statistics associated with [MATH], since the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-2-51-0': 'The cache management algorithm described above is implemented in the main search loop in steps 7-11 (Figure [REF]).', '1804.03392-2-51-1': 'In step 7 the cache is interrogated for a required [MATH]-statistic value [MATH]: if it is in the cache, it is retrieved and utilized (step 8), otherwise it is computed and inserted into the cache (step 9).', '1804.03392-2-51-2': 'In the latter case, the cache is also checked to see if any cache items can be discarded.', '1804.03392-2-51-3': 'Starting with step 10, cache items indexed by [MATH] are retrieved in order of ascending [MATH].', '1804.03392-2-51-4': 'If [MATH], the cache items are discarded (step 11).', '1804.03392-2-51-5': 'Only one cache item is removed at any one time, and therefore the memory usage of the cache will either remain constant, or increase by one item per main search loop iteration.', '1804.03392-2-51-6': 'The cache is implemented using two data structures [CITATION]: a binary heap to rank cache items by [MATH], and a hash table to find cache items indexed by [MATH].', '1804.03392-2-52-0': '# Models of Weave Behavior', '1804.03392-2-53-0': 'This section presents semi-analytic models of the Weave implementation.', '1804.03392-2-53-1': 'It greatly facilitates the practical usage of any search method if its behavior can be characterized a priori as much as possible using a computationally-cheap model.', '1804.03392-2-53-2': 'For example, a model of the distribution of [MATH]-statistic mismatches (Section [REF]) permits the estimation of the sensitivity of a particular search setup [CITATION] which in turn allows the setup to be optimized so as to maximize sensitivity [CITATION].', '1804.03392-2-53-3': 'Similarly, models of the number of coherent and semicoherent templates (Section [REF]) and computational cost (Section [REF]) allow the parameters of the optimal search setup to be estimated [CITATION].', '1804.03392-2-53-4': 'The memory usage (Section [REF]) and input data bandwidth (Section [REF]) required by the implementation are also important properties when implementing a search pipeline.', '1804.03392-2-54-0': 'Each model presented in this section is implemented as an Octave [CITATION] script, and is freely available as part of the OctApps [CITATION] script library.', '1804.03392-2-55-0': '## Distribution of [MATH]-statistic mismatches', '1804.03392-2-56-0': 'The distribution of the mismatch between the [MATH]-statistic computed at an exact signal location, and at the nearest point in the Weave semicoherent template bank, gives an idea of the expected loss in signal-to-noise ratio due to the necessary coarseness of the template bank.', '1804.03392-2-56-1': 'Figure [REF] plots the predicted means and standard deviations of Weave [MATH]-statistic mismatch distributions, against their measured values, for a variety of setups given in Table [REF].', '1804.03392-2-56-2': 'The distributions were measured using software injection studies, where relatively strong ([MATH]) simulated signals are added to Gaussian-distributed noise and then searched for using lalappsWeave.', '1804.03392-2-57-0': 'The predicted means and standard deviations are from the model presented in [CITATION], and are generally conservative: Figure [REF] shows that the model generally overestimates the mean [MATH]-statistic mismatch by [MATH] (Figure [REF]) to [MATH] (Figure [REF]); and the predicted standard deviations imply slightly broader distributions than were measured.', '1804.03392-2-57-1': 'As explored in [CITATION], the relationship between the maximum mismatches of the coherent and semicoherent template banks (which are inputs to lalappsWeave) and the [MATH]-statistic mismatch distribution (which is output by lalappsWeave) is difficult to model when the former are large e.g. [MATH].', '1804.03392-2-58-0': 'In addition, an optimization implemented in Weave but not accounted for in the model of [CITATION] complicates the picture: the coherent and semicoherent template banks are constructed to have equally-spaced templates in the frequency parameter [MATH].', '1804.03392-2-58-1': 'This permits (in step 9 of Figure [REF]) the simultaneous computation of a series of [MATH] values at equally-spaced values of [MATH] across the frequency parameter space, which can be performed efficiently using Fast Fourier Transform-based algorithms (see Section [REF]).', '1804.03392-2-58-2': 'The construction of equal-frequency-spacing coherent and semicoherent template banks is performed by first constructing each bank independently, and then reducing the frequency spacing in all banks to that of the smallest frequency spacing in any bank.', '1804.03392-2-58-3': 'This construction will always reduce the maximum possible mismatch in each grid, but never increase it, and so we would expect the mean [MATH]-statistic mismatch measured by Weave to be smaller than that predicted by the model of [CITATION].', '1804.03392-2-59-0': 'The model of [CITATION] is implemented in the OctApps script WeaveFstatMismatch.m.', '1804.03392-2-60-0': '## Number of templates', '1804.03392-2-61-0': 'Since the Weave coherent and semicoherent template banks are constructed using lattices (see Section [REF]), the number of templates in each is estimated starting from the formula [CITATION] [EQUATION] where [MATH] is the volume of the [MATH]-dimensional parameter space, [MATH] the parameter-space metric, and [MATH] the maximum mismatch.', '1804.03392-2-61-1': 'The normalized thickness [MATH] is a property of the particular lattice used to generate the template bank [CITATION].', '1804.03392-2-62-0': 'The parameter-space volume is given explicitly by the following expressions: [EQUATION]', '1804.03392-2-62-1': 'Here, [MATH] is the vector whose components are the extents of the bounding box of [MATH] in each dimension; it is used to ensure that the volume of the parameter space in each dimension is not smaller than the extent of a single template.', '1804.03392-2-62-2': 'In Eq. [REF], the volume of the sky parameter space may be specified either by a rectangular patch [MATH], or by the number [MATH] of equal-size sky patches (see Section [REF]).', '1804.03392-2-63-0': 'Finally, the total number of coherent and semicoherent templates, [MATH] and [MATH] respectively, are given by: [EQUATION]', '1804.03392-2-63-1': 'The numerical prefactor on the right-hand side of Eq. [REF] is chosen to better match [MATH] to the number of coherent templates actually computed by lalappsWeave: the coherent parameter space is augmented with additional padding along its boundaries to ensure that it encloses the semicoherent parameter space, i.e. that it includes a nearest neighbor for every [MATH].', '1804.03392-2-64-0': 'Equations [REF] and [REF] are used to predict the number of templates computed by lalappsWeave for a variety of search setups detailed in Table [REF].', '1804.03392-2-64-1': 'Figure [REF] plots the predicted [MATH] and [MATH] against the values measured by running lalappsWeave.', '1804.03392-2-64-2': 'Reasonable agreement is achieved between predicted and measured [MATH] (Figure [REF]): while Eq. [REF] sometimes underestimates the number of coherent templates, it rarely does so by more than a factor of a few.', '1804.03392-2-64-3': 'Better agreement is seen between predicted and measured [MATH] (Figure [REF]).', '1804.03392-2-65-0': 'Equations [REF] and [REF] are implemented in the OctApps script WeaveTemplateCount.m.', '1804.03392-2-66-0': '## Computational cost', '1804.03392-2-67-0': 'The total computational cost [MATH] of a particular search setup may be modeled in terms of the number of coherent [MATH] and semicoherent [MATH] templates (see Section [REF]), the number of segments [MATH] and number of detectors [MATH].', '1804.03392-2-67-1': 'Following [CITATION] we write [EQUATION] where [MATH] and [MATH] denote the computational cost of the coherent and semicoherent stages of the search method respectively, and [MATH] denotes any unmodeled computational costs.', '1804.03392-2-68-0': 'The computational cost model takes as input fundamental timing constants which give the time taken to complete certain fundamental computations.', '1804.03392-2-68-1': 'Their values are highly dependent on various properties of the computer hardware used to run lalappsWeave, such as the processor speed and cache sizes, as well as what other programs were using the computer hardware at the same time as lalappsWeave.', '1804.03392-2-68-2': 'Some values are also specific to the search setups detailed in Table [REF].', '1804.03392-2-68-3': 'For the interest of the reader, Table [REF] lists representative values of the fundamental timing constants obtained on a particular computer cluster.', '1804.03392-2-69-0': 'The coherent cost [MATH] is simply the cost of computing the [MATH]-statistic (step 9 of Figure [REF]): [EQUATION]', '1804.03392-2-69-1': 'The fundamental timing constant [MATH] gives the time taken to compute the [MATH]-statistic per template and per detector, and is further described in [CITATION].', '1804.03392-2-69-2': 'Its value depends primarily upon the range of the frequency parameter space [MATH], the coherent segment length [MATH], and the algorithm used to compute the [MATH]-statistic.', '1804.03392-2-69-3': 'Choices for the latter are: the resampling algorithm [CITATION], which computes the [MATH]-statistic over a wide band of frequencies efficiently using the Fast Fourier Transform, and is generally used to performing an initial wide-parameter-space search; and the demodulation algorithm of [CITATION], which uses a Dirichlet kernel to compute the [MATH]-statistic more efficiently at a single frequency or over a narrow frequency band, and is therefore used to perform follow-up searches of localized parameter spaces around interesting candidates.', '1804.03392-2-69-4': 'The additional cost of managing the cache of computed [MATH]-statistic values (steps 8, 10, and 11) is amortized into [MATH].', '1804.03392-2-70-0': 'The semicoherent cost [EQUATION] has a number of components: [(i)] [MATH] is the cost of iterating over the semicoherent template bank (steps 5 and 16 of Figure [REF]); [MATH] is the cost of finding the nearest templates in the coherent template banks (step 6 and 13) and of interrogating the cache of computed [MATH]-statistic values (step 7); [MATH] is the cost of computing [MATH] and, if required, [MATH] (step 12); [MATH] is the cost of computing [MATH] (step 14); [MATH] is the cost of computing [MATH], if required (step 14); and [MATH] is the cost of adding candidates to toplists (step 15).', '1804.03392-2-71-0': 'These components of [MATH] are further defined in terms of [MATH], [MATH], [MATH], and various fundamental timing constants (see Table [REF]) as follows: [EQUATION]', '1804.03392-2-71-1': 'Figure [REF] compares the computational cost model of Eqs. [REF]-[REF] against the measured computational cost of lalappsWeave (see Table [REF]), using the search setups detailed in Table [REF].', '1804.03392-2-71-2': 'The total computational cost of lalappsWeave is generally well-modeled (Figure [REF]) and the unmodeled component of the measured computational cost is low (Figure [REF]).', '1804.03392-2-71-3': 'The coherent computational cost [MATH] of Eq. [REF] and the components of the semicoherent cost [MATH] of Eq. [REF] are also in good agreement (Figures [REF]-[REF]).', '1804.03392-2-72-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveRunTime.m.', '1804.03392-2-73-0': '## Memory usage', '1804.03392-2-74-0': 'The memory usage [MATH] of lalappsWeave is modeled by [EQUATION]', '1804.03392-2-74-1': 'The first term on the right-hand side, [MATH], is the memory usage of the [MATH]-statistic algorithm (which includes the gravitational-wave detector data) and is further described in [CITATION].', '1804.03392-2-74-2': 'The second term, [MATH], is the memory usage of the cache of computed [MATH]-statistic values, and is further given by [EQUATION] where [MATH] is the maximum size of the cache (across all segments), and [MATH] MiB is the memory required to store one [MATH] value as a 4-byte single precision floating-point number.', '1804.03392-2-74-3': 'The maximum cache [MATH] cannot easily be predicted from first principles, i.e. given the search setup, parameter space, and other input arguments to lalappsWeave.', '1804.03392-2-74-4': 'Instead, it is measured by running lalappsWeave in a special mode which simulates the performance of the cache but without computing any [MATH]-statistic or derived values; essentially it follows Figure [REF] but with the first part of step 9, step 12, and step 14 omitted.', '1804.03392-2-75-0': 'Figure [REF] plots the predicted memory usage of Eqs. [REF] and [REF] against the measured memory usage of lalappsWeave, using the search setups detailed in Table [REF].', '1804.03392-2-75-1': 'The [MATH]-statistic is computed using both the resampling and demodulation algorithms: in the former case, both [MATH] and [MATH] are computed, thereby triggering the first case in Eq. [REF]; in the latter case, only [MATH] is computed, thereby triggering the second case in Eq. [REF].', '1804.03392-2-75-2': 'Good agreement between predicted and measured memory usage is seen for both algorithms.', '1804.03392-2-76-0': 'Equations [REF] and [REF] are also implemented in the OctApps script WeaveRunTime.m.', '1804.03392-2-77-0': '## Input data bandwidth', '1804.03392-2-78-0': 'Our final Weave model concerns what bandwidth of the input gravitational-wave detector data is required to search a given frequency range.', '1804.03392-2-78-1': 'For most continuous-wave search pipelines, short (typically 1800 s) contiguous segments of gravitational-wave strain data are Fourier transformed, and the resulting complex spectra stored as Short Fourier Transform (SFT) files.', '1804.03392-2-78-2': 'A continuous-wave search of a large frequency parameter space will generally be divided into smaller jobs, with each job searching a smaller partition of the whole frequency parameter space.', '1804.03392-2-78-3': 'Each job therefore requires that only a small bandwidth out of the full SFT spectra be read into memory.', '1804.03392-2-79-0': 'Given an input frequency parameter space [MATH] and spindown parameter space [MATH], predicting the bandwidth of the SFT spectra required by lalappsWeave proceeds in several steps.', '1804.03392-2-79-1': 'First, the input parameter spaces are augmented to account for extra padding of the Weave template banks: [EQUATION] where [MATH] and [MATH] are empirically chosen.', '1804.03392-2-79-2': 'Next, the maximum frequency range [MATH] is found by evolving the frequency-spindown parameter space [MATH] from the reference time [MATH] to the start and end times of each segment, [MATH] and [MATH] respectively: [EQUATION]', '1804.03392-2-79-3': 'Finally, the SFT bandwidth [MATH] of the SFT spectra which is required by lalappsWeave is given by: [EQUATION]', '1804.03392-2-79-4': 'The [MATH] enlarges [MATH] to account for the maximum frequency-dependent Doppler modulation of a continuous-wave signal due to the sidereal and orbital motions of the Earth, and is given by [EQUATION] where [MATH] is the speed of light, [MATH] is the Earth-Sun distance and [MATH] the radius of the Earth.', '1804.03392-2-79-5': 'Additional padding of [MATH] is also required for use by the chosen [MATH]-statistic algorithm, and is given by [MATH] [CITATION].', '1804.03392-2-80-0': 'Figure [REF] compares the model of Eqs. [REF]-[REF] against the behavior of lalappsWeave when run with the search setups detailed in Table [REF].', '1804.03392-2-80-1': 'Note that the model satisfies [EQUATION] i.e. all circles plotted in Figure [REF] are below the horizontal axis, and [EQUATION] i.e. all crosses plotted in Figure [REF] are above the horizontal axis.', '1804.03392-2-80-2': 'The model is therefore conservative, i.e. it may predict a slightly larger SFT bandwidth than required, but should never predict a smaller SFT bandwidth, which would cause a fatal error in lalappsWeave.', '1804.03392-2-80-3': "The model is generally more conservative at higher frequencies, where the Doppler modulation due to the Earth's motion is higher.", '1804.03392-2-81-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveInputSFTBand.m.', '1804.03392-2-82-0': '# Discussion', '1804.03392-2-83-0': 'This paper details the Weave implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-2-83-1': 'It focuses on all-sky surveys for isolated continuous-wave sources, for which the parameter space is the sky position and frequency evolution of the source.', '1804.03392-2-83-2': 'We note, however, that the implementation is in fact indifferent to the parameter space being searched, as long as the relevant constant parameter-space metric is available.', '1804.03392-2-83-3': 'The implementation could therefore be adapted to search other parameter spaces for continuous-wave sources such as known low-mass X-ray binaries, for which the parameter space includes the evolution parameters of the binary orbit, using the metric of [CITATION].', '1804.03392-2-84-0': 'There is scope to improve the semi-analytic models of the behavior of lalappsWeave presented in Section [REF].', '1804.03392-2-84-1': 'In particular, a more accurate model of the distribution of [MATH]-statistic mismatches than that presented in Section [REF] would allow the sensitivity of a search to be more accurately estimated without resorting to software injection studies.', '1804.03392-2-84-2': 'The memory model of Section [REF] would also be improved if the maximum cache size [MATH] could be predicted from first principles.', '1804.03392-2-85-0': 'In a forthcoming paper [CITATION] we plan to more fully characterize the performance of the Weave implementation, and compare it to an implementation of the method of [CITATION] using a mock data challenge.', '1804.03392-2-86-0': 'We thank Bruce Allen and Heinz-Bernd Eggenstein for valuable discussions.', '1804.03392-2-86-1': 'KW is supported by ARC CE170100004.', '1804.03392-2-86-2': 'Numerical simulations were performed on the Atlas computer cluster of the Max Planck Institute for Gravitational Physics.', '1804.03392-2-86-3': 'This paper has document number LIGO-P1800074-v3.', '1804.03392-2-87-0': '# Properties of equal-area sky patches', '1804.03392-2-88-0': 'The search program lalappsWeave allows the sky search parameter space to be partitioned into [MATH] patches, and a patch selected by an index [MATH].', '1804.03392-2-88-1': 'Tests of this feature found that, provided [MATH] (the number of templates with just one patch), the variation in the number of templates between patches [MATH] is generally small and well-approximated by [EQUATION]', '1804.03392-2-88-2': 'The ratio [MATH] of the number of templates in all [MATH] patches to the number of templates with just one patch is generally [MATH]%.', '1804.03392-2-88-3': 'The union of all templates in a set of [MATH] patches also faithfully reproduces the unpartitioned template bank, i.e. with just one patch.'} | [['1804.03392-1-74-0', '1804.03392-2-75-0'], ['1804.03392-1-74-1', '1804.03392-2-75-1'], ['1804.03392-1-74-2', '1804.03392-2-75-2'], ['1804.03392-1-32-0', '1804.03392-2-33-0'], ['1804.03392-1-32-1', '1804.03392-2-33-1'], ['1804.03392-1-32-2', '1804.03392-2-33-2'], ['1804.03392-1-32-3', '1804.03392-2-33-3'], ['1804.03392-1-44-0', '1804.03392-2-45-0'], 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1804.03392 | {'1804.03392-3-0-0': 'All-sky surveys for isolated continuous gravitational waves present a significant data-analysis challenge.', '1804.03392-3-0-1': 'Semicoherent search methods are commonly used to efficiently perform the computationally-intensive task of searching for these weak signals in the noisy data of gravitational-wave detectors such as LIGO and Virgo.', '1804.03392-3-0-2': 'We present a new implementation of a semicoherent search method, Weave, that for the first time makes full use of a parameter-space metric to generate banks of search templates at the correct resolution, combined with optimal lattices to minimize the required number of templates and hence the computational cost of the search.', '1804.03392-3-0-3': 'We describe the implementation of Weave and associated design choices, and characterize its behavior using semi-analytic models.', '1804.03392-3-1-0': '# Introduction', '1804.03392-3-2-0': 'The detections of short-duration gravitational-wave events from the inspiral and merger of binary black holes [CITATION] and binary neutron stars [CITATION] are enabling advances across astronomy, astrophysics, and cosmology.', '1804.03392-3-2-1': 'As the gravitational-wave detectors LIGO [CITATION], Virgo [CITATION] improve in sensitivity in the coming years, and as new detectors KAGRA [CITATION] and LIGO India [CITATION] come online, it may become possible to detect gravitational radiation from other astrophysical phenomena.', '1804.03392-3-2-2': 'Rapidly-spinning, non-axisymmetrically-deformed neutron stars will emit gravitational waves in the form of continuous quasi-sinusoidal signals, and remain an intriguing prospect for detection with advanced instruments.', '1804.03392-3-2-3': 'Searches for continuous gravitational waves in contemporary LIGO and Virgo data are ongoing [CITATION].', '1804.03392-3-3-0': 'Since the maximum non-axisymmetric deformation of neutron stars is expected to be small [CITATION], continuous waves are expected to be weak relative to the sensitivity of the current generation of interferometric detectors.', '1804.03392-3-3-1': 'Consequentially there has accumulated a significant body of research devoted to the data analysis challenge of extracting such weak signals from the gravitational-wave detector data.', '1804.03392-3-3-2': 'Early results [CITATION] focused on the method of matched filtering the entire dataset against the known continuous-wave signal model; while theoretically optimal (in the Neyman–Pearson sense), this method quickly becomes computationally intractable if some or all of the model parameters are unknown.', '1804.03392-3-3-3': 'Such is the case if one wished to target an interesting sky direction e.g. associated with a supernova remnant [CITATION] or a low-mass X-ray binary [CITATION], or perform an all-sky survey for isolated continuous-wave sources unassociated with known pulsars [CITATION].', '1804.03392-3-3-4': 'It is the latter type of search that is the subject of this paper.', '1804.03392-3-4-0': 'The additional challenge of a practical upper limit on the computational cost of all-sky searches has spurred the development of various sub-optimal but computationally-tractable hierarchical or semicoherent algorithms [CITATION].', '1804.03392-3-4-1': 'They share a common approach: the dataset (which for this example we assume is contiguous) with timespan [MATH] is partitioned into [MATH] segments, each with timespan [MATH].', '1804.03392-3-4-2': 'A fully-coherent matched filter search is then performed individually for each segment.', '1804.03392-3-4-3': 'Most methods then combine segments by incoherently summing the power from [MATH] filters, one from each segment, which together follow a consistent frequency evolution as dictated by the continuous-wave signal model.', '1804.03392-3-4-4': 'The phase evolution need not be continuous over the [MATH] filters, however; nor need the gravitational-wave amplitudes in each segment be consistent.', '1804.03392-3-4-5': 'This loss of complete signal self-consistency comes, however, with a computational benefit: while the computational cost of a fully-coherent matched filter search of the entire dataset scales as [MATH] with [MATH] a high power [MATH] to 6, the cost of a semicoherent method typically scales as [MATH] with [MATH] [CITATION].', '1804.03392-3-4-6': 'The strain sensitivities of a fully-coherent and semicoherent search typically scale as [MATH] and [MATH] respectively, with [MATH] [CITATION]; for the loss of a factor [MATH] in sensitivity, a semicoherent method is able to gain by being able to analyze large (e.g. [MATH] year) datasets, whereas a fully-coherent search would be computationally restricted to a much shorter (e.g. [MATH] year) subset.', '1804.03392-3-5-0': 'An important early advance in the development of semicoherent methods was the adaption of the Hough transform [CITATION], originally created to analyze tracks in bubble chamber photographs, to instead track the frequency evolution of a continuous gravitational-wave signal [CITATION].', '1804.03392-3-5-1': 'A number of variations of the Hough transform have been implemented, which map the signal track in the time-frequency plane to either its sky position at a fixed reference frequency and frequency derivative [CITATION], or conversely to its reference frequency and frequency derivative at a fixed sky position [CITATION].', '1804.03392-3-5-2': 'The detection statistic computed, the number count, sums either zero or one from each segment depending on whether the significance of a filter exceeds a set threshold.', '1804.03392-3-5-3': 'Some variations use short-duration ([MATH]s) segments and incoherently sum power above threshold from each segment; others analyze longer segments, and set a threshold on the [MATH]-statistic [CITATION] which computes the matched filter analytically maximized over the gravitational-wave amplitudes.', '1804.03392-3-5-4': 'Another modification is to weigh each segment by the antenna response function of the detector, and to sum these weights instead of zero or one [CITATION].', '1804.03392-3-6-0': 'Two semicoherent methods which use short-duration segments but which, unlike the Hough transform methods, sum power without thresholding are the StackSlide [CITATION] and PowerFlux [CITATION] methods.', '1804.03392-3-6-1': 'The StackSlide method builds a time-frequency plane, where each column represents a segment.', '1804.03392-3-6-2': 'For each choice of signal parameters, it "slides" each column up and down in frequency so that a signal with those parameters would follow a horizontal line, and then "stacks" (i.e. sums) the columns horizontally to accumulate the signal power over time for each frequency bin.', '1804.03392-3-6-3': '(Due to this intuitive representation of a semicoherent search method, the term StackSlide is often used to refer to semicoherent methods in general [CITATION].)', '1804.03392-3-6-4': "The PowerFlux method follows a similar methodology, and in addition weights the power from each segment by that segment's noise level and antenna response function, so that segments containing transient instrumental noise and/or where the response of the detector is weak are deweighted.", '1804.03392-3-6-5': 'A "loosely coherent" adaption to PowerFlux allows the degree of phase consistency imposed at the semicoherent stage to be controlled explicitly [CITATION].', '1804.03392-3-6-6': 'A third semicoherent method [CITATION] was developed based on the observance of global correlations between search parameters [CITATION] and uses longer segments analyzed with the [MATH]-statistic.', '1804.03392-3-6-7': 'A comprehensive comparison of many of the all-sky search methods described above is performed in [CITATION].', '1804.03392-3-7-0': 'Aside from developments in semicoherent search techniques, two other ideas have played an important role in the development of continuous gravitational-wave data analysis.', '1804.03392-3-7-1': 'First is the use of a parameter-space metric [CITATION], which is used to determine the appropriate resolution of the bank of template signals such that the mismatch, or fractional loss in signal-to-noise ratio between any signal present in the data and its nearest template, never exceeds a prescribed maximum.', '1804.03392-3-7-2': 'The metric of the [MATH]-statistic for continuous-wave signals was first studied rigorously in [CITATION].', '1804.03392-3-7-3': 'An approximate form of the metric was utilized in semicoherent search methods developed by [CITATION], and a related approximation was used in [CITATION].', '1804.03392-3-7-4': 'The latter approximation, however, lead to an underestimation of the number of required templates in the sky parameter space when analyzing long data stretches; an improved approximate metric developed in [CITATION] addresses this limitation.', '1804.03392-3-7-5': 'It was also later realized that a further approximation fundamental to the metric derivation - namely that the prescribed maximum mismatch (as measured by the metric) could be assumed small - generally does not hold under realistic restrictions on computational cost.', '1804.03392-3-7-6': 'This issue was addressed in [CITATION] which computed an empirical relation between the metric-measured mismatch and the true mismatch of the [MATH]-statistic.', '1804.03392-3-8-0': 'A second important idea is the borrowing of results from lattice theory [CITATION] to optimize the geometric placement of templates within the search parameter space, so as to fulfill the maximum prescribed mismatch criteria described above with the smallest possible density of templates [CITATION].', '1804.03392-3-8-1': 'Practical algorithms for generating template banks for continuous-wave searches, using both the parameter-space metric and optimal lattices, were proposed in [CITATION].', '1804.03392-3-8-2': 'An alternative idea studied in [CITATION] is to instead place templates at random, using the parameter-space metric only as a guide as to the relative density of templates; this idea has found utility in searches for radio [CITATION] and X-ray [CITATION] pulsars.', '1804.03392-3-9-0': 'The number of computations that must be performed during an all-sky search, even when utilizing an efficient semicoherent search method, remains formidable.', '1804.03392-3-9-1': 'For example, a recent all-sky search [CITATION] of data from the first Advanced LIGO observing run divided the data into [MATH] segments of timespan [MATH] hours, performed [MATH] matched-filtering operations per segment, and finally performed [MATH] incoherent summations to combine filter power from each segment.', '1804.03392-3-9-2': 'The total computational cost of the search was [MATH] CPU days, although this was distributed over [MATH] computers volunteered through the Einstein@Home distributed computing project [CITATION].', '1804.03392-3-9-3': 'Nevertheless, the significant number of filtering/incoherent summation operations that must be performed during a typical all-sky search emphasizes the need to optimize the construction of the template banks, and thereby minimize the computational cost of the search, as much as practicable.', '1804.03392-3-10-0': 'In this paper we present Weave, an implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-3-10-1': 'This implementation brings together, for the first time, several strands of previous research: the use of a semicoherent method to combine data segments analyzed with the [MATH]-statistic, combined with optimal template placement using the parameter-space metric of [CITATION] and optimal lattices [CITATION].', '1804.03392-3-10-2': 'After a review of relevant background information in Section [REF], the Weave implementation is presented in Section [REF].', '1804.03392-3-10-3': 'In Section [REF] we demonstrate that important behaviors of the Weave implementation can be modeled semi-analytically, thereby enabling characterization and optimization of a search setup without, in the first instance, the need to resort to time-consuming Monte-Carlo simulations.', '1804.03392-3-10-4': 'In Section [REF] we discuss ideas for further improvement and extension.', '1804.03392-3-11-0': '# Background', '1804.03392-3-12-0': 'This section presents background material pertaining to the continuous-wave signal model, parameter-space metric, and template bank generation.', '1804.03392-3-13-0': '## Continuous-wave signals', '1804.03392-3-14-0': 'The phase of a continuous-wave signal [MATH] at time [MATH] at the detector is given by, neglecting relativistic corrections [CITATION], [EQUATION]', '1804.03392-3-14-1': 'The first term on the right-hand side primarily encodes the loss of rotational energy of the neutron star as observed from the Solar System barycenter: [MATH] is the gravitational-wave frequency; and the spindowns [MATH], [MATH], etc. are the 1st-order, 2nd-order, etc. rates of change of the gravitational-wave frequency with time.', '1804.03392-3-14-2': 'All [MATH] parameters are given with respect to a reference time [MATH].', '1804.03392-3-14-3': 'The second term on the right-hand side describes the Doppler modulation of the gravitational waves due to the motion of an Earth-based detector: [MATH] is the detector position relative to the Solar System barycenter, thereby including both the sidereal and orbital motions of the Earth; and [MATH] is a unit vector pointing from the Solar System barycenter to the continuous-wave source.', '1804.03392-3-14-4': 'The value of [MATH] is chosen conservatively to be the maximum of [MATH] over the timespan of the analyzed data.', '1804.03392-3-15-0': 'Together the phase evolution parameters [MATH] parameterize the continuous-wave signal template; additional amplitude parameters [MATH] are analytically maximized over when computing the [MATH]-statistic [CITATION].', '1804.03392-3-15-1': 'In noise the [MATH]-statistic is a central [MATH] statistic with 4 degrees of freedom; when in the vicinity of a signal, the noncentrality parameter [MATH] of the noncentral [MATH] distribution scales as [MATH], where [MATH] is the gravitational-wave amplitude, [MATH] the amount of analyzed data, and [MATH] is the noise power spectral density in the vicinity of the signal frequency [MATH].', '1804.03392-3-16-0': '## Parameter-space metric', '1804.03392-3-17-0': 'The parameter-space metric [MATH] of the [MATH]-statistic is defined by a 2nd-order Taylor expansion of the noncentrality parameter: [EQUATION] with implicit summation over [MATH], and where [EQUATION]', '1804.03392-3-17-1': 'Here [MATH] is the noncentrality parameter of the [MATH]-statistic when perfectly matched to a signal with parameters [MATH], and [MATH] is the noncentrality parameter when computed at some mismatched parameters [MATH].', '1804.03392-3-17-2': 'The mismatch is defined to be [EQUATION]', '1804.03392-3-17-3': 'A very useful approximation to Eq. [REF] is the phase metric [CITATION]; it discards the amplitude modulation of the signal, and thereby the dependence on the known parameters [MATH], retaining only dependence on the phase evolution parameters: [EQUATION]', '1804.03392-3-18-0': '## Optimal template placement', '1804.03392-3-19-0': 'Template placement using optimal lattices is an example of a sphere covering [CITATION]: a collection of lattice-centered [MATH]-dimensional spheres of equal radius.', '1804.03392-3-19-1': 'The radius is chosen to be the smallest value that satisfies the property that each point in the [MATH]-dimensional parameter space is contained in at least one sphere.', '1804.03392-3-19-2': 'A lattice where the ratio of the volume of the sphere to the volume of a lattice cell is minimized generates a minimal sphere covering, i.e. the minimal number of points required to cover a parameter space, which is exactly the property desired for template banks.', '1804.03392-3-19-3': '(For example, in two dimensions the minimal sphere covering is generated by the hexagonal lattice.)', '1804.03392-3-19-4': 'We identify the covering spheres with the metric ellipsoids [MATH], where [MATH] is the prescribed maximum; it follows that the radii of the covering spheres is [MATH].', '1804.03392-3-19-5': 'A matrix transform [MATH] can then be constructed [CITATION] which takes integers in [MATH] to template parameters [MATH] to generate the template bank: [EQUATION] where [MATH] is a function of the metric [MATH], and [MATH] is particular to the lattice being used.', '1804.03392-3-19-6': 'If [MATH] is a lower triangular matrix, an efficient algorithm [CITATION] can be found for generating the template bank.', '1804.03392-3-20-0': '## Reduced supersky metric', '1804.03392-3-21-0': 'In order for Eq. [REF] to preserve the sphere covering property, however, it must be independent of the template parameters [MATH].', '1804.03392-3-21-1': 'Since [MATH] is a function of the metric, we require a metric which is also independent of [MATH]: [MATH].', '1804.03392-3-21-2': 'The phase metric of Eq. [REF] is independent of the frequency and spindown parameters [MATH], but retains a dependence on sky position parameters, e.g. [MATH] in terms of right ascension [MATH] and declination [MATH].', '1804.03392-3-21-3': 'The question of how to derive a useful metric which is independent of the sky position parameters, i.e. [MATH], has stimulated numerous approaches [CITATION].', '1804.03392-3-21-4': 'In [CITATION], a useful [MATH] is derived through the following procedure: [(i)]', '1804.03392-3-22-0': '[MATH] is expressed in terms of the 3 components of [MATH], instead of 2 parameters such as [MATH].', '1804.03392-3-22-1': 'The 3 components of [MATH] are taken to be independent; geometrically this is equivalent to embedding [MATH] into a 3-dimensional supersky parameter space, instead of being restricted to the 2-sphere defined by [MATH].', '1804.03392-3-22-2': 'In the supersky parameter space, [MATH] is independent of the sky position parameters, i.e. we have the desired [MATH], but with the addition of a 3rd unwanted parameter-space dimension.', '1804.03392-3-23-0': 'A linear coordinate transform [MATH] is derived which satisfies: [MATH] is diagonal in the sky position parameters [MATH], i.e. [MATH]; [MATH]; and [MATH].', '1804.03392-3-23-1': 'The last two properties imply that the metric ellipsoids are much longer along the [MATH] axis than along the [MATH] and [MATH] axes.', '1804.03392-3-23-2': 'In computing the coordinate transform, use is made of the well-known correlation between the sky and frequency/spindown parameters of the continuous-wave signal [CITATION].', '1804.03392-3-23-3': 'The correlations arise because, on sufficiently short timescales, the change in phase due to the cyclical sidereal and orbital motions of the Earth may be Taylor expanded as linear, quadratic, etc. changes in phase with time, and thereby are equivalent to changes in the frequency ([MATH]), 1st spindown ([MATH]), etc. parameters.', '1804.03392-3-24-0': 'Since, in the new coordinates [MATH] the mismatch [MATH] is only weakly dependent on [MATH], a useful approximate metric [MATH] is found by discarding the [MATH] dimension.', '1804.03392-3-24-1': 'Geometrically this corresponds to projecting the 3-dimensional supersky parameter space and metric onto the 2-dimensional [MATH]-[MATH] plane.', '1804.03392-3-24-2': 'The resultant reduced supersky parameter-space metric [MATH] and associated coordinates [MATH] has reduced the sky parameter space dimensionality back to 2, while retaining the property that [MATH] is parameter-independent.', '1804.03392-3-25-0': '# Weave Implementation', '1804.03392-3-26-0': 'This section describes the Weave implementation of the semicoherent search method, a schematic of which is shown in Figure [REF].', '1804.03392-3-26-1': 'The implementation is freely available as part of the LALSuite [CITATION] gravitational-wave data analysis library.', '1804.03392-3-27-0': '## Overview', '1804.03392-3-28-0': 'In step 1 the user runs a precursor program lalappsWeaveSetup, which takes as an argument a list of [MATH] segments [MATH] into which the dataset is to be partitioned.', '1804.03392-3-28-1': 'The program computes in step 2 the [MATH] coherent parameter-space metrics [MATH] used to construct template banks within each segment, and the semicoherent parameter-space metric [MATH] used to incoherently combine segments.', '1804.03392-3-28-2': 'The metrics are written to a setup file in the FITS format [CITATION].', '1804.03392-3-28-3': 'Due to the numerical ill-conditionedness of the parameter-space metric [CITATION], this computation involves a bootstrapping process, whereby successively better-conditioned iterations of the supersky metric are computed, before then computing the reduced supersky metric as outlined in Section [REF].', '1804.03392-3-28-4': 'Since this bootstrapping process can be time-consuming for large [MATH], and may give slightly different results on different computer hardware, precomputing the metrics both saves computing time and adds robustness against numerical errors.', '1804.03392-3-28-5': 'Note that, by Eq. [REF], the sky components of the metrics will scale with [MATH]; since its value depends on the search frequency parameter space, which is not known by lalappsWeaveSetup, an arbitrary fiducial value [MATH] is used, and the sky components of the metrics are later rescaled by [MATH].', '1804.03392-3-29-0': 'In step 3 the user runs the main search program lalappsWeave.', '1804.03392-3-29-1': 'The principle arguments to this program are the setup file output by lalappsWeaveSetup, the search parameter space, and the prescribed maximum mismatches [MATH] and [MATH] for the coherent and semicoherent template banks respectively.', '1804.03392-3-29-2': 'The frequency and spindown parameter space is specified by ranges [MATH], where [MATH], 1, etc. as required.', '1804.03392-3-29-3': 'The sky search parameter space may be specified either as a rectangular patch in right ascension and declination [MATH], or alternatively partitioned into [MATH] patches containing approximately equal number of templates (see Appendix [REF]), and a patch selected by an index [MATH], [MATH].', '1804.03392-3-29-4': 'In step 4 various preparatory tasks are performed, such as loading the gravitational-wave detector data into memory, before beginning the main search loop.', '1804.03392-3-30-0': 'The main search loop of a semicoherent search method may be structured in two complementary ways, which differ in the memory each requires to store intermediate results: [(i)]', '1804.03392-3-31-0': 'The semicoherent template bank [MATH] is stored in memory, and the [MATH] segments are processed in sequence.', '1804.03392-3-31-1': 'For each segment [MATH], every coherent template [MATH] is mapped back to the semicoherent template bank, i.e. [MATH].', '1804.03392-3-31-2': 'Because the semicoherent template bank must track the continuous-wave signal over a larger timespan [MATH] than the coherent template banks, it will contain a greater density of templates; the ratio of semicoherent to coherent template bank densities is the refinement factor [MATH] [CITATION].', '1804.03392-3-31-3': 'It follows that the mapping [MATH] will be one-to-many.', '1804.03392-3-32-0': 'As the [MATH] segments are processed, any semicoherent detection statistic associated with [MATH] is then updated based on the corresponding coherent detection statistic associated with [MATH].', '1804.03392-3-32-1': 'For example, it is common to compute the summed [MATH]-statistic [MATH]; here we would then have [MATH].', '1804.03392-3-32-2': 'Once every segment has been processed, computed [MATH] for every [MATH] will exist in memory.', '1804.03392-3-32-3': 'The memory usage of the main search loop will therefore be proportional to the number of semicoherent templates [MATH], where [MATH] is the average number of templates in a coherent template bank.', '1804.03392-3-33-0': 'The [MATH] coherent template banks [MATH] are stored in memory, and the semicoherent template bank is processed in sequence.', '1804.03392-3-33-1': 'Each semicoherent template [MATH] is mapped back to the coherent template bank in each segment [MATH], i.e. [MATH]; since [MATH] in each segment this mapping will be many-to-one.', '1804.03392-3-33-2': 'With these [MATH] mappings in hand, the semicoherent detection statistics may be immediately computed in full, e.g. [MATH].', '1804.03392-3-33-3': 'The memory usage of the main search loop will therefore be proportional to [MATH].', '1804.03392-3-34-0': 'For the parameter-space metric for all-sky searches, [MATH] [CITATION], and therefore the latter structuring given above will have the lower memory requirement; the Weave implementation uses this structuring of the main search loop.', '1804.03392-3-34-1': 'The semicoherent template bank [MATH] is generated one template at a time using the algorithm described in [CITATION].', '1804.03392-3-34-2': 'For each coherent template bank, an efficient lookup table [CITATION] is constructed for the mapping [MATH].', '1804.03392-3-35-0': 'We note an important distinction between the definition of the Weave template banks and the traditional StackSlide picture of a semicoherent search method.', '1804.03392-3-35-1': 'In the latter picture, the frequency and spindown template banks of each segment are defined with respect to individual reference times [MATH], typically the midtime of each segment.', '1804.03392-3-35-2': 'When combining segments, therefore, the frequency and spindown parameters of each coherent template must be adjusted so as to bring the parameters of all segments to a common reference time [MATH]; this is the "sliding" step.', '1804.03392-3-35-3': 'The Weave implementation, however, defines the frequency and spindown templates banks of all segments at the same reference time [MATH], which is also the reference time of the semicoherent bank.', '1804.03392-3-35-4': 'Consequentially, there is no analogy to the "sliding" step of StackSlide.', '1804.03392-3-35-5': 'Instead, the orientation of the metric ellipses in the [MATH] plane changes from segment to segment, as illustrated in Figure [REF].', '1804.03392-3-35-6': 'As the absolute difference [EQUATION] between the midtime of each segment [MATH] and [MATH] increases, both the extent of the ellipses in [MATH] and the correlation between [MATH] and [MATH] also increase.', '1804.03392-3-36-0': 'Steps 5-16 comprise the main search loop; which performs two key tasks: the computation and output of the detection statistics over the semicoherent template bank (steps 5, 6, and 12-17), and the management of an internal cache of required detection statistics computed on each coherent template bank (steps 7-11).', '1804.03392-3-36-1': 'These two tasks are described more fully in the following two sections, and with reference to a diagram of their operation in Figure [REF].', '1804.03392-3-37-0': 'In this section and in Figure [REF] we focus for simplicity on the computation of the semicoherent [MATH]-statistics [MATH] and [MATH].', '1804.03392-3-37-1': 'The computation of other detection statistics is also possible: in particular a family of Bayesian statistics has been developed which weigh the likelihood of a continuous wave signal against that of an instrumental line which appears in all segments [CITATION], or a transient instrumental line which appears only in one segment [CITATION].', '1804.03392-3-37-2': 'Computation of the former statistic, denoted [MATH], is also illustrated in Figure [REF]; it takes as input the multi-detector [MATH] which uses data from all gravitational-wave detectors, as well as the per-detector [MATH] which are computed from each detector [MATH] individually.', '1804.03392-3-38-0': '## Computation of semicoherent statistics', '1804.03392-3-39-0': 'In steps 5 and 16 (Figure [REF]), the main loop of the search method generates successive points [MATH] in the semicoherent template bank.', '1804.03392-3-39-1': 'An example of such a point is indicated in Figure [REF].', '1804.03392-3-39-2': 'Next, in steps 6 and 13, each segment [MATH] is visited and the mapping [MATH] is performed.', '1804.03392-3-39-3': 'The mapping used by Weave is nearest-neighbor interpolation: the [MATH] is expressed in the coherent metric coordinates of the [MATH]th segment, and the nearest (with respect to the metric) coherent template in the respective bank [MATH] is determined.', '1804.03392-3-39-4': 'If the template bank is constructed on a lattice, efficient algorithms exist for determining the nearest point [CITATION].', '1804.03392-3-39-5': 'In Figure [REF], example nearest coherent templates are labeled [MATH], [MATH], and [MATH].', '1804.03392-3-40-0': 'As each nearest point is determined, the coherent [MATH]-statistic in the respective segment is computed (steps 7-11, see Section [REF]), and the value of the semicoherent statistic [MATH] is updated (step 12).', '1804.03392-3-40-1': 'Once all segments have been processed (step 13), additional semicoherent statistics such as [MATH] are computed (step 14), and a candidate comprising the signal parameters together with the computed semicoherent statistics is added (step 15) to one or more toplists which ranks each candidate by a chosen semicoherent statistic.', '1804.03392-3-40-2': 'The size of the toplists is generally of a fixed user-determined size so that only a fixed number of the most promising candidates will be returned.', '1804.03392-3-41-0': 'Once the semicoherent template bank is exhausted (step 16) the toplists are written to an output file in the FITS format, and the search concludes (step 17).', '1804.03392-3-42-0': '## Management of cache of coherent statistics', '1804.03392-3-43-0': 'It is important that the main search loop minimizes its memory usage as much as possible.', '1804.03392-3-43-1': 'Even though in Section [REF] we chose a structuring of the main search loop so as to reduce memory usage, a naive implementation which stores [MATH] coherent statistics would still require a prohibitive amount of memory, given that both [MATH] and [MATH] are typically large.', '1804.03392-3-43-2': 'We therefore implement a per-segment cache which stores only those coherent statistics associated with coherent templates [MATH] accessible from the unprocessed portion of the semicoherent template bank via the mapping [MATH].', '1804.03392-3-43-3': 'Put another way, if a [MATH] can no longer be mapped to by any [MATH] remaining in [MATH], then [MATH] can be safely removed from the cache.', '1804.03392-3-44-0': 'In order to devise a cache management algorithm with the above desired properties, we first define an operator called relevance, denoted [MATH].', '1804.03392-3-44-1': 'The relevance operates on both coherent and semicoherent templates, and should satisfy the following property: [EQUATION]', '1804.03392-3-44-2': 'A definition of [MATH] satisfying this property is derived as follows.', '1804.03392-3-45-0': 'First, take any coherent template (e.g. [MATH] in Figure [REF]) and surround it by its metric ellipsoid at mismatch [MATH].', '1804.03392-3-45-1': 'Then surround the metric ellipsoid in turn by its bounding box, the smallest coordinate box which contains the ellipsoid [CITATION]; the metric ellipse bounding box centered on [MATH] is also shown in Figure [REF].', '1804.03392-3-45-2': 'Now, transform the bounding box into the semicoherent parameter space; practically this may be achieved by expressing the coordinates of each vertex of the bounding box in the semicoherent metric coordinates.', '1804.03392-3-45-3': 'See Figure [REF] for the transformed bounding box of [MATH] in the semicoherent parameter space, which is centered on [MATH].', '1804.03392-3-46-0': 'Note that, by definition, any semicoherent template [MATH] outside of the transformed bounding box centered on [MATH] cannot map to [MATH] under [MATH].', '1804.03392-3-46-1': 'Thus, to determine whether [MATH] is accessible by [MATH], we can compute whether [MATH] is within the transformed bounding box of [MATH].', '1804.03392-3-46-2': 'To be conservative, however, we also surround [MATH] by its bounding box as shown in Figure [REF], and instead compute whether the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-3-47-0': 'To simplify the bounding box intersection calculation, we compare just the coordinates of the bounding boxes of [MATH] and [MATH] in one dimension; for reasons that will soon be apparent, we choose the lowest-dimensional coordinate, [MATH].', '1804.03392-3-47-1': 'First, we define the relevance [MATH] for both coherent and semicoherent templates: [EQUATION]', '1804.03392-3-47-2': 'We now compute [MATH] and [MATH]; in Figure [REF], [MATH] is the [MATH] coordinate of the right-most edge of the transformed bounding box of [MATH], and [MATH] is the [MATH] coordinate of the left-most edge of the bounding box of [MATH].', '1804.03392-3-47-3': 'In this example, [MATH], and it follows from the definition of [MATH] in Eqs. [REF] that the bounding boxes of [MATH] and [MATH] cannot intersect.', '1804.03392-3-48-0': 'On the other hard, let us choose another coherent template [MATH], and examine its relevance [MATH]; here we have [MATH] (see Figure [REF]).', '1804.03392-3-48-1': 'From the simplified bounding box intersection calculation, we conclude that the bounding boxes of [MATH] and [MATH] could potentially intersect, since at least in the [MATH] dimension the bounding boxes overlap (although in this example the bounding boxes do not overlap in the [MATH] dimension).', '1804.03392-3-49-0': 'Finally, if for some [MATH] we have [MATH], then this condition is guaranteed to remain true for all remaining [MATH] in the template bank.', '1804.03392-3-49-1': 'This is simply a consequence of the algorithm used to generate the semicoherent template bank [CITATION], which operates as follows: first, values of [MATH] are generated in a constant range [MATH]; then, for each value of [MATH], values of [MATH] are generated in ranges [MATH] dependent on [MATH], and so on.', '1804.03392-3-49-2': 'It follows that the value of [MATH] can only increase during the generation of the semicoherent template bank, and since [MATH] is defined in terms of [MATH], it too can only increase.', '1804.03392-3-50-0': 'To summarize, the relevance operator [MATH] defined by Eqs. [REF] satisfies the desired property given by Eq. [REF].', '1804.03392-3-50-1': 'In Figure [REF], since [MATH], the cache management algorithm would discard any coherent statistics associated with [MATH] from memory, since they cannot be accessed by [MATH] nor any remaining semicoherent template.', '1804.03392-3-50-2': 'On the other hard, the algorithm would retain any coherent statistics associated with [MATH], since they could still be needed for future semicoherent templates; indeed in Figure [REF] it is clear that the next semicoherent template in the bank, labeled [MATH], could require coherent statistics associated with [MATH], since the bounding boxes of [MATH] and [MATH] intersect.', '1804.03392-3-51-0': 'The cache management algorithm described above is implemented in the main search loop in steps 7-11 (Figure [REF]).', '1804.03392-3-51-1': 'In step 7 the cache is interrogated for a required [MATH]-statistic value [MATH]: if it is in the cache, it is retrieved and utilized (step 8), otherwise it is computed and inserted into the cache (step 9).', '1804.03392-3-51-2': 'In the latter case, the cache is also checked to see if any cache items can be discarded.', '1804.03392-3-51-3': 'Starting with step 10, cache items indexed by [MATH] are retrieved in order of ascending [MATH].', '1804.03392-3-51-4': 'If [MATH], the cache items are discarded (step 11).', '1804.03392-3-51-5': 'Only one cache item is removed at any one time, and therefore the memory usage of the cache will either remain constant, or increase by one item per main search loop iteration.', '1804.03392-3-51-6': 'The cache is implemented using two data structures [CITATION]: a binary heap to rank cache items by [MATH], and a hash table to find cache items indexed by [MATH].', '1804.03392-3-52-0': '# Models of Weave Behavior', '1804.03392-3-53-0': 'This section presents semi-analytic models of the Weave implementation.', '1804.03392-3-53-1': 'It greatly facilitates the practical usage of any search method if its behavior can be characterized a priori as much as possible using a computationally-cheap model.', '1804.03392-3-53-2': 'For example, a model of the distribution of [MATH]-statistic mismatches (Section [REF]) permits the estimation of the sensitivity of a particular search setup [CITATION] which in turn allows the setup to be optimized so as to maximize sensitivity [CITATION].', '1804.03392-3-53-3': 'Similarly, models of the number of coherent and semicoherent templates (Section [REF]) and computational cost (Section [REF]) allow the parameters of the optimal search setup to be estimated [CITATION].', '1804.03392-3-53-4': 'The memory usage (Section [REF]) and input data bandwidth (Section [REF]) required by the implementation are also important properties when implementing a search pipeline.', '1804.03392-3-54-0': 'Each model presented in this section is implemented as an Octave [CITATION] script, and is freely available as part of the OctApps [CITATION] script library.', '1804.03392-3-55-0': '## Distribution of [MATH]-statistic mismatches', '1804.03392-3-56-0': 'The distribution of the mismatch between the [MATH]-statistic computed at an exact signal location, and at the nearest point in the Weave semicoherent template bank, gives an idea of the expected loss in signal-to-noise ratio due to the necessary coarseness of the template bank.', '1804.03392-3-56-1': 'Figure [REF] plots the predicted means and standard deviations of Weave [MATH]-statistic mismatch distributions, against their measured values, for a variety of setups given in Table [REF].', '1804.03392-3-56-2': 'The distributions were measured using software injection studies, where relatively strong ([MATH]) simulated signals are added to Gaussian-distributed noise and then searched for using lalappsWeave.', '1804.03392-3-57-0': 'The predicted means and standard deviations are from the model presented in [CITATION], and are generally conservative: Figure [REF] shows that the model generally overestimates the mean [MATH]-statistic mismatch by [MATH] (Figure [REF]) to [MATH] (Figure [REF]); and the predicted standard deviations imply slightly broader distributions than were measured.', '1804.03392-3-57-1': 'As explored in [CITATION], the relationship between the maximum mismatches of the coherent and semicoherent template banks (which are inputs to lalappsWeave) and the [MATH]-statistic mismatch distribution (which is output by lalappsWeave) is difficult to model when the former are large e.g. [MATH].', '1804.03392-3-58-0': 'In addition, an optimization implemented in Weave but not accounted for in the model of [CITATION] complicates the picture: the coherent and semicoherent template banks are constructed to have equally-spaced templates in the frequency parameter [MATH].', '1804.03392-3-58-1': 'This permits (in step 9 of Figure [REF]) the simultaneous computation of a series of [MATH] values at equally-spaced values of [MATH] across the frequency parameter space, which can be performed efficiently using Fast Fourier Transform-based algorithms (see Section [REF]).', '1804.03392-3-58-2': 'The construction of equal-frequency-spacing coherent and semicoherent template banks is performed by first constructing each bank independently, and then reducing the frequency spacing in all banks to that of the smallest frequency spacing in any bank.', '1804.03392-3-58-3': 'This construction will always reduce the maximum possible mismatch in each grid, but never increase it, and so we would expect the mean [MATH]-statistic mismatch measured by Weave to be smaller than that predicted by the model of [CITATION].', '1804.03392-3-59-0': 'The model of [CITATION] is implemented in the OctApps script WeaveFstatMismatch.m.', '1804.03392-3-60-0': '## Number of templates', '1804.03392-3-61-0': 'Since the Weave coherent and semicoherent template banks are constructed using lattices (see Section [REF]), the number of templates in each is estimated starting from the formula [CITATION] [EQUATION] where [MATH] is the volume of the [MATH]-dimensional parameter space, [MATH] the parameter-space metric, and [MATH] the maximum mismatch.', '1804.03392-3-61-1': 'The normalized thickness [MATH] is a property of the particular lattice used to generate the template bank [CITATION].', '1804.03392-3-62-0': 'The parameter-space volume is given explicitly by the following expressions: [EQUATION]', '1804.03392-3-62-1': 'Here, [MATH] is the vector whose components are the extents of the bounding box of [MATH] in each dimension; it is used to ensure that the volume of the parameter space in each dimension is not smaller than the extent of a single template.', '1804.03392-3-62-2': 'In Eq. [REF], the volume of the sky parameter space may be specified either by a rectangular patch [MATH], or by the number [MATH] of equal-size sky patches (see Section [REF]).', '1804.03392-3-63-0': 'Finally, the total number of coherent and semicoherent templates, [MATH] and [MATH] respectively, are given by: [EQUATION]', '1804.03392-3-63-1': 'The numerical prefactor on the right-hand side of Eq. [REF] is chosen to better match [MATH] to the number of coherent templates actually computed by lalappsWeave: the coherent parameter space is augmented with additional padding along its boundaries to ensure that it encloses the semicoherent parameter space, i.e. that it includes a nearest neighbor for every [MATH].', '1804.03392-3-64-0': 'Equations [REF] and [REF] are used to predict the number of templates computed by lalappsWeave for a variety of search setups detailed in Table [REF].', '1804.03392-3-64-1': 'Figure [REF] plots the predicted [MATH] and [MATH] against the values measured by running lalappsWeave.', '1804.03392-3-64-2': 'Reasonable agreement is achieved between predicted and measured [MATH] (Figure [REF]): while Eq. [REF] sometimes underestimates the number of coherent templates, it rarely does so by more than a factor of a few.', '1804.03392-3-64-3': 'Better agreement is seen between predicted and measured [MATH] (Figure [REF]).', '1804.03392-3-65-0': 'Equations [REF] and [REF] are implemented in the OctApps script WeaveTemplateCount.m.', '1804.03392-3-66-0': '## Computational cost', '1804.03392-3-67-0': 'The total computational cost [MATH] of a particular search setup may be modeled in terms of the number of coherent [MATH] and semicoherent [MATH] templates (see Section [REF]), the number of segments [MATH] and number of detectors [MATH].', '1804.03392-3-67-1': 'Following [CITATION] we write [EQUATION] where [MATH] and [MATH] denote the computational cost of the coherent and semicoherent stages of the search method respectively, and [MATH] denotes any unmodeled computational costs.', '1804.03392-3-68-0': 'The computational cost model takes as input fundamental timing constants which give the time taken to complete certain fundamental computations.', '1804.03392-3-68-1': 'Their values are highly dependent on various properties of the computer hardware used to run lalappsWeave, such as the processor speed and cache sizes, as well as what other programs were using the computer hardware at the same time as lalappsWeave.', '1804.03392-3-68-2': 'Some values are also specific to the search setups detailed in Table [REF].', '1804.03392-3-68-3': 'For the interest of the reader, Table [REF] lists representative values of the fundamental timing constants obtained on a particular computer cluster.', '1804.03392-3-69-0': 'The coherent cost [MATH] is simply the cost of computing the [MATH]-statistic (step 9 of Figure [REF]): [EQUATION]', '1804.03392-3-69-1': 'The fundamental timing constant [MATH] gives the time taken to compute the [MATH]-statistic per template and per detector, and is further described in [CITATION].', '1804.03392-3-69-2': 'Its value depends primarily upon the range of the frequency parameter space [MATH], the coherent segment length [MATH], and the algorithm used to compute the [MATH]-statistic.', '1804.03392-3-69-3': 'Choices for the latter are: the resampling algorithm [CITATION], which computes the [MATH]-statistic over a wide band of frequencies efficiently using the Fast Fourier Transform, and is generally used to performing an initial wide-parameter-space search; and the demodulation algorithm of [CITATION], which uses a Dirichlet kernel to compute the [MATH]-statistic more efficiently at a single frequency or over a narrow frequency band, and is therefore used to perform follow-up searches of localized parameter spaces around interesting candidates.', '1804.03392-3-69-4': 'The additional cost of managing the cache of computed [MATH]-statistic values (steps 8, 10, and 11) is amortized into [MATH].', '1804.03392-3-70-0': 'The semicoherent cost [EQUATION] has a number of components: [(i)] [MATH] is the cost of iterating over the semicoherent template bank (steps 5 and 16 of Figure [REF]); [MATH] is the cost of finding the nearest templates in the coherent template banks (step 6 and 13) and of interrogating the cache of computed [MATH]-statistic values (step 7); [MATH] is the cost of computing [MATH] and, if required, [MATH] (step 12); [MATH] is the cost of computing [MATH] (step 14); [MATH] is the cost of computing [MATH], if required (step 14); and [MATH] is the cost of adding candidates to toplists (step 15).', '1804.03392-3-71-0': 'These components of [MATH] are further defined in terms of [MATH], [MATH], [MATH], and various fundamental timing constants (see Table [REF]) as follows: [EQUATION]', '1804.03392-3-71-1': 'Figure [REF] compares the computational cost model of Eqs. [REF]-[REF] against the measured computational cost of lalappsWeave (see Table [REF]), using the search setups detailed in Table [REF].', '1804.03392-3-71-2': 'The total computational cost of lalappsWeave is generally well-modeled (Figure [REF]) and the unmodeled component of the measured computational cost is low (Figure [REF]).', '1804.03392-3-71-3': 'The coherent computational cost [MATH] of Eq. [REF] and the components of the semicoherent cost [MATH] of Eq. [REF] are also in good agreement (Figures [REF]-[REF]).', '1804.03392-3-72-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveRunTime.m.', '1804.03392-3-73-0': '## Memory usage', '1804.03392-3-74-0': 'The memory usage [MATH] of lalappsWeave is modeled by [EQUATION]', '1804.03392-3-74-1': 'The first term on the right-hand side, [MATH], is the memory usage of the [MATH]-statistic algorithm (which includes the gravitational-wave detector data) and is further described in [CITATION].', '1804.03392-3-74-2': 'The second term, [MATH], is the memory usage of the cache of computed [MATH]-statistic values, and is further given by [EQUATION] where [MATH] is the maximum size of the cache (across all segments), and [MATH] MiB (mebibytes) is the memory required to store one [MATH] value as a 4-byte single precision floating-point number.', '1804.03392-3-74-3': 'The maximum cache [MATH] cannot easily be predicted from first principles, i.e. given the search setup, parameter space, and other input arguments to lalappsWeave.', '1804.03392-3-74-4': 'Instead, it is measured by running lalappsWeave in a special mode which simulates the performance of the cache but without computing any [MATH]-statistic or derived values; essentially it follows Figure [REF] but with the first part of step 9, step 12, and step 14 omitted.', '1804.03392-3-75-0': 'Figure [REF] plots the predicted memory usage of Eqs. [REF] and [REF] against the measured memory usage of lalappsWeave, using the search setups detailed in Table [REF].', '1804.03392-3-75-1': 'The [MATH]-statistic is computed using both the resampling and demodulation algorithms: in the former case, both [MATH] and [MATH] are computed, thereby triggering the first case in Eq. [REF]; in the latter case, only [MATH] is computed, thereby triggering the second case in Eq. [REF].', '1804.03392-3-75-2': 'Good agreement between predicted and measured memory usage is seen for both algorithms.', '1804.03392-3-76-0': 'Equations [REF] and [REF] are also implemented in the OctApps script WeaveRunTime.m.', '1804.03392-3-77-0': '## Input data bandwidth', '1804.03392-3-78-0': 'Our final Weave model concerns what bandwidth of the input gravitational-wave detector data is required to search a given frequency range.', '1804.03392-3-78-1': 'For most continuous-wave search pipelines, short (typically 1800 s) contiguous segments of gravitational-wave strain data are Fourier transformed, and the resulting complex spectra stored as Short Fourier Transform (SFT) files.', '1804.03392-3-78-2': 'A continuous-wave search of a large frequency parameter space will generally be divided into smaller jobs, with each job searching a smaller partition of the whole frequency parameter space.', '1804.03392-3-78-3': 'Each job therefore requires that only a small bandwidth out of the full SFT spectra be read into memory.', '1804.03392-3-79-0': 'Given an input frequency parameter space [MATH] and spindown parameter space [MATH], predicting the bandwidth of the SFT spectra required by lalappsWeave proceeds in several steps.', '1804.03392-3-79-1': 'First, the input parameter spaces are augmented to account for extra padding of the Weave template banks: [EQUATION] where [MATH] and [MATH] are empirically chosen.', '1804.03392-3-79-2': 'Next, the maximum frequency range [MATH] is found by evolving the frequency-spindown parameter space [MATH] from the reference time [MATH] to the start and end times of each segment, [MATH] and [MATH] respectively: [EQUATION]', '1804.03392-3-79-3': 'Finally, the SFT bandwidth [MATH] of the SFT spectra which is required by lalappsWeave is given by: [EQUATION]', '1804.03392-3-79-4': 'The [MATH] enlarges [MATH] to account for the maximum frequency-dependent Doppler modulation of a continuous-wave signal due to the sidereal and orbital motions of the Earth, and is given by [EQUATION] where [MATH] is the speed of light, [MATH] is the Earth-Sun distance and [MATH] the radius of the Earth.', '1804.03392-3-79-5': 'Additional padding of [MATH] is also required for use by the chosen [MATH]-statistic algorithm, and is given by [MATH] [CITATION].', '1804.03392-3-80-0': 'Figure [REF] compares the model of Eqs. [REF]-[REF] against the behavior of lalappsWeave when run with the search setups detailed in Table [REF].', '1804.03392-3-80-1': 'Note that the model satisfies [EQUATION] i.e. all circles plotted in Figure [REF] are below the horizontal axis, and [EQUATION] i.e. all crosses plotted in Figure [REF] are above the horizontal axis.', '1804.03392-3-80-2': 'The model is therefore conservative, i.e. it may predict a slightly larger SFT bandwidth than required, but should never predict a smaller SFT bandwidth, which would cause a fatal error in lalappsWeave.', '1804.03392-3-80-3': "The model is generally more conservative at higher frequencies, where the Doppler modulation due to the Earth's motion is higher.", '1804.03392-3-81-0': 'Equations [REF]-[REF] are implemented in the OctApps script WeaveInputSFTBand.m.', '1804.03392-3-82-0': '# Discussion', '1804.03392-3-83-0': 'This paper details the Weave implementation of a semicoherent search method for continuous gravitational waves.', '1804.03392-3-83-1': 'It focuses on all-sky surveys for isolated continuous-wave sources, for which the parameter space is the sky position and frequency evolution of the source.', '1804.03392-3-83-2': 'We note, however, that the implementation is in fact indifferent to the parameter space being searched, as long as the relevant constant parameter-space metric is available.', '1804.03392-3-83-3': 'The implementation could therefore be adapted to search other parameter spaces for continuous-wave sources such as known low-mass X-ray binaries, for which the parameter space includes the evolution parameters of the binary orbit, using the metric of [CITATION].', '1804.03392-3-84-0': 'There is scope to improve the semi-analytic models of the behavior of lalappsWeave presented in Section [REF].', '1804.03392-3-84-1': 'In particular, a more accurate model of the distribution of [MATH]-statistic mismatches than that presented in Section [REF] would allow the sensitivity of a search to be more accurately estimated without resorting to software injection studies.', '1804.03392-3-84-2': 'The memory model of Section [REF] would also be improved if the maximum cache size [MATH] could be predicted from first principles.', '1804.03392-3-85-0': 'In a forthcoming paper [CITATION] we plan to more fully characterize the performance of the Weave implementation, and compare it to an implementation of the method of [CITATION] using a mock data challenge.', '1804.03392-3-86-0': 'We thank Bruce Allen and Heinz-Bernd Eggenstein for valuable discussions.', '1804.03392-3-86-1': 'KW is supported by ARC CE170100004.', '1804.03392-3-86-2': 'Numerical simulations were performed on the Atlas computer cluster of the Max Planck Institute for Gravitational Physics.', '1804.03392-3-86-3': 'This paper has document number LIGO-P1800074-v4.', '1804.03392-3-87-0': '# Properties of equal-area sky patches', '1804.03392-3-88-0': 'The search program lalappsWeave allows the sky search parameter space to be partitioned into [MATH] patches, and a patch selected by an index [MATH].', '1804.03392-3-88-1': 'Tests of this feature found that, provided [MATH] (the number of templates with just one patch), the variation in the number of templates between patches [MATH] is generally small and well-approximated by [EQUATION]', '1804.03392-3-88-2': 'The ratio [MATH] of the number of templates in all [MATH] patches to the number of templates with just one patch is generally [MATH]%.', '1804.03392-3-88-3': 'The union of all templates in a set of [MATH] patches also faithfully reproduces the unpartitioned template bank, i.e. with just one patch.'} | null | null | null | null |
0712.0959 | {'0712.0959-1-0-0': 'Herbig-Haro jets often show some degree of curvature along their path, in many cases produced by the ram pressure of a side-wind.', '0712.0959-1-0-1': 'We present simulations of both laboratory and astrophysical curved jets and experimental results from laboratory experiments.', '0712.0959-1-0-2': 'We discuss the properties and similarities of the laboratory and astrophysical flow, which show the formation of internal shocks and working surfaces.', '0712.0959-1-0-3': 'In particular the results illustrate how the break-up of the bow-shock and clumps in the flow are produced without invoking jet variability; we also discuss how jet rotation reduces the growth of the Rayleigh-Taylor instability in curved jets.', '0712.0959-1-1-0': '# Introduction', '0712.0959-1-2-0': 'Parsec-scale Herbig-Haro (HH) jets are one of the most spectacular phenomena connected to star formation.', '0712.0959-1-2-1': 'Observations of line emission features reveal an intricate succession of bow shocks and knots along the jet, and a complex and rich dynamics arising from the interaction with the circumstellar environment (see [CITATION] for a review).', '0712.0959-1-2-2': 'In the reflection nebula NGC 1333, in the Perseus molecular cloud, a number of bipolar HH jets exhibit a distinguishing C-shape morphology indicative of a steady bending .', '0712.0959-1-2-3': 'Less regular curvature is also observed in a number of other HH jets; for example in HH 30 a small side drift close to the jet source is followed further away by a sudden bending .', '0712.0959-1-2-4': 'In general, the curvature in jets has been linked either with the motion of the jet sources relative to the ambient medium or with the presence of a widespread outflow; both cases giving rise to an effective transverse ram pressure (cross-wind) which curves the jet.', '0712.0959-1-2-5': 'In fact, numerical simulations of jet bending produced by magnetic fields seem to require field strength that are unrealistically high .', '0712.0959-1-2-6': "For the jets in the Orion Nebula, it was also suggested that the bending may be caused by the 'rocket' effect induced by the photo-ionizing UV radiation from nearby high-mass stars ; recent work by [CITATION] has investigated the influence of an ionizing photon flux on the jet-wind collision in the HH555 jet.", '0712.0959-1-2-7': "The presence of curved jets is also common to other environments, such as in active galactic nuclei where they are observed as narrow-angle-tail radio sources (O'Dea and Owen 1986), or the pulsar jets in the Vela nebula, which are curved under the combined action of the wind within the supernovae remnant and the proper motion of the pulsar (Pavlov, Teter et al. 2003).", '0712.0959-1-2-8': 'Under simplifying assumption the curvature of the jets can be described analytically , however these models cannot detail the shock structures and the intricate dynamics developing in the interaction, and that were observed in a number of numerical studies.', '0712.0959-1-2-9': 'For the narrow-angle-tail sources internal shocks were shown to be responsible for the formation of density enhancements (knots) inside the jet beam which was then disrupted through the combined effect of Kelvin-Helmholtz and Rayleigh-Taylor instabilities.', '0712.0959-1-2-10': 'HH jets were first investigated numerically by [CITATION] for different incident angles of the wind and by [CITATION] for the case of irradiated jets.', '0712.0959-1-2-11': 'More recently the jet-wind collisions have also been studied in the laboratory with supersonic (Mach number [MATH]) jets interacting with a fast, radiatively ablated plasma that acted as a cross-wind.', '0712.0959-1-2-12': 'The laboratory results clearly demonstrated a steady curvature in the jets under the impinging cross-wind and the appearance of internal shocks.', '0712.0959-1-3-0': 'In the present work we combine consistently the laboratory modelling (experiments and simulations) with simulations of curved HH jets, drawing a parallel between the observed dynamics in the two systems and, with in mind future experiments, we also discuss the effects of rotation on curved jets.', '0712.0959-1-4-0': '# Laboratory Jets', '0712.0959-1-5-0': 'Laboratory jets are produced on the MAGPIE pulsed-power generator using a conical cage of micron-sized metallic wires (Al, Fe, W, etc) which is subject to a 1 Mega Ampere current, with a rise-time of 250 ns (see [CITATION] for a review).', '0712.0959-1-5-1': 'Figure 1 shows a schematic of the experimental set-up; the ohmically heated wires rapidly vaporize and turn into streams of plasma accelerated by the j[MATH]B[MATH] force towards the axis of the array.', '0712.0959-1-5-2': 'The collision of the streams forms a standing conical shock, which redirects the plasma axially into a jet.', '0712.0959-1-5-3': 'Radiative cooling rapidly reduces the temperature in the jet which attains high Mach numbers (M[MATH]) and small opening angles.', '0712.0959-1-5-4': 'The interaction of the jet with a side-wind is replicated in the laboratory by placing a plastic foil in the jet propagation region.', '0712.0959-1-5-5': 'XUV radiation from the standing shock radiatively ablates the foil and generates a fast moving plasma ([MATH] km s[MATH]), providing the ram pressure to bend the jet.', '0712.0959-1-5-6': 'By changing the position, angle and length of the foil the characteristic parameters of the impinging wind can be altered, and different regimes of the interaction studied .', '0712.0959-1-5-7': 'Similar to the HH jets, the characteristic cooling time in the laboratory jets is much smaller than the hydrodynamic time, this ratio is [MATH], and radiation losses play a significant role in the energy balance of the jets.', '0712.0959-1-5-8': 'The numerical simulations are performed with the three-dimensional resistive MHD code GORGON , the ion and electron energies are solved separately and include electron and ion thermal conductions.', '0712.0959-1-5-9': 'The coupling of the two energy equations due to collisional heat exchange is included as a sink/source term.', '0712.0959-1-5-10': 'The plasma is assumed to be in local thermodynamic equilibrium (LTE) and the average ionization is calculated by a Thomas-Fermi model.', '0712.0959-1-5-11': 'The plasma is optically thin and a radiation losses term is included in the electron energy equation.', '0712.0959-1-6-0': 'Two-dimensional axis-symmetric MHD simulations of the whole conical wire array were performed to obtain the mass and momentum fluxes of the jet.', '0712.0959-1-6-1': 'The calculated time and spatially varying profiles are then used as boundary conditions to inject the jet into the three-dimensional (3D) computational domain.', '0712.0959-1-6-2': 'Typical axial jet velocities and ion densities vary in the range [MATH] km s[MATH] and [MATH] cm[MATH] respectively ; the jet is injected with an initial temperature [MATH] eV [MATH] K.', '0712.0959-1-6-3': 'The material of the wires, and thus the jet, is tungsten.', '0712.0959-1-6-4': 'In the jet and interaction region the magnetic field is negligible and in the 3D simulations presented here only the gasdynamic equations are integrated in time.', '0712.0959-1-6-5': 'The physical conditions of the cross-wind, produced by photo-ablation of the foil, depend on the incident radiation flux coming from the conical shock and are also time-dependent.', '0712.0959-1-6-6': 'However the simulations show that within the range of the expected and measured wind parameters , the dynamics of the interaction remains substantially the same and a detailed treatment of the time-variation of the wind is not essential.', '0712.0959-1-6-7': 'For the present simulations the injected wind has a constant velocity ([MATH] km s[MATH]) and temperature ([MATH] eV), the characteristic ratio of the jet velocity (which we remind is time-dependent) to the wind velocity [MATH].', '0712.0959-1-6-8': 'Two wind density profiles are discussed, one is constant [MATH] cm[MATH] (case A) and the other is exponentially increasing (case B).', '0712.0959-1-6-9': 'However for both cases the density contrast ratio is in the range [MATH]; the wind is supersonic and pressure gradients play no significant role in bending the jet.', '0712.0959-1-7-0': 'Simulation results for the laboratory jet-wind interaction are shown in Figure 2 as XUV emission maps (photon energies [MATH] eV) integrated along the line of sight.', '0712.0959-1-7-1': 'The emitted radiation includes bound-bound, free-bound and free-free transition and it is calculated assuming LTE within a screened hydrogenic model .', '0712.0959-1-7-2': 'Self-absorption of radiation and scattering are not taken into consideration.', '0712.0959-1-7-3': 'In the images the jet is injected through the lower boundary and the wind is injected on the left boundary.', '0712.0959-1-7-4': 'The collision of the wind with the jet generates a bow shock which envelopes the whole jet.', '0712.0959-1-7-5': 'As the bow shock is advected downstream with the wind, it develops a highly asymmetric shape, with the upwind side of the jet cocoon effectively disappearing.', '0712.0959-1-7-6': 'An oblique shock forms in the jet body which begins to bend and redirect the jet flow sideways.', '0712.0959-1-7-7': 'It is clear that the jet bends considerably more in case A where the time-averaged ram pressure of the wind is about an order magnitude higher than in case B.', '0712.0959-1-7-8': 'As the jet curves, momentum transferred from the jet to the working surface rapidly decreases, until the working surface effectively detaches from the jet; a result described by in the context of simulations of the jet HH 505.', '0712.0959-1-7-9': 'The subsequent propagation of the working surface is ballistic and as it entrains more ambient mass it slows down and its emission rapidly decays.', '0712.0959-1-7-10': 'The presence of internal shocks in the jet beam is visible in both the simulations and experiments.', '0712.0959-1-7-11': 'The experimental XUV self-emission image (Figure 3) shows the formation of a new working surface half-way up the image and enhanced emission inside the jet beam.', '0712.0959-1-7-12': 'These shocks are produced by perturbations developing along the jet-wind interface and their effect is to slow down and redirect the jet sideway, allowing the un-shocked flow ahead to detach.', '0712.0959-1-7-13': 'The effect is particularly pronounced for the case, considered here, where the jet momentum increases in time.', '0712.0959-1-7-14': 'This produces a narrowing of the angle of the main oblique shock (with respect to the z-axis) and causes the jet to bore a new, straighter channel in the ambient medium.', '0712.0959-1-7-15': 'Also a time-varying wind would have the same effect.', '0712.0959-1-7-16': 'Nevertheless the formation of new working surfaces was seen in simulations of uniform laboratory jets and wind, and in the simulations of astrophysical jets presented next.', '0712.0959-1-8-0': '# Jets from young stars', '0712.0959-1-9-0': 'The relevance of laboratory experiments to astrophysical jets, and for the case discussed here the jet-wind interaction, rests on the ability of producing both an adequate representation of the dynamics of the astrophysical system and, more importantly, obtaining dimensionless parameters in the appropriate astrophysical range .', '0712.0959-1-9-1': 'For the present experiments such conditions are well met and we expect much of the dynamics observed in the laboratory to be valid for astrophysical jets.', '0712.0959-1-10-0': 'Expected wind velocities vary from a few km s[MATH] for the jet-wind interaction associated with relative motions of TTauri stars with respect to the surrounding environment, (see for example [CITATION]) to typically higher velocities for irradiated jets: best fits to HH505 H[MATH] emission maps were obtained by [CITATION] for a wind velocity of 15 km s[MATH] and estimates by [CITATION] give wind velocities in the Orion nebula and in NGC1333 of [MATH] km s[MATH].', '0712.0959-1-10-1': 'Here we present simulations of HH jets in a cross-wind but do not model a specific jet.', '0712.0959-1-10-2': 'Instead we maintain the ratios of jet and wind velocities, and densities in a similar range as those obtained in the laboratory system.', '0712.0959-1-10-3': 'Specifically we take for the jet [MATH] km s[MATH], [MATH] cm[MATH] and for the wind [MATH] km s[MATH], [MATH] cm[MATH].', '0712.0959-1-10-4': 'Although these values are consistent with observations, the somewhat larger wind velocity was chosen to produce enough bending in the jet while at the same time maintaining as much as possible of the interesting dynamics within the computational domain.', '0712.0959-1-10-5': 'Lower [MATH] produce qualitatively similar results but because of the smaller curvature radius require considerably more computational resources.', '0712.0959-1-10-6': 'The initial jet radius [MATH] AU and the resolution is 4 AU over the entire Cartesian grid.', '0712.0959-1-10-7': 'The initial temperature in both jet and wind is 5000 K.', '0712.0959-1-10-8': 'The HH jet simulations are performed with an appropriately modified version of our laboratory code, where we now follow the time-dependent ionization of hydrogen.', '0712.0959-1-10-9': 'We take into account the recombination and collisional ionization and use the rate coefficients as tabulated in [CITATION] (and references therein).', '0712.0959-1-10-10': 'For temperatures above 15000 K cooling is implemented by a function appropriate for interstellar gas composition .', '0712.0959-1-10-11': 'However for temperatures below 15000 K cooling is calculated by including collisional excitation, collisional ionization and radiative recombination of hydrogen, and the collisional excitation of O I and O II.', '0712.0959-1-10-12': 'Because of charge exchange the neutral and singly ionized populations of oxygen are assumed to follow closely those of hydrogen ; the atomic abundance of oxygen is taken to be [MATH] that of hydrogen.', '0712.0959-1-11-0': 'The HH jet evolution is shown in Figure 4 in a time series of column density maps spanning 500 years of evolution.', '0712.0959-1-11-1': 'In general the overall jet dynamics is similar to that seen in laboratory jets.', '0712.0959-1-11-2': 'The presence of internal oblique shocks in the jet is apparent both in Figure 4 and also in Figure 5c which plots the H[MATH] emission.', '0712.0959-1-11-3': 'The simulations illustrate the formation of a number of bow-shaped shocks and knots in the resulting flow.', '0712.0959-1-11-4': 'It is worth noting that the clumpiness seen in the jet and the break-up of the bow shock arise entirely from the interaction with the cross-wind and are not imposed by means of a variable jet injection velocity.', '0712.0959-1-11-5': 'The latter is often used in numerical modeling to reproduce the knotty emissions features observed in the jets and which are thought to be caused by small velocity variations in the flow.', '0712.0959-1-11-6': 'The present simulations and more importantly the experiments, suggest that these internal shocks can also be produced by a wind impacting on a jet and are a dynamical observable feature dependent on the environment and not on the jet velocity profile.', '0712.0959-1-11-7': 'In addition to a steady cross-wind, the relative motion of the jet with respect to the interstellar medium, such as due to jet precession or the localized presence of transverse winds, may be also responsible for the formation of some of the knots observed.', '0712.0959-1-11-8': 'The detachment of the main working surface, as the jet body is increasingly curved under the action of the wind, occurs during the first 200 years and it is particularly evident in Figure 5b, which shows a passive tracer of the jet material at 370 years.', '0712.0959-1-11-9': 'The contact discontinuity between the jet and wind plasmas clearly shows perturbations along its surface which are ultimately responsible for producing the internal shocks in the jet.', '0712.0959-1-11-10': "It is also evident that the jet is 'naked': the upstream side of the jet is essentially in contact with the wind and the cocoon is absent.", '0712.0959-1-11-11': 'As pointed out in the work of [CITATION] the jet is liable to the growth of the Rayleigh-Taylor instability (RTI).', '0712.0959-1-11-12': 'We find that at the interface between the jet and the shocked wind, density and pressure gradients are in opposite directions, also the effective acceleration experienced by the jet material is in the direction of the centrifugal force and it is pointing into the density gradient, making the interface unstable.', '0712.0959-1-11-13': 'The development of the RTI in the jet beam is shown in Figures 5e and 5d which plot slices across the computational box (x-y plane) of the atomic number density.', '0712.0959-1-11-14': 'The wind is coming from the left and the jet is traveling out of the page.', '0712.0959-1-11-15': 'The arrows on the left-most panel indicate the height where the slices were taken.', '0712.0959-1-11-16': 'A well defined bow shock around the jet is clearly visible and so is the shock internal to the jet on the upwind side (Figure 5e).', '0712.0959-1-11-17': 'This is where the RTI can be seen to grow.', '0712.0959-1-11-18': 'The initial perturbation develops the characteristic bubble and spike structure (Figure 5d), with the tip of spikes rolling-up on the sides due to Kelvin-Helmholtz instability (KHI).', '0712.0959-1-11-19': 'The RTI tends to split the jet into well defined filaments; in addition as it penetrates deep into the jet the peak emission inside the jet shifts to the down-stream side, resulting into a broadening of the emission across the whole diameter.', '0712.0959-1-11-20': 'However we might expect that any departures from uniform jet, wind and symmetry would modify the dynamics of the RTI.', '0712.0959-1-12-0': 'An example of particular interest is given by the simulation of a rotating jet shown in Figures 5f-j.', '0712.0959-1-12-1': 'The jet is taken to rotate as a solid body and we set the azimuthal velocity at its boundary [MATH], where [MATH] is the initial sound speed in the jet.', '0712.0959-1-12-2': 'The other parameters of the wind and jet are as those of the non-rotating case.', '0712.0959-1-12-3': 'The evolution is indeed similar, however the cross section of the rotating jet is somewhat larger due to the increased radial expansion and the density is correspondingly lower.', '0712.0959-1-12-4': 'The result is that the rotating jet tends to have a higher curvature (see for example Figures 5b and 5g) and a shorter propagation length.', '0712.0959-1-12-5': 'The most interesting difference though, can be seen in the slices across the computational domain (Figure 5i and 5j).', '0712.0959-1-12-6': 'Because of rotation (anti-clockwise in the images) perturbations driven by the RTI are advected around the surface of the jet and are rapidly sheared seeding the KHI; however for the instabilities are confined to a narrower region of the jet.', '0712.0959-1-12-7': 'The H[MATH] emission maps in Figure 5c and 5h show that the internal shock for the rotating jet case remains narrower and the flow laminar for much longer.', '0712.0959-1-12-8': 'Thus rotation, combined with the action of the cross wind, reduces the growth of the RTI and the disruption of the jet.', '0712.0959-1-12-9': 'For the simulations presented here, the rotating jet survives up to [MATH] AU as opposed to [MATH] AU for the non-rotating case, above those heights the inabilities have grown to cover the whole cross sectional area of the jet.', '0712.0959-1-13-0': '# Discussion', '0712.0959-1-14-0': 'This work illustrates how combining laboratory experiments, numerical simulations of these experiments and the simulations of astrophysical systems can offer significant insights into the phenomena studied.', '0712.0959-1-14-1': 'In the case of the interaction of HH jets with a cross-wind, the results show the presence of oblique shocks and knots in curved jets, the formation of working surfaces, the detachment and fragmentation of the bow shock.', '0712.0959-1-14-2': 'In particular some of the observed features in HH jets may arise as a consequence of the curvature and the subsequent instabilities developing in the jet.', '0712.0959-1-14-3': 'However, it remains an open question what would be the minimum curvature required to produce them, especially for large values of the ratio [MATH], where computations become very demanding.', '0712.0959-1-14-4': 'The simulations show the development of the RTI in curved jets and how it may be partially quenched by rotation, which rapidly shears the RT modes.', '0712.0959-1-14-5': 'To determine the observational signatures of such effect and for direct comparison to observation, more realistic angular velocity profiles need to be used (see for example [CITATION]).', '0712.0959-1-15-0': 'A simple estimate of the RTI growth rate is obtained by assuming the plasma to be incompressible, its classical value is then given by [MATH] where [MATH] is the wave number of the perturbation and [MATH] is the Atwood number evaluated with the local densities across the interface.', '0712.0959-1-15-1': 'For the jets we take the effective acceleration [MATH] on the jet fluid to be equal to the centrifugal acceleration: [MATH], where [MATH] and [MATH] is the radius of curvature.', '0712.0959-1-15-2': 'Thus for the simulated HH jets we have: [MATH] m s[MATH], [MATH] AU, [MATH] m[MATH], [MATH]; where we have taken the wavelength of the perturbation to be the jet radius.', '0712.0959-1-15-3': 'The characteristic growth time is then [MATH] years which compares very well with the growth time of the instability seen in the simulations.', '0712.0959-1-15-4': 'Laboratory jets also are susceptible to the RTI.', '0712.0959-1-15-5': 'However while it was seen in simulations, it is not clear if the filamentary structures observed in the experiments (cf. Figure 3) are caused by the growth of the RTI into the jet body.', '0712.0959-1-15-6': 'In general for the experimental jets [MATH] m s[MATH], [MATH] m, [MATH] m[MATH], [MATH]; giving a characteristic growth time, [MATH] ns, which is comparable to the propagation time of the jet in the cross-wind.', '0712.0959-1-16-0': 'A useful relation to estimate the growth rate of the RTI can be simply derived by using the analytical expression of the curvature radius at the stagnation point for an isothermal jet given in [CITATION], [MATH].', '0712.0959-1-16-1': 'The growth rate can then be written as [EQUATION] which also shows that within these simplifying approximations [MATH] does not depend on the jet velocity.', '0712.0959-1-16-2': 'The sound speed [MATH] should be evaluated in the region of the jet body in pressure equilibrium with the shocked wind.', '0712.0959-1-16-3': 'Thus using reasonable values for the perturbed sound speed, [MATH] km s[MATH], we find for the simulated laboratory and astrophysical jets [MATH] ns and [MATH] years respectively; in good agreement with the previously estimated characteristic growth times.', '0712.0959-1-16-4': 'To observe the full development of the RTI in the laboratory jets we expect that experiments with longer interaction times ([MATH]) will be required.', '0712.0959-1-16-5': 'Such work is planned for the future and it will be coupled with the study of the effects of rotation on the jet.', '0712.0959-1-16-6': 'In fact, by using a modification of the conical array described in the present work we have recently demonstrated the possibility of producing rotating jet in the laboratory with ratios of the axial to azimuthal velocity, [MATH], of the same order as those reported in recent observation of HH jets .', '0712.0959-1-17-0': 'The authors wish to thank Fabio de Colle and Sylvie Cabrit for useful discussions.', '0712.0959-1-17-1': "The present work was supported in part by the European Community's Marie Curie Actions - Human Resource and Mobility within the JETSET (Jet Simulations Experiments and Theory) network under contract MRTN-CT-2004 005592.", '0712.0959-1-17-2': 'The authors also wish to acknowledge the London e-Science Centre (LESC) for the provision of computational facilities and support.'} | {'0712.0959-2-0-0': 'Herbig-Haro jets often show some degree of curvature along their path, in many cases produced by the ram pressure of a side-wind.', '0712.0959-2-0-1': 'We present simulations of both laboratory and astrophysical curved jets and experimental results from laboratory experiments.', '0712.0959-2-0-2': 'We discuss the properties and similarities of the laboratory and astrophysical flow, which show the formation of internal shocks and working surfaces.', '0712.0959-2-0-3': 'In particular the results illustrate how the break-up of the bow-shock and clumps in the flow are produced without invoking jet variability; we also discuss how jet rotation reduces the growth of the Rayleigh-Taylor instability in curved jets.', '0712.0959-2-1-0': '# Introduction', '0712.0959-2-2-0': 'Parsec-scale Herbig-Haro (HH) jets are one of the most spectacular phenomena connected to star formation.', '0712.0959-2-2-1': 'Observations of line emission features reveal an intricate succession of bow shocks and knots along the jet, and a complex and rich dynamics arising from the interaction with the circumstellar environment (see [CITATION] for a review).', '0712.0959-2-2-2': 'In the reflection nebula NGC 1333, in the Perseus molecular cloud, a number of bipolar HH jets exhibit a distinguishing C-shape morphology indicative of a steady bending .', '0712.0959-2-2-3': 'Less regular curvature is also observed in a number of other HH jets; for example in HH 30 a small side drift close to the jet source is followed further away by a sudden bending .', '0712.0959-2-2-4': 'In general, the curvature in jets has been linked either with the motion of the jet sources relative to the ambient medium or with the presence of a widespread outflow; both cases giving rise to an effective transverse ram pressure (cross-wind) which curves the jet.', '0712.0959-2-2-5': 'In fact, numerical simulations of jet bending produced by magnetic fields seem to require field strength that are unrealistically high .', '0712.0959-2-2-6': "For the jets in the Orion Nebula, it was also suggested that the bending may be caused by the 'rocket' effect induced by the photo-ionizing UV radiation from nearby high-mass stars ; recent work by [CITATION] has investigated the influence of an ionizing photon flux on the jet-wind collision in the HH555 jet.", '0712.0959-2-2-7': "The presence of curved jets is also common to other environments, such as in active galactic nuclei where they are observed as narrow-angle-tail radio sources (O'Dea and Owen 1986), or the pulsar jets in the Vela nebula, which are curved under the combined action of the wind within the supernovae remnant and the proper motion of the pulsar (Pavlov, Teter et al. 2003).", '0712.0959-2-2-8': 'Under simplifying assumption the curvature of the jets can be described analytically , however these models cannot detail the shock structures and the intricate dynamics developing in the interaction, and that were observed in a number of numerical studies.', '0712.0959-2-2-9': 'For the narrow-angle-tail sources internal shocks were shown to be responsible for the formation of density enhancements (knots) inside the jet beam which was then disrupted through the combined effect of Kelvin-Helmholtz and Rayleigh-Taylor instabilities.', '0712.0959-2-2-10': 'HH jets were first investigated numerically by [CITATION] for different incident angles of the wind and by [CITATION] for the case of irradiated jets.', '0712.0959-2-2-11': 'More recently the jet-wind collisions have also been studied in the laboratory with supersonic (Mach number [MATH]) jets interacting with a fast, radiatively ablated plasma that acted as a cross-wind.', '0712.0959-2-2-12': 'The laboratory results clearly demonstrated a steady curvature in the jets under the impinging cross-wind and the appearance of internal shocks.', '0712.0959-2-3-0': 'In the present work we combine consistently the laboratory modelling (experiments and simulations) with simulations of curved HH jets, drawing a parallel between the observed dynamics in the two systems and, with in mind future experiments, we also discuss the effects of rotation on curved jets.', '0712.0959-2-4-0': '# Laboratory Jets', '0712.0959-2-5-0': 'Laboratory jets are produced on the MAGPIE pulsed-power generator using a conical cage of micron-sized metallic wires (Al, Fe, W, etc) which is subject to a 1 Mega Ampere current, with a rise-time of 250 ns (see [CITATION] for a review).', '0712.0959-2-5-1': 'Figure 1 shows a schematic of the experimental set-up; the ohmically heated wires rapidly vaporize and turn into streams of plasma accelerated by the j[MATH]B[MATH] force towards the axis of the array.', '0712.0959-2-5-2': 'The collision of the streams forms a standing conical shock, which redirects the plasma axially into a jet.', '0712.0959-2-5-3': 'Radiative cooling rapidly reduces the temperature in the jet which attains high Mach numbers (M[MATH]) and small opening angles.', '0712.0959-2-5-4': 'The interaction of the jet with a side-wind is replicated in the laboratory by placing a plastic foil in the jet propagation region.', '0712.0959-2-5-5': 'XUV radiation from the standing shock radiatively ablates the foil and generates a fast moving plasma ([MATH] km s[MATH]), providing the ram pressure to bend the jet.', '0712.0959-2-5-6': 'By changing the position, angle and length of the foil the characteristic parameters of the impinging wind can be altered, and different regimes of the interaction studied .', '0712.0959-2-5-7': 'Similar to the HH jets, the characteristic cooling time in the laboratory jets is much smaller than the hydrodynamic time, this ratio is [MATH], and radiation losses play a significant role in the energy balance of the jets.', '0712.0959-2-5-8': 'The numerical simulations are performed with the three-dimensional resistive MHD code GORGON , the ion and electron energies are solved separately and include electron and ion thermal conductions.', '0712.0959-2-5-9': 'The coupling of the two energy equations due to collisional heat exchange is included as a sink/source term.', '0712.0959-2-5-10': 'The plasma is assumed to be in local thermodynamic equilibrium (LTE) and the average ionization is calculated by a Thomas-Fermi model.', '0712.0959-2-5-11': 'The plasma is optically thin and a radiation losses term is included in the electron energy equation.', '0712.0959-2-6-0': 'Two-dimensional axis-symmetric MHD simulations of the whole conical wire array were performed to obtain the mass and momentum fluxes of the jet.', '0712.0959-2-6-1': 'The calculated time and spatially varying profiles are then used as boundary conditions to inject the jet into the three-dimensional (3D) computational domain.', '0712.0959-2-6-2': 'Typical axial jet velocities and ion densities vary in the range [MATH] km s[MATH] and [MATH] cm[MATH] respectively ; the jet is injected with an initial temperature [MATH] eV [MATH] K.', '0712.0959-2-6-3': 'The material of the wires, and thus the jet, is tungsten.', '0712.0959-2-6-4': 'In the jet and interaction region the magnetic field is negligible and in the 3D simulations presented here only the gasdynamic equations are integrated in time.', '0712.0959-2-6-5': 'The physical conditions of the cross-wind, produced by photo-ablation of the foil, depend on the incident radiation flux coming from the conical shock and are also time-dependent.', '0712.0959-2-6-6': 'However the simulations show that within the range of the expected and measured wind parameters , the dynamics of the interaction remains substantially the same and a detailed treatment of the time-variation of the wind is not essential.', '0712.0959-2-6-7': 'For the present simulations the injected wind has a constant velocity ([MATH] km s[MATH]) and temperature ([MATH] eV), the characteristic ratio of the jet velocity (which we remind is time-dependent) to the wind velocity [MATH].', '0712.0959-2-6-8': 'Two wind density profiles are discussed, one is constant [MATH] cm[MATH] (case A) and the other is exponentially increasing (case B).', '0712.0959-2-6-9': 'However for both cases the density contrast ratio is in the range [MATH]; the wind is supersonic and pressure gradients play no significant role in bending the jet.', '0712.0959-2-7-0': 'Simulation results for the laboratory jet-wind interaction are shown in Figure 2 as XUV emission maps (photon energies [MATH] eV) integrated along the line of sight.', '0712.0959-2-7-1': 'The emitted radiation includes bound-bound, free-bound and free-free transition and it is calculated assuming LTE within a screened hydrogenic model .', '0712.0959-2-7-2': 'Self-absorption of radiation and scattering are not taken into consideration.', '0712.0959-2-7-3': 'In the images the jet is injected through the lower boundary and the wind is injected on the left boundary.', '0712.0959-2-7-4': 'The collision of the wind with the jet generates a bow shock which envelopes the whole jet.', '0712.0959-2-7-5': 'As the bow shock is advected downstream with the wind, it develops a highly asymmetric shape, with the upwind side of the jet cocoon effectively disappearing.', '0712.0959-2-7-6': 'An oblique shock forms in the jet body which begins to bend and redirect the jet flow sideways.', '0712.0959-2-7-7': 'It is clear that the jet bends considerably more in case A where the time-averaged ram pressure of the wind is about an order magnitude higher than in case B.', '0712.0959-2-7-8': 'As the jet curves, momentum transferred from the jet to the working surface rapidly decreases, until the working surface effectively detaches from the jet; a result described by in the context of simulations of the jet HH 505.', '0712.0959-2-7-9': 'The subsequent propagation of the working surface is ballistic and as it entrains more ambient mass it slows down and its emission rapidly decays.', '0712.0959-2-7-10': 'The presence of internal shocks in the jet beam is visible in both the simulations and experiments.', '0712.0959-2-7-11': 'The experimental XUV self-emission image (Figure 3) shows the formation of a new working surface half-way up the image and enhanced emission inside the jet beam.', '0712.0959-2-7-12': 'These shocks are produced by perturbations developing along the jet-wind interface and their effect is to slow down and redirect the jet sideway, allowing the un-shocked flow ahead to detach.', '0712.0959-2-7-13': 'The effect is particularly pronounced for the case, considered here, where the jet momentum increases in time.', '0712.0959-2-7-14': 'This produces a narrowing of the angle of the main oblique shock (with respect to the z-axis) and causes the jet to bore a new, straighter channel in the ambient medium.', '0712.0959-2-7-15': 'Also a time-varying wind would have the same effect.', '0712.0959-2-7-16': 'Nevertheless the formation of new working surfaces was seen in simulations of uniform laboratory jets and wind, and in the simulations of astrophysical jets presented next.', '0712.0959-2-8-0': '# Jets from young stars', '0712.0959-2-9-0': 'The relevance of laboratory experiments to astrophysical jets, and for the case discussed here the jet-wind interaction, rests on the ability of producing both an adequate representation of the dynamics of the astrophysical system and, more importantly, obtaining dimensionless parameters in the appropriate astrophysical range .', '0712.0959-2-9-1': 'For the present experiments such conditions are well met and we expect much of the dynamics observed in the laboratory to be valid for astrophysical jets.', '0712.0959-2-10-0': 'Expected wind velocities vary from a few km s[MATH] for the jet-wind interaction associated with relative motions of TTauri stars with respect to the surrounding environment, (see for example [CITATION]) to typically higher velocities for irradiated jets: best fits to HH505 H[MATH] emission maps were obtained by [CITATION] for a wind velocity of 15 km s[MATH] and estimates by [CITATION] give wind velocities in the Orion nebula and in NGC1333 of [MATH] km s[MATH].', '0712.0959-2-10-1': 'Here we present simulations of HH jets in a cross-wind but do not model a specific jet.', '0712.0959-2-10-2': 'Instead we maintain the ratios of jet and wind velocities, and densities in a similar range as those obtained in the laboratory system.', '0712.0959-2-10-3': 'Specifically we take for the jet [MATH] km s[MATH], [MATH] cm[MATH] and for the wind [MATH] km s[MATH], [MATH] cm[MATH].', '0712.0959-2-10-4': 'Although these values are consistent with observations, the somewhat larger wind velocity was chosen to produce enough bending in the jet while at the same time maintaining as much as possible of the interesting dynamics within the computational domain.', '0712.0959-2-10-5': 'Lower [MATH] produce qualitatively similar results but because of the smaller curvature radius require considerably more computational resources.', '0712.0959-2-10-6': 'The initial jet radius [MATH] AU and the resolution is 4 AU over the entire Cartesian grid.', '0712.0959-2-10-7': 'The initial temperature in both jet and wind is 5000 K.', '0712.0959-2-10-8': 'The HH jet simulations are performed with an appropriately modified version of our laboratory code, where we now follow the time-dependent ionization of hydrogen.', '0712.0959-2-10-9': 'We take into account the recombination and collisional ionization and use the rate coefficients as tabulated in [CITATION] (and references therein).', '0712.0959-2-10-10': 'For temperatures above 15000 K cooling is implemented by a function appropriate for interstellar gas composition .', '0712.0959-2-10-11': 'However for temperatures below 15000 K cooling is calculated by including collisional excitation, collisional ionization and radiative recombination of hydrogen, and the collisional excitation of O I and O II.', '0712.0959-2-10-12': 'Because of charge exchange the neutral and singly ionized populations of oxygen are assumed to follow closely those of hydrogen ; the atomic abundance of oxygen is taken to be [MATH] that of hydrogen.', '0712.0959-2-11-0': 'The HH jet evolution is shown in Figure 4 in a time series of column density maps spanning 500 years of evolution.', '0712.0959-2-11-1': 'In general the overall jet dynamics is similar to that seen in laboratory jets.', '0712.0959-2-11-2': 'The presence of internal oblique shocks in the jet is apparent both in Figure 4 and also in Figure 5c which plots the H[MATH] emission.', '0712.0959-2-11-3': 'The simulations illustrate the formation of a number of bow-shaped shocks and knots in the resulting flow.', '0712.0959-2-11-4': 'It is worth noting that the clumpiness seen in the jet and the break-up of the bow shock arise entirely from the interaction with the cross-wind and are not imposed by means of a variable jet injection velocity.', '0712.0959-2-11-5': 'The latter is often used in numerical modeling to reproduce the knotty emissions features observed in the jets and which are thought to be caused by small velocity variations in the flow.', '0712.0959-2-11-6': 'The present simulations and more importantly the experiments, suggest that these internal shocks can also be produced by a wind impacting on a jet and are a dynamical observable feature dependent on the environment and not on the jet velocity profile.', '0712.0959-2-11-7': 'In addition to a steady cross-wind, the relative motion of the jet with respect to the interstellar medium, such as due to jet precession or the localized presence of transverse winds, may be also responsible for the formation of some of the knots observed.', '0712.0959-2-11-8': 'The detachment of the main working surface, as the jet body is increasingly curved under the action of the wind, occurs during the first 200 years and it is particularly evident in Figure 5b, which shows a passive tracer of the jet material at 370 years.', '0712.0959-2-11-9': 'The contact discontinuity between the jet and wind plasmas clearly shows perturbations along its surface which are ultimately responsible for producing the internal shocks in the jet.', '0712.0959-2-11-10': "It is also evident that the jet is 'naked': the upstream side of the jet is essentially in contact with the wind and the cocoon is absent.", '0712.0959-2-11-11': 'As pointed out in the work of [CITATION] the jet is liable to the growth of the Rayleigh-Taylor instability (RTI).', '0712.0959-2-11-12': 'We find that at the interface between the jet and the shocked wind, density and pressure gradients are in opposite directions, also the effective acceleration experienced by the jet material is in the direction of the centrifugal force and it is pointing into the density gradient, making the interface unstable.', '0712.0959-2-11-13': 'The development of the RTI in the jet beam is shown in Figures 5e and 5d which plot slices across the computational box (x-y plane) of the atomic number density.', '0712.0959-2-11-14': 'The wind is coming from the left and the jet is traveling out of the page.', '0712.0959-2-11-15': 'The arrows on the left-most panel indicate the height where the slices were taken.', '0712.0959-2-11-16': 'A well defined bow shock around the jet is clearly visible and so is the shock internal to the jet on the upwind side (Figure 5e).', '0712.0959-2-11-17': 'This is where the RTI can be seen to grow.', '0712.0959-2-11-18': 'The initial perturbation develops the characteristic bubble and spike structure (Figure 5d), with the tip of spikes rolling-up on the sides due to Kelvin-Helmholtz instability (KHI).', '0712.0959-2-11-19': 'The RTI tends to split the jet into well defined filaments; in addition as it penetrates deep into the jet the peak emission inside the jet shifts to the down-stream side, resulting into a broadening of the emission across the whole diameter.', '0712.0959-2-11-20': 'However we might expect that any departures from uniform jet, wind and symmetry would modify the dynamics of the RTI.', '0712.0959-2-12-0': 'An example of particular interest is given by the simulation of a rotating jet shown in Figures 5f-j.', '0712.0959-2-12-1': 'The jet is taken to rotate as a solid body and we set the azimuthal velocity at its boundary [MATH], where [MATH] is the initial sound speed in the jet.', '0712.0959-2-12-2': 'The other parameters of the wind and jet are as those of the non-rotating case.', '0712.0959-2-12-3': 'The evolution is indeed similar, however the cross section of the rotating jet is somewhat larger due to the increased radial expansion and the density is correspondingly lower.', '0712.0959-2-12-4': 'The result is that the rotating jet tends to have a higher curvature (see for example Figures 5b and 5g) and a shorter propagation length.', '0712.0959-2-12-5': 'The most interesting difference though, can be seen in the slices across the computational domain (Figure 5i and 5j).', '0712.0959-2-12-6': 'Because of rotation (anti-clockwise in the images) perturbations driven by the RTI are advected around the surface of the jet and are rapidly sheared seeding the KHI; however for the instabilities are confined to a narrower region of the jet.', '0712.0959-2-12-7': 'The H[MATH] emission maps in Figure 5c and 5h show that the internal shock for the rotating jet case remains narrower and the flow laminar for much longer.', '0712.0959-2-12-8': 'Thus rotation, combined with the action of the cross wind, reduces the growth of the RTI and the disruption of the jet.', '0712.0959-2-12-9': 'For the simulations presented here, the rotating jet survives up to [MATH] AU as opposed to [MATH] AU for the non-rotating case, above those heights the inabilities have grown to cover the whole cross sectional area of the jet.', '0712.0959-2-13-0': '# Discussion', '0712.0959-2-14-0': 'This work illustrates how combining laboratory experiments, numerical simulations of these experiments and the simulations of astrophysical systems can offer significant insights into the phenomena studied.', '0712.0959-2-14-1': 'In the case of the interaction of HH jets with a cross-wind, the results show the presence of oblique shocks and knots in curved jets, the formation of working surfaces, the detachment and fragmentation of the bow shock.', '0712.0959-2-14-2': 'In particular some of the observed features in HH jets may arise as a consequence of the curvature and the subsequent instabilities developing in the jet.', '0712.0959-2-14-3': 'However, it remains an open question what would be the minimum curvature required to produce them, especially for large values of the ratio [MATH], where computations become very demanding.', '0712.0959-2-14-4': 'The simulations show the development of the RTI in curved jets and how it may be partially quenched by rotation, which rapidly shears the RT modes.', '0712.0959-2-14-5': 'To determine the observational signatures of such effect and for direct comparison to observation, more realistic angular velocity profiles need to be used (see for example [CITATION]).', '0712.0959-2-15-0': 'A simple estimate of the RTI growth rate is obtained by assuming the plasma to be incompressible, its classical value is then given by [MATH] where [MATH] is the wave number of the perturbation and [MATH] is the Atwood number evaluated with the local densities across the interface.', '0712.0959-2-15-1': 'For the jets we take the effective acceleration [MATH] on the jet fluid to be equal to the centrifugal acceleration: [MATH], where [MATH] and [MATH] is the radius of curvature.', '0712.0959-2-15-2': 'Thus for the simulated HH jets we have: [MATH] m s[MATH], [MATH] AU, [MATH] m[MATH], [MATH]; where we have taken the wavelength of the perturbation to be the jet radius.', '0712.0959-2-15-3': 'The characteristic growth time is then [MATH] years which compares very well with the growth time of the instability seen in the simulations.', '0712.0959-2-15-4': 'Laboratory jets also are susceptible to the RTI.', '0712.0959-2-15-5': 'However while it was seen in simulations, it is not clear if the filamentary structures observed in the experiments (cf. Figure 3) are caused by the growth of the RTI into the jet body.', '0712.0959-2-15-6': 'In general for the experimental jets [MATH] m s[MATH], [MATH] m, [MATH] m[MATH], [MATH]; giving a characteristic growth time, [MATH] ns, which is comparable to the propagation time of the jet in the cross-wind.', '0712.0959-2-16-0': 'A useful relation to estimate the growth rate of the RTI can be simply derived by using the analytical expression of the curvature radius at the stagnation point for an isothermal jet given in [CITATION], [MATH].', '0712.0959-2-16-1': 'The growth rate can then be written as [EQUATION] which also shows that within these simplifying approximations [MATH] does not depend on the jet velocity.', '0712.0959-2-16-2': 'The sound speed [MATH] should be evaluated in the region of the jet body in pressure equilibrium with the shocked wind.', '0712.0959-2-16-3': 'Thus using reasonable values for the perturbed sound speed, [MATH] km s[MATH], we find for the simulated laboratory and astrophysical jets [MATH] ns and [MATH] years respectively; in good agreement with the previously estimated characteristic growth times.', '0712.0959-2-16-4': 'To observe the full development of the RTI in the laboratory jets we expect that experiments with longer interaction times ([MATH]) will be required.', '0712.0959-2-16-5': 'Such work is planned for the future and it will be coupled with the study of the effects of rotation on the jet.', '0712.0959-2-16-6': 'In fact, by using a modification of the conical array described in the present work we have recently demonstrated the possibility of producing rotating jet in the laboratory with ratios of the axial to azimuthal velocity, [MATH], of the same order as those reported in recent observation of HH jets .', '0712.0959-2-17-0': 'The authors wish to thank Fabio de Colle and Sylvie Cabrit for useful discussions.', '0712.0959-2-17-1': "The present work was supported in part by the European Community's Marie Curie Actions - Human Resource and Mobility within the JETSET (Jet Simulations Experiments and Theory) network under contract MRTN-CT-2004 005592.", '0712.0959-2-17-2': 'The authors also wish to acknowledge the London e-Science Centre (LESC) for the provision of computational facilities and support.'} | [['0712.0959-1-3-0', '0712.0959-2-3-0'], ['0712.0959-1-14-0', '0712.0959-2-14-0'], ['0712.0959-1-14-1', '0712.0959-2-14-1'], ['0712.0959-1-14-2', '0712.0959-2-14-2'], ['0712.0959-1-14-3', '0712.0959-2-14-3'], ['0712.0959-1-14-4', '0712.0959-2-14-4'], ['0712.0959-1-14-5', '0712.0959-2-14-5'], ['0712.0959-1-7-0', '0712.0959-2-7-0'], ['0712.0959-1-7-1', '0712.0959-2-7-1'], ['0712.0959-1-7-2', '0712.0959-2-7-2'], ['0712.0959-1-7-3', '0712.0959-2-7-3'], ['0712.0959-1-7-4', '0712.0959-2-7-4'], ['0712.0959-1-7-5', '0712.0959-2-7-5'], ['0712.0959-1-7-6', '0712.0959-2-7-6'], ['0712.0959-1-7-7', '0712.0959-2-7-7'], ['0712.0959-1-7-8', '0712.0959-2-7-8'], ['0712.0959-1-7-9', '0712.0959-2-7-9'], ['0712.0959-1-7-10', 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'0712.0959-2-2-1'], ['0712.0959-1-2-2', '0712.0959-2-2-2'], ['0712.0959-1-2-3', '0712.0959-2-2-3'], ['0712.0959-1-2-4', '0712.0959-2-2-4'], ['0712.0959-1-2-5', '0712.0959-2-2-5'], ['0712.0959-1-2-6', '0712.0959-2-2-6'], ['0712.0959-1-2-7', '0712.0959-2-2-7'], ['0712.0959-1-2-8', '0712.0959-2-2-8'], ['0712.0959-1-2-9', '0712.0959-2-2-9'], ['0712.0959-1-2-10', '0712.0959-2-2-10'], ['0712.0959-1-2-11', '0712.0959-2-2-11'], ['0712.0959-1-2-12', '0712.0959-2-2-12'], ['0712.0959-1-9-0', '0712.0959-2-9-0'], ['0712.0959-1-9-1', '0712.0959-2-9-1'], ['0712.0959-1-10-0', '0712.0959-2-10-0'], ['0712.0959-1-10-1', '0712.0959-2-10-1'], ['0712.0959-1-10-2', '0712.0959-2-10-2'], ['0712.0959-1-10-3', '0712.0959-2-10-3'], ['0712.0959-1-10-4', '0712.0959-2-10-4'], ['0712.0959-1-10-5', '0712.0959-2-10-5'], ['0712.0959-1-10-6', '0712.0959-2-10-6'], ['0712.0959-1-10-7', '0712.0959-2-10-7'], ['0712.0959-1-10-8', '0712.0959-2-10-8'], ['0712.0959-1-10-9', '0712.0959-2-10-9'], ['0712.0959-1-10-10', '0712.0959-2-10-10'], ['0712.0959-1-10-11', '0712.0959-2-10-11'], ['0712.0959-1-10-12', '0712.0959-2-10-12'], ['0712.0959-1-5-0', '0712.0959-2-5-0'], ['0712.0959-1-5-1', '0712.0959-2-5-1'], ['0712.0959-1-5-2', '0712.0959-2-5-2'], ['0712.0959-1-5-3', '0712.0959-2-5-3'], ['0712.0959-1-5-4', '0712.0959-2-5-4'], ['0712.0959-1-5-5', '0712.0959-2-5-5'], ['0712.0959-1-5-6', '0712.0959-2-5-6'], ['0712.0959-1-5-7', '0712.0959-2-5-7'], ['0712.0959-1-5-8', '0712.0959-2-5-8'], ['0712.0959-1-5-9', '0712.0959-2-5-9'], ['0712.0959-1-5-10', '0712.0959-2-5-10'], ['0712.0959-1-5-11', '0712.0959-2-5-11'], ['0712.0959-1-16-0', '0712.0959-2-16-0'], ['0712.0959-1-16-1', '0712.0959-2-16-1'], ['0712.0959-1-16-2', '0712.0959-2-16-2'], ['0712.0959-1-16-3', '0712.0959-2-16-3'], ['0712.0959-1-16-4', '0712.0959-2-16-4'], ['0712.0959-1-16-5', '0712.0959-2-16-5'], ['0712.0959-1-16-6', '0712.0959-2-16-6'], ['0712.0959-1-12-0', '0712.0959-2-12-0'], ['0712.0959-1-12-1', '0712.0959-2-12-1'], ['0712.0959-1-12-2', '0712.0959-2-12-2'], ['0712.0959-1-12-3', '0712.0959-2-12-3'], ['0712.0959-1-12-4', '0712.0959-2-12-4'], ['0712.0959-1-12-5', '0712.0959-2-12-5'], ['0712.0959-1-12-6', '0712.0959-2-12-6'], ['0712.0959-1-12-7', '0712.0959-2-12-7'], ['0712.0959-1-12-8', '0712.0959-2-12-8'], ['0712.0959-1-12-9', '0712.0959-2-12-9'], ['0712.0959-1-17-0', '0712.0959-2-17-0'], ['0712.0959-1-17-1', '0712.0959-2-17-1'], ['0712.0959-1-17-2', '0712.0959-2-17-2'], ['0712.0959-1-11-0', '0712.0959-2-11-0'], ['0712.0959-1-11-1', '0712.0959-2-11-1'], ['0712.0959-1-11-2', '0712.0959-2-11-2'], ['0712.0959-1-11-3', '0712.0959-2-11-3'], ['0712.0959-1-11-4', '0712.0959-2-11-4'], ['0712.0959-1-11-5', '0712.0959-2-11-5'], ['0712.0959-1-11-6', '0712.0959-2-11-6'], ['0712.0959-1-11-7', '0712.0959-2-11-7'], ['0712.0959-1-11-8', '0712.0959-2-11-8'], ['0712.0959-1-11-9', '0712.0959-2-11-9'], ['0712.0959-1-11-10', '0712.0959-2-11-10'], ['0712.0959-1-11-11', '0712.0959-2-11-11'], ['0712.0959-1-11-12', '0712.0959-2-11-12'], ['0712.0959-1-11-13', '0712.0959-2-11-13'], ['0712.0959-1-11-14', '0712.0959-2-11-14'], ['0712.0959-1-11-15', '0712.0959-2-11-15'], ['0712.0959-1-11-16', '0712.0959-2-11-16'], ['0712.0959-1-11-17', '0712.0959-2-11-17'], ['0712.0959-1-11-18', '0712.0959-2-11-18'], ['0712.0959-1-11-19', '0712.0959-2-11-19'], ['0712.0959-1-11-20', '0712.0959-2-11-20'], ['0712.0959-1-0-0', '0712.0959-2-0-0'], ['0712.0959-1-0-1', '0712.0959-2-0-1'], ['0712.0959-1-0-2', '0712.0959-2-0-2'], ['0712.0959-1-0-3', '0712.0959-2-0-3'], ['0712.0959-1-15-0', '0712.0959-2-15-0'], ['0712.0959-1-15-1', '0712.0959-2-15-1'], ['0712.0959-1-15-2', '0712.0959-2-15-2'], ['0712.0959-1-15-3', '0712.0959-2-15-3'], ['0712.0959-1-15-4', '0712.0959-2-15-4'], ['0712.0959-1-15-5', '0712.0959-2-15-5'], ['0712.0959-1-15-6', '0712.0959-2-15-6']] | [] | [] | [] | [] | [] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/0712.0959 | null | null | null | null | null |
physics-0510150 | {'physics-0510150-1-0-0': 'Scattering of random surface gravity waves by small amplitude topography in the presence of a uniform current is investigated theoretically.', 'physics-0510150-1-0-1': 'This problem is relevant to ocean waves propagation on shallow continental shelves where tidal currents are often significant.', 'physics-0510150-1-0-2': 'A perturbation expansion of the wave action to second order in powers of the bottom amplitude yields an evolution equation for the wave action spectrum.', 'physics-0510150-1-0-3': 'Based on numerical calculations for sinusoidal bars, a mixed surface-bottom bispectrum that arises in the lowest order evolution equation is unlikely to be significant in most oceanic conditions.', 'physics-0510150-1-0-4': 'Neglecting that term, the present theory yields a closed wave equation with a scattering source term that gives the rate of exchange of the wave action spectrum between wave components, with conservation of the total action at each absolute frequency.', 'physics-0510150-1-0-5': 'With and without current, the scattering term yields reflection coefficients for the amplitudes of waves that converge, in the limit of small bottom amplitudes and small Froude numbers, to the results of previous theories for monochromatic waves propagating in one dimension over sinusoidal bars.', 'physics-0510150-1-0-6': 'In particular, the frequency of the waves that experience the maximum reflection is shifted by the current, as the surface wavenumber [MATH] changes for a fixed absolute frequency.', 'physics-0510150-1-0-7': 'Over sandy continental shelves, tidal currents are known to generate sandwaves with scales comparable to those of surface waves and elevation spectra that roll-off sharply at high wavenumbers.', 'physics-0510150-1-0-8': 'Application of the theory to such a real topography suggests that scattering mainly results in a broadening of the directional wave spectrum, due to forward scattering, while the back-scattering is generally weaker.', 'physics-0510150-1-0-9': 'The current may strongly influence surface gravity wave scattering by selecting different bottom scales with widely different spectral densities due the sharp bottom spectrum roll-off.', 'physics-0510150-1-1-0': '# Introduction', 'physics-0510150-1-2-0': 'Following the early observations of Heathershaw (1982), a considerable body of knowledge has been accumulated on the scattering of small amplitude surface gravity waves by periodic bottom topography.', 'physics-0510150-1-2-1': 'An asymptotic theory for small bottom amplitudes, that reproduces the observed scattering of monochromatic waves over a few sinusoidal bars, was put forward by Mei (1985), leading to practical phase-resolving equations that may be used to model this phenomenon for more general bottom shapes (Kirby 1986).', 'physics-0510150-1-2-2': 'For sinusoidal bottoms of wavenumber [MATH], Mei (1985) proposed an approximate analytical solution.', 'physics-0510150-1-2-3': 'In two dimensions (one horizontal and the vertical) this solution yields simple expressions for the wave amplitude reflection coefficient [MATH], as a function of the mismatch between the wavenumber of the surface waves [MATH] and the resonant value [MATH], for which [MATH] is maximum due to Bragg resonance.', 'physics-0510150-1-2-4': 'Beyond a cut-off value of that mismatch, it was found that the incident and reflected wave amplitudes oscillate in space instead of decreasing monotonically from the incident region.', 'physics-0510150-1-2-5': 'In three dimensions the Bragg resonance condition becomes [MATH] and [MATH], with [MATH], [MATH], and [MATH] the norms of the horizontal wave-vectors [MATH], [MATH], and [MATH].', 'physics-0510150-1-2-6': 'A uniform current was later introduced by Kirby (1988).', 'physics-0510150-1-2-7': 'The resonant condition is modified in that case, with [MATH] if incident and reflected waves propagate at different angles relative to the current direction.', 'physics-0510150-1-2-8': 'Other contributions have shown that higher-order theories are necessary to represent the sub-harmonic resonance observed over a bottom that is a superposition of two components of different wavelengths (Guazzelli, Rey Belzons 1992).', 'physics-0510150-1-2-9': 'Such sub-harmonic resonance was found to have as large an effect as the lowest order resonance for bottom amplitudes of only 25% of the water depth, due to a general stronger reflection for relatively longer waves.', 'physics-0510150-1-2-10': 'However, these methods are still prohibitively expensive for investigating the propagation of random waves over distances larger than about 100 wavelengths, and the details of the bottom are typically not available over large areas.', 'physics-0510150-1-2-11': 'Besides, a consistent phase-averaged wave energy evolution equation is also necessary for the investigation of the long waves associated with short wave groups (Hara Mei 1987).', 'physics-0510150-1-3-0': 'The large scale behaviour of the wave field may rather be represented by the evolution of the wave action spectrum assuming random phases.', 'physics-0510150-1-3-1': 'Such an approach was already proposed by Hasselmann (1966) for wind-wave propagation, and Elter Molyneux (1972) for the calculation of tsunami propagation.', 'physics-0510150-1-3-2': 'A proper theory for the evolution of the wave spectrum can be obtained from a solvability condition, a method similar to that of Mei (1985) and Kirby (1988), but applied to the action spectral densities instead of the amplitudes of monochromatic waves.', 'physics-0510150-1-3-3': 'In the absence of currents the correct form of that equation was first obtained by Ardhuin Herbers (2002, hereinafter referred to as AH) using a two scale approach.', 'physics-0510150-1-3-4': 'They decomposed the water depth [MATH] in a slowly varying depth [MATH], that causes shoaling and refraction, and a rapidly varying perturbation [MATH] with zero mean, that causes scattering.', 'physics-0510150-1-3-5': 'The resulting scattering was shown to be consistent with the dramatic increase of the directional width of the wave spectra observed on the North Carolina continental shelf.', 'physics-0510150-1-3-6': '(Ardhuin et al. 2003a, 2003b).', 'physics-0510150-1-3-7': "Recently, Magne et al. (2005, hereinafter referred to as MAHR) showed that AH's theory gives the same damping of incident waves as a Green function solution applied to any two dimensional topography (random or not) of small amplitude (see also Pihl, Mei Hancock 2002; Mei Hancock 2003).", 'physics-0510150-1-3-8': "Investigating the applicability limits of the scattering term of AH, MAHR also performed numerical calculations, comparing AH's theory to the accurate matched-boundary model of Rey (1995) that uses a decomposition of the bottom in a series of steps, including evanescent modes.", 'physics-0510150-1-3-9': "The numerical results show that AH's theory is generally limited by the relative bottom amplitude [MATH] rather than the bottom slope.", 'physics-0510150-1-4-0': 'The resulting expression of the scattered energy as a Bragg scattering term is consistent with results for scattering of acoustic and electromagnetic waves obtained by the small perturbation method, valid in the limit of small [MATH] with [MATH] the wavenumber of the propagating waves (Rayleigh 1896, see Elfouhaily Guerin 2004 for a review of this and other approximations).', 'physics-0510150-1-4-1': "Since there is no scattering for [MATH], as the waves do not 'feel' the bottom, the small parameter [MATH] may be used in our context, instead of the more general [MATH].", 'physics-0510150-1-4-2': 'The scattering strength is thus entirely determined by the bottom elevation variance spectrum at the bottom scales resonant with the incident waves.', 'physics-0510150-1-4-3': "Based on these results, Mei's (1985) theory should yield the same reflection coefficient as AH's theory in the limit of small bottom amplitudes.", 'physics-0510150-1-4-4': "Yet, AH predict that the wave amplitude in 2D would decay monotonically, which is not compatible with the oscillatory nature of Mei's theory for large detunings from resonance.", 'physics-0510150-1-4-5': "Further, outside of the surf zone and the associated multiple bar systems, the application of AH's theory is most relevant in areas where the bottom topography changes significantly on the scale of the wavelengths of swells.", 'physics-0510150-1-4-6': 'This often corresponds, over sand, to the presence of sandwaves.', 'physics-0510150-1-4-7': 'These sandwaves are generated by currents, and particularly by tidal currents (e.g. Dalrymple Knoght Lambiase 1978; Idier, Erhold Garlan 2002).', 'physics-0510150-1-4-8': 'It is thus logical to seek a theory for the scattering of waves in the presence of currents.', 'physics-0510150-1-4-9': 'A first theory was proposed by Kirby (1988), in the form of an extension of Mei (1985), with waves in a uniform current over a sinusoidal bottom.', 'physics-0510150-1-5-0': "The present paper provides an extension of AH's theory for the case of uniform currents in 2, and a detailed discussion of the differences between this theory and those of Mei (1985) and Kirby (1988) in 3.", 'physics-0510150-1-5-1': 'Finally the oceanographic effects of the current are investigated in 4 using a spectral phase-averaged numerical model, predicting the evolution of the wave action spectrum, and detailed measurements of the topography in the southern North Sea.', 'physics-0510150-1-5-2': 'Conclusions follow in 5.', 'physics-0510150-1-5-3': 'Further details on the source term derivation, and model results may be found in Magne (2005).', 'physics-0510150-1-6-0': '# Theory', 'physics-0510150-1-7-0': '## General formulation', 'physics-0510150-1-8-0': 'The variation in the action spectral density due to wave-bottom scattering is derived following the method of AH, now including the effect of a uniform current.', 'physics-0510150-1-8-1': 'We consider weakly nonlinear random waves propagating over an irregular bottom with a constant mean depth [MATH] and random small-scale topography [MATH], with [MATH] the horizontal position vector, so that the bottom elevation is given by [MATH] where [MATH] is the elevation relative to the mean water level.', 'physics-0510150-1-8-2': 'The free surface is at [MATH].', 'physics-0510150-1-8-3': 'Extension to current and mean depth variations on a large scale is expected to follow from a two-scale approximation, similar to the effect of large scale depth variation (AH).', 'physics-0510150-1-8-4': 'When the depth varies in the flow direction, the current should also vary so that the flows remains non-divergent.', 'physics-0510150-1-8-5': 'Perturbations of the current should thus be of the order of [MATH] with [MATH], and may scatter waves (Bal Chou 2002).', 'physics-0510150-1-8-6': 'As far as this and other effects do not modify the wave-bottom resonance, they should only contribute separate source terms, and shoaling and refraction terms.', 'physics-0510150-1-8-7': 'Further, for a depth-varying current, [MATH] should be regarded as the wave advection velocity (Andrews McIntyre 1978, see Kirby Chen 1989 for practical approximate expressions).', 'physics-0510150-1-9-0': 'The maximum surface slope is caracterized by [MATH] and we shall assume that [MATH] so that the bottom scattering contributions to the wave energy to order [MATH] are much larger than the resonant non-linear four wave interactions (Hasselmann 1962) and may be neglected.', 'physics-0510150-1-9-1': 'For shallow water waves ([MATH]) a stricter inequality is needed to prevent triad wave-wave interactions to enter the energy evolution equation at the same order as bottom scattering.', 'physics-0510150-1-10-0': 'The solution is obtained in a frame of reference moving with the current, so that the current only introduces a modification of the bottom boundary condition.', 'physics-0510150-1-10-1': "The governing wave equations are thus given by Laplace's equation for the wave potential, the bottom kinematic boundary conditions and a combination of Bernouilli's equation with the free surface kinematic boundary condition, [EQUATION]", 'physics-0510150-1-10-2': 'The symbol [MATH] represents the usual gradient operator restricted to the two horizontal dimensions.', 'physics-0510150-1-11-0': 'Following Hasselmann (1962) we shall approximate [MATH] and [MATH] with discrete sums, and take the limit to continuous integrals after deriving expressions for the evolution of the phase-averaged wave energy.', 'physics-0510150-1-11-1': 'The current [MATH] introduced a transformation of the horizontal coordinates [MATH], where [MATH] and [MATH] are the coordinates in the moving and fixed frames, respectively.', 'physics-0510150-1-11-2': 'The bottom elevation thus becomes [EQUATION]', 'physics-0510150-1-11-3': 'We look for a velocity potential solution in the form [EQUATION] where [MATH] and [MATH] are the radian frequencies in the moving and fixed frame, respectively.', 'physics-0510150-1-11-4': '[MATH] and [MATH] are the bottom and surface wavenumber, respectively, [MATH] is a sign index equal to 1 or [MATH], and [MATH].', 'physics-0510150-1-11-5': 'In the moving frame of reference, components with [MATH] propagate in the direction of the vector [MATH], while components with [MATH] propagate in the opposite direction.', 'physics-0510150-1-11-6': 'The amplitudes [MATH] are slowly modulated in time, with a slowness defined by the small parameter [MATH].', 'physics-0510150-1-11-7': 'Because [MATH] is a real quantity we also have [MATH].', 'physics-0510150-1-12-0': 'Numerical calculations by MAHR showed that the reflection coefficient predicted by AH were accurate for any bottom slope, but appeared to be limited by the relative bottom amplitude [MATH].', 'physics-0510150-1-12-1': 'We thus chose to expand the bottom boundary condition and wave potential in powers of [MATH], [EQUATION]', 'physics-0510150-1-12-2': 'The boundary conditions ([REF]) and ([REF]) are expressed at [MATH] and [MATH], respectively, using Taylor series of [MATH] about [MATH] and [MATH].', 'physics-0510150-1-13-0': 'The spectral statistics of free wave components can be expressed in terms of covariances [MATH] of the velocity potential amplitudes, [EQUATION]', 'physics-0510150-1-13-1': 'The contribution of the complex conjugate pairs of components ([MATH]) and ([MATH]) are combined in ([REF]) so that [MATH] is the covariance of all waves with wavenumber magnitude k propagating in the direction of [MATH].', 'physics-0510150-1-13-2': 'In the limit of small wavenumber separation, a continuous slowly-varying cross-spectrum can be defined (e.g. Priestley 1981, ch.11; see also AH), [EQUATION]', 'physics-0510150-1-13-3': 'The definition of all spectral densities are chosen so that the integral over the entire wavenumber plane yields the total covariance of [MATH] and [MATH].', 'physics-0510150-1-13-4': 'The bottom elevation spectrum in discrete form is given by [MATH] and in continuous form by [EQUATION] and verifies, [EQUATION] [MATH] is defined as the [MATH] order depth-integrated wave action contribution from correlation between [MATH] and [MATH] order components with wavenumber [MATH].', 'physics-0510150-1-13-5': 'For freely propagating waves, and following the common usage in non-accelerated reference frames, the gravity [MATH] is left out, so that the action has units of meters squared times second.', 'physics-0510150-1-13-6': 'Accurate to second order in [MATH] and [MATH] (Andrews McIntyre 1978), [MATH] is given from the velocity potential by the linear relation, [EQUATION]', 'physics-0510150-1-13-7': 'The spectral wave action is thus, [EQUATION]', 'physics-0510150-1-13-8': 'We shall now solve for the velocity potential in the frame of reference moving with the current.', 'physics-0510150-1-14-0': '## Zeroth-order solution', 'physics-0510150-1-15-0': 'In the moving frame of reference, the governing equations for [MATH] are unchanged from the case with a current.', 'physics-0510150-1-15-1': 'The solution is thus [EQUATION] where intrinsic frequency [MATH] is the positive root of the linear dispersion relation, [EQUATION]', 'physics-0510150-1-16-0': '## First-order solution', 'physics-0510150-1-17-0': 'The equations at order [MATH] are [EQUATION] and [EQUATION] where the non-linear terms [MATH] force a bound wave solution [MATH] (Hasselmann 1962) that will be neglected here because it does not modify our second order wave energy balance.', 'physics-0510150-1-17-1': 'The small-scale variation [MATH] that causes scattering now appears in the bottom boundary condition.', 'physics-0510150-1-17-2': 'A general solution for the (unchanged) Laplace equation ([REF]) is given by the following superposition of free and bound wave components, with amplitudes [MATH] and [MATH] respectively, [EQUATION]', 'physics-0510150-1-17-3': 'Substitution of ([REF]) in the bottom boundary condition ([REF]) yields [EQUATION] for [MATH].', 'physics-0510150-1-17-4': 'The amplitude of the [MATH] term forced by the [MATH] term in ([REF]) is given by [EQUATION]', 'physics-0510150-1-17-5': 'Replacing now ([REF]) in the surface boundary condition ([REF]), yields an equation for [MATH], [EQUATION] for [MATH], with [EQUATION]', 'physics-0510150-1-17-6': 'For the [MATH] component we have, [EQUATION] with [EQUATION]', 'physics-0510150-1-17-7': 'The forced harmonic oscillator equation ([REF]) leads to [EQUATION] with the function [MATH] defined in Appendix A, and [EQUATION]', 'physics-0510150-1-17-8': 'The components of amplitude [MATH] and [MATH] correspond to stationary waves such as generated by the bottom topography in rivers (e.g. Fredse, 1974).', 'physics-0510150-1-17-9': 'These components do not give rise to resonant interaction except at the critical Froude number, when the current is equal to the phase speed of the waves.', 'physics-0510150-1-17-10': 'This solution is identical to equation (2.9) in Kirby (1988) for monochromatic waves.', 'physics-0510150-1-17-11': 'The second term gives rise to scattered waves and reduces to the form given by AH when [MATH] goes to zero.', 'physics-0510150-1-18-0': '### First order energy', 'physics-0510150-1-19-0': 'The lowest order perturbation of the wave energy by scattering involves [MATH], and because it is a quadratic term, it is found in the first order covariance [EQUATION] with [MATH] denoting the real part.', 'physics-0510150-1-19-1': 'Including only the non-bounded terms, we get [EQUATION]', 'physics-0510150-1-19-2': 'Although this term was assumed to be zero in AH, it is not zero for sinusoidal bottoms with partially standing waves, and may become significant at resonance due to the function [MATH].', 'physics-0510150-1-19-3': 'At this order, another term is needed to balance this energy transfer.', 'physics-0510150-1-19-4': 'In uniform conditions, the time evolution of the wave field requires that the non-stationarity must come into play so that [MATH], the non-stationary term is given by AH (their appendix D), [EQUATION]', 'physics-0510150-1-19-5': 'In order to simplify the discussion, we shall briefly assume that there is no current and that the waves are unidirectional.', 'physics-0510150-1-19-6': 'In that case, [MATH] and [MATH].', 'physics-0510150-1-19-7': 'Replacing ([REF]) in ([REF]) and combining it with ([REF]) yields the action balance [EQUATION] with [MATH] denoting the imaginary part.', 'physics-0510150-1-20-0': 'For a real bottom (e.g. random or consisting of a finite series of sinusoidal bars), the evaluation of ([REF]) is not simple.', 'physics-0510150-1-20-1': 'Indeed, for directionally spread random waves and with a current, using [MATH] and taking the limit to continuous surface and bottom spectra yields [EQUATION] with the mixed surface bottom bispectrum [MATH] defined by [EQUATION] with [MATH] and [MATH].', 'physics-0510150-1-20-2': '[MATH] is identical to a classical bispectrum (e.g. Herbers et al. 2003) with one surface wave amplitude replaced by a bottom amplitude, and a similar expression is fond for a non-zero current.', 'physics-0510150-1-20-3': 'Since we have neglected non-linear effects, only the waves that have the same absolute frequency interact.', 'physics-0510150-1-20-4': 'Thus the phase coupling of all other wave component pairs is random, and the bispectrum is zero for [MATH].', 'physics-0510150-1-20-5': 'The energy balance ([REF]) is not closed, and requires a knowledge of the wave phases that are not available in a phase-averaged model.', 'physics-0510150-1-20-6': 'This contribution of the mixed bispectrum will thus be evaluated below, in order to investigate in which cases it may be neglected or parameterized.', 'physics-0510150-1-20-7': 'It is expected that [MATH] is generally negligible because MAHR have neglected [MATH], and still found a good agreement of the second order energy balance, with an exact numerical solution.', 'physics-0510150-1-21-0': '### Second order energy', 'physics-0510150-1-22-0': 'At the next order, one of the contributions from the covariance of the velocity potential amplitudes is given by [EQUATION]', 'physics-0510150-1-22-1': 'Using ([REF]), ([REF]) can be re-written as [EQUATION] in which [EQUATION] [MATH] is the one-dimension Dirac distribution, infinite where the argument is zero.', 'physics-0510150-1-22-2': 'Taking the limit of ([REF]) when [MATH], and changing variables from [MATH] to [MATH] yields [EQUATION]', 'physics-0510150-1-22-3': 'Only the terms for which [MATH] contribute to the integral.', 'physics-0510150-1-22-4': 'Thus [MATH], with [EQUATION]', 'physics-0510150-1-22-5': 'Using the relation between velocity potential and action given by ([REF]), and evaluating the integral over [MATH], one obtains [EQUATION]', 'physics-0510150-1-23-0': '## Second order solution', 'physics-0510150-1-24-0': 'Because we have computed one second order energy term, we now have to compute all other second order terms in ([REF]) to obtain the solvability condition.', 'physics-0510150-1-24-1': 'This requires solving for the second order potential [MATH], that is a solution of [EQUATION] that simplifies because odd vertical derivatives of [MATH] are zero at [MATH], [EQUATION] and [EQUATION]', 'physics-0510150-1-24-2': 'Ignoring the non-linear contributions [MATH], the solution [MATH] is given by the following form, [EQUATION]', 'physics-0510150-1-24-3': 'The non-stationarity term [MATH] is defined as the solution of the second order equations forced by only the first term on the right-hand side of ([REF]) and is given by AH.', 'physics-0510150-1-24-4': 'Following the method used at first order, substitution of ([REF]) in the bottom boundary condition ([REF]) leads to, for [MATH], [EQUATION]', 'physics-0510150-1-24-5': 'After calculations detailed in Appendix B (see Magne 2005 for further details), [MATH] yields the following contribution to the wave action, [EQUATION] in which [MATH], [MATH] and [MATH].', 'physics-0510150-1-25-0': '## Action and momentum balances', 'physics-0510150-1-26-0': 'The solvability condition for the spectral wave action at second order imposes that all secular terms cancel.', 'physics-0510150-1-26-1': 'Neglecting the first order energy contribution [MATH] given by ([REF]), and using [MATH] one has, [EQUATION] with the spectral action source term, [EQUATION] where [MATH] and [MATH].', 'physics-0510150-1-26-2': 'This interaction rule was already given by Kirby (1988).', 'physics-0510150-1-26-3': 'The only waves that can interact share the same absolute frequency [MATH].', 'physics-0510150-1-26-4': 'For a given [MATH] and without current, the resonant [MATH] and [MATH] lie on circles in the wavenumber plane (see AH).', 'physics-0510150-1-26-5': 'The current slightly modifies this geometric property.', 'physics-0510150-1-26-6': 'For [MATH] the circles become ellipses (Appendix C).', 'physics-0510150-1-27-0': 'For a given value of [MATH], one may obtain the source term integrated over all directions, [EQUATION]', 'physics-0510150-1-27-1': 'This expression is fully symmetric, and is thus unchanged when [MATH] and [MATH] are exchanged.', 'physics-0510150-1-27-2': 'Thus [MATH] is a substraction of two equal terms, so that for any bottom and wave spectra [MATH].', 'physics-0510150-1-27-3': "In other words, the 'source term' is rather an 'exchange term', and conserves the wave action at each absolute frequency.", 'physics-0510150-1-27-4': 'This conservation is consistent with the general wave action conservation theorem proved by Andrews McIntyre (1978), which states that there is no flux of action through an unperturbed boundary (here the bottom).', 'physics-0510150-1-28-0': '[MATH] may be re-written in a form close to that in AH, [EQUATION] with [EQUATION]', 'physics-0510150-1-28-1': 'Finally, we may also write the evolution equation for the wave pseudo-momentum [MATH] (see Andrews McIntyre 1978), where [MATH] is the density of sea water.', 'physics-0510150-1-28-2': 'For slow medium and wave field variations, that do not interfere with the scattering process, except by probably reducing the surface-bottom bispectrum [MATH], one obtains an extension of the equation of Phillips (1977) [EQUATION] with the dummy indices [MATH] and [MATH] denoting dummy horizontal components, and the scattering stress vector, [EQUATION]', 'physics-0510150-1-28-3': 'This stress has dimensions of force per unit length and corresponds to the force necessary to compensate for the divergence of the wave pseudo-momentum flux.', 'physics-0510150-1-28-4': 'Based on the results of Longuet-Higgins (1967) and Hara Mei (1987), this force does not contribute to the mean flow equilibrium with a balance of the radiation stresses divergence by long waves (or wave set-up in stationary conditions), contrary to the initial proposition of Mei (1985).', 'physics-0510150-1-28-5': 'This force is thus provided by a mean non-hydrostatic pressure on the bottom that correlates with the bottom slope, and must arise from the pressure under partial standing waves locked in phase with the bottom undulations.', 'physics-0510150-1-29-0': '# Wave scattering in two dimensions', 'physics-0510150-1-30-0': 'Before considering the full complexity of the 3D wave-bottom scattering in the presence of a current, we first examine the behavior of the source term in the case of 2D sinusoidal seabeds.', 'physics-0510150-1-30-1': 'MAHR have investigated the applicability limits of the source term with [MATH], using [MATH]D test cases.', 'physics-0510150-1-30-2': 'They showed that for small bottom amplitudes the source term yields accurate reflection estimates, even for localized scatterers.', 'physics-0510150-1-30-3': "It is thus expected that this also holds for [MATH], and that the present theory should conform to Kirby's (1988) theory in the limit of small reflection coefficients.", 'physics-0510150-1-31-0': '## Wave evolution equation in [MATH]D', 'physics-0510150-1-32-0': 'We consider here a steady wave field in two dimension with incident and reflected waves propagating along the [MATH]-axis.', 'physics-0510150-1-32-1': 'We shall consider in particular the case of [MATH] sinusoidal bars of amplitude [MATH] and height [MATH], defined by, [EQUATION]', 'physics-0510150-1-32-2': 'This form is identical to that of the bottom profile chosen by Kirby (1988) but differs, for [MATH], by a [MATH] phase shift from the bottom profile chosen by Mei (1985).', 'physics-0510150-1-32-3': 'The bottom spectrum is thus [EQUATION]', 'physics-0510150-1-32-4': 'For this particular bottom [EQUATION] with [EQUATION]', 'physics-0510150-1-32-5': 'Please note that this is a double-sided spectrum, with only half of the bottom variance contained in the range [MATH].', 'physics-0510150-1-32-6': 'For a generic bottom, for which [MATH] does not go to zero at infinity, the spectrum is obtained using standard spectral analysis method, for example, from the Fourier transform of the bottom auto-covariance function (see MAHR).', 'physics-0510150-1-33-0': 'First, replacing ([REF]) in ([REF]) removes the angular integral in the source term.', 'physics-0510150-1-33-1': 'Taking [MATH], we have [MATH], and [EQUATION]', 'physics-0510150-1-33-2': 'Second, assuming now that waves propagate only along the [MATH]-axis, the wave spectral densities are of the form [EQUATION]', 'physics-0510150-1-33-3': 'Integrating over [MATH] removes the singularities on [MATH], and assuming a steady state one obtains [EQUATION] with [EQUATION]', 'physics-0510150-1-33-4': 'Although the present theory is formulated for random waves, the wave spectrum does not need to be continuous because there is no possible coupling between waves of different frequencies.', 'physics-0510150-1-33-5': 'We may thus express the result for monochromatic incident waves, such that, [MATH] with [MATH] and [MATH].', 'physics-0510150-1-33-6': 'The resulting evolution equation is, omitting the 0 subscripts on [MATH] and [MATH], [EQUATION] with a similar equation for [MATH] obtained by exchanging [MATH] and [MATH], and [MATH] and [MATH], from which it is easy to verify that the total action is conserved.', 'physics-0510150-1-34-0': 'The stationary evolution equation only couples two wave components [MATH] and [MATH].', 'physics-0510150-1-34-1': 'For a uniform mean depth [MATH], and uniform bottom spectrum [MATH], as considered here, we thus have a linear system of two differential equations, that may be written in matrix form for any [MATH], [EQUATION]', 'physics-0510150-1-34-2': 'The general solution is thus [EQUATION] and the reflection coefficient for the wave action is found using the boundary condition expressing the absence of incoming waves from beyond the bars, [MATH], giving, [EQUATION]', 'physics-0510150-1-34-3': 'A reflection coefficient for the modulus of the wave amplitude predicted by the source term is thus, [EQUATION]', 'physics-0510150-1-35-0': '## Analytical solution for [MATH]', 'physics-0510150-1-36-0': 'If [MATH] then [MATH], and [EQUATION] with [MATH] the [MATH] component of [MATH].', 'physics-0510150-1-36-1': '[MATH] is not diagonalizable, which would allow a simple way of evaluating the matrix exponential [MATH].', 'physics-0510150-1-36-2': 'However [MATH] so that [MATH], where [MATH] is the identity matrix.', 'physics-0510150-1-36-3': 'The solution is thus simply, [EQUATION]', 'physics-0510150-1-36-4': 'An example of spatial variation of the wave spectrum from [MATH] to [MATH] is shown in Figure [REF], for [MATH], and a uniform (white) incident spectrum.', 'physics-0510150-1-36-5': 'The reflected wave energy (at [MATH] in figure [REF].a', 'physics-0510150-1-36-6': 'a) compensates the loss of energy in the transmitted spectrum (at [MATH] in figure [REF].b', 'physics-0510150-1-37-0': "At resonance, in the limit of small bar amplitudes ([REF]) yields [EQUATION] which is identical to Mei's (1985) equation (3.21)-(3.22) for exact resonance, in the limit of [MATH], and also converges to the result of Davies Heathershaw (1984) for that same limit.", 'physics-0510150-1-37-1': 'For large bar amplitudes, the reflection is significant if the bars occupy a length [MATH] longer than the localization length [MATH].', 'physics-0510150-1-37-2': 'However, the reflection coefficient for the wave amplitude only increases with [MATH] as [MATH], which is slower than the exponential asymptote given by Mei (1985) for sinusoidal bars, and predicted by (Belzons et al. 1988) from the lowest-order theory applied to a random bottom.', 'physics-0510150-1-37-3': 'Our use of higher-order correction may be thought as the representation of multiple reflections that tend to increase the penetration length in the random medium.', 'physics-0510150-1-38-0': 'A deeper understanding of this question is provided by the comparison of numerical estimations of the reflection coefficients for the wave amplitudes [MATH].', 'physics-0510150-1-38-1': 'A benchmark estimation for linear waves is provided by the step-wise model of Rey (1995) using integral matching conditions for the free propagating waves and three evanescent modes at the step boundaries.', 'physics-0510150-1-38-2': "This model is known to converge to the reflection coefficents given by an exact solution of Laplace's equation and the boundary conditions, in the limite of an infinite number of steps and evanescent modes.", 'physics-0510150-1-38-3': 'Calculations are performed here with 70 steps.', 'physics-0510150-1-38-4': 'This number is chosen because a larger number of steps gives indistinguishable results in figure [REF].', 'physics-0510150-1-38-5': 'An analytical expression [MATH] is given by Mei (1985).', 'physics-0510150-1-38-6': '[MATH] for the present second order theory is given by [MATH] ([REF]).', 'physics-0510150-1-39-0': 'We further compare these estimates to the reflection coefficient [MATH] that is deduced from the energy evolution given by Hara Mei (1987) using the approximate solutions of Mei (1985, his equations 3.8-3.23).', 'physics-0510150-1-39-1': "One may prefer to reformulate the energy evolution from the amplitude evolution equations of Kirby (1988) because he used a continuous water depth [MATH], instead of Mei's [MATH] which is discontinuous at [MATH] and [MATH].", 'physics-0510150-1-39-2': "Yet both Mei's and Kirby's equations lead to the same energy exchange between the incident and reflected components.", 'physics-0510150-1-39-3': "Using Mei's (1985) notations, the amplitudes of the incident waves, reflected waves, and bottom undulations are [MATH], [MATH], and [MATH], and the 'cut-off' frequency is [EQUATION]", 'physics-0510150-1-39-4': 'The energy evolution of waves propagating over sinusoidal bars along the [MATH]-axis is given by Hara Mei (1987).', 'physics-0510150-1-39-5': 'The reflected wave energy [MATH] should be a solution of [EQUATION] where [MATH] denotes the complex conjugate of [MATH].', 'physics-0510150-1-39-6': 'This is identical to ([REF]) for a monochromatic bottom except that the imaginary part replaced by a real part.', 'physics-0510150-1-40-0': 'Equation ([REF]) yields a corresponding energy reflection coefficient, given by the fraction of energy lost by the incoming waves, [EQUATION]', 'physics-0510150-1-40-1': "Simple analytical expressions can be obtained at resonance, where Mei's (1985) eq. (3.20)-(3.21) give, [EQUATION] so that [EQUATION] and [EQUATION]", 'physics-0510150-1-40-2': 'It is not surprising that the energy transfer thus computed differs from the energy computed from the amplitude evolution equations.', 'physics-0510150-1-40-3': 'This is typical of small perturbation methods, and was discussed by Hasselmann (1962), among others.', 'physics-0510150-1-40-4': 'Yet, it is remarkable that the ratio of the two is exactly one half.', 'physics-0510150-1-40-5': 'The transfer of energy given by [MATH] in ([REF]) thus correspond to an amplitude reflection coefficient [MATH] that is smaller by a factor [MATH], at resonance, compared to [MATH] (figure 3).', 'physics-0510150-1-40-6': 'This underprediction of the the reflexion of the energy by ([REF]) also has consequences for the analysis and calculation of wave set-up due to wave group propagation over a reflecting bottom.', 'physics-0510150-1-40-7': 'Indeed, the estimation of the scattering stress ([REF]), that contribute to the driving of long waves, was analyzed by Hara Mei (1987) using ([REF]), which is a factor 2 too small.', 'physics-0510150-1-40-8': 'This may explain, in part, their under-prediction of the observed elevation of the long wave travelling with the incident wave group.', 'physics-0510150-1-41-0': '## Effects of wave and bottom relative phases', 'physics-0510150-1-42-0': 'The energy exchange coefficient given by the source term always gives energy to the least energetic components (in the absence of currents), and thus the energy evolution is monotonic.', 'physics-0510150-1-42-1': 'However, the first order term that was neglected so far may have any sign, and thus lead to oscillatory evolutions for the wave amplitudes, as predicted by Mei (1985) and observed by Hara Mei (1987).', 'physics-0510150-1-42-2': 'At resonance, and for [MATH], it can be seen that the first-order energy product [MATH] in ([REF]) is equal to [MATH], in the limit of a large number of bars.', 'physics-0510150-1-42-3': "Based on Mei's (1985) approximate solution, in the absence of waves coming from across the bars, this quantity is purely real so that its imaginary part is zero and the corresponding reflection coefficient [MATH] is zero.", 'physics-0510150-1-42-4': "For [MATH] this property remains as can be seen by replacing Mei's (1985) solution with Kirby's (1988).", 'physics-0510150-1-42-5': 'However, similar correlation terms were also neglected in the second order energy, so that the oscillatory behaviour may occur due to terms of the same order as the scattering source term, including interactions of the sub-harmonic kind (Guazzelli et al. 1992).', 'physics-0510150-1-42-6': 'Further, the bottom-surface bispectrum may become significant in the first order term if there is a large amount of wave energy coming from beyond the bars.', 'physics-0510150-1-42-7': 'This kind of situation, e.g. due to reflection over a beach, was discussed by Yu Mei (2000).', 'physics-0510150-1-43-0': 'In the absence of such a reflection, and away from resonance but for small values of the scattering strength parameter [MATH], the imaginary part of [MATH] is an order [MATH] smaller than the real part and thus contributes a negligible amount to the reflection.', 'physics-0510150-1-44-0': '## Source term and deterministic results for sinusoidal bars', 'physics-0510150-1-45-0': 'For large bar amplitudes, such as [MATH] (figure 3.', 'physics-0510150-1-45-1': 'a), all theories with linearized bottom boundary conditions fail to capture the shift of the reflection pattern to lower wavenumbers.', 'physics-0510150-1-45-2': 'This effect was discussed by Rey (1992), and attributed to the non-linear nature of the dispersion relation and the rapid changes in the water depth.', 'physics-0510150-1-45-3': 'However reflection coefficients are still relatively well estimated.', 'physics-0510150-1-45-4': "For these large amplitudes Mei's (1985) approximate solution is found to be more accurate at resonance compared to the source term.", 'physics-0510150-1-45-5': 'As expected from MAHR, [MATH] and [MATH] become identical as [MATH] goes to zero (figure [REF].', 'physics-0510150-1-45-6': 'b).', 'physics-0510150-1-45-7': "This fact provides a verification that the first order term [MATH] is different from Hara and Mei's (1987) energy transfer term, and only accounts for a small fraction of the reflection, a fraction that goes to zero as [MATH].", 'physics-0510150-1-45-8': "It is also found that for all bottom amplitudes, but away from resonance, the source term expression provides a simple solution that is more accurate than Mei's (1985) approximate solutions (see the sidelobes in figure 3).", 'physics-0510150-1-46-0': '## Effects of currents', 'physics-0510150-1-47-0': 'The basic feature of the solutions with currents is the modification of the resonant condition from [MATH] and [MATH], to [MATH] and [MATH] .', 'physics-0510150-1-47-1': 'Notations here assume that [MATH] is in the direction of the current and [MATH] is opposite to the current.', 'physics-0510150-1-47-2': 'Yet, the introduction of the current makes the solution much more complex.', 'physics-0510150-1-47-3': "It is striking that Kirby's (1988) equations involve a modified cut-off parameter [MATH], which is analytically similar to our source term result except for the rather complicated [MATH] term.", 'physics-0510150-1-47-4': 'At resonance, Kirby (1988) found that the transmission losses [MATH] differ from the case without current.', 'physics-0510150-1-47-5': 'In the long-wave limit this difference is increased by a factor [MATH] (his equation 5.29).', 'physics-0510150-1-47-6': 'However, the present theory predicts an increase by a factor [MATH].', 'physics-0510150-1-47-7': 'Thus both theories only agree in the limit [MATH].', 'physics-0510150-1-47-8': 'To our knowledge there is unfortunately no simple numerical method to arrive at an independent solutions.', 'physics-0510150-1-47-9': 'Besides, observations of that effect require to test relatively large Froude numbers.', 'physics-0510150-1-47-10': 'Some first observations of the shift in resonant frequencies in the presence of currents were only performed at relatively low Froude numbers, and are not accurate enough to test these predictions (Magne, Rey Ardhuin manuscript submitted to Physics of Fluids).', 'physics-0510150-1-48-0': "We now compare reflection coefficients for monochromatic waves, as obtained with the source term using ([REF]), and with Kirby's (1988) analytical approximate solutions for near-resonant waves.", 'physics-0510150-1-48-1': 'An approximation to the reflection coefficient ([REF]) corresponding to the solution of ([REF]) is obtained with a fourth order Taylor expansion of the matrix exponential.', 'physics-0510150-1-48-2': 'Anticipating oceanographic conditions with a water depth of 20 m, a strong 2 m s[MATH] current corresponds to a Froude number of 0.17 only.', 'physics-0510150-1-48-3': "For such a low value of [MATH] in the context of the Davies Heathershaw (1984) laboratory experiments, the difference in peak reflection for [MATH] is of 8% only between the source term and Kirby's (1988) solution, while the reflection coefficient is largely increased due to the general conservation of the wave action flux.", 'physics-0510150-1-48-4': '[MATH] can thus be larger than 1 for currents following the incident waves because it is enhanced by the factor [MATH], compared to the transmission losses [MATH].', 'physics-0510150-1-48-5': 'The overall increase in [MATH] for following waves amounts to about 60% at [MATH], for the laboratory sinusoidal bars of Davies Heathershaw (1984) shown before (figure 4).', 'physics-0510150-1-48-6': 'The horizontal density of reflected wave energy is thus multiplied by a factor 2.5 in this case, for a Froude number of only 0.17.', 'physics-0510150-1-49-0': 'Both solutions agree reasonably well, and we thus expect the source term to represent accurately the scattering of waves over bottom topographies in cases of uniform currents.', 'physics-0510150-1-49-1': "For four sinusoidal bars the energy reflection coefficients was found to be within 20% of the exact solution for [MATH] and [MATH], and this conclusion is expected to hold for [MATH], given the agreement with Kirby's (1988) approximate solution.", 'physics-0510150-1-49-2': 'This accuracy is a factor two better than what was found for a rectangular step with [MATH] (MAHR).', 'physics-0510150-1-49-3': 'The present method has the advantage of a large economy in computing power.', 'physics-0510150-1-49-4': 'This method is also well adapted for natural sea beds, for which continuous bathymetric coverage is only available in restricted areas, and thus only the statistical properties of the bottom topography are accessible, assuming homogeneity.', 'physics-0510150-1-50-0': '# Scattering with current on a realistic topography', 'physics-0510150-1-51-0': '## Sandwaves in the North Sea', 'physics-0510150-1-52-0': 'A real ocean topography, at least on the continental shelf, generally presents a continuous and broad bottom elevation spectrum.', 'physics-0510150-1-52-1': 'Given this bottom spectrum, simple solutions are available for uniform conditions, because the scattering source term is a linear function of the directional spectrum at a given value of the absolute frequency [MATH] (see AH for numerical methods).', 'physics-0510150-1-52-2': 'However, practical situations rather correspond to quasi-stationary conditions with spatial gradients in at least one dimension.', 'physics-0510150-1-52-3': 'In this situation the simple steady solutions found above for 2D topography are not physical.', 'physics-0510150-1-52-4': 'Indeed, a 3D bottom causes scattering along the transversal direction [MATH], and the energy propagating in that direction builds up slowly up to the point where it becomes as large as the incident wave energy.', 'physics-0510150-1-52-5': 'This process can take a time much longer than the typical duration of a storm or swell arrival, and dissipative processes are likely to be important as the wave energy increases (e.g. Ardhuin et al. 2003).', 'physics-0510150-1-53-0': 'Therefore the source term [MATH] was introduced in the version [MATH] of the wave model WAVEWATCH-III (Tolman 2002), based on the wave action evolution equation ([REF]) in which the time derivative on the left hand side is now a Lagrangian derivative following a wave packet in physical and spectral space.', 'physics-0510150-1-53-1': 'Bottom scattering is the only source term introduced in the present calculation.', 'physics-0510150-1-53-2': 'There is thus no transfer of wave action between frequencies.', 'physics-0510150-1-53-3': 'However, the model uses a spectrum is discretized with components at fixed intrinsic frequencies [MATH] and directions [MATH], which is most appropriate for other processes.', 'physics-0510150-1-53-4': 'Thus, the model was run with a typical grid of [MATH] frequencies ranging from [MATH] to [MATH] Hz and a high directional resolution of [MATH].', 'physics-0510150-1-54-0': 'The effects of a mean current on wave scattering are now examined using a real bathymetry spectrum that is estimated from a detailed bathymetric survey of an area centered on the crest of a sand dune, in the southern North Sea (figure [REF]).', 'physics-0510150-1-54-1': 'In this region, tidal currents are known to generate a wide array of bedforms, from large scale tidal Banks to sand dunes and sand waves (e.g. Dyer Huntley 1999; Hulscher van den Brink 2001).', 'physics-0510150-1-54-2': 'Although sand dunes present a threat to navigation and are closely monitored (Idier et al. 2002), dunes are much larger than typical wind seas and swells wavelengths.', 'physics-0510150-1-54-3': 'These dunes, however, are generally covered with shorter sandwaves.', 'physics-0510150-1-54-4': 'In the surveyed area the sandwaves have a peak wavelength of 250 m, and a variance of 1.7 m[MATH], which should lead to strong oblique scattering of waves with periods of 10 s and longer.', 'physics-0510150-1-54-5': 'Over areas of 3 by 3 km the variance can be as large as 3.3 m[MATH] with a better defined spectral peak, so that our chosen spectrum is expected to be representative of the entire region, including high and low variances on dunes crests and troughs, respectively.', 'physics-0510150-1-54-6': 'The southern North Sea is also known for the attenuation of long swells, generated in the Norwegian Sea.', 'physics-0510150-1-54-7': 'This attenuation has been generally attributed to the dissipation of wave energy by bottom friction (Weber 1991).', 'physics-0510150-1-55-0': 'The bottom spectrum of the area that we chose, like the spectra that were obtained by AH from the North Carolina shelf, roll off sharply at high wavenumbers, typically like [MATH] for the two-dimensional spectrum.', 'physics-0510150-1-55-1': 'Here the maximum variance is found for bottom wavelengths of the order or larger than 250 m (figure [REF]).', 'physics-0510150-1-55-2': 'For a typical swell period of 10 s, this corresponds to 2 times the wavelength in 20 m depth, and thus a rather small scattering angle, 30[MATH] off from the incident direction.', 'physics-0510150-1-55-3': 'Swells propagating from a distant storm, with fixed absolute frequency [MATH], should be reflected by bottom undulations with widely different variances as the current changes.', 'physics-0510150-1-56-0': '## Scattering of waves normally incident on the sandwaves', 'physics-0510150-1-57-0': 'To simplify the interpretation of the results, and the processing of the boundary conditions, a one dimensional (East/West) propagation grid is used for the computations, assuming that the wave field, still fully directional, is uniform in the North-South direction.', 'physics-0510150-1-57-1': 'The waves are propagated over a model grid [MATH] km long, with a mean depth of [MATH]m, and a spatial grid step of 5 km (figure [REF].', 'physics-0510150-1-57-2': 'a).', 'physics-0510150-1-57-3': 'As a result, scattering is probably stronger than in real conditions where the mean water depth is typically larger than 20 m.', 'physics-0510150-1-57-4': 'The following results should still provide some understanding of the likely real effects, at least for larger wave periods with similar values of [MATH].', 'physics-0510150-1-58-0': 'A Gaussian incident surface wave spectrum is imposed, with a mean direction from the North-East, a narrow peak directional spread of [MATH], and a peak frequency of [MATH] Hz (figure [REF].', 'physics-0510150-1-58-1': 'b).', 'physics-0510150-1-58-2': 'The source term is integrated with a time step of [MATH] s, and the advection in space uses a third order scheme with a time step of [MATH] s.', 'physics-0510150-1-59-0': 'The scattering source term acts as a diffusion operator with a typical 3-lobe structure, negative at the peak of the wave spectrum, and positive in directions of about 30[MATH] on both sides of the peak.', 'physics-0510150-1-59-1': 'This is identical, but with a larger magnitude, to the effect described by AH.', 'physics-0510150-1-59-2': 'In general the scattering effects are relatively stronger at the lowest frequencies, at least in the range of frequencies used here.', 'physics-0510150-1-59-3': 'For still lower frequencies the scattering coefficient [MATH] decreases (see also AH) so that, on these spatial scales, very little scattering occurs for infra-gravity waves ([MATH] Hz).', 'physics-0510150-1-59-4': 'In addition to this grazing-angle forward scattering the present case shows a significant back-scattering, in particular in the case of following currents.', 'physics-0510150-1-60-0': 'For a wave frequency of [MATH] Hz, the curves followed by the bottom resonant wavenumbers are overlaid on the bottom spectrum (figure [REF].', 'physics-0510150-1-60-1': 'b).', 'physics-0510150-1-60-2': 'The wavenumbers [MATH] along these curves satisfy both the relations [MATH] and [MATH].', 'physics-0510150-1-60-3': 'Without current the curve is exactly a circle, and transforms to an ellipse for weak currents (Appendix C).', 'physics-0510150-1-60-4': 'This approximation is used in the model to compute the source term.', 'physics-0510150-1-60-5': 'The current shifts significantly the resonant configuration for the bottom and surface wavenumbers.', 'physics-0510150-1-60-6': "A current opposed to the waves enlarges the ellipse towards higher wavenumbers, while a following current will lead to a 'sampling' of shorter wave numbers and longer bottom features.", 'physics-0510150-1-60-7': 'Since the bottom topography has the lowest variance at the largest wavenumbers, the scattering is strongest for following currents (figure [REF]).', 'physics-0510150-1-60-8': 'With our choice of parameters, there is about a factor 10 reduction in the bottom variance that causes backscatter as [MATH] is changed from [MATH] m s[MATH] to [MATH] m s[MATH].', 'physics-0510150-1-60-9': 'Besides, as in the 2D case discussed above, the coupling coefficient [MATH] is increased in the case of a following current.', 'physics-0510150-1-60-10': 'Furthermore, a current opposed to the waves decreases the surface waves wavelength and favours resonant configuration such as [MATH] corresponding to forward-scattering, so that the 3 lobes in the source term occupy a broader range of directions in the case of opposing currents.', 'physics-0510150-1-61-0': 'The resulting wave spectra are further modified due to the conservation of the wave action flux.', 'physics-0510150-1-61-1': 'For [MATH] the reflected wave energies are enhanced (figure [REF]).', 'physics-0510150-1-61-2': 'This effect is similar to what was found in the 2D cases considered above, due to the different energy flux velocities [MATH] for the incident waves, and [MATH] for the reflected waves.', 'physics-0510150-1-61-3': 'In all cases investigated here, the narrow incident wave spectrum is significantly broadened in directions, and that effect is most pronounced at the lowest frequencies.', 'physics-0510150-1-61-4': 'Without current or with following currents, spectra at the beginning of the model domain (figure [REF]) contain a large back-scattered energy, which increases the significant wave height and the directional spread on the up-wave side of the sandwave field.', 'physics-0510150-1-61-5': 'This effect should not be very sensitive to the directional spread of the incident wave field and should thus occur for a wide range of sea states.', 'physics-0510150-1-61-6': 'In contrast, it should be noted the initial spectral peak in figure (figure [REF]) is not much modified, because of the relatively short propagation distance from the forcing point (figure [REF].', 'physics-0510150-1-61-7': 'a).', 'physics-0510150-1-61-8': 'Nevertheless, a significant broadening is predicted at the down-wave end of the model domain, with values of the directional spreads [MATH] larger than [MATH] in all cases considered here.', 'physics-0510150-1-61-9': 'That broadening effect is small for relatively broad incident spectra (i.e. directional spreads [MATH]), as found by Ardhuin et al. (2003a) and Ardhuin and Herbers (2005).', 'physics-0510150-1-61-10': 'It was also found that this broadening of the main spectral peak is largest for waves propagating along the main sandwave crest directions (i.e. from the North-West in our case) due to the larger bottom variance at [MATH] with [MATH] (Magne 2005), with a significant modification of the mean direction.', 'physics-0510150-1-62-0': 'Finally, a wave height decrease along the grid is observed, indicating an attenuation due to wave-bottom scattering.', 'physics-0510150-1-62-1': 'In reality, bottom friction would likely induce a stronger decay, and that decay would be stronger than in the absence of scattering.', 'physics-0510150-1-62-2': 'Essentially the scattering increases the average time taken by wave energy to cross the domain, and bottom friction together with scattering would lead to a larger dissipation than friction alone because of that longer time (Ardhuin et al. 2003).', 'physics-0510150-1-63-0': '# Conclusion', 'physics-0510150-1-64-0': 'The effect of a uniform current on the wave-bottom Bragg scattering was investigated theoretically, extending the derivations of Ardhuin Herbers (2002).', 'physics-0510150-1-64-1': 'After Magne et al. (2005) showed that the source term was applicable to non-random topography and accurate in the limit of small bottom amplitudes, it is found here that the source term is also applicable to monochromatic waves.', 'physics-0510150-1-64-2': 'Indeed, there is no process capable of coupling waves with different frequencies.', 'physics-0510150-1-64-3': 'For small bottom amplitudes the source term may only be inaccurate for non-stationary conditions, in which case the generation of the local bound waves requires some energy not explicitly evaluated by the present theory.', 'physics-0510150-1-64-4': 'The two scale approximation was found to hold reasonably well, even for a relatively fast evolution of the wave amplitudes over two wavelengths (figure 3).', 'physics-0510150-1-64-5': "For a sinusoidal bottom and monochromatic waves, the source term converges to Mei's (1985) theory in absence of current and in the limit of the small bottom amplitudes.", 'physics-0510150-1-64-6': "In the presence of a current, monochromatic wave results generally agree with Kirby's (1988) theory, but only converge in the limit of small Froude numbers.", 'physics-0510150-1-64-7': 'In two dimensions, the main effect of a current is a Doppler-like shift of the resonant wave frequencies that undergo maximum reflection.', 'physics-0510150-1-64-8': 'For a given bottom topography this leads to a modification of the wave reflection coefficient that is sensitive to the current strength and direction.', 'physics-0510150-1-65-0': 'In three dimension and over the shallow areas of the southern North Sea, where large sand waves are found with strong tidal currents, wave scattering is expected to be significant, and largely influenced by currents.', 'physics-0510150-1-65-1': 'Over natural topographies, the bottom typically de-correlates over scales shorter than the scattering-induced attenuation scales, so that a modification of the reflection due to a phase locking of the incident and reflected waves with the bottom may be neglected.', 'physics-0510150-1-65-2': 'The wave scattering theory presented in this paper is thus one more piece in the puzzle of wave propagation over shallow continental shelves, and this process may account for a significant part of the observed attenuation of swells in the southern North Sea.', 'physics-0510150-1-65-3': 'Our representation of this phenomenon with a source term in the wave action balance equation is expected to be accurate in many conditions of interest.', 'physics-0510150-1-65-4': 'It is consistent with the wide use of phase-averaged models for engineering and scientific purposes when such large scales are involved.', 'physics-0510150-1-65-5': 'The alternative use of phase-resolving elliptic refraction-diffraction models (e.g. Berkhoff 1972 or Belibassakis et al. 2001), is much more expensive in terms of computer resources, due to the necessity to resolve the wave phase, and may not be much more accurate.', 'physics-0510150-1-66-0': 'This research was supported by a joint grant from CNRS and DGA.', 'physics-0510150-1-66-1': 'Bathymetric data was acquired by the French Hydrographic and Oceanographic Service (SHOM).', 'physics-0510150-1-66-2': 'Discussions with Michael McIntyre, Kostas Belibassakis, Vincent Rey, and Thierry Garlan and gratefully acknowledged.', 'physics-0510150-1-67-0': '# Harmonic oscillator equation for the first order potential', 'physics-0510150-1-68-0': 'The harmonic oscillator equation ([REF]) can be written as a linear superposition of equations of the type [EQUATION]', 'physics-0510150-1-68-1': 'In order to specify a unique solution to ([REF]), initial conditions must be prescribed.', 'physics-0510150-1-68-2': 'In the limit of the large propagations distances, the initial conditions contribute a negligible bounded term to the solution.', 'physics-0510150-1-68-3': 'Following Hasselmann (1962), we choose [MATH] and [MATH], giving the solution [EQUATION]', 'physics-0510150-1-69-0': '# Harmonic oscillator equation and energy for the second order potential', 'physics-0510150-1-70-0': 'Replacing [MATH] ([REF]) in the surface boundary condition ([REF]), [EQUATION] and conserving only the non-bounded terms of [MATH], one obtains [EQUATION]', 'physics-0510150-1-70-1': 'The amplitude [MATH] satisfies a forced harmonic oscillator equation.', 'physics-0510150-1-70-2': 'Anticipating that only [MATH] will give a non-zero contribution due to the correlation [MATH], we may simplify the notations in that equation as, [EQUATION] with [MATH], and [EQUATION]', 'physics-0510150-1-70-3': 'The second order potential potential amplitude must verify the equation [EQUATION] thus we have to solve, [EQUATION]', 'physics-0510150-1-70-4': 'The solution [MATH] may be written as [EQUATION] where [EQUATION]', 'physics-0510150-1-70-5': 'The second order energy contribution from correlation between the zeroth and first order velocity potential is given by, [EQUATION]', 'physics-0510150-1-70-6': 'Then ([REF]) becomes [EQUATION] with [EQUATION]', 'physics-0510150-1-70-7': 'Taking the limit when [MATH], neglecting bounded terms, and anticipating that only [MATH] gives a non null contribution from the bottom variance yields [EQUATION]', 'physics-0510150-1-70-8': 'Changing the spectral coordinates from [MATH] to [MATH], one has [EQUATION]', 'physics-0510150-1-70-9': 'Finally evaluating the integral over [MATH] to remove the Dirac distribution and changing again variables from [MATH] to [MATH], we find, including the bounded terms, [EQUATION]', 'physics-0510150-1-71-0': '# Resonant wavenumber configuration for [MATH]', 'physics-0510150-1-72-0': 'Under the assumption [MATH], and for a current in the [MATH] direction, the resonant conditions [EQUATION] yields the following Taylor expansion to first order in [MATH], [EQUATION]', 'physics-0510150-1-72-1': 'We define, [MATH], [MATH], [MATH], so that [EQUATION] and thus [EQUATION]', 'physics-0510150-1-72-2': 'This is the parametric equation of an ellipse of semi-major axis [MATH], semi-minor axis [MATH], half the foci distance [MATH], and eccentricity [MATH], with [MATH], and [MATH].', 'physics-0510150-1-72-3': 'The interaction between a surface wave with wavenumber [MATH] and a bottom component with wavenumber [MATH] excites a surface wave with the sum wavenumber [MATH].', 'physics-0510150-1-72-4': 'For a fixed [MATH] and current [MATH], in the limit of [MATH] the resonant [MATH] and [MATH] follow ellipses described by their polar equation ([REF]), that reduce to circle for [MATH].'} | {'physics-0510150-2-0-0': 'The scattering of random surface gravity waves by topography of small amplitude, and horizontal scales of the order of the wavelength, is investigated theoretically in the presence of a an almost uniform irrotational current.', 'physics-0510150-2-0-1': 'This problem is relevant to ocean waves propagation on shallow continental shelves where tidal currents are often significant.', 'physics-0510150-2-0-2': 'Defining the small scale bottom amplitude normalized by the mean water depth, [MATH], a perturbation expansion of the wave action to order [MATH] yields an evolution equation for the wave action spectrum.', 'physics-0510150-2-0-3': 'Based on numerical calculations for sinusoidal bars, a mixed surface-bottom bispectrum, that arises at order [MATH], is unlikely to be significant in most oceanic conditions.', 'physics-0510150-2-0-4': 'Neglecting that term, the present theory yields a closed equation with a scattering source term that gives the rate of exchange of action between spectral wave components that have the same absolute frequency.', 'physics-0510150-2-0-5': 'This source term is proportional to the bottom elevation variance at the resonant wavenumbers, and thus represents a Bragg scattering approximation.', 'physics-0510150-2-0-6': 'With current, the source term formally combines a direct effect of the bottom topography with an indirect effect of the bottom through the modulation of the surface current and mean surface elevation.', 'physics-0510150-2-0-7': 'For Froude numbers of the order of 0.6 or less, the bottom topography effects dominate.', 'physics-0510150-2-0-8': 'For all Froude numbers, the reflection coefficients for the wave amplitudes that are inferred from the source term are asymptotically identical, as [MATH] goes to zero, to previous theoretical results for monochromatic waves propagating in one dimension over sinusoidal bars.', 'physics-0510150-2-0-9': 'In particular, the frequency of the waves that experience the maximum reflection is shifted by the current, as the surface wavenumber [MATH] changes for a fixed absolute frequency.', 'physics-0510150-2-0-10': 'Over sandy continental shelves, tidal currents are known to generate sandwaves with scales comparable to those of surface waves, with bottom elevation spectra that roll-off sharply at high wavenumbers.', 'physics-0510150-2-0-11': 'Application of the theory to such a real topography suggests that scattering mainly results in a broadening of the directional wave spectrum, i.e. forward scattering, while back-scattering is generally weaker.', 'physics-0510150-2-0-12': 'The current may strongly influence surface gravity wave scattering by selecting different bottom scales, with widely different spectral densities due the sharp bottom spectrum roll-off.', 'physics-0510150-2-1-0': '# Introduction', 'physics-0510150-2-2-0': 'Following the early observations of Heathershaw (1982), a considerable body of knowledge has been accumulated on the scattering of small amplitude surface gravity waves by periodic bottom topography.', 'physics-0510150-2-2-1': 'An asymptotic theory for small bottom amplitudes, that reproduces the observed scattering of monochromatic waves over a few sinusoidal bars, was put forward by Mei (1985), leading to practical phase-resolving equations that may be used to model this phenomenon for more general bottom shapes (Kirby 1986).', 'physics-0510150-2-2-2': 'For sinusoidal bottoms of wavenumber [MATH], Mei (1985) proposed an approximate analytical solution.', 'physics-0510150-2-2-3': 'In two dimensions (one horizontal and the vertical) this solution yields simple expressions for the wave amplitude reflection coefficient [MATH], as a function of the mismatch between the wavenumber of the surface waves [MATH] and the resonant value [MATH], for which [MATH] is maximum due to Bragg resonance.', 'physics-0510150-2-2-4': 'Beyond a cut-off value of that mismatch, it was found that the incident and reflected wave amplitudes oscillate in space instead of decreasing monotonically from the incident region.', 'physics-0510150-2-2-5': 'In three dimensions the Bragg resonance condition becomes [MATH] and [MATH], with [MATH] and [MATH] the wave radian frequencies corresponding to the wavenumber vectors [MATH] and [MATH] through the linear dispersion relation.', 'physics-0510150-2-3-0': 'Other contributions have shown that higher-order theories are necessary to represent the sub-harmonic resonance observed over a bottom that is a superposition of two components of different wavelengths (Guazzelli, Rey Belzons 1992).', 'physics-0510150-2-3-1': 'Such sub-harmonic resonance was found to have as large an effect as the lowest order resonance for bottom amplitudes of only 25% of the water depth, due to a general stronger reflection for relatively longer waves.', 'physics-0510150-2-3-2': 'However, these amplitude evolution equations are still prohibitively expensive for investigating the propagation of random waves over distances larger than about 100 wavelengths, and the details of the bottom are typically not available over large areas.', 'physics-0510150-2-3-3': 'Besides, a consistent phase-averaged wave action evolution equation is also necessary for the investigation of the long waves associated with short wave groups (Hara Mei 1987).', 'physics-0510150-2-4-0': 'The large scale behaviour of the wave field may rather be represented by the evolution of the wave action spectrum assuming random phases.', 'physics-0510150-2-4-1': 'Such an approach was already proposed by Hasselmann (1966) and Elter Molyneux (1972) for the calculation of wind-wave and tsunami propagation.', 'physics-0510150-2-4-2': 'A proper theory for the evolution of the wave spectrum can be obtained from a solvability condition, a method similar to that of Mei (1985) and Kirby (1988), but applied to the action spectral densities instead of the amplitudes of monochromatic waves.', 'physics-0510150-2-4-3': 'In the absence of currents the correct form of that equation was first obtained by Ardhuin Herbers (2002, hereinafter referred to as AH) using a two scale approach.', 'physics-0510150-2-4-4': 'They decomposed the water depth [MATH] in a slowly varying depth [MATH], that causes shoaling and refraction, and a rapidly varying perturbation [MATH] with zero mean, that causes scattering.', 'physics-0510150-2-4-5': 'This equation is formally similar to general transport equations for waves in random media (e.g. Ryzhik, Papanicolaou Keller 1996), although the waves considered here propagate only in the two horizontal dimensions.', 'physics-0510150-2-4-6': 'The resulting scattering was shown to be consistent with the dramatic increase of the directional width of the wave spectra observed on the North Carolina continental shelf (Ardhuin et al. 2003a, 2003b).', 'physics-0510150-2-5-0': "Recently, Magne et al. (2005, hereinafter referred to as MAHR) showed that AH's theory gives the same damping of incident waves as the Green function solution of Pihl, Mei Hancock (2002), applied to any two dimensional topography, random or not.", 'physics-0510150-2-5-1': "Investigating the applicability limits of the scattering term of AH, MAHR also performed numerical calculations, comparing AH's theory to the accurate matched-boundary model of Rey (1992) that uses a decomposition of the bottom in a series of steps, including evanescent modes.", 'physics-0510150-2-5-2': "The numerical results show that AH's theory is generally limited by the relative bottom amplitude [MATH] rather than the bottom slope.", 'physics-0510150-2-5-3': "In particular, AH's theory predicts accurate reflections, with a relative error of order [MATH], even for isolated steps that have an infinite slope (MAHR).", 'physics-0510150-2-6-0': 'The resulting expression of the Bragg scattering term is consistent with results for scattering of acoustic and electromagnetic waves obtained by the small perturbation method, valid in the limit of small [MATH] where [MATH] is the wavenumber of the propagating waves (Rayleigh 1896, see Elfouhaily Guerin 2004 for a review of this and other approximations).', 'physics-0510150-2-6-1': "Since there is no scattering for [MATH], as the waves do not 'feel' the bottom, the small parameter [MATH] may be used in the context of surface gravity waves, instead of the more general [MATH].", 'physics-0510150-2-6-2': 'For [MATH], the scattering strength is thus entirely determined by the bottom elevation variance spectrum at the bottom scales resonant with the incident waves.', 'physics-0510150-2-7-0': "Based on these results, Mei's (1985) theory should yield the same reflection coefficient as AH's theory in the limit of small bottom amplitudes.", 'physics-0510150-2-7-1': "Yet, AH predict that the wave amplitude in 2D would decay monotonically, which is not compatible with the oscillatory nature of Mei's theory for large detunings from resonance.", 'physics-0510150-2-7-2': "Further, outside of the surf zone and the associated multiple bar systems, the application of AH's theory is most relevant in areas where the bottom topography changes significantly on the scale of the wavelengths of swells.", 'physics-0510150-2-7-3': 'This often corresponds, over sand, to the presence of sandwaves.', 'physics-0510150-2-7-4': 'These sandwaves are generated by currents, and particularly by tidal currents (e.g. Dalrymple Knight Lambiase 1978; Idier, Erhold Garlan 2002).', 'physics-0510150-2-7-5': 'It is thus logical to include the effects of currents in any theory for wave scattering over a random bottom.', 'physics-0510150-2-7-6': "Kirby (1988) developed such a theory for monochromatic waves over a sinusoidal bottom and a slowly varying mean current, extending Mei's (1985) work.", 'physics-0510150-2-7-7': 'The geometry of the resonant wavenumbers is modified in that case, with with incident and reflected waves having the same absolute frequency, but different wavenumber magnitudes if incident and reflected waves propagate at different angles relative to the current direction.', 'physics-0510150-2-7-8': 'Kirby (1988) also considered the short scale fluctuations of the current, due to the sinusoidal bottom, that may be interpreted as a separate scattering mechanism, and generalized further to any irrotational current fluctuations, leading to results similar to those obtained for gravity-capillary waves by Bal Chou (2002).', 'physics-0510150-2-8-0': 'The present paper thus deals with these two questions.', 'physics-0510150-2-8-1': "An extension of AH's theory for surface gravity wave scattering in the presence of irrotational currents with uniform mean velocities is provided in 2, and the differences between this theory and those of Mei (1985) and Kirby (1988) are discussed in detail in 3.", 'physics-0510150-2-8-2': 'Expected oceanographic effects of scattering in the presence of a current are investigated in 4, using a spectral phase-averaged numerical model, predicting the evolution of the wave action spectrum, and detailed measurements of the topography in the southern North Sea.', 'physics-0510150-2-8-3': 'Conclusions follow in 5.', 'physics-0510150-2-9-0': '# Theory', 'physics-0510150-2-10-0': '## General formulation', 'physics-0510150-2-11-0': 'The variation in the action spectral density due to wave-bottom scattering is derived following the method of AH, now including the effect of a uniform mean current.', 'physics-0510150-2-11-1': 'The method is identical to that of Kirby (1988) with the difference that an equation for the spectral wave action is sought instead of one for the wave amplitudes.', 'physics-0510150-2-11-2': 'Thus intermediate results are identical to those of Kirby (1988).', 'physics-0510150-2-11-3': 'Since the wave action is a quadratic function in the wave amplitude, we will naturally consider the wave potential up to second order in the normalized bottom amplitude [MATH], in order to have all wave action terms to order [MATH].', 'physics-0510150-2-11-4': "The only important terms in this type of calculation are the 'secular terms', i.e. the harmonic oscillator solutions for the wave potential forced at resonance, with an amplitude that grows unbounded in time.", 'physics-0510150-2-11-5': 'We shall thus obtain a rate of change of the action from the equality of all the secular terms.', 'physics-0510150-2-11-6': 'The particularity of the random wave approach is also that we will consider all possible couplings between wave components, and not just two wave trains.', 'physics-0510150-2-11-7': 'With random waves, secularity is limited to a sub-space of the wavenumber plane that generally has a zero measure.', 'physics-0510150-2-11-8': 'Thus the near-resonant terms, once integrated across the resonant singularity, are the ones that provide the secular terms for random waves.', 'physics-0510150-2-11-9': 'This integration assumes that the spectral properties are continuous, a real theoretical problem for nonlinear wave-wave interactions (e.g. Benney Saffmann 1966, Onorato et al. 2004).', 'physics-0510150-2-11-10': 'Here we shall see that the only relevant condition is that the bottom spectrum be continuous, at least in one dimension.', 'physics-0510150-2-11-11': 'This is obviously satisfied by any real topography, since a truly infinite sinusoidal bottom of wavelength [MATH], with an infinite spectral density at the wavenumber [MATH], is not to be found, even in the laboratory.', 'physics-0510150-2-12-0': 'We consider weakly nonlinear random waves propagating over an irregular bottom with a constant mean depth [MATH] and mean current [MATH], and random topography [MATH], with [MATH] the horizontal position vector, so that the bottom elevation is given by [MATH] where [MATH] is the elevation relative to the still water level.', 'physics-0510150-2-12-1': 'The bottom undulations cause a stationary random small-scale current fluctuation [MATH] deriving from a potential [MATH].', 'physics-0510150-2-12-2': 'The free surface is at [MATH].', 'physics-0510150-2-12-3': 'Extension to mean current and mean depth variations on a large scale follows from a standard two-scale approximation, identical to that of by Kirby (1988).', 'physics-0510150-2-12-4': 'This is not included in the present derivation for the sake of simplicity.', 'physics-0510150-2-13-0': 'The maximum surface slope is characterized by [MATH] and we shall assume that [MATH] so that the bottom scattering contributions to the wave action to order [MATH] are much larger than the resonant non-linear four wave interactions (Hasselmann 1962) that shall be neglected.', 'physics-0510150-2-13-1': 'Such interactions could also be allowed in the present calculation providing an additional source of scattering with the known form due to cubic non-linearities.', 'physics-0510150-2-13-2': 'For shallow water waves ([MATH]) a stricter inequality is needed to prevent triad wave-wave interactions to enter the action evolution equation at the same order as bottom scattering.', 'physics-0510150-2-14-0': 'The solution is obtained in a frame of reference moving with the mean current vector [MATH], which has the advantage of removing the convective terms due to the mean current velocity.', 'physics-0510150-2-14-1': 'The corresponding transformation of the horizontal coordinates is [MATH], where [MATH] and [MATH] are the coordinates in the moving and fixed frames, respectively.', 'physics-0510150-2-14-2': 'As a result of this transformation, the bottom is moving, and the bottom boundary condition for the velocity potential is modified.', 'physics-0510150-2-14-3': "The governing equations consist of Laplace's equation for the velocity potential, which includes both wave and current motions, the bottom kinematic boundary conditions, and Bernoulli's equation with the free surface kinematic boundary condition.", 'physics-0510150-2-14-4': 'Assuming that the atmospheric pressure is zero for simplicity, and neglecting surface tension, one has [EQUATION] with [MATH] a function of time only, to be determined.', 'physics-0510150-2-14-5': 'The symbol [MATH] represents the usual gradient operator restricted to the two horizontal dimensions.', 'physics-0510150-2-14-6': 'The latter two equations may be combined to remove the linear part in [MATH].', 'physics-0510150-2-14-7': 'Taking [MATH] +[MATH]([REF]), yields, [EQUATION]', 'physics-0510150-2-14-8': 'Following Hasselmann (1962), we approximate [MATH] and [MATH] with discrete sums over Fourier components, and take the limit to continuous integrals after deriving expressions for the evolution of the phase-averaged wave action.', 'physics-0510150-2-14-9': 'We look for a velocity potential solution in the form [EQUATION] where [MATH] is the radian frequencies in the moving frame, [MATH] is the surface wavenumber, with magnitude [MATH], and [MATH] is a sign index equal to 1 or [MATH].', 'physics-0510150-2-14-10': 'In the moving frame of reference, [MATH] for wave components that propagate in the direction of the vector [MATH], and [MATH] for components that propagate in the opposite direction.', 'physics-0510150-2-14-11': 'Thus the radian frequency in the fixed frame is [MATH].', 'physics-0510150-2-14-12': 'The amplitudes [MATH] are slowly modulated in time, with a slowness defined by the small parameter [MATH].', 'physics-0510150-2-14-13': 'Because [MATH] is a real quantity we also have [MATH], where the overbar denotes the complex conjugate.', 'physics-0510150-2-14-14': 'Thus the double decomposition made in ([REF]) in wavenumber [MATH] and propagation direction [MATH] or [MATH] replaces a more general decomposition in wavenumber and frequency that would be necessary if nonlinear dispersive effects were included.', 'physics-0510150-2-14-15': 'Here the frequency [MATH] is always related to [MATH] via the linear dispersion relation.', 'physics-0510150-2-15-0': 'In the alternative decomposition with amplitudes [MATH] that contain the fast time variation, only the part of the solution that has the vertical structure of Airy waves has been given explicitly.', 'physics-0510150-2-15-1': "The other part, represented by '[MATH]', will be found to be negligible for small bottom amplitudes.", 'physics-0510150-2-15-2': 'Our rather archaic use of the [MATH] index to distinguish the wave propagation direction is preferred here to the more modern use of the Hamiltonian variables that combine elevation and potential at the free surface, widely used for wave-wave interaction studies (e.g. Janssen 2004).', 'physics-0510150-2-15-3': 'The complexity of the Hamiltonian variables appears unnecessary for the linear waves considered here.', 'physics-0510150-2-16-0': 'Expanding the bottom boundary condition and wave potential in powers of [MATH], [EQUATION] where each term [MATH] is of order [MATH].', 'physics-0510150-2-16-1': 'The boundary conditions ([REF]) and ([REF]) are expressed at [MATH] and [MATH], respectively, using Taylor series of [MATH] about [MATH] and [MATH].', 'physics-0510150-2-17-0': 'Unless stated otherwise, these potential amplitudes will be random variables.', 'physics-0510150-2-17-1': 'Since we are solving for [MATH] seeking an equation the for wave action [MATH], we must relate [MATH] to [MATH].', 'physics-0510150-2-17-2': 'Accurate to second order in [MATH] and [MATH] (see Andrews McIntyre 1978 for the general expression of [MATH]) we have [MATH] for a monochromatic wave of surface elevation variance [MATH] and intrinsic frequency [MATH], in which, following the common usage in non-accelerated reference frames, the gravity [MATH] is left out, so that the action has units of meters squared times second.', 'physics-0510150-2-17-3': 'For general waves, the variance [MATH] may be written as [EQUATION] where [MATH] denotes an average over flow realizations, and [MATH] is the surface elevation solution of order [MATH], and terms of like order in [MATH] have been grouped.', 'physics-0510150-2-17-4': 'Each of these terms may be expanded in this form [EQUATION]', 'physics-0510150-2-17-5': 'For free wave components, the elevation amplitude is proportional to the velocity potential amplitude [EQUATION] so that the elevation co-variances are proportional to the co-variances [MATH] of the surface velocity potential, [EQUATION]', 'physics-0510150-2-17-6': 'The contribution of the complex conjugate pairs of components ([MATH]) and ([MATH]) are combined in ([REF]) so that the covariance [MATH] correspond to that of all waves with wavenumber magnitude [MATH] propagating in the direction of [MATH].', 'physics-0510150-2-17-7': 'In the limit of small wavenumber separation, a continuous slowly-varying cross-spectrum can be defined (e.g. Priestley 1981, ch.11; see also AH), [EQUATION]', 'physics-0510150-2-17-8': 'The definition of all spectral densities are chosen so that the integral over the entire wavenumber plane yields the total covariance of [MATH] and [MATH].', 'physics-0510150-2-18-0': 'Finally, [MATH] is defined as the [MATH] order depth-integrated wave action contribution from correlation between [MATH] and [MATH] order components with wavenumber [MATH].', 'physics-0510150-2-18-1': 'From ([REF]) and ([REF]) one has, [EQUATION]', 'physics-0510150-2-18-2': 'The spectral wave action is thus, [EQUATION]', 'physics-0510150-2-18-3': 'Defining [MATH] as the amplitude of the Fourier component of wavenumber [MATH], the bottom elevation is given by [EQUATION] with a summation on the entire wavenumber plane.', 'physics-0510150-2-18-4': 'Because [MATH] is real, [MATH].', 'physics-0510150-2-18-5': 'The bottom elevation spectrum in discrete form is given by [MATH] and in continuous form by [EQUATION] and verifies, [EQUATION]', 'physics-0510150-2-18-6': 'Now that the scene is set, we shall solve for the velocity potential [MATH] in the frame of reference moving with the mean current, and use ([REF]) to estimate the action spectra density at each successive order.', 'physics-0510150-2-18-7': "In the course of this calculation, [MATH] will appear as the sum of many terms, some of which are secular (these are the 'resonant terms' in Hasselmann's terminology), i.e. with growing amplitudes in time.", 'physics-0510150-2-18-8': 'Most importantly among these are those that lead to resonant terms in [MATH].', 'physics-0510150-2-18-9': 'All other terms are bounded in time and thus do not contribute to the long-term evolution of the wave spectrum, i.e. on the scale of several wave periods, and shall be neglected (see Hasselmann 1962).', 'physics-0510150-2-19-0': '## Zeroth-order solution', 'physics-0510150-2-20-0': 'In the moving frame of reference, the governing equations for [MATH] are identical to those in the fixed frame in the absence of current.', 'physics-0510150-2-20-1': 'The solution is thus [EQUATION] where the intrinsic frequency [MATH] is the positive root of the linear dispersion relation, [EQUATION]', 'physics-0510150-2-21-0': '## First-order solution', 'physics-0510150-2-22-0': 'Surface non-linearity becomes relevant at first order due to a coupling between the zeroth order solution and current-induced first order terms.', 'physics-0510150-2-22-1': 'Including all powers of [MATH], the expansion of the surface boundary condition to order [MATH] gives, at [MATH], [EQUATION]', 'physics-0510150-2-22-2': 'The equations at order [MATH] are [EQUATION] and, at [MATH], expansion of ([REF]) to first order in [MATH] yields, [EQUATION] where the terms [MATH], not written explicitly (see Hasselmann 1962 eq. 1.11-1.12), are quadratic products of the zeroth-order solution.', 'physics-0510150-2-22-3': 'Since no gravity waves satisfy both [MATH] and [MATH], [MATH] forces a non-resonant wave solution [MATH] that will be neglected because it does not modify our second order wave action balance, thanks to the choice [MATH].', 'physics-0510150-2-22-4': 'The spatially uniform term [MATH] has been incorporated into [MATH] and is also of second order in the wave slope, and does not lead to resonances.', 'physics-0510150-2-22-5': 'That term, omitted by Hasselmann (1962), is responsible for generating microseisms (e.g. Longuet-Higgins 1950).', 'physics-0510150-2-23-0': 'The first-order system of equations is non-linear due to the surface boundary condition ([REF]).', 'physics-0510150-2-23-1': 'However, all the right hand side terms of ([REF]) are of order [MATH], and thus negligible, provided that [MATH] is of order [MATH].', 'physics-0510150-2-23-2': 'Without [MATH] in ([REF]) this would be the case, since the other forcing terms are all proportional to [MATH].', 'physics-0510150-2-23-3': 'However, as suggested by anonymous reviewers, [MATH] introduces an external forcing.', 'physics-0510150-2-23-4': 'We thus first give the solution [MATH] forced by [MATH] only, in the right hand side of ([REF]).', 'physics-0510150-2-23-5': 'This solution is physically identical to mean current perturbation caused by the bottom topography and given by Kirby (1988, his eq. 2.9) for a sinusoidal bottom.', 'physics-0510150-2-23-6': 'With a more general bottom, it is [EQUATION] where [EQUATION] and [EQUATION]', 'physics-0510150-2-23-7': 'The corresponding surface elevation oscillations, given by ([REF]), are second order in the Froude number [MATH], and 180[MATH] out of phase with the bottom oscillations for slow currents when [MATH] (Kirby 1988, eq. 2.10), [EQUATION]', 'physics-0510150-2-23-8': 'From ([REF]), the following expression are derived, [EQUATION]', 'physics-0510150-2-23-9': 'These shall be particularly useful for plugging into ([REF]).', 'physics-0510150-2-24-0': 'We can now obtain the general solution to our equations ([REF])-([REF]) by the following superposition of the previous solution with free and bound (i.e. non-resonant) wave components, with amplitudes [MATH] and [MATH] respectively, [EQUATION] where the last two terms corresponds to the solution to the forcing by all the right hand side terms except for [MATH].', 'physics-0510150-2-24-1': 'Because [MATH] and [MATH] are the only terms that may be larger than [MATH], all others are neglected in the right-hand side of ([REF]).', 'physics-0510150-2-25-0': 'Substitution of ([REF]) in the bottom boundary condition ([REF]) yields [EQUATION]', 'physics-0510150-2-25-1': 'Replacing now ([REF]) in the surface boundary condition ([REF]), yields an equation for [MATH].', 'physics-0510150-2-25-2': 'Using [MATH] and [MATH], it writes [EQUATION] with [EQUATION] where [MATH] is given by all the right-hand side terms in ([REF]) and thus corresponds to the scattering induced by current and current-induced surface elevation variations.', 'physics-0510150-2-25-3': 'Anticipating resonance, we only give the form of [MATH] for [MATH], with [MATH], [EQUATION] in which the term (a) is given by the term (II) in ([REF]), (b) is given by (III) and (IV), (c) is given by (I), and (d) is given by (V)-(VIII).', 'physics-0510150-2-25-4': 'Because we are first solving the problem to order [MATH], it is natural that our solution is a linear superposition of the solutions found by Kirby (1988) for a single bottom component.', 'physics-0510150-2-25-5': 'Indeed, [MATH], with [MATH] the interaction coefficient of Kirby (1988, eq. 4.22b) and [MATH] his bottom amplitude, here [MATH].', 'physics-0510150-2-26-0': 'The solution to the forced harmonic oscillator equation ([REF]) is [EQUATION] where [MATH], and the function [MATH] is defined in Appendix A.', 'physics-0510150-2-27-0': '### First order action', 'physics-0510150-2-28-0': 'The lowest order perturbation of the wave action by scattering involves the order [MATH] covariances [EQUATION] with [MATH] denoting the real part.', 'physics-0510150-2-28-1': 'Including only the secular terms, we get [EQUATION]', 'physics-0510150-2-28-2': 'Although this term was assumed to be zero in AH, it is not zero for sinusoidal bottoms with partially standing waves, and may become significant at resonance due to the function [MATH].', 'physics-0510150-2-28-3': 'In uniform conditions, the time evolution of the wave field requires that the non-stationarity must come into play.', 'physics-0510150-2-28-4': 'Thus [MATH] and the non-stationary term is given by AH (their appendix D), [EQUATION]', 'physics-0510150-2-28-5': 'In order to simplify the discussion, we shall briefly assume that there is no current and that the waves are unidirectional.', 'physics-0510150-2-28-6': 'In that case, [MATH] and [MATH].', 'physics-0510150-2-28-7': 'Replacing ([REF]) in ([REF]) and combining it with ([REF]) yields the action balance [EQUATION] with [MATH] denoting the imaginary part.', 'physics-0510150-2-29-0': 'For directionally spread random waves, with a current, and a real bottom (e.g. random or consisting of a finite series of sinusoidal bars), the evaluation of ([REF]) is not simple.', 'physics-0510150-2-29-1': 'First of all, resonant terms given by [MATH] only occur for [MATH], that is [MATH].', 'physics-0510150-2-29-2': 'Using [MATH] and taking the limit to continuous surface and bottom spectra yields [EQUATION] with the mixed surface bottom bispectrum [MATH] defined by [EQUATION] with [MATH] and [MATH].', 'physics-0510150-2-29-3': '[MATH] is similar to a classical bispectrum (e.g. Herbers et al. 2003) with one surface wave amplitude replaced by a bottom amplitude, and a similar expression is found for a non-zero current.', 'physics-0510150-2-29-4': 'The action balance ([REF]) is generally not closed, and requires a knowledge of the wave phases that are not available in a phase-averaged model.', 'physics-0510150-2-29-5': 'The same type of coupling, although due to the large scale topography, also occurs in the stochastic equations for non-linear wave evolution derived by Janssen, Herbers Battjes (2006).', 'physics-0510150-2-30-0': 'The contribution of the mixed bispectrum will thus be evaluated below, in order to investigate in which cases it may be neglected or parameterized.', 'physics-0510150-2-30-1': 'It is expected that [MATH] is generally negligible because MAHR have neglected [MATH], and still found a good agreement of the second order action balance with exact numerical solutions for the wave amplitude reflection coefficient.', 'physics-0510150-2-31-0': '### Second order action', 'physics-0510150-2-32-0': 'From the expansion ([REF]), the second order action is [MATH].', 'physics-0510150-2-32-1': 'The first term can be estimated from [MATH], using the covariance of the velocity potential amplitudes ([REF]), [EQUATION]', 'physics-0510150-2-32-2': 'Using ([REF]), ([REF]) can be re-written as [EQUATION]', 'physics-0510150-2-32-3': 'Taking the limit of ([REF]) when [MATH], [EQUATION]', 'physics-0510150-2-32-4': 'Due to the singularity in [MATH], and assuming that the rest of the integrand can be approximated by an anlytical function in the neighbourhood of the singularity [MATH], which requires both bottom and surface elevation spectra to be continuous, the integral can be evaluated by using [EQUATION] [MATH] is the one-dimension Dirac distribution, infinite where the argument is zero, and such that [MATH] for any continuous function [MATH].', 'physics-0510150-2-32-5': 'In order to remove that singularity, the argument of [MATH] maye be re-written as [MATH], making explicit all the dependencies on [MATH].', 'physics-0510150-2-32-6': 'Evaluation of the [MATH] function is then performed by changing integration variables [MATH] are changed to [MATH], with a Jacobian [MATH].', 'physics-0510150-2-32-7': 'We thus have [EQUATION]', 'physics-0510150-2-32-8': 'When [MATH], the integrand simplifies.', 'physics-0510150-2-32-9': '[MATH] is equal to [MATH], defined by [EQUATION] with [MATH] given by ([REF]).', 'physics-0510150-2-32-10': 'Using the ([REF]) relation between velocity potential and action, and evaluating the integral over [MATH], one obtains [EQUATION]', 'physics-0510150-2-32-11': 'Again we note the correspondance with the theory of Kirby (1988, eq. 4.21).', 'physics-0510150-2-32-12': "Specifically, one has [MATH], with [MATH] being Kirby's interaction coefficient.", 'physics-0510150-2-33-0': '## Second order potential and corresponding terms in [MATH]', 'physics-0510150-2-34-0': 'In order to estimate the other two terms that contribute to [MATH], the second order potential [MATH] must be obtained.', 'physics-0510150-2-34-1': 'It is a solution of [EQUATION] that simplifies because odd vertical derivatives of [MATH] are zero at [MATH], [EQUATION] and [EQUATION]', 'physics-0510150-2-34-2': 'The terms I-VIII are identical to those in ([REF]) with [MATH], [MATH], [MATH], [MATH] replaced by [MATH], [MATH], [MATH] and [MATH], respectively.', 'physics-0510150-2-34-3': 'All other non-linear terms have been grouped in [MATH].', 'physics-0510150-2-34-4': 'In order to yield contributions to the second order action [MATH], terms must correlate with [MATH] to give second-order terms in [MATH] with non-zero means.', 'physics-0510150-2-34-5': 'For zeroth order components with random phases, inspection shows that [MATH] do not contribute to [MATH] and will thus be neglected.', 'physics-0510150-2-35-0': 'The solution [MATH] is given by the following form, [EQUATION]', 'physics-0510150-2-35-1': 'The non-stationarity term [MATH] leads to the action evolution term ([REF]), now assuming [MATH].', 'physics-0510150-2-35-2': 'Following the method used at first order, substitution of ([REF]) in the bottom boundary condition ([REF]) leads to, [EQUATION]', 'physics-0510150-2-35-3': 'After calculations detailed in Appendix B, [MATH] yields the following contribution to the wave action, [EQUATION] in which [MATH], [MATH], and [MATH].', 'physics-0510150-2-36-0': '## Action and momentum balances', 'physics-0510150-2-37-0': 'We shall neglect the first order action contribution [MATH] given by ([REF]).', 'physics-0510150-2-37-1': 'The solvability condition imposed on the action spectrum is that [MATH] remains an order [MATH] smaller than [MATH] for all times.', 'physics-0510150-2-37-2': 'Thus all secular terms of order [MATH] must cancel.', 'physics-0510150-2-37-3': 'Combining ([REF]), ([REF]), and ([REF]) gives [EQUATION]', 'physics-0510150-2-37-4': 'Since [MATH] and [MATH] remain small, [MATH], and one has, [EQUATION] with the spectral action source term, [EQUATION] where [MATH] and [MATH].', 'physics-0510150-2-37-5': 'This interaction rule was already given by Kirby (1988).', 'physics-0510150-2-37-6': 'The only waves that can interact share the same absolute frequency [MATH].', 'physics-0510150-2-37-7': 'For a given [MATH] and without current, the resonant [MATH] and [MATH] lie on circles in the wavenumber plane (see AH).', 'physics-0510150-2-37-8': 'The current slightly modifies this geometric property.', 'physics-0510150-2-37-9': 'For [MATH] the circles become ellipses (Appendix C).', 'physics-0510150-2-38-0': 'For a given value of [MATH], one may obtain the source term integrated over all directions, [EQUATION]', 'physics-0510150-2-38-1': 'This expression is anti-symmetric, multiplied by -1 when [MATH] and [MATH] are exchanged.', 'physics-0510150-2-38-2': 'Thus [MATH] is a substraction of two equal terms, so that for any bottom and wave spectra [MATH].', 'physics-0510150-2-38-3': "In other words, the 'source term' is rather an 'exchange term', and conserves the wave action at each absolute frequency.", 'physics-0510150-2-38-4': 'This conservation is consistent with the general wave action conservation theorem proved by Andrews McIntyre (1978), which states that there is no flux of action through an unperturbed boundary (here the bottom).', 'physics-0510150-2-38-5': 'It also appears that [MATH] and [MATH] are natural spectral coordinates in which the scattering source term takes a symmetric form.', 'physics-0510150-2-38-6': 'Finally, we may consider the equilibrium spectra that satisfy [MATH] for all [MATH].', 'physics-0510150-2-38-7': 'Without current, an equilibrium exists when either [MATH] or [MATH] is isotropic.', 'physics-0510150-2-38-8': 'With current, the scattering term is uniformly zero if and only if the spectral densities in [MATH]-space, [MATH], are uniform along the curves of constant [MATH].', 'physics-0510150-2-39-0': 'The source term [MATH] may also be re-written in a form corresponding to that in AH, which now appears much less elegant, [EQUATION] with [EQUATION]', 'physics-0510150-2-39-1': 'One may wonder how large is the current-induced scattering represented by [MATH], our eq. ([REF]), compared to the bottom-induced scattering represented by [MATH].', 'physics-0510150-2-39-2': 'Since [MATH], the (a) and (b) terms in the numerator [MATH] almost cancel for small Froude numbers, and the (a)+(b) part is of order [MATH].', 'physics-0510150-2-39-3': 'Thus [MATH] is generally an order [MATH] smaller than [MATH].', 'physics-0510150-2-39-4': 'For [MATH] and [MATH] in opposite directions (i.e. back-scattering), the (a)+(b) part is even smaller, of order [MATH], and exactly zero in the long wave limit [MATH].', 'physics-0510150-2-39-5': 'Thus, for back-scattering, the numerator in [MATH] is itself of the order of (c), i.e. [MATH].', 'physics-0510150-2-39-6': 'Interestingly (c) formally comes from the modulations of the surface elevation [MATH] so that the [MATH] elevation modulation is at least as important as the [MATH] current modulation for this back-scattering situation.', 'physics-0510150-2-39-7': 'In that case, [MATH] is of the order of [MATH].', 'physics-0510150-2-39-8': 'The relative magnitudes of [MATH] and [MATH] thus depend on [MATH] that appears in [MATH].', 'physics-0510150-2-39-9': 'This [MATH]-scale Froude number may be formally close to 1, and thus [MATH] may be larger than [MATH].', 'physics-0510150-2-39-10': 'However, scattering is limited by blocking as no scattered waves can propagate when [MATH].', 'physics-0510150-2-39-11': 'In the long wave limit, [MATH] and for [MATH], one has [MATH].', 'physics-0510150-2-39-12': 'For oblique scattering, the (a)+(b) term may dominate the numerator of [MATH] and the situation is more complex.', 'physics-0510150-2-39-13': 'Nevertheless, for Froude numbers typical of continental shelf situations, say [MATH], [MATH] may be neglected in most situations since its [MATH] correction corresponds to only a few percent of the reflection.', 'physics-0510150-2-39-14': 'Obvious exceptions are cases in which [MATH] is zero, such as when [MATH] and [MATH] are perpendicular.', 'physics-0510150-2-40-0': 'Finally, we may also write the evolution equation for the wave pseudo-momentum [MATH] (see Andrews McIntyre 1978), where [MATH] is the density of sea water.', 'physics-0510150-2-40-1': 'Introducing now the slow medium and wave field variations given by Kirby (1988), that do not interfere with the scattering process, except by probably reducing the surface-bottom bispectrum [MATH], one obtains an extension of the equation of Phillips (1977) [EQUATION] with the dummy indices [MATH] and [MATH] denoting dummy horizontal components, and the scattering stress vector, [EQUATION]', 'physics-0510150-2-40-2': 'This stress has dimensions of force per unit area, and corresponds to a force equal to the the divergence of the wave pseudo-momentum flux.', 'physics-0510150-2-40-3': 'Based on the results of Longuet-Higgins (1967) and Hara Mei (1987), this force does not contribute to the mean flow equilibrium with a balance of the radiation stresses divergence by long waves (or wave set-up in stationary conditions), contrary to the initial proposition of Mei (1985).', 'physics-0510150-2-40-4': 'This force is thus a net flux of momentum through the bottom, arising from a correlation between the non-hydrostatic bottom pressure and the bottom slope.', 'physics-0510150-2-40-5': 'That force is likely related to the pressure under partial standing waves locked in phase with the bottom undulations.', 'physics-0510150-2-40-6': 'Although the part [MATH] of the coupling coefficient [MATH] given by ([REF]) is formally due to scattering by the current modulations [MATH], and associated surface fluctuations [MATH], it should be noted that these motions and related pressures are correlated with the bottom slope in the same way as the part represented by [MATH].', 'physics-0510150-2-40-7': 'Thus both terms contribute to this force [MATH] which acts on the bottom and not on the mean flow.', 'physics-0510150-2-41-0': '# Wave scattering in two dimensions', 'physics-0510150-2-42-0': 'Before considering the full complexity of the 3D wave-bottom scattering in the presence of a current, we first examine the behaviour of the source term in the case of 2D sinusoidal seabeds.', 'physics-0510150-2-42-1': 'Although the bottom spectrum is not continuous along the [MATH]-axis, continuity in [MATH] is sufficient for the use of ([REF]) and the source term can be applied, after proper transformation to remove these singularities.', 'physics-0510150-2-42-2': 'MAHR have investigated the applicability limits of the source term with [MATH].', 'physics-0510150-2-42-3': 'They proved that for small bottom amplitudes the source term yields accurate reflection estimates, even for localized scatterers, and verified this with test cases.', 'physics-0510150-2-42-4': 'It is thus expected that this also holds for [MATH].', 'physics-0510150-2-43-0': '## Wave evolution equation in [MATH]D', 'physics-0510150-2-44-0': 'We consider here a steady wave field in two dimension with incident and reflected waves propagating along the [MATH]-axis.', 'physics-0510150-2-44-1': 'We shall consider in particular the case of [MATH] sinusoidal bars of amplitude [MATH] and height [MATH], with a wavelength [MATH].', 'physics-0510150-2-44-2': 'The bottom elevation is thus [EQUATION]', 'physics-0510150-2-44-3': 'Such a bottom is shown in figure 1 for [MATH].', 'physics-0510150-2-44-4': 'This form is identical to that of the bottom profile chosen by Kirby (1988) but differs, for [MATH], by a [MATH] phase shift from the bottom profile chosen by Mei (1985).', 'physics-0510150-2-44-5': 'The bottom spectrum is of the form [EQUATION] and for the particular bottom given by ([REF]), [EQUATION] with [EQUATION]', 'physics-0510150-2-44-6': 'Note that this is a double-sided spectrum, with only half of the bottom variance contained in the range [MATH].', 'physics-0510150-2-44-7': 'For a generic bottom, for which [MATH] does not go to zero at infinity, the spectrum is obtained using standard spectral analysis methods, for example, from the Fourier transform of the bottom auto-covariance function (see MAHR).', 'physics-0510150-2-44-8': 'In that case [MATH] is equivalent to a Wigner distribution (see e.g. Ryzhik et al. 1996).', 'physics-0510150-2-45-0': 'First, replacing ([REF]) in ([REF]) removes the angular integral in the source term.', 'physics-0510150-2-45-1': 'Taking [MATH], we have [MATH], thus [MATH], and [EQUATION]', 'physics-0510150-2-45-2': 'Second, assuming now that waves propagate only along the [MATH]-axis, the wave spectral densities are of the form [EQUATION] with [MATH] for [MATH] and [MATH] for [MATH].', 'physics-0510150-2-45-3': 'Integrating over [MATH] removes the singularities on [MATH], and assuming a steady state one obtains [EQUATION] with [EQUATION]', 'physics-0510150-2-45-4': 'Although the present theory is formulated for random waves, there is no possible coupling between waves of different frequencies.', 'physics-0510150-2-45-5': 'Mathematically, it is possible to take the limit to an infinitely narrow wave spectrum, such that, [MATH] with [MATH] and [MATH].', 'physics-0510150-2-45-6': 'Using [MATH], the resulting evolution equation is, omitting the 0 subscripts on [MATH] and [MATH], [EQUATION] with a similar equation for [MATH] obtained by exchanging [MATH] and [MATH], and [MATH] and [MATH], from which it is easy to verify that the total action is conserved.', 'physics-0510150-2-46-0': 'The stationary evolution equation ([REF]) only couples two wave components [MATH] and [MATH].', 'physics-0510150-2-46-1': 'For a uniform mean depth [MATH], and uniform bottom spectrum [MATH], as considered here, we thus have a linear system of two differential equations, that may be written in matrix form for any [MATH], [EQUATION] with [EQUATION]', 'physics-0510150-2-46-2': 'Defining [MATH], the action advection velocities [MATH] and [MATH], the terms of the non-dimensional matrix [MATH] are given by [EQUATION] where [MATH] is the [MATH] row and [MATH] column term of [MATH].', 'physics-0510150-2-46-3': 'The general solution is thus [EQUATION]', 'physics-0510150-2-46-4': 'The matrix exponential is classically the infinite series [MATH], in which matrix multiplications are used.', 'physics-0510150-2-46-5': 'The reflection coefficient for the wave action is found using the boundary condition expressing the absence of incoming waves from beyond the bars, [MATH], giving, [EQUATION]', 'physics-0510150-2-46-6': 'A reflection coefficient for the modulus of the wave amplitude predicted by the source term is thus, [EQUATION]', 'physics-0510150-2-46-7': 'The spatial variation of the amplitudes may be linear, oscillatory, or exponential, depending on whether the determinant of [MATH], is zero, negative or positive, respectively.', 'physics-0510150-2-46-8': 'That determinant is [MATH], which is always of the sign of [MATH].', 'physics-0510150-2-47-0': '## Analytical solution for [MATH]', 'physics-0510150-2-48-0': 'In the absence of a mean current, [MATH], and [EQUATION]', 'physics-0510150-2-48-1': 'Thus [MATH] so that its exponential is only the sum of two terms, [MATH], where [MATH] is the identity matrix.', 'physics-0510150-2-48-2': 'The solution to ([REF]) is simply, [EQUATION]', 'physics-0510150-2-48-3': 'An example of spatial variation of the wave spectrum from [MATH] to [MATH] is shown in Figure [REF], for [MATH], and a uniform (white) incident spectrum.', 'physics-0510150-2-48-4': 'The reflected wave energy (at [MATH] in figure [REF].a', 'physics-0510150-2-48-5': 'a) compensates the loss of energy in the transmitted spectrum (at [MATH] in figure [REF].b', 'physics-0510150-2-49-0': "For [MATH], in the limit of small bar amplitudes, and replacing ([REF]) in ([REF]) yields [EQUATION] which is identical to Mei's (1985) equation (3.21)-(3.22) for exact resonance, in the limit of [MATH], and also converges to the result of Davies Heathershaw (1984) for that same limit.", 'physics-0510150-2-49-1': 'For large bar amplitudes, the reflection is significant if the bars occupy a length [MATH] longer than the localization length [MATH].', 'physics-0510150-2-49-2': 'However, the reflection coefficient for the wave amplitude only increases with [MATH] as [MATH], which is slower than the exponential asymptote given by Mei (1985) for sinusoidal bars, and predicted by (Belzons et al. 1988) from the lowest-order theory applied to a random bottom.', 'physics-0510150-2-49-3': 'The present inclusion of the correlations of second-order and zeroth order terms may be thought as the representation of multiple reflections that tend to increase the penetration length in the random medium.', 'physics-0510150-2-50-0': 'A deeper understanding of this question is provided by the comparison of numerical estimations of the reflection coefficients for the wave amplitudes [MATH].', 'physics-0510150-2-50-1': 'A benchmark estimation for linear waves is provided by the step-wise model of Rey (1995) using integral matching conditions for the free propagating waves and three evanescent modes at the step boundaries.', 'physics-0510150-2-50-2': "This model is known to converge to the reflection coefficents given by an exact solution of Laplace's equation and the boundary conditions, in the limit of an infinite number of steps and evanescent modes.", 'physics-0510150-2-50-3': 'Calculations are performed here with 70 steps and 3 evanescent modes.', 'physics-0510150-2-50-4': 'These numbers are chosen because a larger number of steps or evanescent modes gives indistinguishable results in figure [REF].', 'physics-0510150-2-50-5': 'Results of the benchmark model are in good agreement with the measurements of Davies Heathershaw (1984), except for wave components for which the reflection over the beach, not included in the model, is comparable to the reflection over the bars.', 'physics-0510150-2-50-6': 'An analytical expression [MATH] is given by Mei (1985).', 'physics-0510150-2-50-7': '[MATH] for the present second order theory is given by [MATH] ([REF]).', 'physics-0510150-2-51-0': 'We further compare these estimates to the reflection coefficient [MATH] that is deduced from the energy evolution given by Hara Mei (1987), using the approximate solutions of Mei (1985, his equations 3.8-3.23).', 'physics-0510150-2-51-1': "One may prefer to reformulate the energy evolution from the amplitude evolution equations of Kirby (1988) because he used a continuous water depth [MATH], instead of Mei's [MATH] which is discontinuous at [MATH] and [MATH].", 'physics-0510150-2-51-2': "Yet both Mei's and Kirby's equations lead to the same energy exchange between the incident and reflected components.", 'physics-0510150-2-51-3': "Using Mei's (1985) notations, the amplitudes of the incident waves, reflected waves, and bottom undulations are [MATH], [MATH], and [MATH], and the 'cut-off' frequency is [EQUATION]", 'physics-0510150-2-51-4': 'The energy evolution of waves propagating over sinusoidal bars along the [MATH]-axis is given by Hara Mei (1987).', 'physics-0510150-2-51-5': 'The reflected wave energy [MATH] should be a solution of [EQUATION] where [MATH] denotes the complex conjugate of [MATH].', 'physics-0510150-2-51-6': 'This is identical to ([REF]) for a monochromatic bottom except that the imaginary part replaced by a real part.', 'physics-0510150-2-52-0': 'Equation ([REF]) yields a corresponding energy reflection coefficient, given by the fraction of energy lost by the incoming waves, [EQUATION]', 'physics-0510150-2-52-1': "Simple analytical expressions can be obtained at resonance, where Mei's (1985) eq. (3.20)-(3.21) give, [EQUATION] with [MATH], so that [EQUATION] and [EQUATION]", 'physics-0510150-2-52-2': 'It is not surprising that the energy transfer thus computed differs from the energy computed from the amplitude evolution equations.', 'physics-0510150-2-52-3': 'This is typical of small perturbation methods, and was discussed by Hasselmann (1962), among others.', 'physics-0510150-2-52-4': 'Yet, it is remarkable that the ratio of the two is exactly one half.', 'physics-0510150-2-52-5': 'The transfer of energy given by [MATH] in ([REF]) thus correspond to an amplitude reflection coefficient [MATH] that is smaller by a factor [MATH], at resonance, compared to [MATH] (figure 3).', 'physics-0510150-2-52-6': 'This underprediction of the the reflexion of the energy by ([REF]) also has consequences for the analysis and calculation of wave set-up due to wave group propagation over a reflecting bottom.', 'physics-0510150-2-52-7': 'Indeed, the estimation of the scattering stress ([REF]), that contribute to the driving of long waves, was analyzed by Hara Mei (1987) using a calculation similar to ([REF]), which is a factor 2 too small.', 'physics-0510150-2-52-8': 'This may explain, in part, their under-prediction of the observed elevation of the long wave travelling with the incident wave group.', 'physics-0510150-2-52-9': 'However, the present theory, compared to that of Hara Mei (1987), is limited to small bar amplitudes, and fails to reproduce their observation of the transition from oscillatory to exponential decay in the spatial evolution of the wave amplitude.', 'physics-0510150-2-53-0': '## Effects of wave and bottom relative phases', 'physics-0510150-2-54-0': 'The energy exchange coefficient given by the source term always gives energy to the least energetic components (in the absence of currents), and thus the energy evolution is monotonic.', 'physics-0510150-2-54-1': 'The action source term ([REF]) of order [MATH], that was neglected so far, may have any sign, and thus lead to oscillatory evolutions for the wave amplitudes, as predicted by Mei (1985) and observed by Hara Mei (1987).', 'physics-0510150-2-54-2': 'At resonance, and for [MATH], it can be seen that the first-order energy product [MATH] in ([REF]) is equal to [MATH], in the limit of a large number of bars.', 'physics-0510150-2-54-3': "Based on Mei's (1985) approximate solution, in the absence of waves coming from across the bars, this quantity is purely real so that its imaginary part is zero and the corresponding reflection coefficient [MATH] is zero.", 'physics-0510150-2-54-4': "For [MATH] this property remains as can be seen by replacing Mei's (1985) solution with Kirby's (1988).", 'physics-0510150-2-54-5': 'However, similar correlation terms were also neglected in the second order energy (Appendix B), so that the oscillations of the amplitude across the bar field, observed by Hara and Mei (1987) may occur due to terms of the same order as the scattering source term, including interactions of the sub-harmonic kind (Guazzelli et al. 1992).', 'physics-0510150-2-54-6': 'Further, the bottom-surface bispectrum in [MATH] may become significant if there is a large amount of wave energy coming from beyond the bars.', 'physics-0510150-2-54-7': 'This kind of situation, e.g. due to reflection over a beach, was discussed by Yu Mei (2000).', 'physics-0510150-2-55-0': 'In the absence of such a reflection, and away from resonance but for small values of the scattering strength parameter [MATH], the imaginary part of [MATH] is an order [MATH] smaller than the real part and thus contributes a negligible amount to the reflection.', 'physics-0510150-2-56-0': '## Source term and deterministic results for sinusoidal bars', 'physics-0510150-2-57-0': 'For large bar amplitudes, such as [MATH] (figure 3.', 'physics-0510150-2-57-1': 'a), all theories with linearized bottom boundary conditions fail to capture the shift of the reflection pattern to lower wavenumbers.', 'physics-0510150-2-57-2': 'This effect was discussed by Rey (1992), and attributed to the non-linear nature of the dispersion relation and the rapid changes in the water depth.', 'physics-0510150-2-57-3': 'Reflection coefficients are still relatively well estimated.', 'physics-0510150-2-57-4': "For these large amplitudes Mei's (1985) approximate solution is found to be more accurate at resonance compared to the source term.", 'physics-0510150-2-57-5': 'As expected from MAHR and proved here, [MATH] and [MATH] become identical as [MATH] goes to zero (figure [REF].', 'physics-0510150-2-57-6': 'b).', 'physics-0510150-2-57-7': "This fact provides a verification that the first order scattering term [MATH] is different from Hara and Mei's (1987) energy transfer term, and only accounts for a small fraction of the reflection, a fraction that goes to zero as [MATH].", 'physics-0510150-2-57-8': 'It is also found that for all bottom amplitudes, the source term expression provides a simple and accurate solution away from resonance.', 'physics-0510150-2-58-0': 'Nevertheless, the scattering source term cannot give an accurate description of the spatial variation of the wave amplitude over a deterministic bottom, as shown in figure [REF].', 'physics-0510150-2-58-1': 'This is related to the fact that, in MAHR, the present reflection coefficient was obtained from the theory of Pihl et al. (2002) after averaging over the auto-correlation scale of the bottom topography.', 'physics-0510150-2-58-2': 'The present theory can only provide an accurate description of the spatial evolution of the wave field over scales larger than this bottom auto-correlation distance.', 'physics-0510150-2-59-0': '## Effects of currents', 'physics-0510150-2-60-0': 'A prominent feature of solutions with current is the modification of the resonant condition from [MATH] and [MATH], to [MATH] and [MATH], discussed in detail by Kirby (1988).', 'physics-0510150-2-60-1': 'This shift was verified in the laboratory by Magne, Rey Ardhuin (2005).', 'physics-0510150-2-60-2': 'The magnitude of the resonant peak is also largely enhanced for waves against the current, due to a general conservation of the action fluxes and the variation in the action transport velocity, from [MATH] for the incident waves, to [MATH] for the reflected waves.', 'physics-0510150-2-60-3': 'Further, the modulation of the current and the surface elevation also introduce an additional scattering, via the [MATH] term in the coupling coefficent ([REF]).', 'physics-0510150-2-60-4': 'Notations here assume that [MATH] is in the direction of the current and [MATH] is opposite to the current.', 'physics-0510150-2-60-5': 'At resonance, in the limit [MATH], the amplitude reflection coefficient [MATH] given by ([REF]) converges to the reflection coefficient given by Kirby (1988).', 'physics-0510150-2-60-6': 'Using our notations, he obtained [EQUATION] with [EQUATION] and [MATH].', 'physics-0510150-2-60-7': 'Our amplitude reflection coefficient [MATH] is estimated with the approximation [MATH], so that, to first order in [MATH], [EQUATION]', 'physics-0510150-2-60-8': 'Replacing the analytical expression ([REF]) in ([REF]) yields [EQUATION] which is clearly identical to([REF]) at first order in [MATH].', 'physics-0510150-2-61-0': 'For finite values of [MATH], the reflection coefficient ([REF]) corresponding to the solution of ([REF]) is obtained by calculating the proper matrix exponential.', 'physics-0510150-2-61-1': 'Anticipating oceanographic conditions with a water depth of 20 m, a strong 2 m s[MATH] current corresponds to a Froude number of 0.17 only.', 'physics-0510150-2-61-2': "For such a low value of [MATH] in the context of Davies Heathershaw's (1984) laboratory experiments, the convergence of the present theory and that of Kirby (1988) is illustrated in figure 5.", 'physics-0510150-2-61-3': 'The reflection coefficient is largely increased for following currents due to the general conservation of the wave action flux.', 'physics-0510150-2-61-4': 'In that case [MATH] is enhanced by the factor [MATH].', 'physics-0510150-2-61-5': 'The overall increase in [MATH] for following waves amounts to about 60 at [MATH], for the laboratory sinusoidal bars of Davies Heathershaw (1984) shown before (figure 3), with a reflected wave energy multiplied by a factor 2.5, compared to the case without current.', 'physics-0510150-2-62-0': 'For this mild current the contribution of the current fluctuation to the coupling coefficient is small, with a maximum increase of 16 on the action reflection coefficent, 8 for the wave amplitude.', 'physics-0510150-2-62-1': 'However, for larger Froude numbers, this additional scattering may become significant as illustrated by figure [REF].', 'physics-0510150-2-62-2': 'The present theory and that of Kirby (1988) agree reasonably well for finite values of [MATH], and we thus expect the source term to represent accurately the scattering of waves over bottom topographies in cases of uniform currents.', 'physics-0510150-2-63-0': "For [MATH] sinusoidal bars, the energy reflection coefficients was found to be within 10% of the exact solution for over 90% of the wavenumber range shown in figure 3, for [MATH] and [MATH], and this conclusion is expected to hold for [MATH], given the agreement with Kirby's (1988) approximate solution.", 'physics-0510150-2-63-1': 'This accuracy is twice better than what was found for a rectangular step with [MATH] (MAHR).', 'physics-0510150-2-63-2': 'The present method has the advantage of a large economy in computing power.', 'physics-0510150-2-63-3': 'This method is also well adapted for natural sea beds, for which continuous bathymetric coverage is only available in restricted areas, and thus only the statistical properties of the bottom topography are accessible, assuming homogeneity.', 'physics-0510150-2-64-0': '# Scattering with current on a realistic topography', 'physics-0510150-2-65-0': '## Sandwaves in the North Sea', 'physics-0510150-2-66-0': 'A real ocean topography, at least on the continental shelf, generally presents a continuous and broad bottom elevation spectrum.', 'physics-0510150-2-66-1': 'The effects of a mean current on wave scattering are now examined using a bottom spectrum estimated from a detailed bathymetric survey of an area centered on the crest of a sand dune, in the southern North Sea (figure [REF]).', 'physics-0510150-2-66-2': 'In this region, tidal currents are known to generate a wide array of bedforms, from large scale tidal Banks to sand dunes and sand waves (e.g. Dyer Huntley 1999; Hulscher van den Brink 2001).', 'physics-0510150-2-66-3': 'Although sand dunes present a threat to navigation and are closely monitored (Idier et al. 2002), dunes are much larger than typical wind sea and swell wavelengths.', 'physics-0510150-2-66-4': 'These dunes, however, are generally covered with shorter sandwaves.', 'physics-0510150-2-66-5': 'In the surveyed area the sandwaves have a peak wavelength of 250 m, and an elevation variance of 1.7 m[MATH], which should lead to strong oblique scattering of waves with periods of 10 s and longer.', 'physics-0510150-2-66-6': 'Over smaller areas of 3 by 3 km the variance can be as large as 3.3 m[MATH] with a better defined spectral peak, so that our chosen spectrum is expected to be representative of the entire region, including high and low variances on dunes crests and troughs, respectively.', 'physics-0510150-2-66-7': 'The southern North Sea is also known for the attenuation of long swells, generated in the Norwegian Sea.', 'physics-0510150-2-66-8': 'This attenuation has been generally attributed to the dissipation of wave energy by bottom friction (Weber 1991).', 'physics-0510150-2-67-0': 'The bottom spectrum of the chosen area, like the spectra that were obtained by AH from the North Carolina shelf, rolls off sharply at high wavenumbers, typically like [MATH] for the directionally-integrated bottom spectrum [MATH], and proportional to [MATH] for the full spectrum [MATH].', 'physics-0510150-2-67-1': 'Here the maximum variance is found for bottom wavelengths of the order of or larger than 250 m (figure [REF]).', 'physics-0510150-2-67-2': 'For a typical swell period of 10 s, this corresponds to 2 times the wavelength in 20 m depth, and thus a rather small scattering angle, 30[MATH] off from the incident direction.', 'physics-0510150-2-67-3': 'Swells propagating from a distant storm, with fixed absolute frequency [MATH], should be reflected by bottom undulations with widely different variances as the current changes.', 'physics-0510150-2-68-0': 'Given this bottom spectrum and the mean water depth, simple solutions are available for uniform conditions, because the scattering source term is a linear function of the directional spectrum at a given value of the absolute frequency [MATH] (see AH for numerical methods).', 'physics-0510150-2-68-1': 'We consider the wave directional spectrum for a frequency [MATH] and discretize it in [MATH] directions.', 'physics-0510150-2-68-2': 'This spectrum is thus a vector [MATH] in a space with [MATH] dimensions.', 'physics-0510150-2-68-3': 'The square matrix [MATH] such that [MATH] is symmetric and positive, and can thus be diagonalized, which gives [MATH] eigenvalues [MATH] and corresponding eigenvectors [MATH], such that [MATH].', 'physics-0510150-2-68-4': 'Thus the time evolution is easily obtained by a projection of [MATH] on the basis [MATH], giving a decomposition of [MATH] in elementary components.', 'physics-0510150-2-68-5': 'Each of these components of the directional spectrum decays exponentially in time, except for the isotropic part of the spectrum which remains constant because that eigenvector corresponds to [MATH].', 'physics-0510150-2-68-6': 'The eigenvalues thus give interesting timescales for the evolution of the spectrum toward this isotropic state, with a half-life time of each eigenvector given by [MATH].', 'physics-0510150-2-69-0': 'Numerical results are shown here for a mean water depth of 20 m, in order to make the result more visible.', 'physics-0510150-2-69-1': 'For that depth, waves with a period [MATH] s have a dimensionless depth [MATH], which is close the value for which the coupling coefficient [MATH] is maximum (AH).', 'physics-0510150-2-69-2': 'As a result, scattering is probably stronger than in real conditions where the mean water depth is 30 m.', 'physics-0510150-2-69-3': 'The following results should still provide some understanding of the likely real effects, at least for larger wave periods with similar values of [MATH].', 'physics-0510150-2-69-4': 'Without current, if [MATH] is kept constant, the magnitude of the coupling coefficient [MATH] decreases like [MATH] (AH), but it is compounded by a higher bottom elevation spectral density for small values of [MATH].', 'physics-0510150-2-69-5': 'For back-scattering, the bottom wavenumbers are generally in the range where the bottom spectrum rolls off like [MATH] (figure [REF]).', 'physics-0510150-2-69-6': 'Therefore, for these back-scattering directions, the evolution time scale of waves with the same value of [MATH], e.g. [MATH] s in 25 m depth or [MATH] s in 35 m depth, is larger by a factor [MATH] or [MATH], respectively.', 'physics-0510150-2-69-7': 'For incident wave and scattering directions for which the bottom spectrum is more uniform and does not compensate for the reduction in the coupling coefficient, such as forward scattering of waves from the North-West, the time scales increase by [MATH] or [MATH], respectively.', 'physics-0510150-2-70-0': 'With [MATH], corresponding to a directional resolution of [MATH], figure 8 shows that the shortest time scales (large negative values of [MATH]) correspond to directional spectra (eigenvectors) with strong local variations.', 'physics-0510150-2-70-1': 'These eigenvectors are thus associated with scattering at small oblique angles (forward scattering).', 'physics-0510150-2-70-2': 'Only the last 10 eigenvalues have a rather broad support, corresponding to scattering at much larger angles.', 'physics-0510150-2-70-3': 'Besides, the strongest scattering corresponds to a half-life time of 430 s, and mostly affects waves from the North-West or South-East, i.e. propagating in a direction along the sandwave crests.', 'physics-0510150-2-70-4': 'The timescale for waves from the North-East or South-West is about five times larger (the corresponding range of indices is [MATH]).', 'physics-0510150-2-71-0': 'The [MATH] eigenvector corresponds to an exchange of wave energy between waves travelling in opposite directions across the sandwaves, but the corresponding half-life is of 3 hours and 15 minutes.', 'physics-0510150-2-71-1': 'Similar results were found for [MATH] and [MATH] and appear little sensitive to the discretization.', 'physics-0510150-2-72-0': 'Instead of this idealized horizontally uniform situation, practical situations rather correspond to quasi-stationary conditions with spatial gradients in at least one dimension.', 'physics-0510150-2-72-1': 'In this case the simple steady solutions found above for 2D topography are not physical.', 'physics-0510150-2-72-2': 'Indeed, a 3D bottom causes scattering along the transversal direction [MATH], and the energy propagating in that direction builds up slowly up to the point where it becomes as large as the incident wave energy.', 'physics-0510150-2-72-3': 'This process can take a time much longer than the typical duration of a storm or swell arrival, and dissipative processes are likely to be important as the wave energy increases (e.g. Ardhuin et al. 2003).', 'physics-0510150-2-72-4': 'In order to go beyond qualitative statements on time and spatial scales of spectral relaxation, and short of simulating an actual storm in two dimensions, the effects on the wave spectrum are illustrated with a one-dimensional model configuration.', 'physics-0510150-2-73-0': 'The source term [MATH] was introduced in the version [MATH] of the wave model WAVEWATCH III (Tolman 1991, 2002), based on the wave action evolution equation ([REF]) in which the time derivative on the left hand side is now a Lagrangian derivative following a wave packet in physical and spectral space.', 'physics-0510150-2-73-1': 'Bottom scattering is the only source term activated in the present calculation.', 'physics-0510150-2-73-2': 'The model was run with a spectral grid of [MATH] frequencies ranging from [MATH] to [MATH] Hz and a directional resolution of [MATH].', 'physics-0510150-2-73-3': 'Unfortunately the model spectrum is discretized with components at fixed intrinsic frequencies [MATH] and directions [MATH], which is most appropriate for other processes.', 'physics-0510150-2-73-4': 'Therefore a small amount of numerical diffusion leads to a change of action at each absolute frequencies when [MATH], and the total action is only approximately conserved in that case, with a net change of about [MATH] of the integral of the absolute value of the source term for [MATH] m s[MATH], and four orders of magnitudes smaller, i.e. at the round-off error level, for [MATH].', 'physics-0510150-2-73-5': 'We have chosen to show cases with significant back-scatter, corresponding to waves normally incident over the sandwaves.', 'physics-0510150-2-73-6': 'This choice also corresponds to a weaker forward scattering, compared to waves propagating along the the sandwave crests.', 'physics-0510150-2-74-0': '## Scattering of waves normally incident on the sandwaves', 'physics-0510150-2-75-0': 'To simplify the interpretation of the results, and the processing of the boundary conditions, a one dimensional (East-West) propagation grid is used for the computations, assuming that the wave field, still fully directional, is uniform in the North-South direction.', 'physics-0510150-2-75-1': 'The waves are propagated over a model grid [MATH] km long, with a mean depth of [MATH]m, and a spatial grid step of 5 km (figure [REF].', 'physics-0510150-2-75-2': 'a).', 'physics-0510150-2-75-3': 'As discussed above, this water depth is chosen to make the result more visible, and a significant broadening of the incident peak with a (weaker) back-scatter of waves is also found for [MATH] m and [MATH] Hz (not shown).', 'physics-0510150-2-76-0': 'A Gaussian incident surface wave spectrum is imposed, with a mean direction from the North-East, a narrow peak directional spread of [MATH], and a peak frequency of [MATH] Hz (figure [REF].', 'physics-0510150-2-76-1': 'b).', 'physics-0510150-2-76-2': 'The source term is integrated with a time step of [MATH] s, and the advection in space uses a third order scheme with a time step of [MATH] s (Tolman 2002).', 'physics-0510150-2-77-0': 'The scattering source term acts as a diffusion operator with a typical 3-lobe structure, negative at the peak of the wave spectrum, and positive in directions of about 30[MATH] on both sides of the peak.', 'physics-0510150-2-77-1': 'This is identical, but with a larger magnitude, to the effect described by AH.', 'physics-0510150-2-77-2': 'In general the scattering effects are relatively stronger at the lowest frequencies, at least in the range of frequencies used here.', 'physics-0510150-2-77-3': 'For still lower frequencies the scattering coefficient [MATH] decreases (see also AH) so that, on these spatial scales, very little scattering occurs for infra-gravity waves ([MATH] Hz).', 'physics-0510150-2-77-4': 'In addition to this grazing-angle forward scattering, a significant back-scatter is found, in particular in the case of following currents.', 'physics-0510150-2-78-0': 'For an absolute wave frequency of [MATH] Hz, the curves followed by the bottom resonant wavenumbers are overlaid on the bottom spectrum (figure [REF].', 'physics-0510150-2-78-1': 'b).', 'physics-0510150-2-78-2': 'The wavenumbers [MATH] along these curves satisfy both the relations [MATH] and [MATH].', 'physics-0510150-2-78-3': 'Without current the curve is exactly a circle, and transforms to an ellipse for relatively weak currents (Appendix C).', 'physics-0510150-2-78-4': 'This approximation is used in the model to compute the source term.', 'physics-0510150-2-78-5': 'The current imposed here shifts significantly the resonant configuration for the bottom and surface wavenumbers.', 'physics-0510150-2-78-6': "A current opposed to the waves enlarges the ellipse towards higher wavenumbers, while a following current will lead to a 'sampling' of shorter wave numbers, i.e. bottom features of larger scales.", 'physics-0510150-2-78-7': 'Since the bottom topography has the largest variance at low wavenumbers, scattering is strongest for following currents (figure [REF]).', 'physics-0510150-2-78-8': 'With our choice of parameters, there is about a factor 10 reduction in the bottom variance that causes backscatter as [MATH] is changed from [MATH] m s[MATH] to [MATH] m s[MATH].', 'physics-0510150-2-78-9': 'Besides, the coupling coefficient [MATH] is increased in the case of a following current, as discussed above for the 2D cases.', 'physics-0510150-2-79-0': 'The resulting wave spectra are also modified due to the conservation of the wave action flux, enhancing the reflected wave energies for [MATH] (figure [REF]).', 'physics-0510150-2-79-1': 'This effect is similar to what was found in the 2D cases considered above, due to the different energy flux velocities [MATH] for the incident waves, and [MATH] for the reflected waves.', 'physics-0510150-2-79-2': 'In all cases investigated here, the narrow incident wave spectrum is significantly broadened in directions, and that effect is most pronounced for frequencies in the range 0.07-0.10 Hz.', 'physics-0510150-2-79-3': 'Without current or with following currents, spectra in the middle of the model domain exhibit a significant level of back-scattered energy, which increases the significant wave height and the directional spread on the up-wave side of the sandwave field (figure [REF]).', 'physics-0510150-2-79-4': "This effect should not be very sensitive to the directional spread of the incident wave field, because the projection of the directional spectrum on the corresponding 'smooth' eigenvectors of the scattering matrix (figure 8) is insensitive to local variations in the directional spectrum.", 'physics-0510150-2-79-5': 'This reflection should thus occur for a wide range of sea states.', 'physics-0510150-2-79-6': 'At the same time, the incident peak of the wave field broadens in directions as it propagates to the down-wave end of the model domain.', 'physics-0510150-2-79-7': 'This broadening is fast close the the forcing boundary (point F), with values of the peak frequency directional spreads [MATH] larger than [MATH] at a point 5 km inside the domain (not shown), and becomes more gradual as the waves propagate, due to the slower evolution of broad spectra that are associated with smaller eigenvalues in the scattering matrix (see also Ardhuin et al. 2003a, Ardhuin Herbers 2005).', 'physics-0510150-2-79-8': 'It was also verified that this broadening of the main spectral peak is strongest for waves propagating along the main sandwave crest directions (e.g. from the North-West in our case) due to the larger bottom variance at [MATH] with [MATH], resulting in a significant modification of the mean direction (Magne 2005).', 'physics-0510150-2-80-0': 'Finally, a decrease in significant wave height is found along the grid, indicating an attenuation due to wave-bottom scattering.', 'physics-0510150-2-80-1': 'In reality, bottom friction would likely induce a stronger decay, and that decay would be stronger than in the absence of scattering.', 'physics-0510150-2-80-2': 'Essentially the scattering increases the average time taken by wave energy to cross the domain, and, because of that longer time, bottom friction together with scattering would lead to a larger dissipation than friction alone (Ardhuin et al. 2003).', 'physics-0510150-2-81-0': '# Conclusion', 'physics-0510150-2-82-0': 'The effect of a uniform current on the scattering of random surface gravity waves was investigated theoretically, extending the derivations of Ardhuin Herbers (2002).', 'physics-0510150-2-82-1': 'Wave scattering may thus be represented by a scattering source term [MATH] for each wave component [MATH], in a closed spectral action balance equation.', 'physics-0510150-2-82-2': 'That term gives the rate of exchange of wave action between wave components [MATH] and [MATH] that have the same absolute frequency, as a result of both water depth variations on the scale of the surface gravity waves wavelength, and current and mean free surface inhomogeneities induced by the bottom topography.', 'physics-0510150-2-82-3': 'The exchange of action between any two wave component pairs [MATH] and [MATH] is proportional to the bottom elevation spectrum at the wavenumber vector [MATH], which is characteristic of Bragg scattering.', 'physics-0510150-2-82-4': 'The spectral integral of the corresponding wave pseudo-momentum source term [MATH] gives a recoil force exerted by the bottom on the water column, in addition to the hydrostatic pressure force.', 'physics-0510150-2-83-0': 'After Magne et al. (2005a) proved that the source term was applicable to non-random topography and accurate in the limit of small bottom amplitudes, just like Bragg scattering approximations for acoustic or electromagnetic waves (e.g. Elfouhaily Guerin 2004), it is further found here that monochromatic wave results are recovered by taking the limit to narrow incident and reflected wave spectra.', 'physics-0510150-2-83-1': "In absence of current, for a finite sinusoidal bottom and monochromatic waves, the reflection coefficients given by the source term converges to Mei's (1985) theory in the limit of the small bottom amplitudes.", 'physics-0510150-2-83-2': 'The range of maximum reflection and the side lobe pattern of the reflection coefficient as a function of the incident wavenumber is thus a direct consequence of the shape of the bottom spectrum in that case.', 'physics-0510150-2-83-3': 'With this point of view, there is resonance at all wavenumbers but its strength is proportional to the bottom elevation variance at the corresponding scale.', 'physics-0510150-2-83-4': 'In the presence of a current, reflections converge in the same manner to the more general theory of Kirby (1988).', 'physics-0510150-2-83-5': 'In two dimensions, the main effects of a current is an enhancement of reflected wave amplitudes when the incident waves propagate with the current, due to a conservation of the wave action flux, and a Doppler-like shift of the resonant wave frequencies that undergo maximum reflection.', 'physics-0510150-2-83-6': 'The two scale approximation was found to hold very well, even for a relatively fast evolutions of the wave amplitudes over two wavelengths (e.g. figure 3).', 'physics-0510150-2-83-7': 'However, the source term does not give a good representation of the spatial evolution of the wave field on scales shorter that the bottom correlation length, nor can it give reasonable results when another wave train propagates from beyond the bars.', 'physics-0510150-2-83-8': 'In that latter case, a lower order source term must be considered, and a closed action balance cannot be obtained since that extra term depends on the phase relationship between the incident waves, reflected waves and bottom undulations.', 'physics-0510150-2-84-0': 'In three dimension and over the shallow areas of the southern North Sea, where large sand waves are found with strong tidal currents, wave scattering is expected to be significant, and largely influenced by currents.', 'physics-0510150-2-84-1': 'Over natural topographies, the bottom typically de-correlates over scales shorter than the scattering-induced attenuation scales, so that a modification of the reflection due to a phase locking of the incident and reflected waves with the bottom can be neglected.', 'physics-0510150-2-84-2': 'The wave scattering theory presented in this paper is thus one more piece in the puzzle of wave propagation over shallow continental shelves, and this process may account for a significant part of the observed attenuation of swells in the southern North Sea.', 'physics-0510150-2-84-3': 'The representation of this phenomenon with a source term in the wave action balance equation is expected to be accurate in many conditions of interest.', 'physics-0510150-2-84-4': 'It is consistent with the wide use of phase-averaged models for engineering and scientific purposes when such large scales are involved.', 'physics-0510150-2-84-5': 'The alternative use of phase-resolving elliptic refraction-diffraction models (e.g. Belibassakis et al. 2001), is much more expensive in terms of computer resources, due to the necessity to resolve the wave phase and the ellipticity of the problem when back-scattering occurs.', 'physics-0510150-2-84-6': 'For applications to rotational currents, the mean current [MATH] should be regarded as the wave advection velocity (Andrews McIntyre 1978, see Kirby Chen 1989 for practical approximate expressions), but a detailed derivation including scattering by rotational current fluctuations should be the next logical extension of the present theory.', 'physics-0510150-2-84-7': 'This is probably achievable by coupling the rotational part of the flow to the irrotational part, giving a modified Bernoulli equation (e.g. McWilliams et al. 2004).', 'physics-0510150-2-84-8': 'In practice, non-homogeneities in the bottom spectrum will probably have to be addressed due the sharp decrease of the coupling coefficient with water depth, and the generally higher bottom elevation variances in the shallower parts of the sea floor.', 'physics-0510150-2-84-9': 'In particular our limited bathymetric survey shows that sandwaves are modulated by sand dunes, very much like short water waves are modulated by long waves.', 'physics-0510150-2-85-0': 'This research was supported by a joint grant from CNRS and DGA.', 'physics-0510150-2-85-1': 'Bathymetric data was acquired by the French Hydrographic and Oceanographic Service (SHOM).', 'physics-0510150-2-85-2': 'Discussions with Michael McIntyre, Kostas Belibassakis, Vincent Rey, and Thierry Garlan and gratefully acknowledged.', 'physics-0510150-2-85-3': 'The results of the relative effects of current modulations and water depths changes owes much to remarks made by anonymous reviewers, without whom the present paper would have been limited to small Froude numbers.', 'physics-0510150-2-86-0': '# Harmonic oscillator equation for the first order potential', 'physics-0510150-2-87-0': 'The harmonic oscillator equation ([REF]) can be written as a linear superposition of equations of the type [EQUATION]', 'physics-0510150-2-87-1': 'In order to specify a unique solution to ([REF]), initial conditions must be prescribed.', 'physics-0510150-2-87-2': 'In the limit of the large propagations distances, the initial conditions contribute a negligible non-secular term to the solution.', 'physics-0510150-2-87-3': 'Following Hasselmann (1962), we choose [MATH] and [MATH], giving, [EQUATION]', 'physics-0510150-2-88-0': '# Harmonic oscillator equation and energy for the second order potential', 'physics-0510150-2-89-0': 'Replacing [MATH] ([REF]) in the surface boundary condition ([REF]), [EQUATION] and conserving only the resonant terms of [MATH], one obtains [EQUATION] with [MATH].', 'physics-0510150-2-89-1': 'In order to simplify the algebra we assume that the zeroth-order waves are random, with no correlation between [MATH] and [MATH] unless [MATH] and [MATH].', 'physics-0510150-2-89-2': 'Thus the only contributing terms to [MATH] must verify [MATH].', 'physics-0510150-2-89-3': "Only those terms are now written explicitly, the others being grouped in the '[MATH]'.", 'physics-0510150-2-89-4': 'The amplitude [MATH] satisfies the following forced harmonic oscillator equation, [EQUATION]', 'physics-0510150-2-89-5': 'This is a sum of equations of the form, [EQUATION]', 'physics-0510150-2-89-6': 'The solution [MATH] may be written as [EQUATION] where [EQUATION]', 'physics-0510150-2-89-7': 'The second order action contribution from correlation between the zeroth and first order velocity potential is given by, [EQUATION]', 'physics-0510150-2-89-8': "This correlation imposes that all non-zero terms must have [MATH], which removes the '[MATH]' terms, so that ([REF]) becomes [EQUATION] with [EQUATION]", 'physics-0510150-2-89-9': 'Taking the limit when [MATH], and neglecting [MATH] terms yields [EQUATION]', 'physics-0510150-2-89-10': 'Changing the spectral coordinates from [MATH] to [MATH] allows a simple removal of the singularity, [EQUATION]', 'physics-0510150-2-90-0': '# Resonant wavenumber configuration for [MATH]', 'physics-0510150-2-91-0': 'Under the assumption [MATH], and for a current in the [MATH] direction, the resonant conditions [EQUATION] yields the following Taylor expansion to first order in [MATH], [EQUATION]', 'physics-0510150-2-91-1': 'We define, [MATH], [MATH], [MATH], so that [EQUATION] and thus [EQUATION]', 'physics-0510150-2-91-2': 'This is the parametric equation of an ellipse of semi-major axis [MATH], semi-minor axis [MATH], half the foci distance [MATH], and eccentricity [MATH], with [MATH], and [MATH].', 'physics-0510150-2-91-3': 'The interaction between a surface wave with wavenumber [MATH] and a bottom component with wavenumber [MATH] excites a surface wave with the sum wavenumber [MATH].', 'physics-0510150-2-91-4': 'For a fixed [MATH] and current [MATH], in the limit of [MATH] the resonant [MATH] and [MATH] follow ellipses described by their polar equation ([REF]), that reduce to circles for [MATH].'} | [['physics-0510150-1-48-2', 'physics-0510150-2-61-1'], ['physics-0510150-1-36-4', 'physics-0510150-2-48-3'], ['physics-0510150-1-36-5', 'physics-0510150-2-48-4'], ['physics-0510150-1-36-6', 'physics-0510150-2-48-5'], ['physics-0510150-1-66-0', 'physics-0510150-2-85-0'], ['physics-0510150-1-66-1', 'physics-0510150-2-85-1'], ['physics-0510150-1-66-2', 'physics-0510150-2-85-2'], 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1804.03557 | {'1804.03557-1-0-0': '# Introduction', '1804.03557-1-1-0': 'The hadroproduction of heavy-flavoured mesons at the LHC, in particular the D- and B-meson measurements at forward direction [CITATION], has recently attracted a growing interest for its potential to provide information on partonic dynamics at low momentum fractions.', '1804.03557-1-1-1': 'Because of the finite heavy-quark mass [MATH], the perturbative methods are applicable down to zero transverse momentum ([MATH]) of the observed meson, and the measurements provide opportunities e.g. to constrain the collinearly factorized gluon distributions at small momentum fractions in proton [CITATION] or nucleus [CITATION], or to test other scenarios like saturation physics [CITATION], or [MATH] factorization [CITATION].', '1804.03557-1-1-2': 'The D meson production is also of great interest from the viewpoint of neutrino astrophysics as the secondary neutrinos from D mesons produced in scatterings of cosmic rays in the atmosphere form a significant background for the extraterrestrial neutrinos.', '1804.03557-1-1-3': 'Given that the D-meson measurements at the LHCb [CITATION] are kinematically close to the cosmic-ray-on-air scattering, the rates for secondary neutrinos can be constrained by the LHC data [CITATION].', '1804.03557-1-1-4': 'In heavy-ion collisions the measured open heavy-flavour data [CITATION] provides opportunities e.g. to test the so-called dead-cone effect [CITATION] in QCD medium [CITATION].', '1804.03557-1-2-0': 'Theoretically, there are several collinear-factorization-based ways to calculate cross sections for heavy-flavoured mesons in proton-proton (p-p) collisions, see e.g. Refs. [CITATION] for reviews.', '1804.03557-1-2-1': 'On one hand, parton-level heavy-quark cross sections at fixed-flavour-number scheme (FFNS) [CITATION] can be folded with phenomenological, scale-independent parton-to-meson fragmentation functions (FFs), or the parton-level calculation is matched to a parton-shower [CITATION] from a general-purpose Monte-Carlo event generator, such as Pythia 8 [CITATION] or Herwig [CITATION], and the showered event is then hadronized according to the hadronization model of the generator.', '1804.03557-1-2-2': 'Alternatively, one can work fully within the collinear factorization where the fragmentation is described with universal, scale-dependent FFs [CITATION].', '1804.03557-1-2-3': 'In this paper, we will focus on this latter approach.', '1804.03557-1-3-0': 'The general framework in QCD to treat the heavy-quark production is the so-called general-mass variable flavour number scheme (GM-VFNS) [CITATION].', '1804.03557-1-3-1': 'In this framework, at low interaction scales [MATH] the heavy quarks are not treated as partons in PDFs but are considered only as massive objects in the final state.', '1804.03557-1-3-2': 'The full mass dependence is retained in the production cross sections, but the initial-state partons are restricted only to the light ones.', '1804.03557-1-3-3': 'These cross sections contain mass-dependent logarithmic terms which, towards higher interaction scales, will eventually dominate and diverge.', '1804.03557-1-3-4': 'In GM-VFNS these large logarithms are subtracted at a certain transition scale [MATH] - typically the heavy-quark mass threshold - and resummed into the PDFs and scale-dependent FFs.', '1804.03557-1-3-5': 'At asymptotically high interaction scales [MATH] the result reduces (up to finite terms) to the calculation where the quark mass has been put to zero from the outset, the so-called zero-mass variable flavour number scheme (ZM-VFNS).', '1804.03557-1-4-0': 'To obtain a well-behaved description for the heavy-flavoured mesons within GM-VFNS approach from zero to asymptotically large [MATH] has, however, been a bit challenging.', '1804.03557-1-4-1': 'The difficulty is related to the intrinsic freedom in GM-VFNS to use the zero-mass formalism for the processes with heavy-quarks in the initial state or where the fragmenting parton is a light one.', '1804.03557-1-4-2': 'The massless coefficient functions display a divergent behaviour towards low [MATH] and with a typical scale choice [MATH] their contribution dominates the cross sections immediately above [MATH].', '1804.03557-1-4-3': 'Thus, the production cross sections diverge towards [MATH].', '1804.03557-1-5-0': 'A solution was proposed in Ref. [CITATION].', '1804.03557-1-5-1': 'In essence, the idea was to exclude the aforementioned divergent contributions at low [MATH] by retaining the factorization and fragmentation scales at the threshold [MATH] until large-enough [MATH].', '1804.03557-1-5-2': 'Formally, the difference with respect to a more natural choice [MATH] is one order higher in QCD coupling than what one works at, but numerically the effect is large and the cross sections are rendered finite down to [MATH].', '1804.03557-1-5-3': 'A relatively good description of the LHCb data can be obtained by tuning the scales in this manner [CITATION], but the price to pay is that there will be a certain unphysical wiggle in the production cross section near the region where one decides to turn on the heavy-quark PDFs and light-parton FFs, see e.g. Figure 6 in Ref. [CITATION].', '1804.03557-1-5-4': 'An alternative strategy along this line would be to take the transition scale [MATH] to be much higher than the heavy-quark mass [CITATION].', '1804.03557-1-5-5': 'However, this would lead to a discontinuity in the cross sections at the arbitrary point where one decides to make the transition.', '1804.03557-1-5-6': 'Higher-order calculations should decrease the transient effects in both cases, but will not cure them completely.', '1804.03557-1-5-7': 'Clearly, a different solution would be beneficial.', '1804.03557-1-6-0': 'The option we propose here is to make use of the scheme dependence inherent to GM-VFNS.', '1804.03557-1-6-1': 'Physically, our choice of scheme is rooted in the observation that - in the absence of intrinsic charm component - the contributions from heavy-quark PDFs and light-parton FFs are simply an efficient way to resum diagrams where a heavy quark-antiquark ([MATH]) pair is dynamically produced.', '1804.03557-1-6-2': 'Being of the same origin, it is natural to require that these contributions respect the same kinematical constraints as the channels where the pair is explicit produced.', '1804.03557-1-6-3': 'These are formally [MATH] effects and can be included in the definition of a scheme.', '1804.03557-1-6-4': 'However, the contributions from heavy-quark PDFs and light-parton FFs will no longer diverge in the [MATH] limit, but are regulated by the heavy-quark mass.', '1804.03557-1-6-5': 'The production cross sections thus remain finite in the [MATH] limit with arbitrary factorization and fragmentation scales.', '1804.03557-1-7-0': '# Formalism', '1804.03557-1-8-0': 'In this section we will describe our theoretical construction and its numerical implementation.', '1804.03557-1-8-1': 'As the GM-VFNS framework in hadroproduction of heavy quarks has been detailedly discussed in Refs. [CITATION], we will here focus only on the most important features of our approach.', '1804.03557-1-8-2': 'However, enough details are still given so that our results can be reproduced.', '1804.03557-1-9-0': '## General structure and kinematics', '1804.03557-1-10-0': 'The process we study is an inclusive production of a hadron [MATH] with momentum [MATH] in collision of two hadrons [MATH] and [MATH] with momenta [MATH] and [MATH], [EQUATION].', '1804.03557-1-10-1': "In the approximation where the masses of partons and produced hadron are neglected, the cross section differentiated with respect to the produced hadron's transverse momentum [MATH] and rapidity [MATH] can be written in the well-known factorized form [CITATION], [EQUATION] where the fragmenting parton's transverse momentum and rapidity are [MATH] and [MATH].", '1804.03557-1-10-2': 'Here [MATH] are the PDFs for parton species [MATH] in hadron [MATH] and [MATH] is the parton-to-[MATH] FF.', '1804.03557-1-10-3': 'The invariants [MATH] are defined as [EQUATION] where [MATH] and [MATH] are the momenta of the incoming partons, [MATH] is the momentum of the produced, outgoing parton and [MATH] is center-of-mass (c.m.) energy of the collision.', '1804.03557-1-10-4': 'The integration limits are given by [EQUATION]', '1804.03557-1-11-0': '### Partonic kinematics in the presence of mass', '1804.03557-1-12-0': 'When a [MATH] pair is produced from light partons, the zero-mass partonic kinematics above should be adjusted to account for the heavy-quark mass [MATH].', '1804.03557-1-12-1': 'In practice, this amounts to replacing the partonic transverse momentum [MATH] in the [MATH] integration limits and scaling variables [MATH] by the partonic transverse mass [MATH] , [EQUATION]', '1804.03557-1-12-2': 'When the produced parton is a heavy quark, the above replacements follow directly from the momentum conservation.', '1804.03557-1-12-3': 'However, in the case that the fragmenting parton is a light one or when there is a heavy quark in the initial state, these replacements are strictly speaking not necessary, but are part of our choice of scheme (SACOT-[MATH], explained in more detail later).', '1804.03557-1-12-4': 'In the picture where the heavy quarks are generated perturbatively, the heavy-flavour PDFs and light-flavour FFs are merely an efficient way to resum diagrams where a heavy quark-antiquark pair is created.', '1804.03557-1-12-5': 'That is, the production of heavy-flavour pair is implicit in these contributions and motivates the usage of heavy-flavour kinematics.', '1804.03557-1-13-0': '### Massive fragmentation variable', '1804.03557-1-14-0': 'The zero-mass version of the fragmentation scaling variable [MATH] is ill-defined in the presence of massive quarks/hadrons.', '1804.03557-1-14-1': 'Here, we choose to define the scaling variable [MATH] in a Lorentz-invariant way as [EQUATION]', '1804.03557-1-14-2': 'As indicated, in the c.m. frame of the colliding hadrons [MATH] can be interpreted as the fraction of partonic energy carried by the outgoing hadron [CITATION].', '1804.03557-1-14-3': 'Alternatively, the scaling variable could be defined e.g. in terms of light-cone momentum fractions [CITATION].', '1804.03557-1-14-4': 'From the above definition and considering the fragmentation to be collinear in the c.m. frame, we have two equations, [EQUATION] which we can solve for the hadronic transverse momentum and rapidity, [EQUATION]', '1804.03557-1-14-5': 'The cross section corresponding to the above definition of [MATH] can be obtained as [EQUATION] by integrating over [MATH] and [MATH].', '1804.03557-1-14-6': 'Using the relation [EQUATION] we find again Eq. ([REF]) where the partonic transverse momentum and rapidity are now given by [EQUATION] and the hadron mass corrects the lower limit of the [MATH] integration as [EQUATION]', '1804.03557-1-14-7': 'Otherwise the cross-section formula is formally identical to the case of zero-mass partons and hadrons.', '1804.03557-1-15-0': '## Partonic cross sections in SACOT and SACOT-[MATH] schemes', '1804.03557-1-16-0': 'The starting point in our GM-VFNS construction, is the next-to-leading-order (NLO) one-particle inclusive heavy-quark cross section in FFNS [CITATION] where heavy flavour can be produced in three different partonic processes, [EQUATION]', '1804.03557-1-16-1': 'In GM-VFNS, these are the only ways to produce heavy flavour at low interaction scales.', '1804.03557-1-16-2': 'The heavy-quark mass [MATH] is kept finite in these processes and in the high-[MATH] limit, the partonic cross sections develop logarithmic divergences [MATH] coming from kinematic regions where the heavy quarks become collinear with other partons.', '1804.03557-1-16-3': 'These are the first terms in the whole series of large collinear logarithms which, in GM-VFNS framework, are resummed to heavy-quark PDFs and parton-to-hadron FFs when the interaction scale exceeds a chosen transition scale [MATH].', '1804.03557-1-16-4': 'From now on identify [MATH] as the heavy quark mass, [MATH].', '1804.03557-1-16-5': 'To avoid double counting, one has then to subtract the related logarithmic pieces.', '1804.03557-1-16-6': 'In what follows, we will explain what are the added and subtracted terms in our case, using the [MATH] channel as an explicit example.', '1804.03557-1-17-0': 'The gluon-fusion process [MATH] at NLO entails four different sources of collinear divergences, illustrated in Figure [REF]:', '1804.03557-1-18-0': '- one of the two initial-state gluons splits into a collinear heavy quark-antiquark pair,', '1804.03557-1-19-0': '- an outgoing gluon splits into collinear heavy quark-antiquark pair,', '1804.03557-1-20-0': '- an outgoing heavy quark emits a collinear gluon.', '1804.03557-1-21-0': 'A simple way to specify the GM-VFNS subtraction terms at NLO is to take as the starting point the leading-order (LO) contributions from channels where there are heavy quarks in the initial state or the fragmenting parton is a light one.', '1804.03557-1-21-1': 'Let us begin with the former case.', '1804.03557-1-21-2': 'Using Eq. ([REF]), we write the leading-order contribution for process [MATH] as [EQUATION]', '1804.03557-1-21-3': 'This now uniquely determines the subtraction term which cancels the logarithmic term from diagrams like (b) in Figure [REF].', '1804.03557-1-21-4': 'The expression for perturbative heavy-quark PDF, to the first order in strong coupling [MATH], reads [EQUATION] where [MATH] with [MATH], is the leading-order gluon-to-quark splitting function.', '1804.03557-1-21-5': 'Using this expression for [MATH] in Eq. ([REF]) gives our definition of the subtraction term, [EQUATION]', '1804.03557-1-21-6': 'When adding the leading-order contribution of Eq. ([REF]), one must then compensate by subtracting Eq. ([REF]).', '1804.03557-1-21-7': 'The difference contributes at [MATH] and is not considered at an NLO-level [MATH] calculation.', '1804.03557-1-21-8': 'Here, we also plainly see the origin of the scheme dependence in GM-VFNS: The exact form of [MATH] does not really matter as long as its expression is the same in Eq. ([REF]) and Eq. ([REF]).', '1804.03557-1-21-9': 'The only requirement is that in the [MATH] limit [MATH] must tend to its zero-mass expression so as to ensure that the corresponding collinear logarithm from [MATH] process cancels.', '1804.03557-1-21-10': 'Otherwise we can choose it at will.', '1804.03557-1-21-11': 'Similarly, the exact expressions for the integration limits are irrelevant as far as the zero-mass expressions given in Eq. ([REF]) are found in the [MATH] limit.', '1804.03557-1-21-12': 'The simplest option is to use the zero-mass matrix elements and kinematics from the beginning - this choice of scheme is usually dubbed as simplified ACOT, or SACOT scheme [CITATION].', '1804.03557-1-21-13': 'Here, we shall adopt a prescription where we use the zero-mass matrix elements but still retain the kinematic mass dependence.', '1804.03557-1-21-14': 'In other words, the integration limits and the invariants [MATH] and [MATH] are as in Eq. ([REF]), and for the squared matrix element in Eqs. ([REF]) and ([REF]) we take, [EQUATION] where [MATH] is obtained from the zero-mass expression [CITATION], [EQUATION] with [MATH], [MATH], [MATH], and the "massive" Mandelstam variables being now [MATH], and [MATH].', '1804.03557-1-21-15': 'In practice, our prescription amounts to replacing the partonic transverse momentum [MATH] in the zero-mass expressions by the transverse mass [MATH] - hence we shall name the present implementation as SACOT-[MATH] scheme.', '1804.03557-1-22-0': 'The leading-order contribution and subtraction term for the [MATH] channel are defined in a similar manner as above, so let us then discuss the contributions from light-parton fragmentation and the corresponding subtraction terms.', '1804.03557-1-22-1': 'Proceeding as in the case of initial state, we define the leading-order contribution from [MATH] channel, originating from diagrams like (c) in Figure [REF], by [EQUATION]', '1804.03557-1-22-2': 'Together with the the perturbative expression for the gluon fragmentation function (considering that the only non-zero FF at the mass threshold is [MATH]), [EQUATION] this defines the subtraction term [EQUATION] where now [EQUATION]', '1804.03557-1-22-3': 'The subtractions required to cancel the large logarithm originating from diagram (d) in Figure [REF] goes slightly different than the above cases.', '1804.03557-1-22-4': 'The reason is that the contributions from [MATH] channel (part of the inclusive heavy-quark cross sections) that we here use to determine the subtraction terms, are included using the full mass-dependence.', '1804.03557-1-22-5': 'Therefore, the subtraction term required to cancel the large logarithm that occurs when final-state heavy quark emits a collinear gluon is [EQUATION] where [MATH] is the quark-to-quark splitting function, and the matrix element [CITATION], [EQUATION] now carries the full mass dependence.', '1804.03557-1-22-6': 'In order to recover the standard [MATH] zero-mass results at high [MATH] we must still compensate for the fact that the [MATH] limit in the massive calculation does not exactly match that of usual massless [MATH], but some finite differences remain as a relic of a different regularization procedure.', '1804.03557-1-22-7': 'As explained in Ref. [CITATION], this can be effectively achieved by replacing Eq. ([REF]) by [EQUATION] where [MATH] is the partonic fragmentation function [CITATION], [EQUATION]', '1804.03557-1-22-8': 'Also the renormalization procedure applied in FFNS calculations is slightly different than in the purely zero-mass case and in order to revert to the standard [MATH] scheme, we must add a term [EQUATION] as explained in Ref. [CITATION].', '1804.03557-1-23-0': 'The same line of reasoning is applied when defining the subtraction terms for [MATH] and [MATH] channels and the emerging leading-order contributions from [MATH], [MATH], and [MATH] channels.', '1804.03557-1-23-1': 'As in Eq. ([REF]), the definition of the [MATH] subtraction term involves the partonic fragmentation function [MATH], and a term [EQUATION] is added to recover the [MATH] renormalization scheme [CITATION].', '1804.03557-1-23-2': 'In addition, our full results include the contributions from all other partonic subprocesses whose inclusion does not require a preparation of subtraction terms at the perturbative order we work at.', '1804.03557-1-23-3': 'The NLO [MATH] contributions, taken from Ref. [CITATION] are included as well.', '1804.03557-1-23-4': 'We stress that when including these terms in our SACOT-[MATH] scheme, we consistently retain the kinematics which they inherit from [MATH] pair-creation process as explained earlier.', '1804.03557-1-23-5': 'In practice this is done by trading the massless variables [MATH] and [MATH] used in Ref. [CITATION] by their massive counterparts [MATH] and [MATH], see e.g. Sect. 2 of Ref. [CITATION], and imposing the proper integration limits explained in Section [REF].', '1804.03557-1-23-6': 'In this way, we already implicitly define the subtraction terms that would be required at a next-to-NLO (NNLO) -level calculation.', '1804.03557-1-24-0': 'In comparison to the earlier works [CITATION], the most notable advantage of the SACOT-[MATH] scheme is that the cross sections remain finite in the [MATH] limit.', '1804.03557-1-24-1': 'Indeed, in Refs. [CITATION] at least part of the contributions not coming directly from flavour-creation processes are included using purely zero-mass formalism, and give rise to a divergent [MATH] behaviour at [MATH] limit.', '1804.03557-1-24-2': 'The difficulty will not be completely resolved at NNLO either, though the divergences may be a bit "softer".', '1804.03557-1-24-3': 'In Ref. [CITATION] these divergent contributions were excluded at small [MATH] by maintaining the factorization and fragmentation scale at (or below) the heavy-quark mass threshold until large-enough [MATH].', '1804.03557-1-24-4': 'This procedure leads to finite cross sections in the [MATH] limit, but causes certain unphysical slope change near the [MATH] value where the factorization and fragmentation scales go above the mass threshold - we will come back to this in Section [REF] (see also see Fig. 6 in Ref. [CITATION]).', '1804.03557-1-24-5': 'In our case - and this applies also for the fixed-order calculations - the divergent behaviour is regulated by the heavy-quark mass and leads to finite cross sections even at [MATH] (at any perturbative order) without a need to fine tune the scale choices.', '1804.03557-1-24-6': 'Technically, this happens because the lower limits for the scaling variables [MATH] and [MATH] appearing in the squared matrix elements are not zero but limited by the heavy-quark mass.', '1804.03557-1-25-0': '## Numerical implementation', '1804.03557-1-26-0': 'Our numerical realization of the GM-VFNS scheme described above is crafted around the public INCNLO [CITATION] and Mangano-Nason-Ridolfi (MNR) [CITATION] codes.', '1804.03557-1-26-1': 'The former provides the zero-mass matrix elements, and the latter one the one-particle inclusive heavy-quark cross section of Ref. [CITATION].', '1804.03557-1-26-2': 'As already noted in Ref. [CITATION], in order to obtain reliable numerical results from INCNLO at high [MATH] away from the midrapidity [MATH], the numerical stability of the original code has had to be improved, see p.30-32 in Ref. [CITATION] for a detailed explanation.', '1804.03557-1-26-3': 'Schematically, we compute [EQUATION] where the inclusion of charge-conjugate contributions and shuffling between the initial-state partons is implicit.', '1804.03557-1-26-4': 'That is, from the full zero-mass result we subtract the zero-mass contributions of [MATH] channels which we add back using the full mass dependence.', '1804.03557-1-26-5': 'The subtraction terms provide the proper matching.', '1804.03557-1-26-6': 'Towards high [MATH], only the first sum term in Eq. ([REF]) survives - others add up to zero.', '1804.03557-1-26-7': 'In the numerical evaluation we have used NNPDF31nlopchas0118 PDFs [CITATION], which is the latest NNPDF fit assuming no intrinsic charm content in the proton.', '1804.03557-1-26-8': 'The PDFs are interfaced by using LHAPDF 6 library [CITATION].', '1804.03557-1-26-9': 'The introduced framework is applicable to production of any hadrons involving heavy quarks but in this work we consider only D-meson production due to good availability and precision of the experimental data from LHC experiments.', '1804.03557-1-26-10': 'Furthermore, since our SACOT-[MATH] scheme differs from other GM-VFNS implementations mainly in the [MATH] region and only D[MATH] meson data goes down into this region, we restrict the numerical studies to this final state.', '1804.03557-1-26-11': "For D[MATH] mesons there would be more recent FF analyses available [CITATION] but for D[MATH] we use KKKS08 [CITATION] FFs, which is the only available FF set for D[MATH]'s.", '1804.03557-1-26-12': 'The strong coupling [MATH] is taken from LHAPDF 6 and for the charm-quark mass we use [MATH] in accordance with the used PDF set.', '1804.03557-1-26-13': 'The input charm mass in KKKS08 analysis was [MATH] so the pairing with NNPDF3.1 is consistent.', '1804.03557-1-26-14': 'Our default scale choice will be [MATH], and for the D-meson mass we use [MATH].', '1804.03557-1-26-15': 'The small contribution from b-quark fragmentation is retained in the calculation neglecting the finite b-quark mass.', '1804.03557-1-26-16': 'The D mesons from B-meson decays have been excluded from the LHCb and ALICE data we discuss later on, but as the KKKS08 FFs include these feed-down D mesons as well, there is no fully consistent way to exclude them without explicitly evaluating the D[MATH] meson spectra from B-meson decays and subtracting it from the fully inclusive cross section.', '1804.03557-1-26-17': 'However, the contributions from B decays are very small, less than 1% in the integrated inclusive D[MATH]-meson cross section of ALICE [CITATION].', '1804.03557-1-27-0': 'In order to compare with another popular approach, we have used here the Powheg method [CITATION] in which the [MATH] production at FFNS is matched with the Pythia parton shower providing NLO accuracy for the matrix element generation and leading-log resummation from the parton shower.', '1804.03557-1-27-1': 'In practice, we have first generated [MATH] events with the hvq part [CITATION] of Powheg-Box generator [CITATION].', '1804.03557-1-27-2': 'The generated events are then fed into Pythia (version 8.230) [CITATION] which generates the [MATH]-ordered parton shower and hadronizes the events using the implemented Lund string model with parameter values from the default Monash tune [CITATION].', '1804.03557-1-27-3': 'The D[MATH] mesons (and its charge conjugate) are then picked up from the hadronized final state and binned in [MATH] and [MATH].', '1804.03557-1-27-4': 'The same NNPDF3.1 PDFs as for the GM-VFNS calculations have been used for the event generation in Powheg and also in showering within Pythia.', '1804.03557-1-27-5': 'In Powheg generation the default scale choice is [MATH] with [MATH].', '1804.03557-1-27-6': 'We have not explicitly introduced the matching terms, Eqs. ([REF]) and ([REF]), at the heavy-quark mass thresholds as their effect has been found small in the [MATH] range of LHCb data [CITATION].', '1804.03557-1-27-7': 'Indeed, with [MATH] the first matching term in Eq. ([REF]) is zero, and the second term in Eq. ([REF]) is small as the LO contribution of [MATH] channels is small.', '1804.03557-1-27-8': 'As discussed in Ref. [CITATION], Powheg+Pythia yields very similar results as e.g. FONLL [CITATION] or Madgraph5aMC@NLO [CITATION] approaches in the kinematic domain of LHCb.', '1804.03557-1-28-0': '# Results', '1804.03557-1-29-0': 'In this section, we will first illustrate some features of our calculation that we have studied numerically and then compare with the available experimental LHC data.', '1804.03557-1-30-0': '## Consistency checks and other trivia', '1804.03557-1-31-0': 'We begin to fold out the numerical results by showing in the left-hand panel of Figure [REF] contributions from the channels where the [MATH] pair is explicitly produced.', '1804.03557-1-31-1': 'Here, we have taken [MATH] and [MATH].', '1804.03557-1-31-2': 'The solid curves are from the calculation with full mass dependence including the relevant subtraction terms and the dashed ones correspond to the evaluation with zero-mass Wilson coefficients (but still retaining the [MATH] kinematics).', '1804.03557-1-31-3': 'The results are normalized by the full GM-VFNS calculation including all the partonic channels.', '1804.03557-1-31-4': 'At high [MATH] the solid and dashed curves merge which provides a non-trivial, strong check on the consistency of our implementation.', '1804.03557-1-31-5': 'Towards [MATH] the two sets of curves, however, behave completely differently: Whereas all channels of the "massive" calculation yield a positive contribution at [MATH] limit, even the overall result with zero-mass matrix elements remains negative.', '1804.03557-1-32-0': 'As can be seen from the left-hand panel of Figure [REF], the overall contribution from the channels where the [MATH] pair is explicitly produced, is only a few percents from [MATH] onwards.', '1804.03557-1-32-1': 'In fact, almost the entire cross sections in this region accumulates from the partonic subprocesses with heavy quarks in the initial state or gluon fragmentation, around 50% coming from each of these two sources.', '1804.03557-1-32-2': 'This is demonstrated in the right-hand panel of Figure [REF] where we plot the contributions from these channels, normalized to the full GM-VFNS result.', '1804.03557-1-32-3': 'The balance between the contributions shown in the left- and right-hand panels of Figure [REF] depends rather strongly on the scale choices at low [MATH], and the pace at which the contributions in the right-hand panel begin to dominate can be controlled by adjusting the scales.', '1804.03557-1-32-4': 'Indeed, using a lower scale than our default choice, the contributions shown in the right-hand panel would begin to dominate at higher [MATH] than now shown in Figure [REF].', '1804.03557-1-32-5': 'As already mentioned in Section [REF], it was exactly this property that was taken advantage of in Refs. [CITATION] to suppress the divergent contributions at low [MATH].', '1804.03557-1-33-0': 'In the left-hand panel of Figure [REF] we estimate the effects of charm-quark and D-meson masses in our cross sections.', '1804.03557-1-33-1': 'The green curve corresponds to a ZM-VFNS calculation (but still using the aforementioned default scale choice) normalized with the full GM-VFNS result.', '1804.03557-1-33-2': 'In accord with what was seen in Figure [REF], we observe that neglecting the charm mass leads to a lower cross section at low [MATH] due to increasingly negative contributions from [MATH] and [MATH] channels in ZM-VFNS.', '1804.03557-1-33-3': 'The blue dashed curve corresponds to putting [MATH] in Eq. ([REF]) and Eq. ([REF]), that is, ignoring the mass dependence in the fragmentation variable [MATH].', '1804.03557-1-33-4': 'We observe that this manoeuvre leads to increased cross sections.', '1804.03557-1-33-5': 'The origin of the effect can be understood relatively easily on the grounds of Eqs. ([REF]) and ([REF]) from which it follows that [EQUATION] when [MATH].', '1804.03557-1-33-6': 'That is, for fixed [MATH] and [MATH] the partonic cross sections are probed at larger [MATH] and larger [MATH] in comparison to the massless kinematics.', '1804.03557-1-33-7': 'Since the partonic cross sections decrease steeply, particularly with increasing [MATH], also the hadronic cross sections are consequently lower.', '1804.03557-1-33-8': 'In our framework, this explains why the hadronic cross sections are suppressed in the presence of non-zero masses.', '1804.03557-1-33-9': 'This is in contrast to what has been found in Ref. [CITATION] in the case of B mesons, though there a different version of the fragmentation variable [MATH] was used.', '1804.03557-1-33-10': 'Moreover, in Ref. [CITATION] a very similar definition of [MATH] as in Ref. [CITATION] was adopted and there, in turn, the mass effects led to suppressed cross sections (as in our case).', '1804.03557-1-33-11': 'To clear up the systematics of different definitions of the fragmentation variable warrants a separate study which is beyond our scope here.', '1804.03557-1-33-12': 'Nevertheless, the effects of finite hadron and quark masses can be non-negligible up to [MATH] which signifies a considerable source of theoretical uncertainty, given that the definition of fragmentation variable [MATH] is ambiguous.', '1804.03557-1-34-0': 'The validity of our calculation towards low [MATH] could be potentially compromised by the unstable fixed-order NLO scale evolution of the fragmentation functions below [MATH], stemming from singular [MATH] terms in the time-like NLO quark-to-gluon and gluon-to-gluon splitting functions, see e.g. Ref. [CITATION].', '1804.03557-1-34-1': 'The proper treatment of this region requires resummation in both, splitting functions and Wilson coefficients [CITATION].', '1804.03557-1-34-2': 'To exclude contributions from the unstable region we have imposed a condition [MATH] when computing the cross sections.', '1804.03557-1-34-3': 'When doing so we must then make sure that this cut is not overly strict, i.e. that the contribution outside of the introduced cut is negligible.', '1804.03557-1-34-4': 'The reason why the [MATH] limit could pose a problem, can be easily understood: As discussed e.g. in Refs. [CITATION], approximating the convolution of partonic cross sections and PDF by a power law, [EQUATION] where [MATH] and [MATH] does not depend on [MATH], one gets [EQUATION] in the zero-mass approximation.', '1804.03557-1-34-5': 'If the partonic spectrum drops sufficiently strongly in [MATH] (i.e. the exponent [MATH] is large enough), the factor [MATH] efficiently eliminates the contributions from the problematic low-[MATH] domain.', '1804.03557-1-34-6': 'However, in the low-[MATH] region the LHC data [CITATION] show that the hadronic D-meson cross sections tend to level off towards [MATH], see Figure [REF] ahead.', '1804.03557-1-34-7': 'That is, the exponent [MATH] in Eq. ([REF]) decreases and the mechanism above is not as effective in suppressing the small-[MATH] contributions.', '1804.03557-1-34-8': 'In the right-hand panel of Figure [REF] we show [MATH] distributions obtained directly from the full calculation for a few fixed values of [MATH].', '1804.03557-1-34-9': 'Unlike could have been expected on the basis of the above discussion, the cross sections are found practically inert to the small [MATH] region even at very small [MATH].', '1804.03557-1-34-10': 'Here, the explanation seems to be in the form of the D-meson fragmentation functions which at low [MATH] are clearly suppressed in the small-[MATH] region as shown in Figure [REF].', '1804.03557-1-34-11': 'Towards higher [MATH] the small-[MATH] tails go up but then also the probed [MATH] is larger (larger exponent [MATH]) and the contributions are suppressed by virtue of Eq. ([REF]).', '1804.03557-1-34-12': 'The behaviour of the D-meson fragmentation functions are quite different in comparison to e.g. typical pion fragmentation function as demonstrated in Figure [REF] as well.', '1804.03557-1-34-13': 'All in all, the cross sections get hardly any contributions from the small [MATH] region.', '1804.03557-1-34-14': 'At NNLO and beyond, also the PDF evolution becomes similarly unstable at small [MATH], and resummation [CITATION] appears to be required in order to optimally reproduce the small-[MATH] HERA data at low [MATH] [CITATION].', '1804.03557-1-34-15': 'However, at the NLO level we work at, these issues are not yet that pressing.', '1804.03557-1-35-0': 'Before comparing with the data, we wish to shortly discuss the predominant [MATH] ranges sampled by D meson production, in particular the small-[MATH] sensitivity within the LHCb acceptance.', '1804.03557-1-35-1': 'To this end, the upper left-hand panel of Figure [REF] presents examples of [MATH] distributions as obtained from our GM-VFNS calculation.', '1804.03557-1-35-2': 'The distributions are presented for [MATH] with typical LHCb kinematics.', '1804.03557-1-35-3': 'While the lower limits for the [MATH] distributions are always deep in the small-[MATH] domain, the distributions carry a long tail towards large [MATH] - in all of the considered cases there is a clearly non-negligible contribution coming even from the [MATH] region.', '1804.03557-1-35-4': 'In part, this tail originates from the NLO contributions in processes where the [MATH] pair is explicitly produced, but mostly it comes from the new partonic channels that "open" as a result of resumming the collinear logarithms into non-zero heavy-quark PDFs and light-parton FFs.', '1804.03557-1-35-5': 'To corroborate this point, the upper right-hand panel of Figure [REF] shows various [MATH] distributions for the bin [MATH].', '1804.03557-1-35-6': 'The LO calculation with the heavy-quark PDFs and light-parton FFs turned off (only "direct" [MATH]) gets almost no contribution from large [MATH], but when the NLO contributions are switched on, a smallish large-[MATH] tail develops.', '1804.03557-1-35-7': 'The normalized [MATH] spectra from the full LO and NLO calculations are mutually very similar, both getting a significant contribution from [MATH] unlike the contributions from "direct" [MATH] production processes.', '1804.03557-1-35-8': 'This shows that the large-[MATH] tail mostly originates from other than explicit [MATH] processes.', '1804.03557-1-35-9': 'To illustrate this point even further from a different viewpoint we show, in the lower panel of Figure [REF], also predictions from pure LO Pythia simulations.', '1804.03557-1-35-10': 'On one hand, when the origin of the D meson is restricted to underlying [MATH] events, the [MATH] distributions are clearly suppressed at large [MATH].', '1804.03557-1-35-11': 'On the other hand, when the D mesons are allowed to be created from all partonic QCD processes (here we omitted b quarks), the large-[MATH] tail emerges.', '1804.03557-1-35-12': 'All in all, the D-meson production at forward rapidity is sensitive to the PDFs at small-[MATH], but the role of large-[MATH] contributions is still clearly non negligible.', '1804.03557-1-35-13': 'The importance of the large-[MATH] part is something that has maybe been a bit underrated in many recent articles [CITATION], in part because their importance does not show up in fixed-order-based calculations.', '1804.03557-1-35-14': 'We note that a very similar large-[MATH] behaviour is present also in isolated photon production [CITATION].', '1804.03557-1-35-15': 'In inclusive pion production the large-[MATH] tail is even more pronounced [CITATION] due to different behaviour of parton-to-pion FFs, see Figure [REF].', '1804.03557-1-36-0': '## Comparison with LHCb and ALICE data', '1804.03557-1-37-0': 'In this section we will present comparisons of our calculation with the experimental LHCb and ALICE D[MATH] data taken in p-p collisions at [MATH] [CITATION], [MATH] [CITATION] and [MATH] [CITATION].', '1804.03557-1-37-1': 'We will consider two types of theoretical uncertainties, namely those related to the choices of the QCD scales and those related to PDF errors.', '1804.03557-1-37-2': 'In principle, there are also other sources of theory uncertainty in the input variables, like the value picked for the charm-quark mass or the value of [MATH] at the Z-boson pole.', '1804.03557-1-37-3': 'However, to consistently vary these quantities they should be accompanied by PDFs and FFs extracted with the same variation.', '1804.03557-1-37-4': 'The are no error sets (unlike for most modern PDFs) available for the D[MATH] meson fragmentation functions, so the uncertainties in FFs are not considered either.', '1804.03557-1-37-5': 'The definitions of fragmentation variable and heavy-quark scheme are taken here as inherent to the presented calculation.', '1804.03557-1-38-0': 'The PDF uncertainties are evaluated in the standard NNPDF way by computing the variance [EQUATION] where [MATH] is the cross section computed with the [MATH]th member out of the collection of [MATH] PDF replicas, and where [EQUATION] is the central prediction.', '1804.03557-1-38-1': 'The stability of the results against scale variations is quantified as e.g. in Ref. [CITATION] by varying the three scales independently as [EQUATION] where the parameters [MATH] vary between [MATH] and [MATH].', '1804.03557-1-38-2': 'The total scale uncertainty is taken as the maximum and minimum of the 16 cross section found in this way.', '1804.03557-1-38-3': 'With the above choice, the scales remain always above the charm mass-threshold and the potentially large contributions from [MATH] and [MATH] terms are suppressed by limiting the maximal difference of the respective scales to a factor of two.', '1804.03557-1-38-4': 'The results for absolute cross sections are presented in Figure [REF] in the case of [MATH] LHCb p-p data, Figure [REF] in the case of [MATH] LHCb p-p data, Figure [REF] in the case of [MATH] LHCb p-p data, and Figure [REF] in the case of [MATH] ALICE p-p data.', '1804.03557-1-38-5': 'In addition to our GM-VFNS results, we also show the central prediction from the Powheg+Pythia framework in all panels, and separately compare the scale uncertainties of these two different approaches in two rapidity bins at [MATH] in Figure [REF].', '1804.03557-1-39-0': 'In all cases, the data are reproduced very well by the GM-VFNS calculations within the considered theory uncertainties, whereas the central values of the Powheg+Pythia calculations systematically fall short of the data.', '1804.03557-1-39-1': 'Essentially the same hierarchy has been observed e.g. in LHCb/ALICE papers [CITATION] and elsewhere [CITATION], though the GM-VFNS calculations do not extend to zero [MATH] in these references.', '1804.03557-1-40-0': 'We believe that most, or at least a significant part, of the systematic difference between GM-VFNS and Powheg+Pythia setups can be explained by contributions from gluon fragmentation that are resummed in GM-VFNS but that are not accounted for in the Powheg+Pythia calculation.', '1804.03557-1-40-1': 'To better illustrate the point we show, in Figure [REF], a diagram where the upper gluon line radiates one or more gluons before splitting into a [MATH] pair.', '1804.03557-1-40-2': 'Contributions of this type are implicitly included in the GM-VFNS calculation, resummed into the gluon FF [MATH] in collinear configuration.', '1804.03557-1-40-3': 'The Powheg+Pythia framework, however, does not allow for these contributions as the starting point are events in which the [MATH] pair has been produced in the first place.', '1804.03557-1-40-4': 'In other words, the possibility that the [MATH] pair is produced only later on by the Pythia parton shower is possible only for the hard processes where one [MATH] pair has already been produced in matrix-element level.', '1804.03557-1-40-5': 'On the other hand, the standard Pythia simulation with all hard-QCD processes subsumed does include also contributions like those in Figure [REF] and, as was shown in the lower panel of Figure [REF], enabling this possibility changes the [MATH] distributions quite significantly.', '1804.03557-1-40-6': 'In the FONLL approach [CITATION], the contributions like those in Figure [REF] are resummed by partonic FFs, but the contribution of the resummed part is shrouded by a multiplicative factor [MATH] engineered so as to suppress these contributions at low [MATH].', '1804.03557-1-41-0': 'As expected, the scale variation in GM-VFNS calculations, best visible in Figure [REF], is maximal in the low-[MATH] region and diminishes toward larger values.', '1804.03557-1-41-1': 'At low [MATH], the upper limit is set by the calculation with [MATH] and [MATH], and the lower border traces the configuration [MATH] and [MATH].', '1804.03557-1-41-2': 'Neither of these "extreme" cases reproduces the shape of the data particularly well at low [MATH]: either the spectra rise too steeply ([MATH]), or they are too flat ([MATH]).', '1804.03557-1-41-3': "The change of slope of the lower border between [MATH] and [MATH] is a consequence of the heavy-quark PDFs and light-parton FFs becoming 'active' as the scale goes above the mass threshold.", '1804.03557-1-41-4': 'That is, the contributions shown in the right-hand panel of Figure [REF] become suddenly dominant.', '1804.03557-1-41-5': 'Below [MATH] the set condition [MATH] halts the growth of scale uncertainty downwards.', '1804.03557-1-41-6': 'The "natural" choice [MATH] matches much better with the shape of data.', '1804.03557-1-41-7': 'The central prediction goes somewhat above the LHCb data at low [MATH], which could be improved by using a bit lower scale.', '1804.03557-1-41-8': 'Indeed, we have checked that setting the scales as [MATH] with the parameter [MATH] [CITATION], would improve the description with the central prediction.', '1804.03557-1-41-9': 'However, here our intention is not to fine tune the predictions but rather to present the calculation as it is "out of the box" with default settings.', '1804.03557-1-41-10': 'In the Powheg+Pythia case the scales [MATH] and [MATH] are varied independently by a factor of two wrt. central scale within [MATH] in the generation of Powheg [MATH] events.', '1804.03557-1-41-11': 'As can be seen in Figure [REF], the Powheg+Pythia approach is clearly more sensitive to the scale variations than GM-VFNS, especially at [MATH], and the LHCb data still remain within these uncertainties, though at the very upper part of the band.', '1804.03557-1-41-12': 'The scale-uncertainty estimates of Powheg+Pythia method we find here are well in line with the error bands shown in the original data papers of LHCb and ALICE, and e.g. in Ref. [CITATION].', '1804.03557-1-42-0': 'Figure [REF] presents ratios of cross sections measured by LHCb at different [MATH].', '1804.03557-1-42-1': 'Here, a significant part of the theory uncertainties cancel and, sure enough, the central predictions of the both considered methods describe the data rather well even at low [MATH].', '1804.03557-1-42-2': 'However, the Powheg results are systematically below the GM-VFNS predictions in most of the cases, best visible in the [MATH] panel.', '1804.03557-1-42-3': 'That is, the [MATH] dependence is stronger in the GM-VFNS calculation.', '1804.03557-1-42-4': 'We believe that this hierarchy follows mainly from the presence (absence) of gluon fragmentation in GM-VFNS (Powheg+Pyhtia) approach, similarly as in the case of absolute cross sections.', '1804.03557-1-42-5': 'As the Powheg generates events where the [MATH] pair is produced in the hard process or from the hardest emission, part of the increased phase-space for parton shower due to increased [MATH] is not in use for heavy-quark production.', '1804.03557-1-42-6': 'This is consistent also with the observation that the difference between the GM-VFNS and Powheg+Pythia cross sections decreases, though slowly, towards lower [MATH].', '1804.03557-1-43-0': '# Summary', '1804.03557-1-44-0': 'In the present article, we have introduced a novel implementation of the GM-VFNS for hadroproduction of heavy-flavoured mesons.', '1804.03557-1-44-1': 'Here, the novelty amounts to a specific definition of scheme, SACOT-[MATH], which retains the kinematics of heavy quark-antiquark pair production also in contributions where the heavy-flavoured meson formally comes from light-flavour fragmentation or initial-state heavy quarks.', '1804.03557-1-44-2': 'As we have explained, this is physically a natural choice as the origin of these contributions is in the heavy quark-antiquark pair production, and as such it is analogous to the SACOT-[MATH] scheme in deeply inelastic scattering.', '1804.03557-1-44-3': 'Within the SACOT-[MATH] scheme, it is possible to compute the heavy-flavoured meson spectra down to [MATH] with arbitrary choices for renormalization, fragmentation, and renormalization scales.', '1804.03557-1-44-4': 'In earlier works presented in the literature, a finite [MATH] limit could only be achieved by setting the scales in a particular way.', '1804.03557-1-44-5': 'Comparisons with the available D[MATH]-meson data from LHCb and ALICE collaborations indicate that our calculation in SACOT-[MATH] scheme performs well, though it must be admitted that the theoretical uncertainties at low [MATH] are significant.', '1804.03557-1-44-6': 'Here, it is not only the scale, PDF, and FF uncertainties that matter, but also the scheme dependence and ambiguities in treating the heavy-quark and heavy-flavoured meson masses in the fragmentation.', '1804.03557-1-44-7': 'These uncertainties will clearly hamper the usage of low-[MATH] D meson production as a constraint for PDFs.', '1804.03557-1-44-8': 'We have also shown that in all considered rapidities there is a sizeable contribution from large-[MATH] region which, as we have argued, appears to originate from gluon fragmentation.', '1804.03557-1-44-9': 'Thus, estimates based on fixed-order [MATH] pair production overstate the small-[MATH] sensitivity of inclusive D-meson production.', '1804.03557-1-44-10': 'In the future, we plan to extend the SACOT-[MATH] scheme also to the case of intrinsic/fitted charm, B-meson production, as well as nuclear collisions.'} | {'1804.03557-2-0-0': '# Introduction', '1804.03557-2-1-0': 'The hadroproduction of heavy-flavoured mesons at the LHC, in particular the D- and B-meson measurements at forward direction [CITATION], has recently attracted a growing interest for its potential to provide information on partonic dynamics at low momentum fractions.', '1804.03557-2-1-1': 'Because of the finite heavy-quark mass [MATH], the perturbative methods are applicable down to zero transverse momentum ([MATH]) of the observed meson, and the measurements provide opportunities e.g. to constrain the collinearly factorized gluon distributions at small momentum fractions in proton [CITATION] or nucleus [CITATION], or to test other scenarios like saturation physics [CITATION], or [MATH] factorization [CITATION].', '1804.03557-2-1-2': 'The D-meson production is also of great interest from the viewpoint of neutrino astrophysics as the secondary neutrinos from D mesons produced in scatterings of cosmic rays in the atmosphere form a significant background for the extraterrestrial neutrinos.', '1804.03557-2-1-3': 'Given that the D-meson measurements at LHCb [CITATION] are kinematically close to the cosmic-ray-on-air scattering, the rates for secondary neutrinos can be constrained by the LHC data [CITATION].', '1804.03557-2-1-4': 'In heavy-ion collisions the measured open heavy-flavour data [CITATION] provides opportunities e.g. to test the so-called dead-cone effect [CITATION] in QCD medium [CITATION].', '1804.03557-2-2-0': 'Theoretically, there are several collinear-factorization-based ways to calculate cross sections for heavy-flavoured mesons in proton-proton (p-p) collisions, see e.g. Refs. [CITATION] for reviews.', '1804.03557-2-2-1': 'On one hand, parton-level heavy-quark cross sections at fixed-flavour-number scheme (FFNS) [CITATION] can be folded with phenomenological, scale-independent parton-to-meson fragmentation functions (FFs), or the parton-level calculation is matched to a parton-shower [CITATION] from a general-purpose Monte-Carlo event generator, such as Pythia 8 [CITATION] or Herwig [CITATION], and the showered event is then hadronized according to the hadronization model of the generator.', '1804.03557-2-2-2': 'Alternatively, one can work fully within the collinear factorization where the fragmentation is described with universal, scale-dependent FFs [CITATION].', '1804.03557-2-2-3': 'In this paper, we will focus on this latter approach.', '1804.03557-2-3-0': 'The general framework in QCD to treat the heavy-quark production is the so-called general-mass variable flavour number scheme (GM-VFNS) [CITATION].', '1804.03557-2-3-1': 'In this framework, at low interaction scales [MATH] the heavy quarks are not treated as partons in PDFs but are considered only as massive objects in the final state.', '1804.03557-2-3-2': 'The full mass dependence is retained in the production cross sections, but the initial-state partons are restricted only to the light ones.', '1804.03557-2-3-3': 'These cross sections contain mass-dependent logarithmic terms which, towards higher interaction scales, will eventually dominate and diverge.', '1804.03557-2-3-4': 'In GM-VFNS these large logarithms are subtracted at a certain transition scale [MATH] - typically the heavy-quark mass threshold - and resummed into the PDFs and scale-dependent FFs.', '1804.03557-2-3-5': 'At asymptotically high interaction scales [MATH] the result reduces (up to finite terms) to the calculation where the quark mass has been put to zero from the outset, the so-called zero-mass variable flavour number scheme (ZM-VFNS).', '1804.03557-2-4-0': 'To obtain a well-behaved description for the heavy-flavoured mesons within GM-VFNS approach from zero to asymptotically large [MATH] has, however, been a bit challenging.', '1804.03557-2-4-1': 'The difficulty is related to the intrinsic freedom in GM-VFNS to use the zero-mass formalism for the processes with heavy-quarks in the initial state or where the fragmenting parton is a light one.', '1804.03557-2-4-2': 'The massless coefficient functions display a divergent behaviour towards low [MATH] and with a typical scale choice [MATH] their contribution dominates the cross sections immediately above [MATH].', '1804.03557-2-4-3': 'Thus, the production cross sections diverge towards [MATH].', '1804.03557-2-5-0': 'A solution was proposed in Ref. [CITATION].', '1804.03557-2-5-1': 'In essence, the idea was to exclude the aforementioned divergent contributions at low [MATH] by retaining the factorization and fragmentation scales at the threshold [MATH] until large-enough [MATH].', '1804.03557-2-5-2': 'Formally, the difference with respect to a more natural choice [MATH] is one order higher in QCD coupling than what one works at, but numerically the effect is large and the cross sections are rendered finite down to [MATH].', '1804.03557-2-5-3': 'A relatively good description of the LHCb data can be obtained by tuning the scales in this manner [CITATION], but the price to pay is that there will be a certain unphysical wiggle in the production cross section near the region where one decides to turn on the heavy-quark PDFs and light-parton FFs, see e.g. Figure 6 in Ref. [CITATION].', '1804.03557-2-5-4': 'An alternative strategy along this line would be to take the transition scale [MATH] to be much higher than the heavy-quark mass [CITATION].', '1804.03557-2-5-5': 'However, this would lead to a discontinuity in the cross sections at the arbitrary point where one decides to make the transition.', '1804.03557-2-5-6': 'Higher-order calculations should decrease the transient effects in both cases, but will not cure them completely.', '1804.03557-2-5-7': 'Clearly, a different solution would be beneficial.', '1804.03557-2-6-0': 'The option we propose here is to make use of the scheme dependence inherent to GM-VFNS.', '1804.03557-2-6-1': 'Physically, our choice of scheme is rooted in the observation that - in the absence of intrinsic charm component - the contributions from heavy-quark PDFs and light-parton FFs are simply an efficient way to resum diagrams where a heavy quark-antiquark ([MATH]) pair is dynamically produced.', '1804.03557-2-6-2': 'Being of the same origin, it is natural to require that these contributions respect the same kinematical constraints as the channels where the pair is explicit produced.', '1804.03557-2-6-3': 'These are formally [MATH] effects and can be included in the definition of a scheme.', '1804.03557-2-6-4': 'However, the contributions from heavy-quark PDFs and light-parton FFs will no longer diverge in the [MATH] limit, but are regulated by the heavy-quark mass.', '1804.03557-2-6-5': 'The production cross sections thus remain finite in the [MATH] limit with arbitrary factorization and fragmentation scales.', '1804.03557-2-7-0': '# Formalism', '1804.03557-2-8-0': 'In this section we will describe our theoretical construction and its numerical implementation.', '1804.03557-2-8-1': 'As the GM-VFNS framework in hadroproduction of heavy quarks has been detailedly discussed in Refs. [CITATION], we will here focus only on the most important features of our approach.', '1804.03557-2-8-2': 'However, enough details are still given so that our results can be reproduced.', '1804.03557-2-9-0': '## General structure and kinematics', '1804.03557-2-10-0': 'The process we study is an inclusive production of a hadron [MATH] with momentum [MATH] in collision of two hadrons [MATH] and [MATH] with momenta [MATH] and [MATH], [EQUATION].', '1804.03557-2-10-1': "In the approximation where the masses of partons and produced hadron are neglected, the cross section differentiated with respect to the produced hadron's transverse momentum [MATH] and rapidity [MATH] can be written in the well-known factorized form [CITATION], [EQUATION] where the fragmenting parton's transverse momentum and rapidity are [MATH] and [MATH].", '1804.03557-2-10-2': 'Here [MATH] are the PDFs for parton species [MATH] in hadron [MATH] and [MATH] is the parton-to-[MATH] FF.', '1804.03557-2-10-3': 'The invariants [MATH] are defined as [EQUATION] where [MATH] and [MATH] are the momenta of the incoming partons, [MATH] is the momentum of the produced, outgoing parton and [MATH] is center-of-mass (c.m.) energy of the collision.', '1804.03557-2-10-4': 'The integration limits are given by [EQUATION]', '1804.03557-2-11-0': '### Partonic kinematics in the presence of mass', '1804.03557-2-12-0': 'When a [MATH] pair is produced from light partons, the zero-mass partonic kinematics above should be adjusted to account for the heavy-quark mass [MATH].', '1804.03557-2-12-1': 'In practice, this amounts to replacing the partonic transverse momentum [MATH] in the [MATH] integration limits and scaling variables [MATH] by the partonic transverse mass [MATH] , [EQUATION]', '1804.03557-2-12-2': 'These kinematics correspond to the inclusive heavy-quark production.', '1804.03557-2-12-3': 'When the produced parton is a heavy quark, the above replacements follow directly from the momentum conservation.', '1804.03557-2-12-4': 'However, in the case that the fragmenting parton is a light one or when there is a heavy quark in the initial state, these replacements are strictly speaking not necessary, but are part of our choice of scheme (SACOT-[MATH], explained in more detail later).', '1804.03557-2-12-5': 'In the picture where the heavy quarks are generated perturbatively, the heavy-flavour PDFs and light-flavour FFs are merely an efficient way to resum diagrams where a heavy quark-antiquark pair is created.', '1804.03557-2-12-6': 'That is, the production of heavy-flavour pair is implicit in these contributions and motivates the usage of heavy-flavour kinematics.', '1804.03557-2-13-0': '### Massive fragmentation variable', '1804.03557-2-14-0': 'The zero-mass version of the fragmentation scaling variable [MATH] is ill-defined in the presence of massive quarks/hadrons, and the zero-mass relations [MATH] and [MATH] are no longer true.', '1804.03557-2-14-1': 'Here, we choose to define the scaling variable [MATH] in a Lorentz-invariant way as [EQUATION]', '1804.03557-2-14-2': 'As indicated, in the c.m. frame of the colliding hadrons [MATH] can be interpreted as the fraction of partonic energy carried by the outgoing hadron [CITATION].', '1804.03557-2-14-3': 'Alternatively, the scaling variable could be defined e.g. in terms of light-cone momentum fractions [CITATION].', '1804.03557-2-14-4': 'From the above definition and considering the fragmentation to be collinear in the c.m. frame, we have two equations, [EQUATION] where the hadronic transverse mass is defined as [MATH], [MATH] being the mass of the produced hadron.', '1804.03557-2-14-5': 'We can solve these equations for the hadronic transverse momentum and rapidity, [EQUATION]', '1804.03557-2-14-6': 'The cross section corresponding to the above definition of [MATH] can be obtained as [EQUATION] by integrating over [MATH] and [MATH].', '1804.03557-2-14-7': 'Using the relation [EQUATION] we find again Eq. ([REF]) where the partonic transverse momentum and rapidity are now given by [EQUATION] and the hadron mass corrects the lower limit of the [MATH] integration as [EQUATION]', '1804.03557-2-14-8': 'Otherwise the cross-section formula is formally identical to the case of zero-mass partons and hadrons.', '1804.03557-2-15-0': '## Partonic cross sections in SACOT and SACOT-[MATH] schemes', '1804.03557-2-16-0': 'The starting point in our GM-VFNS construction, is the next-to-leading-order (NLO) one-particle inclusive heavy-quark cross section in FFNS [CITATION] where heavy flavour can be produced in three different partonic processes, [EQUATION]', '1804.03557-2-16-1': 'In FFNS (and also GM-VFNS at low interaction scales), these are the only ways to produce heavy flavour.', '1804.03557-2-16-2': 'The heavy-quark mass [MATH] is kept finite in these processes and in the high-[MATH] limit, the partonic cross sections develop logarithmic divergences [MATH] coming from kinematic regions where the heavy quarks become collinear with other partons.', '1804.03557-2-16-3': 'These are the first terms in the whole series of large collinear logarithms which, in GM-VFNS framework, are resummed to heavy-quark PDFs and parton-to-hadron FFs when the interaction scale exceeds a chosen transition scale [MATH].', '1804.03557-2-16-4': 'From now on identify [MATH] as the heavy quark mass, [MATH].', '1804.03557-2-16-5': 'To avoid double counting when including the contributions also from heavy-quark PDFs and using the scale-dependent parton-to-hadron FFs, one has then to subtract these logarithmic pieces from the coefficient functions.', '1804.03557-2-16-6': 'In what follows, we will explain what are the added and subtracted terms in our case, using the [MATH] channel as an explicit example.', '1804.03557-2-17-0': 'The gluon-fusion process [MATH] at NLO entails four different sources of collinear divergences in the [MATH] (or equivalently [MATH]) limit, illustrated in Figure [REF]:', '1804.03557-2-18-0': '- one of the two initial-state gluons splits into a collinear heavy quark-antiquark pair,', '1804.03557-2-19-0': '- an outgoing gluon splits into collinear heavy quark-antiquark pair,', '1804.03557-2-20-0': '- an outgoing heavy quark emits a collinear gluon.', '1804.03557-2-21-0': 'A simple way to specify the GM-VFNS subtraction terms at NLO is to take as the starting point the leading-order (LO) contributions from channels where there are heavy quarks in the initial state or the fragmenting parton is a light one.', '1804.03557-2-21-1': 'Let us begin with the former case.', '1804.03557-2-21-2': 'Using Eq. ([REF]), we write the leading-order contribution for process [MATH] as [EQUATION]', '1804.03557-2-21-3': 'This now uniquely determines the subtraction term which cancels the logarithmic term from diagrams like (b) in Figure [REF].', '1804.03557-2-21-4': 'The expression for perturbative heavy-quark PDF, to the first order in strong coupling [MATH], reads [EQUATION] where [MATH] with [MATH], is the leading-order gluon-to-quark splitting function.', '1804.03557-2-21-5': 'Using this expression for [MATH] in Eq. ([REF]) gives our definition of the subtraction term, [EQUATION]', '1804.03557-2-21-6': 'When adding the leading-order contribution of Eq. ([REF]), one must then compensate by subtracting Eq. ([REF]).', '1804.03557-2-21-7': 'The difference contributes at [MATH] and is not considered at an NLO-level [MATH] calculation.', '1804.03557-2-21-8': 'Here, we also plainly see the origin of the scheme dependence in GM-VFNS: The exact form of [MATH] appearing in Eq. ([REF]) and Eq. ([REF]) is subject to a certain amount of arbitrariness.', '1804.03557-2-21-9': 'Indeed, the only requirement is that in the [MATH] limit [MATH] must tend to the zero-mass expression [MATH] so as to ensure that the corresponding collinear logarithm from [MATH] process cancels.', '1804.03557-2-21-10': 'Otherwise we can choose it at will.', '1804.03557-2-21-11': 'Similarly, the exact expressions for the integration limits are irrelevant as far as the zero-mass expressions given in Eq. ([REF]) are found in the [MATH] limit.', '1804.03557-2-21-12': 'The simplest option is to use the zero-mass matrix elements and kinematics from the beginning - this choice of scheme is usually dubbed as simplified ACOT, or SACOT scheme [CITATION].', '1804.03557-2-21-13': 'Here, we shall adopt a prescription where we use the zero-mass matrix elements but still retain the kinematic mass dependence.', '1804.03557-2-21-14': 'In other words, the integration limits and the invariants [MATH] and [MATH] are as in Eq. ([REF]), and for the squared matrix element in Eqs. ([REF]) and ([REF]) we take, [EQUATION] where [MATH] is obtained from the zero-mass expression [CITATION], [EQUATION] with [MATH], [MATH], [MATH], and the "massive" Mandelstam variables being now [MATH], and [MATH].', '1804.03557-2-21-15': 'In practice, our prescription amounts to replacing the partonic transverse momentum [MATH] in the zero-mass expressions by the transverse mass [MATH] - hence we shall name the present implementation as SACOT-[MATH] scheme.', '1804.03557-2-22-0': 'The leading-order contribution and subtraction term for the [MATH] channel are defined in a similar manner as above, so let us then discuss the contributions from light-parton fragmentation and the corresponding subtraction terms.', '1804.03557-2-22-1': 'Proceeding as in the case of initial state, we define the leading-order contribution from [MATH] channel, originating from diagrams like (c) in Figure [REF], by [EQUATION]', '1804.03557-2-22-2': 'Together with the the perturbative expression for the gluon fragmentation function (considering that the only non-zero FF at the mass threshold is [MATH]), [EQUATION] this defines the subtraction term [EQUATION] where now [EQUATION]', '1804.03557-2-22-3': 'The subtractions required to cancel the large logarithm originating from diagram (d) in Figure [REF] goes slightly different than the above cases.', '1804.03557-2-22-4': 'The reason is that the contributions from [MATH] channel (part of the inclusive heavy-quark cross sections) that we here use to determine the subtraction terms, are included using the full mass-dependence.', '1804.03557-2-22-5': 'Therefore, the subtraction term required to cancel the large logarithm that occurs when final-state heavy quark emits a collinear gluon is [EQUATION] where [MATH] is the quark-to-quark splitting function, and the matrix element [CITATION], [EQUATION] now carries the full mass dependence.', '1804.03557-2-22-6': 'In order to recover the standard [MATH] zero-mass results at high [MATH] we must still compensate for the fact that the [MATH] limit in the massive calculation does not exactly match that of usual massless [MATH], but some finite differences remain as a relic of a different regularization procedure.', '1804.03557-2-22-7': 'As explained in Ref. [CITATION], this can be effectively achieved by replacing Eq. ([REF]) by [EQUATION] where [MATH] is the partonic fragmentation function [CITATION], [EQUATION]', '1804.03557-2-22-8': 'Also the renormalization procedure applied in FFNS calculations is slightly different than in the purely zero-mass case.', '1804.03557-2-22-9': 'Indeed, the FFNS results of Ref. [CITATION] are obtained in a so-called decoupling scheme, where [MATH] runs with only light partons (gluons + 3 light-flavour quarks).', '1804.03557-2-22-10': 'Above the transition scale [MATH], also the heavy-quark is considered as being "active" in the running of [MATH], and the matching between the two schemes induces an additional contribution.', '1804.03557-2-22-11': 'Specifically, we must add a term [EQUATION] as explained in Ref. [CITATION].', '1804.03557-2-23-0': 'The same line of reasoning is applied when defining the subtraction terms for [MATH] and [MATH] channels and the emerging leading-order contributions from [MATH], [MATH], and [MATH] channels.', '1804.03557-2-23-1': 'As in Eq. ([REF]), the definition of the [MATH] subtraction term involves the partonic fragmentation function [MATH], and a term [EQUATION] is added to recover the [MATH] renormalization scheme [CITATION].', '1804.03557-2-23-2': 'In addition, our full results include the contributions from all other partonic subprocesses whose inclusion does not require a preparation of subtraction terms at the perturbative order we work at.', '1804.03557-2-23-3': 'The NLO [MATH] contributions, taken from Ref. [CITATION] are included as well.', '1804.03557-2-23-4': 'We stress that when including these terms in our SACOT-[MATH] scheme, we consistently retain the kinematics which they inherit from [MATH] pair-creation process as explained earlier.', '1804.03557-2-23-5': 'In practice this is done by trading the massless variables [MATH] and [MATH] used in Ref. [CITATION] by their massive counterparts [MATH] and [MATH], see e.g. Sect. 2 of Ref. [CITATION], and imposing the proper integration limits explained in Section [REF].', '1804.03557-2-23-6': 'In this way, we already implicitly define the subtraction terms that would be required at a next-to-NLO (NNLO) -level calculation.', '1804.03557-2-24-0': 'In comparison to the earlier works [CITATION], the most notable advantage of the SACOT-[MATH] scheme is that the cross sections remain finite in the [MATH] limit.', '1804.03557-2-24-1': 'Indeed, in Refs. [CITATION] at least part of the contributions not coming directly from flavour-creation processes are included using purely zero-mass formalism, and give rise to a divergent [MATH] behaviour at [MATH] limit.', '1804.03557-2-24-2': 'The difficulty will not be completely resolved at NNLO either, though the divergences may be a bit "softer".', '1804.03557-2-24-3': 'In Ref. [CITATION] these divergent contributions were excluded at small [MATH] by maintaining the factorization and fragmentation scale at (or below) the heavy-quark mass threshold until large-enough [MATH].', '1804.03557-2-24-4': 'This procedure leads to finite cross sections in the [MATH] limit, but causes certain unphysical slope change near the [MATH] value where the factorization and fragmentation scales go above the mass threshold - we will come back to this in Section [REF] (see also see Fig. 6 in Ref. [CITATION]).', '1804.03557-2-24-5': 'In our case - and this applies also for the fixed-order calculations - the divergent behaviour is regulated by the heavy-quark mass and leads to finite cross sections even at [MATH] (at any perturbative order) without a need to fine tune the scale choices.', '1804.03557-2-24-6': 'Technically, this happens because the lower limits for the scaling variables [MATH] and [MATH] appearing in the squared matrix elements are not zero but limited by the heavy-quark mass.', '1804.03557-2-25-0': '## Numerical implementation', '1804.03557-2-26-0': 'Our numerical realization of the GM-VFNS scheme described above is crafted around the public INCNLO [CITATION] and Mangano-Nason-Ridolfi (MNR) [CITATION] codes.', '1804.03557-2-26-1': 'The former provides the zero-mass matrix elements, and the latter one the one-particle inclusive heavy-quark cross section of Ref. [CITATION].', '1804.03557-2-26-2': 'As already noted in Ref. [CITATION], in order to obtain reliable numerical results from INCNLO at high [MATH] away from the midrapidity [MATH], the numerical stability of the original code has had to be improved, see p.30-32 in Ref. [CITATION] for a detailed explanation.', '1804.03557-2-26-3': 'Schematically, we compute [EQUATION] where the inclusion of charge-conjugate contributions and shuffling between the initial-state partons is implicit.', '1804.03557-2-26-4': 'That is, from the full zero-mass result we subtract the zero-mass contributions of [MATH] channels which we add back using the full mass dependence.', '1804.03557-2-26-5': 'The subtraction terms provide the proper matching.', '1804.03557-2-26-6': 'Towards high [MATH], only the first sum term in Eq. ([REF]) survives - others add up to zero.', '1804.03557-2-26-7': 'In the numerical evaluation we have used NNPDF31nlopchas0118 variable-flavour-number PDFs and the corresponding running strong coupling [MATH] [CITATION].', '1804.03557-2-26-8': 'This is the latest NNPDF fit assuming no intrinsic charm content in the proton.', '1804.03557-2-26-9': 'The PDFs are interfaced by using LHAPDF 6 library [CITATION].', '1804.03557-2-26-10': 'The introduced framework is applicable to production of any hadrons involving heavy quarks but in this work we consider only D-meson production due to good availability and precision of the experimental data from LHC experiments.', '1804.03557-2-26-11': 'In particular, we will focus on D[MATH]-meson production, with the data from LHCb [CITATION], ALICE [CITATION], and CMS [CITATION] (though not yet available) extending to small [MATH] region, which is where our SACOT-[MATH] scheme mostly differs from other GM-VFNS implementations (the LHCb collaboration has also measured D[MATH] at small [MATH]).', '1804.03557-2-26-12': "For D[MATH] mesons there would be more recent FF analyses available [CITATION] but for D[MATH] we use KKKS08 [CITATION] FFs, which is the only available FF set for D[MATH]'s.", '1804.03557-2-26-13': 'For the charm-quark mass we use [MATH] in accordance with the used PDF set.', '1804.03557-2-26-14': 'The input charm mass in KKKS08 analysis was [MATH] so the pairing with NNPDF3.1 is consistent.', '1804.03557-2-26-15': 'Our default scale choice will be [MATH], and for the D-meson mass we use [MATH].', '1804.03557-2-26-16': 'The small contribution from b-quark fragmentation is retained in the calculation neglecting the finite b-quark mass.', '1804.03557-2-26-17': 'The D mesons from B-meson decays have been excluded from the LHCb and ALICE data we discuss later on, but as the KKKS08 FFs include these feed-down D mesons as well, there is no fully consistent way to exclude them without explicitly evaluating the D[MATH] meson spectra from B-meson decays and subtracting it from the fully inclusive cross section.', '1804.03557-2-26-18': 'However, the contributions from B decays are very small, less than 1% in the integrated inclusive D[MATH]-meson cross section of ALICE [CITATION].', '1804.03557-2-27-0': 'In order to compare with another popular approach, we have used here the Powheg method [CITATION] in which the [MATH] production at FFNS is matched with the Pythia parton shower providing NLO accuracy for the matrix element generation and leading-log resummation from the parton shower.', '1804.03557-2-27-1': 'In practice, we have first generated [MATH] events with the hvq part [CITATION] of Powheg-Box generator [CITATION].', '1804.03557-2-27-2': 'The generated events are then fed into Pythia (version 8.230) [CITATION] which generates the [MATH]-ordered parton shower and hadronizes the events using the implemented Lund string model with parameter values from the default Monash tune [CITATION].', '1804.03557-2-27-3': 'The D[MATH] mesons (and its charge conjugate) are then picked up from the hadronized final state and binned in [MATH] and [MATH].', '1804.03557-2-27-4': 'The same NNPDF3.1 PDFs as for the GM-VFNS calculations have been used for the event generation in Powheg and also in showering within Pythia.', '1804.03557-2-27-5': 'The sensitivity of the Pythia shower to PDFs is very mild as they affect only the initial-state emission probabilities and there only ratios of PDFs are involved.', '1804.03557-2-27-6': 'In Powheg generation the default scale choice is [MATH] with [MATH].', '1804.03557-2-27-7': 'We have not explicitly introduced the matching terms, Eqs. ([REF]) and ([REF]), at the heavy-quark mass thresholds as their effect has been found small in the [MATH] range of LHCb data [CITATION].', '1804.03557-2-27-8': 'Indeed, with [MATH] the first matching term in Eq. ([REF]) is zero, and the second term in Eq. ([REF]) is small as the LO contribution of [MATH] channels is small.', '1804.03557-2-27-9': 'As discussed in Ref. [CITATION], Powheg+Pythia yields very similar results as e.g. FONLL [CITATION] or Madgraph5aMC@NLO [CITATION] approaches in the kinematic domain of LHCb.', '1804.03557-2-28-0': '# Results', '1804.03557-2-29-0': 'In this section, we will first illustrate some features of our calculation that we have studied numerically and then compare with the available experimental LHC data.', '1804.03557-2-30-0': '## Consistency checks and other trivia', '1804.03557-2-31-0': 'We begin to fold out the numerical results by showing in the left-hand panel of Figure [REF] contributions from the channels where the [MATH] pair is explicitly produced.', '1804.03557-2-31-1': 'Here, we have taken [MATH] and [MATH].', '1804.03557-2-31-2': 'The solid curves are from the calculation with full mass dependence including the relevant subtraction terms and the dashed ones correspond to the evaluation with zero-mass Wilson coefficients (but still retaining the [MATH] kinematics).', '1804.03557-2-31-3': 'The results are normalized by the full GM-VFNS calculation including all the partonic channels.', '1804.03557-2-31-4': 'At high [MATH] the solid and dashed curves merge which provides a non-trivial, strong check on the consistency of our implementation.', '1804.03557-2-31-5': 'Towards [MATH] the two sets of curves, however, behave completely differently: Whereas all channels of the "massive" calculation yield a positive contribution at [MATH] limit, even the overall result with zero-mass matrix elements remains negative.', '1804.03557-2-32-0': 'As can be seen from the left-hand panel of Figure [REF], the overall contribution from the channels where the [MATH] pair is explicitly produced, is only a few percents from [MATH] onwards.', '1804.03557-2-32-1': 'In fact, almost the entire cross sections in this region accumulates from the partonic subprocesses with heavy quarks in the initial state or gluon fragmentation, around 50% coming from each of these two sources.', '1804.03557-2-32-2': 'This is demonstrated in the right-hand panel of Figure [REF] where we plot the contributions from these channels, normalized to the full GM-VFNS result.', '1804.03557-2-32-3': 'The balance between the contributions shown in the left- and right-hand panels of Figure [REF] depends rather strongly on the scale choices at low [MATH], and the pace at which the contributions in the right-hand panel begin to dominate can be controlled by adjusting the scales.', '1804.03557-2-32-4': 'Indeed, using a lower scale than our default choice, the contributions shown in the right-hand panel would begin to dominate at higher [MATH] than now shown in Figure [REF].', '1804.03557-2-32-5': 'As already mentioned in Section [REF], it was exactly this property that was taken advantage of in Refs. [CITATION] to suppress the divergent contributions at low [MATH].', '1804.03557-2-33-0': 'In the left-hand panel of Figure [REF] we estimate the effects of charm-quark and D-meson masses in our cross sections at [MATH].', '1804.03557-2-33-1': 'The green curve corresponds to a ZM-VFNS calculation (but still using the aforementioned default scale choice) normalized with the full GM-VFNS result.', '1804.03557-2-33-2': 'In accord with what was seen in Figure [REF], we observe that neglecting the charm mass leads to a lower cross section at low [MATH] due to increasingly negative contributions from [MATH] and [MATH] channels in ZM-VFNS.', '1804.03557-2-33-3': 'The blue dashed curve corresponds to putting [MATH] in Eq. ([REF]) and Eq. ([REF]), that is, ignoring the mass dependence in the fragmentation variable [MATH].', '1804.03557-2-33-4': 'We observe that this manoeuvre leads to increased cross sections.', '1804.03557-2-33-5': 'The origin of the effect can be understood relatively easily on the grounds of Eqs. ([REF]) and ([REF]) from which it follows that [EQUATION] when [MATH].', '1804.03557-2-33-6': 'That is, for fixed [MATH] and [MATH] the partonic cross sections are probed at larger [MATH] and larger [MATH] in comparison to the massless kinematics.', '1804.03557-2-33-7': 'Since the partonic cross sections decrease steeply, particularly with increasing [MATH], also the hadronic cross sections are consequently lower.', '1804.03557-2-33-8': 'In our framework, this explains why the hadronic cross sections are suppressed in the presence of non-zero masses.', '1804.03557-2-33-9': 'This is in contrast to what has been found in Ref. [CITATION] in the case of B mesons, though there a different version of the fragmentation variable [MATH] was used.', '1804.03557-2-33-10': 'Moreover, in Ref. [CITATION] a very similar definition of [MATH] as in Ref. [CITATION] was adopted and there, in turn, the mass effects led to suppressed cross sections (as in our case).', '1804.03557-2-33-11': 'To clear up the systematics of different definitions of the fragmentation variable warrants a separate study which is beyond our scope here.', '1804.03557-2-33-12': 'Nevertheless, the effects of finite hadron and quark masses can be non-negligible up to [MATH] which signifies a possibly considerable source of theoretical uncertainty, given that the definition of fragmentation variable [MATH] is ambiguous.', '1804.03557-2-33-13': 'In the context of the present definition of [MATH] these effects will, however, get milder towards larger [MATH].', '1804.03557-2-33-14': 'This can be easily understood from Eq. ([REF]) from which it follows that [MATH] when [MATH].', '1804.03557-2-33-15': 'Thus, with the present definition of the fragmentation variable [MATH], part of the significant effect found at [MATH] can be expected to melt away.', '1804.03557-2-33-16': 'This is demonstrated in the right-hand panel of Figure [REF] where we show the impact of massive fragmentation variable also in the forward direction.', '1804.03557-2-33-17': 'The differences between massless and massive fragmentation variable get clearly suppressed when moving to larger [MATH].', '1804.03557-2-33-18': 'At very large [MATH] the effect starts to rise again as the partonic [MATH] spectrum gets steeper (near [MATH] it is quite flat) and the condition [MATH] of Eq. ([REF]) begins to matter increasingly.', '1804.03557-2-34-0': 'The validity of our calculation towards low [MATH] could be potentially compromised by the unstable fixed-order NLO scale evolution of the fragmentation functions below [MATH], stemming from singular [MATH] terms in the time-like NLO quark-to-gluon and gluon-to-gluon splitting functions, see e.g. Ref. [CITATION].', '1804.03557-2-34-1': 'The proper treatment of this region requires resummation in both, splitting functions and Wilson coefficients [CITATION].', '1804.03557-2-34-2': 'To exclude contributions from the unstable region we have imposed a condition [MATH] when computing the cross sections.', '1804.03557-2-34-3': 'When doing so we must then make sure that this cut is not overly strict, i.e. that the contribution outside of the introduced cut is negligible.', '1804.03557-2-34-4': 'The reason why the [MATH] limit could pose a problem, can be easily understood: As discussed e.g. in Refs. [CITATION], approximating the convolution of partonic cross sections and PDF by a power law, [EQUATION] where [MATH] and [MATH] does not depend on [MATH], one gets [EQUATION] in the zero-mass approximation.', '1804.03557-2-34-5': 'If the partonic spectrum drops sufficiently strongly in [MATH] (i.e. the exponent [MATH] is large enough), the factor [MATH] efficiently eliminates the contributions from the problematic low-[MATH] domain.', '1804.03557-2-34-6': 'However, in the low-[MATH] region the LHC data [CITATION] show that the hadronic D-meson cross sections tend to level off towards [MATH], see Figure [REF] ahead.', '1804.03557-2-34-7': 'That is, the exponent [MATH] in Eq. ([REF]) decreases and the mechanism above is not as effective in suppressing the small-[MATH] contributions.', '1804.03557-2-34-8': 'In Figure [REF] we show [MATH] distributions obtained directly from the full calculation for a few fixed values of [MATH].', '1804.03557-2-34-9': 'Unlike could have been expected on the basis of the above discussion, the cross sections are found practically inert to the small [MATH] region even at very small [MATH].', '1804.03557-2-34-10': 'Here, the explanation seems to be in the form of the D-meson fragmentation functions which at low [MATH] are clearly suppressed in the small-[MATH] region as shown in Figure [REF].', '1804.03557-2-34-11': 'Towards higher [MATH] the small-[MATH] tails go up but then also the probed [MATH] is larger (larger exponent [MATH]) and the contributions are suppressed by virtue of Eq. ([REF]).', '1804.03557-2-34-12': 'The behaviour of the D-meson fragmentation functions are quite different in comparison to e.g. typical pion fragmentation function as demonstrated in Figure [REF] as well.', '1804.03557-2-34-13': 'All in all, the cross sections get hardly any contributions from the small [MATH] region.', '1804.03557-2-34-14': 'At NNLO and beyond, also the PDF evolution becomes similarly unstable at small [MATH], and resummation [CITATION] appears to be required in order to optimally reproduce the small-[MATH] HERA data at low [MATH] [CITATION].', '1804.03557-2-34-15': 'However, at the NLO level we work at, these issues are not yet that pressing.', '1804.03557-2-34-16': 'Indeed, based on Ref. [CITATION], the effects of resummation in PDFs are only modest at NLO, and thus we expect that the small-[MATH] resummation would lead to only subleading effects in comparison to the very large scale uncertainty in low [MATH] D-meson production, see e.g. Figure [REF] ahead.', '1804.03557-2-35-0': 'Before comparing with the data, we wish to shortly discuss the predominant [MATH] ranges sampled by D meson production, in particular the small-[MATH] sensitivity within the LHCb acceptance.', '1804.03557-2-35-1': 'To this end, the upper left-hand panel of Figure [REF] presents examples of [MATH] distributions as obtained from our GM-VFNS calculation.', '1804.03557-2-35-2': 'The distributions are presented for [MATH] with typical LHCb kinematics.', '1804.03557-2-35-3': 'While the lower limits for the [MATH] distributions are always deep in the small-[MATH] domain, the distributions carry a long tail towards large [MATH] - in all of the considered cases there is a clearly non-negligible contribution coming even from the [MATH] region.', '1804.03557-2-35-4': 'In part, this tail originates from the NLO contributions in processes where the [MATH] pair is explicitly produced, but mostly it comes from the new partonic channels that "open" as a result of resumming the collinear logarithms into non-zero heavy-quark PDFs and light-parton FFs.', '1804.03557-2-35-5': 'To corroborate this point, the upper right-hand panel of Figure [REF] shows various [MATH] distributions for the bin [MATH].', '1804.03557-2-35-6': 'The LO calculation with the heavy-quark PDFs and light-parton FFs turned off (only "direct" [MATH]) gets almost no contribution from large [MATH], but when the NLO contributions are switched on, a smallish large-[MATH] tail develops.', '1804.03557-2-35-7': 'The normalized [MATH] spectra from the full LO and NLO calculations are mutually very similar, both getting a significant contribution from [MATH] unlike the contributions from "direct" [MATH] production processes.', '1804.03557-2-35-8': 'This shows that the large-[MATH] tail mostly originates from other than explicit [MATH] processes.', '1804.03557-2-35-9': 'To illustrate this point even further from a different viewpoint we show, in the lower panel of Figure [REF], also predictions from pure LO Pythia simulations.', '1804.03557-2-35-10': 'On one hand, when the origin of the D meson is restricted to underlying [MATH] events, the [MATH] distributions are clearly suppressed at large [MATH].', '1804.03557-2-35-11': 'On the other hand, when the D mesons are allowed to be created from all partonic QCD processes (here we omitted b quarks), the large-[MATH] tail emerges.', '1804.03557-2-35-12': 'All in all, the D-meson production at forward rapidity is sensitive to the PDFs at small-[MATH], but the role of large-[MATH] contributions is still clearly non negligible.', '1804.03557-2-35-13': 'The importance of the large-[MATH] part is something that has maybe been a bit underrated in many recent articles [CITATION], in part because their importance does not show up in fixed-order-based calculations.', '1804.03557-2-35-14': 'We note that a very similar large-[MATH] behaviour is present also in isolated photon production [CITATION].', '1804.03557-2-35-15': 'In inclusive pion production the large-[MATH] tail is even more pronounced [CITATION] due to different behaviour of parton-to-pion FFs, see Figure [REF].', '1804.03557-2-36-0': '## Comparison with LHCb and ALICE data', '1804.03557-2-37-0': 'In this section we will present comparisons of our calculation with the experimental LHCb and ALICE D[MATH] data taken in p-p collisions at [MATH] [CITATION], [MATH] [CITATION] and [MATH] [CITATION].', '1804.03557-2-37-1': 'We will consider two types of theoretical uncertainties, namely those related to the choices of the QCD scales and those related to PDF errors.', '1804.03557-2-37-2': 'In principle, there are also other sources of theory uncertainty in the input variables, like the value picked for the charm-quark mass or the value of [MATH] at the Z-boson pole.', '1804.03557-2-37-3': 'However, to consistently vary these quantities they should be accompanied by PDFs and FFs extracted with the same variation.', '1804.03557-2-37-4': 'The are no error sets (unlike for most modern PDFs) available for the D[MATH] meson fragmentation functions, so the uncertainties in FFs are not considered either.', '1804.03557-2-37-5': 'The definitions of fragmentation variable and heavy-quark scheme are taken here as inherent to the presented calculation.', '1804.03557-2-38-0': 'The PDF uncertainties are evaluated in the standard NNPDF way by computing the variance [EQUATION] where [MATH] is the cross section computed with the [MATH]th member out of the collection of [MATH] PDF replicas, and where [EQUATION] is the central prediction.', '1804.03557-2-38-1': 'The stability of the results against scale variations is quantified as e.g. in Ref. [CITATION] by varying the three scales independently as [EQUATION] where the parameters [MATH] vary between [MATH] and [MATH].', '1804.03557-2-38-2': 'The total scale uncertainty is taken as the maximum and minimum of the 16 cross section found in this way.', '1804.03557-2-38-3': 'With the above choice, the scales remain always above the charm mass-threshold and the potentially large contributions from [MATH] and [MATH] terms are suppressed by limiting the maximal difference of the respective scales to a factor of two.', '1804.03557-2-38-4': 'The results for absolute cross sections are presented in Figure [REF] in the case of [MATH] LHCb p-p data, Figure [REF] in the case of [MATH] LHCb p-p data, Figure [REF] in the case of [MATH] LHCb p-p data, and Figure [REF] in the case of [MATH] ALICE p-p data.', '1804.03557-2-38-5': 'In addition to our GM-VFNS results, we also show the central prediction from the Powheg+Pythia framework in all panels, and separately compare the scale uncertainties of these two different approaches in two rapidity bins at [MATH] in Figure [REF].', '1804.03557-2-39-0': 'In all cases, the data are reproduced very well by the GM-VFNS calculations within the considered theory uncertainties, whereas the central values of the Powheg+Pythia calculations systematically fall short of the data.', '1804.03557-2-39-1': 'Essentially the same hierarchy has been observed e.g. in LHCb/ALICE papers [CITATION] and elsewhere [CITATION], though the GM-VFNS calculations do not extend to zero [MATH] in these references.', '1804.03557-2-40-0': 'We believe that most, or at least a significant part, of the systematic difference between GM-VFNS and Powheg+Pythia setups can be explained by contributions from gluon fragmentation that are resummed in GM-VFNS but that are not accounted for in the Powheg+Pythia calculation.', '1804.03557-2-40-1': 'To better illustrate the point we show, in Figure [REF], a diagram where the upper gluon line radiates one or more gluons before splitting into a [MATH] pair.', '1804.03557-2-40-2': 'Contributions of this type are implicitly included in the GM-VFNS calculation, resummed into the gluon FF [MATH] in collinear configuration.', '1804.03557-2-40-3': 'They correspond to roughly 50% of the [MATH]-meson cross sections in [MATH] and [MATH] region investigated here (ALICE and LHCb acceptance).', '1804.03557-2-40-4': 'The Powheg+Pythia framework, however, does not allow for these contributions as the starting point are events in which the [MATH] pair has been produced in the first place.', '1804.03557-2-40-5': 'In other words, the possibility that the [MATH] pair is produced only later on by the Pythia parton shower is possible only for the hard processes where one [MATH] pair has already been produced in matrix-element level.', '1804.03557-2-40-6': 'On the other hand, the standard Pythia simulation with all hard-QCD processes subsumed does include also contributions like those in Figure [REF] and, as was shown in the lower panel of Figure [REF], enabling this possibility changes the [MATH] distributions quite significantly.', '1804.03557-2-40-7': 'In the FONLL approach [CITATION], the contributions like those in Figure [REF] are resummed by partonic FFs, but the contribution of the resummed part is shrouded by a multiplicative factor [MATH] engineered so as to suppress these contributions at low [MATH].', '1804.03557-2-41-0': 'As expected, the scale variation in GM-VFNS calculations, best visible in Figure [REF], is maximal in the low-[MATH] region and diminishes toward larger values.', '1804.03557-2-41-1': 'At the LHCb acceptance, the scale uncertainty is clearly larger than the variation we found using massless/massive fragmentation variable, see back Figure [REF].', '1804.03557-2-41-2': 'At low [MATH], the upper limit for the scale variation is set by the calculation with [MATH] and [MATH], and the lower border traces the configuration [MATH] and [MATH].', '1804.03557-2-41-3': 'Neither of these "extreme" cases reproduces the shape of the data particularly well at low [MATH]: either the spectra rise too steeply ([MATH]), or they are too flat ([MATH]).', '1804.03557-2-41-4': "The change of slope of the lower border between [MATH] and [MATH] is a consequence of the heavy-quark PDFs and light-parton FFs becoming 'active' as the scale goes above the mass threshold.", '1804.03557-2-41-5': 'That is, the contributions shown in the right-hand panel of Figure [REF] become suddenly dominant.', '1804.03557-2-41-6': 'Below [MATH] the set condition [MATH] halts the growth of scale uncertainty downwards.', '1804.03557-2-41-7': 'The "natural" choice [MATH] matches much better with the shape of data.', '1804.03557-2-41-8': 'The central prediction goes somewhat above the LHCb data at low [MATH], which could be improved by using a bit lower scale.', '1804.03557-2-41-9': 'Indeed, we have checked that setting the scales as [MATH] with the parameter [MATH] [CITATION], would improve the description with the central prediction.', '1804.03557-2-41-10': 'However, here our intention is not to fine tune the predictions but rather to present the calculation as it is "out of the box" with default settings.', '1804.03557-2-41-11': 'In the Powheg+Pythia case the scales [MATH] and [MATH] are varied independently by a factor of two wrt. central scale within [MATH] in the generation of Powheg [MATH] events.', '1804.03557-2-41-12': 'As can be seen in Figure [REF], the Powheg+Pythia approach is clearly more sensitive to the scale variations than GM-VFNS, especially at [MATH], and the LHCb data still remain within these uncertainties, though at the very upper part of the band.', '1804.03557-2-41-13': 'The scale-uncertainty estimates of Powheg+Pythia method we find here are well in line with the error bands shown in the original data papers of LHCb and ALICE, and e.g. in Ref. [CITATION].', '1804.03557-2-42-0': 'Figure [REF] presents ratios of cross sections measured by LHCb at different [MATH].', '1804.03557-2-42-1': 'Here, a significant part of the theory uncertainties cancel and, sure enough, the central predictions of the both considered methods describe the data rather well even at low [MATH].', '1804.03557-2-42-2': 'However, the Powheg results are systematically below the GM-VFNS predictions in most of the cases, best visible in the [MATH] panel.', '1804.03557-2-42-3': 'That is, the [MATH] dependence is stronger in the GM-VFNS calculation.', '1804.03557-2-42-4': 'We believe that this hierarchy follows mainly from the presence (absence) of gluon fragmentation in GM-VFNS (Powheg+Pyhtia) approach, similarly as in the case of absolute cross sections.', '1804.03557-2-42-5': 'As the Powheg generates events where the [MATH] pair is produced in the hard process or from the hardest emission, part of the increased phase-space for parton shower due to increased [MATH] is not in use for heavy-quark production.', '1804.03557-2-42-6': 'This is consistent also with the observation that the difference between the GM-VFNS and Powheg+Pythia cross sections decreases, though slowly, towards lower [MATH].', '1804.03557-2-42-7': 'In Figure [REF] we show the [MATH] case without PDF errors and including also the predictions obtained with the zero-mass version of the fragmentation variable ([MATH], [MATH]) to estimate the theoretical bias of this origin.', '1804.03557-2-42-8': 'We observe that the differences between our default definition and the zero-mass version are still clearly smaller than the scale uncertainties.', '1804.03557-2-43-0': 'Another observable that has been discussed in the recent literature [CITATION] is the normalized cross section [EQUATION] where [MATH] is a fixed reference rapidity.', '1804.03557-2-43-1': 'Also here, large part of the theory uncertainties cancel upon taking the ratio.', '1804.03557-2-43-2': 'Our calculations for this quantity within the LHCb acceptance at [MATH] in GM-VFNS framework are presented in Figure [REF] taking [MATH].', '1804.03557-2-43-3': 'We also compare with the Powheg+Pythia approach and with the GM-VFNS predictions using the zero-mass fragmentation variable.', '1804.03557-2-43-4': 'The shown LHCb data points have been formed from the absolute cross sections adding all the uncertainties in quadrature (as if they were uncorrelated).', '1804.03557-2-43-5': 'Again, the scale variation appears more significant than the effect induced by using the zero-mass fragmentation variable ([MATH], [MATH]).', '1804.03557-2-43-6': 'The Powheg+Pythia results are systematically above the GM-VFNS ones.', '1804.03557-2-43-7': 'As earlier, this seems to follow from the absence of gluon fragmentation in Powheg+Pythia: The "missing" contributions from gluon fragmentation are relatively larger for [MATH] than for the more forward bins (since the phase space is larger for [MATH]), and thus the ratio is higher than in GM-VFNS.', '1804.03557-2-44-0': '# Summary', '1804.03557-2-45-0': 'In the present article, we have introduced a novel implementation of the GM-VFNS for hadroproduction of heavy-flavoured mesons.', '1804.03557-2-45-1': 'Here, the novelty amounts to a specific definition of scheme, SACOT-[MATH], which retains the kinematics of heavy quark-antiquark pair production also in contributions where the heavy-flavoured meson formally comes from light-flavour fragmentation or initial-state heavy quarks.', '1804.03557-2-45-2': 'As we have explained, this is physically a natural choice as the origin of these contributions is in the heavy quark-antiquark pair production, and as such it is analogous to the SACOT-[MATH] scheme in deeply inelastic scattering.', '1804.03557-2-45-3': 'Within the SACOT-[MATH] scheme, it is possible to compute the heavy-flavoured meson spectra down to [MATH] with arbitrary choices for renormalization, fragmentation, and renormalization scales.', '1804.03557-2-45-4': 'In earlier works presented in the literature, a finite [MATH] limit could only be achieved by setting the scales in a particular way.', '1804.03557-2-45-5': 'Comparisons with the available D[MATH]-meson data from LHCb and ALICE collaborations indicate that our calculation in SACOT-[MATH] scheme performs well, though it must be admitted that the theoretical uncertainties at low [MATH] are significant.', '1804.03557-2-45-6': 'Here, it is not only the scale, PDF, and FF uncertainties that matter, but also the scheme dependence and ambiguities in defining the fragmentation variable [MATH] in the presence of finite heavy-quark and heavy-flavoured meson masses.', '1804.03557-2-45-7': 'In particular, as we have shown, the latter can have a significant impact on the shape of the absolute spectra at low [MATH].', '1804.03557-2-45-8': 'Within our definition of the fragmentation variable we have found, however, that the mass effects are suppressed in the forward direction, especially within the LHCb acceptance.', '1804.03557-2-45-9': 'Nevertheless, a particular choice of fragmentation variable may still bias the usage of low-[MATH] D-meson production e.g. as a constraint for PDFs.', '1804.03557-2-45-10': 'We note that this type of uncertainty is not inherent only to the GM-VFNS approach but the very same ambiguity is there also in FFNS calculations when scale-independent FFs are used for the [MATH] transition.', '1804.03557-2-45-11': 'We have also shown that in all considered rapidities there is a sizeable contribution from large-[MATH] region which, as we have argued, appears to originate from gluon fragmentation.', '1804.03557-2-45-12': 'Thus, estimates based on fixed-order [MATH] pair production overstate the small-[MATH] sensitivity of inclusive D-meson production.', '1804.03557-2-45-13': 'Though formally higher order in strong coupling, the effect of gluon fragmentation is numerically large and seems to explain why FFNS-based calculations are typically a factor of two below the LHC data.', '1804.03557-2-45-14': 'In addition, we have observed that an approach (Powheg+Pythia) neglecting large part of the gluon fragmentation deviates systematically from the GM-VFNS predictions also in the case of normalized cross sections and ratios across different [MATH].', '1804.03557-2-45-15': 'The size of these systematic differences competes or exceeds the scale uncertainty of GM-VFNS, though the scale uncertainties in the Powheg-based setup are presumably somewhat larger also for these observables.', '1804.03557-2-45-16': 'In the future, we plan to extend the SACOT-[MATH] scheme also to the case of intrinsic/fitted charm, B-meson production, as well as nuclear collisions.'} | [['1804.03557-1-26-0', '1804.03557-2-26-0'], ['1804.03557-1-26-1', '1804.03557-2-26-1'], ['1804.03557-1-26-2', '1804.03557-2-26-2'], ['1804.03557-1-26-3', '1804.03557-2-26-3'], ['1804.03557-1-26-4', '1804.03557-2-26-4'], ['1804.03557-1-26-5', '1804.03557-2-26-5'], 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'1804.03557-2-40-2'], ['1804.03557-1-40-3', '1804.03557-2-40-4'], ['1804.03557-1-40-4', '1804.03557-2-40-5'], ['1804.03557-1-40-5', '1804.03557-2-40-6'], ['1804.03557-1-40-6', '1804.03557-2-40-7'], ['1804.03557-1-8-0', '1804.03557-2-8-0'], ['1804.03557-1-8-1', '1804.03557-2-8-1'], ['1804.03557-1-8-2', '1804.03557-2-8-2'], ['1804.03557-1-14-1', '1804.03557-2-14-1'], ['1804.03557-1-14-2', '1804.03557-2-14-2'], ['1804.03557-1-14-3', '1804.03557-2-14-3'], ['1804.03557-1-14-5', '1804.03557-2-14-6'], ['1804.03557-1-14-6', '1804.03557-2-14-7'], ['1804.03557-1-14-7', '1804.03557-2-14-8'], ['1804.03557-1-3-0', '1804.03557-2-3-0'], ['1804.03557-1-3-1', '1804.03557-2-3-1'], ['1804.03557-1-3-2', '1804.03557-2-3-2'], ['1804.03557-1-3-3', '1804.03557-2-3-3'], ['1804.03557-1-3-4', '1804.03557-2-3-4'], ['1804.03557-1-3-5', '1804.03557-2-3-5'], ['1804.03557-1-16-0', '1804.03557-2-16-0'], ['1804.03557-1-16-2', '1804.03557-2-16-2'], ['1804.03557-1-16-3', '1804.03557-2-16-3'], ['1804.03557-1-16-4', '1804.03557-2-16-4'], ['1804.03557-1-16-6', '1804.03557-2-16-6']] | [['1804.03557-1-33-0', '1804.03557-2-33-0'], ['1804.03557-1-33-12', '1804.03557-2-33-12'], ['1804.03557-1-1-2', '1804.03557-2-1-2'], ['1804.03557-1-1-3', '1804.03557-2-1-3'], ['1804.03557-1-41-1', '1804.03557-2-41-2'], ['1804.03557-1-21-9', '1804.03557-2-21-9'], ['1804.03557-1-34-8', '1804.03557-2-34-8']] | [] | [['1804.03557-1-26-7', '1804.03557-2-26-7'], ['1804.03557-1-26-7', '1804.03557-2-26-8'], ['1804.03557-1-26-12', '1804.03557-2-26-13'], ['1804.03557-1-44-6', '1804.03557-2-45-6'], ['1804.03557-1-22-8', '1804.03557-2-22-8'], ['1804.03557-1-21-8', '1804.03557-2-21-8'], ['1804.03557-1-14-0', '1804.03557-2-14-0'], ['1804.03557-1-14-4', '1804.03557-2-14-4'], ['1804.03557-1-14-4', '1804.03557-2-14-5'], ['1804.03557-1-16-1', '1804.03557-2-16-1'], ['1804.03557-1-16-5', '1804.03557-2-16-5']] | [] | ['1804.03557-1-17-0', '1804.03557-1-18-0', '1804.03557-1-19-0', '1804.03557-1-20-0', '1804.03557-2-17-0', '1804.03557-2-18-0', '1804.03557-2-19-0', '1804.03557-2-20-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1804.03557 | null | null | null | null | null |
physics-9712056 | {'physics-9712056-1-0-0': 'Although Lorentz symmetry has been tested at low energy with extremely good accuracy, its validity at very high energy is much less well established.', 'physics-9712056-1-0-1': 'If Lorentz symmetry violation (LSV) is energy-dependent (e.g. [MATH]), it can be of order 1 at Planck scale and undetectable at GeV scale or below.', 'physics-9712056-1-0-2': 'Similarly, superluminal particles with positive mass and energy (superbradyons) can exist and be the ultimate building blocks of matter.', 'physics-9712056-1-0-3': 'We discuss a few cosmological consequences of such a scenario, as well as possible experimental tests.', 'physics-9712056-1-1-0': '# Lorentz symmetry violation and superluminal particles', 'physics-9712056-1-2-0': '"Experiment has provided numerous facts justifying the following generalization: absolute motion of matter, or, to be more precise, the relative motion of weighable matter and ether, cannot be disclosed.', 'physics-9712056-1-2-1': 'All that can be done is to reveal the motion of weighable matter with respect to weighable matter" (H. Poincare, 1895)', 'physics-9712056-1-3-0': '"Such a strange property seems to be a real coup de pouce presented by Nature itself, for avoiding the disclosure of absolute motion... I consider quite probable that optical phenomena depend only on the relative motion of the material bodies present, of the sources of light and optical instruments, and this dependence is not accurate... but rigorous.', 'physics-9712056-1-3-1': 'This principle will be confirmed with increasing precision, as measurements become more and more accurate" (H. Poincare, 1901)', 'physics-9712056-1-4-0': '"The interpretation of geometry advocated here cannot be directly applied to submolecular spaces... it might turn out that such an extrapolation is just as incorrect as an extension of the concept of temperature to particles of a solid of molecular dimensions" (A. Einstein, 1921)', 'physics-9712056-1-5-0': '## Status of the Poincare relativity principle', 'physics-9712056-1-6-0': 'The Poincare relativity principle [CITATION] has been confirmed by very accurate low-energy tests [CITATION], but its validity at much higher energies is not obvious [CITATION].', 'physics-9712056-1-6-1': 'The possibility that special relativity could fail at small distance scales was already considered by A. Einstein [CITATION]: it is remarkable that the relativity principle holds at the energies attained by particle accelerators.', 'physics-9712056-1-6-2': 'Experiments devoted to the highest-energy cosmic rays may provide crucial tests of Lorentz symmetry [CITATION].', 'physics-9712056-1-7-0': '## Lorentz symmetry violation (LSV)', 'physics-9712056-1-8-0': 'Lorentz symmetry can be broken introducing a local absolute rest frame (the vacuum rest frame, VRF) and a fundamental distance scale [MATH] [CITATION].', 'physics-9712056-1-8-1': 'If LSV follows a [MATH] law ([MATH] = energy), it can be [MATH] at Planck scale and [MATH] at the [MATH] scale, escaping all low-energy tests of Lorentz symmetry.', 'physics-9712056-1-8-2': 'But a [MATH] LSV at Planck scale can produce [CITATION] obervable effects at the highest cosmic-ray energies ([MATH]).', 'physics-9712056-1-8-3': 'If [MATH] is the wave vector, nonlocal models lead in the VRF [CITATION] to a deformed relativistic kinematics which for [MATH] gives: E p c + m^2 c^3 (2 p)^-1 - p c (k a)^2/2 where [MATH] stands for momentum, [MATH] for mass and [MATH] is a positive constant.', 'physics-9712056-1-9-0': '## Deformed relativistic kinematics (DRK)', 'physics-9712056-1-10-0': 'Contrary to the [MATH] model [CITATION], DRK preserves relativity in the limit [MATH] .', 'physics-9712056-1-10-1': 'A fundamental question is that of the universality of [MATH] : is [MATH] the same for all bodies, or does it depend on the object under consideration?', 'physics-9712056-1-10-2': 'If [MATH] is universal and [MATH] , equation (1) amounts to a relation between [MATH] and [MATH] , as in relativistic kinematics.', 'physics-9712056-1-10-3': 'From a naive soliton model [CITATION], we inferred that: a) [MATH] is expected to be universal up to very small corrections ([MATH]) escaping existing bounds; b) an approximate rule can be to take [MATH] universal for leptons, gauge bosons and light hadrons (pions, nucleons...) and assume a [MATH] law for nuclei and heavier objects, the nucleon mass setting the scale.', 'physics-9712056-1-11-0': '## Cosmic superluminal particles (CSL)', 'physics-9712056-1-12-0': 'If Lorentz symmetry is broken at Planck scale, nothing prevents the existence of particles with positive mass and energy and critical speed in vacuum [MATH] (the subscript [MATH] stands for the [MATH]-th superluminal sector) much larger than the speed of light [MATH] [CITATION].', 'physics-9712056-1-12-1': 'Such particles (superbradyons) could be the ultimate building blocks of matter from which, for instance, strings would be made.', 'physics-9712056-1-12-2': 'They can satisfy the same kinematics as "ordinary" particles, but replacing the speed of light [MATH] by the new critical speed [MATH] , and interact weakly with "ordinary" matter.', 'physics-9712056-1-12-3': 'Nonlocal models at Planck scale may be the limit of an underlying superluminal dynamics in the limit [MATH] .', 'physics-9712056-1-12-4': 'CSL can possibly propagate in vacuum just like photons in a perfectly transparent crystal.', 'physics-9712056-1-13-0': '# Some cosmological implications', 'physics-9712056-1-14-0': 'It was suggested [CITATION], using a different DRK from (1), that DRK could explain the dark matter problem: the non additivity of rest energy for non interacting systems at rest would account for the illusion of a missing mass.', 'physics-9712056-1-14-1': 'But it was later argued [CITATION] that the effect would actually be opposite to observation.', 'physics-9712056-1-14-2': 'However, both authors use a model where the additive quantity, instead of energy, is: F (m , E ) = 2 (m) sinh [2^-1 ^-1 (m) E ] and the constant [MATH] (similar to the parameter [MATH] of our model) has a universal value.', 'physics-9712056-1-14-3': 'There is no fundamental reason for this universality and similar arguments to those developed for our DRK model would suggest [CITATION] [MATH] , restoring the additivity of rest energy for large non interacting systems at rest.', 'physics-9712056-1-15-0': 'A generalization of Friedmann equations in the presence of superluminal sectors of matter can be built [CITATION] and does not present inconsistency with data.', 'physics-9712056-1-15-1': 'Superluminal particles may actually be most of the cosmic (dark) matter.', 'physics-9712056-1-16-0': '# References'} | {'physics-9712056-2-0-0': 'Although Lorentz symmetry has been tested at low energy with extremely good accuracy, its validity at very high energy is much less well established.', 'physics-9712056-2-0-1': 'If Lorentz symmetry violation (LSV) is energy-dependent (e.g. [MATH]), it can be of order 1 at Planck scale and undetectable at GeV scale or below.', 'physics-9712056-2-0-2': 'Similarly, superluminal particles with positive mass and energy (superbradyons) can exist and be the ultimate building blocks of matter.', 'physics-9712056-2-0-3': 'We discuss a few cosmological consequences of such a scenario, as well as possible experimental tests.', 'physics-9712056-2-1-0': '# Lorentz symmetry violation and superluminal particles', 'physics-9712056-2-2-0': '"Experiment has provided numerous facts justifying the following generalization: absolute motion of matter, or, to be more precise, the relative motion of weighable matter and ether, cannot be disclosed.', 'physics-9712056-2-2-1': 'All that can be done is to reveal the motion of weighable matter with respect to weighable matter" (H. Poincare, 1895)', 'physics-9712056-2-3-0': '"Such a strange property seems to be a real coup de pouce presented by Nature itself, for avoiding the disclosure of absolute motion... I consider quite probable that optical phenomena depend only on the relative motion of the material bodies present, of the sources of light and optical instruments, and this dependence is not accurate... but rigorous.', 'physics-9712056-2-3-1': 'This principle will be confirmed with increasing precision, as measurements become more and more accurate" (H. Poincare, 1901)', 'physics-9712056-2-4-0': '"The interpretation of geometry advocated here cannot be directly applied to submolecular spaces... it might turn out that such an extrapolation is just as incorrect as an extension of the concept of temperature to particles of a solid of molecular dimensions" (A. Einstein, 1921)', 'physics-9712056-2-5-0': '## Status of the Poincare relativity principle', 'physics-9712056-2-6-0': 'The Poincare relativity principle [CITATION] has been confirmed by very accurate low-energy tests [CITATION], but its validity at much higher energies is not obvious [CITATION].', 'physics-9712056-2-6-1': 'The possibility that special relativity could fail at small distance scales was already considered by A. Einstein [CITATION]: it is remarkable that the relativity principle holds at the energies attained by particle accelerators.', 'physics-9712056-2-6-2': 'Experiments devoted to the highest-energy cosmic rays may provide crucial tests of Lorentz symmetry [CITATION].', 'physics-9712056-2-7-0': '## Lorentz symmetry violation (LSV)', 'physics-9712056-2-8-0': 'Lorentz symmetry can be broken introducing a local absolute rest frame (the vacuum rest frame, VRF) and a fundamental distance scale [MATH] [CITATION].', 'physics-9712056-2-8-1': 'If LSV follows a [MATH] law ([MATH] = energy), it can be [MATH] at Planck scale and [MATH] at the [MATH] scale, escaping all low-energy tests of Lorentz symmetry.', 'physics-9712056-2-8-2': 'But a [MATH] LSV at Planck scale can produce [CITATION] obervable effects at the highest cosmic-ray energies ([MATH]).', 'physics-9712056-2-8-3': 'If [MATH] is the wave vector, nonlocal models lead in the VRF [CITATION] to a deformed relativistic kinematics which for [MATH] gives: E c (p^2 + m^2 c^2)^1/2 - (c /2) (p k a)^2 (p^2 + m^2 c^2)^-1/2 where [MATH] stands for momentum, [MATH] for mass and [MATH] is a positive constant.', 'physics-9712056-2-9-0': '## Deformed relativistic kinematics (DRK)', 'physics-9712056-2-10-0': 'Contrary to the [MATH] model [CITATION], DRK preserves relativity in the limit [MATH] .', 'physics-9712056-2-10-1': 'A fundamental question is that of the universality of [MATH] : is [MATH] the same for all bodies, or does it depend on the object under consideration?', 'physics-9712056-2-10-2': 'If [MATH] is universal and [MATH] , equation (1) amounts to a relation between [MATH] and [MATH] , as in relativistic kinematics.', 'physics-9712056-2-10-3': 'From a naive soliton model [CITATION], we inferred that: a) [MATH] is expected to be universal up to very small corrections ([MATH]) escaping existing bounds; b) an approximate rule can be to take [MATH] universal for leptons, gauge bosons and light hadrons (pions, nucleons...) and assume a [MATH] law for nuclei and heavier objects, the nucleon mass setting the scale.', 'physics-9712056-2-11-0': '## Cosmic superluminal particles (CSL)', 'physics-9712056-2-12-0': 'If Lorentz symmetry is broken at Planck scale, nothing prevents the existence of particles with positive mass and energy and critical speed in vacuum [MATH] (the subscript [MATH] stands for the [MATH]-th superluminal sector) much larger than the speed of light [MATH] [CITATION].', 'physics-9712056-2-12-1': 'Such particles (superbradyons) could be the ultimate building blocks of matter from which, for instance, strings would be made.', 'physics-9712056-2-12-2': 'They can satisfy the same kinematics as "ordinary" particles, but replacing the speed of light [MATH] by the new critical speed [MATH] , and interact weakly with "ordinary" matter.', 'physics-9712056-2-12-3': 'Nonlocal models at Planck scale may be the limit of an underlying superluminal dynamics in the limit [MATH] .', 'physics-9712056-2-12-4': 'CSL can possibly propagate in vacuum just like photons in a perfectly transparent crystal.', 'physics-9712056-2-13-0': '# Some cosmological implications', 'physics-9712056-2-14-0': 'It was suggested [CITATION], using a different DRK from (1), that DRK could explain the dark matter problem: the non additivity of rest energy for non interacting systems at rest would account for the illusion of a missing mass.', 'physics-9712056-2-14-1': 'But it was later argued [CITATION] that the effect would actually be opposite to observation.', 'physics-9712056-2-14-2': 'However, both authors use a model where the additive quantity, instead of energy, is: F (m , E ) = 2 (m) sinh [2^-1 ^-1 (m) E ] and the constant [MATH] (similar to the parameter [MATH] of our model) has a universal value.', 'physics-9712056-2-14-3': 'There is no fundamental reason for this universality and similar arguments to those developed for our DRK model would suggest [CITATION] [MATH] , restoring the additivity of rest energy for large non interacting systems at rest.', 'physics-9712056-2-15-0': 'A generalization of Friedmann equations in the presence of superluminal sectors of matter can be built [CITATION] and does not present inconsistency with data.', 'physics-9712056-2-15-1': 'Superluminal particles may actually be most of the cosmic (dark) matter.', 'physics-9712056-2-16-0': '# References'} | [['physics-9712056-1-3-0', 'physics-9712056-2-3-0'], ['physics-9712056-1-3-1', 'physics-9712056-2-3-1'], ['physics-9712056-1-6-0', 'physics-9712056-2-6-0'], ['physics-9712056-1-6-1', 'physics-9712056-2-6-1'], ['physics-9712056-1-6-2', 'physics-9712056-2-6-2'], ['physics-9712056-1-14-0', 'physics-9712056-2-14-0'], ['physics-9712056-1-14-1', 'physics-9712056-2-14-1'], ['physics-9712056-1-14-2', 'physics-9712056-2-14-2'], ['physics-9712056-1-14-3', 'physics-9712056-2-14-3'], ['physics-9712056-1-12-0', 'physics-9712056-2-12-0'], ['physics-9712056-1-12-1', 'physics-9712056-2-12-1'], ['physics-9712056-1-12-2', 'physics-9712056-2-12-2'], ['physics-9712056-1-12-3', 'physics-9712056-2-12-3'], ['physics-9712056-1-12-4', 'physics-9712056-2-12-4'], ['physics-9712056-1-10-0', 'physics-9712056-2-10-0'], ['physics-9712056-1-10-1', 'physics-9712056-2-10-1'], ['physics-9712056-1-10-2', 'physics-9712056-2-10-2'], ['physics-9712056-1-10-3', 'physics-9712056-2-10-3'], ['physics-9712056-1-2-0', 'physics-9712056-2-2-0'], ['physics-9712056-1-2-1', 'physics-9712056-2-2-1'], ['physics-9712056-1-4-0', 'physics-9712056-2-4-0'], ['physics-9712056-1-0-0', 'physics-9712056-2-0-0'], ['physics-9712056-1-0-1', 'physics-9712056-2-0-1'], ['physics-9712056-1-0-2', 'physics-9712056-2-0-2'], ['physics-9712056-1-0-3', 'physics-9712056-2-0-3'], ['physics-9712056-1-15-0', 'physics-9712056-2-15-0'], ['physics-9712056-1-15-1', 'physics-9712056-2-15-1'], ['physics-9712056-1-8-0', 'physics-9712056-2-8-0'], ['physics-9712056-1-8-1', 'physics-9712056-2-8-1'], ['physics-9712056-1-8-2', 'physics-9712056-2-8-2'], ['physics-9712056-1-8-3', 'physics-9712056-2-8-3']] | [['physics-9712056-1-3-0', 'physics-9712056-2-3-0'], ['physics-9712056-1-3-1', 'physics-9712056-2-3-1'], ['physics-9712056-1-6-0', 'physics-9712056-2-6-0'], ['physics-9712056-1-6-1', 'physics-9712056-2-6-1'], ['physics-9712056-1-6-2', 'physics-9712056-2-6-2'], ['physics-9712056-1-14-0', 'physics-9712056-2-14-0'], ['physics-9712056-1-14-1', 'physics-9712056-2-14-1'], ['physics-9712056-1-14-2', 'physics-9712056-2-14-2'], ['physics-9712056-1-14-3', 'physics-9712056-2-14-3'], ['physics-9712056-1-12-0', 'physics-9712056-2-12-0'], ['physics-9712056-1-12-1', 'physics-9712056-2-12-1'], ['physics-9712056-1-12-2', 'physics-9712056-2-12-2'], ['physics-9712056-1-12-3', 'physics-9712056-2-12-3'], ['physics-9712056-1-12-4', 'physics-9712056-2-12-4'], ['physics-9712056-1-10-0', 'physics-9712056-2-10-0'], ['physics-9712056-1-10-1', 'physics-9712056-2-10-1'], ['physics-9712056-1-10-2', 'physics-9712056-2-10-2'], ['physics-9712056-1-10-3', 'physics-9712056-2-10-3'], ['physics-9712056-1-2-0', 'physics-9712056-2-2-0'], ['physics-9712056-1-2-1', 'physics-9712056-2-2-1'], ['physics-9712056-1-4-0', 'physics-9712056-2-4-0'], ['physics-9712056-1-0-0', 'physics-9712056-2-0-0'], ['physics-9712056-1-0-1', 'physics-9712056-2-0-1'], ['physics-9712056-1-0-2', 'physics-9712056-2-0-2'], ['physics-9712056-1-0-3', 'physics-9712056-2-0-3'], ['physics-9712056-1-15-0', 'physics-9712056-2-15-0'], ['physics-9712056-1-15-1', 'physics-9712056-2-15-1'], ['physics-9712056-1-8-0', 'physics-9712056-2-8-0'], ['physics-9712056-1-8-1', 'physics-9712056-2-8-1'], ['physics-9712056-1-8-2', 'physics-9712056-2-8-2']] | [] | [] | [['physics-9712056-1-8-3', 'physics-9712056-2-8-3']] | [] | [] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/physics/9712056 | null | null | null | null | null |
1811.09657 | {'1811.09657-1-0-0': 'We study how to implement quantum stochastic differential equations (QSDEs) on a quantum computer.', '1811.09657-1-0-1': 'This is illustrated by an implementation of the QSDE that couples a laser driven two-level atom to the electromagnetic field in the vacuum state on the IBMqx4 Tenerife computer [CITATION].', '1811.09657-1-0-2': 'We compare the resulting master equation and quantum filtering equations to existing theory.', '1811.09657-1-0-3': 'In this way we characterize the performance of the computer.', '1811.09657-1-1-0': '# Introduction', '1811.09657-1-2-0': 'In these short notes, we study a very simple problem with a solution that is universally well-known: spontaneous decay of a laser-driven two-level atom which is coupled to the electromagnetic field in the vacuum state.', '1811.09657-1-2-1': 'The techniques that we use to discretize the quantum stochastic differential equation [CITATION] that describes the interaction between the two-level atom and the laser field are very well-known [CITATION].', '1811.09657-1-2-2': 'The discretized model consists of a repeated unitary interaction of the two-level atom with subsequent field qubits parametrized by a discretization parameter [MATH].', '1811.09657-1-2-3': 'The repeated interaction model easily leads to a quantum stochastic difference equation that has the QSDE we wish to simulate as its limit as [MATH] goes to zero.', '1811.09657-1-2-4': 'Note that unitarity of the interaction is preserved in the discretized model, which is a very desirable feature: e.g. after time evolution probabilities will still always take values between [MATH] and [MATH].', '1811.09657-1-2-5': 'Furthermore, the unitarity allows us to easily map the interaction on unitary gates in a quantum computer.', '1811.09657-1-3-0': 'The motivation for the work we present here is twofold and aimed at a future with computers with more and more reliable qubits:', '1811.09657-1-4-0': 'We wish to emphasize that quantum optics might prove a very fruitful field of application for early quantum computers.', '1811.09657-1-4-1': 'It is well-known how to discretize the type of problems that arise in quantum optics and the resulting quantum stochastic difference equations are easy to implement on a quantum computer.', '1811.09657-1-4-2': 'Moreover, the field of quantum optics historically contains many interesting problems and techniques that can serve as interesting benchmark problems for early quantum computers.', '1811.09657-1-4-3': 'On a quantum computer we can do a fully coherent simulation of a system in interaction with the electromagnetic field.', '1811.09657-1-4-4': 'That is, on a large enough quantum computer, we can simulate the complete unitary that describes the interaction between system and field, putting us past standard analyses using master equations or quantum filtering equations [CITATION] because we also have a complete description of the field to our availability.', '1811.09657-1-4-5': 'This could be very useful if we wish to simulate non-Markovian networks of systems interacting at different points with the same field, possibly containing fully coherent feedback loops [CITATION].', '1811.09657-1-5-0': 'The remainder of these notes is organized as follows.', '1811.09657-1-5-1': 'Section [REF] introduces the QSDE that we wish to simulate on a quantum computer: a laser driven two-level atom in interaction with the vacuum EM-field.', '1811.09657-1-5-2': 'We discuss repeated unitary interaction models, the resulting quantum stochastic difference equations and discuss how to take the limit to obtain quantum stochastic differential equations [CITATION].', '1811.09657-1-5-3': 'Next, we introduce the repeated interaction that leads to the QSDE corresponding to the problem at hand: a laser driven two-level atom in interaction with the vacuum EM-field.', '1811.09657-1-5-4': 'Section [REF] describes how we have implemented the repeated interaction model of Section [REF] on the IBMqx4 Tenerife quantum computer [CITATION].', '1811.09657-1-6-0': 'Section [REF] presents the results of our simulations.', '1811.09657-1-6-1': 'We compute the reduced dynamics of the two-level atom from the simulation results and compare it to the dynamics given by the theoretical master equation, derived from the underlying discrete model.', '1811.09657-1-6-2': 'We also compute the conditional dynamics of the two-level atom conditioned on both counting photons in the field and observing a field quadrature in the field and compare the results to the quantum filtering equations derived from the underlying discrete model [CITATION].', '1811.09657-1-6-3': 'Reproducing the correct quantum filtering equations would give an indication that the simulation also reproduces the correlation between the field and the two-level atom correctly.', '1811.09657-1-7-0': 'In Section [REF], we formulate some conclusions from our results.', '1811.09657-1-7-1': 'It should be noted that currently only very limited simulations can be done due to the small number of reliable qubits in the quantum computers that are currently available.', '1811.09657-1-7-2': 'In a 5 qubit machine such as the IBMqx4, we only have 4 field qubits to our availability, severely limiting our simulation capabilities.', '1811.09657-1-7-3': 'The work in these notes should be seen through the lens of a hoped-for strong increase in the computing capacity in the near future.', '1811.09657-1-8-0': '# Quantum stochastic difference equations', '1811.09657-1-9-0': 'We will now first introduce the problem that we will be studying in this paper.', '1811.09657-1-9-1': 'We consider a two-level atom with a coupling to the electromagnetic field.', '1811.09657-1-9-2': 'We will let the two-level atom be driven by a laser.', '1811.09657-1-9-3': 'We will not model the laser by an additional channel in the field that is in an coherent state, but will directly introduce the Rabi oscillations induced by the laser field as a Hamiltonian term in the QSDE.', '1811.09657-1-9-4': 'We will use the following notation throughout the paper: [MATH] and [MATH] are the standard Pauli matrices.', '1811.09657-1-9-5': 'Furthermore, [MATH] and [MATH] are the raising and lowering operator matrix of the two-level atom, respectively.', '1811.09657-1-9-6': 'The interaction of the laser driven two-level atom with the vacuum electromagnetic field is given by the following quantum stochastic differential equation (QSDE) in the sense of [CITATION] [EQUATION] where [MATH] is the decay rate, [MATH] is the transition frequency of the two-level atom and [MATH] is the frequency of the Rabi oscillations induced by a laser field.', '1811.09657-1-10-0': 'In order to simulate Eqn [REF] on a quantum computer, we first need to introduce the discrete models (see e.g. [CITATION] for a detailed introduction) that in a suitable limit will converge to a quantum stochastic differential equation [CITATION] such as Eqn [REF].', '1811.09657-1-10-1': 'To this end we first define a time interval [MATH].', '1811.09657-1-10-2': 'We divide this time interval into [MATH] equal sub intervals of length [MATH].', '1811.09657-1-10-3': 'We define [MATH], i.e. we have [MATH] sub intervals of length [MATH].', '1811.09657-1-10-4': 'With each sub interval we associate a two-level quantum system representing the slice of the (truncated) field that interacts with the two-level atom at that moment.', '1811.09657-1-10-5': 'In this way we obtain a repeated interaction [EQUATION]', '1811.09657-1-10-6': 'Here [MATH] is a unitary operator that couples the two-level atom and the [MATH]th field slice which is here also represented by a two-level system.', '1811.09657-1-10-7': "Note that we take all [MATH]'s to be identical apart from the fact that they all act on their own slice of the field.", '1811.09657-1-10-8': "Furthermore, we will let the [MATH]'s be a function of [MATH], such that if [MATH] goes to [MATH] (i.e. [MATH] goes to infinity) the [MATH]'s converge to the identity map [MATH].", '1811.09657-1-10-9': 'That is, we will get more and more interactions, with smaller and smaller effect.', '1811.09657-1-11-0': 'We now introduce linear operators [MATH] and [MATH] [CITATION], acting on the two-level atom Hilbert space, in such a way that we have the following decomposition [EQUATION] where the discrete quantum noises (see e.g. [CITATION]) are given by [EQUATION]', '1811.09657-1-11-1': 'Note that this decomposition uniquely defines the coefficients [MATH] and [MATH] and note that these coefficients are all a function of [MATH], which we leave implicit to keep our notation light.', '1811.09657-1-11-2': 'Furthermore, we usually omit the tensor products in Eqn [REF] to keep the notation light.', '1811.09657-1-12-0': 'We can now write Eqn [REF] as the following quantum stochastic difference equation (see e.g. [CITATION]) [EQUATION] where [MATH] and [MATH].', '1811.09657-1-12-1': 'We now have the following theorem due to Parthasarathy [CITATION] (weak convergence), Parthasarathy and Lindsay [CITATION] (weak convergence of the quantum flow) and Attal and Pautrat [CITATION] (strong convergence uniform on compact time intervals).', '1811.09657-1-12-2': 'We state the theorem without giving the precise meaning of the mode of convergence of the repeated interaction model to the unitary solution of the QSDE because we would need to introduce further mathematical details that would make us stray too far from the main narrative of this article (see however [CITATION], or [CITATION]).', '1811.09657-1-13-0': '(Parthasarathy, Lindsay, Attal and Pautrat [CITATION]) Suppose the following limits exist: [EQUATION] then it follows that [MATH] is unitary, [MATH] is the adjoint of [MATH], i.e. [MATH], [MATH] is self-adjoint and [MATH] (where the brackets [ ] stand for rounding down to an integer) converges to a unitary [MATH] given by the following QSDE [EQUATION]', '1811.09657-1-13-1': 'We proceed by guessing an interaction unitary [MATH] in the repeated interaction Eqn [REF] and check via Thm [REF] that it leads to the correct limit coefficients to reproduce in the limit our system of interest which is given Eqn [REF].', '1811.09657-1-13-2': 'Note that there are several [MATH]s that will lead to the correct limit system.', '1811.09657-1-13-3': 'We will take the following [MATH] and will show that it indeed leads to Eqn [REF] in the limit: [EQUATION]', '1811.09657-1-13-4': 'A short calculation then reveals [EQUATION]', '1811.09657-1-13-5': 'Using the definition of [MATH] and [MATH] in Eqn [REF], we find [EQUATION]', '1811.09657-1-13-6': 'That is, we have found a repeated interaction model that converges to the QSDE of Eqn [REF].', '1811.09657-1-14-0': '# The quantum circuit', '1811.09657-1-15-0': 'The various contributions to the interaction between the atom and the field, given in Eqn [REF], can easily be mapped to elementary quantum gates.', '1811.09657-1-15-1': 'The following quantum circuit [CITATION] implements this interaction for a single time slice:', '1811.09657-1-16-0': '@C=1em @R=.7em atom & R_y(^2) & R_z(^2) & 1 & R_y(2) & 1 &', '1811.09657-1-17-0': 'field & & & & -1 & &', '1811.09657-1-18-0': 'For modeling multiple time slices, different qubits are used to describe the field at different times, and coupled to the atom qubit.', '1811.09657-1-18-1': 'Currently, this limits the simulation to a maximum of four time slices on the IBMqx4 computer.', '1811.09657-1-18-2': 'If operations based on classical bits are enabled, we could reuse the same qubit for the field by measuring this qubit, and rotating it back to its [MATH] state when the outcome is [MATH].', '1811.09657-1-18-3': 'Unfortunately, this is currently only implemented in simulators and not in real hardware.', '1811.09657-1-19-0': 'At the end of the simulation, all qubits (both field and atom) were measured.', '1811.09657-1-19-1': 'The atom was measured in the [MATH], [MATH], and [MATH] basis, whereas the field qubits were measured in the [MATH] and [MATH] basis.', '1811.09657-1-19-2': 'Statistics were collected over 10,240 runs for each combination of measurement directions of atom and field.', '1811.09657-1-20-0': '# Results', '1811.09657-1-21-0': '## Master equation', '1811.09657-1-22-0': 'The repeated interaction model given by Eqn [REF] leads to the following discrete time master equation [CITATION] for the state [MATH] of the two-level atom: [EQUATION] where the discretized Lindblad operator is given by [CITATION] [EQUATION]', '1811.09657-1-22-1': 'Here [MATH] and [MATH] are given by Eqn [REF].', '1811.09657-1-22-2': 'Figure [REF] compares the theoretical results given by the master equation Eqn [REF] and the results obtained with the IBMqx4 Tenerife quantum computer.', '1811.09657-1-23-0': '## Homodyne quantum filter', '1811.09657-1-24-0': 'We now turn to the situation where we are not simply tracing over the field, but condition on observations made in the field.', '1811.09657-1-24-1': 'Suppose that for [MATH], we have observed [MATH] in the field: [EQUATION]', '1811.09657-1-24-2': 'Physically, this corresponds to the case where we observe the field with a homodyne detection setup after the interaction with the two-level atom.', '1811.09657-1-24-3': 'We can condition the time evolution of the density matrix on an observed homodyne photo current detection record.', '1811.09657-1-24-4': 'The conditioned density matrix obeys the following discrete quantum filtering equation for homodyne detection [CITATION] [EQUATION] where [MATH] is given by Eqn [REF] and [MATH] and the initial state [MATH] are given by [EQUATION]', '1811.09657-1-24-5': 'Here [MATH] and [MATH] are given by Eqn [REF].', '1811.09657-1-24-6': 'Figures [REF], [REF] and [REF] compare the theoretical results given by the quantum filter equation Eqn [REF] and the results obtained with the IBM Qiskit simulator and the IBMqx4 Tenerife quantum computer.', '1811.09657-1-25-0': '## Counting quantum filter', '1811.09657-1-26-0': 'Finally, we turn to the situation where we condition on having observed [MATH] in the field for [MATH].', '1811.09657-1-26-1': 'That is, we have the following observations: [EQUATION]', '1811.09657-1-26-2': 'Physically, this corresponds to the case where we observe the field with a photo detector after the interaction with the two-level atom.', '1811.09657-1-26-3': 'We can condition the time evolution of the density matrix on an observed photo detection record.', '1811.09657-1-26-4': 'The conditioned density matrix obeys the following discrete quantum filtering equation for photon counting [CITATION] [EQUATION]', '1811.09657-1-26-5': 'Here [MATH] is given by Eqn [REF].', '1811.09657-1-26-6': 'Figures [REF], [REF] and [REF] compare the theoretical results given by the quantum filter equation Eqn [REF] and the results obtained with the IBM Qiskit simulator and the IBMqx4 Tenerife quantum computer.', '1811.09657-1-27-0': '# Conclusion', '1811.09657-1-28-0': 'In these notes we have shown that it is fairly straightforward to implement quantum stochastic differential equations on a quantum computer.', '1811.09657-1-28-1': 'The mathematical theory [CITATION] behind the necessary discretization of the equations is well worked out and easily translated into a quantum circuit.', '1811.09657-1-28-2': 'It is possible with the very limited capacity of the currently available quantum computers to simulate some simple quantum optical features described by a QSDE (e.g. a Rabi oscillation).', '1811.09657-1-29-0': 'We have also seen that the filter equations are to a large extent correctly reproduced on the IBMqx4 Tenerife quantum computer.', '1811.09657-1-29-1': 'This provides confidence that the (quantum) correlations between the atom and the field are accounted for correctly.', '1811.09657-1-29-2': 'This opens the door to fully coherent simulations of systems that interact with the field at different points, even including fully coherent feedback loops [CITATION].', '1811.09657-1-29-3': 'Naturally, this is only possible when quantum computers are available with more and more reliable qubits.', '1811.09657-1-30-0': 'As can be seen in Figures [REF] and [REF], the time evolution between counts in the photon counting scheme seems to be accounted for correctly, however, the jump operation does not seem to be accurate to the theory.', '1811.09657-1-30-1': 'This can be understood though: there are relatively few jumps and there is already quite a bit of noise on the results when the field is in the vacuum state.', '1811.09657-1-30-2': 'This means there is an identity component in the jump operator that leads to a deviation from the theory (in which this noise is not present, but could be modeled).', '1811.09657-1-31-0': 'With respect to the discrete quantum filtering equations [CITATION], we note that they completely coincide with the IBM Qiskit simulator results.', '1811.09657-1-31-1': 'However, they are much less computationally intensive and could still be used if the number of qubits is larger than the simulator can deal with.', '1811.09657-1-31-2': 'Note, however, that in the simulator it is also possible to recycle the field qubits after they have been measured.', '1811.09657-1-31-3': 'This is currently not yet possible on the real IBMQ hardware.', '1811.09657-1-32-0': 'It will be interesting to see future quantum computers simulate more complex benchmark problems originating from quantum optics.'} | {'1811.09657-2-0-0': 'We study how to implement quantum stochastic differential equations (QSDEs) on a quantum computer.', '1811.09657-2-0-1': 'This is illustrated by an implementation of the QSDE that couples a laser driven two-level atom to the electromagnetic field in the vacuum state on the IBMqx4 Tenerife computer [CITATION].', '1811.09657-2-0-2': 'We compare the resulting master equation and quantum filtering equations to existing theory.', '1811.09657-2-0-3': 'In this way we characterize the performance of the computer.', '1811.09657-2-1-0': '# Introduction', '1811.09657-2-2-0': 'In these short notes, we study a very simple problem with a solution that is universally well-known: spontaneous decay of a laser-driven two-level atom which is coupled to the electromagnetic field in the vacuum state.', '1811.09657-2-2-1': 'The techniques that we use to discretize the quantum stochastic differential equation [CITATION] that describes the interaction between the two-level atom and the laser field are very well-known [CITATION].', '1811.09657-2-2-2': 'The discretized model consists of a repeated unitary interaction of the two-level atom with subsequent field qubits parametrized by a discretization parameter [MATH].', '1811.09657-2-2-3': 'The repeated interaction model easily leads to a quantum stochastic difference equation that has the QSDE we wish to simulate as its limit as [MATH] goes to zero.', '1811.09657-2-2-4': 'Note that unitarity of the interaction is preserved in the discretized model, which is a very desirable feature: e.g. after time evolution probabilities will still always take values between [MATH] and [MATH].', '1811.09657-2-2-5': 'Furthermore, the unitarity allows us to easily map the interaction on unitary gates in a quantum computer.', '1811.09657-2-3-0': 'The motivation for the work we present here is twofold and aimed at a future with computers with more and more reliable qubits:', '1811.09657-2-4-0': 'We wish to emphasize that quantum optics might prove a very fruitful field of application for early quantum computers.', '1811.09657-2-4-1': 'It is well-known how to discretize the type of problems that arise in quantum optics and the resulting quantum stochastic difference equations are easy to implement on a quantum computer.', '1811.09657-2-4-2': 'Moreover, the field of quantum optics historically contains many interesting problems and techniques that can serve as interesting benchmark problems for early quantum computers.', '1811.09657-2-4-3': 'On a quantum computer we can do a fully coherent simulation of a system in interaction with the electromagnetic field.', '1811.09657-2-4-4': 'That is, on a large enough quantum computer, we can simulate the complete unitary that describes the interaction between system and field, putting us past standard analyses using master equations or quantum filtering equations [CITATION] because we also have a complete description of the field to our availability.', '1811.09657-2-4-5': 'This could be very useful if we wish to simulate non-Markovian networks of systems interacting at different points with the same field, possibly containing fully coherent feedback loops [CITATION].', '1811.09657-2-5-0': 'The remainder of these notes is organized as follows.', '1811.09657-2-5-1': 'Section [REF] introduces the QSDE that we wish to simulate on a quantum computer: a laser driven two-level atom in interaction with the vacuum EM-field.', '1811.09657-2-5-2': 'We discuss repeated unitary interaction models, the resulting quantum stochastic difference equations and discuss how to take the limit to obtain quantum stochastic differential equations [CITATION].', '1811.09657-2-5-3': 'Next, we introduce the repeated interaction that leads to the QSDE corresponding to the problem at hand: a laser driven two-level atom in interaction with the vacuum EM-field.', '1811.09657-2-5-4': 'Section [REF] describes how we have implemented the repeated interaction model of Section [REF] on the IBMqx4 Tenerife quantum computer [CITATION].', '1811.09657-2-6-0': 'Section [REF] presents the results of our simulations.', '1811.09657-2-6-1': 'We compute the reduced dynamics of the two-level atom from the simulation results and compare it to the dynamics given by the theoretical master equation, derived from the underlying discrete model.', '1811.09657-2-6-2': 'We also compute the conditional dynamics of the two-level atom conditioned on both counting photons in the field and observing a field quadrature in the field and compare the results to the quantum filtering equations derived from the underlying discrete model [CITATION].', '1811.09657-2-6-3': 'Reproducing the correct quantum filtering equations would give an indication that the simulation also reproduces the correlation between the field and the two-level atom correctly.', '1811.09657-2-7-0': 'In Section [REF], we formulate some conclusions from our results.', '1811.09657-2-7-1': 'It should be noted that currently only very limited simulations can be done due to the small number of reliable qubits in the quantum computers that are currently available.', '1811.09657-2-7-2': 'In a 5 qubit machine such as the IBMqx4, we only have 4 field qubits to our availability, severely limiting our simulation capabilities.', '1811.09657-2-7-3': 'The work in these notes should be seen through the lens of a hoped-for strong increase in the computing capacity in the near future.', '1811.09657-2-8-0': '# Quantum stochastic difference equations', '1811.09657-2-9-0': 'We will now first introduce the problem that we will be studying in this paper.', '1811.09657-2-9-1': 'We consider a two-level atom with a coupling to the electromagnetic field.', '1811.09657-2-9-2': 'We will let the two-level atom be driven by a laser.', '1811.09657-2-9-3': 'We will not model the laser by an additional channel in the field that is in an coherent state, but will directly introduce the Rabi oscillations induced by the laser field as a Hamiltonian term in the QSDE.', '1811.09657-2-9-4': 'We will use the following notation throughout the paper: [MATH] and [MATH] are the standard Pauli matrices.', '1811.09657-2-9-5': 'Furthermore, [MATH] and [MATH] are the raising and lowering operator matrix of the two-level atom, respectively.', '1811.09657-2-9-6': 'The interaction of the laser driven two-level atom with the vacuum electromagnetic field is given by the following quantum stochastic differential equation (QSDE) in the sense of [CITATION] [EQUATION] where [MATH] is the decay rate, [MATH] is the transition frequency of the two-level atom and [MATH] is the frequency of the Rabi oscillations induced by a laser field.', '1811.09657-2-10-0': 'In order to simulate Eqn [REF] on a quantum computer, we first need to introduce the discrete models (see e.g. [CITATION] for a detailed introduction) that in a suitable limit will converge to a quantum stochastic differential equation [CITATION] such as Eqn [REF].', '1811.09657-2-10-1': 'To this end we first define a time interval [MATH].', '1811.09657-2-10-2': 'We divide this time interval into [MATH] equal sub intervals of length [MATH].', '1811.09657-2-10-3': 'We define [MATH], i.e. we have [MATH] sub intervals of length [MATH].', '1811.09657-2-10-4': 'With each sub interval we associate a two-level quantum system representing the slice of the (truncated) field that interacts with the two-level atom at that moment.', '1811.09657-2-10-5': 'In this way we obtain a repeated interaction [EQUATION]', '1811.09657-2-10-6': 'Here [MATH] is a unitary operator that couples the two-level atom and the [MATH]th field slice which is here also represented by a two-level system.', '1811.09657-2-10-7': "Note that we take all [MATH]'s to be identical apart from the fact that they all act on their own slice of the field.", '1811.09657-2-10-8': "Furthermore, we will let the [MATH]'s be a function of [MATH], such that if [MATH] goes to [MATH] (i.e. [MATH] goes to infinity) the [MATH]'s converge to the identity map [MATH].", '1811.09657-2-10-9': 'That is, we will get more and more interactions, with smaller and smaller effect.', '1811.09657-2-11-0': 'We now introduce linear operators [MATH] and [MATH] [CITATION], acting on the two-level atom Hilbert space, in such a way that we have the following decomposition [EQUATION] where the discrete quantum noises (see e.g. [CITATION]) are given by [EQUATION]', '1811.09657-2-11-1': 'Note that this decomposition uniquely defines the coefficients [MATH] and [MATH] and note that these coefficients are all a function of [MATH], which we leave implicit to keep our notation light.', '1811.09657-2-11-2': 'Furthermore, we usually omit the tensor products in Eqn [REF] to keep the notation light.', '1811.09657-2-12-0': 'We can now write Eqn [REF] as the following quantum stochastic difference equation (see e.g. [CITATION]) [EQUATION] where [MATH] and [MATH].', '1811.09657-2-12-1': 'We now have the following theorem due to Parthasarathy [CITATION] (weak convergence), Parthasarathy and Lindsay [CITATION] (weak convergence of the quantum flow) and Attal and Pautrat [CITATION] (strong convergence uniform on compact time intervals).', '1811.09657-2-12-2': 'We state the theorem without giving the precise meaning of the mode of convergence of the repeated interaction model to the unitary solution of the QSDE because we would need to introduce further mathematical details that would make us stray too far from the main narrative of this article (see however [CITATION], or [CITATION]).', '1811.09657-2-13-0': '(Parthasarathy, Lindsay, Attal and Pautrat [CITATION]) Suppose the following limits exist: [EQUATION] then it follows that [MATH] is unitary, [MATH] is the adjoint of [MATH], i.e. [MATH], [MATH] is self-adjoint and [MATH] (where the brackets [ ] stand for rounding down to an integer) converges to a unitary [MATH] given by the following QSDE [EQUATION]', '1811.09657-2-13-1': 'We proceed by guessing an interaction unitary [MATH] in the repeated interaction Eqn [REF] and check via Thm [REF] that it leads to the correct limit coefficients to reproduce in the limit our system of interest which is given Eqn [REF].', '1811.09657-2-13-2': 'Note that there are several [MATH]s that will lead to the correct limit system.', '1811.09657-2-13-3': 'We will take the following [MATH] and will show that it indeed leads to Eqn [REF] in the limit: [EQUATION]', '1811.09657-2-13-4': 'A short calculation then reveals [EQUATION]', '1811.09657-2-13-5': 'Using the definition of [MATH] and [MATH] in Eqn [REF], we find [EQUATION]', '1811.09657-2-13-6': 'That is, we have found a repeated interaction model that converges to the QSDE of Eqn [REF].', '1811.09657-2-14-0': '# The quantum circuit', '1811.09657-2-15-0': 'The various contributions to the interaction between the atom and the field, given in Eqn [REF], can easily be mapped to elementary quantum gates.', '1811.09657-2-15-1': 'The following quantum circuit [CITATION] implements this interaction for a single time slice:', '1811.09657-2-16-0': '@C=1em @R=.7em atom & R_y(^2) & R_z(^2) & 1 & R_y(2) & 1 &', '1811.09657-2-17-0': 'field & & & & -1 & &', '1811.09657-2-18-0': 'For modeling multiple time slices, different qubits are used to describe the field at different times, and coupled to the atom qubit.', '1811.09657-2-18-1': 'Currently, this limits the simulation to a maximum of four time slices on the IBMqx4 computer.', '1811.09657-2-18-2': 'If operations based on classical bits are enabled, we could reuse the same qubit for the field by measuring this qubit, and rotating it back to its [MATH] state when the outcome is [MATH].', '1811.09657-2-18-3': 'Unfortunately, this is currently only implemented in simulators and not in real hardware.', '1811.09657-2-19-0': 'At the end of the simulation, all qubits (both field and atom) were measured.', '1811.09657-2-19-1': 'The atom was measured in the [MATH], [MATH], and [MATH] basis, whereas the field qubits were measured in the [MATH] and [MATH] basis.', '1811.09657-2-19-2': 'Statistics were collected over 10,240 runs for each combination of measurement directions of atom and field.', '1811.09657-2-20-0': '# Results', '1811.09657-2-21-0': '## Master equation', '1811.09657-2-22-0': 'The repeated interaction model given by Eqn [REF] leads to the following discrete time master equation [CITATION] for the state [MATH] of the two-level atom: [EQUATION] where the discretized Lindblad operator is given by [CITATION] [EQUATION]', '1811.09657-2-22-1': 'Here [MATH] and [MATH] are given by Eqn [REF].', '1811.09657-2-22-2': 'Figure [REF] compares the theoretical results given by the master equation Eqn [REF] and the results obtained with the IBMqx4 Tenerife quantum computer.', '1811.09657-2-23-0': '## Homodyne quantum filter', '1811.09657-2-24-0': 'We now turn to the situation where we are not simply tracing over the field, but condition on observations made in the field.', '1811.09657-2-24-1': 'Suppose that for [MATH], we have observed [MATH] in the field: [EQUATION]', '1811.09657-2-24-2': 'Physically, this corresponds to the case where we observe the field with a homodyne detection setup after the interaction with the two-level atom.', '1811.09657-2-24-3': 'We can condition the time evolution of the density matrix on an observed homodyne photo current detection record.', '1811.09657-2-24-4': 'The conditioned density matrix obeys the following discrete quantum filtering equation for homodyne detection [CITATION] [EQUATION] where [MATH] is given by Eqn [REF] and [MATH] and the initial state [MATH] are given by [EQUATION]', '1811.09657-2-24-5': 'Here [MATH] and [MATH] are given by Eqn [REF].', '1811.09657-2-24-6': 'Figures [REF], [REF] and [REF] compare the theoretical results given by the quantum filter equation Eqn [REF] and the results obtained with the IBM Qiskit [CITATION] simulator and the IBMqx4 Tenerife quantum computer.', '1811.09657-2-25-0': '## Counting quantum filter', '1811.09657-2-26-0': 'Finally, we turn to the situation where we condition on having observed [MATH] in the field for [MATH].', '1811.09657-2-26-1': 'That is, we have the following observations: [EQUATION]', '1811.09657-2-26-2': 'Physically, this corresponds to the case where we observe the field with a photo detector after the interaction with the two-level atom.', '1811.09657-2-26-3': 'We can condition the time evolution of the density matrix on an observed photo detection record.', '1811.09657-2-26-4': 'The conditioned density matrix obeys the following discrete quantum filtering equation for photon counting [CITATION] [EQUATION]', '1811.09657-2-26-5': 'Here [MATH] is given by Eqn [REF].', '1811.09657-2-26-6': 'Figures [REF], [REF] and [REF] compare the theoretical results given by the quantum filter equation Eqn [REF] and the results obtained with the IBM Qiskit simulator and the IBMqx4 Tenerife quantum computer.', '1811.09657-2-27-0': '# Conclusion', '1811.09657-2-28-0': 'In these notes we have shown that it is fairly straightforward to implement quantum stochastic differential equations on a quantum computer.', '1811.09657-2-28-1': 'The mathematical theory [CITATION] behind the necessary discretization of the equations is well worked out and easily translated into a quantum circuit.', '1811.09657-2-28-2': 'It is possible with the very limited capacity of the currently available quantum computers to simulate some simple quantum optical features described by a QSDE (e.g. a Rabi oscillation).', '1811.09657-2-29-0': 'We have also seen that the filter equations are to a large extent correctly reproduced on the IBMqx4 Tenerife quantum computer.', '1811.09657-2-29-1': 'This provides confidence that the (quantum) correlations between the atom and the field are accounted for correctly.', '1811.09657-2-29-2': 'This opens the door to fully coherent simulations of systems that interact with the field at different points, even including fully coherent feedback loops [CITATION].', '1811.09657-2-29-3': 'Naturally, this is only possible when quantum computers are available with more and more reliable qubits.', '1811.09657-2-30-0': 'As can be seen in Figures [REF] and [REF], the time evolution between counts in the photon counting scheme seems to be accounted for correctly, however, the jump operation does not seem to be accurate to the theory.', '1811.09657-2-30-1': 'This can be understood though: there are relatively few jumps and there is already quite a bit of noise on the results when the field is in the vacuum state.', '1811.09657-2-30-2': 'This means there is an identity component in the jump operator that leads to a deviation from the theory (in which this noise is not present, but could be modeled).', '1811.09657-2-31-0': 'With respect to the discrete quantum filtering equations [CITATION], we note that they completely coincide with the IBM Qiskit simulator results.', '1811.09657-2-31-1': 'However, they are much less computationally intensive and could still be used if the number of qubits is larger than the simulator can deal with.', '1811.09657-2-31-2': 'Note, however, that in the simulator it is also possible to recycle the field qubits after they have been measured.', '1811.09657-2-31-3': 'This is currently not yet possible on the real IBMQ hardware.', '1811.09657-2-32-0': 'It will be interesting to see future quantum computers simulate more complex benchmark problems originating from quantum optics.'} | 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'1811.09657-2-2-2'], ['1811.09657-1-2-3', '1811.09657-2-2-3'], ['1811.09657-1-2-4', '1811.09657-2-2-4'], ['1811.09657-1-2-5', '1811.09657-2-2-5'], ['1811.09657-1-6-0', '1811.09657-2-6-0'], ['1811.09657-1-6-1', '1811.09657-2-6-1'], ['1811.09657-1-6-2', '1811.09657-2-6-2'], ['1811.09657-1-6-3', '1811.09657-2-6-3'], ['1811.09657-1-12-0', '1811.09657-2-12-0'], ['1811.09657-1-12-1', '1811.09657-2-12-1'], ['1811.09657-1-12-2', '1811.09657-2-12-2'], ['1811.09657-1-7-0', '1811.09657-2-7-0'], ['1811.09657-1-7-1', '1811.09657-2-7-1'], ['1811.09657-1-7-2', '1811.09657-2-7-2'], ['1811.09657-1-7-3', '1811.09657-2-7-3']] | [['1811.09657-1-24-6', '1811.09657-2-24-6']] | [] | [] | [] | ['1811.09657-1-3-0', '1811.09657-1-15-1', '1811.09657-1-16-0', '1811.09657-1-17-0', '1811.09657-2-3-0', '1811.09657-2-15-1', '1811.09657-2-16-0', '1811.09657-2-17-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1811.09657 | null | null | null | null | null |
hep-ph-0411385 | {'hep-ph-0411385-1-0-0': 'In a first-order phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-1-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-1-1-0': 'plain', 'hep-ph-0411385-1-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe, and have important cosmological implications [CITATION].', 'hep-ph-0411385-1-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a first-order cosmological phase transition and may act as seeds for the nucleation of the phase transition.', 'hep-ph-0411385-1-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-1-3-2': 'Then if a Q-ball has large enough charge it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-1-4-0': 'This result may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-1-5-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-1-5-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-1-5-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-1-6-0': 'I use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-1-7-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-1-7-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-1-8-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-1-8-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-1-8-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-1-9-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-1-10-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-1-10-1': 'So our final approximations, to be checked later for self-consistency are [EQUATION]', 'hep-ph-0411385-1-10-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-1-11-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-1-11-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-1-11-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-1-11-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-1-11-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-1-11-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-1-11-6': 'We see that they are similar in form to the first order gravitational corrections for bubble nucleation for ordinary false vacuum decay [CITATION].', 'hep-ph-0411385-1-12-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that ([REF]) is equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-1-12-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-1-12-2': 'It is also easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential.', 'hep-ph-0411385-1-13-0': 'We also see that these corrections have the expected effect for the case [MATH], which corresponds to decay in de Sitter space, that is to decrease the critical radius and critical charge of the soliton.', 'hep-ph-0411385-1-13-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | {'hep-ph-0411385-2-0-0': 'In a first-order phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-2-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-2-1-0': 'plain', 'hep-ph-0411385-2-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe, and have important cosmological implications [CITATION].', 'hep-ph-0411385-2-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a first-order cosmological phase transition and may act as seeds for the nucleation of the phase transition.', 'hep-ph-0411385-2-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-2-3-2': 'Then if a Q-ball has large enough charge it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-2-3-3': 'Q-ball stability has also been investigated in other models that may be of cosmological relevance [CITATION].', 'hep-ph-0411385-2-3-4': 'These results may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-2-4-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-2-4-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-2-4-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-2-5-0': 'I use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-2-6-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-2-6-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-2-7-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-2-7-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-2-7-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-2-8-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-2-9-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-2-9-1': 'So our final approximations, to be checked later for self-consistency, are [EQUATION]', 'hep-ph-0411385-2-9-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-2-10-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-2-10-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-2-10-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-2-10-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-2-10-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-2-10-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-2-10-6': 'We see that they are similar in form to the first order gravitational corrections for bubble nucleation for ordinary false vacuum decay [CITATION].', 'hep-ph-0411385-2-11-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that our approximations ([REF]) are equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-2-11-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-2-11-2': 'It is also easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential well below the Planck scale.', 'hep-ph-0411385-2-12-0': 'We also see that these corrections have the expected effect for the case [MATH], which corresponds to decay in de Sitter space, that is to decrease the critical radius and critical charge of the soliton.', 'hep-ph-0411385-2-12-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | [['hep-ph-0411385-1-10-0', 'hep-ph-0411385-2-9-0'], ['hep-ph-0411385-1-10-2', 'hep-ph-0411385-2-9-2'], ['hep-ph-0411385-1-7-0', 'hep-ph-0411385-2-6-0'], ['hep-ph-0411385-1-7-1', 'hep-ph-0411385-2-6-1'], ['hep-ph-0411385-1-9-0', 'hep-ph-0411385-2-8-0'], ['hep-ph-0411385-1-6-0', 'hep-ph-0411385-2-5-0'], ['hep-ph-0411385-1-13-0', 'hep-ph-0411385-2-12-0'], ['hep-ph-0411385-1-13-1', 'hep-ph-0411385-2-12-1'], ['hep-ph-0411385-1-2-0', 'hep-ph-0411385-2-2-0'], ['hep-ph-0411385-1-11-0', 'hep-ph-0411385-2-10-0'], ['hep-ph-0411385-1-11-1', 'hep-ph-0411385-2-10-1'], ['hep-ph-0411385-1-11-2', 'hep-ph-0411385-2-10-2'], ['hep-ph-0411385-1-11-3', 'hep-ph-0411385-2-10-3'], ['hep-ph-0411385-1-11-4', 'hep-ph-0411385-2-10-4'], ['hep-ph-0411385-1-11-5', 'hep-ph-0411385-2-10-5'], ['hep-ph-0411385-1-11-6', 'hep-ph-0411385-2-10-6'], ['hep-ph-0411385-1-12-1', 'hep-ph-0411385-2-11-1'], ['hep-ph-0411385-1-8-0', 'hep-ph-0411385-2-7-0'], ['hep-ph-0411385-1-8-1', 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'hep-ph-0411385-5-3-2'], ['hep-ph-0411385-5-0-0', 'hep-ph-0411385-6-0-0'], ['hep-ph-0411385-5-3-0', 'hep-ph-0411385-6-3-0'], ['hep-ph-0411385-4-11-2', 'hep-ph-0411385-5-15-0'], ['hep-ph-0411385-1-4-0', 'hep-ph-0411385-2-3-4']] | [] | [] | [] | ['hep-ph-0411385-1-1-0', 'hep-ph-0411385-2-1-0', 'hep-ph-0411385-3-1-0', 'hep-ph-0411385-4-1-0', 'hep-ph-0411385-5-1-0', 'hep-ph-0411385-6-1-0'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/', '5': 'http://arxiv.org/licenses/assumed-1991-2003/', '6': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/0411385 | {'hep-ph-0411385-3-0-0': 'In a first-order phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-3-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-3-1-0': 'plain', 'hep-ph-0411385-3-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe, and have important cosmological implications [CITATION].', 'hep-ph-0411385-3-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a first-order cosmological phase transition and may act as seeds for the nucleation of the phase transition.', 'hep-ph-0411385-3-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-3-3-2': 'Then if a Q-ball has large enough charge it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-3-3-3': 'Q-ball stability has also been investigated in other models that may be of cosmological relevance [CITATION].', 'hep-ph-0411385-3-3-4': 'These results may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-3-4-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-3-4-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-3-4-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-3-5-0': 'I use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-3-6-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-3-6-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-3-7-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-3-7-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-3-7-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-3-8-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-3-9-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-3-9-1': 'So our final approximations, to be checked later for self-consistency, are [EQUATION]', 'hep-ph-0411385-3-9-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-3-10-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-3-10-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-3-10-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-3-10-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-3-10-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-3-10-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-3-10-6': 'We see that they are similar in form to the first order gravitational corrections for bubble nucleation for ordinary false vacuum decay [CITATION].', 'hep-ph-0411385-3-11-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that our approximations ([REF]) are equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-3-11-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-3-11-2': 'It is also easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential well below the Planck scale.', 'hep-ph-0411385-3-12-0': 'We also see that these corrections have the expected effect for the case [MATH], which corresponds to decay in de Sitter space, that is to decrease the critical radius and critical charge of the soliton.', 'hep-ph-0411385-3-12-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | {'hep-ph-0411385-4-0-0': 'In a first-order phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-4-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-4-1-0': 'plain', 'hep-ph-0411385-4-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe, and have important cosmological implications [CITATION].', 'hep-ph-0411385-4-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a first-order cosmological phase transition and may act as seeds for the nucleation of the phase transition.', 'hep-ph-0411385-4-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-4-3-2': 'Then if a Q-ball has large enough charge it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-4-3-3': 'Q-ball stability has also been investigated in other models that may be of cosmological relevance [CITATION].', 'hep-ph-0411385-4-3-4': 'These results may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-4-4-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-4-4-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-4-4-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-4-5-0': 'I use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-4-6-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-4-6-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-4-7-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-4-7-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-4-7-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-4-8-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-4-9-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-4-9-1': 'So our final approximations, to be checked later for self-consistency, are [EQUATION]', 'hep-ph-0411385-4-9-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-4-10-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-4-10-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-4-10-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-4-10-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-4-10-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-4-10-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-4-10-6': 'We see that they are similar in form to the first order gravitational corrections for bubble nucleation for ordinary false vacuum decay [CITATION].', 'hep-ph-0411385-4-11-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that our approximations ([REF]) are equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-4-11-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-4-11-2': 'It is also easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential well below the Planck scale.', 'hep-ph-0411385-4-12-0': 'We also see that these corrections have the expected effect for the case [MATH], which corresponds to decay in de Sitter space, that is to decrease the critical radius and critical charge of the soliton.', 'hep-ph-0411385-4-12-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | {'hep-ph-0411385-5-0-0': 'In a first-order phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-5-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-5-1-0': 'plain', 'hep-ph-0411385-5-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe with various mechanisms, and have important cosmological implications [CITATION].', 'hep-ph-0411385-5-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a first-order cosmological phase transition and may act as seeds for the nucleation of the phase transition.', 'hep-ph-0411385-5-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-5-3-2': 'Then, if a Q-ball has large enough charge, it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-5-3-3': 'Q-ball stability has also been investigated in other models that may be of cosmological relevance [CITATION].', 'hep-ph-0411385-5-3-4': 'These results may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-5-4-0': 'In these models, that exhibit a phase transition at or above the TeV scale, the calculation of the tunneling rate and the kinetics of the phase transition is done along the lines of [CITATION] and gives a specific picture of the phase transition, its progress and its completion.', 'hep-ph-0411385-5-4-1': 'However, when the theory admits topological or non-topological defects (solitons, monopoles, or Q-balls as in MSSM) the picture of the phase transition may be quite different [CITATION].', 'hep-ph-0411385-5-4-2': 'These defects may act as nucleations sites for the phase transition and accelerate its progress, facilitating its completion.', 'hep-ph-0411385-5-5-0': 'In the case of Q-balls, we are interested in the critical soliton charge [MATH] (and the corresponding critical soliton radius [MATH]) above which the solitons become unstable and expand, converting space to the true vacuum and accelerating the phase transition [CITATION].', 'hep-ph-0411385-5-6-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-5-6-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-5-6-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-5-7-0': 'In the case of a de Sitter background, the effect of the cosmological expansion that I derive here is to decrease the values of [MATH] and [MATH], as is expected, thereby enhancing further the progress of the phase transition.', 'hep-ph-0411385-5-7-1': 'Although the effect is of higher order, the correction to [MATH] is finite and may be important, especially for small values of the supercooling parameter [MATH], as will be shown here.', 'hep-ph-0411385-5-8-0': 'I will use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I will calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-5-9-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-5-9-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-5-10-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-5-10-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-5-10-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-5-11-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-5-12-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-5-12-1': 'So our final approximations, to be checked later for self-consistency, are [EQUATION]', 'hep-ph-0411385-5-12-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-5-13-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-5-13-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-5-13-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-5-13-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-5-13-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-5-13-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-5-13-6': 'We see that the effect of the cosmological expansion has been to lower the values of the critical charge and the critical radius, as is expected, thereby facilitating the phase transition.', 'hep-ph-0411385-5-14-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that our approximations ([REF]) are equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-5-14-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-5-14-2': 'Although the scale of the specific model (MSSM or other) is fixed, of order TeV or higher, the value of [MATH], which corresponds to the cosmological constant term, is arbitrary, provided it it satisfies ([REF]).', 'hep-ph-0411385-5-15-0': 'It is easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential well below the Planck scale.', 'hep-ph-0411385-5-15-1': 'We can also check that for the values of the parameters that satisfy ([REF]) the soliton radius is much smaller than the radius of the compactified de Sitter space [MATH], so we are indeed in the limit of weak gravity.', 'hep-ph-0411385-5-16-0': 'However, even though we are in the limit of weak gravity, the correction to [MATH] derived here, may be finite, and even large for the cosmological phase transition.', 'hep-ph-0411385-5-16-1': 'The production rate for non-topological solitons with charge [MATH] in a cosmological setting is generically proportional to [MATH] [CITATION] and a change in [MATH] gives important corrections regarding the progress and completion of the phase transition.', 'hep-ph-0411385-5-17-0': 'In order to see this we can consider a generic effective potential [MATH], depending on the temperature [MATH], with the two degenerate minima (at [MATH] and [MATH]) at the critical temperature [MATH].', 'hep-ph-0411385-5-17-1': 'Then, slightly below the critical temperature, we have [CITATION] [EQUATION] with [MATH] a coupling constant of the model and [MATH] the supercooling parameter [MATH].', 'hep-ph-0411385-5-17-2': 'Then we can calculate from ([REF]) the decrease of the critical charge due to gravity [EQUATION]', 'hep-ph-0411385-5-17-3': 'We see that, although our approximation ([REF]) demands the second term in ([REF]) to be small, the result for [MATH] may be large for sufficiently small values of the supercooling parameter [MATH].', 'hep-ph-0411385-5-17-4': 'It is for these cases of a "fast" phase transition that the results presented here are relevant and may give important corrections to the usual calculation of the false vacuum decay rate.', 'hep-ph-0411385-5-18-0': 'In summary we see that the corrections calculated here have the expected effects for the case of non-topological solitons in an expanding de Sitter background, that is to decrease the critical radius and critical charge of the solitons.', 'hep-ph-0411385-5-18-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | {'hep-ph-0411385-6-0-0': 'In a cosmological phase transition in theories that admit Q-balls there is a value of the soliton charge above which the soliton becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling.', 'hep-ph-0411385-6-0-1': 'Here I consider the effects of gravity on these solitons and I calculate the lowest gravitational corrections to the critical radius and charge.', 'hep-ph-0411385-6-1-0': 'plain', 'hep-ph-0411385-6-2-0': 'Non-topological solitons of the Q-ball type, exist in field theories with a global [MATH] symmetry [CITATION], may be formed in the early Universe with various mechanisms, and have important cosmological implications [CITATION].', 'hep-ph-0411385-6-3-0': 'Here I will be interested in the case of Q-balls that are formed in the context of a cosmological phase transition and may act as seeds for its nucleation.', 'hep-ph-0411385-6-3-1': 'In the case of a first-order phase transition, when the symmetric vacuum becomes metastable, the potential energy density in the interior of the Q-balls becomes negative with respect to the potential energy density of the symmetric vacuum.', 'hep-ph-0411385-6-3-2': 'Then, if a Q-ball has large enough charge, it becomes unstable and expands, converting space to the true vacuum, much like a critical bubble in the case of ordinary tunneling [CITATION].', 'hep-ph-0411385-6-3-3': 'Q-ball stability has also been investigated in other models that may be of cosmological relevance [CITATION].', 'hep-ph-0411385-6-3-4': 'These results may be important for cosmological phase transitions in theories that admit non-topological solitons, such as the Minimal Supersymmetric Standard Model (MSSM) [CITATION].', 'hep-ph-0411385-6-4-0': 'In these models, that exhibit a phase transition at or above the TeV scale, the calculation of the tunneling rate and the kinetics of the phase transition is done along the lines of [CITATION] and gives a specific picture of the phase transition, its progress and its completion.', 'hep-ph-0411385-6-4-1': 'However, when the theory admits topological or non-topological defects (solitons, monopoles, or Q-balls as in MSSM) the picture of the phase transition may be quite different [CITATION].', 'hep-ph-0411385-6-4-2': 'These defects may act as nucleations sites for the phase transition and accelerate its progress, facilitating its completion.', 'hep-ph-0411385-6-5-0': 'In the case of Q-balls, we are interested in the critical soliton charge [MATH] (and the corresponding critical soliton radius [MATH]) above which the solitons become unstable and expand, converting space to the true vacuum and accelerating the phase transition [CITATION].', 'hep-ph-0411385-6-6-0': 'Here I will calculate the first-order gravitational corrections to this effect, that can mediate tunneling in curved space-time [CITATION].', 'hep-ph-0411385-6-6-1': 'The effects of gravity to the ordinary bubble nucleation rate can be very distinct [CITATION], due mainly to the finiteness of the compactified de Sitter space.', 'hep-ph-0411385-6-6-2': 'Here, however, I will be interested only in the case where the relevant solitons are much smaller than the radius of the background de Sitter space, and I will calculate the lowest gravitational corrections to these solitons.', 'hep-ph-0411385-6-7-0': 'In the case of a de Sitter background, the effect of the cosmological expansion that I derive here is to decrease the values of [MATH] and [MATH], as is expected, thereby enhancing further the progress of the phase transition.', 'hep-ph-0411385-6-7-1': 'Although the effect is of higher order, the correction to [MATH] is finite and may be important, especially for small values of the supercooling parameter [MATH], as will be shown here.', 'hep-ph-0411385-6-8-0': 'I will use the gravitational equations that describe Q-balls in curved space-time [CITATION] in the thin-wall approximation and I will calculate the gravitational effects to the critical radius and critical charge that these Q-balls must have in order to mediate the phase transition.', 'hep-ph-0411385-6-9-0': 'Consider a quantum field theory of a complex scalar field [MATH] with a [MATH] symmetric scalar potential [MATH].', 'hep-ph-0411385-6-9-1': 'For spherically symmetric configurations with metric [EQUATION] the action is [EQUATION] and for solutions of the Q-ball type [EQUATION] the field equations are [EQUATION] where [EQUATION].', 'hep-ph-0411385-6-10-0': 'Now we consider a potential that has a local minimum [MATH] at [MATH] and a global minimum [MATH] at [MATH], with the height of the barrier much greater than [MATH] the difference between the two minima.', 'hep-ph-0411385-6-10-1': 'Like in the flat case [CITATION] this theory admits non-topological thin-wall solitons of the Q-ball type [CITATION], where the field forms a bubble of radius [MATH], inside the bubble the field is essentially constant and outside zero.', 'hep-ph-0411385-6-10-2': 'For this kind of potential there is no minimum [MATH] since [MATH] is already metastable, so the solutions have [MATH] and inside the bubble [MATH] (these correspond to the type II solitons of [CITATION]).', 'hep-ph-0411385-6-11-0': 'For the configuration inside the bubble we make the ansatz [EQUATION] where, in the weak gravity approximation, [EQUATION] and we get [EQUATION] and our approximation is self-consistent when [EQUATION].', 'hep-ph-0411385-6-12-0': 'Now we calculate the values of the charge [MATH] and the energy [MATH] of this configuration, making the further approximation [MATH], which holds for this kind of potentials.', 'hep-ph-0411385-6-12-1': 'So our final approximations, to be checked later for self-consistency, are [EQUATION]', 'hep-ph-0411385-6-12-2': 'In our further calculations we keep the terms that are of lowest order in [MATH] and [MATH].', 'hep-ph-0411385-6-13-0': 'In this approximation the expression for the total charge gives [EQUATION] and the expression for the soliton energy [MATH] becomes [EQUATION] with [EQUATION] where [EQUATION] and [EQUATION]', 'hep-ph-0411385-6-13-1': 'For the surface term, we assume that the soliton surface has a thickness [MATH] and an average value of the potential [MATH].', 'hep-ph-0411385-6-13-2': 'Minimizing with respect to [MATH] [CITATION] we get [EQUATION]', 'hep-ph-0411385-6-13-3': 'Now the soliton radius can be found minimizing the energy with respect to [MATH], and the critical radius and charge for these solitons are found from the simultaneous solution of [CITATION] [EQUATION]', 'hep-ph-0411385-6-13-4': 'The critical radius in this approximation turns out to be [EQUATION] and for the critical charge we get [EQUATION]', 'hep-ph-0411385-6-13-5': 'These are the first order gravitational corrections to the flat space expressions (that we get for [MATH]).', 'hep-ph-0411385-6-13-6': 'We see that the effect of the cosmological expansion has been to lower the values of the critical charge and the critical radius, as is expected, thereby facilitating the phase transition.', 'hep-ph-0411385-6-14-0': 'Now, in order to check for the validity of our approximations, we take [MATH] where [MATH] is a typical mass scale of the theory, and we see that our approximations ([REF]) are equivalent to [EQUATION] where [MATH] is the Planck mass.', 'hep-ph-0411385-6-14-1': 'So our approximations for this problem are self-consistent.', 'hep-ph-0411385-6-14-2': 'Although the scale of the specific model (MSSM or other) is fixed, of order TeV or higher, the value of [MATH], which corresponds to the cosmological constant term, is arbitrary, provided it it satisfies ([REF]).', 'hep-ph-0411385-6-15-0': 'It is easy to see that the condition for stability, [MATH], is satisfied for generic values of the potential well below the Planck scale.', 'hep-ph-0411385-6-15-1': 'We can also check that for the values of the parameters that satisfy ([REF]) the soliton radius is much smaller than the radius of the compactified de Sitter space [MATH], so we are indeed in the limit of weak gravity.', 'hep-ph-0411385-6-16-0': 'However, even though we are in the limit of weak gravity, the correction to [MATH] derived here, may be finite, and even large for the cosmological phase transition.', 'hep-ph-0411385-6-16-1': 'The production rate for non-topological solitons with charge [MATH] in a cosmological setting is generically proportional to [MATH] [CITATION] and a change in [MATH] gives important corrections regarding the progress and completion of the phase transition.', 'hep-ph-0411385-6-17-0': 'In order to see this we can consider a generic effective potential [MATH], depending on the temperature [MATH], with the two degenerate minima (at [MATH] and [MATH]) at the critical temperature [MATH].', 'hep-ph-0411385-6-17-1': 'Then, slightly below the critical temperature, we have [CITATION] [EQUATION] with [MATH] a coupling constant of the model and [MATH] the supercooling parameter [MATH].', 'hep-ph-0411385-6-17-2': 'Then we can calculate from ([REF]) the decrease of the critical charge due to gravity [EQUATION]', 'hep-ph-0411385-6-17-3': 'We see that, although our approximation ([REF]) demands the second term in ([REF]) to be small, the result for [MATH] may be large for sufficiently small values of the supercooling parameter [MATH].', 'hep-ph-0411385-6-17-4': 'It is for these cases of a "fast" phase transition that the results presented here are relevant and may give important corrections to the usual calculation of the false vacuum decay rate.', 'hep-ph-0411385-6-18-0': 'Generally the presence of solitonic configurations (of the type described here or other) greatly facilitates the progress and completion of the phase transition since they act as seeds for its nucleation, however the actual dynamics and kinematics of the transition are quite complicated since they depend on the interactions of these solitons with the heat bath.', 'hep-ph-0411385-6-18-1': 'For a phase transition that occurs at a scale [MATH], we see from the previous expressions that the corrections calculated here are important when the transition proceeds with a small value of supercooling [MATH].', 'hep-ph-0411385-6-18-2': 'For the lowest scale supersymmetric models that admit such configurations, with [MATH] of order of a few TeV this corresponds to [MATH], that is very close to a second order phase transition.', 'hep-ph-0411385-6-18-3': 'This is the case when these solitons are formed at high temperature, by thermal and charge fluctuations, before the onset of the first order phase transition, and as soon as the symmetric vacuum becomes metastable they become unstable and expand converting the space in the true vacuum.', 'hep-ph-0411385-6-18-4': 'Similar estimates hold when we are interested in an intermediate scale phase transition, provided ([REF]) is satisfied, the actual dynamics of the transition, however, may change these estimates.', 'hep-ph-0411385-6-19-0': 'In summary we see that the corrections calculated here have the expected effects for the case of non-topological solitons in an expanding de Sitter background, that is to decrease the critical radius and critical charge of the solitons.', 'hep-ph-0411385-6-19-1': 'Although I worked in the limit of weak gravity, these effects show that, in the general case of vacuum decay in curved spacetime, such solitonic configurations may be important in the calculation of the rate of the phase transition.'} | null |
1904.01211 | {'1904.01211-1-0-0': 'We introduce the notion of Lowner (ellipsoid) function for a log concave function and show that it is an extension of the Lowner ellipsoid for convex bodies.', '1904.01211-1-0-1': 'We investigate its duality relation to the recently defined John (ellipsoid) function [CITATION].', '1904.01211-1-0-2': 'For convex bodies, John and Lowner ellipsoids are dual to each other.', '1904.01211-1-0-3': 'Interestingly, this need not be the case for the John function and the Lowner function.', '1904.01211-1-1-0': '# Introduction', '1904.01211-1-2-0': 'Asymptotic convex geometry studies the properties of convex bodies with emphasis on the dependence of geometric and analytic invariants on the dimension.', '1904.01211-1-2-1': 'The convexity assumption enforces concentration of volume in a canonical way and it is a main question if under natural normalizations the answers to fundamental questions are independent of the dimension.', '1904.01211-1-2-2': 'The most classical normalizations of convex bodies arise as solutions of extremal problems.', '1904.01211-1-2-3': 'These normalizations include the isotropic position, which arose from classical mechanics of the 19th century and which is related to a famous open problem in convex geometry, the hyperplane conjecture (see, e.g., the survey [CITATION]).', '1904.01211-1-2-4': 'The best results currently available there are due Bourgain [CITATION] and Klartag [CITATION].', '1904.01211-1-2-5': 'Other positions are the John position, also called maximal volume ellipsoid position and the Lowner position, also called minimal volume ellipsoid position.', '1904.01211-1-2-6': 'The right choice of a position is important for the study of affinely invariant quantities and their related isoperimetric inequalities.', '1904.01211-1-2-7': "For instance, John and Lowner position are related to the Brascamp-Lieb inequality and its reverse [CITATION], to K. Ball's sharp reverse isoperimetric inequality [CITATION], to the notion of volume ratio [CITATION], which is defined as the [MATH]-th root of the volume of a convex body divided by the volume of its John ellipsoid and which finds applications in functional analysis and Banach space theory [CITATION].", '1904.01211-1-2-8': 'John and Lowner position are even relevant in quantum information theory [CITATION].', '1904.01211-1-2-9': 'Since a position may be seen as a choice of a special ellipsoid, and since an ellipsoid entails a Euclidean structure of the underlying space, John and Lowner ellipsoids provide a way to measure how far a normed space is from Euclidean space [CITATION].', '1904.01211-1-2-10': 'For a detailed discussion of the John and the Lowner ellipsoid and its connections to functional analysis we refer the reader to [CITATION] and the survey [CITATION].', '1904.01211-1-2-11': 'F. John proved in [CITATION] that among all ellipsoids contained in a convex body [MATH] , there is a unique ellipsoid of maximal volume, now called the John ellipsoid of [MATH] .', '1904.01211-1-2-12': 'The Lowner ellipsoid of [MATH] is the unique ellipsoid of minimal volume containing [MATH] .', '1904.01211-1-2-13': 'These two notions are closely related by polarity (see, e.g., [CITATION]): A [MATH]-symmetric ellipsoid [MATH] is the ellipsoid of maximal volume inside [MATH] if and only if [MATH] is the ellipsoid of minimal volume outside [MATH], where [MATH] is the polar of [MATH].', '1904.01211-1-2-14': 'Probabilisitic methods have become extremely useful in convex geometry.', '1904.01211-1-2-15': 'In this context, log-concave functions arise naturally from the uniform measure on convex bodies.', '1904.01211-1-2-16': 'A function [MATH] is said to be log concave, if it is of the form [MATH] where [MATH] is convex.', '1904.01211-1-2-17': 'Extensive research has been devoted within the last ten years to extend the concepts and inequalities from convex bodies to the setting of functions.', '1904.01211-1-2-18': 'In fact, it was observed early that the Prekopa-Leindler inequality (see, e.g., [CITATION]) is the functional analog of the Brunn-Minkowski inequality (see, e.g., [CITATION]) for convex bodies.', '1904.01211-1-2-19': 'Much progress has been made since and functional analogs of many other geometric inequalities were established.', '1904.01211-1-2-20': 'Among them are the functional Blaschke-Santalo inequality [CITATION] and its reverse [CITATION], a functional affine isoperimetric inequality for log-concave functions which can be viewed as an inverse log-Sobolev inequality for entropy [CITATION] and a theory of valuations, an important concept for convex bodies (e.g., [CITATION]), is currently being developed in the functional setting, e.g., [CITATION].', '1904.01211-1-2-21': 'It was only recently that the notion of a John (ellipsoid) function of a log-concave function was established by Alonso-Gutierrez, Merino, Jimenez, and Villa [CITATION].', '1904.01211-1-2-22': 'However, the notion of a Lowner ellipsoid function for log concave functions has been missing till now.', '1904.01211-1-2-23': 'In this paper we put forward such a notion and we investigate, among other things, its relation to the John ellipsoid function of [CITATION].', '1904.01211-1-2-24': 'Our main result reads as follows.', '1904.01211-1-2-25': 'We denote by [MATH] the set of all invertible affine transformations and by [MATH] denote the Euclidean norm on [MATH].', '1904.01211-1-2-26': 'We say that a function is nondegenerate if [MATH].', '1904.01211-1-3-0': 'Theorem.', '1904.01211-1-3-1': 'Let [MATH] (A_0,t_0) A R [MATH] A_0 [MATH] e^-A_0x_2+t_0 [MATH] f [MATH] L(f) [MATH] 1_K(x) [MATH] KR^n [MATH] L(1_K)1 [MATH] K [MATH] K [MATH](1_K)^[MATH] L(f) [MATH] f [MATH] R^n [MATH] A [MATH] S_+ [MATH] bR^n [MATH] SA(b)=A(b)SA [MATH] A=_bR^n A(b) [MATH] SA=_bR^nSA(b) [MATH] A:R^nR^n [MATH] f:R^nR) is defined as [MATH] .', '1904.01211-1-3-2': 'For [MATH] , let [MATH] be a translation of a function by [MATH] , that is, for a function [MATH] , [EQUATION]', '1904.01211-1-3-3': 'For [MATH] and a function [MATH], we denote by [EQUATION] the level sets of [MATH] .', '1904.01211-1-4-0': '## Log-concave functions', '1904.01211-1-5-0': 'A function [MATH] is said to be log-concave if it is of the form [MATH] where [MATH] is a convex function.', '1904.01211-1-5-1': 'We always consider in this paper log-concave functions [MATH] that are integrable and such that [MATH] is nondegenerate, i.e., the interior of the support of [MATH] is non-empty, [MATH].', '1904.01211-1-5-2': 'This then implies that [MATH].', '1904.01211-1-5-3': 'We will also need the Legendre transform which we recall now.', '1904.01211-1-5-4': 'Let [MATH] and let [MATH] be a convex function.', '1904.01211-1-5-5': 'Then [EQUATION] is the Legendre transform of [MATH] [CITATION].', '1904.01211-1-5-6': 'If [MATH] is log-concave, then [EQUATION] is called the dual or polar function of [MATH] with respect to [MATH].', '1904.01211-1-5-7': 'In particular, when [MATH] , [EQUATION] where [MATH] , also denoted by [MATH][MATH]S_z[MATH] L) and [MATH] are involutions, that is, [MATH] and [MATH].', '1904.01211-1-6-0': '(ii) [MATH].', '1904.01211-1-7-0': '(iii) [MATH].', '1904.01211-1-8-0': 'We now list some basic well known facts on log-concave functions.', '1904.01211-1-8-1': 'A log-concave function is continuous on its support, e.g., [CITATION].', '1904.01211-1-9-0': "We include a proof of the first fact for the reader's convenience.", '1904.01211-1-9-1': 'More on log-concave functions can be found in e.g., [CITATION].', '1904.01211-1-10-0': 'If [MATH] is a nondegenerate integrable log-concave function, then [MATH] is convex and compact for [MATH] .', '1904.01211-1-11-0': 'Let [MATH] .', '1904.01211-1-11-1': 'As [MATH] is convex and as [MATH] is nondegenerate, the level set [EQUATION] is convex and closed for all [MATH].', '1904.01211-1-11-2': 'As [MATH] , it remains to show that [MATH] is bounded for [MATH] .', '1904.01211-1-11-3': 'It follows from Theorem 7.6 of [CITATION] that every level set [MATH] , [MATH] , has the same affine dimension as the support of [MATH] , which has affine dimension [MATH] .', '1904.01211-1-11-4': 'Chebyshev inequality then yields [EQUATION].', '1904.01211-1-11-5': 'Since [MATH] is a full dimensional convex set with finite volume, it is bounded.', '1904.01211-1-11-6': 'Therefore, [MATH] is compact for [MATH] .', '1904.01211-1-12-0': 'The following fact is a direct corollary of the functional Blaschke-Santalo inequality [CITATION] and the functional reverse Santalo inequality[CITATION].', '1904.01211-1-13-0': 'Let [MATH] be a nondegenerate, integrable, log concave function Then [MATH] is again a nondegenerate integrable log concave function or all [MATH] and thus [MATH] .', '1904.01211-1-13-1': 'Furthermore, [MATH] for all [MATH] .', '1904.01211-1-14-0': '# The Lo wner function of a log-concave function', '1904.01211-1-15-0': 'We now define the Lowner function for an integrable, nondegenerate, log-concave function [MATH] .', '1904.01211-1-16-0': '## A minimization problem.', '1904.01211-1-16-1': 'Definition of the Lowner function', '1904.01211-1-17-0': 'We consider the following minimization problem [EQUATION] subject to [EQUATION] where the minimum is taken over all nonsingular affine maps [MATH] and all [MATH]e^-Ax_2+t[MATH]f[MATH]K[MATH]K[MATH] f: R^nR^+ [MATH] f(x)=e^-(x) [MATH]O(n)[MATH] (A_0,t_0) [MATH]t_0[MATH]A_0[MATH] e^-A_0x_2+t_0 [MATH] f [MATH]K R^n[MATH]L(K)[MATH]K[MATH]T_L(K)[MATH]T_L(K) B^n_2= L(K)[MATH]A A[MATH]t R[MATH](x) = Ax_2-t[MATH]K[MATH]A=T+a[MATH]a=0[MATH]Ax_2-t 0[MATH]x K[MATH]L(K)[MATH]K[MATH] t T^-1 B^n_2 = L(K)[MATH] _(T,t)e^tT[MATH]t_0= n[MATH]T_0=n T_L(K)^-1[MATH] L (1_K)(x) = e^-n (T_L(K)^-1 x -1)[MATH]g(x) = e^-x^2_2/2[MATH]f(x) = e^-(x)[MATH][MATH]x[MATH](x) = (x)[MATH]A A[MATH]A_0= a Id[MATH]a[MATH]t_0[MATH] f=e^- [MATH]A=T+a A[MATH] TGL(n) [MATH] T=O R [MATH] R [MATH] OO(n) [MATH]A S A[MATH]A= T+a[MATH] T [MATH]T S_+[MATH]b=T^-1 a[MATH]T S_+[MATH]b R^n[MATH] bR^n [MATH] f=e^- [MATH] R^n [MATH] z=Tx[MATH] f_b=S_-bf [MATH] ( f_b)^[MATH]b[MATH]b=0[MATH]O(n)[MATH] f=e^- [MATH](t_0, T_0) R S_+[MATH] 0<sf_[MATH]_f[MATH]s = e^-t[MATH] s1_TB_2^nfTB_2^nG_f(s) [MATH] s>f_[MATH] T: TB_2^nG_f(s)=[MATH] _s0 _f(s)=0 [MATH] s_f(s) [MATH] f=e^- [MATH] R^n [MATH] s_1,s_2(0,f_] [MATH] _s0 _f(s)=0 [MATH]TS_+:TB_2^nG_f(s)[MATH] T_0, T_1 [MATH] T_2 [MATH] _f(s_1^1-s_2^)=s_1^1-s_2^T_0, _f(s_1)=s_1T_1 [MATH] _f(s_2)=s_2T_2 [MATH]f[MATH] T_0 [(1-)T_1+T_2)] [MATH] T_0(T_1)^1- (T_2)^[MATH]t >0[MATH] _f(t) [MATH] _t0_f(t)=0 [MATH] _f(s)_f(s) [MATH] s [MATH] _s0_f(s)=0 [MATH]K[MATH]L[MATH] K^n [MATH] R^n [MATH] K [MATH] T_K [MATH]T[MATH] 0int(K) [MATH] _T S_+: TB_2^nKT =0 [MATH] K [MATH] 0int(K) [MATH] T_K [MATH] T_K= _TS_+: TB_2^nKT [MATH] K=K(-K) [MATH]K (-K)[MATH]K (-K)[MATH] T_KB_2^n [MATH] J(K) [MATH] K=K(-K) [MATH] T_K [MATH]O(n)[MATH] J(K) [MATH] K [MATH]L[MATH] d_H(K,L)<[MATH] d_H(K, L)< 2[MATH]K K^n[MATH]K J(K)[MATH]>0[MATH][MATH]LK^n[MATH] d_H(K, L)< [MATH] d_H(J(K), J(L))< [MATH]L[MATH] d_H(K,L)</2 [MATH] s_f(s) [MATH] s [MATH] s [MATH] sG_f(s) [MATH] K_TS_+: TB_2^nKT [MATH] s [MATH] _s0_f(s)=0 [MATH] _f(s) [MATH] (0,f_] [MATH] _f(s) [MATH] s_0(0,f_] [MATH] T_0S_+ [MATH] t_0=-s_0 [MATH] T_0 [MATH]O(n)[MATH] s [MATH] T [MATH]O(n)[MATH] s_1,s_2 [MATH] s_1>s_2 [MATH] _f(s_1)=_f(s_2) [MATH]_f[MATH] T_0, T_1 [MATH] T_2 [MATH] T_1=T_2 [MATH] f_m,f [MATH] f_mf [MATH] f_m,f [MATH] G_f(k) [MATH] 0<k < f_[MATH] G_f_m(k) [MATH] 0<k < f_m_[MATH] m1 [MATH]k[MATH]G_f_m(k)G_f(k)[MATH](x_m_j)_j N[MATH]x_m_j G_f_m_j(k)[MATH]j[MATH]G_f(k)[MATH]G_f(k)[MATH](x_m)_N[MATH]x_m G_f_m(k)[MATH]m N[MATH](x_m_j)_j N[MATH]x_m_j G_f_m_j(k)[MATH]j[MATH]x=_j x_m_j [MATH]D = co[x_m_j : j N ][MATH]x_m_j : j N [MATH]D[MATH]f_m_j f[MATH]R^n[MATH]f_m_j f[MATH]D[MATH]j[MATH]f[MATH] f_m_j (x_m_j) f(x)[MATH] f_m_j (x_m_j) k[MATH]f(x) k[MATH]x G_f(k)[MATH]0< k < f_[MATH]l= -k[MATH]G_f_m(k)= E__m(l)[MATH]x E_(l)[MATH](x_m)_N[MATH]x_m E_f_m(k)[MATH]m[MATH](x)=l[MATH]f[MATH]x_0[MATH]R^n[MATH](x_0)= _x R^n (x)[MATH]x_0=0[MATH]x[MATH]e_n+1 = (0, , 1)[MATH]k < f_[MATH]l > (x_0) = (0)[MATH]0 < 2 ^12 < (x) - (0)[MATH]f_m f[MATH]_m [MATH]m m_0[MATH]L[MATH](0, (0)+ )[MATH](x, _m(x))[MATH]x_m[MATH]L[MATH]x_m[MATH]l[MATH]_m(x) - ((0) + ) _m(x) - (0) - ) > 2 ^12 - 2 [MATH]_m[MATH]y[MATH][0,x][MATH]_m(y L(y)[MATH] _m(x ) (x) [MATH]m m_0[MATH]x_m [0,x][MATH]x_m E__m(l)[MATH]m_1 m_0[MATH]_m_1(x) < (x) =l[MATH]x E__m_1(l)[MATH]x_m_1 =x[MATH]m > m_1[MATH]_m(x) (x)[MATH]x_m[MATH]_m(x) < (x)[MATH]x_m=x[MATH] f_m,f [MATH] f_mf [MATH] f_m_f_[MATH]f[MATH] x_0R^n [MATH] f(x_0)=f_[MATH] >0 [MATH] m_1 [MATH] m>m_1 [MATH] f_m_f_m(x_0)f(x_0)- [MATH] m>m_1 [MATH] 0 < < 14 f_[MATH] f [MATH] >0 [MATH] m_2 [MATH] m>m_2 [MATH] f_m(x)<12f_ [MATH] xG_f(12f_)+B_2^n [MATH] m>m_2 [MATH] m>m_2 [MATH] f_m(x)f(x) [MATH] G_f(12f_)+B_2^n [MATH] G_f(12f_)+B_2^n [MATH] f_mf [MATH] G_f(12f_)+B_2^n [MATH] ), there exists [MATH] such that [EQUATION] _xR^n f_m(x)=f_m_f_+.', '1904.01211-1-17-1': '[EQUATION] _0<sf_m__f_m(s)_0<sf__f(s).', '1904.01211-1-17-2': '[EQUATION]', '1904.01211-1-17-3': 'T_m,s=_TS_+: TB_2^nG_f_m(s) T. [EQUATION] _f_m(s_m)= _0 < s f_m_ _f_m(s) = s_m T_m, s_m .', '1904.01211-1-17-4': '[EQUATION] 0 < _R^n f(x) dx=_0^f_vol_n(G_f(s))ds<.', '1904.01211-1-17-5': '[EQUATION] 0 < _0^vol_n(G_f(s))ds<.', '1904.01211-1-17-6': '[EQUATION] 0 < _0^vol_n(G_f(s))ds<_0.', '1904.01211-1-17-7': '[EQUATION]', '1904.01211-1-17-8': 'G_f(s) _TS_+: TB_2^nG_f(s)T = T_0,s [EQUATION]', '1904.01211-1-17-9': 'G_f(s) vol_n(G_f(s)) [EQUATION]', '1904.01211-1-17-10': 'G_f_m(s)G_f(s) [EQUATION] vol_n(G_f_m(s)) vol_n(G_f_m(s_m)) T_m,s_m vol_n(B^n_2) .', '1904.01211-1-17-11': '[EQUATION] vol_n(G_f(f_-_1) f dsf_-_1 10 f ds9 f_, [EQUATION]', '1904.01211-1-17-12': 'T_0,s_0 T_m,s_0 + _1 .', '1904.01211-1-17-13': '[EQUATION] _f(s_0) = s_0T_0,s_0s_mT_m,s_m + s_0 _1 = _f_m(s_m) + s_0 _1, [EQUATION] _m_f_m(s_m)_f(s_0)_m_f_m(s_m), [EQUATION] _m _0<sf_m__f_m(s) = _0<sf__f(s).', '1904.01211-1-17-14': '[EQUATION] (f_b_m)^LS_b_m(f)(y)LS_b(f)(y)=(f_b)^[EQUATION]', '1904.01211-1-17-15': 'I_f:=_R^ne^-Ax_2+tdx: AA,tR, Ax_2-t(x) [EQUATION]', '1904.01211-1-17-16': 'I_f(b):= _R^ne^-Ax_2+tdx: AA(b),tR, Ax_2-t(x) .', '1904.01211-1-17-17': '[EQUATION] d_0vol_n(G_f(d_0))< f (x) dx I_f(b_0).', '1904.01211-1-17-18': '[EQUATION]', '1904.01211-1-17-19': 'K^=xR^n: vol_n( conv[K,x])+ vol_n(K).', '1904.01211-1-17-20': '[EQUATION] conv [z, G_f(d_0)]G_h(d_0).', '1904.01211-1-17-21': '[EQUATION]e^t_1T_1 =e^t_2T_2 , [EQUATION] e^-_2T_1+T_22x+T_1b_1+T_2b_22_2+t_1+t_22.', '1904.01211-1-17-22': '[EQUATION] e^-T_1+T_22x+T_1b_1+T_2b_22_2+t_1+t_22 dx=n!vol_n(B_2^n) e^t_1+t_22(T_1+T_22).', '1904.01211-1-17-23': '[EQUATION] ((T_1+T_22) )^1n 12 ( (T_1) ^1n + (T_2) ^1n), [EQUATION] (T_1+T_22) > 12 ( T_1 + T_2 ).', '1904.01211-1-17-24': '[EQUATION] e^t_1+t_22(T_1+T_22) < e^t_1T_1, [EQUATION] f_1=e^-T_1(x+b_1)_2+t_1 [EQUATION] f_2=e^-T_2(x+b_2)_2+t_2=e^-T_1(x+b_2)_2+t_1.', '1904.01211-1-17-25': '[EQUATION]', '1904.01211-1-17-26': 'G_f(s)G_f_1(s)G_f_2(s).', '1904.01211-1-17-27': '[EQUATION] x=-b_1+b_22+(t_1-s) T_1^-1( u+v2) -b_1+b_22+(t_1-s) T_1^-1B_2^n. [EQUATION]', '1904.01211-1-17-28': 'G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n-b_1+b_22+(t_1-s) T_1^-1B_2^n .', '1904.01211-1-17-29': '[EQUATION] ( G_f_1(s)G_f_2(s)) ( -b_1+b_22+(t_1-s) T_1^-1B_2^n) ^c=, [EQUATION] dist( G_f_1(s)G_f_2(s), ( -b_1+b_22+(t_1-s) T_1^-1B_2^n) ^c) >0, [EQUATION]', '1904.01211-1-17-30': 'G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n [EQUATION]', '1904.01211-1-17-31': 'G_f(s)G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n [EQUATION]', '1904.01211-1-17-32': 'T_1(x+b_1+b_22)_2-t_1 (x).', '1904.01211-1-17-33': '[EQUATION] e^- T_1(x+b_1+b_22)_2+t_1 dx=n!vol(B_2^n) e^t_1T_1< n!vol(B_2^n) e^t_1T_1, [EQUATION]', '1904.01211-1-17-34': 'J_f(b)= s T: sf_, TS_+, s 1_TB_2^n(x)f(x-b) , [EQUATION] _0^() vol_n(G_f(s))ds<.', '1904.01211-1-17-35': '[EQUATION]', '1904.01211-1-17-36': 'J_f(b)_0^() vol_n(G_f(s))ds<.', '1904.01211-1-17-37': '[EQUATION] _bR^n J_f(b)= _bG_f(()) J_f(b).', '1904.01211-1-17-38': '[EQUATION]', '1904.01211-1-17-39': 'J_f(b_1)= t_1T_1 and J_f(b_2)=t_2T_2.', '1904.01211-1-17-40': '[EQUATION] f(T_1v+b_1)t_1 and f(T_2v+b_2)t_2, vB_2^n. [EQUATION] t_1+ T_1= t_2+T_2 .', '1904.01211-1-17-41': '[EQUATION] t_1t_21_T_1+T_22B_2^n+b_1+b_22.', '1904.01211-1-17-42': '[EQUATION] t_1t_21_T_1+T_22B_2^n+b_1+b_22f.', '1904.01211-1-17-43': '[EQUATION] (L(1_K))^= J((1_K)^).', '1904.01211-1-17-44': '[EQUATION] (L (1_K))^= e^-n 1_n (T_L(K)^t)^-1 B_2^n = e^-n 1_n J(K^).', '1904.01211-1-17-45': '[EQUATION]', '1904.01211-1-17-46': 'I_K(x)= 0 & xK', '1904.01211-1-18-0': '& xK.', '1904.01211-1-19-0': '[EQUATION] e^-h_K(y)s h_K(y) -s y(-s )K^.', '1904.01211-1-19-1': '[EQUATION]', '1904.01211-1-19-2': 'J(-s K^)= -s J(K^) = -s (L(K) )^[EQUATION] t 1_E and (-Tx+a_2+t), [EQUATION] (L(f))^-b_0 = e^- L_-b_0( T_0x +a_0_2 +t_0) = e^-t_0 1_T_0 B_2^n -b_0.', '1904.01211-1-19-3': '[EQUATION] n!vol_n(B_2^n) _bR^n e^tT: TS_+, tR, e^-t1_TB_2^n(y)(f_b)^(y) , [EQUATION] (L(f))^e^-T_0 x_2+t_0)^= e^-t_01_T_0B_2^n=J(f^).', '1904.01211-1-19-4': '[EQUATION]', '1904.01211-1-19-5': 'L(f) = e^- 45x- 38 5 - 12 [EQUATION] (L(f))^ 38 5 = e^-12 1_[-45, 45] + 38 5.', '1904.01211-1-19-6': '[EQUATION] (f)^ 38 5 = e^ 385(x- 38 5) - 116 (x- 38 5)^2 1_(- , 385] + e^ 385(x- 38 5) - 14 (x- 38 5)^2 1_[385, ) .', '1904.01211-1-19-7': '[EQUATION] h(s) = s4 5 ( (9 - 80 s)^12 + (9 - 320 s)^12 ).', '1904.01211-1-19-8': '[EQUATION](L(f))^ 38 5 = e^-12 1_[-45, 45] + 38 5= J((f)^ 38 5), [MATH]h[MATH]s=e^-12[MATH]h[MATH]s=e^-12[MATH]h^(e^-12) >0[MATH]'} | {'1904.01211-2-0-0': 'We introduce the notion of Lowner (ellipsoid) function for a log-concave function and show that it is an extension of the Lowner ellipsoid for convex bodies.', '1904.01211-2-0-1': 'We investigate its duality relation to the recently defined John (ellipsoid) function [CITATION].', '1904.01211-2-0-2': 'For convex bodies, John and Lowner ellipsoids are dual to each other.', '1904.01211-2-0-3': 'Interestingly, this need not be the case for the John function and the Lowner function.', '1904.01211-2-1-0': '# Introduction', '1904.01211-2-2-0': 'Asymptotic convex geometry studies the properties of convex bodies with emphasis on the dependence of geometric and analytic invariants on the dimension.', '1904.01211-2-2-1': 'The convexity assumption enforces concentration of volume in a canonical way and it is a main question if under natural normalizations the answers to fundamental questions are independent of the dimension.', '1904.01211-2-2-2': 'The most classical normalizations of convex bodies arise as solutions of extremal problems.', '1904.01211-2-2-3': 'These normalizations include the isotropic position, which arose from classical mechanics of the 19th century and which is related to a famous open problem in convex geometry, the hyperplane conjecture (see, e.g., the survey [CITATION]).', '1904.01211-2-2-4': 'The best results currently available there are due Bourgain [CITATION] and Klartag [CITATION].', '1904.01211-2-2-5': 'Other positions are the John position, also called maximal volume ellipsoid position and the Lowner position, also called minimal volume ellipsoid position.', '1904.01211-2-2-6': 'The right choice of a position is important for the study of affinely invariant quantities and their related isoperimetric inequalities.', '1904.01211-2-2-7': "For instance, John and Lowner position are related to the Brascamp-Lieb inequality and its reverse [CITATION], to K. Ball's sharp reverse isoperimetric inequality [CITATION], to the notion of volume ratio [CITATION], which is defined as the [MATH]-th root of the volume of a convex body divided by the volume of its John ellipsoid and which finds applications in functional analysis and Banach space theory [CITATION].", '1904.01211-2-2-8': 'John and Lowner position are even relevant in quantum information theory [CITATION].', '1904.01211-2-2-9': 'Since a position may be seen as a choice of a special ellipsoid, and since an ellipsoid entails a Euclidean structure of the underlying space, John and Lowner ellipsoids provide a way to measure how far a normed space is from Euclidean space [CITATION].', '1904.01211-2-2-10': 'For a detailed discussion of the John and the Lowner ellipsoid and its connections to functional analysis we refer the reader to [CITATION] and the survey [CITATION].', '1904.01211-2-2-11': 'F. John proved in [CITATION] that among all ellipsoids contained in a convex body [MATH] , there is a unique ellipsoid of maximal volume, now called the John ellipsoid of [MATH] .', '1904.01211-2-2-12': 'The Lowner ellipsoid of [MATH] is the unique ellipsoid of minimal volume containing [MATH] .', '1904.01211-2-2-13': 'These two notions are closely related by polarity (see, e.g., [CITATION]): A [MATH]-symmetric ellipsoid [MATH] is the ellipsoid of maximal volume inside [MATH] if and only if [MATH] is the ellipsoid of minimal volume outside [MATH], where [MATH] is the polar of [MATH].', '1904.01211-2-2-14': 'Probabilisitic methods have become extremely useful in convex geometry.', '1904.01211-2-2-15': 'In this context, log-concave functions arise naturally from the uniform measure on convex bodies.', '1904.01211-2-2-16': 'A function [MATH] is said to be log-concave, if it is of the form [MATH] where [MATH] is convex.', '1904.01211-2-2-17': 'Extensive research has been devoted within the last ten years to extend the concepts and inequalities from convex bodies to the setting of functions.', '1904.01211-2-2-18': 'In fact, it was observed early that the Prekopa-Leindler inequality (see, e.g., [CITATION]) is the functional analog of the Brunn-Minkowski inequality (see, e.g., [CITATION]) for convex bodies.', '1904.01211-2-2-19': 'Much progress has been made since and functional analogs of many other geometric inequalities were established.', '1904.01211-2-2-20': 'Among them are the functional Blaschke-Santalo inequality [CITATION] and its reverse [CITATION], a functional affine isoperimetric inequality for log-concave functions which can be viewed as an inverse log-Sobolev inequality for entropy [CITATION] and a theory of valuations, an important concept for convex bodies (e.g., [CITATION]), is currently being developed in the functional setting, e.g., [CITATION].', '1904.01211-2-2-21': 'It was only recently that the notion of a John (ellipsoid) function of a log-concave function was established by Alonso-Gutierrez, Merino, Jimenez, and Villa [CITATION].', '1904.01211-2-2-22': 'However, the notion of a Lowner ellipsoid function for log-concave functions has been missing till now.', '1904.01211-2-2-23': 'In this paper we put forward such a notion and we investigate, among other things, its relation to the John ellipsoid function of [CITATION].', '1904.01211-2-2-24': 'Our main result reads as follows.', '1904.01211-2-2-25': 'We denote by [MATH] the set of all invertible affine transformations and by [MATH] denote the Euclidean norm on [MATH].', '1904.01211-2-2-26': 'We say that a function is nondegenerate if [MATH].', '1904.01211-2-3-0': 'Theorem.', '1904.01211-2-3-1': 'Let [MATH] (A_0,t_0) A R [MATH] A_0 [MATH] e^-A_0x_2+t_0 [MATH] f [MATH] L(f) [MATH] 1_K(x) [MATH] KR^n [MATH] L(1_K)1 [MATH] K [MATH] K [MATH](1_K)^[MATH] L(f) [MATH] f [MATH] R^n [MATH] A [MATH] S_+ [MATH] bR^n [MATH] SA(b)=A(b)SA [MATH] A=_bR^n A(b) [MATH] SA=_bR^nSA(b) [MATH] A:R^nR^n [MATH] f:R^nR) is defined as [MATH] .', '1904.01211-2-3-2': 'For [MATH] , let [MATH] be a translation of a function by [MATH] , that is, for a function [MATH] , [EQUATION]', '1904.01211-2-3-3': 'For [MATH] and a function [MATH], we denote by [EQUATION] the super-level sets of [MATH] .', '1904.01211-2-4-0': '## Log-concave functions', '1904.01211-2-5-0': 'A function [MATH] is said to be log-concave if it is of the form [MATH] where [MATH] is a convex function.', '1904.01211-2-5-1': 'We always consider in this paper log-concave functions [MATH] that are integrable and such that [MATH] is nondegenerate, i.e., the interior of the support of [MATH] is non-empty, [MATH].', '1904.01211-2-5-2': 'This then implies that [MATH].', '1904.01211-2-5-3': 'We will also need the Legendre transform which we recall now.', '1904.01211-2-5-4': 'Let [MATH] and let [MATH] be a convex function.', '1904.01211-2-5-5': 'Then [EQUATION] is the Legendre transform of [MATH] with respect to [MATH] [CITATION] .', '1904.01211-2-5-6': 'If [MATH] is log-concave, then [EQUATION] is called the dual or polar function of [MATH] with respect to [MATH].', '1904.01211-2-5-7': 'In particular, when [MATH] , [EQUATION] where [MATH] , also denoted by [MATH][MATH]S_z[MATH] L) and [MATH] are involutions, that is, [MATH] and [MATH].', '1904.01211-2-6-0': '(ii) [MATH].', '1904.01211-2-7-0': '(iii) [MATH].', '1904.01211-2-7-1': '(iv) Legendre transform reverses the oder relation, i.e., if [MATH], then [MATH].', '1904.01211-2-8-0': 'We now list some basic well-known facts on log-concave functions.', '1904.01211-2-8-1': 'A log-concave function is continuous on the interior of its support, e.g., [CITATION].', '1904.01211-2-9-0': "We include a proof of the first fact for the reader's convenience.", '1904.01211-2-9-1': 'More on log-concave functions can be found in e.g., [CITATION].', '1904.01211-2-10-0': 'If [MATH] is a nondegenerate integrable log-concave function, then [MATH] is convex and compact for [MATH] .', '1904.01211-2-11-0': 'Let [MATH] .', '1904.01211-2-11-1': 'As [MATH] is convex and as [MATH] is nondegenerate, the super-level set [EQUATION] is convex and closed for all [MATH].', '1904.01211-2-11-2': 'As [MATH] , it remains to show that [MATH] is bounded for [MATH] .', '1904.01211-2-11-3': 'It follows from Theorem 7.6 of [CITATION] that every super-level set [MATH] , [MATH] , has the same affine dimension as the support of [MATH] , which has affine dimension [MATH] .', '1904.01211-2-11-4': 'Chebyshev inequality then yields [EQUATION].', '1904.01211-2-11-5': 'Since [MATH] is a full dimensional convex set with finite volume, it is bounded.', '1904.01211-2-11-6': 'Therefore, [MATH] is compact for [MATH] .', '1904.01211-2-12-0': 'The following fact is a direct corollary of the functional Blaschke-Santalo inequality [CITATION] and the functional reverse Santalo inequality [CITATION].', '1904.01211-2-13-0': 'Let [MATH] be a nondegenerate, integrable, log-concave function such that [MATH] is in the interior of the support of [MATH].', '1904.01211-2-13-1': 'Then [MATH] is again a nondegenerate, integrable log-concave function and thus [MATH] .', '1904.01211-2-13-2': 'Furthermore, [MATH] is again a nondegenerate, integrable log-concave function, i.e., [MATH] , provided that [MATH] is in the interior of [MATH] .', '1904.01211-2-14-0': '# The Lo wner function of a log-concave function', '1904.01211-2-15-0': 'We now define the Lowner function for an integrable, nondegenerate, log-concave function [MATH] .', '1904.01211-2-16-0': '## A minimization problem.', '1904.01211-2-16-1': 'Definition of the Lowner function', '1904.01211-2-17-0': 'We consider the following minimization problem [EQUATION] subject to [EQUATION] where the minimum is taken over all nonsingular affine maps [MATH] and all [MATH]e^-Ax_2+t[MATH]f[MATH]K[MATH]K[MATH] f: R^nR^+ [MATH] f(x)=e^-(x) [MATH]O(n)[MATH] (A_0,t_0) [MATH]t_0[MATH]A_0[MATH] e^-A_0x_2+t_0 [MATH] f [MATH]K R^n[MATH]L(K)[MATH]K[MATH]T_L(K)[MATH]T_L(K) B^n_2= L(K)[MATH]A A[MATH]t R[MATH](x) = Ax_2-t[MATH]K[MATH]A=T+a[MATH]a=0[MATH]Ax_2-t 0[MATH]x K[MATH]L(K)[MATH]K[MATH] t T^-1 B^n_2 = L(K)[MATH] _(T,t)e^tT[MATH]t_0= n[MATH]T_0=n T_L(K)^-1[MATH] L (1_K)(x) = e^-n (T_L(K)^-1 x_2 -1)[MATH]g(x) = e^-x^2_2/2[MATH]f(x) = e^-(x)[MATH][MATH]x[MATH](x) = (x)[MATH]A A[MATH]A_0= a Id[MATH]a[MATH]t_0[MATH] f=e^- [MATH]A=T+a A[MATH] TGL(n) [MATH] T=O R [MATH] R [MATH] OO(n) [MATH]A S A[MATH]A= T+a[MATH] T [MATH]T S_+[MATH]b=T^-1 a[MATH]T S_+[MATH]b R^n[MATH] bR^n [MATH] f=e^- [MATH] R^n [MATH]h:R^nR[MATH]w[MATH]:R^nR[MATH]_sym,w[MATH]w[MATH],_dia,w[MATH]f_sym,w=e^-_sym,w[MATH]f=e^-[MATH]xR^n[MATH][MATH]-b[MATH] z=Tx[MATH] f_b=S_-bf [MATH] ( f_b)^[MATH]b int ( supp f)[MATH]b int ( supp f)[MATH]f[MATH]f_sym,-b[MATH]b[MATH]b=0 int ( supp f)[MATH]f[MATH]f_sym[MATH]O(n)[MATH] f=e^- [MATH](t_0, T_0) R S_+[MATH] 0<sf_[MATH]_f[MATH]s = e^-t[MATH] s1_TB_2^nfTB_2^nG_f(s) [MATH] s>f_[MATH] T: TB_2^nG_f(s)=[MATH] _s0 _f(s)=0 [MATH] s_f(s) [MATH] f=e^- [MATH] R^n [MATH] s_1,s_2(0,f_] [MATH] _s0 _f(s)=0 [MATH]TS_+:TB_2^nG_f(s)[MATH] T_0, T_1 [MATH] T_2 [MATH] _f(s_1^1-s_2^)=s_1^1-s_2^T_0, _f(s_1)=s_1T_1 [MATH] _f(s_2)=s_2T_2 [MATH]f[MATH] T_0 [(1-)T_1+T_2)] [MATH] T_0(T_1)^1- (T_2)^[MATH]t >0[MATH] _f(t) [MATH] _t0_f(t)=0 [MATH] _f(s)_f(s) [MATH] s [MATH] _s0_f(s)=0 [MATH]K[MATH]L[MATH] K^n [MATH] R^n [MATH] K [MATH] T_K [MATH]T[MATH] 0int(K) [MATH]T S_+: TB_2^nK = [MATH] K [MATH] 0int(K) [MATH] T_K [MATH] T_K= _TS_+: TB_2^nKT [MATH] K=K(-K) [MATH]K (-K)[MATH]K (-K)[MATH] T_KB_2^n [MATH] J(K) [MATH] K=K(-K) [MATH] T_K [MATH]O(n)[MATH] J(K) [MATH] K [MATH]L[MATH] d_H(K,L)<[MATH] d_H(K, L)< 2[MATH]K K^n[MATH]K J(K)[MATH]>0[MATH][MATH]LK^n[MATH] d_H(K, L)< [MATH] d_H(J(K), J(L))< [MATH]L[MATH] d_H(K,L)</2 [MATH] s_f(s) [MATH] s [MATH] s [MATH] sG_f(s) [MATH] K_TS_+: TB_2^nKT [MATH] s [MATH] _s0_f(s)=0 [MATH] _f(s) [MATH] (0,f_] [MATH] _f(s) [MATH] s_0(0,f_] [MATH] T_0S_+ [MATH] t_0=-s_0 [MATH] T_0 [MATH]O(n)[MATH] s [MATH] T [MATH]O(n)[MATH] s_1,s_2 [MATH] s_1>s_2 [MATH] _f(s_1)=_f(s_2) [MATH]_f[MATH] T_0, T_1 [MATH] T_2 [MATH] T_1=T_2 [MATH] f_m,f [MATH] f_mf [MATH] f_m,f [MATH] G_f(k) [MATH] 0<k < f_[MATH] G_f_m(k) [MATH] 0<k < f_m_[MATH] m1 [MATH]k[MATH]G_f_m(k)G_f(k)[MATH](x_m_j)_j N[MATH]x_m_j G_f_m_j(k)[MATH]j[MATH]G_f(k)[MATH]G_f(k)[MATH](x_m)_N[MATH]x_m G_f_m(k)[MATH]m N[MATH](x_m_j)_j N[MATH]x_m_j G_f_m_j(k)[MATH]j[MATH]x=_j x_m_j [MATH]D = co[x_m_j : j N ][MATH]x_m_j : j N [MATH]D[MATH]f_m_j f[MATH]R^n[MATH]f_m_j f[MATH]D[MATH]j[MATH]f[MATH] f_m_j (x_m_j) f(x)[MATH] f_m_j (x_m_j) k[MATH]f(x) k[MATH]x G_f(k)[MATH]0< k < f_[MATH]l= -k[MATH]G_f_m(k)= E__m(l)[MATH]x E_(l)[MATH](x_m)_N[MATH]x_m E_f_m(k)[MATH]m[MATH](x)=l[MATH]f[MATH]x_0[MATH]R^n[MATH](x_0)= _x R^n (x)[MATH]x_0=0[MATH]x[MATH]e_n+1 = (0, , 1)[MATH]k < f_[MATH]l > (x_0) = (0)[MATH]0 < 2 ^12 < (x) - (0)[MATH]f_m f[MATH]_m [MATH]m m_0[MATH]L[MATH](0, (0)+ )[MATH](x, _m(x))[MATH]x_m[MATH]L[MATH]x_m[MATH]l[MATH]_m(x) - ((0) + ) = _m(x) - (0) - > 2 ^12 - 2 [MATH]_m[MATH]y[MATH][0,x][MATH]_m(y) L(y)[MATH] _m(x ) (x) [MATH]m m_0[MATH]x_m [0,x][MATH]x_m E__m(l)[MATH]m_1 m_0[MATH]_m_1(x) < (x) =l[MATH]x E__m_1(l)[MATH]x_m_1 =x[MATH]m > m_1[MATH]_m(x) (x)[MATH]x_m[MATH]_m(x) < (x)[MATH]x_m=x[MATH] f_m,f [MATH] f_mf [MATH] f_m_f_[MATH]f[MATH] x_0R^n [MATH] f(x_0)=f_[MATH] >0 [MATH] m_1 [MATH] m>m_1 [MATH] f_m_f_m(x_0)f(x_0)- [MATH] m>m_1 [MATH] 0 < < 14 f_[MATH] f [MATH] >0 [MATH] m_2 [MATH] m>m_2 [MATH] f_m(x)<12f_ [MATH] xG_f(12f_)+B_2^n [MATH] m>m_2 [MATH] m>m_2 [MATH] f_m(x)f(x) [MATH] G_f(12f_)+B_2^n [MATH] G_f(12f_)+B_2^n [MATH] f_mf [MATH] G_f(12f_)+B_2^n [MATH] ), there exists [MATH] such that [EQUATION] _xR^n f_m(x)=f_m_f_+.', '1904.01211-2-17-1': '[EQUATION] _0<sf_m__f_m(s)_0<sf__f(s).', '1904.01211-2-17-2': '[EQUATION]', '1904.01211-2-17-3': 'T_m,s=_TS_+: TB_2^nG_f_m(s) T. [EQUATION] _f_m(s_m)= _0 < s f_m_ _f_m(s) = s_m T_m, s_m .', '1904.01211-2-17-4': '[EQUATION] 0 < _R^n f(x) dx=_0^f_vol_n(G_f(s))ds<.', '1904.01211-2-17-5': '[EQUATION] 0 < _0^vol_n(G_f(s))ds<.', '1904.01211-2-17-6': '[EQUATION] 0 < _0^vol_n(G_f(s))ds<_0.', '1904.01211-2-17-7': '[EQUATION]', '1904.01211-2-17-8': 'G_f(s) _TS_+: TB_2^nG_f(s)T = T_0,s [EQUATION]', '1904.01211-2-17-9': 'G_f(s) vol_n(G_f(s)) [EQUATION]', '1904.01211-2-17-10': 'G_f_m(s)G_f(s) [EQUATION] vol_n(G_f_m(s)) vol_n(G_f_m(s_m)) T_m,s_m vol_n(B^n_2) .', '1904.01211-2-17-11': '[EQUATION] vol_n(G_f(f_-_1) _R^n f(s) dsf_-_1 10 _R^n f(s) ds9 f_, [EQUATION]', '1904.01211-2-17-12': 'T_0,s_0 T_m,s_0 + _1 .', '1904.01211-2-17-13': '[EQUATION] _f(s_0) = s_0T_0,s_0s_mT_m,s_m + s_0 _1 = _f_m(s_m) + s_0 _1, [EQUATION] _m_f_m(s_m)_f(s_0)_m_f_m(s_m), [EQUATION] _m _0<sf_m__f_m(s) = _0<sf__f(s).', '1904.01211-2-17-14': '[EQUATION] xR^n: Tx_2-t(x-b_m) [EQUATION] yR^n: T(y+b_m)_2-t(y).', '1904.01211-2-17-15': '[EQUATION]', '1904.01211-2-17-16': 'B_m =(T,t):T(y+b_m)_2-t(y)yR^n [EQUATION]', '1904.01211-2-17-17': 'B_ =(T,t):T(y+b_)_2-t(y)yR^n .', '1904.01211-2-17-18': '[EQUATION]', '1904.01211-2-17-19': 'T(y+b_m)_2T(y+b)_2-T(b-b_m)_2 [EQUATION]', '1904.01211-2-17-20': 'I_f:=_R^ne^-Ax_2+tdx: AA,tR, Ax_2-t(x) [EQUATION]', '1904.01211-2-17-21': 'I_f(b):= _R^ne^-Ax_2+tdx: AA(b),tR, Ax_2-t(x) .', '1904.01211-2-17-22': '[EQUATION] d_0vol_n(G_f(d_0))< _R^n f (x) dx I_f(b_0).', '1904.01211-2-17-23': '[EQUATION]', '1904.01211-2-17-24': 'K^=xR^n: vol_n( conv[K,x])+ vol_n(K).', '1904.01211-2-17-25': '[EQUATION] conv [z, G_f(d_0)]G_h(d_0).', '1904.01211-2-17-26': '[EQUATION]e^t_1T_1 =e^t_2T_2 , [EQUATION] e^-T_1+T_22x+T_1b_1+T_2b_22_2+t_1+t_22.', '1904.01211-2-17-27': '[EQUATION] _R^n e^-T_1+T_22x+T_1b_1+T_2b_22_2+t_1+t_22 dx=n!vol_n(B_2^n) e^t_1+t_22(T_1+T_22).', '1904.01211-2-17-28': '[EQUATION] ((T_1+T_22) )^1n 12 ( (T_1) ^1n + (T_2) ^1n), [EQUATION] (T_1+T_22) > 12 ( T_1 + T_2 ).', '1904.01211-2-17-29': '[EQUATION] e^t_1+t_22(T_1+T_22) < e^t_1T_1, [EQUATION] f_1(x)=e^-T_1(x+b_1)_2+t_1 [EQUATION] f_2(x)=e^-T_2(x+b_2)_2+t_2=e^-T_1(x+b_2)_2+t_1.', '1904.01211-2-17-30': '[EQUATION]', '1904.01211-2-17-31': 'G_f(s)G_f_1(s)G_f_2(s).', '1904.01211-2-17-32': '[EQUATION] x=-b_1+b_22+(t_1-s) T_1^-1( u+v2) -b_1+b_22+(t_1-s) T_1^-1B_2^n. [EQUATION]', '1904.01211-2-17-33': 'G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n-b_1+b_22+(t_1-s) T_1^-1B_2^n .', '1904.01211-2-17-34': '[EQUATION] ( G_f_1(s)G_f_2(s)) ( -b_1+b_22+(t_1-s) T_1^-1B_2^n) ^c=, [EQUATION] dist( G_f_1(s)G_f_2(s), ( -b_1+b_22+(t_1-s) T_1^-1B_2^n) ^c) >0, [EQUATION]', '1904.01211-2-17-35': 'G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n [EQUATION]', '1904.01211-2-17-36': 'G_f(s)G_f_1(s)G_f_2(s)-b_1+b_22+(t_1-s) T_1^-1B_2^n [EQUATION]', '1904.01211-2-17-37': 'T_1(x+b_1+b_22)_2-t_1 (x).', '1904.01211-2-17-38': '[EQUATION] _R^n e^- T_1(x+b_1+b_22)_2+t_1 dx=n!vol(B_2^n) e^t_1T_1< n!vol(B_2^n) e^t_1T_1, [EQUATION]', '1904.01211-2-17-39': 'J_f(b)= s T: sf_, TS_+, s 1_TB_2^n(x)f(x-b) x R^n , [EQUATION] _0^() vol_n(G_f(s))ds<.', '1904.01211-2-17-40': '[EQUATION]', '1904.01211-2-17-41': 'J_f(b)_0^() vol_n(G_f(s))ds<.', '1904.01211-2-17-42': '[EQUATION] _bR^n J_f(b)= _bG_f(()) J_f(b).', '1904.01211-2-17-43': '[EQUATION]', '1904.01211-2-17-44': 'J_f(b_1)= t_1T_1 and J_f(b_2)=t_2T_2.', '1904.01211-2-17-45': '[EQUATION] f(T_1v+b_1)t_1 and f(T_2v+b_2)t_2, vB_2^n. [EQUATION] t_1+ T_1= t_2+T_2 .', '1904.01211-2-17-46': '[EQUATION] t_1t_21_T_1+T_22B_2^n+b_1+b_22.', '1904.01211-2-17-47': '[EQUATION] t_1t_21_T_1+T_22B_2^n+b_1+b_22f.', '1904.01211-2-17-48': '[EQUATION] (L(1_K))^= J((1_K)^).', '1904.01211-2-17-49': '[EQUATION] (L (1_K))^= e^-n 1_n (T_L(K)^t)^-1 B_2^n = e^-n 1_n J(K^).', '1904.01211-2-17-50': '[EQUATION]', '1904.01211-2-17-51': 'I_K(x)= 0 & xK', '1904.01211-2-18-0': '& xK.', '1904.01211-2-19-0': '[EQUATION] e^-h_K(y)s h_K(y) -s y(-s )K^.', '1904.01211-2-19-1': '[EQUATION]', '1904.01211-2-19-2': 'J(-s K^)= -s J(K^) = -s (L(K) )^[EQUATION] t 1_E and (-Tx+a_2+t), [EQUATION] (L(f))^-b_0 = e^- L_-b_0( T_0x +a_0_2 +t_0) = e^-t_0 1_T_0 B_2^n -b_0.', '1904.01211-2-19-3': '[EQUATION] n!vol_n(B_2^n) _bR^n e^tT: TS_+, tR, e^-t1_TB_2^n(y)(f_b)^(y) , [EQUATION] (L(f))^e^-T_0 x_2+t_0)^= e^-t_01_T_0B_2^n=J(f^).', '1904.01211-2-19-4': '[EQUATION]', '1904.01211-2-19-5': 'L(f) = e^- 45x- 38 5 + 12 [EQUATION] (L(f))^ 38 5 = e^-12 1_[-45, 45] + 38 5.', '1904.01211-2-19-6': '[EQUATION] (f)^ 38 5 = e^ 385(x- 38 5) - 116 (x- 38 5)^2 1_(- , 385] + e^ 385(x- 38 5) - 14 (x- 38 5)^2 1_(385, ) .', '1904.01211-2-19-7': '[EQUATION](L(f))^ 38 5 = e^-12 1_[-45, 45] + 38 5= J((f)^ 38 5), [MATH]h[MATH]s=e^-12 [MATH]h[MATH]s=e^-12[MATH]h^(e^-12)-0.3538<0[MATH]'} | [['1904.01211-1-9-0', '1904.01211-2-9-0'], ['1904.01211-1-9-1', '1904.01211-2-9-1'], ['1904.01211-1-3-2', '1904.01211-2-3-2'], ['1904.01211-1-10-0', '1904.01211-2-10-0'], ['1904.01211-1-5-0', '1904.01211-2-5-0'], ['1904.01211-1-5-1', '1904.01211-2-5-1'], ['1904.01211-1-5-2', '1904.01211-2-5-2'], ['1904.01211-1-5-3', '1904.01211-2-5-3'], ['1904.01211-1-5-4', '1904.01211-2-5-4'], ['1904.01211-1-5-6', '1904.01211-2-5-6'], ['1904.01211-1-5-7', '1904.01211-2-5-7'], ['1904.01211-1-2-0', '1904.01211-2-2-0'], ['1904.01211-1-2-1', '1904.01211-2-2-1'], ['1904.01211-1-2-2', '1904.01211-2-2-2'], ['1904.01211-1-2-3', '1904.01211-2-2-3'], ['1904.01211-1-2-4', '1904.01211-2-2-4'], ['1904.01211-1-2-5', 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'1904.01211-2-15-0'], ['1904.01211-1-11-2', '1904.01211-2-11-2'], ['1904.01211-1-11-4', '1904.01211-2-11-4'], ['1904.01211-1-11-5', '1904.01211-2-11-5'], ['1904.01211-1-11-6', '1904.01211-2-11-6'], ['1904.01211-1-3-3', '1904.01211-2-3-3'], ['1904.01211-1-13-0', '1904.01211-2-13-1'], ['1904.01211-1-12-0', '1904.01211-2-12-0'], ['1904.01211-1-2-16', '1904.01211-2-2-16'], ['1904.01211-1-2-22', '1904.01211-2-2-22'], ['1904.01211-1-0-0', '1904.01211-2-0-0'], ['1904.01211-1-8-0', '1904.01211-2-8-0'], ['1904.01211-1-8-1', '1904.01211-2-8-1'], ['1904.01211-1-11-1', '1904.01211-2-11-1'], ['1904.01211-1-11-3', '1904.01211-2-11-3'], ['1904.01211-1-5-5', '1904.01211-2-5-5']] | [['1904.01211-1-9-0', '1904.01211-2-9-0'], ['1904.01211-1-9-1', '1904.01211-2-9-1'], ['1904.01211-1-3-2', '1904.01211-2-3-2'], ['1904.01211-1-10-0', '1904.01211-2-10-0'], ['1904.01211-1-5-0', '1904.01211-2-5-0'], ['1904.01211-1-5-1', '1904.01211-2-5-1'], ['1904.01211-1-5-2', '1904.01211-2-5-2'], ['1904.01211-1-5-3', '1904.01211-2-5-3'], ['1904.01211-1-5-4', '1904.01211-2-5-4'], ['1904.01211-1-5-6', '1904.01211-2-5-6'], ['1904.01211-1-5-7', '1904.01211-2-5-7'], ['1904.01211-1-2-0', '1904.01211-2-2-0'], ['1904.01211-1-2-1', '1904.01211-2-2-1'], ['1904.01211-1-2-2', '1904.01211-2-2-2'], ['1904.01211-1-2-3', '1904.01211-2-2-3'], ['1904.01211-1-2-4', '1904.01211-2-2-4'], ['1904.01211-1-2-5', '1904.01211-2-2-5'], ['1904.01211-1-2-6', '1904.01211-2-2-6'], ['1904.01211-1-2-7', '1904.01211-2-2-7'], ['1904.01211-1-2-8', '1904.01211-2-2-8'], ['1904.01211-1-2-9', '1904.01211-2-2-9'], ['1904.01211-1-2-10', '1904.01211-2-2-10'], ['1904.01211-1-2-11', '1904.01211-2-2-11'], ['1904.01211-1-2-12', '1904.01211-2-2-12'], ['1904.01211-1-2-13', '1904.01211-2-2-13'], ['1904.01211-1-2-14', '1904.01211-2-2-14'], ['1904.01211-1-2-15', '1904.01211-2-2-15'], ['1904.01211-1-2-17', '1904.01211-2-2-17'], ['1904.01211-1-2-18', '1904.01211-2-2-18'], ['1904.01211-1-2-19', '1904.01211-2-2-19'], ['1904.01211-1-2-20', '1904.01211-2-2-20'], ['1904.01211-1-2-21', '1904.01211-2-2-21'], ['1904.01211-1-2-23', '1904.01211-2-2-23'], ['1904.01211-1-2-24', '1904.01211-2-2-24'], ['1904.01211-1-2-25', '1904.01211-2-2-25'], ['1904.01211-1-2-26', '1904.01211-2-2-26'], ['1904.01211-1-0-1', '1904.01211-2-0-1'], ['1904.01211-1-0-2', '1904.01211-2-0-2'], ['1904.01211-1-0-3', '1904.01211-2-0-3'], ['1904.01211-1-15-0', '1904.01211-2-15-0'], ['1904.01211-1-11-2', '1904.01211-2-11-2'], ['1904.01211-1-11-4', '1904.01211-2-11-4'], ['1904.01211-1-11-5', '1904.01211-2-11-5'], ['1904.01211-1-11-6', '1904.01211-2-11-6']] | [['1904.01211-1-3-3', '1904.01211-2-3-3'], ['1904.01211-1-13-0', '1904.01211-2-13-1'], ['1904.01211-1-12-0', '1904.01211-2-12-0'], ['1904.01211-1-2-16', '1904.01211-2-2-16'], ['1904.01211-1-2-22', '1904.01211-2-2-22'], ['1904.01211-1-0-0', '1904.01211-2-0-0'], ['1904.01211-1-8-0', '1904.01211-2-8-0'], ['1904.01211-1-8-1', '1904.01211-2-8-1'], ['1904.01211-1-11-1', '1904.01211-2-11-1'], ['1904.01211-1-11-3', '1904.01211-2-11-3']] | [] | [['1904.01211-1-5-5', '1904.01211-2-5-5']] | [] | ['1904.01211-1-3-0', '1904.01211-1-3-1', '1904.01211-1-6-0', '1904.01211-1-7-0', '1904.01211-1-11-0', '1904.01211-1-16-0', '1904.01211-1-16-1', '1904.01211-1-17-0', '1904.01211-1-17-1', '1904.01211-1-17-2', '1904.01211-1-17-3', '1904.01211-1-17-4', '1904.01211-1-17-5', '1904.01211-1-17-6', '1904.01211-1-17-7', '1904.01211-1-17-8', '1904.01211-1-17-9', '1904.01211-1-17-10', '1904.01211-1-17-11', '1904.01211-1-17-12', '1904.01211-1-17-13', '1904.01211-1-17-14', '1904.01211-1-17-15', '1904.01211-1-17-16', '1904.01211-1-17-17', '1904.01211-1-17-18', '1904.01211-1-17-19', '1904.01211-1-17-20', '1904.01211-1-17-21', '1904.01211-1-17-22', '1904.01211-1-17-23', '1904.01211-1-17-24', '1904.01211-1-17-25', '1904.01211-1-17-26', '1904.01211-1-17-27', '1904.01211-1-17-28', '1904.01211-1-17-29', '1904.01211-1-17-30', '1904.01211-1-17-31', '1904.01211-1-17-32', '1904.01211-1-17-33', '1904.01211-1-17-34', '1904.01211-1-17-35', '1904.01211-1-17-36', '1904.01211-1-17-37', '1904.01211-1-17-38', '1904.01211-1-17-39', '1904.01211-1-17-40', '1904.01211-1-17-41', '1904.01211-1-17-42', '1904.01211-1-17-43', '1904.01211-1-17-44', '1904.01211-1-17-45', '1904.01211-1-17-46', '1904.01211-1-18-0', '1904.01211-1-19-0', '1904.01211-1-19-1', '1904.01211-1-19-2', '1904.01211-1-19-3', '1904.01211-1-19-4', '1904.01211-1-19-5', '1904.01211-1-19-6', '1904.01211-1-19-7', '1904.01211-1-19-8', '1904.01211-2-3-0', '1904.01211-2-3-1', '1904.01211-2-6-0', '1904.01211-2-7-0', '1904.01211-2-11-0', '1904.01211-2-16-0', '1904.01211-2-16-1', '1904.01211-2-17-0', '1904.01211-2-17-1', '1904.01211-2-17-2', '1904.01211-2-17-3', '1904.01211-2-17-4', '1904.01211-2-17-5', '1904.01211-2-17-6', '1904.01211-2-17-7', '1904.01211-2-17-8', '1904.01211-2-17-9', '1904.01211-2-17-10', '1904.01211-2-17-11', '1904.01211-2-17-12', '1904.01211-2-17-13', '1904.01211-2-17-14', '1904.01211-2-17-15', '1904.01211-2-17-16', '1904.01211-2-17-17', '1904.01211-2-17-18', '1904.01211-2-17-19', '1904.01211-2-17-20', '1904.01211-2-17-21', '1904.01211-2-17-22', '1904.01211-2-17-23', '1904.01211-2-17-24', '1904.01211-2-17-25', '1904.01211-2-17-26', '1904.01211-2-17-27', '1904.01211-2-17-28', '1904.01211-2-17-29', '1904.01211-2-17-30', '1904.01211-2-17-31', '1904.01211-2-17-32', '1904.01211-2-17-33', '1904.01211-2-17-34', '1904.01211-2-17-35', '1904.01211-2-17-36', '1904.01211-2-17-37', '1904.01211-2-17-38', '1904.01211-2-17-39', '1904.01211-2-17-40', '1904.01211-2-17-41', '1904.01211-2-17-42', '1904.01211-2-17-43', '1904.01211-2-17-44', '1904.01211-2-17-45', '1904.01211-2-17-46', '1904.01211-2-17-47', '1904.01211-2-17-48', '1904.01211-2-17-49', '1904.01211-2-17-50', '1904.01211-2-17-51', '1904.01211-2-18-0', '1904.01211-2-19-0', '1904.01211-2-19-1', '1904.01211-2-19-2', '1904.01211-2-19-3', '1904.01211-2-19-4', '1904.01211-2-19-5', '1904.01211-2-19-6', '1904.01211-2-19-7'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1904.01211 | null | null | null | null | null |
1709.04194 | {'1709.04194-1-0-0': 'We consider the weighted Radon transforms [MATH] along hyperplanes in [MATH], with strictly positive weights [MATH].', '1709.04194-1-0-1': 'We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions.', '1709.04194-1-0-2': 'In addition, the related weight [MATH] is infinitely smooth almost everywhere and is bounded.', '1709.04194-1-0-3': 'Our construction is based on the famous example of non-uniqueness of J. Boman (1993) for the weighted Radon transforms in [MATH] and on a recent result of F. Goncharov and R. Novikov (2016).', '1709.04194-1-1-0': 'Keywords: weighted Radon transforms, injectivity, non-injectivity', '1709.04194-1-2-0': 'AMS Mathematics Subject Classification: 44A12, 65R32', '1709.04194-1-3-0': '# Introduction', '1709.04194-1-4-0': 'We consider the weighted Radon transforms [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test function on [MATH].', '1709.04194-1-4-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION] where [MATH] denotes the complex conjugate of [MATH], [MATH] is a constant.', '1709.04194-1-4-2': 'If [MATH], then [MATH] is reduced to the classical Radon transform [MATH] along hyperplanes in [MATH].', '1709.04194-1-4-3': 'This transform is invertible by the classical Radon inversion formulas; see [CITATION].', '1709.04194-1-4-4': 'If [MATH] is strictly positive, [MATH] and [MATH], then in [CITATION] the inversion of [MATH] is reduced to solving a Fredholm type linear integral equation.', '1709.04194-1-4-5': 'Besides, in [CITATION] it was proved that [MATH] is injective (for example, in [MATH]) if [MATH] is real-analytic and strictly positive.', '1709.04194-1-4-6': 'In addition, an example of [MATH] in [MATH] with infinitely smooth strictly positive [MATH] and with non-trivial kernel [MATH] in [MATH] was constructed in [CITATION].', '1709.04194-1-4-7': 'Here [MATH] denote the spaces of functions from [MATH] with compact support, respectively.', '1709.04194-1-4-8': 'In connection with the most recent progress in inversion methods for weighted Radon transforms [MATH], see [CITATION].', '1709.04194-1-4-9': 'We recall also that inversion methods for [MATH] in [MATH] admit applications in the framework of emission tomographies (see [CITATION]).', '1709.04194-1-5-0': 'In the present work we construct an example of [MATH] in [MATH], with non-trivial kernel [MATH] in [MATH].', '1709.04194-1-5-1': 'The related [MATH] satisfies [REF].', '1709.04194-1-5-2': 'In addition, our weight [MATH] is infinitely smooth almost everywhere on [MATH].', '1709.04194-1-5-3': 'In our construction we proceed from results of [CITATION] and [CITATION].', '1709.04194-1-6-0': 'In Section [REF], in particular, we recall the result of [CITATION].', '1709.04194-1-6-1': 'In Section [REF] we recall the result of [CITATION].', '1709.04194-1-6-2': 'In Section [REF] we obtain the main result of the present work.', '1709.04194-1-7-0': '# Relations between the Radon and the ray transforms', '1709.04194-1-8-0': 'We consider also the weighted ray transforms [MATH] in [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test-function on [MATH].', '1709.04194-1-8-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION]', '1709.04194-1-8-2': 'We recall that [MATH] can be interpreted as the set of all oriented rays in [MATH].', '1709.04194-1-8-3': 'In particular, if [MATH], then [EQUATION] where [MATH] gives the orientation of [MATH].', '1709.04194-1-9-0': 'We recall that for [MATH], transforms [MATH] and [MATH] are equivalent up to the following change of variables: [EQUATION] where [MATH] is a test-function on [MATH].', '1709.04194-1-9-1': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas (see [CITATION]): [EQUATION] where [MATH] is some fixed vector from [MATH], [MATH] denotes the standard vector product in [MATH], [MATH] denotes the orthogonality of vectors.', '1709.04194-1-9-2': 'Actually, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH], where [MATH] and [MATH] are related by [REF].', '1709.04194-1-10-0': 'Below we present analogs of [REF]-[REF] for [MATH].', '1709.04194-1-10-1': 'Let [EQUATION]', '1709.04194-1-10-2': 'If [MATH] is not specified otherwise, it is assumed that [MATH] is the standard basis in [MATH].', '1709.04194-1-11-0': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas: [EQUATION] where [MATH], are defined as follows: [EQUATION] and [MATH] are given by [REF], [REF], respectively.', '1709.04194-1-11-1': 'Here, due to the condition [MATH]: [EQUATION]', '1709.04194-1-11-2': 'Formula [REF] is proved in Section [REF].', '1709.04194-1-11-3': 'Note that formulas [REF]-[REF] are also valid for [MATH].', '1709.04194-1-11-4': 'In this case these formulas are reduced to [REF]-[REF], where [MATH].', '1709.04194-1-11-5': 'Note that, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH].', '1709.04194-1-12-0': 'Remark 1.', '1709.04194-1-12-1': 'In [REF] one can also write: [EQUATION] where [MATH]-denotes the Hodge star, [MATH] - is the exterior product in [MATH] (exterior algebra on [MATH]); see, for example, Chapters 2.1.c, 4.1.c of [CITATION].', '1709.04194-1-13-0': 'Note that the value of the integral in the right hand-side of [REF] does not depend on the particular choice of [MATH] of [REF].', '1709.04194-1-13-1': 'Note also that, due to [REF], [REF], [REF], [REF], the weight [MATH] is defined everywhere on [MATH].', '1709.04194-1-13-2': 'In addition, this [MATH] has the same smoothness as [MATH] in [MATH] on [MATH] and in [MATH] on [MATH], where [MATH] is defined in [REF] and has zero Lebesgue measure on [MATH].', '1709.04194-1-14-0': "# Boman's example", '1709.04194-1-15-0': 'For [MATH], in [CITATION] there were constructed a weight [MATH] and a function [MATH], such that: [EQUATION]', '1709.04194-1-15-1': 'In addition, as a corollary of [REF], [REF], [REF]-[REF], we have that [EQUATION] where [EQUATION]', '1709.04194-1-16-0': '# Main results', '1709.04194-1-17-0': 'Let [EQUATION]', '1709.04194-1-17-1': 'Remark 2.', '1709.04194-1-17-2': 'According to [REF], [REF], [MATH] for [MATH] can be specified as follows: [EQUATION] [Proof of Theorem [REF]] We define [EQUATION] where [EQUATION]', '1709.04194-1-17-3': 'From [REF], [REF], [REF]-[REF] it follows that: [EQUATION]', '1709.04194-1-17-4': 'Properties [REF]-[REF] follow from [REF]-[REF], [REF], [REF], [REF], [REF], [REF].', '1709.04194-1-17-5': 'Theorem [REF] is proved.', '1709.04194-1-18-0': '# Proof of formula [REF]', '1709.04194-1-19-0': 'Note that [EQUATION] which implies that the intersection [MATH] is one of the following:', '1709.04194-1-20-0': 'The intersection is the one dimensional line [MATH]: [EQUATION] where [MATH] is an arbitrary point of [MATH], the orientation of [MATH] is chosen such that: [EQUATION]', '1709.04194-1-20-1': 'Condition [REF] fixes uniquely the direction of [MATH] of [REF].', '1709.04194-1-20-2': 'Formulas [REF], [REF], [REF] imply that [REF] can hold if and only if [MATH].', '1709.04194-1-20-3': 'The intersection is the two-dimensional plane [MATH].', '1709.04194-1-20-4': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-1-20-5': 'The intersection is an empty set.', '1709.04194-1-20-6': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-1-20-7': 'Note that [EQUATION]', '1709.04194-1-20-8': 'This completes the proof of formula [REF].'} | {'1709.04194-2-0-0': 'We consider the weighted Radon transforms [MATH] along hyperplanes in [MATH], with strictly positive weights [MATH].', '1709.04194-2-0-1': 'We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions.', '1709.04194-2-0-2': 'In addition, the related weight [MATH] is infinitely smooth almost everywhere and is bounded.', '1709.04194-2-0-3': 'Our construction is based on the famous example of non-uniqueness of J. Boman (1993) for the weighted Radon transforms in [MATH] and on a recent result of F. Goncharov and R. Novikov (2016).', '1709.04194-2-1-0': 'Keywords: weighted Radon transforms, injectivity, non-injectivity', '1709.04194-2-2-0': 'AMS Mathematics Subject Classification: 44A12, 65R32', '1709.04194-2-3-0': '# Introduction', '1709.04194-2-4-0': 'We consider the weighted Radon transforms [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test function on [MATH].', '1709.04194-2-4-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION] where [MATH] denotes the complex conjugate of [MATH], [MATH] is a constant.', '1709.04194-2-4-2': 'If [MATH], then [MATH] is reduced to the classical Radon transform [MATH] along hyperplanes in [MATH].', '1709.04194-2-4-3': 'This transform is invertible by the classical Radon inversion formulas; see [CITATION].', '1709.04194-2-4-4': 'If [MATH] is strictly positive, [MATH] and [MATH], then in [CITATION] the inversion of [MATH] is reduced to solving a Fredholm type linear integral equation.', '1709.04194-2-4-5': 'Besides, in [CITATION] it was proved that [MATH] is injective (for example, in [MATH]) if [MATH] is real-analytic and strictly positive.', '1709.04194-2-4-6': 'In addition, an example of [MATH] in [MATH] with infinitely smooth strictly positive [MATH] and with non-trivial kernel [MATH] in [MATH] was constructed in [CITATION].', '1709.04194-2-4-7': 'Here [MATH] denote the spaces of functions from [MATH] with compact support, respectively.', '1709.04194-2-4-8': 'In connection with the most recent progress in inversion methods for weighted Radon transforms [MATH], see [CITATION].', '1709.04194-2-4-9': 'We recall also that inversion methods for [MATH] in [MATH] admit applications in the framework of emission tomographies (see [CITATION]).', '1709.04194-2-5-0': 'In the present work we construct an example of [MATH] in [MATH], with non-trivial kernel [MATH] in [MATH].', '1709.04194-2-5-1': 'The related [MATH] satisfies [REF].', '1709.04194-2-5-2': 'In addition, our weight [MATH] is infinitely smooth almost everywhere on [MATH].', '1709.04194-2-5-3': 'In our construction we proceed from results of [CITATION] and [CITATION].', '1709.04194-2-6-0': 'In Section [REF], in particular, we recall the result of [CITATION].', '1709.04194-2-6-1': 'In Section [REF] we recall the result of [CITATION].', '1709.04194-2-6-2': 'In Section [REF] we obtain the main result of the present work.', '1709.04194-2-7-0': '# Relations between the Radon and the ray transforms', '1709.04194-2-8-0': 'We consider also the weighted ray transforms [MATH] in [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test-function on [MATH].', '1709.04194-2-8-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION]', '1709.04194-2-8-2': 'We recall that [MATH] can be interpreted as the set of all oriented rays in [MATH].', '1709.04194-2-8-3': 'In particular, if [MATH], then [EQUATION] where [MATH] gives the orientation of [MATH].', '1709.04194-2-9-0': 'We recall that for [MATH], transforms [MATH] and [MATH] are equivalent up to the following change of variables: [EQUATION] where [MATH] is a test-function on [MATH].', '1709.04194-2-9-1': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas (see [CITATION]): [EQUATION] where [MATH] is some fixed vector from [MATH], [MATH] denotes the standard vector product in [MATH], [MATH] denotes the orthogonality of vectors.', '1709.04194-2-9-2': 'Actually, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH], where [MATH] and [MATH] are related by [REF].', '1709.04194-2-10-0': 'Below we present analogs of [REF]-[REF] for [MATH].', '1709.04194-2-10-1': 'Let [EQUATION]', '1709.04194-2-10-2': 'If [MATH] is not specified otherwise, it is assumed that [MATH] is the standard basis in [MATH].', '1709.04194-2-11-0': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas: [EQUATION] where [MATH], are defined as follows: [EQUATION] and [MATH] are given by [REF], [REF], respectively.', '1709.04194-2-11-1': 'Here, due to the condition [MATH]: [EQUATION]', '1709.04194-2-11-2': 'Formula [REF] is proved in Section [REF].', '1709.04194-2-11-3': 'Note that formulas [REF]-[REF] are also valid for [MATH].', '1709.04194-2-11-4': 'In this case these formulas are reduced to [REF]-[REF], where [MATH].', '1709.04194-2-11-5': 'Note that, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH].', '1709.04194-2-12-0': 'Remark 1.', '1709.04194-2-12-1': 'In [REF] one can also write: [EQUATION] where [MATH]-denotes the Hodge star, [MATH] - is the exterior product in [MATH] (exterior algebra on [MATH]); see, for example, Chapters 2.1.c, 4.1.c of [CITATION].', '1709.04194-2-13-0': 'Note that the value of the integral in the right hand-side of [REF] does not depend on the particular choice of [MATH] of [REF].', '1709.04194-2-13-1': 'Note also that, due to [REF], [REF], [REF], [REF], the weight [MATH] is defined everywhere on [MATH].', '1709.04194-2-13-2': 'In addition, this [MATH] has the same smoothness as [MATH] in [MATH] on [MATH] and in [MATH] on [MATH], where [MATH] is defined in [REF] and has zero Lebesgue measure on [MATH].', '1709.04194-2-14-0': "# Boman's example", '1709.04194-2-15-0': 'For [MATH], in [CITATION] there were constructed a weight [MATH] and a function [MATH], such that: [EQUATION]', '1709.04194-2-15-1': 'In addition, as a corollary of [REF], [REF], [REF]-[REF], we have that [EQUATION] where [EQUATION]', '1709.04194-2-16-0': '# Main results', '1709.04194-2-17-0': 'Let [EQUATION]', '1709.04194-2-17-1': 'Remark 2.', '1709.04194-2-17-2': 'According to [REF], [REF], [MATH] for [MATH] can be specified as follows: [EQUATION] [Proof of Theorem [REF]] We define [EQUATION] where [EQUATION]', '1709.04194-2-17-3': 'From [REF], [REF], [REF]-[REF] it follows that: [EQUATION]', '1709.04194-2-17-4': 'Properties [REF]-[REF] follow from [REF]-[REF], [REF], [REF], [REF], [REF], [REF].', '1709.04194-2-17-5': 'Theorem [REF] is proved.', '1709.04194-2-18-0': '# Proof of formula [REF]', '1709.04194-2-19-0': 'Note that [EQUATION] which implies that the intersection [MATH] is one of the following:', '1709.04194-2-20-0': 'The intersection is the one dimensional line [MATH]: [EQUATION] where [MATH] is an arbitrary point of [MATH], the orientation of [MATH] is chosen such that: [EQUATION]', '1709.04194-2-20-1': 'Condition [REF] fixes uniquely the direction of [MATH] of [REF].', '1709.04194-2-20-2': 'Formulas [REF], [REF], [REF] imply that [REF] can hold if and only if [MATH].', '1709.04194-2-20-3': 'The intersection is the two-dimensional plane [MATH].', '1709.04194-2-20-4': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-2-20-5': 'The intersection is an empty set.', '1709.04194-2-20-6': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-2-20-7': 'Note that [EQUATION]', '1709.04194-2-20-8': 'This completes the proof of formula [REF].'} | [['1709.04194-1-4-0', '1709.04194-2-4-0'], ['1709.04194-1-4-1', '1709.04194-2-4-1'], ['1709.04194-1-4-2', '1709.04194-2-4-2'], ['1709.04194-1-4-3', '1709.04194-2-4-3'], ['1709.04194-1-4-4', '1709.04194-2-4-4'], ['1709.04194-1-4-5', '1709.04194-2-4-5'], ['1709.04194-1-4-6', '1709.04194-2-4-6'], ['1709.04194-1-4-7', '1709.04194-2-4-7'], ['1709.04194-1-4-8', '1709.04194-2-4-8'], ['1709.04194-1-4-9', '1709.04194-2-4-9'], ['1709.04194-1-5-0', '1709.04194-2-5-0'], ['1709.04194-1-5-1', '1709.04194-2-5-1'], ['1709.04194-1-5-2', '1709.04194-2-5-2'], ['1709.04194-1-5-3', '1709.04194-2-5-3'], ['1709.04194-1-0-0', '1709.04194-2-0-0'], ['1709.04194-1-0-1', '1709.04194-2-0-1'], ['1709.04194-1-0-2', '1709.04194-2-0-2'], ['1709.04194-1-0-3', '1709.04194-2-0-3'], ['1709.04194-1-13-0', '1709.04194-2-13-0'], ['1709.04194-1-13-1', '1709.04194-2-13-1'], ['1709.04194-1-13-2', '1709.04194-2-13-2'], ['1709.04194-1-9-0', '1709.04194-2-9-0'], ['1709.04194-1-9-1', '1709.04194-2-9-1'], ['1709.04194-1-9-2', '1709.04194-2-9-2'], ['1709.04194-1-12-1', '1709.04194-2-12-1'], ['1709.04194-1-8-0', '1709.04194-2-8-0'], ['1709.04194-1-8-1', '1709.04194-2-8-1'], ['1709.04194-1-8-2', '1709.04194-2-8-2'], ['1709.04194-1-8-3', '1709.04194-2-8-3'], ['1709.04194-1-20-0', '1709.04194-2-20-0'], ['1709.04194-1-20-1', '1709.04194-2-20-1'], ['1709.04194-1-20-2', '1709.04194-2-20-2'], ['1709.04194-1-20-3', '1709.04194-2-20-3'], ['1709.04194-1-20-4', '1709.04194-2-20-4'], ['1709.04194-1-20-5', '1709.04194-2-20-5'], ['1709.04194-1-20-6', '1709.04194-2-20-6'], ['1709.04194-1-20-8', '1709.04194-2-20-8'], ['1709.04194-1-11-0', '1709.04194-2-11-0'], ['1709.04194-1-11-1', '1709.04194-2-11-1'], ['1709.04194-1-11-2', '1709.04194-2-11-2'], ['1709.04194-1-11-3', '1709.04194-2-11-3'], ['1709.04194-1-11-4', '1709.04194-2-11-4'], ['1709.04194-1-11-5', '1709.04194-2-11-5'], ['1709.04194-1-6-0', '1709.04194-2-6-0'], ['1709.04194-1-6-1', '1709.04194-2-6-1'], ['1709.04194-1-6-2', '1709.04194-2-6-2'], ['1709.04194-1-10-0', '1709.04194-2-10-0'], ['1709.04194-1-10-2', '1709.04194-2-10-2'], ['1709.04194-2-10-0', '1709.04194-3-12-0'], ['1709.04194-2-10-2', '1709.04194-3-12-2'], ['1709.04194-2-0-0', '1709.04194-3-0-0'], ['1709.04194-2-0-1', '1709.04194-3-0-1'], ['1709.04194-2-0-2', '1709.04194-3-0-2'], ['1709.04194-2-0-3', '1709.04194-3-0-3'], ['1709.04194-2-11-0', '1709.04194-3-13-0'], ['1709.04194-2-11-1', '1709.04194-3-13-1'], ['1709.04194-2-11-2', '1709.04194-3-13-2'], ['1709.04194-2-11-3', '1709.04194-3-13-3'], ['1709.04194-2-11-4', '1709.04194-3-13-4'], ['1709.04194-2-11-5', '1709.04194-3-13-5'], ['1709.04194-2-4-0', '1709.04194-3-4-0'], ['1709.04194-2-4-1', '1709.04194-3-4-1'], ['1709.04194-2-4-2', '1709.04194-3-4-2'], ['1709.04194-2-4-3', '1709.04194-3-4-3'], ['1709.04194-2-4-4', '1709.04194-3-4-4'], ['1709.04194-2-4-5', '1709.04194-3-4-5'], ['1709.04194-2-4-6', '1709.04194-3-4-6'], ['1709.04194-2-4-7', '1709.04194-3-4-7'], ['1709.04194-2-4-8', '1709.04194-3-4-8'], ['1709.04194-2-4-9', '1709.04194-3-4-9'], ['1709.04194-2-12-1', '1709.04194-3-14-1'], ['1709.04194-2-8-0', '1709.04194-3-9-0'], ['1709.04194-2-8-1', '1709.04194-3-9-1'], ['1709.04194-2-8-2', '1709.04194-3-9-2'], ['1709.04194-2-8-3', '1709.04194-3-9-3'], ['1709.04194-2-13-0', '1709.04194-3-15-0'], ['1709.04194-2-13-1', '1709.04194-3-15-1'], ['1709.04194-2-13-2', '1709.04194-3-15-2'], ['1709.04194-2-6-0', '1709.04194-3-7-0'], ['1709.04194-2-6-1', '1709.04194-3-7-1'], ['1709.04194-2-6-2', '1709.04194-3-7-2'], ['1709.04194-2-5-0', '1709.04194-3-5-0'], ['1709.04194-2-5-1', '1709.04194-3-5-1'], ['1709.04194-2-5-2', '1709.04194-3-5-2'], ['1709.04194-2-5-3', '1709.04194-3-6-0'], ['1709.04194-2-20-0', '1709.04194-3-22-0'], ['1709.04194-2-20-1', '1709.04194-3-22-1'], ['1709.04194-2-20-2', '1709.04194-3-22-2'], ['1709.04194-2-20-3', '1709.04194-3-23-0'], ['1709.04194-2-20-4', '1709.04194-3-23-1'], ['1709.04194-2-20-5', '1709.04194-3-23-2'], ['1709.04194-2-20-6', '1709.04194-3-23-3'], ['1709.04194-2-20-8', '1709.04194-3-23-5'], ['1709.04194-2-9-0', '1709.04194-3-10-0'], ['1709.04194-2-9-1', '1709.04194-3-11-0'], ['1709.04194-2-9-2', '1709.04194-3-11-1']] | [['1709.04194-1-4-0', '1709.04194-2-4-0'], ['1709.04194-1-4-1', '1709.04194-2-4-1'], ['1709.04194-1-4-2', '1709.04194-2-4-2'], ['1709.04194-1-4-3', '1709.04194-2-4-3'], ['1709.04194-1-4-4', '1709.04194-2-4-4'], ['1709.04194-1-4-5', '1709.04194-2-4-5'], ['1709.04194-1-4-6', '1709.04194-2-4-6'], ['1709.04194-1-4-7', '1709.04194-2-4-7'], ['1709.04194-1-4-8', '1709.04194-2-4-8'], ['1709.04194-1-4-9', '1709.04194-2-4-9'], ['1709.04194-1-5-0', '1709.04194-2-5-0'], ['1709.04194-1-5-1', '1709.04194-2-5-1'], ['1709.04194-1-5-2', '1709.04194-2-5-2'], ['1709.04194-1-5-3', '1709.04194-2-5-3'], ['1709.04194-1-0-0', '1709.04194-2-0-0'], ['1709.04194-1-0-1', '1709.04194-2-0-1'], ['1709.04194-1-0-2', '1709.04194-2-0-2'], ['1709.04194-1-0-3', '1709.04194-2-0-3'], ['1709.04194-1-13-0', '1709.04194-2-13-0'], ['1709.04194-1-13-1', '1709.04194-2-13-1'], ['1709.04194-1-13-2', '1709.04194-2-13-2'], ['1709.04194-1-9-0', '1709.04194-2-9-0'], ['1709.04194-1-9-1', '1709.04194-2-9-1'], ['1709.04194-1-9-2', '1709.04194-2-9-2'], ['1709.04194-1-12-1', '1709.04194-2-12-1'], ['1709.04194-1-8-0', '1709.04194-2-8-0'], ['1709.04194-1-8-1', '1709.04194-2-8-1'], ['1709.04194-1-8-2', '1709.04194-2-8-2'], ['1709.04194-1-8-3', '1709.04194-2-8-3'], ['1709.04194-1-20-0', '1709.04194-2-20-0'], ['1709.04194-1-20-1', '1709.04194-2-20-1'], ['1709.04194-1-20-2', '1709.04194-2-20-2'], ['1709.04194-1-20-3', '1709.04194-2-20-3'], ['1709.04194-1-20-4', '1709.04194-2-20-4'], ['1709.04194-1-20-5', '1709.04194-2-20-5'], ['1709.04194-1-20-6', '1709.04194-2-20-6'], ['1709.04194-1-20-8', '1709.04194-2-20-8'], ['1709.04194-1-11-0', '1709.04194-2-11-0'], ['1709.04194-1-11-1', '1709.04194-2-11-1'], ['1709.04194-1-11-2', '1709.04194-2-11-2'], ['1709.04194-1-11-3', '1709.04194-2-11-3'], ['1709.04194-1-11-4', '1709.04194-2-11-4'], ['1709.04194-1-11-5', '1709.04194-2-11-5'], ['1709.04194-1-6-0', '1709.04194-2-6-0'], ['1709.04194-1-6-1', '1709.04194-2-6-1'], ['1709.04194-1-6-2', '1709.04194-2-6-2'], ['1709.04194-1-10-0', '1709.04194-2-10-0'], ['1709.04194-1-10-2', '1709.04194-2-10-2'], ['1709.04194-2-10-0', '1709.04194-3-12-0'], ['1709.04194-2-10-2', '1709.04194-3-12-2'], ['1709.04194-2-0-0', '1709.04194-3-0-0'], ['1709.04194-2-0-1', '1709.04194-3-0-1'], ['1709.04194-2-0-2', '1709.04194-3-0-2'], ['1709.04194-2-0-3', '1709.04194-3-0-3'], ['1709.04194-2-11-0', '1709.04194-3-13-0'], ['1709.04194-2-11-1', '1709.04194-3-13-1'], ['1709.04194-2-11-2', '1709.04194-3-13-2'], ['1709.04194-2-11-3', '1709.04194-3-13-3'], ['1709.04194-2-11-4', '1709.04194-3-13-4'], ['1709.04194-2-11-5', '1709.04194-3-13-5'], ['1709.04194-2-4-0', '1709.04194-3-4-0'], ['1709.04194-2-4-1', '1709.04194-3-4-1'], ['1709.04194-2-4-2', '1709.04194-3-4-2'], ['1709.04194-2-4-3', '1709.04194-3-4-3'], ['1709.04194-2-4-4', '1709.04194-3-4-4'], ['1709.04194-2-4-5', '1709.04194-3-4-5'], ['1709.04194-2-4-6', '1709.04194-3-4-6'], ['1709.04194-2-4-7', '1709.04194-3-4-7'], ['1709.04194-2-4-8', '1709.04194-3-4-8'], ['1709.04194-2-4-9', '1709.04194-3-4-9'], ['1709.04194-2-12-1', '1709.04194-3-14-1'], ['1709.04194-2-8-0', '1709.04194-3-9-0'], ['1709.04194-2-8-1', '1709.04194-3-9-1'], ['1709.04194-2-8-2', '1709.04194-3-9-2'], ['1709.04194-2-8-3', '1709.04194-3-9-3'], ['1709.04194-2-13-0', '1709.04194-3-15-0'], ['1709.04194-2-13-1', '1709.04194-3-15-1'], ['1709.04194-2-13-2', '1709.04194-3-15-2'], ['1709.04194-2-6-0', '1709.04194-3-7-0'], ['1709.04194-2-6-1', '1709.04194-3-7-1'], ['1709.04194-2-6-2', '1709.04194-3-7-2'], ['1709.04194-2-5-0', '1709.04194-3-5-0'], ['1709.04194-2-5-1', '1709.04194-3-5-1'], ['1709.04194-2-5-2', '1709.04194-3-5-2'], ['1709.04194-2-5-3', '1709.04194-3-6-0'], ['1709.04194-2-20-0', '1709.04194-3-22-0'], ['1709.04194-2-20-1', '1709.04194-3-22-1'], ['1709.04194-2-20-2', '1709.04194-3-22-2'], ['1709.04194-2-20-3', '1709.04194-3-23-0'], ['1709.04194-2-20-4', '1709.04194-3-23-1'], ['1709.04194-2-20-5', '1709.04194-3-23-2'], ['1709.04194-2-20-6', '1709.04194-3-23-3'], ['1709.04194-2-20-8', '1709.04194-3-23-5'], ['1709.04194-2-9-0', '1709.04194-3-10-0'], ['1709.04194-2-9-1', '1709.04194-3-11-0'], ['1709.04194-2-9-2', '1709.04194-3-11-1']] | [] | [] | [] | [] | ['1709.04194-1-1-0', '1709.04194-1-2-0', '1709.04194-1-10-1', '1709.04194-1-12-0', '1709.04194-1-15-0', '1709.04194-1-15-1', '1709.04194-1-17-0', '1709.04194-1-17-1', '1709.04194-1-17-2', '1709.04194-1-17-3', '1709.04194-1-17-4', '1709.04194-1-17-5', '1709.04194-1-19-0', '1709.04194-1-20-7', '1709.04194-2-1-0', '1709.04194-2-2-0', '1709.04194-2-10-1', '1709.04194-2-12-0', '1709.04194-2-15-0', '1709.04194-2-15-1', '1709.04194-2-17-0', '1709.04194-2-17-1', '1709.04194-2-17-2', '1709.04194-2-17-3', '1709.04194-2-17-4', '1709.04194-2-17-5', '1709.04194-2-19-0', '1709.04194-2-20-7', '1709.04194-3-1-0', '1709.04194-3-2-0', '1709.04194-3-12-1', '1709.04194-3-14-0', '1709.04194-3-17-0', '1709.04194-3-17-1', '1709.04194-3-19-0', '1709.04194-3-19-1', '1709.04194-3-19-2', '1709.04194-3-19-3', '1709.04194-3-19-4', '1709.04194-3-19-5', '1709.04194-3-21-0', '1709.04194-3-23-4'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1709.04194 | {'1709.04194-3-0-0': 'We consider the weighted Radon transforms [MATH] along hyperplanes in [MATH], with strictly positive weights [MATH].', '1709.04194-3-0-1': 'We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions.', '1709.04194-3-0-2': 'In addition, the related weight [MATH] is infinitely smooth almost everywhere and is bounded.', '1709.04194-3-0-3': 'Our construction is based on the famous example of non-uniqueness of J. Boman (1993) for the weighted Radon transforms in [MATH] and on a recent result of F. Goncharov and R. Novikov (2016).', '1709.04194-3-1-0': 'Keywords: weighted Radon transforms, injectivity, non-injectivity', '1709.04194-3-2-0': 'AMS Mathematics Subject Classification: 44A12, 65R32', '1709.04194-3-3-0': '# Introduction', '1709.04194-3-4-0': 'We consider the weighted Radon transforms [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test function on [MATH].', '1709.04194-3-4-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION] where [MATH] denotes the complex conjugate of [MATH], [MATH] is a constant.', '1709.04194-3-4-2': 'If [MATH], then [MATH] is reduced to the classical Radon transform [MATH] along hyperplanes in [MATH].', '1709.04194-3-4-3': 'This transform is invertible by the classical Radon inversion formulas; see [CITATION].', '1709.04194-3-4-4': 'If [MATH] is strictly positive, [MATH] and [MATH], then in [CITATION] the inversion of [MATH] is reduced to solving a Fredholm type linear integral equation.', '1709.04194-3-4-5': 'Besides, in [CITATION] it was proved that [MATH] is injective (for example, in [MATH]) if [MATH] is real-analytic and strictly positive.', '1709.04194-3-4-6': 'In addition, an example of [MATH] in [MATH] with infinitely smooth strictly positive [MATH] and with non-trivial kernel [MATH] in [MATH] was constructed in [CITATION].', '1709.04194-3-4-7': 'Here [MATH] denote the spaces of functions from [MATH] with compact support, respectively.', '1709.04194-3-4-8': 'In connection with the most recent progress in inversion methods for weighted Radon transforms [MATH], see [CITATION].', '1709.04194-3-4-9': 'We recall also that inversion methods for [MATH] in [MATH] admit applications in the framework of emission tomographies (see [CITATION]).', '1709.04194-3-5-0': 'In the present work we construct an example of [MATH] in [MATH], with non-trivial kernel [MATH] in [MATH].', '1709.04194-3-5-1': 'The related [MATH] satisfies [REF].', '1709.04194-3-5-2': 'In addition, our weight [MATH] is infinitely smooth almost everywhere on [MATH].', '1709.04194-3-6-0': 'In our construction we proceed from results of [CITATION] and [CITATION].', '1709.04194-3-7-0': 'In Section [REF], in particular, we recall the result of [CITATION].', '1709.04194-3-7-1': 'In Section [REF] we recall the result of [CITATION].', '1709.04194-3-7-2': 'In Section [REF] we obtain the main result of the present work.', '1709.04194-3-8-0': '# Relations between the Radon and the ray transforms', '1709.04194-3-9-0': 'We consider also the weighted ray transforms [MATH] in [MATH], defined by the formulas: [EQUATION] where [MATH] is the weight, [MATH] is a test-function on [MATH].', '1709.04194-3-9-1': 'We assume that [MATH] is real valued, bounded and strictly positive, i.e.: [EQUATION]', '1709.04194-3-9-2': 'We recall that [MATH] can be interpreted as the set of all oriented rays in [MATH].', '1709.04194-3-9-3': 'In particular, if [MATH], then [EQUATION] where [MATH] gives the orientation of [MATH].', '1709.04194-3-10-0': 'We recall that for [MATH], transforms [MATH] and [MATH] are equivalent up to the following change of variables: [EQUATION] where [MATH] is a test-function on [MATH].', '1709.04194-3-11-0': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas (see [CITATION]): [EQUATION] where [MATH] is some fixed vector from [MATH], [MATH] denotes the standard vector product in [MATH], [MATH] denotes the orthogonality of vectors.', '1709.04194-3-11-1': 'Actually, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH], where [MATH] and [MATH] are related by [REF].', '1709.04194-3-12-0': 'Below we present analogs of [REF]-[REF] for [MATH].', '1709.04194-3-12-1': 'Let [EQUATION]', '1709.04194-3-12-2': 'If [MATH] is not specified otherwise, it is assumed that [MATH] is the standard basis in [MATH].', '1709.04194-3-13-0': 'For [MATH], the transforms [MATH] and [MATH] are related by the following formulas: [EQUATION] where [MATH], are defined as follows: [EQUATION] and [MATH] are given by [REF], [REF], respectively.', '1709.04194-3-13-1': 'Here, due to the condition [MATH]: [EQUATION]', '1709.04194-3-13-2': 'Formula [REF] is proved in Section [REF].', '1709.04194-3-13-3': 'Note that formulas [REF]-[REF] are also valid for [MATH].', '1709.04194-3-13-4': 'In this case these formulas are reduced to [REF]-[REF], where [MATH].', '1709.04194-3-13-5': 'Note that, formula [REF] gives an expression for [MATH] on [MATH] in terms of [MATH] restricted to the rays [MATH], such that [MATH].', '1709.04194-3-14-0': 'Remark 1.', '1709.04194-3-14-1': 'In [REF] one can also write: [EQUATION] where [MATH]-denotes the Hodge star, [MATH] - is the exterior product in [MATH] (exterior algebra on [MATH]); see, for example, Chapters 2.1.c, 4.1.c of [CITATION].', '1709.04194-3-15-0': 'Note that the value of the integral in the right hand-side of [REF] does not depend on the particular choice of [MATH] of [REF].', '1709.04194-3-15-1': 'Note also that, due to [REF], [REF], [REF], [REF], the weight [MATH] is defined everywhere on [MATH].', '1709.04194-3-15-2': 'In addition, this [MATH] has the same smoothness as [MATH] in [MATH] on [MATH] and in [MATH] on [MATH], where [MATH] is defined in [REF] and has zero Lebesgue measure on [MATH].', '1709.04194-3-16-0': "# Boman's example", '1709.04194-3-17-0': 'For [MATH], in [CITATION] there were constructed a weight [MATH] and a function [MATH], such that: [EQUATION]', '1709.04194-3-17-1': 'In addition, as a corollary of [REF], [REF], [REF]-[REF], we have that [EQUATION] where [EQUATION]', '1709.04194-3-18-0': '# Main results', '1709.04194-3-19-0': 'Let [EQUATION]', '1709.04194-3-19-1': 'Remark 2.', '1709.04194-3-19-2': 'According to [REF], [REF], [MATH] for [MATH] can be specified as follows: [EQUATION] [Proof of Theorem [REF]] We define [EQUATION] where [EQUATION]', '1709.04194-3-19-3': 'From [REF], [REF], [REF]-[REF] it follows that: [EQUATION]', '1709.04194-3-19-4': 'Properties [REF]-[REF] follow from [REF]-[REF], [REF], [REF], [REF], [REF], [REF].', '1709.04194-3-19-5': 'Theorem [REF] is proved.', '1709.04194-3-20-0': '# Proof of formula [REF]', '1709.04194-3-21-0': 'Note that [EQUATION] which implies that the intersection [MATH] is one of the following:', '1709.04194-3-22-0': 'The intersection is the one dimensional line [MATH]: [EQUATION] where [MATH] is an arbitrary point of [MATH], the orientation of [MATH] is chosen such that: [EQUATION]', '1709.04194-3-22-1': 'Condition [REF] fixes uniquely the direction of [MATH] of [REF].', '1709.04194-3-22-2': 'Formulas [REF], [REF], [REF] imply that [REF] can hold if and only if [MATH].', '1709.04194-3-23-0': 'The intersection is the two-dimensional plane [MATH].', '1709.04194-3-23-1': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-3-23-2': 'The intersection is an empty set.', '1709.04194-3-23-3': 'Formulas [REF], [REF] imply that it is the case if and only if [EQUATION]', '1709.04194-3-23-4': 'Note that [EQUATION]', '1709.04194-3-23-5': 'This completes the proof of formula [REF].'} | null | null | null | null |
1204.4249 | {'1204.4249-1-0-0': 'Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design optimal coding schemes for general point-to-point memoryless channels with feedback.', '1204.4249-1-0-1': 'In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC channel with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme.', '1204.4249-1-0-2': "The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder's posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots.", '1204.4249-1-0-3': "It turns out that the well-known Ozarow's encoding scheme, which obtains the capacity of two-user Gaussian channel, is a special case of our extended posterior matching framework like Schalkwijk-Kailath's scheme is a special case of the point-to-point posterior matching mentioned above.", '1204.4249-1-0-4': 'Furthermore, our designed posterior matching also obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC channel.', '1204.4249-1-0-5': 'Besides, the encoding scheme in this paper is designed for the real Gaussian MAC channel, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC channel.', '1204.4249-1-0-6': 'More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels.', '1204.4249-1-1-0': 'Gaussian Multiple Access Channel, Feedback, Posterior Matching.', '1204.4249-1-2-0': 'Introduction', '1204.4249-1-3-0': 'In the early work [9], Shannon proved that feedback couldnt increase the capacity of point-to-point memoryless channel.', '1204.4249-1-3-1': 'However, feedback can improve error performance and simplify the transmission scheme for this kind of channel.', '1204.4249-1-3-2': 'In [10], Hostein proposed a simple sequential transmission scheme, which achieves the capacity of Binary Symmetric Channel (BSC) and provides large error exponents than traditional fixed length block coding.', '1204.4249-1-3-3': 'Besides, Schalkwijk and Kailath also showed that feedback could improve error performance and/or simplify the transmission scheme for point-to-point Gaussian channel [7], [8].', '1204.4249-1-3-4': 'For the Gaussian multiuser channels, the situation is more interesting.', '1204.4249-1-3-5': 'In [12], Gaarder and Wolf proved that feedback can enlarge the capacity region of the multiple access channel, and Ozarow [3] successfully constructed a simple coding scheme for the two user Gaussian MAC channel and reaffirmed that feedback could increase the capacity of this kind of channel.', '1204.4249-1-3-6': 'Furthermore, Kramer devised a code for complex Gaussian channel based on a beautiful property of the circulant matrix that has all columns of the DFT (Discrete Fourier Transform) matrix as its eigenvectors [14].', '1204.4249-1-3-7': 'This code was proved to obtain the linear-feedback sum-capacity of the symmetric Gaussian channel with feedback in [16].', '1204.4249-1-3-8': "Besides, by using the control-theoretic approach to communications with feedback, Ehsan Ardestanizadeh and Massimo Fraceschetii [17] also proposed a linear code that has the same performance as Kramer's code for symmetric Gaussian complex channel.", '1204.4249-1-3-9': "However, as we mentioned, both these codes were designed for Gaussian complex channel and don't work for Gaussian real channel.", '1204.4249-1-4-0': 'Recently, Ofer Shayevitz and Meir Feder [1], [2], and [4] have discovered an underlying principle between the Hostein and Schalkwijk-Kailth schemes in a simple encoding scheme called posterior matching scheme for general point-to-point memoryless channels.', '1204.4249-1-4-1': "The ideas of posterior matching is that the transmitter encapsulates the information that receiver does not know up to present time in one random variable and then transmit that random variable to the receiver in the next transmission to refine the receiver's knowledge.", '1204.4249-1-4-2': 'The distribution of that variable will be selected in a way such that the input constraint is satisfied.', '1204.4249-1-4-3': 'Later, J.H.Bae and A.Anastasopolous have extended this scheme for the finite-state channel with feedbacks [11].', '1204.4249-1-4-4': 'One interesting open problem is to extend the Ofer Shayevitz and Meir Feder posterior matching scheme for multiuser cases.', '1204.4249-1-4-5': 'In this paper, using the same approach as Ofer Shayevitz and Meir Feder used for point-to-point memoryless channels, we propose a posterior matching based encoding and decoding strategy for real Gaussian MAC channels, referred to as a time-varying posterior matching scheme, analyze the error probabilities for all encoding-decoding schemes designed by using these strategies.', '1204.4249-1-5-0': 'We analyze the achievable rate region and error performance of encoding and decoding schemes using these strategies by defining a generalized iterated function systems (GIFS) which has the generalized average contractive property (average contractive in the limit).', '1204.4249-1-5-1': 'Refer our Theorem I for more details.', '1204.4249-1-5-2': 'Note that our imposed constraint is less strict than the constraint that Ofer Shayevitz and Meir Feder imposed to analyze the point-to-point memoryless channels.', '1204.4249-1-5-3': 'Specifically, in Theorem 6 in [4], Ofer Shayevitz and Meir Feder used the relations between the information rates and contraction properties of the iterated function systems (IFS) to analyze the error probability for point-to-point cases.', '1204.4249-1-5-4': 'For the continuous cases, they assumed that the reverse iterated function system (RIFS), generated by the kernel [MATH] and controlled by the identically distributed output sequence [MATH], has the average contractive property to analyze the error performance of their posterior matching schemes.', '1204.4249-1-5-5': 'That assumption requires the distribution at the output of the point-to-point memoryless channel be identically distributed when using their proposed encoding schemes.', '1204.4249-1-5-6': 'This also means that if the output distribution is not identically distributed, the error analysis in Theorem 6 in [4] cannot apply.', '1204.4249-1-5-7': 'For example, this situation happens with our proposed matching schemes for MAC channels in this paper.', '1204.4249-1-6-0': 'Finally, we illustrate our strategies by designing schemes that obtain the optimal performance for Gaussian MAC channels.', '1204.4249-1-6-1': "Specifically, our proposed code for non-symmetric two-user Gaussian channel obtains the same performance as Ozarow's code [3], so it obtains the capacity of this channel.", '1204.4249-1-6-2': "For the case when the number of users greater than 3, our proposed code obtains the same performance as Kramer's code in the sense of sum-rate, so it is optimal among linear code with respect to sum rate capacity.", '1204.4249-1-6-3': 'An interesting thing is that our code works for real Gaussian channel and it is the first code designed that works for the real Gaussian MAC channel for arbitrary number of users.', '1204.4249-1-7-0': 'Channel Model and Preliminaries', '1204.4249-1-8-0': '## Mathematical notations', '1204.4249-1-9-0': 'Upper-case letters, their realizations by corresponding lower-case letters, denotes random variables.', '1204.4249-1-9-1': 'A real-valued random variable [MATH] is associated with a distribution [MATH] defined on the usual Borel [MATH]-algebra over [MATH], and we write [MATH].', '1204.4249-1-9-2': 'The cumulative distribution function (c.d.f.) of [MATH] is given by [MATH], and their inverse c.d.f is defined to be [MATH].', '1204.4249-1-9-3': 'The uniform probability distribution over [MATH] is denoted through [MATH].', '1204.4249-1-9-4': 'The composition function [MATH].', '1204.4249-1-9-5': 'In this paper, we use the following lemma:', '1204.4249-1-10-0': 'Lemma I: Let [MATH] be a continuous random variable with [MATH] and [MATH] be an uniform distribution random variable, i.e. [MATH] be statistical independent.', '1204.4249-1-10-1': 'Then [MATH] and [MATH].', '1204.4249-1-11-0': 'Refer [4] for the proof.', '1204.4249-1-12-0': '## Information Theoretic Formulation of the Gaussian Multiple Access Channel', '1204.4249-1-13-0': 'Consider the communication problem between [MATH] senders and a receiver over a multiple access channel with additive Gaussian noise (AWGN-MAC) when channel outputs are noiselessly fed back to all the senders (Figure 1).', '1204.4249-1-13-1': 'Each sender [MATH] wishes to reliably transmit a message [MATH] in the unit interval, i.e. , [MATH] to the receiver.', '1204.4249-1-13-2': 'At each time [MATH], the output of the channel is [EQUATION]', '1204.4249-1-13-3': 'Where [MATH] is the transmitted symbol by sender [MATH] at time [MATH], [MATH] is the output of the channel, and [MATH] is a discrete-time zero mean white Gaussian noise process with unit average power, i.e., [MATH] and is independent of [MATH].', '1204.4249-1-13-4': 'We assume that output symbols are casually fed back to the sender and the transmitted symbol [MATH] for sender [MATH] at time [MATH] can depend on both the message [MATH] and the previous channel output sequence [MATH].', '1204.4249-1-14-0': 'A transmission scheme for a Gaussian MAC channel is a set of [MATH] sequence of transmission functions [MATH], so that the input to the channel generated by the transmitter is given by [EQUATION]', '1204.4249-1-15-0': 'A decoding rule for a MAC channel is a sequence of measurable mappings [MATH], where [MATH] are the sets of all open intervals in [MATH].', '1204.4249-1-15-1': 'The [MATH]th component of this decoded vector, denoted as [MATH], refers as the decoded interval for the user [MATH].', '1204.4249-1-15-2': 'The error probabilities at time [MATH] associated with a transmission scheme and a decoding rule, is defined as [EQUATION] and the corresponding achievable rate vector at time [MATH] is defined to be [EQUATION]', '1204.4249-1-15-3': 'We say that a transmission scheme together with a decoding rule achieve a rate vector [MATH] over a Gaussian MAC channel if for all [MATH] we have [EQUATION]', '1204.4249-1-15-4': 'The rate vector is achieved within input power constraints [MATH], if in addition [EQUATION]', '1204.4249-1-15-5': 'An optimal fixed rate decoding rule for a MAC channel with rate region [MATH] is one that decodes a vector of fixed length intervals [MATH], whose marginal posteriori probabilities are maximal, i.e., [EQUATION]', '1204.4249-1-15-6': 'An optimal variable rate decoding rule with target error probabilities [MATH] is one that decodes a vector of minimal-length intervals [MATH] with accumulated marginal posteriori probabilities exceeds corresponding targets, i.e., [EQUATION]', '1204.4249-1-15-7': 'Both decoding rules make use of the marginal posterior distribution of the message point [MATH] which can calculate online at the transmitter [MATH] and the receiver.', '1204.4249-1-15-8': 'Refer [4] for more details.', '1204.4249-1-15-9': 'A proof that the achievability in the sense of [MATH] and [MATH] implies that the achievability in the standard framework are in the Appendix.', '1204.4249-1-16-0': 'Lemma II: The achievability in the definition [MATH] and [MATH] implies the achievability in the framework.', '1204.4249-1-17-0': 'Refer to the Appendix.', '1204.4249-1-18-0': '# Time-varying Posterior Matching Scheme', '1204.4249-1-19-0': "## Shayevitz and Feder's Posterior Matching Scheme", '1204.4249-1-20-0': 'In this part, we firstly review the posterior matching scheme proposed by Ofer Shayevitz and Meir Feder for point-to-point channel in [4].', '1204.4249-1-20-1': 'Specifically, the authors argued that after the receiver observed the output sequence [MATH], there is still some "missing information" that can be encapsulated in a random variable [MATH] with the following properties:', '1204.4249-1-21-0': '(i) [MATH] is statistically independent of [MATH].', '1204.4249-1-22-0': '(ii) The message point [MATH] can be a.s. uniquely recovered from [MATH]', '1204.4249-1-23-0': 'With that line of thought, they proposed a principle for generating the next channel input as follow:', '1204.4249-1-24-0': 'The transmission function [MATH] should be selected so that [MATH] is [MATH]- distributed, and is a fixed function of some random variable [MATH] satisfying properties [MATH] and [MATH].', '1204.4249-1-25-0': 'Lemma III: (Posterior Matching Scheme [4]).', '1204.4249-1-25-1': 'The following transmission scheme satisfies the posterior matching principle for any [MATH]: [EQUATION]', '1204.4249-1-25-2': 'Based on the transmission functions, the input to the channel is a sequence of random variables given by [EQUATION]', '1204.4249-1-25-3': 'Refer [4] for the proof.', '1204.4249-1-26-0': '## Time-varying Posterior Matching Scheme', '1204.4249-1-27-0': 'In this section, we propose a posterior matching scheme for additive Gaussian multiple access channel (MAC) with feedback, called time-varying posterior matching.', '1204.4249-1-27-1': 'Our encoding proposal is based on the two following lemmas:', '1204.4249-1-28-0': 'Lemma IV: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the output signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-1-28-1': 'Under the condition that the correlation matrix among transmitted symbols at each time slot [MATH] defined as [EQUATION] is a fixed function of [MATH].', '1204.4249-1-28-2': 'Then the posterior matching scheme in Lemma III at each transmitter [MATH] becomes a time-varying posterior matching which is given by [EQUATION] where the random variable [MATH] encapsulates the input power constraint, and [MATH] is the intended transmitted message at the transmitter [MATH].', '1204.4249-1-29-0': '[EQUATION]', '1204.4249-1-29-1': 'Then we have [EQUATION] where (a) follows the fact that the transmission function is continuous and monotone, and (b) follows that [MATH] is a linear combination with known coefficients of inputs [MATH], therefore the posterior distributions of [MATH] are only dependent on the correlations among transmitted symbols at time [MATH].', '1204.4249-1-29-2': 'Since we assume that the correlation matrix [MATH] is a fixed function of [MATH], hence the distributions of [MATH] are fixed function of [MATH].', '1204.4249-1-29-3': 'In other words, given [MATH], each [MATH] is independent of [MATH].', '1204.4249-1-30-0': 'Besides, from the Lemma I, we know that [MATH], hence [MATH].', '1204.4249-1-31-0': 'We refer the transmission scheme in (4) to as time-varying posterior matching scheme and prove an interesting converse result of the Lemma IV.', '1204.4249-1-32-0': 'Lemma V: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the received signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-1-32-1': 'Assuming that at each transmitter, a time-varying posterior matching encoder is deployed, so the recursive transmitted sequence at the transmitter [MATH] is given by [EQUATION] where [MATH] is the intended transmitted message.', '1204.4249-1-32-2': 'Then the correlation matrix among transmitted symbols at each time slot [MATH] defined as in the Lemma IV is a deterministic function of [MATH], and more importantly the posterior distribution [MATH] can be calculated online at both the transmitter [MATH] and the receiver.', '1204.4249-1-33-0': 'Indeed, we know that [MATH] is a linear function of [MATH] and [MATH] is a function of [MATH], so [MATH] is a function of [MATH].', '1204.4249-1-33-1': 'Therefore, the correlation between [MATH] and [MATH] only depends on the correlations among transmitted symbols at time [MATH], i.e. [MATH].', '1204.4249-1-33-2': 'In other words, [MATH] is a function of [MATH], so the transmitters and receiver can be calculated the matrix [MATH] in advance.', '1204.4249-1-33-3': 'Moreover, from the relation [MATH], we see that the distribution of [MATH] is a function of all elements in the correlation matrix [MATH].', '1204.4249-1-33-4': 'This completes our proof.', '1204.4249-1-34-0': "For a continuous point-to-point memoryless channel, the distribution [MATH] doesn't depend on [MATH], hence we have the posterior matching scheme for this case like the formula (16) in [4].", '1204.4249-1-34-1': 'However, in a MAC channel (for example additive white Gaussian MAC channel), where each received signal is a linear combination of all transmitted signals and Gaussian noise, the distribution [MATH] between the input [MATH] and the output may be dependent on [MATH].', '1204.4249-1-34-2': 'Therefore, time-varying posterior matching scheme may be the solution to overcome this problem.', '1204.4249-1-34-3': 'However, we will see from our proof in the Theorem I below the variable rate decoding rule, or Generalized Reverse Iterated Function System (RIFS), can be applied at receiver to decode signals if and only if all the distributions [MATH] can be calculated online at the corresponding transmitters and receiver.', '1204.4249-1-34-4': 'With the result in the Lemma V, this condition is always satisfied when time-varying posterior matching encoding schemes used at transmitters.', '1204.4249-1-34-5': 'We will show in the next parts that using time-varying posterior matching scheme at transmitters can obtain optimal performances for some known cases.', '1204.4249-1-35-0': '# Error analysis for time-varying posterior matching scheme', '1204.4249-1-36-0': 'In this section, we analyze error performance for Gaussian MAC channel with feedback deployed the time-varying posterior matching scheme at the transmitters and variable-decoding rule at the receiver.', '1204.4249-1-37-0': 'Theorem I: Consider a real Gaussian MAC channel with [MATH] transmitters and one receiver without input power constraints.', '1204.4249-1-37-1': 'Assuming that at each transmitter [MATH], transmitted sequence [MATH] conforms the time-varying posterior matching rule, as following: [EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the Gaussian MAC channel which is a linear combination of all these transmitted signals at time [MATH].', '1204.4249-1-37-2': 'Let [MATH] and [MATH] as the global Lipschitz operator.', '1204.4249-1-37-3': 'Under the conditions that [EQUATION] define: [MATH] Then the rate region [MATH] is achievable by setting the target error probabilities [MATH] under the constraint [EQUATION]', '1204.4249-1-38-0': 'First, observe that all since the distributions [MATH] can be calculated online at both transmitters and receiver by Lemma V, therefore [MATH] can be calculated online at both the transmitters and receiver.', '1204.4249-1-38-1': 'Denote [MATH], referred as to a Generalized Iterated Function System (GIFS) generated by the kernel sequence [MATH].', '1204.4249-1-38-2': 'For each [MATH], select a fixed interval [MATH] as the decoded interval with respect to [MATH].', '1204.4249-1-38-3': 'Define the corresponding interval at the origin to be [MATH] and set them to be the decoded interval for [MATH], and so the decoded interval for [MATH] are set to be [MATH].', '1204.4249-1-38-4': 'Let [EQUATION]', '1204.4249-1-38-5': 'From the condition [MATH] in the theorem, we know that [EQUATION]', '1204.4249-1-38-6': 'Observe that: [EQUATION] where [MATH].', '1204.4249-1-39-0': 'On the other hand, since [MATH], then for any arbitrarily small [MATH], there exists an [MATH] such that [MATH].', '1204.4249-1-39-1': 'Let [MATH], and [MATH].', '1204.4249-1-39-2': "For any fixed number [MATH], from [MATH] we have: [EQUATION] for any [MATH] arbitrarily small, where [MATH] follows the Markov's inequality, [MATH] follows [MATH], and [MATH] is a recursive application of the preceding transitions, [MATH] follows [MATH] above and recursive applications of the preceding transitions.", '1204.4249-1-39-3': 'From [MATH], it is easy to see that a sufficient condition for [MATH] is given by choosing [MATH].', '1204.4249-1-39-4': 'Observe that [MATH] can be made arbitrarily small, therefore, the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-1-39-5': 'Since [MATH] depends only on the length of [MATH], so we can choose [MATH].', '1204.4249-1-39-6': 'Furthermore, from the Lemma I, we are easy to come to conclusion that [MATH].', '1204.4249-1-39-7': 'Using the standard bounds for Gaussian distribution we obtain [EQUATION]', '1204.4249-1-40-0': '# A Posterior Matching Scheme for Gaussian MAC Channel with Feedback', '1204.4249-1-41-0': 'In this section, we consider a real Gaussian MAC channel with [MATH] receivers and input power constraints [MATH] at the transmitters [MATH], respectively as defined in the section II.', '1204.4249-1-41-1': 'Our encoding scheme for this channel as following:', '1204.4249-1-42-0': 'Encoding:', '1204.4249-1-43-0': '[EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the MAC channel.', '1204.4249-1-44-0': 'where [MATH] is the column [MATH] of the Hadamard matrix [MATH] by [MATH], which is a spreading code for CDMA systems.', '1204.4249-1-45-0': "We refer this encoding strategy as Gaussian MAC channel's posterior matching feedback coding and decoding strategy.", '1204.4249-1-46-0': "Theorem II: Using the Gaussian MAC channel's posterior matching feedback coding and decoding strategy above, the rate region [MATH] is achievable for Gaussian MAC channel with feedback, where [EQUATION] by setting the target error probabilities [EQUATION] where [EQUATION]", '1204.4249-1-47-0': 'Applying the Lemma I, we see that for any [MATH] then: [EQUATION]', '1204.4249-1-47-1': 'Observe that, by this transmission strategy, each transmitter [MATH] transmits [MATH] at time [MATH] with [MATH], thus the input power constraints at all transmitters are always satisfied at each transmission time [MATH] Moreover, the output at receiver at time [MATH] will be [EQUATION]', '1204.4249-1-47-2': 'Thus [EQUATION]', '1204.4249-1-47-3': 'Finally, we have: [EQUATION] and [EQUATION]', '1204.4249-1-47-4': 'Therefore, [EQUATION] where [EQUATION] and [EQUATION]', '1204.4249-1-47-5': 'Moreover, from [MATH] we have: [EQUATION] thus, [EQUATION]', '1204.4249-1-47-6': 'From [MATH] and [MATH] we obtain: [EQUATION]', '1204.4249-1-47-7': 'Hence, [EQUATION]', '1204.4249-1-47-8': 'Finally, we have: [EQUATION]', '1204.4249-1-47-9': 'If we can achieve [EQUATION] then [EQUATION]', '1204.4249-1-47-10': 'This means that the condition in the Theorem I is satisfied, which leads to the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-1-47-11': 'Note that this capacity region is obtained by setting the target error probabilities [MATH] under the constraints [EQUATION] which have the well-known double-exponential behavior.', '1204.4249-1-48-0': 'In the following, we will specify achievable rate regions, and error probabilities for two cases: the general two-user Gaussian MAC channel, and the real symmetric Gaussian MAC channel when the number of users is arbitrary.', '1204.4249-1-49-0': 'From the Theorem II, we see that the strategy to design posterior encoding scheme for multiple access channels with feedback is to find the sequences [MATH] such that [EQUATION] for all [MATH].', '1204.4249-1-50-0': 'Case 1: Two-user Gaussian MAC channel with feedback.', '1204.4249-1-51-0': "To show that Ozarow's coding scheme [3] is a special case of our posterior matching framework, we can set [MATH] and later prove that [MATH].", '1204.4249-1-51-1': 'Observe that the constraint [MATH] can be also checked to be satisfied by this setting since [EQUATION] for [MATH] and [EQUATION] for [MATH].', '1204.4249-1-52-0': 'Now we need to find the recursion of [MATH] and [MATH] in this case.', '1204.4249-1-52-1': 'Observe that the output sequence: [EQUATION] where [MATH] is noise process and [MATH].', '1204.4249-1-53-0': 'From (12), (13) we have [EQUATION] and [EQUATION]', '1204.4249-1-53-1': 'Therefore, we obtain: [EQUATION] and [EQUATION]', '1204.4249-1-54-0': 'Finally, the time-varying posterior matching encoding scheme for Gaussian MAC channel with feedback in this special case as following:', '1204.4249-1-55-0': 'Achievable rate region and error analysis:', '1204.4249-1-56-0': 'For this special case, we have [EQUATION] and [EQUATION]', '1204.4249-1-56-1': 'Apply the Theorem II above, we obtain the achievable rate region for Gaussian MAC channel with two users as [MATH] where [EQUATION]', '1204.4249-1-56-2': 'Similarly, [EQUATION] by setting the target error probability to [EQUATION] and [EQUATION]', '1204.4249-1-56-3': 'Much like Ozarow in [MATH], at the reception [MATH], the receiver adds an independent random variable [MATH] before feeding back the first receiver signal to the transmitters 1 and 2 to set [MATH], where [MATH] is the biggest solution in [MATH] of the following equation: [EQUATION]', '1204.4249-1-56-4': 'By this changing, from [MATH] we see that [MATH], so [MATH] as mentioned above.', '1204.4249-1-56-5': 'We also have [MATH], where [MATH] is a positive solution of the equation [MATH].', '1204.4249-1-56-6': 'Replace this result to [MATH] and combine with [MATH], we have: [EQUATION] where [MATH] is defined above.', '1204.4249-1-56-7': "We see that, all the results are the same as Ozarow's results in [3].", '1204.4249-1-56-8': 'So our posterior matching encoding scheme is optimal for Gaussian channel MAC with two users.', '1204.4249-1-57-0': 'Case 2: M-user symmetric Gaussian MAC channel with feedback.', '1204.4249-1-57-1': 'We consider symmetric case, where [MATH].', '1204.4249-1-58-0': 'Achievable rate region and error analysis:', '1204.4249-1-59-0': 'Assuming that all the transmitted messages are statistically independent.', '1204.4249-1-59-1': 'We will prove by induction that the normalized covariance [MATH] have all the columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] are symmetric positive definite for all [MATH]', '1204.4249-1-60-0': 'Indeed, with the assumption all the transmitted information messages are statistically independent, we will have [MATH], which is an identity matrix of size [MATH].', '1204.4249-1-60-1': 'Therefore, it is obvious that all the columns of the Hadamard matrix [MATH] by [MATH] are eigenvectors of the matrix [MATH] and that [MATH] is a positive definite matrix.', '1204.4249-1-61-0': 'Now, assume that [MATH] has all columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is positive definite for some [MATH].', '1204.4249-1-61-1': 'Since we assumed that [MATH] is symmetric positive definite matrix, all its eigenvalues are greater than [MATH].', '1204.4249-1-61-2': 'Denote [MATH] are [MATH] columns of the Hadamard matrix [MATH].', '1204.4249-1-61-3': 'By this encoding scheme, we set the vector [MATH].', '1204.4249-1-61-4': 'Assume that [MATH] is the eigenvalue of [MATH] associated with this eigenvector.', '1204.4249-1-61-5': 'We have [EQUATION]', '1204.4249-1-61-6': 'On the other hand, we also have [EQUATION] where [MATH] is the [MATH]th column of the matrix [MATH].', '1204.4249-1-61-7': 'Note that [MATH], this means that [EQUATION]', '1204.4249-1-61-8': 'Hence, [EQUATION]', '1204.4249-1-61-9': 'Substitute [MATH] into [MATH] we obtain: [EQUATION]', '1204.4249-1-61-10': 'From [MATH] and [MATH], we obtain: [EQUATION] and [EQUATION] for all [MATH].', '1204.4249-1-62-0': 'Observe that from the proof of the Theorem II above, then [EQUATION].', '1204.4249-1-62-1': 'Define the correlation coefficient between [MATH] and [MATH] [EQUATION]', '1204.4249-1-62-2': 'Then, we have [EQUATION]', '1204.4249-1-62-3': 'By replacing all values of [MATH] calculated above, we obtain [EQUATION] for all [MATH].', '1204.4249-1-63-0': 'Since we assumed that [MATH] is symmetric positive definite matrix, hence [MATH], hence [MATH].', '1204.4249-1-63-1': 'Therefore, from [MATH], it is easy to see that we will have [MATH], or [MATH] is also a symmetric matrix.', '1204.4249-1-64-0': 'Moreover, from [MATH], we also obtain [EQUATION]', '1204.4249-1-64-1': 'Denote [MATH].', '1204.4249-1-64-2': 'We see that the columns of [MATH] creates [MATH] linearly independent eigenvectors of the matrix [MATH].', '1204.4249-1-64-3': 'Moreover, we also have [EQUATION]', '1204.4249-1-64-4': 'Note that since all columns of [MATH] are eigenvectors of the matrix [MATH], so all the columns of the matrix [MATH] are also eigenvectors of the matrix [MATH], hence we has the following eigenvalue decomposition [EQUATION] where [MATH] is a diagonal matrix.', '1204.4249-1-65-0': 'Moreover, we have [EQUATION] where [EQUATION]', '1204.4249-1-65-1': 'From (26), (27), and (28), we must have [MATH] must be a diagonal matrix since the right side of [MATH] is a diagonal matrix.', '1204.4249-1-65-2': 'Hence, all columns of the matrix [MATH] are eigenvectors of the matrix [MATH].', '1204.4249-1-66-0': 'Assuming [MATH] are [MATH] eigenvalues corresponding to eigenvectors which are columns of the matrix [MATH].', '1204.4249-1-66-1': 'By this notation, we see that [MATH].', '1204.4249-1-67-0': 'Combining with [MATH] with [MATH], and [MATH] we obtain [EQUATION] for all [MATH].', '1204.4249-1-68-0': 'Since we assumed that all eigenvalues of [MATH] are positive ([MATH] symmetric positive definite), from [MATH] we see that all eigenvalues of [MATH] are also positive.', '1204.4249-1-68-1': 'Note that we also confirmed that [MATH] is symmetric above, therefore [MATH] is a symmetric positive definite matrix.', '1204.4249-1-68-2': 'In short, if [MATH] is a symmetric positive definite matrix and has all columns of the Hadamard matrix as its eigenvectors, then [MATH] has all these properties.', '1204.4249-1-68-3': 'This concludes our proof by induction.', '1204.4249-1-69-0': 'According to the Lemma 1, [MATH], the sequence [MATH] converges to a fixed point [MATH], which is the solution in [MATH] of the following equation [EQUATION] when [EQUATION]', '1204.4249-1-69-1': 'When the condition (32) satisfies, from (22), (23) we obtain [EQUATION]', '1204.4249-1-69-2': 'Observe that [MATH], so [MATH].', '1204.4249-1-69-3': 'Therefore, the constraints in the Theorem II is satisfied.', '1204.4249-1-69-4': 'Applying the result of this theorem, we have any rate less than [EQUATION] is achievable, for all [MATH].', '1204.4249-1-69-5': 'Hence, any sum rate which is less than [EQUATION] is achievable, where [MATH] is solution in the [MATH] of the equation [MATH].', '1204.4249-1-69-6': 'This result coincides with the formula (68) in [14].', '1204.4249-1-69-7': 'The paper [16] proves that this achievable sum rate is optimal for the class of linear feedback coding.', '1204.4249-1-70-0': '# Conclusion', '1204.4249-1-71-0': 'A posterior matching based encoding-decoding strategy for general Gaussian MAC channel with feedback was proposed, and achievable rate region, error performance were drawn.', '1204.4249-1-71-1': 'Finally, we analyzed error performance of the proposed posterior encoding scheme and showed that the time-varying posterior matching scheme and variable rate decoding ideas can be applied to Gaussian MAC channel and obtain optimal performances.', '1204.4249-1-71-2': 'Specifically, the proposed encoding scheme achieves the capacity of two-user feedback Gaussian MAC channel as well as linear-feedback sum-rate for symmetric Gaussian MAC channel with feedback where the number of users is arbitrary.', '1204.4249-1-71-3': "Moreover, by the encoding scheme's structure, which uses the spreading codes like the Hadamard matrix, our encoding scheme can be directly applied to CDMA systems with feedback.", '1204.4249-1-71-4': 'Finally, by analyzing all arguments in the theorem I, the time-varying posterior matching scheme approach in this paper might be applied for other Gaussian and non-Gaussian multiuser channels to achieve optimal performances.', '1204.4249-1-71-5': '[Proof of the Lemma II]', '1204.4249-1-72-0': 'We use the same line argument as Lemma 1 [4].', '1204.4249-1-72-1': 'Assume we are given a transmission scheme with [MATH] transmission functions [MATH] and a decoding rule which are known to achieve the rate vector [MATH].', '1204.4249-1-72-2': 'For simplicity, we assume that the decoding rule is fixed rate [MATH] for all [MATH]), since any variable rate decoding rule can be easily mapped into a fixed rate rule that achieves the same rate vector.', '1204.4249-1-72-3': 'It is easy to see that in order to prove that the above translates into achievability for some rate vector [MATH] in the standard framework, it is enough to show we can find [MATH] sequences [MATH], and such that we have the uniform achievability over [MATH], i.e. , [EQUATION]', '1204.4249-1-72-4': 'We now show how [MATH] can be constructed for any [MATH].', '1204.4249-1-72-5': 'Let [MATH] be the (average) error probability associated with our scheme and the fixed rate vector [MATH].', '1204.4249-1-72-6': 'Define [EQUATION] and write [EQUATION] and so we have that [MATH].', '1204.4249-1-72-7': 'It is now easy to see that if we want to select [MATH] such that [MATH], and also [MATH], then a sufficient condition is that [MATH] for some positive [MATH].', '1204.4249-1-72-8': 'This condition can be written as [EQUATION]', '1204.4249-1-72-9': 'At the same time, we also have by definition [EQUATION]'} | {'1204.4249-2-0-0': 'Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design optimal coding schemes for general point-to-point memoryless channels with feedback.', '1204.4249-2-0-1': 'In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC channel with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme.', '1204.4249-2-0-2': "The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder's posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots.", '1204.4249-2-0-3': "It turns out that the well-known Ozarow's encoding scheme, which obtains the capacity of two-user Gaussian channel, is a special case of our extended posterior matching framework as the Schalkwijk-Kailath's scheme is a special case of the point-to-point posterior matching mentioned above.", '1204.4249-2-0-4': 'Furthermore, our designed posterior matching also obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC channel.', '1204.4249-2-0-5': 'Besides, the encoding scheme in this paper is designed for the real Gaussian MAC channel to obtain that performance, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC channel.', '1204.4249-2-0-6': 'More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels.', '1204.4249-2-1-0': 'Gaussian Multiple Access Channel, Feedback, Posterior Matching.', '1204.4249-2-2-0': 'Introduction', '1204.4249-2-3-0': 'In the early work [9], Shannon proved that feedback could not increase the capacity of point-to-point memoryless channel.', '1204.4249-2-3-1': 'However, feedback can improve error performance and simplify the transmission scheme for this kind of channel.', '1204.4249-2-3-2': 'In [10], Hostein proposed a simple sequential transmission scheme, which achieves the capacity of Binary Symmetric Channel (BSC) and provides large error exponents than traditional fixed length block coding.', '1204.4249-2-3-3': 'Besides, Schalkwijk and Kailath also showed that feedback could improve error performance and/or simplify the transmission scheme for point-to-point Gaussian channel [7], [8].', '1204.4249-2-3-4': 'For the Gaussian multiuser channels, the situation is more interesting.', '1204.4249-2-3-5': 'In [12], Gaarder and Wolf proved that feedback can enlarge the capacity region of the multiple access channel, and Ozarow [3] successfully constructed a simple coding scheme for the two user Gaussian MAC channel and reaffirmed that feedback could increase the capacity of this kind of channel.', '1204.4249-2-3-6': 'Furthermore, Kramer devised a code for complex Gaussian channel based on a beautiful property of the circulant matrix that has all columns of the DFT (Discrete Fourier Transform) matrix as its eigenvectors [14].', '1204.4249-2-3-7': 'This code was proved to obtain the linear-feedback sum-capacity of the symmetric Gaussian channel with feedback in [16].', '1204.4249-2-3-8': "Besides, by using the control-theoretic approach to communications with feedback, Ehsan Ardestanizadeh and Massimo Fraceschetii [17] also proposed a linear code that has the same performance as Kramer's code for symmetric Gaussian complex channel.", '1204.4249-2-3-9': "However, as we mentioned, both these codes were designed for Gaussian complex channel and don't work for Gaussian real channel.", '1204.4249-2-4-0': 'Recently, Ofer Shayevitz and Meir Feder [1], [2], and [4] have discovered an underlying principle between the Hostein and Schalkwijk-Kailth schemes in a simple encoding scheme called posterior matching scheme for general point-to-point memoryless channels.', '1204.4249-2-4-1': "The ideas of posterior matching is that the transmitter encapsulates the information that receiver does not know up to present time in one random variable and then transmit that random variable to the receiver in the next transmission to refine the receiver's knowledge.", '1204.4249-2-4-2': 'The distribution of that variable will be selected in a way such that the input constraint is satisfied.', '1204.4249-2-4-3': 'Later, J.H.Bae and A.Anastasopolous have extended this scheme for the finite-state channel with feedbacks by using another approach [11].', '1204.4249-2-4-4': 'One interesting open problem is to extend the Ofer Shayevitz and Meir Feder posterior matching scheme for multiuser cases.', '1204.4249-2-4-5': 'In this paper, using the same approach as Ofer Shayevitz and Meir Feder used for point-to-point memoryless channels, we propose a posterior matching based encoding and decoding strategy for real Gaussian MAC channels, referred to as a time-varying posterior matching scheme, analyze the error probabilities for all encoding-decoding schemes designed by using these strategies.', '1204.4249-2-5-0': 'We analyze the achievable rate region and error performance of encoding and decoding schemes using these strategies by defining a generalized iterated function systems (GIFS) which has the generalized average contractive property (average contractive in the limit).', '1204.4249-2-5-1': 'Refer our Theorem I for more details.', '1204.4249-2-5-2': 'Note that our imposed constraint is less strict than the constraint that Ofer Shayevitz and Meir Feder imposed to analyze the point-to-point memoryless channels.', '1204.4249-2-5-3': 'Specifically, in Theorem 6 in [4], Ofer Shayevitz and Meir Feder used the relations between the information rates and contraction properties of the iterated function systems (IFS) to analyze the error probability for point-to-point cases.', '1204.4249-2-5-4': 'For the continuous cases, they assumed that the reverse iterated function system (RIFS), generated by the kernel [MATH] and controlled by the identically distributed output sequence [MATH], has the average contractive property to analyze the error performance of their posterior matching schemes.', '1204.4249-2-5-5': 'That assumption requires the distribution at the output of the point-to-point memoryless channel be identically distributed when using their proposed encoding schemes.', '1204.4249-2-5-6': 'This also means that if the output distribution is not identically distributed, the error analysis in Theorem 6 in [4] cannot apply.', '1204.4249-2-5-7': 'For example, this situation happens with our proposed matching schemes for MAC channels in this paper.', '1204.4249-2-6-0': 'Finally, we illustrate our strategies by designing an encoding scheme that obtains optimal performances for Gaussian MAC channels.', '1204.4249-2-6-1': "Specifically, our proposed code obtains the same performance as Ozarow's code [3] for the general two-user Gaussian channel, so it achieves the capacity of this channel.", '1204.4249-2-6-2': "For the case when the number of users is greater than 3, our proposed code obtains the same performance as the Kramer's code in the sense of sum-rate, so it is optimal among linear code with respect to sum rate capacity.", '1204.4249-2-6-3': 'To the best of our knowledge, the time-varying posterior matching in this paper is the first code designed for the real symmetric Gaussian MAC channel to achieve the linear-feedback sum-capacity when the number of users is greater than 3.', '1204.4249-2-7-0': 'Channel Model and Preliminaries', '1204.4249-2-8-0': '## Mathematical notations', '1204.4249-2-9-0': 'Upper-case letters, their realizations by corresponding lower-case letters, denotes random variables.', '1204.4249-2-9-1': 'A real-valued random variable [MATH] is associated with a distribution [MATH] defined on the usual Borel [MATH]-algebra over [MATH], and we write [MATH].', '1204.4249-2-9-2': 'The cumulative distribution function (c.d.f.) of [MATH] is given by [MATH], and their inverse c.d.f is defined to be [MATH].', '1204.4249-2-9-3': 'The uniform probability distribution over [MATH] is denoted through [MATH].', '1204.4249-2-9-4': 'The composition function [MATH].', '1204.4249-2-9-5': 'In this paper, we use the following lemma:', '1204.4249-2-10-0': 'Lemma I: Let [MATH] be a continuous random variable with [MATH] and [MATH] be an uniform distribution random variable, i.e. [MATH] be statistical independent.', '1204.4249-2-10-1': 'Then [MATH] and [MATH].', '1204.4249-2-11-0': 'Refer [4] for the proof.', '1204.4249-2-12-0': '## Gaussian Multiple Access Channel with Feedback', '1204.4249-2-13-0': 'Consider the communication problem between [MATH] senders and a receiver over a multiple access channel with additive Gaussian noise (AWGN-MAC) when channel outputs are noiselessly fed back to all the senders (Figure 1).', '1204.4249-2-13-1': 'Each sender [MATH] wishes to reliably transmit a message [MATH] in the unit interval, i.e., [MATH] to the receiver.', '1204.4249-2-13-2': 'At each time [MATH], the output of the channel is [EQUATION] where [MATH] is the transmitted symbol by sender [MATH] at time [MATH], [MATH] is the output of the channel, and [MATH] is a discrete-time zero mean white Gaussian noise process with unit average power, i.e., [MATH] and is independent of [MATH].', '1204.4249-2-13-3': 'We assume that output symbols are casually fed back to the sender and the transmitted symbol [MATH] for sender [MATH] at time [MATH] can depend on both the message [MATH] and the previous channel output sequence [MATH].', '1204.4249-2-14-0': 'A transmission scheme for a Gaussian MAC channel is a set of [MATH] sequence of transmission functions [MATH], so that the input to the channel generated by the transmitter is given by [EQUATION]', '1204.4249-2-15-0': 'A decoding rule for a MAC channel is a sequence of measurable mappings [MATH], where [MATH] are the sets of all open intervals in [MATH].', '1204.4249-2-15-1': 'The [MATH]th component of this decoded vector, denoted as [MATH], refers as the decoded interval for the user [MATH].', '1204.4249-2-15-2': 'The error probabilities at time [MATH] associated with a transmission scheme and a decoding rule, is defined as [EQUATION] and the corresponding achievable rate vector at time [MATH] is defined to be [EQUATION]', '1204.4249-2-15-3': 'We say that a transmission scheme together with a decoding rule achieve a rate vector [MATH] over a Gaussian MAC channel if for all [MATH] we have [EQUATION]', '1204.4249-2-15-4': 'The rate vector is achieved within input power constraints [MATH], if in addition [EQUATION]', '1204.4249-2-15-5': 'An optimal fixed rate decoding rule for a MAC channel with rate region [MATH] is one that decodes a vector of fixed length intervals [MATH], whose marginal posteriori probabilities are maximal, i.e., [EQUATION]', '1204.4249-2-15-6': 'An optimal variable rate decoding rule with target error probabilities [MATH] is one that decodes a vector of minimal-length intervals [MATH] with accumulated marginal posteriori probabilities exceeds corresponding targets, i.e., [EQUATION]', '1204.4249-2-15-7': 'Both decoding rules make use of the marginal posterior distribution of the message point [MATH] which can calculate online at the transmitter [MATH] and the receiver.', '1204.4249-2-15-8': 'Refer [4] for more details.', '1204.4249-2-15-9': 'A proof that the achievability in the sense of [MATH] and [MATH] implies that the achievability in the standard framework are in the Appendix.', '1204.4249-2-16-0': 'Lemma II: The achievability in the definition [MATH] and [MATH] implies the achievability in the standard framework.', '1204.4249-2-17-0': 'Refer to the Appendix.', '1204.4249-2-18-0': '# Time-varying Posterior Matching Scheme', '1204.4249-2-19-0': "## Shayevitz and Feder's Posterior Matching Scheme", '1204.4249-2-20-0': 'In this part, we firstly review the posterior matching scheme proposed by Ofer Shayevitz and Meir Feder for point-to-point channel in [4].', '1204.4249-2-20-1': 'Specifically, the authors argued that after the receiver observed the output sequence [MATH], there is still some "missing information" that can be encapsulated in a random variable [MATH] with the following properties:', '1204.4249-2-21-0': '(i) [MATH] is statistically independent of [MATH].', '1204.4249-2-22-0': '(ii) The message point [MATH] can be a.s. uniquely recovered from [MATH]', '1204.4249-2-23-0': 'With that line of thought, they proposed a principle for generating the next channel input as follow:', '1204.4249-2-24-0': 'The transmission function [MATH] should be selected so that [MATH] is [MATH]-distributed, and is a fixed function of some random variable [MATH] satisfying properties [MATH] and [MATH].', '1204.4249-2-25-0': 'Lemma III: (Posterior Matching Scheme [4]).', '1204.4249-2-25-1': 'The following transmission scheme satisfies the posterior matching principle for any [MATH]: [EQUATION]', '1204.4249-2-25-2': 'Based on the transmission functions, the input to the channel is a sequence of random variables given by [EQUATION]', '1204.4249-2-25-3': 'Refer [4] for the proof.', '1204.4249-2-26-0': '## Time-varying Posterior Matching Scheme', '1204.4249-2-27-0': 'In this section, we propose a posterior matching scheme for additive Gaussian multiple access channel (MAC) with feedback, called time-varying posterior matching.', '1204.4249-2-27-1': 'Our encoding proposal is based on the two following lemmas:', '1204.4249-2-28-0': 'Lemma IV: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the output signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-2-28-1': 'Under the condition that the correlation matrix among transmitted symbols at each time slot [MATH] defined as [EQUATION] is a fixed function of [MATH].', '1204.4249-2-28-2': 'Then the posterior matching scheme in Lemma III at each transmitter [MATH] becomes a time-varying posterior matching which is given by [EQUATION] where the random variable [MATH] encapsulates the input power constraint, and [MATH] is the intended transmitted message at the transmitter [MATH].', '1204.4249-2-29-0': '[EQUATION]', '1204.4249-2-29-1': 'Then we have [EQUATION] where (a) follows the fact that the transmission function is continuous and monotone, and (b) follows that [MATH] is a linear combination with known coefficients of inputs [MATH], therefore the posterior distributions of [MATH] are only dependent on the correlations among transmitted symbols at time [MATH].', '1204.4249-2-29-2': 'Since we assume that the correlation matrix [MATH] is a fixed function of [MATH], hence the distributions of [MATH] are fixed function of [MATH].', '1204.4249-2-29-3': 'In other words, given [MATH], each [MATH] is independent of [MATH].', '1204.4249-2-30-0': 'Besides, from the Lemma I, we know that [MATH], hence [MATH].', '1204.4249-2-31-0': 'We refer the transmission scheme in (4) to as time-varying posterior matching scheme and prove an interesting converse result of the Lemma IV.', '1204.4249-2-32-0': 'Lemma V: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the received signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-2-32-1': 'Assuming that at each transmitter, a time-varying posterior matching encoder is deployed, so the recursive transmitted sequence at the transmitter [MATH] is given by [EQUATION] where [MATH] is the intended transmitted message.', '1204.4249-2-32-2': 'Then the correlation matrix among transmitted symbols at each time slot [MATH] defined as in the Lemma IV can be calculated online at the transmitters and the receiver, and more importantly the posterior distribution [MATH] can be calculated online at both the transmitter [MATH] and the receiver.', '1204.4249-2-33-0': 'Observe that [MATH] is a linear combination of [MATH].', '1204.4249-2-33-1': 'Moreover, with time-varying posterior matching transmission applied for Gaussian MAC, [MATH] is a linear combination of [MATH] (as we will see in the proof of the Theorem II below), so [MATH] is a linear combination of [MATH].', '1204.4249-2-33-2': 'Therefore, the correlation between [MATH] and [MATH] only depends on the correlations among transmitted symbols at time [MATH], i.e. [MATH].', '1204.4249-2-33-3': 'In other words, [MATH] is a function of [MATH], so the transmitters and receiver can be calculated the matrix [MATH] online at both transmitters and receiver.', '1204.4249-2-33-4': 'Moreover, from the relation [MATH], we see that the distribution of [MATH] is a function of all elements in the correlation matrix [MATH].', '1204.4249-2-33-5': 'That concludes the proof.', '1204.4249-2-34-0': "For a continuous point-to-point memoryless channel, the distribution [MATH] doesn't depend on [MATH], hence we have the posterior matching scheme for this case like the formula (16) in [4].", '1204.4249-2-34-1': 'However, in a MAC channel (for example additive white Gaussian MAC channel), where each received signal is a linear combination of all transmitted signals and Gaussian noise, the distribution [MATH] between the input [MATH] and the output may be dependent on [MATH].', '1204.4249-2-34-2': 'Therefore, time-varying posterior matching scheme may be the solution to overcome this problem.', '1204.4249-2-34-3': 'However, we will see from our proof in the Theorem I below the variable rate decoding rule, or Generalized Reverse Iterated Function System (GRIFS), can be applied at receiver to decode signals if and only if all the distributions [MATH] can be calculated online at the corresponding transmitters and receiver.', '1204.4249-2-34-4': 'With the result in the Lemma V, this condition is always satisfied when time-varying posterior matching encoding schemes used at transmitters.', '1204.4249-2-34-5': 'We will show in the next parts that using time-varying posterior matching scheme at transmitters can obtain optimal performances for some known cases.', '1204.4249-2-35-0': '# Error analysis for time-varying posterior matching scheme', '1204.4249-2-36-0': 'In this section, we analyze error performance for Gaussian MAC channel with feedback deployed the time-varying posterior matching scheme at the transmitters and variable-decoding rule at the receiver.', '1204.4249-2-37-0': 'Theorem I: Consider a real Gaussian MAC channel with [MATH] transmitters and one receiver without input power constraints.', '1204.4249-2-37-1': 'Assuming that at each transmitter [MATH], transmitted sequence [MATH] conforms the time-varying posterior matching rule, as following: [EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the Gaussian MAC channel which is a linear combination of all these transmitted signals at time [MATH].', '1204.4249-2-37-2': 'Let [MATH] and [MATH] as the global Lipschitz operator.', '1204.4249-2-37-3': 'Under the conditions that [EQUATION] define: [MATH] Then the rate region [MATH] is achievable by setting the target error probabilities [MATH] under the constraint [EQUATION]', '1204.4249-2-38-0': 'First, observe that since the distributions [MATH] can be calculated online at both transmitters and receiver by Lemma V, therefore [MATH] can be calculated online at both the transmitters and receiver.', '1204.4249-2-38-1': 'Denote [MATH], referred as to a Generalized Iterated Function System (GIFS) generated by the kernel sequence [MATH].', '1204.4249-2-38-2': 'For each [MATH], select a fixed interval [MATH] as the decoded interval with respect to [MATH].', '1204.4249-2-39-0': 'Define the corresponding interval at the origin to be [MATH] and set them to be the decoded interval for [MATH], and so the decoded interval for [MATH] are set to be [MATH].', '1204.4249-2-39-1': 'Let [EQUATION]', '1204.4249-2-39-2': 'From the condition [MATH] in the theorem, we know that [EQUATION]', '1204.4249-2-39-3': 'Observe that: [EQUATION] where [MATH].', '1204.4249-2-40-0': 'On the other hand, since [MATH], then for any arbitrarily small [MATH], there exists an [MATH] such that [MATH].', '1204.4249-2-40-1': 'Let [MATH], and [MATH].', '1204.4249-2-40-2': "For any fixed number [MATH], from [MATH] we have: [EQUATION] for any [MATH] arbitrarily small, where [MATH] follows the Markov's inequality, [MATH] follows [MATH], and [MATH] is a recursive application of the preceding transitions, [MATH] follows [MATH] above and recursive applications of the preceding transitions.", '1204.4249-2-40-3': 'From [MATH], it is easy to see that a sufficient condition for [MATH] is given by choosing [MATH].', '1204.4249-2-40-4': 'Observe that [MATH] can be made arbitrarily small, therefore, the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-2-40-5': 'Since [MATH] depends only on the length of [MATH], so we can choose [MATH].', '1204.4249-2-40-6': 'Furthermore, from the Lemma I, we are easy to come to conclusion that [MATH].', '1204.4249-2-40-7': 'Using the standard bounds for Gaussian distribution we obtain [EQUATION]', '1204.4249-2-41-0': '# A Posterior Matching Scheme for Gaussian MAC Channel with Feedback', '1204.4249-2-42-0': 'In this section, we consider a real Gaussian MAC channel with [MATH] receivers and input power constraints [MATH] at the transmitters [MATH], respectively as defined in the section II.', '1204.4249-2-42-1': 'Our encoding scheme for this channel as following:', '1204.4249-2-43-0': 'Encoding:', '1204.4249-2-44-0': '[EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the MAC channel.', '1204.4249-2-45-0': 'where [MATH] is the column [MATH] of the Hadamard matrix [MATH] by [MATH], which is a spreading code for CDMA systems.', '1204.4249-2-46-0': "We refer this encoding strategy as Gaussian MAC channel's posterior matching feedback coding and decoding strategy.", '1204.4249-2-47-0': "Theorem II: Using the Gaussian MAC channel's posterior matching feedback coding and decoding strategy above, the rate region [MATH] is achievable for Gaussian MAC channel with feedback, where [EQUATION] by setting the target error probabilities [EQUATION] where [EQUATION]", '1204.4249-2-48-0': 'Applying the Lemma I, we see that for any [MATH] then: [EQUATION]', '1204.4249-2-48-1': 'Observe that, by this transmission strategy, each transmitter [MATH] transmits [MATH] at time [MATH] with [MATH], thus the input power constraints at all transmitters are always satisfied at each transmission time [MATH] Moreover, the output at receiver at time [MATH] will be [EQUATION]', '1204.4249-2-48-2': 'Thus [EQUATION]', '1204.4249-2-48-3': 'Finally, we have: [EQUATION] and [EQUATION]', '1204.4249-2-48-4': 'Therefore, [EQUATION] where [EQUATION] and [EQUATION]', '1204.4249-2-48-5': 'Moreover, from [MATH] we have: [EQUATION] thus, [EQUATION]', '1204.4249-2-48-6': 'From [MATH] and [MATH] we obtain: [EQUATION]', '1204.4249-2-48-7': 'Hence, [EQUATION]', '1204.4249-2-48-8': 'Finally, we have: [EQUATION]', '1204.4249-2-48-9': 'If we can achieve [EQUATION] then [EQUATION]', '1204.4249-2-48-10': 'This means that the condition in the Theorem I is satisfied, which leads to the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-2-48-11': 'Note that this capacity region is obtained by setting the target error probabilities [MATH] under the constraints [EQUATION] which have the well-known double-exponential behavior.', '1204.4249-2-49-0': 'In the following, we will specify achievable rate regions, and error probabilities for two cases: the general two-user Gaussian MAC channel, and the real symmetric Gaussian MAC channel when the number of users is arbitrary.', '1204.4249-2-50-0': 'From the Theorem II, we see that the strategy to design posterior encoding scheme for multiple access channels with feedback is to find the sequences [MATH] such that [EQUATION] for all [MATH].', '1204.4249-2-51-0': 'Case 1: Two-user Gaussian MAC channel with feedback.', '1204.4249-2-52-0': "To show that Ozarow's coding scheme [3] is a special case of our posterior matching framework, we can set [MATH] and later prove that [MATH].", '1204.4249-2-52-1': 'Observe that the constraint [MATH] can be also checked to be satisfied by this setting since [EQUATION] for [MATH] and [EQUATION] for [MATH].', '1204.4249-2-53-0': 'Now we need to find the recursion of [MATH] and [MATH] in this case.', '1204.4249-2-53-1': 'Observe that the output sequence: [EQUATION] where [MATH] is noise process and [MATH].', '1204.4249-2-54-0': 'From (12), (13) we have [EQUATION] and [EQUATION]', '1204.4249-2-54-1': 'Therefore, we obtain: [EQUATION] and [EQUATION]', '1204.4249-2-55-0': 'Finally, the time-varying posterior matching encoding scheme for Gaussian MAC channel with feedback in this special case as following:', '1204.4249-2-56-0': 'Achievable rate region and error analysis:', '1204.4249-2-57-0': 'For this special case, we have [EQUATION] and [EQUATION]', '1204.4249-2-57-1': 'Apply the Theorem II above, we obtain the achievable rate region for Gaussian MAC channel with two users as [MATH] where [EQUATION]', '1204.4249-2-57-2': 'Similarly, [EQUATION] by setting the target error probability to [EQUATION] and [EQUATION]', '1204.4249-2-57-3': 'Much like Ozarow in [MATH], at the reception [MATH], the receiver adds an independent random variable [MATH] before feeding back the first receiver signal to the transmitters 1 and 2 to set [MATH], where [MATH] is the biggest solution in [MATH] of the following equation: [EQUATION]', '1204.4249-2-57-4': 'By this changing, from [MATH] we see that [MATH], so [MATH] as mentioned above.', '1204.4249-2-57-5': 'We also have [MATH], where [MATH] is a positive solution of the equation [MATH].', '1204.4249-2-57-6': 'Replace this result to [MATH] and combine with [MATH], we have: [EQUATION] where [MATH] is defined above.', '1204.4249-2-57-7': "We see that, all the results are the same as Ozarow's results in [3].", '1204.4249-2-57-8': 'So our posterior matching encoding scheme is optimal for Gaussian channel MAC with two users.', '1204.4249-2-58-0': 'Case 2: M-user symmetric Gaussian MAC channel with feedback.', '1204.4249-2-58-1': 'We consider symmetric case, where [MATH].', '1204.4249-2-59-0': 'Achievable rate region and error analysis:', '1204.4249-2-60-0': 'Assuming that all the transmitted messages are statistically independent.', '1204.4249-2-60-1': 'We will prove by induction that the normalized covariance [MATH] have all the columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] are symmetric positive definite for all [MATH]', '1204.4249-2-61-0': 'Indeed, with the assumption all the transmitted information messages are statistically independent, we will have [MATH], which is an identity matrix of size [MATH].', '1204.4249-2-61-1': 'Therefore, it is obvious that all the columns of the Hadamard matrix [MATH] by [MATH] are eigenvectors of the matrix [MATH] and that [MATH] is a positive definite matrix.', '1204.4249-2-62-0': 'Now, assume that [MATH] has all columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is positive definite for some [MATH].', '1204.4249-2-62-1': 'Since we assumed that [MATH] is symmetric positive definite matrix, all its eigenvalues are greater than [MATH].', '1204.4249-2-62-2': 'Denote [MATH] are [MATH] columns of the Hadamard matrix [MATH].', '1204.4249-2-62-3': 'By this encoding scheme, we set the vector [MATH].', '1204.4249-2-62-4': 'Assume that [MATH] is the eigenvalue of [MATH] associated with this eigenvector.', '1204.4249-2-62-5': 'We have [EQUATION]', '1204.4249-2-62-6': 'On the other hand, we also have [EQUATION] where [MATH] is the [MATH]th column of the matrix [MATH].', '1204.4249-2-62-7': 'Note that [MATH], this means that [EQUATION]', '1204.4249-2-62-8': 'Hence, [EQUATION]', '1204.4249-2-62-9': 'Substitute [MATH] into [MATH] we obtain: [EQUATION]', '1204.4249-2-62-10': 'From [MATH] and [MATH], we obtain: [EQUATION] and [EQUATION] for all [MATH].', '1204.4249-2-63-0': 'Observe that from the proof of the Theorem II above, then [EQUATION].', '1204.4249-2-63-1': 'Define the correlation coefficient between [MATH] and [MATH] [EQUATION]', '1204.4249-2-63-2': 'Then, we have [EQUATION]', '1204.4249-2-63-3': 'By replacing all values of [MATH] calculated above, we obtain [EQUATION] for all [MATH].', '1204.4249-2-64-0': 'Since we assumed that [MATH] is symmetric positive definite matrix, hence [MATH], hence [MATH].', '1204.4249-2-64-1': 'Therefore, from [MATH], it is easy to see that we will have [MATH], or [MATH] is also a symmetric matrix.', '1204.4249-2-65-0': 'Moreover, from [MATH], we also obtain [EQUATION]', '1204.4249-2-65-1': 'Denote [MATH].', '1204.4249-2-65-2': 'We see that the columns of [MATH] creates [MATH] linearly independent eigenvectors of the matrix [MATH].', '1204.4249-2-65-3': 'Moreover, we also have [EQUATION]', '1204.4249-2-65-4': 'Note that since all columns of [MATH] are eigenvectors of the matrix [MATH], so all the columns of the matrix [MATH] are also eigenvectors of the matrix [MATH], hence we has the following eigenvalue decomposition [EQUATION] where [MATH] is a diagonal matrix.', '1204.4249-2-66-0': 'Moreover, we have [EQUATION] where [EQUATION]', '1204.4249-2-66-1': 'From (26), (27), and (28), we must have [MATH] must be a diagonal matrix since the right side of [MATH] is a diagonal matrix.', '1204.4249-2-66-2': 'Hence, all columns of the matrix [MATH] are eigenvectors of the matrix [MATH].', '1204.4249-2-67-0': 'Assuming [MATH] are [MATH] eigenvalues corresponding to eigenvectors which are columns of the matrix [MATH].', '1204.4249-2-67-1': 'By this notation, we see that [MATH].', '1204.4249-2-68-0': 'Combining with [MATH] with [MATH], and [MATH] we obtain [EQUATION] for all [MATH].', '1204.4249-2-69-0': 'Since we assumed that all eigenvalues of [MATH] are positive ([MATH] symmetric positive definite), from [MATH] we see that all eigenvalues of [MATH] are also positive.', '1204.4249-2-69-1': 'Note that we also confirmed that [MATH] is symmetric above, therefore [MATH] is a symmetric positive definite matrix.', '1204.4249-2-69-2': 'In short, if [MATH] is a symmetric positive definite matrix and has all columns of the Hadamard matrix as its eigenvectors, then [MATH] has all these properties.', '1204.4249-2-69-3': 'This concludes our proof by induction.', '1204.4249-2-70-0': 'According to the Lemma 1, [MATH], the sequence [MATH] converges to a fixed point [MATH], which is the solution in [MATH] of the following equation [EQUATION] when [EQUATION]', '1204.4249-2-70-1': 'When the condition (32) satisfies, from (22), (23) we obtain [EQUATION]', '1204.4249-2-70-2': 'Observe that [MATH], so [MATH].', '1204.4249-2-70-3': 'Therefore, the constraints in the Theorem II is satisfied.', '1204.4249-2-70-4': 'Applying the result of this theorem, we have any rate less than [EQUATION] is achievable, for all [MATH].', '1204.4249-2-70-5': 'Hence, any sum rate which is less than [EQUATION] is achievable, where [MATH] is solution in the [MATH] of the equation [MATH].', '1204.4249-2-70-6': 'This result coincides with the formula (68) in [14].', '1204.4249-2-70-7': 'The paper [16] proves that this achievable sum rate is optimal for the class of linear feedback coding.', '1204.4249-2-71-0': '# Conclusion', '1204.4249-2-72-0': 'A posterior matching based encoding-decoding strategy for general Gaussian MAC channel with feedback was proposed, and achievable rate region, error performance were drawn.', '1204.4249-2-72-1': 'Finally, we analyzed error performance of the proposed posterior encoding scheme and showed that the time-varying posterior matching scheme and variable rate decoding ideas can be applied to Gaussian MAC channel and obtain optimal performances.', '1204.4249-2-72-2': 'Specifically, the proposed encoding scheme achieves the capacity of two-user feedback Gaussian MAC channel as well as linear-feedback sum-rate for symmetric Gaussian MAC channel with feedback where the number of users is arbitrary.', '1204.4249-2-72-3': "Moreover, by the encoding scheme's structure, which uses the spreading codes like the Hadamard matrix, our encoding scheme can be directly applied to CDMA systems with feedback.", '1204.4249-2-72-4': 'Finally, by analyzing all arguments in the theorem I, the time-varying posterior matching scheme approach in this paper might be applied for other Gaussian and non-Gaussian multiuser channels to achieve optimal performances.', '1204.4249-2-72-5': '[Proof of the Lemma II]', '1204.4249-2-73-0': 'We use the same line argument as Lemma 1 [4].', '1204.4249-2-73-1': 'Assume we are given a transmission scheme with [MATH] transmission functions [MATH] and a decoding rule which are known to achieve the rate vector [MATH].', '1204.4249-2-73-2': 'For simplicity, we assume that the decoding rule is fixed rate [MATH] for all [MATH]), since any variable rate decoding rule can be easily mapped into a fixed rate rule that achieves the same rate vector.', '1204.4249-2-73-3': 'It is easy to see that in order to prove that the above translates into achievability for some rate vector [MATH] in the standard framework, it is enough to show we can find [MATH] sequences [MATH] and such that we have the uniform achievability over [MATH], i.e., [EQUATION]', '1204.4249-2-73-4': 'We now show how [MATH] can be constructed for any [MATH].', '1204.4249-2-73-5': 'Let [MATH] be the (average) error probability associated with our scheme and the fixed rate vector [MATH].', '1204.4249-2-73-6': 'Define [EQUATION] and write [EQUATION] and so we have that [MATH].', '1204.4249-2-73-7': 'It is now easy to see that if we want to select [MATH] such that [MATH], and also [MATH], then a sufficient condition is that [MATH] for some positive [MATH].', '1204.4249-2-73-8': 'This condition can be written as [EQUATION]', '1204.4249-2-73-9': 'At the same time, we also have by definition [EQUATION]'} | [['1204.4249-1-48-0', '1204.4249-2-49-0'], ['1204.4249-1-41-0', '1204.4249-2-42-0'], ['1204.4249-1-64-0', '1204.4249-2-65-0'], ['1204.4249-1-64-2', '1204.4249-2-65-2'], ['1204.4249-1-64-3', '1204.4249-2-65-3'], ['1204.4249-1-64-4', '1204.4249-2-65-4'], ['1204.4249-1-30-0', '1204.4249-2-30-0'], ['1204.4249-1-51-0', '1204.4249-2-52-0'], ['1204.4249-1-51-1', '1204.4249-2-52-1'], ['1204.4249-1-4-0', '1204.4249-2-4-0'], ['1204.4249-1-4-1', '1204.4249-2-4-1'], ['1204.4249-1-4-2', '1204.4249-2-4-2'], ['1204.4249-1-4-4', '1204.4249-2-4-4'], ['1204.4249-1-4-5', '1204.4249-2-4-5'], ['1204.4249-1-47-0', '1204.4249-2-48-0'], ['1204.4249-1-47-1', '1204.4249-2-48-1'], ['1204.4249-1-47-3', '1204.4249-2-48-3'], ['1204.4249-1-47-4', '1204.4249-2-48-4'], ['1204.4249-1-47-5', '1204.4249-2-48-5'], ['1204.4249-1-47-6', 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'1204.4249-2-31-0'], ['1204.4249-1-13-0', '1204.4249-2-13-0'], ['1204.4249-1-13-4', '1204.4249-2-13-3'], ['1204.4249-1-0-0', '1204.4249-2-0-0'], ['1204.4249-1-0-1', '1204.4249-2-0-1'], ['1204.4249-1-0-2', '1204.4249-2-0-2'], ['1204.4249-1-0-4', '1204.4249-2-0-4'], ['1204.4249-1-0-6', '1204.4249-2-0-6'], ['1204.4249-1-32-0', '1204.4249-2-32-0'], ['1204.4249-1-32-1', '1204.4249-2-32-1'], ['1204.4249-1-68-0', '1204.4249-2-69-0'], ['1204.4249-1-68-1', '1204.4249-2-69-1'], ['1204.4249-1-68-2', '1204.4249-2-69-2'], ['1204.4249-1-68-3', '1204.4249-2-69-3'], ['1204.4249-1-69-0', '1204.4249-2-70-0'], ['1204.4249-1-69-1', '1204.4249-2-70-1'], ['1204.4249-1-69-2', '1204.4249-2-70-2'], ['1204.4249-1-69-3', '1204.4249-2-70-3'], ['1204.4249-1-69-4', '1204.4249-2-70-4'], ['1204.4249-1-69-5', '1204.4249-2-70-5'], ['1204.4249-1-69-6', '1204.4249-2-70-6'], ['1204.4249-1-69-7', '1204.4249-2-70-7'], ['1204.4249-1-72-0', '1204.4249-2-73-0'], ['1204.4249-1-72-1', '1204.4249-2-73-1'], ['1204.4249-1-72-2', '1204.4249-2-73-2'], ['1204.4249-1-72-4', '1204.4249-2-73-4'], ['1204.4249-1-72-5', '1204.4249-2-73-5'], ['1204.4249-1-72-6', '1204.4249-2-73-6'], ['1204.4249-1-72-7', '1204.4249-2-73-7'], ['1204.4249-1-72-8', '1204.4249-2-73-8'], ['1204.4249-1-72-9', '1204.4249-2-73-9'], ['1204.4249-2-16-0', '1204.4249-3-16-0'], ['1204.4249-2-27-0', '1204.4249-3-27-0'], ['1204.4249-2-49-0', '1204.4249-3-50-0'], ['1204.4249-2-36-0', '1204.4249-3-36-0'], ['1204.4249-2-58-0', '1204.4249-3-59-0'], ['1204.4249-2-58-1', '1204.4249-3-59-1'], ['1204.4249-2-9-0', '1204.4249-3-9-0'], ['1204.4249-2-9-1', '1204.4249-3-9-1'], ['1204.4249-2-9-2', '1204.4249-3-9-2'], ['1204.4249-2-9-3', '1204.4249-3-9-3'], ['1204.4249-2-9-4', '1204.4249-3-9-4'], ['1204.4249-2-30-0', '1204.4249-3-30-0'], ['1204.4249-2-0-0', '1204.4249-3-0-0'], ['1204.4249-2-0-1', '1204.4249-3-0-1'], ['1204.4249-2-0-2', '1204.4249-3-0-2'], ['1204.4249-2-0-3', '1204.4249-3-0-3'], ['1204.4249-2-0-4', '1204.4249-3-0-4'], ['1204.4249-2-0-5', 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'1204.4249-3-49-1'], ['1204.4249-2-48-3', '1204.4249-3-49-3'], ['1204.4249-2-48-4', '1204.4249-3-49-4'], ['1204.4249-2-48-5', '1204.4249-3-49-5'], ['1204.4249-2-48-6', '1204.4249-3-49-6'], ['1204.4249-2-48-8', '1204.4249-3-49-8'], ['1204.4249-2-48-9', '1204.4249-3-49-9'], ['1204.4249-2-48-10', '1204.4249-3-49-10'], ['1204.4249-2-48-11', '1204.4249-3-49-11'], ['1204.4249-2-13-0', '1204.4249-3-13-0'], ['1204.4249-2-13-1', '1204.4249-3-13-1'], ['1204.4249-2-13-2', '1204.4249-3-13-2'], ['1204.4249-2-13-3', '1204.4249-3-13-3'], ['1204.4249-2-4-0', '1204.4249-3-4-0'], ['1204.4249-2-4-1', '1204.4249-3-4-1'], ['1204.4249-2-4-2', '1204.4249-3-4-2'], ['1204.4249-2-4-4', '1204.4249-3-4-4'], ['1204.4249-2-44-0', '1204.4249-3-45-0'], ['1204.4249-2-60-0', '1204.4249-3-61-0'], ['1204.4249-2-60-1', '1204.4249-3-61-1'], ['1204.4249-2-22-0', '1204.4249-3-22-0'], ['1204.4249-2-14-0', '1204.4249-3-14-0'], ['1204.4249-2-52-0', '1204.4249-3-53-0'], ['1204.4249-2-52-1', '1204.4249-3-53-1'], 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'1204.4249-2-54-0', '1204.4249-2-54-1', '1204.4249-2-55-0', '1204.4249-2-56-0', '1204.4249-2-59-0', '1204.4249-2-62-5', '1204.4249-2-62-8', '1204.4249-2-65-1', '1204.4249-2-68-0', '1204.4249-3-1-0', '1204.4249-3-2-0', '1204.4249-3-7-0', '1204.4249-3-9-5', '1204.4249-3-10-1', '1204.4249-3-11-0', '1204.4249-3-17-0', '1204.4249-3-20-1', '1204.4249-3-21-0', '1204.4249-3-23-0', '1204.4249-3-27-1', '1204.4249-3-29-0', '1204.4249-3-39-1', '1204.4249-3-40-1', '1204.4249-3-43-1', '1204.4249-3-44-0', '1204.4249-3-49-2', '1204.4249-3-49-7', '1204.4249-3-52-0', '1204.4249-3-55-0', '1204.4249-3-55-1', '1204.4249-3-56-0', '1204.4249-3-57-0', '1204.4249-3-60-0', '1204.4249-3-63-5', '1204.4249-3-63-8', '1204.4249-3-66-1', '1204.4249-3-69-0', '1204.4249-4-1-0', '1204.4249-4-2-0', '1204.4249-4-9-5', '1204.4249-4-10-1', '1204.4249-4-11-0', '1204.4249-4-20-0', '1204.4249-4-23-1', '1204.4249-4-24-0', '1204.4249-4-26-0', '1204.4249-4-30-1', '1204.4249-4-33-1', '1204.4249-4-36-1', '1204.4249-4-43-1', '1204.4249-4-44-1', '1204.4249-4-47-1', '1204.4249-4-48-0', '1204.4249-4-52-3', '1204.4249-4-52-8', '1204.4249-4-55-0', '1204.4249-4-58-0', '1204.4249-4-58-1', '1204.4249-4-59-0', '1204.4249-4-60-0', '1204.4249-4-63-0', '1204.4249-4-67-5', '1204.4249-4-67-8', '1204.4249-4-68-0', '1204.4249-4-70-1', '1204.4249-4-73-0', '1204.4249-5-1-0', '1204.4249-5-2-0', '1204.4249-5-9-5', '1204.4249-5-10-1', '1204.4249-5-11-0', '1204.4249-5-20-0', '1204.4249-5-23-1', '1204.4249-5-24-0', '1204.4249-5-26-0', '1204.4249-5-30-1', '1204.4249-5-33-1', '1204.4249-5-45-1', '1204.4249-5-49-1', '1204.4249-5-53-3', '1204.4249-5-53-8', '1204.4249-5-56-0', '1204.4249-5-59-0', '1204.4249-5-59-1', '1204.4249-5-60-0', '1204.4249-5-61-0', '1204.4249-5-64-0', '1204.4249-5-68-5', '1204.4249-5-68-8', '1204.4249-5-68-15', '1204.4249-5-70-1', '1204.4249-5-73-0', '1204.4249-5-75-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1204.4249 | {'1204.4249-3-0-0': 'Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design optimal coding schemes for general point-to-point memoryless channels with feedback.', '1204.4249-3-0-1': 'In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC channel with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme.', '1204.4249-3-0-2': "The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder's posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots.", '1204.4249-3-0-3': "It turns out that the well-known Ozarow's encoding scheme, which obtains the capacity of two-user Gaussian channel, is a special case of our extended posterior matching framework as the Schalkwijk-Kailath's scheme is a special case of the point-to-point posterior matching mentioned above.", '1204.4249-3-0-4': 'Furthermore, our designed posterior matching also obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC channel.', '1204.4249-3-0-5': 'Besides, the encoding scheme in this paper is designed for the real Gaussian MAC channel to obtain that performance, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC channel.', '1204.4249-3-0-6': 'More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels with feedback.', '1204.4249-3-1-0': 'Gaussian Multiple Access Channel, Feedback, Posterior Matching.', '1204.4249-3-2-0': 'Introduction', '1204.4249-3-3-0': 'In the early work [9], Shannon proved that feedback could not increase the capacity of point-to-point memoryless channel.', '1204.4249-3-3-1': 'However, feedback can improve error performance and simplify the transmission scheme for this kind of channel.', '1204.4249-3-3-2': 'In [10], Hostein proposed a simple sequential transmission scheme, which achieves the capacity of Binary Symmetric Channel (BSC) and provides large error exponents than traditional fixed length block coding.', '1204.4249-3-3-3': 'Besides, Schalkwijk and Kailath also showed that feedback could improve error performance and/or simplify the transmission scheme for point-to-point Gaussian channel [7], [8].', '1204.4249-3-3-4': 'For the Gaussian multiuser channels, the situation is more interesting.', '1204.4249-3-3-5': 'In [12], Gaarder and Wolf proved that feedback could enlarge the capacity region of the multiple access channel (MAC).', '1204.4249-3-3-6': 'Ozarow [3] successfully constructed a simple coding scheme for the two user Gaussian MAC channel with feedback and reaffirmed that feedback could increase the capacity of the channel.', '1204.4249-3-3-7': 'Furthermore, Kramer devised a code for the complex Gaussian MAC channel based on a beautiful property of the circulant matrix that has all the columns of the DFT (Discrete Fourier Transform) matrix as its eigenvectors [14].', '1204.4249-3-3-8': 'This code was proved to obtain the linear-feedback sum-capacity of the symmetric Gaussian MAC channel with feedback [16].', '1204.4249-3-3-9': "Besides, by using the control-theoretic approach to communications with feedback, Ehsan Ardestanizadeh and Massimo Fraceschetii [17] also proposed a linear code which has the same performance as Kramer's code for symmetric complex Gaussian MAC channel.", '1204.4249-3-3-10': "However, as we mentioned, both these codes were designed for the complex Gaussian MAC channel and don't work for the real Gaussian MAC channel.", '1204.4249-3-4-0': 'Recently, Ofer Shayevitz and Meir Feder [1], [2], and [4] have discovered an underlying principle between the Hostein and Schalkwijk-Kailth schemes in a simple encoding scheme called posterior matching scheme for general point-to-point memoryless channels.', '1204.4249-3-4-1': "The ideas of posterior matching is that the transmitter encapsulates the information that receiver does not know up to present time in one random variable and then transmit that random variable to the receiver in the next transmission to refine the receiver's knowledge.", '1204.4249-3-4-2': 'The distribution of that variable will be selected in a way such that the input constraint is satisfied.', '1204.4249-3-4-3': 'Later, J.H.Bae and A.Anastasopolous extended this scheme for the finite-state channel with feedback by using another approach [11].', '1204.4249-3-4-4': 'One interesting open problem is to extend the Ofer Shayevitz and Meir Feder posterior matching scheme for multiuser cases.', '1204.4249-3-4-5': 'In this paper, using the same approach as Ofer Shayevitz and Meir Feder used for point-to-point memoryless channels, we propose a posterior matching based encoding and decoding strategy for the real Gaussian MAC channel, referred to as a time-varying posterior matching scheme, analyze the error probabilities for all encoding-decoding schemes designed by using these strategies.', '1204.4249-3-5-0': 'We analyze the achievable rate region and error performance of encoding and decoding schemes using these strategies by defining a generalized iterated function systems (GIFS) which has the generalized average contractive property (average contractive in the limit).', '1204.4249-3-5-1': 'Refer our Theorem I for more details.', '1204.4249-3-5-2': 'Note that our imposed constraint is less strict than the constraint that Ofer Shayevitz and Meir Feder imposed to analyze the point-to-point memoryless channels.', '1204.4249-3-5-3': 'Specifically, in Theorem 6 in [4], Ofer Shayevitz and Meir Feder used the relations between the information rates and contraction properties of the iterated function systems (IFS) to analyze the error probability for point-to-point cases.', '1204.4249-3-5-4': 'For the continuous cases, they assumed that the reverse iterated function system (RIFS), generated by the kernel [MATH] and controlled by the identically distributed output sequence [MATH], has the average contractive property to analyze the error performance of their posterior matching schemes.', '1204.4249-3-5-5': 'That assumption requires the distribution at the output of the point-to-point memoryless channel be identically distributed when using their proposed encoding schemes.', '1204.4249-3-5-6': 'This also means that if the output distribution is not identically distributed, the error analysis in Theorem 6 in [4] cannot apply.', '1204.4249-3-5-7': 'For example, this situation happens with our proposed matching schemes for MAC channels in this paper.', '1204.4249-3-6-0': 'Finally, we illustrate our strategies by designing an encoding scheme that obtains optimal performances for the real Gaussian MAC channel.', '1204.4249-3-6-1': "Specifically, our proposed code obtains the same performance as the Ozarow's code [3] for the general two-user Gaussian channel, so it achieves the capacity of this channel.", '1204.4249-3-6-2': "For the case when the number of users is greater than 3, our proposed code obtains the same performance as the Kramer's code in the sense of sum-rate, so it is optimal among linear codes with respect to sum rate capacity.", '1204.4249-3-6-3': 'To the best of our knowledge, the time-varying posterior matching scheme in this paper is the first code designed for the real symmetric Gaussian MAC channel to achieve the linear-feedback sum-capacity when the number of users is greater than 3.', '1204.4249-3-7-0': 'Channel Model and Preliminaries', '1204.4249-3-8-0': '## Mathematical notations', '1204.4249-3-9-0': 'Upper-case letters, their realizations by corresponding lower-case letters, denotes random variables.', '1204.4249-3-9-1': 'A real-valued random variable [MATH] is associated with a distribution [MATH] defined on the usual Borel [MATH]-algebra over [MATH], and we write [MATH].', '1204.4249-3-9-2': 'The cumulative distribution function (c.d.f.) of [MATH] is given by [MATH], and their inverse c.d.f is defined to be [MATH].', '1204.4249-3-9-3': 'The uniform probability distribution over [MATH] is denoted through [MATH].', '1204.4249-3-9-4': 'The composition function [MATH].', '1204.4249-3-9-5': 'In this paper, we use the following lemma:', '1204.4249-3-10-0': 'Lemma I: Let [MATH] be a continuous random variable with [MATH] and [MATH] be an uniform distribution random variable, i.e. [MATH] be statistical independent.', '1204.4249-3-10-1': 'Then [MATH] and [MATH].', '1204.4249-3-11-0': 'Refer [4] for the proof.', '1204.4249-3-12-0': '## Gaussian Multiple Access Channel with Feedback', '1204.4249-3-13-0': 'Consider the communication problem between [MATH] senders and a receiver over a multiple access channel with additive Gaussian noise (AWGN-MAC) when channel outputs are noiselessly fed back to all the senders (Figure 1).', '1204.4249-3-13-1': 'Each sender [MATH] wishes to reliably transmit a message [MATH] in the unit interval, i.e., [MATH] to the receiver.', '1204.4249-3-13-2': 'At each time [MATH], the output of the channel is [EQUATION] where [MATH] is the transmitted symbol by sender [MATH] at time [MATH], [MATH] is the output of the channel, and [MATH] is a discrete-time zero mean white Gaussian noise process with unit average power, i.e., [MATH] and is independent of [MATH].', '1204.4249-3-13-3': 'We assume that output symbols are casually fed back to the sender and the transmitted symbol [MATH] for sender [MATH] at time [MATH] can depend on both the message [MATH] and the previous channel output sequence [MATH].', '1204.4249-3-14-0': 'A transmission scheme for a Gaussian MAC channel is a set of [MATH] sequence of transmission functions [MATH], so that the input to the channel generated by the transmitter is given by [EQUATION]', '1204.4249-3-15-0': 'A decoding rule for a MAC channel is a sequence of measurable mappings [MATH], where [MATH] are the sets of all open intervals in [MATH].', '1204.4249-3-15-1': 'The [MATH]th component of this decoded vector, denoted as [MATH], refers as the decoded interval for the user [MATH].', '1204.4249-3-15-2': 'The error probabilities at time [MATH] associated with a transmission scheme and a decoding rule, is defined as [EQUATION] and the corresponding achievable rate vector at time [MATH] is defined to be [EQUATION]', '1204.4249-3-15-3': 'We say that a transmission scheme together with a decoding rule achieve a rate vector [MATH] over a Gaussian MAC channel if for all [MATH] we have [EQUATION]', '1204.4249-3-15-4': 'The rate vector is achieved within input power constraints [MATH], if in addition [EQUATION]', '1204.4249-3-15-5': 'An optimal fixed rate decoding rule for a MAC channel with rate region [MATH] is one that decodes a vector of fixed length intervals [MATH], whose marginal posteriori probabilities are maximal, i.e., [EQUATION]', '1204.4249-3-15-6': 'An optimal variable rate decoding rule with target error probabilities [MATH] is one that decodes a vector of minimal-length intervals [MATH] with accumulated marginal posteriori probabilities exceeds corresponding targets, i.e., [EQUATION]', '1204.4249-3-15-7': 'Both decoding rules make use of the marginal posterior distribution of the message point [MATH] which can calculate online at the transmitter [MATH] and the receiver.', '1204.4249-3-15-8': 'Refer [4] for more details.', '1204.4249-3-15-9': 'A proof that the achievability in the sense of [MATH] and [MATH] implies that the achievability in the standard framework are in the Appendix.', '1204.4249-3-16-0': 'Lemma II: The achievability in the definition [MATH] and [MATH] implies the achievability in the standard framework.', '1204.4249-3-17-0': 'Refer to the Appendix.', '1204.4249-3-18-0': '# Time-varying Posterior Matching Scheme', '1204.4249-3-19-0': "## Shayevitz and Feder's Posterior Matching Scheme", '1204.4249-3-20-0': 'In this part, we firstly review the posterior matching scheme proposed by Ofer Shayevitz and Meir Feder for point-to-point channel in [4].', '1204.4249-3-20-1': 'Specifically, the authors argued that after the receiver observed the output sequence [MATH], there is still some "missing information" that can be encapsulated in a random variable [MATH] with the following properties:', '1204.4249-3-21-0': '(i) [MATH] is statistically independent of [MATH].', '1204.4249-3-22-0': '(ii) The message point [MATH] can be a.s. uniquely recovered from [MATH]', '1204.4249-3-23-0': 'With that line of thought, they proposed a principle for generating the next channel input as follow:', '1204.4249-3-24-0': 'The transmission function [MATH] should be selected so that [MATH] is [MATH]-distributed, and is a fixed function of some random variable [MATH] satisfying properties [MATH] and [MATH].', '1204.4249-3-25-0': 'Lemma III: (Posterior Matching Scheme [4]).', '1204.4249-3-25-1': 'The following transmission scheme satisfies the posterior matching principle for any [MATH]: [EQUATION]', '1204.4249-3-25-2': 'Based on the transmission functions, the input to the channel is a sequence of random variables given by [EQUATION]', '1204.4249-3-25-3': 'Refer [4] for the proof.', '1204.4249-3-26-0': '## Time-varying Posterior Matching Scheme', '1204.4249-3-27-0': 'In this section, we propose a posterior matching scheme for additive Gaussian multiple access channel (MAC) with feedback, called time-varying posterior matching.', '1204.4249-3-27-1': 'Our encoding proposal is based on the two following lemmas:', '1204.4249-3-28-0': 'Lemma IV: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the output signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-3-28-1': 'Under the condition that the correlation matrix among transmitted symbols at each time slot [MATH] defined as [EQUATION] is a fixed function of [MATH].', '1204.4249-3-28-2': 'Then the posterior matching scheme in Lemma III at each transmitter [MATH] becomes a time-varying posterior matching which is given by [EQUATION] where the random variable [MATH] encapsulates the input power constraint, and [MATH] is the intended transmitted message at the transmitter [MATH].', '1204.4249-3-29-0': '[EQUATION]', '1204.4249-3-29-1': 'Then we have [EQUATION] where (a) follows the fact that the transmission function is continuous and monotone, and (b) follows that [MATH] is a linear combination with known coefficients of inputs [MATH], therefore the posterior distributions of [MATH] are only dependent on the correlations among transmitted symbols at time [MATH].', '1204.4249-3-29-2': 'Since we assume that the correlation matrix [MATH] is a fixed function of [MATH], hence the distributions of [MATH] are fixed function of [MATH].', '1204.4249-3-29-3': 'In other words, given [MATH], each [MATH] is independent of [MATH].', '1204.4249-3-30-0': 'Besides, from the Lemma I, we know that [MATH], hence [MATH].', '1204.4249-3-31-0': 'We refer the transmission scheme in (4) to as time-varying posterior matching scheme and prove an interesting converse result of the Lemma IV.', '1204.4249-3-32-0': 'Lemma V: For an additive white Gaussian MAC channel with feedback having [MATH] inputs and one output, where the received signal to be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise.', '1204.4249-3-32-1': 'Assuming that at each transmitter, a time-varying posterior matching encoder is deployed, so the recursive transmitted sequence at the transmitter [MATH] is given by [EQUATION] where [MATH] is the intended transmitted message.', '1204.4249-3-32-2': 'Then the correlation matrix among transmitted symbols at each time slot [MATH] defined as in the Lemma IV can be calculated online at the transmitters and the receiver, and more importantly the posterior distribution [MATH] can be calculated online at both the transmitter [MATH] and the receiver.', '1204.4249-3-33-0': 'Observe that [MATH] is a linear combination of [MATH].', '1204.4249-3-33-1': 'Moreover, with time-varying posterior matching transmission applied for Gaussian MAC, [MATH] is a linear combination of [MATH] (as we will see in the proof of the Theorem II below), so [MATH] is a linear combination of [MATH].', '1204.4249-3-33-2': 'Therefore, the correlation between [MATH] and [MATH] only depends on the correlations among transmitted symbols at time [MATH], i.e. [MATH].', '1204.4249-3-33-3': 'In other words, [MATH] is a function of [MATH], so the transmitters and receiver can be calculated the matrix [MATH] online at both transmitters and receiver.', '1204.4249-3-33-4': 'Moreover, from the relation [MATH], we see that the distribution of [MATH] is a function of all elements in the correlation matrix [MATH].', '1204.4249-3-33-5': 'That concludes the proof.', '1204.4249-3-34-0': "For a continuous point-to-point memoryless channel, the distribution [MATH] doesn't depend on [MATH], hence we have the posterior matching scheme for this case like the formula (16) in [4].", '1204.4249-3-34-1': 'However, in a MAC channel (for example additive white Gaussian MAC channel), where each received signal is a linear combination of all transmitted signals and Gaussian noise, the distribution [MATH] between the input [MATH] and the output may be dependent on [MATH].', '1204.4249-3-34-2': 'Therefore, time-varying posterior matching scheme may be the solution to overcome this problem.', '1204.4249-3-34-3': 'However, we will see from our proof in the Theorem I below the variable rate decoding rule, or Generalized Reverse Iterated Function System (GRIFS), can be applied at receiver to decode signals if and only if all the distributions [MATH] can be calculated online at the corresponding transmitters and receiver.', '1204.4249-3-34-4': 'With the result in the Lemma V, this condition is always satisfied when time-varying posterior matching encoding schemes used at transmitters.', '1204.4249-3-34-5': 'We will show in the next parts that using time-varying posterior matching scheme at transmitters can obtain optimal performances for some known cases.', '1204.4249-3-35-0': '# Error analysis for time-varying posterior matching scheme', '1204.4249-3-36-0': 'In this section, we analyze error performance for Gaussian MAC channel with feedback deployed the time-varying posterior matching scheme at the transmitters and variable-decoding rule at the receiver.', '1204.4249-3-37-0': 'Theorem I: Consider a real Gaussian MAC channel with [MATH] transmitters and one receiver without input power constraints.', '1204.4249-3-37-1': 'Assuming that at each transmitter [MATH], transmitted sequence [MATH] conforms the time-varying posterior matching rule, as following: [EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the Gaussian MAC channel which is a linear combination of all these transmitted signals at time [MATH].', '1204.4249-3-37-2': 'Let [MATH] and [MATH] as the global Lipschitz operator.', '1204.4249-3-37-3': 'Under the conditions that [EQUATION] define: [MATH] Then the rate region [MATH] is achievable by setting the target error probabilities [MATH] under the constraint [EQUATION]', '1204.4249-3-38-0': 'First, observe that since the distributions [MATH] can be calculated online at both transmitters and receiver by Lemma V, therefore [MATH] can be calculated online at both the transmitters and receiver.', '1204.4249-3-38-1': 'Denote [MATH], referred as to a Generalized Iterated Function System (GIFS) generated by the kernel sequence [MATH].', '1204.4249-3-38-2': 'For each [MATH], select a fixed interval [MATH] as the decoded interval with respect to [MATH].', '1204.4249-3-39-0': 'Define the corresponding interval at the origin to be [MATH] and set them to be the decoded interval for [MATH], and so the decoded interval for [MATH] are set to be [MATH].', '1204.4249-3-39-1': 'Let [EQUATION]', '1204.4249-3-39-2': 'From the condition [MATH] in the theorem, we know that [EQUATION]', '1204.4249-3-39-3': 'Observe that: [EQUATION] where [MATH].', '1204.4249-3-40-0': 'On the other hand, since [MATH], then for any arbitrarily small [MATH], there exists an [MATH] such that [MATH].', '1204.4249-3-40-1': 'Let [MATH].', '1204.4249-3-40-2': "For any fixed number [MATH], from [MATH] we have: [EQUATION] for any [MATH] arbitrarily small, where [MATH] follows the Markov's inequality, [MATH] follows [MATH], and [MATH] is a recursive application of the preceding transitions, [MATH] follows [MATH] above and recursive applications of the preceding transitions.", '1204.4249-3-41-0': 'From [MATH], it is easy to see that a sufficient condition for [MATH] is given by choosing [MATH].', '1204.4249-3-41-1': 'Observe that [MATH] can be made arbitrarily small, therefore, the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-3-41-2': 'Since [MATH] depends only on the length of [MATH], so we can choose [MATH].', '1204.4249-3-41-3': 'Furthermore, from the Lemma I, we are easy to come to conclusion that [MATH].', '1204.4249-3-41-4': 'Using the standard bounds for Gaussian distribution we obtain [EQUATION]', '1204.4249-3-42-0': '# A Posterior Matching Scheme for Gaussian MAC Channel with Feedback', '1204.4249-3-43-0': 'In this section, we consider a real Gaussian MAC channel with [MATH] receivers and input power constraints [MATH] at the transmitters [MATH], respectively as defined in the section II.', '1204.4249-3-43-1': 'Our encoding scheme for this channel as following:', '1204.4249-3-44-0': 'Encoding:', '1204.4249-3-45-0': '[EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the MAC channel.', '1204.4249-3-46-0': 'where [MATH] is the column [MATH] of the Hadamard matrix [MATH] by [MATH], which is a spreading code for CDMA systems.', '1204.4249-3-47-0': "We refer this encoding strategy as Gaussian MAC channel's posterior matching feedback coding and decoding strategy.", '1204.4249-3-48-0': "Theorem II: Using the Gaussian MAC channel's posterior matching feedback coding and decoding strategy above, the rate region [MATH] is achievable for Gaussian MAC channel with feedback, where [EQUATION] by setting the target error probabilities [EQUATION] where [EQUATION]", '1204.4249-3-49-0': 'Applying the Lemma I, we see that for any [MATH] then: [EQUATION]', '1204.4249-3-49-1': 'Observe that, by this transmission strategy, each transmitter [MATH] transmits [MATH] at time [MATH] with [MATH], thus the input power constraints at all transmitters are always satisfied at each transmission time [MATH] Moreover, the output at receiver at time [MATH] will be [EQUATION]', '1204.4249-3-49-2': 'Thus [EQUATION]', '1204.4249-3-49-3': 'Finally, we have: [EQUATION] and [EQUATION]', '1204.4249-3-49-4': 'Therefore, [EQUATION] where [EQUATION] and [EQUATION]', '1204.4249-3-49-5': 'Moreover, from [MATH] we have: [EQUATION] thus, [EQUATION]', '1204.4249-3-49-6': 'From [MATH] and [MATH] we obtain: [EQUATION]', '1204.4249-3-49-7': 'Hence, [EQUATION]', '1204.4249-3-49-8': 'Finally, we have: [EQUATION]', '1204.4249-3-49-9': 'If we can achieve [EQUATION] then [EQUATION]', '1204.4249-3-49-10': 'This means that the condition in the Theorem I is satisfied, which leads to the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-3-49-11': 'Note that this capacity region is obtained by setting the target error probabilities [MATH] under the constraints [EQUATION] which have the well-known double-exponential behavior.', '1204.4249-3-50-0': 'In the following, we will specify achievable rate regions, and error probabilities for two cases: the general two-user Gaussian MAC channel, and the real symmetric Gaussian MAC channel when the number of users is arbitrary.', '1204.4249-3-51-0': 'From the Theorem II, we see that the strategy to design posterior encoding scheme for multiple access channels with feedback is to find the sequences [MATH] such that [EQUATION] for all [MATH].', '1204.4249-3-52-0': 'Case 1: Two-user Gaussian MAC channel with feedback.', '1204.4249-3-53-0': "To show that Ozarow's coding scheme [3] is a special case of our posterior matching framework, we can set [MATH] and later prove that [MATH].", '1204.4249-3-53-1': 'Observe that the constraint [MATH] can be also checked to be satisfied by this setting since [EQUATION] for [MATH] and [EQUATION] for [MATH].', '1204.4249-3-54-0': 'Now we need to find the recursion of [MATH] and [MATH] in this case.', '1204.4249-3-54-1': 'Observe that the output sequence: [EQUATION] where [MATH] is noise process and [MATH].', '1204.4249-3-55-0': 'From (12), (13) we have [EQUATION] and [EQUATION]', '1204.4249-3-55-1': 'Therefore, we obtain: [EQUATION] and [EQUATION]', '1204.4249-3-56-0': 'Finally, the time-varying posterior matching encoding scheme for Gaussian MAC channel with feedback in this special case as following:', '1204.4249-3-57-0': 'Achievable rate region and error analysis:', '1204.4249-3-58-0': 'For this special case, we have [EQUATION] and [EQUATION]', '1204.4249-3-58-1': 'Apply the Theorem II above, we obtain the achievable rate region for Gaussian MAC channel with two users as [MATH] where [EQUATION]', '1204.4249-3-58-2': 'Similarly, [EQUATION] by setting the target error probability to [EQUATION] and [EQUATION]', '1204.4249-3-58-3': 'Much like Ozarow in [MATH], at the reception [MATH], the receiver adds an independent random variable [MATH] before feeding back the first receiver signal to the transmitters 1 and 2 to set [MATH], where [MATH] is the biggest solution in [MATH] of the following equation: [EQUATION]', '1204.4249-3-58-4': 'By this changing, from [MATH] we see that [MATH], so [MATH] as mentioned above.', '1204.4249-3-58-5': 'We also have [MATH], where [MATH] is a positive solution of the equation [MATH].', '1204.4249-3-58-6': 'Replace this result to [MATH] and combine with [MATH], we have: [EQUATION] where [MATH] is defined above.', '1204.4249-3-58-7': "We see that, all the results are the same as Ozarow's results in [3].", '1204.4249-3-58-8': 'So our posterior matching encoding scheme is optimal for Gaussian channel MAC with two users.', '1204.4249-3-59-0': 'Case 2: M-user symmetric Gaussian MAC channel with feedback.', '1204.4249-3-59-1': 'We consider symmetric case, where [MATH].', '1204.4249-3-60-0': 'Achievable rate region and error analysis:', '1204.4249-3-61-0': 'Assuming that all the transmitted messages are statistically independent.', '1204.4249-3-61-1': 'We will prove by induction that the normalized covariance [MATH] have all the columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] are symmetric positive definite for all [MATH]', '1204.4249-3-62-0': 'Indeed, with the assumption all the transmitted information messages are statistically independent, we will have [MATH], which is an identity matrix of size [MATH].', '1204.4249-3-62-1': 'Therefore, it is obvious that all the columns of the Hadamard matrix [MATH] by [MATH] are eigenvectors of the matrix [MATH] and that [MATH] is a positive definite matrix.', '1204.4249-3-63-0': 'Now, assume that [MATH] has all columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is positive definite for some [MATH].', '1204.4249-3-63-1': 'Since we assumed that [MATH] is symmetric positive definite matrix, all its eigenvalues are greater than [MATH].', '1204.4249-3-63-2': 'Denote [MATH] are [MATH] columns of the Hadamard matrix [MATH].', '1204.4249-3-63-3': 'By this encoding scheme, we set the vector [MATH].', '1204.4249-3-63-4': 'Assume that [MATH] is the eigenvalue of [MATH] associated with this eigenvector.', '1204.4249-3-63-5': 'We have [EQUATION]', '1204.4249-3-63-6': 'On the other hand, we also have [EQUATION] where [MATH] is the [MATH]th column of the matrix [MATH].', '1204.4249-3-63-7': 'Note that [MATH], this means that [EQUATION]', '1204.4249-3-63-8': 'Hence, [EQUATION]', '1204.4249-3-63-9': 'Substitute [MATH] into [MATH] we obtain: [EQUATION]', '1204.4249-3-63-10': 'From [MATH] and [MATH], we obtain: [EQUATION] and [EQUATION] for all [MATH].', '1204.4249-3-64-0': 'Observe that from the proof of the Theorem II above, then [EQUATION].', '1204.4249-3-64-1': 'Define the correlation coefficient between [MATH] and [MATH] [EQUATION]', '1204.4249-3-64-2': 'Then, we have [EQUATION]', '1204.4249-3-64-3': 'By replacing all values of [MATH] calculated above, we obtain [EQUATION] for all [MATH].', '1204.4249-3-65-0': 'Since we assumed that [MATH] is symmetric positive definite matrix, hence [MATH], hence [MATH].', '1204.4249-3-65-1': 'Therefore, from [MATH], it is easy to see that we will have [MATH], or [MATH] is also a symmetric matrix.', '1204.4249-3-66-0': 'Moreover, from [MATH], we also obtain [EQUATION]', '1204.4249-3-66-1': 'Denote [MATH].', '1204.4249-3-66-2': 'We see that the columns of [MATH] creates [MATH] linearly independent eigenvectors of the matrix [MATH].', '1204.4249-3-66-3': 'Moreover, we also have [EQUATION]', '1204.4249-3-66-4': 'Note that since all columns of [MATH] are eigenvectors of the matrix [MATH], so all the columns of the matrix [MATH] are also eigenvectors of the matrix [MATH], hence we has the following eigenvalue decomposition [EQUATION] where [MATH] is a diagonal matrix.', '1204.4249-3-67-0': 'Moreover, we have [EQUATION] where [EQUATION]', '1204.4249-3-67-1': 'From (26), (27), and (28), we must have [MATH] must be a diagonal matrix since the right side of [MATH] is a diagonal matrix.', '1204.4249-3-67-2': 'Hence, all columns of the matrix [MATH] are eigenvectors of the matrix [MATH].', '1204.4249-3-68-0': 'Assuming [MATH] are [MATH] eigenvalues corresponding to eigenvectors which are columns of the matrix [MATH].', '1204.4249-3-68-1': 'By this notation, we see that [MATH].', '1204.4249-3-69-0': 'Combining with [MATH] with [MATH], and [MATH] we obtain [EQUATION] for all [MATH].', '1204.4249-3-70-0': 'Since we assumed that all eigenvalues of [MATH] are positive ([MATH] symmetric positive definite), from [MATH] we see that all eigenvalues of [MATH] are also positive.', '1204.4249-3-70-1': 'Note that we also confirmed that [MATH] is symmetric above, therefore [MATH] is a symmetric positive definite matrix.', '1204.4249-3-70-2': 'In short, if [MATH] is a symmetric positive definite matrix and has all columns of the Hadamard matrix as its eigenvectors, then [MATH] has all these properties.', '1204.4249-3-70-3': 'This concludes our proof by induction.', '1204.4249-3-71-0': 'According to the Lemma 1, [MATH], the sequence [MATH] converges to a fixed point [MATH], which is the solution in [MATH] of the following equation [EQUATION] when [EQUATION]', '1204.4249-3-71-1': 'When the condition (32) satisfies, from (22), (23) we obtain [EQUATION]', '1204.4249-3-71-2': 'Observe that [MATH], so [MATH].', '1204.4249-3-71-3': 'Therefore, the constraints in the Theorem II is satisfied.', '1204.4249-3-71-4': 'Applying the result of this theorem, we have any rate less than [EQUATION] is achievable, for all [MATH].', '1204.4249-3-71-5': 'Hence, any sum rate which is less than [EQUATION] is achievable, where [MATH] is solution in the [MATH] of the equation [MATH].', '1204.4249-3-71-6': 'This result coincides with the formula (68) in [14].', '1204.4249-3-71-7': 'The paper [16] proves that this achievable sum rate is optimal for the class of linear feedback coding.', '1204.4249-3-72-0': '# Conclusion', '1204.4249-3-73-0': 'A posterior matching based encoding-decoding strategy for general Gaussian MAC channel with feedback was proposed, and achievable rate region, error performance were drawn.', '1204.4249-3-73-1': 'Finally, we analyzed error performance of the proposed posterior encoding scheme and showed that the time-varying posterior matching scheme and variable rate decoding ideas can be applied to Gaussian MAC channel and obtain optimal performances.', '1204.4249-3-73-2': 'Specifically, the proposed encoding scheme achieves the capacity of two-user feedback Gaussian MAC channel as well as linear-feedback sum-rate for symmetric Gaussian MAC channel with feedback where the number of users is arbitrary.', '1204.4249-3-73-3': "Moreover, by the encoding scheme's structure, which uses the spreading codes like the Hadamard matrix, our encoding scheme can be directly applied to CDMA systems with feedback.", '1204.4249-3-73-4': 'Finally, by analyzing all arguments in this paper, the time-varying posterior matching scheme approach might be applied for other Gaussian and non-Gaussian multiuser channels to achieve optimal performances.', '1204.4249-3-73-5': '[Proof of the Lemma II]', '1204.4249-3-74-0': 'We use the same line argument as Lemma 1 [4].', '1204.4249-3-74-1': 'Assume we are given a transmission scheme with [MATH] transmission functions [MATH] and a decoding rule which are known to achieve the rate vector [MATH].', '1204.4249-3-74-2': 'For simplicity, we assume that the decoding rule is fixed rate [MATH] for all [MATH]), since any variable rate decoding rule can be easily mapped into a fixed rate rule that achieves the same rate vector.', '1204.4249-3-74-3': 'It is easy to see that in order to prove that the above translates into achievability for some rate vector [MATH] in the standard framework, it is enough to show we can find [MATH] sequences [MATH] and such that we have the uniform achievability over [MATH], i.e., [EQUATION]', '1204.4249-3-74-4': 'We now show how [MATH] can be constructed for any [MATH].', '1204.4249-3-74-5': 'Let [MATH] be the (average) error probability associated with our scheme and the fixed rate vector [MATH].', '1204.4249-3-74-6': 'Define [EQUATION] and write [EQUATION] and so we have that [MATH].', '1204.4249-3-74-7': 'It is now easy to see that if we want to select [MATH] such that [MATH], and also [MATH], then a sufficient condition is that [MATH] for some positive [MATH].', '1204.4249-3-74-8': 'This condition can be written as [EQUATION]', '1204.4249-3-74-9': 'At the same time, we also have by definition [EQUATION]'} | {'1204.4249-4-0-0': 'Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design capacity achieving coding schemes for general point-to-point memoryless channels with feedback.', '1204.4249-4-0-1': 'In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme.', '1204.4249-4-0-2': "The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder's posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots.", '1204.4249-4-0-3': "It turns out that the well-known Ozarow's encoding scheme, which obtains the capacity of two-user Gaussian channel, is a special case of our extended posterior matching framework as the Schalkwijk-Kailath's scheme is a special case of the point-to-point posterior matching mentioned above.", '1204.4249-4-0-4': 'Furthermore, our designed posterior matching also obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC.', '1204.4249-4-0-5': 'Besides, the encoding scheme in this paper is designed for the real Gaussian MAC to obtain that performance, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC.', '1204.4249-4-0-6': 'More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels with feedback.', '1204.4249-4-1-0': 'Gaussian Multiple Access Channel, Feedback, Posterior Matching, Iterated Function Systems.', '1204.4249-4-2-0': 'Introduction', '1204.4249-4-3-0': 'In his early work [9], Shannon proved that feedback could not increase the capacity of a point-to-point memoryless channel.', '1204.4249-4-3-1': 'However, feedback can improve error performance and simplify the transmission scheme for this kind of channel.', '1204.4249-4-3-2': 'In [10], Horstein proposed a simple sequential transmission scheme, which achieves the capacity of Binary Symmetric Channel (BSC) and provides larger error exponents than traditional fixed length block coding.', '1204.4249-4-3-3': 'Besides, Schalkwijk and Kailath also showed that feedback could improve error performance and/or simplify the transmission scheme for the point-to-point Gaussian channel [7], [8].', '1204.4249-4-3-4': 'For Gaussian multiuser channels, the situation is more interesting.', '1204.4249-4-3-5': 'In [12], Gaarder and Wolf proved that feedback can enlarge the capacity region of the multiple access channel, and Ozarow [3] successfully constructed a simple coding scheme for the two user Gaussian MAC with feedback and reaffirmed that feedback could increase the capacity of the channel.', '1204.4249-4-3-6': 'Furthermore, Kramer devised a code for complex Gaussian channel based on a beautiful property of the circulant matrix that has all columns of the DFT (Discrete Fourier Transform) matrix as its eigenvectors [14].', '1204.4249-4-3-7': 'This code was proved to obtain the linear-feedback sum-capacity of the symmetric Gaussian channel with feedback in [16].', '1204.4249-4-3-8': "By using the control-theoretic approach to communications with feedback, Ehsan Ardestanizadeh and Massimo Fraceschetii [17] also proposed a linear code that has the same performance as Kramer's code for symmetric Gaussian complex channels.", '1204.4249-4-4-0': 'Recently, Ofer Shayevitz and Meir Feder [1], [2], and [4] have discovered an underlying principle between the Horstein and Schalkwijk-Kailth schemes in a simple encoding scheme called posterior matching scheme for general point-to-point memoryless channels.', '1204.4249-4-4-1': "The idea of posterior matching is that the transmitter encapsulates the information the receiver does not know up to present time in one random variable and then transmits that random variable to the receiver in the next transmission to refine the receiver's knowledge.", '1204.4249-4-4-2': 'The distribution of that variable will be selected in a way such that the input constraint is satisfied.', '1204.4249-4-4-3': 'Later, J.H.Bae and A.Anastasopolous have extended this scheme for the finite-state channel with feedback by using another approach [11].', '1204.4249-4-4-4': 'One interesting open problem is to extend the Ofer Shayevitz and Meir Feder posterior matching scheme for multiuser cases.', '1204.4249-4-4-5': 'In this paper, using the same approach as Ofer Shayevitz and Meir Feder used for point-to-point memoryless channels, we propose a posterior matching based encoding and decoding strategy for real Gaussian MACs, referred to as a time-varying posterior matching scheme, and analyze the error probabilities for all encoding-decoding schemes designed by using these strategies.', '1204.4249-4-5-0': 'We analyze the achievable rate region and error performance of encoding and decoding schemes using these strategies by defining a generalized iterated function systems (GIFS) which has the generalized average contractive property (average contractive in the limit).', '1204.4249-4-5-1': 'Refer to our Theorem I for more details.', '1204.4249-4-5-2': 'Note that our imposed constraint is less strict than the constraint that Ofer Shayevitz and Meir Feder imposed to analyze the point-to-point memoryless channels.', '1204.4249-4-5-3': 'Specifically, in Theorem 6 in [4], Ofer Shayevitz and Meir Feder used the relations between the information rates and contraction properties of the iterated function system (IFS) to analyze the error probability for point-to-point cases.', '1204.4249-4-5-4': 'For the continuous cases, they assumed that the reverse iterated function system (RIFS), generated by the kernel [MATH] and controlled by the identically distributed output sequence [MATH], has the average contractive property to analyze the error performance of their posterior matching schemes.', '1204.4249-4-5-5': 'That assumption requires the distribution at the output of the point-to-point memoryless channel be identically distributed when using their proposed encoding schemes.', '1204.4249-4-5-6': 'This also means that if the output distribution is not identically distributed, the error analysis in Theorem 6 in [4] cannot apply.', '1204.4249-4-5-7': 'For example, this situation happens with our proposed matching schemes for the Gaussian MAC in this paper.', '1204.4249-4-6-0': 'Finally, we illustrate our strategies by designing an encoding scheme that obtains optimal performance for the Gaussian MAC.', '1204.4249-4-6-1': "Specifically, our proposed code obtains the same performance as Ozarow's code [3] for the general two-user Gaussian channel, so it achieves the capacity of this channel.", '1204.4249-4-6-2': "For the case when the number of users is greater than 3, our proposed code obtains the same performance as the Kramer's code in the sense of sum-rate, so it is optimal among linear code with respect to sum rate capacity.", '1204.4249-4-6-3': 'To the best of our knowledge, the time-varying posterior matching in this paper is the first code designed for the real symmetric Gaussian MAC to achieve the linear-feedback sum-capacity when the number of users is greater than 3.', '1204.4249-4-7-0': 'The rest of this paper is organized as follows.', '1204.4249-4-7-1': 'Section II presents the channel model and some mathematical preliminaries.', '1204.4249-4-7-2': 'Sections III, IV introduce the time-varying posterior matching idea, and perform the error analysis of the encoding-decoding schemes for the Gaussian MAC with feedback constructed by using that idea.', '1204.4249-4-7-3': 'A time-varying encoding-decoding strategy and error analysis for the general two-user white Gaussian MAC and the multiuser symmetric white Gaussian MAC are placed in Section V. Finally, Section VI concludes this paper.', '1204.4249-4-7-4': 'Channel Model and Preliminaries', '1204.4249-4-8-0': '## Mathematical notations', '1204.4249-4-9-0': 'Upper-case letters, their realizations by corresponding lower-case letters, denote random variables.', '1204.4249-4-9-1': 'A real-valued random variable [MATH] is associated with a distribution [MATH] defined on the usual Borel [MATH]-algebra over [MATH], and we write [MATH].', '1204.4249-4-9-2': 'The cumulative distribution function (c.d.f.) of [MATH] is given by [MATH], and their inverse c.d.f is defined to be [MATH].', '1204.4249-4-9-3': 'The uniform probability distribution over [MATH] is denoted through [MATH].', '1204.4249-4-9-4': 'The composition function [MATH].', '1204.4249-4-9-5': 'In this paper, we use the following lemma:', '1204.4249-4-10-0': 'Lemma I: Let [MATH] be a continuous random variable with [MATH] and [MATH] be an uniform distribution random variable, i.e. [MATH] be statistical independent.', '1204.4249-4-10-1': 'Then [MATH] and [MATH].', '1204.4249-4-11-0': 'Refer to [4] for the proof.', '1204.4249-4-12-0': "Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions.", '1204.4249-4-12-1': 'Basically, it tells you how fast a function grows or declines.', '1204.4249-4-12-2': 'For the formal definition, suppose [MATH] and [MATH] are two functions defined on positive integer number.', '1204.4249-4-12-3': 'We write [EQUATION] (or [MATH] for [MATH] to be more precise) if and only if there exists constants [MATH] and [MATH] such that [EQUATION]', '1204.4249-4-12-4': 'Intuitively, this means that [MATH] does not grow faster than [MATH].', '1204.4249-4-13-0': 'In addition to big O notations, another Landau symbol is used in mathematics: the little o. Formally, we write [MATH] for [MATH] if and only if for every [MATH] there exists a real number [MATH] such that for all [MATH] we have [MATH].', '1204.4249-4-13-1': 'If [MATH], this is equivalent to [MATH].', '1204.4249-4-14-0': 'A Hadamard matrix [15] of order [MATH] is an [MATH] matrix of [MATH]s and [MATH]s such that [MATH].', '1204.4249-4-14-1': 'In fact, it is not yet known for which values of [MATH] an [MATH] does exists.', '1204.4249-4-14-2': 'However, we know that if a Hadamard matrix of order [MATH] exists, then [MATH] is [MATH], or a multiple of [MATH].', '1204.4249-4-14-3': 'Moreover, if [MATH] is of the form [MATH], [MATH] a positive integer, we can construct [MATH] by using the Sylvester method.', '1204.4249-4-14-4': 'Besides, the Paley construction, which uses quadratic residues, can be used to construct Hadamard matrices of order [MATH], where [MATH] is of the form [MATH], [MATH] is a prime, and [MATH] is a multiple of [MATH].', '1204.4249-4-15-0': '## Gaussian Multiple Access Channel with Feedback', '1204.4249-4-16-0': 'Consider the communication problem between [MATH] senders and a receiver over a multiple access channel with additive Gaussian noise (AWGN-MAC) when channel outputs are noiselessly fed back to all the senders (Figure 1).', '1204.4249-4-16-1': 'Each sender [MATH] wishes to reliably transmit a random message point [MATH], which is uniformly distributed over the unit interval with its binary expansion representing an infinite independent-identically-distributed (i.i.d.) Bernoulli(1/2) sequence, to the receiver.', '1204.4249-4-16-2': 'At each time [MATH], the output of the channel is [EQUATION] where [MATH] is the transmitted symbol by sender [MATH] at time [MATH], [MATH] is the output of the channel, and [MATH] is a discrete-time zero mean white Gaussian noise process with unit average power, i.e., [MATH] and is independent of [MATH].', '1204.4249-4-16-3': 'We assume that output symbols are casually fed back to the sender and the transmitted symbol [MATH] for sender [MATH] at time [MATH] can depend on both the message [MATH] and the previous channel output sequence [MATH].', '1204.4249-4-17-0': 'A transmission scheme for a Gaussian MAC is a set of [MATH] sequences of transmission functions [MATH] for [MATH], so that the input to the channel generated by the transmitter is given by [EQUATION]', '1204.4249-4-18-0': 'A decoding rule for a MAC is set of sequences of measurable mappings [MATH], where [MATH] is the set of all open intervals in [MATH] and [MATH].', '1204.4249-4-18-1': 'Here, [MATH], refers as to the decoded interval for the user [MATH].', '1204.4249-4-18-2': 'The error probabilities at time [MATH] associated with a transmission scheme and a decoding rule, is defined as [EQUATION] and the corresponding achievable rate vector at time [MATH] is defined to be [EQUATION]', '1204.4249-4-18-3': 'We say that a transmission scheme together with a decoding rule achieve a rate vector [MATH] over a Gaussian MAC if for all [MATH] we have [EQUATION]', '1204.4249-4-18-4': 'The rate vector is achieved within input power constraints [MATH], if in addition [EQUATION]', '1204.4249-4-18-5': 'An optimal fixed rate decoding rule for a MAC with rate region [MATH] is one that decodes a vector of fixed length intervals [MATH], whose marginal posteriori probabilities are maximal, i.e., [EQUATION]', '1204.4249-4-18-6': 'An optimal variable rate decoding rule with target error probabilities [MATH] is one that decodes a vector of minimal-length intervals [MATH] with accumulated marginal posteriori probabilities exceeds corresponding targets, i.e., [EQUATION]', '1204.4249-4-18-7': 'Both decoding rules make use of the marginal posterior distribution of the message point [MATH] which can calculate online at the transmitter [MATH] and the receiver.', '1204.4249-4-18-8': 'Refer [4] for more details.', '1204.4249-4-18-9': 'A proof that the achievability in the sense of [MATH] and [MATH] implies that the achievability in the standard framework are in the Appendix.', '1204.4249-4-19-0': 'Lemma II: The achievability in the definition [MATH] and [MATH] implies the achievability in the standard framework.', '1204.4249-4-20-0': 'Refer to the Appendix.', '1204.4249-4-21-0': '# Time-varying Posterior Matching Scheme', '1204.4249-4-22-0': "## Shayevitz and Feder's Posterior Matching Scheme", '1204.4249-4-23-0': 'In this part, we firstly review the posterior matching scheme proposed by Ofer Shayevitz and Meir Feder for point-to-point channel in [4].', '1204.4249-4-23-1': 'Specifically, the authors argued that after the receiver observed the output sequence [MATH], there is still some "missing information" that can be encapsulated in a random variable [MATH] with the following properties:', '1204.4249-4-24-0': '(i) [MATH] is statistically independent of [MATH].', '1204.4249-4-25-0': '(ii) The message point [MATH] can be a.s. uniquely recovered from [MATH]', '1204.4249-4-26-0': 'With that line of thought, they proposed a principle for generating the next channel input as follow:', '1204.4249-4-27-0': 'The transmission function [MATH] should be selected so that [MATH] is [MATH]-distributed, and is a fixed function of some random variable [MATH] satisfying properties [MATH] and [MATH].', '1204.4249-4-28-0': 'Lemma III: (Posterior Matching Scheme [4]).', '1204.4249-4-28-1': 'The following transmission scheme satisfies the posterior matching principle for any [MATH]: [EQUATION]', '1204.4249-4-28-2': 'Based on the transmission functions defined in section II-B [4], the input to the channel is a sequence of random variables given by [EQUATION]', '1204.4249-4-28-3': 'Refer to [4] for the proof.', '1204.4249-4-29-0': '## Time-varying Posterior Matching Scheme', '1204.4249-4-30-0': 'In this section, we propose a posterior matching scheme for additive Gaussian multiple access channel (MAC) with feedback, called time-varying posterior matching.', '1204.4249-4-30-1': 'Our encoding proposal is based on the following lemma:', '1204.4249-4-31-0': 'Lemma IV: For an additive white Gaussian MAC with feedback having [MATH] inputs and one output, let the output signal be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise and the coefficients [MATH] are part of the coding scheme, but are viewed as part of the channel for the purpose of deriving the posterior matching rule.', '1204.4249-4-31-1': 'Also assume that the covariance matrix among transmitted symbols at each time slot [MATH] defined as [EQUATION]', '1204.4249-4-31-2': 'Then the posterior matching scheme in Lemma III at each transmitter [MATH], i.e. [EQUATION] becomes a time-varying posterior matching given by [EQUATION] where [MATH] is a random variable encapsulating the input power constraint, [MATH], and [MATH] is the intended transmitted message at the transmitter [MATH].', '1204.4249-4-32-0': 'In addition, the correlation matrix among transmitted symbols at each time slot [MATH], i.e. [MATH], can be calculated online at the transmitters and the receiver.', '1204.4249-4-32-1': 'Especially, the posterior distribution [MATH] can be calculated online at both the transmitter [MATH] and the receiver.', '1204.4249-4-33-0': 'To begin with, we show that [MATH] constitutes a Markov chain.', '1204.4249-4-33-1': 'Indeed,', '1204.4249-4-34-0': 'The two observations imply that [MATH] are mutually independent of [MATH], which together with the memoryless property of the channel, implies the Markov chains.', '1204.4249-4-35-0': 'Moreover, by our construction of the multi-letter posterior matching formula, we have [EQUATION]', '1204.4249-4-35-1': 'Since [MATH] are continuous random variables, their c.d.f. and inverse c.d.f. functions are continuous, hence the composite function [MATH] is continuous on [MATH].', '1204.4249-4-35-2': 'Besides, the monotonicity of this function is originated from the monotonicity of the c.d.f. and the inverse c.d.f.', '1204.4249-4-36-0': "Now, let's return to prove the Lemma IV.", '1204.4249-4-36-1': 'Note that [EQUATION]', '1204.4249-4-36-2': 'Then we have [EQUATION] where (a) follows from the fact that the transmission function is continuous and monotone, and (b) follows from the aforementioned fact that [MATH] forms a Markov chain for any [MATH].', '1204.4249-4-37-0': 'Moreover, with time-varying posterior matching transmission applied for Gaussian MAC, [MATH] is a linear combination of [MATH] (as we will see in the proof of the Theorem II below), so [MATH] is a linear combination of [MATH].', '1204.4249-4-37-1': 'Therefore, the correlation between [MATH] and [MATH] only depends on the correlations among transmitted symbols at time [MATH], i.e. [MATH].', '1204.4249-4-37-2': 'In other words, [MATH] is a function of [MATH], so the transmitters and receiver can calculate the matrix [MATH] online at both transmitters and receiver.', '1204.4249-4-37-3': 'Moreover, from the relation [MATH], we see that the distribution of [MATH] is a function of all elements in the correlation matrix [MATH].', '1204.4249-4-37-4': 'That concludes the proof.', '1204.4249-4-38-0': 'We refer the transmission scheme in Lemma IV to as time-varying posterior matching scheme.', '1204.4249-4-38-1': "For a continuous point-to-point memoryless channel, the distribution [MATH] doesn't depend on [MATH], hence we have the posterior matching scheme for this case like the formula (16) in [4].", '1204.4249-4-38-2': 'However, in a MAC (for example additive white Gaussian MAC), where each received signal is a linear combination of all transmitted signals and Gaussian noise, the distribution [MATH] between the input [MATH] and the output may be dependent on [MATH].', '1204.4249-4-38-3': 'Therefore, time-varying posterior matching scheme may be the solution to overcome this problem.', '1204.4249-4-38-4': 'However, we will see from our proof in the Theorem I below the variable rate decoding rule, or Generalized Reverse Iterated Function System (GRIFS), can be applied at receiver to decode signals if and only if all the distributions [MATH] can be calculated online at the corresponding transmitters and receiver.', '1204.4249-4-38-5': 'With the result in the Lemma V, this condition is always satisfied when time-varying posterior matching encoding schemes used at transmitters.', '1204.4249-4-38-6': 'We will show in the next parts that using time-varying posterior matching scheme at transmitters can obtain optimal performances for some known cases.', '1204.4249-4-39-0': '# Error analysis for time-varying posterior matching scheme', '1204.4249-4-40-0': 'In this section, we analyze error performance for Gaussian MAC with feedback employing the time-varying posterior matching scheme at the transmitters and variable-decoding rule at the receiver.', '1204.4249-4-41-0': 'Theorem I: Consider a real Gaussian MAC with [MATH] transmitters and one receiver without input power constraints.', '1204.4249-4-41-1': 'Assuming that at each transmitter [MATH], transmitted sequence [MATH] conforms to the time-varying posterior matching rule, as following: [EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the Gaussian MAC which is a linear combination of all these transmitted signals at time [MATH].', '1204.4249-4-41-2': 'Let [MATH] and [MATH] as the global Lipschitz operator.', '1204.4249-4-41-3': 'Under the conditions that [EQUATION] define: [MATH] Then the rate region [MATH] is achievable and the error probabilities [MATH] double exponentially tend to zero as following [EQUATION]', '1204.4249-4-42-0': 'First, observe that since the distributions [MATH] can be calculated online at both transmitters and receiver by Lemma V, therefore [MATH] can be calculated online at both the transmitters and receiver.', '1204.4249-4-42-1': 'Denote [MATH], referred as to a Generalized Iterated Function System (GIFS) generated by the kernel sequence [MATH].', '1204.4249-4-42-2': 'For each [MATH], select a fixed interval [MATH] as the decoded interval with respect to [MATH].', '1204.4249-4-43-0': 'Define the corresponding interval at the origin to be [MATH] and set them to be the decoded interval for [MATH], and so the decoded interval for [MATH] are set to be [MATH].', '1204.4249-4-43-1': 'Let [EQUATION]', '1204.4249-4-43-2': 'From the condition [MATH] in the theorem, we know that [EQUATION]', '1204.4249-4-43-3': 'Observe that: [EQUATION] where [MATH].', '1204.4249-4-43-4': 'Here, (a) follows from the definition of the instant corresponding achievable rate vector at time [MATH] in the Section II above, and (b) follows from the fact that we set the decoded interval [MATH].', '1204.4249-4-44-0': 'On the other hand, since [MATH], then for any arbitrarily small [MATH], there exists an [MATH] such that [MATH].', '1204.4249-4-44-1': 'Let [MATH].', '1204.4249-4-44-2': "For any fixed number [MATH], from [MATH] we have: [EQUATION] for any [MATH] arbitrarily small, where [MATH] follows from the Markov's inequality, [MATH] follows from [MATH], and [MATH] is a recursive application of the preceding transitions, [MATH] follows from [MATH] above and recursive applications of the preceding transitions.", '1204.4249-4-45-0': 'From [MATH], it is easy to see that a sufficient condition for [MATH] is given by choosing [MATH].', '1204.4249-4-45-1': 'Observe that [MATH] can be made arbitrarily small, therefore, the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-4-45-2': 'Since [MATH] depends only on the length of [MATH], so we can choose [MATH].', '1204.4249-4-45-3': 'Furthermore, from the Lemma I, we are easy to come to conclusion that [MATH].', '1204.4249-4-45-4': 'Denote [MATH] as the well-known tail function of the standard normal distribution and use the Chernoff bound of this function we obtain [EQUATION] as [MATH] since [MATH] as [MATH] if [MATH].', '1204.4249-4-45-5': 'Here, (a) follows from the fact that [MATH] is symmetric [MATH], and (b) follows from the Chernoff bound for the Q-function [MATH].', '1204.4249-4-45-6': 'This means that the error probabilities double exponentially tend to zero as [MATH] as [EQUATION]', '1204.4249-4-46-0': '# A Posterior Matching Scheme for Gaussian MAC with Feedback', '1204.4249-4-47-0': 'In this section, we consider a real Gaussian MAC with [MATH] receivers and input power constraints [MATH] at the transmitters [MATH], respectively as defined in the section II.', '1204.4249-4-47-1': 'Our encoding scheme for this channel as following:', '1204.4249-4-48-0': 'Encoding:', '1204.4249-4-49-0': '[EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the MAC.', '1204.4249-4-50-0': 'where [MATH] is the column [MATH] of the Hadamard matrix [MATH] by [MATH], which is a spreading code for CDMA systems.', '1204.4249-4-51-0': 'We refer this encoding strategy as Gaussian MAC posterior matching feedback coding and decoding strategy.', '1204.4249-4-52-0': 'Theorem II: Using the Gaussian MAC posterior matching feedback coding and decoding strategy above, the rate region [MATH] is achievable for Gaussian MAC with feedback, where [EQUATION] by setting the target error probabilities [EQUATION] where [EQUATION] with [EQUATION].', '1204.4249-4-52-1': 'Applying the Lemma I, we see that for any [MATH] then: [EQUATION]', '1204.4249-4-52-2': 'Observe that, by this transmission strategy, each transmitter [MATH] transmits [MATH] at time [MATH] with [MATH], thus the input power constraints at all transmitters are always satisfied at each transmission time [MATH] Moreover, the output at receiver at time [MATH] will be [EQUATION]', '1204.4249-4-52-3': 'Thus [EQUATION]', '1204.4249-4-52-4': 'Then, we have [EQUATION] and [EQUATION] where [EQUATION] and [EQUATION]', '1204.4249-4-52-5': 'Finally, we obtain [EQUATION] where [EQUATION]', '1204.4249-4-52-6': 'Moreover, from [MATH] we have: [EQUATION] thus, [EQUATION]', '1204.4249-4-52-7': 'Combining with [MATH], we obtain: [EQUATION]', '1204.4249-4-52-8': 'Hence, [EQUATION]', '1204.4249-4-52-9': 'Finally, we have: [EQUATION]', '1204.4249-4-52-10': 'If we can achieve [EQUATION] then [EQUATION]', '1204.4249-4-52-11': 'This means that the condition in the Theorem I is satisfied, which leads to the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-4-52-12': 'Note that this capacity region is obtained by setting the target error probabilities [MATH] under the constraints [EQUATION] which have the well-known double-exponential behavior.', '1204.4249-4-53-0': 'In the following, we will specify achievable rate regions, and error probabilities for two cases: the general two-user Gaussian MAC, and the real symmetric Gaussian MAC when the number of users is arbitrary.', '1204.4249-4-54-0': 'From the Theorem II, we see that the strategy to design posterior encoding scheme for multiple access channels with feedback is to find the sequences [MATH] such that [EQUATION] for all [MATH].', '1204.4249-4-55-0': 'Case 1: Two-user Gaussian MAC with feedback.', '1204.4249-4-56-0': "To show that Ozarow's coding scheme [3] is a special case of our posterior matching framework, we can set [MATH] and later prove that [MATH].", '1204.4249-4-56-1': 'Observe that the constraint [MATH] can be also checked to be satisfied by this setting since [EQUATION] for [MATH] and [EQUATION] for [MATH].', '1204.4249-4-57-0': 'Now we need to find the recursion of [MATH] and [MATH] in this case.', '1204.4249-4-57-1': 'Observe that the output sequence: [EQUATION] where [MATH] is noise process and [MATH].', '1204.4249-4-58-0': 'From [MATH] we have [EQUATION] and [EQUATION]', '1204.4249-4-58-1': 'Therefore, we obtain: [EQUATION] and [EQUATION]', '1204.4249-4-59-0': 'Finally, the time-varying posterior matching encoding scheme for Gaussian MAC with feedback in this special case as following:', '1204.4249-4-60-0': 'Achievable rate region and error analysis:', '1204.4249-4-61-0': 'For this special case, we have [EQUATION] and [EQUATION]', '1204.4249-4-61-1': 'Apply the Theorem II above, we obtain the achievable rate region for Gaussian MAC with two users as [MATH] where [EQUATION]', '1204.4249-4-61-2': 'Similarly, [EQUATION] by setting the target error probability to [EQUATION] and [EQUATION]', '1204.4249-4-61-3': 'Much like Ozarow in [MATH], at the reception [MATH], the receiver adds an independent random variable [MATH] before feeding back the first receiver signal to the transmitters 1 and 2 to set [MATH], where [MATH] is the biggest solution in [MATH] of the following equation: [EQUATION]', '1204.4249-4-61-4': 'By this changing, from [MATH] we see that [MATH], so [MATH] as mentioned above.', '1204.4249-4-61-5': 'We also have [MATH], where [MATH] is a positive solution of the equation [MATH].', '1204.4249-4-61-6': 'Replace this result to [MATH] and combine with [MATH], we have: [EQUATION] where [MATH] is defined above.', '1204.4249-4-61-7': "We see that, all the results are the same as Ozarow's results in [3].", '1204.4249-4-61-8': 'So our posterior matching encoding scheme is optimal for Gaussian channel MAC with two users.', '1204.4249-4-62-0': 'Case 2: M-user symmetric Gaussian MAC with feedback.', '1204.4249-4-62-1': 'We consider symmetric case, where [MATH].', '1204.4249-4-63-0': 'Achievable rate region and error analysis:', '1204.4249-4-64-0': 'Assuming that all the transmitted messages are statistically independent.', '1204.4249-4-64-1': 'Define the normalized covariance matrix by [EQUATION]', '1204.4249-4-64-2': 'Then [EQUATION] where [EQUATION] is the correlation coefficient between [MATH] and [MATH].', '1204.4249-4-65-0': 'We will prove by induction that the normalized covariance have all the columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] are symmetric positive definite for all [MATH]', '1204.4249-4-66-0': 'Indeed, with the assumption all the transmitted information messages are statistically independent, we will have [MATH], which is an identity matrix of size [MATH].', '1204.4249-4-66-1': 'Therefore, it is obvious that all the columns of the Hadamard matrix [MATH] by [MATH] are eigenvectors of the matrix [MATH] and that [MATH] is a positive definite matrix.', '1204.4249-4-67-0': 'Now, assume that [MATH] has all columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is positive definite for some [MATH].', '1204.4249-4-67-1': 'Since we assumed that [MATH] is symmetric positive definite matrix, all its eigenvalues are greater than [MATH].', '1204.4249-4-67-2': 'Denote by [MATH] the [MATH] columns of the Hadamard matrix [MATH].', '1204.4249-4-67-3': 'By this encoding scheme, we set the vector [MATH].', '1204.4249-4-67-4': 'Assume that [MATH] is the eigenvalue of [MATH] associated with this eigenvector.', '1204.4249-4-67-5': 'We have [EQUATION]', '1204.4249-4-67-6': 'On the other hand, we also have [EQUATION] where [MATH] is the [MATH]th column of the matrix [MATH].', '1204.4249-4-67-7': 'Note that [MATH], this means that [EQUATION]', '1204.4249-4-67-8': 'Hence, [EQUATION].', '1204.4249-4-67-9': 'Moreover, observe that [EQUATION]', '1204.4249-4-67-10': 'Combining these results, we obtain: [EQUATION] and [EQUATION]', '1204.4249-4-67-11': 'Substitute [MATH] and [MATH] into [MATH], we obtain: [EQUATION] for all [MATH].', '1204.4249-4-68-0': 'and [EQUATION]', '1204.4249-4-68-1': 'Observe that from the proof of the Theorem II above, then [EQUATION].', '1204.4249-4-68-2': 'Then, we have [EQUATION]', '1204.4249-4-68-3': 'Moreover, from (13) we know that [EQUATION]', '1204.4249-4-68-4': 'Therefore, we obtain [EQUATION]', '1204.4249-4-68-5': 'Replacing the results in [MATH], and [MATH] to this expression, we obtain [EQUATION] for all [MATH].', '1204.4249-4-69-0': 'Since we assumed that [MATH] is symmetric positive definite matrix, hence [MATH], hence [MATH].', '1204.4249-4-69-1': 'Therefore, from [MATH], it is easy to see that [MATH], or [MATH] is also a symmetric matrix.', '1204.4249-4-70-0': 'Moreover, from [MATH], we also obtain [EQUATION]', '1204.4249-4-70-1': 'Denote [MATH].', '1204.4249-4-70-2': 'We see that the columns of [MATH] creates [MATH] linearly independent eigenvectors of the matrix [MATH].', '1204.4249-4-70-3': 'Moreover, we also have [EQUATION]', '1204.4249-4-70-4': 'Note that since all columns of [MATH] are eigenvectors of the matrix [MATH], so all the columns of the matrix [MATH] are also eigenvectors of the matrix [MATH], hence we has the following eigenvalue decomposition [EQUATION] where [MATH] is a diagonal matrix.', '1204.4249-4-71-0': 'Moreover, we have [EQUATION] where [EQUATION]', '1204.4249-4-71-1': 'From [MATH], the matrix [MATH] must be a diagonal one since the right side of [MATH] is a diagonal matrix.', '1204.4249-4-71-2': 'Hence, all columns of the matrix [MATH] are eigenvectors of the matrix [MATH].', '1204.4249-4-72-0': 'Assuming [MATH] are [MATH] eigenvalues corresponding to eigenvectors which are columns of the matrix [MATH].', '1204.4249-4-72-1': 'By this notation, we see that [MATH].', '1204.4249-4-73-0': 'Combining [MATH], and [MATH] we obtain [EQUATION] for all [MATH].', '1204.4249-4-74-0': 'Since we assumed that all eigenvalues of [MATH] are positive ([MATH] symmetric positive definite), from [MATH] we see that all eigenvalues of [MATH] are also positive.', '1204.4249-4-74-1': 'Note that we also confirmed that [MATH] is symmetric above, therefore [MATH] is a symmetric positive definite matrix.', '1204.4249-4-74-2': 'In short, if [MATH] is a symmetric positive definite matrix and has all columns of the Hadamard matrix as its eigenvectors, then [MATH] has all these properties.', '1204.4249-4-74-3': 'This concludes our proof by induction.', '1204.4249-4-75-0': 'According to the Lemma 1, [MATH], the sequence [MATH] converges to a fixed point [MATH], which is the solution in [MATH] of the following equation [EQUATION] when [EQUATION]', '1204.4249-4-75-1': 'When the condition (38) satisfies, from (27), (28) we obtain [EQUATION]', '1204.4249-4-75-2': 'Observe that [MATH], so [MATH].', '1204.4249-4-75-3': 'Therefore, the constraints in the Theorem II is satisfied.', '1204.4249-4-75-4': 'Applying the result of this theorem, we have any rate less than [EQUATION] is achievable, for all [MATH].', '1204.4249-4-75-5': 'Hence, any sum rate which is less than [EQUATION] is achievable, where [MATH] is solution in the [MATH] of the equation [MATH].', '1204.4249-4-75-6': 'This result coincides with the formula (68) in [14].', '1204.4249-4-75-7': 'The paper [16] proves that this achievable sum rate is optimal for the class of linear feedback coding.', '1204.4249-4-76-0': '# Conclusion', '1204.4249-4-77-0': 'A posterior matching based encoding-decoding strategy for general Gaussian MAC with feedback was proposed, and achievable rate region, error performance were drawn.', '1204.4249-4-77-1': 'Finally, we analyzed error performance of the proposed posterior encoding scheme and showed that the time-varying posterior matching scheme and variable rate decoding ideas can be applied to Gaussian MAC and obtain optimal performances.', '1204.4249-4-77-2': 'Specifically, the proposed encoding scheme achieves the capacity of two-user feedback Gaussian MAC as well as linear-feedback sum-rate for symmetric Gaussian MAC with feedback where the number of users is arbitrary.', '1204.4249-4-77-3': "Moreover, by the encoding scheme's structure, which uses the spreading codes like the Hadamard matrix, our encoding scheme can be directly applied to CDMA systems with feedback.", '1204.4249-4-77-4': 'Finally, by analyzing all arguments in the theorem I, the time-varying posterior matching scheme approach in this paper might be applied for other Gaussian and non-Gaussian multiuser channels to achieve optimal performances.', '1204.4249-4-77-5': '[Proof of the Lemma II]', '1204.4249-4-78-0': 'We use the same line argument as Lemma 1 [4].', '1204.4249-4-78-1': 'Assume we are given a transmission scheme with [MATH] transmission functions [MATH] and a decoding rule which are known to achieve the rate vector [MATH].', '1204.4249-4-78-2': 'For simplicity, we assume that the decoding rule is fixed rate [MATH] for all [MATH]), since any variable rate decoding rule can be easily mapped into a fixed rate rule that achieves the same rate vector.', '1204.4249-4-78-3': 'It is easy to see that in order to prove that the above translates into achievability for some rate vector [MATH] in the standard framework, it is enough to show we can find [MATH] sequences [MATH] and such that we have the uniform achievability over [MATH], i.e., [EQUATION]', '1204.4249-4-78-4': 'We now show how [MATH] can be constructed for any [MATH].', '1204.4249-4-78-5': 'Let [MATH] be the (average) error probability associated with our scheme and the fixed rate vector [MATH].', '1204.4249-4-78-6': 'Define [EQUATION] and write [EQUATION] and so we have that [MATH].', '1204.4249-4-78-7': 'It is now easy to see that if we want to select [MATH] such that [MATH], and also [MATH], then a sufficient condition is that [MATH] for some positive [MATH].', '1204.4249-4-78-8': 'This condition can be written as [EQUATION]', '1204.4249-4-78-9': 'At the same time, we also have by definition [EQUATION]'} | {'1204.4249-5-0-0': 'Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design capacity achieving coding schemes for general point-to-point memoryless channels with feedback.', '1204.4249-5-0-1': 'In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme.', '1204.4249-5-0-2': "The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder's posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots.", '1204.4249-5-0-3': "It turns out that the well-known Ozarow's encoding scheme, which obtains the capacity of two-user Gaussian channel, is a special case of our extended posterior matching framework as the Schalkwijk-Kailath's scheme is a special case of the point-to-point posterior matching mentioned above.", '1204.4249-5-0-4': 'Furthermore, our designed posterior matching also obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC.', '1204.4249-5-0-5': 'Besides, the encoding scheme in this paper is designed for the real Gaussian MAC to obtain that performance, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC.', '1204.4249-5-0-6': 'More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels with feedback.', '1204.4249-5-1-0': 'Gaussian Multiple Access Channel, Feedback, Posterior Matching, Iterated Function Systems.', '1204.4249-5-2-0': 'Introduction', '1204.4249-5-3-0': 'In his early work [9], Shannon proved that feedback could not increase the capacity of a point-to-point memoryless channel.', '1204.4249-5-3-1': 'However, feedback could improve error performance and simplify the transmission scheme for this kind of channel.', '1204.4249-5-3-2': 'In [10], Horstein proposed a simple sequential transmission scheme, which achieves the capacity of Binary Symmetric Channel (BSC) and provides larger error exponents than traditional fixed length block coding.', '1204.4249-5-3-3': 'Besides, Schalkwijk and Kailath also showed that feedback could improve error performance and/or simplify the transmission scheme for the point-to-point Gaussian channel [7], [8].', '1204.4249-5-3-4': 'For Gaussian multiuser channels, the situation is more interesting.', '1204.4249-5-3-5': 'In [12], Gaarder and Wolf proved that feedback can enlarge the capacity region of the multiple access channel, and Ozarow [3] successfully constructed a simple coding scheme for the two user Gaussian MAC with feedback and reaffirmed that feedback could increase the capacity of the channel.', '1204.4249-5-3-6': 'Furthermore, Kramer devised a code for complex Gaussian channel based on a beautiful property of the circulant matrix that has all columns of the DFT (Discrete Fourier Transform) matrix as its eigenvectors [14].', '1204.4249-5-3-7': 'This code was proved to obtain the linear-feedback sum-capacity of the symmetric Gaussian channel with feedback in [16].', '1204.4249-5-3-8': "By using the control-theoretic approach to communications with feedback, Ardestanizadeh and Fraceschetii [17] also proposed a linear code that has the same performance as Kramer's code for symmetric Gaussian complex channels.", '1204.4249-5-4-0': 'Recently, Shayevitz and Feder [1], [2], and [4] have discovered an underlying principle between the Horstein and Schalkwijk-Kailth schemes in a simple encoding scheme called posterior matching scheme for general point-to-point memoryless channels.', '1204.4249-5-4-1': "The idea of posterior matching is that the transmitter encapsulates the information the receiver does not know up to present time in one random variable and then transmits that random variable to the receiver in the next transmission to refine the receiver's knowledge.", '1204.4249-5-4-2': 'The distribution of that variable will be selected in a way such that the input constraint is satisfied.', '1204.4249-5-4-3': 'Later, Bae and Anastasopolous extended this scheme for the finite-state channel with feedback by using another approach [11].', '1204.4249-5-4-4': 'Ma and Coleman provided a viewpoint on posterior matching from stochastic control perspectives [18] and generalized this encoding scheme to higher dimension via optimal transportation [19].', '1204.4249-5-4-5': 'One interesting open problem is to extend the Shayevitz and Feder posterior matching scheme for multiuser cases.', '1204.4249-5-4-6': 'In this paper, using the same approach as Shayevitz and Feder used for point-to-point memoryless channels, we propose a posterior matching based encoding and decoding strategy for real Gaussian MACs, referred to as a time-varying posterior matching scheme, and analyze the error probabilities for all encoding-decoding schemes designed by using these strategies.', '1204.4249-5-5-0': 'We analyze the achievable rate region and error performance of encoding and decoding schemes using these strategies by defining a generalized iterated function systems (GIFS) which has the generalized average contractive property (asymptotically average contractive).', '1204.4249-5-5-1': 'Refer to our Theorem I for more details.', '1204.4249-5-5-2': 'Note that our imposed constraint is less strict than the constraint that Shayevitz and Feder imposed to analyze the point-to-point memoryless channels.', '1204.4249-5-5-3': 'Specifically, in Theorem 6 in [4], Shayevitz and Feder used the relations between the information rates and contraction properties of the iterated function system (IFS) to analyze the error probability for point-to-point cases.', '1204.4249-5-5-4': 'For the continuous cases, they assumed that the reverse iterated function system (RIFS), generated by the kernel [MATH] and controlled by the identically distributed output sequence [MATH], has the average contractive property to analyze the error performance of their posterior matching schemes.', '1204.4249-5-5-5': 'That assumption requires the distribution at the output of the point-to-point memoryless channel be identically distributed when using their proposed encoding schemes.', '1204.4249-5-5-6': 'This also means that if the output distribution is not identically distributed, the error analysis in Theorem 6 in [4] cannot apply.', '1204.4249-5-5-7': 'For example, this situation happens with our proposed matching schemes for the Gaussian MAC in this paper.', '1204.4249-5-6-0': 'Finally, we illustrate our strategies by designing an encoding scheme that obtains optimal performance for the Gaussian MAC.', '1204.4249-5-6-1': "Specifically, our proposed code obtains the same performance as Ozarow's code [3] for the general two-user Gaussian channel, so it achieves the capacity of this channel.", '1204.4249-5-6-2': "For the case when the number of users is greater than 3, our proposed code obtains the same performance as the Kramer's code in the sense of sum-rate, so it is optimal among linear code with respect to sum rate capacity.", '1204.4249-5-6-3': 'To the best of our knowledge, the time-varying posterior matching in this paper is the first code designed for the real symmetric Gaussian MAC to achieve the linear-feedback sum-capacity when the number of users is greater than 3.', '1204.4249-5-7-0': 'The rest of this paper is organized as follows.', '1204.4249-5-7-1': 'Section II presents the channel model and some mathematical preliminaries.', '1204.4249-5-7-2': 'Sections III, IV introduce the time-varying posterior matching idea, and perform the error analysis of an encoding-decoding scheme for the Gaussian MAC with feedback constructed by using that idea.', '1204.4249-5-7-3': 'A time-varying encoding-decoding strategy and error analysis for the general two-user white Gaussian MAC and the multiuser symmetric white Gaussian MAC are placed in Section V. Finally, Section VI concludes this paper.', '1204.4249-5-7-4': 'Channel Model and Preliminaries', '1204.4249-5-8-0': '## Mathematical notations', '1204.4249-5-9-0': 'Upper-case letters, their realizations by corresponding lower-case letters, denote random variables.', '1204.4249-5-9-1': 'A real-valued random variable [MATH] is associated with a distribution [MATH] defined on the usual Borel [MATH]-algebra over [MATH], and we write [MATH].', '1204.4249-5-9-2': 'The cumulative distribution function (c.d.f.) of [MATH] is given by [MATH], and their inverse c.d.f is defined to be [MATH].', '1204.4249-5-9-3': 'The uniform probability distribution over [MATH] is denoted through [MATH].', '1204.4249-5-9-4': 'The composition function [MATH].', '1204.4249-5-9-5': 'In this paper, we use the following lemma:', '1204.4249-5-10-0': 'Lemma I: Let [MATH] be a continuous random variable with [MATH] and [MATH] be an uniform distribution random variable, i.e. [MATH] be statistical independent.', '1204.4249-5-10-1': 'Then [MATH] and [MATH].', '1204.4249-5-11-0': 'Refer to [4] for the proof.', '1204.4249-5-12-0': "Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions.", '1204.4249-5-12-1': 'Basically, it tells you how fast a function grows or declines.', '1204.4249-5-12-2': 'For the formal definition, suppose [MATH] and [MATH] are two functions defined on positive integer number.', '1204.4249-5-12-3': 'We write [EQUATION] (or [MATH] for [MATH] to be more precise) if and only if there exists constants [MATH] and [MATH] such that [EQUATION]', '1204.4249-5-12-4': 'Intuitively, this means that [MATH] does not grow faster than [MATH].', '1204.4249-5-13-0': 'In addition to big O notations, another Landau symbol is used in mathematics: the little o. Formally, we write [MATH] for [MATH] if and only if for every [MATH] there exists a real number [MATH] such that for all [MATH] we have [MATH].', '1204.4249-5-13-1': 'If [MATH], this is equivalent to [MATH].', '1204.4249-5-14-0': 'A Hadamard matrix [15] of order [MATH] is an [MATH] matrix of [MATH]s and [MATH]s such that [MATH].', '1204.4249-5-14-1': 'In fact, it is not yet known for which values of [MATH] an [MATH] does exists.', '1204.4249-5-14-2': 'However, we know that if a Hadamard matrix of order [MATH] exists, then [MATH] is [MATH], or a multiple of [MATH].', '1204.4249-5-14-3': 'Moreover, if [MATH] is of the form [MATH], [MATH] a positive integer, we can construct [MATH] by using the Sylvester method.', '1204.4249-5-14-4': 'Besides, the Paley construction, which uses quadratic residues, can be used to construct Hadamard matrices of order [MATH], where [MATH] is of the form [MATH], [MATH] is a prime, and [MATH] is a multiple of [MATH].', '1204.4249-5-15-0': '## Gaussian Multiple Access Channel with Feedback', '1204.4249-5-16-0': 'Consider the communication problem between [MATH] senders and a receiver over a multiple access channel with additive Gaussian noise (AWGN-MAC) when channel outputs are noiselessly fed back to all the senders (Figure 1).', '1204.4249-5-16-1': 'Each sender [MATH] wishes to reliably transmit a random message point [MATH], which is uniformly distributed over the unit interval with its binary expansion representing an infinite independent-identically-distributed (i.i.d.) Bernoulli(1/2) sequence, to the receiver.', '1204.4249-5-16-2': 'At each time [MATH], the output of the channel is [EQUATION] where [MATH] is the transmitted symbol by sender [MATH] at time [MATH], [MATH] is the output of the channel, and [MATH] is a discrete-time zero mean white Gaussian noise process with unit average power, i.e., [MATH] and is independent of [MATH].', '1204.4249-5-16-3': 'We assume that output symbols are casually fed back to the sender and the transmitted symbol [MATH] for sender [MATH] at time [MATH] can depend on both the message [MATH] and the previous channel output sequence [MATH].', '1204.4249-5-17-0': 'A transmission scheme for a Gaussian MAC is a set of [MATH] sequences of transmission functions [MATH] for [MATH], so that the input to the channel generated by the transmitter is given by [EQUATION]', '1204.4249-5-18-0': 'A decoding rule for a MAC is set of sequences of measurable mappings [MATH], where [MATH] is the set of all open intervals in [MATH] and [MATH].', '1204.4249-5-18-1': 'Here, [MATH], refers as to the decoded interval for the user [MATH].', '1204.4249-5-18-2': 'The error probabilities at time [MATH] associated with a transmission scheme and a decoding rule, is defined as [EQUATION] and the corresponding achievable rate vector at time [MATH] is defined to be [EQUATION]', '1204.4249-5-18-3': 'We say that a transmission scheme together with a decoding rule achieve a rate vector [MATH] over a Gaussian MAC if for all [MATH] we have [EQUATION]', '1204.4249-5-18-4': 'The rate vector is achieved within input power constraints [MATH], if in addition [EQUATION]', '1204.4249-5-18-5': 'An optimal fixed rate decoding rule for a MAC with rate region [MATH] is one that decodes a vector of fixed length intervals [MATH], whose marginal posteriori probabilities are maximal, i.e., [EQUATION]', '1204.4249-5-18-6': 'An optimal variable rate decoding rule with target error probabilities [MATH] is one that decodes a vector of minimal-length intervals [MATH] with accumulated marginal posteriori probabilities exceeds corresponding targets, i.e., [EQUATION]', '1204.4249-5-18-7': 'Both decoding rules make use of the marginal posterior distribution of the message point [MATH] which can calculate online at the transmitter [MATH] and the receiver.', '1204.4249-5-18-8': 'Refer [4] for more details.', '1204.4249-5-18-9': 'A proof that the achievability in the sense of [MATH] and [MATH] implies that the achievability in the standard framework are in the Appendix.', '1204.4249-5-19-0': 'Lemma II: The achievability in the definition [MATH] and [MATH] implies the achievability in the standard framework.', '1204.4249-5-20-0': 'Refer to the Appendix.', '1204.4249-5-21-0': '# Time-varying Posterior Matching Scheme', '1204.4249-5-22-0': "## Shayevitz and Feder's Posterior Matching Scheme", '1204.4249-5-23-0': 'In this part, we firstly review the posterior matching scheme proposed by Ofer Shayevitz and Meir Feder for point-to-point channel in [4].', '1204.4249-5-23-1': 'Specifically, the authors argued that after the receiver observed the output sequence [MATH], there is still some "missing information" that can be encapsulated in a random variable [MATH] with the following properties:', '1204.4249-5-24-0': '(i) [MATH] is statistically independent of [MATH].', '1204.4249-5-25-0': '(ii) The message point [MATH] can be a.s. uniquely recovered from [MATH]', '1204.4249-5-26-0': 'With that line of thought, they proposed a principle for generating the next channel input as follow:', '1204.4249-5-27-0': 'The transmission function [MATH] should be selected so that [MATH] is [MATH]-distributed, and is a fixed function of some random variable [MATH] satisfying properties [MATH] and [MATH].', '1204.4249-5-28-0': 'Lemma III: (Posterior Matching Scheme [4]).', '1204.4249-5-28-1': 'The following transmission scheme satisfies the posterior matching principle for any [MATH]: [EQUATION]', '1204.4249-5-28-2': 'Based on the transmission functions defined in section II-B [4], the input to the channel is a sequence of random variables given by [EQUATION]', '1204.4249-5-28-3': 'Refer to [4] for the proof.', '1204.4249-5-29-0': '## Time-varying Posterior Matching Scheme', '1204.4249-5-30-0': 'In this section, we propose a posterior matching scheme for additive Gaussian multiple access channel (MAC) with feedback, called time-varying posterior matching.', '1204.4249-5-30-1': 'Our encoding proposal is based on the following lemma:', '1204.4249-5-31-0': 'Lemma IV: For an additive white Gaussian MAC with feedback having [MATH] inputs and one output, let the output signal be a linear combination with known coefficients of the input signals, i.e. [MATH] where [MATH] is the additive white Gaussian noise and the coefficients [MATH] are part of the coding scheme, but are viewed as part of the channel for the purpose of deriving the posterior matching rule.', '1204.4249-5-31-1': 'Also assume that the covariance matrix among transmitted symbols at each time slot [MATH] defined as [EQUATION]', '1204.4249-5-31-2': 'Then the posterior matching scheme in Lemma III at each transmitter [MATH], i.e. [EQUATION] becomes a time-varying posterior matching given by [EQUATION] where [MATH] is a random variable encapsulating the input power constraint, [MATH], and [MATH] is the intended transmitted message at the transmitter [MATH].', '1204.4249-5-32-0': 'In addition, the correlation matrix among transmitted symbols at each time slot [MATH], i.e. [MATH], can be calculated online at the transmitters and the receiver.', '1204.4249-5-32-1': 'Especially, the posterior distribution [MATH] can be calculated online at both the transmitter [MATH] and the receiver.', '1204.4249-5-33-0': 'To begin with, we show that [MATH] constitutes a Markov chain.', '1204.4249-5-33-1': 'Indeed,', '1204.4249-5-34-0': 'The two observations imply that [MATH] are mutually independent of [MATH], which together with the memoryless property of the channel, implies the Markov chains.', '1204.4249-5-35-0': 'Moreover, by our construction of the multi-letter posterior matching formula, we have [EQUATION]', '1204.4249-5-35-1': 'Since [MATH] are continuous random variables, their c.d.f. and inverse c.d.f. functions are continuous, hence the composite function [MATH] is continuous on [MATH].', '1204.4249-5-35-2': 'Besides, the monotonicity of this function is originated from the monotonicity of the c.d.f. and the inverse c.d.f.', '1204.4249-5-36-0': "Now, let's return to prove the Lemma IV.", '1204.4249-5-36-1': 'Note that [EQUATION] where (a) follows from the fact that the transmission function is continuous and monotone, and (b) follows from the aforementioned fact that [MATH] forms a Markov chain for any [MATH].', '1204.4249-5-37-0': 'Finally, we have [EQUATION]', '1204.4249-5-37-1': 'Moreover, with time-varying posterior matching transmission applied for Gaussian MAC, [MATH] is a linear combination of [MATH] (as we will see in the proof of the Theorem II below), so [MATH] is a linear combination of [MATH].', '1204.4249-5-37-2': 'Therefore, the correlation between [MATH] and [MATH] only depends on the correlations among transmitted symbols at time [MATH], i.e. [MATH].', '1204.4249-5-37-3': 'In other words, [MATH] is a function of [MATH], so the transmitters and receiver can calculate the matrix [MATH] online at both transmitters and receiver.', '1204.4249-5-37-4': 'Moreover, from the relation [MATH], we see that the distribution of [MATH] is a function of all elements in the correlation matrix [MATH].', '1204.4249-5-37-5': 'That concludes the proof.', '1204.4249-5-38-0': 'Remark: We refer the transmission scheme in Lemma IV to as time-varying posterior matching scheme.', '1204.4249-5-38-1': "For a continuous point-to-point memoryless channel, the distribution [MATH] doesn't depend on [MATH], hence we have the posterior matching scheme for this case like the formula (16) in [4].", '1204.4249-5-38-2': 'However, in a MAC (for example additive white Gaussian MAC), where each received signal is a linear combination of all transmitted signals and Gaussian noise, the distribution [MATH] between the input [MATH] and the output may be dependent on [MATH].', '1204.4249-5-38-3': 'Therefore, time-varying posterior matching scheme may be the solution to overcome this problem.', '1204.4249-5-38-4': 'However, we will see from our proof in the Theorem I below the variable rate decoding rule, or Generalized Reverse Iterated Function System (GRIFS), can be applied at receiver to decode signals if and only if all the distributions [MATH] can be calculated online at the corresponding transmitters and receiver.', '1204.4249-5-38-5': 'With the result in the Lemma IV, this condition is always satisfied when time-varying posterior matching encoding schemes used at transmitters.', '1204.4249-5-38-6': 'We will show in the next parts that using time-varying posterior matching scheme at transmitters can obtain optimal performances for some known cases.', '1204.4249-5-39-0': '# Error analysis for time-varying posterior matching scheme', '1204.4249-5-40-0': 'In this section, we analyze error performance for Gaussian MAC with feedback employing the time-varying posterior matching scheme at the transmitters and variable-decoding rule at the receiver.', '1204.4249-5-41-0': 'Theorem I: Consider a real Gaussian MAC with [MATH] transmitters and one receiver without input power constraints.', '1204.4249-5-41-1': 'Assuming that at each transmitter [MATH], transmitted sequence [MATH] conforms to the time-varying posterior matching rule, as following: [EQUATION] where [MATH] is a Gaussian random variable, and [MATH] is the output of the Gaussian MAC which is a linear combination of all these transmitted signals at time [MATH].', '1204.4249-5-41-2': 'Let [MATH] and [MATH] as the global Lipschitz operator.', '1204.4249-5-41-3': 'Under the conditions that [EQUATION] define: [MATH] Then the rate region [MATH] is achievable and the error probabilities [MATH] decay to zero as [EQUATION]', '1204.4249-5-42-0': 'First, observe that since the distributions [MATH] can be calculated online at both transmitters and receiver by Lemma IV, therefore [MATH] can be calculated online at both the transmitters and receiver.', '1204.4249-5-42-1': 'Denote [MATH], referred as to a Generalized Iterated Function System (GIFS) generated by the kernel sequence [MATH].', '1204.4249-5-42-2': 'For each [MATH], select a fixed interval [MATH] as the decoded interval with respect to [MATH].', '1204.4249-5-43-0': 'Define the corresponding interval at the origin to be [MATH] and set them to be the decoded interval for [MATH], and so the decoded interval for [MATH] are set to be [MATH].', '1204.4249-5-44-0': 'It is easy to see that [EQUATION] where [EQUATION]', '1204.4249-5-44-1': 'For any fixed rate [MATH], we can find an [MATH] such that [MATH].', '1204.4249-5-44-2': 'Observe that: [EQUATION] where [MATH].', '1204.4249-5-44-3': 'Here, (a) follows from the definition of the instant corresponding achievable rate vector at time [MATH] in the Section II above, and (b) follows from the fact that we set the decoded interval [MATH].', '1204.4249-5-45-0': 'On the other hand, since [MATH], there exists an [MATH] such that [MATH].', '1204.4249-5-45-1': 'Let [MATH].', '1204.4249-5-45-2': "From (7) we have [EQUATION] where [MATH] follows from the Markov's inequality, [MATH] follows from [MATH], and [MATH] is a recursive application of the preceding transitions, [MATH] follows from [MATH] above and recursive applications of the preceding transitions.", '1204.4249-5-46-0': 'From (8), it is easy to see that a sufficient condition for both [MATH] is given by choosing [MATH].', '1204.4249-5-46-1': 'Since [MATH] depends only on the length of [MATH], so we can choose [MATH].', '1204.4249-5-46-2': 'Furthermore, from the Lemma I, we are easy to come to conclusion that [MATH].', '1204.4249-5-46-3': 'Denote [MATH] as the well-known tail function of the standard normal distribution and use the Chernoff bound of this function we obtain [EQUATION]', '1204.4249-5-46-4': 'Here, (a) follows from the fact that [MATH] is symmetric [MATH], and (b) follows from the Chernoff bound for the Q-function [MATH].', '1204.4249-5-47-0': 'To put it simply, any rate vector [MATH] is achievable, where [EQUATION]', '1204.4249-5-47-1': 'The error probabilities decay to zero as [EQUATION]', '1204.4249-5-47-2': 'Remark: Since we can estimate [MATH] and know our desired rate [MATH] in advance, it is possible to choose [MATH] by target.', '1204.4249-5-47-3': 'This means that the decoding algorithm is technically realizable.', '1204.4249-5-47-4': 'However, there is a tradeoff between the transmission rate [MATH] (the possible values of [MATH]) and the code length [MATH].', '1204.4249-5-47-5': 'If we transmit at the rate [MATH] very close to [MATH], we need to choose [MATH] to be very small.', '1204.4249-5-47-6': 'As a result, the required [MATH] may be very big.', '1204.4249-5-47-7': 'Furthermore, the fact that [MATH] is very lose to [MATH] also makes the error probabilities slowly decayed to zero.', '1204.4249-5-47-8': 'To put it simply, the code length [MATH] may be very large if we transmit at the rate [MATH] is nearly [MATH].', '1204.4249-5-47-9': 'On the contrary, being able to choose quite large [MATH] makes the required [MATH] smaller and the decay of error probabilities faster.', '1204.4249-5-48-0': '# A Posterior Matching Scheme for Gaussian MAC with Feedback', '1204.4249-5-49-0': 'In this section, we consider a real Gaussian MAC with [MATH] receivers and input power constraints [MATH] at the transmitters [MATH], respectively as defined in the section II.', '1204.4249-5-49-1': 'Our encoding scheme for this channel as following:', '1204.4249-5-50-0': '## Encoding', '1204.4249-5-51-0': '## Decoding', '1204.4249-5-52-0': 'We refer this encoding strategy as Gaussian MAC posterior matching feedback coding and decoding strategy.', '1204.4249-5-53-0': 'Theorem II: Using the Gaussian MAC posterior matching feedback coding and decoding strategy above, the rate region [MATH] is achievable for Gaussian MAC with feedback, where [EQUATION] by setting the target error probabilities [EQUATION] where [EQUATION] with [EQUATION].', '1204.4249-5-53-1': 'Applying the Lemma I, we see that for any [MATH] then: [EQUATION]', '1204.4249-5-53-2': 'Observe that, by this transmission strategy, each transmitter [MATH] transmits [MATH] at time [MATH] with [MATH], thus the input power constraints at all transmitters are always satisfied at each transmission time [MATH] Moreover, the output at receiver at time [MATH] will be [EQUATION]', '1204.4249-5-53-3': 'Thus [EQUATION]', '1204.4249-5-53-4': 'Then, we have [EQUATION] and [EQUATION] where [EQUATION] and [EQUATION]', '1204.4249-5-53-5': 'Finally, we obtain [EQUATION] where [EQUATION]', '1204.4249-5-53-6': 'Moreover, from [MATH] we have: [EQUATION] thus, [EQUATION]', '1204.4249-5-53-7': 'Combining with [MATH], we obtain: [EQUATION]', '1204.4249-5-53-8': 'Hence, [EQUATION]', '1204.4249-5-53-9': 'Finally, we have: [EQUATION]', '1204.4249-5-53-10': 'If we can achieve [EQUATION] then [EQUATION]', '1204.4249-5-53-11': 'This means that the condition in the Theorem I is satisfied, which leads to the rate region [MATH] is achievable, where [EQUATION]', '1204.4249-5-53-12': 'Note that this capacity region is obtained by setting the target error probabilities [MATH] under the constraints [EQUATION] which have the well-known double-exponential behavior.', '1204.4249-5-54-0': 'In the following, we will specify achievable rate regions, and error probabilities for two cases: the general two-user Gaussian MAC, and the real symmetric Gaussian MAC when the number of users is arbitrary.', '1204.4249-5-55-0': 'From the Theorem II, we see that the strategy to design posterior encoding scheme for multiple access channels with feedback is to find the sequences [MATH] such that [EQUATION] for all [MATH].', '1204.4249-5-56-0': 'Case 1: Two-user Gaussian MAC with feedback.', '1204.4249-5-57-0': "To show that Ozarow's coding scheme [3] is a special case of our posterior matching framework, we can set [MATH] and later prove that [MATH].", '1204.4249-5-57-1': 'Observe that the constraint [MATH] can be also checked to be satisfied by this setting since [EQUATION] for [MATH] and [EQUATION] for [MATH].', '1204.4249-5-58-0': 'Now we need to find the recursion of [MATH] and [MATH] in this case.', '1204.4249-5-58-1': 'Observe that the output sequence: [EQUATION] where [MATH] is noise process and [MATH].', '1204.4249-5-59-0': 'From [MATH] we have [EQUATION] and [EQUATION]', '1204.4249-5-59-1': 'Therefore, we obtain: [EQUATION] and [EQUATION]', '1204.4249-5-60-0': 'Finally, the time-varying posterior matching encoding scheme for Gaussian MAC with feedback in this special case as following:', '1204.4249-5-61-0': 'Achievable rate region and error analysis:', '1204.4249-5-62-0': 'For this special case, we have [EQUATION] and [EQUATION]', '1204.4249-5-62-1': 'Apply the Theorem II above, we obtain the achievable rate region for Gaussian MAC with two users as [MATH] where [EQUATION]', '1204.4249-5-62-2': 'Similarly, [EQUATION] by setting the target error probability to [EQUATION] and [EQUATION]', '1204.4249-5-62-3': 'Much like Ozarow in [MATH], at the reception [MATH], the receiver adds an independent random variable [MATH] before feeding back the first receiver signal to the transmitters 1 and 2 to set [MATH], where [MATH] is the biggest solution in [MATH] of the following equation: [EQUATION]', '1204.4249-5-62-4': 'By this changing, from [MATH] we see that [MATH], so [MATH] as mentioned above.', '1204.4249-5-62-5': 'We also have [MATH], where [MATH] is a positive solution of the equation [MATH].', '1204.4249-5-62-6': 'Replace this result to [MATH] and combine with [MATH], we have: [EQUATION] where [MATH] is defined above.', '1204.4249-5-62-7': "We see that, all the results are the same as Ozarow's results in [3].", '1204.4249-5-62-8': 'So our posterior matching encoding scheme is optimal for Gaussian channel MAC with two users.', '1204.4249-5-63-0': 'Case 2: M-user symmetric Gaussian MAC with feedback.', '1204.4249-5-63-1': 'We consider symmetric case, where [MATH].', '1204.4249-5-64-0': 'Achievable rate region and error analysis:', '1204.4249-5-65-0': 'Assuming that all the transmitted messages are statistically independent.', '1204.4249-5-65-1': 'Define the normalized covariance matrix by [EQUATION]', '1204.4249-5-65-2': 'Then [EQUATION] where [EQUATION] is the correlation coefficient between [MATH] and [MATH].', '1204.4249-5-66-0': 'We will prove by induction that the normalized covariance has all the columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is symmetric positive definite for all [MATH]', '1204.4249-5-67-0': 'Indeed, with the assumption all the transmitted information messages are statistically independent, we will have [MATH], which is an identity matrix of size [MATH].', '1204.4249-5-67-1': 'Therefore, it is obvious that all the columns of the Hadamard matrix [MATH] by [MATH] are eigenvectors of the matrix [MATH] and that [MATH] is a positive definite matrix.', '1204.4249-5-68-0': 'Now, assume that [MATH] has all columns of the Hadamard matrix [MATH] by [MATH] as its eigenvectors and that [MATH] is positive definite for some [MATH].', '1204.4249-5-68-1': 'Since we assumed that [MATH] is symmetric positive definite matrix, all its eigenvalues are positive.', '1204.4249-5-68-2': 'Denote by [MATH] the [MATH] columns of the Hadamard matrix [MATH].', '1204.4249-5-68-3': 'By this encoding scheme, we set the vector [MATH].', '1204.4249-5-68-4': 'Assume that [MATH] is the eigenvalue of [MATH] associated with the [MATH] eigenvector.', '1204.4249-5-68-5': 'We have [EQUATION]', '1204.4249-5-68-6': 'On the other hand, we also have [EQUATION] where [MATH] is the [MATH]th column of the matrix [MATH].', '1204.4249-5-68-7': 'Note that [MATH], this means that [EQUATION]', '1204.4249-5-68-8': 'Hence, [EQUATION].', '1204.4249-5-68-9': 'Moreover, observe that [EQUATION]', '1204.4249-5-68-10': 'Combining these results, we obtain: [EQUATION] and [EQUATION]', '1204.4249-5-68-11': 'Substitute [MATH] and [MATH] into [MATH], we obtain: [EQUATION]', '1204.4249-5-68-12': 'Observe that from the proof of the Theorem II above, then [EQUATION].', '1204.4249-5-68-13': 'Then, we have [EQUATION]', '1204.4249-5-68-14': 'Moreover, from (13) we know that [EQUATION]', '1204.4249-5-68-15': 'Therefore, [EQUATION]', '1204.4249-5-68-16': 'Replacing the results in [MATH], and [MATH] to this expression, we obtain [EQUATION] for all [MATH].', '1204.4249-5-69-0': 'Since we assumed that [MATH] is symmetric positive definite matrix, hence [MATH], hence [MATH].', '1204.4249-5-69-1': 'Therefore, from [MATH], it is easy to see that [MATH].', '1204.4249-5-69-2': 'In other words, [MATH] is also a symmetric matrix.', '1204.4249-5-70-0': 'Moreover, from [MATH], we also have [EQUATION]', '1204.4249-5-70-1': 'Denote [MATH].', '1204.4249-5-70-2': 'We see that the columns of [MATH] creates [MATH] linearly independent eigenvectors of the matrix [MATH].', '1204.4249-5-70-3': 'Moreover, we also have [EQUATION]', '1204.4249-5-70-4': 'Note that since all columns of [MATH] are eigenvectors of the matrix [MATH], so all the columns of the matrix [MATH] are also eigenvectors of the matrix [MATH], hence we has the following eigenvalue decomposition [EQUATION] where [MATH] is a diagonal matrix.', '1204.4249-5-71-0': 'Moreover, we have [EQUATION] where [EQUATION]', '1204.4249-5-71-1': 'From [MATH], the matrix [MATH] must be a diagonal one since the right side of [MATH] is a diagonal matrix.', '1204.4249-5-71-2': 'Hence, all columns of the matrix [MATH] are eigenvectors of the matrix [MATH].', '1204.4249-5-72-0': 'Assuming [MATH] are [MATH] eigenvalues corresponding to eigenvectors which are columns of the matrix [MATH].', '1204.4249-5-72-1': 'By this notation, we see that [MATH].', '1204.4249-5-73-0': 'Combining [MATH], and [MATH] we obtain [EQUATION] for all [MATH].', '1204.4249-5-74-0': 'Since we assumed that all eigenvalues of [MATH] are positive ([MATH] symmetric positive definite), from [MATH] we see that all eigenvalues of [MATH] are also positive.', '1204.4249-5-74-1': 'Note that we also confirmed that [MATH] is symmetric above, therefore [MATH] is a symmetric positive definite matrix.', '1204.4249-5-74-2': 'In short, if [MATH] is a symmetric positive definite matrix and has all columns of the Hadamard matrix as its eigenvectors, then [MATH] has all these properties.', '1204.4249-5-74-3': 'This concludes our proof by induction.', '1204.4249-5-75-0': 'According to the Lemma 1 [12] (or see the Appendix below), the sequence [MATH] converges (or can be forced to converge) to a fixed point [MATH], which is the biggest positive solution in [MATH] of the following equation [EQUATION]', '1204.4249-5-75-1': 'From (27), (28) we obtain [EQUATION]', '1204.4249-5-75-2': 'Since [MATH], we have [EQUATION]', '1204.4249-5-75-3': 'Therefore, the constraints in the Theorem II is satisfied.', '1204.4249-5-75-4': 'Applying the result of this theorem, we have any rate less than [EQUATION] is achievable, for all [MATH].', '1204.4249-5-75-5': 'Hence, any sum rate which is less than [EQUATION] is achievable, where [MATH] is solution in the [MATH] of the equation [MATH].', '1204.4249-5-75-6': 'This result coincides with the formula (68) in [14].', '1204.4249-5-75-7': 'The paper [16] proves that this achievable sum rate is optimal for the class of linear feedback coding.', '1204.4249-5-76-0': '# Conclusion', '1204.4249-5-77-0': 'A posterior matching based encoding-decoding strategy for general Gaussian MAC with feedback was proposed, and achievable rate region, error performance were drawn.', '1204.4249-5-77-1': 'Finally, we analyzed error performance of the proposed posterior encoding scheme and showed that the time-varying posterior matching scheme and variable rate decoding ideas can be applied to Gaussian MAC and obtain optimal performances.', '1204.4249-5-77-2': 'Specifically, the proposed encoding scheme achieves the capacity of two-user feedback Gaussian MAC as well as linear-feedback sum-rate for symmetric Gaussian MAC with feedback where the number of users is arbitrary.', '1204.4249-5-77-3': "Moreover, by the encoding scheme's structure, which uses the spreading codes like the Hadamard matrix, our encoding scheme can be directly applied to CDMA systems with feedback.", '1204.4249-5-77-4': 'Finally, by analyzing all arguments in the theorem I, the time-varying posterior matching scheme approach in this paper might be applied for other Gaussian and non-Gaussian multiuser channels to achieve optimal performances.', '1204.4249-5-77-5': '[Proof of Lemma II]', '1204.4249-5-78-0': 'We use the same line argument as Lemma 1 [4].', '1204.4249-5-78-1': 'Assume we are given a transmission scheme with [MATH] transmission functions [MATH] and a decoding rule which are known to achieve the rate vector [MATH].', '1204.4249-5-78-2': 'For simplicity, we assume that the decoding rule is fixed rate [MATH] for all [MATH]), since any variable rate decoding rule can be easily mapped into a fixed rate rule that achieves the same rate vector.', '1204.4249-5-78-3': 'It is easy to see that in order to prove that the above translates into achievability for some rate vector [MATH] in the standard framework, it is enough to show we can find [MATH] sequences [MATH] and such that we have the uniform achievability over [MATH], i.e., [EQUATION]', '1204.4249-5-78-4': 'We now show how [MATH] can be constructed for any [MATH].', '1204.4249-5-78-5': 'Let [MATH] be the (average) error probability associated with our scheme and the fixed rate vector [MATH].', '1204.4249-5-78-6': 'Define [EQUATION] and write [EQUATION] and so we have that [MATH].', '1204.4249-5-78-7': 'It is now easy to see that if we want to select [MATH] such that [MATH], and also [MATH], then a sufficient condition is that [MATH] for some positive [MATH].', '1204.4249-5-78-8': 'This condition can be written as [EQUATION]', '1204.4249-5-78-9': 'At the same time, we also have by definition [EQUATION] [Forcing the sequence convergence]', '1204.4249-5-79-0': 'We use the similar arguments as the Appendix A [14].', '1204.4249-5-79-1': 'It is difficult to prove that the recursion (36) is convergence.', '1204.4249-5-79-2': 'However, we know that for [MATH] [EQUATION]', '1204.4249-5-79-3': 'Therefore, if the sequence [MATH] is convergent, it will converges to the solution of the following equation: [EQUATION]', '1204.4249-5-79-4': 'It is easy to show that this equation has at least one solution in the [MATH].', '1204.4249-5-79-5': 'Set [MATH] to be the biggest solution of this equation.', '1204.4249-5-79-6': 'We also set [EQUATION]', '1204.4249-5-79-7': 'We will force the recursion (36) to yield the desired values [MATH] at time [MATH].', '1204.4249-5-80-0': 'To perform this forcing, we replace [MATH] by [MATH] at time [MATH] and apply (36) [MATH] times to get [EQUATION] for all [MATH], where [EQUATION]', '1204.4249-5-80-1': 'We thus have, for [MATH] [EQUATION]', '1204.4249-5-80-2': 'Solving for [MATH], we have [EQUATION] where [MATH] can be computed recursively via [MATH] and [EQUATION]'} | null | null |
1201.3968 | {'1201.3968-1-0-0': "We show that Gribov's cancellation condition in top-quark condensation can be rephrased as the physical condition of tadpole cancellation.", '1201.3968-1-0-1': 'This is naturally implemented as the cancellation between the top-quark, Goldstone-boson and Higgs-boson contributions, and thus an extra condition on the mass of the Higgs boson is obtained.', '1201.3968-1-0-2': 'The Goldstone-boson and Higgs couplings are fixed by the relevant Ward-Takahashi identities.', '1201.3968-1-0-3': "In Gribov's U(1)[MATH] scenario, the new condition yields [MATH] GeV, with large theoretical uncertainty.", '1201.3968-1-0-4': "We also calculated the bosonic correction terms to Gribov's mass formula, which reduces the prediction for [MATH] from [MATH] GeV to 132 GeV.", '1201.3968-1-0-5': 'There being three conditions and three unknowns, we are able to obtain all three masses, [MATH] and [MATH], in [MATH] MeV to [MATH] TeV energy range, as a function of the cut-off scale and the gauge couplings only.', '1201.3968-1-1-0': '# Introduction', '1201.3968-1-2-0': 'As is well known, the Standard Model, with an elementary Higgs doublet field, suffers from the problem of quadratic divergences when radiative corrections, due to the loops of fermions, the top quark in particular, to the mass of the Higgs boson is calculated.', '1201.3968-1-2-1': 'This problem is often referred to as the hierarchy problem, as we require an artificial fine tuning between the induced radiative mass, which is of the order of the cut-off scale, e.g., [MATH], and the counterterm to produce a mass of the order of the electroweak scale.', '1201.3968-1-3-0': 'This leads us to speculate that the Higgs boson may in fact be composite.', '1201.3968-1-3-1': 'The simplest implementation of this idea is top-quark condensation [CITATION].', '1201.3968-1-3-2': 'It turns out, however, that this compositeness by itself is not sufficient to remove the problem of quadratic divergences, and fine tuning is still required in the simpler approaches to top-quark condensation.', '1201.3968-1-4-0': 'The problem of divergences can only be artificial, because the same loop corrections, applied to the mass of the Goldstone bosons, produce a similar quadratic divergence, whereas the Goldstone theorem guarantees that spontaneous symmetry breaking results in massless Goldstone bosons.', '1201.3968-1-4-1': 'This suggests that these quadratic divergences are an artifact, and will vanish if the condition of current conservation is implemented properly.', '1201.3968-1-5-0': 'For example, the approach of Chesterman, King and Ross [CITATION] uses the vanishing of the mass of the Goldstone boson as a consistency check, and is able to obtain sensible values for the mass of the Higgs boson.', '1201.3968-1-6-0': 'A more direct investigation into this point regarding current conservation was made by Gribov in ref. [CITATION], and his results are in quantitative agreement with ref. [CITATION].', '1201.3968-1-6-1': 'In that paper, he implemented the symmetry condition, somewhat by force, by requiring that the Goldstone-boson self-energy vanishes.', '1201.3968-1-6-2': 'The mass of the Higgs boson is then obtained by subtracting off the Goldstone-boson self-energy, leading to the following Pagels-Stokar-type equation for the mass of the Higgs boson: [EQUATION]', '1201.3968-1-6-3': 'This gives the mass [MATH] GeV for the Higgs boson using the top-quark mass 174 GeV and [MATH] as input.', '1201.3968-1-6-4': "Gribov's cut-off [MATH] is given by the [MATH] Landau pole, [MATH] GeV, but the results are relatively insensitive to the value of the cut-off scale.", '1201.3968-1-6-5': 'This value of [MATH] is much less than other values that are obtained in the literature [CITATION] and, in this sense, compares more favourably with the recent LHC results [CITATION] which suggest [MATH] GeV.', '1201.3968-1-7-0': 'Despite this conceptual and qualitative success, Gribov does not explicitly work out the mechanism for the cancellation of the Goldstone-boson self-energy.', '1201.3968-1-8-0': 'In this paper, we show that this cancellation is brought about through the cancellation of the tadpole, which is a physical condition that is required in order that the vacuum is stable.', '1201.3968-1-8-1': 'The analysis is analogous to that carried out in ref. [CITATION] in the context of anti-ferromagnetism.', '1201.3968-1-8-2': 'The cancellation of the Higgs tadpole, between the top-quark, Goldstone-boson and Higgs-boson loops, is responsible for the taming of the quadratic divergences, and this leads to the following extra condition: [EQUATION]', '1201.3968-1-8-3': 'This gives us the value [MATH] GeV, using the same input as above.', '1201.3968-1-8-4': 'This is very close to the LHC value, but this remarkable success should not be taken too seriously, because this condition is sensitive to the renormalization of the top-quark mass at near the Landau pole.', '1201.3968-1-9-0': 'It should be noted that our extra condition on the Higgs mass completely fixes the low energy parameters.', '1201.3968-1-9-1': 'That is, all three masses, [MATH], [MATH] and [MATH], are fixed for given values of the cut-off scale and gauge couplings.', '1201.3968-1-9-2': 'The form of the UV theory does not affect the results, so long as the symmetry is spontaneously broken by a UV condensate which exists only at high energies.', '1201.3968-1-9-3': "A natural candidate is Gribov's U(1)[MATH] scenario.", '1201.3968-1-9-4': 'We discuss the physics of this scenario.', '1201.3968-1-10-0': '# Tadpole cancellation conditions', '1201.3968-1-11-0': 'Tadpole cancellation in bosonic self-energy was, in part, discussed by Gribov himself in the context of chiral symmetry breaking in ref. [CITATION].', '1201.3968-1-11-1': 'However, he imposes tadpole cancellation there by introducing a boson-boson-quark-quark term (e.g., eqn. (132) of ref. [CITATION]) to cancel the anomalous term in the Ward-Takahashi identity associated with Compton scattering.', '1201.3968-1-11-2': 'One problem in doing so is that this will lead to the necessity of introducing an infinite number of [MATH]-boson-quark-quark couplings.', '1201.3968-1-12-0': 'In ref. [CITATION], and in the context of anti-ferromagnetism, we proposed a more economical mechanism which also seems to us to be more natural.', '1201.3968-1-12-1': 'Here, only three sets of couplings are needed: the Goldstone-Higgs-Goldstone couplings, the four-Goldstone couplings, and the Goldstone-Goldstone-Higgs-Higgs couplings.', '1201.3968-1-12-2': 'These couplings are fixed by symmetry conditions such as the Ward-Takahashi identities.', '1201.3968-1-12-3': 'The couplings exist because these Goldstone states are composite.', '1201.3968-1-12-4': 'Otherwise, e.g., in the Standard model, they are zero.', '1201.3968-1-13-0': 'Let us start with the fermionic vertex function for the left-handed SU(2) current.', '1201.3968-1-13-1': 'As discussed in ref. [CITATION], Ward-Takahashi identity is satisfied by the following modified vertex: [EQUATION]', '1201.3968-1-13-2': "Here, the small blob stands for the unmodified vertex [MATH] and the two-point function ([MATH] in Gribov's notation) which follows from that, and the asterisk stands for the vertex that is modified by the inclusion of the Goldstone-boson contribution.", '1201.3968-1-13-3': 'The dashed line stands for one of the three Goldstone bosons [MATH], [MATH].', '1201.3968-1-14-0': 'The Ward-Takahashi identity, applied to the modified vertex, fixes the [MATH] coupling to fermions to be, for example, [EQUATION] where [MATH], the Goldstone-boson form factor, is defined by the two-point function of eqn. ([REF]).', '1201.3968-1-14-1': "[MATH], where [MATH] is the usual 'vacuum-expectation value of the Higgs field'.", '1201.3968-1-14-2': 'Note that the Feynman rules are, as usual, given by [MATH].', '1201.3968-1-14-3': 'This will apply to all couplings that appear in the following.', '1201.3968-1-15-0': 'The same exercise may be repeated for the bosonic vertex: [EQUATION]', '1201.3968-1-15-1': 'The unmodified vertex is necessarily proportional to [MATH] (momenta flows left to right, or more generally [MATH] to [MATH]) in order to satisfy the Ward-Takahashi identity, and the normalization is fixed by the Ward-Takahashi identity applied to the amplitude shown in fig. [REF].', '1201.3968-1-16-0': 'This fixes the [MATH] coupling to be: [EQUATION] when the Yukawa coupling of the Higgs boson to fermions is given by [EQUATION]', '1201.3968-1-16-1': 'Next, we consider the Ward-Takahashi identity in the set of amplitudes which are described by fig. [REF].', '1201.3968-1-16-2': 'This allows us to work out the Goldstone-boson quartic couplings: [EQUATION]', '1201.3968-1-16-3': 'We then turn to the set of amplitudes that are described by fig. [REF], and obtain [EQUATION]', '1201.3968-1-16-4': 'Note that the Higgs-boson self-coupling is not fixed by current conservation conditions but by loops.', '1201.3968-1-16-5': 'As noted in ref. [CITATION], these turn out to be the same as the couplings of the Standard Model: [EQUATION]', '1201.3968-1-16-6': 'This implies [EQUATION]', '1201.3968-1-16-7': 'We notice that the effective Lagrangian for the multi-boson interaction terms can be written as [EQUATION]', '1201.3968-1-16-8': 'We now consider the tadpole cancellation condition: [EQUATION]', '1201.3968-1-16-9': 'Let us neglect the contribution of all fermions other than the top quark.', '1201.3968-1-16-10': 'This yields [EQUATION] where the simplification [MATH] has been made.', '1201.3968-1-16-11': 'The shorthand notation [MATH] is used.', '1201.3968-1-16-12': 'Note that in the Goldstone loop, [MATH] is not corrected by renormalization effects.', '1201.3968-1-16-13': 'Running [MATH] is used in the Higgs boson loop.', '1201.3968-1-16-14': 'This contribution will be negligible since [MATH] will be small at high scales.', '1201.3968-1-16-15': 'We have neglected the renormalization effects to [MATH], which are relatively small, due to the Goldstone and Higgs boson propagators.', '1201.3968-1-17-0': 'Since eqn. ([REF]) is dominated by the large energy region, we then obtain [EQUATION] as discussed in the introduction.', '1201.3968-1-17-1': 'Strictly speaking, the scale on the right-hand side should be slightly below [MATH].', '1201.3968-1-17-2': 'The running of the top quark mass is as given by eqn. (10) of ref. [CITATION]: [EQUATION] where only the QCD part of the evolution is included, and [EQUATION]', '1201.3968-1-17-3': 'The result of eqn. ([REF]) is shown in fig. [REF].', '1201.3968-1-17-4': "Gribov's scenario yields [MATH] GeV using the U(1)[MATH] Landau scale as [MATH].", '1201.3968-1-17-5': 'The Landau scale ([MATH] GeV) is as given by eqns. (45) and (46) of ref. [CITATION]: [EQUATION]', '1201.3968-1-17-6': 'As mentioned in the introduction, this spectacular agreement with the LHC results [CITATION] should be treated with caution and suspicion, because the result is sensitive to renormalization effects at near the cut-off (Landau) scale.', '1201.3968-1-17-7': "Even so, this result provides a support to Gribov's U(1)[MATH] scenario, if the LHC results will be confirmed.", '1201.3968-1-18-0': 'Let us consider the Goldstone-boson self-energy.', '1201.3968-1-18-1': 'This is given by the Feynman graphs of fig. [REF].', '1201.3968-1-19-0': 'At zero external momentum, the contributions of the three diagrams are given by: [EQUATION]', '1201.3968-1-19-1': 'Adding together these three equations yields zero so long as the tadpoles cancel.', '1201.3968-1-19-2': 'For a more formal, all-order treatment of such cancellations in the Goldstone-boson mass, we may, for instance, consider the non-vanishing part of [MATH] in eqn. (29) of ref. [CITATION] and show that this is equivalent to tadpole contributions.', '1201.3968-1-20-0': 'Because the Goldstone-boson self-energy must be equal to [MATH], calculating [MATH] at finite and small external momentum yields [MATH], and this reproduces eqn. (14) of ref. [CITATION]: [EQUATION]', '1201.3968-1-20-1': 'The bosonic contributions do not give logarithmic corrections to this equation.', '1201.3968-1-20-2': 'This equation predicts [MATH] as a function of [MATH], or vice versa and, as a generic problem in top-quark condensation approaches, it is well known that the predicted [MATH] is too high.', '1201.3968-1-20-3': 'For example, Gribov [CITATION] predicts [MATH] GeV for [MATH].', '1201.3968-1-20-4': 'One source of this discrepancy may be new contributions (e.g., gravitational) at high scales, since eqn. ([REF]) is relatively sensitive to the mass evolution at high scales.', '1201.3968-1-21-0': 'Let us now turn to the Higgs-boson self-energy.', '1201.3968-1-21-1': 'These are also given by the three diagrams shown in fig. [REF].', '1201.3968-1-21-2': 'The amplitudes at zero external momentum are now given by [EQUATION]', '1201.3968-1-21-3': 'The quadratic divergences cancel between [MATH] and [MATH], and we obtain [EQUATION]', '1201.3968-1-21-4': 'The second term of eqn. ([REF]) gives a correction to eqn. ([REF]).', '1201.3968-1-21-5': 'We can write the left-hand side of eqn. ([REF]) as [MATH] by setting the lower limit of integration to [MATH] and replacing [MATH] by [MATH].', '1201.3968-1-21-6': "We found, by solving the resulting integral equation numerically, using the same input as Gribov's work, that [MATH] is reduced from 167 GeV to 132 GeV.", '1201.3968-1-22-0': 'Note that the tadpole cancellation condition provides a third condition, in addition to eqn. ([REF]) and eqn. ([REF]).', '1201.3968-1-22-1': 'There being three unknowns, [MATH], [MATH], [MATH], and three equations, we can solve the set of equations with [MATH] as the input.', '1201.3968-1-22-2': 'We may do so by solving the following integral equation: [EQUATION] numerically, with the condition [MATH].', '1201.3968-1-22-3': 'We then search for a value of [MATH] which satisfies [MATH].', '1201.3968-1-22-4': 'We then obtain [EQUATION]', '1201.3968-1-22-5': 'This generates a large hierarchy.', '1201.3968-1-22-6': 'We plot the resulting [MATH] and [MATH] as a function of the cut-off scale [MATH], in fig. [REF].', '1201.3968-1-22-7': '[MATH] is indistinguishable from [MATH] on the logarithmic scale.', '1201.3968-1-23-0': 'Taking [MATH] to be the U(1)[MATH] Landau scale yields [EQUATION] which is only two orders of magnitude away from the top-mass scale.', '1201.3968-1-23-1': 'Note that [MATH] is needed in the U(1)[MATH] scenario.', '1201.3968-1-24-0': 'One should beware of hasty conclusions, from fig. [REF], that the cut-off scale is at [MATH] GeV.', '1201.3968-1-24-1': 'The need to produce a value of [MATH] that is as large as the observed value favours large [MATH].', '1201.3968-1-24-2': 'One possibility is that other, hitherto neglected, corrections will increase [MATH] in eqn. ([REF]).', '1201.3968-1-25-0': '# Discussions', '1201.3968-1-26-0': 'Our result shows that the electroweak scale is not necessarily fixed by the parameters at the UV scale.', '1201.3968-1-26-1': 'Rather, it is fixed by the condition of vacuum stability and the radiative effects.', '1201.3968-1-27-0': 'What conditions are necessary, then, in order that symmetry breaking occurs?', '1201.3968-1-28-0': 'In our opinion, it is sufficient that the symmetry is broken by a chiral condensate which appears at a high scale.', '1201.3968-1-28-1': 'This requires that a coupling constant becomes large.', '1201.3968-1-28-2': 'Gribov has shown [CITATION] that supercriticality and chiral symmetry breaking occurs when the relevant coupling constant satisfies [EQUATION]', '1201.3968-1-28-3': "Gribov's U(1)[MATH] scenario is a natural candidate, but one will then need to show that the gravitational coupling will not satisfy this condition.", '1201.3968-1-28-4': 'A possibility would be that gravitation is described by a weak-coupling theory above the Planck scale, and by asymptotic freedom, or otherwise, it becomes strongly coupled near the Planck scale and reduces to Einstein gravity at low scales.', '1201.3968-1-28-5': '[MATH] gravity might be a possible candidate.', '1201.3968-1-29-0': 'It is interesting to ask, what might be the behaviour of the U(1)[MATH] coupling at high scales?', '1201.3968-1-29-1': 'This question was addressed partially in ref. [CITATION].', '1201.3968-1-29-2': 'We argued that when the U(1)[MATH] coupling grows large, it will decouple as [MATH].', '1201.3968-1-29-3': 'The effective coupling experienced by the electrons is then given by [EQUATION] where [MATH] is the first coefficient of the beta function.', '1201.3968-1-29-4': 'If we can use the U(1)[MATH] beta function here, then the effective coupling will tend to [MATH], which is large compared with [MATH].', '1201.3968-1-29-5': 'Thus chiral symmetry breaking will almost certainly occur by U(1)[MATH].', '1201.3968-1-30-0': 'The condensate which appears at high scales must have decayed to fermions and Goldstone bosons at low scales, from phenomenological reasons.', '1201.3968-1-30-1': 'First, the observed masses of particles are light.', '1201.3968-1-30-2': 'Second, a condensate will lead to a cosmological constant which is much heavier than is observed.', '1201.3968-1-31-0': 'If EWSB is due to the formation of the supercritical condensate at high scales, one must ask the question of whether the supercritical states might affect the running of the parameters.', '1201.3968-1-31-1': 'Our answer is that the Goldstone and the Higgs states are supercritical states, but the other states cannot affect the running, because the Goldstone and the Higgs are the only states whose masses are protected by symmetry from growing large.', '1201.3968-1-31-2': 'The other supercritical states will have masses that are of the order of [MATH].', '1201.3968-1-32-0': '# Conclusions', '1201.3968-1-33-0': "We have shown that Gribov's cancellation condition in top-quark condensation may be rephrased as the cancellation, between the top, Higgs and the Goldstone contributions, of the Higgs-boson tadpole.", '1201.3968-1-33-1': 'This is a physical condition which must be satisfied in order that the ground state is stable.', '1201.3968-1-34-0': 'Tadpole cancellation gives us an extra condition on the mass of the Higgs boson.', '1201.3968-1-34-1': "In Gribov's U(1)[MATH] scenario, we obtain [MATH] GeV.", '1201.3968-1-34-2': 'This is in good agreement with the recent LHC announcement, [MATH] GeV, but our numbers are sensitive to the running of the top-quark mass in the UV region.', '1201.3968-1-34-3': "We have, furthermore, been able to calculate the bosonic contribution to Gribov's mass formula, eqn. ([REF]), which reduces Gribov's prediction of [MATH] GeV to [MATH] GeV.", '1201.3968-1-35-0': 'Our having one extra condition fixes the low-energy parameters completely in terms of the cut-off scale.', '1201.3968-1-35-1': "In Gribov's scenario, we obtain [MATH] TeV, which is remarkably close to the actual scale, starting from [MATH] GeV, with no other input than the values of the dimensionless couplings and their running.", '1201.3968-1-36-0': 'As an extension of our work, we note that using the couplings worked out in this paper, physical processes, for example Goldstone-boson scattering, may be calculated as [EQUATION] where the sum is over all allowed channels.', '1201.3968-1-36-1': 'These may be compared with gauge-boson scattering processes at LHC.'} | {'1201.3968-2-0-0': "We show that Gribov's cancellation condition in top-quark condensation can be rephrased as the physical condition of tadpole cancellation.", '1201.3968-2-0-1': 'This is naturally implemented as the cancellation between the top-quark, Goldstone and Higgs contributions, and thus an extra condition on the mass of the Higgs boson is obtained.', '1201.3968-2-0-2': 'The Goldstone and Higgs couplings are fixed by the relevant Ward-Takahashi identities.', '1201.3968-2-0-3': "In Gribov's U(1)[MATH] scenario, the new condition yields [MATH] GeV, with large theoretical uncertainty.", '1201.3968-2-0-4': "We also calculated the bosonic correction terms to Gribov's mass formula, which reduces the prediction for [MATH] from [MATH] GeV to 132 GeV.", '1201.3968-2-0-5': 'There being three conditions and three unknowns, we are able to obtain all three masses, [MATH] and [MATH], in [MATH] MeV to [MATH] TeV energy range, as a function of the cut-off scale and the gauge couplings only.', '1201.3968-2-1-0': '# Introduction', '1201.3968-2-2-0': 'As is well known, the Standard Model, with an elementary Higgs doublet field, suffers from the problem of quadratic divergences when radiative corrections, due to the loops of fermions, the top quark in particular, to the mass of the Higgs boson is calculated.', '1201.3968-2-2-1': 'This problem is often referred to as the hierarchy problem, as we require an artificial fine tuning between the induced radiative mass, which is of the order of the cut-off scale, e.g., [MATH], and the counterterm to produce a mass of the order of the electroweak scale.', '1201.3968-2-3-0': 'This leads us to speculate that the Higgs boson may in fact be composite.', '1201.3968-2-3-1': 'The simplest implementation of this idea is top-quark condensation [CITATION].', '1201.3968-2-3-2': 'It turns out, however, that this compositeness by itself is not sufficient to remove the problem of quadratic divergences, and fine tuning is still required in the simpler approaches to top-quark condensation.', '1201.3968-2-4-0': 'The problem of divergences can only be artificial, because the same loop corrections, applied to the mass of the Goldstone bosons, produce a similar quadratic divergence, whereas the Goldstone theorem guarantees that spontaneous symmetry breaking results in massless Goldstone bosons.', '1201.3968-2-4-1': 'This suggests that these quadratic divergences are an artifact, and will vanish if the condition of current conservation is implemented properly.', '1201.3968-2-5-0': 'For example, the approach of Chesterman, King and Ross [CITATION] uses the vanishing of the mass of the Goldstone boson as a consistency check, and is able to obtain sensible values for the mass of the Higgs boson.', '1201.3968-2-6-0': 'A more direct investigation into this point regarding current conservation was made by Gribov in ref. [CITATION], and his results are in quantitative agreement with ref. [CITATION].', '1201.3968-2-6-1': 'In that paper, he implemented the symmetry condition, somewhat by force, by requiring that the Goldstone-boson self-energy vanishes in the soft limit.', '1201.3968-2-6-2': 'The mass of the Higgs boson is then obtained by subtracting off the Goldstone-boson self-energy, leading to the following Pagels-Stokar-type equation for the mass of the Higgs boson: [EQUATION]', '1201.3968-2-6-3': 'This gives the mass [MATH] GeV for the Higgs boson using the top-quark mass 174 GeV and [MATH] as input.', '1201.3968-2-6-4': "Gribov's cut-off [MATH] is given by the [MATH] Landau pole, [MATH] GeV, but the results are relatively insensitive to the value of the cut-off scale.", '1201.3968-2-6-5': 'This value of [MATH] is much less than other values that are obtained in the literature [CITATION] and, in this sense, compares more favourably with the recent LHC results [CITATION] which suggest [MATH] GeV.', '1201.3968-2-7-0': 'Despite this conceptual and qualitative success, Gribov does not explicitly work out the mechanism for the cancellation of the Goldstone-boson self-energy.', '1201.3968-2-8-0': 'In this paper, we show that this cancellation is brought about through the cancellation of the tadpole, which is a physical condition that is required in order that the vacuum is stable.', '1201.3968-2-8-1': 'The analysis is analogous to that carried out in ref. [CITATION] in the context of anti-ferromagnetism.', '1201.3968-2-8-2': 'The cancellation of the Higgs tadpole, between the top-quark, Goldstone-boson and Higgs-boson loops, is responsible for the taming of the quadratic divergences, and this leads to the following extra condition: [EQUATION]', '1201.3968-2-8-3': 'This gives us the value [MATH] GeV, using the same input as above.', '1201.3968-2-8-4': 'This is very close to the LHC value, but this remarkable success should not be taken too seriously, because this condition is sensitive to the renormalization of the top-quark mass at near the Landau pole.', '1201.3968-2-9-0': 'It should be noted that our extra condition on the Higgs mass completely fixes the low energy parameters.', '1201.3968-2-9-1': 'That is, all three masses, [MATH], [MATH] and [MATH], are fixed for given values of the cut-off scale and gauge couplings.', '1201.3968-2-9-2': 'The form of the UV theory does not affect the results, so long as the symmetry is spontaneously broken by a UV condensate which exists only at high energies.', '1201.3968-2-9-3': "A natural candidate is Gribov's U(1)[MATH] scenario.", '1201.3968-2-9-4': 'We discuss the physics of this scenario.', '1201.3968-2-10-0': '# Tadpole cancellation conditions', '1201.3968-2-11-0': 'Tadpole cancellation in bosonic self-energy was, in part, discussed by Gribov himself in the context of chiral symmetry breaking in ref. [CITATION].', '1201.3968-2-11-1': 'However, he imposes tadpole cancellation there by introducing a boson-boson-quark-quark term (e.g., eqn. (132) of ref. [CITATION]) to cancel the anomalous term in the Ward-Takahashi identity associated with Compton scattering.', '1201.3968-2-11-2': 'One problem in doing so is that this will lead to the necessity of introducing an infinite number of [MATH]-boson-quark-quark couplings.', '1201.3968-2-12-0': 'In ref. [CITATION], and in the context of anti-ferromagnetism, we proposed a more economical mechanism which also seems to us to be more natural.', '1201.3968-2-12-1': 'Here, only the three-point and four-point couplings among the Goldstone and Higgs bosons are needed, and these couplings are fixed by symmetry conditions such as the Ward-Takahashi identities.', '1201.3968-2-12-2': 'It turns out that these couplings may be rotated away in the resulting electro-weak effective Lagrangian when the Goldstone bosons are absorbed in the massive gauge bosons.', '1201.3968-2-12-3': 'The low-energy phenomenology is then identical to that of the Standard Model.', '1201.3968-2-13-0': 'Let us start with the fermionic vertex function for the left-handed SU(2) current.', '1201.3968-2-13-1': 'As discussed in ref. [CITATION], Ward-Takahashi identity is satisfied by the following modified vertex: [EQUATION]', '1201.3968-2-13-2': "Here, the small blob stands for the unmodified vertex [MATH] and the two-point function ([MATH] in Gribov's notation) which follows from that, and the asterisk stands for the vertex that is modified by the inclusion of the Goldstone-boson contribution.", '1201.3968-2-13-3': 'The dashed line stands for one of the three Goldstone bosons [MATH], [MATH].', '1201.3968-2-14-0': 'The Ward-Takahashi identity, applied to the modified vertex, fixes the [MATH] coupling to fermions to be, for example, [EQUATION] where [MATH], the Goldstone-boson form factor, is defined by the two-point function of eqn. ([REF]).', '1201.3968-2-14-1': "[MATH], where [MATH] is the usual 'vacuum-expectation value of the Higgs field'.", '1201.3968-2-14-2': 'Note that the Feynman rules are, as usual, given by [MATH].', '1201.3968-2-14-3': 'This will apply to all couplings that appear in the following.', '1201.3968-2-15-0': 'The same exercise may be repeated for the bosonic vertex: [EQUATION]', '1201.3968-2-15-1': 'The unmodified vertex is necessarily proportional to [MATH] (momenta flows left to right, or more generally [MATH] to [MATH]) in order to satisfy the Ward-Takahashi identity, and the normalization is fixed by the Ward-Takahashi identity applied to the amplitude shown in fig. [REF].', '1201.3968-2-16-0': 'This fixes the [MATH] coupling to be: [EQUATION] when the Yukawa coupling of the Higgs boson to fermions is given by [EQUATION]', '1201.3968-2-16-1': 'Next, we consider the Ward-Takahashi identity in the set of amplitudes which are described by fig. [REF].', '1201.3968-2-16-2': 'This allows us to work out the Goldstone-boson quartic couplings: [EQUATION]', '1201.3968-2-16-3': 'We then turn to the set of amplitudes that are described by fig. [REF], and obtain [EQUATION]', '1201.3968-2-16-4': 'Note that the Higgs-boson self-coupling is not fixed by current conservation conditions but either by loops or by the symmetry between Goldstone and Higgs bosons.', '1201.3968-2-16-5': 'As noted in ref. [CITATION], these turn out to be the same as the couplings of the Standard Model: [EQUATION]', '1201.3968-2-16-6': 'This implies [EQUATION]', '1201.3968-2-16-7': 'We notice that the effective Lagrangian for the multi-boson interaction terms can be written as [EQUATION]', '1201.3968-2-16-8': 'In terms of the Standard-Model Higgs-doublet field [MATH], this expression is proportional to [MATH].', '1201.3968-2-16-9': 'That is, the mass-generation mechanism of the effective theory is equivalent to that of the Standard Model, but, surprisingly, this symmetry-breaking potential is derived from symmetry considerations in our approach.', '1201.3968-2-17-0': 'We now consider the tadpole cancellation condition: [EQUATION]', '1201.3968-2-17-1': 'Let us neglect the contribution of all fermions other than the top quark.', '1201.3968-2-17-2': 'This yields [EQUATION] where the simplification [MATH] has been made.', '1201.3968-2-17-3': 'The shorthand notation [MATH] is used.', '1201.3968-2-17-4': 'Note that in the Goldstone loop, [MATH] is not corrected by renormalization effects.', '1201.3968-2-17-5': 'Running [MATH] is used in the Higgs boson loop.', '1201.3968-2-17-6': 'This contribution will be negligible since [MATH] will be small at high scales.', '1201.3968-2-17-7': 'We have neglected the renormalization effects to [MATH], which are relatively small, due to the Goldstone and Higgs boson propagators.', '1201.3968-2-18-0': 'Since eqn. ([REF]) is dominated by the large energy region, we then obtain [EQUATION] as discussed in the introduction.', '1201.3968-2-18-1': 'Strictly speaking, the scale on the right-hand side should be slightly below [MATH].', '1201.3968-2-18-2': 'The running of the top quark mass is as given by eqn. (10) of ref. [CITATION]: [EQUATION] where only the QCD part of the evolution is included, and [EQUATION]', '1201.3968-2-18-3': 'The result of eqn. ([REF]) is shown in fig. [REF].', '1201.3968-2-18-4': "Gribov's scenario yields [MATH] GeV using the U(1)[MATH] Landau scale as [MATH].", '1201.3968-2-18-5': 'The Landau scale ([MATH] GeV) is as given by eqns. (45) and (46) of ref. [CITATION]: [EQUATION]', '1201.3968-2-18-6': 'As mentioned in the introduction, this spectacular agreement with the LHC results [CITATION] should be treated with caution and suspicion, because the result is sensitive to renormalization effects at near the cut-off (Landau) scale.', '1201.3968-2-18-7': "Even so, this result provides a support to Gribov's U(1)[MATH] scenario, if the LHC results will be confirmed.", '1201.3968-2-19-0': 'Let us consider the Goldstone-boson self-energy.', '1201.3968-2-19-1': 'This is given by the Feynman graphs of fig. [REF].', '1201.3968-2-20-0': 'At zero external momentum, the contributions of the three diagrams are given by: [EQUATION]', '1201.3968-2-20-1': 'Adding together these three equations yields zero so long as the tadpoles cancel.', '1201.3968-2-20-2': 'In fact, this cancellation occurs in any case if the tadpole contributions are also included in fig. [REF].', '1201.3968-2-20-3': 'For a more formal, all-order treatment of such cancellations in the Goldstone-boson mass, we may, for instance, consider the non-vanishing part of [MATH] in eqn. (29) of ref. [CITATION] and show that this is equivalent to tadpole contributions.', '1201.3968-2-21-0': 'Because the Goldstone-boson self-energy must be equal to [MATH], calculating [MATH] at finite and small external momentum yields [MATH], and this reproduces eqn. (14) of ref. [CITATION]: [EQUATION]', '1201.3968-2-21-1': 'The bosonic contributions do not give logarithmic corrections to this equation.', '1201.3968-2-21-2': 'This equation predicts [MATH] as a function of [MATH], or vice versa and, as a generic problem in top-quark condensation approaches, it is well known that the predicted [MATH] is too high.', '1201.3968-2-21-3': 'For example, Gribov [CITATION] predicts [MATH] GeV for [MATH].', '1201.3968-2-21-4': 'One source of this discrepancy may be new contributions (e.g., gravitational) at high scales, since eqn. ([REF]) is relatively sensitive to the mass evolution at high scales.', '1201.3968-2-22-0': 'Let us now turn to the Higgs-boson self-energy.', '1201.3968-2-22-1': 'These are also given by the three diagrams shown in fig. [REF].', '1201.3968-2-22-2': 'The amplitudes at zero external momentum are now given by [EQUATION]', '1201.3968-2-22-3': 'The quadratic divergences cancel between [MATH] and [MATH], and we obtain [EQUATION]', '1201.3968-2-22-4': 'The second term of eqn. ([REF]) gives a correction to eqn. ([REF]).', '1201.3968-2-22-5': 'We can write the left-hand side of eqn. ([REF]) as [MATH] by changing the lower limit of integration to [MATH] and replacing [MATH] by [MATH].', '1201.3968-2-22-6': "We found, by solving the resulting integral equation numerically, using the same input as in Gribov's work, that [MATH] is reduced from 167 GeV to 132 GeV.", '1201.3968-2-22-7': '[MATH] also becomes more independent of [MATH], and we find that the maximum value [MATH] GeV is obtained for [MATH] GeV.', '1201.3968-2-23-0': 'Note that the tadpole cancellation condition provides a third condition, in addition to eqn. ([REF]) and eqn. ([REF]).', '1201.3968-2-23-1': 'There being three unknowns, [MATH], [MATH], [MATH], and three equations, we can solve the set of equations with [MATH] as the input.', '1201.3968-2-23-2': 'We may do so by solving the following integral equation: [EQUATION] numerically, with the condition [MATH].', '1201.3968-2-23-3': 'We then search for a value of [MATH] that satisfies [MATH].', '1201.3968-2-23-4': 'By doing so, we obtained [EQUATION]', '1201.3968-2-23-5': 'This generates a large hierarchy.', '1201.3968-2-23-6': 'We plot the resulting [MATH] and [MATH] as a function of the cut-off scale [MATH], in fig. [REF].', '1201.3968-2-23-7': '[MATH] is indistinguishable from [MATH] on the logarithmic scale.', '1201.3968-2-24-0': 'Taking [MATH] to be the U(1)[MATH] Landau scale yields [EQUATION] which is only two orders of magnitude away from the top-mass scale.', '1201.3968-2-25-0': 'One should beware of hasty conclusions, from fig. [REF], that the cut-off scale is at [MATH] GeV.', '1201.3968-2-25-1': 'The numbers that are shown are sensitive to possible corrections, e.g., the choice of energy scale, in eqn. ([REF]), and the need to produce a value of [MATH] that is as large as the observed value favours large [MATH].', '1201.3968-2-26-0': '# Discussions', '1201.3968-2-27-0': 'Our result shows that the electroweak scale is not necessarily fixed by the parameters at the UV scale.', '1201.3968-2-27-1': 'Rather, it is fixed by the condition of vacuum stability and the radiative effects.', '1201.3968-2-28-0': 'What conditions are necessary, then, in order that symmetry breaking occurs?', '1201.3968-2-29-0': 'In our opinion, it is sufficient that the symmetry is broken by a chiral condensate which appears at a high scale.', '1201.3968-2-29-1': 'This requires that a coupling constant becomes large.', '1201.3968-2-29-2': 'Gribov has shown [CITATION] that supercriticality and chiral symmetry breaking occurs when the relevant coupling constant satisfies [EQUATION]', '1201.3968-2-29-3': "Gribov's U(1)[MATH] scenario is a natural candidate, but one will then need to show that the gravitational coupling will not satisfy this condition.", '1201.3968-2-29-4': 'A possibility would be that gravitation is described by a weak-coupling theory above the Planck scale, becomes strongly coupled near the Planck scale, and reduces to Einstein gravity at low scales.', '1201.3968-2-29-5': 'For example, it may be that scale generation at the U(1)[MATH] pole is itself the source of scale generation in gravity.', '1201.3968-2-30-0': 'It is interesting to ask, what might be the behaviour of the U(1)[MATH] coupling at high scales?', '1201.3968-2-30-1': 'This question was addressed partially in ref. [CITATION].', '1201.3968-2-30-2': 'We argued that when the U(1)[MATH] coupling grows large, it will decouple as [MATH].', '1201.3968-2-30-3': 'The effective coupling experienced by the electrons is then given by [EQUATION] where [MATH] is the first coefficient of the beta function.', '1201.3968-2-30-4': 'If we can use the U(1)[MATH] beta function here, then the effective coupling will tend to [MATH], which is large compared with [MATH].', '1201.3968-2-30-5': 'Thus chiral symmetry breaking will almost certainly occur by U(1)[MATH].', '1201.3968-2-31-0': 'The condensate which appears at high scales must have decayed to fermions and Goldstone bosons at low scales, from phenomenological reasons.', '1201.3968-2-31-1': 'First, the observed masses of particles are light.', '1201.3968-2-31-2': 'Second, a condensate will lead to a cosmological constant which is much heavier than is observed.', '1201.3968-2-32-0': 'If EWSB is due to the formation of the supercritical condensate at high scales, one must ask the question of whether the supercritical states might affect the running of the parameters.', '1201.3968-2-32-1': 'Our answer is that the Goldstone and the Higgs states are supercritical states, but the other states cannot affect the running, because the Goldstone and the Higgs are the only states whose masses are protected by symmetry from growing large.', '1201.3968-2-32-2': 'The other supercritical states will have masses that are of the order of [MATH].', '1201.3968-2-33-0': '# Conclusions', '1201.3968-2-34-0': "We have shown that Gribov's cancellation condition in top-quark condensation may be rephrased as the cancellation, between the top, Higgs and the Goldstone contributions, of the Higgs-boson tadpole.", '1201.3968-2-34-1': 'This is a physical condition which must be satisfied in order that the ground state is stable.', '1201.3968-2-35-0': 'Tadpole cancellation gives us an extra condition for the mass of the Higgs boson.', '1201.3968-2-35-1': "In Gribov's U(1)[MATH] scenario, we obtain [MATH] GeV.", '1201.3968-2-35-2': 'This is in good agreement with the recent LHC announcement, [MATH] GeV, but our numbers are sensitive to the running of the top-quark mass in the UV region.', '1201.3968-2-35-3': "We have, furthermore, been able to calculate the bosonic contribution to Gribov's mass formula, eqn. ([REF]), which reduces Gribov's prediction of [MATH] GeV to [MATH] GeV.", '1201.3968-2-36-0': 'The low-energy phenomenology is equivalent to that of the Standard Model.', '1201.3968-2-37-0': 'Our having one extra condition fixes the low-energy parameters completely in terms of the cut-off scale.', '1201.3968-2-37-1': "In Gribov's scenario, we obtain [MATH] TeV, which is remarkably close to the actual scale, starting from [MATH] GeV, with no other input than the values of the dimensionless couplings and their running."} | [['1201.3968-1-15-0', '1201.3968-2-15-0'], ['1201.3968-1-15-1', '1201.3968-2-15-1'], ['1201.3968-1-27-0', '1201.3968-2-28-0'], ['1201.3968-1-5-0', '1201.3968-2-5-0'], ['1201.3968-1-34-1', '1201.3968-2-35-1'], ['1201.3968-1-34-2', '1201.3968-2-35-2'], ['1201.3968-1-34-3', '1201.3968-2-35-3'], ['1201.3968-1-23-0', '1201.3968-2-24-0'], ['1201.3968-1-9-0', '1201.3968-2-9-0'], ['1201.3968-1-9-1', '1201.3968-2-9-1'], ['1201.3968-1-9-2', '1201.3968-2-9-2'], 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['1201.3968-1-12-2', '1201.3968-2-12-1'], ['1201.3968-2-27-1', '1201.3968-3-41-1'], ['1201.3968-2-28-0', '1201.3968-3-42-0'], ['1201.3968-2-6-5', '1201.3968-3-6-5'], ['1201.3968-2-21-4', '1201.3968-3-31-5'], ['1201.3968-2-0-0', '1201.3968-3-0-0'], ['1201.3968-2-0-1', '1201.3968-3-0-1'], ['1201.3968-2-0-1', '1201.3968-3-0-4'], ['1201.3968-2-0-3', '1201.3968-3-0-4'], ['1201.3968-2-0-4', '1201.3968-3-0-2'], ['1201.3968-2-0-4', '1201.3968-3-0-3'], ['1201.3968-2-37-0', '1201.3968-3-50-2'], ['1201.3968-2-23-5', '1201.3968-3-37-5'], ['1201.3968-2-16-9', '1201.3968-3-22-9'], ['1201.3968-1-16-4', '1201.3968-2-16-4'], ['1201.3968-2-9-3', '1201.3968-3-14-1'], ['1201.3968-2-18-6', '1201.3968-3-25-5'], ['1201.3968-2-18-7', '1201.3968-3-27-0'], ['1201.3968-2-20-2', '1201.3968-3-30-4']] | [] | ['1201.3968-1-16-6', '1201.3968-2-16-6', '1201.3968-3-22-6', '1201.3968-3-25-4'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1201.3968 | {'1201.3968-3-0-0': 'We show that quadratic divergences in top-quark condensation are cancelled when the tadpoles cancel.', '1201.3968-3-0-1': 'This latter cancellation is naturally implemented as the cancellation among the top-quark, Goldstone and Higgs contributions.', '1201.3968-3-0-2': "We also calculate the bosonic correction terms to Gribov's mass formula for the Higgs boson.", '1201.3968-3-0-3': 'These reduce the prediction for [MATH] from [MATH] GeV to [MATH] GeV.', '1201.3968-3-0-4': "The tadpole cancellation condition by itself is an independent condition on the mass of the Higgs boson which, in Gribov's U(1)[MATH] scenario, yields [MATH] GeV with large theoretical uncertainty.", '1201.3968-3-0-5': 'More generally, we are able to obtain all three masses, [MATH], [MATH] and [MATH], in [MATH] MeV to [MATH] TeV energy range as a function of the cut-off scale and the gauge couplings only.', '1201.3968-3-1-0': '# Introduction', '1201.3968-3-2-0': 'As is well known, the Standard Model, with an elementary Higgs doublet field, suffers from the problem of quadratic divergences when radiative corrections, due to the loops of fermions, the top quark in particular, to the mass of the Higgs boson is calculated.', '1201.3968-3-2-1': 'This problem is often referred to as the hierarchy problem, as we require an artificial fine tuning between the induced radiative mass, which is of the order of the cut-off scale, e.g., [MATH], and the counterterm to produce a mass of the order of the electroweak scale.', '1201.3968-3-3-0': 'This tempts us to speculate that the Higgs boson may in fact be composite.', '1201.3968-3-3-1': 'The simplest implementation of this idea is top-quark condensation [CITATION].', '1201.3968-3-3-2': 'It turns out, however, that this compositeness by itself is not sufficient to remove the problem of quadratic divergences, and fine tuning is still required in the simpler approaches to top-quark condensation.', '1201.3968-3-4-0': 'The problem of divergences can only be artificial, because the same loop corrections, applied to the mass of the Goldstone bosons, produce a similar quadratic divergence, whereas the Goldstone theorem guarantees that spontaneous symmetry breaking results in massless Goldstone bosons.', '1201.3968-3-4-1': 'This suggests that these quadratic divergences are an artifact, and will vanish if the condition of current conservation is implemented properly.', '1201.3968-3-5-0': 'For example, the approach of Chesterman, King and Ross [CITATION] uses the vanishing of the mass of the Goldstone boson as a consistency check, and is able to obtain sensible values for the mass of the Higgs boson.', '1201.3968-3-6-0': 'A more direct investigation into this point regarding current conservation was made by Gribov in ref. [CITATION], and his results are in quantitative agreement with ref. [CITATION].', '1201.3968-3-6-1': 'In that paper, he implemented the symmetry condition, somewhat by force, by requiring that the Goldstone-boson self-energy vanishes in the soft limit.', '1201.3968-3-6-2': 'The mass of the Higgs boson is then obtained by subtracting off the Goldstone-boson self-energy, leading to the following Pagels-Stokar-type equation for the mass of the Higgs boson: [EQUATION]', '1201.3968-3-6-3': 'This gives the mass [MATH] GeV for the Higgs boson using the top-quark mass [MATH] GeV and [MATH] as input.', '1201.3968-3-6-4': "Gribov's cut-off [MATH] is given by the U(1)[MATH] Landau pole, [MATH] GeV, but the results are relatively insensitive to the value of the cut-off scale.", '1201.3968-3-6-5': 'This value of [MATH] is not far from the recent LHC results [CITATION] which suggest [MATH] GeV.', '1201.3968-3-6-6': 'It is worth noting here that eqn. ([REF]) is general and is independent of the physics that causes the cut-off.', '1201.3968-3-7-0': 'Our questions are the following.', '1201.3968-3-7-1': 'First, what may be the justification for this procedure, in particular the mechanism for the cancellation of the divergences?', '1201.3968-3-7-2': 'Second, whether there may be correction terms due to the loops of Goldstone and Higgs bosons in eqn. ([REF]).', '1201.3968-3-8-0': 'Concerning the first question, we shall show in this paper that the quadratic divergences are proportional to the tadpole and therefore vanish when the vacuum is stable.', '1201.3968-3-8-1': 'Although this statement must be independent of the basis, it is most easily verified by defining the Goldstone bosons as the divergence of the weak current so that derivative couplings do not arise.', '1201.3968-3-9-0': 'Our analysis is quite general in the sense that no new interactions are introduced up to the UV cut-off scale.', '1201.3968-3-9-1': 'This procedure has the well-known phenomenological disadvantage that the mass [MATH] of the [MATH] boson is predicted to be too low for fixed [MATH] or, equivalently, [MATH] is predicted to be too large for fixed [MATH].', '1201.3968-3-9-2': "This is partially alleviated by taking large [MATH] such as, as in Gribov's scenario, the U(1)[MATH] Landau pole.", '1201.3968-3-9-3': 'Even this is not sufficient and, as remarked in ref. [CITATION], more UV contribution is required, which may be, one would naturally guess, due to strong dynamics at near and above the cut-off.', '1201.3968-3-10-0': 'The methods of our analysis are analogous to those in ref. [CITATION] that are worked out in the context of anti-ferromagnetism.', '1201.3968-3-10-1': 'The second of the questions above is also addressed within this framework, and we find that the correction to eqn. ([REF]) is large and negative.', '1201.3968-3-11-0': 'As an explanation of our framework, we implement current conservation by requiring the vanishing of the Ward-Takahashi identities.', '1201.3968-3-11-1': 'This fixes the bosonic three-point and four-point couplings.', '1201.3968-3-11-2': 'In other words, the shape of the Higgs potential is fixed by current conservation, and this turns out to be of the same form as that of the Standard Model.', '1201.3968-3-11-3': 'We then implement the vacuum-stability condition, namely that the Higgs [MATH] one-point function, or the tadpole, vanishes.', '1201.3968-3-12-0': 'Note that the cancellation of the tadpole, which is imposed as the cancellation among the top-quark, Goldstone-boson and Higgs-boson loops, is by itself an independent condition on the symmetry-breaking parameters.', '1201.3968-3-12-1': 'This yields [EQUATION] which implies [MATH] GeV in the U(1)[MATH] scenario, if we only include the QCD part of the running.', '1201.3968-3-12-2': 'This is very close to the LHC value, but this remarkable success should not be taken too seriously, because this condition will be affected by the renormalization, due to the U(1)[MATH] interaction, of the top-quark mass at near the Landau pole.', '1201.3968-3-13-0': 'It should be noted that this extra condition on the Higgs mass, in principle, completely fixes the symmetry-breaking parameters.', '1201.3968-3-13-1': 'That is, all three masses, [MATH], [MATH] and [MATH], are fixed for given values of the cut-off scale and gauge couplings.', '1201.3968-3-13-2': 'This provides a natural solution to the hierarchy problem and, indeed, we obtain a large hierarchy that is comparable with the actual hierarchy between the UV and EWSB scales.', '1201.3968-3-14-0': 'The form of the UV theory does not affect our analysis at the present level of approximation, so long as the symmetry is spontaneously broken by a UV condensate which exists only at high energies.', '1201.3968-3-14-1': "This requires a supercritically strong UV interaction, and a natural candidate is Gribov's U(1)[MATH] scenario.", '1201.3968-3-14-2': 'We discuss the physics of this scenario.', '1201.3968-3-15-0': '# Current conservation conditions', '1201.3968-3-16-0': 'Tadpole cancellation in bosonic self-energy was, in part, discussed by Gribov himself in the context of chiral symmetry breaking in ref. [CITATION].', '1201.3968-3-16-1': 'However, he imposes tadpole cancellation there by introducing a boson-boson-quark-quark term (e.g., eqn. (132) of ref. [CITATION]) to cancel the anomalous term in the Ward-Takahashi identity associated with Compton scattering.', '1201.3968-3-16-2': 'One problem in doing so is that this will lead to the necessity of introducing an infinite number of [MATH]-boson-quark-quark couplings.', '1201.3968-3-17-0': 'In ref. [CITATION], and in the context of anti-ferromagnetism, we proposed a more economical mechanism which also seems to us to be more natural.', '1201.3968-3-17-1': 'Here, only the three-point and four-point couplings among the Goldstone and Higgs bosons are needed, and these couplings are fixed by symmetry conditions such as the Ward-Takahashi identities.', '1201.3968-3-18-0': 'We shall demonstrate that the couplings are consistent with the Standard-Model-like quartic Higgs potential.', '1201.3968-3-18-1': 'In the low-energy effective theory, the phenomenology is identical to that of the Standard Model, at least insofar as the couplings to the bosons and the top quark are concerned.', '1201.3968-3-18-2': 'As for the other fermionic Yukawa couplings, it is natural to assume that they are as given by the Standard Model though, strictly speaking, other possibilities cannot be ruled out.', '1201.3968-3-19-0': 'Let us start with the fermionic vertex function for the left-handed SU(2) current.', '1201.3968-3-19-1': 'As discussed in ref. [CITATION], Ward-Takahashi identity is satisfied by the following modified vertex: [EQUATION]', '1201.3968-3-19-2': "Here, the small blob stands for the unmodified vertex [MATH] and the two-point function ([MATH] in Gribov's notation) which follows from that, and the asterisk stands for the vertex that is modified by the inclusion of the Goldstone-boson contribution.", '1201.3968-3-19-3': 'The dashed line stands for one of the three Goldstone bosons [MATH], [MATH].', '1201.3968-3-20-0': 'The Ward-Takahashi identity, applied to the modified vertex, fixes the [MATH] coupling to fermions to be, for example, [EQUATION] where [MATH], the Goldstone-boson form factor, is defined by the two-point function of eqn. ([REF]).', '1201.3968-3-20-1': "[MATH], where [MATH] is the usual 'vacuum-expectation value of the Higgs field'.", '1201.3968-3-20-2': 'Note that the Feynman rules are, as usual, given by [MATH].', '1201.3968-3-20-3': 'This will apply to all couplings that appear in the following.', '1201.3968-3-21-0': 'The same exercise may be repeated for the bosonic vertex: [EQUATION]', '1201.3968-3-21-1': 'The unmodified vertex is necessarily proportional to [MATH] (momenta flows left to right, or more generally [MATH] to [MATH]) in order to satisfy the Ward-Takahashi identity, and the normalization is fixed by the Ward-Takahashi identity applied to the amplitude shown in fig. [REF].', '1201.3968-3-22-0': 'This fixes the [MATH] coupling to be: [EQUATION] when the Yukawa coupling of the Higgs boson to fermions is given by [EQUATION]', '1201.3968-3-22-1': 'Next, we consider the Ward-Takahashi identity in the set of amplitudes that are described by fig. [REF].', '1201.3968-3-22-2': 'This allows us to work out the Goldstone-boson quartic couplings: [EQUATION]', '1201.3968-3-22-3': 'We then turn to the set of amplitudes that are described by fig. [REF], and obtain [EQUATION]', '1201.3968-3-22-4': 'Note that the Higgs-boson self-coupling is not fixed by current conservation conditions but either by loops or by the symmetry between Goldstone and Higgs bosons.', '1201.3968-3-22-5': 'As noted in ref. [CITATION], these turn out to be the same as the couplings of the Standard Model: [EQUATION]', '1201.3968-3-22-6': 'This implies [EQUATION]', '1201.3968-3-22-7': 'We notice that the effective Lagrangian for the multi-boson interaction terms can be written as [EQUATION]', '1201.3968-3-22-8': 'In terms of the Standard-Model Higgs-doublet field [MATH], this expression is proportional to [MATH].', '1201.3968-3-22-9': 'That is, the mass-generation mechanism of the effective theory is equivalent to that of the Standard Model.', '1201.3968-3-23-0': '# Tadpole cancellation', '1201.3968-3-24-0': 'We now consider the tadpole cancellation condition: [EQUATION]', '1201.3968-3-24-1': 'Let us neglect the contribution of all fermions other than the top quark.', '1201.3968-3-24-2': 'This yields [EQUATION] where the simplification [MATH] has been made.', '1201.3968-3-24-3': 'The shorthand notation [MATH] is used.', '1201.3968-3-24-4': 'Note that in the Goldstone loop, [MATH] is not corrected by renormalization effects.', '1201.3968-3-24-5': 'This is in contrast to the Higgs-boson loop, which is suppressed because of the running Higgs mass.', '1201.3968-3-24-6': 'We have neglected the renormalization effects to [MATH], which are relatively small, due to the Goldstone and Higgs boson propagators.', '1201.3968-3-25-0': 'Since eqn. ([REF]) is dominated by the large energy region, we then obtain [EQUATION] as discussed in the introduction.', '1201.3968-3-25-1': 'Strictly speaking, the scale on the right-hand side should be slightly below [MATH].', '1201.3968-3-25-2': 'The running of the top quark mass is as given by eqn. (10) of ref. [CITATION]: [EQUATION] where only the QCD part of the evolution is included, and [EQUATION]', '1201.3968-3-25-3': 'The result of eqn. ([REF]) is shown in fig. [REF].', '1201.3968-3-25-4': 'The U(1)[MATH] scenario yields [MATH] GeV, when [MATH] GeV) is given by eqns. (45) and (46) of ref. [CITATION]: [EQUATION]', '1201.3968-3-25-5': 'As mentioned in the introduction, this spectacular agreement with the LHC results [CITATION] should be treated with caution and suspicion.', '1201.3968-3-25-6': 'The full running [MATH], which includes both QCD and U(1)[MATH] running, formally vanishes at the Landau pole.', '1201.3968-3-25-7': 'The reason for our omitting the latter running is that the effect is much less than the QCD running, except at very near the Landau pole.', '1201.3968-3-25-8': 'Furthermore, at near the Landau pole, the perturbative result for the running mass cannot be trusted.', '1201.3968-3-26-0': 'Having said that, we may, as a means of error estimation, include the U(1)[MATH] running effect in eqn. ([REF]) which is then evaluated literally.', '1201.3968-3-26-1': 'This yields [MATH] GeV.', '1201.3968-3-26-2': 'Thus there is roughly 25 % error in our prediction of [MATH] GeV.', '1201.3968-3-27-0': "Even so, this result provides some support to Gribov's U(1)[MATH] scenario since, as can be seen from fig. [REF], low [MATH] requires large [MATH].", '1201.3968-3-28-0': '# Bosonic self-energies', '1201.3968-3-29-0': 'The three types of Feynman diagram for the self-energy of Goldstone and Higgs bosons are shown in fig. [REF].', '1201.3968-3-29-1': 'We have not shown the tadpole diagrams of the form discussed in the previous section, as these are zero if the tadpole cancellation condition is imposed.', '1201.3968-3-29-2': 'The contributions of figs. [REF]a and c are quadratically divergent, and we shall show that they mutually cancel.', '1201.3968-3-29-3': 'The contribution of fig. [REF]b provides a correction to eqn. ([REF]).', '1201.3968-3-30-0': 'Let us consider the Goldstone-boson self-energy.', '1201.3968-3-30-1': 'At zero external momentum, the contributions of the three diagrams are given by: [EQUATION]', '1201.3968-3-30-2': 'We see that adding together these three equations yields zero so long as the tadpoles cancel.', '1201.3968-3-30-3': 'For a more formal, all-order treatment of such cancellations in the Goldstone-boson mass, we may, for instance, consider the non-vanishing part of [MATH] in eqn. (29) of ref. [CITATION] and show that this is equivalent to tadpole contributions.', '1201.3968-3-30-4': 'As a generalization, we note that the divergences cancel in any case in Goldstone-boson self-energy, regardless of the vanishing of the tadpoles, if the tadpole diagrams are added explicitly to fig. [REF].', '1201.3968-3-31-0': 'Because the Goldstone-boson self-energy must be equal to [MATH], calculating [MATH] at finite and small external momentum allows us to work out [MATH], and this reproduces eqn. (14) of ref. [CITATION]: [EQUATION]', '1201.3968-3-31-1': 'The bosonic contributions do not give logarithmic corrections to this equation.', '1201.3968-3-31-2': 'This equation predicts [MATH] as a function of [MATH], or vice versa and, as a generic problem in top-quark condensation approaches, it is well known that the predicted [MATH] is too high.', '1201.3968-3-31-3': 'For example, Gribov [CITATION] predicts [MATH] GeV for [MATH].', '1201.3968-3-31-4': 'In ref. [CITATION], this discrepancy is attributed to UV contributions near the Landau pole due to the strong dynamics.', '1201.3968-3-31-5': 'This is one possibility, but another source of the discrepancy may be new contributions (e.g., gravitational) at high scales, since eqn. ([REF]) is relatively sensitive to the mass evolution at high scales, unlike eqn. ([REF]) which is affected by [MATH] quartically and is therefore less sensitive to the UV region.', '1201.3968-3-32-0': 'Let us now turn to the Higgs-boson self-energy.', '1201.3968-3-32-1': 'These are also given by the three diagrams shown in fig. [REF].', '1201.3968-3-32-2': 'The amplitudes at zero external momentum are now given by [EQUATION]', '1201.3968-3-32-3': 'The sum of the quadratic divergences of [MATH] and [MATH] is proportional to the tadpole and therefore vanishes.', '1201.3968-3-32-4': 'We then obtain [EQUATION]', '1201.3968-3-32-5': 'The second term of eqn. ([REF]) gives a correction to eqn. ([REF]).', '1201.3968-3-33-0': 'In order to solve this integral, we need the running [MATH].', '1201.3968-3-33-1': 'This is obtained by writing the left-hand side of eqn. ([REF]) as [MATH], changing the lower limit of integration to [MATH], and replacing [MATH] by [MATH].', '1201.3968-3-33-2': 'The low-energy Higgs mass, i.e. [MATH], is then obtained by solving the resulting integral equation numerically.', '1201.3968-3-34-0': 'The result of this calculation is shown in fig. [REF], using [MATH] GeV and [MATH].', '1201.3968-3-34-1': 'We see that in the U(1)[MATH] scenario, [MATH] is reduced from 167 GeV to 132 GeV.', '1201.3968-3-34-2': '[MATH] also becomes more independent of [MATH], and we find that the maximum value [MATH] GeV is obtained for [MATH] GeV.', '1201.3968-3-35-0': 'Above [MATH] GeV, the uncertainty, which may be estimated by varying [MATH] and [MATH], becomes nearly constant.', '1201.3968-3-35-1': 'For example, by varying [MATH] by [MATH] GeV, we obtain [MATH] GeV variation in [MATH], and by varying [MATH] by [MATH], we again obtain [MATH] GeV variation in [MATH].', '1201.3968-3-35-2': 'The combined error is thus [MATH] GeV.', '1201.3968-3-35-3': 'Other sources of uncertainty are: the higher-order corrections which are expected to be small when the running couplings are small; the UV correction near the Landau pole; and other UV interactions which may be present.', '1201.3968-3-36-0': 'Note that the Higgs mass prediction of fig. [REF] is independent of the prediction of fig. [REF].', '1201.3968-3-36-1': 'The former was obtained by the condition that the tadpole vanishes, and the latter was obtained by evaluating the self-energy integrals.', '1201.3968-3-36-2': 'If both predictions are literally true, it follows that the point at which the two predictions match, which comes out to be about [MATH] GeV, is the true scale of the UV cut-off.', '1201.3968-3-37-0': 'Another way of saying the same thing is that given a value of the UV cut-off, and the running coupling constants, we can work out all three of [MATH], [MATH] and [MATH] as opposed to the previous studies in which [MATH] is predicted for given [MATH] and the cut-off, and so on.', '1201.3968-3-37-1': 'This follows because we now have three equations, namely eqns. ([REF]), ([REF]) and ([REF]), as opposed to two.', '1201.3968-3-37-2': 'The combined set of equations may be solved by solving the following integral equation: [EQUATION] numerically, with the condition [MATH].', '1201.3968-3-37-3': 'We then search for a value of [MATH] that satisfies [MATH].', '1201.3968-3-37-4': 'By doing so, we obtain [EQUATION]', '1201.3968-3-37-5': 'This ratio corresponds to a significantly large hierarchy.', '1201.3968-3-37-6': 'We plot the resulting [MATH] and [MATH] as a function of the cut-off scale [MATH], in fig. [REF].', '1201.3968-3-37-7': '[MATH] is indistinguishable from [MATH] on the logarithmic scale.', '1201.3968-3-38-0': 'Taking [MATH] to be the U(1)[MATH] Landau scale yields [EQUATION] which is only two orders of magnitude away from the top-quark mass scale.', '1201.3968-3-38-1': 'In order to cover the two remaining orders of magnitude, we will need to resolve the [MATH] discrepancy discussed below eqn. ([REF]).', '1201.3968-3-38-2': 'We should mention here that the variations of [MATH] and [MATH] cannot explain the two-orders-of-magnitude difference.', '1201.3968-3-39-0': 'One should beware of hasty conclusions, from fig. [REF], that the cut-off scale is at [MATH] GeV.', '1201.3968-3-39-1': 'The numbers that are shown are sensitive to possible corrections, e.g., the choice of energy scale, in eqn. ([REF]), and the need to produce a value of [MATH] that is as large as the observed value favours large [MATH].', '1201.3968-3-40-0': '# Discussions', '1201.3968-3-41-0': 'Our result shows that the electroweak scale is not necessarily fixed by the parameters at the UV scale.', '1201.3968-3-41-1': 'Rather, it is fixed by dimensional transmutation, and this in turn is governed by the condition of vacuum stability and the radiative effects.', '1201.3968-3-42-0': 'What conditions are necessary, then, in order that symmetry breaking occurs in the first place?', '1201.3968-3-43-0': 'In our opinion, it is sufficient that the symmetry is broken by a chiral condensate which appears at a high scale.', '1201.3968-3-43-1': 'This requires that a coupling constant becomes large.', '1201.3968-3-43-2': 'Gribov has shown [CITATION] that supercriticality and chiral symmetry breaking occurs when the relevant coupling constant satisfies [EQUATION]', '1201.3968-3-43-3': "Gribov's U(1)[MATH] scenario is a natural candidate, but one will then need to show that the gravitational coupling will not satisfy this condition.", '1201.3968-3-43-4': 'A possibility would be that gravitation is described by a weak-coupling theory above the Planck scale, becomes strongly coupled near the Planck scale, and reduces to Einstein gravity at low scales.', '1201.3968-3-43-5': 'For example, it may be that scale generation at the U(1)[MATH] pole is itself the source of scale generation in gravity.', '1201.3968-3-44-0': 'It is interesting to ask, what might be the behaviour of the U(1)[MATH] coupling at high scales?', '1201.3968-3-44-1': 'This question was addressed partially in ref. [CITATION].', '1201.3968-3-44-2': 'We argued that when the U(1)[MATH] coupling grows large, it will decouple as [MATH].', '1201.3968-3-44-3': 'The effective coupling experienced by the electrons is then given by [EQUATION] where [MATH] is the first coefficient of the beta function.', '1201.3968-3-44-4': 'If we can use the U(1)[MATH] beta function here, then the effective coupling will tend to [MATH], which is large compared with [MATH].', '1201.3968-3-44-5': 'Thus chiral symmetry breaking will almost certainly occur by U(1)[MATH].', '1201.3968-3-45-0': 'The condensate which appears at high scales must have decayed to fermions and Goldstone bosons at low scales, from phenomenological reasons.', '1201.3968-3-45-1': 'First, the observed masses of particles are light.', '1201.3968-3-45-2': 'Second, a condensate will lead to a cosmological constant which is much heavier than is observed.', '1201.3968-3-46-0': 'If EWSB is due to the formation of the supercritical condensate at high scales, one must ask the question of whether the supercritical states might affect the running of the parameters.', '1201.3968-3-46-1': 'Our answer is that the Goldstone and the Higgs states are supercritical states, but the other states cannot affect the running, because the Goldstone and the Higgs are the only states whose masses are protected by symmetry from growing large.', '1201.3968-3-46-2': 'The other supercritical states will have masses that are of the order of [MATH].', '1201.3968-3-47-0': '# Conclusions', '1201.3968-3-48-0': "We have shown that Gribov's cancellation condition in top-quark condensation may be rephrased as the cancellation, between the top, Higgs and the Goldstone contributions, of the Higgs-boson tadpole.", '1201.3968-3-48-1': 'This is a physical condition which must be satisfied in order that the ground state is stable.', '1201.3968-3-48-2': 'The low-energy phenomenology following from our analysis is equivalent to that of the Standard Model.', '1201.3968-3-49-0': 'Tadpole cancellation gives us an extra condition for the mass of the Higgs boson.', '1201.3968-3-49-1': "In Gribov's U(1)[MATH] scenario, we obtain [MATH] GeV.", '1201.3968-3-49-2': 'This is in good agreement with the recent LHC announcement, [MATH] GeV, but our numbers are subject to [MATH] % error due to the running of the top-quark mass in the UV region.', '1201.3968-3-49-3': "We have, furthermore, been able to calculate the bosonic contribution to Gribov's mass formula, eqn. ([REF]), which reduces Gribov's prediction of [MATH] GeV to [MATH] GeV.", '1201.3968-3-49-4': 'The error here is estimated to be [MATH] GeV.', '1201.3968-3-50-0': 'The two predictions of [MATH] are independent.', '1201.3968-3-50-1': 'We may, by requiring that the two predictions match, work out the cut-off scale, which comes out to be [MATH] GeV.', '1201.3968-3-50-2': 'Alternatively, we may work out the low-energy parameters for a given value of the cut-off scale.', '1201.3968-3-50-3': "In Gribov's scenario, we obtain [MATH] TeV, which is remarkably close to the actual scale, starting from [MATH] GeV, with no other input than the values of the dimensionless couplings and their running."} | null | null | null | null |
1704.06994 | {'1704.06994-1-0-0': '# Introduction', '1704.06994-1-1-0': 'String theory predicts the existence of a vast landscape of vacuum states with diverse properties [CITATION].', '1704.06994-1-1-1': 'In the cosmological context this leads to the picture of an eternally inflating multiverse, where different spacetime regions are occupied by different vacua.', '1704.06994-1-1-2': 'Transitions between the vacua occur through quantum tunneling, with bubbles of daughter vacuum nucleating and expanding in the parent vacuum background.', '1704.06994-1-1-3': 'According to this picture, our local region originated as a result of tunneling from some inflating parent vacuum and then went through a period of slow-roll inflation.', '1704.06994-1-1-4': 'The number of vacua in the landscape is expected to be enormous, so predictions in this kind of model must necessarily be statistical.', '1704.06994-1-1-5': 'The properties of string theory landscape are not well understood, and the approach adopted in much of the recent work is to substitute it by a scalar field model with a random Gaussian potential [CITATION].', '1704.06994-1-2-0': 'In a recent paper [CITATION] we developed analytic and numerical techniques for studying the statistics of slow-roll inflation in random Gaussian landscapes.', '1704.06994-1-2-1': 'We applied these techniques to the simplest case of small-field inflation in a one-dimensional random landscape.', '1704.06994-1-2-2': 'In this case, inflation typically occurs at local maxima or at inflection points of the potential [CITATION].', '1704.06994-1-2-3': 'Focusing mostly on the inflection points, we found the probability distributions for the maximal number of inflationary e-folds [MATH] and for the spectral index of density fluctuations [MATH].', '1704.06994-1-3-0': 'The maximal e-fold number [MATH] depends only on the shape of the potential near the inflection point, but the actual number of e-folds, [MATH], is sensitive to the initial conditions - that is, to the initial value [MATH] of the inflaton field right after it tunnels from the parent vacuum.', '1704.06994-1-3-1': 'If [MATH] is too far away from the inflection point, the field may develop a large velocity and overshoot or it may miss the slow-roll region entirely.', '1704.06994-1-3-2': 'In the present paper we shall use numerical simulations to determine the probability distribution for [MATH] and to investigate its effect on the statistical properties of inflation.', '1704.06994-1-3-3': 'As before, we shall restrict our analysis to the simplest case of one-dimensional potentials.', '1704.06994-1-4-0': 'In the next Section we review some general properties of random Gaussian potentials, and in Sec. 3 we summarize earlier work on inflection-point inflation.', '1704.06994-1-4-1': 'Our numerical simulations and the results for the distribution of [MATH] are presented in Sec. [REF].', '1704.06994-1-4-2': 'In Sec. [REF] we develop a semi-analytic method to study the evolution of the scalar field after tunneling.', '1704.06994-1-4-3': 'We find the probability distribution for the number of e-folds of slow-roll inflation [MATH] and discuss the implications of our results for the prospects of detection of spatial curvature.', '1704.06994-1-4-4': 'Our conclusions are summarized and discussed in Sec. [REF].', '1704.06994-1-4-5': 'Some technical details related to the simulations are relegated to the appendices.', '1704.06994-1-4-6': 'Throughout the paper we use reduced Planck units with [MATH].', '1704.06994-1-5-0': '# Random Gaussian landscapes', '1704.06994-1-6-0': 'Consider a one-dimensional random Gaussian landscape model with a potential [MATH] satisfying the following correlation function: [EQUATION]', '1704.06994-1-6-1': 'We specifically consider a Gaussian-type correlation function defined as [EQUATION] with [MATH] playing the role of the correlation length in the landscape.', '1704.06994-1-6-2': 'Then the spectral function [MATH] is given by [EQUATION]', '1704.06994-1-6-3': 'We define different moments of the spectral function as [EQUATION] where we used Eq. ([REF]) in the second line.', '1704.06994-1-7-0': 'Once we specify a set of points [MATH] and define [MATH], the probability distribution for [MATH] is given by [EQUATION] where the positive definite matrix [MATH] is an [MATH] matrix of correlators defined by [EQUATION]', '1704.06994-1-7-1': 'Since [MATH] is a symmetric matrix, we can diagonalize it by an orthogonal matrix [MATH].', '1704.06994-1-7-2': "Then the probability distribution for variables [MATH] is given by [EQUATION] where [MATH]'s are eigenvalues of the matrix [MATH].", '1704.06994-1-7-3': 'Such random variables can be easily generated in numerical simulations.', '1704.06994-1-8-0': 'When the values of [MATH] are generated at a sufficient density on the [MATH]-axis, we can interpolate them to obtain a smooth potential.', '1704.06994-1-8-1': 'We check in the App.', '1704.06994-1-8-2': '[REF] that this procedure typically saturates at a few points per correlation length [MATH]: the interpolated potential changes very little with the addition of more points.', '1704.06994-1-8-3': 'We shall use this method with four points per correlation length to generate realizations of a random Gaussian landscape.', '1704.06994-1-9-0': 'Assuming that our universe is a result of a bubble nucleation event, we anticipate that the state of the universe after nucleation allows for a prolonged period of inflation.', '1704.06994-1-9-1': 'This is possible for parts of the landscape that resemble Fig. [REF], which includes a parent vacuum and a daughter vacuum separated by a barrier.', '1704.06994-1-9-2': 'We shall refer to parent and daughter vacua as "false" (FV) and "true" (TV) vacua, respectively.', '1704.06994-1-9-3': 'The inflection points where [MATH] are marked by red dots in the figure.', '1704.06994-1-9-4': 'In this example, there are three inflection points between the top of the barrier and the true vacuum.', '1704.06994-1-9-5': 'The potential is rather flat near the middle inflection point ([MATH] and [MATH] are small), and slow-roll inflation can be expected to occur in this region.', '1704.06994-1-10-0': 'We are going to focus on the case when the correlation length of the potential is small compared to the Planck scale, [MATH].', '1704.06994-1-10-1': 'Inflation in this case is typically of the small-field type: the slow-roll occurs in a narrow range [MATH] and the potential during the slow-roll remains approximately constant.', '1704.06994-1-11-0': 'The correlation function ([REF]) implies that the average value of the potential is [MATH], so positive and negative values of [MATH] are equally likely.', '1704.06994-1-11-1': 'In this case, the minima of the potential are predominantly at negative values of [MATH].', '1704.06994-1-11-2': 'This effect is especially pronounced for higher-dimensional random landscapes, where positive-energy vacua may not exist at all [CITATION].', '1704.06994-1-11-3': 'This problem can be alleviated by adding a constant shift term to the potential, [MATH].', '1704.06994-1-11-4': 'One can also consider models where [MATH] is variable, but its characteristic scale of variation is much greater than [MATH].', '1704.06994-1-11-5': 'For example, in axionic landscapes this term can have the form [MATH] with a very small mass [MATH] [CITATION] or [MATH] with [MATH] [CITATION].', '1704.06994-1-11-6': 'Then the local properties of the potential are still determined by the correlator ([REF]), while the extra term provides an additive shift, which is characterized by its own probability distribution.', '1704.06994-1-12-0': 'Here, we are interested in the case when the true vacuum has an almost vanishing vacuum energy.', '1704.06994-1-12-1': 'In a generic landscape, the number of such vacua is extremely small, and trying to find them by random sampling is a hopeless task.', '1704.06994-1-12-2': 'In our numerical simulations we simply added a constant to the generated random potential, so that [MATH] in the true vacuum.', '1704.06994-1-12-3': 'We expect the resulting ensemble of realizations to be similar to what one would get by random sampling in a landscape with a flat distribution of the shift parameter [MATH].', '1704.06994-1-13-0': '# Analytic results for inflection-point inflation', '1704.06994-1-14-0': 'In this section, we review some analytic results obtained in Refs. [CITATION] for inflection-point inflation.', '1704.06994-1-14-1': 'These results will be useful in subsequent sections.', '1704.06994-1-15-0': 'The necessary conditions for slow-roll inflation are [MATH], where [EQUATION]', '1704.06994-1-15-1': 'Note that the values of the slow-roll parameters at a randomly chosen point in the landscape are typically given by [MATH], so inflation can occur only in rare regions of the landscape.', '1704.06994-1-16-0': 'To study inflection-point inflation, we approximate the potential by a third-order Taylor expansion, [EQUATION] where the subscript [MATH] indicates the value at the inflection point [MATH], which is set to be at [MATH]: [MATH].', '1704.06994-1-16-1': 'The slow-roll conditions require that [MATH], but [MATH] does not need to be small, and we shall assume it to be comparable to the rms value, [MATH].', '1704.06994-1-16-2': 'The slow-roll region is then determined mainly by the condition [MATH] and can be specified as [MATH], where [EQUATION]', '1704.06994-1-16-3': 'We assume that [MATH], since otherwise the potential has a shallow high-energy minimum near the inflection point and inflation drives the field into that minimum [CITATION].', '1704.06994-1-17-0': 'It follows from ([REF]),([REF]) that the size of the slow-roll region is typically [MATH].', '1704.06994-1-17-1': 'This justifies the assumption that [MATH] can be approximated by a constant in this region.', '1704.06994-1-18-0': 'The "maximal e-folding number" can be defined as [EQUATION] where we have assumed that [MATH] (which is necessary for extending the integration to [MATH]).', '1704.06994-1-19-0': 'The spectral index [MATH] can be expressed in terms of [MATH] as [EQUATION] where [MATH]-[MATH]) is the e-folding number at which the CMB scale leaves the horizon.', '1704.06994-1-19-1': 'It follows from ([REF]) that [MATH] is greater than [MATH], which is approached in the limit [MATH].', '1704.06994-1-19-2': 'Note that for hilltop inflation [MATH] [CITATION], while the observed value is [MATH] and lies in the inflection-point range.', '1704.06994-1-19-3': 'This provides additional motivation to focus our analysis on inflection-point inflation.', '1704.06994-1-20-0': 'The magnitude of density fluctuations is given by [EQUATION] where in the second step we used Eq. ([REF]) for [MATH] and in the last step we assumed that [MATH] and [MATH] have their typical values.', '1704.06994-1-20-1': 'The observed value of [MATH] can be obtained by adjusting the parameters [MATH] and [MATH]: [EQUATION]', '1704.06994-1-20-2': 'The probability distributions for [MATH] and [MATH] in a random Gaussian landscape have been calculated in Ref. [CITATION].', '1704.06994-1-20-3': 'Here we quote the results: [EQUATION]', '1704.06994-1-20-4': 'The distribution ([REF]) gives the probability that a randomly chosen inflection point in the landscape is characterized by a given value of [MATH].', '1704.06994-1-20-5': 'Note, however, that here we are interested only in inflection points located between a high-energy false vacuum and a zero-energy true vacuum, as shown in Fig. [REF].', '1704.06994-1-20-6': 'As explained in Sec. [REF], we obtain such configurations by adding a constant term to a randomly generated potential.', '1704.06994-1-20-7': 'This procedure changes the value of [MATH] at the inflection point; hence it affects the value of [MATH] in Eq. ([REF]) and may potentially affect the distribution ([REF]).', '1704.06994-1-20-8': 'We shall see, however, that the form of this distribution remains unchanged.', '1704.06994-1-21-0': 'As we mentioned in the Introduction, the actual number of inflationary e-folds [MATH] depends on the initial conditions after tunneling and is generally different from [MATH].', '1704.06994-1-21-1': 'We shall find the probability distributions for the initial conditions and for [MATH] in the following sections.', '1704.06994-1-22-0': '# Initial conditions for inflation', '1704.06994-1-23-0': '## General formalism', '1704.06994-1-24-0': 'Decay of the false vacuum occurs through bubble nucleation, which is a quantum tunneling process.', '1704.06994-1-24-1': 'In the semiclassical approximation, the tunneling is described by an [MATH]-symmetric instanton [MATH], which can be found by solving the Euclidean field equation [CITATION] [EQUATION]', '1704.06994-1-24-2': 'Here we assume that gravitational effects on the tunneling can be neglected, which is usually the case in a small-field landscape.', '1704.06994-1-24-3': 'The boundary conditions for [MATH] are [MATH] and [MATH], where [MATH] is the value of [MATH] in the false vacuum.', '1704.06994-1-24-4': 'The tunneling probability is determined mostly by the exponential factor, [EQUATION] where [MATH] is the Euclidean instanton action [EQUATION] and [MATH].', '1704.06994-1-25-0': 'It will be convenient to introduce dimensionless variables [MATH] and [MATH] as [EQUATION]', '1704.06994-1-25-1': 'In terms of the new variables, Eq. ([REF]) still has the same form, [EQUATION] where the potential [MATH] is now characterized by the correlation function ([REF]) with [MATH].', '1704.06994-1-25-2': 'We note that [EQUATION] where [MATH] is the maximal e-folding number for the rescaled potential [MATH].', '1704.06994-1-25-3': 'We also define the action [MATH] for the rescaled variables: [EQUATION]', '1704.06994-1-25-4': 'The initial value [MATH] of the inflaton field after tunneling is set by the value of [MATH] at the center of the instanton, [EQUATION]', '1704.06994-1-25-5': 'The probability distribution for [MATH] can now be found with the aid of numerical simulations.', '1704.06994-1-26-0': 'It is well known that instanton solutions for multi-dimensional field spaces are not unique, because there may be more than one saddle point between the true vacuum and false vacuum.', '1704.06994-1-26-1': 'As we explain in App.', '1704.06994-1-26-2': '[REF], there may also be multiple instanton solutions in the one-dimensional case, but for a different reason.', '1704.06994-1-26-3': 'For a generic potential, there is typically a single instanton describing tunneling to a close vicinity of the true vacuum.', '1704.06994-1-26-4': 'In the presence of a flat inflection region, additional instantons may appear, corresponding to tunneling to the neighborhood of the inflection point [MATH].', '1704.06994-1-26-5': 'As we make the inflection region flatter, at some point the instanton tunneling to the true vacuum disappears, and only tunneling to the vicinity of inflection point remains possible.', '1704.06994-1-27-0': 'The number and character of the instantons also depend on the shape and height of the potential barrier.', '1704.06994-1-27-1': 'As an illustration we show some examples in Fig. [REF], where the tunneling points [MATH] are indicated by blue squares.', '1704.06994-1-27-2': 'All four potentials in the figure are rather similar, except they have different values of the false vacuum energy density [MATH].', '1704.06994-1-27-3': 'In the upper left frame, [MATH] is almost degenerate with [MATH] and there is a single instanton solution, which brings [MATH] almost all the way to the true vacuum.', '1704.06994-1-27-4': 'In the upper right frame, [MATH] is somewhat higher and additional instantons appear, which describe tunneling with [MATH] close to the inflection point.', '1704.06994-1-27-5': 'As we explain in App.', '1704.06994-1-27-6': '[REF], additional instanton solutions appear in pairs.', '1704.06994-1-27-7': 'As [MATH] gets higher, the middle tunneling point moves towards the true vacuum tunneling point, and eventually the two points "annihilate".', '1704.06994-1-27-8': 'In the lower right frame, [MATH] is still higher, and tunneling is now possible only to the neighborhood of the inflection point.', '1704.06994-1-28-0': 'When several instantons are present, one can compare the instanton actions to determine the dominant decay channel.', '1704.06994-1-28-1': 'We found that tunneling to the true vacuum dominates in most of these cases.', '1704.06994-1-28-2': 'We note, however, that in the present context we are interested in tunnelings that lead to sufficiently long inflation, regardless of their relative rate compared to other tunneling processes.', '1704.06994-1-29-0': 'To illustrate the dependence of the action of different instantons on the shape of the potential, we fix the potential on the right side of the barrier in Fig. [REF] and calculate the instanton actions for different values of [MATH].', '1704.06994-1-29-1': 'The result is shown in Fig. [REF].', '1704.06994-1-29-2': 'The red dashed (green solid) line is the action of the instanton solution that brings [MATH] close to the true vacuum (inflection point).', '1704.06994-1-29-3': 'The blue dotted line is the one with [MATH] between the other two tunneling points.', '1704.06994-1-29-4': 'We see that tunneling to the true vacuum dominates in (almost) the entire range where the corresponding instanton exists.', '1704.06994-1-29-5': 'The blue dotted line is always just above the green solid line; they are so close that they appear to coincide in the left frame of the figure.', '1704.06994-1-29-6': 'When [MATH] increases to a certain threshold, the blue dotted line meets and annihilates with the red dashed line (see the right frame).', '1704.06994-1-29-7': 'Before the annihilation, the red dashed line crosses the green solid line, which means that tunneling to the inflection point becomes dominant.', '1704.06994-1-29-8': 'However, the region where this occurs is so small that we cannot see it in the left frame.', '1704.06994-1-29-9': 'Since the instanton solution with [MATH] between the other two tunneling points is always subdominant, we neglect it in the rest of the paper.', '1704.06994-1-29-10': 'We denote the instanton action for solutions that bring [MATH] close to the true vacuum (inflection point) as [MATH]).', '1704.06994-1-30-0': '## Numerical simulation', '1704.06994-1-31-0': 'We used the procedure outlined in Sec. [REF] to generate [MATH] segments of a random Gaussian landscape, with each segment having length [MATH].', '1704.06994-1-31-1': 'The number of local minima in such a segment is almost always greater than or equal to two.', '1704.06994-1-31-2': 'We set the number of points per correlation length at [MATH], which means that every segment contains 33 points [MATH].', '1704.06994-1-31-3': 'We generate the values of the potential [MATH] at these points according to the probability distribution ([REF]) and use a fifth order spline to interpolate between them.', '1704.06994-1-32-0': 'In each realization of [MATH], we identify all extrema and inflection points.', '1704.06994-1-32-1': 'For any pair of adjacent minima, we refer to the higher- and lower-energy ones as [MATH] and [MATH], respectively, and shift the potential so that [MATH].', '1704.06994-1-32-2': 'We keep only realizations that have an inflection point [MATH] satisfying the following criteria: (i) it is located between the top of the barrier and the true vacuum, (ii) its energy density is lower than that of the false vacuum, [MATH].', '1704.06994-1-32-3': 'This procedure is illustrated in Fig. [REF].', '1704.06994-1-33-0': 'For each of the selected pairs of vacua, we seek instanton solutions of Eq. ([REF]) using the shooting method.', '1704.06994-1-33-1': 'We then use Eq. ([REF]) to find the initial value(s) of the field [MATH] after tunneling.', '1704.06994-1-34-0': '## Distributions for [MATH] and [MATH]', '1704.06994-1-35-0': 'We plotted the distribution of the initial values [MATH] in the left panel of Fig. [REF].', '1704.06994-1-35-1': 'Here and hereafter, the normalization of probability distributions is arbitrary.', '1704.06994-1-35-2': 'In cases where the potential admitted two instanton solutions, we included the values of [MATH] for both of them (disregarding the subdominant "middle" instanton).', '1704.06994-1-35-3': 'The distribution has a sharp peak centered near [MATH] and a somewhat larger and broader peak at [MATH].', '1704.06994-1-35-4': 'These peaks correspond to tunnelings to the vicinity of the inflection point [MATH] and of the true vacuum [MATH], respectively.', '1704.06994-1-35-5': 'In the right panel of Fig. [REF] we plotted the distribution of [MATH] for tunnelings to the inflection point - that is, including only cases where [MATH] is closer to [MATH] than to [MATH].', '1704.06994-1-35-6': 'This distribution is peaked near [MATH] with a width [MATH].', '1704.06994-1-36-0': 'Fig. [REF] shows a histogram of the maximal e-fold number [MATH], evaluated from Eq. ([REF]).', '1704.06994-1-36-1': 'The result is well fitted by the analytic function [EQUATION] which is shown by a red line in the figure.', '1704.06994-1-36-2': 'This shows that the form of the [MATH] distribution is not affected by the shift of the potential to [MATH].', '1704.06994-1-37-0': 'The plot in the left panel of Fig. [REF] is the same as in Fig. [REF], but now the contributions due to different types of instantons are shown separately.', '1704.06994-1-37-1': 'The green (magenta) lines represent the cases where there is only one instanton solution and [MATH] is closer to the inflection point (true vacuum).', '1704.06994-1-37-2': 'The red (blue) lines are for realizations with multiple instantons, where the dominant tunneling is to the inflection point (true vacuum).', '1704.06994-1-37-3': 'The type of instanton is not relevant for the present paper, but our results may be useful in other contexts, so we present them here for completeness.', '1704.06994-1-37-4': 'The figure shows that in the multiple instanton case the dominant tunneling channel is almost always to the true vacuum: the number of realizations with [MATH] is suppressed by more than an order of magnitude.', '1704.06994-1-37-5': 'This is consistent with the discussion of the tunneling action in Sec. [REF].', '1704.06994-1-37-6': 'We note also that the numbers of landscape realizations represented by blue and green curves are nearly the same and are within about a factor of 2 from those represented by the magenta curve.', '1704.06994-1-37-7': 'We have not found any explanation for this surprising fact.', '1704.06994-1-37-8': 'It implies that realizations with large values of [MATH] split into three comparable groups: a group with tunneling only to inflection point, a group with tunneling to the true vacuum, and a group with multiple tunneling channels, where the tunneling to the true vacuum dominates.', '1704.06994-1-38-0': 'The right panel of Fig. [REF] is a scatter plot of realizations in [MATH]-[MATH] plane, with the same color code as in the left panel.', '1704.06994-1-38-1': 'The average values [MATH] are shown as yellow, magenta, and cyan lines for the data indicated by green, red, and blue dots, respectively.', '1704.06994-1-38-2': 'We see that [MATH] is almost constant at [MATH] for all types of instantons, with [MATH] for the single instanton case and [MATH] for the multi-instanton case.', '1704.06994-1-38-3': 'This indicates that [MATH] and [MATH] are essentially uncorrelated in the most interesting regime of large [MATH].', '1704.06994-1-39-0': 'From now on, we shall not distinguish between single and multi-instanton tunnelings and treat all inflection-point tunnelings on equal footing.', '1704.06994-1-39-1': 'In multi-instanton realizations with a dominant true-vacuum instanton, we keep only the inflection-point instanton.', '1704.06994-1-39-2': 'The reason is that true-vacuum tunneling is irrelevant for our discussion, and the tunneling processes described by inflection-point instantons occur regardless of whether or not there is a more probable tunneling channel.', '1704.06994-1-40-0': '## Tunneling action', '1704.06994-1-41-0': 'The distributions for [MATH] and [MATH] that we calculated here are defined as frequencies of occurrence in the landscape.', '1704.06994-1-41-1': 'They should not be confused with the probabilities of occurrence in the multiverse, which can be directly related to observational predictions.', '1704.06994-1-41-2': 'The problem of defining these probabilities is known as the measure problem, which at present remains unresolved.', '1704.06994-1-41-3': '(For a review of the measure problem see [CITATION].)', '1704.06994-1-41-4': 'A number of different measure prescriptions have been suggested in the literature.', '1704.06994-1-41-5': 'Some of them lead to paradoxes or to a glaring conflict with observations and have therefore been ruled out.', '1704.06994-1-41-6': 'This process of elimination has not been enough to fix a unique measure of the multiverse.', '1704.06994-1-41-7': 'However, the measure prescriptions which are not obviously problematic tend to give similar predictions and introduce similar weighting factors for different realizations of the potential.', '1704.06994-1-41-8': 'The scale factor measure [CITATION] can be taken as a representative example of such "acceptable" measures.', '1704.06994-1-42-0': 'For a given measure prescription, probabilities can be calculated by solving the rate equation, which is similar to the Boltzmann equation in the multiverse.', '1704.06994-1-42-1': 'Naively, one might expect that different tunneling realizations in the landscape should be weighted by the tunneling rate, which is proportional to [MATH], where [MATH] is the tunneling action.', '1704.06994-1-42-2': 'However, analysis of the rate equation in the scale-factor measure shows that this expectation is incorrect.', '1704.06994-1-42-3': 'A simple counter-example is a landscape with an everywhere positive potential, [MATH].', '1704.06994-1-42-4': 'It can be shown that in such a landscape the probabilities depend only on the vacuum energy density and are independent of the transition rates [CITATION].', '1704.06994-1-42-5': 'In general, the probability of a given vacuum has a complicated dependence on the transition rates between different vacua in the landscape, not just on the rate of tunneling to this particular vacuum.', '1704.06994-1-42-6': 'One also finds that transitions with a small tunneling action are not generally "rewarded" with a high weighting factor.', '1704.06994-1-42-7': 'The reason can be roughly explained as follows.', '1704.06994-1-42-8': 'The weighting factor for vacuum [MATH] due to tunneling from a false vacuum [MATH] is proportional to [MATH], where [MATH] is the tunneling rate from [MATH] to [MATH] per Hubble volume per Hubble time, and [MATH] is the volume fraction occupied by vacuum [MATH] on constant scale factor surfaces.', '1704.06994-1-42-9': 'If the tunneling rate is very high, this leads to a rapid depletion of the false vacuum, so [MATH] gets very small.', '1704.06994-1-42-10': 'These two effects tend to compensate one another.', '1704.06994-1-42-11': 'On the other hand, tunnelings with a large instanton action tend to be disfavored in the presence of other decay channels with a smaller action.', '1704.06994-1-43-0': 'A study of the rate equation in a random Gaussian landscape would require a complicated statistical analysis, which is beyond the scope of the present paper.', '1704.06994-1-43-1': 'To facilitate such analysis in future work, here we shall analyze possible correlations of the action [MATH] with [MATH] and [MATH].', '1704.06994-1-43-2': 'If present, such correlations may affect probabilities for the initial conditions and for the observational effects of inflation.', '1704.06994-1-44-0': 'We show a scatter plot of [MATH] and [MATH] in the left panel of Fig. [REF].', '1704.06994-1-44-1': 'This includes only tunnelings to the inflection point.', '1704.06994-1-44-2': 'For large values of [MATH], the rescaled instanton action is mostly in the range [MATH].', '1704.06994-1-44-3': 'Note that the full action ([REF]) is much larger than that.', '1704.06994-1-44-4': 'From Eq. ([REF]) we have [EQUATION] and thus [MATH].', '1704.06994-1-44-5': 'The plot suggests that for [MATH], [MATH] is essentially uncorrelated with [MATH].', '1704.06994-1-44-6': 'In particular, the average value of [MATH] is nearly constant at large [MATH].', '1704.06994-1-44-7': 'This is not surprising, since the instanton action depends on the shape of the potential around the top of the barrier, while [MATH] is not sensitive to this shape.', '1704.06994-1-44-8': 'The right panel of Fig. [REF] shows the probability distribution of [MATH] under the condition of [MATH].', '1704.06994-1-44-9': 'The distribution is still well fitted by [MATH].', '1704.06994-1-45-0': 'Figure [REF] is a scatter plot in the [MATH]-[MATH] plane for [MATH].', '1704.06994-1-45-1': 'It exhibits significant correlation between [MATH] and [MATH].', '1704.06994-1-45-2': 'In particular, the number of realizations with large values of [MATH] increases towards small values of [MATH].', '1704.06994-1-45-3': 'This can be understood as the contribution of realizations with [MATH], which correspond to the thin-wall regime.', '1704.06994-1-45-4': 'In the limit [MATH], the tunneling action diverges and [MATH] [CITATION].', '1704.06994-1-45-5': 'On the other hand, the average value of S appears to saturate at a constant [MATH] at [MATH].', '1704.06994-1-45-6': 'In the rest of the paper we shall assume that the correlations between [MATH] and [MATH] have no significant effect on the probability distribution for [MATH].', '1704.06994-1-45-7': 'This issue, however, requires some further study.', '1704.06994-1-46-0': '# Slow-roll inflation after tunneling', '1704.06994-1-47-0': 'After tunneling, the bubble has the interior geometry of an open FRW universe, [EQUATION]', '1704.06994-1-47-1': 'Its evolution is described by the equations [EQUATION] with the initial conditions at [MATH] [EQUATION] where [MATH] is determined from the instanton solution.', '1704.06994-1-47-2': 'Our aim in this section is to find the probability distribution for the e-folding number after tunneling [MATH].', '1704.06994-1-48-0': 'We first note that the results of Sec. [REF] for the initial value [MATH] are independent of [MATH] and can be applied for any value of [MATH].', '1704.06994-1-48-1': 'Thus we expect the tunneling to occur to a point [MATH] with a probability distribution of width [EQUATION] near the inflection point (which we assume to be at [MATH]).', '1704.06994-1-48-2': 'We note also that the range of [MATH] around the inflection point where slow-roll inflation is possible, [MATH], is much smaller than ([REF]) for small [MATH].', '1704.06994-1-48-3': 'Thus, most tunnelings will occur outside of the slow-roll range.', '1704.06994-1-49-0': 'Furthermore, the dynamics of inflation does depend on the value of [MATH].', '1704.06994-1-49-1': 'In order to study this dynamics numerically, one would have to find a large sample of realizations of [MATH] with [MATH].', '1704.06994-1-49-2': 'But since [MATH], it follows from Eq. ([REF]) that the number of such realizations is suppressed by a factor [MATH], so a sufficiently large sample can be found only for [MATH].', '1704.06994-1-49-3': 'We have therefore developed a semi-analytic approach to the problem.', '1704.06994-1-50-0': 'The potential near the inflection point is given by Eq. ([REF]), [EQUATION]', '1704.06994-1-50-1': 'In the tunneling range ([REF]), the cubic term gives a correction to [MATH] of the order [EQUATION] where in the last step we assume [MATH].', '1704.06994-1-50-2': 'For [MATH], the linear term in ([REF]) is much smaller than the cubic term everywhere except in a small vicinity of [MATH].', '1704.06994-1-50-3': 'Hence the potential after tunneling and until the end of inflation is well approximated by [MATH].', '1704.06994-1-51-0': 'The kinetic energy [MATH] does not exceed the cubic term (friction can only reduce it), so it is also negligible during inflation (compared to [MATH]).', '1704.06994-1-51-1': 'Thus the Friedmann equation ([REF]) can be approximated as [EQUATION] where [MATH].', '1704.06994-1-51-2': 'The solution is [EQUATION]', '1704.06994-1-51-3': 'This implies that inflation starts at [MATH], after a brief curvature-dominated period.', '1704.06994-1-52-0': 'With the scale factor ([REF]), Eq. ([REF]) for [MATH] takes the form [EQUATION]', '1704.06994-1-52-1': 'Rescaling the variables as [EQUATION] we have [EQUATION] where dots now stand for derivatives with respect to [MATH] and [EQUATION]', '1704.06994-1-52-2': 'Here, [MATH] is the maximal number of efolds defined by Eq. ([REF]).', '1704.06994-1-52-3': 'The slow roll condition fails at the point where [MATH], which means that the slow roll range is [MATH].', '1704.06994-1-52-4': 'The initial conditions for [MATH] are [EQUATION]', '1704.06994-1-53-0': '## Beginning of slow roll', '1704.06994-1-54-0': 'As we already noted, the tunneling range ([REF]) is much wider than the slow roll range [MATH] for small values of [MATH].', '1704.06994-1-54-1': 'If [MATH] happens to be in this narrow range, then inflation begins at [MATH], right after tunneling.', '1704.06994-1-54-2': 'If [MATH], then clearly inflation does not happen.', '1704.06994-1-54-3': 'For [MATH], the field starts rolling fast at [MATH] and may overshoot part or all of the slow-roll region.', '1704.06994-1-54-4': 'We shall find when (and whether) the slow roll begins assuming that the last term in Eq. ([REF]) can be neglected up to that moment.', '1704.06994-1-54-5': 'This approximation is justified for large values of [MATH], as we shall later verify.', '1704.06994-1-55-0': 'Without the last term, Eq. ([REF]) has no free parameters: [EQUATION]', '1704.06994-1-55-1': 'Hence the only free parameter of the problem is the initial value [MATH].', '1704.06994-1-55-2': 'We solved Eq. ([REF]) numerically to determine the value of [MATH] at the onset of slow roll.', '1704.06994-1-56-0': 'If [MATH] eventually gets into the slow-roll regime, the first term in Eq. ([REF]) becomes negligible compared to the other two terms; then [EQUATION]', '1704.06994-1-56-1': 'For the purpose of our numerical analysis, we rewrite this condition as follows: [EQUATION]', '1704.06994-1-56-2': 'The value of [MATH] when this is first satisfied marks the beginning of slow roll.', '1704.06994-1-56-3': 'We shall denote it by [MATH].', '1704.06994-1-56-4': 'In Fig. [REF] we plotted [MATH] as a function of [MATH].', '1704.06994-1-57-0': 'For [MATH], the condition ([REF]) is never satisfied, indicating that the inflaton field overshoots the entire slow-roll region.', '1704.06994-1-57-1': 'We note also that there is a critical value of [MATH], [MATH], above which [MATH] is negative and slow-roll starts before the inflaton reaches the inflection point.', '1704.06994-1-57-2': 'For later use we also give the slope of the curve in Fig. [REF] at [MATH]: [EQUATION]', '1704.06994-1-58-0': '## The number of e-folds', '1704.06994-1-59-0': 'The number of e-folds of slow-roll inflation can now be found from [EQUATION] where [MATH] is the value of [MATH] corresponding to [MATH].', '1704.06994-1-59-1': 'As before, we replace the upper bound of integration by [MATH] and approximate [MATH] by [MATH] in the numerator.', '1704.06994-1-59-2': 'However, we can no longer neglect the linear term in the potential, since otherwise the integral would diverge at [MATH].', '1704.06994-1-59-3': 'Thus we obtain [EQUATION]', '1704.06994-1-59-4': 'For [MATH] and [MATH], we can use [MATH], which gives [EQUATION]', '1704.06994-1-59-5': 'With [MATH], the second term in ([REF]) can be significant only for [MATH].', '1704.06994-1-59-6': 'From the graph in Fig. [REF] we see that this is satisfied only in a narrow range [MATH] around [MATH].', '1704.06994-1-59-7': 'We thus conclude that [MATH] in most of the range [EQUATION]', '1704.06994-1-59-8': 'We shall call it the target range.', '1704.06994-1-60-0': 'Similarly, for positive [MATH] we find [EQUATION]', '1704.06994-1-60-1': 'In order to have [MATH], we need [MATH], and once again this is satisfied only in a narrow range near [MATH].', '1704.06994-1-61-0': 'We calculated numerically [MATH] as a function of [MATH] for several values of [MATH].', '1704.06994-1-61-1': 'The results are shown in Fig. [REF].', '1704.06994-1-61-2': 'We see that large values of [MATH] are reached only for [MATH].', '1704.06994-1-61-3': 'For such values of [MATH], [MATH] in almost the entire target range and [MATH] is negligibly small outside of this range.', '1704.06994-1-61-4': 'This is in full agreement with the above analysis.', '1704.06994-1-62-0': 'We now comment on the validity of the approximation that we used to neglect the last term in Eq. ([REF]).', '1704.06994-1-62-1': 'This term can be significant when [MATH], which corresponds to a narrow range of [MATH] around [MATH].', '1704.06994-1-62-2': 'Using Eq. ([REF]) we can estimate this range as [EQUATION]', '1704.06994-1-62-3': 'This is precisely the range where [MATH] in Eq. ([REF]) is significantly different from either [MATH] or 0.', '1704.06994-1-62-4': 'For [MATH] in this range, Eq. ([REF]) for [MATH] may not be very accurate, but we expect that [MATH] interpolates between [MATH] and 0 over this range of [MATH], so its qualitative behavior should be well represented by ([REF]).', '1704.06994-1-63-0': '## Relation to earlier work', '1704.06994-1-64-0': 'The issue of overshooting in slow-roll inflation after tunneling has been discussed earlier by a number of authors [CITATION].', '1704.06994-1-64-1': 'Freivogel et al [CITATION] assumed that the exit from tunneling is described by a potential with a linear slope, followed by a flat slow-roll region.', '1704.06994-1-64-2': 'Dutta et al [CITATION] considered power-law exit potentials, [MATH] with [MATH].', '1704.06994-1-64-3': 'The main conclusion following from this work is that if [MATH] starts with some initial value [MATH], it overshoots by an amount [MATH] into the slow-roll region.', '1704.06994-1-65-0': 'In our context this would imply that tunnelings with initial values [MATH] overshoot most, if not all, of the slow-roll region.', '1704.06994-1-65-1': 'We find, however, that there is essentially no overshoot if the bubble nucleates with [MATH] in a much wider range, [MATH].', '1704.06994-1-65-2': 'The reason for this discrepancy may be that both Refs. [CITATION] and [CITATION] assumed that [MATH].', '1704.06994-1-65-3': 'We find that, on the contrary, [MATH] is typically rather close to [MATH], [MATH].', '1704.06994-1-66-0': '## Distributions for [MATH] and [MATH]', '1704.06994-1-67-0': 'For [MATH] the target range ([REF]) is much smaller than the tunneling range ([REF]).', '1704.06994-1-67-1': 'We shall assume that the correlation between the tunneling action and [MATH] in this narrow range does not significantly affect the distribution for [MATH], as suggested by our discussion in Sec. [REF].', '1704.06994-1-67-2': 'Then the distribution of the initial values [MATH] in the target range can be approximated as [EQUATION]', '1704.06994-1-67-3': 'Since [MATH] is not correlated with [MATH] for [MATH] (see Fig. [REF]), we can write the joint probability distribution as [EQUATION] with a normalization factor [EQUATION]', '1704.06994-1-67-4': 'Furthermore, since [MATH] in almost all realizations with a reasonably large value of [MATH], we expect the probability distribution for [MATH] to have the same form as that for [MATH], [EQUATION]', '1704.06994-1-67-5': 'If the observable scales lie within the slow-roll range, the spectral index [MATH] is related to [MATH] by Eq. ([REF]) and its distribution is given by Eq. ([REF]).', '1704.06994-1-67-6': 'The observed value of [MATH] is in the mid-range of this distribution, as discussed in [CITATION].', '1704.06994-1-67-7': 'Hence a random [MATH] Gaussian landscape is consistent with observations.', '1704.06994-1-68-0': '## Spatial curvature', '1704.06994-1-69-0': 'Since the interior geometry of the bubble is an open FRW universe, the curvature parameter is nonzero and may have an observable effect [CITATION].', '1704.06994-1-69-1': 'At the present time this parameter is given by [EQUATION] where the subscripts "end" and "p" represent the values at the end of inflation and at present, respectively.', '1704.06994-1-69-2': 'Hereafter, we assume instantaneous reheating and GUT scale inflation to give reference values.', '1704.06994-1-69-3': 'Then we have [MATH].', '1704.06994-1-70-0': 'The Planck collaboration puts an upper bound on [MATH] at [MATH], which requires [EQUATION]', '1704.06994-1-70-1': 'On the other hand, a detection of spatial curvature is probably possible only if [MATH], or [EQUATION]', '1704.06994-1-70-2': 'Naively, we could use Eq. ([REF]) to calculate the probability for [MATH] to be within the range defined by these bounds.', '1704.06994-1-70-3': 'This would give [EQUATION]', '1704.06994-1-70-4': 'We should note, however, that [MATH] is correlated with [MATH], which is in turn related to the spectral index [MATH] by Eq. ([REF]).', '1704.06994-1-70-5': 'With the observed value of [MATH], we should have [MATH].', '1704.06994-1-70-6': 'Hence, in order to have observable curvature, [MATH] has to be significantly smaller than [MATH].', '1704.06994-1-70-7': 'This is possible only if the tunneling point [MATH] happens to be very close to the critical value [MATH].', '1704.06994-1-71-0': 'For [MATH], we can approximate Eq. ([REF]) as [EQUATION]', '1704.06994-1-71-1': 'The range of [MATH] corresponding to the number of e-folds [MATH] in the range of interest, [MATH], is then given by [EQUATION] where the derivatives are evaluated at [MATH].', '1704.06994-1-71-2': 'Using Eqs. ([REF]) and ([REF]) we find [MATH] and [EQUATION]', '1704.06994-1-71-3': 'We thus see that the probability for the curvature to be smaller than the present upper bound but above the future detection limit is rather small.', '1704.06994-1-72-0': '# Conclusions', '1704.06994-1-73-0': 'We used numerical simulations to study bubble nucleation by quantum tunneling in random Gaussian potentials.', '1704.06994-1-73-1': 'We were particularly interested in slow-roll inflation after the tunneling.', '1704.06994-1-73-2': 'For a potential with a correlation length [MATH], this typically occurs near a flat inflection point [MATH], characterized by [MATH].', '1704.06994-1-73-3': 'We sampled a large number of randomly generated potentials with flat inflection points and found that a substantial fraction of them (about a half) allow for tunneling from the false vacuum to the neighborhood of [MATH].', '1704.06994-1-73-4': 'For each tunneling we found the initial value [MATH] of the inflaton field in the bubble by solving the Euclidean field equation for the instanton.', '1704.06994-1-73-5': 'The resulting distribution is peaked at [MATH], where the inflection point is taken to be at [MATH] and the positive direction of [MATH] is taken to be towards the "true" vacuum.', '1704.06994-1-73-6': 'The width of the distribution is [MATH] and is much larger than the size of the region where slow-roll inflation is possible, [MATH].', '1704.06994-1-73-7': 'This indicates that most of the tunnelings take the field [MATH] outside of the slow-roll range.', '1704.06994-1-74-0': 'We developed a semi-analytic technique to study the evolution of [MATH] after tunneling.', '1704.06994-1-74-1': 'Our main conclusions can be stated as follows.', '1704.06994-1-74-2': 'If the bubble nucleates with [MATH] outside the slow-roll range, the field starts rolling fast (after a brief curvature-dominated period) and may overshoot part or all of the slow-roll region, or it may miss this region altogether.', '1704.06994-1-74-3': 'We find that if [MATH] is in the range [EQUATION] where [MATH], then the field slows down and undergoes a nearly maximal number of inflationary e-folds, [MATH].', '1704.06994-1-74-4': 'On either side of the range ([REF]), [MATH] drops towards zero within a distance [MATH].', '1704.06994-1-74-5': 'The probability distribution for [MATH] has the same power-law form as that for [MATH], [EQUATION]', '1704.06994-1-74-6': 'The distribution for the spectral index [MATH] is then given by Eq. ([REF]) and is consistent with the observed value.', '1704.06994-1-75-0': 'We also discussed the prospects for observational detection of nonzero spatial curvature [MATH], which is related to the number of e-folds [MATH].', '1704.06994-1-75-1': 'These prospects are not very good, because the observational lower bound on [MATH] is pretty close to the upper bound that would make observational detection possible.', '1704.06994-1-75-2': 'Using the distribution ([REF]), we found that the probability for a random observer in the multiverse to detect spatial curvature between these two bounds is [MATH].', '1704.06994-1-75-3': 'This estimate changes, however, if we take into account one more data point that is available to us: the measurement of the spectral index of density perturbations: [MATH].', '1704.06994-1-75-4': 'In a small-field Gaussian landscape, [MATH] is rigidly related to [MATH], and the observed value implies [MATH].', '1704.06994-1-75-5': 'On the other hand, the bound for future detectability requires that [MATH], which is significantly smaller than [MATH].', '1704.06994-1-75-6': 'This situation is possible only if the tunneling point [MATH] is very close to [MATH] (within a range [MATH], where [MATH] interpolates between [MATH] and 0).', '1704.06994-1-75-7': 'Our estimate for the probability of this to happen is [MATH].', '1704.06994-1-75-8': 'The bottom line is that the probability of detecting spatial curvature is pretty low if we live in a small-field random Gaussian landscape.', '1704.06994-1-76-0': 'We finally comment on some limitations of our analysis.', '1704.06994-1-76-1': 'The distribution for [MATH] may have some additional factors, due to the probability measure of the multiverse.', '1704.06994-1-76-2': 'These factors may depend on the tunneling transition rates between different vacua and on the correlation between the tunneling action and [MATH].', '1704.06994-1-76-3': 'This issue is entangled with the measure problem, which at present has no definitive solution and may require some new ideas to be resolved.', '1704.06994-1-77-0': 'Another serious limitation is that we restricted our analysis to a one-dimensional landscape.', '1704.06994-1-77-1': 'On the other hand, in multi-dimensional models parts of the landscape where slow-roll inflation is possible may be effectively one-dimensional, with only one of the fields having a nearly flat potential, due to an approximate shift symmetry.', '1704.06994-1-77-2': 'In lieu of detailed information about the landscape, it may then be a reasonable approximation to consider such effective [MATH] potentials as samples of a random Gaussian field.', '1704.06994-1-77-3': 'We note also that the methods we used here may have a wider applicability.', '1704.06994-1-77-4': 'In Ref. [CITATION] we indicated some directions in which these methods can be extended to multi-dimensional landscapes.'} | {'1704.06994-2-0-0': '# Introduction', '1704.06994-2-1-0': 'String theory predicts the existence of a vast landscape of vacuum states with diverse properties [CITATION].', '1704.06994-2-1-1': 'In the cosmological context this leads to the picture of an eternally inflating multiverse, where different spacetime regions are occupied by different vacua.', '1704.06994-2-1-2': 'Transitions between the vacua occur through quantum tunneling, with bubbles of daughter vacuum nucleating and expanding in the parent vacuum background.', '1704.06994-2-1-3': 'According to this picture, our local region originated as a result of tunneling from some inflating parent vacuum and then went through a period of slow-roll inflation.', '1704.06994-2-1-4': 'The number of vacua in the landscape is expected to be enormous, so predictions in this kind of model must necessarily be statistical.', '1704.06994-2-1-5': 'The properties of string theory landscape are not well understood, and the approach adopted in much of the recent work is to substitute it by a scalar field model with a random Gaussian potential [CITATION].', '1704.06994-2-2-0': 'In a recent paper [CITATION] we developed analytic and numerical techniques for studying the statistics of slow-roll inflation in random Gaussian landscapes.', '1704.06994-2-2-1': 'We applied these techniques to the simplest case of small-field inflation in a one-dimensional random landscape.', '1704.06994-2-2-2': 'In this case, inflation typically occurs at local maxima or at inflection points of the potential [CITATION].', '1704.06994-2-2-3': 'Focusing mostly on the inflection points, we found the probability distributions for the maximal number of inflationary e-folds [MATH] and for the spectral index of density fluctuations [MATH].', '1704.06994-2-3-0': 'The maximal e-fold number [MATH] depends only on the shape of the potential near the inflection point, but the actual number of e-folds, [MATH], is sensitive to the initial conditions - that is, to the initial value [MATH] of the inflaton field right after it tunnels from the parent vacuum.', '1704.06994-2-3-1': 'If [MATH] is too far away from the inflection point, the field may develop a large velocity and overshoot or it may miss the slow-roll region entirely.', '1704.06994-2-3-2': 'In the present paper we shall use numerical simulations to determine the probability distribution for [MATH] and to investigate its effect on the statistical properties of inflation.', '1704.06994-2-3-3': 'As before, we shall restrict our analysis to the simplest case of one-dimensional potentials.', '1704.06994-2-4-0': 'In the next Section we review some general properties of random Gaussian potentials, and in Sec. 3 we summarize earlier work on inflection-point inflation.', '1704.06994-2-4-1': 'Our numerical simulations and the results for the distribution of [MATH] are presented in Sec. [REF].', '1704.06994-2-4-2': 'In Sec. [REF] we develop a semi-analytic method to study the evolution of the scalar field after tunneling.', '1704.06994-2-4-3': 'We find the probability distribution for the number of e-folds of slow-roll inflation [MATH] and discuss the implications of our results for the prospects of detection of spatial curvature.', '1704.06994-2-4-4': 'Our conclusions are summarized and discussed in Sec. [REF].', '1704.06994-2-4-5': 'Some technical details related to the simulations are relegated to the appendices.', '1704.06994-2-4-6': 'Throughout the paper we use reduced Planck units with [MATH].', '1704.06994-2-5-0': '# Random Gaussian landscapes', '1704.06994-2-6-0': 'Consider a one-dimensional random Gaussian landscape model with a potential [MATH] satisfying the following correlation function: [EQUATION]', '1704.06994-2-6-1': 'We specifically consider a Gaussian-type correlation function defined as [EQUATION] with [MATH] playing the role of the correlation length in the landscape.', '1704.06994-2-6-2': 'Then the spectral function [MATH] is given by [EQUATION]', '1704.06994-2-6-3': 'We define different moments of the spectral function as [EQUATION] where we used Eq. ([REF]) in the second line.', '1704.06994-2-7-0': 'Once we specify a set of points [MATH] and define [MATH], the probability distribution for [MATH] is given by [EQUATION] where the positive definite matrix [MATH] is an [MATH] matrix of correlators defined by [EQUATION]', '1704.06994-2-7-1': 'Since [MATH] is a symmetric matrix, we can diagonalize it by an orthogonal matrix [MATH].', '1704.06994-2-7-2': "Then the probability distribution for variables [MATH] is given by [EQUATION] where [MATH]'s are eigenvalues of the matrix [MATH].", '1704.06994-2-7-3': 'Such random variables can be easily generated in numerical simulations.', '1704.06994-2-8-0': 'When the values of [MATH] are generated at a sufficient density on the [MATH]-axis, we can interpolate them to obtain a smooth potential.', '1704.06994-2-8-1': 'We check in the App.', '1704.06994-2-8-2': '[REF] that this procedure typically saturates at a few points per correlation length [MATH]: the interpolated potential changes very little with the addition of more points.', '1704.06994-2-8-3': 'We shall use this method with four points per correlation length to generate realizations of a random Gaussian landscape.', '1704.06994-2-9-0': 'Assuming that our universe is a result of a bubble nucleation event, we anticipate that the state of the universe after nucleation allows for a prolonged period of inflation.', '1704.06994-2-9-1': 'This is possible for parts of the landscape that resemble Fig. [REF], which includes a parent vacuum and a daughter vacuum separated by a barrier.', '1704.06994-2-9-2': 'We shall refer to parent and daughter vacua as "false" (FV) and "true" (TV) vacua, respectively.', '1704.06994-2-9-3': 'The inflection points where [MATH] are marked by red dots in the figure.', '1704.06994-2-9-4': 'In this example, there are three inflection points between the top of the barrier and the true vacuum.', '1704.06994-2-9-5': 'The potential is rather flat near the middle inflection point ([MATH] and [MATH] are small), and slow-roll inflation can be expected to occur in this region.', '1704.06994-2-10-0': 'We are going to focus on the case when the correlation length of the potential is small compared to the Planck scale, [MATH].', '1704.06994-2-10-1': 'Inflation in this case is typically of the small-field type: the slow-roll occurs in a narrow range [MATH] and the potential during the slow-roll remains approximately constant.', '1704.06994-2-11-0': 'The correlation function ([REF]) implies that the average value of the potential is [MATH], so positive and negative values of [MATH] are equally likely.', '1704.06994-2-11-1': 'In this case, the minima of the potential are predominantly at negative values of [MATH].', '1704.06994-2-11-2': 'This effect is especially pronounced for higher-dimensional random landscapes, where positive-energy vacua may not exist at all [CITATION].', '1704.06994-2-11-3': 'This problem can be alleviated by adding a constant shift term to the potential, [MATH].', '1704.06994-2-11-4': 'One can also consider models where [MATH] is variable, but its characteristic scale of variation is much greater than [MATH].', '1704.06994-2-11-5': 'For example, in axionic landscapes this term can have the form [MATH] with a very small mass [MATH] [CITATION] or [MATH] with [MATH] [CITATION].', '1704.06994-2-11-6': 'Then the local properties of the potential are still determined by the correlator ([REF]), while the extra term provides an additive shift, which is characterized by its own probability distribution.', '1704.06994-2-12-0': 'Here, we are interested in the case when the true vacuum has an almost vanishing vacuum energy.', '1704.06994-2-12-1': 'In a generic landscape, the number of such vacua is extremely small, and trying to find them by random sampling is a hopeless task.', '1704.06994-2-12-2': 'In our numerical simulations we simply added a constant to the generated random potential, so that [MATH] in the true vacuum.', '1704.06994-2-12-3': 'We expect the resulting ensemble of realizations to be similar to what one would get by random sampling in a landscape with a flat distribution of the shift parameter [MATH].', '1704.06994-2-13-0': '# Analytic results for inflection-point inflation', '1704.06994-2-14-0': 'Inflection-point inflation was first studied by Baumann et al [CITATION].', '1704.06994-2-14-1': 'Here we review some of their results, which will be useful in subsequent sections.', '1704.06994-2-15-0': 'The necessary conditions for slow-roll inflation are [MATH], where [EQUATION]', '1704.06994-2-15-1': 'Note that the values of the slow-roll parameters at a randomly chosen point in the landscape are typically given by [MATH], so inflation can occur only in rare regions of the landscape.', '1704.06994-2-16-0': 'To study inflection-point inflation, we approximate the potential by a third-order Taylor expansion, [EQUATION] where the subscript [MATH] indicates the value at the inflection point [MATH], which is set to be at [MATH]: [MATH].', '1704.06994-2-16-1': 'The slow-roll conditions require that [MATH], but [MATH] does not need to be small, and we shall assume it to be comparable to the rms value, [MATH].', '1704.06994-2-16-2': 'The slow-roll region is then determined mainly by the condition [MATH] and can be specified as [MATH], where [EQUATION]', '1704.06994-2-16-3': 'We assume that [MATH], since otherwise the potential has a shallow high-energy minimum near the inflection point and inflation drives the field into that minimum [CITATION].', '1704.06994-2-17-0': 'It follows from ([REF]),([REF]) that the size of the slow-roll region is typically [MATH].', '1704.06994-2-17-1': 'This justifies the assumption that [MATH] can be approximated by a constant in this region.', '1704.06994-2-18-0': 'The "maximal e-folding number" can be defined as [EQUATION] where we have assumed that [MATH] (which is necessary for extending the integration to [MATH]).', '1704.06994-2-19-0': 'The spectral index [MATH] can be expressed in terms of [MATH] as [EQUATION] where [MATH]-[MATH]) is the e-folding number at which the CMB scale leaves the horizon.', '1704.06994-2-19-1': 'It follows from ([REF]) that [MATH] is greater than [MATH], which is approached in the limit [MATH].', '1704.06994-2-19-2': 'Note that for hilltop inflation [MATH] [CITATION], while the observed value is [MATH] and lies in the inflection-point range.', '1704.06994-2-19-3': 'This provides additional motivation to focus our analysis on inflection-point inflation.', '1704.06994-2-20-0': 'The magnitude of density fluctuations is given by [EQUATION] where in the second step we used Eq. ([REF]) for [MATH] and in the last step we assumed that [MATH] and [MATH] have their typical values.', '1704.06994-2-20-1': 'The observed value of [MATH] can be obtained by adjusting the parameters [MATH] and [MATH]: [EQUATION]', '1704.06994-2-20-2': 'The probability distributions for [MATH] and [MATH] in a random Gaussian landscape have been calculated in Ref. [CITATION].', '1704.06994-2-20-3': 'Here we quote the results: [EQUATION]', '1704.06994-2-20-4': 'The distribution ([REF]) gives the probability that a randomly chosen inflection point in the landscape is characterized by a given value of [MATH].', '1704.06994-2-20-5': 'Note, however, that here we are interested only in inflection points located between a high-energy false vacuum and a zero-energy true vacuum, as shown in Fig. [REF].', '1704.06994-2-20-6': 'As explained in Sec. [REF], we obtain such configurations by adding a constant term to a randomly generated potential.', '1704.06994-2-20-7': 'This procedure changes the value of [MATH] at the inflection point; hence it affects the value of [MATH] in Eq. ([REF]) and may potentially affect the distribution ([REF]).', '1704.06994-2-20-8': 'We shall see, however, that the form of this distribution remains unchanged.', '1704.06994-2-21-0': 'As we mentioned in the Introduction, the actual number of inflationary e-folds [MATH] depends on the initial conditions after tunneling and is generally different from [MATH].', '1704.06994-2-21-1': 'We shall find the probability distributions for the initial conditions and for [MATH] in the following sections.', '1704.06994-2-22-0': '# Initial conditions for inflation', '1704.06994-2-23-0': '## General formalism', '1704.06994-2-24-0': 'Decay of the false vacuum occurs through bubble nucleation, which is a quantum tunneling process.', '1704.06994-2-24-1': 'In the semiclassical approximation, the tunneling is described by an [MATH]-symmetric instanton [MATH], which can be found by solving the Euclidean field equation [CITATION] [EQUATION]', '1704.06994-2-24-2': 'Here we assume that gravitational effects on the tunneling can be neglected, which is usually the case in a small-field landscape.', '1704.06994-2-24-3': 'The boundary conditions for [MATH] are [MATH] and [MATH], where [MATH] is the value of [MATH] in the false vacuum.', '1704.06994-2-24-4': 'The tunneling probability is determined mostly by the exponential factor, [EQUATION] where [MATH] is the Euclidean instanton action [EQUATION] and [MATH].', '1704.06994-2-25-0': 'It will be convenient to introduce dimensionless variables [MATH] and [MATH] as [EQUATION]', '1704.06994-2-25-1': 'In terms of the new variables, Eq. ([REF]) still has the same form, [EQUATION] where the potential [MATH] is now characterized by the correlation function ([REF]) with [MATH].', '1704.06994-2-25-2': 'We note that [EQUATION] where [MATH] is the maximal e-folding number for the rescaled potential [MATH].', '1704.06994-2-25-3': 'We also define the action [MATH] for the rescaled variables: [EQUATION]', '1704.06994-2-25-4': 'The initial value [MATH] of the inflaton field after tunneling is set by the value of [MATH] at the center of the instanton, [EQUATION]', '1704.06994-2-25-5': 'The probability distribution for [MATH] can now be found with the aid of numerical simulations.', '1704.06994-2-26-0': 'It is well known that instanton solutions for multi-dimensional field spaces are not unique, because there may be more than one saddle point between the true vacuum and false vacuum.', '1704.06994-2-26-1': 'As we explain in App.', '1704.06994-2-26-2': '[REF], there may also be multiple instanton solutions in the one-dimensional case, but for a different reason.', '1704.06994-2-26-3': 'For a generic potential, there is typically a single instanton describing tunneling to a close vicinity of the true vacuum.', '1704.06994-2-26-4': 'In the presence of a flat inflection region, additional instantons may appear, corresponding to tunneling to the neighborhood of the inflection point [MATH].', '1704.06994-2-26-5': 'As we make the inflection region flatter, at some point the instanton tunneling to the true vacuum disappears, and only tunneling to the vicinity of inflection point remains possible.', '1704.06994-2-27-0': 'The number and character of the instantons also depend on the shape and height of the potential barrier.', '1704.06994-2-27-1': 'As an illustration we show some examples in Fig. [REF], where the tunneling points [MATH] are indicated by blue squares.', '1704.06994-2-27-2': 'All four potentials in the figure are rather similar, except they have different values of the false vacuum energy density [MATH].', '1704.06994-2-27-3': 'In the upper left frame, [MATH] is almost degenerate with [MATH] and there is a single instanton solution, which brings [MATH] almost all the way to the true vacuum.', '1704.06994-2-27-4': 'In the upper right frame, [MATH] is somewhat higher and additional instantons appear, which describe tunneling with [MATH] close to the inflection point.', '1704.06994-2-27-5': 'As we explain in App.', '1704.06994-2-27-6': '[REF], additional instanton solutions appear in pairs.', '1704.06994-2-27-7': 'As [MATH] gets higher, the middle tunneling point moves towards the true vacuum tunneling point, and eventually the two points "annihilate".', '1704.06994-2-27-8': 'In the lower right frame, [MATH] is still higher, and tunneling is now possible only to the neighborhood of the inflection point.', '1704.06994-2-28-0': 'When several instantons are present, one can compare the instanton actions to determine the dominant decay channel.', '1704.06994-2-28-1': 'We found that tunneling to the true vacuum dominates in most of these cases.', '1704.06994-2-28-2': 'We note, however, that in the present context we are interested in tunnelings that lead to sufficiently long inflation, regardless of their relative rate compared to other tunneling processes.', '1704.06994-2-29-0': 'To illustrate the dependence of the action of different instantons on the shape of the potential, we fix the potential on the right side of the barrier in Fig. [REF] and calculate the instanton actions for different values of [MATH].', '1704.06994-2-29-1': 'The result is shown in Fig. [REF].', '1704.06994-2-29-2': 'The red dashed (green solid) line is the action of the instanton solution that brings [MATH] close to the true vacuum (inflection point).', '1704.06994-2-29-3': 'The blue dotted line is the one with [MATH] between the other two tunneling points.', '1704.06994-2-29-4': 'We see that tunneling to the true vacuum dominates in (almost) the entire range where the corresponding instanton exists.', '1704.06994-2-29-5': 'The blue dotted line is always just above the green solid line; they are so close that they appear to coincide in the left frame of the figure.', '1704.06994-2-29-6': 'When [MATH] increases to a certain threshold, the blue dotted line meets and annihilates with the red dashed line (see the right frame).', '1704.06994-2-29-7': 'Before the annihilation, the red dashed line crosses the green solid line, which means that tunneling to the inflection point becomes dominant.', '1704.06994-2-29-8': 'However, the region where this occurs is so small that we cannot see it in the left frame.', '1704.06994-2-29-9': 'Since the instanton solution with [MATH] between the other two tunneling points is always subdominant, we neglect it in the rest of the paper.', '1704.06994-2-29-10': 'We denote the instanton action for solutions that bring [MATH] close to the true vacuum (inflection point) as [MATH]).', '1704.06994-2-30-0': '## Numerical simulation', '1704.06994-2-31-0': 'We used the procedure outlined in Sec. [REF] to generate [MATH] segments of a random Gaussian landscape, with each segment having length [MATH].', '1704.06994-2-31-1': 'The number of local minima in such a segment is almost always greater than or equal to two.', '1704.06994-2-31-2': 'We set the number of points per correlation length at [MATH], which means that every segment contains 33 points [MATH].', '1704.06994-2-31-3': 'We generate the values of the potential [MATH] at these points according to the probability distribution ([REF]) and use a fifth order spline to interpolate between them.', '1704.06994-2-32-0': 'In each realization of [MATH], we identify all extrema and inflection points.', '1704.06994-2-32-1': 'For any pair of adjacent minima, we refer to the higher- and lower-energy ones as [MATH] and [MATH], respectively, and shift the potential so that [MATH].', '1704.06994-2-32-2': 'We keep only realizations that have an inflection point [MATH] satisfying the following criteria: (i) it is located between the top of the barrier and the true vacuum, (ii) its energy density is lower than that of the false vacuum, [MATH].', '1704.06994-2-32-3': 'This procedure is illustrated in Fig. [REF].', '1704.06994-2-33-0': 'For each of the selected pairs of vacua, we seek instanton solutions of Eq. ([REF]) using the shooting method.', '1704.06994-2-33-1': 'We then use Eq. ([REF]) to find the initial value(s) of the field [MATH] after tunneling.', '1704.06994-2-34-0': '## Distributions for [MATH] and [MATH]', '1704.06994-2-35-0': 'We plotted the distribution of the initial values [MATH] in the left panel of Fig. [REF].', '1704.06994-2-35-1': 'Here and hereafter, the normalization of probability distributions is arbitrary.', '1704.06994-2-35-2': 'In cases where the potential admitted two instanton solutions, we included the values of [MATH] for both of them (disregarding the subdominant "middle" instanton).', '1704.06994-2-35-3': 'The distribution has a sharp peak centered near [MATH] and a somewhat larger and broader peak at [MATH].', '1704.06994-2-35-4': 'These peaks correspond to tunnelings to the vicinity of the inflection point [MATH] and of the true vacuum [MATH], respectively.', '1704.06994-2-35-5': 'In the right panel of Fig. [REF] we plotted the distribution of [MATH] for tunnelings to the inflection point - that is, including only cases where [MATH] is closer to [MATH] than to [MATH].', '1704.06994-2-35-6': 'This distribution is peaked near [MATH] with a width [MATH].', '1704.06994-2-36-0': 'Fig. [REF] shows a histogram of the maximal e-fold number [MATH], evaluated from Eq. ([REF]).', '1704.06994-2-36-1': 'The result is well fitted by the analytic function [EQUATION] which is shown by a red line in the figure.', '1704.06994-2-36-2': 'This shows that the form of the [MATH] distribution is not affected by the shift of the potential to [MATH].', '1704.06994-2-37-0': 'The plot in the left panel of Fig. [REF] is the same as in Fig. [REF], but now the contributions due to different types of instantons are shown separately.', '1704.06994-2-37-1': 'The green (magenta) lines represent the cases where there is only one instanton solution and [MATH] is closer to the inflection point (true vacuum).', '1704.06994-2-37-2': 'The red (blue) lines are for realizations with multiple instantons, where the dominant tunneling is to the inflection point (true vacuum).', '1704.06994-2-37-3': 'The type of instanton is not relevant for the present paper, but our results may be useful in other contexts, so we present them here for completeness.', '1704.06994-2-37-4': 'The figure shows that in the multiple instanton case the dominant tunneling channel is almost always to the true vacuum: the number of realizations with [MATH] is suppressed by more than an order of magnitude.', '1704.06994-2-37-5': 'This is consistent with the discussion of the tunneling action in Sec. [REF].', '1704.06994-2-37-6': 'We note also that the numbers of landscape realizations represented by blue and green curves are nearly the same and are within about a factor of 2 from those represented by the magenta curve.', '1704.06994-2-37-7': 'We have not found any explanation for this surprising fact.', '1704.06994-2-37-8': 'It implies that realizations with large values of [MATH] split into three comparable groups: a group with tunneling only to inflection point, a group with tunneling to the true vacuum, and a group with multiple tunneling channels, where the tunneling to the true vacuum dominates.', '1704.06994-2-38-0': 'The right panel of Fig. [REF] is a scatter plot of realizations in [MATH]-[MATH] plane, with the same color code as in the left panel.', '1704.06994-2-38-1': 'The average values [MATH] are shown as yellow, magenta, and cyan lines for the data indicated by green, red, and blue dots, respectively.', '1704.06994-2-38-2': 'We see that [MATH] is almost constant at [MATH] for all types of instantons, with [MATH] for the single instanton case and [MATH] for the multi-instanton case.', '1704.06994-2-38-3': 'This indicates that [MATH] and [MATH] are essentially uncorrelated in the most interesting regime of large [MATH].', '1704.06994-2-39-0': 'From now on, we shall not distinguish between single and multi-instanton tunnelings and treat all inflection-point tunnelings on equal footing.', '1704.06994-2-39-1': 'In multi-instanton realizations with a dominant true-vacuum instanton, we keep only the inflection-point instanton.', '1704.06994-2-39-2': 'The reason is that true-vacuum tunneling is irrelevant for our discussion, and the tunneling processes described by inflection-point instantons occur regardless of whether or not there is a more probable tunneling channel.', '1704.06994-2-40-0': '## Tunneling action', '1704.06994-2-41-0': 'The distributions for [MATH] and [MATH] that we calculated here are defined as frequencies of occurrence in the landscape.', '1704.06994-2-41-1': 'They should not be confused with the probabilities of occurrence in the multiverse, which can be directly related to observational predictions.', '1704.06994-2-41-2': 'The problem of defining these probabilities is known as the measure problem, which at present remains unresolved.', '1704.06994-2-41-3': '(For a review of the measure problem see [CITATION].)', '1704.06994-2-41-4': 'A number of different measure prescriptions have been suggested in the literature.', '1704.06994-2-41-5': 'Some of them lead to paradoxes or to a glaring conflict with observations and have therefore been ruled out.', '1704.06994-2-41-6': 'This process of elimination has not been enough to fix a unique measure of the multiverse.', '1704.06994-2-41-7': 'However, the measure prescriptions which are not obviously problematic tend to give similar predictions and introduce similar weighting factors for different realizations of the potential.', '1704.06994-2-41-8': 'The scale factor measure [CITATION] can be taken as a representative example of such "acceptable" measures.', '1704.06994-2-42-0': 'For a given measure prescription, probabilities can be calculated by solving the rate equation, which is similar to the Boltzmann equation in the multiverse.', '1704.06994-2-42-1': 'Naively, one might expect that different tunneling realizations in the landscape should be weighted by the tunneling rate, which is proportional to [MATH], where [MATH] is the tunneling action.', '1704.06994-2-42-2': 'However, analysis of the rate equation in the scale-factor measure shows that this expectation is incorrect.', '1704.06994-2-42-3': 'A simple counter-example is a landscape with an everywhere positive potential, [MATH].', '1704.06994-2-42-4': 'It can be shown that in such a landscape the probabilities depend only on the vacuum energy density and are independent of the transition rates [CITATION].', '1704.06994-2-42-5': 'In general, the probability of a given vacuum has a complicated dependence on the transition rates between different vacua in the landscape, not just on the rate of tunneling to this particular vacuum.', '1704.06994-2-42-6': 'One also finds that transitions with a small tunneling action are not generally "rewarded" with a high weighting factor.', '1704.06994-2-42-7': 'The reason can be roughly explained as follows.', '1704.06994-2-42-8': 'The weighting factor for vacuum [MATH] due to tunneling from a false vacuum [MATH] is proportional to [MATH], where [MATH] is the tunneling rate from [MATH] to [MATH] per Hubble volume per Hubble time, and [MATH] is the volume fraction occupied by vacuum [MATH] on constant scale factor surfaces.', '1704.06994-2-42-9': 'If the tunneling rate is very high, this leads to a rapid depletion of the false vacuum, so [MATH] gets very small.', '1704.06994-2-42-10': 'These two effects tend to compensate one another.', '1704.06994-2-42-11': 'On the other hand, tunnelings with a large instanton action tend to be disfavored in the presence of other decay channels with a smaller action.', '1704.06994-2-43-0': 'A study of the rate equation in a random Gaussian landscape would require a complicated statistical analysis, which is beyond the scope of the present paper.', '1704.06994-2-43-1': 'To facilitate such analysis in future work, here we shall analyze possible correlations of the action [MATH] with [MATH] and [MATH].', '1704.06994-2-43-2': 'If present, such correlations may affect probabilities for the initial conditions and for the observational effects of inflation.', '1704.06994-2-44-0': 'We show a scatter plot of [MATH] and [MATH] in the left panel of Fig. [REF].', '1704.06994-2-44-1': 'This includes only tunnelings to the inflection point.', '1704.06994-2-44-2': 'For large values of [MATH], the rescaled instanton action is mostly in the range [MATH].', '1704.06994-2-44-3': 'Note that the full action ([REF]) is much larger than that.', '1704.06994-2-44-4': 'From Eq. ([REF]) we have [EQUATION] and thus [MATH].', '1704.06994-2-44-5': 'The plot suggests that for [MATH], [MATH] is essentially uncorrelated with [MATH].', '1704.06994-2-44-6': 'In particular, the average value of [MATH] is nearly constant at large [MATH].', '1704.06994-2-44-7': 'This is not surprising, since the instanton action depends on the shape of the potential around the top of the barrier, while [MATH] is not sensitive to this shape.', '1704.06994-2-44-8': 'The right panel of Fig. [REF] shows the probability distribution of [MATH] under the condition of [MATH].', '1704.06994-2-44-9': 'The distribution is still well fitted by [MATH].', '1704.06994-2-45-0': 'Figure [REF] is a scatter plot in the [MATH]-[MATH] plane for [MATH].', '1704.06994-2-45-1': 'It exhibits significant correlation between [MATH] and [MATH].', '1704.06994-2-45-2': 'In particular, the number of realizations with large values of [MATH] increases towards small values of [MATH].', '1704.06994-2-45-3': 'This can be understood as the contribution of realizations with [MATH], which correspond to the thin-wall regime.', '1704.06994-2-45-4': 'In the limit [MATH], the tunneling action diverges and [MATH] [CITATION].', '1704.06994-2-45-5': 'On the other hand, the average value of S appears to saturate at a constant [MATH] at [MATH].', '1704.06994-2-45-6': 'In the rest of the paper we shall assume that the correlations between [MATH] and [MATH] have no significant effect on the probability distribution for [MATH].', '1704.06994-2-45-7': 'This issue, however, requires some further study.', '1704.06994-2-46-0': '# Slow-roll inflation after tunneling', '1704.06994-2-47-0': 'After tunneling, the bubble has the interior geometry of an open FRW universe, [EQUATION]', '1704.06994-2-47-1': 'Its evolution is described by the equations [EQUATION] with the initial conditions at [MATH] [EQUATION] where [MATH] is determined from the instanton solution.', '1704.06994-2-47-2': 'Our aim in this section is to find the probability distribution for the e-folding number after tunneling [MATH].', '1704.06994-2-48-0': 'We first note that the results of Sec. [REF] for the initial value [MATH] are independent of [MATH] and can be applied for any value of [MATH].', '1704.06994-2-48-1': 'Thus we expect the tunneling to occur to a point [MATH] with a probability distribution of width [EQUATION] near the inflection point (which we assume to be at [MATH]).', '1704.06994-2-48-2': 'We note also that the range of [MATH] around the inflection point where slow-roll inflation is possible, [MATH], is much smaller than ([REF]) for small [MATH].', '1704.06994-2-48-3': 'Thus, most tunnelings will occur outside of the slow-roll range.', '1704.06994-2-49-0': 'Furthermore, the dynamics of inflation does depend on the value of [MATH].', '1704.06994-2-49-1': 'In order to study this dynamics numerically, one would have to find a large sample of realizations of [MATH] with [MATH].', '1704.06994-2-49-2': 'But since [MATH], it follows from Eq. ([REF]) that the number of such realizations is suppressed by a factor [MATH], so a sufficiently large sample can be found only for [MATH].', '1704.06994-2-49-3': 'We have therefore developed a semi-analytic approach to the problem.', '1704.06994-2-50-0': 'The potential near the inflection point is given by Eq. ([REF]), [EQUATION]', '1704.06994-2-50-1': 'In the tunneling range ([REF]), the cubic term gives a correction to [MATH] of the order [EQUATION] where in the last step we assume [MATH].', '1704.06994-2-50-2': 'For [MATH], the linear term in ([REF]) is much smaller than the cubic term everywhere except in a small vicinity of [MATH].', '1704.06994-2-50-3': 'Hence the potential after tunneling and until the end of inflation is well approximated by [MATH].', '1704.06994-2-51-0': 'The kinetic energy [MATH] does not exceed the cubic term (friction can only reduce it), so it is also negligible during inflation (compared to [MATH]).', '1704.06994-2-51-1': 'Thus the Friedmann equation ([REF]) can be approximated as [EQUATION] where [MATH].', '1704.06994-2-51-2': 'The solution is [EQUATION]', '1704.06994-2-51-3': 'This implies that inflation starts at [MATH], after a brief curvature-dominated period.', '1704.06994-2-52-0': 'With the scale factor ([REF]), Eq. ([REF]) for [MATH] takes the form [EQUATION]', '1704.06994-2-52-1': 'Rescaling the variables as [EQUATION] we have [EQUATION] where dots now stand for derivatives with respect to [MATH] and [EQUATION]', '1704.06994-2-52-2': 'Here, [MATH] is the maximal number of efolds defined by Eq. ([REF]).', '1704.06994-2-52-3': 'The slow roll condition fails at the point where [MATH], which means that the slow roll range is [MATH].', '1704.06994-2-52-4': 'The initial conditions for [MATH] are [EQUATION]', '1704.06994-2-53-0': '## Beginning of slow roll', '1704.06994-2-54-0': 'As we already noted, the tunneling range ([REF]) is much wider than the slow roll range [MATH] for small values of [MATH].', '1704.06994-2-54-1': 'If [MATH] happens to be in this narrow range, then inflation begins at [MATH], right after tunneling.', '1704.06994-2-54-2': 'If [MATH], then clearly inflation does not happen.', '1704.06994-2-54-3': 'For [MATH], the field starts rolling fast at [MATH] and may overshoot part or all of the slow-roll region.', '1704.06994-2-54-4': 'We shall find when (and whether) the slow roll begins assuming that the last term in Eq. ([REF]) can be neglected up to that moment.', '1704.06994-2-54-5': 'This approximation is justified for large values of [MATH], as we shall later verify.', '1704.06994-2-55-0': 'Without the last term, Eq. ([REF]) has no free parameters: [EQUATION]', '1704.06994-2-55-1': 'Hence the only free parameter of the problem is the initial value [MATH].', '1704.06994-2-55-2': 'We solved Eq. ([REF]) numerically to determine the value of [MATH] at the onset of slow roll.', '1704.06994-2-56-0': 'If [MATH] eventually gets into the slow-roll regime, the first term in Eq. ([REF]) becomes negligible compared to the other two terms; then [EQUATION]', '1704.06994-2-56-1': 'For the purpose of our numerical analysis, we rewrite this condition as follows: [EQUATION]', '1704.06994-2-56-2': 'The value of [MATH] when this is first satisfied marks the beginning of slow roll.', '1704.06994-2-56-3': 'We shall denote it by [MATH].', '1704.06994-2-56-4': 'In Fig. [REF] we plotted [MATH] as a function of [MATH].', '1704.06994-2-57-0': 'For [MATH], the condition ([REF]) is never satisfied, indicating that the inflaton field overshoots the entire slow-roll region.', '1704.06994-2-57-1': 'We note also that there is a critical value of [MATH], [MATH], above which [MATH] is negative and slow-roll starts before the inflaton reaches the inflection point.', '1704.06994-2-57-2': 'For later use we also give the slope of the curve in Fig. [REF] at [MATH]: [EQUATION]', '1704.06994-2-58-0': '## The number of e-folds', '1704.06994-2-59-0': 'The number of e-folds of slow-roll inflation can now be found from [EQUATION] where [MATH] is the value of [MATH] corresponding to [MATH].', '1704.06994-2-59-1': 'As before, we replace the upper bound of integration by [MATH] and approximate [MATH] by [MATH] in the numerator.', '1704.06994-2-59-2': 'However, we can no longer neglect the linear term in the potential, since otherwise the integral would diverge at [MATH].', '1704.06994-2-59-3': 'Thus we obtain [EQUATION]', '1704.06994-2-59-4': 'For [MATH] and [MATH], we can use [MATH], which gives [EQUATION]', '1704.06994-2-59-5': 'With [MATH], the second term in ([REF]) can be significant only for [MATH].', '1704.06994-2-59-6': 'From the graph in Fig. [REF] we see that this is satisfied only in a narrow range [MATH] around [MATH].', '1704.06994-2-59-7': 'We thus conclude that [MATH] in most of the range [EQUATION]', '1704.06994-2-59-8': 'We shall call it the target range.', '1704.06994-2-60-0': 'Similarly, for positive [MATH] we find [EQUATION]', '1704.06994-2-60-1': 'In order to have [MATH], we need [MATH], and once again this is satisfied only in a narrow range near [MATH].', '1704.06994-2-61-0': 'We calculated numerically [MATH] as a function of [MATH] for several values of [MATH].', '1704.06994-2-61-1': 'The results are shown in Fig. [REF].', '1704.06994-2-61-2': 'We see that large values of [MATH] are reached only for [MATH].', '1704.06994-2-61-3': 'For such values of [MATH], [MATH] in almost the entire target range and [MATH] is negligibly small outside of this range.', '1704.06994-2-61-4': 'This is in full agreement with the above analysis.', '1704.06994-2-62-0': 'We now comment on the validity of the approximation that we used to neglect the last term in Eq. ([REF]).', '1704.06994-2-62-1': 'This term can be significant when [MATH], which corresponds to a narrow range of [MATH] around [MATH].', '1704.06994-2-62-2': 'Using Eq. ([REF]) we can estimate this range as [EQUATION]', '1704.06994-2-62-3': 'This is precisely the range where [MATH] in Eq. ([REF]) is significantly different from either [MATH] or 0.', '1704.06994-2-62-4': 'For [MATH] in this range, Eq. ([REF]) for [MATH] may not be very accurate, but we expect that [MATH] interpolates between [MATH] and 0 over this range of [MATH], so its qualitative behavior should be well represented by ([REF]).', '1704.06994-2-63-0': '## Relation to earlier work', '1704.06994-2-64-0': 'The issue of overshooting in slow-roll inflation after tunneling has been discussed earlier by a number of authors [CITATION].', '1704.06994-2-64-1': 'Freivogel et al [CITATION] assumed that the exit from tunneling is described by a potential with a linear slope, followed by a flat slow-roll region.', '1704.06994-2-64-2': 'Dutta et al [CITATION] considered power-law exit potentials, [MATH] with [MATH].', '1704.06994-2-64-3': 'The main conclusion following from this work is that if [MATH] starts with some initial value [MATH], it overshoots by an amount [MATH] into the slow-roll region.', '1704.06994-2-65-0': 'In our context this would imply that tunnelings with initial values [MATH] overshoot most, if not all, of the slow-roll region.', '1704.06994-2-65-1': 'We find, however, that there is essentially no overshoot if the bubble nucleates with [MATH] in a much wider range, [MATH].', '1704.06994-2-65-2': 'The reason for this discrepancy may be that both Refs. [CITATION] and [CITATION] assumed that [MATH].', '1704.06994-2-65-3': 'We find that, on the contrary, [MATH] is typically rather close to [MATH], [MATH].', '1704.06994-2-66-0': '## Probability of inflation', '1704.06994-2-67-0': 'We can now estimate the probability for a randomly chosen inflection point [MATH] in the landscape to be a site of slow-roll inflation.', '1704.06994-2-67-1': 'Inflation is possible only if the field tunnels out of the false vacuum to a point within the target range of [MATH].', '1704.06994-2-67-2': 'The size of the target range is [MATH], while the distribution of the nucleation points has the width [MATH].', '1704.06994-2-67-3': 'For [MATH], we have [MATH], and the probability for [MATH] to be in the target range is [EQUATION]', '1704.06994-2-67-4': 'For most of inflection points, the tunneling then occurs outside the target range, so the inflaton overshoots and inflation does not happen.', '1704.06994-2-67-5': 'On the other hand, for inflection points where [MATH] is in the target range, inflation typically occurs over the entire slow-roll region, so [MATH].', '1704.06994-2-67-6': 'Since [MATH] is uncorrelated with with [MATH] for [MATH], we expect that the probability distribution for [MATH] is (almost) the same as that for [MATH].', '1704.06994-2-67-7': 'The latter distribution has been calculated in Ref. [CITATION], [EQUATION]', '1704.06994-2-67-8': 'Using Eqs. ([REF]) and ([REF]), we can now find the probability that a given inflection point supports inflation with [MATH] e-folds, [EQUATION]', '1704.06994-2-67-9': 'With [MATH] and [MATH], this probability is rather small.', '1704.06994-2-67-10': 'In our view, however, this is not an argument against the random Gaussian landscape model.', '1704.06994-2-68-0': 'Observational predictions of the model are based on landscape realizations with a large number of e-folds, sufficient to solve the flatness problem and to allow for structure formation.', '1704.06994-2-68-1': 'All such realizations have [MATH] in the target range.', '1704.06994-2-68-2': 'Most of them have [MATH] and exhibit universal behavior.', '1704.06994-2-68-3': 'If the observable scales lie within the slow-roll range, the spectral index [MATH] is related to [MATH] by Eq. ([REF]) and its distribution is given by Eq. ([REF]).', '1704.06994-2-68-4': 'The observed value of [MATH] is in the mid-range of this distribution, as discussed in [CITATION].', '1704.06994-2-68-5': 'Hence a random [MATH] Gaussian landscape is consistent with observations.', '1704.06994-2-69-0': '## Spatial curvature', '1704.06994-2-70-0': 'Since the interior geometry of the bubble is an open FRW universe, the curvature parameter is nonzero and may have an observable effect [CITATION].', '1704.06994-2-70-1': 'At the present time this parameter is given by [EQUATION] where the subscripts "end" and "p" represent the values at the end of inflation and at present, respectively.', '1704.06994-2-70-2': 'Hereafter, we assume instantaneous reheating and GUT scale inflation to give reference values.', '1704.06994-2-70-3': 'Then we have [MATH].', '1704.06994-2-71-0': 'The Planck collaboration puts an upper bound on [MATH] at [MATH], which requires [EQUATION]', '1704.06994-2-71-1': 'On the other hand, a detection of spatial curvature is probably possible only if [MATH], or [EQUATION]', '1704.06994-2-71-2': 'Naively, we could use Eq. ([REF]) to calculate the probability for [MATH] to be within the range defined by these bounds.', '1704.06994-2-71-3': 'This would give [EQUATION]', '1704.06994-2-71-4': 'We should note, however, that [MATH] is correlated with [MATH], which is in turn related to the spectral index [MATH] by Eq. ([REF]).', '1704.06994-2-71-5': 'With the observed value of [MATH], we should have [MATH].', '1704.06994-2-71-6': 'Hence, in order to have observable curvature, [MATH] has to be significantly smaller than [MATH].', '1704.06994-2-71-7': 'This is possible only if the tunneling point [MATH] happens to be very close to the critical value [MATH].', '1704.06994-2-72-0': 'For [MATH], we can approximate Eq. ([REF]) as [EQUATION]', '1704.06994-2-72-1': 'The range of [MATH] corresponding to the number of e-folds [MATH] in the range of interest, [MATH], is then given by [EQUATION] where the derivatives are evaluated at [MATH].', '1704.06994-2-72-2': 'Using Eqs. ([REF]) and ([REF]) we find [MATH] and [EQUATION]', '1704.06994-2-72-3': 'We thus see that the probability for the curvature to be smaller than the present upper bound but above the future detection limit is rather small.', '1704.06994-2-73-0': '# Conclusions', '1704.06994-2-74-0': 'We used numerical simulations to study bubble nucleation by quantum tunneling in random Gaussian potentials.', '1704.06994-2-74-1': 'We were particularly interested in slow-roll inflation after the tunneling.', '1704.06994-2-74-2': 'For a potential with a correlation length [MATH], this typically occurs near a flat inflection point [MATH], characterized by [MATH].', '1704.06994-2-74-3': 'We sampled a large number of randomly generated potentials with flat inflection points and found that a substantial fraction of them (about a half) allow for tunneling from the false vacuum to the neighborhood of [MATH].', '1704.06994-2-74-4': 'For each tunneling we found the initial value [MATH] of the inflaton field in the bubble by solving the Euclidean field equation for the instanton.', '1704.06994-2-74-5': 'The resulting distribution is peaked at [MATH], where the inflection point is taken to be at [MATH] and the positive direction of [MATH] is taken to be towards the "true" vacuum.', '1704.06994-2-74-6': 'The width of the distribution is [MATH] and is much larger than the size of the region where slow-roll inflation is possible, [MATH].', '1704.06994-2-74-7': 'This indicates that most of the tunnelings take the field [MATH] outside of the slow-roll range.', '1704.06994-2-75-0': 'We developed a semi-analytic technique to study the evolution of [MATH] after tunneling.', '1704.06994-2-75-1': 'Our main conclusions can be stated as follows.', '1704.06994-2-75-2': 'If the bubble nucleates with [MATH] outside the slow-roll range, the field starts rolling fast (after a brief curvature-dominated period) and may overshoot part or all of the slow-roll region, or it may miss this region altogether.', '1704.06994-2-75-3': 'We find that if [MATH] is in the range [EQUATION] where [MATH], then the field slows down and undergoes a nearly maximal number of inflationary e-folds, [MATH].', '1704.06994-2-75-4': 'On either side of the range ([REF]), [MATH] drops towards zero within a distance [MATH].', '1704.06994-2-75-5': 'The probability distribution for [MATH] has the same power-law form as that for [MATH], [EQUATION]', '1704.06994-2-75-6': 'The distribution for the spectral index [MATH] is then given by Eq. ([REF]) and is consistent with the observed value.', '1704.06994-2-76-0': 'We also discussed the prospects for observational detection of nonzero spatial curvature [MATH], which is related to the number of e-folds [MATH].', '1704.06994-2-76-1': 'These prospects are not very good, because the observational lower bound on [MATH] is pretty close to the upper bound that would make observational detection possible.', '1704.06994-2-76-2': 'Using the distribution ([REF]), we found that the probability for a random observer in the multiverse to detect spatial curvature between these two bounds is [MATH].', '1704.06994-2-76-3': 'This estimate changes, however, if we take into account one more data point that is available to us: the measurement of the spectral index of density perturbations: [MATH].', '1704.06994-2-76-4': 'In a small-field Gaussian landscape, [MATH] is rigidly related to [MATH], and the observed value implies [MATH].', '1704.06994-2-76-5': 'On the other hand, the bound for future detectability requires that [MATH], which is significantly smaller than [MATH].', '1704.06994-2-76-6': 'This situation is possible only if the tunneling point [MATH] is very close to [MATH] (within a range [MATH], where [MATH] interpolates between [MATH] and 0).', '1704.06994-2-76-7': 'Our estimate for the probability of this to happen is [MATH].', '1704.06994-2-76-8': 'The bottom line is that the probability of detecting spatial curvature is pretty low if we live in a small-field random Gaussian landscape.', '1704.06994-2-77-0': 'We finally comment on some limitations of our analysis.', '1704.06994-2-77-1': 'The distribution for [MATH] may have some additional factors, due to the probability measure of the multiverse.', '1704.06994-2-77-2': 'These factors may depend on the tunneling transition rates between different vacua and on the correlation between the tunneling action and [MATH].', '1704.06994-2-77-3': 'This issue is entangled with the measure problem, which at present has no definitive solution and may require some new ideas to be resolved.', '1704.06994-2-78-0': 'Another serious limitation is that we restricted our analysis to a one-dimensional landscape.', '1704.06994-2-78-1': 'On the other hand, in multi-dimensional models parts of the landscape where slow-roll inflation is possible may be effectively one-dimensional, with only one of the fields having a nearly flat potential, due to an approximate shift symmetry.', '1704.06994-2-78-2': 'In lieu of detailed information about the landscape, it may then be a reasonable approximation to consider such effective [MATH] potentials as samples of a random Gaussian field.', '1704.06994-2-78-3': 'We note also that the methods we used here may have a wider applicability.', '1704.06994-2-78-4': 'In Ref. [CITATION] we indicated some directions in which these methods can be extended to multi-dimensional landscapes.'} | [['1704.06994-1-2-0', '1704.06994-2-2-0'], ['1704.06994-1-2-1', '1704.06994-2-2-1'], ['1704.06994-1-2-2', '1704.06994-2-2-2'], ['1704.06994-1-2-3', '1704.06994-2-2-3'], ['1704.06994-1-27-0', '1704.06994-2-27-0'], ['1704.06994-1-27-1', '1704.06994-2-27-1'], ['1704.06994-1-27-2', '1704.06994-2-27-2'], ['1704.06994-1-27-3', '1704.06994-2-27-3'], ['1704.06994-1-27-4', '1704.06994-2-27-4'], ['1704.06994-1-27-5', '1704.06994-2-27-5'], ['1704.06994-1-27-6', '1704.06994-2-27-6'], ['1704.06994-1-27-7', '1704.06994-2-27-7'], ['1704.06994-1-27-8', '1704.06994-2-27-8'], ['1704.06994-1-11-0', '1704.06994-2-11-0'], ['1704.06994-1-11-1', '1704.06994-2-11-1'], ['1704.06994-1-11-2', '1704.06994-2-11-2'], ['1704.06994-1-11-3', '1704.06994-2-11-3'], ['1704.06994-1-11-4', '1704.06994-2-11-4'], ['1704.06994-1-11-5', '1704.06994-2-11-5'], ['1704.06994-1-11-6', '1704.06994-2-11-6'], 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'1704.06994-2-24-1'], ['1704.06994-1-24-2', '1704.06994-2-24-2'], ['1704.06994-1-24-3', '1704.06994-2-24-3'], ['1704.06994-1-24-4', '1704.06994-2-24-4'], ['1704.06994-1-48-0', '1704.06994-2-48-0'], ['1704.06994-1-48-1', '1704.06994-2-48-1'], ['1704.06994-1-48-2', '1704.06994-2-48-2'], ['1704.06994-1-48-3', '1704.06994-2-48-3'], ['1704.06994-1-17-0', '1704.06994-2-17-0'], ['1704.06994-1-17-1', '1704.06994-2-17-1'], ['1704.06994-1-51-0', '1704.06994-2-51-0'], ['1704.06994-1-51-1', '1704.06994-2-51-1'], ['1704.06994-1-51-2', '1704.06994-2-51-2'], ['1704.06994-1-51-3', '1704.06994-2-51-3'], ['1704.06994-1-7-0', '1704.06994-2-7-0'], ['1704.06994-1-7-1', '1704.06994-2-7-1'], ['1704.06994-1-7-2', '1704.06994-2-7-2'], ['1704.06994-1-7-3', '1704.06994-2-7-3'], ['1704.06994-1-26-0', '1704.06994-2-26-0'], ['1704.06994-1-26-1', '1704.06994-2-26-1'], ['1704.06994-1-26-2', '1704.06994-2-26-2'], ['1704.06994-1-26-3', '1704.06994-2-26-3'], ['1704.06994-1-26-4', '1704.06994-2-26-4'], ['1704.06994-1-26-5', '1704.06994-2-26-5'], ['1704.06994-1-56-0', '1704.06994-2-56-0'], ['1704.06994-1-56-1', '1704.06994-2-56-1'], ['1704.06994-1-56-2', '1704.06994-2-56-2'], ['1704.06994-1-56-3', '1704.06994-2-56-3'], ['1704.06994-1-56-4', '1704.06994-2-56-4'], ['1704.06994-1-33-0', '1704.06994-2-33-0'], ['1704.06994-1-33-1', '1704.06994-2-33-1'], ['1704.06994-1-36-0', '1704.06994-2-36-0'], ['1704.06994-1-36-1', '1704.06994-2-36-1'], ['1704.06994-1-36-2', '1704.06994-2-36-2'], ['1704.06994-1-3-0', '1704.06994-2-3-0'], ['1704.06994-1-3-1', '1704.06994-2-3-1'], ['1704.06994-1-3-2', '1704.06994-2-3-2'], ['1704.06994-1-3-3', '1704.06994-2-3-3'], ['1704.06994-1-45-0', '1704.06994-2-45-0'], ['1704.06994-1-45-1', '1704.06994-2-45-1'], ['1704.06994-1-45-2', '1704.06994-2-45-2'], ['1704.06994-1-45-3', '1704.06994-2-45-3'], ['1704.06994-1-45-4', '1704.06994-2-45-4'], ['1704.06994-1-45-5', '1704.06994-2-45-5'], ['1704.06994-1-45-6', '1704.06994-2-45-6'], ['1704.06994-1-45-7', '1704.06994-2-45-7']] | [] | [] | [] | [] | ['1704.06994-1-71-2', '1704.06994-2-72-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1704.06994 | null | null | null | null | null |
1907.10475 | {'1907.10475-1-0-0': 'We use density-functional band-structure calculations to explore the origin of the up-up-down-down (UUDD) magnetic order in Cu[MATH]GeO[MATH] with the frustrated [MATH] spin chains coupled into layers within the spinel-like crystal structure.', '1907.10475-1-0-1': 'In contrast to earlier studies, we find that the nearest-neighbor coupling [MATH] may be negligibly small, owing to a nearly perfect compensation of the ferromagnetic direct exchange and antiferromagnetic superexchange.', '1907.10475-1-0-2': 'Under this condition, weak symmetric anisotropy of the exchange couplings gives rise to the UUDD order observed experimentally and also elucidates the non-trivial ordering pattern between the layers, whereas a small Dzyaloshinsky-Moriya interaction causes a spin canting that may generate local electric polarization.', '1907.10475-1-0-3': 'We argue that the buckling of the copper chains plays a crucial role in the suppression of [MATH] in Cu[MATH]GeO[MATH] and sets this compound apart from other [MATH] chain magnets.', '1907.10475-1-1-0': '# Introduction', '1907.10475-1-2-0': 'Copper oxides built by chains of edge-sharing CuO[MATH] plaquettes serve as material prototypes of frustrated spin-[MATH] chains with competing nearest-neighbor and next-nearest-neighbor interactions [MATH] and [MATH], respectively.', '1907.10475-1-2-1': 'This simple spin model received ample attention [CITATION] triggered by the prospects of chiral, multipolar, and spin-nematic phases that may occur therein [CITATION].', '1907.10475-1-2-2': 'Whereas long-range order does not take place in one dimension (1D), interchain couplings in real materials will usually cause three-dimensional (3D) collinear or non-collinear order depending on the [MATH] ratio.', '1907.10475-1-2-3': 'On the classical level, incommensurate spiral order appears for [MATH], whereas at [MATH] the second-neighbor coupling is not strong enough to tilt the spins, and the collinear ferromagnetic or up-down-up-down antiferromagnetic order form depending on the sign of [MATH].', '1907.10475-1-2-4': 'Quantum effects preserve the spiral state in the case of ferromagnetic (FM) [MATH] [CITATION], but destroy the order and open a spin gap for antiferromagnetic (AFM) [MATH] at [MATH] [CITATION].', '1907.10475-1-3-0': 'Real-world prototypes of the [MATH] spin chains will typically follow one of these scenarios.', '1907.10475-1-3-1': 'The majority of quasi-1D copper oxides reveal [MATH] with FM [MATH] and develop the incommensurate spiral order [CITATION].', '1907.10475-1-3-2': 'Li[MATH]CuO[MATH] [CITATION] and CuAs[MATH]O[MATH] [CITATION] are notable exceptions, where [MATH] is also FM, but [MATH] renders spin alignment along the chains purely ferromagnetic.', '1907.10475-1-3-3': 'Spin-chain compounds with AFM [MATH] are more rare, although tentative indications of the spin-gap formation at [MATH] have been reported [CITATION].', '1907.10475-1-4-0': 'One puzzling case in this series is Cu[MATH]GeO[MATH] [CITATION] that reveals an unanticipated antiferromagnetic up-up-down-down (UUDD) order [CITATION] despite the prediction of FM [MATH] and AFM [MATH], both of the same magnitude [CITATION].', '1907.10475-1-4-1': 'This parameter regime would normally lead to the incommensurate spiral order, similar to LiCuVO[MATH], CuCl[MATH], and other [MATH] cuprates.', '1907.10475-1-4-2': 'Here, we address this discrepancy and first analyze whether additional terms beyond [MATH] and [MATH] could destabilize the incommensurate order and give way to the UUDD state.', '1907.10475-1-4-3': 'This appears not to be the case, but instead [MATH] is unusually weak in Cu[MATH]GeO[MATH] and underlies the UUDD ground state of this compound.', '1907.10475-1-5-0': 'The remainder of this paper is organized as follows.', '1907.10475-1-5-1': 'In Sec. [REF], we review the crystal structure of Cu[MATH]GeO[MATH] and experimental information available for this material.', '1907.10475-1-5-2': 'Sec. [REF] covers methodological aspects.', '1907.10475-1-5-3': 'In Sec. [REF], we estimate both isotropic and anisotropic exchange interactions in Cu[MATH]GeO[MATH], and in Sec. [REF] analyze the ensuing magnetic ground state.', '1907.10475-1-5-4': 'Ferromagnetic direct exchange appears to be crucial and merits further analysis presented in Sec. [REF] followed by the analysis of experimental magnetic susceptibility in Sec. [REF] and a brief discussion and summary in Sec. [REF].', '1907.10475-1-6-0': '# Structure and properties of Cu[MATH]GeO[MATH]', '1907.10475-1-7-0': 'Cu[MATH]GeO[MATH] adopts a distorted spinel structure, where the Jahn-Teller effect inherent to Cu[MATH] transforms CuO[MATH] octahedra into CuO[MATH] plaquettes [CITATION].', '1907.10475-1-7-1': 'The backbone of the structure is then formed by infinite chains of edge-shared plaquettes linked into a 3D network via the non-magnetic GeO[MATH] tetrahedra (Fig. [REF]a).', '1907.10475-1-8-0': 'Magnetic susceptibility measurements revealed a broad maximum around 80K followed by an antiferromagnetic transition at [MATH]K [CITATION].', '1907.10475-1-8-1': 'This behavior is typical of low-dimensional and frustrated magnetism.', '1907.10475-1-8-2': 'In the case of Cu[MATH]GeO[MATH], strong magnetic interactions are expected in the [MATH] plane, both along the chains of the Cu atoms ([MATH]) and perpendicular to the chains ([MATH]), see Fig. [REF]b.', '1907.10475-1-8-3': 'The interactions [MATH] between the planes are at least one order of magnitude weaker and form triangular loops together with [MATH].', '1907.10475-1-8-4': 'This tentative magnetic model was confirmed by density-functional (DFT) band-structure calculations that yield [MATH]meV (FM) and [MATH]meV (AFM) as well as [MATH]meV.', '1907.10475-1-8-5': 'Even if the leading coupling [MATH] runs perpendicular to the copper chains, magnetic order along these chains is still determined by the competition between [MATH] and [MATH], similar to the 1D [MATH] model.', '1907.10475-1-8-6': 'Detailed numerical analysis confirmed the stability of the spiral order along the copper chains as well as the collinear spin arrangement perpendicular to the chains, where no significant frustration occurs [CITATION].', '1907.10475-1-9-0': 'Surprisingly, neutron diffraction data [CITATION] did not support this scenario and pinpointed the collinear UUDD order along the [MATH] chains (Fig. [REF]b).', '1907.10475-1-9-1': 'This spin configuration is uncommon for cuprates and has never been seen in the [MATH] compounds before.', '1907.10475-1-9-2': 'Biquadratic exchange was considered as the driving force of this unusual order [CITATION] and may explain it indeed [CITATION], but appears irrelevant to Cu[MATH]GeO[MATH], because biquadratic terms do not exist for spin-[MATH] (they can be re-written as standard bilinear terms in the Hamiltonian [CITATION], see Appendix [REF]).', '1907.10475-1-9-3': 'Additionally, dielectric measurements revealed a clear anomaly in the permittivity at [MATH], as well as a non-zero electric polarization that appears below [MATH] in this formally centrosymmetric ([MATH]) crystal structure [CITATION].', '1907.10475-1-9-4': 'In the absence of spiral order that is typically associated with the electric polarization in chain cuprates [CITATION], the origin of ferroelectricity in Cu[MATH]GeO[MATH] remains controversial [CITATION].', '1907.10475-1-10-0': 'Here, we seek to throw some light on this problem from the ab initio perspective.', '1907.10475-1-10-1': 'The conclusion of Ref. [CITATION] on the spiral order was based on the study of an isotropic spin Hamiltonian, so it is natural to suspect, following Ref. [CITATION], that non-Heisenberg terms stabilize the UUDD order.', '1907.10475-1-10-2': 'We calculate such terms but find them to be small and largely irrelevant.', '1907.10475-1-10-3': 'On the other hand, isotropic exchange couplings of Ref. [CITATION] have to be revised, eventually giving a clue to the formation of the UUDD order in Cu[MATH]GeO[MATH].', '1907.10475-1-11-0': '# Methods', '1907.10475-1-12-0': 'In Ref. [CITATION], the magnetic behavior of Cu[MATH]GeO[MATH] was analyzed on the level of the Heisenberg spin Hamiltonian, [EQUATION]', '1907.10475-1-12-1': 'With [MATH]meV and [MATH]meV [CITATION], it leads to the spiral order along the copper chains at odds with the experiment.', '1907.10475-1-12-2': 'To account for the experimental UUDD order, additional terms may be invoked as follows, [EQUATION] where [MATH] are Dzyaloshinskii-Moriya (DM) vectors and [MATH] are symmetric anisotropy tensors.', '1907.10475-1-12-3': 'The latter favor collinear spins and, therefore, may stabilize the UUDD order over the spiral one, whereas the former do not stabilize collinear spin configurations per se, but may act against the spiral state.', '1907.10475-1-12-4': 'Specifically, in Cu[MATH]GeO[MATH] the alternating directions of [MATH] (Fig. [REF]b) are incompatible with the continuous spin rotation in the spiral.', '1907.10475-1-12-5': 'Biquadratic and other higher-order corrections do not appear as independent terms in the spin-[MATH] Hamiltonian [CITATION], see also Appendix [REF].', '1907.10475-1-13-0': 'The parameters of Eq. [REF] are obtained from DFT calculations performed within the generalized gradient approximation (GGA) [CITATION] implemented in Vienna ab initio Simulation Package (VASP) [CITATION].', '1907.10475-1-13-1': 'Additionally, the pseudopotential Quantum Espresso [CITATION], as well as full-potential FPLO [CITATION] and ELK [CITATION] codes were used.', '1907.10475-1-13-2': 'The crystal structure given in Ref. [CITATION] was employed in all calculation.', '1907.10475-1-14-0': 'In the absence of electronic correlations, Cu[MATH]GeO[MATH] features a metallic band structure with several bands crossing the Fermi level.', '1907.10475-1-14-1': 'The complex of four bands between [MATH] and 0.6eV corresponds to four Cu atoms in the primitive cell and arises from [MATH] orbitals that are half-filled in Cu[MATH].', '1907.10475-1-14-2': 'Electronic correlations split these bands and open a gap.', '1907.10475-1-14-3': 'The effect of correlations is modeled on the DFT+[MATH]+SO level, with all parameters of the spin Hamiltonian, Eq. [REF], extracted from total energies of ordered spin configurations using the mapping procedure [CITATION].', '1907.10475-1-14-4': 'Alternatively, we perform a model analysis based on hopping parameters of the uncorrelated band structure and additionally calculate ferromagnetic contribution to the exchange from the overlap of Wannier functions, as further explained in Sec. [REF].', '1907.10475-1-15-0': "The DFT+[MATH]+SO method relies on the empirical parametrization of the Coulomb and Hund's exchange interactions [MATH] and [MATH], respectively.", '1907.10475-1-15-1': 'These parameters were obtained via the linear-response approach [CITATION] that yields [MATH]eV.', '1907.10475-1-15-2': 'Assuming [MATH]eV, we find [MATH]eV, which is similar to the parametrization that is typically used for copper oxides [CITATION] in conjunction with the double-counting correction in the fully localized limit (FLL) that we applied throughout this work too.', '1907.10475-1-16-0': 'Magnetic ground state of the spin Hamiltonian is obtained from the Luttinger-Tisza (LT) method considering spins as classical moments [CITATION], [EQUATION] where [MATH] is the Fourier transform of the spin: [EQUATION]', '1907.10475-1-16-1': 'Diagonalization of Eq. [REF] yields [CITATION] [EQUATION] where [MATH], [MATH], and [MATH] are corresponding eigenvalues and eigenvectors of [MATH].', '1907.10475-1-16-2': 'The LT mode [MATH] with the most negative eigenvalue [MATH] is considered as an "optimal" mode with the wave vector [MATH].', '1907.10475-1-16-3': 'If the constructed spin state [MATH] is the linear combination of the optimal LT modes and complies with the "strong constraint" of [MATH], it can be considered as a ground state [CITATION].', '1907.10475-1-17-0': 'We also calculate magnetic susceptibility of Cu[MATH]GeO[MATH] using the loop algorithm [CITATION] of the ALPS simulation package [CITATION].', '1907.10475-1-17-1': 'To this end, finite lattices with up to [MATH] sites and periodic boundary conditions were used.', '1907.10475-1-18-0': '# Results', '1907.10475-1-19-0': '## Microscopic magnetic model', '1907.10475-1-20-0': 'Isotropic exchange couplings of the Heisenberg spin Hamiltonian, Eq. [REF], are listed in Table [REF].', '1907.10475-1-20-1': 'DFT calculations were performed in four different codes that delivered largely consistent results for [MATH] but not for [MATH] that varies between [MATH]meV in VASP and [MATH]meV in FPLO, with QE and ELK returning intermediate values.', '1907.10475-1-20-2': 'The variation of [MATH] is somewhat similar in magnitude but clearly less acute, because this coupling is not frustrated and simply leads to the antiparallel spin alignment perpendicular to the copper chains, no matter how strong the coupling is.', '1907.10475-1-20-3': 'On the other hand, the large spread of [MATH] implies that the competition between [MATH] and [MATH] is either strong (FPLO) or nearly non-existent (VASP and QE).', '1907.10475-1-21-0': 'This ambiguity may be partly related to the fact that different DFT codes use different basis sets, wherein [MATH] and [MATH] of DFT+[MATH] entail different correlation strength, thus affecting the magnetic couplings.', '1907.10475-1-21-1': 'However, even an intentional variation of [MATH] within FPLO did not bring [MATH] toward the VASP result, and neither the VASP result for [MATH] approached the one from FPLO when [MATH] was varied within the reasonable range of [MATH]eV.', '1907.10475-1-21-2': 'Therefore, the ab initio determination of [MATH] is rather ambiguous, reflecting the general problem [CITATION] of calculating short-range exchange interactions within DFT+[MATH].', '1907.10475-1-21-3': 'This problem is rooted in the subtle interplay between the kinetic (superexchange) and potential (direct exchange) contributions that can be equal in magnitude and have to be evaluated with high accuracy.', '1907.10475-1-21-4': 'We return to this problem in Sec. [REF] but first consider anisotropic, non-Heisenberg terms that may also affect the ground state.', '1907.10475-1-21-5': 'These terms are obtained in VASP, because it delivers the most realistic estimate of [MATH], as we show below.'} | {'1907.10475-2-0-0': 'We use density-functional band-structure calculations to explore the origin of the up-up-down-down (UUDD) magnetic order in Cu[MATH]GeO[MATH] with the frustrated [MATH] spin chains coupled into layers within the spinel-like crystal structure.', '1907.10475-2-0-1': 'In contrast to earlier studies, we find that the nearest-neighbor coupling [MATH] should be negligibly small, owing to a nearly perfect compensation of the ferromagnetic direct exchange and antiferromagnetic superexchange.', '1907.10475-2-0-2': 'Under this condition, weak symmetric anisotropy of the exchange couplings gives rise to the UUDD order observed experimentally and also elucidates the non-trivial ordering pattern between the layers, whereas a small Dzyaloshinsky-Moriya interaction causes a spin canting that may generate local electric polarization.', '1907.10475-2-0-3': 'We argue that the buckling of the copper chains plays a crucial role in the suppression of [MATH] in Cu[MATH]GeO[MATH] and sets this compound apart from other [MATH] chain magnets.', '1907.10475-2-1-0': '# Introduction', '1907.10475-2-2-0': 'Copper oxides built by chains of edge-sharing CuO[MATH] plaquettes serve as material prototypes of frustrated spin-[MATH] chains with competing nearest-neighbor and next-nearest-neighbor interactions [MATH] and [MATH], respectively.', '1907.10475-2-2-1': 'This simple spin model received ample attention [CITATION] triggered by the prospects of chiral, multipolar, and spin-nematic phases that may occur therein [CITATION].', '1907.10475-2-2-2': 'Whereas long-range order does not take place in one dimension (1D), interchain couplings in real materials will usually cause three-dimensional (3D) collinear or non-collinear order depending on the [MATH] ratio.', '1907.10475-2-2-3': 'On the classical level, incommensurate spiral order appears for [MATH], whereas at [MATH] the second-neighbor coupling is not strong enough to tilt the spins, and the collinear ferromagnetic or up-down-up-down antiferromagnetic order form depending on the sign of [MATH].', '1907.10475-2-2-4': 'Quantum effects preserve the spiral state in the case of ferromagnetic (FM) [MATH] [CITATION], but destroy the order and open a spin gap for antiferromagnetic (AFM) [MATH] at [MATH] [CITATION].', '1907.10475-2-3-0': 'Real-world prototypes of the [MATH] spin chains will typically follow one of these scenarios.', '1907.10475-2-3-1': 'The majority of quasi-1D copper oxides develop incommensurate spiral order [CITATION].', '1907.10475-2-3-2': 'Li[MATH]CuO[MATH] [CITATION], Ca[MATH]Y[MATH]Cu[MATH]O[MATH], [CITATION], and CuAs[MATH]O[MATH] [CITATION] are notable exceptions, where [MATH] is also FM, but spin alignment along the chains is purely ferromagnetic, owing to a smaller [MATH].', '1907.10475-2-3-3': 'Spin-chain compounds with AFM [MATH] are more rare, although tentative indications of the spin-gap formation at [MATH] have been reported [CITATION].', '1907.10475-2-4-0': 'One puzzling case in this series is Cu[MATH]GeO[MATH] [CITATION] that reveals an unanticipated antiferromagnetic up-up-down-down (UUDD) order [CITATION] despite the prediction of FM [MATH] and AFM [MATH], both of the same magnitude [CITATION].', '1907.10475-2-4-1': 'This parameter regime would normally lead to the incommensurate spiral order, similar to LiCuVO[MATH], CuCl[MATH], and other [MATH] cuprates.', '1907.10475-2-4-2': 'Here, we address this discrepancy and first analyze whether additional terms beyond [MATH] and [MATH] could destabilize the incommensurate order and give way to the UUDD state.', '1907.10475-2-4-3': 'This appears not to be the case, but instead [MATH] is unusually weak in Cu[MATH]GeO[MATH] and underlies the UUDD ground state of this compound.', '1907.10475-2-5-0': 'The remainder of this paper is organized as follows.', '1907.10475-2-5-1': 'In Sec. [REF], we review the crystal structure of Cu[MATH]GeO[MATH] and experimental information available for this material.', '1907.10475-2-5-2': 'Sec. [REF] covers methodological aspects.', '1907.10475-2-5-3': 'In Sec. [REF], we estimate both isotropic and anisotropic exchange interactions in Cu[MATH]GeO[MATH], and in Sec. [REF] analyze the ensuing magnetic ground state.', '1907.10475-2-5-4': 'Ferromagnetic direct exchange appears to be crucial and merits further analysis presented in Sec. [REF] followed by the analysis of experimental magnetic susceptibility in Sec. [REF] and a brief discussion and summary in Sec. [REF].', '1907.10475-2-6-0': '# Structure and properties of Cu[MATH]GeO[MATH]', '1907.10475-2-7-0': 'Cu[MATH]GeO[MATH] adopts a distorted spinel structure, where the Jahn-Teller effect inherent to Cu[MATH] transforms CuO[MATH] octahedra into CuO[MATH] plaquettes [CITATION].', '1907.10475-2-7-1': 'The backbone of the structure is then formed by infinite chains of edge-shared plaquettes linked into a 3D network via the non-magnetic GeO[MATH] tetrahedra (Fig. [REF]a).', '1907.10475-2-8-0': 'Magnetic susceptibility measurements revealed a broad maximum around 80K followed by an antiferromagnetic transition at [MATH]K [CITATION].', '1907.10475-2-8-1': 'This behavior is typical of low-dimensional and frustrated magnetism.', '1907.10475-2-8-2': 'In the case of Cu[MATH]GeO[MATH], strong magnetic interactions are expected in the [MATH] plane, both along the chains of the Cu atoms ([MATH]) and perpendicular to the chains ([MATH]), see Fig. [REF]b.', '1907.10475-2-8-3': 'The interactions [MATH] between the planes are at least one order of magnitude weaker and form triangular loops together with [MATH].', '1907.10475-2-8-4': 'This tentative magnetic model was confirmed by density-functional (DFT) band-structure calculations that yield [MATH]meV (FM) and [MATH]meV (AFM) as well as [MATH]meV.', '1907.10475-2-8-5': 'Even if the leading coupling [MATH] runs perpendicular to the copper chains, magnetic order along these chains is still determined by the competition between [MATH] and [MATH], similar to the 1D [MATH] model.', '1907.10475-2-8-6': 'Detailed numerical analysis confirmed the stability of the spiral order along the copper chains as well as the collinear spin arrangement perpendicular to the chains, where no significant frustration occurs [CITATION].', '1907.10475-2-9-0': 'Surprisingly, neutron diffraction data [CITATION] did not support this scenario and pinpointed the collinear UUDD order along the [MATH] chains (Fig. [REF]b).', '1907.10475-2-9-1': 'This spin configuration is uncommon for cuprates and has never been seen in the [MATH] compounds before.', '1907.10475-2-9-2': 'Biquadratic exchange was considered as the driving force of this unusual order [CITATION] and may explain it indeed [CITATION], but appears irrelevant to Cu[MATH]GeO[MATH], because biquadratic terms do not exist for spin-[MATH] (they can be re-written as standard bilinear terms in the Hamiltonian [CITATION], see Appendix [REF]).', '1907.10475-2-9-3': 'Additionally, dielectric measurements revealed a clear anomaly in the permittivity at [MATH], as well as a non-zero electric polarization that appears below [MATH] in this formally centrosymmetric ([MATH]) crystal structure [CITATION].', '1907.10475-2-9-4': 'In the absence of spiral magnetic order that is typically associated with the electric polarization in chain cuprates [CITATION], the origin of ferroelectricity in Cu[MATH]GeO[MATH] remains controversial [CITATION].', '1907.10475-2-10-0': 'Here, we seek to throw some light on this problem from the ab initio perspective.', '1907.10475-2-10-1': 'The conclusion of Ref. [CITATION] on the spiral order was based on the parametrization of an isotropic spin Hamiltonian, so it is natural to suspect, following Ref. [CITATION], that non-Heisenberg terms act against the spiral order and stabilize the UUDD one.', '1907.10475-2-10-2': 'We calculate such terms but find them to be small and affecting spin directions in the ordered state but not the nature of the ordered state itself.', '1907.10475-2-10-3': 'On the other hand, isotropic exchange couplings of Ref. [CITATION] have to be revised, eventually giving a clue to the formation of the UUDD order in Cu[MATH]GeO[MATH].', '1907.10475-2-11-0': '# Methods', '1907.10475-2-12-0': 'In Ref. [CITATION], the magnetic behavior of Cu[MATH]GeO[MATH] was analyzed on the level of the Heisenberg spin Hamiltonian, [EQUATION]', '1907.10475-2-12-1': 'With [MATH]meV and [MATH]meV [CITATION], it leads to the spiral order along the copper chains at odds with the experiment.', '1907.10475-2-12-2': 'To account for the experimental UUDD order, additional terms may be invoked as follows, [EQUATION] where [MATH] are Dzyaloshinskii-Moriya (DM) vectors and [MATH] are symmetric anisotropy tensors.', '1907.10475-2-12-3': 'The latter favor collinear spins and, therefore, may stabilize the UUDD order over the spiral one, whereas the former do not stabilize collinear spin configurations per se, but may act against the spiral state.', '1907.10475-2-12-4': 'Specifically, in Cu[MATH]GeO[MATH] the alternating directions of [MATH] (Fig. [REF]b) are incompatible with the continuous spin rotation in the spiral.', '1907.10475-2-12-5': 'Biquadratic and other higher-order corrections do not appear as independent terms in the spin-[MATH] Hamiltonian [CITATION], see also Appendix [REF].', '1907.10475-2-13-0': 'Magnetic exchange parameters are obtained from DFT calculations performed within the generalized gradient approximation (GGA) [CITATION] implemented in Vienna ab initio Simulation Package (VASP) [CITATION].', '1907.10475-2-13-1': 'Additionally, the full-potential FPLO [CITATION] and ELK [CITATION] codes were used.', '1907.10475-2-13-2': 'The crystal structure given in Ref. [CITATION] was employed in all calculations, similar to Ref. [CITATION].', '1907.10475-2-14-0': 'In the absence of electronic correlations, Cu[MATH]GeO[MATH] features a metallic band structure with several bands crossing the Fermi level.', '1907.10475-2-14-1': 'The complex of four bands between [MATH] and 0.6eV corresponds to four Cu atoms in the primitive cell and arises from [MATH] orbitals that are half-filled in Cu[MATH].', '1907.10475-2-14-2': 'Electronic correlations split these bands and open a gap, in agreement with the insulating behavior of Cu[MATH]GeO[MATH] expected from the green sample color [CITATION].', '1907.10475-2-14-3': 'The effect of correlations is modeled on the DFT+[MATH]+SO level, with all parameters of the spin Hamiltonian, Eq. [REF], extracted from total energies of ordered spin configurations using the mapping procedure [CITATION].', '1907.10475-2-14-4': 'Alternatively, we perform a model analysis based on hopping parameters of the uncorrelated band structure and additionally calculate ferromagnetic contribution to the exchange from the overlap of Wannier functions, as further explained in Sec. [REF].', '1907.10475-2-14-5': 'Similar methodology has been used in the previous DFT study [CITATION], but several technical details were different and proved to be crucial, as we show below.', '1907.10475-2-15-0': "The DFT+[MATH]+SO method relies on the parametrization of the Coulomb and Hund's exchange interactions [MATH] and [MATH], respectively.In Ref. [CITATION], [MATH]eV and [MATH]eV were chosen empirically along with the around-mean-field (AMF) double-counting correction scheme that is more suitable for correlated metals.", '1907.10475-2-15-1': 'Here, we use the double-counting correction in the fully localized limit appropriate for insulators and obtain [MATH]eV from the linear-response method [CITATION].', '1907.10475-2-15-2': 'Assuming [MATH]eV, we find [MATH]eV, which is similar to the parametrization that is typically used for copper oxides [CITATION] in conjunction with the FLL flavor of the double-counting correction.', '1907.10475-2-16-0': 'Magnetic ground state of the spin Hamiltonian is obtained from the Luttinger-Tisza (LT) method considering spins as classical moments [CITATION], [EQUATION] where [MATH] is the Fourier transform of the spin: [EQUATION]', '1907.10475-2-16-1': 'Diagonalization of Eq. [REF] yields [CITATION] [EQUATION] where [MATH], [MATH], and [MATH] are corresponding eigenvalues and eigenvectors of [MATH].', '1907.10475-2-16-2': 'The LT mode [MATH] with the most negative eigenvalue [MATH] is considered as an "optimal" mode with the wave vector [MATH].', '1907.10475-2-16-3': 'If the constructed spin state [MATH] is the linear combination of the optimal LT modes and complies with the "strong constraint" of [MATH], it can be considered as a ground state [CITATION].', '1907.10475-2-17-0': 'We also calculate magnetic susceptibility of Cu[MATH]GeO[MATH] using the loop algorithm [CITATION] of the ALPS simulation package [CITATION].', '1907.10475-2-17-1': 'To this end, finite lattices with up to [MATH] sites and periodic boundary conditions were used.', '1907.10475-2-18-0': '# Results', '1907.10475-2-19-0': '## Microscopic magnetic model', '1907.10475-2-20-0': 'Isotropic exchange couplings of the Heisenberg spin Hamiltonian, Eq. [REF], are listed in Table [REF].', '1907.10475-2-20-1': 'DFT calculations were performed in three different codes that delivered largely consistent results for [MATH] but not for [MATH] that varies between [MATH]meV in VASP and [MATH]meV in FPLO, with ELK returning an intermediate value.', '1907.10475-2-20-2': 'This large spread of [MATH] leads to a highly ambiguous physical picture, because the competition between [MATH] and [MATH] may be either strong (FPLO) or nearly non-existent (VASP).', '1907.10475-2-21-0': 'Other exchange couplings, including [MATH] and [MATH], are largely consistent between the different band-structure codes.', '1907.10475-2-21-1': 'This alone indicates that the ambiguity of [MATH] does not stem from numerical inaccuracies, but reflects a complex nature of the coupling, which is short-range and combines dissimilar contributions of the direct exchange and superexchange, as opposed to the long-range couplings [MATH] and [MATH] dominated by the superexchange.', '1907.10475-2-21-2': 'To further exclude any technical issues related to the basis sets or energy convergence, we performed spin-polarized DFT calculations with [MATH] and arrived at [MATH] 40meV in all three codes (Table [REF]), thus confirming that the ambiguity of [MATH] arises not from the different basis sets and not from the different treatment of the crystal potential, but from the way the DFT+[MATH] correction is applied in each code.', '1907.10475-2-21-3': 'We also note that a variation of [MATH] within the reasonable range of [MATH]eV does not improve the consistency between VASP and FPLO.'} | [['1907.10475-1-10-0', '1907.10475-2-10-0'], ['1907.10475-1-10-3', '1907.10475-2-10-3'], ['1907.10475-1-7-0', '1907.10475-2-7-0'], ['1907.10475-1-7-1', '1907.10475-2-7-1'], ['1907.10475-1-12-0', '1907.10475-2-12-0'], ['1907.10475-1-12-1', '1907.10475-2-12-1'], ['1907.10475-1-12-2', '1907.10475-2-12-2'], ['1907.10475-1-12-3', '1907.10475-2-12-3'], ['1907.10475-1-12-4', 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'1907.10475-2-16-0'], ['1907.10475-1-16-1', '1907.10475-2-16-1'], ['1907.10475-1-16-2', '1907.10475-2-16-2'], ['1907.10475-1-16-3', '1907.10475-2-16-3'], ['1907.10475-1-5-0', '1907.10475-2-5-0'], ['1907.10475-1-5-1', '1907.10475-2-5-1'], ['1907.10475-1-5-2', '1907.10475-2-5-2'], ['1907.10475-1-5-3', '1907.10475-2-5-3'], ['1907.10475-1-5-4', '1907.10475-2-5-4'], ['1907.10475-1-0-0', '1907.10475-2-0-0'], ['1907.10475-1-0-2', '1907.10475-2-0-2'], ['1907.10475-1-0-3', '1907.10475-2-0-3'], ['1907.10475-1-20-0', '1907.10475-2-20-0'], ['1907.10475-1-4-0', '1907.10475-2-4-0'], ['1907.10475-1-4-1', '1907.10475-2-4-1'], ['1907.10475-1-4-2', '1907.10475-2-4-2'], ['1907.10475-1-4-3', '1907.10475-2-4-3'], ['1907.10475-1-8-0', '1907.10475-2-8-0'], ['1907.10475-1-8-1', '1907.10475-2-8-1'], ['1907.10475-1-8-2', '1907.10475-2-8-2'], ['1907.10475-1-8-3', '1907.10475-2-8-3'], ['1907.10475-1-8-4', '1907.10475-2-8-4'], ['1907.10475-1-8-5', '1907.10475-2-8-5'], ['1907.10475-1-8-6', '1907.10475-2-8-6'], ['1907.10475-1-9-0', '1907.10475-2-9-0'], ['1907.10475-1-9-1', '1907.10475-2-9-1'], ['1907.10475-1-9-2', '1907.10475-2-9-2'], ['1907.10475-1-9-3', '1907.10475-2-9-3'], ['1907.10475-1-17-0', '1907.10475-2-17-0'], ['1907.10475-1-17-1', '1907.10475-2-17-1'], ['1907.10475-1-2-0', '1907.10475-2-2-0'], ['1907.10475-1-2-1', '1907.10475-2-2-1'], ['1907.10475-1-2-2', '1907.10475-2-2-2'], ['1907.10475-1-2-3', '1907.10475-2-2-3'], ['1907.10475-1-2-4', '1907.10475-2-2-4'], ['1907.10475-1-14-0', '1907.10475-2-14-0'], ['1907.10475-1-14-1', '1907.10475-2-14-1'], ['1907.10475-1-14-3', '1907.10475-2-14-3'], ['1907.10475-1-14-4', '1907.10475-2-14-4']] | [['1907.10475-1-10-1', '1907.10475-2-10-1'], ['1907.10475-1-3-1', '1907.10475-2-3-1'], ['1907.10475-1-0-1', '1907.10475-2-0-1'], ['1907.10475-1-20-1', '1907.10475-2-20-1'], ['1907.10475-1-9-4', '1907.10475-2-9-4'], ['1907.10475-1-13-0', '1907.10475-2-13-0']] | [] | [['1907.10475-1-10-2', '1907.10475-2-10-2'], ['1907.10475-1-3-2', '1907.10475-2-3-2'], ['1907.10475-1-15-0', '1907.10475-2-15-0'], ['1907.10475-1-15-2', '1907.10475-2-15-2'], ['1907.10475-1-20-3', '1907.10475-2-20-2'], ['1907.10475-1-13-1', '1907.10475-2-13-1'], ['1907.10475-1-13-2', '1907.10475-2-13-2'], ['1907.10475-1-14-2', '1907.10475-2-14-2']] | [] | [] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1907.10475 | null | null | null | null | null |
cond-mat-0610066 | {'cond-mat-0610066-1-0-0': 'The structural and electronic properties of the wurtzite phase of the InAs and GaAs compounds are, for the first time, studied within an all-electron approach based on Density Functional Theory (DFT).', 'cond-mat-0610066-1-0-1': 'We used the full-potential linearized augmented plane wave (LAPW) method and the local density approximation (LDA) for the exchange-correlation potential.', 'cond-mat-0610066-1-1-0': 'From the structural optimization of the wurtzite polymorph of the InAs we found that the [MATH] ratio is somewhat greater than the ideal one and that the internal parameter [MATH] has a value slightly smaller than the ideal one.', 'cond-mat-0610066-1-1-1': 'The wurtzite polymorph has a smaller equilibrium volume per InAs pair and a higher binding energy when compared to the zinc-blende phase.', 'cond-mat-0610066-1-1-2': 'The latter property is at odds with the well-known stability of the zinc-blende phase.', 'cond-mat-0610066-1-1-3': 'The influence of the relativistic effects on the band structure is investigated.', 'cond-mat-0610066-1-1-4': 'The bands are calculated within a non-relativistic and within a scalar-relativistic approach including or not including spin-orbit interaction.', 'cond-mat-0610066-1-1-5': 'We find that the energy gaps of both polymorphs are positive when obtained from a non-relativistic calculation and negative otherwise.', 'cond-mat-0610066-1-2-0': 'The electronic properties of GaAs in the wurtzite phase (at the experimental lattice constants) are also investigated within the scalar relativistic approximation with and without spin-orbit interaction.', 'cond-mat-0610066-1-2-1': 'For both semiconductors, we determine the spin-orbit splittings for the zinc-blende and the wurtzite phases as well as the crystal-field splittings for the new wurtzite polymorphs.', 'cond-mat-0610066-1-3-0': '# Introduction', 'cond-mat-0610066-1-4-0': 'Nanowires (NW:s) are attracting an increasing amount of attention in the scientific community because they offer the possibility of investigating fundamental physical properties at the nano scale and for the numerous possible applications in electronic and photonic devices.', 'cond-mat-0610066-1-4-1': 'In particular, InAs-based nanowires have already been used to improve electronics in 1D as, for instance, in the case of resonant tunneling diodes [CITATION] and single-electron transistors [CITATION].', 'cond-mat-0610066-1-4-2': 'These NW:s, typically grown on an (111)B InAs substrate via Chemical Beam Epitaxy (CBE), have a wurtzite crystal structure in contrast to the stable phase at normal pressure and temperature which is the zinc-blende structure ([MATH]) with the space group [MATH]).', 'cond-mat-0610066-1-4-3': 'Despite the successes achieved so far with InAs-based NW:s, it has recently been realized that their wurtzite structure is an important factor in the interpretation of experimental data obtained from measurements of photo-luminescence (PL) [CITATION], photo-current (PC) [CITATION] or other electronic properties [CITATION].', 'cond-mat-0610066-1-5-0': 'A similar situation occurs for GaAs NW:s: when the (111)B GaAs surface is chosen as substrate, the resulting NW:s are constituted of alternating wurtzite and zinc-blend segments [CITATION].', 'cond-mat-0610066-1-5-1': 'Hence, both segments contribute to the optical and electronic properties of such a NW.', 'cond-mat-0610066-1-6-0': 'Theoretical studies based on total-energy calculations [CITATION] predict that the wurtzite polymorph ([MATH]), having space group [MATH]), is a metastable high-pressure modification of both the InAs and GaAs compounds.', 'cond-mat-0610066-1-6-1': 'Hence, we are faced with new materials, InAs and GaAs in the wurtzite phase, for which no previous experimental study exists and only little theoretical work has been done.', 'cond-mat-0610066-1-7-0': 'In this article we investigate the structural and electronic properties of these compounds by using first principles one-electron calculations.', 'cond-mat-0610066-1-7-1': 'Indeed, this approach does not require a knowledge of material parameters and enable us to compute numerous properties ranging from the equilibrium lattice constant to the band structure of the system.', 'cond-mat-0610066-1-7-2': 'Density Functional Theory (DFT) [CITATION] in the Local Density Approximation (LDA) [CITATION] provides an accurate description of the ground state electronic structure of solids and such calculations are adequate tools for structural studies.', 'cond-mat-0610066-1-7-3': 'The comparison of the calculated band structure to experimental excitation energies is strictly speaking not allowed but, nevertheless, provides important informations on the electronic properties of the material.', 'cond-mat-0610066-1-7-4': 'It is, for instance, well known that the experimental band gaps are usually underestimated by DFT.', 'cond-mat-0610066-1-7-5': 'The excited states properties of a many-electron system, such as the band structure, are better described within Many-Body perturbation theory as, for instance, by the GW approximation [CITATION] to the quasiparticle self-energy [MATH].', 'cond-mat-0610066-1-7-6': 'Still, a DFT calculation is a necessary step toward the GW calculation, since the LDA eigenenergies and eigenfunctions are usually taken to be the zero:th order approximation for the perturbative expansion.', 'cond-mat-0610066-1-7-7': 'In particular, the LDA provides an accurate Hartree potential which always must be treated to infinite order in an extended system.', 'cond-mat-0610066-1-8-0': 'Related studies reported in the literature are performed within DFT, use pseudopotentials [CITATION], and include the In 4d and the Ga 3d electrons among the core states.', 'cond-mat-0610066-1-8-1': 'These approximations lead to an underestimation of the equilibrium lattice parameters and to a less accurate description of the band-structure.', 'cond-mat-0610066-1-8-2': 'Here, we focus on the study of InAs and GaAs in the wurtzite phase at the DFT/LDA level with a full potential description of the atomic species.', 'cond-mat-0610066-1-8-3': 'We also refer the reader to another paper in which we report on a quasiparticle calculation of the band structure of these compounds [CITATION].', 'cond-mat-0610066-1-9-0': 'The article is organized as follows.', 'cond-mat-0610066-1-9-1': 'After the description of the theoretical background underlying the calculations, we present the results concerning the structure optimization of both the zinc-blende and the wurtzite phases of InAs.', 'cond-mat-0610066-1-9-2': 'Then, we discuss the band structure obtained in the non-relativistic and in the scalar relativistic approximations, the latter with and without the inclusion of the spin-orbit (S-O) interaction.', 'cond-mat-0610066-1-9-3': 'In the GaAs case, instead, we use the experimental lattice constants taken from TEM measurements on GaAs NW:s [CITATION] to allow for a direct comparison with experiment.', 'cond-mat-0610066-1-9-4': 'Indeed, since the deformation potential for the GaAs gap is very large [CITATION], small changes in the lattice constant have a strong influence on the band gap.', 'cond-mat-0610066-1-9-5': 'Finally, we present the band structures for the GaAs compound calculated in the scalar relativistic approximation with and without the spin-orbit interaction.', 'cond-mat-0610066-1-10-0': '# Methods of investigation', 'cond-mat-0610066-1-11-0': 'The electronic structure calculations presented here are performed within DFT using the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method [CITATION] as implemented in the WIEN2k software package [CITATION].', 'cond-mat-0610066-1-11-1': 'For the exchange-correlation energy in the Local Spin Density Approximation (LSDA) [CITATION] we have chosen the electron gas data of Ceperley and Alder [CITATION] as parametrized by Perdew and Wang [CITATION].', 'cond-mat-0610066-1-12-0': 'The APW method [CITATION] is a procedure to solve the Kohn-Sham equation based on the division of the unit cell into two spatial regions: non-overlapping muffin-tin spheres centered at the atomic sites of radius [MATH] and a remaining interstitial region.', 'cond-mat-0610066-1-12-1': 'The basis set used consist of atomic partial waves and plane waves in the two regions, respectively.', 'cond-mat-0610066-1-12-2': 'By denoting [MATH] the cut-off wave number for the plane waves, the number of basis functions is determined by the product [MATH], where [MATH] is the smallest muffin-tin radius in the unit cell.', 'cond-mat-0610066-1-12-3': 'The angular-momentum cut-off of the expansion of the wave functions in spherical harmonics within the atomic spheres is called [MATH].', 'cond-mat-0610066-1-13-0': 'Starting from the original APW method, several improved versions of the method were developed as, for instance, the Linearized APW method (LAPW) [CITATION], and, with the addition of local orbitals to the basis functions, the LAPW+LO [CITATION], and the APW+lo [CITATION] methods.', 'cond-mat-0610066-1-13-1': 'Within the WIEN2k code it is possible to use a mixed LAPW+LO and APW+lo basis [CITATION] to get the advantage of both schemes.', 'cond-mat-0610066-1-13-2': 'For the expansion inside the atomic spheres we have chosen the APW+lo basis for [MATH], the LAPW+LO for higher [MATH]:s and we have used [MATH] as the highest [MATH] value.', 'cond-mat-0610066-1-14-0': 'To ensure that no charge leaks outside the atomic spheres we have chosen [MATH] Ry for InAs and [MATH] Ry for GaAs as the energy which separates the core and the valence states.', 'cond-mat-0610066-1-14-1': 'In the InAs case, this choice guarantees that the In [MATH], As [MATH] and all higher-energy states are treated as band states, i.e., 36 valence electrons per InAs pair.', 'cond-mat-0610066-1-14-2': 'In the GaAs, instead, the valence band states start from the Ga [MATH] and the As [MATH], leading to 28 valence electrons per GaAs pair.', 'cond-mat-0610066-1-15-0': 'For both InAs and GaAs polymorphs we have performed convergence studies to determine the optimum value of [MATH] and the number of [MATH]-points in the whole Brillouin zone (BZ).', 'cond-mat-0610066-1-15-1': 'We have found that for [MATH] the total energy is converged to within less than 10 meV for both phases.', 'cond-mat-0610066-1-15-2': 'The number of [MATH]-points which ensure convergence is 1000 ([MATH]) in the zb and 405 ([MATH]) in the wz cases, respectively.', 'cond-mat-0610066-1-15-3': 'The chosen number of [MATH]-points corresponds to 73 [MATH]-points in the irreducible wedge of the Brillouin zone (IBZ) for the zb and to 95 [MATH]-points for the wz phase.', 'cond-mat-0610066-1-15-4': 'These [MATH]-meshes are used in a modified tetrahedron integration scheme [CITATION] to determine which states are occupied.', 'cond-mat-0610066-1-15-5': 'The Fourier expansion of the electron density in the interstitial region (i.e. valence charge density) is cut-off at [MATH], leading to 913 and 6524 plane waves for the [MATH] and the [MATH] phases, respectively.', 'cond-mat-0610066-1-16-0': '# Structure Optimization', 'cond-mat-0610066-1-17-0': 'The parameters obtained in the convergence study were used to carry out the structure optimization of both InAs polymorphs in the way described below.', 'cond-mat-0610066-1-18-0': 'Due to the cubic symmetry, the equilibrium lattice constant [MATH] of the zinc-blende compound is determined by the minimization of the total energy of the system with respect to the volume of the unit cell.', 'cond-mat-0610066-1-18-1': 'The energy vs volume curve is then fitted to the Murnaghan equation of state [CITATION] giving the equilibrium parameters for the volume [MATH] (and hence one lattice constant), the binding energy [MATH] per pair of atoms, the bulk modulus [MATH], and its pressure coefficient [MATH] .', 'cond-mat-0610066-1-19-0': 'On the other hand, the wurtzite phase is characterized by two lattice parameters, namely [MATH] and [MATH], and by the internal parameter [MATH].', 'cond-mat-0610066-1-19-1': 'Consequently, the structure optimization of the wurtzite polymorph is carried out in four steps ordered according to the importance of the parameter under investigation.', 'cond-mat-0610066-1-19-2': 'At first, the equilibrium volume is determined by assuming the ideal [MATH] ratio.', 'cond-mat-0610066-1-19-3': 'This gives a first estimate of the equilibrium [MATH] which is then used in the determination of the deviation of [MATH] from the ideal value.', 'cond-mat-0610066-1-19-4': 'Then, the volume optimization is repeated using the new [MATH] and [MATH] finally producing the equilibrium values.', 'cond-mat-0610066-1-19-5': 'As the last step, the internal parameter [MATH] is determined via force minimization.', 'cond-mat-0610066-1-19-6': 'We obtained [MATH], which is very close to the ideal value (0.375).', 'cond-mat-0610066-1-20-0': 'The deviation from the ideal [MATH] ratio is obtained by calculating the total energy of the wz polymorph for different values of the ratio and then fitting the so obtained curve to a [MATH] order polynomial (Fig. [REF]).', 'cond-mat-0610066-1-20-1': 'In this way, we found [MATH], which is slightly larger (by [MATH]) than the ideal value [MATH].', 'cond-mat-0610066-1-20-2': 'The resulting equilibrium value for [MATH] is 7.00118 AA.', 'cond-mat-0610066-1-20-3': 'The fact that the equilibrium value of [MATH] is larger than the ideal one suggests that the wurtzite phase is not the stable polymorph of InAs, in agreement with the rule stated in Ref. [CITATION].', 'cond-mat-0610066-1-21-0': 'The coefficients of the Murnaghan fit of the energy vs volume curves are summarized in Table [REF] for both polymorphs.', 'cond-mat-0610066-1-21-1': 'By comparing the results obtained for two phases we find that the equilibrium volume per cation-anion pair of the wurtzite structure is smaller than that of the zinc-blende by 2.3703 a.u.[MATH], corresponding to a volume reduction of 0.637 %.', 'cond-mat-0610066-1-22-0': 'The result obtained for the binding energy is somewhat surprising.', 'cond-mat-0610066-1-22-1': 'Contrary to the observed stability of the zb phase, we find that the binding energy of an InAs pair is larger in the wz than in the zb case by 3 meV.', 'cond-mat-0610066-1-22-2': 'This is clearly seen in the energy vs volume curves, normalized to one InAs pair, shown in Figure [REF] for both phases.', 'cond-mat-0610066-1-22-3': 'We have checked this result by also calculating the total energy in the generalized gradient approximation (GGA) by Perdew-Burke-Ernzerhof (PBE) [CITATION].', 'cond-mat-0610066-1-22-4': 'In the GGA calculation the same lattice constant and convergence parameter as in the LDA were used.', 'cond-mat-0610066-1-22-5': 'The wz total energy was found to be lower than the zb by 28 meV.', 'cond-mat-0610066-1-22-6': 'We note that the difference in the equilibrium total energies calculated within the LDA is actually smaller than the expected accuracy of these calculations, i.e. [MATH] meV.', 'cond-mat-0610066-1-22-7': 'We thus conclude that the relative stability of the two phases cannot be determined by our present calculations.', 'cond-mat-0610066-1-23-0': 'The calculated equilibrium lattice constants are displayed in Table [REF] together with the experimental values and those resulting from plane wave calculations based on DFT, LDA and Ultra Soft (US) pseudopotentials.', 'cond-mat-0610066-1-23-1': 'The latter calculations are done by treating the In 4d states as either valence or core states.', 'cond-mat-0610066-1-23-2': 'The details of these pseudopotential calculations are given in Ref. [CITATION].', 'cond-mat-0610066-1-23-3': 'The experimental values for the wurtzite lattice constants are taken from TEM measurements [CITATION].', 'cond-mat-0610066-1-23-4': 'For the zinc-blende structure the experimental parameter measured at room temperature [CITATION] has been extrapolated to zero degrees Kelvin.', 'cond-mat-0610066-1-23-5': 'We thus find that the all-electron calculations and the pseudopotential calculations with the In 4d treated as valence electrons give very similar results and they slightly underestimate ([MATH] AA) the experimental values for the lattice parameters, as is usually the case within the LDA.', 'cond-mat-0610066-1-23-6': 'The results of the pseudopotential calculation with the In 4d electrons frozen into the core are, instead, rather far from the experimental values.', 'cond-mat-0610066-1-23-7': 'This shows the importance of treating the d-electrons as valence states in order to obtain a realistic description of the InAs compound.', 'cond-mat-0610066-1-23-8': 'In the wz case the results reported in the literature [CITATION] are calculated using HGH pseudopotentials [CITATION] constructed by treating the d electrons as core states.', 'cond-mat-0610066-1-23-9': 'The results are: [MATH] , [MATH], [MATH] .', 'cond-mat-0610066-1-23-10': 'Also results for the zb case are reported in the same article using the same approach, leading to [MATH] AA.', 'cond-mat-0610066-1-23-11': 'Other calculations based on the all-electron approach have given the results [MATH] AA [CITATION] and [MATH] AA [CITATION] in the case of InAs in the zb phase.', 'cond-mat-0610066-1-24-0': 'The last step in the structural optimization is the search for that value of the internal parameter [MATH] for which the force on the nuclei vanishes.', 'cond-mat-0610066-1-24-1': 'We searched for this point starting from the wz structure with the equilibrium [MATH] and [MATH] lattice constants.', 'cond-mat-0610066-1-24-2': 'The search was carried out by means of a minimization routine from the PORT library [CITATION] choosing 0.5 mRy/Bohr as a tolerance on the force.', 'cond-mat-0610066-1-24-3': 'We obtained [MATH], i.e. 0.05% smaller than the ideal value.', 'cond-mat-0610066-1-25-0': '# Band structures', 'cond-mat-0610066-1-26-0': 'All the LDA band structures presented in this article are calculated by solving the Kohn-Sham equation [CITATION] at the experimental lattice parameters, along lines in [MATH]-space which include high symmetry points.', 'cond-mat-0610066-1-26-1': 'We have performed such calculations for both the wz and the zb polymorphs of InAs and GaAs.', 'cond-mat-0610066-1-27-0': 'The WIEN2k code allows for the inclusion of relativistic effects at different levels of approximation: non-relativistic, scalar relativistic without spin-orbit interaction and approximate fully-relativistic with the inclusion of the spin-orbit interaction.', 'cond-mat-0610066-1-27-1': 'The core states are treated fully relativistically [CITATION], unless otherwise specified.', 'cond-mat-0610066-1-27-2': 'In the case of the valence states the relativistic effects are included via a scalar relativistic approximation [CITATION] whereafter the spin-orbit interaction can be added by using the scalar relativistic eigenfunctions as a basis [CITATION].', 'cond-mat-0610066-1-27-3': 'In order to overcome the problem associated with the averaging procedure inherent in the scalar relativistic approximation additional local orbitals of definite [MATH] character have been added to the basis set [CITATION].', 'cond-mat-0610066-1-28-0': 'When the spin-orbit coupling is taken into account, the corresponding Hamiltonian is diagonalized using scalar relativistic eigenfunctions in a range of 20 Ry centered around the top of the valence band.', 'cond-mat-0610066-1-28-1': 'This is necessary because the S-O splitting in InAs has a strength comparable to the energy gap and can, thus, not be considered as a "small" correction to the scalar relativistic bands.', 'cond-mat-0610066-1-28-2': 'In the case of GaAs the S-O spitting is smaller than the energy gap (at low temperature, for the zb phase: 0.33 eV and 1.519 eV, respectively [CITATION]), but still the effect is important and so we use the same approximation as for the InAs.', 'cond-mat-0610066-1-29-0': '## InAs band structures', 'cond-mat-0610066-1-30-0': 'At first, the band structures of InAs in both the zb and the wz phases were calculated within the non-relativistic approximation.', 'cond-mat-0610066-1-30-1': 'We then found that InAs is correctly predicted to be a semiconductor with energy gaps of 0.395 eV and 0.444 eV in the zb and the wz cases, respectively.', 'cond-mat-0610066-1-30-2': 'We are grateful to Prof. Almbladh for corroborating these results using his own LAPW code [CITATION].', 'cond-mat-0610066-1-30-3': 'Hence, within the LDA, the energy gap of the wz polymorph is 49 meV larger that that of the zb.', 'cond-mat-0610066-1-30-4': 'In the zb case, the three [MATH]-like states constituting the valence band maximum (VBM) are degenerate at the [MATH] point.', 'cond-mat-0610066-1-30-5': 'In the wz case two of these states show a crystal field splitting of 91 meV.', 'cond-mat-0610066-1-30-6': 'The resulting band structure for the wz polymorph is displayed in Fig. [REF].', 'cond-mat-0610066-1-31-0': 'The scalar relativistic band structures of the two polymorphs were then calculated without including the spin-orbit interaction.', 'cond-mat-0610066-1-31-1': 'Within this approach the InAs energy gap is not present anymore.', 'cond-mat-0610066-1-31-2': 'Instead, we found a "negative gap" or "wrong band ordering" for both polymorphs.', 'cond-mat-0610066-1-31-3': 'Indeed, in the zb case, at the [MATH] point, the three [MATH]-like degenerate states at the top of the valence band have a higher energy than the [MATH]-like conduction band minimum (CBM), resulting in an energy gap of [MATH] eV.', 'cond-mat-0610066-1-31-4': 'The wz phase exhibit the same problem and the calculated energy gap is [MATH] eV.', 'cond-mat-0610066-1-31-5': 'The crystal field splitting can be identified as a splitting of [MATH] meV between the the [MATH] and [MATH] valence band levels.', 'cond-mat-0610066-1-31-6': 'The scalar relativistic band structure for InAs in the wz phase is shown in Fig. [REF].', 'cond-mat-0610066-1-32-0': 'The main reason for this un-physical result has to be ascribed to the notorious limitations of the LDA in reproducing band structures.', 'cond-mat-0610066-1-32-1': 'Since the calculated energy gaps are typically underestimated, this may result in the "negative gap" problem when the semiconductor under investigation has a small energy gap.', 'cond-mat-0610066-1-32-2': 'This is indeed the case in InAs, for which the experimental energy gap of the zb polymorph at zero Kelvin is 0.415 eV [CITATION].', 'cond-mat-0610066-1-33-0': 'We should also consider the fact that the binding energy of the In 4d states is under-estimated by the LDA [CITATION].', 'cond-mat-0610066-1-33-1': 'As a consequence the d states are too close to the VBM, resulting in a too large repulsion between the p and the d states.', 'cond-mat-0610066-1-33-2': 'This has the effect of shifting the VBM toward higher energies, hence contributing to the reduction of the energy gap.', 'cond-mat-0610066-1-34-0': 'Another source of error can be found in the scalar relativistic treatment of InAs.', 'cond-mat-0610066-1-34-1': 'Indeed, in this semiconductor, the relativistic effects are of the same order of magnitude as the energy gap, as can be seen by considering the fact that the spin-orbit splitting at zero Kelvin is 0.38 eV [CITATION].', 'cond-mat-0610066-1-35-0': 'We have, therefore, improved our relativistic description of the system by including the spin-orbit interaction as a correction to the scalar relativistic band structure.', 'cond-mat-0610066-1-35-1': 'This approach led to spin-orbit splittings in the zb and the wz cases of 0.355 eV and 0.336 eV, respectively.', 'cond-mat-0610066-1-35-2': 'In addition, we obtained a crystal field splitting of 48 meV in the wz polymorph.', 'cond-mat-0610066-1-35-3': 'Unfortunately, the inclusion of spin-orbit interaction did not solve the problem with the "wrong band ordering".', 'cond-mat-0610066-1-35-4': 'We obtained the "negative energy gaps" [MATH] eV and [MATH] eV for the zb and the wz phases, respectively.', 'cond-mat-0610066-1-35-5': 'The band structure with the inclusion of the spin-orbit interaction is shown in Fig. [REF] for the wz polymorph.', 'cond-mat-0610066-1-35-6': 'The results for the InAs case are collected in Tab. [REF].', 'cond-mat-0610066-1-36-0': '## GaAs band structures', 'cond-mat-0610066-1-37-0': 'The band structure calculations for GaAs were performed by using the values of lattice constants measured via TEM [CITATION] on single NW:s having a diameter of 110 nm.', 'cond-mat-0610066-1-37-1': 'We used the lattice constants [MATH] , [MATH] , [MATH] AA and [MATH], i.e. the ideal [MATH] value.', 'cond-mat-0610066-1-37-2': 'As mentioned in the introduction, since the deformation potential of GaAs is very strong the calculated band gap depends in a critical way on the lattice constant.', 'cond-mat-0610066-1-37-3': 'Hence, in order to allow a better comparison with experiment, we choose to work with the experimental lattice parameters.', 'cond-mat-0610066-1-38-0': 'We tested this choice by calculating the relativistic band structure (without taking into account the spin-orbit interaction) of GaAs in the zb phase with both the TEM lattice constant and the equilibrium lattice constant [MATH] AA taken from Ref. [CITATION].', 'cond-mat-0610066-1-38-1': 'The resulting band gaps are 280 meV and 493 meV, respectively.', 'cond-mat-0610066-1-38-2': 'Hence, a decrease in the lattice constant of 0.94% results in an increase of the band gap of 43.2%.', 'cond-mat-0610066-1-39-0': 'The scalar relativistic band structures of both the zb and the wz phases of GaAs were calculated using the same procedure as in the case of InAs.', 'cond-mat-0610066-1-39-1': 'The main difference with respect to InAs is that GaAs is correctly predicted to be a semiconductor, even though the LDA gap is very small compared to experiment.', 'cond-mat-0610066-1-39-2': 'This behavior can be explained with arguments analogous to those used in the discussion of the InAs band structures.', 'cond-mat-0610066-1-39-3': 'The band structures calculated for the wz polymorph without and with the inclusion of the spin-orbit interaction are shown in Figs. [REF] and [REF], respectively.', 'cond-mat-0610066-1-40-0': 'Without the S-O interaction we find that the band structure of the wz phase is characterized by a band gap of 327 meV and a crystal field splitting of 131 meV.', 'cond-mat-0610066-1-40-1': 'When the S-O interaction is taken into account, the band gap is decreased to 167 meV for the zb and to 211 meV for the wz phase.', 'cond-mat-0610066-1-40-2': 'The calculated spin-orbit splitting is 339 meV and 340 meV in the zb and the wz cases, respectively, and the crystal field splitting in the wz band structure is 79 meV.', 'cond-mat-0610066-1-40-3': 'All these results are collected in Tab. [REF].', 'cond-mat-0610066-1-41-0': '# SUMMARY AND CONCLUSIONS', 'cond-mat-0610066-1-42-0': 'We have reported full-potential all-electron studies of InAs and GaAs in both the wurtzite and the zinc-blende phases.', 'cond-mat-0610066-1-42-1': 'The structural optimization of both polymorphs of InAs is carried out and result is good agreement with the TEM measured lattice parameters.', 'cond-mat-0610066-1-42-2': 'We have found that, within the accuracy of the calculations, it is not possible to decide which of the phases of InAs is the stable one.', 'cond-mat-0610066-1-42-3': 'Our total energy difference between the phases is [MATH] meV and much smaller than the numerical accuracy of the code.', 'cond-mat-0610066-1-43-0': 'In the case of GaAs, the band structures were calculated at the experimental lattice constant measured via TEM on NW:s having partially wurtzite structure.', 'cond-mat-0610066-1-44-0': 'The LDA band structures have been computed for both polymorphs of InAs and GaAs within different relativistic approximations.', 'cond-mat-0610066-1-44-1': 'At first, results from the non-relativistic approach have been presented for InAs.', 'cond-mat-0610066-1-44-2': 'Then we performed calculations using an approximate relativistic model in which the core states are treated fully relativistically while the valence states are described by a scalar relativistic model.', 'cond-mat-0610066-1-44-3': 'Spin-orbit interaction was then added as a correction.', 'cond-mat-0610066-1-44-4': 'When the bands are calculated relativistically we find that InAs is wrongly predicted to be a metal while InAs is a semiconductor within the non-relativistic approximation.', 'cond-mat-0610066-1-44-5': 'GaAs, instead, is correctly predicted to be a semiconductor in the scalar relativistic approximation, even tough the calculated band gaps are considerably smaller as compared to experiment.', 'cond-mat-0610066-1-45-0': 'Spin-orbit splittings and crystal field splittings for the new polymorph of InAs and GaAs have been calculated.', 'cond-mat-0610066-1-46-0': 'We thank the Nanoquanta network of Excellence (contract number NMP4-CT-2004-500198) for support.', 'cond-mat-0610066-1-46-1': 'We also thank the Photon-Mediated Phenomena (PMP) Research Training Network (contract number HPRN-CT-2002-00298) for supporting this research.', 'cond-mat-0610066-1-47-0': 'Biork2002 M.T. Bjork, B.J. Ohlsson, C. Thelander, A.I. Persson, K. Deppert, L.R. Wallenberg, L. Samuelson, Appl.', 'cond-mat-0610066-1-48-0': 'Thelander2003 C. Thelander, T. Maartensson, M.T. Bjork, B.J. Ohlsson, M.W. Larsson, L.R. Wallenberg, L. Samuelson, Appl.', 'cond-mat-0610066-1-49-0': 'Pistol, L.E. Froberg, and L. Samuelson, Journal of Physics: Condensed Matter (to be published).', 'cond-mat-0610066-1-50-0': 'Tragardh2006 J. Tragaardh, A.I. Persson, J.B. Wagner, D. Hessman, and L. Samuelson (to be published).', 'cond-mat-0610066-1-51-0': 'Bjork2005 M.T. Bjork, A. Fuhrer, A.E. Hansen, M.W. Larsson, L.E. Froberg, L. Samuelson, Phys.', 'cond-mat-0610066-1-52-0': 'Soshnikov2005 I.P. Soshnikov, G.E. Cirlin, A.A. Tonkikh, Yu.B.', 'cond-mat-0610066-1-52-1': 'Samsonenko, V.G. Dubovskii, V.M. Ustinov, O.M. Gorbenko, D. Litvinov, and D. Gerthsen, Physics of the Solid State 47, 2213 (2005).', 'cond-mat-0610066-1-53-0': 'Crain1994 J. Crain, R.O. Piltz, G.J. Ackland, S.J. Clark, M.C. Payne, V. Millman, S.J. Lin, P.D. Hatton, and Y.H. Nam, Phys.', 'cond-mat-0610066-1-54-0': 'Hedin1969 L. Hedin and S. Lundqvist, in Solid State Physics, Vol.', 'cond-mat-0610066-1-54-1': '23, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1969), p.1', 'cond-mat-0610066-1-55-0': 'WIEN2k P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, (Karlheinz Schwarz, Techn.', 'cond-mat-0610066-1-56-0': 'Physics of Group IV elements and III-V Compounds, edited by K. H. Hellwege and O. Madelung, Landolt-Bornstein, New Series, Group III, Vol.', 'cond-mat-0610066-1-57-0': 'Massidda1990 S. Massidda, A. Continenza, A.J. Freeman, T.M. de Pascale, F. Meloni, M. Serra, Phys.', 'cond-mat-0610066-1-58-0': 'Singh1994 D. Singh, "Plane waves, pseudopotentials and the LAPW method", Kluwer Academic (1994).', 'cond-mat-0610066-1-59-0': 'Kunes2001 J. Kunes, P. Novak, R. Schmid, P. Blaha, and K. 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Louie, Phys.'} | {'cond-mat-0610066-2-0-0': 'The structural and electronic properties of the wurtzite phase of the InAs and GaAs compounds are, for the first time, studied within the framework of Density Functional Theory (DFT).', 'cond-mat-0610066-2-0-1': 'We used the full-potential linearized augmented plane wave (LAPW) method and the local density approximation (LDA) for exchange and correlation and compared the results to the corresponding pseudopotential calculations.', 'cond-mat-0610066-2-1-0': 'From the structural optimization of the wurtzite polymorph of InAs we found that the [MATH] ratio is somewhat greater than the ideal one and that the internal parameter [MATH] has a value slightly smaller than the ideal one.', 'cond-mat-0610066-2-1-1': 'In the all-electron approach the wurtzite polymorph has a smaller equilibrium volume per InAs pair and a higher binding energy when compared to the zinc-blende phase whereas the situation is reversed in the pseudo treatment.', 'cond-mat-0610066-2-1-2': 'The energy differences are, however, smaller than the accuracy of standard density-functional codes ([MATH] meV) and a theoretical prediction of the relative stability of the two phases cannot be made.', 'cond-mat-0610066-2-2-0': 'In order to investigate the possibility of using an LDA calculation as a starting point for many-body calculations of excitations properties, we here also present the band-structures of these materials.', 'cond-mat-0610066-2-2-1': 'The bands are calculated with and without relativistic effects.', 'cond-mat-0610066-2-2-2': 'In InAs we find that the energy gaps of both polymorphs are positive when obtained from a non-relativistic calculation and negative otherwise.', 'cond-mat-0610066-2-3-0': 'For both semiconductors, we determine the spin-orbit splittings for the zinc-blende and the wurtzite phases as well as the crystal-field splittings for the new wurtzite polymorphs.', 'cond-mat-0610066-2-4-0': '# Introduction', 'cond-mat-0610066-2-5-0': 'Nanowires (NW:s) are attracting an increasing amount of attention in the scientific community because they offer the possibility of investigating fundamental physical properties at the nano scale and because of the numerous possible applications in electronic and photonic devices.', 'cond-mat-0610066-2-5-1': 'In particular, InAs-based nanowires have already been used to improve electronics in 1D as, for instance, in the case of resonant tunneling diodes [CITATION] and single-electron transistors [CITATION].', 'cond-mat-0610066-2-5-2': 'These NW:s, typically grown on an (111)B InAs substrate via Chemical Beam Epitaxy (CBE), have a wurtzite crystal structure in contrast to the stable phase at normal pressure and temperature which is the zinc-blende structure ([MATH]) with the space group [MATH]).', 'cond-mat-0610066-2-5-3': 'Despite the successes achieved so far with InAs-based NW:s, it has recently been realized that their wurtzite structure is an important factor in the interpretation of experimental data obtained from measurements of photo-luminescence (PL) [CITATION], photo-current (PC) [CITATION] or other electronic properties [CITATION].', 'cond-mat-0610066-2-6-0': 'The situation is similar in the case of GaAs NW:s.', 'cond-mat-0610066-2-6-1': 'When the (111)B GaAs surface is chosen as a substrate, the resulting NW:s consist of alternating wurtzite and zinc-blend segments [CITATION].', 'cond-mat-0610066-2-6-2': 'Hence, both segments contribute to the optical and electronic properties of such a NW.', 'cond-mat-0610066-2-7-0': 'Theoretical studies based on total-energy calculations [CITATION] predict that the wurtzite polymorph ([MATH]), having space group [MATH]), is a metastable high-pressure modification of both the InAs and GaAs compounds.', 'cond-mat-0610066-2-7-1': 'Hence, we are faced with new materials, InAs and GaAs in the wurtzite phase, for which no previous experimental study exists and only little theoretical work has been done.', 'cond-mat-0610066-2-8-0': 'In this article we investigate the structural and electronic properties of these compounds by using all-electron calculations based on Density-Functional Theory (DFT) [CITATION].', 'cond-mat-0610066-2-8-1': 'Indeed, this approach does not require a knowledge of material parameters and enables us to compute numerous properties ranging from the equilibrium lattice constant to the cohesive energy of the system.', 'cond-mat-0610066-2-9-0': 'DFT in the Local Density Approximation (LDA) [CITATION] provides an accurate description of the ground state electronic structure of solids and such calculations are adequate tools for structural studies.', 'cond-mat-0610066-2-9-1': 'The comparison of the calculated band structure to experimental excitation energies is, strictly speaking, not allowed but, nevertheless, provides important information on the electronic properties of the material.', 'cond-mat-0610066-2-9-2': 'It is, for instance, well known that the experimental band gaps are usually underestimated by DFT.', 'cond-mat-0610066-2-9-3': 'The excited-state properties of a many-electron system, such as the band structure, are better described within Many-Body Perturbation Theory (MBPT) as, for instance, by the GW approximation [CITATION] to the quasiparticle self-energy [MATH].', 'cond-mat-0610066-2-9-4': 'Still, a DFT calculation is a necessary step toward the GW calculation, since the LDA eigenenergies and eigenfunctions are usually taken to be the zero:th order approximation for the perturbative expansion.', 'cond-mat-0610066-2-9-5': 'In particular, the LDA provides an accurate Hartree potential which always must be treated to infinite order in an extended system.', 'cond-mat-0610066-2-9-6': 'Hence, the present band structure calculations allow us to investigate how well the LDA can work as a starting point for subsequent many-body calculations.', 'cond-mat-0610066-2-10-0': 'Although the LDA cannot describe the excited-state properties, it can nevertheless provide a good description of the properties of the occupied states [CITATION].', 'cond-mat-0610066-2-10-1': 'This means that we can extract useful informations from the valence part of the band structure as, for instance, the spin-orbit and - for the wurtzite compounds - the crystal field splittings.', 'cond-mat-0610066-2-11-0': 'Moreover, the all-electron band structures, calculated for the first time for the wurtzite phase, offer benchmarks for the corresponding pseudopotential calculations.', 'cond-mat-0610066-2-12-0': 'Related studies reported in the literature are performed within DFT, use pseudopotentials [CITATION], and include the In 4d and the Ga 3d electrons among the core states.', 'cond-mat-0610066-2-12-1': 'These approximations lead to an underestimation of the equilibrium lattice parameters and to a less accurate description of the band-structure.', 'cond-mat-0610066-2-12-2': 'Here, we focus on a study of the ground state properties of InAs and GaAs in the wurtzite phase at the DFT/LDA level with a full potential description of the atomic species.', 'cond-mat-0610066-2-12-3': 'For the calculations of quasiparticle band structures of these compounds we refer the reader to a subsequent paper [CITATION].', 'cond-mat-0610066-2-13-0': 'The article is organized as follows.', 'cond-mat-0610066-2-13-1': 'After the description of the theoretical background underlying the calculations, we present the results concerning the structure optimization of both the zinc-blende and the wurtzite phases of InAs.', 'cond-mat-0610066-2-13-2': 'Then, we discuss the band structure obtained in the non-relativistic and in the scalar relativistic approximations, the latter with and without the inclusion of the spin-orbit (S-O) interaction.', 'cond-mat-0610066-2-13-3': 'In the GaAs case, instead, we use the experimental lattice constants taken from TEM measurements on GaAs NW:s [CITATION] to allow for a direct comparison with experiment.', 'cond-mat-0610066-2-13-4': 'Indeed, since the deformation potential for the GaAs gap is very large [CITATION], small changes in the lattice constant have a strong influence on the band gap.', 'cond-mat-0610066-2-13-5': 'Finally, we present the band structures for the GaAs compound calculated in the scalar relativistic approximation with and without the spin-orbit interaction.', 'cond-mat-0610066-2-14-0': '# Methods of investigation', 'cond-mat-0610066-2-15-0': 'The electronic structure calculations presented here are performed within DFT using the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method [CITATION] as implemented in the WIEN2k software package [CITATION].', 'cond-mat-0610066-2-15-1': 'For the exchange-correlation energy in the Local Spin Density Approximation (LSDA) [CITATION] we have chosen the electron gas data of Ceperley and Alder [CITATION] as parametrized by Perdew and Wang [CITATION].', 'cond-mat-0610066-2-16-0': 'The APW method [CITATION] is a procedure to solve the Kohn-Sham equation based on the division of the unit cell into two spatial regions: non-overlapping muffin-tin spheres centered at the atomic sites of radius [MATH] and a remaining interstitial region.', 'cond-mat-0610066-2-16-1': 'The basis set used consist of atomic partial waves and plane waves in the two regions, respectively.', 'cond-mat-0610066-2-16-2': 'Denoting the cut-off wave number of the plane waves by [MATH], the number of basis functions is determined by the product [MATH], where [MATH] is the smallest muffin-tin radius in the unit cell.', 'cond-mat-0610066-2-16-3': 'The angular-momentum cut-off of the expansion of the wave functions in spherical harmonics within the atomic spheres is called [MATH].', 'cond-mat-0610066-2-17-0': 'Starting from the original APW method, several improved versions of the method were developed as, for instance, the Linearized APW method (LAPW) [CITATION], and, with the addition of local orbitals to the basis functions, the LAPW+LO [CITATION], and the APW+lo [CITATION] methods.', 'cond-mat-0610066-2-17-1': 'Within the WIEN2k code it is possible to use a mixed LAPW+LO and APW+lo basis [CITATION] to get the advantage of both schemes.', 'cond-mat-0610066-2-17-2': 'For the expansion inside the atomic spheres we have chosen the APW+lo basis for [MATH], the LAPW+LO for higher [MATH]:s and we have used [MATH] as the highest [MATH] value.', 'cond-mat-0610066-2-18-0': 'To ensure that no charge leaks outside the atomic spheres we have chosen [MATH] Ry for InAs and [MATH] Ry for GaAs as the energy which separates the core and the valence states.', 'cond-mat-0610066-2-18-1': 'In the InAs case, this choice guarantees that the In [MATH], As [MATH] and all higher-energy states are treated as band states, i.e., 36 valence electrons per InAs pair.', 'cond-mat-0610066-2-18-2': 'Instead, in GaAs, the valence band states start from the Ga [MATH] and the As [MATH], leading to 28 valence electrons per GaAs pair.', 'cond-mat-0610066-2-19-0': 'For both the InAs and the GaAs polymorphs we have performed convergence studies to determine the optimum value of [MATH] and the number of [MATH]-points in the whole Brillouin zone (BZ).', 'cond-mat-0610066-2-19-1': 'We have found that for [MATH] the total energy is converged to within less than 10 meV for both phases.', 'cond-mat-0610066-2-19-2': 'The number of [MATH]-points which ensures convergence is 1000 ([MATH]) in the zb and 405 ([MATH]) in the wz cases, respectively.', 'cond-mat-0610066-2-19-3': 'The chosen number of [MATH]-points corresponds to 73 [MATH]-points in the irreducible wedge of the Brillouin zone (IBZ) for the zb structure and to 95 [MATH]-points for the wz structure.', 'cond-mat-0610066-2-19-4': 'These [MATH]-meshes are used in a modified tetrahedron integration scheme [CITATION] to determine which states are occupied.', 'cond-mat-0610066-2-19-5': 'The Fourier expansion of the electron density in the interstitial region (i.e. valence charge density) is cut-off at [MATH], leading to 913 and 6524 plane waves for the [MATH] and the [MATH] structures, respectively.', 'cond-mat-0610066-2-20-0': '# Structure Optimization', 'cond-mat-0610066-2-21-0': 'The parameters obtained in the convergence study were used to carry out the structure optimization of both InAs polymorphs in the way described below.', 'cond-mat-0610066-2-22-0': 'Due to the cubic symmetry, the equilibrium lattice constant [MATH] of the zinc-blende compound is determined by the minimization of the total energy of the system with respect to the volume of the unit cell.', 'cond-mat-0610066-2-22-1': 'The energy vs volume curve is then fitted to the Murnaghan equation of state [CITATION] giving the equilibrium parameters for the volume [MATH] (and hence one lattice constant), the binding energy [MATH] per pair of atoms, the bulk modulus [MATH], and its pressure coefficient [MATH] .', 'cond-mat-0610066-2-23-0': 'On the other hand, the wurtzite phase is characterized by two lattice parameters, namely [MATH] and [MATH], and by the internal parameter [MATH].', 'cond-mat-0610066-2-23-1': 'Consequently, the structure optimization of the wurtzite polymorph is carried out in four steps ordered according to the importance of the parameter under investigation.', 'cond-mat-0610066-2-23-2': 'At first, the equilibrium volume is determined by assuming the ideal [MATH] ratio.', 'cond-mat-0610066-2-23-3': 'This gives a first estimate of the equilibrium [MATH] which is then used in the determination of the deviation of [MATH] from the ideal value.', 'cond-mat-0610066-2-23-4': 'Then, the volume optimization is repeated using the new [MATH] and [MATH] finally producing the equilibrium values.', 'cond-mat-0610066-2-23-5': 'As the last step, the internal parameter [MATH] is determined via force minimization.', 'cond-mat-0610066-2-23-6': 'We obtained [MATH], which is very close to the ideal value (0.375).', 'cond-mat-0610066-2-24-0': 'The deviation from the ideal [MATH] ratio is obtained by calculating the total energy of the wz polymorph for different values of the ratio and then fitting the so obtained curve to a [MATH] order polynomial (Fig. [REF]).', 'cond-mat-0610066-2-24-1': 'In this way, we found [MATH], which is slightly larger (by [MATH]) than the ideal value [MATH].', 'cond-mat-0610066-2-24-2': 'The resulting equilibrium value for [MATH] is 7.00118 AA.', 'cond-mat-0610066-2-24-3': 'The fact that the equilibrium value of [MATH] is larger than the ideal one suggests that the wurtzite phase is not the stable polymorph of InAs, in agreement with the rule stated in Ref. [CITATION].', 'cond-mat-0610066-2-25-0': 'The coefficients of the Murnaghan fit of the energy vs volume curves are summarized in Table [REF] for both polymorphs.', 'cond-mat-0610066-2-25-1': 'By comparing the results obtained for two phases we find that the equilibrium volume per cation-anion pair of the wurtzite structure is smaller than that of the zinc-blende by 2.3703 a.u.[MATH], corresponding to a volume reduction of 0.637 %.', 'cond-mat-0610066-2-26-0': 'The result obtained for the binding energy is somewhat surprising.', 'cond-mat-0610066-2-26-1': 'Contrary to the observed stability of the zb phase, we find that the binding energy of an InAs pair is larger in the wz than in the zb case by 3 meV.', 'cond-mat-0610066-2-26-2': 'This is clearly seen in the energy vs volume curves, normalized to one InAs pair, shown in Figure [REF] for both phases.', 'cond-mat-0610066-2-26-3': 'We have checked this result by also calculating the total energy in the generalized gradient approximation (GGA) by Perdew-Burke-Ernzerhof (PBE) [CITATION].', 'cond-mat-0610066-2-26-4': 'In the GGA calculation we used the same lattice constant and convergence parameter as in the case of the LDA.', 'cond-mat-0610066-2-26-5': 'The wz total energy was found to be 28 meV lower than that of the zb.', 'cond-mat-0610066-2-26-6': 'We note that the difference in the equilibrium total energies calculated within the LDA is actually smaller than the typical accuracy of calculations carried out within standard DFT codes, i.e. [MATH] meV.', 'cond-mat-0610066-2-26-7': 'We thus conclude that the relative stability of the two phases cannot be determined within the present accuracy of the calculations.', 'cond-mat-0610066-2-27-0': 'As a further check, we have also performed the structure optimization by using Ultra Soft (US) pseudopotentials and plane waves [CITATION], as implemented in the VASP code [CITATION].', 'cond-mat-0610066-2-27-1': 'The latter calculations are done by treating the In 4d states as either valence (d-val) or core (d-core) states.', 'cond-mat-0610066-2-27-2': 'In the pseudopotential calculations we have used a Monkhorst-Pack [CITATION] grid centered on the [MATH] point with a [MATH] and [MATH] mesh in reciprocal space for the zb and the wz phase, respectively.', 'cond-mat-0610066-2-27-3': 'The convergence study gives the following kinetic energy cutoffs: 262 eV (d-val, for both polymorphs) and 222 eV and 202 eV (d-core) for the zb and wz cases, respectively.', 'cond-mat-0610066-2-27-4': 'The structure optimization was performed by minimization of the total energy.', 'cond-mat-0610066-2-27-5': 'In the wurtzite case we have assumed the ideal values for [MATH] and [MATH].', 'cond-mat-0610066-2-27-6': 'We have thus found that the zb phase, having a binding energy lower than the wz by [MATH] meV, is correctly predicted to be the stable phase of InAs.', 'cond-mat-0610066-2-27-7': 'We also found that the equilibrium volume per InAs pair of the wz phase is smaller than the zb by 0.08%.', 'cond-mat-0610066-2-28-0': 'Clearly, the differing results is connected to the different method of calculation, i.e. all-electron versus pseudopotential calculations.', 'cond-mat-0610066-2-28-1': 'These tests thus provide an indication of the accuracy obtainable from calculations based on Ultra-Soft paseudopotentials.', 'cond-mat-0610066-2-28-2': 'In our opinion, both sets of results must be considered correct in the sense that the total energy difference between them is of the same magnitude as the overall accuracy in the total energies from standard DFT codes.', 'cond-mat-0610066-2-29-0': 'The calculated equilibrium lattice constants (in both the full and pseudo potential method) are displayed in Table [REF] together with the experimental values.', 'cond-mat-0610066-2-29-1': 'The experimental values for the wurtzite lattice constants are taken from TEM measurements [CITATION].', 'cond-mat-0610066-2-29-2': 'For the zinc-blende structure the experimental parameter measured at room temperature [CITATION] has been extrapolated to zero degrees Kelvin.', 'cond-mat-0610066-2-29-3': 'We thus find that the all-electron calculations and the pseudopotential calculations with the In 4d treated as valence electrons give very similar results and they slightly underestimate ([MATH] AA) the experimental values for the lattice parameters, as is usually the case within the LDA.', 'cond-mat-0610066-2-29-4': 'The results of the pseudopotential calculation with the In 4d electrons frozen into the core are, instead, rather far from the experimental values.', 'cond-mat-0610066-2-29-5': 'This shows the importance of treating the d-electrons as valence states in order to obtain a realistic description of the InAs compound.', 'cond-mat-0610066-2-29-6': 'In the wz case the results reported in the literature [CITATION] are calculated using HGH pseudopotentials [CITATION] constructed by treating the d electrons as core states.', 'cond-mat-0610066-2-29-7': 'The results are: [MATH] , [MATH], [MATH] .', 'cond-mat-0610066-2-29-8': 'Also results for the zb case are reported in the same article using the same approach, leading to [MATH] AA.', 'cond-mat-0610066-2-29-9': 'Other calculations based on the all-electron approach have given the results [MATH] AA [CITATION] and [MATH] AA [CITATION] in the case of InAs in the zb phase.', 'cond-mat-0610066-2-30-0': 'The last step in the structural optimization is the search for that value of the internal parameter [MATH] for which the force on the nuclei vanishes.', 'cond-mat-0610066-2-30-1': 'We searched for this point starting from the wz structure with the equilibrium [MATH] and [MATH] lattice constants.', 'cond-mat-0610066-2-30-2': 'The search was carried out by means of a minimization routine from the PORT library [CITATION] choosing 0.5 mRy/Bohr as a tolerance on the force.', 'cond-mat-0610066-2-30-3': 'We obtained [MATH], i.e. 0.05% smaller than the ideal value.', 'cond-mat-0610066-2-31-0': '# Band structures', 'cond-mat-0610066-2-32-0': 'All the LDA band structures presented in this article are calculated by solving the Kohn-Sham equation [CITATION] at the experimental lattice parameters, along lines in [MATH]-space which include high symmetry points.', 'cond-mat-0610066-2-32-1': 'We have performed such calculations for both the wz and the zb polymorphs of InAs and GaAs.', 'cond-mat-0610066-2-33-0': 'The motivations for calculating the full-potential LDA band structures are as follows.', 'cond-mat-0610066-2-33-1': 'The all-electron bands (i) allow us to investigate how well the LDA can function as a zero:th order approximation for the perturbation expansions of MBPT; (ii) allow us to assess the effects of different relativistic approximations; (iii) provide a benchmark for pseudopotential calculations based on LDA; (iv) provide a good approximation to the electronic structure of the occupied states (i.e. valence band states), thus allowing us to estimate the spin-orbit and the crystal field splittings for the new wurtzite structures.', 'cond-mat-0610066-2-34-0': 'The WIEN2k code allows for the inclusion of relativistic effects at different levels of approximation: non-relativistic, scalar relativistic without spin-orbit interaction and approximate fully-relativistic with the inclusion of the spin-orbit interaction.', 'cond-mat-0610066-2-34-1': 'The core states are treated fully relativistically [CITATION], unless otherwise specified.', 'cond-mat-0610066-2-34-2': 'In the case of the valence states the relativistic effects are included via a scalar relativistic approximation [CITATION] whereafter the spin-orbit interaction can be added by using the scalar relativistic eigenfunctions as a basis [CITATION].', 'cond-mat-0610066-2-34-3': 'In order to overcome the problem associated with the averaging procedure inherent in the scalar relativistic approximation additional local orbitals of definite [MATH] character have been added to the basis set [CITATION].', 'cond-mat-0610066-2-35-0': 'When the spin-orbit coupling is taken into account, the corresponding Hamiltonian is diagonalized using scalar relativistic eigenfunctions in a range of 20 Ry centered around the top of the valence band.', 'cond-mat-0610066-2-35-1': 'This is necessary because the S-O splitting in InAs has a strength comparable to the energy gap and can, thus, not be considered as a "small" correction to the scalar relativistic bands.', 'cond-mat-0610066-2-35-2': 'In the case of GaAs the S-O spitting is smaller than the energy gap (at low temperature, for the zb phase: 0.33 eV and 1.519 eV, respectively [CITATION]), but still the effect is important and so we use the same approximation as for the InAs.', 'cond-mat-0610066-2-36-0': '## InAs band structures', 'cond-mat-0610066-2-37-0': 'At first, the band structures of InAs in both the zb and the wz phases were calculated within the non-relativistic approximation.', 'cond-mat-0610066-2-37-1': 'We then found that InAs is correctly predicted to be a semiconductor with energy gaps of 0.395 eV and 0.444 eV in the zb and the wz cases, respectively.', 'cond-mat-0610066-2-37-2': 'We are grateful to Prof. Almbladh for corroborating these results using his own LAPW code [CITATION].', 'cond-mat-0610066-2-37-3': 'Hence, within the LDA, the energy gap of the wz polymorph is 49 meV larger that that of the zb.', 'cond-mat-0610066-2-37-4': 'In the zb case, the three [MATH]-like states constituting the valence band maximum (VBM) are degenerate at the [MATH] point.', 'cond-mat-0610066-2-37-5': 'In the wz case two of these states show a crystal field splitting of 91 meV.', 'cond-mat-0610066-2-37-6': 'The resulting band structure for the wz polymorph is displayed in Fig. [REF]a.', 'cond-mat-0610066-2-38-0': 'The scalar relativistic band structures of the two polymorphs were then calculated without including the spin-orbit interaction.', 'cond-mat-0610066-2-38-1': 'Within this approach the InAs energy gap is not present anymore.', 'cond-mat-0610066-2-38-2': 'Instead, we found a "negative gap" or "wrong band ordering" for both polymorphs.', 'cond-mat-0610066-2-38-3': 'Indeed, in the zb case, at the [MATH] point, the three [MATH]-like degenerate states at the top of the valence band have a higher energy than the [MATH]-like conduction band minimum (CBM), resulting in an energy gap of [MATH] eV.', 'cond-mat-0610066-2-38-4': 'The wz phase exhibit the same problem and the calculated energy gap is [MATH] eV.', 'cond-mat-0610066-2-38-5': 'The crystal field splitting can be identified as a splitting of [MATH] meV between the the [MATH] and [MATH] valence band levels.', 'cond-mat-0610066-2-38-6': 'The scalar relativistic band structure for InAs in the wz phase is shown in Fig. [REF]b.', 'cond-mat-0610066-2-39-0': 'The main reason for this un-physical result has to be ascribed to the notorious limitations of the LDA in reproducing band structures.', 'cond-mat-0610066-2-39-1': 'Since the calculated energy gaps are typically underestimated, this may result in the "negative gap" problem when the semiconductor under investigation has a small energy gap.', 'cond-mat-0610066-2-39-2': 'This is indeed the case in InAs, for which the experimental energy gap of the zb polymorph at zero Kelvin is 0.415 eV [CITATION].', 'cond-mat-0610066-2-40-0': 'We should also consider the fact that the binding energy of the In 4d states is under-estimated by the LDA [CITATION].', 'cond-mat-0610066-2-40-1': 'As a consequence the d states are too close to the VBM, resulting in a too large repulsion between the p and the d states.', 'cond-mat-0610066-2-40-2': 'This has the effect of shifting the VBM toward higher energies, hence contributing to the reduction of the energy gap.', 'cond-mat-0610066-2-41-0': 'Another source of error can be found in the scalar relativistic treatment of InAs.', 'cond-mat-0610066-2-41-1': 'Indeed, in this semiconductor, the relativistic effects are of the same order of magnitude as the energy gap, as can be seen by considering the fact that the spin-orbit splitting at zero Kelvin is 0.38 eV [CITATION].', 'cond-mat-0610066-2-42-0': 'We have, therefore, improved our relativistic description of the system by including the spin-orbit interaction as a correction to the scalar relativistic band structure.', 'cond-mat-0610066-2-42-1': 'This approach led to spin-orbit splittings in the zb and the wz cases of 0.355 eV and 0.336 eV, respectively.', 'cond-mat-0610066-2-42-2': 'In addition, we obtained a crystal field splitting of 48 meV in the wz polymorph.', 'cond-mat-0610066-2-42-3': 'Unfortunately, the inclusion of spin-orbit interaction did not solve the problem with the "wrong band ordering".', 'cond-mat-0610066-2-42-4': 'We obtained the "negative energy gaps" [MATH] eV and [MATH] eV for the zb and the wz phases, respectively.', 'cond-mat-0610066-2-42-5': 'The band structure with the inclusion of the spin-orbit interaction is shown in Fig. [REF]c for the wz polymorph.', 'cond-mat-0610066-2-42-6': 'The results for the InAs case are collected in Tab. [REF].', 'cond-mat-0610066-2-43-0': '## GaAs band structures', 'cond-mat-0610066-2-44-0': 'The band structure calculations for GaAs were performed by using the values of lattice constants measured via TEM [CITATION] on single NW:s having a diameter of 110 nm.', 'cond-mat-0610066-2-44-1': 'We used the lattice constants [MATH] , [MATH] , [MATH] AA and [MATH], i.e. the ideal [MATH] value.', 'cond-mat-0610066-2-44-2': 'As mentioned in the introduction, since the deformation potential of GaAs is very strong the calculated band gap depends in a critical way on the lattice constant.', 'cond-mat-0610066-2-44-3': 'Hence, in order to allow a better comparison with experiment, we choose to work with the experimental lattice parameters.', 'cond-mat-0610066-2-45-0': 'We tested this choice by calculating the relativistic band structure (without taking into account the spin-orbit interaction) of GaAs in the zb phase with both the TEM lattice constant and the equilibrium lattice constant [MATH] AA taken from Ref. [CITATION].', 'cond-mat-0610066-2-45-1': 'The resulting band gaps are 280 meV and 493 meV, respectively.', 'cond-mat-0610066-2-45-2': 'Hence, a decrease in the lattice constant of 0.94% results in an increase of the band gap of 43.2%.', 'cond-mat-0610066-2-46-0': 'The scalar relativistic band structures of both the zb and the wz phases of GaAs were calculated using the same procedure as in the case of InAs.', 'cond-mat-0610066-2-46-1': 'The main difference with respect to InAs is that GaAs is correctly predicted to be a semiconductor, even though the LDA gap is very small compared to experiment.', 'cond-mat-0610066-2-46-2': 'This behavior can be explained with arguments analogous to those used in the discussion of the InAs band structures.', 'cond-mat-0610066-2-46-3': 'The band structures calculated for the wz polymorph without and with the inclusion of the spin-orbit interaction are shown in Figs. [REF] and [REF], respectively.', 'cond-mat-0610066-2-47-0': 'Without the S-O interaction we find that the band structure of the wz phase is characterized by a band gap of 327 meV and a crystal field splitting of 131 meV.', 'cond-mat-0610066-2-47-1': 'When the S-O interaction is taken into account, the band gap is decreased to 167 meV for the zb and to 211 meV for the wz phase.', 'cond-mat-0610066-2-47-2': 'The calculated spin-orbit splitting is 339 meV and 340 meV in the zb and the wz cases, respectively, and the crystal field splitting in the wz band structure is 79 meV.', 'cond-mat-0610066-2-47-3': 'All these results are collected in Tab. [REF].', 'cond-mat-0610066-2-48-0': '# SUMMARY AND CONCLUSIONS', 'cond-mat-0610066-2-49-0': 'We have reported full-potential all-electron studies of InAs and GaAs in both the wurtzite and the zinc-blende phases.', 'cond-mat-0610066-2-49-1': 'The structural optimization of both polymorphs of InAs is carried out and the result is in good agreement with the TEM measured lattice parameters.', 'cond-mat-0610066-2-49-2': 'We have found that, within the present accuracy of standard density functional calculations, it is not possible to decide which of the phases of InAs is the stable one.', 'cond-mat-0610066-2-49-3': 'Our total energy difference between the phases is [MATH] meV and much smaller than the numerical accuracy of the code ([MATH] meV).', 'cond-mat-0610066-2-50-0': 'In the case of GaAs, the band structures were calculated at the experimental lattice constants measured via TEM on NW:s having partially wurtzite structure.', 'cond-mat-0610066-2-51-0': 'The LDA band structures have been computed for both polymorphs of InAs and GaAs using different relativistic approximations.', 'cond-mat-0610066-2-51-1': 'At first, results from the non-relativistic approach have been presented for InAs.', 'cond-mat-0610066-2-51-2': 'Then we performed calculations using an approximate relativistic model in which the core states are treated fully relativistically while the valence states are described by a scalar relativistic model.', 'cond-mat-0610066-2-51-3': 'Spin-orbit interaction was then added as a correction.', 'cond-mat-0610066-2-51-4': 'When the bands are calculated relativistically we find that InAs is wrongly predicted to be a zero-gap semiconductor while InAs is a finite gap semiconductor within the non-relativistic approximation.', 'cond-mat-0610066-2-51-5': 'Instead GaAs is correctly predicted to be a semiconductor in the scalar relativistic approximation, even tough the calculated band gaps are considerably smaller than the experimental one.', 'cond-mat-0610066-2-52-0': 'Spin-orbit splittings and crystal field splittings for the new polymorph of InAs and GaAs have been calculated for the first time.', 'cond-mat-0610066-2-53-0': 'We thank the Nanoquanta network of Excellence (contract number NMP4-CT-2004-500198) for support.', 'cond-mat-0610066-2-53-1': 'We also thank the Photon-Mediated Phenomena (PMP) Research Training Network (contract number HPRN-CT-2002-00298) for supporting this research.', 'cond-mat-0610066-2-54-0': 'Biork2002 M. 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Louie, Phys.'} | [['cond-mat-0610066-1-2-1', 'cond-mat-0610066-2-3-0'], ['cond-mat-0610066-1-21-0', 'cond-mat-0610066-2-25-0'], ['cond-mat-0610066-1-21-1', 'cond-mat-0610066-2-25-1'], ['cond-mat-0610066-1-40-0', 'cond-mat-0610066-2-47-0'], ['cond-mat-0610066-1-40-1', 'cond-mat-0610066-2-47-1'], ['cond-mat-0610066-1-40-2', 'cond-mat-0610066-2-47-2'], ['cond-mat-0610066-1-40-3', 'cond-mat-0610066-2-47-3'], ['cond-mat-0610066-1-24-0', 'cond-mat-0610066-2-30-0'], ['cond-mat-0610066-1-24-1', 'cond-mat-0610066-2-30-1'], ['cond-mat-0610066-1-24-2', 'cond-mat-0610066-2-30-2'], ['cond-mat-0610066-1-24-3', 'cond-mat-0610066-2-30-3'], ['cond-mat-0610066-1-44-1', 'cond-mat-0610066-2-51-1'], ['cond-mat-0610066-1-44-2', 'cond-mat-0610066-2-51-2'], ['cond-mat-0610066-1-44-3', 'cond-mat-0610066-2-51-3'], ['cond-mat-0610066-1-6-0', 'cond-mat-0610066-2-7-0'], ['cond-mat-0610066-1-6-1', 'cond-mat-0610066-2-7-1'], ['cond-mat-0610066-1-26-0', 'cond-mat-0610066-2-32-0'], ['cond-mat-0610066-1-26-1', 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'cond-mat-0610066-2-21-0'], ['cond-mat-0610066-1-28-0', 'cond-mat-0610066-2-35-0'], ['cond-mat-0610066-1-28-1', 'cond-mat-0610066-2-35-1'], ['cond-mat-0610066-1-28-2', 'cond-mat-0610066-2-35-2'], ['cond-mat-0610066-1-58-0', 'cond-mat-0610066-2-70-0'], ['cond-mat-0610066-1-42-0', 'cond-mat-0610066-2-49-0'], ['cond-mat-0610066-1-34-0', 'cond-mat-0610066-2-41-0'], ['cond-mat-0610066-1-34-1', 'cond-mat-0610066-2-41-1'], ['cond-mat-0610066-1-32-0', 'cond-mat-0610066-2-39-0'], ['cond-mat-0610066-1-32-1', 'cond-mat-0610066-2-39-1'], ['cond-mat-0610066-1-32-2', 'cond-mat-0610066-2-39-2'], ['cond-mat-0610066-1-13-0', 'cond-mat-0610066-2-17-0'], ['cond-mat-0610066-1-13-1', 'cond-mat-0610066-2-17-1'], ['cond-mat-0610066-1-13-2', 'cond-mat-0610066-2-17-2'], ['cond-mat-0610066-1-37-0', 'cond-mat-0610066-2-44-0'], ['cond-mat-0610066-1-37-1', 'cond-mat-0610066-2-44-1'], ['cond-mat-0610066-1-37-2', 'cond-mat-0610066-2-44-2'], ['cond-mat-0610066-1-37-3', 'cond-mat-0610066-2-44-3'], 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'cond-mat-0610066-2-5-0'], ['cond-mat-0610066-1-12-2', 'cond-mat-0610066-2-16-2'], ['cond-mat-0610066-1-22-7', 'cond-mat-0610066-2-26-7'], ['cond-mat-0610066-1-0-0', 'cond-mat-0610066-2-0-0'], ['cond-mat-0610066-1-0-1', 'cond-mat-0610066-2-0-1'], ['cond-mat-0610066-1-14-2', 'cond-mat-0610066-2-18-2'], ['cond-mat-0610066-1-15-0', 'cond-mat-0610066-2-19-0'], ['cond-mat-0610066-1-15-2', 'cond-mat-0610066-2-19-2'], ['cond-mat-0610066-1-15-3', 'cond-mat-0610066-2-19-3'], ['cond-mat-0610066-1-15-5', 'cond-mat-0610066-2-19-5'], ['cond-mat-0610066-1-50-0', 'cond-mat-0610066-2-57-0'], ['cond-mat-0610066-1-8-2', 'cond-mat-0610066-2-12-2'], ['cond-mat-0610066-1-30-6', 'cond-mat-0610066-2-37-6'], ['cond-mat-0610066-1-5-0', 'cond-mat-0610066-2-6-1'], ['cond-mat-0610066-1-43-0', 'cond-mat-0610066-2-50-0'], ['cond-mat-0610066-1-45-0', 'cond-mat-0610066-2-52-0'], ['cond-mat-0610066-1-42-1', 'cond-mat-0610066-2-49-1'], ['cond-mat-0610066-1-42-2', 'cond-mat-0610066-2-49-2'], ['cond-mat-0610066-1-42-3', 'cond-mat-0610066-2-49-3'], ['cond-mat-0610066-1-31-6', 'cond-mat-0610066-2-38-6'], ['cond-mat-0610066-1-54-1', 'cond-mat-0610066-2-61-1'], ['cond-mat-0610066-1-7-1', 'cond-mat-0610066-2-8-1'], ['cond-mat-0610066-1-7-2', 'cond-mat-0610066-2-9-0'], ['cond-mat-0610066-1-7-3', 'cond-mat-0610066-2-9-1'], ['cond-mat-0610066-1-7-5', 'cond-mat-0610066-2-9-3'], ['cond-mat-0610066-1-1-0', 'cond-mat-0610066-2-1-0'], ['cond-mat-0610066-1-1-5', 'cond-mat-0610066-2-2-2']] | [] | [['cond-mat-0610066-1-52-0', 'cond-mat-0610066-2-59-0'], ['cond-mat-0610066-1-52-1', 'cond-mat-0610066-2-59-1'], ['cond-mat-0610066-1-44-5', 'cond-mat-0610066-2-51-5'], ['cond-mat-0610066-1-22-4', 'cond-mat-0610066-2-26-4'], ['cond-mat-0610066-1-22-5', 'cond-mat-0610066-2-26-5'], ['cond-mat-0610066-1-22-6', 'cond-mat-0610066-2-26-6'], ['cond-mat-0610066-1-53-0', 'cond-mat-0610066-2-60-0'], ['cond-mat-0610066-1-8-3', 'cond-mat-0610066-2-12-3'], ['cond-mat-0610066-1-23-0', 'cond-mat-0610066-2-29-0'], ['cond-mat-0610066-1-49-0', 'cond-mat-0610066-2-56-0'], ['cond-mat-0610066-1-51-0', 'cond-mat-0610066-2-58-0'], ['cond-mat-0610066-1-48-0', 'cond-mat-0610066-2-55-0'], ['cond-mat-0610066-1-47-0', 'cond-mat-0610066-2-54-0'], ['cond-mat-0610066-1-60-0', 'cond-mat-0610066-2-72-0'], ['cond-mat-0610066-1-7-0', 'cond-mat-0610066-2-8-0'], ['cond-mat-0610066-1-1-1', 'cond-mat-0610066-2-1-1'], ['cond-mat-0610066-1-1-3', 'cond-mat-0610066-2-2-1'], ['cond-mat-0610066-1-1-4', 'cond-mat-0610066-2-2-1']] | [] | ['cond-mat-0610066-1-23-9', 'cond-mat-0610066-2-29-7'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/cond-mat/0610066 | null | null | null | null | null |
1702.03414 | {'1702.03414-1-0-0': 'LP[MATH] is a three-valued paraconsistent propositional logic which is essentially the same as J3.', '1702.03414-1-0-1': 'It has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-1-0-2': 'However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-1-0-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-1-0-4': 'As one of the bonuses of focussing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction.', '1702.03414-1-1-0': '03B53, 03B50.', '1702.03414-1-2-0': '# Introduction', '1702.03414-1-3-0': 'A set of propositions is contradictory if there exists a proposition such that both that proposition and the negation of that proposition can be deduced from it.', '1702.03414-1-3-1': 'In classical propositional logic, every proposition can be deduced from every contradictory set of propositions.', '1702.03414-1-3-2': 'In a paraconsistent propositional logic, this is not the case.', '1702.03414-1-4-0': 'LP[MATH] is the three-valued paraconsistent propositional logic LP [CITATION] enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-1-4-1': 'This logic, which is essentially the same as J3 [CITATION], the propositional fragment of CLuNs [CITATION] without bi-implication, and LFI1 [CITATION], has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-1-4-2': 'However, it follows easily from results presented in [CITATION] that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-1-4-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-1-5-0': 'It turns out that only 32 of those 8192 logics are logics of which the logical equivalence relation satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction; and only 16 of them are logics of which the logical equivalence relation additionally satisfies the double negation law.', '1702.03414-1-5-1': 'LP[MATH] is one of those 16 logics.', '1702.03414-1-5-2': 'Three additional classical laws of logical equivalence turn out to be sufficient to distinguish LP[MATH] completely from the others.', '1702.03414-1-6-0': 'The desirable properties of reasonable paraconsistent propositional logics referred to above concern the logical consequence relation of a logic.', '1702.03414-1-6-1': 'Unlike in classical propositional logic, the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction do not follow from those properties in a three-valued paraconsistent propositional logic.', '1702.03414-1-6-2': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies these laws.', '1702.03414-1-6-3': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-1-7-0': 'The structure of this note is as follows.', '1702.03414-1-7-1': 'First, we give a survey of the paraconsistent propositional logic LP[MATH] (Section [REF]).', '1702.03414-1-7-2': 'Next, we discuss the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic (Section [REF]).', '1702.03414-1-7-3': 'After that, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the other three-valued paraconsistent propositional logics with the properties discussed earlier (Section [REF]).', '1702.03414-1-7-4': 'Finally, we make some concluding remarks (Section [REF]).', '1702.03414-1-8-0': 'It is relevant to realize that the work presented in this note is restricted to three-valued paraconsistent propositional logics that are truth-functional three-valued logics.', '1702.03414-1-9-0': 'There is text overlap between this note and [CITATION].', '1702.03414-1-9-1': 'This note generalizes and elaborates Section 2 of that paper in such a way that it may be of independent importance to the area of paraconsistent logics.', '1702.03414-1-10-0': '# The Paraconsistent Logic LP[MATH]', '1702.03414-1-11-0': 'A set of propositions [MATH] is contradictory if there exists a proposition [MATH] such that both [MATH] and [MATH] can be deduced from [MATH].', '1702.03414-1-11-1': 'In classical propositional logic, every proposition can be deduced from a contradictory set of propositions.', '1702.03414-1-11-2': 'A paraconsistent propositional logic is a propositional logic in which not every proposition can be deduced from every contradictory set of propositions.', '1702.03414-1-12-0': 'In [CITATION], Priest proposed the paraconsistent propositional logic LP (Logic of Paradox).', '1702.03414-1-12-1': 'The logic introduced in this section is LP enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-1-12-2': 'This logic, called LP[MATH], is in fact the propositional fragment of CLuNs [CITATION] without bi-implications.', '1702.03414-1-13-0': 'LP[MATH] has the following logical constants and connectives: a falsity constant [MATH], a unary negation connective [MATH], a binary conjunction connective [MATH], a binary disjunction connective [MATH], and a binary implication connective [MATH].', '1702.03414-1-13-1': 'Truth and bi-implication are defined as abbreviations: [MATH] stands for [MATH] and [MATH] stands for [MATH].', '1702.03414-1-14-0': 'A Hilbert-style formulation of LP[MATH] is given in Table [REF].', '1702.03414-1-15-0': 'In this formulation, which is taken from [CITATION], [MATH], [MATH], and [MATH] are used as meta-variables ranging over all formulas of LP[MATH].', '1702.03414-1-15-1': 'The axiom schemas on the left-hand side of Table [REF] and the single inference rule (modus ponens) constitute a Hilbert-style formulation of the positive fragment of classical propositional logic.', '1702.03414-1-15-2': 'The first four axiom schemas on the right-hand side of Table [REF] allow for the negation connective to be moved inward.', '1702.03414-1-15-3': 'The fifth axiom schema on the right-hand side of Table [REF] is the law of the excluded middle.', '1702.03414-1-15-4': 'This axiom schema can be thought of as saying that, for every proposition, the proposition or its negation is true, while leaving open the possibility that both are true.', '1702.03414-1-15-5': 'If we add the axiom schema [MATH], which says that any proposition follows from a contradiction, to the given Hilbert-style formulation of LP[MATH], then we get a Hilbert-style formulation of classical propositional logic (see e.g. [CITATION]).', '1702.03414-1-15-6': 'We use the symbol [MATH] without decoration to denote the syntactic logical consequence relation induced by the axiom schemas and inference rule of LP[MATH].', '1702.03414-1-16-0': 'The following outline of the semantics of LP[MATH] is based on [CITATION].', '1702.03414-1-16-1': 'Like in the case of classical propositional logic, meanings are assigned to the formulas of LP[MATH] by means of valuations.', '1702.03414-1-16-2': 'However, in addition to the two classical truth values [MATH] (true) and [MATH] (false), a third meaning [MATH] (both true and false) may be assigned.', '1702.03414-1-16-3': 'A valuation for LP[MATH] is a function [MATH] from the set of all formulas of LP[MATH] to the set [MATH] such that for all formulas [MATH] and [MATH] of LP[MATH]: [EQUATION]', '1702.03414-1-16-4': 'The classical truth-conditions and falsehood-conditions for the logical connectives are retained.', '1702.03414-1-16-5': 'Except for implications, a formula is classified as both-true-and-false exactly when it cannot be classified as true or false by the classical truth-conditions and falsehood-conditions.', '1702.03414-1-16-6': 'The definition of a valuation given above shows that the logical connectives of LP[MATH] are (three-valued) truth-functional, which means that each [MATH]-ary connective represents a function from [MATH] to [MATH].', '1702.03414-1-17-0': 'For LP[MATH], the semantic logical consequence relation, denoted by [MATH], is based on the idea that a valuation [MATH] satisfies a formula [MATH] if [MATH].', '1702.03414-1-17-1': 'It is defined as follows: [MATH] iff for every valuation [MATH], either [MATH] for some [MATH] or [MATH].', '1702.03414-1-17-2': 'We have that the Hilbert-style formulation of LP[MATH] is strongly complete with respect to its semantics, i.e. [MATH] iff [MATH] (see e.g. [CITATION]).', '1702.03414-1-18-0': 'A formula of LP[MATH] in which the falsity constant [MATH] does not occur is called a [MATH]-free formula of LP[MATH].', '1702.03414-1-19-0': 'For all formulas [MATH] of LP[MATH] in which [MATH] does not occur, for all formulas [MATH] of LP[MATH] in which no propositional variable occurs that occurs in [MATH], [MATH] if [MATH] (see e.g. [CITATION]).', '1702.03414-1-20-0': 'Moreover, [MATH] is an extension of classical negation, i.e., for all valuations [MATH] and all formulas [MATH] of LP[MATH]: (a) [MATH] if [MATH], (b) [MATH] if [MATH].', '1702.03414-1-20-1': 'Hence, LP[MATH] is a paraconsistent logic.', '1702.03414-1-21-0': 'For LP[MATH], the logical equivalence relation [MATH] is defined as for classical propositional logic: [MATH] iff for every valuation [MATH], [MATH].', '1702.03414-1-21-1': 'Unlike in classical propositional logic, we do not have that [MATH] iff [MATH].', '1702.03414-1-22-0': 'For LP[MATH], the consistency property is defined as to be expected: [MATH] is consistent iff for every valuation [MATH], [MATH].', '1702.03414-1-23-0': '# Known Properties of LP[MATH]', '1702.03414-1-24-0': 'In this section, the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are presented.', '1702.03414-1-24-1': 'Each of the properties in question has to do with logical consequence relations.', '1702.03414-1-24-2': 'Like above, the symbol [MATH] is used to denote the logical consequence relation of LP[MATH].', '1702.03414-1-24-3': 'The symbol [MATH] is used to denote the logical consequence relation of classical propositional logic.', '1702.03414-1-25-0': 'The known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are:', '1702.03414-1-26-0': '(b) [(a)] containment in classical logic: [MATH]; [(b)] proper basic connectives: for all sets [MATH] of formulas of LP[MATH] and all formulas [MATH], [MATH], and [MATH] of LP[MATH]:', '1702.03414-1-27-0': '(c) [(b[MATH])] [MATH][MATH][MATH] iff [MATH], [(b[MATH])] [MATH],[MATH] iff [MATH] and [MATH], [(b[MATH])] [MATH] iff [MATH] and [MATH];', '1702.03414-1-28-0': '[(c)] weakly maximal paraconsistency relative to classical logic: for all formulas [MATH] of LP[MATH] with [MATH] and [MATH], for the minimal consequence relation [MATH] such that [MATH] and [MATH], for all formulas [MATH] of LP[MATH], [MATH] iff [MATH]; [(d)] strongly maximal absolute paraconsistency: for all propositional logics [MATH] with the same logical constants and connectives as LP[MATH] and a consequence relation [MATH] such that [MATH], [MATH] is not paraconsistent; [(e)] internalized notion of consistency: [MATH] is consistent iff [MATH]; [(f)] internalized notion of logical equivalence: [MATH] iff [MATH].', '1702.03414-1-29-0': 'Properties (a)-(f) have been mentioned relatively often in the literature (see e.g. [CITATION]).', '1702.03414-1-29-1': 'Properties (a), (b[MATH]), (c), and (d) make LP[MATH] an ideal paraconsistent logic in the sense made precise in [CITATION].', '1702.03414-1-29-2': 'By property (e), LP[MATH] is also a logic of formal inconsistency in the sense made precise in [CITATION].', '1702.03414-1-30-0': 'Properties (a)-(c) indicate that LP[MATH] retains much of classical propositional logic.', '1702.03414-1-30-1': 'Actually, property (c) can be strengthened to the following property: for all formulas [MATH] of LP[MATH], [MATH] iff [MATH].', '1702.03414-1-30-2': 'In [CITATION], properties (e) and (f) are considered essential and desirable, respectively, for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-1-31-0': 'From Theorem 4.42 in [CITATION], we know that there are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-1-31-1': 'From Theorem 2 in [CITATION], we know that properties (c) and (d) are common properties of all three-valued paraconsistent propositional logics with properties (a) and (b[MATH]).', '1702.03414-1-31-2': 'From Fact 103 in [CITATION], we know that property (f) is a common property of all three-valued paraconsistent propositional logics with properties (a), (b) and (e).', '1702.03414-1-31-3': 'Moreover, it is easy to see that that property (e) is a common property of all three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-1-31-4': 'Hence, each three-valued paraconsistent propositional logic with properties (a) and (b) has properties (c)-(f) as well.', '1702.03414-1-32-0': 'From Corollary 106 in [CITATION], we know that LP[MATH] is the strongest three-valued paraconsistent propositional logic with properties (a) and (b) in the sense that for each three-valued paraconsistent propositional logic with properties (a) and (b), there exists a translation into LP[MATH] that preserves and reflects its logical consequence relation.', '1702.03414-1-33-0': '# Characterizing LP[MATH] by Laws of Logical Equivalence', '1702.03414-1-34-0': 'There are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-1-34-1': 'This means that these properties, which concern the logical consequence relation of a logic, have little discriminating power.', '1702.03414-1-34-2': 'Properties (c)-(f), which also concern the logical consequence relation of a logic, do not offer additional discrimination because each of the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) has these properties as well.', '1702.03414-1-35-0': 'In this section, properties concerning the logical equivalence relation of a logic are used for additional discrimination.', '1702.03414-1-35-1': 'It turns out that 12 classical laws of logical equivalence, of which at least 9 are considered to belong to the most basic ones, are sufficient to distinguish LP[MATH] completely from the other 8191 three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-1-36-0': 'The logical equivalence relation of LP[MATH] satisfies all laws given in Table [REF].', '1702.03414-1-37-0': 'The logical equivalence relation of LP[MATH] satisfies laws (1)-(12) from Table [REF].', '1702.03414-1-38-0': 'The proof is very easy by constructing, for each of the laws concerned, truth tables for both sides.', '1702.03414-1-39-0': 'Moreover, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), LP[MATH] is the only one whose logical equivalence relation satisfies all laws given in Table [REF].', '1702.03414-1-40-0': 'There is exactly one three-valued paraconsistent propositional logic with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(12) from Table [REF].', '1702.03414-1-41-0': 'We confine ourselves to a brief outline of the proof.', '1702.03414-1-41-1': "Because 'non-deterministic truth tables' that uniquely characterize the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) are given in [CITATION],", '1702.03414-1-42-0': 'the theorem can be proved by showing that, for each of the connectives, only one of the ordinary truth tables represented by the non-deterministic truth table for that connective is compatible with the laws given in Table [REF].', '1702.03414-1-42-1': 'It can be shown by short routine case analyses that only one of the 8 ordinary truth tables represented by the non-deterministic truth tables for conjunction is compatible with laws (1), (3), (5), and (7), only one of the 32 ordinary truth tables represented by the non-deterministic truth tables for disjunction is compatible with laws (2), (4), (6), and (8), and only one of the 2 ordinary truth tables represented by the non-deterministic truth table for negation is compatible with law (9).', '1702.03414-1-42-2': 'Given the ordinary truth table for conjunction, disjunction, and negation so obtained, it can be shown by slightly longer routine case analyses that only one of the 16 ordinary truth tables represented by the non-deterministic truth table for implication is compatible with laws (10)-(12).', '1702.03414-1-43-0': 'The next two corollaries follow immediately from the proof of Theorem [REF].', '1702.03414-1-44-0': 'There are exactly 16 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(9) from Table [REF].', '1702.03414-1-45-0': 'There are exactly 32 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8) from Table [REF].', '1702.03414-1-46-0': 'From a paraconsistent propositional logic with properties (a) and (b), it is only to be expected, because of paraconsistency and property (b[MATH]), that its negation connective and its implication connective deviate clearly from their counterpart in classical propositional logic.', '1702.03414-1-46-1': 'Corollary [REF] shows that, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), there are 8160 logics whose logical equivalence relation does not even satisfy the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction (laws (1)-(8) from Table [REF]).', '1702.03414-1-47-0': "It turns out that the logical equivalence relation of LP[MATH] does not only satisfy the eight above-mentioned basic classical laws of logical equivalence but also other basic classical laws of logical equivalence, including the associative, distributive, and de Morgan's laws for conjunction and disjunction (laws (13)-(18) from Table [REF]).", '1702.03414-1-47-1': 'Indeed, the logical equivalence relation of LP[MATH] satisfies all laws given in Tables [REF] and [REF].', '1702.03414-1-48-0': 'Laws (10)-(12) and (19)-(20), like laws (1)-(9) and (13)-(18), are also satisfied by the logical equivalence relation of classical propositional logic.', '1702.03414-1-48-1': 'We have that [MATH] is satisfied by the logical equivalence relation of classical propositional logic iff it follows from laws (1)-(9) and (13)-(18) and the laws', '1702.03414-1-49-0': '(21) A B A B (22) A A (23) A A .', '1702.03414-1-50-0': 'Laws (10)-(12) and (19)-(20) do not follow from laws (1)-(9) and (13)-(18) alone, but laws (21)-(23) are not satisfied by the logical equivalence relation of LP[MATH].', '1702.03414-1-51-0': '# Concluding Remarks', '1702.03414-1-52-0': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish the logic LP[MATH] from the other logics that belong to the 8192 three-valued paraconsistent propositional logics that have properties (a)-(f) from Section [REF].', '1702.03414-1-52-1': 'These 8192 logics are considered potentially interesting because properties (a)-(f) are generally considered desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-1-53-0': 'Properties (a)-(f) concern the logical consequence relation of a logic.', '1702.03414-1-53-1': 'Unlike in classical propositional logic, we do not have [MATH] iff [MATH] and [MATH] in a three-valued paraconsistent propositional logic.', '1702.03414-1-53-2': 'As a consequence, the classical laws of logical equivalence that follow from properties (a) and (b) in classical propositional logic, viz. laws (1)-(8) and (13)-(16) from Section [REF], do not follow from properties (a) and (b) in a three-valued paraconsistent propositional logic.', '1702.03414-1-53-3': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies laws (1)-(8) and (13)-(16).', '1702.03414-1-54-0': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-1-55-0': 'From Fact 217 in [CITATION], we know that LP[MATH] is Blok-Pigozzi algebraizable.', '1702.03414-1-55-1': 'Although there must exist one, a conditional-equational axiomatization of the algebras concerned in the case of LP[MATH] has not yet been devised.'} | {'1702.03414-2-0-0': 'LP[MATH] is a three-valued paraconsistent propositional logic which is essentially the same as J3.', '1702.03414-2-0-1': 'It has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-2-0-2': 'However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-2-0-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-2-0-4': 'As one of the bonuses of focussing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction.', '1702.03414-2-1-0': '03B53, 03B50.', '1702.03414-2-2-0': '# Introduction', '1702.03414-2-3-0': 'A set of propositions is contradictory if there exists a proposition such that both that proposition and the negation of that proposition can be deduced from it.', '1702.03414-2-3-1': 'In classical propositional logic, every proposition can be deduced from every contradictory set of propositions.', '1702.03414-2-3-2': 'In a paraconsistent propositional logic, this is not the case.', '1702.03414-2-4-0': 'LP[MATH] is the three-valued paraconsistent propositional logic LP [CITATION] enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-2-4-1': 'This logic, which is essentially the same as J3 [CITATION], the propositional fragment of CLuNs [CITATION] without bi-implication, and LFI1 [CITATION], has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-2-4-2': 'However, it follows easily from results presented in [CITATION] that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-2-4-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-2-5-0': 'It turns out that only 32 of those 8192 logics are logics of which the logical equivalence relation satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction; and only 16 of them are logics of which the logical equivalence relation additionally satisfies the double negation law.', '1702.03414-2-5-1': 'LP[MATH] is one of those 16 logics.', '1702.03414-2-5-2': 'Three additional classical laws of logical equivalence turn out to be sufficient to distinguish LP[MATH] completely from the others.', '1702.03414-2-6-0': 'The desirable properties of reasonable paraconsistent propositional logics referred to above concern the logical consequence relation of a logic.', '1702.03414-2-6-1': 'Unlike in classical propositional logic, the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction do not follow from those properties in a three-valued paraconsistent propositional logic.', '1702.03414-2-6-2': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies these laws.', '1702.03414-2-6-3': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-2-7-0': 'The structure of this note is as follows.', '1702.03414-2-7-1': 'First, we give a survey of the paraconsistent propositional logic LP[MATH] (Section [REF]).', '1702.03414-2-7-2': 'Next, we discuss the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic (Section [REF]).', '1702.03414-2-7-3': 'After that, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the other three-valued paraconsistent propositional logics with the properties discussed earlier (Section [REF]).', '1702.03414-2-7-4': 'Finally, we make some concluding remarks (Section [REF]).', '1702.03414-2-8-0': 'It is relevant to realize that the work presented in this note is restricted to three-valued paraconsistent propositional logics that are truth-functional three-valued logics.', '1702.03414-2-9-0': 'There is text overlap between this note and [CITATION].', '1702.03414-2-9-1': 'This note generalizes and elaborates Section 2 of that paper in such a way that it may be of independent importance to the area of paraconsistent logics.', '1702.03414-2-10-0': '# The Paraconsistent Logic LP[MATH]', '1702.03414-2-11-0': 'A set of propositions [MATH] is contradictory if there exists a proposition [MATH] such that both [MATH] and [MATH] can be deduced from [MATH].', '1702.03414-2-11-1': 'In classical propositional logic, every proposition can be deduced from a contradictory set of propositions.', '1702.03414-2-11-2': 'A paraconsistent propositional logic is a propositional logic in which not every proposition can be deduced from every contradictory set of propositions.', '1702.03414-2-12-0': 'In [CITATION], Priest proposed the paraconsistent propositional logic LP (Logic of Paradox).', '1702.03414-2-12-1': 'The logic introduced in this section is LP enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-2-12-2': 'This logic, called LP[MATH], is in fact the propositional fragment of CLuNs [CITATION] without bi-implications.', '1702.03414-2-13-0': 'LP[MATH] has the following logical constants and connectives: a falsity constant [MATH], a unary negation connective [MATH], a binary conjunction connective [MATH], a binary disjunction connective [MATH], and a binary implication connective [MATH].', '1702.03414-2-13-1': 'Truth and bi-implication are defined as abbreviations: [MATH] stands for [MATH] and [MATH] stands for [MATH].', '1702.03414-2-14-0': 'A Hilbert-style formulation of LP[MATH] is given in Table [REF].', '1702.03414-2-15-0': 'In this formulation, which is taken from [CITATION], [MATH], [MATH], and [MATH] are used as meta-variables ranging over all formulas of LP[MATH].', '1702.03414-2-15-1': 'The axiom schemas on the left-hand side of Table [REF] and the single inference rule (modus ponens) constitute a Hilbert-style formulation of the positive fragment of classical propositional logic.', '1702.03414-2-15-2': 'The first four axiom schemas on the right-hand side of Table [REF] allow for the negation connective to be moved inward.', '1702.03414-2-15-3': 'The fifth axiom schema on the right-hand side of Table [REF] is the law of the excluded middle.', '1702.03414-2-15-4': 'This axiom schema can be thought of as saying that, for every proposition, the proposition or its negation is true, while leaving open the possibility that both are true.', '1702.03414-2-15-5': 'If we add the axiom schema [MATH], which says that any proposition follows from a contradiction, to the given Hilbert-style formulation of LP[MATH], then we get a Hilbert-style formulation of classical propositional logic (see e.g. [CITATION]).', '1702.03414-2-15-6': 'We use the symbol [MATH] without decoration to denote the syntactic logical consequence relation induced by the axiom schemas and inference rule of LP[MATH].', '1702.03414-2-16-0': 'The following outline of the semantics of LP[MATH] is based on [CITATION].', '1702.03414-2-16-1': 'Like in the case of classical propositional logic, meanings are assigned to the formulas of LP[MATH] by means of valuations.', '1702.03414-2-16-2': 'However, in addition to the two classical truth values [MATH] (true) and [MATH] (false), a third meaning [MATH] (both true and false) may be assigned.', '1702.03414-2-16-3': 'A valuation for LP[MATH] is a function [MATH] from the set of all formulas of LP[MATH] to the set [MATH] such that for all formulas [MATH] and [MATH] of LP[MATH]: [EQUATION]', '1702.03414-2-16-4': 'The classical truth-conditions and falsehood-conditions for the logical connectives are retained.', '1702.03414-2-16-5': 'Except for implications, a formula is classified as both-true-and-false exactly when it cannot be classified as true or false by the classical truth-conditions and falsehood-conditions.', '1702.03414-2-16-6': 'The definition of a valuation given above shows that the logical connectives of LP[MATH] are (three-valued) truth-functional, which means that each [MATH]-ary connective represents a function from [MATH] to [MATH].', '1702.03414-2-17-0': 'For LP[MATH], the semantic logical consequence relation, denoted by [MATH], is based on the idea that a valuation [MATH] satisfies a formula [MATH] if [MATH].', '1702.03414-2-17-1': 'It is defined as follows: [MATH] iff for every valuation [MATH], either [MATH] for some [MATH] or [MATH].', '1702.03414-2-17-2': 'We have that the Hilbert-style formulation of LP[MATH] is strongly complete with respect to its semantics, i.e. [MATH] iff [MATH] (see e.g. [CITATION]).', '1702.03414-2-18-0': 'A formula of LP[MATH] in which the falsity constant [MATH] does not occur is called a [MATH]-free formula of LP[MATH].', '1702.03414-2-19-0': 'For all formulas [MATH] of LP[MATH] in which [MATH] does not occur, for all formulas [MATH] of LP[MATH] in which no propositional variable occurs that occurs in [MATH], [MATH] if [MATH] (see e.g. [CITATION]).', '1702.03414-2-20-0': 'Moreover, [MATH] is an extension of classical negation, i.e., for all valuations [MATH] and all formulas [MATH] of LP[MATH]: (a) [MATH] if [MATH], (b) [MATH] if [MATH].', '1702.03414-2-20-1': 'Hence, LP[MATH] is a paraconsistent logic.', '1702.03414-2-21-0': 'For LP[MATH], the logical equivalence relation [MATH] is defined as for classical propositional logic: [MATH] iff for every valuation [MATH], [MATH].', '1702.03414-2-21-1': 'Unlike in classical propositional logic, we do not have that [MATH] iff [MATH].', '1702.03414-2-22-0': 'For LP[MATH], the consistency property is defined as to be expected: [MATH] is consistent iff for every valuation [MATH], [MATH].', '1702.03414-2-23-0': '# Known Properties of LP[MATH]', '1702.03414-2-24-0': 'In this section, the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are presented.', '1702.03414-2-24-1': 'Each of the properties in question has to do with logical consequence relations.', '1702.03414-2-24-2': 'Like above, the symbol [MATH] is used to denote the logical consequence relation of LP[MATH].', '1702.03414-2-24-3': 'The symbol [MATH] is used to denote the logical consequence relation of classical propositional logic.', '1702.03414-2-25-0': 'The known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are:', '1702.03414-2-26-0': '(b) [(a)] containment in classical logic: [MATH]; [(b)] proper basic connectives: for all sets [MATH] of formulas of LP[MATH] and all formulas [MATH], [MATH], and [MATH] of LP[MATH]:', '1702.03414-2-27-0': '(c) [(b[MATH])] [MATH][MATH][MATH] iff [MATH], [(b[MATH])] [MATH],[MATH] iff [MATH] and [MATH], [(b[MATH])] [MATH] iff [MATH] and [MATH];', '1702.03414-2-28-0': '[(c)] weakly maximal paraconsistency relative to classical logic: for all formulas [MATH] of LP[MATH] with [MATH] and [MATH], for the minimal consequence relation [MATH] such that [MATH] and [MATH], for all formulas [MATH] of LP[MATH], [MATH] iff [MATH]; [(d)] strongly maximal absolute paraconsistency: for all propositional logics [MATH] with the same logical constants and connectives as LP[MATH] and a consequence relation [MATH] such that [MATH], [MATH] is not paraconsistent; [(e)] internalized notion of consistency: [MATH] is consistent iff [MATH]; [(f)] internalized notion of logical equivalence: [MATH] iff [MATH].', '1702.03414-2-29-0': 'Properties (a)-(f) have been mentioned relatively often in the literature (see e.g. [CITATION]).', '1702.03414-2-29-1': 'Properties (a), (b[MATH]), (c), and (d) make LP[MATH] an ideal paraconsistent logic in the sense made precise in [CITATION].', '1702.03414-2-29-2': 'By property (e), LP[MATH] is also a logic of formal inconsistency in the sense made precise in [CITATION].', '1702.03414-2-30-0': 'Properties (a)-(c) indicate that LP[MATH] retains much of classical propositional logic.', '1702.03414-2-30-1': 'Actually, property (c) can be strengthened to the following property: for all formulas [MATH] of LP[MATH], [MATH] iff [MATH].', '1702.03414-2-30-2': 'In [CITATION], properties (e) and (f) are considered desirable and essential, respectively, for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-2-31-0': 'From Theorem 4.42 in [CITATION], we know that there are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-2-31-1': 'From Theorem 2 in [CITATION], we know that properties (c) and (d) are common properties of all three-valued paraconsistent propositional logics with properties (a) and (b[MATH]).', '1702.03414-2-31-2': 'From Fact 103 in [CITATION], we know that property (f) is a common property of all three-valued paraconsistent propositional logics with properties (a), (b) and (e).', '1702.03414-2-31-3': 'Moreover, it is easy to see that that property (e) is a common property of all three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-2-31-4': 'Hence, each three-valued paraconsistent propositional logic with properties (a) and (b) has properties (c)-(f) as well.', '1702.03414-2-32-0': 'From Corollary 106 in [CITATION], we know that LP[MATH] is the strongest three-valued paraconsistent propositional logic with properties (a) and (b) in the sense that for each three-valued paraconsistent propositional logic with properties (a) and (b), there exists a translation into LP[MATH] that preserves and reflects its logical consequence relation.', '1702.03414-2-33-0': '# Characterizing LP[MATH] by Laws of Logical Equivalence', '1702.03414-2-34-0': 'There are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-2-34-1': 'This means that these properties, which concern the logical consequence relation of a logic, have little discriminating power.', '1702.03414-2-34-2': 'Properties (c)-(f), which also concern the logical consequence relation of a logic, do not offer additional discrimination because each of the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) has these properties as well.', '1702.03414-2-35-0': 'In this section, properties concerning the logical equivalence relation of a logic are used for additional discrimination.', '1702.03414-2-35-1': 'It turns out that 12 classical laws of logical equivalence, of which at least 9 are considered to belong to the most basic ones, are sufficient to distinguish LP[MATH] completely from the other 8191 three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-2-36-0': 'The logical equivalence relation of LP[MATH] satisfies all laws given in Table [REF].', '1702.03414-2-37-0': 'The logical equivalence relation of LP[MATH] satisfies laws (1)-(12) from Table [REF].', '1702.03414-2-38-0': 'The proof is very easy by constructing, for each of the laws concerned, truth tables for both sides.', '1702.03414-2-39-0': 'Moreover, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), LP[MATH] is the only one whose logical equivalence relation satisfies all laws given in Table [REF].', '1702.03414-2-40-0': 'There is exactly one three-valued paraconsistent propositional logic with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(12) from Table [REF].', '1702.03414-2-41-0': 'We confine ourselves to a brief outline of the proof.', '1702.03414-2-41-1': "Because 'non-deterministic truth tables' that uniquely characterize the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) are given in [CITATION],", '1702.03414-2-42-0': 'the theorem can be proved by showing that, for each of the connectives, only one of the ordinary truth tables represented by the non-deterministic truth table for that connective is compatible with the laws given in Table [REF].', '1702.03414-2-42-1': 'It can be shown by short routine case analyses that only one of the 8 ordinary truth tables represented by the non-deterministic truth tables for conjunction is compatible with laws (1), (3), (5), and (7), only one of the 32 ordinary truth tables represented by the non-deterministic truth tables for disjunction is compatible with laws (2), (4), (6), and (8), and only one of the 2 ordinary truth tables represented by the non-deterministic truth table for negation is compatible with law (9).', '1702.03414-2-42-2': 'Given the ordinary truth table for conjunction, disjunction, and negation so obtained, it can be shown by slightly longer routine case analyses that only one of the 16 ordinary truth tables represented by the non-deterministic truth table for implication is compatible with laws (10)-(12).', '1702.03414-2-43-0': 'The next two corollaries follow immediately from the proof of Theorem [REF].', '1702.03414-2-44-0': 'There are exactly 16 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(9) from Table [REF].', '1702.03414-2-45-0': 'There are exactly 32 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8) from Table [REF].', '1702.03414-2-46-0': 'From a paraconsistent propositional logic with properties (a) and (b), it is only to be expected, because of paraconsistency and property (b[MATH]), that its negation connective and its implication connective deviate clearly from their counterpart in classical propositional logic.', '1702.03414-2-46-1': 'Corollary [REF] shows that, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), there are 8160 logics whose logical equivalence relation does not even satisfy the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction (laws (1)-(8) from Table [REF]).', '1702.03414-2-47-0': "It turns out that the logical equivalence relation of LP[MATH] does not only satisfy the eight above-mentioned basic classical laws of logical equivalence but also other basic classical laws of logical equivalence, including the associative, distributive, and de Morgan's laws for conjunction and disjunction (laws (13)-(18) from Table [REF]).", '1702.03414-2-47-1': 'Indeed, the logical equivalence relation of LP[MATH] satisfies all laws given in Tables [REF] and [REF].', '1702.03414-2-48-0': 'Laws (10)-(12) and (19)-(20), like laws (1)-(9) and (13)-(18), are also satisfied by the logical equivalence relation of classical propositional logic.', '1702.03414-2-48-1': 'We have that [MATH] is satisfied by the logical equivalence relation of classical propositional logic iff it follows from laws (1)-(9) and (13)-(18) and the laws', '1702.03414-2-49-0': '(21) A B A B (22) A A (23) A A .', '1702.03414-2-50-0': 'Laws (10)-(12) and (19)-(20) do not follow from laws (1)-(9) and (13)-(18) alone, but laws (21)-(23) are not satisfied by the logical equivalence relation of LP[MATH].', '1702.03414-2-51-0': '# Concluding Remarks', '1702.03414-2-52-0': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish the logic LP[MATH] from the other logics that belong to the 8192 three-valued paraconsistent propositional logics that have properties (a)-(f) from Section [REF].', '1702.03414-2-52-1': 'These 8192 logics are considered potentially interesting because properties (a)-(f) are generally considered desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-2-53-0': 'Properties (a)-(f) concern the logical consequence relation of a logic.', '1702.03414-2-53-1': 'Unlike in classical propositional logic, we do not have [MATH] iff [MATH] and [MATH] in a three-valued paraconsistent propositional logic.', '1702.03414-2-53-2': 'As a consequence, the classical laws of logical equivalence that follow from properties (a) and (b) in classical propositional logic, viz. laws (1)-(8) and (13)-(16) from Section [REF], do not follow from properties (a) and (b) in a three-valued paraconsistent propositional logic.', '1702.03414-2-53-3': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies laws (1)-(8) and (13)-(16).', '1702.03414-2-54-0': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-2-55-0': 'In [CITATION], satisfaction of laws (1)-(8) and (10)-(12) is considered essential for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-2-55-1': 'It follows easily from Theorem [REF] and the proof of Theorem [REF] that LP[MATH] is one of only four three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8) and (10)-(12).', '1702.03414-2-56-0': 'From Fact 217 in [CITATION], we know that LP[MATH] is Blok-Pigozzi algebraizable.', '1702.03414-2-56-1': 'Although there must exist one, a conditional-equational axiomatization of the algebras concerned in the case of LP[MATH] has not yet been devised.'} | [['1702.03414-1-30-0', '1702.03414-2-30-0'], ['1702.03414-1-30-1', '1702.03414-2-30-1'], ['1702.03414-1-55-0', '1702.03414-2-56-0'], 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'1702.03414-3-19-0'], ['1702.03414-2-9-0', '1702.03414-3-9-0'], ['1702.03414-2-9-1', '1702.03414-3-9-1'], ['1702.03414-2-8-0', '1702.03414-3-8-0'], ['1702.03414-2-38-0', '1702.03414-3-38-0'], ['1702.03414-2-36-0', '1702.03414-3-36-0'], ['1702.03414-2-46-0', '1702.03414-3-46-0'], ['1702.03414-2-46-1', '1702.03414-3-46-1'], ['1702.03414-2-35-0', '1702.03414-3-35-0'], ['1702.03414-2-43-0', '1702.03414-3-43-0'], ['1702.03414-2-15-0', '1702.03414-3-15-0'], ['1702.03414-2-15-1', '1702.03414-3-15-1'], ['1702.03414-2-15-2', '1702.03414-3-15-2'], ['1702.03414-2-15-3', '1702.03414-3-15-3'], ['1702.03414-2-15-4', '1702.03414-3-15-4'], ['1702.03414-2-15-5', '1702.03414-3-15-5'], ['1702.03414-2-15-6', '1702.03414-3-15-6'], ['1702.03414-2-11-0', '1702.03414-3-11-0'], ['1702.03414-2-11-1', '1702.03414-3-11-1'], ['1702.03414-2-11-2', '1702.03414-3-11-2'], ['1702.03414-2-31-0', '1702.03414-3-31-0'], ['1702.03414-2-31-1', '1702.03414-3-31-1'], ['1702.03414-2-31-2', '1702.03414-3-31-2'], ['1702.03414-2-31-3', '1702.03414-3-31-3'], ['1702.03414-2-31-4', '1702.03414-3-31-4'], ['1702.03414-3-7-0', '1702.03414-4-7-0'], ['1702.03414-3-7-1', '1702.03414-4-7-1'], ['1702.03414-3-7-2', '1702.03414-4-7-2'], ['1702.03414-3-7-4', '1702.03414-4-7-5'], ['1702.03414-3-35-0', '1702.03414-4-35-0'], ['1702.03414-3-35-1', '1702.03414-4-35-1'], ['1702.03414-3-15-1', '1702.03414-4-15-1'], ['1702.03414-3-15-2', '1702.03414-4-15-2'], ['1702.03414-3-15-3', '1702.03414-4-15-3'], ['1702.03414-3-15-4', '1702.03414-4-15-4'], ['1702.03414-3-15-5', '1702.03414-4-15-5'], ['1702.03414-3-15-6', '1702.03414-4-15-6'], ['1702.03414-3-30-0', '1702.03414-4-30-0'], ['1702.03414-3-30-1', '1702.03414-4-30-1'], ['1702.03414-3-30-2', '1702.03414-4-30-2'], ['1702.03414-3-38-0', '1702.03414-4-38-0'], ['1702.03414-3-8-0', '1702.03414-4-8-0'], ['1702.03414-3-37-0', '1702.03414-4-37-0'], ['1702.03414-3-31-0', '1702.03414-4-31-0'], ['1702.03414-3-31-1', '1702.03414-4-31-1'], ['1702.03414-3-31-2', '1702.03414-4-31-2'], ['1702.03414-3-31-3', '1702.03414-4-31-3'], ['1702.03414-3-31-4', '1702.03414-4-31-4'], ['1702.03414-3-47-1', '1702.03414-4-48-1'], ['1702.03414-3-24-0', '1702.03414-4-24-0'], ['1702.03414-3-24-1', '1702.03414-4-24-1'], ['1702.03414-3-24-2', '1702.03414-4-24-2'], ['1702.03414-3-24-3', '1702.03414-4-24-3'], ['1702.03414-3-16-0', '1702.03414-4-16-0'], ['1702.03414-3-16-1', '1702.03414-4-16-1'], ['1702.03414-3-16-2', '1702.03414-4-16-2'], ['1702.03414-3-16-3', '1702.03414-4-16-3'], ['1702.03414-3-16-4', '1702.03414-4-16-4'], ['1702.03414-3-16-5', '1702.03414-4-16-5'], ['1702.03414-3-16-6', '1702.03414-4-16-6'], ['1702.03414-3-28-0', '1702.03414-4-28-0'], ['1702.03414-3-17-0', '1702.03414-4-17-0'], ['1702.03414-3-17-1', '1702.03414-4-17-1'], ['1702.03414-3-17-2', '1702.03414-4-17-2'], ['1702.03414-3-5-0', '1702.03414-4-5-0'], ['1702.03414-3-5-1', '1702.03414-4-5-1'], ['1702.03414-3-5-2', '1702.03414-4-5-2'], ['1702.03414-3-42-0', '1702.03414-4-42-0'], ['1702.03414-3-42-1', '1702.03414-4-42-1'], ['1702.03414-3-42-2', '1702.03414-4-42-2'], ['1702.03414-3-45-0', '1702.03414-4-45-0'], ['1702.03414-3-39-0', '1702.03414-4-39-0'], ['1702.03414-3-22-0', '1702.03414-4-22-0'], ['1702.03414-3-19-0', '1702.03414-4-19-0'], ['1702.03414-3-53-0', '1702.03414-4-56-0'], ['1702.03414-3-53-1', '1702.03414-4-56-1'], ['1702.03414-3-9-0', '1702.03414-4-9-0'], ['1702.03414-3-9-1', '1702.03414-4-9-1'], ['1702.03414-3-21-0', '1702.03414-4-21-0'], ['1702.03414-3-21-1', '1702.03414-4-21-1'], ['1702.03414-3-18-0', '1702.03414-4-18-0'], ['1702.03414-3-46-0', '1702.03414-4-46-0'], ['1702.03414-3-46-1', '1702.03414-4-46-1'], ['1702.03414-3-36-0', '1702.03414-4-36-0'], ['1702.03414-3-4-0', '1702.03414-4-4-0'], ['1702.03414-3-4-1', '1702.03414-4-4-1'], ['1702.03414-3-4-2', '1702.03414-4-4-2'], ['1702.03414-3-4-3', '1702.03414-4-4-3'], ['1702.03414-3-34-0', '1702.03414-4-34-0'], ['1702.03414-3-34-1', '1702.03414-4-34-1'], ['1702.03414-3-34-2', '1702.03414-4-34-2'], ['1702.03414-3-12-0', '1702.03414-4-12-0'], ['1702.03414-3-12-1', '1702.03414-4-12-1'], ['1702.03414-3-12-2', '1702.03414-4-12-2'], ['1702.03414-3-52-0', '1702.03414-4-55-0'], ['1702.03414-3-52-1', '1702.03414-4-55-1'], ['1702.03414-3-6-0', '1702.03414-4-6-0'], ['1702.03414-3-6-1', '1702.03414-4-6-1'], ['1702.03414-3-6-2', '1702.03414-4-6-2'], ['1702.03414-3-6-3', '1702.03414-4-6-3'], ['1702.03414-3-44-0', '1702.03414-4-44-0'], ['1702.03414-3-13-0', '1702.03414-4-13-0'], ['1702.03414-3-13-1', '1702.03414-4-13-1'], ['1702.03414-3-54-0', '1702.03414-4-57-0'], ['1702.03414-3-56-0', '1702.03414-4-59-0'], ['1702.03414-3-56-1', '1702.03414-4-59-1'], ['1702.03414-3-40-0', '1702.03414-4-40-0'], ['1702.03414-3-32-0', '1702.03414-4-32-0'], ['1702.03414-3-11-0', '1702.03414-4-11-0'], ['1702.03414-3-11-1', '1702.03414-4-11-1'], ['1702.03414-3-11-2', '1702.03414-4-11-2'], ['1702.03414-3-20-0', '1702.03414-4-20-0'], ['1702.03414-3-20-1', '1702.03414-4-20-1'], ['1702.03414-3-3-0', '1702.03414-4-3-0'], ['1702.03414-3-3-1', '1702.03414-4-3-1'], ['1702.03414-3-3-2', '1702.03414-4-3-2'], ['1702.03414-3-29-0', '1702.03414-4-29-0'], ['1702.03414-3-29-1', '1702.03414-4-29-1'], ['1702.03414-3-29-2', '1702.03414-4-29-2'], ['1702.03414-3-0-0', '1702.03414-4-0-0'], ['1702.03414-3-0-1', '1702.03414-4-0-1'], ['1702.03414-3-0-2', '1702.03414-4-0-2'], ['1702.03414-3-0-3', '1702.03414-4-0-3'], ['1702.03414-3-0-4', '1702.03414-4-0-4']] | [['1702.03414-1-30-2', '1702.03414-2-30-2'], ['1702.03414-2-53-2', '1702.03414-3-53-2'], ['1702.03414-2-53-3', '1702.03414-3-53-3'], ['1702.03414-2-42-2', '1702.03414-3-42-2'], ['1702.03414-2-47-0', '1702.03414-3-47-0'], ['1702.03414-2-40-0', '1702.03414-3-40-0'], ['1702.03414-2-5-2', '1702.03414-3-5-2'], ['1702.03414-2-48-0', '1702.03414-3-48-0'], ['1702.03414-2-48-1', '1702.03414-3-48-1'], ['1702.03414-2-55-0', '1702.03414-3-55-0'], ['1702.03414-2-55-1', '1702.03414-3-55-1'], ['1702.03414-2-37-0', '1702.03414-3-37-0'], ['1702.03414-2-35-1', '1702.03414-3-35-1'], ['1702.03414-3-7-3', '1702.03414-4-7-3'], ['1702.03414-3-15-0', '1702.03414-4-15-0'], ['1702.03414-3-48-1', '1702.03414-4-49-3'], ['1702.03414-3-47-0', '1702.03414-4-48-0'], ['1702.03414-3-43-0', '1702.03414-4-43-1'], ['1702.03414-3-53-2', '1702.03414-4-56-2'], ['1702.03414-3-53-3', '1702.03414-4-56-3'], ['1702.03414-3-55-0', '1702.03414-4-58-0'], ['1702.03414-3-55-1', '1702.03414-4-58-1']] | [] | [['1702.03414-3-48-0', '1702.03414-4-49-2']] | [] | ['1702.03414-1-1-0', '1702.03414-1-14-0', '1702.03414-1-25-0', '1702.03414-1-26-0', '1702.03414-1-27-0', '1702.03414-1-41-0', '1702.03414-1-41-1', '1702.03414-1-49-0', '1702.03414-1-50-0', '1702.03414-2-1-0', '1702.03414-2-14-0', '1702.03414-2-25-0', '1702.03414-2-26-0', '1702.03414-2-27-0', '1702.03414-2-41-0', '1702.03414-2-41-1', '1702.03414-2-49-0', '1702.03414-2-50-0', '1702.03414-3-1-0', '1702.03414-3-14-0', '1702.03414-3-25-0', '1702.03414-3-26-0', '1702.03414-3-27-0', '1702.03414-3-41-0', '1702.03414-3-41-1', '1702.03414-3-49-0', '1702.03414-3-50-0', '1702.03414-4-1-0', '1702.03414-4-14-0', '1702.03414-4-25-0', '1702.03414-4-26-0', '1702.03414-4-27-0', '1702.03414-4-41-0', '1702.03414-4-41-1', '1702.03414-4-49-0', '1702.03414-4-50-0', '1702.03414-4-52-0', '1702.03414-4-52-1', '1702.03414-4-52-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1702.03414 | {'1702.03414-3-0-0': 'LP[MATH] is a three-valued paraconsistent propositional logic which is essentially the same as J3.', '1702.03414-3-0-1': 'It has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-3-0-2': 'However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-3-0-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-3-0-4': 'As one of the bonuses of focussing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction.', '1702.03414-3-1-0': '03B53, 03B50.', '1702.03414-3-2-0': '# Introduction', '1702.03414-3-3-0': 'A set of propositions is contradictory if there exists a proposition such that both that proposition and the negation of that proposition can be deduced from it.', '1702.03414-3-3-1': 'In classical propositional logic, every proposition can be deduced from every contradictory set of propositions.', '1702.03414-3-3-2': 'In a paraconsistent propositional logic, this is not the case.', '1702.03414-3-4-0': 'LP[MATH] is the three-valued paraconsistent propositional logic LP [CITATION] enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-3-4-1': 'This logic, which is essentially the same as J3 [CITATION], the propositional fragment of CLuNs [CITATION] without bi-implication, and LFI1 [CITATION], has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-3-4-2': 'However, it follows easily from results presented in [CITATION] that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-3-4-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-3-5-0': 'It turns out that only 32 of those 8192 logics are logics of which the logical equivalence relation satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction; and only 16 of them are logics of which the logical equivalence relation additionally satisfies the double negation law.', '1702.03414-3-5-1': 'LP[MATH] is one of those 16 logics.', '1702.03414-3-5-2': 'Two additional classical laws of logical equivalence turn out to be sufficient to distinguish LP[MATH] completely from the others.', '1702.03414-3-6-0': 'The desirable properties of reasonable paraconsistent propositional logics referred to above concern the logical consequence relation of a logic.', '1702.03414-3-6-1': 'Unlike in classical propositional logic, the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction do not follow from those properties in a three-valued paraconsistent propositional logic.', '1702.03414-3-6-2': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies these laws.', '1702.03414-3-6-3': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-3-7-0': 'The structure of this note is as follows.', '1702.03414-3-7-1': 'First, we give a survey of the paraconsistent propositional logic LP[MATH] (Section [REF]).', '1702.03414-3-7-2': 'Next, we discuss the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic (Section [REF]).', '1702.03414-3-7-3': 'After that, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the other three-valued paraconsistent propositional logics with the properties discussed earlier (Section [REF]).', '1702.03414-3-7-4': 'Finally, we make some concluding remarks (Section [REF]).', '1702.03414-3-8-0': 'It is relevant to realize that the work presented in this note is restricted to three-valued paraconsistent propositional logics that are truth-functional three-valued logics.', '1702.03414-3-9-0': 'There is text overlap between this note and [CITATION].', '1702.03414-3-9-1': 'This note generalizes and elaborates Section 2 of that paper in such a way that it may be of independent importance to the area of paraconsistent logics.', '1702.03414-3-10-0': '# The Paraconsistent Logic LP[MATH]', '1702.03414-3-11-0': 'A set of propositions [MATH] is contradictory if there exists a proposition [MATH] such that both [MATH] and [MATH] can be deduced from [MATH].', '1702.03414-3-11-1': 'In classical propositional logic, every proposition can be deduced from a contradictory set of propositions.', '1702.03414-3-11-2': 'A paraconsistent propositional logic is a propositional logic in which not every proposition can be deduced from every contradictory set of propositions.', '1702.03414-3-12-0': 'In [CITATION], Priest proposed the paraconsistent propositional logic LP (Logic of Paradox).', '1702.03414-3-12-1': 'The logic introduced in this section is LP enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-3-12-2': 'This logic, called LP[MATH], is in fact the propositional fragment of CLuNs [CITATION] without bi-implications.', '1702.03414-3-13-0': 'LP[MATH] has the following logical constants and connectives: a falsity constant [MATH], a unary negation connective [MATH], a binary conjunction connective [MATH], a binary disjunction connective [MATH], and a binary implication connective [MATH].', '1702.03414-3-13-1': 'Truth and bi-implication are defined as abbreviations: [MATH] stands for [MATH] and [MATH] stands for [MATH].', '1702.03414-3-14-0': 'A Hilbert-style formulation of LP[MATH] is given in Table [REF].', '1702.03414-3-15-0': 'In this formulation, which is taken from [CITATION], [MATH], [MATH], and [MATH] are used as meta-variables ranging over all formulas of LP[MATH].', '1702.03414-3-15-1': 'The axiom schemas on the left-hand side of Table [REF] and the single inference rule (modus ponens) constitute a Hilbert-style formulation of the positive fragment of classical propositional logic.', '1702.03414-3-15-2': 'The first four axiom schemas on the right-hand side of Table [REF] allow for the negation connective to be moved inward.', '1702.03414-3-15-3': 'The fifth axiom schema on the right-hand side of Table [REF] is the law of the excluded middle.', '1702.03414-3-15-4': 'This axiom schema can be thought of as saying that, for every proposition, the proposition or its negation is true, while leaving open the possibility that both are true.', '1702.03414-3-15-5': 'If we add the axiom schema [MATH], which says that any proposition follows from a contradiction, to the given Hilbert-style formulation of LP[MATH], then we get a Hilbert-style formulation of classical propositional logic (see e.g. [CITATION]).', '1702.03414-3-15-6': 'We use the symbol [MATH] without decoration to denote the syntactic logical consequence relation induced by the axiom schemas and inference rule of LP[MATH].', '1702.03414-3-16-0': 'The following outline of the semantics of LP[MATH] is based on [CITATION].', '1702.03414-3-16-1': 'Like in the case of classical propositional logic, meanings are assigned to the formulas of LP[MATH] by means of valuations.', '1702.03414-3-16-2': 'However, in addition to the two classical truth values [MATH] (true) and [MATH] (false), a third meaning [MATH] (both true and false) may be assigned.', '1702.03414-3-16-3': 'A valuation for LP[MATH] is a function [MATH] from the set of all formulas of LP[MATH] to the set [MATH] such that for all formulas [MATH] and [MATH] of LP[MATH]: [EQUATION]', '1702.03414-3-16-4': 'The classical truth-conditions and falsehood-conditions for the logical connectives are retained.', '1702.03414-3-16-5': 'Except for implications, a formula is classified as both-true-and-false exactly when it cannot be classified as true or false by the classical truth-conditions and falsehood-conditions.', '1702.03414-3-16-6': 'The definition of a valuation given above shows that the logical connectives of LP[MATH] are (three-valued) truth-functional, which means that each [MATH]-ary connective represents a function from [MATH] to [MATH].', '1702.03414-3-17-0': 'For LP[MATH], the semantic logical consequence relation, denoted by [MATH], is based on the idea that a valuation [MATH] satisfies a formula [MATH] if [MATH].', '1702.03414-3-17-1': 'It is defined as follows: [MATH] iff for every valuation [MATH], either [MATH] for some [MATH] or [MATH].', '1702.03414-3-17-2': 'We have that the Hilbert-style formulation of LP[MATH] is strongly complete with respect to its semantics, i.e. [MATH] iff [MATH] (see e.g. [CITATION]).', '1702.03414-3-18-0': 'A formula of LP[MATH] in which the falsity constant [MATH] does not occur is called a [MATH]-free formula of LP[MATH].', '1702.03414-3-19-0': 'For all formulas [MATH] of LP[MATH] in which [MATH] does not occur, for all formulas [MATH] of LP[MATH] in which no propositional variable occurs that occurs in [MATH], [MATH] if [MATH] (see e.g. [CITATION]).', '1702.03414-3-20-0': 'Moreover, [MATH] is an extension of classical negation, i.e., for all valuations [MATH] and all formulas [MATH] of LP[MATH]: (a) [MATH] if [MATH], (b) [MATH] if [MATH].', '1702.03414-3-20-1': 'Hence, LP[MATH] is a paraconsistent logic.', '1702.03414-3-21-0': 'For LP[MATH], the logical equivalence relation [MATH] is defined as for classical propositional logic: [MATH] iff for every valuation [MATH], [MATH].', '1702.03414-3-21-1': 'Unlike in classical propositional logic, we do not have that [MATH] iff [MATH].', '1702.03414-3-22-0': 'For LP[MATH], the consistency property is defined as to be expected: [MATH] is consistent iff for every valuation [MATH], [MATH].', '1702.03414-3-23-0': '# Known Properties of LP[MATH]', '1702.03414-3-24-0': 'In this section, the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are presented.', '1702.03414-3-24-1': 'Each of the properties in question has to do with logical consequence relations.', '1702.03414-3-24-2': 'Like above, the symbol [MATH] is used to denote the logical consequence relation of LP[MATH].', '1702.03414-3-24-3': 'The symbol [MATH] is used to denote the logical consequence relation of classical propositional logic.', '1702.03414-3-25-0': 'The known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are:', '1702.03414-3-26-0': '(b) [(a)] containment in classical logic: [MATH]; [(b)] proper basic connectives: for all sets [MATH] of formulas of LP[MATH] and all formulas [MATH], [MATH], and [MATH] of LP[MATH]:', '1702.03414-3-27-0': '(c) [(b[MATH])] [MATH][MATH][MATH] iff [MATH], [(b[MATH])] [MATH],[MATH] iff [MATH] and [MATH], [(b[MATH])] [MATH] iff [MATH] and [MATH];', '1702.03414-3-28-0': '[(c)] weakly maximal paraconsistency relative to classical logic: for all formulas [MATH] of LP[MATH] with [MATH] and [MATH], for the minimal consequence relation [MATH] such that [MATH] and [MATH], for all formulas [MATH] of LP[MATH], [MATH] iff [MATH]; [(d)] strongly maximal absolute paraconsistency: for all propositional logics [MATH] with the same logical constants and connectives as LP[MATH] and a consequence relation [MATH] such that [MATH], [MATH] is not paraconsistent; [(e)] internalized notion of consistency: [MATH] is consistent iff [MATH]; [(f)] internalized notion of logical equivalence: [MATH] iff [MATH].', '1702.03414-3-29-0': 'Properties (a)-(f) have been mentioned relatively often in the literature (see e.g. [CITATION]).', '1702.03414-3-29-1': 'Properties (a), (b[MATH]), (c), and (d) make LP[MATH] an ideal paraconsistent logic in the sense made precise in [CITATION].', '1702.03414-3-29-2': 'By property (e), LP[MATH] is also a logic of formal inconsistency in the sense made precise in [CITATION].', '1702.03414-3-30-0': 'Properties (a)-(c) indicate that LP[MATH] retains much of classical propositional logic.', '1702.03414-3-30-1': 'Actually, property (c) can be strengthened to the following property: for all formulas [MATH] of LP[MATH], [MATH] iff [MATH].', '1702.03414-3-30-2': 'In [CITATION], properties (e) and (f) are considered desirable and essential, respectively, for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-3-31-0': 'From Theorem 4.42 in [CITATION], we know that there are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-3-31-1': 'From Theorem 2 in [CITATION], we know that properties (c) and (d) are common properties of all three-valued paraconsistent propositional logics with properties (a) and (b[MATH]).', '1702.03414-3-31-2': 'From Fact 103 in [CITATION], we know that property (f) is a common property of all three-valued paraconsistent propositional logics with properties (a), (b) and (e).', '1702.03414-3-31-3': 'Moreover, it is easy to see that that property (e) is a common property of all three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-3-31-4': 'Hence, each three-valued paraconsistent propositional logic with properties (a) and (b) has properties (c)-(f) as well.', '1702.03414-3-32-0': 'From Corollary 106 in [CITATION], we know that LP[MATH] is the strongest three-valued paraconsistent propositional logic with properties (a) and (b) in the sense that for each three-valued paraconsistent propositional logic with properties (a) and (b), there exists a translation into LP[MATH] that preserves and reflects its logical consequence relation.', '1702.03414-3-33-0': '# Characterizing LP[MATH] by Laws of Logical Equivalence', '1702.03414-3-34-0': 'There are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-3-34-1': 'This means that these properties, which concern the logical consequence relation of a logic, have little discriminating power.', '1702.03414-3-34-2': 'Properties (c)-(f), which also concern the logical consequence relation of a logic, do not offer additional discrimination because each of the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) has these properties as well.', '1702.03414-3-35-0': 'In this section, properties concerning the logical equivalence relation of a logic are used for additional discrimination.', '1702.03414-3-35-1': 'It turns out that 11 classical laws of logical equivalence, of which at least 9 are considered to belong to the most basic ones, are sufficient to distinguish LP[MATH] completely from the other 8191 three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-3-36-0': 'The logical equivalence relation of LP[MATH] satisfies all laws given in Table [REF].', '1702.03414-3-37-0': 'The logical equivalence relation of LP[MATH] satisfies laws (1)-(11) from Table [REF].', '1702.03414-3-38-0': 'The proof is very easy by constructing, for each of the laws concerned, truth tables for both sides.', '1702.03414-3-39-0': 'Moreover, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), LP[MATH] is the only one whose logical equivalence relation satisfies all laws given in Table [REF].', '1702.03414-3-40-0': 'There is exactly one three-valued paraconsistent propositional logic with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(11) from Table [REF].', '1702.03414-3-41-0': 'We confine ourselves to a brief outline of the proof.', '1702.03414-3-41-1': "Because 'non-deterministic truth tables' that uniquely characterize the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) are given in [CITATION],", '1702.03414-3-42-0': 'the theorem can be proved by showing that, for each of the connectives, only one of the ordinary truth tables represented by the non-deterministic truth table for that connective is compatible with the laws given in Table [REF].', '1702.03414-3-42-1': 'It can be shown by short routine case analyses that only one of the 8 ordinary truth tables represented by the non-deterministic truth tables for conjunction is compatible with laws (1), (3), (5), and (7), only one of the 32 ordinary truth tables represented by the non-deterministic truth tables for disjunction is compatible with laws (2), (4), (6), and (8), and only one of the 2 ordinary truth tables represented by the non-deterministic truth table for negation is compatible with law (9).', '1702.03414-3-42-2': 'Given the ordinary truth table for conjunction, disjunction, and negation so obtained, it can be shown by short routine case analyses that only one of the 16 ordinary truth tables represented by the non-deterministic truth table for implication is compatible with laws (10) and (11).', '1702.03414-3-43-0': 'The next two corollaries follow immediately from the proof of Theorem [REF].', '1702.03414-3-44-0': 'There are exactly 16 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(9) from Table [REF].', '1702.03414-3-45-0': 'There are exactly 32 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8) from Table [REF].', '1702.03414-3-46-0': 'From a paraconsistent propositional logic with properties (a) and (b), it is only to be expected, because of paraconsistency and property (b[MATH]), that its negation connective and its implication connective deviate clearly from their counterpart in classical propositional logic.', '1702.03414-3-46-1': 'Corollary [REF] shows that, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), there are 8160 logics whose logical equivalence relation does not even satisfy the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction (laws (1)-(8) from Table [REF]).', '1702.03414-3-47-0': "It turns out that the logical equivalence relation of LP[MATH] does not only satisfy the eight above-mentioned basic classical laws of logical equivalence for conjunction and disjunction but also other basic classical laws of logical equivalence for conjunction and disjunction, including the associative, distributive, and de Morgan's laws (laws (12)-(17) from Table [REF]).", '1702.03414-3-47-1': 'Indeed, the logical equivalence relation of LP[MATH] satisfies all laws given in Tables [REF] and [REF].', '1702.03414-3-48-0': 'Laws (10), (11), and (18)-(20), like laws (1)-(9) and (12)-(17), are also satisfied by the logical equivalence relation of classical propositional logic.', '1702.03414-3-48-1': 'We have that [MATH] is satisfied by the logical equivalence relation of classical propositional logic iff it follows from laws (1)-(9) and (12)-(17) and the laws', '1702.03414-3-49-0': '(21) A A (22) A A (23) A B A B .', '1702.03414-3-50-0': 'Laws (10), (11), and (18)-(20) do not follow from laws (1)-(9) and (12)-(17) alone, but laws (21)-(23) are not satisfied by the logical equivalence relation of LP[MATH].', '1702.03414-3-51-0': '# Concluding Remarks', '1702.03414-3-52-0': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish the logic LP[MATH] from the other logics that belong to the 8192 three-valued paraconsistent propositional logics that have properties (a)-(f) from Section [REF].', '1702.03414-3-52-1': 'These 8192 logics are considered potentially interesting because properties (a)-(f) are generally considered desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-3-53-0': 'Properties (a)-(f) concern the logical consequence relation of a logic.', '1702.03414-3-53-1': 'Unlike in classical propositional logic, we do not have [MATH] iff [MATH] and [MATH] in a three-valued paraconsistent propositional logic.', '1702.03414-3-53-2': 'As a consequence, the classical laws of logical equivalence that follow from properties (a) and (b) in classical propositional logic, viz. laws (1)-(8) and (12)-(15) from Section [REF], do not follow from properties (a) and (b) in a three-valued paraconsistent propositional logic.', '1702.03414-3-53-3': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies laws (1)-(8) and (12)-(15).', '1702.03414-3-54-0': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-3-55-0': 'In [CITATION], satisfaction of laws (1)-(8), (11)-(15), and (18)-(20) is considered essential for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-3-55-1': 'It follows easily from Theorem [REF] and the proof of Theorem [REF] that LP[MATH] is one of only four three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8), (11)-(15), and (18)-(20).', '1702.03414-3-56-0': 'From Fact 217 in [CITATION], we know that LP[MATH] is Blok-Pigozzi algebraizable.', '1702.03414-3-56-1': 'Although there must exist one, a conditional-equational axiomatization of the algebras concerned in the case of LP[MATH] has not yet been devised.'} | {'1702.03414-4-0-0': 'LP[MATH] is a three-valued paraconsistent propositional logic which is essentially the same as J3.', '1702.03414-4-0-1': 'It has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-4-0-2': 'However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-4-0-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-4-0-4': 'As one of the bonuses of focussing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction.', '1702.03414-4-1-0': '03B53, 03B50.', '1702.03414-4-2-0': '# Introduction', '1702.03414-4-3-0': 'A set of propositions is contradictory if there exists a proposition such that both that proposition and the negation of that proposition can be deduced from it.', '1702.03414-4-3-1': 'In classical propositional logic, every proposition can be deduced from every contradictory set of propositions.', '1702.03414-4-3-2': 'In a paraconsistent propositional logic, this is not the case.', '1702.03414-4-4-0': 'LP[MATH] is the three-valued paraconsistent propositional logic LP [CITATION] enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-4-4-1': 'This logic, which is essentially the same as J3 [CITATION], the propositional fragment of CLuNs [CITATION] without bi-implication, and LFI1 [CITATION], has most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-4-4-2': 'However, it follows easily from results presented in [CITATION] that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned.', '1702.03414-4-4-3': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the others.', '1702.03414-4-5-0': 'It turns out that only 32 of those 8192 logics are logics of which the logical equivalence relation satisfies the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction; and only 16 of them are logics of which the logical equivalence relation additionally satisfies the double negation law.', '1702.03414-4-5-1': 'LP[MATH] is one of those 16 logics.', '1702.03414-4-5-2': 'Two additional classical laws of logical equivalence turn out to be sufficient to distinguish LP[MATH] completely from the others.', '1702.03414-4-6-0': 'The desirable properties of reasonable paraconsistent propositional logics referred to above concern the logical consequence relation of a logic.', '1702.03414-4-6-1': 'Unlike in classical propositional logic, the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction do not follow from those properties in a three-valued paraconsistent propositional logic.', '1702.03414-4-6-2': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies these laws.', '1702.03414-4-6-3': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-4-7-0': 'The structure of this note is as follows.', '1702.03414-4-7-1': 'First, we give a survey of the paraconsistent propositional logic LP[MATH] (Section [REF]).', '1702.03414-4-7-2': 'Next, we discuss the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic (Section [REF]).', '1702.03414-4-7-3': 'Then, properties concerning the logical equivalence relation of a logic are used to distinguish LP[MATH] from the other three-valued paraconsistent propositional logics with the properties discussed earlier (Section [REF]).', '1702.03414-4-7-4': 'After that, we examine the logical equivalence relation of LP[MATH] further and show its key role in the algebraization of LP[MATH] (Section [REF]).', '1702.03414-4-7-5': 'Finally, we make some concluding remarks (Section [REF]).', '1702.03414-4-8-0': 'It is relevant to realize that the work presented in this note is restricted to three-valued paraconsistent propositional logics that are truth-functional three-valued logics.', '1702.03414-4-9-0': 'There is text overlap between this note and [CITATION].', '1702.03414-4-9-1': 'This note generalizes and elaborates Section 2 of that paper in such a way that it may be of independent importance to the area of paraconsistent logics.', '1702.03414-4-10-0': '# The Paraconsistent Logic LP[MATH]', '1702.03414-4-11-0': 'A set of propositions [MATH] is contradictory if there exists a proposition [MATH] such that both [MATH] and [MATH] can be deduced from [MATH].', '1702.03414-4-11-1': 'In classical propositional logic, every proposition can be deduced from a contradictory set of propositions.', '1702.03414-4-11-2': 'A paraconsistent propositional logic is a propositional logic in which not every proposition can be deduced from every contradictory set of propositions.', '1702.03414-4-12-0': 'In [CITATION], Priest proposed the paraconsistent propositional logic LP (Logic of Paradox).', '1702.03414-4-12-1': 'The logic introduced in this section is LP enriched with an implication connective for which the standard deduction theorem holds and a falsity constant.', '1702.03414-4-12-2': 'This logic, called LP[MATH], is in fact the propositional fragment of CLuNs [CITATION] without bi-implications.', '1702.03414-4-13-0': 'LP[MATH] has the following logical constants and connectives: a falsity constant [MATH], a unary negation connective [MATH], a binary conjunction connective [MATH], a binary disjunction connective [MATH], and a binary implication connective [MATH].', '1702.03414-4-13-1': 'Truth and bi-implication are defined as abbreviations: [MATH] stands for [MATH] and [MATH] stands for [MATH].', '1702.03414-4-14-0': 'A Hilbert-style formulation of LP[MATH] is given in Table [REF].', '1702.03414-4-15-0': 'In this formulation, which is taken from [CITATION], [MATH], [MATH], and [MATH] are used as meta-variables ranging over the set of all formulas of LP[MATH].', '1702.03414-4-15-1': 'The axiom schemas on the left-hand side of Table [REF] and the single inference rule (modus ponens) constitute a Hilbert-style formulation of the positive fragment of classical propositional logic.', '1702.03414-4-15-2': 'The first four axiom schemas on the right-hand side of Table [REF] allow for the negation connective to be moved inward.', '1702.03414-4-15-3': 'The fifth axiom schema on the right-hand side of Table [REF] is the law of the excluded middle.', '1702.03414-4-15-4': 'This axiom schema can be thought of as saying that, for every proposition, the proposition or its negation is true, while leaving open the possibility that both are true.', '1702.03414-4-15-5': 'If we add the axiom schema [MATH], which says that any proposition follows from a contradiction, to the given Hilbert-style formulation of LP[MATH], then we get a Hilbert-style formulation of classical propositional logic (see e.g. [CITATION]).', '1702.03414-4-15-6': 'We use the symbol [MATH] without decoration to denote the syntactic logical consequence relation induced by the axiom schemas and inference rule of LP[MATH].', '1702.03414-4-16-0': 'The following outline of the semantics of LP[MATH] is based on [CITATION].', '1702.03414-4-16-1': 'Like in the case of classical propositional logic, meanings are assigned to the formulas of LP[MATH] by means of valuations.', '1702.03414-4-16-2': 'However, in addition to the two classical truth values [MATH] (true) and [MATH] (false), a third meaning [MATH] (both true and false) may be assigned.', '1702.03414-4-16-3': 'A valuation for LP[MATH] is a function [MATH] from the set of all formulas of LP[MATH] to the set [MATH] such that for all formulas [MATH] and [MATH] of LP[MATH]: [EQUATION]', '1702.03414-4-16-4': 'The classical truth-conditions and falsehood-conditions for the logical connectives are retained.', '1702.03414-4-16-5': 'Except for implications, a formula is classified as both-true-and-false exactly when it cannot be classified as true or false by the classical truth-conditions and falsehood-conditions.', '1702.03414-4-16-6': 'The definition of a valuation given above shows that the logical connectives of LP[MATH] are (three-valued) truth-functional, which means that each [MATH]-ary connective represents a function from [MATH] to [MATH].', '1702.03414-4-17-0': 'For LP[MATH], the semantic logical consequence relation, denoted by [MATH], is based on the idea that a valuation [MATH] satisfies a formula [MATH] if [MATH].', '1702.03414-4-17-1': 'It is defined as follows: [MATH] iff for every valuation [MATH], either [MATH] for some [MATH] or [MATH].', '1702.03414-4-17-2': 'We have that the Hilbert-style formulation of LP[MATH] is strongly complete with respect to its semantics, i.e. [MATH] iff [MATH] (see e.g. [CITATION]).', '1702.03414-4-18-0': 'A formula of LP[MATH] in which the falsity constant [MATH] does not occur is called a [MATH]-free formula of LP[MATH].', '1702.03414-4-19-0': 'For all formulas [MATH] of LP[MATH] in which [MATH] does not occur, for all formulas [MATH] of LP[MATH] in which no propositional variable occurs that occurs in [MATH], [MATH] if [MATH] (see e.g. [CITATION]).', '1702.03414-4-20-0': 'Moreover, [MATH] is an extension of classical negation, i.e., for all valuations [MATH] and all formulas [MATH] of LP[MATH]: (a) [MATH] if [MATH], (b) [MATH] if [MATH].', '1702.03414-4-20-1': 'Hence, LP[MATH] is a paraconsistent logic.', '1702.03414-4-21-0': 'For LP[MATH], the logical equivalence relation [MATH] is defined as for classical propositional logic: [MATH] iff for every valuation [MATH], [MATH].', '1702.03414-4-21-1': 'Unlike in classical propositional logic, we do not have that [MATH] iff [MATH].', '1702.03414-4-22-0': 'For LP[MATH], the consistency property is defined as to be expected: [MATH] is consistent iff for every valuation [MATH], [MATH].', '1702.03414-4-23-0': '# Known Properties of LP[MATH]', '1702.03414-4-24-0': 'In this section, the known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are presented.', '1702.03414-4-24-1': 'Each of the properties in question has to do with logical consequence relations.', '1702.03414-4-24-2': 'Like above, the symbol [MATH] is used to denote the logical consequence relation of LP[MATH].', '1702.03414-4-24-3': 'The symbol [MATH] is used to denote the logical consequence relation of classical propositional logic.', '1702.03414-4-25-0': 'The known properties of LP[MATH] that have been proposed as desirable properties of a reasonable paraconsistent propositional logic are:', '1702.03414-4-26-0': '(b) [(a)] containment in classical logic: [MATH]; [(b)] proper basic connectives: for all sets [MATH] of formulas of LP[MATH] and all formulas [MATH], [MATH], and [MATH] of LP[MATH]:', '1702.03414-4-27-0': '(c) [(b[MATH])] [MATH][MATH][MATH] iff [MATH], [(b[MATH])] [MATH],[MATH] iff [MATH] and [MATH], [(b[MATH])] [MATH] iff [MATH] and [MATH];', '1702.03414-4-28-0': '[(c)] weakly maximal paraconsistency relative to classical logic: for all formulas [MATH] of LP[MATH] with [MATH] and [MATH], for the minimal consequence relation [MATH] such that [MATH] and [MATH], for all formulas [MATH] of LP[MATH], [MATH] iff [MATH]; [(d)] strongly maximal absolute paraconsistency: for all propositional logics [MATH] with the same logical constants and connectives as LP[MATH] and a consequence relation [MATH] such that [MATH], [MATH] is not paraconsistent; [(e)] internalized notion of consistency: [MATH] is consistent iff [MATH]; [(f)] internalized notion of logical equivalence: [MATH] iff [MATH].', '1702.03414-4-29-0': 'Properties (a)-(f) have been mentioned relatively often in the literature (see e.g. [CITATION]).', '1702.03414-4-29-1': 'Properties (a), (b[MATH]), (c), and (d) make LP[MATH] an ideal paraconsistent logic in the sense made precise in [CITATION].', '1702.03414-4-29-2': 'By property (e), LP[MATH] is also a logic of formal inconsistency in the sense made precise in [CITATION].', '1702.03414-4-30-0': 'Properties (a)-(c) indicate that LP[MATH] retains much of classical propositional logic.', '1702.03414-4-30-1': 'Actually, property (c) can be strengthened to the following property: for all formulas [MATH] of LP[MATH], [MATH] iff [MATH].', '1702.03414-4-30-2': 'In [CITATION], properties (e) and (f) are considered desirable and essential, respectively, for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-4-31-0': 'From Theorem 4.42 in [CITATION], we know that there are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-4-31-1': 'From Theorem 2 in [CITATION], we know that properties (c) and (d) are common properties of all three-valued paraconsistent propositional logics with properties (a) and (b[MATH]).', '1702.03414-4-31-2': 'From Fact 103 in [CITATION], we know that property (f) is a common property of all three-valued paraconsistent propositional logics with properties (a), (b) and (e).', '1702.03414-4-31-3': 'Moreover, it is easy to see that that property (e) is a common property of all three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-4-31-4': 'Hence, each three-valued paraconsistent propositional logic with properties (a) and (b) has properties (c)-(f) as well.', '1702.03414-4-32-0': 'From Corollary 106 in [CITATION], we know that LP[MATH] is the strongest three-valued paraconsistent propositional logic with properties (a) and (b) in the sense that for each three-valued paraconsistent propositional logic with properties (a) and (b), there exists a translation into LP[MATH] that preserves and reflects its logical consequence relation.', '1702.03414-4-33-0': '# Characterizing LP[MATH] by Laws of Logical Equivalence', '1702.03414-4-34-0': 'There are exactly 8192 different three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-4-34-1': 'This means that these properties, which concern the logical consequence relation of a logic, have little discriminating power.', '1702.03414-4-34-2': 'Properties (c)-(f), which also concern the logical consequence relation of a logic, do not offer additional discrimination because each of the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) has these properties as well.', '1702.03414-4-35-0': 'In this section, properties concerning the logical equivalence relation of a logic are used for additional discrimination.', '1702.03414-4-35-1': 'It turns out that 11 classical laws of logical equivalence, of which at least 9 are considered to belong to the most basic ones, are sufficient to distinguish LP[MATH] completely from the other 8191 three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-4-36-0': 'The logical equivalence relation of LP[MATH] satisfies all laws given in Table [REF].', '1702.03414-4-37-0': 'The logical equivalence relation of LP[MATH] satisfies laws (1)-(11) from Table [REF].', '1702.03414-4-38-0': 'The proof is very easy by constructing, for each of the laws concerned, truth tables for both sides.', '1702.03414-4-39-0': 'Moreover, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), LP[MATH] is the only one whose logical equivalence relation satisfies all laws given in Table [REF].', '1702.03414-4-40-0': 'There is exactly one three-valued paraconsistent propositional logic with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(11) from Table [REF].', '1702.03414-4-41-0': 'We confine ourselves to a brief outline of the proof.', '1702.03414-4-41-1': "Because 'non-deterministic truth tables' that uniquely characterize the 8192 three-valued paraconsistent propositional logics with properties (a) and (b) are given in [CITATION],", '1702.03414-4-42-0': 'the theorem can be proved by showing that, for each of the connectives, only one of the ordinary truth tables represented by the non-deterministic truth table for that connective is compatible with the laws given in Table [REF].', '1702.03414-4-42-1': 'It can be shown by short routine case analyses that only one of the 8 ordinary truth tables represented by the non-deterministic truth tables for conjunction is compatible with laws (1), (3), (5), and (7), only one of the 32 ordinary truth tables represented by the non-deterministic truth tables for disjunction is compatible with laws (2), (4), (6), and (8), and only one of the 2 ordinary truth tables represented by the non-deterministic truth table for negation is compatible with law (9).', '1702.03414-4-42-2': 'Given the ordinary truth table for conjunction, disjunction, and negation so obtained, it can be shown by short routine case analyses that only one of the 16 ordinary truth tables represented by the non-deterministic truth table for implication is compatible with laws (10) and (11).', '1702.03414-4-43-0': 'It follow immediately from the proof of Theorem [REF] that all proper subsets of the laws from Table [REF] are insufficient to the distinguish LP[MATH] completely from the other three-valued paraconsistent propositional logics with properties (a) and (b).', '1702.03414-4-43-1': 'More particular, the next two corollaries follow immediately from the proof of Theorem [REF].', '1702.03414-4-44-0': 'There are exactly 16 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(9) from Table [REF].', '1702.03414-4-45-0': 'There are exactly 32 three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8) from Table [REF].', '1702.03414-4-46-0': 'From a paraconsistent propositional logic with properties (a) and (b), it is only to be expected, because of paraconsistency and property (b[MATH]), that its negation connective and its implication connective deviate clearly from their counterpart in classical propositional logic.', '1702.03414-4-46-1': 'Corollary [REF] shows that, among the 8192 three-valued paraconsistent propositional logics with properties (a) and (b), there are 8160 logics whose logical equivalence relation does not even satisfy the identity, annihilation, idempotent, and commutative laws for conjunction and disjunction (laws (1)-(8) from Table [REF]).', '1702.03414-4-47-0': '# More on the Logical Equivalence Relation of LP[MATH]', '1702.03414-4-48-0': "It turns out that the logical equivalence relation of LP[MATH] does not only satisfy the identity, annihilation, idempotent, and commutative laws of logical equivalence for conjunction and disjunction but also other basic classical laws of logical equivalence for conjunction and disjunction, including the absorption, associative, distributive, and de Morgan's laws (laws (12)-(19) from Table [REF]).", '1702.03414-4-48-1': 'Indeed, the logical equivalence relation of LP[MATH] satisfies all laws given in Tables [REF] and [REF].', '1702.03414-4-49-0': 'Laws (1)-(9) and (12)-(21) axiomatize normal i-lattices [CITATION].', '1702.03414-4-49-1': 'Laws (10)-(11) and (22)-(24) are laws concerning the implication connective.', '1702.03414-4-49-2': 'Laws (10)-(11) and (22)-(24), like laws (1)-(9) and (12)-(21), are also satisfied by the logical equivalence relation of classical propositional logic.', '1702.03414-4-49-3': '[MATH] is satisfied by the logical equivalence relation of classical propositional logic iff it follows from laws (1)-(9) and (12)-(21) and the laws', '1702.03414-4-50-0': '(25) A A (26) A A (27) A B A B .', '1702.03414-4-51-0': 'However, laws (25)-(27) are not satisfied by the logical equivalence relation of LP[MATH].', '1702.03414-4-52-0': 'Because [MATH] iff [MATH], [MATH] stands for [MATH], and [MATH] iff [MATH] and [MATH], we have that [MATH] iff [MATH] and [MATH] and [MATH] and [MATH].', '1702.03414-4-52-1': 'Moreover, writing [MATH] for [MATH], we easily find that (i) [MATH], (ii) [MATH], (iii) [MATH], (iv) [MATH] for [MATH], and (v) [MATH] and [MATH].', '1702.03414-4-52-2': 'Therefore, by Corollary 3.6 from [CITATION] and the fact that LP[MATH] is a finitary logic, LP[MATH] is finitely algebraizable', '1702.03414-4-53-0': 'with the equivalence formulas [MATH] and the single defining equation [MATH] and [MATH] are variables).', '1702.03414-4-53-1': 'The algebraization concerned is the quasi-variety generated by the expansion of the 3-element normal i-lattice obtained by adding the unique binary operation [MATH] that satisfies [MATH] and [MATH].', '1702.03414-4-54-0': '# Concluding Remarks', '1702.03414-4-55-0': 'In this note, properties concerning the logical equivalence relation of a logic are used to distinguish the logic LP[MATH] from the other logics that belong to the 8192 three-valued paraconsistent propositional logics that have properties (a)-(f) from Section [REF].', '1702.03414-4-55-1': 'These 8192 logics are considered potentially interesting because properties (a)-(f) are generally considered desirable properties of a reasonable paraconsistent propositional logic.', '1702.03414-4-56-0': 'Properties (a)-(f) concern the logical consequence relation of a logic.', '1702.03414-4-56-1': 'Unlike in classical propositional logic, we do not have [MATH] iff [MATH] and [MATH] in a three-valued paraconsistent propositional logic.', '1702.03414-4-56-2': 'As a consequence, the classical laws of logical equivalence that follow from properties (a) and (b) in classical propositional logic, viz. laws (1)-(8) and (12)-(17) from Section [REF], do not follow from properties (a) and (b) in a three-valued paraconsistent propositional logic.', '1702.03414-4-56-3': 'Therefore, if closeness to classical propositional logic is considered important, it should be a desirable property of a reasonable paraconsistent propositional logic to have a logical equivalence relation that satisfies laws (1)-(8) and (12)-(17).', '1702.03414-4-57-0': 'This would reduce the potentially interesting three-valued paraconsistent propositional logics from 8192 to 32.', '1702.03414-4-58-0': 'In [CITATION], satisfaction of laws (1)-(8), (11), (14)-(17), and (22)-(24) is considered essential for a paraconsistent propositional logic on which a process algebra that allows for dealing with contradictory states is built.', '1702.03414-4-58-1': 'It follows easily from Theorem [REF] and the proof of Theorem [REF] that LP[MATH] is one of only four three-valued paraconsistent propositional logics with properties (a) and (b) of which the logical equivalence relation satisfies laws (1)-(8), (11), (14)-(17), and (22)-(24).', '1702.03414-4-59-0': 'From Fact 217 in [CITATION], we know that LP[MATH] is Blok-Pigozzi algebraizable.', '1702.03414-4-59-1': 'Although there must exist one, a conditional-equational axiomatization of the algebras concerned in the case of LP[MATH] has not yet been devised.'} | null | null | null |
1905.04226 | {'1905.04226-1-0-0': 'We explore multi-layer autoregressive Transformer models in language modeling for speech recognition.', '1905.04226-1-0-1': 'We focus on two aspects.', '1905.04226-1-0-2': 'First, we revisit Transformer model configurations specifically for language modeling.', '1905.04226-1-0-3': 'We show that well configured Transformer models outperform our baseline models based on the shallow stack of LSTM recurrent neural network layers.', '1905.04226-1-0-4': 'We carry out experiments on the open-source LibriSpeech 960hr task, for both 200K vocabulary word-level and 10K byte-pair encoding subword-level language modeling.', '1905.04226-1-0-5': 'We apply our word-level models to conventional hybrid speech recognition by lattice rescoring, and the subword-level models to attention based encoder-decoder models by shallow fusion.', '1905.04226-1-0-6': 'Second, we show that deep Transformer language models do not require positional encoding.', '1905.04226-1-0-7': 'The positional encoding is an essential augmentation for the self-attention mechanism which is invariant to sequence ordering.', '1905.04226-1-0-8': 'However, in autoregressive setup, as is the case for language modeling, the amount of information increases along the position dimension, which is a positional signal by its own.', '1905.04226-1-0-9': 'The analysis of attention weights shows that deep autoregressive self-attention models can automatically make use of such positional information.', '1905.04226-1-0-10': 'We find that removing the positional encoding even slightly improves the performance of these models.', '1905.04226-1-1-0': 'Index Terms: language modeling, self-attention, Transformer, speech recognition', '1905.04226-1-2-0': '# Introduction', '1905.04226-1-3-0': 'Transformer encoder-decoder models [CITATION] have become popular in natural language processing.', '1905.04226-1-3-1': 'The Transformer architecture allows to successfully train a deep stack of self-attention layers [CITATION] via residual connections [CITATION] and layer normalization [CITATION].', '1905.04226-1-3-2': 'The positional encodings [CITATION], typically based on sinusoidal functions, are used to provide the self-attention with the sequence order information.', '1905.04226-1-3-3': 'Across various applications, systematic improvements have been reported over the standard, multi-layer long short-term memory (LSTM) [CITATION] recurrent neural network based models.', '1905.04226-1-3-4': 'While originally designed as an encoder-decoder architecture in machine translation, the encoder (e.g., [CITATION]) and the decoder (e.g., [CITATION]) components are also separately used in corresponding problems depending on whether the problem disposes the whole sequence for prediction or not.', '1905.04226-1-4-0': 'A number of recent works have also shown impressive performance in language modeling using the Transformer decoder component [CITATION].', '1905.04226-1-4-1': 'The earliest example can be found in [CITATION] where such models are investigated for text generation.', '1905.04226-1-4-2': 'Recent works on training larger and deeper models [CITATION] have shown further potential of the Transformer in language modeling.', '1905.04226-1-4-3': 'On the other hand, an obvious limitation of the Transformers is that their memory requirement linearly increases in terms of number of tokens in the sequence, which requires to work with a limited context window (basically a [MATH]-gram model where the typical number for [MATH] is 512) for tasks dealing with long sequences such as character-level language modeling [CITATION].', '1905.04226-1-4-4': 'Dai et al. [CITATION] has introduced a segment-level recurrence and relative positional encoding in the Transformer language model to be able to potentially handle unlimited context.', '1905.04226-1-5-0': 'In this work, we investigate deep autoregressive Transformers for language modeling in speech recognition.', '1905.04226-1-5-1': 'To be specific, we focus on two aspects.', '1905.04226-1-5-2': 'First, we revisit the parameter configurations of Transformers, originally engineered for the sequence-to-sequence problem [CITATION], specifically for language modeling.', '1905.04226-1-5-3': 'We conduct experiments on the LibriSpeech automatic speech recognition (ASR) task [CITATION] for both word-level conventional speech recognition and byte-pair encoding (BPE) [CITATION] level end-to-end speech recognition [CITATION].', '1905.04226-1-5-4': 'We apply our word-level models to hybrid speech recognition by lattice rescoring [CITATION], and the BPE-level models to end-to-end models by shallow fusion [CITATION].', '1905.04226-1-5-5': 'We show that well configured Transformer language models outperform models based on the simple stack of LSTM RNN layers in terms of both perplexity and word error rate (WER).', '1905.04226-1-6-0': 'Second, we experimentally show that the positional encoding is not needed for multi-layer autoregressive self-attention models.', '1905.04226-1-6-1': 'The visualization of the attention weights shows that when the sinusoidal positional encoding is provided with the input, the first layer of the Transformers learns to extract [MATH]-gram features (therefore making use of positional information).', '1905.04226-1-6-2': 'However, in the autoregressive problem where a new token is provided to the model at each time step, the amount of information the model has access to strictly increases from left to right at the lowest level of the network, which should provide some positional information by its own.', '1905.04226-1-6-3': 'We observe that deep Transformer language models without positional encoding automatically make use of such information, and even give slight improvements over models with positional encodings.', '1905.04226-1-7-0': '# Related Work', '1905.04226-1-8-0': "The first part of our work follows the spirits of Al-Rfou et al.'s work [CITATION] and Radford et al.'s work [CITATION] in investigating larger and deeper Transformers for language modeling.", '1905.04226-1-8-1': 'We show that deep Transformer language models can be successfully applied to speech recognition and give good performance.', '1905.04226-1-8-2': 'The second part of this work concerns the positional encoding, which is a crucial component in the original Transformer.', '1905.04226-1-8-3': 'There are active investigations on positional encoding variants to improve self-attention (e.g., [CITATION]).', '1905.04226-1-8-4': 'Previous works in Transformer language models systematically use positional encoding, either jointly learned one or the sinusoidal one (both cases are reported to give similar performance in [CITATION]).', '1905.04226-1-8-5': 'We show that the deep autoregressive self-attention models do not require any explicit model for encoding positions to give the best performance.', '1905.04226-1-9-0': '# Autoregressive Self-Attention', '1905.04226-1-10-0': 'The language model we consider is based on the decoder component of the Transformer architecture [CITATION].', '1905.04226-1-10-1': 'Similar to previous work [CITATION], we define layer as a stack of two modules: self-attention and feed-forward modules.', '1905.04226-1-11-0': 'The autoregressive self-attention module in the [MATH]-th layer transforms the input [MATH] at position [MATH] as follows: [EQUATION] where [MATH], [MATH], [MATH], respectively denote query, key, value projection matrices, [MATH] denotes layer normalization [CITATION], [MATH] denotes the scaled multi-head dot product self-attention [CITATION], and [MATH] denotes the projection matrix for the residual connection [CITATION].', '1905.04226-1-12-0': 'The output [MATH] is then fed to the feed-forward module: [EQUATION] where [MATH] is rectifier [CITATION], Gaussian error linear unit (GELU) [CITATION], or gated linear unit (GLU) [CITATION] in this work.', '1905.04226-1-12-1': 'The final model is build by stacking these layers multiple times.', '1905.04226-1-13-0': 'The input of the network consists of the sum of the token embedding (word or BPE in this work) and the sinusoidal positional encoding as specified in [CITATION].', '1905.04226-1-13-1': 'The output softmax layer gives the probability distribution for the next token.', '1905.04226-1-13-2': 'As shown in the equations above, [MATH] can be seen as states of the Transformer model (whose size, as opposed to the standard RNN states, linearly grows as we progress along the position dimension).', '1905.04226-1-13-3': 'During inference, these states are stored to avoid redundant computation.', '1905.04226-1-13-4': 'During training, the computation along the position dimension is parallelized for speed-up.', '1905.04226-1-14-0': '# LibriSpeech Dataset', '1905.04226-1-15-0': '## Language Modeling Data Descriptions', '1905.04226-1-16-0': 'The LibriSpeech datasets [CITATION] for language modeling consists of 800M-word text only data and 960hr of audio transcriptions which corresponds to 10M-word text data.', '1905.04226-1-16-1': 'Based on analysis of count model perplexities, we observe that the audio transcription part does not contain special domain signal which matches the development set.', '1905.04226-1-16-2': 'Therefore, we simply merge the two datasets to form one training dataset for language model training.', '1905.04226-1-16-3': 'The average sentence length in the resulting training data is 21 words with the maximum length of 600 words.', '1905.04226-1-16-4': 'The development and test sets respectively have two parts: dev-clean, dev-other, test-clean, and test-other.', '1905.04226-1-16-5': 'This separation is based on the audio-level characteristics, therefore it has no special meaning for language modeling.', '1905.04226-1-16-6': 'In the experimental section, we denote by "Dev" and "Test" the concatenation of clean and other parts of the respective data.', '1905.04226-1-16-7': 'Both datasets consist of about 110K running words with average of 20 words per sentence.', '1905.04226-1-16-8': 'The word-level vocabulary contains 200K words.', '1905.04226-1-17-0': '## 4-gram count and LSTM-RNN Baselines', '1905.04226-1-18-0': 'We use the official 4-gram count language model provided with the LibriSpeech dataset [CITATION].', '1905.04226-1-18-1': 'No improvement in perplexity is observed when going up to 5-grams.', '1905.04226-1-18-2': 'For LSTM-RNN language models [CITATION], we first train our base configuration; the model has 2 LSTM-RNN layers with dimension 2048 and the input projection layer of 128, where the dropout with a rate of 0.2 is applied between each layer.', '1905.04226-1-18-3': 'Since we observe that this model underfits the LibriSpeech training set, we further train two models; the same model without dropout, and the one with 4 LSTM layers stacked without dropout.', '1905.04226-1-18-4': 'Both these changes effectively give better perplexity, though our current experiments did not show improvements from simply stacking more LSTM layers.', '1905.04226-1-18-5': 'The perplexities of these models are summarized in Table [REF].', '1905.04226-1-18-6': 'We observe that a good relative improvements of greater than 57% is obtained by LSTM language models over the 4-gram model.', '1905.04226-1-19-0': '# Text based Experiments', '1905.04226-1-20-0': 'We carry out experiments for both word-level and BPE-level language modeling.', '1905.04226-1-20-1': 'We first focus on the word-level one.', '1905.04226-1-21-0': '## Hyper-parameters in Transformers', '1905.04226-1-22-0': 'The Transformer architecture presents a new search space Odyssey [CITATION].', '1905.04226-1-22-1': 'The exhaustive model hyper-parameters for Transformer language models specified by the equations in Sec. [REF] are the number of layers and the dimension of the residual connection, and for each layer the number of attention heads, the dimension of the key and query, the dimension of the value, and the dimension of the feed-forward layer.', '1905.04226-1-23-0': 'In our experiments, we use the same dimension for key, query and value, as well as the residual connection.', '1905.04226-1-23-1': 'We use the same dimensionality across all layers.', '1905.04226-1-23-2': 'Therefore, our models can be fully specified by the tuple (number of layers [MATH], feed-forward dimension [MATH], residual dimension [MATH], number of heads [MATH]).', '1905.04226-1-23-3': 'We do not apply any regularization method including dropout.', '1905.04226-1-23-4': 'We train all models using the plain stochastic gradient descent and new-bob learning rate tuning on a single GPU.', '1905.04226-1-23-5': 'We define our training sub-epoch (for new-bob) as the 10th of the full training data.', '1905.04226-1-23-6': 'All our implementations are based on the Tensorflow [CITATION] based open-source toolkit RETURNN [CITATION].', '1905.04226-1-24-0': '## Hyper-parameter Tuning', '1905.04226-1-25-0': 'Given the amount of LibriSpeech training data (810M words), it is unreasonable to train all model variants until full convergence.', '1905.04226-1-25-1': 'The earlier stage of the training already consistently indicates the performance of the models.', '1905.04226-1-25-2': 'Therefore, we carry out comparisons between models with different configuration at the equal, large enough, but reasonable number of updates.', '1905.04226-1-26-0': 'The first set of comparison investigates the effect of depth and width.', '1905.04226-1-26-1': 'The perplexity results can be found in Table [REF].', '1905.04226-1-26-2': 'All models in the table use 8 attention heads.', '1905.04226-1-26-3': 'Other parameters are specified in the table.', '1905.04226-1-26-4': 'The table is organized in two parts.', '1905.04226-1-26-5': 'The upper part of Table [REF] shows the effect of number of layers; we observe that increasing number of layers (therefore the number of parameters) from 1 to 42 gradually improves the perplexity.', '1905.04226-1-26-6': 'In the lower part of Table [REF] , we vary both the number of layers, feed-forward dimension, and the residual dimension.', '1905.04226-1-26-7': 'First of all, the 12-layer [MATH] model outperforms the 6-layer [MATH] model, while having similar number of parameters, which seems to indicate that the depth effectively benefits Transformer language models.', '1905.04226-1-26-8': 'We also train an extreme model which has only 2 layers with wide dimensions [MATH].', '1905.04226-1-26-9': 'The number of parameters in fact blows up because of the large value of [MATH] which results in a large matrix in the output softmax layer with 200K vocabulary.', '1905.04226-1-26-10': 'We observe that such wide but shallow models do not perform well.', '1905.04226-1-27-0': 'However, the softmax bottleneck dimension for language modeling typically needs to be large for the best performance [CITATION].', '1905.04226-1-27-1': 'In Transformers, the bottleneck dimension corresponds to the residual connection dimension which is typically kept rather small (typically 512 or 1024).', '1905.04226-1-27-2': 'As a control experiment, we also train a model in which we insert an additional projection layer with a large dimension before the softmax layer to give larger bottleneck capacity.', '1905.04226-1-27-3': 'Table [REF] shows the comparison conducted on the (12, 2048, 512, 8) model.', '1905.04226-1-27-4': 'We observe that simply enlarging the bottleneck dimension does not improve Transformer models.', '1905.04226-1-28-0': 'Table [REF] shows the effect of number of attention heads.', '1905.04226-1-28-1': '16 heads which is the largest number we try in this setup give the best performance.', '1905.04226-1-28-2': 'In addition, we examine the type of activation function (Table [REF]).', '1905.04226-1-28-3': 'As opposed to previous work on feed-forward language models using GLUs [CITATION], we do not observe faster convergence.', '1905.04226-1-28-4': 'As we observe that the impact of choice of activation functions on the perplexity is overall limited, all our other models use the standard ReLU.', '1905.04226-1-28-5': 'As reported in the original Transformer, we confirm that both layer normalization and residual connections are needed for these models for stable training.', '1905.04226-1-29-0': 'Finally, we train models with the best configurations until convergence.', '1905.04226-1-29-1': 'Table [REF] shows the perplexities which are better than those obtained by our LSTM based models (Table [REF]).', '1905.04226-1-30-0': '## Parameter Tying', '1905.04226-1-31-0': 'Dehghani et al. [CITATION] reports Universal Transformers to perform particularly well for language modeling.', '1905.04226-1-31-1': 'This motivates us to experiment with Transformer models which share the parameters across layers.', '1905.04226-1-31-2': 'For a Universal Transformer to have comparable number of parameters with the standard deep Transformers, the dimensions in each layer must be increased, which results in slower training; here we simply investigate the effect of number of recurrence.', '1905.04226-1-31-3': 'Table [REF] shows the perplexity results.', '1905.04226-1-31-4': 'First of all, we observe that the model performance is behind that of the standard Transformer.', '1905.04226-1-31-5': 'However, we clearly observe that increasing the number of layers from 3 to 6 consistently improves the perplexity.', '1905.04226-1-31-6': 'This improvement without additional parameters motivates future work to investigate further parameter sharing strategies for Transformers.', '1905.04226-1-32-0': '# ASR Experiments', '1905.04226-1-33-0': '## Lattice Rescoring Results', '1905.04226-1-34-0': 'We apply our word-level Transformer language models to conventional hybrid speech recognition by lattice rescoring.', '1905.04226-1-34-1': 'The standard push-forward lattice rescoring algorithm [CITATION] for long-span language models can be directly applied to self-attention based models.', '1905.04226-1-34-2': 'The only modifications from the RNN version is to define the "state" as all hidden states ([MATH] in Sec. [REF]) in all layers from all predecessor positions and the current position ([MATH]; for position encoding).', '1905.04226-1-34-3': 'Table [REF] shows the WERs and perplexities (PPL).', '1905.04226-1-34-4': 'Our baseline acoustic model is based on multi-layer bi-directional LSTM [CITATION].', '1905.04226-1-34-5': 'Further descriptions of our baseline acoustic model can be found in [CITATION].', '1905.04226-1-34-6': 'We obtain consistent improvements in WER over the LSTM baselines.', '1905.04226-1-35-0': '## End-to-End ASR Shallow Fusion Results', '1905.04226-1-36-0': 'We train 10K BPE-level Transformer language models to be combined with attention-based encoder-decoder speech model by shallow fusion [CITATION].', '1905.04226-1-36-1': 'The 10K BPE level training data has a longer average length of 24 tokens per sentence with the longest sentence length of 1343, which is still manageable without any truncation for self-attention.', '1905.04226-1-36-2': 'We use the Transformer architecture of (24, 2048, 512, 8).', '1905.04226-1-36-3': 'The two-layer LSTM architecture is also the same as described in [REF] without dropout.', '1905.04226-1-36-4': 'We refer to our previous work [CITATION] for the description of the baseline attention model; the better baseline WERs than our previous work [CITATION] are obtained by improved curriculum learning and longer training.', '1905.04226-1-36-5': 'Table [REF] shows both perplexities and WERs .', '1905.04226-1-36-6': 'Again, we obtain consistent improvements over the LSTM baseline.', '1905.04226-1-36-7': 'These results are slightly better than previously reported best WERs [CITATION] for end-to-end models without data augmentation.', '1905.04226-1-37-0': '# Analysis', '1905.04226-1-38-0': 'Compared with hidden states in RNNs, attention weights are easier to be visualized, which gives opportunity for analysis.', '1905.04226-1-38-1': 'In particular, we focus on the comparison of the Transformer language models with and without positional encoding.', '1905.04226-1-39-0': '## Transformer LM without positional encoding', '1905.04226-1-40-0': 'In the autoregressive problem where a new token is provided to the model at each time step, the amount of information the model has access to strictly increases from left to right at the lowest level of the network; the deeper layers should be able to recognize this structure which should provide the model with some positional information by its own.', '1905.04226-1-40-1': 'To check this hypothesis, we train models without any positional encoding.', '1905.04226-1-40-2': 'First, we observe that they give better perplexities than the models with sinusoidal positional encoding (Table [REF]).', '1905.04226-1-41-0': '## First layer', '1905.04226-1-42-0': 'The attention in the first layer is the most straightforward for interpretation because the feature at each position exactly corresponds to the word at the position (while deeper layers can potentially shuffle the feature content).', '1905.04226-1-42-1': 'The attention weights in the first layer of 24-layer Transformer language models with and without positional encodings are visualized in Figure [REF].', '1905.04226-1-42-2': 'We observe that the first layer of the model with positional encoding (Figure [REF](a)) learns to create n-gram features (roughly 2 or 3-gram), which indicates that the positional information is directly used.', '1905.04226-1-42-3': 'In contrast, the first layer of the model without positional encoding learns to focus on the new input token as can be seen as the diagonal in Figure [REF](b) (interestingly, we also see that it ignores some functional words such as "the", "and", "to" which might be modeled by some off-set values, therefore attending to the beginning of sentence token instead), which demonstrates that the model is aware of the position of the new input.', '1905.04226-1-43-0': '## Other layers', '1905.04226-1-44-0': 'We observe that the behavior of other layers are rather similar for both Transformer models with and without positional encoding.', '1905.04226-1-44-1': 'We find 3 categories of layers in the other 23 layers; the second and third layers are "blur" layers as shown in Figure [REF](c), which seems to roughly average over all positions (while we can also see that some heads focus on difficult words, here "verandah").', '1905.04226-1-44-2': 'Layer 4 to 9 are window layers which focus on the local n-gram.', '1905.04226-1-44-3': 'A representative example is show in Figure [REF](d).', '1905.04226-1-44-4': 'Finally, we find the top layers 10 to 23 to be more structured, attending to some specific patterns; an example is shown in Figure [REF](e).', '1905.04226-1-45-0': '# Conclusion', '1905.04226-1-46-0': 'We apply deep Transformer language models for speech recognition.', '1905.04226-1-46-1': 'We show that such models outperform the shallow stack of LSTM-RNNs on both word-level and BPE-level modeling.', '1905.04226-1-46-2': 'Future work investigates application of crucial components of deep Transformers (such as layer normalization) to deeper LSTM models; e.g., the RNMT+ decoder architecture [CITATION] for language modeling.', '1905.04226-1-46-3': 'Furthermore, we do not apply any regularization on models for the large LibriSpeech task, as no overfitting is observed in the range of model sizes we experimented with (for the word-level models).', '1905.04226-1-46-4': 'We can possibly still improve our models simply by scaling up their size and using regularization.'} | {'1905.04226-2-0-0': 'We explore deep autoregressive Transformer models in language modeling for speech recognition.', '1905.04226-2-0-1': 'We focus on two aspects.', '1905.04226-2-0-2': 'First, we revisit Transformer model configurations specifically for language modeling.', '1905.04226-2-0-3': 'We show that well configured Transformer models outperform our baseline models based on the shallow stack of LSTM recurrent neural network layers.', '1905.04226-2-0-4': 'We carry out experiments on the open-source LibriSpeech 960hr task, for both 200K vocabulary word-level and 10K byte-pair encoding subword-level language modeling.', '1905.04226-2-0-5': 'We apply our word-level models to conventional hybrid speech recognition by lattice rescoring, and the subword-level models to attention based encoder-decoder models by shallow fusion.', '1905.04226-2-0-6': 'Second, we show that deep Transformer language models do not require positional encoding.', '1905.04226-2-0-7': 'The positional encoding is an essential augmentation for the self-attention mechanism which is invariant to sequence ordering.', '1905.04226-2-0-8': 'However, in autoregressive setup, as is the case for language modeling, the amount of information increases along the position dimension, which is a positional signal by its own.', '1905.04226-2-0-9': 'The analysis of attention weights shows that deep autoregressive self-attention models can automatically make use of such positional information.', '1905.04226-2-0-10': 'We find that removing the positional encoding even slightly improves the performance of these models.', '1905.04226-2-1-0': 'Index Terms: language modeling, self-attention, Transformer, speech recognition', '1905.04226-2-2-0': '# Introduction', '1905.04226-2-3-0': 'Transformer encoder-decoder models [CITATION] have become popular in natural language processing.', '1905.04226-2-3-1': 'The Transformer architecture allows to successfully train a deep stack of self-attention layers [CITATION] via residual connections [CITATION] and layer normalization [CITATION].', '1905.04226-2-3-2': 'The positional encodings [CITATION], typically based on sinusoidal functions, are used to provide the self-attention with the sequence order information.', '1905.04226-2-3-3': 'Across various applications, systematic improvements have been reported over the standard, multi-layer long short-term memory (LSTM) [CITATION] recurrent neural network based models.', '1905.04226-2-3-4': 'While originally designed as an encoder-decoder architecture in machine translation, the encoder (e.g., [CITATION]) and the decoder (e.g., [CITATION]) components are also separately used in corresponding problems depending on whether the problem disposes the whole sequence for prediction or not.', '1905.04226-2-4-0': 'A number of recent works have also shown impressive performance in language modeling using the Transformer decoder component [CITATION].', '1905.04226-2-4-1': 'The earliest example can be found in [CITATION] where such models are investigated for text generation.', '1905.04226-2-4-2': 'Recent works on training larger and deeper models [CITATION] have shown further potential of the Transformer in language modeling.', '1905.04226-2-4-3': 'On the other hand, an obvious limitation of the Transformers is that their memory requirement linearly increases in terms of number of tokens in the sequence, which requires to work with a limited context window (basically a [MATH]-gram model where the typical number for [MATH] is 512) for tasks dealing with long sequences such as character-level language modeling [CITATION].', '1905.04226-2-4-4': 'Dai et al. [CITATION] has introduced a segment-level recurrence and relative positional encoding in the Transformer language model to be able to potentially handle unlimited context.', '1905.04226-2-5-0': 'In this work, we investigate deep autoregressive Transformers for language modeling in speech recognition.', '1905.04226-2-5-1': 'To be specific, we focus on two aspects.', '1905.04226-2-5-2': 'First, we revisit the parameter configurations of Transformers, originally engineered for the sequence-to-sequence problem [CITATION], specifically for language modeling.', '1905.04226-2-5-3': 'We conduct experiments on the LibriSpeech automatic speech recognition (ASR) task [CITATION] for both word-level conventional speech recognition and byte-pair encoding (BPE) [CITATION] level end-to-end speech recognition [CITATION].', '1905.04226-2-5-4': 'We apply our word-level models to hybrid speech recognition by lattice rescoring [CITATION], and the BPE-level models to end-to-end models by shallow fusion [CITATION].', '1905.04226-2-5-5': 'We show that well configured Transformer language models outperform models based on the simple stack of LSTM RNN layers in terms of both perplexity and word error rate (WER).', '1905.04226-2-6-0': 'Second, we experimentally show that the positional encoding is not needed for multi-layer autoregressive self-attention models.', '1905.04226-2-6-1': 'The visualization of the attention weights shows that when the sinusoidal positional encoding is provided with the input, the first layer of the Transformers learns to extract [MATH]-gram features (therefore making use of positional information).', '1905.04226-2-6-2': 'However, in the autoregressive problem where a new token is provided to the model at each time step, the amount of information the model has access to strictly increases from left to right at the lowest level of the network, which should provide some positional information by its own.', '1905.04226-2-6-3': 'We observe that deep Transformer language models without positional encoding automatically make use of such information, and even give slight improvements over models with positional encodings.', '1905.04226-2-7-0': '# Related Work', '1905.04226-2-8-0': "The first part of our work follows the spirits of Al-Rfou et al.'s work [CITATION] and Radford et al.'s work [CITATION] in investigating larger and deeper Transformers for language modeling.", '1905.04226-2-8-1': 'We show that deep Transformer language models can be successfully applied to speech recognition and give good performance.', '1905.04226-2-8-2': 'The second part of this work concerns the positional encoding, which is a crucial component in the original Transformer.', '1905.04226-2-8-3': 'A number of previous work investigated positional encoding variants to improve self-attention (e.g., [CITATION]).', '1905.04226-2-8-4': 'Previous works in Transformer language models systematically use positional encoding, either jointly learned one or the sinusoidal one (both cases are reported to give similar performance in [CITATION]).', '1905.04226-2-8-5': 'We show that the deep autoregressive self-attention models do not require any explicit model for encoding positions to give the best performance.', '1905.04226-2-9-0': '# Autoregressive Self-Attention', '1905.04226-2-10-0': 'The language model we consider is based on the decoder component of the Transformer architecture [CITATION].', '1905.04226-2-10-1': 'Similar to previous work [CITATION], we define layer as a stack of two components: self-attention and feed-forward modules.', '1905.04226-2-11-0': 'The autoregressive self-attention module in the [MATH]-th layer transforms the input [MATH] at position [MATH] as follows: [EQUATION] where [MATH], [MATH], [MATH], respectively denote query, key, value projection matrices, [MATH] denotes layer normalization [CITATION], [MATH] denotes the scaled multi-head dot product self-attention [CITATION], and [MATH] denotes the projection matrix for the residual connection [CITATION].', '1905.04226-2-12-0': 'The output [MATH] is then fed to the feed-forward module: [EQUATION] where [MATH] is rectifier [CITATION], Gaussian error linear unit (GELU) [CITATION], or gated linear unit (GLU) [CITATION] in this work.', '1905.04226-2-12-1': 'The final model is build by stacking these layers multiple times.', '1905.04226-2-13-0': 'The input of the network consists of the sum of the token embedding (word or BPE in this work) and the sinusoidal positional encoding as specified in [CITATION].', '1905.04226-2-13-1': 'The output softmax layer gives the probability distribution for the next token.', '1905.04226-2-13-2': 'As shown in the equations above, [MATH] can be seen as states of the Transformer model (whose size, as opposed to the RNN states, linearly grows along the position dimension).', '1905.04226-2-13-3': 'During inference, these states are stored to avoid redundant computation.', '1905.04226-2-13-4': 'During training, the computation along the position dimension is parallelized for speed-up.', '1905.04226-2-14-0': '# LibriSpeech Dataset', '1905.04226-2-15-0': '## Language Modeling Data Descriptions', '1905.04226-2-16-0': 'The LibriSpeech datasets [CITATION] for language modeling consists of 800M-word text only data and 960hr of audio transcriptions which corresponds to 10M-word text data.', '1905.04226-2-16-1': 'Based on analysis of count model perplexities, we observe that the audio transcription part does not contain special domain signal which matches the development set.', '1905.04226-2-16-2': 'Therefore, we simply merge the two datasets to form a single dataset for language model training.', '1905.04226-2-16-3': 'The average sentence length in the resulting training data is 21 words with the maximum length of 600 words.', '1905.04226-2-16-4': 'The development and test sets respectively have two parts[CITATION]: dev-clean, dev-other, test-clean, and test-other.', '1905.04226-2-16-5': 'This separation is based on the audio-level characteristics, therefore it has no special meaning for language modeling.', '1905.04226-2-16-6': 'In the experimental section, we denote by "Dev" and "Test" the concatenation of clean and other parts of the respective data.', '1905.04226-2-16-7': 'Both datasets consist of about 110K running words with average of 20 words per sentence.', '1905.04226-2-16-8': 'The word-level vocabulary contains 200K words.', '1905.04226-2-16-9': 'We report all perplexities without making use of contexts beyond the sentence boundary.', '1905.04226-2-17-0': '## 4-gram count and LSTM-RNN Baselines', '1905.04226-2-18-0': 'We use the official 4-gram count language model provided with the LibriSpeech dataset [CITATION].', '1905.04226-2-18-1': 'No improvement in perplexity is observed when going up to 5-grams.', '1905.04226-2-18-2': 'For LSTM-RNN language models [CITATION], we first train our base configuration; the model has 2 LSTM-RNN layers with 2048 nodes and the input projection layer of 128, where the dropout with a rate of 0.2 is applied between each layer.', '1905.04226-2-18-3': 'Since we observe that this model underfits the LibriSpeech training set, we remove the dropout and further increase the model size, which effectively give better perplexities as shown in Table [REF].', '1905.04226-2-18-4': 'We find that improvements from simply stacking layers saturate at 4 layers even without overfitting.', '1905.04226-2-18-5': 'Introducing a small linear bottleneck layer (size 512 here) before the output layer can make the models compact but with a loss in performance.', '1905.04226-2-18-6': 'The best model we obtain has 2 layers with 4096 nodes.', '1905.04226-2-18-7': 'Relative improvements greater than 58% are obtained by the LSTM over the 4-gram language model.', '1905.04226-2-19-0': '# Text based Experiments', '1905.04226-2-20-0': 'We carry out experiments for both word-level and BPE-level language modeling.', '1905.04226-2-20-1': 'We first focus on the word-level one.', '1905.04226-2-21-0': '## Hyper-parameters in Transformers', '1905.04226-2-22-0': 'The Transformer architecture is a new search space Odyssey [CITATION].', '1905.04226-2-22-1': 'The exhaustive model hyper-parameters for Transformer language models specified by the equations in Sec. [REF] are the input token embedding size, the number of layers, the dimension of the residual connection, and for each layer the number of attention heads, the dimension of the key and query, the dimension of the value, and the dimension of the feed-forward layer.', '1905.04226-2-23-0': 'In our experiments, we use the same dimension for key, query and value, as well as the residual connection.', '1905.04226-2-23-1': 'We use the same dimensionality across all layers.', '1905.04226-2-23-2': 'Therefore, our models can be fully specified by the tuple (number of layers [MATH], feed-forward dimension [MATH], residual dimension [MATH], number of heads [MATH]).', '1905.04226-2-23-3': 'We do not apply any regularization method including dropout.', '1905.04226-2-23-4': 'We train all models using the plain stochastic gradient descent and new-bob learning rate tuning on a single GPU.', '1905.04226-2-23-5': 'We define our training sub-epoch (for new-bob) as the 10th of the full training data.', '1905.04226-2-23-6': 'All our implementations are based on the Tensorflow [CITATION] based open-source toolkit RETURNN [CITATION].', '1905.04226-2-24-0': '## Hyper-parameter Tuning', '1905.04226-2-25-0': 'Given the amount of LibriSpeech training data (810M words), it is unreasonable to train all model variants until full convergence.', '1905.04226-2-25-1': 'The earlier stage of the training already consistently indicates the potential performance of the models.', '1905.04226-2-25-2': 'Therefore, we first carry out comparisons between models with different configuration at the equal, large enough, but reasonable number of updates.', '1905.04226-2-26-0': 'The first set of comparison investigates the effect of depth and width.', '1905.04226-2-26-1': 'The perplexity results can be found in Table [REF].', '1905.04226-2-26-2': 'All models in the table use 8 attention heads.', '1905.04226-2-26-3': 'Other parameters are specified in the table.', '1905.04226-2-26-4': 'The table is organized in three parts.', '1905.04226-2-26-5': 'The upper part of Table [REF] shows the effect of number of layers; we observe that increasing number of layers (therefore the number of parameters) from 1 to 42 gradually improves the perplexity.', '1905.04226-2-26-6': 'In the middle part of Table [REF] , we vary both the number of layers, feed-forward dimension, and the residual dimension.', '1905.04226-2-26-7': 'First of all, the 12-layer [MATH] model outperforms the 6-layer [MATH] model, while having similar number of parameters, which seems to indicate that the depth effectively benefits Transformer language models.', '1905.04226-2-26-8': 'We also train an extreme model which has only 2 layers with wide dimensions [MATH].', '1905.04226-2-26-9': 'The number of parameters in fact blows up because of the large value of [MATH] which results in a large matrix in the output softmax layer with 200K vocabulary.', '1905.04226-2-26-10': 'We observe that such wide but shallow models do not perform well.', '1905.04226-2-26-11': 'Finally, the lower part of Table [REF] shows deeper models with a smaller input dimension.', '1905.04226-2-27-0': 'Table [REF] shows the effect of number of attention heads.', '1905.04226-2-27-1': '16 heads which is the largest number we try in this setup give the best performance.', '1905.04226-2-27-2': 'In addition, we examine the type of activation function (Table [REF]).', '1905.04226-2-27-3': 'As opposed to previous work on feed-forward language models using GLUs [CITATION], we do not observe faster convergence.', '1905.04226-2-27-4': 'As we observe that the impact of choice of activation functions on the perplexity is overall limited, all our other models use the standard ReLU.', '1905.04226-2-27-5': 'As reported in the original Transformer, we confirm that both layer normalization and residual connections are needed for these models for stable training.', '1905.04226-2-28-0': 'Finally, we train models with the best configurations for longer.', '1905.04226-2-28-1': 'Table [REF] shows the perplexities which are better than those obtained by our LSTM based models (Table [REF]).', '1905.04226-2-29-0': '## Parameter Tying', '1905.04226-2-30-0': 'Dehghani et al. [CITATION] reports Universal Transformers to perform particularly well for language modeling.', '1905.04226-2-30-1': 'This motivates us to experiment with parameter sharing across layers.', '1905.04226-2-30-2': 'For such models to have comparable number of parameters with the standard deep Transformers, the dimensions in each layer must be increased, which results in slower training; here we simply investigate the effect of number of recurrence.', '1905.04226-2-30-3': 'Table [REF] shows the perplexity results.', '1905.04226-2-30-4': 'First of all, we observe that the model performance is behind that of the standard Transformer (Table [REF]).', '1905.04226-2-30-5': 'However, we clearly observe that increasing the number of layers from 3 to 12 consistently improves the perplexity.', '1905.04226-2-30-6': 'This improvement without additional parameters motivates future work to investigate further parameter sharing strategies for Transformers.', '1905.04226-2-31-0': '# ASR Experiments', '1905.04226-2-32-0': '## Lattice Rescoring Results', '1905.04226-2-33-0': 'We apply our word-level Transformer language models to conventional hybrid speech recognition by lattice rescoring.', '1905.04226-2-33-1': 'The standard push-forward lattice rescoring algorithm [CITATION] for long-span language models can be directly applied to self-attention based models.', '1905.04226-2-33-2': 'The only modifications from the RNN version is to define the "state" as all hidden states ([MATH] in Sec. [REF]) in all layers from all predecessor positions and the current position ([MATH]; for position encoding).', '1905.04226-2-33-3': 'Table [REF] shows the WERs and perplexities (PPL).', '1905.04226-2-33-4': 'Our baseline acoustic model is based on multi-layer bi-directional LSTM [CITATION].', '1905.04226-2-33-5': 'Further descriptions of our baseline acoustic model can be found in [CITATION].', '1905.04226-2-33-6': 'We obtain consistent improvements in terms of WER over the LSTM baselines.', '1905.04226-2-34-0': '## End-to-End ASR Shallow Fusion Results', '1905.04226-2-35-0': 'We train 10K BPE-level Transformer language models to be combined with an attention-based encoder-decoder speech model by shallow fusion [CITATION].', '1905.04226-2-35-1': 'The 10K BPE level training data has a longer average length of 24 tokens per sentence with the longest sentence length of 1343, which is still manageable without any truncation for self-attention.', '1905.04226-2-35-2': 'We use the Transformer architecture of (24, 4096, 1024, 8).', '1905.04226-2-35-3': 'The LSTM model has 4 layers with 2048 nodes.', '1905.04226-2-35-4': 'We refer to our previous work [CITATION] for the description of the baseline attention model; the baseline WERs better than our previous work [CITATION] are obtained by improved curriculum learning and longer training.', '1905.04226-2-35-5': 'Table [REF] shows both perplexities and WERs.', '1905.04226-2-35-6': 'Following [CITATION], we introduce an end-of-sentence penalty in shallow fusion to benefit from a large beam size of 64.', '1905.04226-2-35-7': 'Again, we obtain consistent improvements over the LSTM baseline.', '1905.04226-2-35-8': 'These results are better than previously reported WERs [CITATION] for end-to-end models without data augmentation [CITATION].', '1905.04226-2-36-0': '# Analysis', '1905.04226-2-37-0': 'Compared with hidden states in RNNs, attention weights are easier to be visualized, which gives opportunity for analysis.', '1905.04226-2-37-1': 'In particular, we focus on the comparison of the Transformer language models with and without positional encoding.', '1905.04226-2-38-0': '## Transformer LM without positional encoding', '1905.04226-2-39-0': 'In the autoregressive problem where a new token is provided to the model at each time step, the amount of information the model has access to strictly increases from left to right at the lowest level of the network; the deeper layers should be able to recognize this structure which should provide the model with some positional information by its own.', '1905.04226-2-39-1': 'To check this hypothesis, we train models without any positional encoding.', '1905.04226-2-39-2': 'First, we observe that they give better perplexities than the models with sinusoidal positional encoding (Table [REF]).', '1905.04226-2-40-0': '## First layer', '1905.04226-2-41-0': 'The attention in the first layer is the most straightforward for interpretation because the feature at each position exactly corresponds to the word at the position (while deeper layers can potentially shuffle the feature content).', '1905.04226-2-41-1': 'The attention weights in the first layer of 24-layer Transformer language models with and without positional encodings are visualized in Figure [REF].', '1905.04226-2-41-2': 'We observe that the first layer of the model with positional encoding (Figure [REF](a)) learns to create n-gram features (roughly 2 or 3-gram), which indicates that the positional information is directly used.', '1905.04226-2-41-3': 'In contrast, the first layer of the model without positional encoding learns to focus on the new input token as can be seen as the diagonal in Figure [REF](b) (interestingly, we also see that it ignores some functional words such as "the", "and", "to" which might be modeled by some off-set values, therefore attending to the beginning of sentence token instead), which demonstrates that the model is aware of the position of the new input.', '1905.04226-2-42-0': '## Other layers', '1905.04226-2-43-0': 'We observe that the behavior of other layers are rather similar for both Transformer models with and without positional encoding.', '1905.04226-2-43-1': 'We find 3 categories of layers in the other 23 layers; the second and third layers are "blur" layers as shown in Figure [REF](c), which seems to roughly average over all positions (while we can also see that some heads focus on difficult words, here "verandah").', '1905.04226-2-43-2': 'Layer 4 to 9 are window layers which focus on the local n-gram.', '1905.04226-2-43-3': 'A representative example is show in Figure [REF](d).', '1905.04226-2-43-4': 'Finally, we find the top layers 10 to 24 to be more structured, attending to some specific patterns; an example is shown in Figure [REF](e).', '1905.04226-2-44-0': '# Conclusion', '1905.04226-2-45-0': 'We apply deep Transformer language models for speech recognition.', '1905.04226-2-45-1': 'We show that such models outperform the shallow stack of LSTM-RNNs on both word-level and BPE-level modeling.', '1905.04226-2-45-2': 'Future work investigates application of crucial components of deep Transformers (such as layer normalization) to deeper LSTM models; e.g., the RNMT+ decoder architecture [CITATION] for language modeling.', '1905.04226-2-45-3': 'Furthermore, we do not apply any regularization on models for the LibriSpeech task, as no overfitting is observed in the range of model sizes we experimented with (for the word-level models).', '1905.04226-2-45-4': 'We can possibly still improve our models simply by scaling up their size and using regularization.'} | [['1905.04226-1-18-0', '1905.04226-2-18-0'], ['1905.04226-1-18-1', '1905.04226-2-18-1'], ['1905.04226-1-38-0', '1905.04226-2-37-0'], ['1905.04226-1-38-1', '1905.04226-2-37-1'], ['1905.04226-1-40-0', '1905.04226-2-39-0'], ['1905.04226-1-40-1', '1905.04226-2-39-1'], ['1905.04226-1-40-2', '1905.04226-2-39-2'], ['1905.04226-1-25-0', '1905.04226-2-25-0'], 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'1905.04226-2-3-0'], ['1905.04226-1-3-1', '1905.04226-2-3-1'], ['1905.04226-1-3-2', '1905.04226-2-3-2'], ['1905.04226-1-3-3', '1905.04226-2-3-3'], ['1905.04226-1-3-4', '1905.04226-2-3-4'], ['1905.04226-1-13-0', '1905.04226-2-13-0'], ['1905.04226-1-13-1', '1905.04226-2-13-1'], ['1905.04226-1-13-3', '1905.04226-2-13-3'], ['1905.04226-1-13-4', '1905.04226-2-13-4'], ['1905.04226-1-34-0', '1905.04226-2-33-0'], ['1905.04226-1-34-1', '1905.04226-2-33-1'], ['1905.04226-1-34-2', '1905.04226-2-33-2'], ['1905.04226-1-34-3', '1905.04226-2-33-3'], ['1905.04226-1-34-4', '1905.04226-2-33-4'], ['1905.04226-1-34-5', '1905.04226-2-33-5'], ['1905.04226-1-16-0', '1905.04226-2-16-0'], ['1905.04226-1-16-1', '1905.04226-2-16-1'], ['1905.04226-1-16-3', '1905.04226-2-16-3'], ['1905.04226-1-16-5', '1905.04226-2-16-5'], ['1905.04226-1-16-6', '1905.04226-2-16-6'], ['1905.04226-1-16-7', '1905.04226-2-16-7'], ['1905.04226-1-16-8', '1905.04226-2-16-8'], ['1905.04226-1-6-0', '1905.04226-2-6-0'], ['1905.04226-1-6-1', '1905.04226-2-6-1'], ['1905.04226-1-6-2', '1905.04226-2-6-2'], ['1905.04226-1-6-3', '1905.04226-2-6-3'], ['1905.04226-1-44-0', '1905.04226-2-43-0'], ['1905.04226-1-44-1', '1905.04226-2-43-1'], ['1905.04226-1-44-2', '1905.04226-2-43-2'], ['1905.04226-1-44-3', '1905.04226-2-43-3'], ['1905.04226-1-26-0', '1905.04226-2-26-0'], ['1905.04226-1-26-1', '1905.04226-2-26-1'], ['1905.04226-1-26-2', '1905.04226-2-26-2'], ['1905.04226-1-26-3', '1905.04226-2-26-3'], ['1905.04226-1-26-5', '1905.04226-2-26-5'], ['1905.04226-1-26-7', '1905.04226-2-26-7'], ['1905.04226-1-26-8', '1905.04226-2-26-8'], ['1905.04226-1-26-9', '1905.04226-2-26-9'], ['1905.04226-1-26-10', '1905.04226-2-26-10'], ['1905.04226-1-10-0', '1905.04226-2-10-0'], ['1905.04226-1-0-1', '1905.04226-2-0-1'], ['1905.04226-1-0-2', '1905.04226-2-0-2'], ['1905.04226-1-0-3', '1905.04226-2-0-3'], ['1905.04226-1-0-4', '1905.04226-2-0-4'], ['1905.04226-1-0-5', '1905.04226-2-0-5'], ['1905.04226-1-0-6', '1905.04226-2-0-6'], ['1905.04226-1-0-7', '1905.04226-2-0-7'], ['1905.04226-1-0-8', '1905.04226-2-0-8'], ['1905.04226-1-0-9', '1905.04226-2-0-9'], ['1905.04226-1-0-10', '1905.04226-2-0-10'], ['1905.04226-1-42-0', '1905.04226-2-41-0'], ['1905.04226-1-42-1', '1905.04226-2-41-1'], ['1905.04226-1-42-2', '1905.04226-2-41-2'], ['1905.04226-1-42-3', '1905.04226-2-41-3'], ['1905.04226-1-46-0', '1905.04226-2-45-0'], ['1905.04226-1-46-1', '1905.04226-2-45-1'], ['1905.04226-1-46-2', '1905.04226-2-45-2'], ['1905.04226-1-46-4', '1905.04226-2-45-4'], ['1905.04226-1-29-1', '1905.04226-2-28-1'], ['1905.04226-1-8-0', '1905.04226-2-8-0'], ['1905.04226-1-8-1', '1905.04226-2-8-1'], ['1905.04226-1-8-2', '1905.04226-2-8-2'], ['1905.04226-1-8-4', '1905.04226-2-8-4'], ['1905.04226-1-8-5', '1905.04226-2-8-5'], ['1905.04226-1-4-0', '1905.04226-2-4-0'], ['1905.04226-1-4-1', '1905.04226-2-4-1'], ['1905.04226-1-4-2', '1905.04226-2-4-2'], ['1905.04226-1-4-3', '1905.04226-2-4-3'], ['1905.04226-1-4-4', '1905.04226-2-4-4'], ['1905.04226-1-11-0', '1905.04226-2-11-0'], ['1905.04226-1-20-0', '1905.04226-2-20-0'], ['1905.04226-1-20-1', '1905.04226-2-20-1']] | [['1905.04226-1-18-2', '1905.04226-2-18-2'], ['1905.04226-1-25-1', '1905.04226-2-25-1'], ['1905.04226-1-25-2', '1905.04226-2-25-2'], ['1905.04226-1-31-1', '1905.04226-2-30-1'], ['1905.04226-1-31-2', '1905.04226-2-30-2'], ['1905.04226-1-31-4', '1905.04226-2-30-4'], ['1905.04226-1-31-5', '1905.04226-2-30-5'], ['1905.04226-1-36-0', '1905.04226-2-35-0'], ['1905.04226-1-36-4', '1905.04226-2-35-4'], ['1905.04226-1-36-7', '1905.04226-2-35-8'], ['1905.04226-1-13-2', '1905.04226-2-13-2'], ['1905.04226-1-22-0', '1905.04226-2-22-0'], ['1905.04226-1-22-1', '1905.04226-2-22-1'], ['1905.04226-1-34-6', '1905.04226-2-33-6'], ['1905.04226-1-16-2', '1905.04226-2-16-2'], ['1905.04226-1-16-4', '1905.04226-2-16-4'], ['1905.04226-1-44-4', '1905.04226-2-43-4'], ['1905.04226-1-26-4', '1905.04226-2-26-4'], ['1905.04226-1-26-6', '1905.04226-2-26-6'], ['1905.04226-1-10-1', '1905.04226-2-10-1'], ['1905.04226-1-0-0', '1905.04226-2-0-0'], ['1905.04226-1-46-3', '1905.04226-2-45-3']] | [] | [['1905.04226-1-18-3', '1905.04226-2-18-3'], ['1905.04226-1-18-6', '1905.04226-2-18-7'], ['1905.04226-1-36-5', '1905.04226-2-35-5'], ['1905.04226-1-29-0', '1905.04226-2-28-0'], ['1905.04226-1-8-3', '1905.04226-2-8-3']] | [] | ['1905.04226-1-1-0', '1905.04226-1-36-2', '1905.04226-2-1-0', '1905.04226-2-35-2'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1905.04226 | null | null | null | null | null |
1407.3510 | {'1407.3510-1-0-0': 'We show that cells in a fibrous matrix induce deformation fields that propagate over a longer range than predicted by linear elasticity.', '1407.3510-1-0-1': 'Synthetic, linear elastic hydrogels used in many mechanotransduction studies fail to capture this effect.', '1407.3510-1-0-2': 'We develop a nonlinear microstructural finite element model for a fiber network to simulate localized deformations induced by cells.', '1407.3510-1-0-3': 'The model captures measured cell-induced matrix displacements from experiments and identifies an important mechanism for long range cell mechanosensing: loss of compression stiffness due to microbuckling of individual fibers.', '1407.3510-1-0-4': 'We show evidence that cells sense each other through the formation of localized intercellular bands of tensile deformations caused by this mechanism.', '1407.3510-1-1-0': 'Introduction.', '1407.3510-1-1-1': 'Physical cues control cell behavior through various mechanisms collectively referred to as mechanotransduction [CITATION].', '1407.3510-1-1-2': "For example, the stiffness of a cell's environment controls cellular morphology, migration, and development [CITATION].", '1407.3510-1-1-3': 'Equally important is the response of cells to direct physical forces either through cell-cell adhesions [CITATION] or through the extracellular matrix [CITATION].', '1407.3510-1-1-4': 'While most previous work on mechanotransduction has used synthetic, linear elastic gels [CITATION], nonlinear constitutive properties of biological gels can have a dramatic effect on cell response [CITATION].', '1407.3510-1-1-5': 'Natural fibrous matrices exhibit strain stiffening [CITATION], negative normal strains under shear [CITATION], negative effective bulk modulus (called "negative compressibility") [CITATION], and lower stiffness in compression than in tension [CITATION].', '1407.3510-1-1-6': 'Various models have simulated these observations, but most have focused on homogeneous shearing [CITATION] or uniaxial tension [CITATION] of the bulk material; relatively few studies have considered local, non-uniform deformations in a nonlinear medium [CITATION].', '1407.3510-1-2-0': 'By contracting and changing shape, cells apply localized forces to their surroundings, causing inhomogeneous stress and deformation fields in the matrix.', '1407.3510-1-2-1': 'We measure 3D cell-induced matrix displacements experimentally and report two unexpected findings: (i) displacements decay much slower with distance from the cell than predicted by linear elasticity; (ii) multiple cells cause localized matrix densification and fiber alignment in tether-like bands joining them.', '1407.3510-1-2-2': 'We hypothesize that the mechanism responsible for these phenomena is loss of compression strength due to microbuckling of individual fibers.', '1407.3510-1-2-3': 'To test this claim, we develop a microstructural finite element (FE) network model of the fibrin matrix.', '1407.3510-1-2-4': 'Buckling of individual fibers is modeled by a loss of stiffness in compression for network elements.', '1407.3510-1-2-5': 'Our model agrees with previous experimental observations for fibrin, and predicts both the slow decay of displacements and localization in intercellular tethers.', '1407.3510-1-2-6': 'Variants of the model without loss of stiffness in compression fail to predict these effects.', '1407.3510-1-2-7': 'The long range of cell-generated displacements and stresses, and the localization into intercellular tensile tethers, allow cells to sense each other and their surroundings over larger distances through a fibrous matrix, than through homogeneous hydrogels with linear elastic behavior.', '1407.3510-1-2-8': 'We show that cells respond to localized tension by growing protrusions towards one another, guided by the dense aligned fibers in tethers.', '1407.3510-1-2-9': 'This points to fiber microbuckling as an important mechanism responsible for enhanced range of cell mechanosensing in fibrous matrix environments.', '1407.3510-1-3-0': 'Cell-Induced Matrix Displacements.', '1407.3510-1-3-1': 'We motivate our model by first quantifying cell-induced displacements within a 3D fibrin matrix during initial cell spreading.', '1407.3510-1-3-2': 'A cell seeded in a 3D matrix initially applies tensile tractions to the matrix by undergoing uniform isotropic contraction while in an essentially spherical state.', '1407.3510-1-4-0': "This suggests Eshelby's solution for a contracting spherical inclusion in a homogeneous, linear elastic, infinite medium [CITATION], as a simple analytical model for cell-induced matrix deformation.", '1407.3510-1-4-1': 'In this solution, the displacement magnitude [MATH] scales as [MATH] with distance [MATH] from the cell center.', '1407.3510-1-4-2': 'Stress components, e.g., the radial component [MATH], scale as [MATH].', '1407.3510-1-4-3': 'A spreading, elongated, ellipsoidal cell with polarized alignment, applies tractions equivalent to equal and opposite forces at its poles, i.e. a dipole [CITATION], in view of force equilibrium.', '1407.3510-1-4-4': 'Displacements due to a dipole in a linearly elastic continuum also scale as [MATH].', '1407.3510-1-4-5': 'One would thus expect displacements induced by a spreading cell in a 3D matrix to scale similarly.', '1407.3510-1-5-0': 'We measure displacements induced by isolated fibroblast cells embedded in a 3D fibrin matrix (Fig. 1a) using confocal microscopy and digital volume correlation [CITATION] (see Appendix for more details).', '1407.3510-1-5-1': 'Experimental data from different cells are plotted in a logarithmic scale in Fig. 1b.', '1407.3510-1-5-2': "Fits of the form [MATH] for the constants [MATH] and [MATH] yield [MATH] (mean over data from 6 cells during multiple time points), indicating that displacements decay much slower than predicted by Eshelby's solution.", '1407.3510-1-5-3': 'The ratio of the RMS errors of fits to [MATH] and [MATH], is [MATH] (mean [MATH] standard deviation), hence the scaling [MATH] describes cell-induced displacements in a fibrin matrix much better than the linear elastic scaling [MATH].', '1407.3510-1-6-0': 'A striking difference between fibrin networks and homogeneous gel matrices is the decreased ability of the former to sustain compressive stresses.', '1407.3510-1-6-1': 'Fibrin exhibits a larger stiffness in tension than compression [CITATION] due to buckling of individual fibers under compression [CITATION].', '1407.3510-1-6-2': 'Is this nonlinearity responsible for the discord between the observed displacement scaling and the prediction based on a linear elastic matrix assumption?', '1407.3510-1-6-3': 'A simple theoretical argument for this follows.', '1407.3510-1-6-4': 'Since the cell exerts radial contractile forces, the stress tensor in the matrix has a tensile (positive) radial component in 3D spherical coordinates, and two contractile (negative) hoop (angular) components.', '1407.3510-1-6-5': 'Assuming the individual fibers of the fibrin matrix buckle under a small compressive load, the contractile hoop components of the stress tensor are small and can be neglected.', '1407.3510-1-6-6': 'This assumption reduces the radial equilibrium equation [CITATION] to: [EQUATION]', '1407.3510-1-6-7': 'Solving Eq. (1) gives [MATH].', '1407.3510-1-6-8': "Thus, stress due to cell contraction is transmitted over a longer range than under the scaling [MATH] predicted by Eshelby's solution.", '1407.3510-1-6-9': 'Assuming piecewise linear stress-strain relations with zero stiffness in compression, [MATH] is proportional to the radial strain [MATH] which gives [MATH] .', '1407.3510-1-6-10': 'This is closer to the observed scaling [MATH] than to the linear elastic one [MATH], with slower decay than the latter.', '1407.3510-1-7-0': 'Model.', '1407.3510-1-7-1': 'The previous plausibility argument shows the right trend but ignores the inhomogeneous and discrete nature of the fibrin network.', '1407.3510-1-7-2': 'To account for these factors, we develop a FE-based microstructural model consisting of a 2D/3D network of 1D elements representing fibers.', '1407.3510-1-7-3': 'Each element undergoes uniaxial tension/compression (and rotations) without bending.', '1407.3510-1-7-4': 'We model buckling of fibers as a loss of stiffness in compression in the stress-strain relation of individual elements.', '1407.3510-1-7-5': 'This agrees qualitatively with observed behavior in similar systems [CITATION].', '1407.3510-1-8-0': 'In the context of the FE model, "microbuckling" will refer to elements obeying a stress-strain relation where the stiffness (slope) under compression is [MATH] the stiffness under tension; see Fig. 2a.', '1407.3510-1-8-1': 'While the factor of 1/10 is arbitrary, we find that any positive (for stability reasons) ratio of stiffnesses significantly less than 1/4 yields very similar results.', '1407.3510-1-8-2': 'In contrast, "no microbuckling" will refer to elements with a linear stress-strain relation without a reduced compression stiffness.', '1407.3510-1-8-3': 'For most simulations, networks comprise elements with a bi-linear stress-strain relationship (Fig. 2a, different slopes in tension and compression).', '1407.3510-1-8-4': 'We will also account for the possibility of entropic elasticity by employing a wormlike chain-type (WLC) stress-strain relationship [CITATION] , where the stiffness increases with strain in tension (Fig. 2a).', '1407.3510-1-8-5': 'The elements connect an array of nodes as in Fig. 2b.', '1407.3510-1-8-6': 'Randomness is added to nodal positions to simulate the random array of fibers of different lengths typical of a fibrous network (Fig. 2c).', '1407.3510-1-9-0': 'Another important aspect of actual fibrin networks is their low connectivity, or coordination number [MATH], i.e., the average number of fibers meeting at a node.', '1407.3510-1-9-1': 'The network of Fig. 2b,c has [MATH], while actual fibrin often has a typical value of [MATH] [CITATION].', '1407.3510-1-9-2': 'This is below the critical value for rigidity, [MATH] or [MATH] for 3D and 2D networks, respectively.', '1407.3510-1-9-3': 'As a result, fibrin is typically a "floppy" network, and this affects its mechanical properties [CITATION].', '1407.3510-1-9-4': 'To obtain a model network with lower connectivity (such as [MATH] in Fig. 2d), we removed elements at random from the original [MATH] network of Fig. 2c.', '1407.3510-1-9-5': 'As in [CITATION], deleted elements were replaced by weak elements, whose stiffness was six orders of magnitude less than that of the remaining ones; this ensured stability of numerical calculations.', '1407.3510-1-10-0': 'The behavior of the model in homogeneous deformations, such as simple shear and uniaxial tension, is consistent with previous experimental work [CITATION] (see Fig. 5 for details).', '1407.3510-1-11-0': 'Simulations.', '1407.3510-1-11-1': 'In contrast to previous models that focus on the macroscale behavior of a fiber network [CITATION], we simulate the inhomogeneous, localized displacements induced in a fibrin matrix by an embedded cell.', '1407.3510-1-11-2': 'We performed 2D FE simulations where the cell is modeled as a contracting circle.', '1407.3510-1-11-3': 'The matrix occupies the region [MATH], where [MATH] is distance from the cell center; here [MATH] is the cell radius, [MATH].', '1407.3510-1-11-4': 'The outside boundary [MATH] is free of forces.', '1407.3510-1-11-5': 'The cell boundary [MATH] undergoes a radial contractile displacement [MATH].', '1407.3510-1-11-6': 'Simulations were performed for 9 different connectivities in the interval [MATH] for bilinear-element networks with microbuckling or without.', '1407.3510-1-11-7': 'The displacement magnitude was plotted (Fig. 6) and fitted to [MATH] for the constants [MATH] and [MATH].', '1407.3510-1-11-8': 'Results, [MATH] plotted versus connectivity [MATH], are shown in Fig. 3.', '1407.3510-1-12-0': 'In general, the decay power [MATH] for networks with microbuckling (Fig. 3, white circles) is substantially lower, by at least [MATH], than for networks without microbuckling of fibers (Fig. 3, black circles).', '1407.3510-1-12-1': 'This is true for a wide range of connectivities, with the exception of the critical value [MATH]; for this value [MATH] in both types of networks.', '1407.3510-1-12-2': 'We observe larger spatial inhomogeneities of displacement at the scale of individual fibers in networks with [MATH] than in those with both subcritical and supercritical connectivity (Fig. 6), in accordance with [CITATION].', '1407.3510-1-12-3': 'For the case without buckling, since individual elements have linear stress-strain behavior, we compare the displacement to the linear elastic 2D solution, [MATH], by also fitting simulation data to [MATH] for the constants [MATH], [MATH], [MATH] (black squares, Fig. 3).', '1407.3510-1-12-4': 'This gives [MATH] (mean [MATH] standard deviation, essentially independent of [MATH] over all connectivities except the critical case [MATH], for which [MATH]).', '1407.3510-1-12-5': 'This value is close to the linear elastic solution ([MATH]=1).', '1407.3510-1-12-6': 'The difference between values of [MATH] from the fit to [MATH] with and without buckling (white vs. black squares in Fig. 3) once again exceeds 0.4 as for the fit to [MATH].', '1407.3510-1-12-7': 'Microbuckling is thus crucial for the slow decay of displacements.', '1407.3510-1-12-8': 'Connectivity does not appear to play a major role in displacement decay (except possibly near the critical value).', '1407.3510-1-13-0': 'The long range of cell induced displacements has been attributed to strain stiffening [CITATION], but this has been disputed [CITATION].', '1407.3510-1-13-1': 'To help settle this we repeated our simulations with elements whose stress-strain curve is of WLC type (blue curve, Fig. 2a) and stiffens in tension.', '1407.3510-1-13-2': 'A discontinuous slope at zero strain (10 times smaller for small compressive strain than the tangent stiffness for small tensile strain) models microbuckling.', '1407.3510-1-13-3': 'A continuous slope at the origin was used for non-buckling WLC elements.', '1407.3510-1-13-4': 'In all cases, values of the decay exponent [MATH] from fits for WLC networks were within 5% of those for bilinear networks of the same connectivity and same (buckling or non-buckling) type.', '1407.3510-1-13-5': 'This supports the conclusion that the tension-stiffening nonlinearity in the absence of microbuckling is not the cause of observed slow displacement decay.', '1407.3510-1-14-0': 'We also performed 3D simulations (contracting spherical cell), with similar conclusions.', '1407.3510-1-14-1': 'We recall that the 3D linear elastic solution predicts [MATH], the theoretical argument based on Eq. (1) gives [MATH], while a fit to our experiments yields [MATH].', '1407.3510-1-14-2': 'For 3D networks (microbuckling bilinear elements) with [MATH], a fit to [MATH] gives [MATH].', '1407.3510-1-14-3': 'For [MATH] (below the critical value for rigidity [MATH]) we found [MATH].', '1407.3510-1-15-0': 'These results combine to show that microbuckling of fibers is the key mechanism responsible for the longer range of cell-induced deformations in a fibrin matrix.', '1407.3510-1-16-0': 'Tethers.', '1407.3510-1-17-0': 'Can cells exploit the long propagation range of matrix deformations they themselves induce for sensing the presence of other cells?', '1407.3510-1-17-1': 'We use confocal microscopy to visualize both the matrix and multiple fibroblast cells embedded in it.', '1407.3510-1-17-2': 'We observe that cells whose distance from each other is of the order of 10 cell diameters are connected to each other by linear bands (Fig. 4a) consisting of aligned and densely packed matrix fibers.', '1407.3510-1-17-3': 'Within these "tethers" fibers appear to be in tension in the direction joining the cells.', '1407.3510-1-17-4': 'We investigate the physical mechanism of matrix fiber alignment and densification through our FE network model with microbuckling, by simulating a pair of contracting cells.', '1407.3510-1-17-5': 'A symmetric boundary condition is imposed at the bottom of the square region containing a circle of radius [MATH] contracting with a radial displacement of [MATH] (the other sides are free of applied tractions).', '1407.3510-1-17-6': 'By symmetry this is equivalent to a pair of identical contracting cells in the fibrin matrix.', '1407.3510-1-17-7': 'Tensile strains in the model network (Fig. 4c) occur almost entirely in the band between the two cells, along aligned linear paths formed by elements in tension.', '1407.3510-1-17-8': 'Compressive strains (Fig. 4d) localize perpendicular to these tensile tethers.', '1407.3510-1-17-9': 'Due to low compression stiffness (microbuckling), the magnitude of compressive strains in elements roughly perpendicular to the tether is more than twice the magnitude of the tensile strains.', '1407.3510-1-17-10': 'Thus within the tether, the trace of the strain tensor, or the volumetric strain, is negative, consistent with the observation that matrix fiber density increases between pairs of cells (Fig. 4a).', '1407.3510-1-17-11': 'When simulating networks without microbuckling, we found no such tethers forming; instead, tensile strains had a nearly radially symmetric distribution around each cell (Fig. 7).', '1407.3510-1-17-12': 'Localization of deformation caused by multiple cells in tensile, tether-like regions joining the cells occurs because of microbuckling (of fibers normal to the tether).', '1407.3510-1-18-0': 'Cells actually respond to tether formation by changing shape.', '1407.3510-1-18-1': 'In Fig. 4a, each cell has started forming a pointed protrusion toward the other cell.', '1407.3510-1-18-2': 'These protrusions grow along the tethers by several cell diameters, sometimes eventually joining two cells (Fig. 4b).', '1407.3510-1-18-3': 'This suggests a mechanism whereby cells can sense each other in a fibrin matrix (mechanosensing).', '1407.3510-1-18-4': 'Even if each cell is initially spherical and contracts isotropically, the tether formation mechanism just described results in deformation that is highly polarized in the direction of other cells.', '1407.3510-1-18-5': 'By growing protrusions in directions of large tension, cells have a higher chance of approaching one another.', '1407.3510-1-19-0': 'Conclusions.', '1407.3510-1-19-1': 'Together, our simulations and experiments reveal that microbuckling of fibrin enables cells to induce displacements that propagate along linear, tether-like paths that lead to other cells, over a dramatically longer range than in a linear material.', '1407.3510-1-19-2': 'The fact that cells change shape and grow along such tethers is strong evidence that they use this mechanism to sense and even approach their neighbors.', '1407.3510-1-19-3': 'Our FE-based microstructural model for fibrin is consistent with previous experimental work on fibrin [CITATION] and collagen [CITATION].', '1407.3510-1-19-4': "The results are independent of the tensile portion of the individual fibers' stress-strain relation: we have captured the same scalings for both linear and strain-stiffening stress-strain relations.", '1407.3510-1-20-0': 'Our observations highlight the need to account for nonlinear properties of fibrous materials by developing new constitutive laws for them.', '1407.3510-1-20-1': 'Such new models will play a key role in improving our understanding of cell mechanosensing in biologically relevant fibrous matrices.', '1407.3510-1-21-0': '*', '1407.3510-1-22-0': '# Methods', '1407.3510-1-23-0': '## Cell culture and matrix preparation', '1407.3510-1-24-0': '3T3 fibroblast cells stably expressing a green fluorescent protein-actin fusion protein were cultured in DMEM medium supplemented with 10% fetal bovine serum and 1[MATH] non-essential amino acids.', '1407.3510-1-24-1': 'Fibrin was fluorescently labeled by mixing fibrinogen (Omrix Biopharmaceuticals, Israel) and 546 Alexa Fluor (Life Technologies, Carlsbad, CA, USA) for 1 hour before filtering with a HiTrap desalting column (GE Healthcare, Milwaukee, WI, USA).', '1407.3510-1-24-2': 'Cell-fibrin constructs were created by suspending the cells in 20 U/mL thrombin solution (Omrix), mixing with 5 mg/mL labeled fibrinogen solution, and placing on a 1.5 coverslip.', '1407.3510-1-25-0': '## Microscopy and cell-induced matrix displacements', '1407.3510-1-26-0': 'Within 1 hour of seeding, cell-matrix constructs were transferred to a custom built 5% CO[MATH], 37[MATH]C microscope enclosure.', '1407.3510-1-26-1': 'Volume stacks of the cells and fibrin matrix were obtained with a Swept Field confocal microscope using a 40[MATH] NA 1.15 water immersion objective (Nikon Instruments, Melville, NY, USA).', '1407.3510-1-26-2': '3D matrix displacements were computed directly from the images of the labeled fibrin using digital volume correlation [CITATION] with the initial volume stack (before cell spreading) taken as a reference for the correlation.', '1407.3510-1-26-3': 'Propagation of cell-induced matrix displacements was quantified by computing displacement magnitudes along multiple linear paths propagating outward from the center of each initially rounded cell.', '1407.3510-1-26-4': 'To reduce errors caused by inhomogeneities within the matrix, displacements were averaged over different paths and over time for each cell.', '1407.3510-1-26-5': 'After averaging, the standard deviation of the noise level was found to be 0.04 [MATH]m.', '1407.3510-1-27-0': 'This work was funded in part by a grant from the National Science Foundation (Division of Materials Research No. 0520565) through the Center for the Science and Engineering of Materials at the California Institute of Technology, and in part by National Science Foundation Grant No. DMR-1206121.', '1407.3510-1-27-1': 'J.N. was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469.'} | {'1407.3510-2-0-0': 'Biological cells sense and respond to mechanical forces, but how such a mechanosensing proccess takes place in a nonlinear inhomogeneous fibrous matrix remains unknown.', '1407.3510-2-0-1': 'We show that cells in a fibrous matrix induce deformation fields that propagate over a longer range than predicted by linear elasticity.', '1407.3510-2-0-2': 'Synthetic, linear elastic hydrogels used in many mechanotransduction studies fail to capture this effect.', '1407.3510-2-0-3': 'We develop a nonlinear microstructural finite element model for a fiber network to simulate localized deformations induced by cells.', '1407.3510-2-0-4': 'The model captures measured cell-induced matrix displacements from experiments and identifies an important mechanism for long range cell mechanosensing: loss of compression stiffness due to microbuckling of individual fibers.', '1407.3510-2-0-5': 'We show evidence that cells sense each other through the formation of localized intercellular bands of tensile deformations caused by this mechanism.', '1407.3510-2-1-0': '# Key Words', '1407.3510-2-2-0': 'Fibrous matrix, Buckling, 3D traction force, Cell mechanics', '1407.3510-2-3-0': '# Introduction', '1407.3510-2-4-0': 'Physical cues control cell behavior through various mechanisms collectively referred to as mechanotransduction [CITATION].', '1407.3510-2-4-1': "For example, the stiffness of a cell's environment controls cellular morphology, migration, and development [CITATION].", '1407.3510-2-4-2': 'Equally important is the response of cells to direct physical forces either through cell-cell adhesions [CITATION] or through the extracellular matrix [CITATION].', '1407.3510-2-4-3': 'Nearly all previous work on mechanotransduction has used synthetic, linear elastic gels [CITATION].', '1407.3510-2-4-4': 'The mechanical properties of physiological extracellular environment, however, deviate entirely from simple homogeneous linear elasticity.', '1407.3510-2-4-5': 'Natural fibrous matrices exhibit strain stiffening [CITATION], tensile normal strains under shear loading [CITATION], negative compressibility [CITATION], and lower stiffness in compression than in tension [CITATION].', '1407.3510-2-4-6': 'These nonlinear properties of biological gels can have a dramatic effect on behaviors like cell spreading [CITATION].', '1407.3510-2-5-0': 'Various models have simulated nonlinearity of fibrous biological materials, but relatively few have considered local, non-uniform deformations in such nonlinear inhomogeneous materials [CITATION].', '1407.3510-2-5-1': 'Instead, nearly all previous studies have focused on homogeneous shearing [CITATION] or uniaxial tension [CITATION] of the bulk material.', '1407.3510-2-5-2': 'These studies of uniform deformations have revealed novel constitutive behavior of fibrous materials, but they fail to simulate deformations similar to those applied by a cell.', '1407.3510-2-5-3': 'By contracting and changing shape, cells apply localized forces to their surroundings, resulting in inhomogeneous stress and deformation fields in the matrix.', '1407.3510-2-5-4': 'Given the lack of theoretical and experimental studies of cell-matrix interactions at the local scale, there remains a need to quantify cell-generated forces and displacements and to discern how cells respond to nonlinear properties of fibrous materials at the scale sensed by the cell.', '1407.3510-2-6-0': 'Here we experimentally measure 3D cell-induced matrix displacements and report two findings: (i) displacements decay much slower with distance from the cell than predicted by linear elasticity; (ii) multiple cells cause localized matrix densification and fiber alignment in tether-like bands joining them.', '1407.3510-2-6-1': 'We hypothesize that the mechanism responsible for these phenomena is loss of compression strength due to microbuckling of individual fibers.', '1407.3510-2-6-2': 'To test this claim, we develop a microstructural finite element (FE) network model of the fibrin matrix.', '1407.3510-2-6-3': 'Buckling of individual fibers is modeled by a loss of stiffness in compression for network elements.', '1407.3510-2-6-4': 'Our model agrees with previous experimental observations for fibrin, and it predicts both the slow decay of displacements and localization of intercellular tethers.', '1407.3510-2-6-5': 'Variants of the model without loss of stiffness in compression fail to predict these effects.', '1407.3510-2-6-6': 'The long range of cell-generated displacements and stresses, and the localization into intercellular tensile tethers, allow cells to sense each other and their surroundings over larger distances through a fibrous matrix than through homogeneous hydrogels with linear elastic behavior.', '1407.3510-2-6-7': 'We show evidence that cells respond to localized tension by growing protrusions towards one another, guided by the dense aligned fibers in tethers.', '1407.3510-2-6-8': 'This points to fiber microbuckling as an important mechanism responsible for enhanced range of cell mechanosensing in fibrous matrix environments.', '1407.3510-2-7-0': '# Results', '1407.3510-2-8-0': '## Cell-Induced Matrix Displacements.', '1407.3510-2-9-0': 'We motivate our model by first considering cell-induced displacements within a 3D fibrous matrix during initial cell spreading.', '1407.3510-2-9-1': 'A cell seeded in a 3D matrix initially applies tensile tractions to the fibers by undergoing uniform isotropic contraction while in an essentially spherical state.', '1407.3510-2-9-2': "This suggests Eshelby's solution for a contracting spherical inclusion in a homogeneous, linear elastic, infinite medium [CITATION], as a simple analytical model for cell-induced matrix deformation.", '1407.3510-2-9-3': 'In this solution, the displacement magnitude [MATH] scales as [MATH] with distance [MATH] from the cell center.', '1407.3510-2-9-4': 'Stress components, e.g., the radial component [MATH], scale as [MATH].', '1407.3510-2-9-5': 'A spreading, elongated, ellipsoidal cell with polarized alignment, applies tractions equivalent to equal and opposite forces at its poles, i.e. a dipole [CITATION], in view of force equilibrium.', '1407.3510-2-9-6': 'Displacements due to a dipole in a 3D linear elastic continuum also scale as [MATH].', '1407.3510-2-9-7': 'One would thus expect displacements induced by a spreading cell in a 3D matrix to scale similarly.', '1407.3510-2-10-0': 'Using confocal microscopy and digital volume correlation [CITATION], we measure displacements induced by isolated fibroblast cells embedded in a 3D fibrin matrix.', '1407.3510-2-10-1': 'Displacements induced by the cells are largest near to the cell and decrease with distance from the cell (Fig. 1a).', '1407.3510-2-10-2': 'We quantify the rate at which displacements decay over distance by computing displacements along linear paths starting at the the center of cell and ending [MATH]100 m away (Fig. 1a, white line).', '1407.3510-2-10-3': 'Experimental data from multiple different cells are plotted on a logarithmic scale in Fig. 1b.', '1407.3510-2-10-4': 'Data are fit to the form [MATH].', '1407.3510-2-10-5': 'Here [MATH] and [MATH] are constants; [MATH] is a decay power.', '1407.3510-2-10-6': 'The larger the value of [MATH], the faster the displacement [MATH] decays with distance [MATH] from the cell center.', '1407.3510-2-10-7': 'Fits of the experimental data yield [MATH] (mean over data from 6 cells during multiple time points), indicating that displacements decay much slower than predicted by the linear elastic solution [MATH].', '1407.3510-2-10-8': 'The ratio of the RMS errors of fits to [MATH] and [MATH], is [MATH] (mean [MATH] standard deviation), hence the scaling [MATH] describes cell-induced displacements in a fibrin matrix far better than the 3D linear elastic scaling [MATH].', '1407.3510-2-11-0': 'A striking difference between fibrin networks and homogeneous gel matrices is the phenomenon of buckling of individual fibers under compression; an example is shown in Fig. 5 of ref. [CITATION].', '1407.3510-2-11-1': 'This is directly responsible for the decreased ability of fibrin networks to sustain compressive stresses.', '1407.3510-2-11-2': 'Each fiber has very low resistance to bending, much like a flexible string [CITATION].', '1407.3510-2-11-3': 'If one pulls at the ends of a string, it resists tension.', '1407.3510-2-11-4': 'If one pushes the ends of a string towards each other, the string bends easily without resisting compression (i.e., it buckles), and this buckling can change the mechanical response of a network [CITATION].', '1407.3510-2-11-5': 'Fibrin exhibits a larger stiffness in tension than compression [CITATION] due to buckling of individual fibers under compression [CITATION].', '1407.3510-2-11-6': 'Is this nonlinearity responsible for the discord between the observed displacement scaling and the prediction based on a linear elastic matrix assumption?', '1407.3510-2-11-7': 'While it may be possible to address this via a continuum model for a material with lower stiffness in compression [CITATION], here we present a simple theoretical continuum argument, which we will investigate in detail using a discrete model.', '1407.3510-2-11-8': 'Since the cell exerts radial contractile traction forces, the stress tensor in the matrix has a tensile (positive) radial component in 3D spherical coordinates, and two contractile (negative) hoop (angular) components.', '1407.3510-2-11-9': 'Assuming the individual fibers of the fibrin matrix buckle under a small compressive load, the contractile hoop components of the stress tensor are small and can be neglected.', '1407.3510-2-11-10': 'This assumption reduces the radial equilibrium equation [CITATION] to [EQUATION]', '1407.3510-2-11-11': 'Solving Eq. (1) gives [MATH].', '1407.3510-2-11-12': 'Thus, stress due to cell contraction is transmitted over a longer range than under the scaling [MATH] predicted by linear elasticity.', '1407.3510-2-11-13': 'Assuming piecewise linear stress-strain relations with zero stiffness in compression, [MATH] is proportional to the radial strain [MATH] which gives [MATH].', '1407.3510-2-11-14': '(Coupling between [MATH] and the hoop strains [MATH], [MATH] vanishes due to hyperelastic reciprocity: [MATH] since [MATH] in the compressive regime.', '1407.3510-2-11-15': 'For more details on a hyperelastic material model that leads to Eq. (1) as a special case, see ref. [CITATION].)', '1407.3510-2-11-16': 'The scaling from this simple analysis, [MATH], points towards displacements that propagate over a longer range than the 3D linear elastic scaling, [MATH].', '1407.3510-2-11-17': 'Furthermore, the scaling from the theoretical analysis is closer to the experimentally observed scaling, [MATH] than to the linear elastic one.', '1407.3510-2-11-18': 'This plausibility argument shows the right trend, but ignores the inhomogeneous and discrete nature of the fibrin network.', '1407.3510-2-11-19': 'To account for these factors, we turn to a microstructural network model.', '1407.3510-2-12-0': '## Model.', '1407.3510-2-12-1': 'We develop a FE-based microstructural model consisting of a 2D/3D network of linear elements representing fibers.', '1407.3510-2-12-2': 'This model expands on one that we have recently developed [CITATION].', '1407.3510-2-12-3': 'Each element undergoes uniaxial tension/compression and rotates with no resistance.', '1407.3510-2-12-4': 'We model buckling of fibers as a loss of stiffness in compression in the stress-strain relation of individual elements.', '1407.3510-2-12-5': 'This agrees qualitatively with observed behavior in similar systems [CITATION].', '1407.3510-2-12-6': 'In the context of our model, "microbuckling" will refer to elements obeying a stress-strain relation where the stiffness (slope) under compression is smaller than the stiffness under tension; see Fig. 2a, blue line.', '1407.3510-2-12-7': 'In the following simulations, we use a ratio of stiffness in compression to stiffness in tension [MATH].', '1407.3510-2-12-8': 'While the choice of [MATH] is arbitrary, we find that any positive ratio of stiffnesses [MATH] significantly less than unity yields similar results.', '1407.3510-2-12-9': 'In contrast, "no microbuckling" will refer to elements with [MATH], i.e. elements with a linear stress-strain relation without a reduced compression stiffness.', '1407.3510-2-12-10': 'For most simulations, networks comprise elements with a bi-linear stress-strain relationship (Fig. 2a, different slopes in tension and compression).', '1407.3510-2-12-11': 'We will also account for the possibility of entropic elasticity by employing a wormlike chain-type (WLC) stress-strain relationship [CITATION], where the stiffness increases with strain in tension (Fig. 2b).', '1407.3510-2-12-12': 'The elements connect an array of nodes as in Fig. 2c.', '1407.3510-2-12-13': 'Randomness is added to nodal positions to simulate the random array of fibers of different lengths typical of a fibrous network (Fig. 2d).', '1407.3510-2-13-0': 'Another important aspect of actual fibrin networks is their low connectivity, or coordination number [MATH], i.e. the average number of fibers meeting at a node.', '1407.3510-2-13-1': 'The network of Fig. 2c,d has [MATH], while actual fibrin often has a typical value of [MATH] [CITATION].', '1407.3510-2-13-2': 'This is below the critical value for rigidity, [MATH] or [MATH] for 3D and 2D networks, respectively.', '1407.3510-2-13-3': 'As a result, fibrin is typically a "floppy" network, and this affects its mechanical properties [CITATION].', '1407.3510-2-13-4': 'To obtain a model network with lower connectivity (such as [MATH] in Fig. 2e), we removed elements at random from the original [MATH] network of Fig. 2d.', '1407.3510-2-13-5': 'As in ref. [CITATION], deleted elements were replaced by weak elements, whose stiffness was six orders of magnitude less than that of the deleted ones; this ensured stability of numerical calculations.', '1407.3510-2-14-0': 'In contrast to previous models that focus on the macroscale behavior of a fiber network [CITATION], we simulate the inhomogeneous, localized displacements induced in a fibrin matrix by an embedded cell.', '1407.3510-2-14-1': 'We begin with 2D FE simulations where the cell is modeled as a contracting circle.', '1407.3510-2-14-2': 'The matrix occupies the region [MATH], where [MATH] is distance from the cell center; here [MATH] is the cell radius, and [MATH].', '1407.3510-2-14-3': 'The outside boundary [MATH] is free (a zero traction boundary condition is imposed).', '1407.3510-2-14-4': 'The cell boundary [MATH] undergoes a radial contractile displacement [MATH].', '1407.3510-2-14-5': 'Simulations were performed for different connectivities in the interval [MATH] for bilinear element networks with microbuckling and without.', '1407.3510-2-14-6': 'The displacement magnitude [MATH] was computed (Fig. 3a), averaged around the circular region (Fig. 3b, Supplemental Fig. S1a), and fit against distance from the center of the circular region [MATH] to [MATH] for the constants [MATH] and [MATH].', '1407.3510-2-14-7': 'Results, [MATH] plotted versus connectivity [MATH], are shown in Fig. 3c.', '1407.3510-2-14-8': 'In general, the decay power [MATH] for networks with microbuckling (Fig. 3c, open circles) is substantially lower, by at least [MATH], than for networks without microbuckling of fibers (Fig. 3c, black circles).', '1407.3510-2-14-9': 'This is true for a wide range of connectivities, with an exception near the critical value [MATH]; for these values [MATH] in both types of networks.', '1407.3510-2-14-10': 'We observe larger spatial inhomogeneities of displacement at the scale of individual fibers in networks with [MATH] than in those with both subcritical and supercritical connectivity (Fig. 3b, Supplemental Fig. S2).', '1407.3510-2-14-11': 'These fluctuations are due to the change in phase from subcritical to supercritical connectivity as detailed elsewhere [CITATION].', '1407.3510-2-14-12': 'For the case without microbuckling (i.e. with linear stress-strain relation) since individual elements have linear stress-strain behavior, we compare the displacement to the linear elastic 2D solution [MATH] for the constants [MATH], [MATH], and [MATH].', '1407.3510-2-14-13': 'Except near [MATH], we find [MATH] (mean [MATH] standard deviation, essentially independent of [MATH] over all connectivities).', '1407.3510-2-14-14': 'This value of [MATH] is close to the 2D linear elastic solution [MATH].', '1407.3510-2-14-15': 'Connectivity does not appear to play a major role in displacement decay except near the critical value.', '1407.3510-2-14-16': 'We find no change in these conclusions when the zero traction boundary condition is replaced by a zero displacement condition fixing the external boundary; see Supplemental Fig. S3.', '1407.3510-2-14-17': 'Thus we conclude microbuckling is crucial for the slow decay of displacements.', '1407.3510-2-15-0': 'The long range of cell-induced displacements has been previously attributed to strain stiffening [CITATION], but this has been disputed [CITATION].', '1407.3510-2-15-1': 'We observe that the experiments of ref. [CITATION] were performed on fibrin, which exhibits microbuckling.', '1407.3510-2-15-2': 'Also, ref. [CITATION] provides evidence against strain stiffening as the underlying mechanism, but does not seem to propose an alternative.', '1407.3510-2-15-3': 'To help settle this, we repeated our simulations with elements whose stress-strain curve is of WLC type and stiffens in tension (Fig. 2b).', '1407.3510-2-15-4': 'Two versions of stiffening WLC stress-strain curves were compared.', '1407.3510-2-15-5': 'A curve whose slope is continuous at zero strain and increases smoothly in tension models a tension-stiffening material that does not undergo microbuckling (black dashed line in Fig. 2b).', '1407.3510-2-15-6': 'The alternative stress-strain curve has a discontinuous slope at zero strain (10 times smaller than the tangent stiffness for small tensile strain).', '1407.3510-2-15-7': 'It models microbuckling (red solid line in Fig. 2b) combined with tension stiffening.', '1407.3510-2-15-8': 'In all cases, values of the decay exponent [MATH] from fits for WLC networks (Fig. 3d,e,f, Supplemental Fig. S1c) agreed well with fits for bilinear networks of the same connectivity and same (buckling or non-buckling) type (Fig. 3c).', '1407.3510-2-15-9': 'This provides strong evidence that the tension-stiffening nonlinearity in the absence of microbuckling is not the cause of the slow displacement decay that we observe.', '1407.3510-2-16-0': 'Until now, we have considered round cell geometries, which do not capture the elongated shape of spread cells.', '1407.3510-2-16-1': "For an anisotropic cell contracting along its long axis, an ellipsoid more accurately captures the cell's shape.", '1407.3510-2-16-2': 'For this geometry, linear elasticity predicts that displacements far from the cell scale as [MATH] where [MATH] in three dimensions, [MATH] in two dimensions; [MATH] is the distance along the major axis from the center of the ellipsoid (or ellipse in two dimensions), and [MATH] is the displacement in the [MATH] direction [CITATION].', '1407.3510-2-16-3': 'To compare with the linear elastic solution, we placed in our fibrous network model an ellipse with a ratio of semi-major and semi-minor axes [MATH].', '1407.3510-2-16-4': 'As with the contracting circle, the matrix occupied a circular region of radius [MATH] with the nodes on the boundary [MATH] free and [MATH] defined for the ellipse as [MATH].', '1407.3510-2-16-5': 'Contractile displacements were applied on the boundary of the ellipse, with nonzero component [MATH] (along the long axis of the ellipse).', '1407.3510-2-16-6': "This is equivalent to subjecting the ellipse to a negative uniaxial strain along the ellipse's long axis.", '1407.3510-2-16-7': 'The largest magnitude of contractile displacement is [MATH] (at the ellipse tip), the same value as for the contracting circle.', '1407.3510-2-16-8': 'Displacements along the axis of the ellipse (Fig. 4a) appear to scale similarly to the displacements induced by the contracting circle (Fig. 3b).', '1407.3510-2-16-9': 'Indeed, the fittings to [MATH] show decay powers [MATH] that are significantly smaller for networks with microbuckling (Fig. 4b, [MATH]) than without (Fig. 4b, [MATH], Supplemental Fig. S1b).', '1407.3510-2-16-10': 'Like the contracting circle, the ellipse exhibits an exception at the critical connectivity [MATH].', '1407.3510-2-16-11': 'The trend shown in Figs. 3 and 4 is clear: microbuckling results in cell-induced displacements that propagate over a longer range than predicted by linear elasticity for both a contracting circle and a contractile ellipse.', '1407.3510-2-17-0': 'We also performed 3D simulations (contracting spherical cell), with similar conclusions.', '1407.3510-2-17-1': 'We recall that the 3D linear elastic solution predicts [MATH].', '1407.3510-2-17-2': 'The theoretical argument based on Eq. (1) gives [MATH], while a fit to our experiments yields [MATH].', '1407.3510-2-17-3': 'For 3D networks (microbuckling bilinear elements) with [MATH], a fit to [MATH] gives [MATH].', '1407.3510-2-17-4': 'For [MATH] (below the critical value for rigidity [MATH]) we found [MATH] (Supplemental Fig. S4).', '1407.3510-2-17-5': 'These results combine to show that microbuckling of fibers is the key mechanism responsible for the longer range of cell-induced deformations in a fibrin matrix.', '1407.3510-2-18-0': '## Tethers.', '1407.3510-2-19-0': 'Can cells exploit the long propagation range of matrix deformations they themselves induce for sensing the presence of other cells?', '1407.3510-2-19-1': 'We use confocal microscopy to visualize both the matrix and multiple fibroblast cells embedded in it.', '1407.3510-2-19-2': 'We observe that cells whose distance from each other is of the order of 10 cell diameters are connected to each other by linear bands consisting of aligned and densely packed matrix fibers (Fig. 5).', '1407.3510-2-19-3': 'Within these "tethers" fibers appear to be in tension in the direction joining the cells.', '1407.3510-2-19-4': 'These tethers also occur between multicellular explants in a fibrous matrix [CITATION], but the mechanism for their formation remains unknown.', '1407.3510-2-19-5': "The tethers extend far beyond a single cell's protrusion (Fig. 5).", '1407.3510-2-19-6': 'Matrix remodeling by degradation or deposition cannot be responsible for alignment at such a large distance from the cell.', '1407.3510-2-19-7': 'This leads us to examine the hypothesis that tethers form due to tensile forces.', '1407.3510-2-20-0': 'A previous model has shown that a point force in a fibrous medium induces forces which propagate through tether-like paths [CITATION].', '1407.3510-2-20-1': 'The point force loading of this previous model was not intended to simulate forces due to cells, which maintain force equilibrium while pulling on the matrix.', '1407.3510-2-20-2': 'To investigate the physical mechanism of tether formation, we used our FE network model with microbuckling to simulate a pair of contracting cells.', '1407.3510-2-20-3': 'A symmetric boundary condition is imposed at the bottom of a square region containing a circle of radius [MATH] (the other boundaries are free of applied tractions).', '1407.3510-2-20-4': 'By symmetry this is equivalent to a pair of identical contracting circular cells in the fibrous matrix.', '1407.3510-2-20-5': 'We apply an isotropic inward radial displacement of [MATH] to the circular region.', '1407.3510-2-20-6': 'For a cell with a radius of [MATH] m, this value of [MATH] corresponds to 1 m, in agreement with the experimental data (Fig. 1b).', '1407.3510-2-21-0': 'A different model [CITATION] requires cell displacements nearly an order of magnitude higher than the experimentally observed value of 1 m in order to predict appreciable interaction between cells.', '1407.3510-2-21-1': 'The simulated tensile strains in our network occur almost entirely in the band between the two cells, along aligned linear paths formed by elements in tension (Fig. 6a).', '1407.3510-2-21-2': 'Compressive strains localize perpendicular to these tensile tethers (Fig. 6b).', '1407.3510-2-21-3': 'Due to low compression stiffness (microbuckling), the magnitude of compressive strain in elements roughly perpendicular to the tether is more than twice the magnitude of the tensile strain.', '1407.3510-2-21-4': 'Thus within the tether, the trace of the strain tensor, or the volumetric strain, is negative, consistent with the observation that matrix fiber density increases between pairs of cells (Fig. 5).', '1407.3510-2-21-5': 'When simulating networks without microbuckling, we found no such tethers forming; instead, tensile strains had a nearly radially symmetric distribution around each cell (Supplemental Fig. S5).', '1407.3510-2-21-6': 'Thus we conclude that localization of matrix deformation caused by multiple cells in the tensile, tether-like regions joining those cells occurs because of microbuckling of fibers normal to the tethers.', '1407.3510-2-22-0': '# Discussion', '1407.3510-2-23-0': 'We have shown that cells embedded within a fibrin matrix exert forces that cause matrix displacements to propagate over a longer range than predicted by linear elasticity.', '1407.3510-2-23-1': 'The long range propagation of displacements has been previously observed for cells on a flat, 2D fibrous substrate [CITATION] and for multicellular constructs in a 3D matrix [CITATION].', '1407.3510-2-23-2': 'Our observations, first reported in ref. [CITATION], confirm this result for single cells in a 3D system.', '1407.3510-2-23-3': 'Here, we further quantify the spatial decay of displacements by fitting to a power law resulting in displacements scaling as [MATH].', '1407.3510-2-23-4': 'While the propagation of displacements over a long range is now apparent, the precise mechanism is still unclear.', '1407.3510-2-23-5': 'Recent studies have argued for [CITATION] and against [CITATION] the hypothesis that long range propagation of displacements results from strain stiffening.', '1407.3510-2-23-6': 'When we included strain stiffening in the behavior of fibers, but suppressed compression weakening due to buckling, long range propagation was not observed in simulations of our model.', '1407.3510-2-23-7': 'Thus we conclude that fiber buckling-rather than strain stiffening in tension-explains the long range propagation of displacements observed in the experiments.', '1407.3510-2-24-0': 'To simulate buckling, we used a model that does not resist changes in angle between the elements.', '1407.3510-2-24-1': 'Previous work [CITATION] has pointed to bending as an important mechanism that controls the mechanical response of fibrous materials.', '1407.3510-2-24-2': 'However, the model of ref. [CITATION] does not allow for buckling or even bending of individual fibers.', '1407.3510-2-24-3': 'Instead it models bending by penalizing changes in angle between initially co-linear elements that meet at a node.', '1407.3510-2-24-4': 'Moreover, even for models that include bending, forces are dominated not by transverse bending displacements but by axial ones [CITATION].', '1407.3510-2-24-5': 'To address here the question of microbuckling, we focus on these axial displacements.', '1407.3510-2-24-6': 'We assume fibers buckle immediately after a compressive load is applied, i.e., we assume the fiber buckling load (equivalently the buckling strain) is equal to zero.', '1407.3510-2-24-7': 'Is this assumption reasonable?', '1407.3510-2-24-8': "A typical fibrin fiber with a length of 1 m, diameter of 0.2 m, persistence length of 40 m [CITATION], and Young's modulus of 15 MPa [CITATION] will buckle at a compressive strain of [MATH].", '1407.3510-2-24-9': 'This value is small compared to typical strains experienced in the matrix ([MATH]), so our choice of setting the transition point between different stiffnesses at the onset of compression (vanishing buckling load) is justified.', '1407.3510-2-25-0': 'Besides long range propagation of displacements, fibrous materials exhibit what is termed in ref. [CITATION] a "negative compressibility" in uniaxial tension, i.e., a negative ratio of the trace of the stress tensor and the trace of the strain tensor during a uniaxial tension experiment.', '1407.3510-2-25-1': 'Fibrous materials also exhibit tensile normal stresses under prescribed shear deformation [CITATION].', '1407.3510-2-25-2': 'This is in essence equivalent to negative normal (compressive) strains when the material is subjected to external tangential forces (prescribed shear stress), but not constrained to expand or contract.', '1407.3510-2-25-3': 'To test whether our fiber model is consistent with these experimental observations, we simulated homogeneous uniaxial tension.', '1407.3510-2-25-4': 'We found that when fibers buckle, the model exhibits negative compressibility in tension.', '1407.3510-2-25-5': 'In addition, under applied tangential forces equivalent to an external shear stress, the model responded with negative normal strains in shear.', '1407.3510-2-25-6': 'When microbuckling is removed from the model, neither of the aforementioned behaviors occurs (Supplemental Fig. S6).', '1407.3510-2-25-7': 'Thus, our model with fiber microbuckling is consistent with previous experimental work on fibrin [CITATION] and collagen [CITATION].', '1407.3510-2-25-8': 'Certainly fibrous materials exhibit nonlinear behaviors besides microbuckling in compression, but our model points to microbuckling as being both consistent with previous experimental work and of major importance to the mechanical response of fibrous materials.', '1407.3510-2-26-0': 'Together, our simulations and experiments reveal that microbuckling of fibrin enables cells to induce displacements that follow linear, tether-like paths that lead to other cells.', '1407.3510-2-26-1': 'These displacements propagate over a dramatically longer range than in a linear material.', '1407.3510-2-26-2': 'A remaining question of biological relevance is whether cells physically respond to the formation of tethers.', '1407.3510-2-26-3': 'In our experiments, we have observed pairs of cells forming pointed protrusions along these tethers and subsequently growing toward one another by several cell diameters (Fig. 5), sometimes eventually joining two cells (Supplemental Fig. S7).', '1407.3510-2-26-4': 'A different model indicated that elongated cells initially pointed toward one another may sense displacements induced by their neighbors [CITATION], but it did not answer the question of how cells break their initial spherical symmetry to spread toward one another as observed in our experiments.', '1407.3510-2-26-5': 'Our model, which we present here and have described previously [CITATION], suggests a mechanism whereby cells can sense one another during the initially spherical state.', '1407.3510-2-26-6': 'Even if each cell is initially spherical and contracts isotropically, the tether formation mechanism that we describe results in greater tension and fiber density that is highly polarized in the direction of neighboring cells (Fig. 5).', '1407.3510-2-26-7': 'Both tension and fiber density may provide a directional signal: by growing protrusions along the direction of the tethers, cells have a higher chance of approaching one another.', '1407.3510-2-26-8': 'The fact that cells change shape and grow along such tethers supports the hypothesis that they use this very same mechanism to sense and even approach their neighbors.', '1407.3510-2-26-9': 'We expect future work to further clarify how cells sense the mechanical properties of fibrous materials and how we can better design artificial cell culture platforms to better control cellular response to forces within the extracellular matrix.', '1407.3510-2-27-0': '# Methods', '1407.3510-2-28-0': '## Cell culture and matrix preparation.', '1407.3510-2-28-1': "3T3 fibroblast cells stably expressing a green fluorescent protein-actin fusion protein were cultured in Dulbecco's Modified Eagle Medium supplemented with 10% fetal bovine serum and 1[MATH] non-essential amino acids.", '1407.3510-2-28-2': 'Fibrin was fluorescently labeled by mixing fibrinogen (Omrix Biopharmaceuticals, Israel) and 546 Alexa Fluor (Life Technologies, Carlsbad, CA, USA) for 1 hour before filtering with a HiTrap desalting column (GE Healthcare, Milwaukee, WI, USA).', '1407.3510-2-28-3': 'Cell-fibrin constructs were created by suspending the cells in 20 U/mL thrombin solution (Omrix), mixing with 5 mg/mL labeled fibrinogen solution, and placing on a 1.5 coverslip.', '1407.3510-2-29-0': '## Microscopy and cell-induced matrix displacements.', '1407.3510-2-29-1': 'Within 1 hour of seeding, cell-matrix constructs were transferred to a custom built 5% CO[MATH], 37[MATH]C microscope enclosure.', '1407.3510-2-29-2': 'Imaging was performed with a Swept Field confocal microscope using a 40[MATH] NA 1.15 water immersion objective (Nikon Instruments, Melville, NY, USA).', '1407.3510-2-29-3': 'Volume stacks of the cells and fibrin matrix were captured every 15 minutes over time periods of several hours.', '1407.3510-2-30-0': '3D matrix displacements were computed directly from the images of the labeled fibrin using digital volume correlation [CITATION] with the initial volume stack (before cell spreading) taken as a reference for the correlation.', '1407.3510-2-30-1': 'The digital volume correlation software, written in Matlab (The Mathworks), is freely available online [CITATION].', '1407.3510-2-30-2': 'Propagation of cell-induced matrix displacements was quantified by computing displacement magnitudes along multiple linear paths propagating outward from the center of each initially rounded cell.', '1407.3510-2-30-3': 'To reduce errors caused by inhomogeneities within the matrix, displacements were averaged over [MATH] different paths and over [MATH] time points for each cell.', '1407.3510-2-30-4': 'After averaging, the standard deviation of the noise level was found to be 0.04 m.', '1407.3510-2-31-0': '## Microstructural model.', '1407.3510-2-31-1': 'The microstructural model was developed in the FE software Abaqus 6.10 (Dassault Systemes, Waltham, MA).', '1407.3510-2-31-2': 'Rod elements supporting tension and compression but not bending were connected as shown in Fig. 2c.', '1407.3510-2-31-3': 'Elements were randomly deleted to reduce the network connectivity.', '1407.3510-2-31-4': 'Removed elements were replaced by elements with stiffness six orders of magnitude smaller than the original elements.', '1407.3510-2-31-5': 'The choice of using weak elements with stiffness six orders of magnitude smaller than the original elements came after a series of convergence studies showed that further reduction in the stiffness of the weak elements had no effect on the displacements.', '1407.3510-2-31-6': 'Under tension, both a linear (Fig. 2a) and a strain-stiffening WLC relationship were investigated (Fig. 2b).', '1407.3510-2-31-7': 'Under compression a linear stress-strain relationship was used with slope given by [MATH] times the slope at small tensile strains with [MATH] for microbuckling and [MATH] for no buckling.', '1407.3510-2-31-8': 'Strains within each element (as plotted in Fig. 5, Supplemental Fig. S5) were computed by taking the natural logarithm of the stretch ratio, defined as the final element length divided by the initial element length.', '1407.3510-2-31-9': 'The 3D model used element connectivity as shown in Supplemental Fig. S4 and the bilinear stress-strain relationship.', '1407.3510-2-31-10': 'Uniaxial tension was simulated in a square region by applying displacements on the top side, a symmetric boundary on the bottom side, and traction free boundaries on the right and left sides.', '1407.3510-2-31-11': 'Shear loading was simulated by applying horizontal displacements to the top of a thin rectangular region (aspect ratio 1/10) with a fixed bottom boundary and traction free conditions on the right an left.', '1407.3510-2-31-12': 'Apparent strains were computed by numerically computing the displacement gradients using a linear fitting.', '1407.3510-2-31-13': "Effective Poisson's ratio was defined as the opposite of the ratio of apparent strains in the transverse and axial directions.", '1407.3510-2-32-0': '# Competing Interests Statement', '1407.3510-2-33-0': 'We have no competing interests.', '1407.3510-2-34-0': '# Author Contributions', '1407.3510-2-35-0': 'J.N. and A.L. performed the experiments.', '1407.3510-2-35-1': 'J.N. performed the simulations.', '1407.3510-2-35-2': 'J.N. and P.R. wrote the manuscript.', '1407.3510-2-35-3': 'All authors discussed the results and gave approval for publication.', '1407.3510-2-36-0': '# Funding Statement', '1407.3510-2-37-0': 'This work was funded in part by a grant from the National Science Foundation (Division of Materials Research No. 0520565) through the Center for the Science and Engineering of Materials at the California Institute of Technology, and in part by National Science Foundation Grant No. DMR-1206121.', '1407.3510-2-37-1': 'J.N. was supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469.', '1407.3510-2-38-0': '# Figures', '1407.3510-2-39-0': 'Figure 1: Experimentally measured displacements induced by isolated cells embedded within a 3D fibrous matrix.', '1407.3510-2-39-1': '(a) The colored quivers plot 3D matrix displacement vectors applied by a cell to a 3D fibrin matrix.', '1407.3510-2-39-2': 'Paths (white) are chosen proceeding outward from the cell body.', '1407.3510-2-39-3': '(b) Displacement magnitudes along the paths are averaged for multiple time points and plotted.', '1407.3510-2-39-4': 'Each curve is for a different cell.', '1407.3510-2-39-5': 'The blue curve shows displacements for the cell in (a).', '1407.3510-2-39-6': 'The gray shading behind the blue curve shows typical error of the displacement measurement after averaging.', '1407.3510-2-39-7': 'Data used to generate these curves is in the Supplemental Data.', '1407.3510-2-40-0': 'Figure 2: Finite element model network details.', '1407.3510-2-40-1': '(a) Stress-strain curves for bilinear model.', '1407.3510-2-40-2': "Stress [MATH] is normalized by Young's modulus [MATH].", '1407.3510-2-40-3': 'Dashed black: linear without microbuckling ([MATH]); solid blue: bilinear with microbuckling ([MATH]).', '1407.3510-2-40-4': '(b) Normalized stress-strain curves for the strain-stiffening model, which exhibits WLC-like behavior in tension.', '1407.3510-2-40-5': 'For this model [MATH] is defined as the slope upon approaching the origin from the left divided by the slope upon approaching the origin from the right.', '1407.3510-2-40-6': 'A continuous slope at the origin ([MATH]) was used to simulate non-buckling elements (dashed black line) and a discontinuous slope ([MATH]) was used to simulate microbuckling elements (solid red line).', '1407.3510-2-40-7': '(c) Network array.', '1407.3510-2-40-8': '(d) Randomized network, [MATH].', '1407.3510-2-40-9': '(e) Network with reduced connectivity, [MATH].', '1407.3510-2-41-0': 'Figure 3: Long range propagation of displacements is due to microbuckling.', '1407.3510-2-41-1': '(a) Inhomogeneous forces, like those applied by a cell, are modeled by a circle of radius [MATH] contracting in a circular region with radius [MATH].', '1407.3510-2-41-2': 'Contours of normalized displacement [MATH] are shown here for the case of the bilinear model (Fig. 2a) with microbuckling ([MATH]) and [MATH].', '1407.3510-2-41-3': 'For a cell of radius 10 m, the applied displacement [MATH] would correspond to 1 m. (b) Displacements are averaged around a circle of radius [MATH] about the center of the model and plotted for simulations that used different connectivities ranging from [MATH] to [MATH].', '1407.3510-2-41-4': 'All curves show long range propagation of displacements with slopes of [MATH].', '1407.3510-2-41-5': 'At the critical connectivity, [MATH], displacements exhibit spatial inhomogeneities, resulting in fluctuations.', '1407.3510-2-41-6': '(c) Decay power [MATH] vs. connectivity [MATH].', '1407.3510-2-41-7': 'Circles show fits to [MATH]; squares show fits to [MATH].', '1407.3510-2-41-8': 'Solid black symbols represent fibers that do not buckle ([MATH]); open symbols represent fibers that do buckle ([MATH]).', '1407.3510-2-41-9': 'Most powers [MATH] for the case of microbuckling [MATH] are [MATH], in agreement with the slope of [MATH] observed in panel (b).', '1407.3510-2-41-10': 'The value of [MATH] indicates displacement propagate over a longer range than predicted by linear elasticity, for which [MATH] in two dimensions.', '1407.3510-2-41-11': 'Simulations are repeated for the strain stiffening WLC-type relationship (Fig. 2b).', '1407.3510-2-41-12': '(d) Contours of displacement [MATH] for the strain stiffening relationship with microbuckling ([MATH]).', '1407.3510-2-41-13': '(e) Averaged displacements and (f) decay powers [MATH] for the strain stiffening relationship.', '1407.3510-2-41-14': 'As in (c), circles show fits to [MATH]; squares show fits to [MATH].', '1407.3510-2-41-15': 'Solid black symbols represent fibers that do not buckle ([MATH]); open symbols represent fibers that do buckle ([MATH]).', '1407.3510-2-42-0': 'Figure 4: Simulated displacements due to an elongated cell.', '1407.3510-2-42-1': 'An ellipse with a ratio of semi-major to semi-minor axes [MATH] is simulated contracting along its long axis in a circular region of radius [MATH] where [MATH].', '1407.3510-2-42-2': '(a) Displacements along the major axis [MATH] are plotted against distance along the axis from the center of the ellipse [MATH] for connectivities ranging from [MATH] to [MATH].', '1407.3510-2-42-3': 'As in Fig. 3, fluctuations are observed for connectivites near the critical value [MATH].', '1407.3510-2-42-4': '(b) Displacements far from the cell (i.e. for [MATH]) are fit to [MATH].', '1407.3510-2-42-5': 'Solid black circles represent decay powers [MATH] for fibers that do not buckle ([MATH]); open blue circles represent [MATH] for fibers that do buckle ([MATH]).', '1407.3510-2-42-6': 'For simulations with buckling, except near the critical connectivity [MATH], decay powers are smaller than the linear elastic solution [MATH] and smaller than simulations without buckling.', '1407.3510-2-43-0': 'Figure 5: Pairs of cells spread toward one another along tethers.', '1407.3510-2-43-1': 'Panels in the left column show the cells (green), and panels in the right column show the cells (green) with the matrix (gray/white) at the same time point.', '1407.3510-2-43-2': 'The cells apply tensile force to the fibrin matrix resulting in matrix tethers connecting the cells (white).', '1407.3510-2-43-3': 'These tethers have a high density of matrix fibers, as apparent by the bright fluorescent signal in the space between the cells.', '1407.3510-2-43-4': 'The cells then spread along these tethers.', '1407.3510-2-43-5': 'Times are hours after the cells were seeded in the fibrin matrix.', '1407.3510-2-44-0': 'Figure 6: Fiber alignment and densification provide a mechanism for long-range cell mechanosensing.', '1407.3510-2-44-1': 'Mechanical interactions between cells are simulated using the FE model with a contracting circle and a symmetric boundary (dashed line).', '1407.3510-2-44-2': 'Plots show tensile (a) and compressive (b) strains within fibers.', '1407.3510-2-44-3': 'Fibers under tension (a) form intercellular tethers.', '1407.3510-2-44-4': 'Compressed fibers (b) are roughly perpendicular to tensile ones.', '1407.3510-2-44-5': 'The strains below the dashed line are the reflection of the strains above the dashed line.', '1407.3510-2-45-0': '# Supplemental Figures', '1407.3510-2-46-0': 'Figure S1: Simulated displacements for fibers that support compression, [MATH].', '1407.3510-2-46-1': '(a) Displacements due to a contracting circle are computed with the bilinear (Fig. 2a) model with [MATH].', '1407.3510-2-46-2': 'The radial displacement component is averaged around circles of radius [MATH] from the origin and plotted.', '1407.3510-2-46-3': 'Results show displacement [MATH] vs. distance [MATH] for simulations that used connectivities ranging from [MATH] to [MATH].', '1407.3510-2-46-4': 'See Fig. 3b for the case of microbuckling, [MATH].', '1407.3510-2-46-5': '(b) Displacements due to an ellipse with a ratio of semi-major to semi-minor axes [MATH] with [MATH].', '1407.3510-2-46-6': 'The displacements [MATH] along the major axis are plotted against distance along the axis [MATH] for connectivities ranging from [MATH] to [MATH].', '1407.3510-2-46-7': 'See Fig. 4a for the case of microbuckling, [MATH].', '1407.3510-2-46-8': '(c) Displacements due to a contracting circle computed with the strain stiffening model (Fig. 2b) with [MATH].', '1407.3510-2-46-9': 'As in (a), the radial displacement component is averaged around circles of radius [MATH] from the origin and plotted.', '1407.3510-2-46-10': 'See Fig. 3e for the case of microbuckling, [MATH].', '1407.3510-2-46-11': 'For all cases, typical slopes are [MATH] on logarithmic axes, indicate displacements scale according to the 2D linear elastic solution, [MATH].', '1407.3510-2-47-0': 'Figure S2: Displacements induced by a contracting circle and a contracting ellipse.', '1407.3510-2-47-1': 'A circle of radius [MATH] (top row) or an ellipse with semi-major and semi-minor axes [MATH] and [MATH] (bottom row) is simulated in a circular region with radius [MATH].', '1407.3510-2-47-2': '(For the ellipse, [MATH].)', '1407.3510-2-47-3': 'For the circle, contractile displacements are applied uniformly around the perimeter, [MATH] at [MATH] where [MATH].', '1407.3510-2-47-4': 'For the ellipse, contractile displacements are applied only along the major axis, [MATH].', '1407.3510-2-47-5': 'The outer boundary [MATH] is free of applied tractions.', '1407.3510-2-47-6': 'Microbuckling is simulated using the bilinear model with [MATH].', '1407.3510-2-47-7': 'Displacement magnitudes normalized by [MATH] are shown for the contracting circle and ellipse for connectivities [MATH] of 3, 4, and 8.', '1407.3510-2-47-8': 'Near the critical connectivity [MATH] large fluctuations in displacements occur for both the contracting circle and the ellipse, in agreement with previous models [CITATION].', '1407.3510-2-47-9': 'These fluctuations are not present at lower ([MATH]) or higher ([MATH]) connectivities, where displacement fields are smoother.', '1407.3510-2-48-0': 'Figure S3: Effect of boundary conditions on simulation results.', '1407.3510-2-48-1': 'The simulations of Fig. 3 are repeated with fixed boundaries instead of free.', '1407.3510-2-48-2': '(a) For elements with microbuckling ([MATH]), displacements for all connectivities [MATH] have a slope of [MATH] on logarithmic axes.', '1407.3510-2-48-3': '(b) The simulations are repeated for elements without microbuckling ([MATH]), and displacements have a slope of [MATH] on a logarithmic scale.', '1407.3510-2-48-4': '(c) For each connectivity, and for simulations with microbuckling ([MATH]) and without ([MATH]), displacements are fit to the linear elastic solution for a circular region of finite radius, [MATH].', '1407.3510-2-48-5': '(No fitting is performed to [MATH], because, as shown in (a) and (b), the fixed boundary affects the propagation of displacements for [MATH].)', '1407.3510-2-48-6': 'The fit power [MATH] is plotted for all cases.', '1407.3510-2-48-7': 'Similar to simulations with free boundaries (Fig. 3), simulations with microbuckling ([MATH]) have lower powers of [MATH], indicating displacements propagate over a long range when microbuckling is present.', '1407.3510-2-49-0': 'Figure S4: The displacements due to a contracting sphere in a fibrous matrix are simulated using a 3D model.', '1407.3510-2-49-1': '[MATH] nodes are used in a [MATH] region, where [MATH] is the radius of the sphere.', '1407.3510-2-49-2': 'A symmetric boundary is used at the bottom of the cubic region ([MATH]), and other boundaries are free.', '1407.3510-2-49-3': 'An inward displacement of [MATH] is applied to the nodes located at [MATH].', '1407.3510-2-49-4': '(a) Fiber connectivity.', '1407.3510-2-49-5': "Each cube of [MATH] nodes is connected along the the cube's edges.", '1407.3510-2-49-6': 'Additionally, elements connect the diagonals as shown in the sketch.', '1407.3510-2-49-7': 'To simplify visualization, the sketch shows connections between only [MATH] diagonals, but the model connects all [MATH] diagonals with elements.', '1407.3510-2-49-8': 'As with the 2D models, lower connectivity is simulated by randomly selecting elements to delete.', '1407.3510-2-49-9': 'Deleted elements are replaced by weak elements with stiffness six orders of magnitude lower than that of the deleted elements.', '1407.3510-2-49-10': '(b) Displacements due to the contracting sphere are averaged along circles of radius [MATH] from the center of the sphere in the [MATH] plane and plotted against radial distance for connectivities [MATH] of [MATH] (below the critical value of [MATH]) and [MATH] (full connectivity).', '1407.3510-2-49-11': 'Fits to [MATH] give [MATH] and [MATH] for [MATH] and [MATH], respectively.', '1407.3510-2-50-0': 'Figure S5: Tethers do not form when the matrix resists compression.', '1407.3510-2-50-1': 'Plots show tensile (a) and compressive (b) strains for the same simulation as in Fig. 5, but for fibers with equal stiffness in compression and tension ([MATH]).', '1407.3510-2-50-2': 'Tensile strains (a) propagate radially outward from the contracting circle with no preferred directionality, and therefore no tethers form.', '1407.3510-2-50-3': 'Compressive strains (b) are roughly perpendicular to tensile strains.', '1407.3510-2-51-0': 'Figure S6: Microbuckling induces negative compressibility in tension and negative normal strains in shear.', '1407.3510-2-51-1': "(a) The effective (engineering) Poisson's ratio is calculated in a uniaxial tension simulation for a model with matrix fibers that support compression ([MATH], solid red line) or buckle ([MATH], dashed blue line).", '1407.3510-2-51-2': "In two dimensions, Poisson's ratios greater than 1 (dashed black line) indicate negative compressibility, in agreement with an experimental study on fibrin [CITATION].", '1407.3510-2-51-3': '(b) Negative normal strains are observed in a simulation under shear loading for matrix fibers that support compression ([MATH], solid red line) or buckle ([MATH] dashed blue line).', '1407.3510-2-51-4': 'Similar to [CITATION], negative normal strains are significantly larger for elements that simulate buckling.', '1407.3510-2-51-5': 'Simulations in this figure use the bilinear stress-strain relationship, but results are nearly identical for the WLC relationship.', '1407.3510-2-51-6': 'Results shown are for connectivity of [MATH].', '1407.3510-2-52-0': 'Figure S7: Pairs of fibroblast cells (green) in a 3D fibrin matrix (white) occasionally spread until they touch their neighbors.', '1407.3510-2-52-1': 'Image was captured with confocal microscopy 14 hrs after seeding the cells in the fibrin matrix.', '1407.3510-2-53-0': '# Supplemental Data', '1407.3510-2-54-0': 'Experimental data of 3D cell-induced displacements (Fig. 1b) for [MATH] cells are below.', '1407.3510-2-54-1': 'The data report the magnitude of cell-induced matrix displacements after averaging along different paths outward from the cell over time.', '1407.3510-2-54-2': 'For more details on how these cell-induced displacements are measured, see the methods section.', '1407.3510-2-54-3': "The data set for each cell is displayed as an array of ordered pairs, ([MATH],[MATH]) where each pair gives the distance from the cell's center [MATH] and the magnitude of the cell-induced matrix displacement [MATH].", '1407.3510-2-55-0': '## Cell 1.', '1407.3510-2-55-1': '(25,0.68) (29,0.64) (33,0.61) (38,0.57) (42,0.53) (46,0.50) (50,0.47) (55,0.43) (59,0.40) (63,0.37) (67,0.35) (72,0.35) (76,0.38) (80,0.35)', '1407.3510-2-56-0': '## Cell 2.', '1407.3510-2-56-1': '(15,1.91) (20,1.78) (24,1.60) (28,1.42) (32,1.25) (36,1.15) (40,1.08) (44,1.01) (48,0.97) (52,0.90) (56,0.81) (60,0.69) (64,0.58) (68,0.53) (72,0.54)', '1407.3510-2-57-0': '## Cell 3.', '1407.3510-2-57-1': '(16,1.14) (21,1.04) (26,0.95) (31,0.89) (35,0.83) (40,0.77) (45,0.73) (49,0.70) (54,0.67) (59,0.66) (64,0.65) (68,0.64) (73,0.65)', '1407.3510-2-58-0': '## Cell 4.', '1407.3510-2-58-1': '(17,1.33) (22,1.22) (26,1.14) (31,1.07) (36,1.00) (41,0.95) (46,0.91) (51,0.87) (55,0.84)', '1407.3510-2-59-0': '## Cell 5.', '1407.3510-2-59-1': '(14,1.89) (18,1.60) (21,1.40) (25,1.30) (29,1.25) (32,1.21) (36,1.19) (40,1.17) (43,1.16) (47,1.14) (51,1.12)', '1407.3510-2-60-0': '## Cell 6.', '1407.3510-2-60-1': '(12,1.76) (16,1.66) (20,1.34) (24,1.25) (27,1.07) (31,1.18) (35,1.12) (39,1.08) (43,1.04) (46,0.99) (50,0.95) (54,0.92) (58,0.89) (61,0.86) (65,0.83)'} | [['1407.3510-1-15-0', '1407.3510-2-17-5'], ['1407.3510-1-9-2', '1407.3510-2-13-2'], ['1407.3510-1-9-3', '1407.3510-2-13-3'], ['1407.3510-1-2-2', '1407.3510-2-6-1'], ['1407.3510-1-2-3', '1407.3510-2-6-2'], ['1407.3510-1-2-4', '1407.3510-2-6-3'], ['1407.3510-1-2-6', '1407.3510-2-6-5'], ['1407.3510-1-2-9', '1407.3510-2-6-8'], ['1407.3510-1-6-1', '1407.3510-2-11-5'], ['1407.3510-1-6-2', '1407.3510-2-11-6'], ['1407.3510-1-6-5', '1407.3510-2-11-9'], ['1407.3510-1-0-0', '1407.3510-2-0-1'], ['1407.3510-1-0-1', '1407.3510-2-0-2'], ['1407.3510-1-0-2', '1407.3510-2-0-3'], ['1407.3510-1-0-3', '1407.3510-2-0-4'], ['1407.3510-1-0-4', '1407.3510-2-0-5'], ['1407.3510-1-1-1', '1407.3510-2-4-0'], ['1407.3510-1-1-2', '1407.3510-2-4-1'], ['1407.3510-1-1-3', '1407.3510-2-4-2'], ['1407.3510-1-24-1', '1407.3510-2-28-2'], ['1407.3510-1-24-2', '1407.3510-2-28-3'], ['1407.3510-1-4-0', '1407.3510-2-9-2'], ['1407.3510-1-4-1', '1407.3510-2-9-3'], ['1407.3510-1-4-2', '1407.3510-2-9-4'], ['1407.3510-1-4-3', '1407.3510-2-9-5'], ['1407.3510-1-4-5', '1407.3510-2-9-7'], ['1407.3510-1-7-4', '1407.3510-2-12-4'], ['1407.3510-1-7-5', '1407.3510-2-12-5'], ['1407.3510-1-8-3', '1407.3510-2-12-10'], ['1407.3510-1-5-1', '1407.3510-2-10-3'], ['1407.3510-1-5-3', '1407.3510-2-10-8'], ['1407.3510-1-9-0', '1407.3510-2-13-0'], ['1407.3510-1-9-1', '1407.3510-2-13-1'], ['1407.3510-1-9-4', '1407.3510-2-13-4'], ['1407.3510-1-9-5', '1407.3510-2-13-5'], ['1407.3510-1-2-1', '1407.3510-2-6-0'], ['1407.3510-1-2-5', '1407.3510-2-6-4'], ['1407.3510-1-2-7', '1407.3510-2-6-6'], ['1407.3510-1-2-8', '1407.3510-2-6-7'], ['1407.3510-1-6-4', '1407.3510-2-11-8'], ['1407.3510-1-6-6', '1407.3510-2-11-10'], ['1407.3510-1-6-8', '1407.3510-2-11-12'], ['1407.3510-1-6-9', '1407.3510-2-11-13'], ['1407.3510-1-24-0', '1407.3510-2-28-1'], ['1407.3510-1-3-1', '1407.3510-2-9-0'], ['1407.3510-1-3-2', '1407.3510-2-9-1'], ['1407.3510-1-4-4', '1407.3510-2-9-6'], ['1407.3510-1-8-0', '1407.3510-2-12-6'], ['1407.3510-1-8-2', '1407.3510-2-12-9'], ['1407.3510-1-8-4', '1407.3510-2-12-11'], ['1407.3510-1-8-5', '1407.3510-2-12-12'], ['1407.3510-1-8-6', '1407.3510-2-12-13'], ['1407.3510-1-5-0', '1407.3510-2-10-0'], ['1407.3510-1-5-2', '1407.3510-2-10-7'], ['1407.3510-1-6-0', '1407.3510-2-11-0'], ['1407.3510-1-6-0', '1407.3510-2-11-1'], ['1407.3510-1-6-10', '1407.3510-2-11-17'], ['1407.3510-1-1-4', '1407.3510-2-4-3'], ['1407.3510-1-1-4', '1407.3510-2-4-6'], ['1407.3510-1-1-5', '1407.3510-2-4-5'], ['1407.3510-1-7-2', '1407.3510-2-12-1'], ['1407.3510-1-7-3', '1407.3510-2-12-3'], ['1407.3510-1-8-1', '1407.3510-2-12-8'], ['1407.3510-1-11-1', '1407.3510-2-14-0'], ['1407.3510-1-11-2', '1407.3510-2-14-1'], ['1407.3510-1-11-3', '1407.3510-2-14-2'], ['1407.3510-1-11-4', '1407.3510-2-47-5'], ['1407.3510-1-11-4', '1407.3510-2-14-3'], ['1407.3510-1-11-5', '1407.3510-2-14-4'], ['1407.3510-1-11-6', '1407.3510-2-14-5'], ['1407.3510-1-11-8', '1407.3510-2-14-7'], ['1407.3510-1-12-0', '1407.3510-2-14-8'], ['1407.3510-1-12-1', '1407.3510-2-14-9'], ['1407.3510-1-12-2', '1407.3510-2-14-10'], ['1407.3510-1-12-3', '1407.3510-2-14-12'], ['1407.3510-1-12-4', '1407.3510-2-14-13'], ['1407.3510-1-12-5', '1407.3510-2-14-14'], ['1407.3510-1-12-7', '1407.3510-2-14-17'], ['1407.3510-1-12-8', '1407.3510-2-14-15'], ['1407.3510-1-13-0', '1407.3510-2-15-0'], ['1407.3510-1-13-1', '1407.3510-2-15-3'], ['1407.3510-1-13-2', '1407.3510-2-15-6'], ['1407.3510-1-13-4', '1407.3510-2-15-8'], ['1407.3510-1-13-5', '1407.3510-2-15-9'], ['1407.3510-1-14-0', '1407.3510-2-17-0'], ['1407.3510-1-14-1', '1407.3510-2-17-2'], ['1407.3510-1-14-2', '1407.3510-2-17-3'], ['1407.3510-1-17-0', '1407.3510-2-19-0'], ['1407.3510-1-17-1', '1407.3510-2-19-1'], ['1407.3510-1-17-2', '1407.3510-2-19-2'], ['1407.3510-1-17-3', '1407.3510-2-19-3'], ['1407.3510-1-17-4', '1407.3510-2-20-2'], ['1407.3510-1-17-5', '1407.3510-2-20-3'], ['1407.3510-1-17-6', '1407.3510-2-20-4'], ['1407.3510-1-17-7', '1407.3510-2-21-1'], ['1407.3510-1-17-8', '1407.3510-2-21-2'], ['1407.3510-1-17-9', '1407.3510-2-21-3'], ['1407.3510-1-17-10', '1407.3510-2-21-4'], ['1407.3510-1-17-11', '1407.3510-2-21-5'], ['1407.3510-1-17-12', '1407.3510-2-21-6'], ['1407.3510-1-18-4', '1407.3510-2-26-6'], ['1407.3510-1-18-5', '1407.3510-2-26-7'], ['1407.3510-1-19-1', '1407.3510-2-26-0'], ['1407.3510-1-19-2', '1407.3510-2-26-8'], ['1407.3510-1-19-3', '1407.3510-2-25-7'], ['1407.3510-1-26-0', '1407.3510-2-29-1'], ['1407.3510-1-26-1', '1407.3510-2-29-2'], ['1407.3510-1-26-2', '1407.3510-2-30-0'], ['1407.3510-1-26-3', '1407.3510-2-30-2'], ['1407.3510-1-26-4', '1407.3510-2-30-3'], ['1407.3510-1-26-5', '1407.3510-2-30-4']] | 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'1407.3510-2-9-5'], ['1407.3510-1-4-5', '1407.3510-2-9-7'], ['1407.3510-1-7-4', '1407.3510-2-12-4'], ['1407.3510-1-7-5', '1407.3510-2-12-5'], ['1407.3510-1-8-3', '1407.3510-2-12-10']] | [['1407.3510-1-5-1', '1407.3510-2-10-3'], ['1407.3510-1-5-3', '1407.3510-2-10-8'], ['1407.3510-1-9-0', '1407.3510-2-13-0'], ['1407.3510-1-9-1', '1407.3510-2-13-1'], ['1407.3510-1-9-4', '1407.3510-2-13-4'], ['1407.3510-1-9-5', '1407.3510-2-13-5'], ['1407.3510-1-2-1', '1407.3510-2-6-0'], ['1407.3510-1-2-5', '1407.3510-2-6-4'], ['1407.3510-1-2-7', '1407.3510-2-6-6'], ['1407.3510-1-2-8', '1407.3510-2-6-7'], ['1407.3510-1-6-4', '1407.3510-2-11-8'], ['1407.3510-1-6-6', '1407.3510-2-11-10'], ['1407.3510-1-6-8', '1407.3510-2-11-12'], ['1407.3510-1-6-9', '1407.3510-2-11-13'], ['1407.3510-1-24-0', '1407.3510-2-28-1'], ['1407.3510-1-3-1', '1407.3510-2-9-0'], ['1407.3510-1-3-2', '1407.3510-2-9-1'], ['1407.3510-1-4-4', '1407.3510-2-9-6'], ['1407.3510-1-8-0', '1407.3510-2-12-6'], ['1407.3510-1-8-2', '1407.3510-2-12-9'], ['1407.3510-1-8-4', '1407.3510-2-12-11'], ['1407.3510-1-8-5', '1407.3510-2-12-12'], ['1407.3510-1-8-6', '1407.3510-2-12-13']] | [] | [['1407.3510-1-5-0', '1407.3510-2-10-0'], ['1407.3510-1-5-2', '1407.3510-2-10-7'], ['1407.3510-1-6-0', '1407.3510-2-11-0'], ['1407.3510-1-6-0', '1407.3510-2-11-1'], ['1407.3510-1-6-10', '1407.3510-2-11-17'], ['1407.3510-1-1-4', '1407.3510-2-4-3'], ['1407.3510-1-1-4', '1407.3510-2-4-6'], ['1407.3510-1-1-5', '1407.3510-2-4-5'], ['1407.3510-1-7-2', '1407.3510-2-12-1'], ['1407.3510-1-7-3', '1407.3510-2-12-3'], ['1407.3510-1-8-1', '1407.3510-2-12-8']] | [['1407.3510-1-11-1', '1407.3510-2-14-0'], ['1407.3510-1-11-2', '1407.3510-2-14-1'], ['1407.3510-1-11-3', '1407.3510-2-14-2'], ['1407.3510-1-11-4', '1407.3510-2-47-5'], ['1407.3510-1-11-4', '1407.3510-2-14-3'], ['1407.3510-1-11-5', '1407.3510-2-14-4'], ['1407.3510-1-11-6', '1407.3510-2-14-5'], ['1407.3510-1-11-8', '1407.3510-2-14-7'], ['1407.3510-1-12-0', '1407.3510-2-14-8'], ['1407.3510-1-12-1', '1407.3510-2-14-9'], ['1407.3510-1-12-2', '1407.3510-2-14-10'], ['1407.3510-1-12-3', '1407.3510-2-14-12'], ['1407.3510-1-12-4', '1407.3510-2-14-13'], ['1407.3510-1-12-5', '1407.3510-2-14-14'], ['1407.3510-1-12-7', '1407.3510-2-14-17'], ['1407.3510-1-12-8', '1407.3510-2-14-15'], ['1407.3510-1-13-0', '1407.3510-2-15-0'], ['1407.3510-1-13-1', '1407.3510-2-15-3'], ['1407.3510-1-13-2', '1407.3510-2-15-6'], ['1407.3510-1-13-4', '1407.3510-2-15-8'], ['1407.3510-1-13-5', '1407.3510-2-15-9'], ['1407.3510-1-14-0', '1407.3510-2-17-0'], ['1407.3510-1-14-1', '1407.3510-2-17-2'], ['1407.3510-1-14-2', '1407.3510-2-17-3'], ['1407.3510-1-17-0', '1407.3510-2-19-0'], ['1407.3510-1-17-1', '1407.3510-2-19-1'], ['1407.3510-1-17-2', '1407.3510-2-19-2'], ['1407.3510-1-17-3', '1407.3510-2-19-3'], ['1407.3510-1-17-4', '1407.3510-2-20-2'], ['1407.3510-1-17-5', '1407.3510-2-20-3'], ['1407.3510-1-17-6', '1407.3510-2-20-4'], ['1407.3510-1-17-7', '1407.3510-2-21-1'], ['1407.3510-1-17-8', '1407.3510-2-21-2'], ['1407.3510-1-17-9', '1407.3510-2-21-3'], ['1407.3510-1-17-10', '1407.3510-2-21-4'], ['1407.3510-1-17-11', '1407.3510-2-21-5'], ['1407.3510-1-17-12', '1407.3510-2-21-6'], ['1407.3510-1-18-4', '1407.3510-2-26-6'], ['1407.3510-1-18-5', '1407.3510-2-26-7'], ['1407.3510-1-19-1', '1407.3510-2-26-0'], ['1407.3510-1-19-2', '1407.3510-2-26-8'], ['1407.3510-1-19-3', '1407.3510-2-25-7'], ['1407.3510-1-26-0', '1407.3510-2-29-1'], ['1407.3510-1-26-1', '1407.3510-2-29-2'], ['1407.3510-1-26-2', '1407.3510-2-30-0'], ['1407.3510-1-26-3', '1407.3510-2-30-2'], ['1407.3510-1-26-4', '1407.3510-2-30-3'], ['1407.3510-1-26-5', '1407.3510-2-30-4']] | ['1407.3510-1-1-0', '1407.3510-1-3-0', '1407.3510-1-6-7', '1407.3510-1-7-0', '1407.3510-1-11-0', '1407.3510-1-16-0', '1407.3510-1-19-0', '1407.3510-1-21-0', '1407.3510-1-27-0', '1407.3510-1-27-1', '1407.3510-2-2-0', '1407.3510-2-11-11', '1407.3510-2-33-0', '1407.3510-2-37-0', '1407.3510-2-37-1', '1407.3510-2-40-7', '1407.3510-2-49-4', '1407.3510-2-55-1', '1407.3510-2-56-1', '1407.3510-2-57-1', '1407.3510-2-58-1', '1407.3510-2-59-1', '1407.3510-2-60-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1407.3510 | null | null | null | null | null |
1906.03073 | {'1906.03073-1-0-0': 'Exceptional points, at which two or more eigenfunctions of a Hamiltonian coalesce, occur in non-Hermitian systems and lead to surprising physical effects.', '1906.03073-1-0-1': 'In particular, the behaviour of a system under parameter variation can differ significantly from the familiar Hermitian case in the presence of exceptional points.', '1906.03073-1-0-2': 'Here we analytically derive the probability of a non-adiabatic transition in a two-level system driven through two consecutive exceptional points at finite speed.', '1906.03073-1-0-3': 'The system is Hermitian far away from the exceptional points.', '1906.03073-1-0-4': 'In the adiabatic limit an equal redistribution between the states coalescing in the exceptional point is observed, which can be interpreted as a loss of information when passing through the exceptional point.', '1906.03073-1-0-5': 'For finite parameter variation this gets modified.', '1906.03073-1-0-6': 'We demonstrate how the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations.', '1906.03073-1-1-0': 'The intriguing properties of open quantum systems described by non-Hermitian and PT-symmetric Hamiltonians have opened up a new area of research [CITATION].', '1906.03073-1-1-1': 'The difference between Hermitian and non-Hermitian quantum physics is most pronounced in the presence of exceptional points, at which two or more eigenstates of a system coalesce.', '1906.03073-1-1-2': 'They lead to a number of counterintuitive features and in recent years have attracted a large amount of research interest, both theoretically and experimentally [CITATION].', '1906.03073-1-1-3': 'A number of these studies have investigated the behaviour of the wave function when parameters are varied cyclically around such a point [CITATION].', '1906.03073-1-1-4': 'In Hermitian quantum physics the adiabatic theorem ensures that a system prepared in an eigenstate remains in an instantaneous eigenstate, when parameters are varied sufficiently slowly.', '1906.03073-1-1-5': 'The situation is more involved in non-Hermitian systems.', '1906.03073-1-1-6': 'Here the exponential relative decay between different eigenstates competes with the exponentially small non-adiabatic corrections, which can lead to apparently non-adiabatic behaviour even in the adiabatic limit of infinitely slow parameter variation [CITATION].', '1906.03073-1-1-7': 'This leads to asymmetric behaviour when adiabatically encircling exceptional points.', '1906.03073-1-2-0': 'Here we go one step further and consider the behaviour of a system driven directly through an exceptional point.', '1906.03073-1-2-1': 'In particular, we study the dynamics of a two-level system driven through two consecutive exceptional points, at finite speed and in the adiabatic limit.', '1906.03073-1-2-2': 'The system is close to being Hermitian at the beginning and the end of the parameter sweep, making it meaningful to study the ratio of transmitted population between the instantaneous eigenstates far away from the exceptional points.', '1906.03073-1-2-3': 'We derive an analytic expression for the transmission probability.', '1906.03073-1-2-4': 'For adiabatic parameter variations this predicts a loss of information leading to an equal redistribution of the population between the states coalescing in the exceptional point.', '1906.03073-1-2-5': 'In the fast driving limit the familiar quantum quench behaviour is recovered.', '1906.03073-1-2-6': 'We demonstrate how this could be directly probed using Bloch oscillations in a [MATH]-symmetric lattice in a realistic experimental setup.', '1906.03073-1-2-7': 'This provides the exciting opportunity to observe the loss of information when driving adiabatically through an exceptional point in an optical experiment.', '1906.03073-1-2-8': 'A similar effect of partial transitions between Bloch bands at exceptional points has recently been observed numerically in a more complicated lattice structure and experimentally using optical fibre loops [CITATION].', '1906.03073-1-2-9': 'The quantitative description provided here explains these effects and opens up new opportunities for the control of optical beams.', '1906.03073-1-3-0': 'Let us begin by considering a [MATH]-symmetric two-level system described by the Hamiltonian [EQUATION] where [MATH] and [MATH] are real parameters and we set [MATH] without loss of generality.', '1906.03073-1-3-1': 'This Hamiltonian models two states coupled with strength [MATH], where the population decays exponentially with rate [MATH] in one level and grows exponentially with the same rate in the other.', '1906.03073-1-3-2': 'The eigenvalues [EQUATION] in dependence on [MATH] for a fixed value of [MATH] are depicted in the left panel of Fig. [REF].', '1906.03073-1-3-3': 'The right panel of the same figure shows the overlap between the two (right) eigenstates in dependence on [MATH].', '1906.03073-1-3-4': 'For values of [MATH] the eigenvalues are real and for very large values of [MATH] the eigenstates are almost orthogonal, leading the system to behave in an essentially Hermitian way.', '1906.03073-1-3-5': 'For intermediate values of [MATH] the non-Hermiticity is more apparent in the non-vanishing overlap of the eigenstates.', '1906.03073-1-3-6': 'The system has two exceptional points located at [MATH], at which both eigenvalues coalesce and the Hamiltonian has only a single eigenstate.', '1906.03073-1-3-7': 'For [MATH] the eigenvalues are complex conjugate and there is one exponentially growing and one exponentially decaying mode.', '1906.03073-1-4-0': 'Suppose that the system is initially in one of the eigenstates, for very large negative [MATH], and [MATH] is increased adiabatically.', '1906.03073-1-4-1': 'One expects the state to closely follow the instantaneous eigenstate it originated from up until the exceptional point.', '1906.03073-1-4-2': 'At this point the initial instantaneous eigenstate coalesces with the other and it cannot be inferred in which of the two states the system originated.', '1906.03073-1-4-3': 'Thus, the population of the two instantaneous eigenstates is expected to be equal immediately after the exceptional point.', '1906.03073-1-4-4': 'However, this argument does not take into account that once the parameters have passed through the first exceptional point, one of the states decays exponentially while the other grows exponentially.', '1906.03073-1-4-5': 'If [MATH] is varied infinitely slowly, the system has enough time to "switch" to the configuration where the entire remaining population is in the exponentially growing state.', '1906.03073-1-4-6': 'When the system later passes through the second exceptional point the population is again equally distributed between the two eigenstates.', '1906.03073-1-4-7': 'Now both the states are stable and for large positive values of [MATH] the population is expected to be equally distributed between the two instantaneous eigenstates.', '1906.03073-1-4-8': 'This behaviour occurs independently of the initial state.', '1906.03073-1-4-9': 'That is, the information about the initial state is lost entirely when the system is driven adiabatically through the exceptional points.', '1906.03073-1-5-0': 'This is indeed observed in numerical simulations.', '1906.03073-1-5-1': 'In Fig. [REF] the relative population of the instantaneous eigenstates (calculated as the normalised projection onto the left eigenstate) and the overall norm of the wave function are plotted as a function of time.', '1906.03073-1-5-2': 'Here [MATH] is slowly varied from a large negative initial value to a large positive final value for an initial eigenstate (top) and a randomly selected initial state (bottom).', '1906.03073-1-5-3': 'The norm, depicted on the right, grows approximately exponentially in the region between the two exceptional points and continues to oscillate after the second exceptional point due to the lack of orthogonality of the eigenstates.', '1906.03073-1-5-4': 'The oscillations die off at large values of [MATH] where the eigenstates become approximately orthogonal.', '1906.03073-1-6-0': 'In practice, however, parameters are not always varied slowly, and we are thus interested in the behaviour of the system when [MATH] is varied non-adiabatically.', '1906.03073-1-6-1': "This can in fact be understood analytically by closely following Zener's derivation of the famous Landau-Zener-Majorana-Stuckelberg formula [CITATION].", '1906.03073-1-6-2': 'To simplify the analysis we rotate the frame of reference to bring the Hamiltonian into a form similar to the Landau-Zener problem [EQUATION] where [MATH], [MATH] and [MATH] runs from minus to plus infinity.', '1906.03073-1-6-3': 'At [MATH] the eigenstates are given by the two uncoupled levels, i.e. the standard basis.', '1906.03073-1-6-4': 'We assume that the system is initially in the eigenstate [EQUATION]', '1906.03073-1-6-5': 'We want to deduce the transmission probability into the same diabatic state at [MATH], given by [EQUATION]', '1906.03073-1-6-6': 'For this purpose we start from the Schrodinger equation for [MATH] [EQUATION] and transform ([REF]) into the second order differential equation [EQUATION]', '1906.03073-1-6-7': 'Applying the transformation [MATH] converts this equation into the Weber equation [EQUATION] with [MATH] and [MATH].', '1906.03073-1-6-8': 'A solution satisfying the boundary condition ([REF]) is given by [EQUATION] where [MATH] is a normalisation factor and [MATH] is a Weber function [CITATION].', '1906.03073-1-6-9': 'The normalisation factor is determined from the asymptotic value [EQUATION] with [MATH].', '1906.03073-1-6-10': 'Inserting this into the equation of motion ([REF]) provides an asymptotic expression for [MATH] [EQUATION] which, together with the boundary conditions, yields [MATH].', '1906.03073-1-6-11': 'Making use of the asymptotic value [EQUATION] and well-known properties of the gamma function, leads to the amplitude [EQUATION]', '1906.03073-1-6-12': 'For unitary time-evolution [MATH] can be obtained from [MATH] due to the conservation of probability.', '1906.03073-1-6-13': 'However, for the non-Hermitian dynamics considered here the total probability is no longer conserved.', '1906.03073-1-6-14': 'The amplitude [MATH] must be calculated by other means, starting with equation ([REF]) and following a similar procedure to the one just used.', '1906.03073-1-6-15': 'The detailed calculation, yielding [EQUATION] is provided in the Appendix.', '1906.03073-1-7-0': 'Inserting ([REF]) and ([REF]) into the definition of the transmission probability ([REF]) finally yields [EQUATION]', '1906.03073-1-7-1': 'As expected this approaches [MATH] in the adiabatic limit.', '1906.03073-1-7-2': 'On the other hand, in the limit of fast driving, the usual quantum quench behaviour is observed, i.e. [MATH] and [MATH] for all times.', '1906.03073-1-7-3': 'For intermediate values of [MATH] the transmission probability monotonically increases with [MATH], interpolating between the two limits.', '1906.03073-1-7-4': 'Figure [REF] depicts the transmission probability as a function of [MATH] for various values of [MATH].', '1906.03073-1-8-0': 'In the following we demonstrate that the model ([REF]) very accurately describes the band transitions in a PT-symmetric lattice with an applied static force.', '1906.03073-1-8-1': 'This allows for a direct observation of the transition through a series of exceptional points in dependence on the adiabatic parameter, which can be tuned via the static force.', '1906.03073-1-8-2': 'Let us consider a [MATH]-symmetric tight-binding Hamiltonian of the form [EQUATION] with a gain and loss rate [MATH].', '1906.03073-1-8-3': 'This model can be realised, for example, as a chain of waveguides with absorption in every other waveguide and optical gain of an equal strength in the waveguides in between.', '1906.03073-1-8-4': 'Some of the properties of this model have previously been discussed in [CITATION].', '1906.03073-1-8-5': 'A passively [MATH]-symmetric version of this model has been implemented experimentally in [CITATION].', '1906.03073-1-9-0': 'It is convenient to study the system in the quasimomentum representation, that is, in the basis of the Bloch states [EQUATION] where the quasimomentum [MATH] is confined to the region [MATH].', '1906.03073-1-9-1': 'The Bloch states are orthogonal and normalised to the [MATH]-periodic delta comb [MATH].', '1906.03073-1-9-2': 'In a similar spirit to [CITATION] we introduce the two-component function [MATH], with [MATH] and [MATH].', '1906.03073-1-9-3': 'The time-evolution may then be written as the two-level Schrodinger equation [EQUATION] where the [MATH] are Pauli matrices and the Bloch Hamiltonian is defined as [EQUATION]', '1906.03073-1-9-4': 'The [MATH]-dependent eigenvalues of [MATH] define the dispersion relation of the two band system [EQUATION]', '1906.03073-1-9-5': 'The band structure is complex for arbitrarily small values of [MATH] at the band edge.', '1906.03073-1-9-6': 'For values of [MATH] there are exceptional points at [MATH] and for [MATH]-values such that [MATH] the energy is imaginary.', '1906.03073-1-9-7': 'For [MATH] the bands are purely imaginary and there are no exceptional points.', '1906.03073-1-9-8': 'Here we focus on [MATH] values well below this critical point.', '1906.03073-1-9-9': 'An example of the band structure for [MATH] is depicted in the top row of Fig. [REF].', '1906.03073-1-9-10': 'The bottom row of the same figure shows the modulus squared of the components of the eigenstates.', '1906.03073-1-9-11': 'The eigenstates are close to the standard basis vectors at [MATH] and [MATH].', '1906.03073-1-10-0': 'If a static force is applied to the lattice a term [MATH] is added to the Hamiltonian ([REF]).', '1906.03073-1-10-1': 'An initial state [MATH] that is localised in a single band around a quasimomentum value [MATH] will perform a non-Hermitian version of the famous Bloch oscillations.', '1906.03073-1-10-2': 'This is accompanied by transitions between the bands, which appear as a splitting of the beam in real space.', '1906.03073-1-10-3': 'Some examples of the resulting dynamics can be seen in Fig. [REF].', '1906.03073-1-10-4': 'Similar behaviour has been observed experimentally in a system of optical fibre loops [CITATION].', '1906.03073-1-10-5': 'The populations of the two bands, i.e. the relative amplitudes of the two beams, are modelled by the Landau-Zener-type Hamiltonian ([REF]).', '1906.03073-1-10-6': 'This can be understood in the following way.', '1906.03073-1-11-0': 'The static force introduces a term [MATH] into the two-component Hamiltonian ([REF]) [EQUATION] where [MATH] is canonically conjugate to [MATH] with [MATH].', '1906.03073-1-11-1': 'The expectation value of [MATH] evaluated in the two-component state [MATH] at time [MATH] is [EQUATION] where we have defined the (non-unitary) time-evolution operator [MATH] and the integrals are over the interval [MATH].', '1906.03073-1-11-2': 'The Zassenhaus formula enables the time-evolution operator to be factorised into [MATH], with some matrix operator [MATH] that is independent of [MATH], so that [EQUATION]', '1906.03073-1-11-3': 'Assuming that [MATH] is well localised in [MATH]-space, such that [MATH], we find [EQUATION]', '1906.03073-1-11-4': 'This is the famous acceleration theorem that is well-known for Hermitian systems.', '1906.03073-1-11-5': 'In the non-Hermitian case this is only an approximation and relies on the initial quasimomentum uncertainty being negligible.', '1906.03073-1-11-6': 'Now consider the two intervals [MATH] and [MATH].', '1906.03073-1-11-7': 'We note that if [MATH] then [MATH] and the initial state is in the band [MATH].', '1906.03073-1-11-8': 'While if [MATH], then [MATH] and initial state is in the band [MATH].', '1906.03073-1-12-0': 'In summary, if a static force is applied to the PT-symmetric chain ([REF]), the dynamics of a wave packet that is initially well localised in momentum space can be described by the effectively time-dependent Bloch Hamiltonian ([REF]), with the operator [MATH] replaced by its time-dependent expectation value [MATH].', '1906.03073-1-12-1': 'We can further Taylor expand the effective two-level system ([REF]) around the band edge [MATH].', '1906.03073-1-12-2': 'The eigenvalues of the Taylor expanded Hamiltonian are depicted in the top panel of Fig. [REF], in comparison to the exact eigenvalues for [MATH].', '1906.03073-1-12-3': 'We observe a good agreement of the eigenvalues close to the exceptional points.', '1906.03073-1-12-4': 'The resulting Hamiltonian is of the form ([REF]), with the replacements [MATH] and [MATH], and the instantaneous eigenstates represent the two quasimomentum bands of the system.', '1906.03073-1-12-5': 'It follows from ([REF]) that the transmission probability is given by [EQUATION]', '1906.03073-1-12-6': 'This result agrees well with numerical calculations, as demonstrated in Fig. [REF] for an initial broad Gaussian beam.', '1906.03073-1-12-7': 'The time-evolution of the renormalised density [MATH] on each site [MATH] is plotted for [MATH] and three values of [MATH].', '1906.03073-1-12-8': 'We observe that the Bloch oscillations sweep through the band structure depicted in Fig. [REF].', '1906.03073-1-12-9': 'The data points in the bottom right panel were obtained by evolving the same initial state to half the Bloch period [MATH], for various values of [MATH].', '1906.03073-1-12-10': 'The population in the upper band of the beam in position space was then calculated at [MATH], when the two beams are maximally separated.', '1906.03073-1-12-11': 'The agreement between the numerically observed transmission probability and the approximative formula ([REF]) is astonishing.', '1906.03073-1-12-12': 'This behaviour is observed for a wide range of parameters, as long as the two beams splitting at the exceptional point do not overlap and can be meaningfully distinguished.', '1906.03073-1-12-13': 'We expect the experimental observation of this transition due to sweeping through exceptional points to be entirely within reach.', '1906.03073-1-13-0': 'We have investigated a two-level system driven through two consecutive exceptional points adiabatically and at finite speed.', '1906.03073-1-13-1': 'In the adiabatic limit this leads to behaviour having no analogue in the Hermitian case, where the population is equally distributed between the states coalescing in the exceptional point and a corresponding loss of information of the initial state.', '1906.03073-1-13-2': 'In the limit of fast driving the familiar quantum quench behaviour is recovered.', '1906.03073-1-13-3': 'We have derived an analytic expression for the population transfer for arbitrary speed of parameter variation, interpolating between these two extremes.', '1906.03073-1-13-4': 'We have further demonstrated how this can be experimentally investigated in a PT-symmetric lattice using Bloch oscillations, such as an optical waveguide setup, providing new opportunities for engineering beam dynamics.', '1906.03073-1-14-0': 'E.M.G. acknowledges support from the Royal Society (Grant.', '1906.03073-1-14-1': "UF130339) and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 758453).", '1906.03073-1-15-0': '# APPENDIX', '1906.03073-1-16-0': '## Derivation of the amplitude [MATH]', '1906.03073-1-17-0': 'Here we provide the derivation of equation ([REF]) in the main text.', '1906.03073-1-17-1': 'Performing the transformation [MATH] converts the equation of motion for the component [MATH] [EQUATION] into the Weber equation [EQUATION] where [MATH] with [MATH].', '1906.03073-1-17-2': 'Due to the boundary condition [MATH] the solution should be non-vanishing as [MATH].', '1906.03073-1-17-3': 'Furthermore, the solution should have asymptotic behaviour consistent with equation ([REF]).', '1906.03073-1-17-4': 'Thus, the solution is of the form [MATH].', '1906.03073-1-17-5': 'Matching the asymptotic value [EQUATION] to ([REF]) yields the normalisation factor [MATH].', '1906.03073-1-17-6': 'This result, together with the asymptotic result for the Weber function [EQUATION] leads to the amplitude [EQUATION]'} | {'1906.03073-2-0-0': 'Exceptional points, at which two or more eigenfunctions of a Hamiltonian coalesce, occur in non-Hermitian systems and lead to surprising physical effects.', '1906.03073-2-0-1': 'In particular, the behaviour of a system under parameter variation can differ significantly from the familiar Hermitian case in the presence of exceptional points.', '1906.03073-2-0-2': 'Here we analytically derive the probability of a non-adiabatic transition in a two-level system driven through two consecutive exceptional points at finite speed.', '1906.03073-2-0-3': 'The system is Hermitian far away from the exceptional points.', '1906.03073-2-0-4': 'In the adiabatic limit an equal redistribution between the states coalescing in the exceptional point is observed, which can be interpreted as a loss of information when passing through the exceptional point.', '1906.03073-2-0-5': 'For finite parameter variation this gets modified.', '1906.03073-2-0-6': 'We demonstrate how the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations.', '1906.03073-2-1-0': '# Introduction', '1906.03073-2-2-0': 'The intriguing properties of open quantum systems described by non-Hermitian and PT-symmetric Hamiltonians have opened up a new area of research [CITATION].', '1906.03073-2-2-1': 'The difference between Hermitian and non-Hermitian quantum physics is most pronounced in the presence of exceptional points, at which two or more eigenstates of a system coalesce.', '1906.03073-2-2-2': 'They lead to a number of counterintuitive features and have attracted a large amount of research interest, both theoretically and experimentally [CITATION].', '1906.03073-2-2-3': 'A number of these studies have investigated the behaviour of the wave function when parameters are varied cyclically around such a point [CITATION].', '1906.03073-2-2-4': 'In Hermitian quantum physics the adiabatic theorem ensures that a system prepared in an eigenstate remains in an instantaneous eigenstate, when parameters are varied sufficiently slowly.', '1906.03073-2-2-5': 'The situation is more involved in non-Hermitian systems.', '1906.03073-2-2-6': 'Here the exponential relative decay between different eigenstates competes with the exponentially small non-adiabatic corrections, which can lead to apparently non-adiabatic behaviour even in the adiabatic limit of infinitely slow parameter variation [CITATION].', '1906.03073-2-2-7': 'This causes asymmetric behaviour when encircling exceptional points.', '1906.03073-2-3-0': 'Here we go one step further and consider the behaviour of a system driven directly through an exceptional point.', '1906.03073-2-3-1': 'In particular, we study a two-level system driven through two consecutive exceptional points, at finite speed and in the adiabatic limit.', '1906.03073-2-3-2': 'The system is close to being Hermitian at the beginning and the end of the parameter sweep, making it meaningful to study the ratio of transmitted population between the instantaneous eigenstates far away from the exceptional points.', '1906.03073-2-3-3': 'We derive an analytic expression for the transmission probability.', '1906.03073-2-3-4': 'For adiabatic parameter variations this predicts a loss of information, leading to an equal redistribution of the population between the states coalescing in the exceptional point.', '1906.03073-2-3-5': 'In the fast driving limit quantum quench behaviour is recovered.', '1906.03073-2-3-6': 'We demonstrate how this could be observed using Bloch oscillations in a PT-symmetric lattice in a realistic experimental setup.', '1906.03073-2-3-7': 'Interestingly, the Hamiltonian of this system itself does not have any exceptional points.', '1906.03073-2-3-8': 'A similar effect of partial transitions between Bloch bands at exceptional points has recently been observed numerically in a more complicated lattice structure and experimentally using optical fibre loops [CITATION].', '1906.03073-2-3-9': 'The quantitative description provided here explains these effects and opens up new avenues for the control of optical beams.', '1906.03073-2-4-0': '# Non-adiabatic transitions in a two-mode system', '1906.03073-2-5-0': 'Consider the PT-symmetric Hamiltonian [EQUATION] where [MATH] and [MATH] are real parameters and we set [MATH] without loss of generality.', '1906.03073-2-5-1': 'This Hamiltonian describes two states with an energy difference of [MATH] and an asymmetric coupling [MATH].', '1906.03073-2-5-2': 'Its direct implementation in a two-waveguide setup is nontrivial, due to the nonreciprocal coupling between the two modes.', '1906.03073-2-5-3': 'Nonreciprocal coupling between two resonators has been discussed for example in [CITATION].', '1906.03073-2-5-4': 'In this work we shall consider a perhaps slightly less obvious implementation of the Hamiltonian ([REF]), as the Bloch Hamiltonian of a PT-symmetric tight-binding lattice.', '1906.03073-2-5-5': 'This setup allows for a direct visual observation of the population transfer between the two modes when the system is driven through the exceptional points.', '1906.03073-2-6-0': 'The eigenvalues of the Hamiltonian ([REF]) [EQUATION] in dependence on [MATH] for a fixed value of [MATH] are depicted in the left panel of Fig. [REF].', '1906.03073-2-6-1': 'The right panel shows the overlap between the two (right) eigenstates in dependence on [MATH].', '1906.03073-2-6-2': 'For values of [MATH] the eigenvalues are real and for very large values of [MATH] the eigenstates are almost orthogonal (and tend towards the standard basis vectors), leading the system to behave in an essentially Hermitian way.', '1906.03073-2-6-3': 'For intermediate values of [MATH] the non-Hermiticity is more apparent in the non-vanishing overlap of the eigenstates.', '1906.03073-2-6-4': 'The system has two exceptional points located at [MATH], at which both eigenvalues coalesce and the Hamiltonian has only a single eigenstate.', '1906.03073-2-6-5': 'For [MATH] the eigenvalues are complex conjugate and there is one exponentially growing and one exponentially decaying mode.', '1906.03073-2-7-0': 'Suppose that the system is initially in one of the eigenstates, for very large negative [MATH], and [MATH] is increased adiabatically.', '1906.03073-2-7-1': 'One expects the state to closely follow the instantaneous eigenstate it originated from up until the exceptional point.', '1906.03073-2-7-2': 'At this point the initial instantaneous eigenstate coalesces with the other and it cannot be inferred in which of the two states the system originated.', '1906.03073-2-7-3': 'Thus, the population of the two instantaneous eigenstates is expected to be equal immediately after the exceptional point.', '1906.03073-2-7-4': 'However, this argument does not take into account that once the parameters have passed through the first exceptional point, one of the states decays exponentially while the other grows exponentially.', '1906.03073-2-7-5': 'If [MATH] is varied adiabatically, the system has enough time to "switch" to the configuration where the entire remaining population is in the exponentially growing state.', '1906.03073-2-7-6': 'When the system later passes through the second exceptional point the population is again equally distributed between the two eigenstates.', '1906.03073-2-7-7': 'Now both the states are stable and for large positive values of [MATH] the population is expected to be equally distributed between the two instantaneous eigenstates.', '1906.03073-2-7-8': 'That is, information about the initial state is lost when the system is driven adiabatically through the exceptional points.', '1906.03073-2-7-9': 'After driving the system though the pair of exceptional points the state of the system is given by [MATH] in the basis of eigenstates.', '1906.03073-2-7-10': 'In the adiabatic limit we have [MATH].', '1906.03073-2-7-11': 'Thus, while the initial state has two degrees of freedom (for example the relative phase between the two components and an amplitude), the final state is parameterised by a single variable, which is a relative phase between the two components.', '1906.03073-2-7-12': 'As there is clearly no one-to-one mapping from a one-dimensional space to a two-dimensional space, one cannot time-reverse [MATH] to obtain the initial state.', '1906.03073-2-7-13': 'Of course, this observation is rather academic in nature, as any experiment will not be truly adiabatic.', '1906.03073-2-7-14': 'Thus, one might in principle be able to obtain the initial state by time reversing the dynamics.', '1906.03073-2-7-15': 'However, our numerical simulations show that the backwards time evolution can be very sensitive to small perturbations, which could make it practically very difficult to time-reverse the final state to obtain the initial state.', '1906.03073-2-8-0': 'This behaviour is indeed observed in numerical simulations and occurs independently of the initial state.', '1906.03073-2-8-1': 'In Fig. [REF] the relative population of the instantaneous eigenstates (calculated as the normalised projection onto the left eigenstate) and the overall norm of the wave function are plotted as a function of time.', '1906.03073-2-8-2': 'Here [MATH] is slowly varied from a large negative initial value to a large positive final value for an initial eigenstate (top) and a randomly selected initial state (bottom).', '1906.03073-2-8-3': 'The norm, depicted on the right, grows approximately exponentially in the region between the two exceptional points and continues to oscillate after the second exceptional point due to the lack of orthogonality of the eigenstates.', '1906.03073-2-8-4': 'The oscillations die off at large values of [MATH] where the eigenstates become approximately orthogonal.', '1906.03073-2-9-0': 'In practice, however, parameters are not always varied slowly, and we are thus interested in the behaviour of the system when [MATH] is varied non-adiabatically.', '1906.03073-2-9-1': "This can be understood analytically by closely following Zener's derivation of the famous Landau-Zener-Majorana-Stuckelberg formula [CITATION].", '1906.03073-2-9-2': 'We assume the parameter [MATH] to vary linearly in time, that is, [MATH], [MATH], where [MATH] runs from minus to plus infinity.', '1906.03073-2-9-3': 'At [MATH] the eigenstates are given by the two uncoupled levels, i.e. the standard basis.', '1906.03073-2-9-4': 'We assume that the system is initially in the eigenstate [EQUATION]', '1906.03073-2-9-5': 'We want to deduce the transmission probability into the same diabatic state at [MATH], given by [EQUATION]', '1906.03073-2-9-6': 'For this purpose we start from the Schrodinger equation for [MATH] [EQUATION] and transform ([REF]) into the second order differential equation [EQUATION]', '1906.03073-2-9-7': 'Applying the transformation [MATH] converts this equation into the Weber equation [EQUATION] with [MATH] and [MATH].', '1906.03073-2-9-8': 'A solution satisfying the initial conditions ([REF]) is given by [EQUATION] where [MATH] is a normalisation factor and [MATH] is a Weber (or parabolic cylinder) function [CITATION].', '1906.03073-2-9-9': 'The normalisation factor is determined from the asymptotic value [EQUATION] with [MATH].', '1906.03073-2-9-10': 'Inserting this into the equation of motion ([REF]) provides an asymptotic expression for [MATH] [EQUATION] which, together with the initial conditions, yields [MATH].', '1906.03073-2-9-11': 'Making use of the asymptotic value [EQUATION] and well-known properties of the gamma function, leads to the amplitude [EQUATION]', '1906.03073-2-9-12': 'For unitary time evolution [MATH] can be obtained from [MATH] due to the conservation of probability.', '1906.03073-2-9-13': 'However, for the non-Hermitian dynamics considered here the total probability is no longer conserved.', '1906.03073-2-9-14': 'The amplitude [MATH] must be calculated by other means, starting with equation ([REF]) and following a similar procedure to the one just used.', '1906.03073-2-9-15': 'Performing the transformation [MATH] converts the equation of motion for the component [MATH] [EQUATION] into the Weber equation [EQUATION] where [MATH] with [MATH].', '1906.03073-2-9-16': 'Due to the initial condition [MATH] the solution should be non-vanishing as [MATH].', '1906.03073-2-9-17': 'Furthermore, the solution should have asymptotic behaviour consistent with equation ([REF]).', '1906.03073-2-9-18': 'Thus, the solution is of the form [MATH].', '1906.03073-2-9-19': 'Making use of the asymptotic expansion [EQUATION] and the initial conditions ([REF]) yields the normalisation factor [MATH].', '1906.03073-2-9-20': 'This, together with the asymptotic result for the Weber function [EQUATION] leads to the amplitude [EQUATION]', '1906.03073-2-9-21': 'Inserting ([REF]) and ([REF]) into the definition of the transmission probability ([REF]) finally yields [EQUATION]', '1906.03073-2-9-22': 'As expected, this approaches [MATH] in the adiabatic limit.', '1906.03073-2-9-23': 'On the other hand, in the limit of fast driving, the usual quantum quench behaviour is observed, i.e. [MATH] and [MATH] for all times.', '1906.03073-2-9-24': 'For intermediate values of [MATH] the transmission probability monotonically increases with [MATH], interpolating between the two limits.', '1906.03073-2-9-25': 'Figure [REF] depicts the transmission probability as a function of [MATH] for various values of [MATH].', '1906.03073-2-10-0': '# The two-mode model as the Bloch Hamiltonian of a PT-symmetric tight binding lattice', '1906.03073-2-11-0': 'We now demonstrate that the model ([REF]) accurately describes the band transitions in a PT-symmetric lattice with an applied static force.', '1906.03073-2-11-1': 'This allows for a direct observation of the transition through a series of exceptional points in dependence on the adiabatic parameter, which can be tuned via the static force.', '1906.03073-2-11-2': 'Let us consider a PT-symmetric tight-binding Hamiltonian of the form [EQUATION] with a gain and loss rate [MATH].', '1906.03073-2-11-3': 'This model can be realised, for example, as a chain of waveguides with absorption in every other waveguide and optical gain of an equal strength in the waveguides in between.', '1906.03073-2-11-4': 'Some of the properties of this model have previously been discussed in [CITATION].', '1906.03073-2-11-5': 'A passively PT-symmetric version of this model has been implemented experimentally in [CITATION].', '1906.03073-2-12-0': 'It is convenient to study the system in the quasimomentum representation, that is, in the basis of the Bloch states [EQUATION] where the quasimomentum [MATH] is confined to the region [MATH].', '1906.03073-2-12-1': 'The Bloch states are orthogonal and normalised to the [MATH]-periodic delta comb [MATH].', '1906.03073-2-12-2': 'In a similar spirit to [CITATION] we introduce the two-component function [MATH], with [MATH] and [MATH].', '1906.03073-2-12-3': 'The time evolution may then be written as the two-level Schrodinger equation [EQUATION] where [MATH] are Pauli matrices and the Bloch Hamiltonian is defined as [EQUATION]', '1906.03073-2-12-4': 'The [MATH]-dependent eigenvalues of [MATH] define the dispersion relation of the two band system [EQUATION]', '1906.03073-2-12-5': 'The band structure is complex for arbitrarily small values of [MATH].', '1906.03073-2-12-6': 'For values of [MATH] there are exceptional points at [MATH] and when [MATH] the energy is imaginary.', '1906.03073-2-12-7': 'For [MATH] the bands are purely imaginary and there are no exceptional points.', '1906.03073-2-12-8': 'Here we focus on [MATH] values well below this critical point.', '1906.03073-2-12-9': 'An example of the band structure for [MATH] is depicted in the top row of Fig. [REF].', '1906.03073-2-12-10': 'The bottom row of the same figure shows the modulus squared of the components of the eigenstates.', '1906.03073-2-12-11': 'The eigenstates are close to the standard basis vectors at [MATH] and [MATH].', '1906.03073-2-12-12': 'However, when [MATH] is increased this is no longer the case, as illustrated in Fig. [REF] for [MATH].', '1906.03073-2-13-0': 'We now show that the two-component function [MATH] of a broad Gaussian beam in position space is an approximate eigenstate of the Bloch Hamiltonian, when the Bloch states are close to the standard basis vectors.', '1906.03073-2-13-1': 'Consider a Gaussian in real space representation given by [EQUATION] where [MATH] is the lattice index, [MATH] is the centre of the Gaussian, [MATH] is the initial momentum, [MATH] is the width parameter and [MATH] is a normalisation constant, chosen so that [MATH].', '1906.03073-2-13-2': 'The quasimomentum representation of this state is found to be [EQUATION] where [MATH], [MATH] and the Jacobi theta function [MATH] is defined as [CITATION] [EQUATION]', '1906.03073-2-13-3': 'Thus, [MATH] is the product of a Gaussian distribution in quasimomentum space and a term involving Jacobi theta functions.', '1906.03073-2-13-4': 'Using the properties of the theta function it is straightforward to check that [MATH] for any integer [MATH].', '1906.03073-2-14-0': 'In order to obtain the two-component function [MATH] we need to calculate [MATH].', '1906.03073-2-14-1': 'From ([REF]) it follows that [EQUATION] where the theta function [MATH] is defined as [EQUATION] and we made use of the relationship [EQUATION]', '1906.03073-2-14-2': 'Thus, [MATH] may be written as [EQUATION]', '1906.03073-2-14-3': 'In the broad Gaussian limit [MATH], [MATH] tends to infinity along the imaginary axis and the two-components become [MATH].', '1906.03073-2-14-4': 'The two-component function [MATH] then becomes a standard basis vector multiplied by a Gaussian wave packet that is highly localised around [MATH].', '1906.03073-2-14-5': 'So, for example, a broad Gaussian beam in position space with momentum [MATH]) yields a [MATH] that is approximately one of the eigenstates of the Bloch Hamiltonian at [MATH]) depicted in Fig. [REF].', '1906.03073-2-15-0': 'If a static force is applied to the lattice a term [MATH] is added to the Hamiltonian ([REF]).', '1906.03073-2-15-1': 'An initial state [MATH] that is approximately an eigenstate of the Bloch Hamiltonian will then perform a non-Hermitian version of the famous Bloch oscillations.', '1906.03073-2-15-2': 'This is accompanied by transitions between the bands, which appear as a splitting of the beam in real space.', '1906.03073-2-15-3': 'Some examples of the resulting dynamics can be seen in Fig. [REF].', '1906.03073-2-15-4': 'Similar behaviour has been observed experimentally in a system of optical fibre loops [CITATION].', '1906.03073-2-15-5': 'The populations of the two bands, i.e. the relative amplitudes of the two beams, are approximated by the Landau-Zener-type Hamiltonian ([REF]).', '1906.03073-2-15-6': 'This can be understood in the following way.', '1906.03073-2-16-0': 'The static force introduces a term [MATH] into the two-component Hamiltonian ([REF]) [EQUATION] where [MATH] is canonically conjugate to [MATH] with [MATH].', '1906.03073-2-16-1': 'The expectation value of [MATH] evaluated in the two-component state [MATH] is [EQUATION] where we have defined the (non-unitary) time-evolution operator [MATH] and the integrals are over the interval [MATH].', '1906.03073-2-16-2': 'The Zassenhaus formula enables the factorisation [MATH], with some matrix operator [MATH] that is independent of [MATH], such that [EQUATION]', '1906.03073-2-16-3': 'For a Gaussian in position space the two-component function [MATH] is given by equation ([REF]), and in the broad beam limit the expectation value ([REF]) reduces to [EQUATION]', '1906.03073-2-16-4': 'This is the acceleration theorem that is well known for Hermitian systems.', '1906.03073-2-16-5': 'In the non-Hermitian case this is only an approximation and relies on the initial quasimomentum uncertainty being negligible.', '1906.03073-2-16-6': 'It follows from the properties of the delta function that [MATH] when [MATH] or [MATH].', '1906.03073-2-17-0': 'In summary, if a static force is applied to the PT-symmetric chain ([REF]), and [MATH] is approximately a Bloch state, then the dynamics can be described by the effectively time-dependent Bloch Hamiltonian ([REF]), with the operator [MATH] replaced by its time-dependent expectation value [MATH].', '1906.03073-2-17-1': 'We can further Taylor expand the effective two-level system ([REF]) around the band edge [MATH].', '1906.03073-2-17-2': 'The eigenvalues of the Taylor expanded Hamiltonian are depicted in the top panel of Fig. [REF], in comparison to the exact eigenvalues for [MATH].', '1906.03073-2-17-3': 'We observe a good agreement of the eigenvalues close to the exceptional points.', '1906.03073-2-17-4': 'The resulting Hamiltonian is of the form ([REF]), with [MATH] and [MATH], and the instantaneous eigenstates represent the two quasimomentum bands of the system.', '1906.03073-2-17-5': 'It follows from ([REF]) that the transmission probability is [EQUATION]', '1906.03073-2-17-6': 'Thus, if [MATH] is initialised in an approximate eigenstate of the Bloch Hamiltonian at [MATH] or [MATH], then after half a Bloch period [MATH] (with [MATH]) the populations of the two bands are approximated by ([REF]).', '1906.03073-2-18-0': 'This result agrees well with numerical calculations, as demonstrated in Fig. [REF] for an initial broad Gaussian beam.', '1906.03073-2-18-1': 'The time evolution of the renormalised density [MATH] on each site [MATH] is plotted for [MATH] and three values of [MATH].', '1906.03073-2-18-2': 'We observe that the Bloch oscillations sweep through the band structure depicted in Fig. [REF].', '1906.03073-2-18-3': 'The data points in the bottom right panel were obtained by evolving the same initial state up to half the Bloch period for various values of [MATH].', '1906.03073-2-18-4': 'Each point corresponds to the population in the upper branch of the beam in position space at [MATH] for a particular [MATH] value.', '1906.03073-2-18-5': 'The agreement between the numerically observed transmission probability and the approximative formula ([REF]) is excellent.', '1906.03073-2-18-6': 'This behaviour is observed for a wide range of parameters, as long as the two beams splitting at the exceptional point can be meaningfully distinguished.', '1906.03073-2-18-7': 'We expect the experimental observation of this transition due to sweeping through exceptional points to be entirely within reach.', '1906.03073-2-19-0': '# Summary', '1906.03073-2-20-0': 'We have investigated a two-level system driven through two consecutive exceptional points adiabatically and at finite speed.', '1906.03073-2-20-1': 'In the adiabatic limit this leads to behaviour having no analogue in the Hermitian case.', '1906.03073-2-20-2': 'The population is equally distributed between the states coalescing in the exceptional point, corresponding to a loss of information of the initial state.', '1906.03073-2-20-3': 'In the limit of fast driving the familiar quantum quench behaviour is recovered.', '1906.03073-2-20-4': 'We have derived an analytic expression for the population transfer for arbitrary speed of parameter variation, interpolating between these two extremes.', '1906.03073-2-20-5': 'We have further demonstrated how this can be experimentally investigated in a PT-symmetric lattice using Bloch oscillations, such as an optical waveguide setup, providing new opportunities for engineering beam dynamics.'} | [['1906.03073-1-10-0', '1906.03073-2-15-0'], ['1906.03073-1-10-2', '1906.03073-2-15-2'], ['1906.03073-1-10-3', '1906.03073-2-15-3'], ['1906.03073-1-10-4', '1906.03073-2-15-4'], ['1906.03073-1-10-6', '1906.03073-2-15-6'], ['1906.03073-1-0-0', '1906.03073-2-0-0'], ['1906.03073-1-0-1', '1906.03073-2-0-1'], ['1906.03073-1-0-2', '1906.03073-2-0-2'], ['1906.03073-1-0-3', '1906.03073-2-0-3'], ['1906.03073-1-0-4', '1906.03073-2-0-4'], ['1906.03073-1-0-5', '1906.03073-2-0-5'], ['1906.03073-1-0-6', '1906.03073-2-0-6'], ['1906.03073-1-8-1', '1906.03073-2-11-1'], ['1906.03073-1-8-3', '1906.03073-2-11-3'], ['1906.03073-1-8-4', '1906.03073-2-11-4'], ['1906.03073-1-2-0', '1906.03073-2-3-0'], ['1906.03073-1-2-2', '1906.03073-2-3-2'], ['1906.03073-1-2-3', '1906.03073-2-3-3'], ['1906.03073-1-2-8', '1906.03073-2-3-8'], ['1906.03073-1-3-5', '1906.03073-2-6-3'], ['1906.03073-1-3-6', 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'1906.03073-2-16-2'], ['1906.03073-1-13-1', '1906.03073-2-20-1'], ['1906.03073-1-13-1', '1906.03073-2-20-2']] | [['1906.03073-1-17-1', '1906.03073-2-9-15'], ['1906.03073-1-17-2', '1906.03073-2-9-16'], ['1906.03073-1-17-3', '1906.03073-2-9-17'], ['1906.03073-1-17-4', '1906.03073-2-9-18'], ['1906.03073-1-17-5', '1906.03073-2-9-19'], ['1906.03073-1-17-6', '1906.03073-2-9-20']] | [] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1906.03073 | null | null | null | null | null |
cond-mat-0106236 | {'cond-mat-0106236-1-0-0': 'The one-dimensional Ising model with nearest neighbour interactions and the zero-temperature dynamics recently considered by Lefevre and Dean -J. Phys.', 'cond-mat-0106236-1-0-1': 'A: Math.', 'cond-mat-0106236-1-0-2': 'Gen. 34, L213 (2001)- is investigated.', 'cond-mat-0106236-1-0-3': 'By introducing a particle-hole description, in which the holes are associated to the domain walls of the Ising model, an analytical solution is obtained.', 'cond-mat-0106236-1-0-4': 'The result for the asymptotic energy agrees with that found in the mean field approximation.', 'cond-mat-0106236-1-1-0': '# Introduction', 'cond-mat-0106236-1-2-0': 'Ising systems are often used as simple models in many different fields of statistical physics.', 'cond-mat-0106236-1-2-1': 'In particular, the one-dimensional Ising model with nearest neighbour interactions and Glauber dynamics [CITATION] is ubiquitous.', 'cond-mat-0106236-1-2-2': 'It shows glass-like behaviour, including non-exponential relaxation [CITATION], a laboratory glass transition in cooling processes [CITATION], hysteresis effects in thermal cycling experiments [CITATION], and aging effects at low temperatures [CITATION].', 'cond-mat-0106236-1-2-3': 'When a weak oscillating external field is applied, the amplitude of the induced magnetization presents a maximum as a function of the temperature, a behaviour resembling stochastic resonance [CITATION].', 'cond-mat-0106236-1-3-0': 'Other kind of important Ising systems are those with "facilitated" dynamics.', 'cond-mat-0106236-1-3-1': 'They were introduced in a pioneering work by Fredrickson and Andersen [CITATION], in the context of the relaxation of structural glasses.', 'cond-mat-0106236-1-3-2': 'In these models, a given spin can only flip if its nearest neighbours are in a certain subset of all their possible configurations.', 'cond-mat-0106236-1-3-3': 'The one-dimensional Ising model with facilitated dynamics (1SFM) has been extensively studied [CITATION].', 'cond-mat-0106236-1-3-4': 'Although, in general, it is not possible to find an analytical solution of these facilitated Ising models, the 1SFM has been exactly solved at zero temperature [CITATION].', 'cond-mat-0106236-1-3-5': 'Very recently, the 1SFM has been applied to study the compaction of vibrated granular systems.', 'cond-mat-0106236-1-3-6': 'A hole (particle) is associated to any spin in its excited (ground) state.', 'cond-mat-0106236-1-3-7': 'Trying to mimic what is done in real experiments [CITATION], the tapping process is simulated in the following way.', 'cond-mat-0106236-1-3-8': 'First, the system freely relaxes to a metastable state ([MATH]) following the dynamics at zero temperature, in which the processes decreasing the density of particles are forbidden and any configuration with all the holes being isolated is an absorbent state of the dynamics [CITATION].', 'cond-mat-0106236-1-3-9': 'Then, the system is tapped, allowing the density to decrease.', 'cond-mat-0106236-1-3-10': 'Afterwards, a new free relaxation at zero temperature is done, reaching another metastable state ([MATH]).', 'cond-mat-0106236-1-3-11': 'By repeating this process, a chain ([MATH]) of metastable configurations is obtained, with the density of the system increasing as a function of [MATH].', 'cond-mat-0106236-1-4-0': 'An analogous approach to the problem of granular compaction has been carried out by Lefevre and Dean [CITATION].', 'cond-mat-0106236-1-4-1': 'Their model is again the one-dimensional Ising system with nearest neighbour interactions.', 'cond-mat-0106236-1-4-2': 'Nevertheless, the usual Glauber dynamics [CITATION] is not a good choice for simulating tapping processes.', 'cond-mat-0106236-1-4-3': 'This is because there are no metastable states, and at zero temperature the system always reaches the perfectly ordered ferromagnetic phase.', 'cond-mat-0106236-1-4-4': 'Then, other zero temperature single spin flip dynamics was considered, in which only the elementary events lowering the energy are possible.', 'cond-mat-0106236-1-4-5': 'In this way, all the states composed of domains of [MATH] parallel spins with length [MATH] are metastable, i. e., they are absorbent states for this dynamics.', 'cond-mat-0106236-1-4-6': 'In such states, there is no spin antiparallel to both of its nearest neighbours, which are the only spins being able to flip.', 'cond-mat-0106236-1-4-7': 'With this falling dynamics, the tapping process is simulated in the same way as described above.', 'cond-mat-0106236-1-5-0': 'In ref. [CITATION], the authors claimed that the new zero temperature dynamics does not seem amenable to analytic solution.', 'cond-mat-0106236-1-5-1': 'In fact, the usual way of solving the Glauber model, by constructing the hierarchy of equations for the two-spin moments, does not work.', 'cond-mat-0106236-1-5-2': 'The aim of this paper is to show that it is possible to solve analytically the time evolution of the model by going to an equivalent particle-hole description of the Ising system.', 'cond-mat-0106236-1-5-3': 'The holes are associated to the domain walls separating arrays of parallel spins (particles).', 'cond-mat-0106236-1-5-4': 'Within this picture, the metastable states are those with all the holes being isolated, as in the one-dimensional facilitated Ising model.', 'cond-mat-0106236-1-6-0': 'The paper is organized as follows.', 'cond-mat-0106236-1-6-1': 'In section [REF] the Ising model is introduced, as well as the equivalent particle-hole description.', 'cond-mat-0106236-1-6-2': 'Section [REF] is devoted to the analytical solution of the dynamics.', 'cond-mat-0106236-1-6-3': 'A closed hierarchy of equations is derived for the probabilities [MATH] of finding [MATH] consecutive holes.', 'cond-mat-0106236-1-6-4': 'The solution of this hierarchy is obtained by means of a generating function method.', 'cond-mat-0106236-1-6-5': 'In particular, the asymptotic density of holes and energy in the metastable state are exactly calculated.', 'cond-mat-0106236-1-6-6': 'The last section contains some final remarks.', 'cond-mat-0106236-1-7-0': '# The model', 'cond-mat-0106236-1-8-0': 'We consider the one-dimensional Ising model with nearest neighbour interactions and periodic boundary conditions.', 'cond-mat-0106236-1-8-1': 'The hamiltonian of the system is [EQUATION] where [MATH] is the coupling constant, having dimensions of energy, [MATH] is the number of spins, and [MATH].', 'cond-mat-0106236-1-8-2': 'The time evolution of the system is described by a Markov process with single spin flip dynamics.', 'cond-mat-0106236-1-8-3': 'Then, the probability [MATH] of finding the system in configuration [MATH] at time [MATH] obeys the master equation [EQUATION]', 'cond-mat-0106236-1-8-4': 'Here [MATH] is the configuration obtained from [MATH] by flipping the [MATH]-th spin, and [MATH] is the transition rate for that process.', 'cond-mat-0106236-1-8-5': 'Following Lefevre and Dean [CITATION], we consider a zero temperature dynamics in which only the spin flips lowering the energy are permitted.', 'cond-mat-0106236-1-8-6': 'Namely, we take [EQUATION] i. e., the transition rate equals [MATH] if the [MATH]-th spin is antiparallel to both of its nearest neighbours, and vanishes otherwise.', 'cond-mat-0106236-1-8-7': 'While an exact solution of the usual Glauber dynamics can be found for all temperatures from the hierarchy of equations for the two-spin moments [CITATION], the same procedure applied to this "falling" dynamics leads to a non-closed set of equations, since more complex moments, involving three and four spins, appear in them.', 'cond-mat-0106236-1-9-0': 'For the above reason, it is convenient to go to an equivalent particle-hole description of the Ising model.', 'cond-mat-0106236-1-9-1': 'For each site [MATH], we define a new variable [EQUATION]', 'cond-mat-0106236-1-9-2': 'When [MATH] we will say that site [MATH] is occupied by a hole, while if [MATH] we will refer to site [MATH] as being occupied by a particle.', 'cond-mat-0106236-1-9-3': 'Thus, there is a hole at site [MATH] if the spins [MATH] and [MATH] are antiparallel, while a particle corresponds to spins [MATH] and [MATH] being parallel.', 'cond-mat-0106236-1-9-4': 'It follows that holes are associated to the domain walls separating arrays of parallel spins (particles).', 'cond-mat-0106236-1-9-5': 'It is important to note that the number of holes is even for any configuration with periodic boundary conditions.', 'cond-mat-0106236-1-9-6': 'In terms of the new variables, the hamiltonian of the system reads [EQUATION]', 'cond-mat-0106236-1-9-7': 'We define the average dimensionless energy per spin [MATH] as [EQUATION] where the angular brackets [MATH] denote average with [MATH].', 'cond-mat-0106236-1-10-0': 'In the particle-hole description, the elementary events involve two adjacent sites.', 'cond-mat-0106236-1-10-1': 'When the [MATH]-th spin flips, both [MATH] and [MATH] change their state.', 'cond-mat-0106236-1-10-2': 'From Eq. ([REF]), the transition rate [MATH] for this process is [EQUATION]', 'cond-mat-0106236-1-10-3': 'As [MATH] only determines the arbitrary time scale, we will set [MATH] in the following.', 'cond-mat-0106236-1-10-4': 'The master equation for the probability [MATH] of finding the system in configuration [MATH] at time [MATH] reads [EQUATION]', 'cond-mat-0106236-1-10-5': 'In the dynamics given by the transition rates in Eq. ([REF]), only two nearest neighbour holes can turn into two particles.', 'cond-mat-0106236-1-10-6': 'Therefore, it is clear that, after a long enough time period, the system will become stuck in a "metastable" state with all the holes being isolated, i. e., surrounded by two particles.', 'cond-mat-0106236-1-10-7': 'Of course, the reached metastable state will depend on the initial configuration.', 'cond-mat-0106236-1-10-8': 'This behaviour is reminiscent of the one showed by the 1SFM at [MATH].', 'cond-mat-0106236-1-10-9': 'In the latter, the system also reaches a metastable state with all the holes isolated, the specific final state depending on the initial condition [CITATION].', 'cond-mat-0106236-1-10-10': 'Nevertheless, the model considered here and the 1SFM at zero temperature are not equivalent.', 'cond-mat-0106236-1-10-11': 'The elementary dynamical events occurring in each of them are different.', 'cond-mat-0106236-1-10-12': 'While a particle can be adsorbed on any empty site with at least one nearest neighbour hole in the 1SFM, in the present model two particles must be adsorbed simultaneously on two adjacent empty sites of the one-dimensional lattice.', 'cond-mat-0106236-1-11-0': '# Analytical solution of the dynamics', 'cond-mat-0106236-1-12-0': 'In order to analyze the dynamics of the model we focus on the set of moments [EQUATION] with [MATH].', 'cond-mat-0106236-1-12-1': 'The local density of holes is given by the first term of this hierarchy, [EQUATION] and it is directly related with the energy of the system.', 'cond-mat-0106236-1-12-2': 'From Eq. ([REF]), it is [EQUATION]', 'cond-mat-0106236-1-12-3': 'Note that we are restricting ourselves to spatially homogeneous situations, so these moments [MATH] do not depend on their first site [MATH].', 'cond-mat-0106236-1-12-4': 'From its own definition, the moment [MATH] gives the probability of finding [MATH] consecutive holes starting from a given arbitrary site of the lattice.', 'cond-mat-0106236-1-12-5': 'Using the master equation ([REF]) with the transition rates ([REF]) one gets [EQUATION] for all [MATH].', 'cond-mat-0106236-1-12-6': 'This hierarchy can be solved by introducing the generating function [EQUATION] from which all the moments [MATH] are easily obtained, [EQUATION]', 'cond-mat-0106236-1-12-7': 'The hierarchy of equations ([REF]) is equivalent to the following first order partial differential equation for the generating function: [EQUATION] which has to be solved with the initial condition [EQUATION]', 'cond-mat-0106236-1-12-8': 'By using standard techniques it is easily obtained [EQUATION]', 'cond-mat-0106236-1-12-9': 'For large times the solution approaches the limit [EQUATION] and taking into account Eq. ([REF]), [EQUATION]', 'cond-mat-0106236-1-12-10': 'This shows that all the holes are isolated in the metastable state reached by the system in the long time limit.', 'cond-mat-0106236-1-12-11': 'The probability of finding [MATH] consecutive holes, which is equal to [MATH], vanishes for [MATH].', 'cond-mat-0106236-1-12-12': 'On the other hand, the asymptotic value of the density of holes [MATH] does depend on the initial state, being always smaller than its initial value.', 'cond-mat-0106236-1-12-13': 'In this sense, it is worth noting that the hierarchy of equations ([REF]) has as a stationary solution any constant value of [MATH], and [MATH] for [MATH].', 'cond-mat-0106236-1-13-0': 'Now, we are going to particularize the above results for a specific initial condition.', 'cond-mat-0106236-1-13-1': 'Let us consider that the one-dimensional Ising model is initially at equilibrium at a certain temperature [MATH].', 'cond-mat-0106236-1-13-2': "Then, the probability that spins [MATH] and [MATH] are antiparallel is given by a function of the temperature [MATH], [EQUATION] where [MATH] is Boltzmann's constant.", 'cond-mat-0106236-1-13-3': 'The value [MATH] corresponds to [MATH], the system is at [MATH] in a completely random configuration.', 'cond-mat-0106236-1-13-4': 'From the form of the hamiltonian in the particle-hole description, Eq. ([REF]), it follows that at equilibrium the [MATH] variables are statistically independent, and [EQUATION]', 'cond-mat-0106236-1-13-5': 'By using Eq. ([REF]), one gets the initial condition for the generating function, [EQUATION] and Eq. ([REF]) yields [EQUATION]', 'cond-mat-0106236-1-13-6': 'Now the expressions for the moments [MATH] can be directly calculated from Eq. ([REF]), [EQUATION]', 'cond-mat-0106236-1-13-7': 'As discussed in the previous paragraph, all the moments [MATH] with [MATH] vanish in the long time limit.', 'cond-mat-0106236-1-13-8': 'Moreover, the asymptotic density of holes is [EQUATION] which depends on the initial temperature of the system.', 'cond-mat-0106236-1-13-9': 'Equivalently, from Eq. ([REF]) the specific dimensionless energy [MATH] in the metastable state reads [EQUATION]', 'cond-mat-0106236-1-13-10': 'The mean field calculation presented by Lefevre and Dean [CITATION] leads to the same expression.', 'cond-mat-0106236-1-13-11': 'The asymptotic energy [MATH] is maximal for [MATH], i. e., when the system starts from the completely random configuration.', 'cond-mat-0106236-1-13-12': 'This is in contrast with the behaviour found for the 1SFM.', 'cond-mat-0106236-1-13-13': 'In the latter, if the initial condition is taken as the equilibrium state for a given value of the temperature [MATH], the asymptotic density of holes is a monotonic function of the initial density of holes [CITATION].', 'cond-mat-0106236-1-14-0': '# Final remarks', 'cond-mat-0106236-1-15-0': 'In this work we have studied the time evolution of the one-dimensional Ising model with the zero temperature dynamics recently considered by Lefevre and Dean [CITATION].', 'cond-mat-0106236-1-15-1': 'In this dynamics only transitions decreasing the energy of the system are allowed.', 'cond-mat-0106236-1-15-2': 'The difference with the usual zero temperature Glauber dynamics [CITATION] is that flips of spins parallel to one of its neighbours and antiparallel to the other one are forbidden.', 'cond-mat-0106236-1-15-3': 'In other words, the possibility of a diffusive motion of the domain walls is eliminated.', 'cond-mat-0106236-1-16-0': 'When trying to compute analytically the evolution of the system, the usual procedure of constructing the hierarchy of differential equations for the two spin moments is not useful.', 'cond-mat-0106236-1-16-1': 'We have introduced a particle-hole description of the Ising model, in which the holes correspond to the domain walls (or "defects") between arrays of parallel spins.', 'cond-mat-0106236-1-16-2': 'The energy of the system is directly related to the density of holes.', 'cond-mat-0106236-1-16-3': 'Within this description, the present model has some similarities with the one-dimensional facilitated Ising model at zero temperature.', 'cond-mat-0106236-1-16-4': 'Although the possible elementary dynamical events are different, both systems get eventually stuck in a metastable state characterized by all the holes being isolated, i. e., surrounded by two particles.', 'cond-mat-0106236-1-17-0': 'The particle-hole description allows us to solve analytically the time evolution of the system.', 'cond-mat-0106236-1-17-1': 'A closed hierarchy of equations can be written for the probability distribution functions [MATH] of finding [MATH] consecutive holes in the system.', 'cond-mat-0106236-1-17-2': 'The general solution of this hierarchy has been derived.', 'cond-mat-0106236-1-17-3': 'In the long time limit, the system reaches a state in which [MATH] for all [MATH], i. e., all the holes are isolated.', 'cond-mat-0106236-1-17-4': 'Moreover, the asymptotic value of the density of holes is not unique, but it depends on the initial configuration.', 'cond-mat-0106236-1-18-0': 'Another similarity between the present model and the 1SFM shows up in the context of granular media.', 'cond-mat-0106236-1-18-1': "When both models are applied to study tapping processes, the steady state reached by the system is consistent with Edward's thermodynamic theory [CITATION].", 'cond-mat-0106236-1-18-2': "This has been shown numerically in Lefevre and Dean's model [CITATION], and analytically for the 1SFM [CITATION].", 'cond-mat-0106236-1-18-3': 'In the latter, an effective dynamics for the tapping process has been derived.', 'cond-mat-0106236-1-18-4': "This allows to find explicitly the steady distribution function of the system, as well as the relationship between the vibration intensity and Edwards' compactivity.", 'cond-mat-0106236-1-18-5': 'Therefore, it seems worth trying to derive the corresponding effective dynamics for the tapping process in the model considered in this paper [CITATION].'} | {'cond-mat-0106236-2-0-0': 'The one-dimensional Ising model with nearest neighbour interactions and the zero-temperature dynamics recently considered by Lefevre and Dean -J. Phys.', 'cond-mat-0106236-2-0-1': 'A: Math.', 'cond-mat-0106236-2-0-2': 'Gen. 34, L213 (2001)- is investigated.', 'cond-mat-0106236-2-0-3': 'By introducing a particle-hole description, in which the holes are associated to the domain walls of the Ising model, an analytical solution is obtained.', 'cond-mat-0106236-2-0-4': 'The result for the asymptotic energy agrees with that found in the mean field approximation.', 'cond-mat-0106236-2-1-0': '# Introduction', 'cond-mat-0106236-2-2-0': 'Ising systems are often used as simple models in many different fields of statistical physics.', 'cond-mat-0106236-2-2-1': 'In particular, the one-dimensional Ising model with nearest neighbour interactions and Glauber dynamics [CITATION] is ubiquitous.', 'cond-mat-0106236-2-2-2': 'It shows glass-like behaviour, including non-exponential relaxation [CITATION], a laboratory glass transition in cooling processes [CITATION], hysteresis effects in thermal cycling experiments [CITATION], and aging effects at low temperatures [CITATION].', 'cond-mat-0106236-2-2-3': 'When a weak oscillating external field is applied, the amplitude of the induced magnetization presents a maximum as a function of the temperature, a behaviour resembling stochastic resonance [CITATION].', 'cond-mat-0106236-2-3-0': 'Other kind of important Ising systems are those with "facilitated" dynamics.', 'cond-mat-0106236-2-3-1': 'They were introduced in a pioneering work by Fredrickson and Andersen [CITATION], in the context of the relaxation of structural glasses.', 'cond-mat-0106236-2-3-2': 'In these models, a given spin can only flip if its nearest neighbours are in a certain subset of all their possible configurations.', 'cond-mat-0106236-2-3-3': 'The one-dimensional Ising model with facilitated dynamics (1SFM) has been extensively studied [CITATION].', 'cond-mat-0106236-2-3-4': 'Although, in general, it is not possible to find an analytical solution of these facilitated Ising models, the 1SFM has been exactly solved at zero temperature [CITATION].', 'cond-mat-0106236-2-3-5': 'Very recently, the 1SFM has been applied to study the compaction of vibrated granular systems.', 'cond-mat-0106236-2-3-6': 'A hole (particle) is associated to any spin in its excited (ground) state.', 'cond-mat-0106236-2-3-7': 'Trying to mimic what is done in real experiments [CITATION], the tapping process is simulated in the following way.', 'cond-mat-0106236-2-3-8': 'First, the system freely relaxes to a metastable state ([MATH]) following the dynamics at zero temperature, in which the processes decreasing the density of particles are forbidden and any configuration with all the holes being isolated is an absorbent state of the dynamics [CITATION].', 'cond-mat-0106236-2-3-9': 'Then, the system is tapped, allowing the density to decrease.', 'cond-mat-0106236-2-3-10': 'Afterwards, a new free relaxation at zero temperature is done, reaching another metastable state ([MATH]).', 'cond-mat-0106236-2-3-11': 'By repeating this process, a chain ([MATH]) of metastable configurations is obtained, with the density of the system increasing as a function of [MATH].', 'cond-mat-0106236-2-4-0': 'An analogous approach to the problem of granular compaction has been carried out by Lefevre and Dean [CITATION].', 'cond-mat-0106236-2-4-1': 'Their model is again the one-dimensional Ising system with nearest neighbour interactions.', 'cond-mat-0106236-2-4-2': 'Nevertheless, the usual Glauber dynamics [CITATION] is not a good choice for simulating tapping processes.', 'cond-mat-0106236-2-4-3': 'This is because there are no metastable states, and at zero temperature the system always reaches the perfectly ordered ferromagnetic phase.', 'cond-mat-0106236-2-4-4': 'Then, other zero temperature single spin flip dynamics was considered, in which only the elementary events lowering the energy are possible.', 'cond-mat-0106236-2-4-5': 'In this way, all the states composed of domains of [MATH] parallel spins with length [MATH] are metastable, i. e., they are absorbent states for this dynamics.', 'cond-mat-0106236-2-4-6': 'In such states, there is no spin antiparallel to both of its nearest neighbours, which are the only spins being able to flip.', 'cond-mat-0106236-2-4-7': 'With this falling dynamics, the tapping process is simulated in the same way as described above.', 'cond-mat-0106236-2-5-0': 'In ref. [CITATION], the authors claimed that the new zero temperature dynamics does not seem amenable to analytic solution.', 'cond-mat-0106236-2-5-1': 'In fact, the usual way of solving the Glauber model, by constructing the hierarchy of equations for the two-spin moments, does not work.', 'cond-mat-0106236-2-5-2': 'The aim of this paper is to show that it is possible to solve analytically the time evolution of the model by going to an equivalent particle-hole description of the Ising system.', 'cond-mat-0106236-2-5-3': 'The holes are associated to the domain walls separating arrays of parallel spins (particles).', 'cond-mat-0106236-2-5-4': 'Within this picture, the metastable states are those with all the holes being isolated, as in the one-dimensional facilitated Ising model.', 'cond-mat-0106236-2-6-0': 'The paper is organized as follows.', 'cond-mat-0106236-2-6-1': 'In section [REF] the Ising model is introduced, as well as the equivalent particle-hole description.', 'cond-mat-0106236-2-6-2': 'Section [REF] is devoted to the analytical solution of the dynamics.', 'cond-mat-0106236-2-6-3': 'A closed hierarchy of equations is derived for the probabilities [MATH] of finding [MATH] consecutive holes.', 'cond-mat-0106236-2-6-4': 'The solution of this hierarchy is obtained by means of a generating function method.', 'cond-mat-0106236-2-6-5': 'In particular, the asymptotic density of holes and energy in the metastable state are exactly calculated.', 'cond-mat-0106236-2-6-6': 'The last section contains some final remarks.', 'cond-mat-0106236-2-7-0': '# The model', 'cond-mat-0106236-2-8-0': 'We consider the one-dimensional Ising model with nearest neighbour interactions and periodic boundary conditions.', 'cond-mat-0106236-2-8-1': 'The hamiltonian of the system is [EQUATION] where [MATH] is the coupling constant, having dimensions of energy, [MATH] is the number of spins, and [MATH].', 'cond-mat-0106236-2-8-2': 'The time evolution of the system is described by a Markov process with single spin flip dynamics.', 'cond-mat-0106236-2-8-3': 'Then, the probability [MATH] of finding the system in configuration [MATH] at time [MATH] obeys the master equation [EQUATION]', 'cond-mat-0106236-2-8-4': 'Here [MATH] is the configuration obtained from [MATH] by flipping the [MATH]-th spin, and [MATH] is the transition rate for that process.', 'cond-mat-0106236-2-8-5': 'Following Lefevre and Dean [CITATION], we consider a zero temperature dynamics in which only the spin flips lowering the energy are permitted.', 'cond-mat-0106236-2-8-6': 'Namely, we take [EQUATION] i. e., the transition rate equals [MATH] if the [MATH]-th spin is antiparallel to both of its nearest neighbours, and vanishes otherwise.', 'cond-mat-0106236-2-8-7': 'While an exact solution of the usual Glauber dynamics can be found for all temperatures from the hierarchy of equations for the two-spin moments [CITATION], the same procedure applied to this "falling" dynamics leads to a non-closed set of equations, since more complex moments, involving three and four spins, appear in them.', 'cond-mat-0106236-2-9-0': 'For the above reason, it is convenient to go to an equivalent particle-hole description of the Ising model.', 'cond-mat-0106236-2-9-1': 'For each site [MATH], we define a new variable [EQUATION]', 'cond-mat-0106236-2-9-2': 'When [MATH] we will say that site [MATH] is occupied by a hole, while if [MATH] we will refer to site [MATH] as being occupied by a particle.', 'cond-mat-0106236-2-9-3': 'Thus, there is a hole at site [MATH] if the spins [MATH] and [MATH] are antiparallel, while a particle corresponds to spins [MATH] and [MATH] being parallel.', 'cond-mat-0106236-2-9-4': 'It follows that holes are associated to the domain walls separating arrays of parallel spins (particles).', 'cond-mat-0106236-2-9-5': 'It is important to note that the number of holes is even for any configuration with periodic boundary conditions.', 'cond-mat-0106236-2-9-6': 'In terms of the new variables, the hamiltonian of the system reads [EQUATION]', 'cond-mat-0106236-2-9-7': 'We define the average dimensionless energy per spin [MATH] as [EQUATION] where the angular brackets [MATH] denote average with [MATH].', 'cond-mat-0106236-2-10-0': 'In the particle-hole description, the elementary events involve two adjacent sites.', 'cond-mat-0106236-2-10-1': 'When the [MATH]-th spin flips, both [MATH] and [MATH] change their state.', 'cond-mat-0106236-2-10-2': 'From Eq. ([REF]), the transition rate [MATH] for this process is [EQUATION]', 'cond-mat-0106236-2-10-3': 'As [MATH] only determines the arbitrary time scale, we will set [MATH] in the following.', 'cond-mat-0106236-2-10-4': 'The master equation for the probability [MATH] of finding the system in configuration [MATH] at time [MATH] reads [EQUATION]', 'cond-mat-0106236-2-10-5': 'In the dynamics given by the transition rates in Eq. ([REF]), only two nearest neighbour holes can turn into two particles.', 'cond-mat-0106236-2-10-6': 'Therefore, it is clear that, after a long enough time period, the system will become stuck in a "metastable" state with all the holes being isolated, i. e., surrounded by two particles.', 'cond-mat-0106236-2-10-7': 'Of course, the reached metastable state will depend on the initial configuration.', 'cond-mat-0106236-2-10-8': 'This behaviour is reminiscent of the one showed by the 1SFM at [MATH].', 'cond-mat-0106236-2-10-9': 'In the latter, the system also reaches a metastable state with all the holes isolated, the specific final state depending on the initial condition [CITATION].', 'cond-mat-0106236-2-10-10': 'Nevertheless, the model considered here and the 1SFM at zero temperature are not equivalent.', 'cond-mat-0106236-2-10-11': 'The elementary dynamical events occurring in each of them are different.', 'cond-mat-0106236-2-10-12': 'While a particle can be adsorbed on any empty site with at least one nearest neighbour hole in the 1SFM, in the present model two particles must be adsorbed simultaneously on two adjacent empty sites of the one-dimensional lattice.', 'cond-mat-0106236-2-11-0': '# Analytical solution of the dynamics', 'cond-mat-0106236-2-12-0': 'In order to analyze the dynamics of the model we focus on the set of moments [EQUATION] with [MATH].', 'cond-mat-0106236-2-12-1': 'The local density of holes is given by the first term of this hierarchy, [EQUATION] and it is directly related with the energy of the system.', 'cond-mat-0106236-2-12-2': 'From Eq. ([REF]), it is [EQUATION]', 'cond-mat-0106236-2-12-3': 'Note that we are restricting ourselves to spatially homogeneous situations, so these moments [MATH] do not depend on their first site [MATH].', 'cond-mat-0106236-2-12-4': 'From its own definition, the moment [MATH] gives the probability of finding [MATH] consecutive holes starting from a given arbitrary site of the lattice.', 'cond-mat-0106236-2-12-5': 'Using the master equation ([REF]) with the transition rates ([REF]) one gets [EQUATION] for all [MATH].', 'cond-mat-0106236-2-12-6': 'This hierarchy can be solved by introducing the generating function [EQUATION] from which all the moments [MATH] are easily obtained, [EQUATION]', 'cond-mat-0106236-2-12-7': 'The hierarchy of equations ([REF]) is equivalent to the following first order partial differential equation for the generating function: [EQUATION] which has to be solved with the initial condition [EQUATION]', 'cond-mat-0106236-2-12-8': 'By using standard techniques it is easily obtained [EQUATION]', 'cond-mat-0106236-2-12-9': 'For large times the solution approaches the limit [EQUATION] and taking into account Eq. ([REF]), [EQUATION]', 'cond-mat-0106236-2-12-10': 'This shows that all the holes are isolated in the metastable state reached by the system in the long time limit.', 'cond-mat-0106236-2-12-11': 'The probability of finding [MATH] consecutive holes, which is equal to [MATH], vanishes for [MATH].', 'cond-mat-0106236-2-12-12': 'On the other hand, the asymptotic value of the density of holes [MATH] does depend on the initial state, being always smaller than its initial value.', 'cond-mat-0106236-2-12-13': 'In this sense, it is worth noting that the hierarchy of equations ([REF]) has as a stationary solution any constant value of [MATH], and [MATH] for [MATH].', 'cond-mat-0106236-2-13-0': 'Now, we are going to particularize the above results for a specific initial condition.', 'cond-mat-0106236-2-13-1': 'Let us consider that the one-dimensional Ising model is initially at equilibrium at a certain temperature [MATH].', 'cond-mat-0106236-2-13-2': "Then, the probability that spins [MATH] and [MATH] are antiparallel is given by a function of the temperature [MATH], [EQUATION] where [MATH] is Boltzmann's constant.", 'cond-mat-0106236-2-13-3': 'The value [MATH] corresponds to [MATH], the system is at [MATH] in a completely random configuration.', 'cond-mat-0106236-2-13-4': 'From the form of the hamiltonian in the particle-hole description, Eq. ([REF]), it follows that at equilibrium the [MATH] variables are statistically independent, and [EQUATION]', 'cond-mat-0106236-2-13-5': 'By using Eq. ([REF]), one gets the initial condition for the generating function, [EQUATION] and Eq. ([REF]) yields [EQUATION]', 'cond-mat-0106236-2-13-6': 'Now the expressions for the moments [MATH] can be directly calculated from Eq. ([REF]), [EQUATION]', 'cond-mat-0106236-2-13-7': 'As discussed in the previous paragraph, all the moments [MATH] with [MATH] vanish in the long time limit.', 'cond-mat-0106236-2-13-8': 'Moreover, the asymptotic density of holes is [EQUATION] which depends on the initial temperature of the system.', 'cond-mat-0106236-2-13-9': 'Equivalently, from Eq. ([REF]) the specific dimensionless energy [MATH] in the metastable state reads [EQUATION]', 'cond-mat-0106236-2-13-10': 'The mean field calculation presented by Lefevre and Dean [CITATION] leads to the same expression (see also [CITATION]).', 'cond-mat-0106236-2-13-11': 'The asymptotic energy [MATH] is maximal for [MATH], i. e., when the system starts from the completely random configuration.', 'cond-mat-0106236-2-13-12': 'This is in contrast with the behaviour found for the 1SFM.', 'cond-mat-0106236-2-13-13': 'In the latter, if the initial condition is taken as the equilibrium state for a given value of the temperature [MATH], the asymptotic density of holes is a monotonic function of the initial density of holes [CITATION].', 'cond-mat-0106236-2-14-0': '# Final remarks', 'cond-mat-0106236-2-15-0': 'In this work we have studied the time evolution of the one-dimensional Ising model with the zero temperature dynamics recently considered by Lefevre and Dean [CITATION].', 'cond-mat-0106236-2-15-1': 'In this dynamics only transitions decreasing the energy of the system are allowed.', 'cond-mat-0106236-2-15-2': 'The difference with the usual zero temperature Glauber dynamics [CITATION] is that flips of spins parallel to one of its neighbours and antiparallel to the other one are forbidden.', 'cond-mat-0106236-2-15-3': 'In other words, the possibility of a diffusive motion of the domain walls is eliminated.', 'cond-mat-0106236-2-16-0': 'When trying to compute analytically the evolution of the system, the usual procedure of constructing the hierarchy of differential equations for the two spin moments is not useful.', 'cond-mat-0106236-2-16-1': 'We have introduced a particle-hole description of the Ising model, in which the holes correspond to the domain walls (or "defects") between arrays of parallel spins.', 'cond-mat-0106236-2-16-2': 'The energy of the system is directly related to the density of holes.', 'cond-mat-0106236-2-16-3': 'Within this description, the present model has some similarities with the one-dimensional facilitated Ising model at zero temperature.', 'cond-mat-0106236-2-16-4': 'Although the possible elementary dynamical events are different, both systems get eventually stuck in a metastable state characterized by all the holes being isolated, i. e., surrounded by two particles.', 'cond-mat-0106236-2-17-0': 'The particle-hole description allows us to solve analytically the time evolution of the system.', 'cond-mat-0106236-2-17-1': 'A closed hierarchy of equations can be written for the probability distribution functions [MATH] of finding [MATH] consecutive holes in the system.', 'cond-mat-0106236-2-17-2': 'The general solution of this hierarchy has been derived.', 'cond-mat-0106236-2-17-3': 'In the long time limit, the system reaches a state in which [MATH] for all [MATH], i. e., all the holes are isolated.', 'cond-mat-0106236-2-17-4': 'Moreover, the asymptotic value of the density of holes is not unique, but it depends on the initial configuration.', 'cond-mat-0106236-2-18-0': 'Another similarity between the present model and the 1SFM shows up in the context of granular media.', 'cond-mat-0106236-2-18-1': "When both models are applied to study tapping processes, the steady state reached by the system is consistent with Edward's thermodynamic theory [CITATION].", 'cond-mat-0106236-2-18-2': "This has been shown numerically in Lefevre and Dean's model [CITATION], and analytically for the 1SFM [CITATION].", 'cond-mat-0106236-2-18-3': 'In the latter, an effective dynamics for the tapping process has been derived.', 'cond-mat-0106236-2-18-4': "This allows to find explicitly the steady distribution function of the system, as well as the relationship between the vibration intensity and Edwards' compactivity.", 'cond-mat-0106236-2-18-5': 'Therefore, it seems worth trying to derive the corresponding effective dynamics for the tapping process in the model considered in this paper [CITATION].'} | 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'cond-mat-0106236-2-0-2', 'cond-mat-0106236-2-12-2'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/cond-mat/0106236 | null | null | null | null | null |
hep-ph-9911337 | {'hep-ph-9911337-1-0-0': 'We address the question of the relation between supersymmetry breaking and the origin of flavor in the context of CP violating phenomena.', 'hep-ph-9911337-1-0-1': 'We prove that, in the absence of the Cabibbo-Kobayashi-Maskawa phase, a general Minimal Supersymmetric Standard Model with all possible phases in the soft-breaking terms, but no new flavor structure beyond the usual Yukawa matrices, can never give a sizeable contribution to [MATH], [MATH] or hadronic [MATH] CP asymmetries.', 'hep-ph-9911337-1-0-2': 'Observation of supersymmetric contributions to CP asymmetries in B decays would hint at a non-flavor blind mechanism of supersymmetry breaking.', 'hep-ph-9911337-1-1-0': 'In the near future, new experimental information on CP violation will be available.', 'hep-ph-9911337-1-1-1': 'Not only the new [MATH] factories will start measuring CP violation effects in [MATH] CP asymmetries, but also the experimental sensitivity to the electric dipole moment (EDM) of the neutron and the electron will be substantially improved.', 'hep-ph-9911337-1-1-2': 'These experiments may provide the first sign of physics beyond the Standard Model.', 'hep-ph-9911337-1-2-0': 'If new results do appear and we interpret them in the context of Supersymmetry, both experiments have very different implications on the structure of the soft-breaking terms at the supersymmetry breaking scale.', 'hep-ph-9911337-1-2-1': 'The finding of a non-zero EDM for the neutron would simply indicate the presence of new non-negligible flavor independent susy phases [CITATION].', 'hep-ph-9911337-1-2-2': 'However, a new result in the non-leptonic [MATH] CP asymmetries would be a direct prove of the existence of a completely new flavor structure in the soft-breaking terms.', 'hep-ph-9911337-1-2-3': 'We can rephrase this sentence in the form of a strict no-go theorem: "In the absence of the Cabibbo-Kobayashi-Maskawa (CKM) phase, a general Minimal Supersymmetric Standard Model (MSSM) with possible phases in the soft-breaking terms, but no new flavor structure beyond the usual Yukawa matrices, can never give a sizeable contribution to [MATH], [MATH] or hadronic [MATH] CP asymmetries".', 'hep-ph-9911337-1-3-0': 'Let us first analyze in more detail this strong statement.', 'hep-ph-9911337-1-3-1': 'Indeed, we are going to show that these contributions are at least two orders of magnitude smaller that the required experimental values of [MATH], [MATH], or, in the case of [MATH] CP asymmetries, the expected experimental sensitivity.', 'hep-ph-9911337-1-3-2': 'Moreover, we always take a vanishing phase in the CKM matrix, i.e. [MATH], as a way to isolate the effects of the new supersymmetric phases.', 'hep-ph-9911337-1-3-3': 'We do not include in this no-go theorem other CP violation experiments in rare [MATH] decays, as for instance [MATH], where the contribution from chirality changing operators is important (see discussion below).', 'hep-ph-9911337-1-3-4': 'This "theorem" applies to any MSSM, i.e. with the minimal supersymmetric particle content, and general complex soft-breaking terms, but with a flavor structure strictly given by the two familiar Yukawa matrices or any matrix strictly proportional to them.', 'hep-ph-9911337-1-3-5': 'In these conditions the most general allowed structure of the soft-breaking terms at the large scale, that we call [MATH], is, [EQUATION] where all the allowed phases are explicitly written and one of them can be removed by an R-rotation.', 'hep-ph-9911337-1-3-6': 'All other numbers or matrices in this equation are always real.', 'hep-ph-9911337-1-3-7': 'Notice that this structure covers, not only the Constrained MSSM (CMSSM) [CITATION], but also most of Type I string motivated models considered so far [CITATION], gauge mediated models [CITATION], minimal effective supersymmetry models [CITATION], etc.', 'hep-ph-9911337-1-3-8': 'However, as recently emphasized [CITATION], as soon as one introduces some new flavor structure in the soft Susy-breaking sector, even if the CP violating phases are flavor independent, it is indeed possible to get sizeable CP contribution for large Susy phases and [MATH].', 'hep-ph-9911337-1-4-0': 'Experiments of CP violation in the [MATH] or [MATH] systems only involve supersymmetric particles as virtual particles in the loops.', 'hep-ph-9911337-1-4-1': 'This means that the phases in the soft-breaking terms can only appear though the mass matrices of the susy particles.', 'hep-ph-9911337-1-5-0': 'The key point in our discussion is the absence of any new flavor structure, and its role in the low-energy sparticle mass matrices.', 'hep-ph-9911337-1-5-1': 'Once you have any susy phase that can generate CP violation effects the flavor-change will be necessarily given by a product of Yukawa elements.', 'hep-ph-9911337-1-5-2': 'This fact is completely independent of the presence of only one phase or the 5 phases in Eq. ([REF]) plus the additional [MATH] phase.', 'hep-ph-9911337-1-5-3': 'It is well-known that the Yukawa Renormalization Group Evolution (RGE) is completely independent of all soft-breaking terms [CITATION].', 'hep-ph-9911337-1-5-4': 'In fact, we can solve the Yukawa RGEs for a given value of [MATH] independently of all soft-breaking terms, and the size of all Yukawa matrix elements does not change more than a factor [MATH] from the electroweak scale to the string or susy breaking scale.', 'hep-ph-9911337-1-5-5': 'Then, a typical estimate for the element [MATH] in the [MATH]-[MATH] down squark mass matrix at the electroweak scale would necessarily be (see [CITATION] for details), [EQUATION]', 'hep-ph-9911337-1-5-6': 'The presence of imaginary parts is a slightly more delicate issue, though, in any case Eq. ([REF]) will always be an approximate upper bound.', 'hep-ph-9911337-1-5-7': 'As explained in [CITATION], the RGE equations of all soft-breaking terms are a set of linear differential equations, and thus can be solved as a linear function of the initial conditions, [EQUATION] where [MATH] refers to any scalar, [MATH] to the different gauginos, [MATH] to any tri-linear coupling and the different [MATH] matrices are [MATH] matrices, strictly real.', 'hep-ph-9911337-1-5-8': 'In this equation all the allowed phases have been explicitly written.', 'hep-ph-9911337-1-5-9': 'Regarding the imaginary parts, we can see from Eq. ([REF]) that any imaginary part will always be associated to the non-symmetric part of the [MATH], [MATH] or [MATH] matrices independently of the presence of a single phase or an arbitrary number of them in the initial conditions.', 'hep-ph-9911337-1-5-10': 'This is always true in our general framework, and hence the need of large non-symmetric parts in these matrices on the top of large phases is very clear.', 'hep-ph-9911337-1-5-11': 'To estimate the size of these anti-symmetric parts, we can go to the RGE equations for the scalar mass matrices, where we use the same conventions and notation as in [CITATION].', 'hep-ph-9911337-1-5-12': 'Taking advantage of the linearity of these equations we can directly write the evolution of the anti-symmetric parts, [MATH] as, [EQUATION] where, due to the reality of Yukawa matrices, we have used [MATH], and following [CITATION] a tilde over the couplings ([MATH], [MATH], ...) denotes a re-scaling by a factor [MATH].', 'hep-ph-9911337-1-5-13': 'In the evolution of the [MATH]-[MATH] squark mass matrices, [MATH] and [MATH], only one of the two Yukawa matrices, the one with equal isospin to the squarks, is directly involved.', 'hep-ph-9911337-1-5-14': 'Then, it is easy to understand that these matrices are in a very good approximation diagonal in the SCKM basis once you start with the initial conditions given in Eq. ([REF]).', 'hep-ph-9911337-1-5-15': 'Hence, we can safely neglect the last two terms in Eq. ([REF]) and forget about [MATH] and [MATH].', 'hep-ph-9911337-1-5-16': 'From Eq. ([REF]), the initial conditions for these anti-symmetric parts at [MATH] are identically zero.', 'hep-ph-9911337-1-5-17': 'So, the only source for [MATH] is necessarily [MATH].', 'hep-ph-9911337-1-5-18': 'Now, we can analyze the RGE for [MATH], [EQUATION] with an equivalent equation for [MATH].', 'hep-ph-9911337-1-5-19': 'It is clear that given the general initial conditions in Eq. ([REF]), [MATH] is complex at any scale.', 'hep-ph-9911337-1-5-20': 'However, we are interested in the imaginary parts of [MATH].', 'hep-ph-9911337-1-5-21': 'At [MATH] this combination is exactly real, but this is not true any more at a different scale.', 'hep-ph-9911337-1-5-22': 'From Eq. ([REF]), we can immediately see that these imaginary parts are extremely small.', 'hep-ph-9911337-1-5-23': 'Let us, for a moment, neglect the terms involving [MATH] or [MATH] from the above equation.', 'hep-ph-9911337-1-5-24': 'Then, the only flavor structure appearing in Eq. ([REF]) at [MATH] is [MATH].', 'hep-ph-9911337-1-5-25': 'We can always go to the basis where [MATH] is diagonal and then we will have [MATH] exactly diagonal at any scale.', 'hep-ph-9911337-1-5-26': 'In particular this means that [MATH] would always exactly vanish.', 'hep-ph-9911337-1-5-27': 'The same reasoning applies to [MATH] and [MATH].', 'hep-ph-9911337-1-5-28': 'Hence, simply taking into account the flavor structure, our conclusion is that any non-vanishing element of [MATH] and hence of [MATH] must be necessarily proportional to [MATH].', 'hep-ph-9911337-1-5-29': 'So, we can expect them to be, [EQUATION] where [MATH], with [MATH] the vacuum expectation value of the Higgs, [MATH] and [MATH] is a proportionality constant that includes the effects of the running from [MATH] to [MATH].', 'hep-ph-9911337-1-5-30': 'To estimate this constant we have to keep in mind that the imaginary parts of [MATH] are generated through the RGE running and then these imaginary parts generate [MATH] as a second order effect.', 'hep-ph-9911337-1-5-31': 'This means that roughly [MATH] times a combination of initial conditions as in Eq. ([REF]).', 'hep-ph-9911337-1-5-32': 'So, we estimate these matrix elements to be [MATH] times initial conditions.', 'hep-ph-9911337-1-5-33': 'This was exactly the result we found for the [MATH]-[MATH] terms in [CITATION].', 'hep-ph-9911337-1-5-34': 'In fact, now it is clear that this is the same for all the terms in Eq. ([REF]), [MATH]-[MATH], [MATH]-[MATH] and [MATH]-[MATH], irrespectively of the presence of an arbitrary number of new phases.', 'hep-ph-9911337-1-6-0': 'As we have already said before, the situation in the [MATH]-[MATH] matrices is clearly worse because the RGE of these matrices involves only the corresponding Yukawa matrix and hence, in the SCKM, they are always diagonal and real in extremely good approximation.', 'hep-ph-9911337-1-7-0': 'Hence, so far, we have shown that the [MATH]-[MATH] or [MATH]-[MATH] squark mass matrices are still essentially real.', 'hep-ph-9911337-1-8-0': 'The only complex matrices, then, will still be the [MATH]-[MATH] matrices that include, from the very beginning, the phases [MATH] and [MATH].', 'hep-ph-9911337-1-8-1': 'Once more, the size of these entries is determined by the Yukawa elements with these two phases providing the complex structure.', 'hep-ph-9911337-1-8-2': 'However, this situation is not new for these more general MSSM models and it was already present even in the CMSSM.', 'hep-ph-9911337-1-9-0': 'From here we can start the analysis of the effects of supersymmetric phases in the CP observables.', 'hep-ph-9911337-1-9-1': 'We have already seen that the structure of the sfermion mass matrices remains the same as in the CMSSM case.', 'hep-ph-9911337-1-9-2': 'This is simply due to our dependence to the Yukawa matrices to get any flavor change.', 'hep-ph-9911337-1-9-3': 'On the other hand, the new gaugino phases enter the chargino and neutralino mass matrices.', 'hep-ph-9911337-1-9-4': 'However, in all our previous works [CITATION] we have always ignored the EDM bounds, which means that [MATH] could take any value and large phases in the mixing matrices were already present.', 'hep-ph-9911337-1-9-5': 'So, the inclusion of the new gaugino phases does not lead to new effects apart from those already accounted for varying [MATH].', 'hep-ph-9911337-1-10-0': 'In first place, we will consider indirect CP violation both in the [MATH] and [MATH] systems, refering to [CITATION] for a complete analysis.', 'hep-ph-9911337-1-10-1': 'In the case of the gluino or neutralino, it is well-known that the CMSSM satisfies widely all the constraints imposed by flavor changing experiments [CITATION].', 'hep-ph-9911337-1-10-2': 'Hence, this still holds true in this more general case, where we have shown that the sfermion mass matrices are still of the same size as in the CMSSM.', 'hep-ph-9911337-1-10-3': 'This means then, that all possible mass insertions are always roughly two orders of magnitude bellow the required values to saturate flavor changing observables, (see second part of Ref [CITATION]).', 'hep-ph-9911337-1-10-4': 'Notice that this is true even for CP conserving flavor changing observables and the situation for the CP violating observables with chirality conserving operators, Eq. ([REF]), is still much worse.', 'hep-ph-9911337-1-10-5': 'Also chargino contributions can be comparable in general.', 'hep-ph-9911337-1-10-6': 'This was the main subject of paper [CITATION] where we showed the different constraints in the chirality conserving, [MATH]-[MATH], and chirality changing, [MATH]-[MATH], transitions.', 'hep-ph-9911337-1-10-7': 'From [CITATION] it is clear that chargino chirality changing transitions are directly constrained by the [MATH] decay to be more that three orders of magnitude smaller than the corresponding chirality conserving transitions.', 'hep-ph-9911337-1-10-8': 'And finally, on the other side, we already showed in [CITATION] that chirality conserving transitions were real to a very good approximation.', 'hep-ph-9911337-1-10-9': 'These arguments allow us to discard measurable CP violation in both [MATH] and indirect CP violation in the [MATH] system.', 'hep-ph-9911337-1-11-0': 'Finally we have to consider also direct CP violation in non-leptonic [MATH] decays.', 'hep-ph-9911337-1-11-1': 'Essentially, the only difference with our discussion on indirect CP violation is the presence of the penguins.', 'hep-ph-9911337-1-11-2': 'Once more, in the gluino case chirality conserving transitions are real to a very good approximation, and, in any case, well below the phenomenological bounds [CITATION].', 'hep-ph-9911337-1-11-3': 'The chirality changing transitions on the other hand are suppressed by light quark masses, where we call light even the [MATH] quark, and again below the bounds.', 'hep-ph-9911337-1-11-4': 'Hence, our conclusion for the gluino is necessarily the same.', 'hep-ph-9911337-1-11-5': 'So, we are left with chargino.', 'hep-ph-9911337-1-11-6': '[MATH]-[MATH] transitions are real to a very good approximation, for the very same reasons used in the indirect CP violation case.', 'hep-ph-9911337-1-11-7': 'And now the relation of [MATH] with the chirality changing penguins is even more transparent if possible.', 'hep-ph-9911337-1-11-8': 'This completes the proof of our Theorem.', 'hep-ph-9911337-1-12-0': 'To conclude we would like to discuss the implications of our result in the search for supersymmetric CP violation.', 'hep-ph-9911337-1-12-1': 'In the presence of large supersymmetric phases [CITATION], the EDMs of the electron and the neutron must be very close to the experimental bounds.', 'hep-ph-9911337-1-12-2': 'However, as we have shown in this letter, the presence of these phases is not enough to generate a sizeable contribution to [MATH], [MATH] or [MATH] CP asymmetries.', 'hep-ph-9911337-1-12-3': 'Here a completely new flavor structure in the soft breaking terms is required to get sizeables effects.', 'hep-ph-9911337-1-12-4': 'In this sense, CP experiments in a supersymmetric theory are a direct probe on any additional flavor structure in the soft-breaking terms.', 'hep-ph-9911337-1-13-0': 'Hence, in the absence of new flavor structures, only pure chirality changing observables (EDMs or [MATH]) or observables where, in any case, the chirality flip operators are relevant ([MATH], [MATH]), can show the effects of new supersymmetric phases [CITATION].', 'hep-ph-9911337-1-14-0': 'We thank S. Bertolini, T. Kobayashi and S. Khalil for enlightening discussions.', 'hep-ph-9911337-1-14-1': 'The work of A.M. was partially supported by the European TMR Project "Beyond the Standard Model" contract N. ERBFMRX CT96 0090; O.V. acknowledges financial support from a Marie Curie EC grant (TMR-ERBFMBI CT98 3087).', 'hep-ph-9911337-1-15-0': 'CPcons D.A. Demir, A. Masiero and O. Vives, SISSA report n. SISSA/107/99/EP, accepted for publication in Phys.'} | {'hep-ph-9911337-2-0-0': 'We address the question of the relation between supersymmetry breaking and the origin of flavor in the context of CP violating phenomena.', 'hep-ph-9911337-2-0-1': 'We prove that, in the absence of the Cabibbo-Kobayashi-Maskawa phase, a general Minimal Supersymmetric Standard Model with all possible phases in the soft-breaking terms, but no new flavor structure beyond the usual Yukawa matrices, can never give a sizeable contribution to [MATH], [MATH] or hadronic [MATH] CP asymmetries.', 'hep-ph-9911337-2-0-2': 'Observation of supersymmetric contributions to CP asymmetries in B decays would hint at a non-flavor blind mechanism of supersymmetry breaking.', 'hep-ph-9911337-2-1-0': 'In the near future, new experimental information on CP violation will be available.', 'hep-ph-9911337-2-1-1': 'Not only the new [MATH] factories will start measuring CP violation effects in [MATH] CP asymmetries, but also the experimental sensitivity to the electric dipole moment (EDM) of the neutron and the electron will be substantially improved.', 'hep-ph-9911337-2-1-2': 'These experiments may provide the first sign of physics beyond the Standard Model.', 'hep-ph-9911337-2-2-0': 'If new results do appear and we interpret them in the context of Supersymmetry, both experiments have very different implications on the structure of the soft-breaking terms at the supersymmetry breaking scale.', 'hep-ph-9911337-2-2-1': 'The finding of a non-zero EDM for the neutron would simply indicate the presence of new non-negligible flavor independent susy phases [CITATION].', 'hep-ph-9911337-2-2-2': 'However, a new result in the non-leptonic [MATH] CP asymmetries would be a direct prove of the existence of a completely new flavor structure in the soft-breaking terms.', 'hep-ph-9911337-2-2-3': 'We can rephrase this sentence in the form of a strict no-go theorem: "In the absence of the Cabibbo-Kobayashi-Maskawa (CKM) phase, a general Minimal Supersymmetric Standard Model (MSSM) with possible phases in the soft-breaking terms, but no new flavor structure beyond the usual Yukawa matrices, can never give a sizeable contribution to [MATH], [MATH] or hadronic [MATH] CP asymmetries".', 'hep-ph-9911337-2-3-0': 'Let us first analyze in more detail this strong statement.', 'hep-ph-9911337-2-3-1': 'Indeed, we are going to show that these contributions are at least two orders of magnitude smaller that the required experimental values of [MATH], [MATH], or, in the case of [MATH] CP asymmetries, the expected experimental sensitivity.', 'hep-ph-9911337-2-3-2': 'Moreover, we always take a vanishing phase in the CKM matrix, i.e. [MATH], as a way to isolate the effects of the new supersymmetric phases.', 'hep-ph-9911337-2-3-3': 'We do not include in this no-go theorem other CP violation experiments in rare [MATH] decays, as for instance [MATH], where the contribution from chirality changing operators is important (see discussion below).', 'hep-ph-9911337-2-3-4': 'This "theorem" applies to any MSSM, i.e. with the minimal supersymmetric particle content, and general complex soft-breaking terms, but with a flavor structure strictly given by the two familiar Yukawa matrices or any matrix strictly proportional to them.', 'hep-ph-9911337-2-3-5': 'In these conditions the most general allowed structure of the soft-breaking terms at the large scale, that we call [MATH], is, [EQUATION] where all the allowed phases are explicitly written and one of them can be removed by an R-rotation.', 'hep-ph-9911337-2-3-6': 'All other numbers or matrices in this equation are always real.', 'hep-ph-9911337-2-3-7': 'Notice that this structure covers, not only the Constrained MSSM (CMSSM) [CITATION], but also most of Type I string motivated models considered so far [CITATION], gauge mediated models [CITATION], minimal effective supersymmetry models [CITATION], etc.', 'hep-ph-9911337-2-3-8': 'However, as recently emphasized [CITATION], as soon as one introduces some new flavor structure in the soft Susy-breaking sector, even if the CP violating phases are flavor independent, it is indeed possible to get sizeable CP contribution for large Susy phases and [MATH].', 'hep-ph-9911337-2-4-0': 'Experiments of CP violation in the [MATH] or [MATH] systems only involve supersymmetric particles as virtual particles in the loops.', 'hep-ph-9911337-2-4-1': 'This means that the phases in the soft-breaking terms can only appear though the mass matrices of the susy particles.', 'hep-ph-9911337-2-5-0': 'The key point in our discussion is the absence of any new flavor structure, and its role in the low-energy sparticle mass matrices.', 'hep-ph-9911337-2-5-1': 'Once you have any susy phase that can generate CP violation effects the flavor-change will be necessarily given by a product of Yukawa elements.', 'hep-ph-9911337-2-5-2': 'This fact is completely independent of the presence of only one phase or the 5 phases in Eq. ([REF]) plus the additional [MATH] phase.', 'hep-ph-9911337-2-5-3': 'It is well-known that the Yukawa Renormalization Group Evolution (RGE) is completely independent of all soft-breaking terms [CITATION].', 'hep-ph-9911337-2-5-4': 'In fact, we can solve the Yukawa RGEs for a given value of [MATH] independently of all soft-breaking terms, and the size of all Yukawa matrix elements does not change more than a factor [MATH] from the electroweak scale to the string or susy breaking scale.', 'hep-ph-9911337-2-5-5': 'Then, a typical estimate for the element [MATH] in the [MATH]-[MATH] down squark mass matrix at the electroweak scale would necessarily be (see [CITATION] for details), [EQUATION]', 'hep-ph-9911337-2-5-6': 'The presence of imaginary parts is a slightly more delicate issue, though, in any case Eq. ([REF]) will always be an approximate upper bound.', 'hep-ph-9911337-2-5-7': 'As explained in [CITATION], the RGE equations of all soft-breaking terms are a set of linear differential equations, and thus can be solved as a linear function of the initial conditions, [EQUATION] where [MATH] refers to any scalar, [MATH] to the different gauginos, [MATH] to any tri-linear coupling and the different [MATH] matrices are [MATH] matrices, strictly real.', 'hep-ph-9911337-2-5-8': 'In this equation all the allowed phases have been explicitly written.', 'hep-ph-9911337-2-5-9': 'Regarding the imaginary parts, we can see from Eq. ([REF]) that any imaginary part will always be associated to the non-symmetric part of the [MATH], [MATH] or [MATH] matrices independently of the presence of a single phase or an arbitrary number of them in the initial conditions.', 'hep-ph-9911337-2-5-10': 'This is always true in our general framework, and hence the need of large non-symmetric parts in these matrices on the top of large phases is very clear.', 'hep-ph-9911337-2-5-11': 'To estimate the size of these anti-symmetric parts, we can go to the RGE equations for the scalar mass matrices, where we use the same conventions and notation as in [CITATION].', 'hep-ph-9911337-2-5-12': 'Taking advantage of the linearity of these equations we can directly write the evolution of the anti-symmetric parts, [MATH] as, [EQUATION] where, due to the reality of Yukawa matrices, we have used [MATH], and following [CITATION] a tilde over the couplings ([MATH], [MATH], ...) denotes a re-scaling by a factor [MATH].', 'hep-ph-9911337-2-5-13': 'In the evolution of the [MATH]-[MATH] squark mass matrices, [MATH] and [MATH], only one of the two Yukawa matrices, the one with equal isospin to the squarks, is directly involved.', 'hep-ph-9911337-2-5-14': 'Then, it is easy to understand that these matrices are in a very good approximation diagonal in the SCKM basis once you start with the initial conditions given in Eq. ([REF]).', 'hep-ph-9911337-2-5-15': 'Hence, we can safely neglect the last two terms in Eq. ([REF]) and forget about [MATH] and [MATH].', 'hep-ph-9911337-2-5-16': 'From Eq. ([REF]), the initial conditions for these anti-symmetric parts at [MATH] are identically zero.', 'hep-ph-9911337-2-5-17': 'So, the only source for [MATH] is necessarily [MATH].', 'hep-ph-9911337-2-5-18': 'Now, we can analyze the RGE for [MATH], [EQUATION] with an equivalent equation for [MATH].', 'hep-ph-9911337-2-5-19': 'It is clear that given the general initial conditions in Eq. ([REF]), [MATH] is complex at any scale.', 'hep-ph-9911337-2-5-20': 'However, we are interested in the imaginary parts of [MATH].', 'hep-ph-9911337-2-5-21': 'At [MATH] this combination is exactly real, but this is not true any more at a different scale.', 'hep-ph-9911337-2-5-22': 'From Eq. ([REF]), we can immediately see that these imaginary parts are extremely small.', 'hep-ph-9911337-2-5-23': 'Let us, for a moment, neglect the terms involving [MATH] or [MATH] from the above equation.', 'hep-ph-9911337-2-5-24': 'Then, the only flavor structure appearing in Eq. ([REF]) at [MATH] is [MATH].', 'hep-ph-9911337-2-5-25': 'We can always go to the basis where [MATH] is diagonal and then we will have [MATH] exactly diagonal at any scale.', 'hep-ph-9911337-2-5-26': 'In particular this means that [MATH] would always exactly vanish.', 'hep-ph-9911337-2-5-27': 'The same reasoning applies to [MATH] and [MATH].', 'hep-ph-9911337-2-5-28': 'Hence, simply taking into account the flavor structure, our conclusion is that any non-vanishing element of [MATH] and hence of [MATH] must be necessarily proportional to [MATH].', 'hep-ph-9911337-2-5-29': 'So, we can expect them to be, [EQUATION] where [MATH], with [MATH] the vacuum expectation value of the Higgs, [MATH] and [MATH] is a proportionality constant that includes the effects of the running from [MATH] to [MATH].', 'hep-ph-9911337-2-5-30': 'To estimate this constant we have to keep in mind that the imaginary parts of [MATH] are generated through the RGE running and then these imaginary parts generate [MATH] as a second order effect.', 'hep-ph-9911337-2-5-31': 'This means that roughly [MATH] times a combination of initial conditions as in Eq. ([REF]).', 'hep-ph-9911337-2-5-32': 'So, we estimate these matrix elements to be [MATH] times initial conditions.', 'hep-ph-9911337-2-5-33': 'This was exactly the result we found for the [MATH]-[MATH] terms in [CITATION].', 'hep-ph-9911337-2-5-34': 'In fact, now it is clear that this is the same for all the terms in Eq. ([REF]), [MATH]-[MATH], [MATH]-[MATH] and [MATH]-[MATH], irrespectively of the presence of an arbitrary number of new phases.', 'hep-ph-9911337-2-6-0': 'As we have already said before, the situation in the [MATH]-[MATH] matrices is clearly worse because the RGE of these matrices involves only the corresponding Yukawa matrix and hence, in the SCKM, they are always diagonal and real in extremely good approximation.', 'hep-ph-9911337-2-7-0': 'Hence, so far, we have shown that the [MATH]-[MATH] or [MATH]-[MATH] squark mass matrices are still essentially real.', 'hep-ph-9911337-2-8-0': 'The only complex matrices, then, will still be the [MATH]-[MATH] matrices that include, from the very beginning, the phases [MATH] and [MATH].', 'hep-ph-9911337-2-8-1': 'Once more, the size of these entries is determined by the Yukawa elements with these two phases providing the complex structure.', 'hep-ph-9911337-2-8-2': 'However, this situation is not new for these more general MSSM models and it was already present even in the CMSSM.', 'hep-ph-9911337-2-9-0': 'From here we can start the analysis of the effects of supersymmetric phases in the CP observables.', 'hep-ph-9911337-2-9-1': 'We have already seen that the structure of the sfermion mass matrices remains the same as in the CMSSM case.', 'hep-ph-9911337-2-9-2': 'This is simply due to our dependence to the Yukawa matrices to get any flavor change.', 'hep-ph-9911337-2-9-3': 'On the other hand, the new gaugino phases enter the chargino and neutralino mass matrices.', 'hep-ph-9911337-2-9-4': 'However, in all our previous works [CITATION] we have always ignored the EDM bounds, which means that [MATH] could take any value and large phases in the mixing matrices were already present.', 'hep-ph-9911337-2-9-5': 'So, the inclusion of the new gaugino phases does not lead to new effects apart from those already accounted for varying [MATH].', 'hep-ph-9911337-2-10-0': 'In first place, we will consider indirect CP violation both in the [MATH] and [MATH] systems, refering to [CITATION] for a complete analysis.', 'hep-ph-9911337-2-10-1': 'In the case of the gluino or neutralino, it is well-known that the CMSSM satisfies widely all the constraints imposed by flavor changing experiments [CITATION].', 'hep-ph-9911337-2-10-2': 'Hence, this still holds true in this more general case, where we have shown that the sfermion mass matrices are still of the same size as in the CMSSM.', 'hep-ph-9911337-2-10-3': 'This means then, that all possible mass insertions are always roughly two orders of magnitude bellow the required values to saturate flavor changing observables, (see second part of Ref [CITATION]).', 'hep-ph-9911337-2-10-4': 'Notice that this is true even for CP conserving flavor changing observables and the situation for the CP violating observables with chirality conserving operators, Eq. ([REF]), is still much worse.', 'hep-ph-9911337-2-10-5': 'Also chargino contributions can be comparable in general.', 'hep-ph-9911337-2-10-6': 'This was the main subject of paper [CITATION] where we showed the different constraints in the chirality conserving, [MATH]-[MATH], and chirality changing, [MATH]-[MATH], transitions.', 'hep-ph-9911337-2-10-7': 'From [CITATION] it is clear that chargino chirality changing transitions are directly constrained by the [MATH] decay to be more that three orders of magnitude smaller than the corresponding chirality conserving transitions.', 'hep-ph-9911337-2-10-8': 'And finally, on the other side, we already showed in [CITATION] that chirality conserving transitions were real to a very good approximation.', 'hep-ph-9911337-2-10-9': 'These arguments allow us to discard measurable CP violation in both [MATH] and indirect CP violation in the [MATH] system.', 'hep-ph-9911337-2-11-0': 'Finally we have to consider also direct CP violation in non-leptonic [MATH] decays.', 'hep-ph-9911337-2-11-1': 'Essentially, the only difference with our discussion on indirect CP violation is the presence of the penguins.', 'hep-ph-9911337-2-11-2': 'Once more, in the gluino case chirality conserving transitions are real to a very good approximation, and, in any case, well below the phenomenological bounds [CITATION].', 'hep-ph-9911337-2-11-3': 'The chirality changing transitions on the other hand are suppressed by light quark masses, where we call light even the [MATH] quark, and again below the bounds.', 'hep-ph-9911337-2-11-4': 'Hence, our conclusion for the gluino is necessarily the same.', 'hep-ph-9911337-2-11-5': 'So, we are left with chargino.', 'hep-ph-9911337-2-11-6': '[MATH]-[MATH] transitions are real to a very good approximation, for the very same reasons used in the indirect CP violation case.', 'hep-ph-9911337-2-11-7': 'And now the relation of [MATH] with the chirality changing penguins is even more transparent if possible.', 'hep-ph-9911337-2-11-8': 'This completes the proof of our Theorem.', 'hep-ph-9911337-2-12-0': 'To conclude we would like to discuss the implications of our result in the search for supersymmetric CP violation.', 'hep-ph-9911337-2-12-1': 'In the presence of large supersymmetric phases [CITATION], the EDMs of the electron and the neutron must be very close to the experimental bounds.', 'hep-ph-9911337-2-12-2': 'However, as we have shown in this letter, the presence of these phases is not enough to generate a sizeable contribution to [MATH], [MATH] or [MATH] CP asymmetries.', 'hep-ph-9911337-2-12-3': 'Here a completely new flavor structure in the soft breaking terms is required to get sizeables effects.', 'hep-ph-9911337-2-12-4': 'In this sense, CP experiments in a supersymmetric theory are a direct probe on any additional flavor structure in the soft-breaking terms.', 'hep-ph-9911337-2-13-0': 'Hence, in the absence of new flavor structures, only pure chirality changing observables (EDMs or [MATH]) or observables where, in any case, the chirality flip operators are relevant ([MATH], [MATH]), can show the effects of new supersymmetric phases [CITATION].', 'hep-ph-9911337-2-14-0': 'We thank S. Bertolini, T. Kobayashi and S. Khalil for enlightening discussions.', 'hep-ph-9911337-2-14-1': 'The work of A.M. was partially supported by the European TMR Project "Beyond the Standard Model" contract N. ERBFMRX CT96 0090; O.V. acknowledges financial support from a Marie Curie EC grant (TMR-ERBFMBI CT98 3087).', 'hep-ph-9911337-2-15-0': 'CPcons D.A. Demir, A. Masiero and O. Vives, SISSA report n. SISSA/107/99/EP, accepted for publication in Phys.'} | [['hep-ph-9911337-1-9-0', 'hep-ph-9911337-2-9-0'], ['hep-ph-9911337-1-9-1', 'hep-ph-9911337-2-9-1'], ['hep-ph-9911337-1-9-2', 'hep-ph-9911337-2-9-2'], ['hep-ph-9911337-1-9-3', 'hep-ph-9911337-2-9-3'], ['hep-ph-9911337-1-9-4', 'hep-ph-9911337-2-9-4'], ['hep-ph-9911337-1-9-5', 'hep-ph-9911337-2-9-5'], ['hep-ph-9911337-1-11-0', 'hep-ph-9911337-2-11-0'], ['hep-ph-9911337-1-11-1', 'hep-ph-9911337-2-11-1'], ['hep-ph-9911337-1-11-2', 'hep-ph-9911337-2-11-2'], ['hep-ph-9911337-1-11-3', 'hep-ph-9911337-2-11-3'], ['hep-ph-9911337-1-11-4', 'hep-ph-9911337-2-11-4'], ['hep-ph-9911337-1-11-5', 'hep-ph-9911337-2-11-5'], ['hep-ph-9911337-1-11-6', 'hep-ph-9911337-2-11-6'], ['hep-ph-9911337-1-11-7', 'hep-ph-9911337-2-11-7'], ['hep-ph-9911337-1-11-8', 'hep-ph-9911337-2-11-8'], ['hep-ph-9911337-1-2-0', 'hep-ph-9911337-2-2-0'], ['hep-ph-9911337-1-2-1', 'hep-ph-9911337-2-2-1'], ['hep-ph-9911337-1-2-2', 'hep-ph-9911337-2-2-2'], ['hep-ph-9911337-1-2-3', 'hep-ph-9911337-2-2-3'], ['hep-ph-9911337-1-1-0', 'hep-ph-9911337-2-1-0'], ['hep-ph-9911337-1-1-1', 'hep-ph-9911337-2-1-1'], ['hep-ph-9911337-1-1-2', 'hep-ph-9911337-2-1-2'], ['hep-ph-9911337-1-13-0', 'hep-ph-9911337-2-13-0'], ['hep-ph-9911337-1-0-0', 'hep-ph-9911337-2-0-0'], ['hep-ph-9911337-1-0-1', 'hep-ph-9911337-2-0-1'], ['hep-ph-9911337-1-0-2', 'hep-ph-9911337-2-0-2'], ['hep-ph-9911337-1-3-0', 'hep-ph-9911337-2-3-0'], ['hep-ph-9911337-1-3-1', 'hep-ph-9911337-2-3-1'], ['hep-ph-9911337-1-3-2', 'hep-ph-9911337-2-3-2'], ['hep-ph-9911337-1-3-3', 'hep-ph-9911337-2-3-3'], ['hep-ph-9911337-1-3-4', 'hep-ph-9911337-2-3-4'], ['hep-ph-9911337-1-3-5', 'hep-ph-9911337-2-3-5'], ['hep-ph-9911337-1-3-6', 'hep-ph-9911337-2-3-6'], ['hep-ph-9911337-1-3-7', 'hep-ph-9911337-2-3-7'], ['hep-ph-9911337-1-3-8', 'hep-ph-9911337-2-3-8'], ['hep-ph-9911337-1-6-0', 'hep-ph-9911337-2-6-0'], ['hep-ph-9911337-1-7-0', 'hep-ph-9911337-2-7-0'], ['hep-ph-9911337-1-10-0', 'hep-ph-9911337-2-10-0'], 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'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/9911337 | null | null | null | null | null |
1408.6491 | {'1408.6491-1-0-0': "To partly address people's concerns over web tracking, Google has created the Ad Settings webpage to provide information about and some choice over the profiles Google creates on users.", '1408.6491-1-0-1': "We present AdFisher, an automated tool that explores how user behaviors, Google's ads, and Ad Settings interact.", '1408.6491-1-0-2': 'Our tool uses a rigorous experimental design and analysis to ensure the statistical significance of our results.', '1408.6491-1-0-3': 'It uses machine learning to automate the selection of a statistical test.', '1408.6491-1-0-4': "We use AdFisher to find that Ad Settings is opaque about some features of a user's profile, that it does provide some choice on ads, and that these choices can lead to seemingly discriminatory ads.", '1408.6491-1-0-5': 'In particular, we found that visiting webpages associated with substance abuse will change the ads shown but not the settings page.', '1408.6491-1-0-6': 'We also found that setting the gender to female results in getting fewer instances of an ad related to high paying jobs than setting it to male.', '1408.6491-1-1-0': '# Introduction', '1408.6491-1-2-0': 'Problem and Overview Online advertisers target ads to Internet users based on their browsing behaviors (e.g., [CITATION]).', '1408.6491-1-2-1': "Many websites providing content, such as news, outsource their advertising operations to large third-party ad networks, such as Google's DoubleClick.", '1408.6491-1-2-2': "These networks embed tracking code into webpages across many sites providing the network with a more global view of each user's behaviors.", '1408.6491-1-3-0': 'People are concerned about behavioral marketing on the web (e.g., [CITATION]).', '1408.6491-1-3-1': 'They would like to know what information companies use to determine the ads they show to them.', '1408.6491-1-3-2': 'To increase transparency into these issues, Google provides the Ad Settings webpage, which is dynamically generated for each user (as approximated with tracking cookies).', '1408.6491-1-3-3': 'A user can see the page for himself at https://www.google.com/settings/ads.', '1408.6491-1-3-4': 'Figure [REF] provides a screenshot.', '1408.6491-1-4-0': 'Yahoo and Microsoft also offer personalized ad settings.', '1408.6491-1-5-0': 'However, they provide little information about how these pages operate.', '1408.6491-1-5-1': 'We would like to know how completely the settings describe their profiles of users.', '1408.6491-1-5-2': "In particular, we want to know how a user's behaviors, either directly with the advertiser or with content providers, alter the ads and settings shown to the user and whether these changes are in harmony.", '1408.6491-1-5-3': 'We also want to know how changing settings affects ads.', '1408.6491-1-5-4': 'To explain these matters, we ran experiments where automated agents simulating users interact with Google and content providers and we measure how these interactions alter the ads and settings that Google shows.', '1408.6491-1-6-0': 'To automate these experiments, we created AdFisher, a tool that automates randomized, controlled experiments for learning about online tracking.', '1408.6491-1-6-1': 'Our tool offers a combination of automation, statistical rigor, scalability, and explanation for determining the use of information by web advertising algorithms and by personalized ad settings, such as Google Ad Settings.', '1408.6491-1-6-2': 'The tool automates the simulation of having a particular interest or attribute by visiting webpages associated with that interest or by altering the ad settings provided by Google.', '1408.6491-1-6-3': 'It automates the collection of ads shown to the simulated users and the settings that Google provides.', '1408.6491-1-6-4': 'It automatically analyzes the data to determine whether statistically significant differences between groups of agents exist.', '1408.6491-1-6-5': 'To do so, AdFisher uses machine learning to automatically detect differences and then executes a test of significance specialized for the difference it might have found.', '1408.6491-1-7-0': 'The only inputs the operator has to provide to our tool are the behaviors the two groups are to perform and the measurements to make over the agents after they perform those behaviors.', '1408.6491-1-7-1': 'Thus, we could easily run multiple experiments exploring the causal connections between the ads and settings that Google shows.', '1408.6491-1-8-0': "Motivating Experiments For example, in one experiment, we explored whether visiting websites related to substance abuse has an impact on Google's ads or settings.", '1408.6491-1-8-1': 'We created an experimental group and a control group of agents.', '1408.6491-1-8-2': 'The agents in the experimental group visited such websites while the agents in the control group simply waited idly.', '1408.6491-1-8-3': 'Then, both groups of agents visited the Times of India, a content providing webpage that uses Google for advertising.', '1408.6491-1-8-4': 'We collected the ads shown to the agents.', '1408.6491-1-9-0': 'Having run the experiment and collected the data, we had to determine whether any difference existed in the outputs shown to the agents.', '1408.6491-1-9-1': 'One way to check would be to intuit what the difference could be and test for that difference.', '1408.6491-1-9-2': 'For example, we might have suspected that the experimental group would have received more ads containing the word "alcohol".', '1408.6491-1-9-3': 'Thus, we could use the test statistic that counts the number of instances of "alcohol" in the ads of the experimental group and subtracts from it the number of times it appears in the control group.', '1408.6491-1-9-4': 'If no difference exists, we would expect this test statistic would have a value close to zero.', '1408.6491-1-9-5': 'If our suspicion is correct, we would expect a large positive value.', '1408.6491-1-10-0': 'However, developing the intuition that "alcohol" is a difference between the groups can take considerable effort.', '1408.6491-1-10-1': 'Thus, we instead use machine learning to automatically find such patterns in a training subset of the data.', '1408.6491-1-10-2': 'AdFisher found a classifier that could, using just the ads shown to an agent, accurately determine whether the agent visited the webpages related to substance abuse.', '1408.6491-1-10-3': 'We used a separate test subset of the data to verify that the classifier found a statistically significant difference between the ads shown to each group of agents.', '1408.6491-1-11-0': 'We also measured the settings that Google provided to each agent at its Ad Settings page after the experimental group of agents visited the webpages associated with substance abuse.', '1408.6491-1-11-1': 'We found no differences (significant or otherwise) between the pages for the agents.', '1408.6491-1-11-2': 'Thus, we have determined that Google is using information related to visiting webpages associated with substance abuse but not reporting this use.', '1408.6491-1-11-3': 'Rather than providing transparency, the ad settings are opaque.', '1408.6491-1-12-0': 'In another experiment we examined whether the settings provided choice to the user.', '1408.6491-1-12-1': 'We found that removing interests from Google Ad Settings page changes the ads that a user sees.', '1408.6491-1-12-2': 'In particular, we had both groups of agents visit a site related to online dating.', '1408.6491-1-12-3': 'Then, only one of the groups removed the interest related to online dating.', '1408.6491-1-12-4': 'Thereafter, the top ads shown to the group that kept the interest were related to dating but not the top ads shown to the other group.', '1408.6491-1-12-5': 'Thus, the ad settings do offer the users a degree of choice over the ads they see.', '1408.6491-1-13-0': 'In another experiment, we found evidence suggestive of discrimination.', '1408.6491-1-13-1': "We set the agents' gender to female or male on Google's Ad Settings page.", '1408.6491-1-13-2': 'We then had both the female and male groups of agents visit webpages associated with employment.', '1408.6491-1-13-3': 'Again using the ads on the Times of India, we established that Google used this gender information to select ads, as one would expect.', '1408.6491-1-13-4': 'In this case, the interesting result was how the ads differed as learned by the classifier: Google showed the simulated males certain ads that promised large salaries more frequently than the simulated females, a finding suggestive of discrimination.', '1408.6491-1-14-0': "While neither of our findings of opacity or discrimination are clear violations of Google's privacy policy, they are both concerning.", '1408.6491-1-15-0': 'Contributions In addition to the experimental findings highlighted above, we provide AdFisher, a tool for running such experiments.', '1408.6491-1-15-1': 'AdFisher automates the running of the experiment and the analysis of the data.', '1408.6491-1-16-0': 'AdFisher offers rigor by carefully selecting our experimental design and statistical analysis to not make questionable assumptions about the data we collect.', '1408.6491-1-16-1': 'In particular, we base our design and analysis on a prior proposal that makes no assumptions about the data being independent or identically distributed [CITATION].', '1408.6491-1-16-2': "We find such assumptions, common in statistical analyses, problematic for studying online advertisers since advertisers update their behavior continuously in response to unobserved inputs (such as online ad auctions) and the experimenters' own actions.", '1408.6491-1-16-3': 'Indeed, in practice, the distribution of ads changes over time and agents interfere with one another [CITATION].', '1408.6491-1-17-0': 'Our automation, experimental design, and statistical analyses allow us to scale to handling large numbers of agents for finding subtle differences.', '1408.6491-1-17-1': 'In particular, we modify the prior analysis of Tschantz et al. [CITATION] to allow for experiments running over long periods of time.', '1408.6491-1-17-2': 'We do so by using blocking (e.g., [CITATION]), a nested statistical analysis not previously applied to understanding web advertising.', '1408.6491-1-17-3': 'The blocking analysis ensures that agents are only compared to the agents that start out like it and then aggregates together the comparisons across blocks of agents.', '1408.6491-1-17-4': 'Thus, AdFisher may run agents in batches spread out over time while only comparing those agents running simultaneously to one another.', '1408.6491-1-18-0': 'Our tool provides an explanation as to how Google alters its behaviors in response to different user actions.', '1408.6491-1-18-1': 'We use machine learning to determine ads that accurately distinguish the groups of agents.', '1408.6491-1-18-2': 'We allow the experimenter to find differences between how Google treats the agents with ads and how Google represents agents on the Ad Settings page.', '1408.6491-1-19-0': 'To maintain statistical rigor, we carefully circumscribe our claims.', '1408.6491-1-19-1': 'We only claim statistical soundness in the sense that upon reporting that the difference in agent behavior causes an effect, such an effect really was caused by the difference in behavior with high likelihood (made quantitative by a p-value).', '1408.6491-1-19-2': 'We do not claim that we will always find a difference if one exists, nor that the differences we find are typical of those experienced by users.', '1408.6491-1-19-3': 'Furthermore, while we can characterize the differences, we cannot assign blame for them since either Google or the advertisers working with Google could be responsible.', '1408.6491-1-20-0': 'Contents After covering prior work next, we present, in Section [REF], privacy properties that our tool AdFisher can check: nondiscrimination, transparency, and choice.', '1408.6491-1-20-1': 'Section [REF] explains the methodology we use to ensure sound conclusions from using AdFisher.', '1408.6491-1-20-2': 'Section [REF] presents the design of AdFisher.', '1408.6491-1-20-3': "Section [REF] discusses our use of AdFisher to study Google's ads and settings.", '1408.6491-1-20-4': 'We end with conclusions and future work.', '1408.6491-1-20-5': 'Appendix [REF] provides details about our methodology.', '1408.6491-1-21-0': 'Source code for AdFisher, raw data, and additional details about our experiments can be found at', '1408.6491-1-22-0': 'http://www.cs.cmu.edu/ mtschant/ife/', '1408.6491-1-23-0': '# Prior Work', '1408.6491-1-24-0': 'We are not the first to study how Google uses information.', '1408.6491-1-24-1': 'The work with the closest subject of study to ours is by Wills and Tatar [CITATION].', '1408.6491-1-24-2': 'They studied both the ads shown by Google and the behavior of Google\'s Ad Settings (then called the "Ad Preference Manager").', '1408.6491-1-24-3': 'Like us, they find the presence of opacity: various interests impacted the ads and settings shown to the user and that ads could change without a corresponding change in Ad Settings.', '1408.6491-1-24-4': 'Unlike our study, theirs was mostly manual, small scale, lacked any statistical analysis, and did not follow a rigorous experimental design.', '1408.6491-1-24-5': 'Furthermore, we additionally study choice and discrimination.', '1408.6491-1-25-0': 'The other related works differ from us in both goals and methods.', '1408.6491-1-25-1': 'They all focus on how visiting webpages change the ads seen.', '1408.6491-1-25-2': 'While we examine such changes in our work, we do so as part of a larger analysis of the interactions between ads and personalized ad settings, a topic they do not study.', '1408.6491-1-26-0': 'Barford et al. come the closest in that their recent study looked at both ads and ad settings [CITATION].', '1408.6491-1-26-1': 'They do so in their study of the "adscape", an attempt to understand each ad on the Internet.', '1408.6491-1-26-2': 'They study each ad individually and cast a wide net to analyze many ads from many websites while simulating many different interests.', '1408.6491-1-26-3': 'They only examine the ad settings to determine whether they successfully induced an interest.', '1408.6491-1-26-4': 'We rigorously study how the settings affects the ads shown (choice) and how behaviors can affect ads without affecting the settings (opacity).', '1408.6491-1-26-5': "Furthermore, we use focused collections of data and an analysis that considers all ads collectively to find subtle causal effects within Google's advertising ecosystem.", '1408.6491-1-26-6': 'We also use a different randomized experimental design and analysis to ensure that our results imply causation without making an assumption of independence between ads, which appears to not hold in our setting [CITATION].', '1408.6491-1-27-0': 'The usage study closest to ours in statistical methodology is that of Tschantz et al. [CITATION].', '1408.6491-1-27-1': 'They were only interested in finding a rigorous methodology for determining whether a system like Google uses information.', '1408.6491-1-27-2': 'Due to limitations of their methodology, they only ran small-scale studies.', '1408.6491-1-27-3': 'While they observed that browsing behaviors could affect Ad Settings, they did not study how this related to the ads received.', '1408.6491-1-27-4': 'Furthermore, while we build upon their methodology, we automate it by using machine learning to select an appropriate test statistic whereas they manually selected test statistics.', '1408.6491-1-28-0': 'The usage study closest to ours in terms of implementation is that of Liu et al. in that they also use machine learning [CITATION].', '1408.6491-1-28-1': 'Their goal is to determine whether an ad was selected due to the content of a page, by using behavioral profiling, or from a previous webpage visit.', '1408.6491-1-28-2': 'Thus, rather than use machine learning to select a statistical test for finding causal relations, they do so to detect whether an ad on a webpage matches the content on the page to make a case for the first possibility.', '1408.6491-1-28-3': 'Thus, they have a separate classifier for each interest a webpage might cover.', '1408.6491-1-28-4': "Rather than perform a statistical analysis to determine whether treatment groups have a statistically significant difference, they use their classifiers to judge the ratio of ads on a page unrelated to the page's content, which they presume indicates that the ads were the result of behavioral targeting.", '1408.6491-1-29-0': 'Lecuyer et al. present XRay, a tool that looks for correlations between the data that web services have about users and the ads shown to users [CITATION].', '1408.6491-1-29-1': 'Their tool looks through many changes to a type of input to determine whether any of them has a correlation with the frequency of a single ad.', '1408.6491-1-29-2': 'Our tool instead looks for causal relations from a single change to an input by checking many different possible changes over many ads.', '1408.6491-1-29-3': 'To enable XRay examining many possible inputs using a small number of user accounts, which are expensive to create, they adopt assumptions including independence between ads.', '1408.6491-1-29-4': 'We do not assume independence since it appears to not hold in our setting [CITATION], a setting that does not require creating accounts.', '1408.6491-1-30-0': 'Guha et al. compares ads seen by three agents to see whether Google treats differently the one that behaves differently from the other two [CITATION].', '1408.6491-1-30-1': 'We adopt their suggestion of focusing on the title and URL displayed on ads when comparing ads to avoid noise from other less stable parts of the ad.', '1408.6491-1-30-2': 'Our work differs by studying the ad settings in addition to the ads and by using larger numbers of agents.', '1408.6491-1-30-3': 'Furthermore, we use rigorous statistical analyses whereas they used an ad hoc metric.', '1408.6491-1-30-4': "Balebako et al. run experiments similar to Guha et al.'s to study the effectiveness of privacy tools [CITATION].", '1408.6491-1-31-0': 'Sweeney manually ran an experiment to determine that searching for names associated with African-Americans produce more search ads containing the word "arrest" than names associated with European-Americans [CITATION].', '1408.6491-1-31-1': 'Not only did hand collecting the data for her study take a considerable amount of time, but her study also required considerable insight to determine that the word "arrest" was a key difference.', '1408.6491-1-31-2': 'AdFisher can automate not just the collection of the ads, but also the identification of such key differences by using its machine learning capabilities.', '1408.6491-1-31-3': 'Indeed, it found on its own that simulated males were more often than simulated females shown ads encouraging the user to seek high paying jobs.', '1408.6491-1-32-0': '# Privacy Properties', '1408.6491-1-33-0': 'Motivating our methodology for finding causal relationships, we present some properties of ad networks that we can check with such a methodology in place.', '1408.6491-1-33-1': 'As a fundamental limitation of science, we can only prove the existence of a causal effect; we cannot prove that one does not exist (see Section [REF]).', '1408.6491-1-33-2': 'Thus, experiments can only demonstrate violations of nondiscrimination and transparency, which require effects.', '1408.6491-1-33-3': 'On the other hand, we can experimentally demonstrate that effectful choice and ad choice are complied with in the cases that we test since compliance follows from the existence of an effect.', '1408.6491-1-33-4': 'Table [REF] summarizes these properties.', '1408.6491-1-34-0': '## Discrimination', '1408.6491-1-35-0': 'At its core, discrimination between two classes of individuals (e.g., one race vs. another) occurs when the attribute distinguishing those two classes causes a change in behavior toward those two classes.', '1408.6491-1-35-1': 'In our case, discrimination occurs when membership in a class causes a change in ads.', '1408.6491-1-35-2': 'Such discrimination is not always bad (e.g., many would be comfortable with men and women receiving different clothing ads) and we will make no attempt in this paper to determine the acceptability of discrimination.', '1408.6491-1-36-0': "Determining whether class membership causes a change in ads can be difficult since many factors not under the experimenter's control or even observable to the experimenter may also cause changes.", '1408.6491-1-36-1': 'However, our experimental methodology can determine when membership in certain classes causes changes in ads by comparing many instances of each class.', '1408.6491-1-37-0': 'We are limited in the classes we can consider since we cannot create actual people that vary by the traditional subjects of discrimination, such as race or gender.', '1408.6491-1-37-1': 'Instead, we look at classes that function as surrogates for those classes of interest.', '1408.6491-1-37-2': "For example, rather than directly looking at how gender affects people's ads, we instead look at how altering a gender setting affects ads or at how visiting websites associated with each gender affects ads.", '1408.6491-1-38-0': '## Transparency', '1408.6491-1-39-0': 'Ad users should have some understanding of the information that ad networks collect and use about them.', '1408.6491-1-39-1': 'However, ad networks should not offer complete transparency in the sense of sharing all the information it has with users.', '1408.6491-1-39-2': 'To do so would create privacy violations since it would involve sharing information with one user about all the other users.', '1408.6491-1-39-3': 'Even sharing all the information that the network uses to decide the ads that it shows to a user can be problematic since networks often use complex models built using information from other users.', '1408.6491-1-40-0': 'Thus, we study a more constrained form of transparency that focuses on information about the user in question.', '1408.6491-1-40-1': 'Individual data use transparency requires that the ad network shares the information that it uses about a user for selecting ads for him with him.', '1408.6491-1-40-2': 'Ad privacy settings satisfy this requirement if two people with the same settings who are accessing a webpage in the same manner receives the same distribution of ads.', '1408.6491-1-40-3': 'That is, the ad network does not discriminate between users with the same settings on the basis of any class memberships.', '1408.6491-1-40-4': 'Thus, we may test for transparency by looking for effects caused by factors not recorded in the ad settings.', '1408.6491-1-41-0': '## Choice', '1408.6491-1-42-0': 'Ad privacy settings offer the prospect of choice for users.', '1408.6491-1-42-1': "In particular, some of them, such as Google's Ad Settings, offer users the option of changing the settings inferred about them.", '1408.6491-1-42-2': 'However, the exact nature of how these changes impact the ad network is unclear.', '1408.6491-1-42-3': 'Thus, we examine two notions of choice.', '1408.6491-1-43-0': 'A very coarse form we consider is effectful choice, which requires that altering the settings has some effect on the ads seen by the user.', '1408.6491-1-43-1': 'This shows that altering settings is not merely a "placebo button": it has a real effect on the network\'s ads.', '1408.6491-1-43-2': 'Ideally, the effect would be meaningful and related to the altered setting.', '1408.6491-1-44-0': 'One way such an effect would be meaningful, in the case of removing an inferred interest, is for the change to decrease the number of ads related to the interest seen by the user.', '1408.6491-1-44-1': 'We call this requirement ad choice.', '1408.6491-1-44-2': "(We focus on removing interests since that's the more privacy related form of choice.", '1408.6491-1-44-3': 'Much of what we write will also hold for adding interests but with the proper negations.)', '1408.6491-1-44-4': 'In the extreme case, the user no longer receives any ads related to that interest.', '1408.6491-1-44-5': 'This standard raises questions about defining when an ad is related to an interest.', '1408.6491-1-45-0': 'One way to judge whether an ad is relevant is to check it for keywords associated with the removed interest.', '1408.6491-1-45-1': 'We will conclude that an ad network has ad choice when removing interests causes a statistically significant decrease in the number of ads with keywords related to that interest seen by the user.', '1408.6491-1-46-0': 'We cannot find all violations of ad choice since when the difference in ads is small, we are unable to determine whether it represents a true difference between two distributions or just noise from sampling a finite number of ads from one underlying distribution.', '1408.6491-1-46-1': 'However, we can test for a statistically significant increase in the number of related ads to find egregious violations.', '1408.6491-1-46-2': 'Thus, by requiring the effect to have a fixed direction, we can find both compliance and violations of ad choice.', '1408.6491-1-47-0': 'Note that many of the above conditions can be reformulated without causation.', '1408.6491-1-47-1': 'For example, rather then checking whether removing an interest results in no longer getting ads related to that interest, we could instead simply check whether those who have that interest removed do not get such ads.', '1408.6491-1-47-2': 'Only the second weaker definition will be satisfied in the case that the ad network would not show that person such ads in either case.', '1408.6491-1-47-3': 'We find the causal forms of these definitions to better capture choice since the non-causal forms are satisfied when no real choices are possible: the person does not see such ads regardless of their choices.', '1408.6491-1-48-0': '# Methodology', '1408.6491-1-49-0': 'The goal of our methodology is to establish that a certain type of input to a system causes an effect on a certain type of output of the system.', '1408.6491-1-49-1': 'For example, in our experiments, we study the system of Google.', '1408.6491-1-49-2': "The inputs we study are visits to content providing websites and users' interactions with Google Ad Settings webpage.", '1408.6491-1-49-3': 'The outputs we study are the settings and ads shown to the users by Google.', '1408.6491-1-49-4': 'However, nothing in our methodology limits ourselves to these particular topics; it is appropriate for determining I/O properties of any web system.', '1408.6491-1-50-0': 'Here we present just an overview of our methodology.', '1408.6491-1-50-1': 'Appendix [REF] provides details of the statistical analysis.', '1408.6491-1-51-0': '## Background: Hypothesis Testing', '1408.6491-1-52-0': "To establish causation, we start with the approach of Fisher (our tool's namesake) to randomized hypothesis testing [CITATION] as specialized by Tschantz et al. for the setting of online systems [CITATION].", '1408.6491-1-52-1': 'The experimenter considers a null hypothesis, in our case, that the inputs do not affect the outputs.', '1408.6491-1-52-2': 'To test this hypothesis the experimenter selects two values that the type of inputs could take on, typically called the control and experimental treatments.', '1408.6491-1-53-0': 'The experimenter applies the treatments to experimental units.', '1408.6491-1-53-1': 'In our setting, the units are the agents, that is, simulated users.', '1408.6491-1-53-2': 'The experimental units should be exchangeable, that is, initially identical as far as the inputs and outputs in question are concerned.', '1408.6491-1-53-3': 'For example, an agent created with the Firefox browser would not be exchangeable to one created with the Internet Explorer browser since Google can detect the browser used.', '1408.6491-1-54-0': 'The experimenter randomly partitions the agents to a control group and an experimental group.', '1408.6491-1-54-1': '(See Figure [REF].)', '1408.6491-1-54-2': "To each group, the experimenter applies the group's respective treatment by having the agents perform actions producing inputs to Google.", '1408.6491-1-54-3': 'Next, the experimenter takes measurements of the outputs Google sends to the agents, such as ads.', '1408.6491-1-54-4': 'At this point, the experiment proper is complete and data analysis begins.', '1408.6491-1-55-0': 'Data analysis starts by computing a test statistic over the measurements.', '1408.6491-1-55-1': 'The experimenter selects a test statistic that she suspects will take on a high value when the outputs to the two groups differ.', '1408.6491-1-55-2': 'She then uses the permutation test to determine whether the value the test statistic actually took on is higher than what one would expect by chance unless the groups actually differ.', '1408.6491-1-55-3': 'The permutation test compares the actual value of the test statistic to each hypothetical value it would have taken on if the random assignment of agents to groups had occurred differently and the null hypothesis is true.', '1408.6491-1-55-4': 'Since the null hypothesis is that the inputs have no effect and the units start exchangeable, the random assignment should have no effect on the value of the test statistic.', '1408.6491-1-55-5': 'Thus, under the null hypothesis, it is unlikely to see the actual value of the test statistic being larger than the vast majority of hypothetical values.', '1408.6491-1-56-0': 'The probability of seeing the test statistic take on a value as high as it did under the null hypothesis is called the p-value.', '1408.6491-1-56-1': 'If the value of the test statistic is so high that under the null hypothesis it would only take on as high of a value in [MATH] of the random assignments, then we conclude that the value is statistically significant (at the [MATH] level) and that causation is likely.', '1408.6491-1-56-2': 'Even lower p-values suggest that the results are even more statistically significant.', '1408.6491-1-57-0': '## Blocking', '1408.6491-1-58-0': 'In practice, the above methodology can be difficult to use since creating a large number of exchangeable agents might not be possible.', '1408.6491-1-58-1': 'In our case, we found that we could only run ten agents in parallel given our hardware and network connection.', '1408.6491-1-58-2': 'Agents running at different times are not exchangeable since Google can determine the time at which an agent interacts with it.', '1408.6491-1-58-3': 'Thus, under the above methodology, we were limited to just ten exchangeable units.', '1408.6491-1-58-4': "Since some effects that the inputs have on Google's outputs can be probabilistic and subtle, they might be missed looking at just ten agents.", '1408.6491-1-59-0': 'To avoid this limitation, we extended the above methodology to handle nonexchangeable units using blocking [CITATION].', '1408.6491-1-59-1': 'To use blocking, we created blocks of exchangeable agents running in parallel.', '1408.6491-1-59-2': "Each block's agents were randomly partitioned into the control and experimental groups.", '1408.6491-1-59-3': 'Running these blocks in a staged fashion, the experiment proceeds on block after block.', '1408.6491-1-59-4': 'A modified permutation test now only compares the actual value of the test statistic to hypothetical values computed by reassignments of agents that respect the blocking structure.', '1408.6491-1-60-0': 'Using blocking, we can scale to any number of agents by running as many blocks as needed.', '1408.6491-1-60-1': 'However, the computation of the permutation test increases exponentially with the number of blocks.', '1408.6491-1-60-2': 'Thus, rather than compute the exact p-value, we estimate it by randomly sampling the possible reassignments.', '1408.6491-1-60-3': 'We can use a confidence interval to characterize the quality of the estimation [CITATION].', '1408.6491-1-60-4': 'The p-values we report are actually the upper bounds of the [MATH] confidence intervals of the p-values.', '1408.6491-1-61-0': '## Selecting Test Statistics', '1408.6491-1-62-0': 'The above methodology leaves open the question of how to select the test statistic.', '1408.6491-1-62-1': 'In some cases, the experimenter might be interested in a particular test statistic.', '1408.6491-1-62-2': 'For example, an experimenter testing ad choice could use a test statistic that counts the number of ads related to the removed interest.', '1408.6491-1-62-3': 'In other cases, the experimenter might be looking for any effect.', '1408.6491-1-62-4': 'Thus, AdFisher offers the ability to automatically select a test statistic.', '1408.6491-1-62-5': 'To do so, it collects an additional data set used for training a classifier with machine learning.', '1408.6491-1-62-6': "Figure [REF] shows an overview of AdFisher's workflow.", '1408.6491-1-63-0': 'To select a classifier, AdFisher uses cross validation on the training data to select among several possibilities.', '1408.6491-1-63-1': 'It then converts the classifier that is most accurate on the training data into a test statistic by treating it as a function over the ads seen by the agents to the number of correctly classified agents.', '1408.6491-1-63-2': 'If there is a difference between the ads shown to the two groups of agents, we would expect that the number correctly classified to be high.', '1408.6491-1-63-3': 'If no difference exists, then we would expect the number to be near to the guessing rate of [MATH].', '1408.6491-1-64-0': 'To avoid the possibility of seeing a high accuracy due to overfitting, AdFisher evaluates the accuracy of the classifier on a testing data set that is disjoint from the training data set.', '1408.6491-1-64-1': 'That is, in the language of statistics, we form our hypothesis about the test statistic being able to distinguish the groups before seeing the data on which we test it to ensure that it has predictive power.', '1408.6491-1-64-2': "AdFisher uses the permutation test to determine whether the degree to which the classifier's accuracy on the test data surpasses the guessing rate is statistically significant.", '1408.6491-1-64-3': 'That is, it calculates the p-value that measures the probability of seeing the observed accuracy given that the classifier is just guessing.', '1408.6491-1-64-4': 'If the p-value is below [MATH], we conclude that it is unlikely that classifier is guessing and that it must be making use of some difference between the ads shown to the two groups.', '1408.6491-1-65-0': '## Avoiding Pitfalls', '1408.6491-1-66-0': 'The above methodology avoids some pitfalls.', '1408.6491-1-66-1': 'Most fundamentally, we use a statistical analysis whose assumptions matches those of our experimental design.', '1408.6491-1-66-2': 'Assumptions required by many statistical analyses appear unjustifiable in our setting.', '1408.6491-1-66-3': 'For example, many analyses assume that the agents do not interact or that the ads are independent and identically distributed (e.g., [CITATION]).', '1408.6491-1-66-4': "However, given that all the agents receive ads from the same pool of possible ads governed by the same advertisers' budgets, these assumptions appear unlikely to hold.", '1408.6491-1-66-5': 'Indeed, empirical evidence suggests that it does not [CITATION].', '1408.6491-1-66-6': 'Our use of the permutation test, which does not require this assumption, allows us to ensure the statistical soundness of our analysis without making these assumptions [CITATION].', '1408.6491-1-67-0': 'Our use of randomization implies that many factors that could be confounding factors in unrandomized design become noise in our design (e.g., [CITATION]).', '1408.6491-1-67-1': 'While such noise may require us to use a large sample size to find an effect, it does not affect the soundness of our analysis.', '1408.6491-1-68-0': 'Our use of two data sets, one for training the classifier to select the test statistic and one for hypothesis testing ensures that we do not engage in overfitting, data dredging, or multiple hypothesis testing (e.g., [CITATION]) .', '1408.6491-1-68-1': 'All these problems result from looking for so many possible patterns that one is found by chance.', '1408.6491-1-68-2': 'While we look for many patterns in the training data, we only check for one in the testing data.', '1408.6491-1-69-0': 'Relatedly, by reporting a p-value, we provide a quantitative measure of the confidence we have that the observed effect is genuine and not just by chance [CITATION].', '1408.6491-1-69-1': 'Reporting simply the classifier accuracy or that some difference occurred fails to quantify the possibility that the result was a fluke.', '1408.6491-1-70-0': '## Scope', '1408.6491-1-71-0': 'We restrict the scope of our methodology to making claims that an effect exists with high likelihood as quantified by the p-value.', '1408.6491-1-71-1': 'That is, we expect our methodology to only rarely suggest that an effect exists when one does not.', '1408.6491-1-72-0': 'We do not claim the converse of "completeness" or "power": we might fail to detect some use of information.', '1408.6491-1-72-1': 'For example, Google might turn off some usage upon detecting that our tool does not behave like a normal user.', '1408.6491-1-72-2': 'Despite this limitation, we found interesting instances of usage.', '1408.6491-1-73-0': 'Furthermore, we do not claim that our results generalize to all users.', '1408.6491-1-73-1': 'To do so, we would need to a take a random sample of all users, their IP addresses, browsers, and behaviors, which is prohibitively expensive.', '1408.6491-1-73-2': 'For example, instead of turning off some usage upon detecting our tool, Google might turn it on.', '1408.6491-1-73-3': 'While the detected usage would be real (and our method sound), it might not be experienced by normal users.', '1408.6491-1-73-4': 'However, it would be odd if Google purposefully performs questionable behaviors only in front of those attempting to find it.', '1408.6491-1-74-0': 'While we use webpages associated with various interests to simulate users with those interests, we cannot establish that having the interest itself caused the change to ads.', '1408.6491-1-74-1': 'It is possible that other features of the visited webpages causes change, a form of confounding called "profile contamination" [CITATION], since the pages cover other topics as well.', '1408.6491-1-74-2': 'Nevertheless, we have determined that visiting webpages associated with the interest does result in seeing a change, which should give pause to those visiting webpages associated with sensitive interests.', '1408.6491-1-75-0': 'Lastly, we do not attempt to determine how the information was used.', '1408.6491-1-75-1': 'It could have been used by Google directly for targeting or it could have been used by advertisers to place their bids.', '1408.6491-1-75-2': 'We cannot assign blame.', '1408.6491-1-75-3': 'We hope future work will shed light on these issues, but given that we cannot observe the interactions between Google and advertisers, we are unsure whether it can be done.', '1408.6491-1-76-0': '# AdFisher', '1408.6491-1-77-0': 'We provide AdFisher, a tool implementing our methodology.', '1408.6491-1-77-1': 'AdFisher makes it easy to run experiments using the above methodology for the set of treatments, measurements, and classifiers (test statistics) we have implemented.', '1408.6491-1-77-2': 'AdFisher is also extensible allowing the experimenter to implement additional treatments, measurements, or test statistics.', '1408.6491-1-77-3': 'For example, an experimenter without any interest in methodological concerns could use AdFisher to run a study on an online system we have not considered by writing just the code needed to interact with that system; she will not need to reimplement the code running and analyzing the experiment.', '1408.6491-1-78-0': 'AdFisher is structured as a Python API providing functions for setting up, running, and analyzing experiments.', '1408.6491-1-78-1': 'We use the SciPy [CITATION] library for implementing the statistical analyses of the core methodology.', '1408.6491-1-78-2': 'For browser automation, we used Selenium 2.39.0 and its bindings for Python 2.7 to drive Firefox browsers.', '1408.6491-1-79-0': 'To simulate a new person on the network, AdFisher creates each agent from a fresh browser instance with no browsing history, cookies, or other personalization.', '1408.6491-1-79-1': 'AdFisher randomly assigns each agent to a group and applies the appropriate treatment, such as having the browser visit webpages.', '1408.6491-1-79-2': 'Next, AdFisher makes measurements of the agent, such as collecting the ads shown to the browser upon visiting another webpage.', '1408.6491-1-79-3': 'All of the agents within a block executes and finishes the treatments before moving on to the measurements to remove time as a factor.', '1408.6491-1-79-4': 'AdFisher runs all the agents on the same machine to prevent differences based on location or IP-address between agents.', '1408.6491-1-80-0': 'We detail the particular treatments, measurements, and test statistics that we have implemented below.', '1408.6491-1-80-1': 'We also discuss how AdFisher aids the experimenter in understanding the results.', '1408.6491-1-80-2': 'Additional information about parameters that the experimenter may vary can be found in Appendix [REF].', '1408.6491-1-81-0': 'Treatments AdFisher automatically applies the treatments assigned to each group.', '1408.6491-1-81-1': 'Typically, these treatments involve invoking the Selenium WebDriver to make the agent interact with webpages.', '1408.6491-1-82-0': 'AdFisher makes it easy to perform common treatments by providing ready-made implementations of these treatments.', '1408.6491-1-82-1': "The simplest stock treatments we provide set interests in Google's Ad Settings.", '1408.6491-1-82-2': 'We also provide ready-made treatments for setting the gender and age range.', '1408.6491-1-82-3': 'Another stock treatment is to visit a list of webpages.', '1408.6491-1-83-0': 'To make it easy to see whether websites associated with a particular interest causes a change in behavior, we have provided the ability to create lists of webpages associated with each interest tracked by Alexa.', '1408.6491-1-83-1': 'For each category, Alexa tracks the top websites sorted according to their traffic rank measure (a combination of the number of users and page views).', '1408.6491-1-83-2': 'The function takes the URL of an interest page on Alexa and downloads the URLs of the top webpages Alexa associates with that interest.', '1408.6491-1-83-3': 'By default, it downloads [MATH].', '1408.6491-1-83-4': 'The list can then be converted into the treatment of visiting those webpages.', '1408.6491-1-83-5': 'While these treatments do not correspond directly to having such an interest, it allows us to study how Google responds to people visiting webpages associated with those interests.', '1408.6491-1-84-0': 'Often in our experiments, we compared the effects of a certain treatment applied to the experimental group against the null treatment applied to the control group.', '1408.6491-1-84-1': 'Under the null treatment, agents do nothing while agents under a different treatment complete their respective treatment phase.', '1408.6491-1-85-0': "Measurements Our tool currently measures the values set in Google's Ad Settings page and the ads shown to the agents after the treatments.", '1408.6491-1-85-1': 'We provide stock measurements for collecting and analyzing text ads.', '1408.6491-1-85-2': 'Experimenters can add functionality for image, video, and flash ads as desired.', '1408.6491-1-86-0': 'To find a good default webpage to visit for ad collection, we looked to news sites since they generally show many ads.', '1408.6491-1-86-1': 'We looked at the five of the top [MATH] news websites on alexa.com that also served text ads by Google: theguardian.com/us, timesofindia.indiatimes.com, bbc.com/news, reuters.com/news/us and bloomberg.com.', '1408.6491-1-86-2': 'Among these, we selected the Times of India since, at [MATH], it serves the most text ads per page reload.', '1408.6491-1-87-0': 'We provide functionality that will visit any of the above listed news websites and reload the page a fixed number of times (default [MATH]).', '1408.6491-1-87-1': 'For each page reload, it parses the page to find the ads shown by Google and stores the ads.', '1408.6491-1-87-2': 'The experimenter can add parsers for other webpages.', '1408.6491-1-88-0': 'Classification While the experimenter can provide AdFisher with a test statistic to use on all the collected data, AdFisher is also capable of splitting the data set into training and testing subsets.', '1408.6491-1-88-1': 'AdFisher automatically examines a training subset of the collected measurements to select a classifier that distinguishes between the measurements taken from each group.', '1408.6491-1-88-2': 'From the point of view of machine learning, each of the sets of ads collected by each agent corresponds to an instance of the concept the classifier is attempting to learn.', '1408.6491-1-89-0': 'The machine learning algorithms operate over sets of features.', '1408.6491-1-89-1': 'AdFisher has functions for converting the text ads seen by an agent into three different feature sets.', '1408.6491-1-89-2': 'The URL feature set consists of the URLs displayed by the ads (or occasionally some other text if the ad displays it where URLs normally go).', '1408.6491-1-89-3': "Under this feature set, the feature vector representing an agent's data has a value of [MATH] in the [MATH]th entry iff the agent received [MATH] ads that display the [MATH]th URL where the order is fixed but arbitrary.", '1408.6491-1-90-0': 'The URL+Title feature set looks at both the displayed URL and the title of the ad jointly.', '1408.6491-1-90-1': "It represents an agent's data as a vector where the [MATH]th entry is [MATH] iff the agent received [MATH] ads containing the [MATH]th pair of a URL and title.", '1408.6491-1-91-0': 'The third feature set AdFisher has implemented is the word feature set.', '1408.6491-1-91-1': 'This set is based on word stems, the main part of the word with suffixes such as "ed" or "ing" removed in a manner similar to the work of Balebako et al. [CITATION].', '1408.6491-1-91-2': 'Each word stem that appeared in an ad is assigned a unique id.', '1408.6491-1-91-3': "The [MATH]th entry in the feature vector is the number of times that words with the [MATH]th stem appeared in the agent's ads.", '1408.6491-1-92-0': 'We explored using a variety of classification algorithms provided by the scikit-learn library [CITATION].', '1408.6491-1-92-1': 'We found that logistic regression with an L2 penalty over the URL+title feature set consistently performed well compared to the others.', '1408.6491-1-92-2': 'Thus, we have disabled the others by default.', '1408.6491-1-93-0': 'At its core, logistic regression predicts a class given a feature vector by multiplying each of the entries of the vector by its own weighting coefficient (e.g., [CITATION]).', '1408.6491-1-93-1': 'It then takes a the sum of all these products.', '1408.6491-1-93-2': 'If the sum is positive, it predicts one class; if negative, it predicts the other.', '1408.6491-1-93-3': 'Thus, since our feature vectors contain only [MATH]s and [MATH]s, coefficients of large positive value imply that that feature is associated with being in the first group; negative values are associated with the second group; values near zero are uninformative.', '1408.6491-1-94-0': 'While using logistic regression, the training stage consists of selecting the coefficients assigned to each feature to predict the training data well.', '1408.6491-1-94-1': 'Selecting coefficients requires balancing the training-accuracy of the model with avoiding overfitting the data with an overly complex model.', '1408.6491-1-94-2': 'To do so, we apply [MATH]-fold cross-validation on the training data to select how complex of a model to use.', '1408.6491-1-94-3': 'By default, we split the data into training and test sets by using the last [MATH] of the data collected for testing.', '1408.6491-1-95-0': 'Explanations To explain how the learned classifier distinguished between the groups, we explored several methods.', '1408.6491-1-95-1': 'We found the most informative to be the model produced by the classifier itself.', '1408.6491-1-95-2': 'Recall that logistic regression weighted the various features of the instances with coefficients reflecting how predictive they are of each group.', '1408.6491-1-95-3': 'Thus, with the URL+title feature set, examining the features with the most extreme coefficients identifies the URL+title pair most used to predict the group to which agents receiving an ad with that URL+title belongs.', '1408.6491-1-96-0': 'We also explored using the proportion of times an ad was served to agents in one group to the total number of times observed by all groups.', '1408.6491-1-96-1': 'However, this did not provide much insight since the proportion typically reached its maximum value of [MATH] from ads that only appeared once.', '1408.6491-1-96-2': 'Another choice we explored was to compute the difference in the number of times an ad appears between the groups.', '1408.6491-1-96-3': 'However, this metric is also highly influenced by how common the ad is across all groups.', '1408.6491-1-97-0': '# Experiments', '1408.6491-1-98-0': 'In this section, we discuss experiments that we carried out using AdFisher.', '1408.6491-1-98-1': 'We studied each of the properties of interest.', '1408.6491-1-98-2': 'We found violations of nondiscrimination and data transparency and cases of compliance with effectful and ad choice.', '1408.6491-1-98-3': 'Table [REF] summarizes our findings.', '1408.6491-1-99-0': "We start by providing additional background on Google's Ad Settings page.", '1408.6491-1-100-0': '## Background on Google Ad Settings', '1408.6491-1-101-0': 'Google defines its Ad Settings page as "a Google tool that helps you control the ads you see on Google services and on websites that partner with Google".', '1408.6491-1-101-1': 'This page contains demographics like gender, age, language, and interests.', '1408.6491-1-101-2': 'There are two classes for each of these demographics: (1) for ads on Google services like Search, Gmail, YouTube and Maps; and (2) for ads across the web.', '1408.6491-1-101-3': 'The demographics for ads on Google services show up only when a user is logged in, and follow the user irrespective of which browser the user logs in from.', '1408.6491-1-101-4': 'In this work, we do not analyze ads on Google services, or the corresponding demographics on Ad Settings.', '1408.6491-1-101-5': 'The demographics for ads across the web follow the browser irrespective of who is logged in (or not).', '1408.6491-1-101-6': 'They show up as "N/A" if the user has either opted out of interest-based Google ads across the web, or if he has cleared all cookies.', '1408.6491-1-101-7': 'In either case, the user has the option to enable interest-based ads.', '1408.6491-1-101-8': 'Once enabled, Google places a cookie on the browser to keep track of the browsing activities.', '1408.6491-1-101-9': 'A user can manually edit these demographics on their browser.', '1408.6491-1-102-0': "If the demographics on Google's Ad Settings are left untouched, Google infers them over time.", '1408.6491-1-102-1': 'In the initial state, immediately after a DoubleClick cookie is placed on the browser, all the demographics are displayed as "unknown".', '1408.6491-1-102-2': 'The page mentions that all demographics are inferred "based on websites you\'ve visited".', '1408.6491-1-102-3': 'As the user browses websites on the Internet, Google infers these demographics and displays them on this page.', '1408.6491-1-102-4': 'By giving users the option to edit these demographics, Google claims to have given choice to users about the ads Google serves them.', '1408.6491-1-102-5': 'So, the Ad Settings can function both as input to and output from Google.', '1408.6491-1-102-6': 'In our experiments, we explore these interactions.', '1408.6491-1-103-0': '## Discrimination', '1408.6491-1-104-0': 'AdFisher can demonstrate discrimination.', '1408.6491-1-104-1': 'If AdFisher finds a statistically significant difference in how Google treats two experimental groups, one consisting of members of the class in question and one whose members are not, then the experimenter has strong evidence that Google discriminates on that class.', '1408.6491-1-104-2': "In particular, we use AdFisher's ability to automatically select a test statistic to check for possible differences to test the null hypothesis that the two experimental groups have no differences in the ads they received.", '1408.6491-1-105-0': 'For the treatments, we were interested in how providing information to Ad Settings would alter the ads seen.', '1408.6491-1-105-1': "In particular, we explored whether Google's reactions were appropriate.", '1408.6491-1-105-2': 'In a series of experiments, we set the gender of one group to female and the other to male.', '1408.6491-1-105-3': 'In one of the experiments, the agents went straight to collecting ads; in the others, they simulated an interest in jobs.', '1408.6491-1-105-4': 'In all but one experiment, they collected ads from the Times of India (TOI); in the exception, they collected ads from the Guardian.', '1408.6491-1-105-5': 'In one experiment, they also visited the top 10 websites for the U.S. according to Alexa.com to fill out their interests.', '1408.6491-1-106-0': 'Table [REF] summarizes the results.', '1408.6491-1-107-0': 'AdFisher found a statistically significant effect for the agents that first visited the job-related sites in May, 2014.', '1408.6491-1-107-1': 'We also found evidence of discrimination in the nature of the effect.', '1408.6491-1-107-2': 'In particular, in the gender and jobs experiment, we found that females received fewer instances of an ad encouraging the taking of high paying jobs than males.', '1408.6491-1-107-3': 'AdFisher did not find any statistically significant differences among the agents that did not visit the job-related pages or those operating in July, 2014.', '1408.6491-1-107-4': 'We detail the experiment finding a violation before discussing why we think the other results were insignificant.', '1408.6491-1-108-0': 'Gender and Jobs In this experiment, we examine how changing the gender demographic on Google Ad Settings affects the ads served and interests inferred for agents browsing employment related websites.', '1408.6491-1-108-1': 'We setup AdFisher to have the agents in one group visit the Google Ad Settings page and set the gender bit to female while agents in the other group set theirs to male.', '1408.6491-1-108-2': 'All the agents then visited the top [MATH] websites listed under the Employment category of Alexa.', '1408.6491-1-108-3': 'The agents then collected ads from the Times of India.', '1408.6491-1-109-0': 'AdFisher ran [MATH] blocks of [MATH] agents each.', '1408.6491-1-109-1': '(We used blocks of size [MATH] in all our experiments.)', '1408.6491-1-109-2': 'AdFisher used the ads of [MATH] agents ([MATH] from each group) for training a classifier using the URL+title feature set.', '1408.6491-1-109-3': "AdFisher used the remaining [MATH] agents' ads for testing.", '1408.6491-1-109-4': 'The learned classifier attained a test-accuracy of [MATH], suggesting that Google did in fact treat the genders differently.', '1408.6491-1-110-0': 'To test whether this response was statistically significant, AdFisher computed a p-value by running the permutation test on a million randomly selected block-respecting permutations of the data.', '1408.6491-1-110-1': 'The significance test yielded a p-value of [MATH].', '1408.6491-1-111-0': 'We then examined the model learned by AdFisher to explain the nature of the difference.', '1408.6491-1-111-1': 'Table [REF] shows the five URL+title pairs that the model identifies as the strongest indication of being from the female or male group.', '1408.6491-1-111-2': 'How ads for identifying the two groups differ is concerning.', '1408.6491-1-111-3': 'The two URL+title pairs with the highest coefficients for indicating a male were for a career coaching service for "[MATH]k+" executive positions.', '1408.6491-1-111-4': 'Google showed the ads [MATH] times to the male group but just [MATH] times to the female group.', '1408.6491-1-111-5': 'The top two URL+title pairs for the female group was for a generic job posting service and for an auto dealer.', '1408.6491-1-112-0': 'We believe these results suggest the possibility of discrimination, intentional or otherwise.', '1408.6491-1-112-1': 'Encouraging only male candidates to seek high-paying jobs could further the current gender pay gap (e.g., [CITATION]).', '1408.6491-1-112-2': 'Google or the advertiser might have purposely targeted just males.', '1408.6491-1-112-3': 'Alternatively, males might be more likely to click on an ad for a high-paying job than females, and Google may have picked up this signal from past interactions with other users.', '1408.6491-1-112-4': "Google's policies allow it to serve different ads based on gender, but dissimilarities of this kind are concerning even if unintentional and allowed.", '1408.6491-1-113-0': 'The found discrimination in this experiment was predominately from a pair of job-related ads for the same service making the finding highly sensitive to changes in the serving of these ads.', '1408.6491-1-113-1': 'A closer examination of the ads from the same experimental setup ran in July showed that the frequency of these ads reduced from [MATH] to just [MATH], with one of the ads completely disappearing.', '1408.6491-1-113-2': 'These [MATH] ads were only shown to males, continuing the pattern of discrimination.', '1408.6491-1-113-3': 'This pattern was recognized by the machine learning algorithm, which selected the ad as the second most useful for identifying males.', '1408.6491-1-113-4': '(See Table [REF].)', '1408.6491-1-113-5': 'However, they were too infrequent to establish statistical significance.', '1408.6491-1-113-6': 'A longer running experiment with more blocks might have succeeded.', '1408.6491-1-114-0': '## Transparency', '1408.6491-1-115-0': 'AdFisher can demonstrate violations of individual data use transparency.', '1408.6491-1-115-1': 'AdFisher tests the null hypothesis that two groups of agents with the same ad settings receives ads from the same distribution despite being subjected to different experimental treatments.', '1408.6491-1-115-2': 'Rejecting the null hypothesis implies that some difference exists in the ads that is not documented by the ad settings.', '1408.6491-1-116-0': "In particular, we ran a series of experiments to examine how much transparency Google's Ad Settings provided.", '1408.6491-1-116-1': 'We checked whether visiting webpages associated with some interest could cause a change in the ads shown that is not reflected in the settings.', '1408.6491-1-117-0': 'We ran such experiments for five interests: substance abuse, disabilities, infertility, mental disorders, and "adult" websites.', '1408.6491-1-117-1': 'Results from statistical analysis of whether the ads changed are shown in Table [REF].', '1408.6491-1-118-0': 'We examined the interests found in the settings for the two cases where we found a statistically significant difference in ads, substance abuse and disability.', '1408.6491-1-118-1': 'We found that settings did not change at all for substance abuse and changed in an unexpected manner for disabilities.', '1408.6491-1-118-2': 'Thus, we detail these two experiments below.', '1408.6491-1-119-0': "Substance Abuse We were interested in whether Google's outputs would change in response to visiting webpages associated with substance abuse, a highly sensitive topic.", '1408.6491-1-119-1': 'Thus, we ran an experiment in which the experimental group visited such websites while the control group idled.', '1408.6491-1-119-2': 'Then, we collected the Ad Settings and the Google ads shown to the agents at the Times of India.', '1408.6491-1-119-3': 'For the webpages associated with substance abuse, we used the top [MATH] websites on the Alexa list for substance abuse.', '1408.6491-1-120-0': 'AdFisher ran [MATH] blocks of [MATH] agents each.', '1408.6491-1-120-1': 'At the end of visiting the webpages associated with substance abuse, none of the [MATH] agents in the experimental group had interests listed on their Ad Settings pages.', '1408.6491-1-120-2': '(None of the agents in the control group did either since the settings start out empty.)', '1408.6491-1-120-3': 'Thus, one might expect that Google did not factor the visits to the webpages into its ad selection algorithms.', '1408.6491-1-121-0': 'However, the ads collected from the Times of India told a different story.', '1408.6491-1-121-1': 'The learned classifier attained a test-accuracy of [MATH], suggesting that Google did in fact respond to the page visits.', '1408.6491-1-121-2': 'Indeed, using the permutation test, AdFisher found a p-value of [MATH].', '1408.6491-1-121-3': "Thus, we conclude that the differences are statistically significant: Google's ads changed in response to visiting the webpages associated with substance abuse.", '1408.6491-1-121-4': 'Despite this change being significant, the Ad Settings pages provided no hint of its existence: the transparency tool is opaque!', '1408.6491-1-122-0': 'We looked at the URL+title pairs with the highest coefficients for identifying the experimental group that visited the websites related to substance abuse.', '1408.6491-1-122-1': 'Table [REF] provides information on coefficients and URL+titles learned.', '1408.6491-1-122-2': 'The three highest were for "Watershed Rehab".', '1408.6491-1-122-3': 'The top two had URLs for this drug and alcohol rehab center.', '1408.6491-1-122-4': 'The third lacked a URL and had other text in its place.', '1408.6491-1-122-5': "Figure [REF] shows one of Watershed's ads.", '1408.6491-1-123-0': 'The experimental group saw these ads a total of [MATH] times ([MATH] of the ads); the control group never saw any of them nor contained any ads with the word "rehab" or "rehabilitation".', '1408.6491-1-123-1': 'None of the top five URL+title pairs for identifying the control group had any discernible relationship with rehab or substance abuse.', '1408.6491-1-124-0': 'These results remain robust across variations on this design with statistical significance in three variations.', '1408.6491-1-124-1': 'For example, two of these ads remain the top two ads for identifying the agents that visited the substance abuse websites in July using ads collected from the Guardian (Table [REF]).', '1408.6491-1-125-0': "The fact that no interests are displayed on Google Ad Settings in this experiment despite Google's ads differing for those possibly looking for rehab shows that the tool can be opaque.", '1408.6491-1-125-1': 'Users cannot see that this interest has been inferred or remove it.', '1408.6491-1-125-2': 'We cannot, however, determine whether this was an instance of deliberate targeting or by accident through some other feature of the visited webpages.', '1408.6491-1-126-0': 'Disabilities This experiment was nearly identical in setup but used websites related to disabilities instead of substance abuse.', '1408.6491-1-126-1': 'We used the top [MATH] websites on Alexa on the topic.', '1408.6491-1-127-0': 'For this experiment, AdFisher found a classifier with a test-accuracy of [MATH].', '1408.6491-1-127-1': 'It found a statistically significant difference with a p-value of [MATH].', '1408.6491-1-128-0': 'Looking at the top ads for identifying agents that visited the webpages associated with disabilities, we see that the top two ads have the URL www.abilitiesexpo.com and the titles "Mobility Lifter" and "Standing Wheelchairs".', '1408.6491-1-128-1': 'They were shown a total of [MATH] times to the experimental group but never to the control group.', '1408.6491-1-128-2': '(See Table [REF].)', '1408.6491-1-129-0': 'This time, Google did change the settings in response to the agents visiting the websites.', '1408.6491-1-129-1': 'Figure [REF] shows the interests selected for the experimental group.', '1408.6491-1-129-2': '(The control group, which did nothing, had no interests selected.)', '1408.6491-1-130-0': 'None of them are directly related to disabilities suggesting that Google might have focused on other aspects of the visited pages.', '1408.6491-1-130-1': 'However, it remains difficult to explain the presence of the disability-related ads in just the experimental group.', '1408.6491-1-131-0': '## Effectful Choice', '1408.6491-1-132-0': 'We tested whether making changes to Ad Settings has an effect on the ads seen, thereby giving the users a degree of choice over the ads.', '1408.6491-1-132-1': 'In particular, AdFisher tests the null hypothesis that changing some ad setting has no effect on the ads.', '1408.6491-1-133-0': 'First, we tested whether opting out of tracking actually had an effect by comparing the ads shown to agents that opted out after visiting car-related websites to ads from those that did not opt out.', '1408.6491-1-133-1': 'We found a statistically significant difference.', '1408.6491-1-134-0': 'We also tested whether removing interests from the settings page actually had an effect.', '1408.6491-1-134-1': 'To do so, we set AdFisher to have both groups of agents simulate some interest.', '1408.6491-1-134-2': "AdFisher then had the agents in one of the groups remove interests from Google's Ad Settings related to the induced interest.", '1408.6491-1-134-3': 'We found statistically significant differences between the ads both groups collected from the Times of India for two induced interests: online dating and weight loss.', '1408.6491-1-134-4': 'Table [REF] summarizes the results.', '1408.6491-1-135-0': 'We describe one in detail below.', '1408.6491-1-136-0': 'Online Dating We simulated an interest in online dating by visiting the website www.midsummerseve.com/, a website we choose since it sets Google\'s ad setting for "Dating Personals".', '1408.6491-1-136-1': 'AdFisher then had just the agents in the experimental group remove the interest "Dating Personals" (the only one containing the keyword "dating").', '1408.6491-1-136-2': 'All the agents then collected ads from the Times of India.', '1408.6491-1-137-0': 'AdFisher found statistically significant differences between the groups with a classifier accuracy of 74% and a p-value of [MATH].', '1408.6491-1-138-0': 'Furthermore, the effect appears related to the interests removed.', '1408.6491-1-138-1': 'The top ad for identifying agents that kept the romantic interests has the title "Are You Single?"', '1408.6491-1-138-2': 'and the second ad\'s title is "Why can\'t I find a date?"', '1408.6491-1-138-3': 'None of the top five for the control group that removed the interests were related to dating (Table [REF]).', '1408.6491-1-138-4': 'Thus, the ad settings appear to actually give users the ability to avoid ads they might dislike or find embarrassing.', '1408.6491-1-138-5': 'In the next set of experiments, we explicitly test for this ability.', '1408.6491-1-139-0': 'We repeated this experiment in July using the websites relationshipsurgery.com and datemypet.com, which also have an effect on Ad Settings, but did not find statistically significant differences.', '1408.6491-1-140-0': '## Ad Choice', '1408.6491-1-141-0': 'Whereas the other experiments tested merely for the presence of an effect, testing for ad choice requires determining whether the effect is an increase or decrease in the number of ads seen.', '1408.6491-1-141-1': 'Fortunately, since AdFisher uses a one-sided permutation test, it tests for either an increase or a decrease, but not for both simultaneously, making it usable for this purpose.', '1408.6491-1-141-2': 'In particular, after removing an interest, we check for a decrease to test for compliance using the null hypothesis that either no change or an increase occurred, since rejecting this hypothesis would imply that a decrease in the number of related ads occurred.', '1408.6491-1-141-3': 'To check for a violation, we test the null hypothesis that either no change or a decrease occurred.', '1408.6491-1-142-0': 'Due to testing two hypotheses, we use an adjustment to the p-value cutoff considered significant to avoid finding significant results simply from testing multiple hypotheses.', '1408.6491-1-142-1': 'In particular, we use the standard Bonferroni correction, which calls for a p-value of [MATH] or less (e.g., [CITATION]).', '1408.6491-1-143-0': 'We ran three experiments checking for ad choice.', '1408.6491-1-143-1': 'The experiments followed the same setup as the effectful choice ones, but this time we used all the blocks for testing a given test statistic.', '1408.6491-1-143-2': 'The test statistic counted the number of ads containing keywords.', '1408.6491-1-143-3': 'In the first, we again test online dating using relationshipsurgery.com and datemypet.com.', '1408.6491-1-143-4': 'In the next two we tested weight loss websites again but using two different sets of keywords.', '1408.6491-1-143-5': 'Table [REF] summarizes the results.', '1408.6491-1-144-0': 'In particular, we found that removing online dating did in fact decrease the number of ads containing related keywords.', '1408.6491-1-144-1': 'We detail the inconclusive results for weight loss below.', '1408.6491-1-145-0': 'Weight Loss We induced an interest in weight loss by visiting dietingsucks.blogspot.com.', '1408.6491-1-145-1': 'Afterwards, the agents in the experimental group removed the interests "Fitness" and "Fitness Equipment and Accessories", the only ones related to weight loss.', '1408.6491-1-145-2': 'We then used a test statistic that counted the number of ads containing the keyword "fitness".', '1408.6491-1-145-3': 'Interestingly, the test statistic was higher on the group with the interests removed, although not to a statistically significant degree.', '1408.6491-1-145-4': 'We repeated the process with a longer keyword list and found that removing interests decreased test statistic this time, but also not to a statistically significant degree.', '1408.6491-1-146-0': '# Conclusions', '1408.6491-1-147-0': 'Using AdFisher, we conducted 20 experiments using 16,570 agents and that collected 570,000 ads.', '1408.6491-1-147-1': 'Our experiments found the presence of discrimination, opacity, and choice in targeted ads of Google.', '1408.6491-1-147-2': 'Discrimination, is at some level, inherent to profiling: the point of profiling is to treat some people differently.', '1408.6491-1-147-3': 'While customization can be helpful, we highlight a case where the customization appears inappropriate taking on the negative connotations of discrimination.', '1408.6491-1-147-4': 'In particular, we found that males were shown ads encouraging the seeking of high paying jobs more than females (Section [REF]).', '1408.6491-1-147-5': 'We know of no justification for such customization.', '1408.6491-1-148-0': 'Opacity occurs when a tool designed to provide transparency into how ads are selected and the profile kept on a person actually fails to provide such transparency.', '1408.6491-1-148-1': 'Our experiment on substance abuse showed an extreme case in which the tool failed to show any profiling but the ad distributions were significantly different in response to behavior (Section [REF]).', '1408.6491-1-148-2': 'In particular, our experiment achieved a p-value of [MATH] - 9000 times more significant than the standard [MATH] cutoff for statistical significance.', '1408.6491-1-148-3': 'This experiment remained robust to variations showing a pattern of such opacity.', '1408.6491-1-149-0': 'Ideally, tools, such as Ad Settings, would provide a complete representation of the profile kept on a person, or at least the portion of the profile that is used to select ads shown to the person.', '1408.6491-1-149-1': 'Two people with identical profiles might continue to receive different ads due to other factors affecting the choice of ads such as A/B testing or the time of day.', '1408.6491-1-149-2': 'However, systematic differences between ads shown at the same time and in the same context, such as those we found, would not exist for such pairs of people.', '1408.6491-1-150-0': 'Lastly, we found that Google Ad Settings does provide the user with a degree of choice about the ads shown.', '1408.6491-1-150-1': 'In this aspect, the transparency tool operated as we expected.', '1408.6491-1-150-2': 'In at least some cases, removing interests from the settings behaved in a manner consistent with expectations and removed ads related to the removed interests.', '1408.6491-1-151-0': 'Our work has made it possible to automate these experiments.', '1408.6491-1-151-1': "Our tool, AdFisher, makes it easy to run additional experiments exploring the relations between Google's ads and settings.", '1408.6491-1-151-2': 'It can be extended to study other systems.', '1408.6491-1-151-3': "It's design ensures that it can run and analyze large scale experiments to find subtle differences.", '1408.6491-1-151-4': 'It automatically finds differences between large data sets produced by different groups of agents and explains the nature of those differences.', '1408.6491-1-151-5': 'By completely automating the data analysis, we ensure that an appropriate statistical analysis determines whether these differences are statistically significant, ensuring sound conclusions.', '1408.6491-1-152-0': '# Future Work', '1408.6491-1-153-0': 'Some future work would be extensions of AdFisher.', '1408.6491-1-153-1': 'For example, enabling the analysis of image ads or the study of other Google services such as Gmail.', '1408.6491-1-154-0': 'Other future work is more conceptual.', '1408.6491-1-154-1': 'For example, ideally we would like to know whether Google profiles people with substance abuse issues in violation of their policy promising not to market based on health information.', '1408.6491-1-154-2': 'However, our experiments check whether going to websites associated with substance abuse affects the ads Google shows.', '1408.6491-1-154-3': 'These two concerns diverge in two ways.', '1408.6491-1-155-0': 'First, it is possible that Google profiling did not cause the changes; profiling by the advertisers working with Google could be the cause.', '1408.6491-1-155-1': 'Such advertisers may have found combinations of interests to bid on that allows them to target those with substance abuse problems without Google attempting to enable such targeting.', '1408.6491-1-155-2': 'Before we can assign blame, we must have a better understanding of the interactions between Google and the advertisers.', '1408.6491-1-155-3': 'Nevertheless, these results are useful in that web advertising companies can use AdFisher for internal auditing of their systems, whose complexity, scale, and dynamic models put them beyond the current capabilities of code analysis.', '1408.6491-1-155-4': 'Our results offer regulators the evidence needed to investigate companies.', '1408.6491-1-155-5': 'They also increase public awareness.', '1408.6491-1-155-6': 'Knowing of the existence of these concerning behaviors lets regulators and the public hold companies accountable.', '1408.6491-1-156-0': 'Second, we cannot send a pure signal to Google that a person has substance abuse issues.', '1408.6491-1-156-1': 'We must simulate the behavior of such a person by visiting webpages we believe someone with a substance abuse issue might visit.', '1408.6491-1-156-2': 'However, not only might these pages do a poor job simulating such a person, but they also contain confounding factors, such as content unrelated to substance abuse.', '1408.6491-1-156-3': 'Thus, while we can soundly say visiting those webpages had an effect, we cannot assign that cause to substance abuse or explain what content on those pages caused the effect.', '1408.6491-1-156-4': 'We would like to find ways of controlling for these confounding factors.', '1408.6491-1-157-0': 'For these reasons, we cannot claim that Google has violated its policies.', '1408.6491-1-157-1': 'In fact, we consider it more likely that Google has lost control over its massive, automated profiling system.', '1408.6491-1-157-2': 'Large scale machine learning can behave in unexpected ways.', '1408.6491-1-157-3': 'With this in mind, we hope future research will examine how to produce machine learning algorithms that automatically avoid discriminating against users in unacceptable ways and automatically provide transparency to users.', '1408.6491-1-157-4': 'We suspect our tool could be used to test or even guide such algorithms.'} | {'1408.6491-2-0-0': "To partly address people's concerns over web tracking, Google has created the Ad Settings webpage to provide information about and some choice over the profiles Google creates on users.", '1408.6491-2-0-1': "We present AdFisher, an automated tool that explores how user behaviors, Google's ads, and Ad Settings interact.", '1408.6491-2-0-2': 'AdFisher can run browser-based experiments and analyze data using machine learning and significance tests.', '1408.6491-2-0-3': 'Our tool uses a rigorous experimental design and statistical analysis to ensure the statistical soundness of our results.', '1408.6491-2-0-4': "We use AdFisher to find that the Ad Settings was opaque about some features of a user's profile, that it does provide some choice on ads, and that these choices can lead to seemingly discriminatory ads.", '1408.6491-2-0-5': 'In particular, we found that visiting webpages associated with substance abuse changed the ads shown but not the settings page.', '1408.6491-2-0-6': 'We also found that setting the gender to female resulted in getting fewer instances of an ad related to high paying jobs than setting it to male.', '1408.6491-2-0-7': 'We cannot determine who caused these findings due to our limited visibility into the ad ecosystem, which includes Google, advertisers, websites, and users.', '1408.6491-2-0-8': 'Nevertheless, these results can form the starting point for deeper investigations by either the companies themselves or by regulatory bodies.', '1408.6491-2-1-0': '# Introduction', '1408.6491-2-2-0': 'Problem and Overview With the advancement of tracking technologies and the growth of online data aggregators, data collection on the Internet has become a serious privacy concern.', '1408.6491-2-2-1': 'Colossal amounts of collected data are used, sold, and resold for serving targeted content, notably advertisements, on websites (e.g., [CITATION]).', '1408.6491-2-2-2': "Many websites providing content, such as news, outsource their advertising operations to large third-party ad networks, such as Google's DoubleClick.", '1408.6491-2-2-3': "These networks embed tracking code into webpages across many sites providing the network with a more global view of each user's behaviors.", '1408.6491-2-3-0': 'People are concerned about behavioral marketing on the web (e.g., [CITATION]).', '1408.6491-2-3-1': 'To increase transparency and control, Google provides Ad Settings, which is "a Google tool that helps you control the ads you see on Google services and on websites that partner with Google" [CITATION].', '1408.6491-2-3-2': "It displays inferences Google has made about a user's demographics and interests based on his browsing behavior.", '1408.6491-2-3-3': 'Users can view and edit these settings at', '1408.6491-2-4-0': 'http://www.google.com/settings/ads Figure [REF] provides a screenshot.', '1408.6491-2-4-1': 'Yahoo [CITATION] and Microsoft [CITATION] also offer personalized ad settings.', '1408.6491-2-5-0': "r10.1cm Screenshot of Google's Ad Settings webpage", '1408.6491-2-6-0': 'However, they provide little information about how these pages operate, leaving open the question of how completely these settings describe the profile they have about a user.', '1408.6491-2-6-1': "In this study, we explore how a user's behaviors, either directly with the settings or with content providers, alter the ads and settings shown to the user and whether these changes are in harmony.", '1408.6491-2-6-2': 'In particular, we study the degree to which the settings provides transparency and choice as well as checking for the presence of discrimination.', '1408.6491-2-6-3': 'Transparency is important for people to understand how the use of data about them affects the ads they see.', '1408.6491-2-6-4': 'Choice allows users to control how this data gets used, enabling them to protect the information they find sensitive.', '1408.6491-2-6-5': 'Discrimination is an increasing concern about machine learning systems and one reason people like to keep information private [CITATION].', '1408.6491-2-7-0': 'To conduct these studies, we developed AdFisher, a tool for automating randomized, controlled experiments for studying online tracking.', '1408.6491-2-7-1': 'Our tool offers a combination of automation, statistical rigor, scalability, and explanation for determining the use of information by web advertising algorithms and by personalized ad settings, such as Google Ad Settings.', '1408.6491-2-7-2': 'The tool can simulate having a particular interest or attribute by visiting webpages associated with that interest or by altering the ad settings provided by Google.', '1408.6491-2-7-3': 'It collects ads served by Google and also the settings that Google provides to the simulated users.', '1408.6491-2-7-4': 'It automatically analyzes the data to determine whether statistically significant differences between groups of agents exist.', '1408.6491-2-7-5': 'AdFisher uses machine learning to automatically detect differences and then executes a test of significance specialized for the difference it found.', '1408.6491-2-8-0': 'Someone using AdFisher to study behavioral targeting only has to provide the behaviors the two groups are to perform (e.g., visiting websites) and the measurements (e.g., which ads) to collect afterwards.', '1408.6491-2-8-1': "AdFisher can easily run multiple experiments exploring the causal connections between users' browsing activities, and the ads and settings that Google shows.", '1408.6491-2-9-0': 'The advertising ecosystem is a vast, distributed, and decentralized system with several players including the users consuming content, the advertisers, the publishers of web content, and ad networks.', '1408.6491-2-9-1': 'With the exception of the user, we treat the entire ecosystem as a blackbox.', '1408.6491-2-9-2': "We measure simulated users' interactions with this blackbox including page views, ads, and ad settings.", '1408.6491-2-9-3': 'Without knowledge of the internal workings of the ecosystem, we cannot assign responsibility for our findings to any single player within it nor rule out that they are unintended consequences of interactions between players.', '1408.6491-2-9-4': 'However, our results show the presence of concerning effects illustrating the existence of issues that could be investigated more deeply by either the players themselves or by regulatory bodies with the power to see the internal dynamics of the ecosystem.', '1408.6491-2-10-0': "Motivating Experiments In one experiment, we explored whether visiting websites related to substance abuse has an impact on Google's ads or settings.", '1408.6491-2-10-1': 'We created an experimental group and a control group of agents.', '1408.6491-2-10-2': 'The browser agents in the experimental group visited websites on substance abuse while the agents in the control group simply waited.', '1408.6491-2-10-3': 'Then, both groups of agents collected ads served by Google on a news website.', '1408.6491-2-11-0': 'Having run the experiment and collected the data, we had to determine whether any difference existed in the outputs shown to the agents.', '1408.6491-2-11-1': 'One way would be to intuit what the difference could be (e.g. more ads containing the word "alcohol") and test for that difference.', '1408.6491-2-11-2': 'However, developing this intuition can take considerable effort.', '1408.6491-2-11-3': 'Moreover, it does not help find unexpected differences.', '1408.6491-2-11-4': 'Thus, we instead used machine learning to automatically find differentiating patterns in the data.', '1408.6491-2-11-5': 'Specifically, AdFisher finds a classifier that can predict which group an agent belonged to, from the ads shown to an agent.', '1408.6491-2-11-6': 'The classifier is trained on a subset of the data.', '1408.6491-2-11-7': 'A separate test subset is used to determine whether the classifier found a statistically significant difference between the ads shown to each group of agents.', '1408.6491-2-11-8': 'In this experiment, AdFisher found a classifier that could distinguish between the two groups of agents by using the fact that only the agents that visited the substance abuse websites received ads for Watershed Rehab.', '1408.6491-2-12-0': 'We also measured the settings that Google provided to each agent on its Ad Settings page after the experimental group of agents visited the webpages associated with substance abuse.', '1408.6491-2-12-1': 'We found no differences (significant or otherwise) between the pages for the agents.', '1408.6491-2-12-2': 'Thus, information about visits to these websites is indeed being used to serve ads, but the Ad Settings page does not reflect this use in this case.', '1408.6491-2-12-3': 'Rather than providing transparency, in this instance, the ad settings were opaque as to the impact of this factor.', '1408.6491-2-13-0': 'In another experiment, we examined whether the settings provide choice to users.', '1408.6491-2-13-1': 'We found that removing interests from the Google Ad Settings page changes the ads that a user sees.', '1408.6491-2-13-2': 'In particular, we had both groups of agents visit a site related to online dating.', '1408.6491-2-13-3': 'Then, only one of the groups removed the interest related to online dating.', '1408.6491-2-13-4': 'Thereafter, the top ads shown to the group that kept the interest were related to dating but not the top ads shown to the other group.', '1408.6491-2-13-5': 'Thus, the ad settings do offer the users a degree of choice over the ads they see.', '1408.6491-2-14-0': 'We also found evidence suggestive of discrimination from another experiment.', '1408.6491-2-14-1': "We set the agents' gender to female or male on Google's Ad Settings page.", '1408.6491-2-14-2': 'We then had both the female and male groups of agents visit webpages associated with employment.', '1408.6491-2-14-3': 'We established that Google used this gender information to select ads, as one might expect.', '1408.6491-2-14-4': 'The interesting result was how the ads differed between the groups: during this experiment, Google showed the simulated males ads from a certain career coaching agency that promised large salaries more frequently than the simulated females, a finding suggestive of discrimination.', '1408.6491-2-14-5': 'Ours is the first study that provides statistically significant evidence of an instance of discrimination in online advertising when demographic information is supplied via a transparency-control mechanism (i.e., the Ad Settings page).', '1408.6491-2-15-0': "While neither of our findings of opacity or discrimination are clear violations of Google's privacy policy [CITATION] and we do not claim these findings to generalize or imply widespread issues, we find them concerning and warranting further investigation by those with visibility into the ad ecosystem.", '1408.6491-2-15-1': 'Furthermore, while our finding of discrimination in the non-normative sense of the word is on firm statistical footing, we acknowledge that people may disagree about whether we found discrimination in the normative sense of the word.', '1408.6491-2-15-2': 'We defer discussion of whether our findings suggest unjust discrimination until Section [REF].', '1408.6491-2-16-0': 'Contributions In addition to the experimental findings highlighted above, we provide AdFisher, a tool for automating such experiments.', '1408.6491-2-16-1': 'AdFisher is structured as a Python API providing functions for setting up, running, and analyzing experiments.', '1408.6491-2-16-2': 'We use Selenium to drive Firefox browsers and the scikit-learn library [CITATION] for implementations of classification algorithms.', '1408.6491-2-16-3': 'We use the SciPy library [CITATION] for implementing the statistical analyses of the core methodology.', '1408.6491-2-17-0': 'AdFisher offers rigor by performing a carefully designed experiment.', '1408.6491-2-17-1': 'The statistical analyses techniques applied do not make questionable assumptions about the collected data.', '1408.6491-2-17-2': 'We base our design and analysis on a prior proposal that makes no assumptions about the data being independent or identically distributed [CITATION].', '1408.6491-2-17-3': "Since advertisers update their behavior continuously in response to unobserved inputs (such as ad auctions) and the experimenters' own actions, such assumptions may not always hold.", '1408.6491-2-17-4': 'Indeed, in practice, the distribution of ads changes over time and simulated users, or agents, interfere with one another [CITATION].', '1408.6491-2-18-0': 'Our automation, experimental design, and statistical analyses allow us to scale to handling large numbers of agents for finding subtle differences.', '1408.6491-2-18-1': 'In particular, we modify the prior analysis of Tschantz et al. [CITATION] to allow for experiments running over long periods of time.', '1408.6491-2-18-2': 'We do so by using blocking (e.g., [CITATION]), a nested statistical analysis not previously applied to understanding web advertising.', '1408.6491-2-18-3': 'The blocking analysis ensures that agents are only compared to the agents that start out like it and then aggregates together the comparisons across blocks of agents.', '1408.6491-2-18-4': 'Thus, AdFisher may run agents in batches spread out over time while only comparing those agents running simultaneously to one another.', '1408.6491-2-19-0': 'AdFisher also provides explanations as to how Google alters its behaviors in response to different user actions.', '1408.6491-2-19-1': 'It uses the trained classifier model to find which features were most useful for the classifier to make its predictions.', '1408.6491-2-19-2': 'It provides the top features from each group to provide the experimenter/analyst with a qualitative understanding of how the ads differed between the groups.', '1408.6491-2-20-0': 'To maintain statistical rigor, we carefully circumscribe our claims.', '1408.6491-2-20-1': 'We only claim statistical soundness of our results: if our techniques detect an effect of the browsing activities on the ads, then there is indeed one with high likelihood (made quantitative by a p-value).', '1408.6491-2-20-2': 'We do not claim that we will always find a difference if one exists, nor that the differences we find are typical of those experienced by users.', '1408.6491-2-20-3': 'Furthermore, while we can characterize the differences, we cannot assign blame for them since either Google or the advertisers working with Google could be responsible.', '1408.6491-2-21-0': 'Contents After covering prior work next, we present, in Section [REF], privacy properties that our tool AdFisher can check: nondiscrimination, transparency, and choice.', '1408.6491-2-21-1': 'Section [REF] explains the methodology we use to ensure sound conclusions from using AdFisher.', '1408.6491-2-21-2': 'Section [REF] presents the design of AdFisher.', '1408.6491-2-21-3': "Section [REF] discusses our use of AdFisher to study Google's ads and settings.", '1408.6491-2-21-4': 'We end with conclusions and future work.', '1408.6491-2-22-0': 'Raw data and additional details about AdFisher and our experiments can be found at', '1408.6491-2-23-0': 'http://www.cs.cmu.edu/ mtschant/ife/ AdFisher is freely available at', '1408.6491-2-24-0': 'https://github.com/tadatitam/info-flow-experiments/', '1408.6491-2-25-0': '# Prior Work', '1408.6491-2-26-0': 'We are not the first to study how Google uses information.', '1408.6491-2-26-1': 'The work with the closest subject of study to ours is by Wills and Tatar [CITATION].', '1408.6491-2-26-2': 'They studied both the ads shown by Google and the behavior of Google\'s Ad Settings (then called the "Ad Preferences").', '1408.6491-2-26-3': 'Like us, they find the presence of opacity: various interests impacted the ads and settings shown to the user and that ads could change without a corresponding change in Ad Settings.', '1408.6491-2-26-4': 'Unlike our study, theirs was mostly manual, small scale, lacked any statistical analysis, and did not follow a rigorous experimental design.', '1408.6491-2-26-5': 'Furthermore, we additionally study choice and discrimination.', '1408.6491-2-27-0': 'Other related works differ from us in both goals and methods.', '1408.6491-2-27-1': 'They all focus on how visiting webpages change the ads seen.', '1408.6491-2-27-2': 'While we examine such changes in our work, we do so as part of a larger analysis of the interactions between ads and personalized ad settings, a topic they do not study.', '1408.6491-2-28-0': 'Barford et al. come the closest in that their recent study looked at both ads and ad settings [CITATION].', '1408.6491-2-28-1': 'They do so in their study of the "adscape", an attempt to understand each ad on the Internet.', '1408.6491-2-28-2': 'They study each ad individually and cast a wide net to analyze many ads from many websites while simulating many different interests.', '1408.6491-2-28-3': 'They only examine the ad settings to determine whether they successfully induced an interest.', '1408.6491-2-28-4': 'We rigorously study how the settings affects the ads shown (choice) and how behaviors can affect ads without affecting the settings (transparency).', '1408.6491-2-28-5': "Furthermore, we use focused collections of data and an analysis that considers all ads collectively to find subtle causal effects within Google's advertising ecosystem.", '1408.6491-2-28-6': 'We also use a randomized experimental design and analysis to ensure that our results imply causation.', '1408.6491-2-29-0': 'The usage study closest to ours in statistical methodology is that of Tschantz et al. [CITATION].', '1408.6491-2-29-1': 'They developed a rigorous methodology for determining whether a system like Google uses information.', '1408.6491-2-29-2': 'Due to limitations of their methodology, they only ran small-scale studies.', '1408.6491-2-29-3': 'While they observed that browsing behaviors could affect Ad Settings, they did not study how this related to the ads received.', '1408.6491-2-29-4': 'Furthermore, while we build upon their methodology, we automate the selection of an appropriate test statistic by using machine learning whereas they manually selected test statistics.', '1408.6491-2-30-0': 'The usage study closest to ours in terms of implementation is that of Liu et al. in that they also use machine learning [CITATION].', '1408.6491-2-30-1': 'Their goal is to determine whether an ad was selected due to the content of a page, by using behavioral profiling, or from a previous webpage visit.', '1408.6491-2-30-2': 'Thus, rather than using machine learning to select a statistical test for finding causal relations, they do so to detect whether an ad on a webpage matches the content on the page to make a case for the first possibility.', '1408.6491-2-30-3': 'Thus, they have a separate classifier for each interest a webpage might cover.', '1408.6491-2-30-4': "Rather than perform a statistical analysis to determine whether treatment groups have a statistically significant difference, they use their classifiers to judge the ratio of ads on a page unrelated to the page's content, which they presume indicates that the ads were the result of behavioral targeting.", '1408.6491-2-31-0': 'Lecuyer et al. present XRay, a tool that looks for correlations between the data that web services have about users and the ads shown to users [CITATION].', '1408.6491-2-31-1': 'While their tool may check many changes to a type of input to determine whether any of them has a correlation with the frequency of a single ad, it does not check for causation, as ours does.', '1408.6491-2-32-0': 'Englehardt et al. study filter bubbles with an analysis that assumes independence between observations [CITATION], an assumption we are uncomfortable making.', '1408.6491-2-32-1': '(See Section [REF].)', '1408.6491-2-33-0': 'Guha et al. compare ads seen by three agents to see whether Google treats differently the one that behaves differently from the other two [CITATION].', '1408.6491-2-33-1': 'We adopt their suggestion of focusing on the title and URL displayed on ads when comparing ads to avoid noise from other less stable parts of the ad.', '1408.6491-2-33-2': 'Our work differs by studying the ad settings in addition to the ads and by using larger numbers of agents.', '1408.6491-2-33-3': 'Furthermore, we use rigorous statistical analyses.', '1408.6491-2-33-4': 'Balebako et al. run similar experiments to study the effectiveness of privacy tools [CITATION].', '1408.6491-2-34-0': 'Sweeney ran an experiment to determine that searching for names associated with African-Americans produced more search ads suggestive of an arrest record than names associated with European-Americans [CITATION].', '1408.6491-2-34-1': 'Her study required considerable insight to determine that suggestions of an arrest was a key difference.', '1408.6491-2-34-2': 'AdFisher can automate not just the collection of the ads, but also the identification of such key differences by using its machine learning capabilities.', '1408.6491-2-34-3': 'Indeed, it found on its own that simulated males were more often shown ads encouraging the user to seek coaching for high paying jobs than simulated females.', '1408.6491-2-35-0': '# Privacy Properties', '1408.6491-2-36-0': 'Motivating our methodology for finding causal relationships, we present some properties of ad networks that we can check with such a methodology in place.', '1408.6491-2-36-1': 'As a fundamental limitation of science, we can only prove the existence of a causal effect; we cannot prove that one does not exist (see Section [REF]).', '1408.6491-2-36-2': 'Thus, experiments can only demonstrate violations of nondiscrimination and transparency, which require effects.', '1408.6491-2-36-3': 'On the other hand, we can experimentally demonstrate that effectful choice and ad choice are complied with in the cases that we test since compliance follows from the existence of an effect.', '1408.6491-2-36-4': 'Table [REF] summarizes these properties.', '1408.6491-2-37-0': 'lL5cmL5cml Property Name & Requirement & Causal Test & Finding', '1408.6491-2-38-0': 'Nondiscrimination & Users differing only on protected attributes are treated similarly & Find that presence of protected attribute causes a change in ads & Violation', '1408.6491-2-39-0': 'Transparency & User can view all data about him used for ad selection & Find attribute that causes a change in ads, not in settings & Violation', '1408.6491-2-40-0': 'Effectful choice & Changing a setting has an effect on ads & Find that changing a setting causes a change in ads & Compliance', '1408.6491-2-41-0': 'Ad choice & Removing an interest decreases the number ads related to that interest & Find setting causes a decease in relevant ads & Compliance', '1408.6491-2-42-0': "Privacy Properties Tested on Google's Ad Settings", '1408.6491-2-43-0': '## Discrimination', '1408.6491-2-44-0': 'At its core, discrimination between two classes of individuals (e.g., one race vs. another) occurs when the attribute distinguishing those two classes causes a change in behavior toward those two classes.', '1408.6491-2-44-1': 'In our case, discrimination occurs when membership in a class causes a change in ads.', '1408.6491-2-44-2': 'Such discrimination is not always bad (e.g., many would be comfortable with men and women receiving different clothing ads).', '1408.6491-2-44-3': 'We limit our discussion of whether the discrimination we found is unjust to the discussion section ([REF]) and do not claim to have a scientific method of determining the morality of discrimination.', '1408.6491-2-45-0': "Determining whether class membership causes a change in ads is difficult since many factors not under the experimenter's control or even observable to the experimenter may also cause changes.", '1408.6491-2-45-1': 'Our experimental methodology determines when membership in certain classes causes significant changes in ads by comparing many instances of each class.', '1408.6491-2-46-0': 'We are limited in the classes we can consider since we cannot create actual people that vary by the traditional subjects of discrimination, such as race or gender.', '1408.6491-2-46-1': 'Instead, we look at classes that function as surrogates for those classes of interest.', '1408.6491-2-46-2': "For example, rather than directly looking at how gender affects people's ads, we instead look at how altering a gender setting affects ads or at how visiting websites associated with each gender affects ads.", '1408.6491-2-47-0': '## Transparency', '1408.6491-2-48-0': 'Transparency tools like Google Ad Settings provide online consumers with some understanding of the information that ad networks collect and use about them.', '1408.6491-2-48-1': 'By displaying to users what the ad network may have learned about the interests and demographics of a user, such tools attempt to make targeting mechanisms more transparent.', '1408.6491-2-49-0': 'However the technique for studying transparency is not clear.', '1408.6491-2-49-1': 'One cannot expect an ad network to be completely transparent to a user.', '1408.6491-2-49-2': "This would involve the tool displaying all other users' interests as well.", '1408.6491-2-49-3': 'A more reasonable expectation is for the ad network to display any inferred interests about that user.', '1408.6491-2-49-4': 'So, if an ad network has inferred some interest about a user and is serving ads relevant to that interest, then that interest should be displayed on the transparency tool.', '1408.6491-2-49-5': 'However, even this notion of transparency cannot be checked precisely as the ad network may serve ads about some other interest correlated with the original inferred interest, but not display the correlated interest on the transparency tool.', '1408.6491-2-50-0': 'Thus, we only study the extreme case of the lack of transparency - opacity, and leave complex notions of transparency open for future research.', '1408.6491-2-50-1': 'We say that a transparency tool has opacity if some browsing activity results in a significant effect on the ads served, but has no effect on the ad settings.', '1408.6491-2-50-2': 'If there is a difference in the ads, we can argue that prior browsing activities must have been tracked and used by the ad network to serve relevant ads.', '1408.6491-2-50-3': 'However, if this use does not show up on the transparency tool, we have found at least one example which demonstrates a lack of transparency.', '1408.6491-2-51-0': '## Choice', '1408.6491-2-52-0': 'The Ad Settings page offers users the option of editing the interests and demographics inferred about them.', '1408.6491-2-52-1': 'However, the exact nature of how these edits impact the ad network is unclear.', '1408.6491-2-52-2': 'We examine two notions of choice.', '1408.6491-2-53-0': 'A very coarse form is effectful choice, which requires that altering the settings has some effect on the ads seen by the user.', '1408.6491-2-53-1': 'This shows that altering settings is not merely a "placebo button": it has a real effect on the network\'s ads.', '1408.6491-2-53-2': 'However, effectful choice does not capture whether the effect on ads is meaningful.', '1408.6491-2-53-3': 'For example, even if a user adds interests for cars and starts receiving fewer ads for cars, effectful choice is satisfied.', '1408.6491-2-53-4': 'Moreover, we cannot find violations of effectful choice.', '1408.6491-2-53-5': 'If we find no differences in the ads, we cannot conclude that users do not have effectful choice since it could be the result of the ad repository lacking ads relevant to the interest.', '1408.6491-2-54-0': 'Ideally, the effect on ads after altering a setting would be meaningful and related to the changed setting.', '1408.6491-2-54-1': 'One way such an effect would be meaningful, in the case of removing an inferred interest, is a decrease in the number of ads related to the removed interest.', '1408.6491-2-54-2': 'We call this requirement ad choice.', '1408.6491-2-54-3': 'One way to judge whether an ad is relevant is to check it for keywords associated with the interest.', '1408.6491-2-54-4': 'If upon removing an interest, we find a statistically significant decrease in the number of ads containing some keywords, then we will conclude that the choice was respected.', '1408.6491-2-54-5': 'In addition to testing for compliance in ad choice, we can also test for a violation by checking for a statistically significant increase in the number of related ads to find egregious violations.', '1408.6491-2-54-6': 'By requiring the effect to have a fixed direction, we can find both compliance and violations of ad choice.', '1408.6491-2-55-0': '# Methodology', '1408.6491-2-56-0': 'The goal of our methodology is to establish that a certain type of input to a system causes an effect on a certain type of output of the system.', '1408.6491-2-56-1': 'For example, in our experiments, we study the system of Google.', '1408.6491-2-56-2': "The inputs we study are visits to content providing websites and users' interactions with the Ad Settings page.", '1408.6491-2-56-3': 'The outputs we study are the settings and ads shown to the users by Google.', '1408.6491-2-56-4': 'However, nothing in our methodology limits ourselves to these particular topics; it is appropriate for determining I/O properties of any web system.', '1408.6491-2-56-5': 'Here, we present an overview of our methodology; Appendix [REF] provides details of the statistical analysis.', '1408.6491-2-57-0': '## Background: Significance Testing', '1408.6491-2-58-0': "To establish causation, we start with the approach of Fisher (our tool's namesake) for significance testing [CITATION] as specialized by Tschantz et al. for the setting of online systems [CITATION].", '1408.6491-2-58-1': 'Significance testing examines a null hypothesis, in our case, that the inputs do not affect the outputs.', '1408.6491-2-58-2': 'To test this hypothesis the experimenter selects two values that the inputs could take on, typically called the control and experimental treatments.', '1408.6491-2-58-3': 'The experimenter applies the treatments to experimental units.', '1408.6491-2-58-4': 'In our setting, the units are the browser agents, that is, simulated users.', '1408.6491-2-58-5': 'To avoid noise, the experimental units should initially be as close to identical as possible as far as the inputs and outputs in question are concerned.', '1408.6491-2-58-6': 'For example, an agent created 8.1cm Experimental setup to carry out significance testing on eight browser agents comparing the effects of two treatments.', '1408.6491-2-58-7': 'Each agent is randomly assigned a treatment which specifies what actions to perform on the web.', '1408.6491-2-58-8': 'After these actions are complete, they collect measurements which are used for significance testing.', '1408.6491-2-59-0': 'with the Firefox browser should not be compared to one created with the Internet Explorer browser since Google can detect the browser used.', '1408.6491-2-60-0': 'The experimenter randomly applies the experimental (control) treatment to half of the agents, which form the experimental (control) group.', '1408.6491-2-60-1': '(See Figure [REF].)', '1408.6491-2-60-2': 'Each agent carries out actions specified in the treatment applied to it.', '1408.6491-2-60-3': 'Next, the experimenter takes measurements of the outputs Google sends to the agents, such as ads.', '1408.6491-2-60-4': 'At this point, the experiment is complete and data analysis begins.', '1408.6491-2-61-0': 'Data analysis starts by computing a test statistic over the measurements.', '1408.6491-2-61-1': 'The experimenter selects a test statistic that she suspects will take on a high value when the outputs to the two groups differ.', '1408.6491-2-61-2': 'That is, the statistic is a measure of distance between the two groups.', '1408.6491-2-61-3': 'She then uses the permutation test to determine whether the value the test statistic actually took on is higher than what one would expect by chance unless the groups actually differ.', '1408.6491-2-61-4': 'The permutation test randomly permutes the labels (control and experimental) associated with each observation, and recomputes a hypothetical test statistic.', '1408.6491-2-61-5': 'Since the null hypothesis is that the inputs have no effect, the random assignment should have no effect on the value of the test statistic.', '1408.6491-2-61-6': 'Thus, under the null hypothesis, it is unlikely that the actual value of the test statistic is larger than the vast majority of hypothetical values.', '1408.6491-2-62-0': 'The p-value of the permutation test is the proportion of the permutations where the test statistic was greater than or equal to the actual observed statistic.', '1408.6491-2-62-1': 'If the value of the test statistic is so high that under the null hypothesis it would take on as high of a value in less than [MATH] of the random assignments, then we conclude that the value is statistically significant (at the [MATH] level) and that causation is likely.', '1408.6491-2-63-0': '## Blocking', '1408.6491-2-64-0': 'In practice, the above methodology can be difficult to use since creating a large number of nearly identical agents might not be possible.', '1408.6491-2-64-1': 'In our case, we could only run ten agents in parallel given our hardware and network limitations.', '1408.6491-2-64-2': 'Comparing agents running at different times can result in additional noise since ads served to an agent change over time.', '1408.6491-2-64-3': 'Thus, with the above methodology, we were limited to just ten comparable units.', '1408.6491-2-64-4': "Since some effects that the inputs have on Google's outputs can be probabilistic and subtle, they might be missed looking at so few agents.", '1408.6491-2-65-0': 'To avoid this limitation, we extended the above methodology to handle varying units using blocking [CITATION].', '1408.6491-2-65-1': 'To use blocking, we created blocks of nearly identical agents running in parallel.', '1408.6491-2-65-2': 'These agents differ in terms their identifiers (e.g., process id) and location in memory.', '1408.6491-2-65-3': "Despite the agents running in parallel, the operating system's scheduler determines the exact order in which the agents operate.", '1408.6491-2-65-4': "Each block's agents were randomly partitioned into the control and experimental groups.", '1408.6491-2-65-5': 'This randomization ensures that the minor differences between agents noted above should have no systematic impact upon the results: these differences become noise that probably disappears as the sample size increases.', '1408.6491-2-65-6': 'Running these blocks in a staged fashion, the experiment proceeds on block after block.', '1408.6491-2-65-7': 'A modified permutation test now only compares the actual value of the test statistic to hypothetical values computed by reassignments of agents that respect the blocking structure.', '1408.6491-2-65-8': 'These reassignments do not permute labels across blocks of observations.', '1408.6491-2-66-0': 'Using blocking, we can scale to any number of agents by running as many blocks as needed.', '1408.6491-2-66-1': 'However, the computation of the permutation test increases exponentially with the number of blocks.', '1408.6491-2-66-2': 'Thus, rather than compute the exact p-value, we estimate it by randomly sampling the possible reassignments.', '1408.6491-2-66-3': '8.1cm Our experimental setup with training and testing blocks.', '1408.6491-2-66-4': 'Measurements from the training blocks are used to build a classifier.', '1408.6491-2-66-5': 'The trained classifier is used to compute the test statistic on the measurements from the testing blocks for significance testing.', '1408.6491-2-67-0': 'We can use a confidence interval to characterize the quality of the estimation [CITATION].', '1408.6491-2-67-1': 'The p-values we report are actually the upper bounds of the [MATH] confidence intervals of the p-values (details in Appendix [REF]).', '1408.6491-2-68-0': '## Selecting Test Statistics', '1408.6491-2-69-0': 'The above methodology leaves open the question of how to select the test statistic.', '1408.6491-2-69-1': 'In some cases, the experimenter might be interested in a particular test statistic.', '1408.6491-2-69-2': 'For example, an experimenter testing ad choice could use a test statistic that counts the number of ads related to the removed interest.', '1408.6491-2-69-3': 'In other cases, the experimenter might be looking for any effect.', '1408.6491-2-69-4': 'AdFisher offers the ability to automatically select a test statistic.', '1408.6491-2-69-5': 'To do so, it partitions the collected data into training and testing subsets, and uses the training data to train a classifier.', '1408.6491-2-69-6': "Figure [REF] shows an overview of AdFisher's workflow.", '1408.6491-2-70-0': 'To select a classifier, AdFisher uses [MATH]-fold cross validation on the training data to select among several possible parameters.', '1408.6491-2-70-1': 'The classifier predicts which treatment an agent received, only from the ads that get served to that agent.', '1408.6491-2-70-2': 'If the classifier is able to make this prediction with high accuracy, it suggests a systematic difference between the ads served to the two groups that the classifier was able to learn.', '1408.6491-2-70-3': 'If no difference exists, then we would expect the number to be near the guessing rate of [MATH].', '1408.6491-2-70-4': 'AdFisher uses the accuracy of this classifier as its test statistic.', '1408.6491-2-71-0': 'To avoid the possibility of seeing a high accuracy due to overfitting, AdFisher evaluates the accuracy of the classifier on a testing data set that is disjoint from the training data set.', '1408.6491-2-71-1': 'That is, in the language of statistics, we form our hypothesis about the test statistic being able to distinguish the groups before seeing the data on which we test it to ensure that it has predictive power.', '1408.6491-2-71-2': "AdFisher uses the permutation test to determine whether the degree to which the classifier's accuracy on the test data surpasses the guessing rate is statistically significant.", '1408.6491-2-71-3': 'That is, it calculates the p-value that measures the probability of seeing the observed accuracy given that the classifier is just guessing.', '1408.6491-2-71-4': 'If the p-value is below [MATH], we conclude that it is unlikely that classifier is guessing and that it must be making use of some difference between the ads shown to the two groups.', '1408.6491-2-72-0': '## Avoiding Pitfalls', '1408.6491-2-73-0': 'The above methodology avoids some pitfalls.', '1408.6491-2-73-1': 'Most fundamentally, we use a statistical analysis whose assumptions match those of our experimental design.', '1408.6491-2-73-2': 'Assumptions required by many statistical analyses appear unjustifiable in our setting.', '1408.6491-2-73-3': 'For example, many analyses assume that the agents do not interact or that the ads are independent and identically distributed (e.g., [CITATION]).', '1408.6491-2-73-4': "Given that all agents receive ads from the same pool of possible ads governed by the same advertisers' budgets, these assumptions appear unlikely to hold.", '1408.6491-2-73-5': 'Indeed, empirical evidence suggests that it does not [CITATION].', '1408.6491-2-73-6': 'The permutation test, which does not require this assumption, allows us to ensure statistical soundness of our analysis without making these assumptions [CITATION].', '1408.6491-2-74-0': 'Our use of randomization implies that many factors that could be confounding factors in an unrandomized design become noise in our design (e.g., [CITATION]).', '1408.6491-2-74-1': 'While such noise may require us to use a large sample size to find an effect, it does not affect the soundness of our analysis.', '1408.6491-2-75-0': 'Our use of two data sets, one for training the classifier to select the test statistic and one for hypothesis testing ensures that we do not engage in overfitting, data dredging, or multiple hypothesis testing (e.g., [CITATION]).', '1408.6491-2-75-1': 'All these problems result from looking for so many possible patterns that one is found by chance.', '1408.6491-2-75-2': 'While we look for many patterns in the training data, we only check for one in the testing data.', '1408.6491-2-76-0': 'Relatedly, by reporting a p-value, we provide a quantitative measure of the confidence we have that the observed effect is genuine and not just by chance [CITATION].', '1408.6491-2-76-1': 'Reporting simply the classifier accuracy or that some difference occurred fails to quantify the possibility that the result was a fluke.', '1408.6491-2-77-0': '## Scope', '1408.6491-2-78-0': 'We restrict the scope of our methodology to making claims that an effect exists with high likelihood as quantified by the p-value.', '1408.6491-2-78-1': 'That is, we expect our methodology to only rarely suggest that an effect exists when one does not.', '1408.6491-2-79-0': 'We do not claim "completeness" or "power": we might fail to detect some use of information.', '1408.6491-2-79-1': 'For example, Google might not serve different ads upon detecting that all the browser agents in our experiment are running from the same IP address.', '1408.6491-2-79-2': 'Despite this limitation in our experiments, we found interesting instances of usage.', '1408.6491-2-80-0': 'Furthermore, we do not claim that our results generalize to all users.', '1408.6491-2-80-1': 'To do so, we would need to a take a random sample of all users, their IP addresses, browsers, and behaviors, which is prohibitively expensive.', '1408.6491-2-80-2': 'We cannot generalize our results if for example, instead of turning off some usage upon detecting our experiments, Google turns it on.', '1408.6491-2-80-3': 'While our experiments would detect this usage, it might not be experienced by normal users.', '1408.6491-2-80-4': 'However, it would be odd if Google purposefully performs questionable behaviors only with those attempting to find it.', '1408.6491-2-81-0': 'While we use webpages associated with various interests to simulate users with those interests, we cannot establish that having the interest itself caused the ads to change.', '1408.6491-2-81-1': 'It is possible that other features of the visited webpages causes change - a form of confounding called "profile contamination" [CITATION], since the pages cover other topics as well.', '1408.6491-2-81-2': 'Nevertheless, we have determined that visiting webpages associated with the interest does result in seeing a change, which should give pause to users visiting webpages associated with sensitive interests.', '1408.6491-2-82-0': 'Lastly, we do not attempt to determine how the information was used.', '1408.6491-2-82-1': 'It could have been used by Google directly for targeting or it could have been used by advertisers to place their bids.', '1408.6491-2-82-2': 'We cannot assign blame.', '1408.6491-2-82-3': 'We hope future work will shed light on these issues, but given that we cannot observe the interactions between Google and advertisers, we are unsure whether it can be done.', '1408.6491-2-83-0': '# AdFisher', '1408.6491-2-84-0': 'In this section, we describe AdFisher, a tool implementing our methodology.', '1408.6491-2-84-1': 'AdFisher makes it easy to run experiments using the above methodology for a set of treatments, measurements, and classifiers (test statistics) we have implemented.', '1408.6491-2-84-2': 'AdFisher is also extensible allowing the experimenter to implement additional treatments, measurements, or test statistics.', '1408.6491-2-84-3': 'For example, an experimenter interested in studying a different online platform only needs to add code to perform actions and collect measurements on that platform.', '1408.6491-2-84-4': 'They need not modify methods that randomize the treatments, carry out the experiment, or perform the data analysis.', '1408.6491-2-85-0': 'To simulate a new person on the network, AdFisher creates each agent from a fresh browser instance with no browsing history, cookies, or other personalization.', '1408.6491-2-85-1': 'AdFisher randomly assigns each agent to a group and applies the appropriate treatment, such as having the browser visit webpages.', '1408.6491-2-85-2': 'Next, AdFisher makes measurements of the agent, such as collecting the ads shown to the browser upon visiting another webpage.', '1408.6491-2-85-3': 'All of the agents within a block execute and finish the treatments before moving on to collect the measurements to remove time as a factor.', '1408.6491-2-85-4': 'AdFisher runs all the agents on the same machine to prevent differences based on location, IP address, operating system, or other machine specific differences between agents.', '1408.6491-2-86-0': 'Next, we detail the particular treatments, measurements, and test statistics that we have implemented in AdFisher.', '1408.6491-2-86-1': 'We also discuss how AdFisher aids an experimenter in understanding the results.', '1408.6491-2-87-0': 'Treatments A treatment specifies what actions are to be performed by a browser agent.', '1408.6491-2-87-1': 'AdFisher automatically applies treatments assigned to each agent.', '1408.6491-2-87-2': 'Typically, these treatments involve invoking the Selenium WebDriver to make the agent interact with webpages.', '1408.6491-2-88-0': 'AdFisher makes it easy to carry out common treatments by providing ready-made implementations.', '1408.6491-2-88-1': "The simplest stock treatments we provide set interests, gender, and age range in Google's Ad Settings.", '1408.6491-2-88-2': 'Another stock treatment is to visit a list of webpages stored on a file.', '1408.6491-2-89-0': 'To make it easy to see whether websites associated with a particular interest causes a change in behavior, we have provided the ability to create lists of webpages associated with a category on Alexa.', '1408.6491-2-89-1': 'For each category, Alexa tracks the top websites sorted according to their traffic rank measure (a combination of the number of users and page views) [CITATION].', '1408.6491-2-89-2': 'The experimenter can use AdFisher to download the URLs of the top webpages Alexa associates with an interest.', '1408.6491-2-89-3': 'By default, it downloads the top [MATH] URLs.', '1408.6491-2-89-4': 'A treatment can then specify that agents visit this list of websites.', '1408.6491-2-89-5': 'While these treatments do not correspond directly to having such an interest, it allows us to study how Google responds to people visiting webpages associated with those interests.', '1408.6491-2-90-0': 'Often in our experiments, we compared the effects of a certain treatment applied to the experimental group against the null treatment applied to the control group.', '1408.6491-2-90-1': 'Under the null treatment, agents do nothing while agents under a different treatment complete their respective treatment phase.', '1408.6491-2-91-0': "Measurements AdFisher can currently measure the values set in Google's Ad Settings page and the ads shown to the agents after the treatments.", '1408.6491-2-91-1': 'It comes with stock functionality for collecting and analyzing text ads.', '1408.6491-2-91-2': 'Experimenters can add methods for image, video, and flash ads.', '1408.6491-2-92-0': 'To find a reasonable website for ad collection, we looked to news sites since they generally show many ads.', '1408.6491-2-92-1': 'Among the top [MATH] news websites on alexa.com, only five displayed text ads served by Google: theguardian.com/us, timesofindia.indiatimes.com, bbc.com/news, reuters.com/news/us and bloomberg.com.', '1408.6491-2-92-2': 'AdFisher comes with the built-in functionality to collect ads from any of these websites.', '1408.6491-2-92-3': 'One can also specify for how many reloads ads are to collected (default [MATH]), or how long to wait between successive reloads (default [MATH]s).', '1408.6491-2-92-4': 'For each page reload, AdFisher parses the page to find the ads shown by Google and stores the ads.', '1408.6491-2-92-5': 'The experimenter can add parsers to collect ads from other websites.', '1408.6491-2-93-0': 'We run most of our experiments on Times of India as it serves the most (five) text ads per page reload.', '1408.6491-2-93-1': 'We repeat some experiments on the Guardian (three ads per reload) to demonstrate that our results are not specific to one site.', '1408.6491-2-94-0': 'Classification While the experimenter can provide AdFisher with a test statistic to use on the collected data, AdFisher is also capable of automatically selecting a test statistic using machine learning.', '1408.6491-2-94-1': 'It splits the entire data set into training and testing subsets, and examines a training subset of the collected measurements to select a classifier that distinguishes between the measurements taken from each group.', '1408.6491-2-94-2': 'From the point of view of machine learning, the set of ads collected by an agent corresponds to an instance of the concept the classifier is attempting to learn.', '1408.6491-2-95-0': 'Machine learning algorithms operate over sets of features.', '1408.6491-2-95-1': 'AdFisher has functions for converting the text ads seen by an agent into three different feature sets.', '1408.6491-2-95-2': 'The URL feature set consists of the URLs displayed by the ads (or occasionally some other text if the ad displays it where URLs normally go).', '1408.6491-2-95-3': "Under this feature set, the feature vector representing an agent's data has a value of [MATH] in the [MATH]th entry iff the agent received [MATH] ads that display the [MATH]th URL where the order is fixed but arbitrary.", '1408.6491-2-96-0': 'The URL+Title feature set looks at both the displayed URL and the title of the ad jointly.', '1408.6491-2-96-1': "It represents an agent's data as a vector where the [MATH]th entry is [MATH] iff the agent received [MATH] ads containing the [MATH]th pair of a URL and title.", '1408.6491-2-97-0': 'The third feature set AdFisher has implemented is the word feature set.', '1408.6491-2-97-1': 'This set is based on word stems, the main part of the word with suffixes such as "ed" or "ing" removed in a manner similar to the work of Balebako et al. [CITATION].', '1408.6491-2-97-2': 'Each word stem that appeared in an ad is assigned a unique id.', '1408.6491-2-97-3': "The [MATH]th entry in the feature vector is the number of times that words with the [MATH]th stem appeared in the agent's ads.", '1408.6491-2-98-0': 'We explored a variety of classification algorithms provided by the scikit-learn library [CITATION].', '1408.6491-2-98-1': 'We found that logistic regression with an L2 penalty over the URL+title feature set consistently performed well compared to the others.', '1408.6491-2-98-2': 'At its core, logistic regression predicts a class given a feature vector by multiplying each of the entries of the vector by its own weighting coefficient (e.g., [CITATION]).', '1408.6491-2-98-3': 'It then takes a the sum of all these products.', '1408.6491-2-98-4': 'If the sum is positive, it predicts one class; if negative, it predicts the other.', '1408.6491-2-99-0': 'While using logistic regression, the training stage consists of selecting the coefficients assigned to each feature to predict the training data.', '1408.6491-2-99-1': 'Selecting coefficients requires balancing the training-accuracy of the model with avoiding overfitting the data with an overly complex model.', '1408.6491-2-99-2': 'We apply [MATH]-fold cross-validation on the training data to select the regularization parameter of the logistic regression classifier.', '1408.6491-2-99-3': 'By default, AdFisher splits the data into training and test sets by using the last [MATH] of the data collected for testing.', '1408.6491-2-100-0': 'Explanations To explain how the learned classifier distinguished between the groups, we explored several methods.', '1408.6491-2-100-1': 'We found the most informative to be the model produced by the classifier itself.', '1408.6491-2-100-2': 'Recall that logistic regression weighted the various features of the instances with coefficients reflecting how predictive they are of each group.', '1408.6491-2-100-3': 'Thus, with the URL+title feature set, examining the features with the most extreme coefficients identifies the URL+title pair most used to predict the group to which agents receiving an ad with that URL+title belongs.', '1408.6491-2-101-0': 'We also explored using simple metrics for providing explanations, like ads with the highest frequency in each group.', '1408.6491-2-101-1': 'However, some generic ads gets served in huge numbers to both groups.', '1408.6491-2-101-2': 'We also looked at the proportion of times an ad was served to agents in one group to the total number of times observed by all groups.', '1408.6491-2-101-3': 'However, this did not provide much insight since the proportion typically reached its maximum value of [MATH] from ads that only appeared once.', '1408.6491-2-101-4': 'Another choice we explored was to compute the difference in the number of times an ad appears between the groups.', '1408.6491-2-101-5': 'However, this metric is also highly influenced by how common the ad is across all groups.', '1408.6491-2-102-0': '# Experiments', '1408.6491-2-103-0': 'In this section, we discuss experiments that we carried out using AdFisher.', '1408.6491-2-103-1': 'In total, we ran 21 experiments, each of which created its own testing data sets using independent random assignments of treatments to agents.', '1408.6491-2-103-2': 'We analyze each test data set only once and report the results of each experiment separately.', '1408.6491-2-103-3': 'Thus, we do not test multiple hypotheses on any of our test data sets ensuring that the probability of false positives (p-value) are independent with the exception of our analyses for ad choice.', '1408.6491-2-103-4': 'In that case, we apply a Bonferroni correction.', '1408.6491-2-104-0': 'Each experiment examines one of the properties of interest from Table [REF].', '1408.6491-2-104-1': 'We found violations of nondiscrimination and data transparency and cases of compliance with effectful and ad choice.', '1408.6491-2-104-2': 'Since these summaries each depend upon more than one experiment, they are the composite of multiple hypotheses.', '1408.6491-2-104-3': 'To prevent false positives for these summaries, for each property, we report p-values adjusted by the number of experiments used to explore that property.', '1408.6491-2-104-4': 'We use the Holm-Bonferroni method for our adjustments, which is uniformly more powerful than the commonly used Bonferroni correction [CITATION].', '1408.6491-2-104-5': 'This method orders the component hypotheses by their unadjusted p-values applying a different correction to each until reaching a hypothesis whose adjusted value is too large to reject.', '1408.6491-2-104-6': 'This hypothesis and all remaining hypotheses are rejected regardless of their p-values.', '1408.6491-2-104-7': 'Appendix [REF] provides details.', '1408.6491-2-105-0': 'Table [REF] summarizes the results.', '1408.6491-2-106-0': 'lllllrrl Property & Treatment & Other Actions & Source & When & Length (hrs) & ads & Result', '1408.6491-2-107-0': 'Nondiscrimination & Gender & - & TOI & May & [MATH] & [MATH] & Inconclusive', '1408.6491-2-108-0': '& Gender & Jobs & TOI & May & [MATH] & [MATH] & Violation', '1408.6491-2-109-0': '& Gender & Jobs & TOI & July & [MATH] & [MATH] & Inconclusive', '1408.6491-2-110-0': '& Gender & Jobs & Guardian & July & [MATH] & [MATH] & Inconclusive', '1408.6491-2-111-0': '& Gender & Jobs Top 10 & TOI & July & [MATH] & [MATH] & Inconclusive', '1408.6491-2-112-0': 'Data use transparency & Substance abuse & - & TOI & May & [MATH] & [MATH] & Violation', '1408.6491-2-113-0': '& Substance abuse & - & TOI & July & [MATH] & [MATH] & Violation', '1408.6491-2-114-0': '& Substance abuse & - & Guardian & July & [MATH] & [MATH] & Violation', '1408.6491-2-115-0': '& Substance abuse & Top 10 & TOI & July & [MATH] & [MATH] & Violation', '1408.6491-2-116-0': '& Disability & - & TOI & May & [MATH] & [MATH] & Violation', '1408.6491-2-117-0': '& Mental disorder & - & TOI & May & [MATH] & [MATH] & Inconclusive', '1408.6491-2-118-0': '& Infertility & - & TOI & May & [MATH] & [MATH] & Inconclusive', '1408.6491-2-119-0': '& Adult websites & - & TOI & May & [MATH] & [MATH] & Inconclusive', '1408.6491-2-120-0': 'Effectful choice & Opting out & - & TOI & May & [MATH] & [MATH] & Compliance', '1408.6491-2-121-0': '& Dating interest & - & TOI & May & [MATH] & [MATH] & Compliance', '1408.6491-2-122-0': '& Dating interest & - & TOI & July & [MATH] & [MATH] & Inconclusive', '1408.6491-2-123-0': '& Weight loss interest & - & TOI & May & 15 & [MATH] & Compliance', '1408.6491-2-124-0': '& Weight loss interest & - & TOI & July & 15 & [MATH] & Inconclusive', '1408.6491-2-125-0': 'Ad choice & Dating interest & - & TOI & July & 1 & [MATH] & Compliance', '1408.6491-2-126-0': '& Weight loss interest & - & TOI & July & 1 & [MATH] & Inconclusive', '1408.6491-2-127-0': '& Weight loss interest & - & TOI & July & 1 & [MATH] & Inconclusive', '1408.6491-2-128-0': 'Summary of our experimental results.', '1408.6491-2-128-1': 'Ads are collected from the Times of India (TOI) or the Guardian.', '1408.6491-2-128-2': 'We report how long each experiment took, how many ads were collected for it, and what result we concluded.', '1408.6491-2-129-0': '## Nondiscrimination', '1408.6491-2-130-0': 'We use AdFisher to demonstrate a violation in the nondiscrimination property.', '1408.6491-2-130-1': 'If AdFisher finds a statistically significant difference in how Google treats two experimental groups, one consisting of members having a protected attribute and one whose members do not, then the experimenter has strong evidence that Google discriminates on that attribute.', '1408.6491-2-130-2': "In particular, we use AdFisher's ability to automatically select a test statistic to check for possible differences to test the null hypothesis that the two experimental groups have no differences in the ads they receive.", '1408.6491-2-131-0': 'As mentioned before, it is difficult to send a clear signal about any attribute by visiting related webpages since they may have content related to other attributes.', '1408.6491-2-131-1': 'The only way to send a clear signal is via Ad Settings.', '1408.6491-2-131-2': 'Thus, we focus on attributes that can be set on the Ad Settings page.', '1408.6491-2-131-3': 'In a series of experiments, we set the gender of one group to female and the other to male.', '1408.6491-2-131-4': 'In one of the experiments, the agents went straight to collecting ads; in the others, they simulated an interest in jobs.', '1408.6491-2-131-5': 'In all but one experiment, they collected ads from the Times of India (TOI); in the exception, they collected ads from the Guardian.', '1408.6491-2-131-6': 'In one experiment, they also visited the top 10 websites for the U.S. according to alexa.com to fill out their interests.', '1408.6491-2-131-7': 'Table [REF] summarizes results from these experiments.', '1408.6491-2-132-0': 'lllccccll 2*Treatment & 2*Other visits & 2*Measurement & 2*Blocks & 2c ads ( unique ads) & 2*Accuracy & 2*[EQUATION]', '1408.6491-2-133-0': '5-6 & & & & female & male & &', '1408.6491-2-134-0': 'Gender & Jobs & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-135-0': 'Gender & Jobs & Guardian, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-136-0': 'Gender & Jobs Top 10 & TOI, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-137-0': 'Gender & Jobs & TOI, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-138-0': 'Gender & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-139-0': 'Results from the discrimination experiments sorted by unadjusted p-value.', '1408.6491-2-139-1': 'TOI stands for Times of India.', '1408.6491-2-139-2': '[MATH] denotes statistically significant results under the Holm-Bonferroni method.', '1408.6491-2-140-0': 'AdFisher found a statistically significant difference in the ads for male and female agents that simulated an interest in jobs in May, 2014.', '1408.6491-2-140-1': 'It also found evidence of discrimination in the nature of the effect.', '1408.6491-2-140-2': 'In particular, it found that females received fewer instances of an ad encouraging the taking of high paying jobs than males.', '1408.6491-2-140-3': 'AdFisher did not find any statistically significant differences among the agents that did not visit the job-related pages or those operating in July, 2014.', '1408.6491-2-140-4': 'We detail the experiment finding a violation before discussing why we think the other experiments did not result in significant results.', '1408.6491-2-141-0': 'Gender and Jobs In this experiment, we examine how changing the gender demographic on Google Ad Settings affects the ads served and interests inferred for agents browsing employment related websites.', '1408.6491-2-141-1': 'We set up AdFisher to have the agents in one group visit the Google Ad Settings page and set the gender bit to female while agents in the other group set theirs to male.', '1408.6491-2-141-2': 'All the agents then visited the top [MATH] websites listed under the Employment category of Alexa .', '1408.6491-2-141-3': 'The agents then collect ads from Times of India.', '1408.6491-2-142-0': 'AdFisher ran [MATH] blocks of [MATH] agents each.', '1408.6491-2-142-1': '(We used blocks of size [MATH] in all our experiments.)', '1408.6491-2-142-2': "AdFisher used the ads of [MATH] agents ([MATH] from each group) for training a classifier using the URL+title feature set, and used the remaining [MATH] agents' ads for testing.", '1408.6491-2-142-3': 'The learned classifier attained a test-accuracy of [MATH], suggesting that Google did in fact treat the genders differently.', '1408.6491-2-142-4': 'To test whether this response was statistically significant, AdFisher computed a p-value by running the permutation test on a million randomly selected block-respecting permutations of the data.', '1408.6491-2-142-5': 'The significance test yielded an adjusted p-value of [MATH].', '1408.6491-2-143-0': 'We then examined the model learned by AdFisher to explain the nature of the difference.', '1408.6491-2-143-1': 'Table [REF] shows the five URL+title pairs that the model identifies as the strongest indicators of being from the female or male group.', '1408.6491-2-143-2': 'How ads for identifying the two groups differ is concerning.', '1408.6491-2-143-3': 'The two URL+title pairs with the highest coefficients for indicating a male were for a career coaching service for "[MATH]k+" executive positions.', '1408.6491-2-143-4': 'Google showed the ads [MATH] times to the male group but just [MATH] times to the female group.', '1408.6491-2-143-5': 'The top two URL+title pairs for the female group was for a generic job posting service and for an auto dealer.', '1408.6491-2-144-0': 'llrrrcrr 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-145-0': '4-5 7-8 & & & female & male && female & male', '1408.6491-2-146-0': '8cTop ads for identifying the simulated female group', '1408.6491-2-147-0': 'Jobs (Hiring Now) & www.jobsinyourarea.co & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-148-0': '4Runner Parts Service & www.westernpatoyotaservice.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-149-0': 'Criminal Justice Program & www3.mc3.edu/Criminal+Justice & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-150-0': 'Goodwill - Hiring & goodwill.careerboutique.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-151-0': 'UMUC Cyber Training & www.umuc.edu/cybersecuritytraining & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-152-0': '8cTop ads for identifying agents in the simulated male group', '1408.6491-2-153-0': '200k+ Jobs - Execs Only & careerchange.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-154-0': 'Find Next 200k+ Job & careerchange.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-155-0': 'Become a Youth Counselor & www.youthcounseling.degreeleap.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-156-0': 'CDL-A OTR Trucking Jobs & www.tadrivers.com/OTRJobs & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-157-0': 'Free Resume Templates & resume-templates.resume-now.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-158-0': 'Top URL+titles for the gender and jobs experiment on the Times of India in May.', '1408.6491-2-159-0': 'The found discrimination in this experiment was predominately from a pair of job-related ads for the same service making the finding highly sensitive to changes in the serving of these ads.', '1408.6491-2-159-1': 'A closer examination of the ads from the same experimental setup ran in July, 2014, showed that the frequency of these ads reduced from [MATH] to just [MATH], with one of the ads completely disappearing (Table [REF]).', '1408.6491-2-159-2': 'These [MATH] ads were only shown to males, continuing the pattern of discrimination.', '1408.6491-2-159-3': 'This pattern was recognized by the machine learning algorithm, which selected the ad as the second most useful for identifying males.', '1408.6491-2-159-4': 'However, they were too infrequent to establish statistical significance.', '1408.6491-2-159-5': 'A longer running experiment with more blocks might have succeeded.', '1408.6491-2-160-0': '@llrrrcrr@ 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-161-0': '4-5 7-8 & & & female & male && female & male', '1408.6491-2-162-0': '8cTop ads for identifying the simulated male group', '1408.6491-2-163-0': 'Truck Driving Jobs & www.bestpayingdriverjobs.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-164-0': '200k+ Jobs - Execs Only & careerchange.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-165-0': 'Aircraft Technician Program & pia.edu & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-166-0': 'Paid CDL Training & pamtransport.greatcdltraining.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-167-0': 'Unique Bridal Necklaces & margaretelizabeth.com/Bridal & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-168-0': '8cTop ads for identifying agents in the simulated female group', '1408.6491-2-169-0': 'Business Loans for Women & topsbaloans.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-170-0': 'Post Your Classified Ad & indeed.com/Post-Jobs & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-171-0': 'American Laser Skincare & americanlaser.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-172-0': 'Dedicated Drivers Needed & warrentransport.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-173-0': 'Earn Your Nursing Degree & nursing-colleges.courseadvisor.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-174-0': 'Top URL+titles for the gender and jobs experiment (July).', '1408.6491-2-175-0': '## Transparency', '1408.6491-2-176-0': 'AdFisher can demonstrate violations of individual data use transparency.', '1408.6491-2-176-1': 'AdFisher tests the null hypothesis that two groups of agents with the same ad settings receives ads from the same distribution despite being subjected to different experimental treatments.', '1408.6491-2-176-2': 'Rejecting the null hypothesis implies that some difference exists in the ads that is not documented by the ad settings.', '1408.6491-2-177-0': "In particular, we ran a series of experiments to examine how much transparency Google's Ad Settings provided.", '1408.6491-2-177-1': 'We checked whether visiting webpages associated with some interest could cause a change in the ads shown that is not reflected in the settings.', '1408.6491-2-178-0': 'We ran such experiments for five interests: substance abuse, disabilities, infertility, mental disorders, and adult websites.', '1408.6491-2-178-1': 'Results from statistical analysis of these experiments are shown in Table [REF].', '1408.6491-2-179-0': 'lllcccll 2*Treatment & 2*Other visits & 2*Measurement & 2c ads ( unique ads) & 2*Accuracy & 2*[EQUATION]', '1408.6491-2-180-0': '4-5 & & & experimental & control & &', '1408.6491-2-181-0': 'Substance abuse & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-182-0': 'Substance abuse & - & TOI, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-183-0': 'Substance abuse & Top 10 & TOI, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-184-0': 'Disability & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-185-0': 'Substance abuse & - & Guardian, July & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-186-0': 'Mental disorder & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-187-0': 'Infertility & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-188-0': 'Adult websites & - & TOI, May & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-189-0': 'Results from transparency experiments.', '1408.6491-2-189-1': 'TOI stands for Times of India.', '1408.6491-2-189-2': 'Every experiment for this property ran with [MATH] blocks.', '1408.6491-2-189-3': '[MATH] denotes statistically significant results under the Holm-Bonferroni method.', '1408.6491-2-190-0': 'We examined the interests found in the settings for the two cases where we found a statistically significant difference in ads, substance abuse and disability.', '1408.6491-2-190-1': 'We found that settings did not change at all for substance abuse and changed in an unexpected manner for disabilities.', '1408.6491-2-190-2': 'Thus, we detail these two experiments below.', '1408.6491-2-191-0': "Substance Abuse We were interested in whether Google's outputs would change in response to visiting webpages associated with substance abuse, a highly sensitive topic.", '1408.6491-2-191-1': 'Thus, we ran an experiment in which the experimental group visited such websites while the control group idled.', '1408.6491-2-191-2': 'Then, we collected the Ad Settings and the Google ads shown to the agents at the Times of India.', '1408.6491-2-191-3': 'For the webpages associated with substance abuse, we used the top [MATH] websites on the Alexa list for substance abuse.', '1408.6491-2-192-0': 'AdFisher ran [MATH] blocks of [MATH] agents each.', '1408.6491-2-192-1': 'At the end of visiting the webpages associated with substance abuse, none of the [MATH] agents in the experimental group had interests listed on their Ad Settings pages.', '1408.6491-2-192-2': '(None of the agents in the control group did either since the settings start out empty.)', '1408.6491-2-192-3': 'If one expects the Ad Settings page to reflect all learned inferences, then he would not anticipate ads relevant to those website visits given the lack of interests listed.', '1408.6491-2-193-0': 'However, the ads collected from the Times of India told a different story.', '1408.6491-2-193-1': 'The learned classifier attained a test-accuracy of [MATH], suggesting that Google did in fact respond to the page visits.', '1408.6491-2-193-2': 'Indeed, using the permutation test, AdFisher found an adjusted p-value of [MATH].', '1408.6491-2-193-3': "Thus, we conclude that the differences are statistically significant: Google's ads changed in response to visiting the webpages associated with substance abuse.", '1408.6491-2-193-4': 'Despite this change being significant, the Ad Settings pages provided no hint of its existence: the transparency tool is opaque!', '1408.6491-2-194-0': 'We looked at the URL+title pairs with the highest coefficients for identifying the experimental group that visited the websites related to substance abuse.', '1408.6491-2-194-1': 'Table [REF] provides information on coefficients and URL+titles learned.', '1408.6491-2-194-2': 'The three highest were for "Watershed Rehab".', '1408.6491-2-194-3': 'The top two had URLs for this drug and alcohol rehab center.', '1408.6491-2-194-4': 'The third lacked a URL and had other text in its place.', '1408.6491-2-194-5': "Figure [REF] shows one of Watershed's ads.", '1408.6491-2-195-0': 'llrrrcrr 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-196-0': '4-5 7-8 & & & control & experi.', '1408.6491-2-196-1': '&& control & experi.', '1408.6491-2-197-0': '8cTop ads for identifying agents in the experimental group (visited websites associated with substance abuse)', '1408.6491-2-198-0': 'The Watershed Rehab & www.thewatershed.com/Help & [MATH] & 0 & 280 && 0 & 2276', '1408.6491-2-199-0': 'Watershed Rehab & www.thewatershed.com/Rehab & [MATH] & 0 & 51 && 0 & 362', '1408.6491-2-200-0': 'The Watershed Rehab & Ads by Google & [MATH] & 0 & 258 && 0 & 771', '1408.6491-2-201-0': 'Veteran Home Loans & www.vamortgagecenter.com & [MATH] & 13 & 15 && 22 & 33', '1408.6491-2-202-0': 'CAD Paper Rolls & paper-roll.net/Cad-Paper & [MATH] & 0 & 4 && 0 & 21', '1408.6491-2-203-0': '8cTop ads for identifying agents in control group', '1408.6491-2-204-0': 'Alluria Alert & www.bestbeautybrand.com & [MATH] & 2 & 0 && 9 & 0', '1408.6491-2-205-0': 'Best Dividend Stocks & dividends.wyattresearch.com & [MATH] & 20 & 10 && 54 & 24', '1408.6491-2-206-0': '10 Stocks to Hold Forever & www.streetauthority.com & [MATH] & 51 & 44 && 118 & 76', '1408.6491-2-207-0': 'Delivery Drivers Wanted & get.lyft.com/drive & [MATH] & 22 & 6 && 54 & 14', '1408.6491-2-208-0': 'VA Home Loans Start Here & www.vamortgagecenter.com & [MATH] & 23 & 6 && 41 & 9', '1408.6491-2-209-0': 'Top URL+titles for substance abuse experiment on the Times of India in May.', '1408.6491-2-210-0': 'The experimental group saw these ads a total of [MATH] times ([MATH] of the ads); the control group never saw any of them nor contained any ads with the word "rehab" or "rehabilitation".', '1408.6491-2-210-1': 'None of the top five URL+title pairs for identifying the control group had any discernible relationship with rehab or substance abuse.', '1408.6491-2-211-0': 'These results remain robust across variations on this design with statistical significance in three variations.', '1408.6491-2-211-1': 'In July, we repeated the aforementioned experiment, conducted a variation using the Guardian instead of the Times of India, and conducted a variation that involved all the agents first visiting the top 10 websites in addition to the agents in the experimental group also visiting the substance abuse websites.', '1408.6491-2-211-2': 'In each of these experiments, there were two of Watershed ads that were the top two ads for identifying the agents that visited the substance abuse websites (Tables [REF], [REF], and [REF]).', '1408.6491-2-212-0': '@llrrrcrr@ 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-213-0': '4-5 7-8 & & & control & experi.', '1408.6491-2-213-1': '&& control & experi.', '1408.6491-2-214-0': '8cTop ads for identifying agents in the experimental group (visited websites associated with substance abuse)', '1408.6491-2-215-0': 'The Watershed Rehab & www.thewatershed.com/Help & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-216-0': 'The Watershed Rehab & Ads by Google & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-217-0': '2014 Diabetes Risk Survey & prediabetescenters.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-218-0': 'Honda CR-V Clearance 2014 & honda-clearance-sale.autosite.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-219-0': 'Considering an eMBA?', '1408.6491-2-219-1': '& gsb.stanford.edu/eMBAAlternative & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-220-0': '8cTop ads for identifying agents in control group', '1408.6491-2-221-0': 'Best Dividend Stocks & dividends.wyattresearch.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-222-0': 'Luxury Villas in Gurgaon & tatahousing.in/Arabella_EnquireNow & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-223-0': 'Apply for Discover[MATH] it & www.discovercard.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-224-0': 'Man Cheats Credit Score & www.thecreditsolutionprogram.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-225-0': 'Diabetes Signs Symptoms & prediabetescenters.com/Symptoms & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-226-0': 'Top URL+titles for substance abuse experiment on the Times of India in July.', '1408.6491-2-227-0': '@llrrrcrr@ 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-228-0': '4-5 7-8 & & & control & experi.', '1408.6491-2-228-1': '&& control & experi.', '1408.6491-2-229-0': '8cTop ads for identifying agents in the experimental group (visited websites associated with substance abuse)', '1408.6491-2-230-0': 'The Watershed Rehab & www.thewatershed.com/Help & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-231-0': 'Watershed Rehab & www.thewatershed.com/Rehab & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-232-0': 'Generator Sets Parts & www.mtspowerproducts.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-233-0': 'The AntiChrist: Free Book & voiceofelijah.org/Rapture & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-234-0': 'Israel at War & www.joelrosenberg.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-235-0': '8cTop ads for identifying agents in control group', '1408.6491-2-236-0': 'The Sound of Dear Voices & www.telephonebangladesh.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-237-0': '5-15 Day Ireland Vacation & www.exploringvacations.com/Ireland & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-238-0': '1 Best Selling Blocker & plugnblock.com/Sentry-Call-Blocker & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-239-0': 'Dow Average Over 17,000 & economyandmarkets.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-240-0': 'Block Annoying Phone Call & plugnblock.com/Sentry-Call-Blocker & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-241-0': 'Top URL+titles for substance abuse experiment on the Guardian in July', '1408.6491-2-242-0': '@llrrrcrr@ 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-243-0': '4-5 7-8 & & & control & experi.', '1408.6491-2-243-1': '&& control & experi.', '1408.6491-2-244-0': '8cTop ads for identifying agents in the experimental group (visited websites associated with substance abuse)', '1408.6491-2-245-0': 'The Watershed Rehab & www.thewatershed.com/Help & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-246-0': 'The Watershed Rehab & Ads by Google & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-247-0': '3 veggies keeping you fat & www.beyonddiet.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-248-0': 'Flexible Jobs Available & get.lyft.com/drive & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-249-0': 'Accord Clearance 2014 & honda-clearance-sale.autosite.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-250-0': '8cTop ads for identifying agents in control group', '1408.6491-2-251-0': 'Lung Cancer Symptoms & symptomfind.com/LungCancer & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-252-0': "Women's Gardening Clothes & duluthtrading.com/Women & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]", '1408.6491-2-253-0': 'Avacor[MATH] - Official Site & www.avacor.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-254-0': '5-15 Day Ireland Vacation & www.exploringvacations.com/Ireland & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-255-0': 'Prostate Cancer Treatment & prostrcision.com/Prostate_Cancer & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-256-0': 'Top URL+titles for substance abuse experiment on the Times of India in July.', '1408.6491-2-257-0': "One possible reason why Google served Watershed's ads could be remarketing, a marketing strategy that encourages users to return to previously visited websites [CITATION].", '1408.6491-2-257-1': "The website thewatershed.com features among the top [MATH] websites about substance-abuse on Alexa, and agents visiting that site may be served Watershed's ads as part of remarketing.", '1408.6491-2-257-2': 'However, these users cannot see any changes on Google Ad Settings despite Google having learnt some characteristic (visited thewatershed.com) about them and serving ads relevant to that characteristic.', '1408.6491-2-258-0': '9.1cm For each interest selected for the agents that visited webpages associated with disabilities, the number of agents with that interest selected', '1408.6491-2-259-0': 'Disabilities This experiment was nearly identical in setup but used websites related to disabilities instead of substance abuse.', '1408.6491-2-259-1': 'We used the top [MATH] websites on Alexa on the topic.', '1408.6491-2-260-0': 'For this experiment, AdFisher found a classifier with a test-accuracy of [MATH].', '1408.6491-2-260-1': 'It found a statistically significant difference with an adjusted p-value of less than [MATH].', '1408.6491-2-261-0': 'Looking at the top ads for identifying agents that visited the webpages associated with disabilities, we see that the top two ads have the URL www.abilitiesexpo.com and the titles "Mobility Lifter" and "Standing Wheelchairs".', '1408.6491-2-261-1': 'They were shown a total of [MATH] times to the experimental group but never to the control group.', '1408.6491-2-261-2': '(See Table [REF].)', '1408.6491-2-261-3': 'Table [REF] showing them for the disability experiment.', '1408.6491-2-262-0': 'llrrrcrr 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-263-0': '4-5 7-8 & & & control & experi.', '1408.6491-2-263-1': '&& control & experi.', '1408.6491-2-264-0': '8cTop ads for identifying agents in the experimental group (visited websites associated with disability)', '1408.6491-2-265-0': 'Mobility Lifter & www.abilitiesexpo.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-266-0': 'Standing Wheelchairs & www.abilitiesexpo.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-267-0': 'Smoking MN Healthcare & www.stillaproblem.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-268-0': 'Bike Prices & www.bikesdirect.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-269-0': '19 Car Insurance - New & auto-insurance.quotelab.com/MN & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-270-0': '8cTop ads for identifying agents in control group', '1408.6491-2-271-0': 'Beautiful Women in Kiev & anastasiadate.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-272-0': 'Melucci DDS & Ads by Google & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-273-0': '17.2% 2013 Annuity Return & advisorworld.com/CompareAnnuities & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-274-0': '3 Exercises To Never Do & homeworkoutrevolution.net & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-275-0': 'Find CNA Schools Near You & cna-degrees.courseadvisor.com & [MATH] & [MATH] & [MATH] & & [MATH] & [MATH]', '1408.6491-2-276-0': 'Top URL+titles for disability experiment on the Times of India in May.', '1408.6491-2-277-0': 'This time, Google did change the settings in response to the agents visiting the websites.', '1408.6491-2-277-1': 'Figure [REF] shows the interests selected for the experimental group.', '1408.6491-2-277-2': '(The control group, which did nothing, had no interests selected.)', '1408.6491-2-277-3': 'None of them are directly related to disabilities suggesting that Google might have focused on other aspects of the visited pages.', '1408.6491-2-277-4': 'Once again, we believe that the top ads were served due to remarketing, as abilitiesexpo.com was among the top [MATH] websites related to disabilities.', '1408.6491-2-278-0': '## Effectful Choice', '1408.6491-2-279-0': 'We tested whether making changes to Ad Settings has an effect on the ads seen, thereby giving the users a degree of choice over the ads.', '1408.6491-2-279-1': 'In particular, AdFisher tests the null hypothesis that changing some ad setting has no effect on the ads.', '1408.6491-2-280-0': 'First, we tested whether opting out of tracking actually had an effect by comparing the ads shown to agents that opted out after visiting car-related websites to ads from those that did not opt out.', '1408.6491-2-280-1': 'We found a statistically significant difference.', '1408.6491-2-281-0': 'We also tested whether removing interests from the settings page actually had an effect.', '1408.6491-2-281-1': 'We set AdFisher to have both groups of agents simulate some interest.', '1408.6491-2-281-2': "AdFisher then had the agents in one of the groups remove interests from Google's Ad Settings related to the induced interest.", '1408.6491-2-281-3': 'We found statistically significant differences between the ads both groups collected from the Times of India for two induced interests: online dating and weight loss.', '1408.6491-2-281-4': 'Table [REF] summarizes the results.', '1408.6491-2-281-5': 'We describe one in detail below.', '1408.6491-2-282-0': 'lrrrrcll 2*Experiment & 2*blocks & 3c ads ( unique ads) & 2*accuracy & 2*[EQUATION]', '1408.6491-2-283-0': '3-5 & & removed/opt-out & keep/opt-in & total & &', '1408.6491-2-284-0': 'Opting out & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-285-0': 'Dating (May) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-286-0': 'Weight Loss (May) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH]', '1408.6491-2-287-0': 'Dating (July) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-288-0': 'Weight Loss (July) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-289-0': 'Results from effectful choice experiments using the Times of India sorted by unadjusted p-value.', '1408.6491-2-289-1': '[MATH] denotes statistically significant results under the Holm-Bonferroni method.', '1408.6491-2-290-0': 'Online Dating', '1408.6491-2-291-0': 'We simulated an interest in online dating by visiting the website www.midsummerseve.com/, a website we choose since it sets Google\'s ad setting for "Dating Personals" (this site no longer affects the setting).', '1408.6491-2-291-1': 'AdFisher then had just the agents in the experimental group remove the interest "Dating Personals" (the only one containing the keyword "dating").', '1408.6491-2-291-2': 'All the agents then collected ads from the Times of India.', '1408.6491-2-292-0': 'AdFisher found statistically significant differences between the groups with a classifier accuracy of 74% and an adjusted p-value of [MATH].', '1408.6491-2-292-1': 'Furthermore, the effect appears related to the interests removed.', '1408.6491-2-292-2': 'The top ad for identifying agents that kept the romantic interests has the title "Are You Single?"', '1408.6491-2-292-3': 'and the second ad\'s title is "Why can\'t I find a date?"', '1408.6491-2-292-4': 'None of the top five for the control group that removed the interests were related to dating (Table [REF]).', '1408.6491-2-293-0': 'llrrrcrr 2*Title & 2*URL & 2*Coefficient & 2cappears in agents && 2ctotal appearances', '1408.6491-2-294-0': '4-5 7-8 & & & kept & removed && kept & removed', '1408.6491-2-295-0': '8cTop ads for identifying the group that kept dating interests', '1408.6491-2-296-0': 'Are You Single?', '1408.6491-2-296-1': '& www.zoosk.com/Dating & [MATH] & 367 & 33 && 2433 & 78', '1408.6491-2-297-0': 'Top 5 Online Dating Sites & www.consumer-rankings.com/Dating & [MATH] & 116 & 10 && 408 & 13', '1408.6491-2-298-0': "Why can't I find a date?", '1408.6491-2-298-1': '& www.gk2gk.com & [MATH] & 18 & 3 && 51 & 5', '1408.6491-2-299-0': 'Latest Breaking News & www.onlineinsider.com & [MATH] & 2 & 1 && 6 & 1', '1408.6491-2-300-0': 'Gorgeous Russian Ladies & anastasiadate.com & [MATH] & 11 & 0 && 21 & 0', '1408.6491-2-301-0': '8cTop ads for identifying agents in the group that removed dating interests', '1408.6491-2-302-0': 'Car Loans w/ Bad Credit & www.car.com/Bad-Credit-Car-Loan & [MATH] & 5 & 13 && 8 & 37', '1408.6491-2-303-0': 'Individual Health Plans & www.individualhealthquotes.com & [MATH] & 7 & 9 && 21 & 46', '1408.6491-2-304-0': 'Crazy New Obama Tax & www.endofamerica.com & [MATH] & 19 & 31 && 22 & 51', '1408.6491-2-305-0': 'Atrial Fibrillation Guide & www.johnshopkinshealthalerts.com & [MATH] & 0 & 6 && 0 & 25', '1408.6491-2-306-0': 'Free 5 - 25 Gift Cards & swagbucks.com & [MATH] & 4 & 11 && 5 & 32', '1408.6491-2-307-0': 'Top URL+titles for the dating experiment on Times of India in May.', '1408.6491-2-308-0': 'Thus, the ad settings appear to actually give users the ability to avoid ads they might dislike or find embarrassing.', '1408.6491-2-308-1': 'In the next set of experiments, we explicitly test for this ability.', '1408.6491-2-309-0': 'We repeated this experiment in July, 2014, using the websites relationshipsurgery.com and datemypet.com, which also had an effect on Ad Settings, but did not find statistically significant differences.', '1408.6491-2-310-0': '## Ad Choice', '1408.6491-2-311-0': 'Whereas the other experiments tested merely for the presence of an effect, testing for ad choice requires determining whether the effect is an increase or decrease in the number of relevant ads seen.', '1408.6491-2-311-1': 'Fortunately, since AdFisher uses a one-sided permutation test, it tests for either an increase or a decrease, but not for both simultaneously, making it usable for this purpose.', '1408.6491-2-311-2': 'In particular, after removing an interest, we check for a decrease to test for compliance using the null hypothesis that either no change or an increase occurred, since rejecting this hypothesis would imply that a decrease in the number of related ads occurred.', '1408.6491-2-311-3': 'To check for a violation, we test for the null hypothesis that either no change or a decrease occurred.', '1408.6491-2-311-4': 'Due to testing two hypotheses, we use an adjustment to the p-value cutoff considered significant to avoid finding significant results simply from testing multiple hypotheses.', '1408.6491-2-311-5': 'In particular, we use the standard Bonferroni correction, which calls for multiplying the p-value by [MATH] (e.g., [CITATION]).', '1408.6491-2-312-0': 'We ran three experiments checking for ad choice.', '1408.6491-2-312-1': 'The experiments followed the same setup as the effectful choice ones, but this time we used all the blocks for testing a given test statistic.', '1408.6491-2-312-2': 'The test statistic counted the number of ads containing keywords.', '1408.6491-2-312-3': 'In the first, we again test online dating using relationshipsurgery.com and datemypet.com.', '1408.6491-2-312-4': 'Table [REF] summarizes the experimental setups and Table [REF] summarizes the results.', '1408.6491-2-312-5': 'In particular, we found that removing online dating resulted in a significant decrease (p-value adjusted for all six experiments: [MATH]) in the number of ads containing related keywords (from [MATH] to [MATH]).', '1408.6491-2-312-6': 'We detail the inconclusive results for weight loss below.', '1408.6491-2-313-0': 'llrrcrr 2*Experiment & 2*Keywords & 2c ads ( unique ads) && 2cappearances', '1408.6491-2-314-0': '3-4 6-7 & & removed & kept && removed & kept', '1408.6491-2-315-0': 'Dating & dating, romance, relationship & [MATH] & [MATH] && [MATH] & [MATH]', '1408.6491-2-316-0': 'Weight Loss (1) & fitness & [MATH] & [MATH] && [MATH] & [MATH]', '1408.6491-2-317-0': 'Weight Loss (2) & fitness, health, fat, diet, exercise & [MATH] & [MATH] && [MATH] & [MATH]', '1408.6491-2-318-0': 'Setup for and ads from ad choice experiments.', '1408.6491-2-318-1': 'All experiments used 10 blocks.', '1408.6491-2-318-2': 'The same keywords are used to remove ad interests, as well as create the test statistic for permutation test.', '1408.6491-2-319-0': 'lllllll 2*Experiment & 2*[MATH] & 2*[MATH] & 2*[MATH] & 2*[MATH] & 2*[MATH] & 2*[MATH]', '1408.6491-2-320-0': 'Dating & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-321-0': 'Weight Loss (2) & [MATH] & [MATH] & [MATH] & [MATH] & [MATH] & n/a', '1408.6491-2-322-0': 'Weight Loss (1) & [MATH] & [MATH] & n/a & [MATH] & [MATH] & n/a', '1408.6491-2-323-0': 'P-values from ad choice experiments sorted by the (unflipped) p-value.', '1408.6491-2-323-1': 'The Bonferroni adjusted p-value is only adjusted for the two hypotheses tested within a single experiment (row).', '1408.6491-2-323-2': 'The Holm-Bonferroni adjusts for all [MATH] hypotheses.', '1408.6491-2-323-3': '[MATH] denotes statistically significant results under the Holm-Bonferroni method.', '1408.6491-2-324-0': 'Weight Loss We induced an interest in weight loss by visiting dietingsucks.blogspot.com.', '1408.6491-2-324-1': 'Afterwards, the agents in the experimental group removed the interests "Fitness" and "Fitness Equipment and Accessories", the only ones related to weight loss.', '1408.6491-2-324-2': 'We then used a test statistic that counted the number of ads containing the keyword "fitness".', '1408.6491-2-324-3': 'Interestingly, the test statistic was higher on the group with the interests removed, although not to a statistically significant degree.', '1408.6491-2-324-4': 'We repeated the process with a longer keyword list and found that removing interests decreased test statistic this time, but also not to a statistically significant degree.', '1408.6491-2-325-0': '# Discussion and Conclusion', '1408.6491-2-326-0': 'Using AdFisher, we conducted 21 experiments using 17,370 agents that collected over 600,000 ads.', '1408.6491-2-326-1': 'Our experiments found instances of discrimination, opacity, and choice in targeted ads of Google.', '1408.6491-2-326-2': 'Discrimination, is at some level, inherent to profiling: the point of profiling is to treat some people differently.', '1408.6491-2-326-3': 'While customization can be helpful, we highlight a case where the customization appears inappropriate taking on the negative connotations of discrimination.', '1408.6491-2-326-4': 'In particular, we found that males were shown ads encouraging the seeking of coaching services for high paying jobs more than females ([REF]).', '1408.6491-2-327-0': 'We do not, however, claim that any laws or policies were broken.', '1408.6491-2-327-1': "Indeed, Google's policies allow it to serve different ads based on gender.", '1408.6491-2-327-2': 'Furthermore, we cannot determine whether Google, the advertiser, or complex interactions among them and others caused the discrimination ([REF]).', '1408.6491-2-327-3': 'Even if we could, the discrimination might have resulted unintentionally from algorithms optimizing click-through rates or other metrics free of bigotry.', '1408.6491-2-327-4': 'Given the pervasive structural nature of gender discrimination in society at large, blaming one party may ignore context and correlations that make avoiding such discrimination difficult.', '1408.6491-2-327-5': 'More generally, we believe that no scientific study can demonstrate discrimination in the sense of unjust discrimination since science cannot demonstrate normative statements (e.g., [CITATION])', '1408.6491-2-328-0': 'Nevertheless, we are comfortable describing the results as "discrimination".', '1408.6491-2-328-1': 'From a strictly scientific view point, we have shown discrimination in the non-normative sense of the word.', '1408.6491-2-328-2': 'Personally, we also believe the results show discrimination in the normative sense of the word.', '1408.6491-2-328-3': 'Male candidates getting more encouragement to seek coaching services for high-paying jobs could further the current gender pay gap (e.g., [CITATION]).', '1408.6491-2-328-4': 'Thus, we do not see the found discrimination in our vision of a just society even if we are incapable of blaming any particular parties for this outcome.', '1408.6491-2-329-0': 'Furthermore, we know of no justification for such customization of the ads in question.', '1408.6491-2-329-1': 'Indeed, our concern about this outcome does not depend upon how the ads were selected.', '1408.6491-2-329-2': 'Even if this decision was made solely for economic reasons, it would continue to be discrimination [CITATION].', '1408.6491-2-329-3': 'In particular, we would remain concerned if the cause of the discrimination was an algorithm ran by Google and/or the advertiser automatically determining that males are more likely than females to click on the ads in question.', '1408.6491-2-329-4': 'The amoral status of an algorithm does not negate its effects on society.', '1408.6491-2-330-0': 'However, we also recognize the possibility that no party is at fault and such unjust effects may be inadvertent and difficult to prevent.', '1408.6491-2-330-1': 'We encourage research developing tools that ad networks and advertisers can use to prevent such unacceptable outcomes (e.g., [CITATION]).', '1408.6491-2-331-0': 'Opacity occurs when a tool for providing transparency into how ads are selected and the profile kept on a person actually fails to provide such transparency.', '1408.6491-2-331-1': 'Our experiment on substance abuse showed an extreme case in which the tool failed to show any profiling but the ad distributions were significantly different in response to behavior ([REF]).', '1408.6491-2-331-2': 'In particular, our experiment achieved an adjusted p-value of [MATH], which is 1000 times more significant than the standard [MATH] cutoff for statistical significance.', '1408.6491-2-331-3': 'This experiment remained robust to variations showing a pattern of such opacity.', '1408.6491-2-332-0': 'Ideally, tools, such as Ad Settings, would provide a complete representation of the profile kept on a person, or at least the portion of the profile that is used to select ads shown to the person.', '1408.6491-2-332-1': 'Two people with identical profiles might continue to receive different ads due to other factors affecting the choice of ads such as A/B testing or the time of day.', '1408.6491-2-332-2': 'However, systematic differences between ads shown at the same time and in the same context, such as those we found, would not exist for such pairs of people.', '1408.6491-2-333-0': 'In our experiments testing transparency, we suspect that Google served the top ads as part of remarketing, but our blackbox experiments do not determine whether this is the case.', '1408.6491-2-333-1': 'While such remarketing may appear less concerning than Google inferring a substance abuse issue about a person, its highly targeted nature is worrisome particularly in settings with shared computers or shoulder surfing.', '1408.6491-2-333-2': 'There is a need for a more inclusive transparency/control mechanism which encompasses remarketed ads as well.', '1408.6491-2-333-3': 'Additionally, Google states that "we prohibit advertisers from remarketing based on sensitive information, such as health information" [CITATION].', '1408.6491-2-333-4': 'Although Google does not specify what they consider to be "health information", we view the ads as in violation of Google\'s policy, thereby raising the question of how Google should enforce its policies.', '1408.6491-2-334-0': 'Lastly, we found that Google Ad Settings does provide the user with a degree of choice about the ads shown.', '1408.6491-2-334-1': 'In this aspect, the transparency/control tool operated as we expected.', '1408.6491-2-335-0': "Our tool, AdFisher, makes it easy to run additional experiments exploring the relations between Google's ads and settings.", '1408.6491-2-335-1': 'It can be extended to study other systems.', '1408.6491-2-335-2': "It's design ensures that it can run and analyze large scale experiments to find subtle differences.", '1408.6491-2-335-3': 'It automatically finds differences between large data sets produced by different groups of agents and explains the nature of those differences.', '1408.6491-2-335-4': 'By completely automating the data analysis, we ensure that an appropriate statistical analysis determines whether these differences are statistically significant and sound conclusions.', '1408.6491-2-336-0': 'AdFisher may have cost advertisers a small sum of money.', '1408.6491-2-336-1': 'AdFisher never clicked on any ads to avoid per click fees, which can run over [MATH] [CITATION].', '1408.6491-2-336-2': 'Its experiments may have caused per-impression fees, which run about [MATH] [CITATION].', '1408.6491-2-336-3': 'In the billion dollar ad industry, its total effect was about [MATH].', '1408.6491-2-337-0': '# Future Work', '1408.6491-2-338-0': 'We would like to extend AdFisher to study information flow on other advertising systems like Facebook, Bing, or Gmail.', '1408.6491-2-338-1': 'We would also like to analyze other kinds of ads like image or flash ads.', '1408.6491-2-338-2': 'We also plan to use the tool to detect price discrimination on sites like Amazon or Kayak, or find differences in suggested posts on blogs and news websites, based on past user behavior.', '1408.6491-2-338-3': 'We have already mentioned the interesting problem of how ad networks can ensure that their policies are respected by advertisers ([REF]).', '1408.6491-2-339-0': 'We also like to assign blame where it is due.', '1408.6491-2-339-1': 'However, doing so is often difficult.', '1408.6491-2-339-2': 'For example, our view on blame varies based on why females were discriminated against in our gender and jobs experiment.', '1408.6491-2-339-3': 'If Google allowed the advertiser to easily discriminate, we would blame both.', '1408.6491-2-339-4': "If the advertiser circumvented Google's efforts to prevent such discrimination by targeting correlates of gender, we would blame just the advertiser.", '1408.6491-2-339-5': 'If Google decided to target just males with the ad on its own, we would blame just Google.', '1408.6491-2-339-6': 'While we lack the access needed to make this determination, both Google and the advertiser have enough information to audit the other with our tool.', '1408.6491-2-340-0': 'As another example, consider the results of opacity after visiting substance abuse websites.', '1408.6491-2-340-1': "While we suspect, remarketing is the cause, it is also possible that Google is targeting users without the rehab center's knowledge.", '1408.6491-2-340-2': 'In this case, it would remain unclear as to whether Google is targeting users as substance abusers or due to some other content correlated with the webpages we visited to simulate an interest in substance abuse.', '1408.6491-2-340-3': 'We would like to find ways of controlling for these confounding factors.', '1408.6491-2-341-0': 'For these reasons, we cannot claim that Google has violated its policies.', '1408.6491-2-341-1': 'In fact, we consider it more likely that Google has lost control over its massive, automated advertising system.', '1408.6491-2-341-2': 'Even without advertisers placing inappropriate bids, large-scale machine learning can behave in unexpected ways.', '1408.6491-2-341-3': 'With this in mind, we hope future research will examine how to produce machine learning algorithms that automatically avoid discriminating against users in unacceptable ways and automatically provide transparency to users.'} | [['1408.6491-1-35-0', '1408.6491-2-44-0'], ['1408.6491-1-35-1', '1408.6491-2-44-1'], ['1408.6491-1-43-1', '1408.6491-2-53-1'], ['1408.6491-1-133-0', '1408.6491-2-280-0'], 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'1408.6491-2-215-0', '1408.6491-2-216-0', '1408.6491-2-217-0', '1408.6491-2-218-0', '1408.6491-2-219-0', '1408.6491-2-219-1', '1408.6491-2-220-0', '1408.6491-2-221-0', '1408.6491-2-223-0', '1408.6491-2-225-0', '1408.6491-2-228-0', '1408.6491-2-228-1', '1408.6491-2-230-0', '1408.6491-2-231-0', '1408.6491-2-232-0', '1408.6491-2-234-0', '1408.6491-2-235-0', '1408.6491-2-239-0', '1408.6491-2-243-0', '1408.6491-2-243-1', '1408.6491-2-245-0', '1408.6491-2-246-0', '1408.6491-2-248-0', '1408.6491-2-249-0', '1408.6491-2-250-0', '1408.6491-2-251-0', '1408.6491-2-252-0', '1408.6491-2-253-0', '1408.6491-2-255-0', '1408.6491-2-261-2', '1408.6491-2-263-0', '1408.6491-2-263-1', '1408.6491-2-265-0', '1408.6491-2-266-0', '1408.6491-2-267-0', '1408.6491-2-268-0', '1408.6491-2-269-0', '1408.6491-2-270-0', '1408.6491-2-272-0', '1408.6491-2-273-0', '1408.6491-2-283-0', '1408.6491-2-284-0', '1408.6491-2-285-0', '1408.6491-2-286-0', '1408.6491-2-287-0', '1408.6491-2-288-0', '1408.6491-2-290-0', 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1610.00441 | {'1610.00441-1-0-0': 'The Weil-Kostant integrality theorem states that given a smooth manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold with curvature the given 2-form.', '1610.00441-1-0-1': 'It also characterises the moduli space of line bundles with connection that arise in this way.', '1610.00441-1-0-2': 'This theorem was extended to the case of [MATH]-forms by Gajer in [CITATION].', '1610.00441-1-0-3': 'In this paper we provide a generalization of this theorem where we replace the original manifold by a derived smooth Artin stack.', '1610.00441-1-0-4': 'Our derived Artin stacks are geometric stacks on the etale [MATH]-site of affine derived smooth manifolds.', '1610.00441-1-0-5': 'We introduce the notion of a [MATH]-shifted [MATH]-preplectic derived smooth Artin stack in analogy with the algebraic case constructed by Pantev-Toen-Vaquie-Vezzosi in [CITATION].', '1610.00441-1-0-6': 'This is a derived smooth Artin stack endowed with a complex closed [MATH]-form which has been cohomologically shifted by degree [MATH].', '1610.00441-1-0-7': 'It is a far reaching generalization of a [MATH]-preplectic manifold which includes orbifolds and other highly singular objects.', '1610.00441-1-0-8': 'We then show that when its [MATH]-shifted [MATH]-preplectic form is integral, then there exists a [MATH]-gerbe with [MATH]-connection data and curvature corresponding to the original [MATH]-preplectic form.', '1610.00441-1-0-9': 'We also provide the characterization of the moduli stack of gerbes with connections arising in this context.', '1610.00441-1-0-10': 'We construct a canonical functor from the [MATH]-category of integral [MATH]-shifted [MATH]-preplectic derived smooth Artin stacks to the [MATH]-category of linear [MATH]-categories.', '1610.00441-1-0-11': 'When [MATH] and [MATH], this functor can be thought of like a cohomology functor in that it associates to a derived presymplectic smooth Artin stack a linear invariant in the form of a differential graded module.', '1610.00441-1-0-12': 'In the general case we obtain higher prequantum categories which requires the machinery of linear [MATH]-categories.', '1610.00441-1-1-0': '# Introduction', '1610.00441-1-2-0': 'In this paper we construct linear invariants of certain derived stacks of a smooth nature.', '1610.00441-1-2-1': 'Our first task is to prove a version of the Weil-Kostant integrality theorem in the setting of derived smooth geometry.', '1610.00441-1-2-2': 'The Weil-Kostant integrality theorem states that given a manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold such that the curvature of this connection coincides with the original 2-form.', '1610.00441-1-2-3': 'It also characterises the moduli space of line bundles with connection that arise in this way.', '1610.00441-1-3-0': 'More precisely we have the following theorem (see for example [CITATION] or [CITATION][CITATION] for the original references).', '1610.00441-1-4-0': '[Weil-Kostant] Let [MATH] be a smooth manifold endowed with a complex closed [MATH]-form [MATH].', '1610.00441-1-5-0': 'If [MATH] is integral, ie.', '1610.00441-1-5-1': 'the class [MATH] lies in the image of the map [EQUATION] where [MATH], then there exists a pair [MATH] consisting of a line bundle [MATH] on [MATH] with connection [MATH] such that [MATH] is the curvature of [MATH].', '1610.00441-1-5-2': 'The set of isomorphism classes of pairs [MATH] with curvature [MATH] form a torsor for the group [MATH] of isomorphism classes of flat line bundles over [MATH].', '1610.00441-1-6-0': 'This theorem was generalized to integral complex closed [MATH]-forms by Gajer in [CITATION].', '1610.00441-1-6-1': 'However, many objects of interest to us, for example non-transverse intersections of manifolds or quotients of a manifold by a Lie group with nonfree action, are not contained in these theorems.', '1610.00441-1-6-2': 'We need to introduce a category of more general objects which includes these examples and which contains the category of manifolds as a full subcategory.', '1610.00441-1-6-3': 'The first objective of this article is to provide a proof of this theorem in the case where our objects are derived smooth Artin stacks.', '1610.00441-1-7-0': 'We start by embedding the category of manifolds into the category of derived manifolds.', '1610.00441-1-7-1': 'These are a generalization of the quasi-smooth derived manifolds introduced in [CITATION].', '1610.00441-1-7-2': 'One can think of a derived [MATH]-manifold, where [MATH] is the field of real or complex numbers, as a dg-ringed topological space with extra structure, ie.', '1610.00441-1-7-3': 'a topological space endowed with a sheaf of commutative differential graded [MATH]-algebras with structure enabling one to "compose with smooth functions" ([MATH] or holomorphic functions) which is moreover, locally given by a finite limit of [MATH]-manifolds.', '1610.00441-1-7-4': 'The foundations of the general theory of structured spaces was laid out in [CITATION].', '1610.00441-1-8-0': 'In considering derived [MATH]-manifolds instead of ordinary [MATH]-manifolds we gain, in addition to the inclusion of far more general spaces, better formal properties of the [MATH]-category of such objects.', '1610.00441-1-8-1': 'For example, the [MATH]-category of derived [MATH]-manifolds is closed under finite limits.', '1610.00441-1-8-2': 'The finite limits in the category of [MATH]-manifolds that are correct, for example transverse intersections, are preserved by the fully faithful functor from [MATH]-manifolds to derived [MATH]-manifolds.', '1610.00441-1-9-0': 'Still further examples are not contained in the [MATH]-category of derived [MATH]-manifolds.', '1610.00441-1-9-1': 'For example the category of derived [MATH]-manifolds is not closed under arbitrary colimits and we would like to include possibly singular quotients of manifolds in our theorem.', '1610.00441-1-9-2': 'We build from the [MATH]-category of derived [MATH]-manifolds the notion of a derived [MATH]-smooth stack.', '1610.00441-1-9-3': 'A derived [MATH]-smooth stack will be defined as a sheaf of spaces on the [MATH]-site of affine derived [MATH]-manifolds with respect to the etale topology.', '1610.00441-1-9-4': 'An affine derived [MATH]-manifold is a local model for a derived [MATH]-manifold.', '1610.00441-1-10-0': 'We show that the [MATH]-site of affine derived [MATH]-manifolds with the etale topology is subcanonical by showing that the presheaf of [MATH]-categories sending an affine derived [MATH]-manifold to its [MATH]-category of modules is a sheaf of [MATH]-categories.', '1610.00441-1-10-1': 'As usual, objects in the essential image of the Yoneda embedding from the [MATH]-category of affine derived [MATH]-manifolds to the [MATH]-category of derived [MATH]-smooth stacks will be called affine derived [MATH]-smooth stacks.', '1610.00441-1-11-0': 'We then define what it means for a derived [MATH]-smooth stack to be Artin.', '1610.00441-1-11-1': 'One can roughly think of a derived [MATH]-smooth Artin stack as a presheaf of spaces on the [MATH]-category of affine derived [MATH]-manifolds which is a sheaf for the etale topology and which is locally representable by an affine derived [MATH]-manifold with respect to the smooth topology.', '1610.00441-1-11-2': 'This uses the theory of geometries outlined in [CITATION] which we recall.', '1610.00441-1-11-3': 'We also discuss examples of derived [MATH]-smooth Artin stacks.', '1610.00441-1-11-4': 'They can be presented as quotients by derived Lie groupoid actions.', '1610.00441-1-11-5': 'The main reason for restricting to the collection of Artin stacks is that it includes all the examples of interest to us whilst guaranteeing the existence of a cotangent complex for such objects.', '1610.00441-1-11-6': 'This is necessary for studying presymplectic geometry in our context in the subsequent sections.', '1610.00441-1-12-0': 'In summary, we will introduce three [MATH]-categories which lie to the right of the category of [MATH]-manifolds in a chain of inclusions [EQUATION] with obvious notation, in order to deal with examples whose structure is inaccessible from the first category.', '1610.00441-1-12-1': 'This chain can be compared with the algebraic setting from smooth varieties and derived schemes up to derived (Artin) stacks.', '1610.00441-1-12-2': 'A similar chain holds in the complex analytic setting.', '1610.00441-1-12-3': 'From the discussion above, derived [MATH]-smooth Artin stacks often arise as solutions to derived moduli problems in the smooth setting.', '1610.00441-1-13-0': 'The analogue of a smooth manifold endowed with a complex closed [MATH]-form, or what one may call a [MATH]-preplectic manifold, in our setting is a derived [MATH]-shifted [MATH]-preplectic smooth Artin stack.', '1610.00441-1-13-1': 'This is a derived smooth Artin stack (over [MATH]) endowed with a complex closed [MATH]-form that has been cohomologically shifted by degree [MATH].', '1610.00441-1-13-2': 'In the case where the derived [MATH]-shifted [MATH]-preplectic smooth Artin stack is simply a smooth manifold endowed with a zero shifted 2-form, we recover the theory of presymplectic [MATH]-manifolds.', '1610.00441-1-13-3': 'However, zero shifted [MATH]-forms exist on spaces containing singularities and so our definition is a natural extension of [MATH]-preplectic structures and can be utilized in many more general examples.', '1610.00441-1-14-0': 'We also define what it means for a complex closed [MATH]-form on a derived smooth Artin stack to be integral.', '1610.00441-1-14-1': 'In analogy with the standard definition, it will mean that its cohomology class is the image of an integral class.', '1610.00441-1-15-0': 'The first main result of this paper, a derived version of the Weil-Kostant integrality theorem, is stated as follows (see Section [REF], Theorem [REF]).', '1610.00441-1-16-0': 'There exists a [MATH]-gerbe on [MATH] with curvature [MATH] if and only if [MATH] is integral.', '1610.00441-1-16-1': 'The space of [MATH]-gerbes on [MATH] with curvature [MATH] is parametrized by the space of flat [MATH]-gerbes.', '1610.00441-1-17-0': 'A [MATH]-gerbe on a derived smooth stack is a [MATH]-gerbe with [MATH]-connections, where [MATH] ranges from [MATH] to [MATH], on the derived smooth stack (see Definition [REF]).', '1610.00441-1-17-1': 'We also use the terminology [MATH]-gerbe with [MATH]-connection data.', '1610.00441-1-17-2': 'It is an extension of the notion of a complex line bundle with connection on a smooth manifold.', '1610.00441-1-17-3': 'Our theorem subsumes the classical Weil-Kostant integrality theorem, reconstructing it when the derived smooth Artin stack is an integral ([MATH]-shifted) [MATH]-preplectic [MATH]-manifold.', '1610.00441-1-17-4': 'In this case a [MATH]-gerbe is simply a complex line bundle with connection.', '1610.00441-1-17-5': 'When our derived smooth stack is a (singular) smooth space endowed with an integral complex closed [MATH]-form, this [MATH]-gerbe is understood as a complex line bundle with connection in a derived sense, ie.', '1610.00441-1-17-6': 'the line bundle is a bundle of complexes of [MATH]-modules which encodes how the form differs from being smooth.', '1610.00441-1-17-7': 'The notion of derived geometry arising when one deals with singular spaces is well known and exemplified in this result.', '1610.00441-1-18-0': 'One application of the classical Weil-Kostant integrality theorem is to studying linear invariants of smooth spaces.', '1610.00441-1-18-1': 'Classically, linear invariants of integral presymplectic smooth manifolds arise by considering the complex vector space of sections of the complex line bundle arising from the Weil-Kostant integrality theorem.', '1610.00441-1-18-2': 'This construction is used for example in the theory of geometric quantization, where it is sometimes referred to as the associated prequantum vector space, and is useful in understanding the quantization of classical mechanical systems.', '1610.00441-1-18-3': 'The functor which associates to an integral presymplectic smooth manifold a prequantum vector space is like a cohomology functor but satisfies different functorial properties.', '1610.00441-1-19-0': 'Similarly, one of the main utilities of the derived Weil-Kostant integrality theorem is to the subject of derived geometric quantization.', '1610.00441-1-19-1': 'This is useful in understanding the quantization of classical field theories in an extended sense, ie.', '1610.00441-1-19-2': 'as extended quantum field theories in which higher categorical data is associated to manifolds of greater codimension than one.', '1610.00441-1-19-3': 'In this case, the prequantization functor supplies, in addition to a prequantum vector space (or more generally, a prequantum complex of vector spaces) certain prequantum linear higher categories.', '1610.00441-1-20-0': 'The collection of pairs [MATH] consisting of a derived smooth Artin stack together with an integral [MATH]-shifted complex closed [MATH]-form [MATH] on [MATH] form an [MATH]-category which we denote by [MATH].', '1610.00441-1-20-1': 'The objects in this [MATH]-category will be called integral [MATH]-shifted [MATH]-preplectic derived smooth Artin stacks.', '1610.00441-1-20-2': 'The collection of [MATH]-linear [MATH]-categories also form an [MATH]-category denoted [MATH].', '1610.00441-1-20-3': 'Using the derived Weil-Kostant integrality theorem we prove the following second main result of this paper (see Section [REF], Corollary [REF]).', '1610.00441-1-21-0': 'The application of this theorem to derived Artin stacks arising from moduli problems in classical field theory will appear elsewhere.', '1610.00441-1-22-0': '# Notation', '1610.00441-1-23-0': 'An [MATH]-category will refer to an [MATH]-category, the theory of which is contained in [CITATION] and [CITATION].', '1610.00441-1-23-1': 'The opposite of an [MATH]-category [MATH] will be denoted [MATH].', '1610.00441-1-23-2': 'Appendix A contains a summary of the theory of [MATH]-categories, based on the approach in [CITATION], which is sufficient for our purposes.', '1610.00441-1-23-3': 'More advanced structures in the formal theory of [MATH]-categories, needed in the main text, have been relegated to Appendix B.', '1610.00441-1-24-0': 'Given a simplicial model category [MATH], the [MATH]-category arising from the localization of [MATH] with respect to its class of weak equivalences will always be denoted [MATH].', '1610.00441-1-24-1': 'In the setting of [CITATION], this corresponds to the Dwyer-Kan simplicial localization and in the setting of [CITATION], it corresponds the homotopy coherent nerve of the category [MATH] of fibrant-cofibrant objects in [MATH].', '1610.00441-1-25-0': 'All [MATH]-categories of special note will be written in boldface.', '1610.00441-1-25-1': 'In particular, we let [MATH] denote the [MATH]-category [EQUATION] of spaces given by the localization of the category [MATH] of simplicial sets endowed with the Kan model structure.', '1610.00441-1-25-2': 'This is equivalent to the [MATH]-category [MATH] of topological spaces where the category [MATH] of topological spaces is endowed with its standard model structure.', '1610.00441-1-25-3': 'The [MATH]-category of functors between two objects [MATH] and [MATH] in the model category of [MATH]-categories will be denoted [MATH].', '1610.00441-1-26-0': 'Finally, questions about categorical size will be neglected throughout and can be addressed through the implementation of universes.'} | {'1610.00441-2-0-0': 'The Weil-Kostant integrality theorem states that given a smooth manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold with curvature the given 2-form.', '1610.00441-2-0-1': 'It also characterises the moduli space of line bundles with connection that arise in this way.', '1610.00441-2-0-2': 'This theorem was extended to the case of [MATH]-forms by Gajer in [CITATION].', '1610.00441-2-0-3': 'In this paper we provide a generalization of this theorem where we replace the original manifold by a derived smooth Artin stack.', '1610.00441-2-0-4': 'Our derived Artin stacks are geometric stacks on the etale [MATH]-site of affine derived smooth manifolds.', '1610.00441-2-0-5': 'We introduce the notion of a [MATH]-shifted [MATH]-preplectic derived smooth Artin stack in analogy with the algebraic case constructed by Pantev-Toen-Vaquie-Vezzosi in [CITATION].', '1610.00441-2-0-6': 'This is a derived smooth Artin stack endowed with a complex closed [MATH]-form which has been cohomologically shifted by degree [MATH].', '1610.00441-2-0-7': 'It is a far reaching generalization of a [MATH]-preplectic manifold which includes orbifolds and other highly singular objects.', '1610.00441-2-0-8': 'We then show that when its [MATH]-shifted [MATH]-preplectic form is integral, then there exists a [MATH]-gerbe with [MATH]-connection data and curvature corresponding to the original [MATH]-preplectic form.', '1610.00441-2-0-9': 'We also provide the characterization of the moduli stack of gerbes with connections arising in this context.', '1610.00441-2-0-10': 'We construct a canonical functor from the [MATH]-category of integral [MATH]-shifted [MATH]-preplectic derived smooth Artin stacks to the [MATH]-category of linear [MATH]-categories.', '1610.00441-2-0-11': 'When [MATH] and [MATH], this functor can be thought of like a cohomology functor in that it associates to a derived presymplectic smooth Artin stack a linear invariant in the form of a differential graded module.', '1610.00441-2-0-12': 'In the general case we obtain higher prequantum categories which requires the machinery of linear [MATH]-categories.', '1610.00441-2-1-0': '# Introduction', '1610.00441-2-2-0': 'In this paper we construct linear invariants of certain derived stacks of a smooth nature.', '1610.00441-2-2-1': 'Our first task is to prove a version of the Weil-Kostant integrality theorem in the setting of derived smooth geometry.', '1610.00441-2-2-2': 'The Weil-Kostant integrality theorem states that given a manifold endowed with an integral complex closed 2-form, then there exists a line bundle with connection on this manifold such that the curvature of this connection coincides with the original 2-form.', '1610.00441-2-2-3': 'It also characterises the moduli space of line bundles with connection that arise in this way.', '1610.00441-2-3-0': 'More precisely we have the following theorem (see for example [CITATION] or [CITATION][CITATION] for the original references).', '1610.00441-2-4-0': '[Weil-Kostant] Let [MATH] be a smooth manifold endowed with a complex closed [MATH]-form [MATH].', '1610.00441-2-5-0': 'If [MATH] is integral, ie.', '1610.00441-2-5-1': 'the class [MATH] lies in the image of the map [EQUATION] where [MATH], then there exists a pair [MATH] consisting of a line bundle [MATH] on [MATH] with connection [MATH] such that [MATH] is the curvature of [MATH].', '1610.00441-2-5-2': 'The set of isomorphism classes of pairs [MATH] with curvature [MATH] form a torsor for the group [MATH] of isomorphism classes of flat line bundles over [MATH].', '1610.00441-2-6-0': 'This theorem was generalized to integral complex closed [MATH]-forms by Gajer in [CITATION].', '1610.00441-2-6-1': 'However, many objects of interest to us, for example non-transverse intersections of manifolds or quotients of a manifold by a Lie group with nonfree action, are not contained in these theorems.', '1610.00441-2-6-2': 'We need to introduce a category of more general objects which includes these examples and which contains the category of manifolds as a full subcategory.', '1610.00441-2-6-3': 'The first objective of this article is to provide a proof of this theorem in the case where our objects are derived smooth Artin stacks.', '1610.00441-2-7-0': 'We start by embedding the category of manifolds into the category of derived manifolds.', '1610.00441-2-7-1': 'These are a generalization of the quasi-smooth derived manifolds introduced in [CITATION].', '1610.00441-2-7-2': 'One can think of a derived [MATH]-manifold, where [MATH] is the field of real or complex numbers, as a dg-ringed topological space with extra structure, ie.', '1610.00441-2-7-3': 'a topological space endowed with a sheaf of commutative differential graded [MATH]-algebras with structure enabling one to "compose with smooth functions" ([MATH] or holomorphic functions) which is moreover, locally given by a finite limit of [MATH]-manifolds.', '1610.00441-2-7-4': 'The foundations of the general theory of structured spaces was laid out in [CITATION].', '1610.00441-2-8-0': 'In considering derived [MATH]-manifolds instead of ordinary [MATH]-manifolds we gain, in addition to the inclusion of far more general spaces, better formal properties of the [MATH]-category of such objects.', '1610.00441-2-8-1': 'For example, the [MATH]-category of derived [MATH]-manifolds is closed under finite limits.', '1610.00441-2-8-2': 'The finite limits in the category of [MATH]-manifolds that are correct, for example transverse intersections, are preserved by the fully faithful functor from [MATH]-manifolds to derived [MATH]-manifolds.', '1610.00441-2-9-0': 'Still further examples are not contained in the [MATH]-category of derived [MATH]-manifolds.', '1610.00441-2-9-1': 'For example the category of derived [MATH]-manifolds is not closed under arbitrary colimits and we would like to include possibly singular quotients of manifolds in our theorem.', '1610.00441-2-9-2': 'We build from the [MATH]-category of derived [MATH]-manifolds the notion of a derived [MATH]-smooth stack.', '1610.00441-2-9-3': 'A derived [MATH]-smooth stack will be defined as a sheaf of spaces on the [MATH]-site of affine derived [MATH]-manifolds with respect to the etale topology.', '1610.00441-2-9-4': 'An affine derived [MATH]-manifold is a local model for a derived [MATH]-manifold.', '1610.00441-2-10-0': 'We show that the [MATH]-site of affine derived [MATH]-manifolds with the etale topology is subcanonical by showing that the presheaf of [MATH]-categories sending an affine derived [MATH]-manifold to its [MATH]-category of modules is a sheaf of [MATH]-categories.', '1610.00441-2-10-1': 'As usual, objects in the essential image of the Yoneda embedding from the [MATH]-category of affine derived [MATH]-manifolds to the [MATH]-category of derived [MATH]-smooth stacks will be called affine derived [MATH]-smooth stacks.', '1610.00441-2-11-0': 'We then define what it means for a derived [MATH]-smooth stack to be Artin.', '1610.00441-2-11-1': 'One can roughly think of a derived [MATH]-smooth Artin stack as a presheaf of spaces on the [MATH]-category of affine derived [MATH]-manifolds which is a sheaf for the etale topology and which is locally representable by an affine derived [MATH]-manifold with respect to the smooth topology.', '1610.00441-2-11-2': 'This uses the theory of geometries outlined in [CITATION] which we recall.', '1610.00441-2-11-3': 'We also discuss examples of derived [MATH]-smooth Artin stacks.', '1610.00441-2-11-4': 'They can be presented as quotients by derived Lie groupoid actions.', '1610.00441-2-11-5': 'The main reason for restricting to the collection of Artin stacks is that it includes all the examples of interest to us whilst guaranteeing the existence of a cotangent complex for such objects.', '1610.00441-2-11-6': 'This is necessary for studying presymplectic geometry in our context in the subsequent sections.', '1610.00441-2-12-0': 'In summary, we will introduce three [MATH]-categories which lie to the right of the category of [MATH]-manifolds in a chain of inclusions [EQUATION] with obvious notation, in order to deal with examples whose structure is inaccessible from the first category.', '1610.00441-2-12-1': 'This chain can be compared with the algebraic setting from smooth varieties and derived schemes up to derived (Artin) stacks.', '1610.00441-2-12-2': 'A similar chain holds in the complex analytic setting.', '1610.00441-2-12-3': 'From the discussion above, derived [MATH]-smooth Artin stacks often arise as solutions to derived moduli problems in the smooth setting.', '1610.00441-2-13-0': 'The analogue of a smooth manifold endowed with a complex closed [MATH]-form, or what one may call a [MATH]-preplectic manifold, in our setting is a derived [MATH]-shifted [MATH]-preplectic smooth Artin stack.', '1610.00441-2-13-1': 'This is a derived smooth Artin stack (over [MATH]) endowed with a complex closed [MATH]-form that has been cohomologically shifted by degree [MATH].', '1610.00441-2-13-2': 'In the case where the derived [MATH]-shifted [MATH]-preplectic smooth Artin stack is simply a smooth manifold endowed with a zero shifted 2-form, we recover the theory of presymplectic [MATH]-manifolds.', '1610.00441-2-13-3': 'However, zero shifted [MATH]-forms exist on spaces containing singularities and so our definition is a natural extension of [MATH]-preplectic structures and can be utilized in many more general examples.', '1610.00441-2-14-0': 'We also define what it means for a complex closed [MATH]-form on a derived smooth Artin stack to be integral.', '1610.00441-2-14-1': 'In analogy with the standard definition, it will mean that its cohomology class is the image of an integral class.', '1610.00441-2-15-0': 'The first main result of this paper, a derived version of the Weil-Kostant integrality theorem, is stated as follows (see Section [REF], Theorem [REF]).', '1610.00441-2-16-0': 'There exists a [MATH]-gerbe on [MATH] with curvature [MATH] if and only if [MATH] is integral.', '1610.00441-2-16-1': 'The space of [MATH]-gerbes on [MATH] with curvature [MATH] is parametrized by the space of flat [MATH]-gerbes.', '1610.00441-2-17-0': 'A [MATH]-gerbe on a derived smooth stack is a [MATH]-gerbe with [MATH]-connections, where [MATH] ranges from [MATH] to [MATH], on the derived smooth stack (see Definition [REF]).', '1610.00441-2-17-1': 'We also use the terminology [MATH]-gerbe with [MATH]-connection data.', '1610.00441-2-17-2': 'It is an extension of the notion of a complex line bundle with connection on a smooth manifold.', '1610.00441-2-17-3': 'Our theorem subsumes the classical Weil-Kostant integrality theorem, reconstructing it when the derived smooth Artin stack is an integral ([MATH]-shifted) [MATH]-preplectic [MATH]-manifold.', '1610.00441-2-17-4': 'In this case a [MATH]-gerbe is simply a complex line bundle with connection.', '1610.00441-2-17-5': 'When our derived smooth stack is a (singular) smooth space endowed with an integral complex closed [MATH]-form, this [MATH]-gerbe is understood as a complex line bundle with connection in a derived sense, ie.', '1610.00441-2-17-6': 'the line bundle is a bundle of complexes of [MATH]-modules which encodes how the form differs from being smooth.', '1610.00441-2-17-7': 'The notion of derived geometry arising when one deals with singular spaces is well known and exemplified in this result.', '1610.00441-2-18-0': 'One application of the classical Weil-Kostant integrality theorem is to studying linear invariants of smooth spaces.', '1610.00441-2-18-1': 'Classically, linear invariants of integral presymplectic smooth manifolds arise by considering the complex vector space of sections of the complex line bundle arising from the Weil-Kostant integrality theorem.', '1610.00441-2-18-2': 'This construction is used for example in the theory of geometric quantization, where it is sometimes referred to as the associated prequantum vector space, and is useful in understanding the quantization of classical mechanical systems.', '1610.00441-2-18-3': 'The functor which associates to an integral presymplectic smooth manifold a prequantum vector space is like a cohomology functor but satisfies different functorial properties.', '1610.00441-2-19-0': 'Similarly, one of the main utilities of the derived Weil-Kostant integrality theorem is to the subject of derived geometric quantization.', '1610.00441-2-19-1': 'This is useful in understanding the quantization of classical field theories in an extended sense, ie.', '1610.00441-2-19-2': 'as extended quantum field theories in which higher categorical data is associated to manifolds of greater codimension than one.', '1610.00441-2-19-3': 'In this case, the prequantization functor supplies, in addition to a prequantum vector space (or more generally, a prequantum complex of vector spaces) certain prequantum linear higher categories.', '1610.00441-2-20-0': 'The collection of pairs [MATH] consisting of a derived smooth Artin stack together with an integral [MATH]-shifted complex closed [MATH]-form [MATH] on [MATH] form an [MATH]-category which we denote by [MATH].', '1610.00441-2-20-1': 'The objects in this [MATH]-category will be called integral [MATH]-shifted [MATH]-preplectic derived smooth Artin stacks.', '1610.00441-2-20-2': 'The collection of [MATH]-linear [MATH]-categories also form an [MATH]-category denoted [MATH].', '1610.00441-2-20-3': 'Using the derived Weil-Kostant integrality theorem we prove the following second main result of this paper (see Section [REF], Corollary [REF]).', '1610.00441-2-21-0': 'The application of this theorem to derived Artin stacks arising from moduli problems in classical field theory will appear elsewhere.', '1610.00441-2-22-0': '# Notation', '1610.00441-2-23-0': 'An [MATH]-category will refer to an [MATH]-category, the theory of which is contained in [CITATION] and [CITATION].', '1610.00441-2-23-1': 'The opposite of an [MATH]-category [MATH] will be denoted [MATH].', '1610.00441-2-23-2': 'Appendix A contains a summary of the theory of [MATH]-categories, based on the approach in [CITATION], which is sufficient for our purposes.', '1610.00441-2-23-3': 'More advanced structures in the formal theory of [MATH]-categories, needed in the main text, have been relegated to Appendix B.', '1610.00441-2-24-0': 'Given a simplicial model category [MATH], the [MATH]-category arising from the localization of [MATH] with respect to its class of weak equivalences will always be denoted [MATH].', '1610.00441-2-24-1': 'In the setting of [CITATION], this corresponds to the Dwyer-Kan simplicial localization and in the setting of [CITATION], it corresponds the homotopy coherent nerve of the category [MATH] of fibrant-cofibrant objects in [MATH].', '1610.00441-2-25-0': 'All [MATH]-categories of special note will be written in boldface.', '1610.00441-2-25-1': 'In particular, we let [MATH] denote the [MATH]-category [EQUATION] of spaces given by the localization of the category [MATH] of simplicial sets endowed with the Kan model structure.', '1610.00441-2-25-2': 'This is equivalent to the [MATH]-category [MATH] of topological spaces where the category [MATH] of topological spaces is endowed with its standard model structure.', '1610.00441-2-25-3': 'The [MATH]-category of functors between two objects [MATH] and [MATH] in the model category of [MATH]-categories will be denoted [MATH].', '1610.00441-2-26-0': 'Finally, questions about categorical size will be neglected throughout and can be addressed through the implementation of universes.'} | [['1610.00441-1-14-0', '1610.00441-2-14-0'], ['1610.00441-1-14-1', '1610.00441-2-14-1'], ['1610.00441-1-3-0', '1610.00441-2-3-0'], ['1610.00441-1-10-0', '1610.00441-2-10-0'], ['1610.00441-1-10-1', '1610.00441-2-10-1'], ['1610.00441-1-11-0', '1610.00441-2-11-0'], ['1610.00441-1-11-1', '1610.00441-2-11-1'], ['1610.00441-1-11-2', '1610.00441-2-11-2'], ['1610.00441-1-11-3', '1610.00441-2-11-3'], ['1610.00441-1-11-4', '1610.00441-2-11-4'], ['1610.00441-1-11-5', '1610.00441-2-11-5'], ['1610.00441-1-11-6', '1610.00441-2-11-6'], 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'1610.00441-2-13-0'], ['1610.00441-1-13-1', '1610.00441-2-13-1'], ['1610.00441-1-13-2', '1610.00441-2-13-2'], ['1610.00441-1-13-3', '1610.00441-2-13-3'], ['1610.00441-1-12-0', '1610.00441-2-12-0'], ['1610.00441-1-12-1', '1610.00441-2-12-1'], ['1610.00441-1-12-2', '1610.00441-2-12-2'], ['1610.00441-1-12-3', '1610.00441-2-12-3'], ['1610.00441-1-4-0', '1610.00441-2-4-0'], ['1610.00441-1-19-0', '1610.00441-2-19-0'], ['1610.00441-1-19-1', '1610.00441-2-19-1'], ['1610.00441-1-19-2', '1610.00441-2-19-2'], ['1610.00441-1-19-3', '1610.00441-2-19-3'], ['1610.00441-1-24-0', '1610.00441-2-24-0'], ['1610.00441-1-24-1', '1610.00441-2-24-1'], ['1610.00441-1-25-0', '1610.00441-2-25-0'], ['1610.00441-1-25-1', '1610.00441-2-25-1'], ['1610.00441-1-25-2', '1610.00441-2-25-2'], ['1610.00441-1-25-3', '1610.00441-2-25-3'], ['1610.00441-1-23-0', '1610.00441-2-23-0'], ['1610.00441-1-23-1', '1610.00441-2-23-1'], ['1610.00441-1-23-2', '1610.00441-2-23-2'], ['1610.00441-1-23-3', '1610.00441-2-23-3'], ['1610.00441-1-0-0', '1610.00441-2-0-0'], ['1610.00441-1-0-1', '1610.00441-2-0-1'], ['1610.00441-1-0-2', '1610.00441-2-0-2'], ['1610.00441-1-0-3', '1610.00441-2-0-3'], ['1610.00441-1-0-4', '1610.00441-2-0-4'], ['1610.00441-1-0-5', '1610.00441-2-0-5'], ['1610.00441-1-0-6', '1610.00441-2-0-6'], ['1610.00441-1-0-7', '1610.00441-2-0-7'], ['1610.00441-1-0-8', '1610.00441-2-0-8'], ['1610.00441-1-0-9', '1610.00441-2-0-9'], ['1610.00441-1-0-10', '1610.00441-2-0-10'], ['1610.00441-1-0-11', '1610.00441-2-0-11'], ['1610.00441-1-0-12', '1610.00441-2-0-12']] | [] | [] | [] | [] | [] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1610.00441 | null | null | null | null | null |
0911.1722 | {'0911.1722-1-0-0': 'We compute binding energies and root mean square radii for weakly bound [MATH] and [MATH] identical bosons.', '0911.1722-1-0-1': 'Ground and first excited states appear below the threshold for binding the system with [MATH] particles.', '0911.1722-1-0-2': 'Their root mean square radii approach constants as the respective binding energies vanish.', '0911.1722-1-0-3': 'Their probablility distributions are on average located in non-classical regions of space which result in universal structures.', '0911.1722-1-0-4': 'The ground states for more than five particles are probably non-universal whereas their excited states still may be universal.', '0911.1722-1-0-5': 'The radius decreases with increasing particle number.', '0911.1722-1-1-0': 'Introduction.', '0911.1722-1-2-0': 'Efimov physics could be defined as physics where Universality and Scale Invariance apply [CITATION].', '0911.1722-1-2-1': 'In the crudest version, Universality means that the two-body scattering length [MATH] determines all properties of the [MATH]-body system.', '0911.1722-1-2-2': 'Scale Invariance means that the same properties appear for any length scale.', '0911.1722-1-2-3': 'These conditions are rather restrictive but a number of systems are known to exist within this window [CITATION].', '0911.1722-1-2-4': 'The great advantage is that one theory is sufficient to explain properties without any detailed knowledge of the interactions [CITATION].', '0911.1722-1-2-5': 'Furthermore, properties in different subfields of physics are described as manifestations of the same underlying theory.', '0911.1722-1-3-0': 'The range of validity for such a global theory is only well described for two and three particles [CITATION].', '0911.1722-1-3-1': 'There has been recent attempts for [MATH] with claims of universality of both ground and first excited states [CITATION].', '0911.1722-1-3-2': 'Universality is claimed for both systems in contrast to [CITATION] where the "disentanglement" of the regulators (cutoffs) in three and four-body equations gives rise to a dependence of the four-body ground state on interaction details.', '0911.1722-1-3-3': 'Then a four-body scale is needed.', '0911.1722-1-4-0': 'Very little is known for five particles with fully correlated solutions obtained as dictated by the interaction.', '0911.1722-1-4-1': 'However, with specific assumptions about only [MATH]-waves and essentially no correlations it was concluded in [CITATION] that halos cannot exist for [MATH].', '0911.1722-1-4-2': 'Since halos are universal structures this claim is in apparent conflict with the recent results for the [MATH] system [CITATION].', '0911.1722-1-5-0': 'It was concluded many years ago that Efimov states do not exist for [MATH] [CITATION] and furthermore for three particles only for dimensions [MATH] between [MATH] and [MATH] [CITATION].', '0911.1722-1-5-1': 'However, by restricting to two-body correlations within the [MATH]-body system, a series of (highly) excited [MATH]-body states were found with the characteristic Efimov scaling of energies and radii [CITATION].', '0911.1722-1-5-2': 'Whether they maintain their identity and the universal character when more correlations are allowed in the solutions remains to be seen.', '0911.1722-1-6-0': 'Two obvious limits to universality are apparent.', '0911.1722-1-6-1': 'The first appears for large binding energy where the resulting small radii locate the system within the range of the potentials and sensitivity to details must appear.', '0911.1722-1-6-2': 'The second limit is for excitation energies above the threshold for binding subsystems with fewer particles.', '0911.1722-1-6-3': 'Then the state is smeared out over many continuum states and the subsystems with [MATH] particles can be well bound and outside the universal window.', '0911.1722-1-7-0': 'Even for four particles where the claim is that universality applies [CITATION], a number of questions are still unanswered.', '0911.1722-1-7-1': 'For five and more particles the information becomes very scarce.', '0911.1722-1-7-2': 'A novel study has appeared for particle number less than 40 [CITATION] where various structure properties as well as the critical strength for binding is determined as function of mass.', '0911.1722-1-7-3': 'Universality is found near thresholds for a realistic finite range potential.', '0911.1722-1-8-0': 'The purpose of the present paper is to explore the boundaries for universality preferentially leading to general conclusions applicable to systems of [MATH]-particles.', '0911.1722-1-8-1': 'We first discuss qualitative features and basic properties, then extract numerical results for [MATH] and [MATH] particles very close to thresholds of binding, and relate to classically allowed regions.', '0911.1722-1-8-2': 'We only investigate super Borromean or Brunnian systems where no subsystem is bound.', '0911.1722-1-9-0': 'Qualitative considerations.', '0911.1722-1-10-0': 'For two particles the infinite scattering length corresponds to a bound state at zero energy.', '0911.1722-1-10-1': 'Variation of [MATH] around zero produces either a bound state of spatial extension [MATH] or a continuum state where the correlations range also is [MATH], corresponding to an interaction effectively of range [MATH].', '0911.1722-1-10-2': 'This asymptotic correspondence between large [MATH] and small binding energy is unique for two particles and lost for [MATH].', '0911.1722-1-11-0': 'For three particles the Efimov effect appears, i.e. for the same interaction, [MATH]), infinitely many three-body bound states emerge with progressively smaller binding and correspondingly larger radii [CITATION].', '0911.1722-1-11-1': 'By decreasing the attraction these states one by one cease to be bound.', '0911.1722-1-11-2': 'For sufficiently small attraction there is no bound three-body states at all.', '0911.1722-1-11-3': 'The trajectory followed by these states when we increase the absolute value of the two-body energy depends on the nature of the two-body subsystems: if one of them is bound the three-body bound state passes through the atom-dimer elastic scattering cut becoming virtual [CITATION]; in the Borromean case the three-body bound states goes to the continuum turning into a resonance [CITATION].', '0911.1722-1-11-4': 'This behavior holds even for particles with different masses [CITATION].', '0911.1722-1-12-0': 'The ratios of threshold strengths for several two-body potentials were derived in [CITATION].', '0911.1722-1-12-1': 'These thresholds for binding the first state can be characterized by a threshold value of [MATH] [CITATION], and infinitely many bound three-body states would now appear one by one as [MATH] is changed from the three-body threshold for binding to the threshold for two-body binding [MATH].', '0911.1722-1-13-0': 'All these three-body states from a certain energy and up are universal.', '0911.1722-1-13-1': 'However, this is not an a priori obvious conclusion but nevertheless true because two effects work together, i.e., for [MATH] the system is large for the excited Efimov states and for finite [MATH] the binding is weak and the radius diverges at least logarithmically with binding [CITATION].', '0911.1722-1-13-2': 'In both cases the system is much larger than the range of the interaction.', '0911.1722-1-13-3': 'The continuous connection of these states between these limits is therefore also in the universal region.', '0911.1722-1-14-0': 'The recent results for four particles were that each three-body state has two four-body states at larger binding energy than this three-body state [CITATION].', '0911.1722-1-14-1': 'They have too little energy to decay into this three-body state.', '0911.1722-1-14-2': 'These four-body states are both described as having universal features unambiguously related to corresponding three-body states for interactions of both positive and negative scattering lengths.', '0911.1722-1-14-3': 'Detailed information of structure, correlations, and posssible limits to universality are not available.', '0911.1722-1-15-0': 'The one-to-one correspondence between the two four-body states and one Efimov state can perhaps be extended.', '0911.1722-1-15-1': 'It is tempting to speculate that two [MATH]-body Efimov states obtained with only two-body correlations [CITATION] can be related to one of the [MATH]-Efimov states.', '0911.1722-1-15-2': 'This seems to be rather systematic with only two-body correlations.', '0911.1722-1-15-3': 'If furthermore the [MATH]-body Efimov states remain after extension of the Hilbert space to allow all correlations, we can more generally expect two universal [MATH]-body bound states below the threshold for binding [MATH] states.', '0911.1722-1-15-4': 'This conservation of identity is most likely to be preserved for [MATH] and it may be increasingly worse with less attraction.', '0911.1722-1-15-5': 'However, ground and lowest excited states may be outside the universal region and the correspondence may only appear for even higher excited states.', '0911.1722-1-15-6': 'In any case the scaling properties are different for the states in [CITATION] and [CITATION].', '0911.1722-1-16-0': 'The basic reason for the difficulties in finding detailed and general answers is related to the fact that the thresholds for binding are moving monotonously towards less attraction with [MATH] [CITATION].', '0911.1722-1-16-1': 'For [MATH] weak binding and large scattering length is synonymous.', '0911.1722-1-16-2': 'Already for [MATH] this connection is broken but the weak binding still causes the size to diverge [CITATION].', '0911.1722-1-16-3': 'The indications are that for [MATH] the size remains finite even in the limit of zero binding.', '0911.1722-1-17-0': 'Basic properties.', '0911.1722-1-18-0': 'We consider a system of [MATH] identical bosons each of mass [MATH].', '0911.1722-1-18-1': 'They are confined by a harmonic trap of frequency [MATH] corresponding to a length parameter [MATH].', '0911.1722-1-18-2': 'The particles interact pairwise through a potential [MATH] of short range [MATH].', '0911.1722-1-18-3': 'We shall use the gaussian shape [MATH].', '0911.1722-1-18-4': 'The chosen values of [MATH], [MATH], and [MATH] lead to a two-body scattering length [MATH] and an effective range [MATH].', '0911.1722-1-18-5': 'The solution to the Schrodinger equation is approximately found by the stochastic variational method [CITATION].', '0911.1722-1-18-6': 'The results are energies and root mean square radii.', '0911.1722-1-19-0': 'For two-body systems we know that the [MATH]th radial moment only diverge at threshold of binding when the angular momentum [MATH], see [CITATION].', '0911.1722-1-19-1': 'The equality sign implies a logarithmic divergence with binding [MATH] in contrast to the normal power law [MATH].', '0911.1722-1-19-2': 'For the mean square radius this implies divergence for [MATH].', '0911.1722-1-19-3': 'For an [MATH]-body system with all contributions entirely from [MATH]-waves we can generalize these rigorous results from two-body systems [CITATION].', '0911.1722-1-19-4': 'The number of degrees of freedom is [MATH] and the generalized centrifugal barrier is obtained with an effective angular momentum [MATH].', '0911.1722-1-19-5': 'Divergent root mean square radius is then expected when [MATH] or equivalently when [MATH] or [MATH].', '0911.1722-1-19-6': 'If this result holds, four-body systems should have finite root mean square radii even at the threshold of binding.', '0911.1722-1-20-0': 'The size of the system is measured by the square root of mean square radius, [MATH], which is expressed in units of the "natural" size of the systems which is the range of the potential holding the system together.', '0911.1722-1-20-1': 'The dimensionless unit of the binding energy [MATH] of the system is [MATH].', '0911.1722-1-20-2': 'Both Universality and Scale Invariance is therefore detected by inspection of these quantities as functions of the parameters and the shape of the potentials.', '0911.1722-1-20-3': 'In the regions where the same curve emerges no matter which potential changes are made we conclude that the properties are universal and scale invariant.', '0911.1722-1-20-4': 'On the other hand when different curves appear this set of parameters is not within the universal window.', '0911.1722-1-21-0': 'To test this in calculations for different potential shapes the range has to be expressed in terms of the range of a standard potential.', '0911.1722-1-21-1': 'This may in itself introduce an unnecessary inaccuracy.', '0911.1722-1-21-2': 'To avoid this one can investigate mean square radius and mass times binding energy with the inconvenience of using dimensionful quantities.', '0911.1722-1-22-0': 'Clasical allowed region.', '0911.1722-1-23-0': 'Universal properties can intuitively only appear when the structures are outside the potentials because otherwise any small modification would have an effect on the wavefunction.', '0911.1722-1-23-1': 'Consequently the property would be dependent on these details in conflict with the assumption of universality.', '0911.1722-1-23-2': 'For two-body systems the relative wavefunction is therefore universal only if the largest probability is found outside the potential.', '0911.1722-1-23-3': 'This means that the classically forbidden region is occupied.', '0911.1722-1-23-4': 'The system is extremely quantum mechanical and very far from obeying the laws of classical physics.', '0911.1722-1-24-0': 'To investigate the relation between universality and the classical forbidden regions for [MATH] particles we need to compare features of universality with occupation of classical forbidden regions.', '0911.1722-1-24-1': 'For two-body systems this is straightforward since the coordinate of the wavefunction and the potential is the same.', '0911.1722-1-24-2': 'The probability of finding the system where the energy is smaller than the potential energy is then easy to compute as a simple integral.', '0911.1722-1-25-0': 'For more than two particles the problem is well defined but the classically forbidden regions themselves are difficult to locate.', '0911.1722-1-25-1': 'We attempt a crude estimate which at best can only be valid on average.', '0911.1722-1-25-2': 'The energy is computed by adding kinetic and potential energy, i.e. [EQUATION] where we choose an arbitrary particle [MATH] to get the kinetic part and a set of particles [MATH] and [MATH] to get the potential energy.', '0911.1722-1-25-3': 'The classical region is defined by having positive kinetic energy.', '0911.1722-1-25-4': 'For a two-body gaussian potential we then obtain an estimate for the classical radius [MATH] from [EQUATION]', '0911.1722-1-25-5': 'If the distance between two particles is larger than [MATH] we should be in the universal region.', '0911.1722-1-25-6': 'This value can then be compared to the size obtained from the average distance between two particles, [MATH], computed in the [MATH]-body system from the mean square radius [CITATION], i.e. [EQUATION]', '0911.1722-1-25-7': 'Thus in the classical forbidden region [MATH] from Eq.([REF]) should be smaller than [MATH] from Eq.([REF]).', '0911.1722-1-26-0': 'We show size versus binding energy for [MATH] in Fig.[REF].', '0911.1722-1-26-1': 'The variation arises by change of the strength, [MATH], of the attractive gaussian.', '0911.1722-1-26-2': 'We show results for two trap sizes deviating by an order of magnitude and larger than the interaction range [MATH] is the Bohr radius) by a factor of 20 and 200, respectively.', '0911.1722-1-26-3': 'For large binding in the lower right corner the results for the ground state is independent of trap size.', '0911.1722-1-26-4': 'When the probability extends by about a factor of [MATH] further out than [MATH] the effect of the small trap can be seen.', '0911.1722-1-26-5': 'The tail of the distribution then extends out to [MATH] even though the mean square is 10 times smaller.', '0911.1722-1-27-0': 'In the limit of very small binding energy the radius approaches a constant independent of the binding.', '0911.1722-1-27-1': 'The trap size has to be increased to [MATH] before the trap has no influence which implies that the probability distribution is entirely within that distance when the threshold for zero binding is reached.', '0911.1722-1-27-2': 'We could as well have removed the external field which in turn means that this converged radius is an intrinsic property of the four-body system at threshold.', '0911.1722-1-27-3': 'The size does not diverge as for three particles.', '0911.1722-1-27-4': 'The converged size is about [MATH] for the ground state.', '0911.1722-1-27-5': 'Somewhat surprisingly also the first excited state, which also is below the energy of the three-body state, has converged to a value, [MATH], independent of the trap size.', '0911.1722-1-27-6': 'Both states are at the threshold on average very much smaller than both traps.', '0911.1722-1-27-7': 'Nevertheless the smallest trap would still influence the tail of the distribution.', '0911.1722-1-28-0': 'In Fig. [REF] we also showed the estimated classical average distance between pairs of particles within the [MATH]-body system.', '0911.1722-1-28-1': 'This curve is above the ground state radius for large binding.', '0911.1722-1-28-2': 'Here the probability is mostly found in the classical region within the potential which we believe is in the non-universal region.', '0911.1722-1-28-3': 'Another potential shape would then move these curves in this region.', '0911.1722-1-28-4': 'The classical and root mean square radius crosses each other when the size is slightly larger than the range [MATH].', '0911.1722-1-28-5': 'This limit for universality is similar to the halo condition for universality established in [CITATION].', '0911.1722-1-28-6': 'At smaller binding energy the classical radius becomes less than the size of the system and the probability is on average located outside the potential in the non-classical and universal region.', '0911.1722-1-29-0': 'For the extremely small binding energies close to the threshold our estimate of the classical radius diverges logarithmically with binding energy.', '0911.1722-1-29-1': 'Thus at some point it has to exceed the size of the system which we already concluded converge to a finite value for zero binding.', '0911.1722-1-29-2': 'This is simply due to the character of the gaussian potential which approaches zero for large radii.', '0911.1722-1-29-3': 'Zero energy must then be matched by an infinite radius.', '0911.1722-1-29-4': 'However, this gaussian tail is too small to obstruct the convergence of the probability distribution to a finite size.', '0911.1722-1-29-5': 'This cannot destroy universality because the tail has no influence on the wavefunction in this region far outside the range of the potential.', '0911.1722-1-29-6': 'For universality only the binding energy is decisive as one can see explicitly for the two-body system.', '0911.1722-1-29-7': 'For [MATH]-body systems the same result follows from the asymptotic large distance behavior of the wavefunction expressed in hyperspherical coordinates [CITATION].', '0911.1722-1-29-8': 'Thus the classical average radius argument fails for these extreme energies when the probability has settled outside the range of the short-range potential.', '0911.1722-1-30-0': 'Five-body system.', '0911.1722-1-31-0': 'In Fig. [REF] we also show results for [MATH] where convergence is reached for the trap size of [MATH].', '0911.1722-1-31-1': 'The sizes for both ground and excited states increase again with decreasing binding energy [MATH] and approach finite values when [MATH].', '0911.1722-1-31-2': 'These limiting radii of about [MATH] and [MATH] are substantially smaller than corresponding values for four particles.', '0911.1722-1-31-3': 'Still the largest probability is found outside the potential providing the binding.', '0911.1722-1-31-4': 'This strongly indicates that also these structures are in the universal region.', '0911.1722-1-31-5': 'Comparison to the classical radius supports this conclusion as seen in Fig. [REF].', '0911.1722-1-31-6': 'The classical radius is always smaller than the radius of the excited state and comparable to the radius of the ground state.', '0911.1722-1-31-7': 'As argued for four particles the largest binding for the ground state corresponds to non-universal structure.', '0911.1722-1-31-8': 'When the binding energy is about [MATH] in the dimensionless units on the figure the universal structure appears.', '0911.1722-1-31-9': 'This happens at about the same energy as for four particles.', '0911.1722-1-31-10': 'In both cases the probability is pushed outside the potential and universality is expected for smaller binding energies even for these ground states.', '0911.1722-1-32-0': 'Conclusions.', '0911.1722-1-33-0': 'We have investigated the behavior near threshold for binding of super Borromean systems.', '0911.1722-1-33-1': 'Ground and first excited state for four and five identical bosons appear below the threshold for binding three and four particles, respectively.', '0911.1722-1-33-2': 'Their radii are for small binding energies larger than the range of the potential holding them together.', '0911.1722-1-33-3': 'The largest part of the probability is found in non-classical regions resulting in universal structures.', '0911.1722-1-33-4': 'For six and more particles the ground states would be located inside the potential and thus of non-universal structures.', '0911.1722-1-33-5': 'Excited states are larger and may still be universal but already for seven or eight particles also the first excited state is expected to be non-universal.'} | {'0911.1722-2-0-0': 'We compute binding energies and root mean square radii for weakly bound [MATH] and [MATH] identical bosons.', '0911.1722-2-0-1': 'Ground and first excited states appear below the threshold for binding the system with [MATH] particles.', '0911.1722-2-0-2': 'Their root mean square radii approach constants as their binding energies vanish.', '0911.1722-2-0-3': 'Their probability distributions are on average located in non-classical regions of space which result in universal structures.', '0911.1722-2-0-4': 'Radii decrease with increasing particle number.', '0911.1722-2-0-5': 'The ground states for more than five particles are probably non-universal whereas excited states may be universal.', '0911.1722-2-1-0': 'Introduction.', '0911.1722-2-2-0': 'Efimov physics could be defined as quantum physics where Universality and Scale Invariance apply [CITATION].', '0911.1722-2-2-1': 'Universality means independence of the shape of the interparticle potential.', '0911.1722-2-2-2': 'Scale Invariance means independence of the length scale of the system.', '0911.1722-2-2-3': 'These conditions are rather restrictive but a number of systems are known to exist within this window [CITATION].', '0911.1722-2-2-4': 'The great advantage is that one theory is sufficient to explain properties without any detailed knowledge of the interactions [CITATION].', '0911.1722-2-2-5': 'Furthermore, properties in different subfields of physics are described as manifestations of the same underlying theory.', '0911.1722-2-3-0': 'The range of validity for such a global theory is only well described for two and three particles [CITATION].', '0911.1722-2-3-1': 'For [MATH] two states were found in the zero-range inherently universal effective field model [CITATION].', '0911.1722-2-3-2': 'These states also appeared as universal in finite-range models in connection with each Efimov state [CITATION].', '0911.1722-2-3-3': 'This is in contrast to [CITATION] where the "disentanglement" of the regulators (cutoffs) in three and four-body equations gives rise to a dependence of the four-body ground state on interaction details.', '0911.1722-2-3-4': 'Then a four-body scale is needed.', '0911.1722-2-4-0': 'Very little is known for five particles with fully correlated solutions obtained as dictated by the interaction.', '0911.1722-2-4-1': 'However, with specific assumptions about only [MATH]-waves and essentially no correlations it was concluded in [CITATION] that halos cannot exist for [MATH].', '0911.1722-2-4-2': 'Since halos are universal structures this claim is in apparent conflict with the recent results for the [MATH] system [CITATION].', '0911.1722-2-5-0': 'It was concluded in [CITATION] that Efimov states do not exist for [MATH] and furthermore for three particles exist only for dimensions between [MATH] and [MATH] [CITATION].', '0911.1722-2-5-1': 'However, by restricting to two-body correlations within the [MATH]-body system, a series of (highly) excited [MATH]-body states were found with the characteristic Efimov scaling of energies and radii [CITATION].', '0911.1722-2-5-2': 'Whether they maintain their identity and the universal character, when more correlations are allowed in the solutions, remains to be seen.', '0911.1722-2-6-0': 'Two limits to universality are apparent.', '0911.1722-2-6-1': 'The first appears for large binding energy where the resulting small radii locate the system within the range of the potentials and sensitivity to details must appear.', '0911.1722-2-6-2': 'The less strict second limit is for excitation energies above the threshold for binding subsystems with fewer particles.', '0911.1722-2-6-3': 'Structures with such energies are necessarily continuum states which may, or more often may not, be classified as universal states depending on their structures and the final states reached after the decay.', '0911.1722-2-7-0': 'Even for four particles where universality is found [CITATION], a number of questions are still unanswered.', '0911.1722-2-7-1': 'For five and more particles the information becomes very scarce.', '0911.1722-2-7-2': 'A novel study claiming universality for ground states of a Van der Waals potential has appeared for particle number less than 40 [CITATION].', '0911.1722-2-7-3': 'The critical mass is found as a substitute for the critical strength, but the computed radii at threshold cannot be reliably extracted.', '0911.1722-2-8-0': 'The purpose of the present paper is to explore the boundaries for universality preferentially leading to general conclusions applicable to systems of [MATH]-particles.', '0911.1722-2-8-1': 'We first discuss qualitative features and basic properties, then extract numerical results for [MATH] and [MATH] particles very close to thresholds of binding, and relate to classically allowed regions.', '0911.1722-2-8-2': 'We only investigate Brunnian ([MATH]-body Borromean) systems [CITATION] where no subsystem is bound.', '0911.1722-2-9-0': 'Qualitative considerations.', '0911.1722-2-10-0': 'For two particles the infinite scattering length corresponds to a bound state at zero energy.', '0911.1722-2-10-1': 'Variation of [MATH] around zero produces either a bound state of spatial extension [MATH] or a continuum state where the correlation range also is [MATH], corresponding to an effective interaction of range [MATH].', '0911.1722-2-11-0': 'For three particles the Efimov effect appears, i.e. for the same interaction, [MATH]), infinitely many three-body bound states emerge with progressively smaller binding and correspondingly larger radii [CITATION].', '0911.1722-2-11-1': 'The ratios of two and three-body threshold strengths for several potentials were derived in [CITATION].', '0911.1722-2-11-2': 'These thresholds for binding the first state can be characterized by a value of [MATH] [CITATION].', '0911.1722-2-11-3': 'Infinitely many bound three-body states appear one by one as [MATH] is changed from the three-body threshold for binding to the threshold for two-body binding [MATH].', '0911.1722-2-11-4': 'Moving opposite by decreasing the attraction these states one by one cease to be bound.', '0911.1722-2-11-5': 'They move into the continuum and continue as resonances [CITATION].', '0911.1722-2-11-6': 'For asymmetric systems with a bound two-body subsystem the three-body bound state passes through the particle-dimer elastic scattering cut becoming virtual [CITATION].', '0911.1722-2-11-7': 'This behavior holds even for particles with different masses [CITATION].', '0911.1722-2-12-0': 'All three-body [MATH]-wave states from a certain energy and up are universal.', '0911.1722-2-12-1': 'However, this is not an a priori obvious conclusion but nevertheless true because two effects work together, i.e., for [MATH] the system is large for the excited Efimov states and for finite [MATH] the binding is weak and the radius diverges at least logarithmically with binding [CITATION].', '0911.1722-2-12-2': 'Both Efimov states and weakly bound states are much larger than the range of the interaction.', '0911.1722-2-12-3': 'The continuous connection of these states is therefore also in the universal region.', '0911.1722-2-13-0': 'The recent results for four particles were that each three-body state has two four-body states attached with larger binding energy [CITATION].', '0911.1722-2-13-1': 'These four-body states are both described as having universal features unambiguously related to the corresponding three-body states for interactions of both positive and negative scattering lengths.', '0911.1722-2-13-2': 'Detailed information of structure, correlations, and posssible limits to universality are not available.', '0911.1722-2-14-0': 'The one-to-one correspondence between the two four-body states and one three-body state can perhaps be extended such that two weakly bound [MATH]-body states appear below the ground state of the [MATH]-body state.', '0911.1722-2-14-1': 'This seems to be rather systematic for [MATH]-body Efimov states obtained with only two-body correlations [CITATION].', '0911.1722-2-14-2': 'If these [MATH]-body Efimov states remain after extension of the Hilbert space to allow all correlations, we can expect these sequences to be continued to the thresholds for binding by decreasing the attraction.', '0911.1722-2-14-3': 'However, ground and lowest excited states may be outside the universal region but the sequences may still exist.', '0911.1722-2-14-4': 'In any case the scaling properties are different for the states in [CITATION] and [CITATION].', '0911.1722-2-15-0': 'The basic reason for the difficulties in finding detailed and general answers is related to the fact that the thresholds for binding are moving monotonously towards less attraction with [MATH] [CITATION].', '0911.1722-2-15-1': 'For [MATH] weak binding and large scattering length is synonymous.', '0911.1722-2-15-2': 'Already for [MATH] this connection is broken but the weak binding still causes the size to diverge [CITATION].', '0911.1722-2-15-3': 'The indications are that for [MATH] the size remains finite even in the limit of zero binding.', '0911.1722-2-16-0': 'Basic properties.', '0911.1722-2-17-0': 'We consider a system of [MATH] identical bosons each of mass [MATH].', '0911.1722-2-17-1': 'They are confined by a harmonic trap of frequency [MATH] corresponding to a length parameter [MATH].', '0911.1722-2-17-2': 'The particles interact pairwise through a potential [MATH] of short range [MATH].', '0911.1722-2-17-3': 'We shall use the gaussian shape [MATH].', '0911.1722-2-17-4': 'The chosen values of [MATH], [MATH], and [MATH] lead to a two-body scattering length [MATH] and an effective range [MATH].', '0911.1722-2-17-5': 'The solution to the Schrodinger equation is approximately found by the stochastic variational method [CITATION].', '0911.1722-2-17-6': 'The results are energies and root mean square radii.', '0911.1722-2-18-0': 'For two-body systems we know that the [MATH]th radial moment only diverge at threshold of binding when the angular momentum [MATH], see [CITATION].', '0911.1722-2-18-1': 'The equality sign implies a logarithmic divergence with binding [MATH] in contrast to the normal power law [MATH].', '0911.1722-2-18-2': 'For the mean square radius this implies divergence for [MATH].', '0911.1722-2-18-3': 'For an [MATH]-body system with all contributions entirely from [MATH]-waves we can generalize these rigorous results from two-body systems [CITATION].', '0911.1722-2-18-4': 'The number of degrees of freedom is [MATH] and the generalized centrifugal barrier is obtained with an effective angular momentum [MATH].', '0911.1722-2-18-5': 'Divergent root mean square radius is then expected when [MATH] or equivalently when [MATH] or [MATH].', '0911.1722-2-18-6': 'If this result holds, four-body systems should have finite root mean square radii even at the threshold of binding.', '0911.1722-2-19-0': 'The size of the system is measured by the square root of the mean square radius, [MATH], which is expressed in units of the "natural" size of the systems, i.e. the range of the binding potential.', '0911.1722-2-19-1': 'The dimensionless unit of the binding energy [MATH] of the system is [MATH].', '0911.1722-2-19-2': 'Both Universality and Scale Invariance is therefore detected by inspection of these quantities as functions of parameters and shape of the potentials.', '0911.1722-2-19-3': 'In regions where the curves are proportional independent of the changes, we conclude that the properties are universal and scale invariant.', '0911.1722-2-19-4': 'Results for different potential shapes can be expressed in terms of a standard potential by scaling the range.', '0911.1722-2-19-5': 'Then individual curves become truly universal.', '0911.1722-2-20-0': 'Clasical allowed region.', '0911.1722-2-21-0': 'Universal properties can intuitively only appear when the structures are outside the potentials because otherwise any small modification would have an effect on the wavefunction.', '0911.1722-2-21-1': 'Consequently the property would be dependent on these details in conflict with the assumption of universality.', '0911.1722-2-21-2': 'For two-body systems the relative wavefunction is therefore universal only if the largest probability is found outside the potential.', '0911.1722-2-21-3': 'This means that the classically forbidden region is occupied.', '0911.1722-2-21-4': 'The system is extremely quantum mechanical and very far from obeying the laws of classical physics.', '0911.1722-2-22-0': 'To investigate the relation between universality and the classical forbidden regions for [MATH] particles we need to compare features of universality with occupation of classical forbidden regions.', '0911.1722-2-22-1': 'For two-body systems this is straightforward since the coordinate of the wavefunction and the potential is the same.', '0911.1722-2-22-2': 'The probability of finding the system where the energy is smaller than the potential energy is then easy to compute as a simple integral.', '0911.1722-2-23-0': 'For more than two particles the problem is well defined but the classically forbidden regions themselves are difficult to locate.', '0911.1722-2-23-1': 'We attempt a crude estimate which at best can only be valid on average.', '0911.1722-2-23-2': 'The energy is computed by adding kinetic and potential energy, i.e. [EQUATION] where we choose an arbitrary particle [MATH] to get the kinetic part and a set of particles [MATH] and [MATH] to get the potential energy.', '0911.1722-2-23-3': 'The classical region is defined by having positive kinetic energy.', '0911.1722-2-23-4': 'For a two-body gaussian potential we then obtain an estimate for the classical radius [MATH] from [EQUATION]', '0911.1722-2-23-5': 'If the distance between two particles is larger than [MATH] we should be in the universal region.', '0911.1722-2-23-6': 'This value can then be compared to the size obtained from the average distance between two particles, [MATH], computed in the [MATH]-body system from the mean square radius [CITATION], i.e. [EQUATION]', '0911.1722-2-23-7': 'Thus in the classical forbidden region [MATH] from Eq.([REF]) should be smaller than [MATH] from Eq.([REF]).', '0911.1722-2-24-0': 'We show size versus binding energy for [MATH] in Fig.[REF].', '0911.1722-2-24-1': 'The variation arises by change of the strength, [MATH], of the attractive gaussian.', '0911.1722-2-24-2': 'The system is for numerical convenience confined by an external one-body field.', '0911.1722-2-24-3': 'However, we are only interested in structures independent of that field, i.e. intrinsic properties of the four-body system.', '0911.1722-2-24-4': 'We therefore increase the trap size until the states are converged and located at distances much smaller than the confining walls.', '0911.1722-2-24-5': 'We now know that this happens for four particles in contrast to the three-body system where the size diverges when the binding energy approaches zero.', '0911.1722-2-25-0': 'In Fig. [REF] we show results for two trap sizes deviating by an order of magnitude and larger than the interaction range [MATH] is the Bohr radius) by a factor of 20 and 200, respectively.', '0911.1722-2-25-1': 'For large binding in the lower right corner the results for the ground state is independent of trap size.', '0911.1722-2-25-2': 'When the probability extends by about a factor of [MATH] further out than [MATH] the effect of the small trap can be seen.', '0911.1722-2-25-3': 'The tail of the distribution then extends out to [MATH] even though the mean square is 10 times smaller.', '0911.1722-2-26-0': 'In the limit of very small binding energy the radius approaches a constant independent of the binding.', '0911.1722-2-26-1': 'The trap size has to be increased to [MATH] before the trap has no influence which implies that the probability distribution is entirely within that distance when the threshold for zero binding is reached.', '0911.1722-2-26-2': 'The converged size is about [MATH] for the ground state.', '0911.1722-2-26-3': 'Somewhat surprisingly also the first excited state, which also is below the energy of the three-body state, has converged to a value, [MATH], independent of the trap size.', '0911.1722-2-26-4': 'A shape different from a gaussian would again lead to constants related through specific properties of the potentials, but the ratio would remain unchanged.', '0911.1722-2-26-5': 'This is precisely as found in two dimensions for three particles [CITATION].', '0911.1722-2-26-6': 'Both states are at the threshold on average very much smaller than both traps.', '0911.1722-2-26-7': 'Nevertheless the smallest trap would still influence the tail of the distribution.', '0911.1722-2-27-0': 'In Fig. [REF] we also show the estimated classical average distance between pairs of particles within the [MATH]-body system.', '0911.1722-2-27-1': 'This curve is above the ground state radius for large binding.', '0911.1722-2-27-2': 'Here the probability is mostly found in the classical region within the potential, i.e. in the non-universal region.', '0911.1722-2-27-3': 'Another potential shape would then move these curves.', '0911.1722-2-27-4': 'The classical and root mean square radius cross each other when the size is slightly larger than the range [MATH].', '0911.1722-2-27-5': 'This limit for universality is similar to the halo condition for universality established in [CITATION].', '0911.1722-2-27-6': 'At smaller binding energy the classical radius becomes less than the size of the system and the probability is on average located outside the potential in the non-classical, universal region.', '0911.1722-2-28-0': 'For the extremely small binding energies close to the threshold our estimate of the classical radius diverges logarithmically with binding energy.', '0911.1722-2-28-1': 'Thus at some point it has to exceed the size of the system which we concluded converge to a finite value for zero binding.', '0911.1722-2-28-2': 'This is simply due to the character of the gaussian potential which approaches zero for large radii.', '0911.1722-2-28-3': 'Zero energy must then be matched by an infinite radius.', '0911.1722-2-28-4': 'However, this gaussian tail is too small to obstruct the convergence of the probability distribution to a finite size.', '0911.1722-2-28-5': 'This cannot destroy universality because the tail has no influence on the wavefunction in this region far outside the range of the potential.', '0911.1722-2-28-6': 'For universality only the binding energy is decisive as one can see explicitly for the two-body system.', '0911.1722-2-28-7': 'For [MATH]-body systems the same result follows from the asymptotic large distance behavior of the wavefunction expressed in hyperspherical coordinates [CITATION].', '0911.1722-2-28-8': 'Thus the classical average radius argument fails for these extreme energies when the probability has settled outside the range of the short-range potential.', '0911.1722-2-29-0': 'Five-body system.', '0911.1722-2-30-0': 'In Fig. [REF] we also show results for [MATH] where convergence is reached for the trap size of [MATH].', '0911.1722-2-30-1': 'The two lowest states are both more bound than the four-body ground state and the third state is close to the four-body threshold.', '0911.1722-2-30-2': 'The sizes for both ground and excited states increase again with decreasing binding energy [MATH] and approach finite values when [MATH].', '0911.1722-2-30-3': 'These limiting radii of about [MATH] and [MATH] are substantially smaller than corresponding values for four particles.', '0911.1722-2-30-4': 'Still the largest probability is found outside the potential providing the binding.', '0911.1722-2-30-5': 'This strongly indicates that also these structures are in the universal region.', '0911.1722-2-30-6': 'Again their ratio is anticipated to be essentially independent of potential shape.', '0911.1722-2-30-7': 'This conclusion is supported by the comparison in Fig. [REF] to the classical radius which always is smaller than the radius of the excited state and comparable to the radius of the ground state.', '0911.1722-2-30-8': 'As argued for four particles the largest binding for the ground state corresponds to non-universal structure.', '0911.1722-2-30-9': 'When the binding energy is about [MATH] in the dimensionless units on the figure the universal structure appears.', '0911.1722-2-30-10': 'This happens at about the same energy as for four particles.', '0911.1722-2-30-11': 'In both cases the probability is pushed outside the potential and universality is expected for smaller binding energies.', '0911.1722-2-31-0': 'Conclusions.', '0911.1722-2-32-0': 'We have investigated the behavior near threshold for binding of Brunnian systems.', '0911.1722-2-32-1': 'Ground and first excited state for four and five identical bosons appear below the threshold for binding three and four particles, respectively.', '0911.1722-2-32-2': 'Their radii are for small binding energies larger than the range of the potential holding them together.', '0911.1722-2-32-3': 'The largest part of the probability is found in non-classical regions resulting in universal structures.', '0911.1722-2-32-4': 'For six and more particles the ground states would be located inside the potential and thus of non-universal structures.', '0911.1722-2-32-5': 'Excited states are larger and may still be universal but already for seven or eight particles also the first excited state is expected to be non-universal.'} | [['0911.1722-1-18-0', '0911.1722-2-17-0'], ['0911.1722-1-18-1', '0911.1722-2-17-1'], ['0911.1722-1-18-2', '0911.1722-2-17-2'], ['0911.1722-1-18-3', '0911.1722-2-17-3'], ['0911.1722-1-18-4', '0911.1722-2-17-4'], ['0911.1722-1-18-5', '0911.1722-2-17-5'], ['0911.1722-1-18-6', '0911.1722-2-17-6'], ['0911.1722-1-27-0', '0911.1722-2-26-0'], ['0911.1722-1-27-1', '0911.1722-2-26-1'], ['0911.1722-1-27-4', '0911.1722-2-26-2'], ['0911.1722-1-27-5', '0911.1722-2-26-3'], ['0911.1722-1-27-6', '0911.1722-2-26-6'], ['0911.1722-1-27-7', 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'0911.1722-1-30-0', '0911.1722-1-32-0', '0911.1722-2-1-0', '0911.1722-2-9-0', '0911.1722-2-16-0', '0911.1722-2-20-0', '0911.1722-2-29-0', '0911.1722-2-31-0', '0911.1722-3-1-0', '0911.1722-3-11-0', '0911.1722-3-18-0', '0911.1722-3-22-0', '0911.1722-3-31-0', '0911.1722-3-33-0'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/0911.1722 | {'0911.1722-3-0-0': 'We compute binding energies and root mean square radii for weakly bound systems of [MATH] and [MATH] identical bosons.', '0911.1722-3-0-1': 'Ground and first excited states of an [MATH]-body system appear below the threshold for binding the system with [MATH] particles.', '0911.1722-3-0-2': 'Their root mean square radii approach constants in the limit of weak binding.', '0911.1722-3-0-3': 'Their probability distributions are on average located in non-classical regions of space which result in universal structures.', '0911.1722-3-0-4': 'Radii decrease with increasing particle number.', '0911.1722-3-0-5': 'The ground states for more than five particles are probably non-universal whereas excited states may be universal.', '0911.1722-3-1-0': 'Introduction.', '0911.1722-3-2-0': 'The Efimov effect was predicted in [CITATION] as the appearance of a series of intimately related excited three-body states when at least two scattering lengths are infinitely large.', '0911.1722-3-2-1': 'These states can appear at all length scales and their properties are independent of the details of the potentials.', '0911.1722-3-2-2': 'This effect has in recent years been studied intensively and extended to a wider group of physical phenomena, beginning to be known as Efimov physics.', '0911.1722-3-2-3': 'In general we define Efimov physics as quantum physics where Universality and Scale Invariance apply.', '0911.1722-3-2-4': 'Universality means independence of the shape of the interparticle potential.', '0911.1722-3-2-5': 'Scale Invariance means independence of the length scale of the system.', '0911.1722-3-2-6': 'These conditions are rather restrictive but a number of systems are known to exist within this window [CITATION].', '0911.1722-3-2-7': 'The great advantage is that one theory is sufficient to explain properties without any detailed knowledge of the interactions [CITATION].', '0911.1722-3-2-8': 'Furthermore, properties in different subfields of physics are described as manifestations of the same underlying theory.', '0911.1722-3-3-0': 'Our physical definition of scale invariance originates from the halo physics first realized and discussed in nuclei [CITATION], but quickly observed as applicable also to small molecules like the helium trimers [CITATION].', '0911.1722-3-3-1': 'This original definition of scale invariance, that the concept applies to any length scale is obviously continuous as exemplified by nuclei, atoms and molecules.', '0911.1722-3-3-2': 'Often the notion of scale invariance is used in a different mathematical sense where the spatial extension of the structures in one given system repeats itself in discrete steps like the factor 22.7 for identical particles [CITATION].', '0911.1722-3-3-3': 'This is a result of the independence of potential details and here precisely defining our meaning with the notion of universality.', '0911.1722-3-3-4': 'As the concepts can be defined in different ways we will use throughout the paper this original physical meaning of scale invariance.', '0911.1722-3-3-5': 'Together, these two concepts constitute our meaning of Efimov Physics which to the best of our knowledge has been left undefined in all previous publications.', '0911.1722-3-4-0': 'The range of validity for such a global theory is only well described for two and three particles [CITATION].', '0911.1722-3-4-1': 'For [MATH] two states were found in the zero-range, inherently universal, effective field model [CITATION].', '0911.1722-3-4-2': 'These states also appeared as universal in finite-range models in connection with each Efimov state [CITATION].', '0911.1722-3-4-3': 'This is in contrast to [CITATION] where the disentanglement of the scales used to regularize the three and four-body zero-range Faddeev-Yakubovsky equations gives rise to a dependence of the four-body ground state on interaction details.', '0911.1722-3-4-4': 'Then a four-body scale is needed in analogy to the three-body scale appearing independently on top of the two-body properties.', '0911.1722-3-4-5': 'This apparent discrepancy between Refs. [CITATION] and [CITATION] is not yet resolved.', '0911.1722-3-5-0': 'Recently, three experiments evidenced two four-body bound states connected to an Efimov trimer [CITATION] in accordance with the theoretical predictions of Ref. [CITATION].', '0911.1722-3-5-1': 'In two of these experiments were also observed deviations from universality [CITATION].', '0911.1722-3-5-2': 'Surprisingly, the greatest deviation were observed for large scattering lengths ([MATH]) - exactly at the region where universality should apply [CITATION].', '0911.1722-3-5-3': 'This requires a theoretical explanation where something should be added in the universal model.', '0911.1722-3-6-0': 'Very little is known for five particles with complete solutions containing all correlations as dictated by the interaction.', '0911.1722-3-6-1': 'With specific assumptions about only [MATH]-waves and essentially no correlations it was concluded in [CITATION] that ground state halos cannot exist for [MATH].', '0911.1722-3-6-2': 'These assumptions are rather extreme and could be wrong or only partly correct.', '0911.1722-3-6-3': 'However, if halos exist they have universal structures as the [MATH] states obtained in [CITATION].', '0911.1722-3-6-4': 'These results can only be reconciled by wrong assumptions in the halo discussion or by impermissible comparison between halo ground states and excited states.', '0911.1722-3-7-0': 'It was concluded in [CITATION] that Efimov states do not exist for [MATH] and furthermore for three particles exist only for dimensions between [MATH] and [MATH] [CITATION].', '0911.1722-3-7-1': 'However, by restricting to two-body correlations within the [MATH]-body system, a series of (highly) excited [MATH]-body states were found with the characteristic Efimov scaling of energies and radii [CITATION].', '0911.1722-3-7-2': 'Whether they maintain their identity and the universal character, when more correlations are allowed in the solutions, remains to be seen.', '0911.1722-3-8-0': 'Two limits to the universality are apparent.', '0911.1722-3-8-1': 'The first appears for large binding energy where the resulting small radii locate the system within the range of the potentials and sensitivity to details must appear.', '0911.1722-3-8-2': 'The less strict second limit is for excitation energies above the threshold for binding subsystems with fewer particles.', '0911.1722-3-8-3': 'Structures with such energies are necessarily continuum states which may, or more often may not, be classified as universal states depending on their structures and the final states reached after the decay.', '0911.1722-3-9-0': 'Even for four particles where universality is found [CITATION], a number of questions are still unanswered.', '0911.1722-3-9-1': 'For five and more particles the information becomes very scarce.', '0911.1722-3-9-2': 'A novel study claiming universality for ground states of a Van der Waals potential has appeared for particle number less than 40 [CITATION].', '0911.1722-3-9-3': 'The critical mass is found as a substitute for the critical strength, but the computed radii at threshold cannot be reliably extracted.', '0911.1722-3-10-0': 'The purpose of the present paper is to explore the window for Efimov physics.', '0911.1722-3-10-1': 'We shall investigate the boundaries for universality preferentially leading to general conclusions applicable to systems of [MATH]-particles.', '0911.1722-3-10-2': 'We first discuss qualitative features and basic properties, then extract numerical results for [MATH] and [MATH] particles very close to thresholds of binding, and relate to classically allowed regions.', '0911.1722-3-10-3': 'We only investigate Brunnian ([MATH]-body Borromean) systems [CITATION] where no subsystem is bound.', '0911.1722-3-11-0': 'Qualitative considerations.', '0911.1722-3-12-0': 'For two particles the infinite scattering length corresponds to a bound state at zero energy.', '0911.1722-3-12-1': 'Variation of [MATH] around zero produces either a bound state of spatial extension [MATH] or a continuum state corresponding to spatial configurations correlated over the radius [MATH].', '0911.1722-3-13-0': 'For three particles the Efimov effect appears, i.e. for the same interaction, [MATH]), infinitely many three-body bound states emerge with progressively smaller binding and correspondingly larger radii [CITATION].', '0911.1722-3-13-1': 'The ratios of two and three-body threshold strengths for several potentials were derived in [CITATION].', '0911.1722-3-13-2': 'These thresholds for binding one state can be characterized by a value of [MATH] [CITATION].', '0911.1722-3-13-3': 'Infinitely many bound three-body states appear one by one as [MATH] is changed from the three-body threshold for binding to the threshold for two-body binding [MATH].', '0911.1722-3-13-4': 'Moving opposite by decreasing the attraction these states one by one cease to be bound.', '0911.1722-3-13-5': 'They move into the continuum and continue as resonances [CITATION].', '0911.1722-3-13-6': 'For asymmetric systems with a bound two-body subsystem the three-body bound state passes through the particle-dimer threshold becoming a virtual state [CITATION].', '0911.1722-3-13-7': 'This behavior holds even for particles with different masses [CITATION].', '0911.1722-3-14-0': 'All three-body [MATH]-wave states from a certain energy and up are universal.', '0911.1722-3-14-1': 'However, this is not an a priori obvious conclusion but nevertheless true because two effects work together, i.e., for [MATH] the system is large for the excited Efimov states and for finite [MATH] the binding is weak and the radius diverges with binding [CITATION].', '0911.1722-3-14-2': 'Both Efimov states and weakly bound states are much larger than the range of the interaction.', '0911.1722-3-14-3': 'The continuous connection of these bound states and resonances is therefore also in the universal region.', '0911.1722-3-15-0': 'The recent results for four particles were that each three-body state has two four-body states attached with larger binding energy [CITATION].', '0911.1722-3-15-1': 'These four-body states are both described as having universal features unambiguously related to the corresponding three-body states for interactions of both positive and negative scattering lengths.', '0911.1722-3-15-2': 'Detailed information of structure, correlations, and posssible limits to universality are not available.', '0911.1722-3-16-0': 'The one-to-one correspondence between the two four-body states and one three-body state can perhaps be extended such that two weakly bound [MATH]-body states appear below the ground state of the [MATH]-body state.', '0911.1722-3-16-1': 'This seems to be rather systematic for [MATH]-body Efimov states obtained with only two-body correlations [CITATION].', '0911.1722-3-16-2': 'If these [MATH]-body Efimov states remain after extension of the Hilbert space to allow all correlations, we can expect these sequences to be continued to the thresholds for binding by decreasing the attraction.', '0911.1722-3-16-3': 'However, ground and lowest excited states may be outside the universal region but the sequences may still exist.', '0911.1722-3-16-4': 'In any case the scaling properties are different for the [MATH]-body Efimov states in [CITATION] and the universal four-body states in [CITATION].', '0911.1722-3-17-0': 'The basic reason for the difficulties in finding detailed and general answers is related to the fact that the thresholds for binding are moving monotonously towards less attraction with [MATH] [CITATION].', '0911.1722-3-17-1': 'For [MATH] weak binding and large scattering length is synonymous.', '0911.1722-3-17-2': 'Already for [MATH] this connection is broken but the weak binding still causes the size to diverge [CITATION].', '0911.1722-3-17-3': 'The indications are that for [MATH] the size remains finite even in the limit of zero binding.', '0911.1722-3-18-0': 'Basic properties.', '0911.1722-3-19-0': 'We consider a system of [MATH] identical bosons each of mass [MATH].', '0911.1722-3-19-1': 'They are confined by a harmonic trap of frequency [MATH] corresponding to a length parameter [MATH].', '0911.1722-3-19-2': 'The particles interact pairwise through a potential [MATH] of short range [MATH].', '0911.1722-3-19-3': 'We shall use the gaussian shape [MATH].', '0911.1722-3-19-4': 'The chosen values of [MATH], [MATH], and [MATH] lead to a two-body scattering length [MATH] and an effective range [MATH].', '0911.1722-3-19-5': 'The solution to the Schrodinger equation is approximately found by the stochastic variational method [CITATION].', '0911.1722-3-19-6': 'The results are energies and root mean square radii.', '0911.1722-3-20-0': 'For two-body systems we know that the [MATH]th radial moment only diverge at threshold of binding when the angular momentum [MATH], see [CITATION].', '0911.1722-3-20-1': 'The equality sign implies a logarithmic divergence with binding [MATH] in contrast to the normal power law [MATH].', '0911.1722-3-20-2': 'For the mean square radius this implies divergence for [MATH].', '0911.1722-3-20-3': 'For an [MATH]-body system with all contributions entirely from [MATH]-waves we can generalize these rigorous results from two-body systems [CITATION].', '0911.1722-3-20-4': 'The number of degrees of freedom is [MATH] and the generalized centrifugal barrier is obtained with an effective angular momentum [MATH].', '0911.1722-3-20-5': 'Divergent root mean square radius is then expected when [MATH] or equivalently when [MATH] or [MATH].', '0911.1722-3-20-6': 'If this result holds, four-body systems should have finite root mean square radii even at the threshold of binding.', '0911.1722-3-21-0': 'The size of the system is measured by the square root of the mean square radius, [MATH], which is expressed in units of the "natural" size of the systems, i.e. the range of the binding potential.', '0911.1722-3-21-1': 'The dimensionless unit of the binding energy [MATH] of the system is [MATH].', '0911.1722-3-21-2': 'Both Universality and Scale Invariance is therefore detected by inspection of these quantities as functions of parameters and shape of the potentials.', '0911.1722-3-21-3': 'In regions where the curves are proportional, we conclude that the properties are universal and scale invariant.', '0911.1722-3-21-4': 'Results for different potential shapes can be expressed in terms of a standard potential by scaling the range.', '0911.1722-3-21-5': 'Then the individual curves would fall on top of each other in the universal regime.', '0911.1722-3-22-0': 'Clasical allowed region.', '0911.1722-3-23-0': 'Universal properties can intuitively only appear when the structures are outside the potentials because otherwise any small modification would have an effect on the wavefunction.', '0911.1722-3-23-1': 'Consequently the property would be dependent on these details in conflict with the assumption of universality.', '0911.1722-3-23-2': 'For two-body systems the relative wavefunction is therefore universal only if the largest probability is found outside the potential.', '0911.1722-3-23-3': 'This means that this classically forbidden region is occupied.', '0911.1722-3-23-4': 'The system is extremely quantum mechanical and very far from obeying the laws of classical physics.', '0911.1722-3-24-0': 'To investigate the relation between universality and the classical forbidden regions for [MATH] particles we need to compare features of universality with occupation of classical forbidden regions.', '0911.1722-3-24-1': 'For two-body systems this is straightforward since the coordinate of the wavefunction and the potential is the same.', '0911.1722-3-24-2': 'The probability of finding the system where the energy is smaller than the potential energy is then easy to compute as a simple spatial integral over absolute square of the wavefunction.', '0911.1722-3-25-0': 'For more than two particles the problem is well defined but the classically forbidden regions (total energy is smaller than the potential energy) themselves are difficult to locate.', '0911.1722-3-25-1': 'We attempt a crude estimate which at best can only be valid on average.', '0911.1722-3-25-2': 'The energy is computed by adding kinetic and potential energy, i.e. [EQUATION] where we choose an arbitrary particle [MATH] to get the kinetic part and a set of particles [MATH] and [MATH] to get the potential energy.', '0911.1722-3-25-3': 'The classical region is defined by having positive kinetic energy.', '0911.1722-3-25-4': 'For a two-body gaussian potential we then obtain an estimate of an average, [MATH], for the classical radius from [EQUATION]', '0911.1722-3-25-5': 'If the distance between two particles is larger than [MATH] we should be in the universal region.', '0911.1722-3-25-6': 'This value can then be compared to the size obtained from the average distance between two particles, [MATH], computed in the [MATH]-body system from the mean square radius [CITATION], i.e. [EQUATION]', '0911.1722-3-25-7': 'Thus in the classical forbidden region [MATH] from Eq.([REF]) should be smaller than [MATH] from Eq.([REF]).', '0911.1722-3-26-0': 'We show size versus binding energy for [MATH] in Fig.[REF].', '0911.1722-3-26-1': 'The variation arises by change of the strength, [MATH], of the attractive gaussian.', '0911.1722-3-26-2': 'The system is for numerical convenience confined by an external one-body field.', '0911.1722-3-26-3': 'However, we are only interested in structures independent of that field, i.e. intrinsic properties of the four-body system.', '0911.1722-3-26-4': 'We therefore increase the trap size until the states are converged and located at distances much smaller than the confining walls.', '0911.1722-3-26-5': 'We now know that this happens for four particles in contrast to the three-body system where the size diverges when the binding energy approaches zero.', '0911.1722-3-27-0': 'In Fig. [REF] we show results for two trap sizes deviating by an order of magnitude and larger than the interaction range [MATH] is the Bohr radius) by a factor of 20 and 200, respectively.', '0911.1722-3-27-1': 'For large binding in the lower right corner the results for the ground state is independent of trap size.', '0911.1722-3-27-2': 'When the probability extends by about a factor of [MATH] further out than [MATH] the effect of the small trap can be seen.', '0911.1722-3-27-3': 'The tail of the distribution then extends out to [MATH] even though the mean square is 10 times smaller.', '0911.1722-3-28-0': 'In the limit of very small binding energy the radius approaches a constant independent of the binding.', '0911.1722-3-28-1': 'The trap size has to be increased to [MATH] before the trap has no influence which implies that the probability distribution is entirely within that distance when the threshold for zero binding is reached.', '0911.1722-3-28-2': 'The converged size is about [MATH] for the ground state.', '0911.1722-3-28-3': 'Somewhat surprisingly also the first excited state, which also is below the energy of the three-body state, has converged to a value, [MATH], independent of the trap size.', '0911.1722-3-28-4': 'A shape different from a gaussian would again lead to constants related through specific properties of the potentials, but the ratio would remain unchanged.', '0911.1722-3-28-5': 'This is precisely as found in two dimensions for three particles [CITATION].', '0911.1722-3-28-6': 'Both states are at the threshold on average very much smaller than both traps.', '0911.1722-3-28-7': 'Nevertheless the smallest trap would still influence the tail of the distribution.', '0911.1722-3-29-0': 'In Fig. [REF] we also show the estimated classical average distance between pairs of particles within the [MATH]-body system.', '0911.1722-3-29-1': 'This curve is above the ground state radius for large binding.', '0911.1722-3-29-2': 'Here the probability is mostly found in the classical region within the potential, i.e. in the non-universal region.', '0911.1722-3-29-3': 'Another potential shape would then move these curves.', '0911.1722-3-29-4': 'The classical and root mean square radius cross each other when the size is slightly larger than the range [MATH].', '0911.1722-3-29-5': 'This limit for universality is similar to the halo condition for universality established in [CITATION].', '0911.1722-3-29-6': 'At smaller binding energy the classical radius becomes less than the size of the system and the probability is on average located outside the potential in the non-classical, universal region.', '0911.1722-3-30-0': 'For the extremely small binding energies close to the threshold our estimate of the classical radius diverges logarithmically with binding energy.', '0911.1722-3-30-1': 'Thus at some point it has to exceed the size of the system which we concluded converge to a finite value for zero binding.', '0911.1722-3-30-2': 'This is simply due to the character of the gaussian potential which approaches zero for large radii.', '0911.1722-3-30-3': 'Zero energy must then be matched by an infinite radius.', '0911.1722-3-30-4': 'However, this gaussian tail is too small to obstruct the convergence of the probability distribution to a finite size.', '0911.1722-3-30-5': 'This cannot destroy universality because the tail has no influence on the wavefunction in this region far outside the range of the potential.', '0911.1722-3-30-6': 'For universality only the binding energy is decisive as one can see explicitly for the two-body system.', '0911.1722-3-30-7': 'For [MATH]-body systems the same result follows from the asymptotic large distance behavior of the wavefunction expressed in hyperspherical coordinates [CITATION].', '0911.1722-3-30-8': 'Thus the classical average radius argument fails for these extreme energies when the probability has settled outside the range of the short-range potential.', '0911.1722-3-31-0': 'Five-body system.', '0911.1722-3-32-0': 'In Fig. [REF] we also show results for [MATH] where convergence is reached for the trap size of [MATH].', '0911.1722-3-32-1': 'We found two pentamers with energies -0.0281[MATH] and -0.0113[MATH] below the four-body threshold (-0.0103[MATH]) for infinite scattering length.', '0911.1722-3-32-2': 'The sizes for both ground and excited states increase again with decreasing binding energy [MATH] and approach finite values when [MATH].', '0911.1722-3-32-3': 'These limiting radii of about [MATH] and [MATH] are substantially smaller than corresponding values for four particles.', '0911.1722-3-32-4': 'Still the largest probability is found outside the potential providing the binding.', '0911.1722-3-32-5': 'This strongly indicates that also these structures are in the universal region.', '0911.1722-3-32-6': 'Again their ratio is anticipated to be essentially independent of potential shape.', '0911.1722-3-32-7': 'This conclusion is supported by the comparison in Fig. [REF] to the classical radius which always is smaller than the radius of the excited state and comparable to the radius of the ground state.', '0911.1722-3-32-8': 'As argued for four particles the largest binding for the ground state corresponds to non-universal structure.', '0911.1722-3-32-9': 'When the binding energy is about [MATH] in the dimensionless units on the figure the universal structure appears.', '0911.1722-3-32-10': 'This happens at about the same energy as for four particles.', '0911.1722-3-32-11': 'In both cases the probability is pushed outside the potential and universality is expected for smaller binding energies.', '0911.1722-3-33-0': 'Conclusions.', '0911.1722-3-34-0': 'We have investigated the behavior of Brunnian systems near threshold for binding.', '0911.1722-3-34-1': 'Ground and first excited state for four and five identical bosons appear below the threshold for binding three and four particles, respectively.', '0911.1722-3-34-2': 'Their radii are for small binding energies larger than the range of the potential holding them together.', '0911.1722-3-34-3': 'The largest part of the probability is found in non-classical regions resulting in universal structures.', '0911.1722-3-34-4': 'For six and more particles the ground states would be located inside the potential and thus of non-universal structures.', '0911.1722-3-34-5': 'Excited states are larger and may still be universal but already for seven or eight particles also the first excited state is expected to be non-universal.', '0911.1722-3-34-6': 'The numerical results are obtained for a two-body gaussian potential but the features originating from wavefunctions in non-classical regions of space are expected to be independent of the potential shape.'} | null | null | null | null |
1412.5672 | {'1412.5672-1-0-0': 'In a recent work we have proposed a perturbative approach for the study of the phase transition of pure Yang-Mills theories at finite temperature.', '1412.5672-1-0-1': 'This is based on a simple massive extension of background field methods in the Landau-DeWitt gauge, where the gluon mass term is related to the existence of Gribov ambiguities.', '1412.5672-1-0-2': 'We have shown that a one-loop calculation of the background field effective potential describes well the phase structure of the SU([MATH]) and SU([MATH]) theories.', '1412.5672-1-0-3': 'Here, we present the calculation of the next-to-leading order contribution in perturbation theory for the SU([MATH]) case.', '1412.5672-1-0-4': 'In particular, we compute the background field effective potential at two-loop order and the corresponding Polyakov loop, a gauge invariant order parameter of the transition, at one-loop order.', '1412.5672-1-0-5': 'We show that the two-loop correction brings the critical temperature closer to its actual value as compared to the previous one-loop result.', '1412.5672-1-0-6': 'We also compute the thermodynamic pressure as a function of the temperature and show that two-loop contributions play an important role in the vicinity of the phase transition.', '1412.5672-1-1-0': '# Introduction', '1412.5672-1-2-0': 'The deconfinement transition of hadronic matter into a plasma of quarks and gluons at high temperature is a remarkable phenomenon.', '1412.5672-1-2-1': 'It is thought to have played a role in the early Universe and it is the major subject of investigation of ultrarelativistic heavy ion collisions at CERN and at RHIC.', '1412.5672-1-2-2': 'Establishing firmly the existence of this transition in QCD and characterizing its properties is a formidable task which has only been possible thanks to more than three decades of dedicated lattice studies [CITATION]; see Refs. [CITATION] for recent reviews.', '1412.5672-1-2-3': 'Early calculations clearly established the existence of a phase transition in pure SU([MATH]) Yang-Mills theories, related to the spontaneous breaking of the center ([MATH]) of the gauge group [CITATION].', '1412.5672-1-2-4': 'The inclusion of dynamical quarks, first with heavy masses and then with physical masses, has been a central issue in the field for years.', '1412.5672-1-2-5': 'Only recently have precise calculations of the thermodynamic properties of QCD with physical quark masses been achieved, with the result that the phase transition of the pure gauge theory becomes a crossover in QCD [CITATION].', '1412.5672-1-3-0': 'Thanks to asymptotic freedom, standard perturbative approaches make sense at high temperatures and an intense activity has been concerned with computing the thermodynamics and transport properties of the quark-gluon plasma by means of (semi-)analytical methods [CITATION].', '1412.5672-1-3-1': 'In this regime, infrared divergences call for the resummation of infinite subclasses of perturbative diagrams, the so-called hard thermal loops [CITATION].', '1412.5672-1-3-2': 'Such high temperature approaches reproduce the thermodynamic properties of the deconfined plasma down to a few times the transition temperature [CITATION] but fail to capture the physics of the phase transition.', '1412.5672-1-3-3': 'It is commonly accepted that the low temperature confining phase cannot be described by means of perturbation theory because of the existence of a Landau pole at low energy, where the running coupling diverges.', '1412.5672-1-4-0': 'Existing nonperturbative continuum descriptions of the transition region are based on truncations of Dyson-Schwinger equations (DSE), nonperturbative/functional renormalization group techniques [CITATION], the Hamiltonian approach of [CITATION], or two-particle-irreducible (2PI) inspired approaches [CITATION].', '1412.5672-1-4-1': 'These have the advantage over lattice calculations that they can easily be used at finite chemical potential [CITATION] and/or for computing real-time quantities [CITATION].', '1412.5672-1-4-2': 'To be trustable in these situations though, they have to be tested against lattice calculations in situations where the latter is well under control.', '1412.5672-1-4-3': 'Functional renormalization group (FRG) methods have been shown to correctly describe the phase structure of pure gauge theories, with transition temperatures in agreement with lattice results [CITATION].', '1412.5672-1-4-4': 'Such approaches also provide nontrivial insight concerning dynamical aspects of the deconfinement transition.', '1412.5672-1-4-5': 'For instance, an interesting connection between the confinement of static quarks and the infrared (IR) behavior of gluon and ghost correlators has been pointed out [CITATION].', '1412.5672-1-4-6': "Of course, neither the DSE nor the FRG can be solved exactly and one's ability to perform actual calculations relies on several assumptions which, even when these are well-motivated, can be difficult to check explicitly.", '1412.5672-1-4-7': 'A general criticism that many nonperturbative approaches have to face is that they do not always involve a systematic approximation scheme and it is often difficult to compute corrections to the obtained results.', '1412.5672-1-4-8': 'Still, the DSE/FRG provide the most powerful nonperturbative tools-apart from lattice techniques-to investigate the physics of the deconfinement transition directly at the level of the basic degrees of freedom of the theory.', '1412.5672-1-5-0': 'In a series of recent works [CITATION], yet a different route for the study of the infrared dynamics of Yang-Mills fields has been proposed.', '1412.5672-1-5-1': 'This is based on simple massive extensions of the standard Faddeev-Popov Lagrangian in the Landau gauge and in the Landau-DeWitt gauge.', '1412.5672-1-5-2': 'This is motivated by the observation that perturbative calculations of Yang-Mills correlators in the vacuum [CITATION] and at finite temperature [CITATION] in the massive extension of the Landau gauge action agree well with lattice data down to deep IR momenta.', '1412.5672-1-5-3': 'It is worth emphasizing that, if the typical value of the gauge coupling [MATH] required for such comparisons is of a few units, the relevant expansion parameter at zero temperature is [MATH].', '1412.5672-1-5-4': 'An important feature of the massive theory is that there exist IR safe renormalization group trajectories, with no Landau pole [CITATION].', '1412.5672-1-5-5': 'Moreover, it has been argued in [CITATION] that such a massive extension naturally arises as an effective theory for Yang-Mills correlators in a new one-parameter family of Landau gauges which, unlike the standard Faddeev-Popov construction, takes into account the existence of Gribov ambiguities.', '1412.5672-1-5-6': 'In this approach, the gluon mass term appears as a gauge fixing parameter which lifts the degeneracy between Gribov copies.', '1412.5672-1-6-0': 'This has been extended to the Landau-DeWitt gauge in the context of background field methods in [CITATION].', '1412.5672-1-6-1': 'There, we have shown that a calculation of the background field potential at first nontrivial (one-loop) order in perturbation theory correctly reproduces the phase structure of SU([MATH]) theories: one finds a confining phase at low temperature and a transition to a deconfined phase at high temperature which is second order for [MATH] and first order for [MATH], with transition temperatures in qualitative agreement with known values from lattice calculations.', '1412.5672-1-6-2': 'Our one-loop results are similar to those of the FRG studies of [CITATION] and actually corroborate the related findings concerning the relation between the IR behavior of gluon and ghost propagators and the existence of a confining phase at low temperature.', '1412.5672-1-6-3': 'Definite advantages of such a perturbative approach are, first, that low order calculations are technically very simple and, second, that they can be systematically improved by computing higher orders.', '1412.5672-1-6-4': 'If at asymptotically high temperatures, the expansion parameter is [MATH] due to collective infrared effects which necessitate the resummation of hard thermal loops [CITATION], near the transition region and below, the effective gluon mass tames some of the infrared problems of the perturbative series and it is not clear what is the relevant expansion parameter.', '1412.5672-1-6-5': 'It is the purpose of the present article to study the importance of such higher-order terms by computing the background field potential, the Polyakov loop and the thermodynamic pressure at next-to-leading order in the perturbative expansion.', '1412.5672-1-7-0': 'Before embarking in actual calculations, let us make some general comments concerning the massive extension of the Faddeev-Popov action in the class of (Landau or Landau-DeWitt) gauges considered here.', '1412.5672-1-7-1': 'What usually prevents a mass term in the gauged-fixed action is obviously not gauge-invariance, but BRST symmetry.', '1412.5672-1-7-2': 'The latter is a property of the Faddeev-Popov action, which is known to be valid at best in the high energy perturbative regime.', '1412.5672-1-7-3': 'But the Faddeev-Popov construction is certainly wrong in general since it ignores the existence of Gribov copies and, hence, does not completely fix the gauge.', '1412.5672-1-7-4': 'In fact, it is well-known that the only truly nonperturbative formulation of the gauge-fixed theory known so far, that is the lattice, cannot accomodate the BRST symmetry without leading to undefined zero over zero ratios [CITATION].', '1412.5672-1-8-0': 'A consistent quantization procedure, free of Gribov ambiguities, is likely to break the BRST symmetry.', '1412.5672-1-8-1': 'A well-known example is the minimal Landau gauge on the lattice, where one picks up a random Gribov copy on each gauge orbit [CITATION].', '1412.5672-1-8-2': 'Examples in the continuum include the so-called (refined) Gribov-Zwanziger approach [CITATION] or the averaging procedure of [CITATION], already mentioned.', '1412.5672-1-8-3': 'In the latter case, the bare gluon mass originates from the averaging procedure and is simply a gauge-fixing parameter which explicitly breaks the BRST symmetry.', '1412.5672-1-8-4': 'Such gauge-fixing procedures provide efficient starting points for perturbative calculations of Yang-Mills correlators and, for the latter, of the phase structure of the theory at finite temperature.', '1412.5672-1-8-5': 'In both approaches the BRST breaking is soft and the gauge-fixed actions present modified (non-nilpotent) BRST symmetries, which ensure their renormalizability.', '1412.5672-1-8-6': 'At this point, it is worth emphasizing that the continuum approaches mentioned above also introduce, in one way or another, a BRST-breaking ingredient.', '1412.5672-1-8-7': 'This typically appears through choices of boundary conditions and/or ultraviolet subtractions in the context of DSE [CITATION], or through the infrared regulator in FRG approaches.', '1412.5672-1-9-0': 'The plan of the paper is as follows.', '1412.5672-1-9-1': 'Section [REF] sets the scene and briefly recalls the basics of the [MATH] transition and of static quark confinement.', '1412.5672-1-9-2': 'In Section [REF] we present the massive extension of the Landau-DeWitt gauge and derive the corresponding Feynman rules.', '1412.5672-1-9-3': 'In Section [REF], we summarize the calculation of the two-loop correction to the background field effective potential and in Section [REF], we give the corresponding expression for the one-loop correction to the Polyakov loop.', '1412.5672-1-9-4': 'Finally, in Sec. [REF], we present our results concerning the order of the phase transition and the value of the transition temperature as well as the temperature dependence of the Polyakov loop and of the thermodynamic pressure.', '1412.5672-1-9-5': 'The essential steps of our calculations are presented in the main text while the technical details are gathered in the Appendices.', '1412.5672-1-10-0': 'Although we specify to the case [MATH] throughout this work, some formulas are valid for arbitrary [MATH].', '1412.5672-1-10-1': 'An important observation is that many steps of our calculations are similar to those in the massive extension of the Landau gauge, i.e., at vanishing background field.', '1412.5672-1-10-2': 'In fact, when expressed in an appropriate color basis, the Feynman rules of the theory have the same form as those in the Landau gauge, with the only difference that the momenta get shifted by an amount proportional to the background field.', '1412.5672-1-10-3': 'The key point is that, because of the residual global color symmetry, these shifts are conserved at the interaction vertices.', '1412.5672-1-10-4': 'This allows us to use various simplifying manipulations, detailed in Appendix [REF], and to reduce all the two-loop diagrams contributing to the background field potential to simple scalar-like sum-integrals; see Appendix [REF].', '1412.5672-1-10-5': 'The evaluation of the corresponding Matsubara sums is presented in Appendix [REF], which allows us to write the background field potential in a rather simple form in Appendix [REF].', '1412.5672-1-10-6': 'In Appendix [REF], we detail the calculation of the Polyakov loop at one-loop order in the presence of the nontrivial background field.', '1412.5672-1-10-7': 'In particular, this demonstrates explicitly that the Polyakov loop vanishes if and only if the minimum of the background field potential takes particular, [MATH]-symmetric values.', '1412.5672-1-10-8': 'This confirms, at this order of approximation, that the background field itself can be used as an order parameter for confinement, as advocated in [CITATION].', '1412.5672-1-10-9': 'Finally, we provide, in Appendix [REF], a general proof that the Polyakov loop vanishes when the background field takes [MATH]-symmetric values.', '1412.5672-1-10-10': 'We do not have a similar proof for the converse.', '1412.5672-1-11-0': '# Confinement-deconfinement transition in Yang-Mills theory', '1412.5672-1-12-0': 'We consider the Euclidean Yang-Mills action in [MATH] dimensions [EQUATION] where [MATH], with [MATH] the bare coupling constant and [MATH] the bare gauge field, with [MATH] being an element of the Lie algebra of SU([MATH]).', '1412.5672-1-12-1': 'Here, we have defined [MATH], with [MATH] the inverse temperature.', '1412.5672-1-13-0': 'Let us recall some basic aspects of the deconfinement transition to be considered below.', '1412.5672-1-13-1': 'The free-energy [MATH] of an isolated static quark in a thermal gluon bath is directly related to the expectation value of the traced Polyakov loop-which we refer to as the Polyakov loop for short in what follows-as [CITATION]: [EQUATION] where [MATH] is the free energy in the absence of quark.', '1412.5672-1-13-2': 'Here, [MATH] orders the matrix fields [MATH] from left to right with decreasing value of their time arguments and the brackets refer to the average in the theory defined by the action ([REF]).', '1412.5672-1-13-3': 'A vanishing [MATH] signals an infinite free energy, hence a quark confining phase, while [MATH] is interpreted as a phase where isolated static quarks are energetically allowed.', '1412.5672-1-14-0': 'It is well known [CITATION] that the Polyakov loop gets multiplied by a phase under generalized gauge transformations that leave the Yang-Mills action at finite temperature invariant and which are [MATH]-periodic in imaginary time, up to an element of the center of the group.', '1412.5672-1-14-1': 'This means that the deconfined phase is necessarily a phase where this symmetry group, or more precisely the quotient of the group of generalized gauge transformations by the subgroup of standard gauge transformations-which is isomorphic to [MATH]-is spontanously broken.', '1412.5672-1-14-2': 'Note that the converse is not necessarily true: Although this is not the expected behavior, one could, in principle, imagine a situation where the center is spontaneously broken but where the Polyakov loop still vanishes, the breaking being only manifest at the level of higher order correlations.', '1412.5672-1-14-3': 'This emphasizes the fact that the confined or deconfined nature of the system, in the sense described above, is not the breaking of the [MATH] symmetry itself but really the zero or nonzero value of the Polyakov loop (or of any equivalent order parameter).', '1412.5672-1-14-4': 'In what follows, we shall compute the Polyakov loop at one-loop order within the massive extension of the Landau-DeWitt gauge put forward in [CITATION].', '1412.5672-1-15-0': '# The massive Landau-DeWitt action', '1412.5672-1-16-0': '## General set-up', '1412.5672-1-17-0': 'We quantize the theory using the background field method [CITATION], where we introduce an a priori arbitrary background field configuration [MATH] and define the fluctuating field [MATH].', '1412.5672-1-17-1': 'The Landau-DeWitt gauge condition reads [EQUATION] where [MATH] for any field [MATH] in the Lie algebra of the gauge group.', '1412.5672-1-17-2': 'Our gauge-fixed action reads [CITATION] [EQUATION] with [MATH] a (real) Nakanishi-Lautrup field and [MATH] and [MATH] the Faddeev-Popov ghosts and antighost fields.', '1412.5672-1-17-3': 'In terms of the field [MATH], we have [EQUATION] with [MATH] the field strength tensor evaluated at [MATH], and [EQUATION]', '1412.5672-1-17-4': 'The action ([REF]) has the obvious property [EQUATION] where [MATH], [MATH] is a local SU([MATH]) matrix and [EQUATION]', '1412.5672-1-17-5': 'At the level of the (quantum) effective action [MATH] this implies [CITATION]: [EQUATION] where the fields [MATH] are now to be understood as average values in the presence of sources.', '1412.5672-1-18-0': 'In principle, to evaluate physical observables at zero sources, one should minimize [MATH] with respect to [MATH] at a given [MATH].', '1412.5672-1-18-1': 'It can be argued, however, that one can alternatively minimize the functional [EQUATION] with respect to the background field [MATH] [CITATION].', '1412.5672-1-18-2': 'This functional obeys the background gauge symmetry [EQUATION] which is trivially preserved in perturbation theory.', '1412.5672-1-18-3': 'The Polyakov loop can be obtained as [EQUATION] with [MATH] and where the brackets stand for an average in the gauge-fixed theory ([REF]).', '1412.5672-1-18-4': 'The right-hand side of Eq. ([REF]) must be evaluated at any absolute minimum [MATH] of [MATH].', '1412.5672-1-18-5': 'Because the Polyakov loop involves only the temporal component of the background field and because [MATH] (since [MATH] by construction) is [MATH]-independent, it is sufficient to consider homogeneous background fields in the temporal direction [MATH].', '1412.5672-1-18-6': 'Moreover, the Hermitian matrix [MATH] can be diagonalized by means of a global SU([MATH]) rotation and one can thus, with no loss of generality, restrict [MATH] to the Cartan subalgebra of the color group.', '1412.5672-1-18-7': 'We shall write [MATH], where [MATH] are the SU([MATH]) generators in the Cartan subalgebra.', '1412.5672-1-18-8': 'We thus have to minimize the background field effective potential [EQUATION] where [MATH], [MATH] is the spatial volume, and where we have subtracted the value of the potential at zero temperature [MATH].', '1412.5672-1-18-9': 'As we shall see below, the latter is independent of [MATH].', '1412.5672-1-18-10': 'Finally, the thermodynamic pressure is simply given by [EQUATION] where [MATH].', '1412.5672-1-19-0': 'The symmetry ([REF]) implies that the potential ([REF]) is invariant under the group of center transformations mentioned earlier.', '1412.5672-1-19-1': 'This translates into periodicity properties of [MATH] in the variables [MATH].', '1412.5672-1-19-2': 'Combined with the invariance under global color rotations and charge conjugation, which we assume are not spontaneously broken, this fully constrains the possible values of [MATH] where the [MATH] symmetry is manifest [CITATION].', '1412.5672-1-19-3': 'In the SU([MATH]) case, this is given by [MATH].', '1412.5672-1-19-4': 'In appendix [REF], we provide a general proof that this value is such that [MATH] and thus confining.', '1412.5672-1-19-5': 'This supports the general expectation [CITATION] that the background field itself can be used as an order parameter.', '1412.5672-1-19-6': 'Our proof is, however, not complete because the converse, i.e. [MATH], is missing.', '1412.5672-1-19-7': 'In the present work, we shall, however, explicitly show that the equivalence holds at next-to-leading order in the loop expansion.', '1412.5672-1-20-0': '## Renormalization', '1412.5672-1-21-0': 'We introduce renormalized parameters and fields, related to the corresponding bare quantities in the usual way: [EQUATION] and [EQUATION] where we have kept the same notation for bare (left) and renormalized (right) fields for simplicity.', '1412.5672-1-21-1': 'Notice that the background field [MATH] and the fluctuating field [MATH] have different renormalization factors [CITATION].', '1412.5672-1-21-2': 'The background field gauge symmetry ([REF]) implies that the product [MATH] is finite [CITATION].', '1412.5672-1-21-3': 'In the following we impose the renormalization condition [EQUATION] for the finite parts as well, so that [MATH].', '1412.5672-1-21-4': 'From here on, we only consider renormalized quantities unless explicitly stated.', '1412.5672-1-21-5': 'The values of the parameters [MATH] and [MATH] must be fixed from some external input, e.g., lattice data.', '1412.5672-1-22-0': 'To set the value of the renormalized mass [MATH], one would ideally use the value of a physical observable such as the glueball mass.', '1412.5672-1-22-1': 'An easier possibility in practice is to fix this parameter by employing lattice results for gauge dependent quantities such as the the Yang-Mills correlators.', '1412.5672-1-22-2': 'In principle, this requires lattice results in the same gauge as described above, involving an average of Gribov copies.', '1412.5672-1-22-3': 'Instead, existing gauge-fixed lattice calculations typically select a particular Gribov copy in the so-called first Gribov region, where the Faddeev-Popov operator is positive definite.', '1412.5672-1-22-4': 'Still, explicit calculations in the massive extension of the Faddeev-Popov Lagrangian in the Landau gauge show that there exists a value of the renormalized mass which allows one to quantitatively reproduce the lattice data for the Yang-Mills correlators at vanishing temperature [CITATION].', '1412.5672-1-23-0': 'No lattice calculation exists so far in the Landau-DeWitt gauge with homogeneous temporal background field as considered here.', '1412.5672-1-23-1': 'However, at zero temperature the background field vanishes and the latter reduces to the standard Landau gauge.', '1412.5672-1-23-2': 'In the present work, we shall thus use the values of [MATH] and [MATH] inferred from lattice calculations in the Landau gauge at vanishing temperature.', '1412.5672-1-23-3': 'To be consistent with the approximation order considered here, we need the one-loop expressions of the vacuum propagators.', '1412.5672-1-23-4': 'These have been computed in [CITATION].', '1412.5672-1-23-5': 'Using the renormalization conditions [EQUATION] where [MATH] and [MATH] denote respectively the renormalized ghost and transverse gluon self-energies in the vacuum, the best fits to lattice data give [MATH] MeV and [MATH], with [MATH] GeV for SU([MATH]) in [MATH].', '1412.5672-1-24-0': '## Feynman rules', '1412.5672-1-25-0': 'For the homogeneous background fields considered here, [MATH], the curvature term vanishes: [MATH].', '1412.5672-1-25-1': 'As emphasized above, the background field [MATH] can be taken in the Cartan subalgebra of the gauge group.', '1412.5672-1-25-2': 'For SU([MATH]), the latter has only one direction which we choose along the third axis in color space.', '1412.5672-1-25-3': 'It is convenient to work with the basis of generators [EQUATION] which satisfy [EQUATION] where [MATH] is the completely antisymmetric tensor, with [MATH].', '1412.5672-1-25-4': 'We denote any element of the gauge group Lie algebra as [MATH].', '1412.5672-1-25-5': 'The components of the background covariant derivatives are [EQUATION] where our convention for the Fourier transform is [EQUATION] such that [MATH].', '1412.5672-1-25-6': 'Here, we defined the shifted momentum [EQUATION] which satisfies [EQUATION]', '1412.5672-1-25-7': 'The background field breaks the global color group but there remains a residual symmetry under those color rotations that leave it invariant.', '1412.5672-1-25-8': 'In the SU([MATH]) theory, this is the group of SO([MATH] U([MATH]) transformations, under which [MATH] and [MATH], with [MATH] a constant phase.', '1412.5672-1-25-9': 'Accordingly, we refer to [MATH] as the "neutral" component and to [MATH] as the "charged" components.', '1412.5672-1-25-10': 'The (temporal) background field plays the role of an external Abelian field coupled to the charged components, which leads to a simple shift of momentum [MATH].', '1412.5672-1-25-11': 'Equivalently, it can be seen as an imaginary chemical potential associated with the conserved U([MATH]) charge.', '1412.5672-1-25-12': 'The residual color symmetry guarantees that the corresponding charge is conserved upon propagation and at the interaction vertices.', '1412.5672-1-26-0': 'It is an easy exercice to compute the Feynman rules of the theory in the basis ([REF]).', '1412.5672-1-26-1': 'To each propagator and to each leg of an interaction vertex is associated a flow of color charge, which we define to follow the flow of momentum.', '1412.5672-1-26-2': 'The tree-level ghost and gluon propagators for momentum [MATH] and charge state [MATH], represented in Fig. [REF], are given by [EQUATION] where we denote the scalar propagator of mass [MATH] by [EQUATION] and where [MATH].', '1412.5672-1-26-3': 'Note the identities [MATH] and [MATH], which follow from ([REF]) and which simply reflect the fact that the choice of orientation of the momentum/charge flow in the diagrams of Fig. [REF] is arbitrary.', '1412.5672-1-27-0': 'The cubic (derivative) vertices are represented on Fig. [REF], with the convention that all momenta and color charges are outgoing.', '1412.5672-1-27-1': 'The ghost-antighost-gluon vertex is given by [EQUATION] and the three-gluon vertex reads [EQUATION] where the various momenta, color charges and spacetime indices are organized as in Fig. [REF].', '1412.5672-1-27-2': 'The structure constant [MATH] guarantees that the charge is conserved at the vertex: [MATH].', '1412.5672-1-27-3': 'Finally, the four-gluon vertex is [EQUATION] where color charge and spacetime indices are organized as in Fig. [REF].', '1412.5672-1-27-4': 'Again, the combinations of the group structure constants guarantee that the color charge is conserved: [MATH].', '1412.5672-1-28-0': 'Written in the basis ([REF]), the Feynman rules ([REF])-([REF]) in presence of the background field are very similar to the standard ones, usually written in the Cartesian color basis.', '1412.5672-1-28-1': 'The essential difference stems in the different structure constants, which can be traced back to the commutation relations ([REF]), and the role of the background field is simply to replace all momenta by shifted ones according to the corresponding color charges.', '1412.5672-1-28-2': 'As we shall discuss in a future work, these remarks generalize to any group SU([MATH])-in fact to any compact Lie group with a semi-simple Lie algebra.', '1412.5672-1-28-3': 'For [MATH], one has the identities [EQUATION] and [EQUATION]', '1412.5672-1-28-4': 'Using ([REF]) and ([REF]), one checks that the vertices ([REF])-([REF]) are unchanged if all momenta and color charges are incoming.', '1412.5672-1-29-0': '# The background field potential at [MATH]', '1412.5672-1-30-0': 'We consider the loop expansion of the background field potential, which corresponds to a perturbative expansion in powers of the renormalized coupling [MATH] with [MATH].', '1412.5672-1-30-1': 'We write the corresponding series as [EQUATION] with [MATH] the [MATH]-loop order contribution.', '1412.5672-1-30-2': 'The classical action ([REF]) evaluated at [MATH], [MATH], and [MATH] vanishes identically since [MATH].', '1412.5672-1-30-3': 'The tree-level contribution to the term [MATH] in ([REF]) is thus trivially independent of the background field and the tree-level potential is trivial [EQUATION]', '1412.5672-1-30-4': 'The one-loop contribution has been obtained in [CITATION] and the relevant two-loop diagrams are shown in Fig. [REF].', '1412.5672-1-31-0': '## The one-loop contribution', '1412.5672-1-32-0': 'We briefly recall the result of [CITATION].', '1412.5672-1-32-1': 'Introducing the function [EQUATION] which is such that, for [MATH], [EQUATION] the one-loop background field potential can be written as [EQUATION] where the first term on the right-hand side is the contribution from the massive gluons and the second one is due to the massless ghosts.', '1412.5672-1-32-2': 'It reduces to the well-known Weiss potential [CITATION] in the high temperature limit : [EQUATION] whereas it gives a confining, inverted Weiss potential at low temperature, where the contribution from massive modes is supressed [CITATION]: [EQUATION]', '1412.5672-1-33-0': '## Two-loop diagrams', '1412.5672-1-34-0': 'Let us start with the ghost-gluon sunset diagram (second diagram of Fig. [REF]).', '1412.5672-1-34-1': 'A straightforward application of the Feynman rules derived above yields [EQUATION] with [MATH] and [MATH].', '1412.5672-1-34-2': 'We employ the general notations [MATH], with [MATH], and [EQUATION] with [MATH] an arbitrary mass scale and we recall that [MATH].', '1412.5672-1-34-3': 'We have also introduced the completely symmetric tensor [EQUATION] whose components vanish unless [MATH] and are equal to one otherwise.', '1412.5672-1-34-4': 'It follows from Eqs. ([REF]) and ([REF]) that [EQUATION] which imply ([MATH]) [EQUATION]', '1412.5672-1-34-5': 'For a vanishing background field, the term within brackets in the right-hand side of Eq. ([REF]) is nothing but the expression of the ghost-gluon sunset in the massive extension of the Landau gauge, up to the corresponding color factor [MATH].', '1412.5672-1-34-6': 'The latter is recovered using ([REF]).', '1412.5672-1-34-7': 'The expression ([REF]) illustrates that the two-loop perturbative diagram at nonvanishing background can be obtained from the corresponding one at [MATH] as follows: First, one writes the momentum integrals in the massive Landau gauge in terms of three momenta [MATH], [MATH], and [MATH], up to the color factor [MATH]; Then, one shifts the momenta to [MATH], [MATH] and [MATH] and averages with the weight [MATH].', '1412.5672-1-34-8': 'This property can be anticipated from the Feynman rules described above and actually generalizes to any closed diagram, with an appropriate weight factor.', '1412.5672-1-34-9': 'The conservation of color charge at the interaction vertices implies the conservation of the shifted momenta under the loop integrals, e.g., [MATH] in the two-loop case above.', '1412.5672-1-34-10': 'This leads to important simplifications of the calculations in the Landau-DeWitt gauge, as detailed in the Appendices.', '1412.5672-1-34-11': 'Let us finally mention that the expression ([REF]) and the above remarks actually generalize to SU([MATH]), where the tensor ([REF]) involves the appropriate structure constant.', '1412.5672-1-34-12': 'If Eq. ([REF]) is specific to the case [MATH], the properties ([REF]) are true for arbitrary [MATH].', '1412.5672-1-35-0': 'For later use, we rewrite Eq. ([REF]) in a more compact form.', '1412.5672-1-35-1': 'Using the conservation of shifted momenta under the sum in ([REF]), we have [EQUATION]', '1412.5672-1-35-2': 'Furthermore, writing [EQUATION] we obtain [EQUATION] in terms of the two-loop integrals [EQUATION] which is needed only for the case [MATH].', '1412.5672-1-35-3': 'In Appendix [REF], we rewrite these integrals in terms of simpler scalar-like loop integrals; see Eq. ([REF]) below.', '1412.5672-1-35-4': 'Here, we simply notice for later use that [MATH] is invariant under the simultaneous permutation of the upper and lower indices, when the corresponding shifts of momenta add up to zero: [EQUATION]', '1412.5672-1-35-5': 'The two diagrams with purely gluonic loops can be treated in a similar way.', '1412.5672-1-35-6': 'We give more details in Appendix [REF] and simply state the results here.', '1412.5672-1-35-7': 'The double tadpole diagram (first diagram of Fig. [REF]) yields [EQUATION] where we have defined the tadpole integrals [EQUATION] and [EQUATION]', '1412.5672-1-35-8': 'As for the gluon sunset diagram (third diagram of Fig. [REF]), we obtain [EQUATION] where ([REF]) has been used.', '1412.5672-1-36-0': '## Counterterm contribution', '1412.5672-1-37-0': 'The expressions derived in the previous section contain ultraviolet (UV) divergences.', '1412.5672-1-37-1': 'These are cancelled by the counterterms from the original action, as we show explicitly in Appendix [REF].', '1412.5672-1-37-2': 'Here, we compute the relevant couterterm contributions to the background field potential.', '1412.5672-1-38-0': 'Writing the renormalization factors as [MATH], [MATH], the counterterm action reads [EQUATION] where [MATH] and the dots denote terms involving the coupling counterterm [MATH], which are not needed in the present work.', '1412.5672-1-39-0': 'Using ([REF]), we obtain, for the ghost counterterm loop, [EQUATION] where the last integral vanishes in dimensional regularization.', '1412.5672-1-39-1': 'The gluon counterterm loop reads [EQUATION]', '1412.5672-1-39-2': 'Then, up to an integral which vanishes in dimensional regularization, the total counterterm contribution can be written in terms of the tadpole integral ([REF]) as [EQUATION]', '1412.5672-1-40-0': '## The two-loop contribution', '1412.5672-1-41-0': 'Adding together the various two-loop pieces, we obtain (notice that the contributions from the integral ([REF]) cancel) [EQUATION]', '1412.5672-1-41-1': 'For a vanishing background field, Eq. ([REF]) is nothing but the two-loop free energy density computed in the massive extension of the Landau gauge.', '1412.5672-1-41-2': 'As detailed in Appendix [REF], the integral ([REF]) can be expressed in terms of the tadpole integrals ([REF]) and [EQUATION] and of the scalar sunset [EQUATION]', '1412.5672-1-41-3': 'Using Eqs. ([REF])-([REF]), our final expression for the two-loop contribution to the background field potential is [EQUATION]', '1412.5672-1-41-4': 'The calculation of the various Matsubara sums and momentum integrals involved in this expression is detailed in Appendix [REF].', '1412.5672-1-41-5': 'Moreover, we show in Appendix [REF] that it is UV finite (up to an overall, temperature and background independent divergence) using the counterterm [MATH] obtained from the renormalization conditions ([REF]).', '1412.5672-1-41-6': 'We also reduce this expression to a sum of one and two-dimensional (radial) momentum integrals involving thermal, Bose-Einstein distribution functions in presence of the background field.', '1412.5672-1-41-7': 'Our final result for the thermal part (the only one that depends on the background) of the two-loop contribution to the background field potential is given in Eq. ([REF]).', '1412.5672-1-42-0': '# The Polyakov loop at [MATH]', '1412.5672-1-43-0': 'We similarly expand the Polyakov loop ([REF]) as [EQUATION] with the [MATH]-loop contribution [MATH].', '1412.5672-1-43-1': 'This is obtained by expanding the path-ordered exponential in Eq. ([REF]) in powers of the coupling with [MATH].', '1412.5672-1-43-2': 'The tree-level and one-loop contributions are evaluated in Appendix [REF] for arbitrary [MATH] and for fields in an arbitrary representation of the gauge group.', '1412.5672-1-43-3': 'For the fundamental representation of SU(2), we obtain [EQUATION] and [EQUATION] where [MATH] is the absolute minimum of the two-loop background field potential and [EQUATION]', '1412.5672-1-43-4': 'It is easily checked that [MATH], hence [EQUATION]', '1412.5672-1-43-5': 'This confirms, at this order, that the background field [MATH] itself is a good order parameter for static quark confinement [CITATION].', '1412.5672-1-44-0': '# Results', '1412.5672-1-45-0': 'As already mentioned, we shall use the values of [MATH] and [MATH] inferred from lattice calculations of ghost and gluon propagators in the Landau gauge at vanishing temperature.', '1412.5672-1-45-1': 'For the present order of approximation, we fit these data to the one-loop expressions obtained in the massive Landau gauge.', '1412.5672-1-45-2': 'The best fitting values are [MATH] MeV and [MATH] with [MATH] GeV.', '1412.5672-1-45-3': 'We shall compare our findings to our earlier leading-order results [CITATION] where the mass parameter was obtained by fitting the data to the tree-level propagators with the best fitting value [MATH] MeV.', '1412.5672-1-45-4': 'There is of course some error related to the determination of the parameters.', '1412.5672-1-45-5': 'We have checked that the results to be presented below do not change qualitatively as we change the parameters within the error bars.', '1412.5672-1-45-6': 'Let us finally mention that we shall not consider the possible temperature dependence of the parameters.', '1412.5672-1-45-7': 'This and the implementation of renormalization-group improvement is deferred to a future work.', '1412.5672-1-46-0': '## Background field potential', '1412.5672-1-47-0': 'In Fig. [REF], we show the rescaled two-loop background field potential [MATH] in the range [MATH] for various values of the temperature.', '1412.5672-1-47-1': 'As the temperature is increased, there is clearly a transition from a confining phase, where the minimum of the potential lies at its confining value [MATH], to a deconfined phase where the location of the minimum departs from [MATH].', '1412.5672-1-47-2': 'The transition is second order and the corresponding critical temperature is obtained by requiring the vanishing of the curvature of the potential at the confining point.', '1412.5672-1-47-3': 'The rescaled curvature [MATH] at [MATH] is plotted as a function of [MATH] in the left panel of Fig. [REF] at one- and two-loop orders.', '1412.5672-1-47-4': 'The two-loop correction leads to a larger critical temperature, [MATH], as compared to our previous one-loop result [CITATION], [MATH].', '1412.5672-1-47-5': 'A typical lattice result is [CITATION] [MATH] MeV.', '1412.5672-1-47-6': 'Although such comparison must be taken with care due to the issue of properly setting the scale, this shows that the two-loop corrections indeed improve the one-loop result.', '1412.5672-1-48-0': 'It is also interesting to compare with other continuum approaches.', '1412.5672-1-48-1': 'The FRG and DSE/2PI calculations of Ref. [CITATION] give [MATH] and [MATH], respectively, which lie in the same ballpark as our one-loop result.', '1412.5672-1-48-2': 'The improved value of the critical temperature obtained above suggests that the present two-loop calculation efficiently captures some of the effects which have been discarded in those calculations.', '1412.5672-1-48-3': 'For instance, although the fully resummed propagators are included, some explicit two-loop contributions to the DSE for the background field potential have been neglected.', '1412.5672-1-48-4': 'As for the FRG calculation, the authors of Ref. [CITATION] mention that their result is modified to [MATH] when some backreaction effects-neglected in their main study-are included.', '1412.5672-1-49-0': 'For completeness, we compare in Fig. [REF] the one- and two-loop potentials at their respective critical temperatures.', '1412.5672-1-49-1': 'It is also instructive to plot the (rescaled) curvature of the potential at the origin as a function of [MATH], see the right panel of Fig. [REF].', '1412.5672-1-49-2': 'For the values of parameters studied here, we observe that, at one-loop order, there exists a temperature [MATH] above which the minimum of the potential is exactly located at [MATH].', '1412.5672-1-49-3': 'This does not seem to be the case at two-loop order where the curvature of the potential at the origin remains negative.', '1412.5672-1-50-0': 'To have a better analytical control on our results, we have considered the formal limit [MATH].', '1412.5672-1-50-1': 'We show in Appendix [REF] that the rescaled potential [MATH] is a polynomial in the range [MATH]: [EQUATION] which minimum is located at [EQUATION]', '1412.5672-1-50-2': 'We discuss the consequences of the different behavior between the one- and two-loop results below.', '1412.5672-1-51-0': '## Polyakov loop', '1412.5672-1-52-0': 'In Fig. [REF], we compare the temperature dependence of the Polyakov loop at leading and next-to-leading order.', '1412.5672-1-52-1': 'At leading order, it saturates to its upper bound [MATH] at the temperature [MATH] discussed previously, above which [MATH] remains the absolute minimum of the potential; see Fig. [REF].', '1412.5672-1-52-2': 'The Polyakov loop is singular at [MATH] because its second derivative is discontinous.', '1412.5672-1-52-3': 'This has to do with the particular form of the Weiss potential, Eq. ([REF]).', '1412.5672-1-52-4': 'For small positive values of [MATH], the latter behaves as [EQUATION] and is thus non-analytic in [MATH].', '1412.5672-1-52-5': 'This is to be contrasted with the massive version [MATH] of the same function which shows a regular expansion in powers of [MATH] when [MATH].', '1412.5672-1-52-6': 'In the vicinity of [MATH] and for [MATH] close to [MATH], the rescaled potential is of the form [MATH], where [MATH] with [MATH] and [MATH].', '1412.5672-1-52-7': 'It follows, that for [MATH] approaching [MATH] from below, [MATH] and [MATH] for [MATH].', '1412.5672-1-52-8': 'Thus, the first derivative of the background with respect to the temperature is discontinuous at [MATH].', '1412.5672-1-52-9': 'This singularity propagates to thermodynamic observables.', '1412.5672-1-52-10': 'For instance the third derivative of the free energy density with respect to the temperature is discontinous.', '1412.5672-1-52-11': 'This is, however, a spurious discontinuity.', '1412.5672-1-53-0': 'As already discussed above, at two-loop order the curvature of the potential at the origin remains negative.', '1412.5672-1-53-1': 'In this range, the Polyakov loop does not show any additional singularity, other than the one at [MATH].', '1412.5672-1-53-2': 'We also observe that, as compared to the one-loop result, the Polyakov loop at two-loop order overshoots its asymptotic ([MATH]) value [EQUATION] with [MATH] given in ([REF]), as computed in Appendix [REF].', '1412.5672-1-54-0': '## Pressure and entropy', '1412.5672-1-55-0': 'The thermodynamic pressure ([REF]) is shown in Fig. [REF] as a function of the temperature in the one- and two-loop approximations.', '1412.5672-1-55-1': 'In both cases, we observe that the pressure is increasing at small temperatures, indicating a positive entropy ([MATH]).', '1412.5672-1-55-2': 'This is a welcome result although it may be surprising at first sight because the ghosts dominate in this regime and one would naively think that they contribute negatively to the entropy.', '1412.5672-1-56-0': 'The positivity of the entropy at low temperatures is ensured by the fact that, in the confining phase with [MATH], those ghosts which feel the background effectively behave as physical fermions, giving a positive contribution to the entropy.', '1412.5672-1-56-1': 'To illustrate this point more precisely, we note that the low temperature behavior of the background field potential is dominated by the one-loop contribution, as discussed in Appendix [REF].', '1412.5672-1-56-2': 'This is directly visible in Fig. [REF].', '1412.5672-1-56-3': 'At one-loop order, in the confined phase, the entropy contribution [MATH] of a ghost with charge [MATH] is [EQUATION]', '1412.5672-1-56-4': 'So we have either a bosonic-like contribution ([MATH]) [EQUATION] with a standard negative (ghost-like) sign from the neutral modes, or fermionic-like contributions [EQUATION] with a positive sign for charged modes due to their coupling to the background.', '1412.5672-1-56-5': 'The total ghost contribution, [EQUATION] is then positive, as announced.', '1412.5672-1-57-0': 'As one increases the temperature, the one-loop result violates the positivity of the entropy, slightly before reaching [MATH], as can be clearly seen in the inset plot of Fig. [REF], where the thermal pressure changes its monotony and even becomes slightly negative before [MATH].', '1412.5672-1-57-1': 'The reason for this behavior is again the change of effective statistics of the relevant degrees of freedom in the presence of the background.', '1412.5672-1-57-2': 'As the temperature is increased, the massive gluons start contributing to the pressure.', '1412.5672-1-57-3': 'However, in the confined phase, the charged gluons, which feel the presence of the background, behave like "wrong" fermions, contributing negatively to the entropy [CITATION].', '1412.5672-1-57-4': 'Remarkably this behavior is completely washed out by the two-loop correction and at two-loop order the entropy is positive (the pressure is monotonically increasing with the temperature) in the whole range of temperatures studied here, see Fig. [REF].', '1412.5672-1-58-0': 'Finally, let us comment on the [MATH] behavior of the pressure at low temperature-see Fig. [REF]-which is at odds with the exponential suppression seen in lattice results [CITATION].', '1412.5672-1-58-1': 'As discussed above, this originates from the fact that massless (ghost) modes directly contribute to the pressure, already at leading order.', '1412.5672-1-58-2': 'We mention that this is likely to be a general problem for continuum approaches, which are essentially based on using (resummed) propagators.', '1412.5672-1-58-3': 'For instance, in the Landau gauge, lattice results for the propagators [CITATION] show that, if the gluon becomes effectively massive for infrared momenta, the ghost remains massless.', '1412.5672-1-58-4': 'This generically produces [MATH] contributions in a leading-order-like-i.e., trace-log-expression for the thermodynamic pressure.', '1412.5672-1-58-5': 'The correct treatment of such unphysical massless degrees of freedom is a serious issue that needs to be further investigated.', '1412.5672-1-59-0': '## Comparison with the massive Landau gauge', '1412.5672-1-60-0': 'It is interesting to compare our results for the pressure to those obtained in the massive Landau gauge (which corresponds to a vanishing background).', '1412.5672-1-60-1': 'The corresponding one- and two-loop pressure curves are shown in Fig. [REF] and compared to those discussed in the previous subsection.', '1412.5672-1-60-2': 'We observe again that the one-loop pressure contains [MATH] contributions at low temperatures but this time those come with a negative prefactor which yields a negative entropy at low temperatures.', '1412.5672-1-60-3': 'The reason for this is simple: in the absence of background all ghosts contribute negatively to the entropy.', '1412.5672-1-60-4': 'In this case the two-loop correction is of no help because it does not contain any [MATH] contribution at small temperature.', '1412.5672-1-60-5': 'In fact it seems that the two-loop term makes the problem even worse since the entropy remains negative in the range of temperature shown here.', '1412.5672-1-60-6': 'Furthermore, we observe in Fig. [REF] that the two-loop correction is smaller in the Landau-DeWitt case than in the Landau case, indicating a better convergence.', '1412.5672-1-61-0': 'The above remarks illustrate the importance of taking into account the order parameter of the [MATH] transition in the description.', '1412.5672-1-61-1': 'Our two-loop results show that the value [MATH]-which would correspond to the Landau gauge-is never a physical point, i.e., an absolute minimum of the background field potential.', '1412.5672-1-61-2': 'In a sense, the perturbative expansion in the (massive) Landau gauge appears as an expansion around an unstable situation.', '1412.5672-1-61-3': 'It would take infinite resummations to correctly describe the physics near the stable physical point.We believe that this remark outranges the present framework and holds for other continuum approaches as well.', '1412.5672-1-62-0': '# Conclusions', '1412.5672-1-63-0': 'To summarize, we have proposed in [CITATION] a perturbative approach to describe the static quark confinement-deconfinement transition in Yang-Mills theories, based on a modified (massive) gauge-fixed action in the Landau-DeWitt gauge.', '1412.5672-1-63-1': 'This describes well the phase structure of SU([MATH]) theories at leading order and gives qualitatively good results for the transition temperatures of the [MATH] and [MATH] cases in [MATH].', '1412.5672-1-63-2': 'However, this leads to a [MATH] behavior of the thermodynamic pressure at low temperatures and to negative entropy and negative pressure near the transition temperature.', '1412.5672-1-63-3': 'Also the leading order calculation leads to a spurious singularity at a temperature [MATH] above which the physical background field vanishes.', '1412.5672-1-64-0': 'With the present work, we wish to demonstrate the interest and feasibility of a next-to-leading order calculation in this modified perturbative scheme.', '1412.5672-1-64-1': 'We investigate the SU([MATH]) theory at next-to-leading order, which shows that the correction to the critical temperature goes in the right direction and actually brings the estimated value close to the lattice result-although we stress again that such comparison must be taken with care due to the issue of scale setting.', '1412.5672-1-64-2': 'The next-to-leading order corrections also cure the negative entropy issue and remove the spurious singularity at [MATH].', '1412.5672-1-64-3': 'In fact, we find that the physical point always corresponds to a nonvanishing background field.', '1412.5672-1-64-4': 'There remains the issue of unphysical [MATH] contributions to the pressure from the massless (ghost) degrees of freedom, which needs to be investigated.', '1412.5672-1-65-0': 'In a future work, we shall extend the present two-loop study to the SU([MATH]) theory and investigate the effect of renormalization group improvement.', '1412.5672-1-65-1': 'Another interesting extension would be to include quark degrees of freedom at nonzero chemical potential.', '1412.5672-1-65-2': 'Finally, the present work points the importance of taking into account a nonvanishing background field near the phase transition.', '1412.5672-1-65-3': 'We plan to extend our previous study [CITATION] of the Landau gauge ghost and gluon correlators at finite temperature to the Landau-DeWitt gauge.'} | {'1412.5672-2-0-0': 'In a recent work we have proposed a perturbative approach for the study of the phase transition of pure Yang-Mills theories at finite temperature.', '1412.5672-2-0-1': 'This is based on a simple massive extension of background field methods in the Landau-DeWitt gauge, where the gluon mass term is related to the existence of Gribov ambiguities.', '1412.5672-2-0-2': 'We have shown that a one-loop calculation of the background field effective potential describes well the phase structure of the SU([MATH]) and SU([MATH]) theories.', '1412.5672-2-0-3': 'Here, we present the calculation of the next-to-leading-order contribution in perturbation theory for the SU([MATH]) case.', '1412.5672-2-0-4': 'In particular, we compute the background field effective potential at two-loop order and the corresponding Polyakov loop, a gauge invariant order parameter of the transition, at one-loop order.', '1412.5672-2-0-5': 'We show that the two-loop correction brings the critical temperature closer to its actual value as compared to the previous one-loop result.', '1412.5672-2-0-6': 'We also compute the thermodynamic pressure as a function of the temperature and show that two-loop contributions play an important role in the vicinity of the phase transition.', '1412.5672-2-1-0': '# Introduction', '1412.5672-2-2-0': 'The deconfinement transition of hadronic matter into a plasma of quarks and gluons at high temperature is a remarkable phenomenon.', '1412.5672-2-2-1': 'It is thought to have played a role in the early Universe and it is the major subject of investigation of ultrarelativistic heavy ion collisions at CERN and at RHIC.', '1412.5672-2-2-2': 'Establishing firmly the existence of this transition in QCD and characterizing its properties is a formidable task which has only been possible thanks to more than three decades of dedicated lattice studies [CITATION]; see Refs. [CITATION] for recent reviews.', '1412.5672-2-2-3': 'Early calculations clearly established the existence of a phase transition in pure SU([MATH]) Yang-Mills theories, related to the spontaneous breaking of the center ([MATH]) of the gauge group [CITATION].', '1412.5672-2-2-4': 'The inclusion of dynamical quarks, first with heavy masses and then with physical masses, has been a central issue in the field for years.', '1412.5672-2-2-5': 'Only recently have precise calculations of the thermodynamic properties of QCD with physical quark masses been achieved, with the result that the phase transition of the pure gauge theory becomes a crossover in QCD [CITATION].', '1412.5672-2-3-0': 'Thanks to asymptotic freedom, standard perturbative approaches make sense at high temperatures and an intense activity has been concerned with computing the thermodynamics and transport properties of the quark-gluon plasma by means of (semi)analytical methods [CITATION].', '1412.5672-2-3-1': 'In this regime, infrared divergences call for the resummation of infinite subclasses of perturbative diagrams, the so-called hard thermal loops [CITATION].', '1412.5672-2-3-2': 'Such high temperature approaches reproduce the thermodynamic properties of the deconfined plasma down to a few times the transition temperature [CITATION] but fail to capture the physics of the phase transition.', '1412.5672-2-3-3': 'It is commonly accepted that the low temperature confining phase cannot be described by means of perturbation theory because of the existence of a Landau pole at low energy, where the running coupling diverges.', '1412.5672-2-4-0': 'Existing nonperturbative continuum descriptions of the transition region are based on truncations of Dyson-Schwinger equations (DSE), nonperturbative/functional renormalization-group techniques [CITATION], the Hamiltonian approach of [CITATION], or two-particle-irreducible (2PI) inspired approaches [CITATION].', '1412.5672-2-4-1': 'These have the advantage over lattice calculations that they can easily be used at finite chemical potential [CITATION] and/or for computing real-time quantities [CITATION].', '1412.5672-2-4-2': 'To be trustable in these situations though, they have to be tested against lattice calculations in situations where the latter is well under control.', '1412.5672-2-4-3': 'Functional renormalization-group (FRG) methods have been shown to correctly describe the phase structure of pure gauge theories, with transition temperatures in agreement with lattice results [CITATION].', '1412.5672-2-4-4': 'Such approaches also provide nontrivial insight concerning dynamical aspects of the deconfinement transition.', '1412.5672-2-4-5': 'For instance, an interesting connection between the confinement of static quarks and the infrared (IR) behavior of gluon and ghost correlators has been pointed out [CITATION].', '1412.5672-2-4-6': "Of course, neither the DSE nor the FRG can be solved exactly and one's ability to perform actual calculations relies on several assumptions which, even when these are well motivated, can be difficult to check explicitly.", '1412.5672-2-4-7': 'A general criticism that many nonperturbative approaches have to face is that they do not always involve a systematic approximation scheme and it is often difficult to compute corrections to the obtained results.', '1412.5672-2-4-8': 'Still, the DSE/FRG provide the most powerful nonperturbative tools-apart from lattice techniques-to investigate the physics of the deconfinement transition directly at the level of the basic degrees of freedom of the theory.', '1412.5672-2-5-0': 'In a series of recent works [CITATION], yet a different route for the study of the infrared dynamics of Yang-Mills fields has been proposed.', '1412.5672-2-5-1': 'This is based on simple massive extensions of the standard Faddeev-Popov Lagrangian in the Landau gauge and in the Landau-DeWitt gauge.', '1412.5672-2-5-2': 'This is motivated by the observation that perturbative calculations of Yang-Mills correlators in the vacuum [CITATION] and at finite temperature [CITATION] in the massive extension of the Landau gauge action agree well with lattice data down to deep IR momenta.', '1412.5672-2-5-3': 'It is worth emphasizing that, if the typical value of the gauge coupling [MATH] required for such comparisons is of a few units, the relevant expansion parameter at zero temperature is [MATH].', '1412.5672-2-5-4': 'An important feature of the massive theory is that there exist IR safe renormalization-group trajectories, with no Landau pole [CITATION].', '1412.5672-2-5-5': 'Moreover, it has been argued in [CITATION] that such a massive extension naturally arises as an effective theory for Yang-Mills correlators in a new one-parameter family of Landau gauges which, unlike the standard Faddeev-Popov construction, takes into account the existence of Gribov ambiguities.', '1412.5672-2-5-6': 'In this approach, the gluon mass term appears as a gauge-fixing parameter which lifts the degeneracy between Gribov copies.', '1412.5672-2-6-0': 'This has been extended to the Landau-DeWitt gauge in the context of background field methods in [CITATION].', '1412.5672-2-6-1': 'There, we have shown that a calculation of the background field potential at first nontrivial (one-loop) order in perturbation theory correctly reproduces the phase structure of SU([MATH]) theories: one finds a confining phase at low temperature and a transition to a deconfined phase at high temperature which is second order for [MATH] and first order for [MATH], with transition temperatures in qualitative agreement with known values from lattice calculations.', '1412.5672-2-6-2': 'Our one-loop results are similar to those of the FRG studies of [CITATION] and actually corroborate the related findings concerning the relation between the IR behavior of gluon and ghost propagators and the existence of a confining phase at low temperature.', '1412.5672-2-6-3': 'Definite advantages of such a perturbative approach are, first, that low-order calculations are technically very simple and, second, that they can be systematically improved by computing higher orders.', '1412.5672-2-6-4': 'If at asymptotically high temperatures, the expansion parameter is [MATH] due to collective infrared effects which necessitate the resummation of hard thermal loops [CITATION], near the transition region and below, the effective gluon mass tames some of the infrared problems of the perturbative series and it is not clear what is the relevant expansion parameter.', '1412.5672-2-6-5': 'It is the purpose of the present article to study the importance of such higher-order terms by computing the background field potential, the Polyakov loop and the thermodynamic pressure at next-to-leading order in the perturbative expansion.', '1412.5672-2-7-0': 'Before embarking in actual calculations, let us make some general comments concerning the massive extension of the Faddeev-Popov action in the class of (Landau or Landau-DeWitt) gauges considered here.', '1412.5672-2-7-1': 'What usually prevents a mass term in the gauged-fixed action is obviously not gauge invariance, but BRST symmetry.', '1412.5672-2-7-2': 'The latter is a property of the Faddeev-Popov action, which is known to be valid at best in the high energy perturbative regime.', '1412.5672-2-7-3': 'But the Faddeev-Popov construction is certainly wrong in general since it ignores the existence of Gribov copies and, hence, does not completely fix the gauge.', '1412.5672-2-7-4': 'In fact, it is well known that the only truly nonperturbative formulation of the gauge-fixed theory known so far, that is the lattice, cannot accommodate the BRST symmetry without leading to undefined zero over zero ratios [CITATION].', '1412.5672-2-8-0': 'A consistent quantization procedure, free of Gribov ambiguities, is likely to break the BRST symmetry.', '1412.5672-2-8-1': 'A well-known example is the minimal Landau gauge on the lattice, where one picks up a random Gribov copy on each gauge orbit [CITATION].', '1412.5672-2-8-2': 'Examples in the continuum include the so-called (refined) Gribov-Zwanziger approach [CITATION] or the averaging procedure of [CITATION], already mentioned.', '1412.5672-2-8-3': 'In the latter case, the bare gluon mass originates from the averaging procedure and is simply a gauge-fixing parameter which explicitly breaks the BRST symmetry.', '1412.5672-2-8-4': 'Such gauge-fixing procedures provide efficient starting points for perturbative calculations of Yang-Mills correlators and, for the latter, of the phase structure of the theory at finite temperature.', '1412.5672-2-8-5': 'In both approaches the BRST breaking is soft and the gauge-fixed actions present modified (non-nilpotent) BRST symmetries, which ensure their renormalizability.', '1412.5672-2-8-6': 'At this point, it is worth emphasizing that the continuum approaches mentioned above also introduce, in one way or another, a BRST-breaking ingredient.', '1412.5672-2-8-7': 'This typically appears through choices of boundary conditions and/or ultraviolet subtractions in the context of DSE [CITATION], or through the infrared regulator in FRG approaches.', '1412.5672-2-9-0': 'The plan of the paper is as follows.', '1412.5672-2-9-1': 'Section [REF] sets the scene and briefly recalls the basics of the [MATH] transition and of static quark confinement.', '1412.5672-2-9-2': 'In Sec. [REF] we present the massive extension of the Landau-DeWitt gauge and derive the corresponding Feynman rules.', '1412.5672-2-9-3': 'In Sec. [REF], we summarize the calculation of the two-loop correction to the background field effective potential and in Sec. [REF], we give the corresponding expression for the one-loop correction to the Polyakov loop.', '1412.5672-2-9-4': 'Finally, in Sec. [REF], we present our results concerning the order of the phase transition and the value of the transition temperature as well as the temperature dependence of the Polyakov loop and of the thermodynamic pressure.', '1412.5672-2-9-5': 'The essential steps of our calculations are presented in the main text while the technical details are gathered in the Appendixes.', '1412.5672-2-10-0': 'Although we specify to the case [MATH] throughout this work, some formulas are valid for arbitrary [MATH].', '1412.5672-2-10-1': 'An important observation is that many steps of our calculations are similar to those in the massive extension of the Landau gauge, i.e., at vanishing background field.', '1412.5672-2-10-2': 'In fact, when expressed in an appropriate color basis, the Feynman rules of the theory have the same form as those in the Landau gauge, with the only difference that the momenta get shifted by an amount proportional to the background field.', '1412.5672-2-10-3': 'The key point is that, because of the residual global color symmetry, these shifts are conserved at the interaction vertices.', '1412.5672-2-10-4': 'This allows us to use various simplifying manipulations, detailed in Appendix [REF], and to reduce all the two-loop diagrams contributing to the background field potential to simple scalarlike sum-integrals; see Appendix [REF].', '1412.5672-2-10-5': 'The evaluation of the corresponding Matsubara sums is presented in Appendix [REF], which allows us to write the background field potential in a rather simple form in Appendix [REF].', '1412.5672-2-10-6': 'In Appendix [REF], we detail the calculation of the Polyakov loop at one-loop order in the presence of the nontrivial background field.', '1412.5672-2-10-7': 'In particular, this demonstrates explicitly that the Polyakov loop vanishes if and only if the minimum of the background field potential takes particular, [MATH]-symmetric values.', '1412.5672-2-10-8': 'This confirms, at this order of approximation, that the background field itself can be used as an order parameter for confinement, as advocated in [CITATION].', '1412.5672-2-10-9': 'Finally, we provide, in Appendix [REF], a general proof that the Polyakov loop vanishes when the background field takes [MATH]-symmetric values.', '1412.5672-2-10-10': 'We do not have a similar proof for the converse.', '1412.5672-2-11-0': '# Confinement-deconfinement transition in Yang-Mills theory', '1412.5672-2-12-0': 'We consider the Euclidean Yang-Mills action in [MATH] dimensions [EQUATION] where [MATH], with [MATH] the bare coupling constant and [MATH] the bare gauge field, [MATH] being an element of the Lie algebra of SU([MATH]).', '1412.5672-2-12-1': 'We have also defined [MATH], with [MATH] the inverse temperature.', '1412.5672-2-13-0': 'Let us recall some basic aspects of the deconfinement transition to be considered below.', '1412.5672-2-13-1': 'The free energy [MATH] of an isolated static quark in a thermal gluon bath is directly related to the expectation value of the traced Polyakov loop-which we refer to as the Polyakov loop for short in what follows-as [CITATION] [EQUATION] where [MATH] is the free energy in the absence of quark.', '1412.5672-2-13-2': 'Here, [MATH] orders the matrix fields [MATH] from left to right with decreasing value of their time arguments and the brackets refer to the average in the theory defined by the action ([REF]).', '1412.5672-2-13-3': 'A vanishing [MATH] signals an infinite free energy, hence a quark confining phase, while [MATH] is interpreted as a phase where isolated static quarks are energetically allowed.', '1412.5672-2-14-0': 'It is well known [CITATION] that the Polyakov loop gets multiplied by a phase under generalized gauge transformations that leave the Yang-Mills action at finite temperature invariant and which are [MATH] periodic in imaginary time, up to an element of the center of the group.', '1412.5672-2-14-1': 'This means that the deconfined phase is necessarily a phase where this symmetry group, or more precisely the quotient of the group of generalized gauge transformations by the subgroup of standard gauge transformations-which is isomorphic to [MATH]-is spontaneously broken.', '1412.5672-2-14-2': 'Note that the converse is not necessarily true: Although this is not the expected behavior, one could, in principle, imagine a situation where the center is spontaneously broken but where the Polyakov loop still vanishes, the breaking being only manifest at the level of higher-order correlations.', '1412.5672-2-14-3': 'This emphasizes the fact that the confined or deconfined nature of the system, in the sense described above, is not the breaking of the [MATH] symmetry itself but really the zero or nonzero value of the Polyakov loop (or of any equivalent order parameter).', '1412.5672-2-14-4': 'In what follows, we shall compute the Polyakov loop at one-loop order within the massive extension of the Landau-DeWitt gauge put forward in [CITATION].', '1412.5672-2-15-0': '# The massive Landau-DeWitt action', '1412.5672-2-16-0': '## General setup', '1412.5672-2-17-0': 'We quantize the theory using the background field method [CITATION], where we introduce an a priori arbitrary background field configuration [MATH] and define the fluctuating field [MATH].', '1412.5672-2-17-1': 'The Landau-DeWitt gauge condition reads [EQUATION] where [MATH] for any field [MATH] in the Lie algebra of the gauge group.', '1412.5672-2-17-2': 'Our gauge-fixed action reads [CITATION] [EQUATION] with [MATH] a (real) Nakanishi-Lautrup field and [MATH] and [MATH] the Faddeev-Popov ghosts and antighost fields.', '1412.5672-2-17-3': 'In terms of the field [MATH], we have [EQUATION] with [MATH] the field strength tensor evaluated at [MATH], and [EQUATION]', '1412.5672-2-17-4': 'The action ([REF]) has the obvious property [EQUATION] where [MATH], [MATH] is a local SU([MATH]) matrix, and [EQUATION]', '1412.5672-2-17-5': 'At the level of the (quantum) effective action [MATH] this implies [CITATION] [EQUATION] where the fields [MATH] are now to be understood as average values in the presence of sources.', '1412.5672-2-18-0': 'In principle, to evaluate physical observables at zero sources, one should minimize [MATH] with respect to [MATH] at a given [MATH].', '1412.5672-2-18-1': 'It can be argued, however, that one can alternatively minimize the functional [EQUATION] with respect to the background field [MATH] [CITATION].', '1412.5672-2-18-2': 'This functional obeys the background gauge symmetry [EQUATION] which is trivially preserved in perturbation theory.', '1412.5672-2-18-3': 'The Polyakov loop can be obtained as [EQUATION] with [MATH] and where the brackets stand for an average in the gauge-fixed theory ([REF]).', '1412.5672-2-18-4': 'The right-hand side of Eq. ([REF]) is evaluated at an absolute minimum [MATH] of [MATH].', '1412.5672-2-18-5': 'Because the Polyakov loop involves only the temporal component of the background field and because [MATH] (since [MATH] by construction) is [MATH] independent, it is sufficient to consider homogeneous background fields in the temporal direction [MATH].', '1412.5672-2-18-6': 'Moreover, the Hermitian matrix [MATH] can be diagonalized by means of a global SU([MATH]) rotation and one can thus, with no loss of generality, restrict [MATH] to the Cartan subalgebra of the color group.', '1412.5672-2-18-7': 'We shall write [MATH], where [MATH] are the SU([MATH]) generators in the Cartan subalgebra.', '1412.5672-2-18-8': 'We thus have to minimize the background field effective potential [EQUATION] where [MATH] and [MATH] is the spatial volume.', '1412.5672-2-18-9': 'We have subtracted [MATH], the value of the potential at zero temperature which is independent of [MATH], as we shall see below.', '1412.5672-2-18-10': 'Finally, the thermodynamic pressure is simply given by [EQUATION] where [MATH].', '1412.5672-2-19-0': 'The symmetry ([REF]) implies that the potential ([REF]) is invariant under gauge transformations of the form [EQUATION] where [MATH] is such that [MATH].', '1412.5672-2-19-1': 'This implies that the potential is periodic along certain directions in the Cartan subalgebra.', '1412.5672-2-19-2': 'Together with the invariance under global color rotations and charge conjugation, this implies that some of the extrema of the potential have specific locations [CITATION].', '1412.5672-2-19-3': 'In the SU([MATH]) case, these considerations show that the potential is [MATH]-periodic in [MATH] and has extremas at [MATH].', '1412.5672-2-19-4': 'In Appendix [REF], we provide a general proof that, among those, the values [MATH] are such that [MATH] and thus correspond to a confined phase.', '1412.5672-2-19-5': 'This supports the general expectation [CITATION] that the background field itself can be used as an order parameter.', '1412.5672-2-19-6': 'Our proof is, however, not complete because the converse, i.e., [MATH], is missing.', '1412.5672-2-19-7': 'In the present work, we shall, however, explicitly show that the equivalence holds at next-to-leading order in the loop expansion.', '1412.5672-2-20-0': '## Renormalization', '1412.5672-2-21-0': 'We introduce renormalized parameters and fields, related to the corresponding bare quantities in the usual way: [EQUATION] and [EQUATION] where we have kept the same notation for bare (left) and renormalized (right) fields for simplicity.', '1412.5672-2-21-1': 'Notice that the background field [MATH] and the fluctuating field [MATH] have different renormalization factors [CITATION].', '1412.5672-2-21-2': 'The background field gauge symmetry ([REF]) implies that the product [MATH] is finite [CITATION].', '1412.5672-2-21-3': 'In the following we impose the renormalization condition [EQUATION] for the finite parts as well, so that [MATH].', '1412.5672-2-21-4': 'From here on, we only consider renormalized quantities unless explicitly stated.', '1412.5672-2-21-5': 'The values of the parameters [MATH] and [MATH] must be fixed from some external input, e.g., lattice data.', '1412.5672-2-22-0': 'To set the value of the renormalized mass [MATH], one would ideally use the value of a physical observable such as a glueball mass.', '1412.5672-2-22-1': 'An easier possibility in practice is to fix this parameter by employing lattice results for gauge-dependent quantities such as the the Yang-Mills correlators.', '1412.5672-2-22-2': 'In principle, this requires lattice results in the same gauge as described above, involving an average of Gribov copies.', '1412.5672-2-22-3': 'Instead, existing gauge-fixed lattice calculations typically select a particular Gribov copy in the so-called first Gribov region, where the Faddeev-Popov operator is positive definite.', '1412.5672-2-22-4': 'Still, explicit calculations in the massive extension of the Faddeev-Popov Lagrangian in the Landau gauge show that there exists a value of the renormalized mass which allows one to quantitatively reproduce the lattice data for the Yang-Mills correlators at vanishing temperature [CITATION].', '1412.5672-2-23-0': 'No lattice calculation exists so far in the Landau-DeWitt gauge with homogeneous temporal background field as considered here.', '1412.5672-2-23-1': 'However, at zero temperature the background field vanishes and the latter reduces to the standard Landau gauge.', '1412.5672-2-23-2': 'In the present work, we shall thus use the values of [MATH] and [MATH] inferred from lattice calculations in the Landau gauge at vanishing temperature.', '1412.5672-2-23-3': 'To be consistent with the approximation order considered here, we need the one-loop expressions of the vacuum propagators.', '1412.5672-2-23-4': 'These have been computed in [CITATION].', '1412.5672-2-23-5': 'Using the renormalization conditions [EQUATION] where [MATH] and [MATH] denote, respectively, the renormalized ghost and transverse gluon self-energies in the vacuum, the best fits to lattice data give [MATH] MeV and [MATH], with [MATH] GeV for SU([MATH]) in [MATH].', '1412.5672-2-24-0': '## Feynman rules', '1412.5672-2-25-0': 'For the homogeneous background fields considered here, [MATH], the curvature term vanishes: [MATH].', '1412.5672-2-25-1': 'Moreover, as emphasized above, the background field [MATH] can be taken in the Cartan subalgebra of the gauge group.', '1412.5672-2-25-2': 'For SU([MATH]), the latter has only one direction which we choose along the third axis in color space.', '1412.5672-2-25-3': 'It is convenient to work with the basis of generators [EQUATION] which satisfy [EQUATION] where [MATH] is the completely antisymmetric tensor, with [MATH].', '1412.5672-2-25-4': 'We denote any element of the gauge group Lie algebra as [MATH].', '1412.5672-2-25-5': 'The components of the background covariant derivative are [EQUATION] where our convention for the Fourier transform is [EQUATION] such that [MATH].', '1412.5672-2-25-6': 'Here, we defined the shifted momentum [EQUATION] which satisfies [EQUATION]', '1412.5672-2-25-7': 'The background field breaks the global color group but there remains a residual symmetry under those color rotations that leave it invariant.', '1412.5672-2-25-8': 'In the SU([MATH]) theory, this is the group of SO([MATH] U([MATH]) transformations, under which [MATH] and [MATH], with [MATH] a constant phase.', '1412.5672-2-25-9': 'Accordingly, we refer to [MATH] as the "neutral" component and to [MATH] as the "charged" components.', '1412.5672-2-25-10': 'The (temporal) background field plays the role of an external Abelian field coupled to the charged components, which leads to a simple shift of momentum [MATH].', '1412.5672-2-25-11': 'Equivalently, it can be seen as an imaginary chemical potential associated with the conserved U([MATH]) charge.', '1412.5672-2-25-12': 'The residual color symmetry guarantees that the corresponding charge is conserved upon propagation and at the interaction vertices.', '1412.5672-2-26-0': 'It is an easy exercise to compute the Feynman rules of the theory in the basis ([REF]).', '1412.5672-2-26-1': 'To each propagator and to each leg of an interaction vertex is associated a flow of color charge, which we define to follow the flow of momentum.', '1412.5672-2-26-2': 'The tree-level ghost and gluon propagators for momentum [MATH] and charge state [MATH], represented in Fig. [REF], are given by [EQUATION] where we denote the scalar propagator of mass [MATH] by [EQUATION] and where [MATH].', '1412.5672-2-26-3': 'Note the identities [MATH] and [MATH], which follow from ([REF]) and which simply reflect the fact that the choice of orientation of the momentum/charge flow in the diagrams of Fig. [REF] is arbitrary.', '1412.5672-2-27-0': 'The cubic (derivative) vertices are represented on Fig. [REF], with the convention that all momenta and color charges are outgoing.', '1412.5672-2-27-1': 'The ghost-antighost-gluon vertex is given by [EQUATION] and the three-gluon vertex reads [EQUATION] where the various momenta, color charges, and spacetime indices are organized as in Fig. [REF].', '1412.5672-2-27-2': 'The structure constant [MATH] guarantees that the charge is conserved at the vertex: [MATH].', '1412.5672-2-27-3': 'Finally, the four-gluon vertex is [EQUATION] where color charge and spacetime indices are organized as in Fig. [REF].', '1412.5672-2-27-4': 'Again, the combinations of the group structure constants guarantee that the color charge is conserved: [MATH].', '1412.5672-2-28-0': 'Written in the basis ([REF]), the Feynman rules ([REF])-([REF]) in the presence of the background field are very similar to the standard ones, usually written in the Cartesian color basis.', '1412.5672-2-28-1': 'The essential difference stems in the different structure constants, which can be traced back to the commutation relations ([REF]), and the role of the background field is simply to replace all momenta by shifted ones according to the corresponding color charges.', '1412.5672-2-28-2': 'As we shall discuss in a future work, these remarks generalize to any group SU([MATH])-in fact to any compact Lie group with a semisimple Lie algebra.', '1412.5672-2-28-3': 'For [MATH], one has the identities [EQUATION] and [EQUATION]', '1412.5672-2-28-4': 'Using ([REF]) and ([REF]), one checks that the vertices ([REF])-([REF]) are unchanged if all momenta and color charges are incoming.', '1412.5672-2-29-0': '# The background field potential at [MATH]', '1412.5672-2-30-0': 'We consider the loop expansion of the background field potential, which corresponds to a perturbative expansion in powers of the renormalized coupling [MATH] with [MATH].', '1412.5672-2-30-1': 'We write the corresponding series as [EQUATION] with [MATH] the [MATH]-loop order contribution.', '1412.5672-2-30-2': 'The classical action ([REF]) evaluated at [MATH], [MATH], and [MATH] vanishes identically since [MATH].', '1412.5672-2-30-3': 'The tree-level contribution to the term [MATH] in ([REF]) is thus trivially independent of the background field and the tree-level potential is trivial [EQUATION]', '1412.5672-2-30-4': 'The one-loop contribution has been obtained in [CITATION] and the relevant two-loop diagrams are shown in Fig. [REF].', '1412.5672-2-31-0': '## The one-loop contribution', '1412.5672-2-32-0': 'We briefly recall the result of [CITATION].', '1412.5672-2-32-1': 'Introducing the function [EQUATION] which is such that, for [MATH], [EQUATION] the one-loop background field potential can be written as [EQUATION] where the first term on the right-hand side is the contribution from the massive gluons and the second one is due to the massless ghosts.', '1412.5672-2-32-2': 'It reduces to the well-known Weiss potential [CITATION] in the high temperature limit : [EQUATION] whereas it gives a confining, inverted Weiss potential at low temperature, where the contribution from massive modes is suppressed [CITATION]: [EQUATION]', '1412.5672-2-33-0': '## Two-loop diagrams', '1412.5672-2-34-0': 'Let us start with the ghost-gluon sunset diagram (second diagram of Fig. [REF]).', '1412.5672-2-34-1': 'A straightforward application of the Feynman rules derived above yields [EQUATION] with [MATH] and [MATH].', '1412.5672-2-34-2': 'We employ the general notations [MATH], with [MATH], and [EQUATION] with [MATH] an arbitrary mass scale and we recall that [MATH].', '1412.5672-2-34-3': 'We have also introduced the completely symmetric tensor [EQUATION] whose components vanish unless [MATH] and are equal to one otherwise.', '1412.5672-2-34-4': 'It follows from Eqs. ([REF]) and ([REF]) that [EQUATION] which imply ([MATH]) [EQUATION]', '1412.5672-2-34-5': 'For a vanishing background field, the term within brackets in the right-hand side of Eq. ([REF]) is nothing but the expression of the ghost-gluon sunset in the massive extension of the Landau gauge, up to the corresponding color factor [MATH].', '1412.5672-2-34-6': 'The latter is recovered using ([REF]).', '1412.5672-2-34-7': 'The expression ([REF]) illustrates that the two-loop perturbative diagram at nonvanishing background can be obtained from the corresponding one at [MATH] as follows: First, one writes the momentum integrals in the massive Landau gauge in terms of three momenta [MATH], [MATH], and [MATH], up to the color factor [MATH]; then, one shifts the momenta to [MATH], [MATH], and [MATH] and averages with the weight [MATH].', '1412.5672-2-34-8': 'This property can be anticipated from the Feynman rules described above and actually generalizes to any closed diagram, with an appropriate weight factor.', '1412.5672-2-34-9': 'The conservation of color charge at the interaction vertices implies the conservation of the shifted momenta under the loop integrals, e.g., [MATH] in the two-loop case above.', '1412.5672-2-34-10': 'This leads to important simplifications of the calculations in the Landau-DeWitt gauge, as detailed in the Appendixes.', '1412.5672-2-34-11': 'Let us finally mention that the expression ([REF]) and the above remarks actually generalize to SU([MATH]), where the tensor ([REF]) involves the appropriate structure constant.', '1412.5672-2-34-12': 'If Eq. ([REF]) is specific to the case [MATH], the properties ([REF]) are true for arbitrary [MATH].', '1412.5672-2-35-0': 'For later use, we rewrite Eq. ([REF]) in a more compact form.', '1412.5672-2-35-1': 'Using the conservation of shifted momenta under the sum in ([REF]), we have [EQUATION]', '1412.5672-2-35-2': 'Furthermore, writing [EQUATION] we obtain [EQUATION] in terms of the two-loop integral [EQUATION] which is needed only for the case [MATH].', '1412.5672-2-35-3': 'In Appendix [REF], we rewrite these integrals in terms of simpler scalarlike loop integrals; see Eq. ([REF]) below.', '1412.5672-2-35-4': 'Here, we simply notice for later use that [MATH] is invariant under the simultaneous permutation of the upper and lower indices, when the corresponding shifts of momenta add up to zero: [EQUATION]', '1412.5672-2-35-5': 'The two diagrams with purely gluonic loops can be treated in a similar way.', '1412.5672-2-35-6': 'We give more details in Appendix [REF] and simply state the results here.', '1412.5672-2-35-7': 'The double tadpole diagram (first diagram of Fig. [REF]) yields [EQUATION] where we have defined the tadpole integrals [EQUATION] and [EQUATION]', '1412.5672-2-35-8': 'As for the gluon sunset diagram (third diagram of Fig. [REF]), we obtain [EQUATION] where ([REF]) has been used.', '1412.5672-2-36-0': '## Counterterm contribution', '1412.5672-2-37-0': 'The expressions derived in the previous section contain ultraviolet (UV) divergences.', '1412.5672-2-37-1': 'These are canceled by the counterterms from the original action, as we show explicitly in Appendix [REF].', '1412.5672-2-37-2': 'Here, we compute the relevant counterterm contributions to the background field potential.', '1412.5672-2-38-0': 'Writing the renormalization factors as [MATH], [MATH], the counterterm action reads [EQUATION] where [MATH] and the dots denote terms involving the coupling counterterm [MATH], which are not needed in the present work.', '1412.5672-2-39-0': 'Using ([REF]), we obtain, for the ghost counterterm loop, [EQUATION] where the last integral vanishes in dimensional regularization.', '1412.5672-2-39-1': 'The gluon counterterm loop reads [EQUATION]', '1412.5672-2-39-2': 'Then, up to an integral which vanishes in dimensional regularization, the total counterterm contribution can be written in terms of the tadpole integral ([REF]) as [EQUATION]', '1412.5672-2-40-0': '## The two-loop contribution', '1412.5672-2-41-0': 'Adding together the various two-loop pieces, we obtain [notice that the contributions from the integral ([REF]) cancel] [EQUATION]', '1412.5672-2-41-1': 'For a vanishing background field, Eq. ([REF]) is nothing but the two-loop free energy density computed in the massive extension of the Landau gauge.', '1412.5672-2-41-2': 'As detailed in Appendix [REF], the integral ([REF]) can be expressed in terms of the tadpole integrals ([REF]) and [EQUATION] and of the scalar sunset [EQUATION]', '1412.5672-2-41-3': 'Using Eqs. ([REF])-([REF]), our final expression for the two-loop contribution to the background field potential is [EQUATION]', '1412.5672-2-41-4': 'The calculation of the various Matsubara sums and momentum integrals involved in this expression is detailed in Appendix [REF].', '1412.5672-2-41-5': 'Moreover, we show in Appendix [REF] that it is UV finite (up to an overall, temperature- and background-independent divergence) once the counterterm [MATH] has been fixed from the renormalization conditions ([REF]).', '1412.5672-2-41-6': 'We also reduce this expression to a sum of one- and two-dimensional (radial) momentum integrals involving thermal, Bose-Einstein distribution functions in the presence of the background field.', '1412.5672-2-41-7': 'Our final result for the thermal part (the only one that depends on the background) of the two-loop contribution to the background field potential is given in Eq. ([REF]).', '1412.5672-2-42-0': '# The Polyakov loop at [MATH]', '1412.5672-2-43-0': 'We similarly expand the Polyakov loop ([REF]) as [EQUATION] with the [MATH]-loop contribution [MATH].', '1412.5672-2-43-1': 'This is obtained by expanding the path-ordered exponential in Eq. ([REF]) in powers of the coupling with [MATH].', '1412.5672-2-43-2': 'The tree-level and one-loop contributions are evaluated in Appendix [REF] for arbitrary [MATH] and for fields in an arbitrary representation of the gauge group.', '1412.5672-2-43-3': 'For the fundamental representation of SU(2), we obtain [EQUATION] and [EQUATION] where [MATH] is the absolute minimum of the two-loop background field potential and [EQUATION]', '1412.5672-2-43-4': 'It is easily checked that [MATH]; hence [EQUATION]', '1412.5672-2-43-5': 'This confirms, at this order, that the background field [MATH] itself is a good order parameter for static quark confinement [CITATION].', '1412.5672-2-44-0': '# Results', '1412.5672-2-45-0': 'As already mentioned, we shall use the values of [MATH] and [MATH] inferred from lattice calculations of ghost and gluon propagators in the Landau gauge at vanishing temperature.', '1412.5672-2-45-1': 'For the present order of approximation, we fit these data to the one-loop expressions obtained in the massive Landau gauge.', '1412.5672-2-45-2': 'The best fitting values are [MATH] MeV and [MATH] with [MATH] GeV.', '1412.5672-2-45-3': 'This sets the scale of the present calculation.', '1412.5672-2-45-4': 'We shall compare our findings to our earlier leading-order results [CITATION] where the mass parameter was obtained by fitting the data to the tree-level propagators with the best fitting value [MATH] MeV.', '1412.5672-2-45-5': 'There is of course some error related to the determination of the parameters.', '1412.5672-2-45-6': 'We have checked that the results to be presented below do not change qualitatively as we change the parameters within the error bars.', '1412.5672-2-45-7': 'Let us finally mention that we shall not consider the possible temperature dependence of the parameters.', '1412.5672-2-45-8': 'This and the implementation of renormalization-group improvement are deferred to a future work.', '1412.5672-2-46-0': '## Background field potential', '1412.5672-2-47-0': 'In Fig. [REF], we show the rescaled two-loop background field potential [MATH] in the range [MATH] for various values of the temperature.', '1412.5672-2-47-1': 'As the temperature is increased, there is clearly a transition from a confining phase, where the minimum of the potential lies at its confining value [MATH], to a deconfined phase where the location of the minimum departs from [MATH].', '1412.5672-2-47-2': 'The transition is second order and the corresponding critical temperature is obtained by requiring the vanishing of the curvature of the potential at the confining point.', '1412.5672-2-47-3': 'The rescaled curvature [MATH] at [MATH] is plotted as a function of [MATH] in the left panel of Fig. [REF] at one- and two-loop orders.', '1412.5672-2-47-4': 'The two-loop correction leads to a larger critical temperature, [MATH], as compared to our previous one-loop result [CITATION], [MATH].', '1412.5672-2-47-5': 'A typical lattice result is [CITATION] [MATH] MeV.', '1412.5672-2-47-6': 'Although such comparison must be taken with care due to the issue of properly setting the scale, this shows that the two-loop corrections indeed improve the one-loop result.', '1412.5672-2-48-0': 'It is also interesting to compare with other continuum approaches.', '1412.5672-2-48-1': 'The FRG and DSE/2PI calculations of Ref. [CITATION] give [MATH] and [MATH], respectively, which lie in the same ballpark as our one-loop result.', '1412.5672-2-48-2': 'The improved value of the critical temperature obtained above suggests that the present two-loop calculation efficiently captures some of the effects which have been discarded in those calculations.', '1412.5672-2-48-3': 'For instance, although the fully resummed propagators are included, some explicit two-loop contributions to the DSE for the background field potential have been neglected.', '1412.5672-2-48-4': 'As for the FRG calculation, the authors of Ref. [CITATION] mention that their result is modified to [MATH] when some backreaction effects-neglected in their main study-are included.', '1412.5672-2-49-0': 'For completeness, we compare in Fig. [REF] the one- and two-loop potentials at their respective critical temperatures.', '1412.5672-2-49-1': 'It is also instructive to plot the (rescaled) curvature of the potential at the origin as a function of [MATH]; see the right panel of Fig. [REF].', '1412.5672-2-49-2': 'For the values of parameters studied here, we observe that, at one-loop order, there exists a temperature [MATH] above which the minimum of the potential is exactly located at [MATH].', '1412.5672-2-49-3': 'This does not seem to be the case at two-loop order where the curvature of the potential at the origin remains negative.', '1412.5672-2-50-0': 'To have a better analytical control on our results, we have considered the formal limit [MATH].', '1412.5672-2-50-1': 'We show in Appendix [REF] that the rescaled potential [MATH] is a polynomial in the range [MATH]: [EQUATION] which minimum is located at [EQUATION]', '1412.5672-2-50-2': 'We discuss the consequences of the different behavior between the one- and two-loop results below.', '1412.5672-2-51-0': '## Polyakov loop', '1412.5672-2-52-0': 'In Fig. [REF], we compare the temperature dependence of the Polyakov loop at leading and next-to-leading order.', '1412.5672-2-52-1': 'At leading order, it saturates to its upper bound [MATH] at the temperature [MATH] discussed previously, above which [MATH] remains the absolute minimum of the potential; see Fig. [REF].', '1412.5672-2-52-2': 'The Polyakov loop is singular at [MATH] because its second derivative is discontinuous.', '1412.5672-2-52-3': 'This has to do with the particular form of the Weiss potential, Eq. ([REF]).', '1412.5672-2-52-4': 'For small positive values of [MATH], the latter behaves as [EQUATION] and is thus nonanalytic in [MATH].', '1412.5672-2-52-5': 'This is to be contrasted with the massive version [MATH] of the same function which shows a regular expansion in powers of [MATH] when [MATH].', '1412.5672-2-52-6': 'In the vicinity of [MATH] and for [MATH] close to [MATH], the rescaled potential is of the form [MATH], where [MATH] with [MATH] and [MATH].', '1412.5672-2-52-7': 'It follows, that for [MATH] approaching [MATH] from below, [MATH] and [MATH] for [MATH].', '1412.5672-2-52-8': 'Thus, the first derivative of the background with respect to the temperature is discontinuous at [MATH].', '1412.5672-2-52-9': 'This singularity propagates to thermodynamic observables.', '1412.5672-2-52-10': 'For instance the third derivative of the free energy density with respect to the temperature is discontinuous.', '1412.5672-2-52-11': 'This is, however, a spurious discontinuity.', '1412.5672-2-53-0': 'As already discussed above, at two-loop order the curvature of the potential at the origin remains negative.', '1412.5672-2-53-1': 'In this range, the Polyakov loop does not show any additional singularity, other than the one at [MATH].', '1412.5672-2-53-2': 'We also observe that, as compared to the one-loop result, the Polyakov loop at two-loop order overshoots its asymptotic ([MATH]) value [EQUATION] with [MATH] given in ([REF]), as computed in Appendix [REF].', '1412.5672-2-54-0': '## Pressure and entropy', '1412.5672-2-55-0': 'The thermodynamic pressure ([REF]) is shown in Fig. [REF] as a function of the temperature in the one- and two-loop approximations.', '1412.5672-2-55-1': 'In both cases, we observe that the pressure is increasing at small temperatures, indicating a positive entropy ([MATH]).', '1412.5672-2-55-2': 'This is a welcome result although it may be surprising at first sight because the ghosts dominate in this regime and one would naively think that they contribute negatively to the entropy.', '1412.5672-2-56-0': 'The positivity of the entropy at low temperatures is ensured by the fact that, in the confining phase with [MATH], those ghosts which feel the background effectively behave as physical fermions, giving a positive contribution to the entropy.', '1412.5672-2-56-1': 'To illustrate this point more precisely, we note that the low temperature behavior of the background field potential is dominated by the one-loop contribution, as discussed in Appendix [REF].', '1412.5672-2-56-2': 'This is directly visible in Fig. [REF].', '1412.5672-2-56-3': 'At one-loop order, in the confined phase, the entropy contribution [MATH] of a ghost with charge [MATH] is [EQUATION]', '1412.5672-2-56-4': 'So we have either a bosoniclike contribution ([MATH]) [EQUATION] with a standard negative (ghostlike) sign from the neutral modes, or fermioniclike contributions [EQUATION] with a positive sign for charged modes due to their coupling to the background.', '1412.5672-2-56-5': 'The total ghost contribution, [EQUATION] is then positive, as announced.', '1412.5672-2-57-0': 'As one increases the temperature, the one-loop result violates the positivity of the entropy, slightly before reaching [MATH], as can be clearly seen in the inset plot of Fig. [REF], where the thermal pressure changes its monotony and even becomes slightly negative before [MATH].', '1412.5672-2-57-1': 'The reason for this behavior is again the change of effective statistics of the relevant degrees of freedom in the presence of the background.', '1412.5672-2-57-2': 'As the temperature is increased, the massive gluons start contributing to the pressure.', '1412.5672-2-57-3': 'However, in the confined phase, the charged gluons, which feel the presence of the background, behave like "wrong" fermions, contributing negatively to the entropy [CITATION].', '1412.5672-2-57-4': 'Remarkably this behavior is completely washed out by the two-loop correction and at two-loop order the entropy is positive (the pressure is monotonically increasing with the temperature) in the whole range of temperatures studied here; see Fig. [REF].', '1412.5672-2-58-0': 'Finally, let us comment on the [MATH] behavior of the pressure at low temperature-see Fig. [REF]-which is at odds with the exponential suppression seen in lattice results [CITATION].', '1412.5672-2-58-1': 'As discussed above, this originates from the fact that massless (ghost) modes directly contribute to the pressure, already at leading order.', '1412.5672-2-58-2': 'We mention that this is likely to be a general problem for continuum approaches, which are essentially based on using (resummed) propagators.', '1412.5672-2-58-3': 'For instance, in the Landau gauge, lattice results for the propagators [CITATION] show that, if the gluon becomes effectively massive for infrared momenta, the ghost remains massless.', '1412.5672-2-58-4': 'This generically produces [MATH] contributions in a leading-order-like-i.e., trace-log-expression for the thermodynamic pressure.', '1412.5672-2-58-5': 'The correct treatment of such unphysical massless degrees of freedom is a serious issue that needs to be further investigated.', '1412.5672-2-59-0': '## Comparison with the massive Landau gauge', '1412.5672-2-60-0': 'It is interesting to compare our results for the pressure to those obtained in the massive Landau gauge (which corresponds to a vanishing background).', '1412.5672-2-60-1': 'The corresponding one- and two-loop pressure curves are shown in Fig. [REF] and compared to those discussed in the previous subsection.', '1412.5672-2-60-2': 'We observe again that the one-loop pressure contains [MATH] contributions at low temperatures but this time those come with a negative prefactor which yields a negative entropy at low temperatures.', '1412.5672-2-60-3': 'The reason for this is simple: in the absence of background all ghosts contribute negatively to the entropy.', '1412.5672-2-60-4': 'In this case the two-loop correction is of no help because it does not contain any [MATH] contribution at small temperature.', '1412.5672-2-60-5': 'In fact it seems that the two-loop term makes the problem even worse since the entropy remains negative in the range of temperature shown here.', '1412.5672-2-60-6': 'Furthermore, we observe in Fig. [REF] that the two-loop correction is smaller in the Landau-DeWitt case than in the Landau case, indicating a better convergence.', '1412.5672-2-61-0': 'The above remarks illustrate the importance of taking into account the order parameter of the [MATH] transition in the description.', '1412.5672-2-61-1': 'Our two-loop results show that the value [MATH]-which would correspond to the Landau gauge-is never a physical point, i.e., an absolute minimum of the background field potential.', '1412.5672-2-61-2': 'In a sense, the perturbative expansion in the (massive) Landau gauge appears as an expansion around an unstable situation.', '1412.5672-2-61-3': 'It would take infinite resummations to correctly describe the physics near the stable physical point.We believe that this remark outranges the present framework and holds for other continuum approaches as well.', '1412.5672-2-62-0': '# Conclusions', '1412.5672-2-63-0': 'To summarize, we have proposed in [CITATION] a perturbative approach to describe the static quark confinement-deconfinement transition in Yang-Mills theories, based on a modified (massive) gauge-fixed action in the Landau-DeWitt gauge.', '1412.5672-2-63-1': 'This describes well the phase structure of SU([MATH]) theories at leading order and gives qualitatively good results for the transition temperatures of the [MATH] and [MATH] cases in [MATH].', '1412.5672-2-63-2': 'However, this leads to a [MATH] behavior of the thermodynamic pressure at low temperatures and to negative entropy and negative pressure near the transition temperature.', '1412.5672-2-63-3': 'Also the leading-order calculation leads to a spurious singularity at a temperature [MATH] above which the physical background field vanishes.', '1412.5672-2-64-0': 'With the present work, we wish to demonstrate the interest and feasibility of a next-to-leading-order calculation in this modified perturbative scheme.', '1412.5672-2-64-1': 'We investigate the SU([MATH]) theory at next-to-leading order, which shows that the correction to the critical temperature goes in the right direction and actually brings the estimated value close to the lattice result-although we stress again that such a comparison must be taken with care due to the issue of scale setting.', '1412.5672-2-64-2': 'The next-to-leading-order corrections also cure the negative entropy issue and remove the spurious singularity at [MATH].', '1412.5672-2-64-3': 'In fact, we find that the physical point always corresponds to a nonvanishing background field.', '1412.5672-2-64-4': 'There remains the issue of unphysical [MATH] contributions to the pressure from the massless (ghost) degrees of freedom, which needs to be investigated.', '1412.5672-2-65-0': 'In a future work, we shall extend the present two-loop study to the SU([MATH]) theory and investigate the effect of renormalization-group improvement.', '1412.5672-2-65-1': 'Another interesting extension would be to include quark degrees of freedom at nonzero chemical potential.', '1412.5672-2-65-2': 'Finally, the present work points out the importance of taking into account a nonvanishing background field near the phase transition.', '1412.5672-2-65-3': 'We plan to extend our previous study [CITATION] of the Landau gauge ghost and gluon correlators at finite temperature to the Landau-DeWitt gauge.', '1412.5672-2-66-0': '# ACKNOWLEDGEMENTS', '1412.5672-2-67-0': 'We thank J. M. Pawlowski for interesting discussions.', '1412.5672-2-67-1': 'We acknowledge financial support from the PEDECIBA and ECOS programs.', '1412.5672-2-67-2': 'U.R., J.S., and M.T. thank the IFFI, and N.W. the LPTMC for their hospitality.', '1412.5672-2-68-0': '# Gluon loops', '1412.5672-2-69-0': 'Here, we detail the derivation of Eqs. ([REF]) and ([REF]) for the purely gluonic two-loop diagrams.', '1412.5672-2-69-1': 'A straightforward application of the Feynman rules of Sec. [REF] yields, for the double gluon tadpole diagram (first diagram of Fig. [REF]), [EQUATION]', '1412.5672-2-69-2': 'The contribution from the first line in the brackets vanishes and the two other lines contribute the same.', '1412.5672-2-69-3': 'Setting [MATH] and [MATH] in the third line and using ([REF]) and [MATH], we obtain [EQUATION] where we renamed [MATH] and we used the definition ([REF]).', '1412.5672-2-69-4': 'This rewrites as Eq. ([REF]) using the integrals ([REF]) and ([REF]).', '1412.5672-2-69-5': 'As mentioned in Sec. [REF] for the ghost-gluon sunset diagram, we observe that the expression ([REF]) can be obtained from the corresponding diagram at vanishing background field by replacing the loop momenta by shifted ones and by averaging with the weight [MATH], where, here, the index [MATH] is redundant since it is not associated to any internal momentum.', '1412.5672-2-70-0': 'A similar writing applies to the gluon sunset diagram (third diagram of Fig. [REF]).', '1412.5672-2-70-1': 'Applying the Feynman rules of Sec. [REF], we obtain, after simple manipulations, [EQUATION] where [MATH] and [EQUATION]', '1412.5672-2-70-2': 'As emphasized previously, because [MATH] for [MATH], the sum of shifted momenta in ([REF]) vanishes: [MATH].', '1412.5672-2-70-3': 'This allows us to use a number of manipulations similar to those used in the case of a vanishing background field.', '1412.5672-2-70-4': 'First, we symmetrize the function [MATH] under the momentum integral in ([REF]) and use the identity [EQUATION] where "perm."', '1412.5672-2-70-5': 'denotes all possible permutations of the momenta [MATH], [MATH], and [MATH].', '1412.5672-2-70-6': 'Note that, for [MATH], the last factor on the right-hand side is symmetric in [MATH]; that is, [EQUATION]', '1412.5672-2-70-7': 'Next, we "desymmetrize" the right-hand side of Eq. ([REF]) under the integral in ([REF]), which amounts to replacing [EQUATION] in ([REF]).', '1412.5672-2-70-8': 'The momentum integral in Eq. ([REF]) can thus be written [EQUATION] where we used [MATH].', '1412.5672-2-70-9': 'Using the trick ([REF]) as well as the symmetry properties of the tensor ([REF]) and of the integral ([REF]), we obtain Eq. ([REF]), after some simple manipulations.', '1412.5672-2-71-0': '# Reduction of the integral [MATH] to the integrals [MATH], [MATH], and [MATH]', '1412.5672-2-72-0': 'Here, we show how the integral [MATH], Eq. ([REF]), can be expressed in terms of the scalar tadpole and sunset loop integrals [MATH], [MATH] and [MATH] given in Eqs. ([REF]), ([REF]), and ([REF]), respectively.', '1412.5672-2-72-1': 'To do so, we work out the various pieces of the integrand in Eq. ([REF]).', '1412.5672-2-72-2': 'We first write [EQUATION] and, using the identity [EQUATION] we also have [EQUATION]', '1412.5672-2-72-3': 'Combining the above identities, we obtain [EQUATION]', '1412.5672-2-72-4': 'The term in curly brackets on the right-hand side can be rewritten as [EQUATION] where we used the fact that [MATH], which, as already emphasized, holds as well for the shifted momenta in Eq. ([REF]) under the average with weight [MATH].', '1412.5672-2-72-5': 'Inserting ([REF]) in ([REF]), we obtain the following, manifestly symmetric expression for the integrand in ([REF]): [EQUATION]', '1412.5672-2-72-6': 'It is then immediate to obtain the desired relation: [EQUATION] where we used [EQUATION]', '1412.5672-2-72-7': 'The relation ([REF]) reads, explicitly, for the various cases of interest, [EQUATION]', '1412.5672-2-72-8': 'Using these relations in Eq. ([REF]), we obtain Eq. ([REF]).', '1412.5672-2-73-0': '# Matsubara sums', '1412.5672-2-74-0': 'In this section, we perform explicitly the various sums over Matsubara frequencies and angular momentum integrations involved in the one- and two-loop scalar integrals derived in the previous section and we extract the UV divergent parts using dimensional regularization.', '1412.5672-2-74-1': 'In what follows, we note [MATH], such that shifted momenta read [MATH].', '1412.5672-2-75-0': '## Tadpoles', '1412.5672-2-76-0': 'We begin with the tadpole integral [EQUATION]', '1412.5672-2-76-1': 'Note that [MATH].', '1412.5672-2-76-2': 'Standard contour integration techniques yield [EQUATION] where [MATH], [MATH], and [MATH] is the Bose-Einstein distribution function, which satisfies [MATH].', '1412.5672-2-76-3': 'Here, we extracted explicitly a zero temperature, background-field-independent contribution [MATH].', '1412.5672-2-76-4': 'For later use, we introduce the following notation [EQUATION] which emphasizes the number of thermal factors (i.e., Bose-Einstein distribution functions) appearing in each term on the right-hand side of ([REF]).', '1412.5672-2-76-5': 'In particular, [MATH].', '1412.5672-2-76-6': 'In dimensional regularization [MATH] and [EQUATION] where [MATH], with [MATH] the Euler constant.', '1412.5672-2-77-0': 'The other tadpole integral which appears in Eq. ([REF]) is [EQUATION]', '1412.5672-2-77-1': 'The Matsubara sum in this expression is not absolutely convergent but it can be defined as the limit of the symmetric summation [MATH] for [MATH].', '1412.5672-2-77-2': 'Alternatively, we can make it an absolutely convergent sum by writing [EQUATION] where the added term vanishes by symmetry.', '1412.5672-2-77-3': 'We can then use standard contour techniques to obtain [EQUATION] which satisfies [MATH] and, in particular, [MATH].', '1412.5672-2-77-4': 'The zero temperature contribution vanishes identically, [MATH], and we also note that [MATH] when [MATH] is a multiple of [MATH].', '1412.5672-2-77-5': 'The integral ([REF]) is finite and never multiplies a divergent contribution, so we can safely set [MATH] there.', '1412.5672-2-78-0': '## The scalar sunset', '1412.5672-2-79-0': 'We now treat the scalar two-loop integral [EQUATION] where [MATH], for the relevant case of conserved shifts, [MATH], which implies [MATH].', '1412.5672-2-79-1': 'We extend the approach of [CITATION].', '1412.5672-2-79-2': 'It proves useful to introduce the spectral representation of the (free) propagators [EQUATION] where [MATH], [MATH] is the shifted Matsubara frequency, and [EQUATION] with [MATH].', '1412.5672-2-79-3': 'The double Matsubara sum in ([REF]) yields [EQUATION] where we have used the identity [MATH] and [MATH].', '1412.5672-2-79-4': 'The second line of this equation makes it clear that ([REF]) is well defined for all [MATH], [MATH], and [MATH], including the limiting case [MATH], for which both the numerator and the denominator vanish linearly.', '1412.5672-2-79-5': 'In the following, we shall decompose the fraction in the third line of ([REF]) in different pieces whose numerators do not vanish at [MATH], thus making the corresponding contribution to ([REF]) formally divergent in this limit.', '1412.5672-2-79-6': 'To avoid this problem, we regulate the denominator in ([REF]) as [EQUATION]', '1412.5672-2-79-7': 'Now, we use the identities [EQUATION] in the third line of ([REF]) to rewrite Eq. ([REF]) as [EQUATION] where [MATH] is an unimportant vacuum contribution, independent of the temperature and of the background field, [MATH], [MATH], [EQUATION] and "perm."', '1412.5672-2-79-8': 'denotes the circular permutations of the pairs of indices [MATH], [MATH], and [MATH] in the two integrals that appear explicitly in ([REF]).', '1412.5672-2-79-9': 'The function [MATH] has been defined in ([REF]) and the function [MATH] is related to the vacuum one-loop integral [here [MATH]] [EQUATION]', '1412.5672-2-79-10': 'In obtaining Eq. ([REF]), we have used [MATH] and [MATH], and we have set [MATH] in the second, UV finite line.', '1412.5672-2-79-11': 'In contrast, one has to keep [MATH] arbitrary in the second term on the right-hand side of Eq. ([REF]) since it contains UV divergent contributions, arising from the zero temperature loop ([REF]).', '1412.5672-2-80-0': 'It is now an easy matter to perform explicitly the frequency and, for the double integral on the second line, the angular integrations.', '1412.5672-2-80-1': 'As before, we decompose the result according to the number of thermal factors [MATH] in each contribution as [EQUATION]', '1412.5672-2-80-2': 'We obtain [EQUATION] and [MATH].', '1412.5672-2-80-3': 'As mentioned before, the contribution [MATH] is UV finite but the contribution [MATH] contains UV divergent terms which explicitly depend on the temperature and on the background field.', '1412.5672-2-80-4': 'We shall check in Appendix [REF] that such contributions cancel after renormalization.', '1412.5672-2-80-5': 'For this purpose, it is useful to note that ([REF]) rewrites [EQUATION] where [MATH] is defined in ([REF]) and ([REF]) and where we used the fact, owing to the O([MATH]) invariance of the Euclidean integral ([REF]), [MATH] depends only on [MATH].', '1412.5672-2-80-6': 'The expression for [MATH] can be found for instance in [CITATION] and reads, up to corrections of [MATH], [EQUATION] where [MATH] (which should not be mistaken with [MATH]) is given by [EQUATION] with [EQUATION]', '1412.5672-2-80-7': 'Using this formula and the definition ([REF]), we get [EQUATION]', '1412.5672-2-81-0': '## Massless integrals at finite temperature', '1412.5672-2-82-0': 'When discussing the (formal) large temperature [MATH] behavior of our results, we encounter the following integrals [EQUATION] with [MATH] and [MATH].', '1412.5672-2-82-1': 'For [MATH], we have [EQUATION]', '1412.5672-2-82-2': 'It follows that [EQUATION] where [MATH] is the integer part of [MATH].', '1412.5672-2-82-3': 'In order to obtain [MATH] for any [MATH] from the knowledge of [MATH], Eq. ([REF]) has to be supplemented by the conditions [MATH] and [MATH] which are easily checked from the definitions ([REF]) and ([REF]).', '1412.5672-2-83-0': 'To compute [MATH], we note that, upon expanding the Bose-Einstein factor [MATH] as a geometric series, [MATH], it can be rewritten as [EQUATION] a Fourier series whose sum is nothing but [MATH] in the interval [MATH].', '1412.5672-2-83-1': "This implies in particular that the [MATH]'s are polynomials in the interval [MATH] and later we shall need the first of them: [EQUATION]", '1412.5672-2-84-0': '# Final expression of [MATH]', '1412.5672-2-85-0': 'We put together the material derived in the previous sections to obtain a final, explicitly finite expression for the two-loop contribution to the background field potential in terms of one- and two-dimensional (radial) momentum integrals.', '1412.5672-2-86-0': '## Renormalization', '1412.5672-2-87-0': 'We perform the renormalization at [MATH] in which case [MATH] and the Landau-DeWitt gauge coincides with the Landau gauge.', '1412.5672-2-87-1': 'The mass counterterm [MATH] has been computed in [CITATION] and reads [EQUATION] where the finite part [MATH] depends on the renormalization scheme.', '1412.5672-2-87-2': 'With the renormalization conditions ([REF]), one obtains [CITATION] [EQUATION] with [MATH] and [MATH].', '1412.5672-2-87-3': 'Using Eqs. ([REF]), ([REF]), ([REF])-([REF]), and ([REF]), one explicitly checks that the temperature- and background-field-dependent divergences cancel out.', '1412.5672-2-87-4': 'That no divergences are generated by the background field can be understood from the background gauge invariance [see Eq. ([REF])] as we shall discuss in a future work.', '1412.5672-2-87-5': 'Setting [MATH] in the remaining finite expression, we obtain, for the thermal contribution, [EQUATION] where we used the symmetry of the tensor [MATH] and we defined [EQUATION] and [EQUATION]', '1412.5672-2-87-6': 'Finally, we emphasize that the results derived so far do not rely on the explicit form of the tensor [MATH] and only use the conservation of color charge (and hence of shifted momenta) at the interaction vertices.', '1412.5672-2-87-7': 'All formulas thus hold for a general group SU([MATH]).', '1412.5672-2-87-8': 'Note that, in general, [MATH] must be understood as a vector in the [MATH]-dimensional Cartan subalgebra.', '1412.5672-2-88-0': '## Finite contributions', '1412.5672-2-89-0': 'Here, we specify to the case [MATH] and use the values of the tensor [MATH] to obtain a more explicit formula for [MATH].', '1412.5672-2-89-1': 'We have, for any quantity [MATH], [EQUATION]', '1412.5672-2-89-2': 'Using Eq. ([REF]), we obtain [EQUATION] where "perm."', '1412.5672-2-89-3': 'denotes the cyclic permutations of [MATH].', '1412.5672-2-89-4': 'It is convenient to reorganize the Bose-Einstein distribution functions in such a way that only one contains a complex argument.', '1412.5672-2-89-5': 'Using the identities [MATH], we have [EQUATION]', '1412.5672-2-89-6': 'Putting everything together, we finally obtain [EQUATION] where we used the fact that [MATH], [MATH], and [MATH], which implies that [MATH], [MATH], [MATH], [MATH], and [MATH].', '1412.5672-2-89-7': 'We recall that the relevant tadpole integrals are given in Eqs. ([REF]) and ([REF]).', '1412.5672-2-89-8': 'Finally, one has [EQUATION]', '1412.5672-2-90-0': '## High and low temperature behavior of [MATH]', '1412.5672-2-91-0': 'Let us first show that, at low temperatures, [MATH] does not receive any two-loop contribution.', '1412.5672-2-91-1': 'Indeed, up to exponentially suppressed terms, we have [EQUATION] where we have rescaled the integration variables as [MATH] and [MATH] and we have introduced [MATH] and [MATH].', '1412.5672-2-91-2': 'In the limit [MATH], each logarithm contributes [MATH].', '1412.5672-2-91-3': 'We can thus combine the two integrals and, after some simple manipulations that undo ([REF]) and ([REF]), we arrive at [EQUATION]', '1412.5672-2-91-4': 'This shows, in particular, that the two-loop corrections to the background field potential do not yield any [MATH] contribution to the pressure at low temperatures.', '1412.5672-2-92-0': 'To discuss the high temperature behavior, we note that the only contributions that survive in this regime, of order [MATH], all come from the first line of ([REF]).', '1412.5672-2-92-1': 'One obtains [EQUATION]', '1412.5672-2-92-2': 'The first line is nothing but the Weiss potential [MATH]-see Eq. ([REF])-which we have rewritten in a more convenient way using an integration by parts.', '1412.5672-2-92-3': 'The last line comes from the contribution [MATH] in ([REF]).', '1412.5672-2-92-4': 'We can conveniently rewrite the corresponding integral as [EQUATION]', '1412.5672-2-92-5': 'We then get, in terms of the polynomials [MATH] and [MATH] introduced in Appendix [REF], [EQUATION]', '1412.5672-2-92-6': 'The extrema of [MATH] obey the following equation [EQUATION] which admits the solutions [MATH] and [EQUATION]', '1412.5672-2-92-7': 'The curvature of [MATH] at [MATH] is [MATH] and the curvature at [MATH] is [MATH].', '1412.5672-2-92-8': 'There is another extremum at the boundary [MATH] but it is a maximum since the derivative of [MATH] at this point is [MATH].', '1412.5672-2-92-9': 'So the absolute minimum is located at [MATH].', '1412.5672-2-93-0': '# The Polyakov loop', '1412.5672-2-94-0': 'We compute the expectation value of the traced Polyakov loop matrix field at one-loop order.', '1412.5672-2-94-1': 'We first consider the general SU([MATH]) case with an arbitrary representation and specialize to the case of interest in this work-namely, the fundamental representation of SU([MATH])-at the end of the section.', '1412.5672-2-94-2': 'The Polyakov loop matrix field in the presence of a nonvanishing background [MATH] is defined as [EQUATION] where the (bare) field [MATH] belongs to a representation [MATH] of dimension [MATH] of the SU([MATH]) Lie algebra and [MATH] is restricted to the corresponding Cartan subalgebra.', '1412.5672-2-94-3': 'Here, we have emphasized that the coupling and fields appearing in the expression of the Polyakov loop matrix are the bare ones.', '1412.5672-2-94-4': 'Finally, [MATH] orders these fields from left to right, with decreasing value of their time argument.', '1412.5672-2-94-5': 'The expectation value of the traced Polyakov loop is defined as [EQUATION] where the right-hand side is to be evaluated at [MATH] which means, in particular, that [MATH].', '1412.5672-2-95-0': 'In order to compute the loop expansion of [MATH] in the presence of the nontrivial background field, we have to expand Eq. ([REF]) in powers of [MATH], with [MATH].', '1412.5672-2-95-1': 'For this purpose it is useful to rewrite Eq. ([REF]) in the following way.', '1412.5672-2-95-2': 'Consider the time-dependent gauge transformation (written in terms of bare field and coupling) [EQUATION] under which [see Eq. ([REF])] [EQUATION] and [EQUATION] such that [MATH].', '1412.5672-2-95-3': 'Using the standard transformation law of the Wilson line under gauge transformations [CITATION], we deduce that [EQUATION]', '1412.5672-2-95-4': 'The loop expansion in the presence of the background field simply amounts to expanding the path-ordered exponential in Eq. ([REF]) in powers of [MATH].', '1412.5672-2-96-0': 'Up to here, we have only considered bare fields and couplings.', '1412.5672-2-96-1': 'Turning to renormalized quantities, one has to carefully take into account the fact that the background and fluctuating fields [MATH] and [MATH] renormalize differently; see Eq. ([REF]).', '1412.5672-2-96-2': 'With our choice of renormalization condition ([REF]), we have, in terms of renormalized quantities, [MATH] for the background field contribution, and [MATH] for the fluctuating part.', '1412.5672-2-96-3': 'However, at the order of approximation considered here, i.e., [MATH], the renormalization factors only appear in the correction term and can thus be safely ignored.', '1412.5672-2-96-4': 'We thus directly use renormalized quantities in the rest of this section.', '1412.5672-2-97-0': 'In order to compute [MATH] at one-loop order, we expand the path-ordered exponential in Eq. ([REF]) up to quadratic order in [MATH].', '1412.5672-2-97-1': 'The leading- (zeroth-) order contribution is simply [EQUATION]', '1412.5672-2-97-2': 'The [MATH] contribution, linear in [MATH], gives a vanishing contribution to [MATH], because of the condition [MATH].', '1412.5672-2-97-3': 'Finally, the [MATH] (one-loop) contribution, quadratic in [MATH], yields, for the trace, [EQUATION]', '1412.5672-2-97-4': 'Writing [MATH], renaming [MATH] and [MATH], and using the periodicity property [EQUATION]', '1412.5672-2-97-5': 'Eq. ([REF]) rewrites as [EQUATION]', '1412.5672-2-97-6': 'The half sum of ([REF]) and ([REF]) thus yields, upon renaming [MATH], [EQUATION]', '1412.5672-2-97-7': 'We thus get, for the corresponding [MATH] contribution to the average of the traced Polyakov loop, [EQUATION] where [MATH].', '1412.5672-2-97-8': 'Equation ([REF]) holds for an arbitrary group SU([MATH]) and for any representation.', '1412.5672-2-98-0': 'We now specify to the fundamental representation of the SU(2) group: [MATH] and [MATH], with [MATH] and [MATH], where [MATH] are the Pauli matrices.', '1412.5672-2-98-1': 'Introducing standard up and down states [MATH] such that [MATH], [MATH], and [MATH], we have [EQUATION] and thus [EQUATION]', '1412.5672-2-98-2': 'It follows that [EQUATION] which is trivially evaluated.', '1412.5672-2-98-3': 'For the cases of interest in Eq. ([REF]), the trace ([REF]) gives [EQUATION]', '1412.5672-2-98-4': 'We then obtain, with [MATH], [EQUATION] where we have used that [MATH] and the change of variables [MATH].', '1412.5672-2-98-5': 'Equivalently, with [MATH], [EQUATION] in terms of the propagators in Fourier space.', '1412.5672-2-98-6': 'With the conventions of Sec. [REF] for the propagators, we have [EQUATION] where [MATH] is defined in ([REF]).', '1412.5672-2-99-0': 'Now, using Eqs. ([REF]) and ([REF]), we have [EQUATION] and [EQUATION] and the evaluation of the Matsubara sum in ([REF]) is straightforward.', '1412.5672-2-99-1': 'We obtain [EQUATION]', '1412.5672-2-99-2': 'Using dimensional regularization and ([REF]), we arrive at the following expression for the relative [MATH] correction to the average of the traced Polyakov loop [EQUATION]', '1412.5672-2-99-3': 'It is positive and we check that the Polyakov loop at one-loop order vanishes if and only if [MATH].', '1412.5672-2-99-4': 'Moreover, in the limit [MATH] the average of the traced Polyakov loop reads [EQUATION] where [MATH] and [MATH] were introduced in Appendix [REF], [MATH] given in Appendix [REF], and we have used an integration by parts.', '1412.5672-2-100-0': '# Proof that [MATH] at [MATH]', '1412.5672-2-101-0': 'We show that the expectation value of the traced Polyakov loop vanishes for a background field [MATH] in the SU([MATH]) theory.', '1412.5672-2-101-1': 'We consider bare quantities throughout this section.', '1412.5672-2-101-2': 'It proves useful to introduce the following generalization of Eq. ([REF]): [EQUATION] where the left-hand side denotes the average with the gauge-fixed action ([REF]), evaluated at an arbitrary background [MATH], with the constraint [MATH] (this can be imposed by an appropriate source term).', '1412.5672-2-101-3': 'The physical expectation value of the traced Polyakov loop is obtained as [MATH].', '1412.5672-2-102-0': 'Now, we make use of the various transformations already invoked in Sec. [REF] to discuss the symmetry properties of the potential [MATH].', '1412.5672-2-102-1': 'We first perform a global rotation of angle [MATH] around the axis 1 in color space, both for the integration variables [MATH] under the path integral and for the background field.', '1412.5672-2-102-2': 'For the latter, this amounts to [MATH].', '1412.5672-2-102-3': 'The integration measure-including the (gauge-fixed) action and the constraint [MATH]-and the Polyakov loop in the brackets of ([REF]) being invariant under global color rotations, we conclude that [EQUATION]', '1412.5672-2-102-4': 'Next, we perform a gauge transformation of the form ([REF]) with a local SU([MATH]) matrix [MATH], with [MATH] the Pauli matrices.', '1412.5672-2-102-5': 'This matrix is periodic in the time direction up to an element of the center, i.e., [MATH].', '1412.5672-2-102-6': 'This produces a shift of the background field [MATH] but leaves the integration measure, again including the (gauge-fixed) action and the constraint [MATH], invariant, while the Polyakov loop gets multiplied by a phase [MATH].', '1412.5672-2-102-7': 'We thus get [EQUATION]', '1412.5672-2-102-8': 'Combining the above identities, we get [EQUATION] from which we conclude that [MATH].', '1412.5672-2-102-9': 'It follows that the physical expectation value of the Polyakov loop [MATH] if the minimum of the background field potential is [MATH].', '1412.5672-2-102-10': 'This proof can easily be generalized, along similar lines to SU([MATH]).'} | [['1412.5672-1-64-3', '1412.5672-2-64-3'], ['1412.5672-1-64-4', '1412.5672-2-64-4'], ['1412.5672-1-6-0', '1412.5672-2-6-0'], ['1412.5672-1-6-1', '1412.5672-2-6-1'], ['1412.5672-1-6-2', '1412.5672-2-6-2'], ['1412.5672-1-6-4', '1412.5672-2-6-4'], ['1412.5672-1-6-5', '1412.5672-2-6-5'], ['1412.5672-1-17-0', '1412.5672-2-17-0'], ['1412.5672-1-17-1', '1412.5672-2-17-1'], ['1412.5672-1-17-2', '1412.5672-2-17-2'], ['1412.5672-1-17-3', '1412.5672-2-17-3'], ['1412.5672-1-43-0', '1412.5672-2-43-0'], ['1412.5672-1-43-1', '1412.5672-2-43-1'], ['1412.5672-1-43-2', '1412.5672-2-43-2'], ['1412.5672-1-43-3', '1412.5672-2-43-3'], ['1412.5672-1-43-5', '1412.5672-2-43-5'], ['1412.5672-1-19-5', '1412.5672-2-19-5'], ['1412.5672-1-19-7', '1412.5672-2-19-7'], ['1412.5672-1-58-0', '1412.5672-2-58-0'], ['1412.5672-1-58-1', '1412.5672-2-58-1'], ['1412.5672-1-58-2', '1412.5672-2-58-2'], ['1412.5672-1-58-3', '1412.5672-2-58-3'], ['1412.5672-1-58-4', '1412.5672-2-58-4'], ['1412.5672-1-58-5', '1412.5672-2-58-5'], ['1412.5672-1-39-0', '1412.5672-2-39-0'], ['1412.5672-1-39-1', '1412.5672-2-39-1'], ['1412.5672-1-39-2', '1412.5672-2-39-2'], ['1412.5672-1-57-0', '1412.5672-2-57-0'], ['1412.5672-1-57-1', '1412.5672-2-57-1'], ['1412.5672-1-57-2', '1412.5672-2-57-2'], ['1412.5672-1-57-3', '1412.5672-2-57-3'], ['1412.5672-1-47-0', '1412.5672-2-47-0'], ['1412.5672-1-47-1', '1412.5672-2-47-1'], ['1412.5672-1-47-2', '1412.5672-2-47-2'], ['1412.5672-1-47-3', '1412.5672-2-47-3'], ['1412.5672-1-47-4', '1412.5672-2-47-4'], ['1412.5672-1-47-5', '1412.5672-2-47-5'], ['1412.5672-1-47-6', '1412.5672-2-47-6'], ['1412.5672-1-8-0', '1412.5672-2-8-0'], ['1412.5672-1-8-1', '1412.5672-2-8-1'], ['1412.5672-1-8-2', '1412.5672-2-8-2'], ['1412.5672-1-8-3', '1412.5672-2-8-3'], ['1412.5672-1-8-4', '1412.5672-2-8-4'], ['1412.5672-1-8-5', '1412.5672-2-8-5'], ['1412.5672-1-8-6', '1412.5672-2-8-6'], ['1412.5672-1-8-7', '1412.5672-2-8-7'], ['1412.5672-1-55-0', '1412.5672-2-55-0'], ['1412.5672-1-55-1', '1412.5672-2-55-1'], ['1412.5672-1-55-2', '1412.5672-2-55-2'], ['1412.5672-1-26-1', '1412.5672-2-26-1'], ['1412.5672-1-26-2', '1412.5672-2-26-2'], ['1412.5672-1-26-3', '1412.5672-2-26-3'], ['1412.5672-1-61-0', '1412.5672-2-61-0'], ['1412.5672-1-61-1', '1412.5672-2-61-1'], ['1412.5672-1-61-2', '1412.5672-2-61-2'], ['1412.5672-1-61-3', '1412.5672-2-61-3'], ['1412.5672-1-37-0', '1412.5672-2-37-0'], ['1412.5672-1-52-0', '1412.5672-2-52-0'], ['1412.5672-1-52-1', '1412.5672-2-52-1'], ['1412.5672-1-52-3', '1412.5672-2-52-3'], ['1412.5672-1-52-5', '1412.5672-2-52-5'], ['1412.5672-1-52-6', '1412.5672-2-52-6'], ['1412.5672-1-52-7', '1412.5672-2-52-7'], ['1412.5672-1-52-8', '1412.5672-2-52-8'], ['1412.5672-1-52-9', '1412.5672-2-52-9'], ['1412.5672-1-52-11', '1412.5672-2-52-11'], ['1412.5672-1-4-1', '1412.5672-2-4-1'], ['1412.5672-1-4-2', '1412.5672-2-4-2'], ['1412.5672-1-4-4', '1412.5672-2-4-4'], ['1412.5672-1-4-5', '1412.5672-2-4-5'], ['1412.5672-1-4-7', '1412.5672-2-4-7'], ['1412.5672-1-4-8', '1412.5672-2-4-8'], ['1412.5672-1-35-0', '1412.5672-2-35-0'], ['1412.5672-1-35-1', '1412.5672-2-35-1'], ['1412.5672-1-35-4', '1412.5672-2-35-4'], ['1412.5672-1-35-5', '1412.5672-2-35-5'], ['1412.5672-1-35-6', '1412.5672-2-35-6'], ['1412.5672-1-35-7', '1412.5672-2-35-7'], ['1412.5672-1-35-8', '1412.5672-2-35-8'], ['1412.5672-1-25-0', '1412.5672-2-25-0'], ['1412.5672-1-25-2', '1412.5672-2-25-2'], ['1412.5672-1-25-3', '1412.5672-2-25-3'], ['1412.5672-1-25-4', '1412.5672-2-25-4'], ['1412.5672-1-25-6', '1412.5672-2-25-6'], ['1412.5672-1-25-7', '1412.5672-2-25-7'], ['1412.5672-1-25-8', '1412.5672-2-25-8'], ['1412.5672-1-25-9', '1412.5672-2-25-9'], ['1412.5672-1-25-10', '1412.5672-2-25-10'], ['1412.5672-1-25-11', '1412.5672-2-25-11'], ['1412.5672-1-25-12', '1412.5672-2-25-12'], ['1412.5672-1-41-1', '1412.5672-2-41-1'], ['1412.5672-1-41-2', '1412.5672-2-41-2'], ['1412.5672-1-41-3', '1412.5672-2-41-3'], ['1412.5672-1-41-4', '1412.5672-2-41-4'], ['1412.5672-1-41-7', '1412.5672-2-41-7'], ['1412.5672-1-30-0', '1412.5672-2-30-0'], ['1412.5672-1-30-1', '1412.5672-2-30-1'], ['1412.5672-1-30-2', '1412.5672-2-30-2'], ['1412.5672-1-30-3', '1412.5672-2-30-3'], ['1412.5672-1-30-4', '1412.5672-2-30-4'], ['1412.5672-1-23-0', '1412.5672-2-23-0'], ['1412.5672-1-23-1', '1412.5672-2-23-1'], ['1412.5672-1-23-2', '1412.5672-2-23-2'], ['1412.5672-1-23-3', '1412.5672-2-23-3'], ['1412.5672-1-23-4', '1412.5672-2-23-4'], ['1412.5672-1-0-0', '1412.5672-2-0-0'], ['1412.5672-1-0-1', '1412.5672-2-0-1'], ['1412.5672-1-0-2', '1412.5672-2-0-2'], ['1412.5672-1-0-4', '1412.5672-2-0-4'], ['1412.5672-1-0-5', '1412.5672-2-0-5'], ['1412.5672-1-0-6', '1412.5672-2-0-6'], ['1412.5672-1-28-1', '1412.5672-2-28-1'], ['1412.5672-1-28-3', '1412.5672-2-28-3'], ['1412.5672-1-28-4', '1412.5672-2-28-4'], ['1412.5672-1-45-0', '1412.5672-2-45-0'], ['1412.5672-1-45-1', '1412.5672-2-45-1'], ['1412.5672-1-45-2', '1412.5672-2-45-2'], ['1412.5672-1-45-3', '1412.5672-2-45-4'], ['1412.5672-1-45-4', '1412.5672-2-45-5'], ['1412.5672-1-45-5', '1412.5672-2-45-6'], ['1412.5672-1-45-6', '1412.5672-2-45-7'], ['1412.5672-1-53-0', '1412.5672-2-53-0'], ['1412.5672-1-53-1', '1412.5672-2-53-1'], ['1412.5672-1-53-2', '1412.5672-2-53-2'], ['1412.5672-1-56-0', '1412.5672-2-56-0'], ['1412.5672-1-56-1', '1412.5672-2-56-1'], ['1412.5672-1-56-2', '1412.5672-2-56-2'], ['1412.5672-1-56-3', '1412.5672-2-56-3'], ['1412.5672-1-56-5', '1412.5672-2-56-5'], ['1412.5672-1-49-0', '1412.5672-2-49-0'], ['1412.5672-1-49-2', '1412.5672-2-49-2'], ['1412.5672-1-49-3', '1412.5672-2-49-3'], ['1412.5672-1-18-0', '1412.5672-2-18-0'], ['1412.5672-1-18-1', '1412.5672-2-18-1'], ['1412.5672-1-18-2', '1412.5672-2-18-2'], ['1412.5672-1-18-3', '1412.5672-2-18-3'], ['1412.5672-1-18-6', '1412.5672-2-18-6'], ['1412.5672-1-18-7', '1412.5672-2-18-7'], ['1412.5672-1-18-10', '1412.5672-2-18-10'], ['1412.5672-1-22-2', '1412.5672-2-22-2'], ['1412.5672-1-22-3', '1412.5672-2-22-3'], ['1412.5672-1-22-4', '1412.5672-2-22-4'], ['1412.5672-1-10-0', '1412.5672-2-10-0'], ['1412.5672-1-10-1', '1412.5672-2-10-1'], ['1412.5672-1-10-2', '1412.5672-2-10-2'], ['1412.5672-1-10-3', '1412.5672-2-10-3'], ['1412.5672-1-10-5', '1412.5672-2-10-5'], ['1412.5672-1-10-6', '1412.5672-2-10-6'], ['1412.5672-1-10-7', '1412.5672-2-10-7'], ['1412.5672-1-10-8', '1412.5672-2-10-8'], ['1412.5672-1-10-9', '1412.5672-2-10-9'], ['1412.5672-1-10-10', '1412.5672-2-10-10'], ['1412.5672-1-34-0', '1412.5672-2-34-0'], ['1412.5672-1-34-1', '1412.5672-2-34-1'], 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'1412.5672-2-49-3'], ['1412.5672-1-18-0', '1412.5672-2-18-0'], ['1412.5672-1-18-1', '1412.5672-2-18-1'], ['1412.5672-1-18-2', '1412.5672-2-18-2'], ['1412.5672-1-18-3', '1412.5672-2-18-3'], ['1412.5672-1-18-6', '1412.5672-2-18-6'], ['1412.5672-1-18-7', '1412.5672-2-18-7'], ['1412.5672-1-18-10', '1412.5672-2-18-10'], ['1412.5672-1-22-2', '1412.5672-2-22-2'], ['1412.5672-1-22-3', '1412.5672-2-22-3'], ['1412.5672-1-22-4', '1412.5672-2-22-4'], ['1412.5672-1-10-0', '1412.5672-2-10-0'], ['1412.5672-1-10-1', '1412.5672-2-10-1'], ['1412.5672-1-10-2', '1412.5672-2-10-2'], ['1412.5672-1-10-3', '1412.5672-2-10-3'], ['1412.5672-1-10-5', '1412.5672-2-10-5'], ['1412.5672-1-10-6', '1412.5672-2-10-6'], ['1412.5672-1-10-7', '1412.5672-2-10-7'], ['1412.5672-1-10-8', '1412.5672-2-10-8'], ['1412.5672-1-10-9', '1412.5672-2-10-9'], ['1412.5672-1-10-10', '1412.5672-2-10-10'], ['1412.5672-1-34-0', '1412.5672-2-34-0'], ['1412.5672-1-34-1', '1412.5672-2-34-1'], ['1412.5672-1-34-2', '1412.5672-2-34-2'], ['1412.5672-1-34-3', '1412.5672-2-34-3'], ['1412.5672-1-34-4', '1412.5672-2-34-4'], ['1412.5672-1-34-5', '1412.5672-2-34-5'], ['1412.5672-1-34-6', '1412.5672-2-34-6'], ['1412.5672-1-34-8', '1412.5672-2-34-8'], ['1412.5672-1-34-9', '1412.5672-2-34-9'], ['1412.5672-1-34-11', '1412.5672-2-34-11'], ['1412.5672-1-34-12', '1412.5672-2-34-12'], ['1412.5672-1-32-0', '1412.5672-2-32-0'], ['1412.5672-1-32-1', '1412.5672-2-32-1'], ['1412.5672-1-21-0', '1412.5672-2-21-0'], ['1412.5672-1-21-1', '1412.5672-2-21-1'], ['1412.5672-1-21-2', '1412.5672-2-21-2'], ['1412.5672-1-21-3', '1412.5672-2-21-3'], ['1412.5672-1-21-4', '1412.5672-2-21-4'], ['1412.5672-1-21-5', '1412.5672-2-21-5'], ['1412.5672-1-5-0', '1412.5672-2-5-0'], ['1412.5672-1-5-1', '1412.5672-2-5-1'], ['1412.5672-1-5-2', '1412.5672-2-5-2'], ['1412.5672-1-5-3', '1412.5672-2-5-3'], ['1412.5672-1-5-5', '1412.5672-2-5-5'], ['1412.5672-1-65-1', '1412.5672-2-65-1'], ['1412.5672-1-65-3', '1412.5672-2-65-3'], ['1412.5672-1-7-0', '1412.5672-2-7-0'], ['1412.5672-1-7-2', '1412.5672-2-7-2'], ['1412.5672-1-7-3', '1412.5672-2-7-3'], ['1412.5672-1-13-0', '1412.5672-2-13-0'], ['1412.5672-1-13-2', '1412.5672-2-13-2'], ['1412.5672-1-13-3', '1412.5672-2-13-3'], ['1412.5672-1-27-0', '1412.5672-2-27-0'], ['1412.5672-1-27-2', '1412.5672-2-27-2'], ['1412.5672-1-27-3', '1412.5672-2-27-3'], ['1412.5672-1-27-4', '1412.5672-2-27-4'], ['1412.5672-1-63-0', '1412.5672-2-63-0'], ['1412.5672-1-63-1', '1412.5672-2-63-1'], ['1412.5672-1-63-2', '1412.5672-2-63-2'], ['1412.5672-1-2-0', '1412.5672-2-2-0'], ['1412.5672-1-2-1', '1412.5672-2-2-1'], ['1412.5672-1-2-2', '1412.5672-2-2-2'], ['1412.5672-1-2-3', '1412.5672-2-2-3'], ['1412.5672-1-2-4', '1412.5672-2-2-4'], ['1412.5672-1-2-5', '1412.5672-2-2-5'], ['1412.5672-1-60-0', '1412.5672-2-60-0'], ['1412.5672-1-60-1', '1412.5672-2-60-1'], ['1412.5672-1-60-2', '1412.5672-2-60-2'], ['1412.5672-1-60-3', '1412.5672-2-60-3'], ['1412.5672-1-60-4', '1412.5672-2-60-4'], ['1412.5672-1-60-5', '1412.5672-2-60-5'], ['1412.5672-1-60-6', '1412.5672-2-60-6'], ['1412.5672-1-3-1', '1412.5672-2-3-1'], ['1412.5672-1-3-2', '1412.5672-2-3-2'], ['1412.5672-1-3-3', '1412.5672-2-3-3'], ['1412.5672-1-9-0', '1412.5672-2-9-0'], ['1412.5672-1-9-1', '1412.5672-2-9-1'], ['1412.5672-1-9-4', '1412.5672-2-9-4'], ['1412.5672-1-14-3', '1412.5672-2-14-3'], ['1412.5672-1-14-4', '1412.5672-2-14-4'], ['1412.5672-1-38-0', '1412.5672-2-38-0'], ['1412.5672-1-50-0', '1412.5672-2-50-0'], ['1412.5672-1-50-1', '1412.5672-2-50-1'], ['1412.5672-1-50-2', '1412.5672-2-50-2'], ['1412.5672-1-48-0', '1412.5672-2-48-0'], ['1412.5672-1-48-1', '1412.5672-2-48-1'], ['1412.5672-1-48-2', '1412.5672-2-48-2'], ['1412.5672-1-48-3', '1412.5672-2-48-3'], ['1412.5672-1-48-4', '1412.5672-2-48-4']] | [['1412.5672-1-64-0', '1412.5672-2-64-0'], ['1412.5672-1-64-1', '1412.5672-2-64-1'], ['1412.5672-1-64-2', '1412.5672-2-64-2'], ['1412.5672-1-6-3', '1412.5672-2-6-3'], ['1412.5672-1-17-4', '1412.5672-2-17-4'], ['1412.5672-1-17-5', '1412.5672-2-17-5'], ['1412.5672-1-43-4', '1412.5672-2-43-4'], ['1412.5672-1-19-6', '1412.5672-2-19-6'], ['1412.5672-1-57-4', '1412.5672-2-57-4'], ['1412.5672-1-26-0', '1412.5672-2-26-0'], ['1412.5672-1-37-1', '1412.5672-2-37-1'], ['1412.5672-1-37-2', '1412.5672-2-37-2'], ['1412.5672-1-52-2', '1412.5672-2-52-2'], ['1412.5672-1-52-4', '1412.5672-2-52-4'], ['1412.5672-1-52-10', '1412.5672-2-52-10'], ['1412.5672-1-4-0', '1412.5672-2-4-0'], ['1412.5672-1-4-3', '1412.5672-2-4-3'], ['1412.5672-1-4-6', '1412.5672-2-4-6'], ['1412.5672-1-35-2', '1412.5672-2-35-2'], ['1412.5672-1-35-3', '1412.5672-2-35-3'], ['1412.5672-1-25-1', '1412.5672-2-25-1'], ['1412.5672-1-25-5', '1412.5672-2-25-5'], ['1412.5672-1-12-0', '1412.5672-2-12-0'], ['1412.5672-1-12-1', '1412.5672-2-12-1'], ['1412.5672-1-41-0', '1412.5672-2-41-0'], ['1412.5672-1-41-5', '1412.5672-2-41-5'], ['1412.5672-1-41-6', '1412.5672-2-41-6'], ['1412.5672-1-23-5', '1412.5672-2-23-5'], ['1412.5672-1-0-3', '1412.5672-2-0-3'], ['1412.5672-1-28-0', '1412.5672-2-28-0'], ['1412.5672-1-28-2', '1412.5672-2-28-2'], ['1412.5672-1-45-7', '1412.5672-2-45-8'], ['1412.5672-1-56-4', '1412.5672-2-56-4'], ['1412.5672-1-49-1', '1412.5672-2-49-1'], ['1412.5672-1-18-4', '1412.5672-2-18-4'], ['1412.5672-1-18-5', '1412.5672-2-18-5'], ['1412.5672-1-22-0', '1412.5672-2-22-0'], ['1412.5672-1-22-1', '1412.5672-2-22-1'], ['1412.5672-1-10-4', '1412.5672-2-10-4'], ['1412.5672-1-34-7', '1412.5672-2-34-7'], ['1412.5672-1-34-10', '1412.5672-2-34-10'], ['1412.5672-1-32-2', '1412.5672-2-32-2'], ['1412.5672-1-5-4', '1412.5672-2-5-4'], ['1412.5672-1-5-6', '1412.5672-2-5-6'], ['1412.5672-1-65-0', '1412.5672-2-65-0'], ['1412.5672-1-65-2', '1412.5672-2-65-2'], ['1412.5672-1-7-1', '1412.5672-2-7-1'], ['1412.5672-1-7-4', '1412.5672-2-7-4'], ['1412.5672-1-13-1', '1412.5672-2-13-1'], ['1412.5672-1-27-1', '1412.5672-2-27-1'], ['1412.5672-1-63-3', '1412.5672-2-63-3'], ['1412.5672-1-3-0', '1412.5672-2-3-0'], ['1412.5672-1-9-2', '1412.5672-2-9-2'], ['1412.5672-1-9-3', '1412.5672-2-9-3'], ['1412.5672-1-9-5', '1412.5672-2-9-5'], ['1412.5672-1-14-0', '1412.5672-2-14-0'], ['1412.5672-1-14-1', '1412.5672-2-14-1'], ['1412.5672-1-14-2', '1412.5672-2-14-2']] | [] | [['1412.5672-1-19-0', '1412.5672-2-19-0'], ['1412.5672-1-19-2', '1412.5672-2-19-2'], ['1412.5672-1-19-4', '1412.5672-2-19-4'], ['1412.5672-1-18-8', '1412.5672-2-18-8'], ['1412.5672-1-18-9', '1412.5672-2-18-9']] | [] | ['1412.5672-2-76-1', '1412.5672-2-76-5', '1412.5672-2-92-1', '1412.5672-2-99-1'] | {'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'} | https://arxiv.org/abs/1412.5672 | null | null | null | null | null |
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